Paper Title: A phase-field model for wet snow metamorphism.





§ ABSTRACT 
The microstructure of snow determines its fundamental properties such as the mechanical strength, reflectivity, or the thermo-hydraulic properties.
Snow undergoes continuous microstructural changes due to local gradients in temperature, humidity or curvature, in a process known as snow metamorphism.
In this work, we focus on wet snow metamorphism, which occurs when temperature is close to the melting point and involves phase transitions amongst liquid water, water vapor, and solid ice. 
We propose a pore-scale phase-field model that simultaneously captures the three relevant phase-change phenomena: sublimation (deposition), evaporation (condensation), and melting (solidification).
The phase-field formulation allows one to track the temperature evolution amongst the three phases and the water vapor concentration in the air. 
Our three-phase model recovers the corresponding two-phase transition model when one phase is not present in the system. 2D simulations of the model unveils the impact of humidity and temperature on the dynamics of wet snow metamorphism at the pore scale. We also explore the role of liquid melt content in controlling the dynamics of snow metamorphism in contrast to the dry regime, before percolation onsets. 
The model can be readily extended to incorporate two-phase flow and may be the basis for investigating other problems involving water phase transitions in a vapor-solid-liquid system such as airplane icing or thermal spray coating.








§ INTRODUCTION

Snow and firn are heterogenous porous material composed of two components, water and air, distributed amongst three phases: solid ice, liquid water, and air with water vapor. The microstructure of snow and firn dictates their mechanical strength, reflectivity, and thermo-hydraulic properties, which govern important processes such as snow avalanches, snowpack and glacial hydrology, radar remote sensing, and the performance of snow vehicles <cit.>.

However, these properties constantly evolve because snow is a thermodynamically active material that undergoes continuous microstructural changes caused by phase transitions between ice, liquid water, and vapor, in a process known as snow metamorphism <cit.>. Ultimately, snow metamorphism leads to a denser snow composed of coarser and rounder ice grains. Based on whether liquid water is present, snow metamorphism is considered in two types —dry and wet. Colbeck and other authors published a series of pioneer papers in the 70's and 80's <cit.> that derived the theoretical foundations to understand the mechanisms driving each type of snow metamorphism.

Dry snow metamorphism <cit.> results from sublimation and deposition and is driven by the transport of water vapor between regions with different vapor pressure, i.e., regions with different temperature and/or grain curvature, according to the Gibbs-Thomson condition <cit.>. 
On the other side, wet snow metamorphism occurs when the temperature is close to the melting point. The presence of liquid water fundamentally alters the mechanisms of metamorphism by increasing the thermodynamic activity of snow and strengthening the mechanical and thermal connection between ice grains through capillary bridging <cit.>.
Thus, compared to dry metamorphism, wet snow metamorphism displays accelerated coarsening and densification.

Colbeck proposed two different regimes for wet metamorphism depending on the liquid water content: the pendular and the funicular regimes <cit.>. 
In the funicular regime, the liquid water content is high so that the water completely surrounds the ice grains.
In the pendular regime, the liquid water content is low and most of the ice surface is in contact with air.
Thus, the pendular regime displays lower heat flow, grain growth, and snow densification, and larger capillary forces and grain-to-grain mechanical strength, compared to the funicular regime.
In both cases, vapor kinetics plays an important role <cit.>. While Colbeck's work provided mechanistic understanding of snow metamorphism,  it focused on theoretical snow configurations under ideal environmental conditions.

Subsequent experimental works used X-ray microtomography (microCT) to study the evolution of the dry snow microstructure in realistic scenarios under isothermal  conditions <cit.> and temperature gradients <cit.>. Many authors leveraged the microCT data to perform a comprehensive analysis of local crystal growth laws <cit.> and validate mean field models <cit.>.

The first experiments of wet metamorphism focused on fully saturated snow <cit.>.
Improved experimental techniques enable studies of wet metamorphism on samples with low liquid water content <cit.>. These works analyzed snow wetness and grain size and derived empirical laws for the snow properties under wet conditions <cit.>. However, more advanced techniques are still being developed to robustly visualize liquid water within snow samples, such as the use of hyperspectral imaging to map out liquid water content in snow <cit.>.


Mathematical models may complement experimental development to provide insights into the wet snow metamorphism process. To this end, wet snow is modeled as a three-phase system composed of liquid water, vapor and solid ice, and thus can be grouped with other class problems such as snowflake growth and water drop icing in a humidity-controlled atmosphere. In this broader group of literature, some have proposed simplified 1D model to describe the role of humidity on the evolution dynamics <cit.>.
More elaborated 2D continuum models resorted to the phase-field method <cit.> or the level-set method <cit.>, which have been successful in reproducing the phase transitions between two phases of water, i.e., solidification <cit.>, sublimation <cit.>, and evaporation <cit.>. However, most of these continuum models are limited to the two-phase regime, and do not simultaneously capture the transitions amongst the three water phases; thus, these models do not address the wet snow metamorphism problem.


Here, we propose a phase-field model that describes the non-equilibrium evolution of the three-phase mixture composed of liquid water, water vapor and solid ice.
The modeling choice is inspired by recent applications of phase-field methods in porous media problems with phase transitions, including the modeling of microstructural evolution of porous materials and multicomponent fluid mixtures <cit.>, phase transitions during two-phase flow in porous media<cit.> and crystallization flow in porous media <cit.>. The three-phase model proposed in this work reduces to the classic two-phase model in the absence of a third phase, and the resulting two-phase models readily capture the Gibbs-Thomson conditions. 
The model allows us to analyze the pore-scale dynamics of that dictates wet snow metamorphism.  Currently, the model does not account for granular compaction, melt film imbibition or melt percolation. Rather, the primary focus of the model is to elucidate the role of thermodynamic-driven phase transitions on microstructure evolution of a quiescent wet snow mixture. 



§ A PHASE-FIELD MODEL FOR WET SNOW MIXTURE 


The phase-field method <cit.> is a mathematical technique that is based on a free energy description of multi-phase mixtures, and is well-suited for problems with moving interfaces. Within the phase-field framework, one defines a dimensionless phase field variable, denoted ϕ_l ∈ [0,1], which represents the volume fraction of the phase l at any given point in the domain. The phase variable smoothly transitions across the interface between two distinct phases (Fig. <ref>). With the appropriate evolution equations, the moving boundary problem is reformulated in a fixed domain, which avoids most of the numerical issues caused by the moving interfaces.

Here, we propose a non-variational phase-field model that captures the phase transitions between ice, liquid water, and water vapor, i.e., solidification, sublimation, evaporation, and the opposite transitions. For simplicity, from here on we denote the liquid water as water and the water vapor as vapor. We assume that the air is at the atmospheric pressure. We assume that ice and water have equal density for the phase evolution equations, which introduces small mass conservation error. Although fluid flow is not considered in the current model, future work that incorporates fluid flow and unequal density may mitigate the mass conservation issue.

We note that in the spirit of numerical robustness and physical consistency, a  variational phase-field formulation <cit.> is often more desirable to ensure non-negative entropy production and the convergence towards equilibrium <cit.>. However, due to the close proximity in the density of the ice and liquid water phase, we find that formulating a free energy for the wet snow system based on the density field introduces numerical singularities in the formulation. 








 §.§ Model variables

Our model unknowns are the phase field variables for ice ϕ_i(,t), water ϕ_w(,t), and air ϕ_a(,t) phases (see Fig. <ref>), as well as variables for the temperature T(,t) (units ^∘C) and the vapor density ρ_v(,t) (units kg/m^3). All variables are defined point-wise in the problem domain Ω (Fig. <ref>). 
Note that even though temperature and vapor density are not phase field variables, here we leverage the formulation to specify phase-dependent temperature evolution (see Section <ref>), and to localize the vapor dynamics onto the air phase (see Section <ref>). 







 §.§ Model equations


In this section, we will introduce the five partial differential equations that describe the co-evolution of our model variables.



  §.§.§ The phase evolution equations


The basic form of the phase evolution equations for the three phase field variables are expressed as:


    ∂ϕ_i/∂ t =  -M_0/Σ_iδℱ^tri(ϕ)/δϕ_i  - α_solϕ_i^2ϕ_w^2T-T_melt/L_sol/c_p,w + α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i +θ_nucl(ϕ,T), 
    
            ∂ϕ_w/∂ t = -M_0/Σ_wδℱ^tri(ϕ)/δϕ_w + α_solϕ_i^2ϕ_w^2T-T_melt/L_sol/c_p,w +  α_evaϕ_w^2ϕ_a^2ρ_v-ρ_vs^W(T)/ρ_w -θ_nucl(ϕ,T), 
    
            ∂ϕ_a/∂ t = -M_0/Σ_aδℱ^tri(ϕ)/δϕ_a - α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i - α_evaϕ_w^2ϕ_a^2ρ_v-ρ_vs^W(T)/ρ_w,


where δ·/δ· denotes the variational derivative and we use the notation ϕ={ϕ_i,ϕ_w,ϕ_a}. Note that Eqs.(<ref>)-(<ref>) account for total mass conservation by assuming equal density of ice and water (ρ_i=ρ_w).


The first terms in the right-hand side of these three equations capture the Allen-Cahn kinetics of phase-field variables <cit.>, where M_0 is the phase-dependent mobility defined below in Sect. <ref> and  ℱ^tri is the classical energy functional of a ternary mixture <cit.>:


    ℱ^tri(ϕ)=∫( 3/ε F^tri(ϕ) + 3/2ε[Σ_i(ϕ_i)^2 +Σ_w(ϕ_w)^2 +Σ_a(ϕ_a)^2 ] +β(1-ϕ_i-ϕ_w-ϕ_a) )Ω.


The parameter ε represents the width of the phase-field interface and β is the Lagrange multiplier (see Section <ref>). The function F^tri is the triple-well potential, defined as: 


    F^tri(ϕ)=1/2Σ_iϕ_i^2(1-ϕ_i)^2+1/2Σ_wϕ_w^2(1-ϕ_w)^2+1/2Σ_aϕ_a^2(1-ϕ_a)^2 +Λϕ_i^2ϕ_w^2ϕ_a^2,


where the Λ-term improves the dynamic consistency of the model<cit.> to mitigate spurious effect along phase-phase boundaries. 
The parameters Σ_i, Σ_w, and Σ_a are related to the surface tensions (σ_lm) between phases l and m such that 


    Σ_i = σ_iw +σ_ia -σ_wa,  Σ_w = σ_iw +σ_wa -σ_ia,   Σ_a = σ_wa +σ_ia -σ_iw.








We extend the Allen-Cahn kinetics with the α-terms in Eqs. (<ref>) to (<ref>), which account for the different phase transitions. In particular, the α_sol term accounts for solidification or melting, the α_sub term accounts for sublimation or deposition, and the α_eva term accounts for evaporation or condensation. 

The estimation of α's parameter values is described in the Supporting Information (Section <ref> and Table <ref>).

The parameters L_sol, c_p,w, ρ_w and ρ_i are the solidification latent heat, water specific heat capacity, water density, and ice density, respectively.

We follow <cit.> and express the saturated vapor density of ice (ρ_vs^I) and water (ρ_vs^W) as

    ρ_vs^J(T)=ρ_a 0.62P_vs^J(T)/P_a-P_vs^J(T),


where the index J stands for I and W, ρ_a is the dry air density, P_a is the atmospheric pressure (units Pa), and P_vs^J is the saturated vapor pressure of phase J (units Pa). The saturated vapor pressure of ice and water are defined as <cit.>


    P_vs^I(T)=exp( ∑_j=0^4 k_j(T+T_0)^j-1 + k_5 ln(T+T_0)),
       P_vs^W(T)=exp( ∑_j=0^6 g_j(T+T_0)^j-2 + g_7 ln(T+T_0)),


respectively, where T_0=273.15, and the values of the parameters k_j and g_j are listed in Table <ref> in the Supporting Information.




The θ_nucl terms in Eqs. (<ref>)–(<ref>) account for ice and water nucleation, which are necessary in order to trigger the nucleation of a third phase when only two phases exist initially. In particular, here we account for the nucleation of water when T increases above T_melt in an ice-air system and the nucleation of ice when T decreases below the ice nucleation temperature T_nucl in a water-air system.

We propose a phenomenological nucleation term

    θ_nucl(ϕ,T) = α_nuclM_0ϕ_a^2 [ϕ_w^2 N_i(T)-ϕ_i^2 N_w(T)],


where α_nucl is a parameter controlling the strength of the nucleation and the ice and water nucleation functions N_i(T) and N_w(T), respectively, are expressed as

    N_i(T) = 0.5-0.5 tanh[20(T-T_nucl+0.1)],     N_w(T) = 0.5+0.5 tanh[20(T-T_melt-0.1)].

Note that N_i becomes active when T<T_nucl, which allows the presence of undercooled water in the range T_nucl<T<T_melt.




According to Eq. (<ref>), ice nucleation occurs on the water-air  interface when T<T_nucl (note the term ϕ_a^2ϕ_w^2 multiplying N_i).

Once ice nucleation advances, the water-air interface splits into the water-ice and ice-air interfaces, resulting in ϕ_a^2ϕ_w^2=0 and thus θ_nucl stays zero even if T<T_nucl.
Water nucleation exhibits an analogous behavior.

Note that θ_nucl only needs to be active until the system enters into the spinodal decomposition region of the ternary mixture.







  §.§.§ Energy conservation equation
 

Here, we formulate conservation of energy in terms of temperature, expressed as


    ρ(ϕ)c_p(ϕ)∂ T/∂ t = ÷[ K(ϕ) T ] + ρ(ϕ)L_sol(∂ϕ_a/∂ t + ∂ϕ_i/∂ t) -ρ(ϕ)L_sub∂ϕ_a/∂ t.


Eq. (<ref>) accounts for thermal diffusion and the latent heat released and/or absorbed during the different phase transitions. L_sub is the sublimation latent heat. The latent heat of evaporation is then captured through L_eva=L_sub-L_sol. The phase-dependent density (ρ), specific heat capacity (c_p), and thermal conductivity (K) are defined as:

    ρ(ϕ)=∑_l=i,w,aϕ_lρ_l;    
        c_p(ϕ)=∑_l=i,w,aϕ_l c_p,l;   
        K(ϕ)=∑_l=i,w,aϕ_l K_l,

where ρ_l, c_p,l, and K_l are the density, specific heat capacity, and thermal conductivity, respectively, of phase l. We assume that ρ_l, c_p,l, and K_l are constant in time. Note that here we violate our primary assumption that ρ_i=ρ_w used in Eqs. (<ref>)-(<ref>), which incurs mass conservation error; instead, we consider different density for the ice and water phase for the energy equation in order to adopt the correct heat capacity for each phase. Although doing so does not eliminate errors in energy conservation due to errors in mass conservation, it improves the thermal consistency of the model.












  §.§.§ Mass conservation of vapor
 

The vapor mass conservation equation is expressed as 


    ∂(ϕ_aρ_v)/∂ t=÷[ ϕ_a D_v(T)ρ_v ] + ρ_SE∂ϕ_a/∂ t,


which accounts for vapor diffusion in the air and mass transfer of water during sublimation and evaporation (and the opposite transitions). 
Here, we formulate a phase-dependent density ρ_SE(ϕ) such that ρ_SE=ρ_i in case of sublimation and ρ_SE=ρ_w in case of evaporation. The specific form for ρ_SE(ϕ) is discussed and provided in Eq. (<ref>) of Section <ref>. 

D_v is the vapor diffusion coefficient, whose dependence on T (in ) follows the expression: <cit.>


    D_v(T)=D_v0(T+T_0/T_0)^1.81,

where D_v0 is the vapor diffusion coefficient at the freezing point and T_0=273.15.






 §.§ Phase-dependent functions
 
The model equations introduced above rely on a few parameters that are phase-dependent. For material property parameters such as ρ, c_p, K, we simply use phase-weighted averages as shown in Eq. (<ref>). However, parameters such as the mobility function M_0 in Eqs. (<ref>)–(<ref>) and ρ_SE in Eq. (<ref>) are related to phase change dynamics at phase-phase boundaries or triple junction regions, and thus require more careful mathematical formulation. Here we resort to the ternary diagram to illustrate these modeling choices.

A ternary diagram is the common tool to represent three-phase systems. 
In a ternary diagram, the vertex l represents the point ϕ_l=1, while the opposite side of that vertex represents the points ϕ_l=0 (for l={i,w,a}). The interior of the diagram represents points where the three phases coexist. Thus, any valid {ϕ_i,ϕ_w,ϕ_a} configuration is represented by a point in the ternary diagram. In particular, evaporation takes place on the side ϕ_i=0, sublimation on the side ϕ_w=0, and solidification on the side ϕ_a=0.

At the boundaries of the ternary diagram (i.e. ϕ_a=0, ϕ_w=0, and ϕ_i=0), the mobility function M_0 must take the values M_sol, M_sub, and M_eva that correspond to the kinetics of solidification, sublimation and evaporation, respectively. The exact expressions for these parameters are provided in Eqs. (<ref>), (<ref>), and (<ref>) of the Supporting Information. 

Within the interior region of the ternary diagram, however, M_0 needs to be interpolated based on M_sol, M_sub, and M_eva. A common choice is the cubic interpolation, which presents issues for two reasons<cit.>: (1) the triple junction is a region, not a point, where the three phases coexist; (2) the values of M_sol, M_sub, and M_eva range several orders of magnitude in this region (see Table <ref> in the Supporting Information).
Thus, a cubic interpolation of M_0 would result in large gradients in mobility within the interior of the ternary diagram, leading to severe numerical challenges. 
In addition, a non-unique value of M_0 within the triple junction is not physically consistent and result in unrealistic behavior of the triple junction.

To avoid these two issues, we propose a piecewise constant interpolation of M_0, similar to that proposed in <cit.>, where M_0 takes a constant value near each of the three sides and in the interior region of the ternary diagram:


    M_0(ϕ)={[                                      M_sol if ϕ_a<=ϕ_i,  ϕ_a<=ϕ_w,   and  ϕ_a < 0.01,;                                      M_eva  if ϕ_i<ϕ_a,  ϕ_i<=ϕ_w,   and  ϕ_i < 0.01,;                                      M_sub   if ϕ_w<ϕ_a,  ϕ_w<ϕ_i,   and  ϕ_w < 0.01,;                                       M_av                                 otherwise, ].


where we take M_av as the geometric mean (i.e., M_av=√(M_subM_solM_eva)). We find that Eq. (<ref>), compared to a cubic interpolation, improves the numerical efficiency and allows us to simulate triple junctions while respecting the actual kinetics of the different phase transitions. 

The density ρ_SE must take the value ρ_i in case of sublimation and ρ_w in case of evaporation (i.e., ϕ_w=0 and  ϕ_i=0, respectively, in the ternary diagram). The value of ρ_SE along the side ϕ_a=0 is irrelevant because ρ_SE∂ϕ_a/∂ t (see Eq. (<ref>)) is zero on that side. We propose a cubic interpolation between the sides ϕ_w=0 and ϕ_i=0 in the ternary diagram, such that the derivatives of ρ_SE(ϕ) are zero on the sides ϕ_w=0 and ϕ_i=0. We define the function ρ_SE as


    ρ_SE(ϕ)={[ ρ_i+ρ_w/2+ρ_w-ρ_i/4x_SE(3-x_SE^2)                        if ϕ_a≠ 1,;                         ρ_i+ρ_w/2                       if ϕ_a = 1, ]. 
          with   x_SE=1-ϕ_a-2ϕ_i/1-ϕ_a.


Equation (<ref>) displays a discontinuity at ϕ_a=1. From a physical and numerical point of view, this discontinuity does not represent an issue because the term ρ_SE∂ϕ_a/∂ t is  zero if ϕ_a=1.
Nevertheless, we implement a regularization of Eq. (<ref>) (see Section <ref> in the Supporting Information).



 §.§ The Gibbs-Thomson condition

The Gibbs-Thomson effect describes the deviations in the equilibrium chemical potential at a two-phase boundary due to the curvature of the interface. Such effect readily applies to the three types of interface considered in wet snow and influences both the kinetics and equilibrium conditions experienced by curved interfaces. 
In Section <ref> of the Supporting Information, we show that our model is equivalent to the two-phase models for solidification, sublimation, and evaporation when only two phases are present, and the resulting reduced model equations recover the Gibbs-Thomson for two-phase boundaries under certain conditions. In particular, our model reproduces, under certain parameter regimes, the kinetics defined by the Gibbs-Thomson condition for ice-liquid interfaces:

    T-T_melt/L_sol/c_p,w=-d_solχ-β_solv_n,

where T is the interface temperature, β_sol is the kinetic attachment coefficient, d_sol is the capillary length, χ is the curvature of the interface (positive for spherical ice grains), and v_n is the normal velocity of the interface (positive for ice growth).
Note that, at thermodynamic equilibrium (i.e., v_n=0), Eq. (<ref>) simplifies to the classic Gibbs-Thomson equation:

    T^equil=T_melt-L_sold_sol/c_p,wχ,

which states that the actual equilibrium temperature for solidification is lower than T_melt=0^∘C for spherical ice grains. Similarly, our model captures, under certain parameter regimes, the kinetics defined by the Gibbs-Thomson condition for air-ice interfaces:

    ρ_v-ρ_vs^I(T)/ρ_vs^I(T)=d_subχ+β_subv_n,

and for air-liquid interfaces:

    ρ_v-ρ_vs^W(T)/ρ_vs^W(T)=d_evaχ+β_evav_n,

where β_sub and d_sub are the Gibbs-Thomson coefficients for sublimation and β_eva and d_eva are the Gibbs-Thomson coefficients for evaporation. The model's versatility in capturing the Gibbs-Thomson conditions for all three types of two-phase boundaries rely on the phase-dependent formulation of M_0 as discussed in Section <ref>.

Note that because we do not assume equal Σ_α's, spurious phase (nonphysical third phase) may appear along phase-phase boundaries, leading to inaccuracies in the equilibrium interface solution. We mitigate this issue using the Λ-term in the free energy formulation (Eq. (<ref>)). In addition, we note that the Gibbs-Thomson condition is only recovered in phase-field models under certain parameter conditions <cit.> or mathematical constructions through asymptotic analysis <cit.>. Our current formulation does not entail the mathematical additions needed to recover the Gibbs-Thomson condition for all interfaces for all parameter regimes; however, under certain parameter regimes, the model recovers these conditions. More details are provided in Section <ref> and Supplementary Information.



§ NUMERICAL IMPLEMENTATION
 

We first apply the phase constraint ϕ_w=1-ϕ_i-ϕ_a to Eqs. (<ref>)–(<ref>), which allows us to compute the Lagrange multiplier β in Eq. (<ref>) and eliminate one phase evolution equation:

    β = 1/1/Σ_a+1/Σ_w+1/Σ_i3/ε( 1/Σ_i∂ F^tri/∂ϕ_i + 1/Σ_w∂ F^tri/∂ϕ_w + 1/Σ_a∂ F^tri/∂ϕ_a)

The resulting system of four coupled partial differential equations are:


    ∂ϕ_i/∂ t       =         - 3 M_0(ϕ)/εΣ_T[ (Σ_w+Σ_a)∂ F^tri/∂ϕ_i -Σ_a∂ F^tri/∂ϕ_w - Σ_w∂ F^tri/∂ϕ_a] + 3M_0(ϕ)ε∇^2ϕ_i 
                - α_solϕ_i^2ϕ_w^2T-T_melt/L_sol/c_p,w + α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i
               +α_nuclM_0(ϕ)ϕ_a^2 [ϕ_w^2 N_i(T)-ϕ_i^2 N_w(T)],
    ∂ϕ_a/∂ t      =        - 3 M_0(ϕ)/εΣ_T[ -Σ_w∂ F^tri/∂ϕ_i -Σ_i∂ F^tri/∂ϕ_w + (Σ_i+Σ_w)∂ F^tri/∂ϕ_a] + 3M_0(ϕ)ε∇^2ϕ_a 
                - α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i - α_evaϕ_w^2ϕ_a^2ρ_v-ρ_vs^W(T)/ρ_w, 
    1/ξ_Tρ(ϕ)c_p(ϕ)∂ T/∂ t      =      ÷[ K(ϕ) T ] + ρ(ϕ)L_sol(∂ϕ_a/∂ t + ∂ϕ_i/∂ t) -ρ(ϕ)L_sub∂ϕ_a/∂ t,
    1/ξ_v∂(ϕ_aρ_v)/∂ t      =      ÷[ ϕ_a D_v(T)ρ_v ] + ρ_SE(ϕ) ∂ϕ_a/∂ t,


where F^tri, ρ_SE and M_0 are defined in Eqs. (<ref>), (<ref>), and (<ref>), respectively. 
The saturated vapor densities ρ^I_vs and ρ^W_vs are defined in Eqs. (<ref>)–(<ref>).
The parameter Σ_T=Σ_iΣ_w+Σ_iΣ_a+Σ_wΣ_a, where Σ_l's are defined in Eq. (<ref>).
The nucleation functions N_i and N_w are defined in Eq. (<ref>).
The phase-dependent functions ρ, c_p, and K are defined through Eq. (<ref>) and D_v follows Eq. (<ref>). The time scaling parameters ξ_T and ξ_v are introduced in the next section (Section <ref>).
The rest of the model parameters are listed in Tables <ref> and <ref> in the Supporting Information.




 §.§ Temporal scaling

Wet snow metamorphism involves multiple coupled processes at different time scales. Assuming a characteristic length scale of L=10^-4m, the characteristic times for different processes can range from ∼ 10^-4 s for vapor diffusion to ∼ 10^4 s for sublimation (see Section <ref> in the Supporting Information for detailed calculations). 
A monolithic numerical solver requires the use of time steps that resolve the fastest process (i.e., vapor diffusion at ∼ 10^-4 s ), which would significantly increase the computational cost. Here, we leverage the procedure explained in <cit.> to speed up the simulations. This approach leverages the fact that T and ρ_v are quasi-steady compared to the phase transition kinetics. 
The procedure consists of multiplying the right-hand side of the T and ρ_v equations (Eqs. (<ref>) and (<ref>), respectively) by a time-scale factor (ξ_T and ξ_v respectively), so that the process of vapor and thermal diffusion is numerically slowed down by orders of magnitude while still maintaining a quasi-steady state in T and ρ_v. 
This approach enables the use of larger time steps without introducing noticeable errors in the simulations. In this work, we take ξ_v=10^-3 and ξ_T=1 or ξ_T=10^-2 depending on whether or not solidification occurs in any parts of the domain.
More details on this procedure are described in Section <ref> of the Supporting Information.





 §.§ Kinetic parameters constraints

In order to accurately capture the interface kinetics set by the Gibbs-Thomson conditions, certain model parameters need to be constrained by the interface physics. To this end, the relations between the Gibbs-Thomson parameters (β_j,d_j) and our model parameters (M_j,α_j) (for j={sol,sub,eva}; see Section <ref>) can be formulated from asymptotic analysis <cit.> but are only valid under certain conditions. In particular, we note that these asymptotic analysis do not readily apply to the wet snow problem as they are originally developed for compositional problems with symmetric <cit.> and asymmetric diffusivities <cit.>. Solutions using anti-trapping current have been devised for the thermal problems, but only with symmetric diffusivities <cit.>. Thus, in order to robustly capture the Gibbs-Thomson condition under all parameter regimes for the wet snow problem, further mathematical additions remain to be developed.

We note, however, in the earlier and similar work on dry snow metamorphism by <cit.>, it is shown that if we impose ε<Dβ', where D are the different thermal or vapor diffusivities and β' is the scaled kinetic coefficient, then no anti-trapping current is needed to correctly recover the sharp-interface limit. In particular, for the wet snow problem, this corresponds to the following sets of constraints: 

    ε < β_solK_w/ρ_w c_p,w≈ 10^-5m;  (
    for water-ice interface)
       ε < β_subρ_vs/ρ_iK_a/ρ_a c_p,a≈ 10^-7m;  (
    for air-ice interface)
       ε < β_evaρ_vs/ρ_iK_w/ρ_w c_p,w≈ 10^-8m.  (
    for air-water interface)

These conditions impose a restriction on ε, which represents the diffusive interface width at phase-phase boundaries. The value of ε dictates the spatial discretization of our model, which must be fine enough to resolve the interface width with sufficient numerical elements. Given that the most restrictive conditions of the above requires ε<10^-8 m, this severely restricts the domain size we can simulate. In order to speed up the simulations, we consider a value ε<10^-6 m, which allows us to use a coarser spatial discretization. As a result, our simulation are less accurate in capturing interface kinetics along the air-ice and air-water interfaces. However, considering the phase change kinetics involving the air phase is slow compared to the solidification/melting processes along the water-ice interface, we suspect the overall error on wet snow processes is small. 




 §.§ Numerical methods


We perform both 1D and 2D simulations of our model equations using the finite element method. In particular, we use a uniform mesh composed of bilinear (linear in 1D) basis functions. For the time integration, we use a semi-implicit algorithm based on the generalized-α method <cit.>, where we treat implicitly all the terms except the function M_0 and the parameter ξ_T, which are constant during each time step. We use the Newton-Raphson method with adaptive time stepping scheme to solve the resulting nonlinear system. To avoid singularities and numerical issues we regularize the functions ρ, c_p, K, and ρ_SE (see Section <ref> in the Supporting Information). 


We impose no-flux boundary conditions for the phase variables and Dirichlet boundary condition for the temperature.  The vapor boundary condition is either Dirichlet or no-flux depending on specific case studies. In most simulations, we initialize the vapor density with the saturated vapor density for ice at the prescribed temperature.









§ RESULTS



 §.§ Directional solidification and nucleation of a third phase

In this section, we demonstrate the ability of our model to capture directional solidification as imposed by a temperature gradient, and the nucleation of a third phase in a system that is initially composed of two phases only. In particular, we present three 1D simulations in Fig. <ref> that show (a) the complete freezing of water in an initial ice-water-air system (Fig. <ref>A), (b) water nucleation in an initial ice-air system (Fig. <ref>B), and (c) ice nucleation in an initial water-air system (Fig. <ref>C).  To perform these 1D simulations, we impose fixed temperatures and no-flux boundary condition for the vapor phase on the two ends of the domain, which measures 1 in length and is discretized with 2000 elements.



In the freezing simulation (Fig. <ref>A), the initially imposed temperature gradient results in the directional solidification of water towards the air phase. At each time step, the temperature distribution appears piecewise linear, which confirms the quasi-steady behavior of T as discussed in Section <ref>.
Although the ρ_v dynamics are also quasi-steady, its distribution is not always linear (see t=27 and 27.5 in Fig. <ref>A). This nonlinear response is primarily triggered when a new ice-air interface is established upon complete freezing (∼27.5), and a new equilibrium is being established to satisfy the saturated vapor density in the presence of ice, prescribed by Eq. (<ref>).





The nucleation simulations demonstrate significantly more nonlinear dynamics. In the water nucleation simulation (Fig. <ref>B), we impose 0 and 6 on the left and right boundaries of the domain respectively to induce melting at the initial ice-air interface. In the ice nucleation simulation (Fig. <ref>C), the water-air system is initially at -2 uniformly and we consider the ice nucleation temperature T_nucl=-1. 
Ice nucleation onsets, preferentially towards the water phase, well before the temperature on the water-air interface increases to T_nucl  (Fig. <ref>C, t=0.2). At t=0.3, when T>T_nucl on the interface, the ice phase has fully developed. 
Both types of nucleation described here are highly energetic events, leading to rapid changes in temperature and saturated vapor density at the nucleation region. 
As a result, the T and ρ_v profiles are no longer linear, in contrast to the directional freezing problem (Fig. <ref>A). 
We note that here, nucleation only occurs at the phase-phase interface due to its mathematical formulation in Eq. (<ref>).





We note that the phase field variables maintain the tanh-profile (quasi-equilibrium profile) during the freezing simulation (insets of Fig. <ref>A), but such profiles are temporarily lost during the nucleation simulations (insets Figs. <ref>B, C). 
In the latter cases, the loss of the tanh-profile is due to changes in mobility M_0 (Eq. (<ref>)) during the dynamic nucleation events, and could result in small errors in curvature calculations and thus influence the kinetics of phase change through the Gibbs-Thomson condition. Nevertheless, the tanh-profile is recovered shortly after nucleation (e.g., at t=1 in Figs. <ref>B, C); thus the undesired loss of the tanh-profile is temporary (occurs only during nucleation) and for this reason, we assume its impact on the overall kinetics of the problem is negligible.




 §.§ Simulation  of dry snow metamorphism in 2D

Under thermal conditions when liquid water phase is absent (ϕ_w=0), our model reduces to the two-phase model for dry snow metamorphism  first proposed in <cit.>:

    ∂ϕ_i/∂ t   = -3 M_0( 1/εϕ_i(1-ϕ_i)(1-2ϕ_i)   - ε∇^2ϕ_i ) + α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i
    1/ξ_Tρ(ϕ)c_p(ϕ)∂ T/∂ t   = ÷[ K(ϕ) T ]  -ρ(ϕ)L_sub∂ϕ_a/∂ t,
    1/ξ_v∂(ϕ_aρ_v)/∂ t   =÷[ ϕ_a D_v(T)ρ_v ] + ρ_SE∂ϕ_a/∂ t,

with ∂ϕ_a/∂ t=-∂ϕ_i/∂ t. 
Here, we show a 2D example of the dry metamorphism model using a simple snow geometry composed of four circular ice grains in contact (see t_0 in Fig. <ref>A).
The domain measures 0.2×0.2^2 and is meshed with 400× 400 elements.
We impose a fixed temperature T_B=-5 and no-flux of ρ_v on all boundaries of the domain. Over the 90 minute time span of the simulation, we plot the time evolution of the ice phase ϕ_i and vapor concentration ρ_v (Fig. <ref>A), along with the T (Fig. <ref>B) and the total ice interface length S_i (Figs. <ref>C and D).
Also known as coarsening, dry snow metamorphism under isothermal condition is a curvature-driven process, which minimizes the total interfacial length and the curvature of the system. The process penalizes regions of high curvature and is driven by vapor mass transfer between regions of different curvatures <cit.>.
Our simulation readily illustrates the above mechanism. We observe that the ice grains quickly sinter (before t_1=40) and the coarsening process gradually slows down, as measured by the decay in slope in Figs. <ref>C and D. Accompanying these geometric changes are the dynamics in T distribution due to latent heat generation in regions of higher curvature (see Fig. <ref>B). 
The simulation also captures the entrapment of an air bubble during the coarsening process.
At steady state (e.g., t=3 in Fig. <ref>D), the vapor concentration inside the bubble is set by its curvature according to the Gibbs-Thomson condition (Eq. (<ref>)).





 §.§ Simulations of wet snow dynamics in 2D


In this section, we perform a suite of numerical simulations of our full model to investigate wet snow dynamics as it undergoes phase transition. Due to high computational cost of the model, here we only focus on small sized domains (∼mm) in 2D.




  §.§.§ Influence of vapor concentration


In this part, we study the impact of vapor concentration on the kinetics of melt generation when the initially dry material is subjected to melting. To achieve this, we consider the same four-grain geometry as the initial dry snow simulation in Fig. <ref>, we impose a boundary temperature of T_B=1 and run two simulations with different vapor concentrations on the boundary, denoted ρ_v,B. In particular, we impose an undersaturated vapor density, ρ_v,B=0.8ρ^I_vs(1), in the first simulation (Fig. <ref>A) and a saturated vapor density, ρ_v,B=ρ^I_vs(1), in the second one (Fig. <ref>B).

We quantify these two simulations by plotting the time evolution of the spatially averaged temperature T, volume of ice phase V_i and water phase V_w in Figs. <ref>A and C. The results illustrate that, when the initial vapor phase is undersaturated, melt nucleation is delayed and the kinetics of melt generation is slower (Fig. <ref>C). We explain this from the perspectives of thermal balance. Because both sublimation and melt generation requires energy input, an unsaturated vapor phase will reduce the amount of energy available for melting. This is evident when we observe that the average temperature decrease is stronger in the unsaturated vapor case (Fig. <ref>A, dashed line). 
This lower temperature also causes a delay in water nucleation (Figs. <ref>A, Figs. <ref>C). We also compare the final snapshots of the two simulations, taken at different times (t=45 for the first and t=16 for the second simulation) but both correspond to a state when ice no longer exists (Fig. <ref>B). It is evident that there is significantly less melt water in the first simulation, which is caused by the significant sublimation that claims part of the ice in order to saturate the vapor phase.





Finally, we want to remark that while water nucleation occurs in many parts of the ice interface, melting into the ice phase only advances in some regions of the interface because the process is limited by the thermal energy influx (Figs. <ref> and <ref>). In particular, we observe that melting usually starts at the grain contact regions because larger curvatures induce higher temperatures, as already shown in the case with dry snow (Fig. <ref>B).







  §.§.§ Influence of temperature

In this part, we study the impact of temperature on wet snow evolution. Using the same initial dry snow geometry (Fig. <ref>), here we fix the vapor concentration on the boundary such that ρ_v,B=ρ^I_vs(1) and consider two different boundary temperature T_B's, namely, T_B=1 (Fig. <ref>A) and T_B=2 (Fig. <ref>B).
We further quantify these two simulations with the time evolution of V_i and V_w (Fig. <ref>C) and the ice-water interface length S_iw (Fig. <ref>D). As expected, we observe that melting is faster at a higher boundary temperature (Fig. <ref>C). 
In particular, our results illustrate that a higher temperature accelerates melting by promoting melt nucleation and thus increases the amount of ice-water interface available for melt advancing (Fig. <ref>D). In particular, we observe that a film of nucleated water completely surrounds the ice phase at the higher temperature (Fig. <ref>B), while water film only appears at some regions of the ice interface at the lower temperature(Fig. <ref>A). 
After nucleation, we observe that melting only advances at the three grain contact locations for low T_B, while melting progresses in a few more locations, some away from grain contacts, at high T_B (Fig. <ref>B).

Finally, upon complete melting, the two simulations display a small difference in V_w (Fig. <ref>C, at t=10). This difference is caused by the imposed T and ρ_v boundary conditions. Since we impose ρ_v,B=ρ^I_vs(1) for both simulations, vapor concentration on the boundary is out of equilibrium for the case of T_B=2. This induces a vapor outflux through the boundary which is reflected in final value of V_w via sublimation/evaporation. 







  §.§.§ Influence of liquid water content




In this section, we study the influence of the liquid water content (LWC) on wet snow metamorphism. We consider a simplified snow geometry composed of circular ice grains in a domain of 0.2×0.2^2 meshed with 800× 800 elements (Fig. <ref>A, t=0s).  We run three simulations with different initial LWC (LWC_0) that correspond to dry (LWC_0=0), partially wet (LWC_0=0.15), and fully wet (LWC_0=0.46) snow. The initial ice phase distribution is the same for all the three cases. For the partially wet snow, we impose a water film of 2.6 thickness uniformly surrounding the ice grains. For the fully wet snow, we fill the pore space entirely with the liquid phase. We adopt the parameter value ε=2×10^-7m. To keep the system in quasi-isothermal conditions, we impose a fixed temperature T_B=T_melt and no-flux in terms of vapor density ρ_v on all boundaries.


The simulations results (Figs. <ref> and <ref>) illustrate the influence of LWC_0 on the detailed dynamics of metamorphism. 
In particular, when comparing against dry snow, we find that partially wet snow evolves towards larger and fewer ice clusters faster (e.g., LWC_0=0 and LWC_0=0.15 at t=10 in Fig. <ref>A). While this accelerated coarsening in partially wet snow is visually apparent, plot of the normalized interface length S_a does not readily capture this difference (Fig. <ref>B, black and red lines). Note that S_a in the partially wet snow accounts for the interface of the combined phase ice and water (ϕ_i+ϕ_w). Meanwhile, the coarsening speed of the fully wet snow is much faster, as captured visually in Fig. <ref>A and also in terms of the normalized ice interface length S_i (Fig. <ref>B, blue line).


Wet snow also experiences a broader range of dynamic temperatures than dry snow during metamorphism (Fig. <ref>). This may be explained by the fact that in wet snow, the thermal fluctuations are dominated by the latent heat released/absorbed during freezing/melting process, which is orders of magnitude faster than the process of deposition/sublimation in dry snow (see Table <ref> in the Supporting Information). The temperature distribution also reflects the Gibbs-Thomson effect: because the equilibrium temperature (T^equil, Eq. (<ref>)) decreases with increasing ice grain curvature, smaller ice grains require colder temperature (T^equil<T_melt) to be stable (Fig. <ref>). Because we impose a temperature T_melt at the boundaries, which is higher than the equilibrium temperature for curved grains, we find that all ice grains eventually melt away in the case of fully saturated snow (Figs. <ref>A, C, LWC=0.46). Note that here, if we initialize the simulation with concave grain shapes, this will lead to T^equil>T_melt and complete freezing of the water in the case of fully wet snow. 

A key characteristic of dry snow metamorphism is the reduction in microstructure surface area while maintaining an almost constant ice volume (Fig. <ref>B-C, black curves). Interestingly, we find that for the partially wet snow case investigated here, the volume of the air phase is also roughly constant throughout the simulation (not shown here). We attribute this to the fact that water storage capacity of the air is limited (the initial vapor concentration is already close to the saturated vapor density). Nevertheless, we do observe a small mass exchange between the ice and water phases for partially wet snow, as evidenced by the changes in V_i for LWC_0=0.15 in Fig. <ref>C.






















§ NUMERICAL CHALLENGES AND MODEL IMPROVEMENTS





We have so far presented a series of 1D and 2D simulations of wet snow in domains of limited sizes (∼ 0.2mm). The reason we do not explore larger problems in 3D is due to the high computational cost associated with these simulations. This limits our ability to compare with experimental studies, which are always in 3D and done on samples that are ∼cm in size (e.g., in <cit.>). Here we summarize these computational challenges as well as points for model improvements in future work.

In order to accurately capture the kinetics defined by the Gibbs-Thomson conditions, our model requires that the numerical interface width parameter ε<10^-8m (see Section <ref>). The consequence of this requirement is the need for a rather fine numerical mesh that resolves such interface, making it computationally expensive to simulate larger problems. An adaptive mesh refinement algorithm, rather than the uniform mesh used in this work, would allow one to run mm- and cm-scale simulations in reasonable computational times. Additionally, mathematical additions to our model based on an asymptotic analysis for the thermal problem with asymmetric diffusivities could further improve robustness of our model at the interface.

Another cause for high computational cost is the need to resolve the kinetics of different processes. In this work, we resort to a monolithic scheme, where our time step size is limited by the fastest process. We have alleviated this issue partially using the time-scaling strategy explained in Section <ref>. Another way to speed up time integration is to neglect the phase transitions related to the vapor phase during melting/freezing scenarios. The errors introduced with this approach would be small as long as melting/freezing occurs for short time intervals compared to the total time of interest.

Finally, our model is currently not suited for studying the dynamics of the triple junction —the region where the air, liquid and ice phases contact— when the system is near the triple point. 
At the triple point, defined here as {T,ρ_v}={T_melt,ρ^I_vs(T_melt)}, the triple junction is expected to display an equilibrium configuration defined by the surface tensions in Eq. (<ref>). In our model, an initially non-equilibrium triple junction configuration will evolve towards equilibrium through the co-movement of the water-air, air-ice and water-ice interfaces. However, because each interface moves at different speed, as defined by the phase-dependent mobility (Eq. (<ref>)), a true equilibrium may not be possible. Instead, the model produces oscillations around the triple junction that impede the achievement of the equilibrium configuration <cit.>. 
Far away from the triple point, phase change kinetics around the triple junction dominate over the surface tension kinetics and these oscillations do not appear.


For future work, we will focus on more realistic representation of the nucleation process that includes randomized Gaussian noise <cit.>, which would produce a more realistic behavior of the wet snow metamorphism problem. The current model also does not consider the density difference between the solid ice and liquid water phase, which would introduce small mass conservation error (the model is volume conserved). Our model might be expanded based on <cit.> to account for mass conservation during melting/freezing, but further research is needed.




§ CONCLUSIONS

We propose a non-variational phase-field model for wet snow metamorphism.
The model accounts for the ice, water, and air phases, along with the temperature and vapor dynamics, and captures the actual kinetics of solidification, sublimation, and evaporation (and the opposite transitions). 
To the best of our knowledge, this is the first phase-field model that simultaneously reproduces the different phase transitions of water.

The model results unveil the intimate coupling amongst the various transport and phase change processes involved in wet snow metamorphism. Our results show that local humidity conditions affect snow melting and melt refreezing.

The results also reveal the differences between dry and wet snow metamorphism when water flow is not considered: partially and fully wet snow experience larger thermal fluctuations due to the dominant freeze/melt processes and the coarsening rate is higher for fully wet snow compared to dry snow as expected. However, our observations are based on numerically generated partial wet snow where a liquid film of constant thickness is imposed over all ice grain surfaces. We recognize that this way of initializing the simulation may not represent actual liquid film distribution in wet snow and thus may result in unrealistic dynamics. In addition, because we do not consider thin film flow and other capillary-driven processes of the liquid phase, our model may be missing key fluid mechanical processes that influence how wet snow evolves. 


The current results focus on 1D and 2D problems, due to the high computational cost of the model equations. Future modification, extension and application of this model in 3D may be used to quantitatively analyze the evolution of the snow pore structure during wet snow metamorphism, which dictates the thermo-mechanical and hydraulic properties of the snow at larger scales. The proposed modeling framework has proven useful to study wet snow metamorphism and may be the basis to investigate other mm- or cm-scale problems involving water phase transitions such as, e.g., the freezing of a water droplet on a surface <cit.>, the human-induced thermo-mechanical changes of icy planetary surfaces, or water spray cooling.





The authors acknowledge the partial support from the Resnick Sustainability Institute at California Institute of Technology, the National Science Foundation under Grant No. EAR-2243631 and the ACS Petroleum Research Fund Doctoral New Investigator Grant No. 66867-DNI9. The authors also acknowledge insightful discussions with Dr. Quirine Krol.












§ EQUIVALENCE TO MODELS OF TWO-PHASE TRANSITION


In this section we show that our wet snow metamorphism model is equivalent to the two-phase models for solidification, sublimation, and evaporation when only two phases are present in the system. 
Only for this section, we assume that θ_nucl=0 (see Eqs. (<ref>)–(<ref>) in the main text).

To prove the equivalency, we first need to show that a two-phase system remains as a two-phase system and the non-existing phase does not appear in Ω.
It is trivial to show that the phase-fields equations in our model, Eqs. (<ref>)–(<ref>) in the main text, have the property ϕ_l(,t)=0 ∀ t if ϕ_l(,0)=0 and θ_nucl=0, which proves this first condition.

The second condition constitutes the actual equivalence, which we analyze for each phase transition in the following paragraphs.



 §.§ Solidification


Solidification and melting occurs when ϕ_a=0.
We can rewrite the ϕ_i evolution equation (Eq. (<ref>) in the main text) imposing ϕ_a=0 as


    ∂ϕ_i/∂ t = -3 M_0( 1/εϕ_i(1-ϕ_i)(1-2ϕ_i)   - ε∇^2ϕ_i ) - α_solϕ_i^2ϕ_w^2T-T_melt/L_sol/c_p,w.


The Lagrange multiplier in our formulation guarantees that ϕ_i+ϕ_w=0, which implies that ∂ϕ_w/∂ t=-∂ϕ_i/∂ t.

Imposing ϕ_a=0, the temperature evolution equation (Eq. (<ref>) in the main text) is expressed as:


    ρ(ϕ)c_p(ϕ)∂ T/∂ t = ÷[ K(ϕ) T ] + ρ(ϕ)L_sol∂ϕ_i/∂ t.


We can disregard the vapor density equation since there is no air phase.

Equations (<ref>) and (<ref>) constitute a two-phase model for solidification (and melting) known as the generalized Stefan problem <cit.>. 
Under certain parameter regimes, this model reproduces the kinetics defined by the Gibbs-Thomson condition for the ice-liquid water interface, which reads


    T-T_melt/L_sol/c_p,w=-d_solχ-β_solv_n,


where T is the interface temperature, β_sol is the kinetic attachment coefficient, d_sol is the capillary length, χ is the curvature of the interface (positive for spherical ice grains), and v_n is the normal velocity of the interface (positive for ice growth). Karma and Rappel <cit.> showed that the two-phase model captures the Gibbs-Thomson condition kinetics when (i) ε→ 0 and (ii) the model parameters M_0 and α_sol follow the relations

    M_0=M_sol=ε/3τ_sol ,   α_sol = λ_sol/τ_sol,   
        d_sol=a_1ε/λ_sol,   β_sol=a_1(τ_sol/ελ_sol - a_2ε/D^*_iw),


where a_1≈ 5, a_2≈0.1581 [The a_1 and a_2 values are computed by following the same procedure as in <cit.>, applied to Eqs. (<ref>) and (<ref>).], and D^*_iw≈ (D_i+D_w)/2 represents thermal diffusivity, such that D_j=K_j/(ρ_j c_p,j) for j={i,w}. 
However, we note that the above relations assume equal thermal diffusivity around the freezing/melting interface, which is not true for ice and water in our case. Asymptotic analysis assuming asymmetric diffusivities have been conducted for compositional problems <cit.>, but the same type of analysis for the thermal problems remain to be developed. Nevertheless, <cit.> has shown that the phase-field solidification model recovers the sharp-interface solution if ε < β_solD^*_iw≈ 10^-5m, which is true in the simulations presented in this paper.



 §.§ Sublimation


Sublimation and deposition occurs when ϕ_w=0.
After imposing ϕ_w=0, Eqs. (<ref>), (<ref>), and (<ref>) in the main text are expressed as


    ∂ϕ_i/∂ t = -3 M_0( 1/εϕ_i(1-ϕ_i)(1-2ϕ_i)   - ε∇^2ϕ_i ) + α_subϕ_i^2ϕ_a^2ρ_v-ρ_vs^I(T)/ρ_i
       ρ(ϕ)c_p(ϕ)∂ T/∂ t = ÷[ K(ϕ) T ]  -ρ(ϕ)L_sub∂ϕ_a/∂ t,
       ∂(ϕ_aρ_v)/∂ t=÷[ ϕ_a D_v(T)ρ_v ] + ρ_SE∂ϕ_a/∂ t,


with ∂ϕ_a/∂ t=-∂ϕ_i/∂ t. 

These equations constitute the two-phase sublimation model <cit.> and capture the kinetics defined by the Gibbs-Thomson condition


    ρ_v-ρ_vs^I(T)/ρ_vs^I(T)=d_subχ+β_subv_n,


as long as the parameters M_0, α_sub, and ρ_SE follow the relations <cit.>:

    M_0=M_sub=ε/3τ_sub ,   α_sub = λ_sub/τ_sub,   ρ_SE=ρ_i,
       d_subρ_vs^I/ρ_i=a_1ε/λ_sub,   β_subρ_vs^I/ρ_i=a_1(τ_sub/ελ_sub - a_2ε/D^*_ia- a_2ε/D_v0),


where β_sub and d_sub are the Gibbs-Thomson coefficients for sublimation and D^*_ia represents the average thermal diffusivity of ice and air. Again, we note that the above relations assume equal thermal diffusivity around the interface, which is not true for ice and air in our case. Asymptotic analysis assuming for this exact problem remains to be developed. Meanwhile, <cit.> has shown that the above phase-field model recovers the sharp-interface solution if ε < β_subρ_vs^I/ρ_iD^*_ia and ε < β_subρ_vs^I/ρ_iD_v0. These two conditions yield the requirement that ε < 10^-7m, which we do not satisfy in the simulations presented. Thus, our numerical results do not strictly recover the Gibbs-Thomson condition for air-ice interfaces.



 §.§ Evaporation


The derivation for the evaporation case (ϕ_i=0) is analogous to the sublimation case. The final two-phase model for evaporation (without flow) is analogous to the 2-phase sublimation model (Eqs. (<ref>)–(<ref>)), where the evaporation latent heat is expressed as (L_eva=L_sub-L_sol). Here, we simply list the final relations between the model parameters M_0, α_eva, ρ_SE, and the Gibbs-Thomson coefficients  β_eva and d_eva, which read


    M_0=M_eva=ε/3τ_eva ,   α_eva = λ_eva/τ_eva,   ρ_SE=ρ_w,
       d_evaρ_vs^W/ρ_w=a_1ε/λ_eva,   β_evaρ_vs^W/ρ_w=a_1(τ_eva/ελ_eva - a_2ε/D^*_wa- a_2ε/D_v0),


where D^*_wa represents the average thermal diffusivity of water and air. We note again that the above relations assume equal thermal diffusivity around the interface, which is not true for water and air in our case. Asymptotic analysis assuming for this exact problem remains to be developed. Meanwhile, <cit.> has shown that the above phase-field model recovers the sharp-interface solution if ε < β_evaρ_vs^W/ρ_wD^*_wa and ε < β_evaρ_vs^I/ρ_wD_v0. These two conditions yield the requirement that ε < 10^-8m, which we do not satisfy in the simulations presented. Thus, our numerical results do not strictly recover the Gibbs-Thomson condition for air-water interfaces.







§ ADDITIONAL DETAILS FOR NUMERICAL IMPLEMENTATION



 §.§ Determining temporal scaling coefficients

Wet snow metamorphism involves processes at different time scales.

The characteristic times for solidification (t_sol), evaporation (t_eva), sublimation (t_sub), thermal diffusion (t_T,i, t_T,w, and t_T,a), and vapor diffusion (t_v), assuming a characteristic length scale of L=10^-4m, are

    t_sol≡ L/v_n,sol∼ 1 s,   t_sub≡ L/v_n,sub∼ 10^4 s,   t_eva≡ L/v_n,eva∼ 10^3 s,   t_T,i≡ L^2/D_i ∼ 10^-2 s, 
        t_T,w≡ L^2/D_w ∼ 10^-1 s,   t_T,a≡ L^2/D_a ∼ 10^-3 s,  t_v ≡ L^2/D_v0∼ 10^-4 s,

where thermal diffusivities are calculated as D_l=K_l/(ρ_l c_p,l), for phase l, and we consider typical interface velocities v_n,j, for j={sol,sub,eva} (see Table <ref>).

These values show that sublimation and evaporation are several orders of magnitude slower than solidification and any diffusion process. 
A monolithic numerical solver requires the use of time steps that capture the lower characteristic time (i.e., t_v), which would significantly increase the computational times. 

Here, we leverage the procedure explained in <cit.> to speed up the simulations. 
This approach leverages the fact that T and ρ_v exhibit a quasi-steady state compared to the phase transition kinetics. 
The procedure consists of multiplying the right-hand side of the T and ρ_v equations (Eqs. (<ref>) and (<ref>) in the main text, respectively) by a time-scale factor, so that the characteristic times for vapor and thermal diffusion increase some orders of magnitude while T and ρ_v keep displaying a quasi-steady state.
This approach enables the use of larger time steps without introducing noticeable errors in the simulations.

We multiply the right-hand side of the ρ_v equation by the time-scale factor ξ_v<1, which provokes an increase of the vapor diffusion characteristic time to t^*_v=t_v/ξ_v.
As long as t^*_v does not exceed t_sol, t_sub, and t_eva,

vapor concentration displays a quasi-steady state and the overall problem kinetics is not affected. This condition leads to ξ_v>10^-4.

For the T equation we consider the largest thermal diffusion characteristic time, t_T,w.
In Eq. (<ref>) we observe that t_T,w∼ t_sol, while t_T,w< t_sub and t_T,w< t_eva. Thus, T displays a quasi-steady state when solidification does not occur. 
We follow the same procedure and multiply the right-hand side of the T equation by the time-scale factor ξ_T<1 when solidification does not take place. 
As long as the new characteristic time t^*_T,w=t_T,w/ξ_T does not exceed t_sub and t_eva, temperature displays a quasi-steady state and the problem kinetics is not affected. This condition leads to ξ_T>10^-4.
In this work, we take ξ_v=10^-3 and ξ_T=1 or ξ_T=10^-2 depending on whether or not solidification occurs in any parts of the domain. 





 §.§ Kinetic parameters constraints on grid resolution


If we assume equal diffusivity for all phases, then the relations between the Gibbs-Thomson parameters (β_j,d_j) and our model parameters (M_j,α_j) (for j={sol,sub,eva}) can be derived from asymptotic analysis <cit.> and are valid under the following conditions:


    1. χε<1,     2. ε v_n,j/D_k<1,      3. λ_j u_j<1,      4. τ_j v_n,j / ε <1,


for k={i,w,a,v0}, where ε represents the interface width, D_k is the thermal or vapor diffusivity coefficient, v_n,j is the normal velocity of the interface, λ_j and τ_j are parameters defined in Section <ref> in the Supporting Information, and u_j is the phase-change factor that multiplies the terms α_jϕ_l^2ϕ_m^2 in the phase-field equations (for {l,m}={i,w,a}; see Eqs. (<ref>) and  (<ref>) in the main text).
Condition 1 imposes an interface width smaller than the curvature radius,
condition 2 guarantees that thermal and vapor diffusion are faster than the interface motion, and
conditions 3 and 4 ensure the preservation of a tanh-profile (i.e., quasi-equilibrium profile) across the interface. We use these conditions to guide the choice of ε in our model; however, we recognize that these conditions are only valid for equal diffusivity problems, which is not true for the wet snow problem.

These conditions impose a restriction on ε, which represents the numerical interface width and is an adjustable model parameter. 




To find a valid range for ε, we first estimate the Gibbs-Thomson parameters β_j and d_j from the literature <cit.> (see Table <ref>). 
Note that β_j may range several orders of magnitude depending on ambient conditions and growth type <cit.>.

We next estimate the typical values of u_j and v_n,j (Table <ref>), based on T and ρ_v distributions observed during the different phase transitions. 
We use the Gibbs-Thomson conditions defined in Section <ref> of the Supporting Information to relate u_j and v_n,j, under the assumption that the curvature χ≈0.

In order to express λ_j and τ_j as a function of β_j and d_j (see Eqs. (<ref>), (<ref>), and (<ref>)), we assume ρ^I_vs/ρ_i=ρ^W_vs/ρ_w= 5×10^-6, which is the approximate value at the freezing point.
For condition 1, we consider typical ice curvatures χ∼ 10^4 m^-1.
With this information, we solve for ε in Eq. (<ref>).
We indicate the less favorable valid range of ε in Table <ref>.

We find that condition 3 is the most restrictive, such that ε<10^-8 m. 

In order to speed up the simulations, we consider a value of d_sol, d_eva and d_sub one or two orders of magnitude higher than its actual value (see bottom rows of Table <ref>).
This is because the T- and ρ_v-terms in the Gibbs-Thomson conditions (Eqs. (<ref>) and (<ref>)) dominate the interface motion compared to the curvature terms (d_j) for χ∼ 10^4 m^-1 in most cases.

Doing so, the ε constraint in Eq. (<ref>) relaxes to ε<10^-6 m, which allows us to use a coarser spatial discretization, whereas the wet snow metamorphism kinetics is not significantly affected. 

If compared against experiments, deviations in our modeled kinetics from the observed kinetics could be attributed to this numerical treatment and, hence, the actual values of 
d_sol, d_eva and d_sub must be considered.



 §.§ Parameter values


We list the value of the parameters used in our model at the bottom of Table <ref> and in Table <ref>.
For simplicity, we assume that the values of these parameters are constant.
The values of the parameters k_j and g_j in the saturated vapor pressures are listed in Table <ref>.
The parameter ξ_T is defined as


    ξ_T={[             1 if  Γ_iw > ϵ,;         10^-2   if  Γ_iw≤ϵ, ].


where Γ_iw is the length (in 2D) of the ice-water interface and the parameter ϵ is close to zero. In particular, we compute the interface as Γ_iw=∫_Ωϕ_i^2ϕ_w^2 dx and we take ϵ=2× 10^-4ε in 1D and ϵ=2× 10^-6ε√(S_Ω) in 2D, where S_Ω is the area of Ω. 









 §.§ Initial and boundary conditions


For the phase fields and the temperature we consider the boundary conditions


    ϕ_i·=0,   ϕ_a·=0,   T=T_B(,t)  on ∂Ω,


where ∂Ω is the boundary of Ω and the boundary temperature T_B may take different values on each boundary and may evolve in time.
For the vapor concentration, we consider a fixed concentration on the boundary or we impose no-flux through the boundary, depending on each simulation, which are expressed as


    ρ_v=ρ_v,B  or  ρ_v·=0  on ∂Ω,


where ρ_v,B is a constant value.

The initial conditions of our problem may be written as


    ϕ_l(,0)=0.5(1+tanh(d_l()/2ε)),   T(,0)=T_0(),   ρ_v(,0)=ρ^I_vs(T_0),


where l={i,a}, d_l is the signed distance to the interface of phase ϕ_l and T_0() is a linear function consistent with the temperature boundary condition T_B. 
The distance d_l depends on the geometry considered in each simulation. 




 §.§ Spatial, time discretization and variable regularization


We use the Finite Element Method to solve our model. In particular, we use a uniform mesh composed of bilinear (linear in 1D) basis functions. 

We compute the weak form by multiplying the model equations (Eqs. (<ref>)–(<ref>) in the main text) by weighting functions, integrating over Ω, and integrating by parts taking into account the boundary conditions defined in Eqs. (<ref>) and (<ref>).
To obtain the Galerkin form, we substitute the problem unknowns and the weighting functions by the basis functions.

The phase field variables may take values ϕ_l<0 and ϕ_l>1 during transient states, with l={i,w,a}.
To avoid singularities and numerical issues we regularize the functions ρ, c_p, K, and ρ_SE: 
We restrict the value of ϕ_l between 0 and 1 only when we compute the functions ρ, c_p, and K.

For the regularization of ρ_SE we
(1) implement an additional cubic interpolation locally around ϕ_a=0 which smooths out the discontinuity at that point, and
(2) extend ρ_SE off the ternary diagram constant in the normal direction to the diagram sides.




 Paper Title: Simulating Vibrationally-Resolved X-ray Photoelectron Spectra of Flexible Molecules: Linear Alkanes C$_{n}$H$_{2n+2}$ ($n$=1-8).






 MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China
Hefei National Laboratory for Physical Science at the Microscale, University of Science and Technology of China, 230026 Hefei, China
Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal Institute of Technology, SE-106 91, Stockholm, Sweden



MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China



Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, China








wjhua@njust.edu.cn
 MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China


§ ABSTRACT  
We integrated full core-hole density functional theory with Franck-Condon calculations, considering Duschinsky rotation, to simulate vibrationally-resolved C1s X-ray photoelectron spectra (XPS) of eight linear alkanes, from methane to octane (C_nH_2n+2, n=1–8). Results align excellently with experimental absolute binding energies and profiles. The spectrum of ethane serves as a “spectral seed”, with each longer alkane's atom-specific spectrum displaying similar characteristics, albeit with shifts and slight intensity adjustments. Detailed assignments clarify the distinct spectra in short alkanes (n=1–4) and their stabilization in long alkanes (n=5–8). Carbons are classified as central or distal (C_1 and C_2), with central carbons contributing nearly identically to the lowest-energy feature A, while distal carbons contribute to the second lowest-energy feature B (both are 0-0 transitions). Increasing molecule size adds more central carbons, enhancing feature A and weakening feature B. Our analysis identifies two dominant Franck-Condon-active vibrations: C^*–H stretching (∼3400-3500 cm^-1) and bending (∼1400-1500 cm^-1) modes. Structural analysis shows that core ionization minimally affects alkane geometry, less than in ring compounds from previous studies. This work extends our protocol from rigid ring compounds to flexible molecules, contributing to building a high-resolution theoretical XPS library and enhancing the understanding of vibronic coupling.

 

Simulating Vibrationally-Resolved X-ray Photoelectron Spectra of Flexible Molecules: Linear Alkanes C_nH_2n+2 (n=1-8)
    Weijie Hua
    August 5, 2024
=====================================================================================================================



§ INTRODUCTION


X-ray photoelectron spectroscopy (XPS) is a powerful analytical technique that provides detailed information about the chemical composition and electronic structure of molecules and materials.<cit.> High-resolution XPS spectra allow for investigation of fine structures, which predominantly originate from the vibronic coupling effect. The sub-eV fine structures carry crucial insights into the interplay between nuclear and electronic motions, unveiling the fundamental physics underpinning core hole ionization dynamics and alterations in potential energy surfaces (PESs) between ground and core-ionized states. Historically, vibronic fine structures in XPS spectra were observed as early as 1974 by Siegbahn and coworkers<cit.> while studying the spectra of CH_4. The continuous advancements in synchrotron radiation technologies have propelled XPS to higher resolutions, unlocking more comprehensive chemical information about the chemical systems.<cit.> 

XPS databases have been helpful in interpreting the structure-spectroscopy relation. However, existing experimental XPS databases<cit.> primarily focus on binding energy (BE) values, often omitting intensity data, limiting the depth of analysis. Even for BEs, different experiments sometimes report values that differ over 1 eV. An ultimate goal is to create a comprehensive, high-resolution theoretical library for XPS spectra with both accurate BEs and profiles. Preliminary studies were conducted to investigate a family of molecules (azines<cit.>, indoles<cit.>, polycyclic aromatic hydrocarbons (PAHs)<cit.>, and some other five- and six-membered ring molecules<cit.>, biomolecules<cit.>) with similar structures, by combing density functional theory (DFT) with the full core-hole (FCH) approximation and Franck-Condon (FC) simulations which incorporate the Duschinsky rotation (DR) effect<cit.> (denoted as the FCH-DR method for simplicity).<cit.> Grounded on the harmonic oscillation (HO) approximation, the calculations yielded good agreement with experiments.<cit.> The comparative analysis on theoretical and experimental high-resolution spectra offers a more rigorous evaluation, extending beyond solely assessing binding energies, for the accuracy of the chosen theoretical methodology, validating prediction of spectra of related molecules with no experimental data or no high-resolution experimental data. The studies provide  data on equal footing, serving as a reliable stand point for analyzing vibronic properties within a specific family and edge. Extensive investigations were performed including influences on functionals, basis sets, temperature effects, and tautomeric effects, as well as analyses on core-ionization induced structural changes, active vibronal modes, etc. Additionally, the experiment-theory agreement also validated the HO approximation used which is less sensitive for polyatomic molecules than diatomic systems.<cit.>

However, all previous calculations on polyatomic systems have predominantly focused on ring compounds which exhibit relatively rigid structures including PAHs. It would be natural to investigate the performance on flexible molecules. In this study, we  extend our research to a series of straight-chain alkanes (C_nH_2n+2, n=1-8; see Fig. <ref>): methane (CH_4), ethane (C_2H_6), propane (C_3H_8), n-butane (C_4H_10), n-pentane (C_5H_12), n-hexane (C_6H_14), n-heptane (C_7H_16), and n-octane (C_8H_18).  They represent the simplest organic molecule family, composed solely of sp^3-hybridized carbon atoms and hydrogen atoms. Gradually increasing the –CH_2 groups provide an ideal set for investigating size effects. There are access to high-resolution experimental spectra<cit.> (except heptane), facilitating the performance evaluation of the theoretical method. Technically, as linear molecules generally possess greater flexibility than their ring counterparts, the geometry optimization of core-ionized states becomes a more intricate task than rigid molecules. 

Alkanes are naturally occurring hydrocarbons found in crude oil, serving as high-value petrochemical feedstock materials for the petroleum industry<cit.>. They also play a prominent role as a major component of non-methane volatile organic compounds in vehicle emissions and urban atmospheric environments<cit.>. Methane, propane, and butane are clean energy sources for domestic and industrial applications, with methane dominating liquefied natural gas and propane/butane used in liquefied petroleum gas. Ethane, a minor natural gas component, holds significant value as a petrochemical feedstock<cit.>, primarily for ethylene production. Pentane is used as a polystyrene foam blowing agent and nonpolar solvent, while hexane is a common affordable, safe, and rapidly evaporable non-polar organic solvent. Heptane serves in various industrial applications, including paints, coatings, and pharmaceuticals, and is a significant gasoline component. Octane, another vital gasoline component, is often blended with the highly branched isomer iso-octane to establish a standard reference for measuring the fuel's resistance to engine knock. Methane exhibits T_d symmetry, while ethane possesses D_3d symmetry. Longer straight-chain alkanes  have either C_2v (propane, pentane, and heptane) or C_2h (butane, hexane, and octane) symmetry, as labeled in Fig. <ref> together with all non-equivalent carbon atoms.

It is necessary to acknowledge early theoretical and experimental studies on vibrationally-resolved XPS of alkanes, particularly the work by Svensson and coworkers.<cit.> They employed equivalent-core hole (ECH) DFT, replacing the core-ionized carbon (C^*) with a nitrogen atom,<cit.>  and Franck-Condon simulations to compute spectra for methane, ethane, and other small hydrocarbons, achieving good agreement with experiments.<cit.>  Subsequently, they transitioned to another electronic structure method, using a nitrogen basis set for C^* with scaled exponents and an effective core potential (ECP) with adjusted parameters to account for one electron in the C1s shell.<cit.> This approach was applied to simulate larger linear alkanes (n=1-6,8) and extrapolated to alkanes with n=1000, providing valuable insights into the vibronic coupling properties of alkanes in XPS.

However, these calculations have several limitations. The ECH-DFT approach<cit.> does not yield accurate absolute binding energies. While theoretical spectra can be adjusted to align with experimental values, this method is unsuitable for predictions when no corresponding experiments exist (e.g., heptane). Additionally, our earlier calculations<cit.> indicated that ECH produces less accurate vibronic profiles compared to FCH, particularly regarding the intensity ratios of the 0-0 and 0-1 peaks, F_01/F_00. In the alternative approach<cit.> using a nitrogen basis set and ECP, scaling issues arise with the basis sets, ECPs, and vibrational frequencies (with different factors for different modes), leading to empirical adjustments that may not be valid for other systems. Furthermore, the linear coupling method (product of Poisson distributions) does not account for mode mixing effects (i.e., Duschinsky rotation). Lastly, while analyses have been conducted on core hole-induced structural changes and potential energy curves, there is a lack of comprehensive examination of active vibrational modes.

The goal of this study is twofold. First, we aim to evaluate the performance of a more accurate FCH-DR approach<cit.> for alkane molecules. This computational protocol has been validated<cit.> to accurately predict both binding energies (within 1 eV) and vibronic fine structures for XPS of cyclic compounds, including PAHs. The computational procedure employs only standard basis sets and model core potentials (MCPs) without any empirical parameter scaling. Second, with accurate theoretical spectra at hand, we will investigate general vibronic coupling rules in gaseous linear alkanes. This comprehensive analysis of aliphatic hydrocarbons, alongside our study of PAHs,<cit.> will enhance our understanding of hydrocarbon systems and contribute to our ongoing goal of constructing a high-resolution XPS library.  




§ COMPUTATIONAL METHODS


The FCH-DR method<cit.> was employed for the XPS computations in this work. Details of the methodology have been outlined previously. Briefly, DFT-based electronic structure calculations were performed using the B3LYP functional<cit.> and the GAMESS software package<cit.>. A double basis set approach<cit.> was utilized for geometry optimization and frequency calculations, involving distinct basis sets for the ground state (χ _GS) and core-ionized state (χ _FCH). Specifically, the cc-pVTZ basis set was used for χ _GS, while for χ _FCH, the IGLO-III basis set<cit.> was used for core-ionized carbon atoms (C^*) and the MCP/NOSeC-V-TZP basis set<cit.> was used for non-excited carbon atoms, complemented by the corresponding MCPs. This choice of basis sets has been validated for its accuracy in spectral simulations of various cyclic molecules<cit.>. Fine integration grid and a stringent gradient convergence tolerance of 1.0 × 10^-7 Hartree/Bohr were used.  The optimized structures and vibrational modes for both the ground and FCH states were used as input to simulate FCFs using our in-house software package, DynaVib<cit.>,  modified<cit.> to extend its capabilities from the optical to the X-ray regimes. The DR method<cit.> was employed in the computation of FCFs, which characterizes the differences between the normal coordinates of the initial and final states. The ⟨ 0|n ⟩ amplitudes were computed through an iterative procedure<cit.> initiated with ⟨ 0|0 ⟩. Lorentzian broadening with a half width at half maximum (HWHM) γ=0.05 eV (i.e., the C1s core hole lifetime<cit.>) was applied to all C1s ionizations, and a small uniform shift δ_rel=0.2 eV<cit.> was added to account for the scalar relativistic effect. The total XPS profile of each alkane is a weighted combination of the XPS spectrum from its nonequivalent carbon atoms. To better compare with the experimental data,<cit.> theoretical spectrum of each alkane molecules was shifted by δ=-0.27 eV, except that a slightly different value of δ=-0.33 eV was applied for methane.



§ RESULTS




 §.§ Overview and notations for peak labels


Figure <ref> presents the simulated vibrationally-resolved C1s XPS spectra for all alkanes under study, compared to available experimental data<cit.> (note that no experimental data is available for heptane). As shown in Figs. <ref>–<ref>, more detailed analyses–including atom-specific spectra, interpretations of vibronic transitions, and identifications of active modes–are provided for each pair of molecules with the same number of non-equivalent carbons, C_nH_2n+2 and C_n+1H_2n+4 (where n=1, 3, 5, 7).  Figure <ref> consolidates the atomic-specific contributions from the edge carbons C_1 and C_2 to highlight patterns in spectral evolution.


For methane and ethane, each containing only one non-equivalent carbon, we use Roman numerals i–iii and i-v to label their major peaks, respectively. The same numeral indicates the same vibronic assignment and is transferable between the two molecules. For other alkanes with multiple non-equivalent carbons, we use Latin letters A, B, C, etc., to label features in ascending order of energy in the total spectrum. These labels are molecule-specific and not necessarily transferable among all molecules, although they are indeed transferable for long alkanes with n=5–8.



 §.§ Methane and ethane


The T_d symmetry molecule methane is the simplest alkane with a single carbon atom. Its experimental spectrum [Fig. <ref>(a)] exhibits three prominent peaks (i-iii) that are well-reproduced by theory [Fig. <ref>(b)]: peaks i (experimental, 290.70 eV; theory, 290.71 eV ), ii (experimental, 291.10 eV; theory, 291.12 eV), and iii (experimental, 291.51 eV; theory, 291.55 eV). One can see that the experiment-theory deviation in peak energies is less than 0.1 eV. These peaks correspond to the 0-0, 0-1, and 0-2 transitions of the totally-symmetric (A_1) C^*–H stretching mode ν(6) (3383 cm^-1) at the final-state geometry min FCH, which is the only Franck-Condon-active mode [Fig. <ref>(a,c)].  For clarity, we label peaks ii and iii as 0-16 and 0-26, respectively [Fig. <ref>(a)], with the mode index indicated as a subscript.<cit.> Beyond these peaks, no additional features are clearly identifiable in the experimental spectrum [Fig. <ref>(a), bottom], while the 0-3_6 peak is predicted at approximately 292.00 eV [Fig. <ref>(b), bottom; Fig. <ref>(a)].
 
The D_3d symmetry molecule ethane also contains one single non-equivalent carbon atom. Its theoretical spectrum [Fig. <ref>(b)] closely matches experiment [Fig. <ref>(a)]. The main peak i (0-0 peak) exhibits a red shift of ca. 0.15 eV compared to methane. Peaks ii-iii are again observed, assigned to the C^*–H stretching mode ν(16) (3428 cm^-1) [Figs. <ref>(d), right]. This mode is essentially similar to the only active mode of methane, and the vibrational frequency is slightly (45 cm^-1) larger than that of methane. Besides, two additional peaks, iv (experimental, 290.73 eV; theory, 290.76 eV) and v (experimental, 291.36 eV; theory, 291.19 eV), are resolved and primarily attributed to the C^*–H bending (scissoring) mode ν(7) (1473 cm^-1), which also includes contributions from the bending vibrations of the other –CH_3 group of the non-excited carbon [Fig. <ref>(d), left]. Specifically, peaks iv and v are assigned as the 0→1_7 and 0→1_7,1_16 transitions, respectively.



 §.§ Propane and butane


Propane with C_2v symmetry contains two non-equivalent carbons, distal carbon C_1 and central carbon C_2, in a 2:1 ratio. In the total theoretical spectrum [Fig. <ref>(b)], it now contains rich features (labeled by A-H), since C_1 and C_2 curves are similar in shape but with different “phase” [see also Fig. <ref>(a-b)]. Feature A at 290.39 eV and feature B at 290.50 eV are 0-0 transitions by C_1  and  C_2, respectively. Corresponding experimental values  are 290.40 and 290.50 eV, respectively, almost equivalent to our predictions.  Features C and D come from 0-1_12 and 0-1_14 transitions for C^*-H bending modes ν(12) [1412 cm^-1, Fig. <ref>(e)] for C_1 and ν(14) [1488 cm^-1, Fig. <ref>(f)] for C_2, respectively. Meanwhile, peak features E and F come from  0-1_25 and 0-1_26 transitions for C^*-H stretching modes ν(25) [3430 cm^-1, Fig. <ref>(e)] for C_1 and ν(26) [3465 cm^-1, Fig. <ref>(f)] for C_2, respectively. Features G and H are both assigned as 0-2 transitions with contributions of both carbons. Feature G is assigned as mixed 0-1 transitions for both the bending and stretching modes; while feature H is assigned as pure 0-2 transitions of the stretching mode.

Similarly, butane with C_2h symmetry also has two non-equivalent carbons, C_1 and C_2, in a 2:2 ratio. Compared to propane, the fine structure looks simper because in the C_1 and C_2 contributions, all peaks almost coincide with each other [see also Fig. <ref>(c-d)]. The active mode analyses for butane [Fig. <ref>(g-h)] mirror the discussions for propane, identifying a C^*-H stretching and a C^*-H bending modes for each carbon. Peak A is contributed by  0-0 transitions; peaks B and C are 0-1 transitions for the bending and stretching modes, respectively; peaks D and E are 0-2 transitions, either mixed 0-1 transitions for both the bending and stretching modes (feature D) or pure 0-2 transitions of the stretching mode.





 §.§ Pentane and hextane


Pentane (C_2v) and hexane (C_2h) both have three non-equivalent carbon atoms, C_1–C_3, with weight ratios of 2:2:1 and 2:2:2, respectively. Both molecules exhibit similar spectra with six major features A-F. Peak A is assigned as 0-0 transition by the central carbon C_3. Peak B is from 0-0 transitions of C_1 and C_2 as well as 0-1 transition of C_3 for a  C^*-H bending mode (Fig. <ref>). Peaks C and D are assigned as 0-1 transitions, and Peaks E and F as 0-2 transitions. Active modes of the two molecules are analyzed in Fig. <ref> (g)-(i) and Fig. <ref> (j)-(l). Each atom-specific spectrum is identified with a medium-frequency (1400–1500 cm^-1) C^*–H bending mode and a high-frequency (3400-3500 cm^-1) C^*–H stretching mode.



 §.§ Heptane and octane


Heptane (C_2v) and octane (C_2h) have four non-equivalent carbon atoms, with weight ratios of 2:2:2:1 and 2:2:2:2, respectively. Although one additional non-equivalent carbon is present as compared to pentane and hextane, the total spectra do not change much, also exhibiting six features A-F. In other words, when n changes from 5 to 8, the vibronic profile almost converges into a consistent pattern. Concerning binding energies, the spectral series exhibit a slight red shift with increasing n. Despite the lack of experimental spectrum for heptane, our predicted theoretical spectrum matches the overall evolutionary trend.

The underlying reason is that in  heptane and  octane with relatively long length, spectra contributed by the two central carbons C_3 and C_4 almost coincide with each other, indicating their similar local chemical environment. Their 0-0 transitions both contribute to feature A in the total spectrum. 0-0 transitions of distal carbons C_1 and C_2 make contributions to feature B some 0.2 eV beyond. The deviation of 0-0 transition energies for C_1 and C_2 makes peak B broader than peak A. In a even longer linear alkane, it is reasonable to yield that the spectral contributions for two neighboring carbons near the terminal deviates more than two neighboring carbons in the central region.  Based on each atom-specific spectrum  of both molecules [Fig. <ref> (a)-(h)], each was identified with two active modes, including one  medium-frequency (1400–1500 cm^-1) C^*–H bending  mode and a high-frequency (3400-3500 cm^-1) C^*–H stretching mode [Fig. <ref> (i)-(p)].



§ DISCUSSION




 §.§ Red shift of the lowest peak A


In the total theoretical spectra [Fig. <ref>(b)], the lowest-energy peak, peak A, exhibits a red shift, with energies of 290.70 eV (methane), 290.58 eV (ethane), 290.38 eV (propane), 290.32 eV (butane), 290.12 eV (pentane), 290.09 eV (hexane), 290.07 eV (heptane), and 290.05 eV (octane), as indicated by the numbers at the top of this panel. The magnitude of the red shift is relatively large for alkanes with n=1 to 4 but becomes significantly smaller starting from n=5. As discussed earlier, for n=5 to 8, peak A for each molecule shares the same assignment (from central carbons). The energy differences between neighboring alkanes are only 0.02 to 0.03 eV (approximately 1 milli-Hartree), indicating that the values are approaching convergence.




 §.§ Ratio of distal and central carbons controls intensity ratios of peaks A and B


Above analyses have shown stable spectral profiles of six features A-F for linear alkanes at n=5-8. Peak A and B are 0-0 peaks of central (C_3 when n=5-6; C_3, C_4 when n=7-8) and distal (C_1, C_2) carbons, respectively. As n increases, the intensity ratio of peak A to peak B increases due to the increase of more central carbons. This can be explained by the locality of core excitations and the alkane structure (without delocalized π structure), which make the atom-specific spectra dominantly determined by the nearest neighboring units. In a long alkane molecule, C_1 directly links to a –CH_2 group, C_2 directly bonds to a -CH_3 and –CH_2 groups on each side, and all other non-equivalent carbons all bond with two –CH_2 groups, which are classified as central carbons. It is worth noting that different central carbons contribute closely similar spectra, as verified by the coincidence of C_3 and C_4 spectra in heptane and octane. Therefore, in an even longer linear alkane, peak A will be extremely high compared to peak B. The total spectra can be considered as contributed from three types of carbons: C_1, C_2, and central carbons. This explains the stable features in longer linear alkanes, with their relative peak intensities adjusted by the ratios of different types of carbons and their relative BE differences.



 §.§ Evolution of distal carbons C_1 and C_2


Figure <ref> clearly demonstrates the evolution of vibrationally-resolved XPS profiles for the two distal carbon atoms C_1 (orange) and C_2 (lightblue). The concept of distal carbons is evident for long alkanes with n=5–8, while all carbons are considered as distal carbons for short alkanes with n=1–4, providing a comprehensive and smooth picture. Methane shows three features i-iii, attributed to the C^*–H stretching vibration. Since ethane, an additional C^*–H bending vibration is involved, contributing to peaks vi (pure 0-2 bending mode excitation) and v (mixed 0-1 excitations with both modes). All other atom-specific spectral profiles show great similarity to the ethane spectrum, with relative peak intensities of the five peaks preserved almost the same. Figure <ref> highlights the evolution of peaks i-iii. These results indicate that the spectrum of ethane can be considered as a “spectral seed”, as different shifted spectra of ethane approximately generate the total spectra of a longer linear alkane. Additionally, a switch of BE order is found for C_1 and C_2. Carbon C_2 starts since n=3, exhibiting a higher binding energy of ca. 0.1 eV than C_1. This energy order is quickly reversed since n=4, and at n=8, the binding energy of C_2 is ca. 0.1 eV lower than C_1.



 §.§ Structural changes induced by core ionizations


Structurally, the linear alkanes are generated by gradually adding methylene groups (–CH_2–) to methane. Each molecule is constructed by linking C-C and C-H single bonds, and studying the structural change induced by the C1s core hole is crucial. In alkanes, the typical C–C bond length hovers around 1.529 Å, with a tiny difference between distal (1.528 Å) and central carbons (1.529–1.530 Å), as shown in Fig. <ref>. The structural changes near the excited carbon are summarized in Table <ref>, revealing discernible patterns. Specifically, the bond lengths C^*-C_r and C^*-C_l in the FCH state exhibit minimal changes (not exceeding 0.02 Å) compared to the ground state structure, indicating marginal influence due to the presence of a C1s core hole. Here, C_l (C_r) is the left (right) carbon atom bonded to C^*  (see relative atomic positions in Fig. <ref>). The angle ∠C_r-C^*-C_l, typically close to 114^∘, experiences an increment of ca. 1^∘ from the ground- to the final-state geometry. This angle at C^* experiences slightly larger bending for central carbons compared to distal carbons. All C^*–H bonds exhibit virtually identical characteristics in both the ground and the final states, with minor shortening of 0.05–0.07 Å.



§ SUMMARY AND CONCLUSIONS


To summarize, we simulated high-resolution, vibrationally-resolved C1s X-ray photoelectron spectra (XPS) for a series of linear alkanes, ranging from methane to octane (C_nH_2n+2, with n=1–8), by combining full core-hole density functional theory with Franck-Condon calculations that account for Duschinsky rotation. Our theoretical spectra align excellently with all available experimental results, and we predicted the spectra of heptane, showing reasonable spectral evolution in this series. This approach has facilitated reliable peak assignments and enabled an investigation into the spectral evolution with increasing chain length, as well as the underlying vibronic coupling rules governing this molecular family.

The spectrum of ethane serves as a “spectral seed”, with the vibronic profile of each longer alkane exhibiting similar characteristics, albeit with shifts in peak positions and slight intensity adjustments. The carbon atoms in these longer alkanes are categorized as central or distal (C_1 and C_2), with central carbons contributing nearly identically to the lowest-energy feature A, while distal carbons contribute to the second-lowest feature B. Detailed assignments elucidate the distinct spectra observed in short alkanes (n=1–4) and their stabilization in longer alkanes (n=5–8). Through comprehensive assignments, we tracked the evolution of specific spectral features across the molecular series. We investigated core ionization-induced structural changes and identified Franck-Condon-active vibrational modes, highlighting two key active modes present in each atom-specific spectrum: the C^*–H symmetric stretching (ca. 3400–3500 cm^-1) and the C^*–H bending (ca. 1400–1500 cm^-1) modes. 

This work extends our FCH-DR approach to flexible molecular systems, enhancing our understanding of vibronic coupling phenomena. The good performance of our protocol in these flexible systems should benefit from the minimal structural changes induced by core ionization. It also proves that the harmonic oscillator approximation works effectively for alkane systems. The insights gained from this analysis of simple alkane systems will be valuable for studying more complex hydrocarbons and other flexible molecules. Ultimately, this study contributes to our ongoing research goal of constructing a comprehensive, high-resolution theoretical XPS library.




§ ACKNOWLEDGMENTS


We thank Prof. Yi Luo for helpful discussions. Financial support from the National Natural Science Foundation of China (Grant No. 12274229)  is greatly acknowledged. M.W. thanks to Fund for Fostering Talented Doctoral Students of Nanjing University of Science and Technology. 


































46
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[van der Heide(2011)]van_der_heide_x-ray_2011
  author author P. van der Heide, https://doi.org/10.1002/9781118162897 title X-Ray Photoelectron Spectroscopy: An
  Introduction to Principles and Practices (publisher
  John Wiley & Sons, Inc., address Hoboken, NJ, USA, year 2011)NoStop
[Gelius et al.(1974)Gelius,
  Svensson, Siegbahn, Basilier,
  Faxälv, and Siegbahn]gelius_vibrational_1974
  author author U. Gelius, author S. Svensson,
  author H. Siegbahn, author E. Basilier, author
  Å. Faxälv, and author
  K. Siegbahn, @noop  journal journal Chem. Phys. Lett. volume 28, pages 1 (year 1974)NoStop
[Carravetta et al.(2022)Carravetta, Couto, and Ågren]carravetta_x-ray_2022
  author author V. Carravetta, author R. C. Couto, and author H. Ågren, https://doi.org/10.1088/1361-648X/ac7d2a journal journal J. Phys.: Condens. Matter volume 34, pages 363002 (year
  2022)NoStop
[Hergenhahn(2004)]hergenhahn_vibrational_2004
  author author U. Hergenhahn, https://doi.org/10.1088/0953-4075/37/12/R01
  journal journal J. Phys. B: At. Mol. Opt. Phys.  volume 37, pages R89 (year
  2004)NoStop
[Svensson(2005)]svensson_soft_2005
  author author S. Svensson, https://doi.org/10.1088/0953-4075/38/9/024 journal journal J. Phys. B: At. Mol. Opt. Phys. volume 38, pages S821 (year
  2005)NoStop
[Siegbahn(1969)]book_ESCA_molecules
  author author K. Siegbahn, @noop  title ESCA applied to free
  molecules (publisher North-Holland Publishing, year 1969)NoStop
[Ueda(2003)]ueda_high-resolution_2003
  author author K. Ueda, https://doi.org/10.1088/0953-4075/36/4/201 journal journal J. Phys. B: At. Mol. Opt. Phys. volume 36, pages R1 (year
  2003)NoStop
[Mendolicchio et al.(2019)Mendolicchio, Baiardi, Fronzoni,
  Stener, Grazioli, de Simone,  and Barone]mendolicchio_theory_2019
  author author M. Mendolicchio, author A. Baiardi, author G. Fronzoni,
  author M. Stener, author C. Grazioli, author
  M. de Simone, and author
  V. Barone, https://doi.org/10.1063/1.5122310 journal journal J. Chem. Phys. volume 151, pages 124105 (year 2019)NoStop
[Montorsi et al.(2022)Montorsi, Segatta, Nenov, Mukamel, and Garavelli]montorsi_soft_2022
  author author F. Montorsi, author F. Segatta,
  author A. Nenov, author S. Mukamel, and author
  M. Garavelli, https://doi.org/10.1021/acs.jctc.1c00566 journal journal J. Chem. Theory Comput. volume 18,  pages 1003 (year 2022)NoStop
[Rubensson et al.(2012)Rubensson, Pietzsch, and Hennies]rubensson_vibrationally_2012
  author author J.-E. Rubensson, author A. Pietzsch, and author F. Hennies, https://doi.org/10.1016/j.elspec.2012.05.003 journal journal J. Electron Spectros. Relat. Phenomena  volume 185, pages 294 (year
  2012)NoStop
[Miron and Morin(2011)]quack_high-resolution_2011
  author author C. Miron and author P. Morin,  in https://doi.org/10.1002/9780470749593.hrs066 booktitle Handbook of High-resolution Spectroscopy, editor edited by editor M. Quack and editor F. Merkt  (publisher John Wiley & Sons, Ltd, address
  Chichester, UK, year 2011) p. pages
  hrs066NoStop
[Rumble Jr. et al.(1992)Rumble Jr., Bickham, and Powell]Rumble1992NIST
  author author J. R. Rumble Jr., author D. M. Bickham, and author C. J. Powell, @noop  journal journal Surf.
  Interface Anal. volume 19, pages 241
  (year 1992)NoStop
[Wagner(1979)]wagner_1979_handbook
  author author C. D. Wagner, @noop  title Handbook of x-ray
  photoelectron spectroscopy: a reference book of standard data for use in
  x-ray photoelectron spectroscopy (publisher Perkin-Elmer,  address Shelton, CT, year 1979)NoStop
[Jolly et al.(1984)Jolly,
  Bomben, and Eyermann]jolly_core-electron_1984
  author author W. Jolly, author K. Bomben, and  author C. Eyermann, @noop
   journal journal Atom. Data Nucl. Data  volume 31, pages 433 (year
  1984)NoStop
[xps()]xpsdatabase
  @noop  note Https://xpsdatabase.com/, Accessed on
  2024-7-20.Stop
[sas()]sasj
  @noop  note Http://www.sasj.jp/, Accessed on
  2024-7-20.Stop
[LaS()]LaSurface
  @noop  note Http://www.lasurface.com/xps/index.php/,
  Accessed on 2024-7-20.Stop
[Wei et al.(2022)Wei,
  Cheng, Zhang, Zhang,
  Wang, Ge, Tian, and  Hua]wei_vibronic_2022
  author author M. Wei, author X. Cheng, author L. Zhang, author
  J.-R. Zhang, author
  S.-Y. Wang, author
  G. Ge, author G. Tian, and author W. Hua, https://doi.org/10.1103/PhysRevA.106.022811 journal journal Phys. Rev. A volume 106, pages 022811 (year 2022)NoStop
[Wei et al.(2023)Wei,
  Zhang, Tian, and Hua]wei_vibronic_2023
  author author M. Wei, author L. Zhang, author G. Tian, and author
  W. Hua, @noop  journal journal Phys. Rev. A volume
  108, pages 022816 (year 2023)NoStop
[Cheng et al.(2022)Cheng,
  Wei, Tian, Luo, and  Hua]cheng_resolved_2022
  author author X. Cheng, author M. Wei, author G. Tian, author
  Y. Luo, and author
  W. Hua, https://doi.org/10.1021/acs.jpca.2c04426 journal journal J. Phys. Chem. A volume 126, pages 5582 (year 2022)NoStop
[Hua et al.(2020)Hua,
  Tian, and Luo]hua_theoretical_2020
  author author W. Hua, author G. Tian, and  author Y. Luo, https://doi.org/10.1039/D0CP02970J journal journal Phys. Chem. Chem. Phys. volume 22,  pages 20014 (year 2020)NoStop
[Wei et al.(2024a)Wei, Zuo, Tian, and Hua]wei_vibronic_2024
  author author M. Wei, author J. Zuo, author G. Tian, and author
  W. Hua, https://doi.org/10.1103/PhysRevA.109.022820 journal journal Phys. Rev. A volume 109, pages 022820 (year 2024a)NoStop
[Wei et al.(2024b)Wei, Zuo, Tian, and Hua]wei_jcp_2024
  author author M. Wei, author J. Zuo, author G. Tian, and author
  W. Hua, @noop  journal journal arXiv preprint arXiv:2403.09109  (year 2024b)NoStop
[Duschinsky(1937)]duschinsky_1937
  author author F. Duschinsky, @noop  journal journal
  Acta Physicochim. URSS volume 7, pages
  551 (year 1937), note and its English
  translation by C. W. Müler available from
  http://www.chem.purdue.edu/zwier/pubs/Duschinsky.pdfNoStop
[Zhang et al.(2024)Zhang,
  Wei, Ge, and Hua]zhanglu_pra_2024
  author author L. Zhang, author M. Wei, author G. Ge, and author
  W. Hua, https://doi.org/10.1103/PhysRevA.109.032815 journal journal Phys. Rev. A volume 109, pages 032815 (year 2024)NoStop
[Karlsen et al.(2002)Karlsen, Børve, Sæthre, Wiesner, Bässler, and Svensson]karlsen_toward_2002
  author author T. Karlsen, author K. J. Børve,
  author L. J. Sæthre, author K. Wiesner, author
  M. Bässler, and author
  S. Svensson, https://doi.org/10.1021/ja010649j journal journal J. Am. Chem. Soc. volume 124, pages 7866 (year 2002)NoStop
[Stauffer et al.(2008)Stauffer, Dolan, and Newman]stauffer_review_2008
  author author E. Stauffer, author J. A. Dolan, and author R. Newman, in https://doi.org/10.1016/B978-012663971-1.50007-5
  booktitle Fire Debris Analysis (publisher Elsevier, year 2008) pp. pages
  49–83NoStop
[Atkinson et al.(2008)Atkinson, Arey, and Aschmann]atkinson_atmospheric_2008
  author author R. Atkinson, author J. Arey,  and author S. M. Aschmann,  https://doi.org/https://doi.org/10.1016/j.atmosenv.2007.08.040
  journal journal Atmos. Environ. volume 42, pages 5859 (year
  2008)NoStop
[Xiang et al.(2018)Xiang,
  Wang, Cheng, and Matsubu]xiang_progress_2018
  author author Y. Xiang, author H. Wang,
  author J. Cheng, and author J. Matsubu, https://doi.org/10.1039/C7CY01878A journal journal Catal. Sci. Technol. volume 8, pages 1500 (year 2018)NoStop
[Gaffney and Mason(2017)]gaffney_ethylene_2017
  author author A. M. Gaffney and author O. M. Mason, https://doi.org/10.1016/j.cattod.2017.01.020 journal journal Catal. Today volume
  285, pages 159 (year 2017)NoStop
[Thomas et al.(1998)Thomas,
  Saethre, Sorensen, and Svensson]thomas_vibrational_1998
  author author T. D. Thomas, author L. J. Saethre,
  author S. L. Sorensen, and  author S. Svensson, https://doi.org/10.1063/1.476646 journal journal
  J. Chem. Phys. volume 109, pages
  1041 (year 1998)NoStop
[Jolly and Hendrickson(1970)]jolly_thermodynamic_1970
  author author W. L. Jolly and author D. N. Hendrickson, @noop  journal journal
  J. Am. Chem. Soc. volume 92, pages
  1863 (year 1970)NoStop
[Davis and Shirley(1972)]davis_relaxation_1972
  author author D. W. Davis and author D. A. Shirley, @noop  journal journal Chem.
  Phys. Lett. volume 15, pages 185
  (year 1972)NoStop
[Plashkevych et al.(2000)Plashkevych, Privalov, Ågren,
  Carravetta, and Ruud]plashkevych_validity_2000
  author author O. Plashkevych, author T. Privalov, author H. Ågren,
  author V. Carravetta, and  author K. Ruud, https://doi.org/10.1016/S0301-0104(00)00171-3 journal
  journal Chem. Phys. volume 260,  pages 11 (year 2000)NoStop
[Becke(1993)]becke_densityfunctional_1993
  author author A. D. Becke, https://doi.org/10.1063/1.464913 journal
  journal J. Chem. Phys. volume 98,  pages 5648 (year 1993)NoStop
[Stephens et al.(1994)Stephens, Devlin, Chabalowski, and  Frisch]stephens_ab_1994
  author author P. J. Stephens, author F. J. Devlin, author C. F. Chabalowski, and author M. J. Frisch, https://doi.org/10.1021/j100096a001 journal journal J. Phys. Chem. volume
  98, pages 11623 (year 1994)NoStop
[Barca et al.(2020)Barca,
  Bertoni, Carrington, Datta,
  De Silva, Deustua, Fedorov,
  Gour, Gunina, Guidez,
  Harville, Irle, Ivanic,
  Kowalski, Leang, Li,
  Li, Lutz, Magoulas,
  Mato, Mironov, Nakata,
  Pham, Piecuch, Poole,
  Pruitt, Rendell, Roskop,
  Ruedenberg, Sattasathuchana, Schmidt, Shen, Slipchenko, Sosonkina, Sundriyal, Tiwari, Galvez Vallejo, Westheimer, Wloch,
  Xu, Zahariev, and Gordon]GAMESS
  author author G. M. J.  Barca, author C. Bertoni, author L. Carrington,
  author D. Datta, author N. De Silva, author
  J. E. Deustua, author
  D. G. Fedorov, author
  J. R. Gour, author
  A. O. Gunina, author
  E. Guidez, author T. Harville, author S. Irle, author J. Ivanic, author K. Kowalski,
  author S. S. Leang, author H. Li, author
  W. Li, author J. J. Lutz, author I. Magoulas, author J. Mato, author V. Mironov, author H. Nakata,
  author B. Q. Pham, author P. Piecuch, author
  D. Poole, author S. R. Pruitt, author A. P.  Rendell, author L. B.  Roskop, author K. Ruedenberg, author T. Sattasathuchana, author M. W. Schmidt, author J. Shen,
  author L. Slipchenko, author M. Sosonkina, author
  V. Sundriyal, author
  A. Tiwari, author J. L. Galvez Vallejo, author B. Westheimer, author M. Wloch, author P. Xu, author F. Zahariev, and author M. S. Gordon, https://doi.org/10.1063/5.0005188 journal
  journal J. Chem. Phys. volume 152,  pages 154102 (year 2020)NoStop
[Kutzelnigg et al.(1990)Kutzelnigg, Fleischer, and Schindler]kutzelnigg1990iglo
  author author W. Kutzelnigg, author U. Fleischer, and author M. Schindler, in @noop  booktitle Deuterium and
  shift calculation (publisher Springer, year
  1990) pp. pages 165–262NoStop
[Sakai et al.(1997)Sakai,
  Miyoshi, Klobukowski, and Huzinaga]sakai_model_1997
  author author Y. Sakai, author E. Miyoshi,
  author M. Klobukowski, and  author S. Huzinaga, @noop
   journal journal J. Chem. Phys. volume 106, pages 8084 (year
  1997)NoStop
[Noro et al.(1997)Noro,
  Sekiya, and Koga]noro_contracted_1997
  author author T. Noro, author M. Sekiya, and  author T. Koga, @noop 
  journal journal Theor. Chem. Acc. volume 98, pages 25 (year
  1997)NoStop
[bsj()]bsjp
  @noop  note
  Http://sapporo.center.ims.ac.jp/sapporo/.Stop
[Tian et al.()Tian,
  Duan, Hua, and Luo]DynaVib
  author author G. Tian, author S. Duan, author W. Hua, and author
  Y. Luo, @noop  title
  DynaVib, version 1.0, note
  Royal Institute of Technology: Sweden, 2012NoStop
[Ruhoff(1994)]ruhoff_recursion_1994
  author author P. T. Ruhoff, https://doi.org/10.1016/0301-0104(94)00173-1 journal journal Chem. Phys. volume
  186, pages 355 (year 1994)NoStop
[Ruhoff and Ratner(2000)]ruhoff_algorithms_2000
  author author P. T. Ruhoff and author M. A. Ratner, https://doi.org/10.1002/(SICI)1097-461X(2000)77:1<383::AID-QUA38>3.0.CO;2-0
  journal journal Int. J. Quant. Chem. volume 77, pages 383 (year
  2000)NoStop
[Nicolas and Miron(2012)]nicolas_lifetime_2012
  author author C. Nicolas and author C. Miron, https://doi.org/10.1016/j.elspec.2012.05.008 journal journal J. Electron. Spectrosc. Relat. Phenom.  volume 185, pages 267 (year
  2012)NoStop
[Triguero et al.(1999)Triguero, Plashkevych, Pettersson, and  Ågren]triguero_separate_1999
  author author L. Triguero, author O. Plashkevych, author L. Pettersson, and author H. Ågren, https://doi.org/10.1016/S0368-2048(99)00008-0 journal journal J. Electron Spectros. Relat. Phenomena  volume 104, pages 195 (year
  1999)NoStop




 Paper Title: Enhancement of the molecular electron chirality by electronic excitation.






Department of Micro Engineering, Kyoto University, Kyoto 615-8540, Japan

senami@me.kyoto-u.ac.jp
Department of Micro Engineering, Kyoto University, Kyoto 615-8540, Japan




§ ABSTRACT 

Previously, it was predicted that the parity-violating energy difference (PVED) between enantiomers 
and the molecular electron chirality (MEC),
which is the integrated value of the electron chirality density over the entire molecule
are enhanced for ionization or electronic excitation.
Following the confirmation of the enhancement of the PVED by electronic excitation in our previous papers,
the present work confirms that electronic excitation enhances 
the MEC for H_2Te_2 and H_2Se_2.
The PVED in the first excited state 
is explained by the contribution from the highest occupied molecular orbital
if the PVED contribution from the highest occupied molecular orbital in the ground state
dominates over any other contributions.
In this paper, 
it is checked whether
this explanation can also be applicable to the MEC.
It is also reported that the MEC is not the simple sum of contributions from each atom 
and 
the mixing by the superposition between atomic orbitals of different atoms is important.




Enhancement of the molecular electron chirality by electronic excitation

    Masato Senami
    August 5, 2024
==========================================================================




§ INTRODUCTION



It is well-known that some molecules have chiral structures,
while the electron also has its chirality,
i.e. the handedness of the electron can be defined.
A free electron does not have a specific charge distribution
and the handedness of the electron cannot be defined by the distribution.
However, the electron has the spin,
and the inner product of the spin and momentum
can define one type of handedness quantity of the electron.
This definition of the handedness is the helicity.
When the external field is negligible,
this helicity is a conserved quantity in time,
since the helicity commutes with the Hamiltonian.
However, the helicity has a different ratio of the electron handedness in a different inertial frame
and is not Lorentz invariant definition.
The chirality of the electron is defined 
as the same handedness ration in any inertial frame.
This is a relativistic extension of the helicity
as explained in the next section.

Nature chooses the chirality as the handedness quantity of electrons.
First, 
the weak interaction, which is one of the four fundamental interactions in particle physics,
distinguish left- and right-handed electrons by the chirality.
For example, 
this interaction is mediated by W and Z gauge bosons,
and the W boson has the coupling with left-handed electrons
and not with right-handed ones.
In another example of the distinction,
the torque density for the electron spin in quantum field theory is 
predicted to be dependent on the electron chirality density
<cit.>.
In this theoretical prediction,
the local torque contribution, which is named the zeta force,
is given as the gradient of the electron chirality density.
Although this contribution of the torque density
has not been observed in experiments,
this is confirmed by numerical simulations <cit.>.

Although the physical property of the electron is different in its chirality,
free electrons lose their polarization in chirality sooner or later
due to the chirality flip by the electron mass.
For example, if electrons are initially polarized more in left-handed chirality,
it attenuates to no polarization state.
The left-handed component flips to the right-handed one
more than the inverse process (from right-handed to left-handed),
since the ratio of the left-handed component is larger than the right-handed one.

However, electrons in chiral molecules, which are parity-violating objects,
are considered to be polarized in the chirality,
since a state polarized in the electron chirality may be the most stable 
in the steady state due to the spin-orbit interaction.
Even though the electron mass flips the electron chirality,
the attenuation goal is the polarized state in the chirality.

One phenomenon related to the electron chirality 
is the parity-violating energy difference (PVED) between enantiomers.
This energy difference is induced by 
the interaction of Z-boson exchange between nuclei and electrons
and is proportional to the electron chirality density difference at nuclei between enantiomers.
Although this energy difference has not been confirmed in observational experiments
despite hard efforts <cit.>,
theoretical works with quantum chemistry computations
confirm nonzero PVED in many chiral molecules
<cit.>.
Nevertheless, this energy difference is related to only the electron chirality at nuclei.
The electron chirality for the entire molecule of H_2Te_2 is reported to be nonzero
<cit.>.
Then, in Ref. <cit.>,
it is systematically shown that 
chiral molecules have generally a nonzero integrated value of electron chirality density over the entire molecule.
The existence of this chirality is named Electron Chirality in Chiral Molecules (ECCM) <cit.>.

The value of ECCM is not large,
and phenomena related to ECCM are not so many.
Nevertheless, some phenomena are studied by our group.
One topic is the homochirality in nature <cit.>.
In an enantiomeric pair,
the number of the left-handed electron is different,
though the total electron number is the same.
Hence, one enantiomer 
has a larger reaction rate of weak interaction 
than the other one.
As a result, one type of an enantiomeric pair is more lost
by collisions with particles with weak charge 
in space or on the earth <cit.>.
Another example is 
the Chirality-Induced Spin Selectivity (CISS)
<cit.>.
CISS is the strong spin polarization 
when electrons pass through chiral organic molecules.
It has not been clarified what makes large spin polarization.
For this puzzle, 
the study from the viewpoint of the torque may be a key to the solution
instead of the spin-orbit interaction, which is in the concept of energy.
The torque for the electron spin is dependent on the electron chirality density,
and the torques induced by the left- and right-handed electrons are opposite.
ECCM predicts opposite electron chirality distribution in enantiomeric pair molecules.
The zeta force may be the driving torque for the CISS.
The correlation between the spin polarization of the CISS and the magnitude of the zeta force
was confirmed for alanine and helicene molecules <cit.>.

In the following, 
the electron chirality of the entire molecule, 
which is the integrated value of the chirality density over the entire molecule,
is named Molecular Electron Chirality (MEC).
In Ref. <cit.>,
it was predicted that PVED and MEC are enhanced 
for ionized or electronic excited states.[
In another viewpoint,
the enhancement of the PVED in electronic excited states
was discussed before our prediction <cit.>. ]
In the ground state of H_2Te_2,
the single contribution from a valence orbital to the PVED (MEC)
is larger than the sum of contributions from all orbitals,
and
the small value of the PVED (MEC)
is derived as a result of the cancellation among large contributions.
In ionized or excited states,
this cancellation may be broken,
and the PVED and MEC may be much enhanced from the value in the ground state.
This cancellation spectrum of the PVED and MEC
is seen for many other molecules <cit.>.
In Refs. <cit.>,
the prediction regarding the enhancement in the PVED of
H_2X_2 (X= O, S, Se, Te), CHFClBr, CHFClI, and CHFBrI has been confirmed.
In the study <cit.>,
the cancellation-breaking enhancement (CBE) hypothesis was proposed.
The PVED in the first electronic excited state 
is significantly enhanced compared to the ground state,
if the PVED contribution from the highest occupied molecular orbital (HOMO) in the ground state
is larger than the sum of contributions from all occupied orbitals.
Furthermore, 
if the HOMO contribution dominates over any other contribution,
the CBE hypothesis predicts the PVED value in the first excited state
from only the HOMO contribution.
For the enhancement of the PVED by the ionization, 
ionized states of CHFBrI derivatives such as CHDBrI^+ and CHCaBrI^+
are recently studied 
from the viewpoint of the experimental detection of parity-violation in vibrational transitions <cit.>.

In the present paper,
it is confirmed that the enhancement by electronic excitation 
also occurs for the MEC.
As target molecules,
H_2Te_2 and H_2Se_2 are chosen.
In Refs. <cit.>,
H_2O_2 and H_2S_2 are also studied for the enhancement of the PVED.
For the MEC,
the computation of molecules including only light elements with small spin-orbit interaction
such as H_2O_2 and H_2S_2
requires very large basis sets and high cost <cit.>.
Hence, H_2O_2 and H_2S_2 are omitted in the present study for the excited states.
However, we consider that H_2O_2 and H_2S_2 will show results similar to H_2Te_2 and H_2Se_2 
due to the similar patterns of the total chirality in the ground state <cit.>
and the enhancement of the PVED <cit.>.
After the enhancement of the MEC is confirmed,
it is studied whether the CBE hypothesis can also be applicable to the enhancement of the MEC.

This paper is organized as follows.
In the next section, 
the chirality of the electron is introduced,
and then
the computational method and details are explained in Sec. <ref>.
In Sec. <ref>, 
our results of the enhancement of the MEC are shown.
First, the enhancement of the MEC by electronic excitation is confirmed 
and then the CBE hypothesis is applicable to the enhancement of the MEC.
The last section is devoted to the summary.






§ THEORY 




The MEC, the integrated value of the electron chirality density 
over the entire molecule,
is defined by the following formula,


    ∫ d^3 x ψ^† (x) γ_5 ψ (x) ,


where γ_5 ≡ i γ^0 γ^1 γ^2 γ^3
with the gamma matrices, γ^μ (μ=0-3)
and ψ is the electron wave function in a four-component representation.
The electron chirality density, ψ^† (x) γ_5 ψ (x),
can be reduced to the combination form of 
the left- and right-handed electron components,


    ψ^† (x) γ_5 ψ (x)
    =
    ψ_R^† (x) ψ_R (x) 
    - ψ_L^† (x) ψ_L (x).


the left- and right-handed electron wave functions are 
defined as


    ψ_R,L (x) = P_R,Lψ (x),


with the projection operators,


    P_R = 1 + γ_5/2,
        
    P_L = 1 - γ_5/2.



The Lorentz transformation commutes with γ_5 <cit.>
and the chirality i.e. the ratio between the left- and right-handed electrons
are not changed in a different inertial frame.
From γ_5 operator, 
it is seen that the chirality is a relativistic extension of the helicity.
This operator is cast into another form,
which is the product of the velocity and spin operators,


    γ_5 
    = α_x Σ_x = α_y Σ_y = α_z Σ_z 
    = 1/3∑_i=1^3 α^i Σ^i .


Here, α^i = γ^0 γ^i,
and Σ^i = γ^0 γ^i γ_5,
which is the 4 × 4 extension of the Pauli matrices.
Therefore, the latter operator is the spin operator.
The velocity operator is known to be defined as 
v̂^i = c α^i with α^i matrices <cit.>.
Hence, γ_5 operator is the product of the velocity (divided by c) and spin operators
in relativistic quantum theory.
However, the observable of the chirality density is not 
the product of the observed velocity and spin, 
ψ^† c α^i ψψ^†Σ^i ψ,
and the observable of the chirality density is ψ^†γ_5 ψ.
The former is not Lorentz invariant.



In this paper, the CBE hypothesis <cit.> is checked for the MEC.
In this paragraph, the CBE hypothesis is reviewed.
This hypothesis requires the following conditions:
the PVED is derived as the result of the cancellation between contributions from the HOMO and other orbitals,
the electron in the HOMO in the ground state is dominantly excited,
and other occupied orbitals in the ground state are not affected by the excitation.


These conditions are satisfied in the first electronic excitation state in many molecules <cit.>.
If these conditions are satisfied,
the PVED in the first excited state is much larger than the PVED in the ground state.
In addition, if the HOMO contribution to the PVED dominates over any other contributions,
the PVED value is roughly estimated only with the contribution from the HOMO,


    E_ PV ( CBE)
    =
    - E_ PV,HOMO .


Here E_ PV,HOMO is the contribution from one electron in the Kramers pair of the HOMO to E_ PV.
(Two electrons in the same Kramers pair give the same contribution to the electron chirality.)
As a more improved estimate, contributions from other orbitals are taken into account,

    E_ PV ( IE)
    =
     ∑_i^ unocc r_i^U E_ PV,  i 
    - ∑_i^ occ r_i^O E_ PV,  i ,

where E_ PV,  i is the contribution from one electron in the i-th occupied or unoccupied orbital,
and r_i^O (r_i^U) is 
the excitation ratio of the electron in the i-th occupied orbital
(to the i-th unoccupied orbital).
The former simple estimate is quite useful,
since only Hartree-Fock (HF) or density functional theory computations are needed,
while the improved estimate (IE) requires a computation beyond self-consistent field computations.

While this CBE hypothesis is about the enhancement of the PVED,
the enhancement of the MEC is considered to occur in the same cancellation-breaking mechanism.
For the CBE hypothesis about the MEC, 
the conditions are the same as the PVED.
The estimate of the MEC is 
given as the HOMO contribution to the MEC,
that is,


    - ∫ d^3 x ψ_ HOMO^† (x) γ_5 ψ_ HOMO (x) ,


where ψ_ HOMO is the wave function of the HOMO.
The improved estimate with other orbital contributions
is given in the following expression,


    ∑_i^ unocc r_i^U ∫ d^3 x ψ_i^† (x) γ_5 ψ_i (x) 
    - ∑_i^ occ r_i^O ∫ d^3 x ψ_i^† (x) γ_5 ψ_i (x) ,


where ψ_i is the wave function of one of the Kramers pair of the i-th orbital.
In this paper, the CBE hypothesis is checked for the MEC.





§ COMPUTATIONAL DETAIL 




The enhancement of the PVED of H_2Te_2 and H_2Se_2
was confirmed in our previous works <cit.>.
In this work, 
the MEC of H_2Te_2 and H_2Se_2 in electronic excited states
is studied with quantum chemistry computations
and 
the enhancement of it is confirmed.
Then 
the enhancement mechanism of the MEC
is compared with that of the PVED by using these molecules.


The MEC of these molecules in the ground state
was studied in previous works <cit.>.

The structures of these molecules are chosen to be 
the same as our previous work <cit.>.
The difference in structure between ground and excited states 
is ignored for simplicity,
since the purpose of this paper is
the confirmation of the enhancement of the MEC by electronic excitation
and to study the mechanism of the enhancement.
The structures of H_2Te_2 and H_2Se_2 were determined 
by geometrical optimization computations 
with the DIRAC19 program package <cit.>.
These computations were performed with the HF method with Dirac-Coulomb Hamiltonian
and the dyall.ae2z basis set <cit.>.
After the geometrical optimization,
the dihedral angle ϕ is taken as a free parameter
and the structure dependence of the enhancement is studied for the change of ϕ.

Computations of relativistic four-component wave functions 
are carried out with DIRAC19.
Computations of excited states are based on 
Equation-of-Motion Coupled-Cluster (EOM-CC) theory.
The MEC in excited states is calculated 
by using the Finite-Field Perturbation Theory (FFPT) <cit.> as explained later.
These computations are also supported by 
the comparison with results by the Z-vector method of Coupled Cluster Singles and Doubles (CCSD).
Electron chirality of a specific molecular orbital
is calculated with the HF method.
For HF computations,
the Dirac-Coulomb-Gaunt Hamiltonian is used,
and for CCSD and EOM-CCSD computations using the RELCCSD module <cit.>,
molecular mean-field approximations to the Dirac-Coulomb-Gaunt Hamiltonian
are used <cit.>.
In all computations,
the Gaussian nuclei density distribution is used <cit.>.

In computations of H_2Te_2 and H_2Se_2, 
C_2 symmetry is adopted, and 
the dyall.acv3z basis set <cit.> is utilized.
In CCSD and EOM-CCSD computations of H_2Te_2 and H_2Se_2,
electrons in 4s4p4d (Te) and 3s3p3d (Se) orbitals are chosen for frozen core electrons.
The upper limit cutoff of the active space is 
70 Hartree (H_2Te_2) and 100 Hartree (H_2Se_2).
The threshold of the convergence in EOM-CCSD computations is 10^-10. 

In our computations, 
the calculation of the MEC in excited states is done with the FFPT <cit.>.
The MEC is derived by
the perturbation calculation 
with the perturbation Hamiltonian H_5 = λγ_5,
where λ is the perturbation parameter.
The expectation value of this operator is 


    ⟨ H_5 ⟩
    =
    λ∫ d^3x ψ^† (x) γ_5 ψ (x) 
    =
    λ∫ d^3x ⟨γ_5 ⟩ .



For this expectation value, 
the following relation is given with the Hellmann-Feynman theorem,


    . ∂ E ( λ) /∂λ|_λ = 0   =
    ⟨ψ|
    ∂ H /∂λ| ψ⟩
    
    = 
    ∫ d^3x ψ^† (x) γ_5 ψ (x) ,


where H is the total Hamiltonian
and ⟨ψ | and | ψ⟩ are bra and ket.
Using the finite difference method,
∂ E / ∂λ |_λ = 0 can be computed 
as 

    . ∂ E /∂λ|_λ = 0≃ E (λ) - E(-λ) / 2λ .

This computation of E (±λ) is done with the EOM-CCSD in the Dirac program.
For the finite difference method,
accurate computations require adequate value of λ.
For too large λ, effects from second or higher-order derivatives cannot be neglected,
while for too small λ, 
the effect of the perturbation is invisible,
since the effect is buried in a numerical error.
To search for an adequate value of λ,
the values of the MEC in the ground states are 
compared with those derived with CCSD computations.
Once the value of λ is determined, 
the same value of λ is used for excited states.
In the present work, 
λ =0.1 is adopted for both H_2Te_2 and H_2Se_2
as explained later.











§ RESULTS




In this section, 
the atomic units are used for the electron chirality, particularly, the MEC.








Figure <ref> shows
the electron chirality in excited states
of (a) H_2Te_2 and (b) H_2Se_2 as a function of the dihedral angle.
The notation of excited states, nx, means the n-th excited state obeying the x-symmetry. 
The electron chirality in excited states
have opposite sign and much larger values
compared to that of the ground state (GS).
The enhancement of the MEC as well as the PVED
is confirmed.
While the MEC in the ground state on the dihedral angle 
is a sigmoidal curve,
in excited states a different pattern of the dependence is seen.
All curves of the MEC have similar distributions.
In their patterns,
the MEC increases to 90^∘
and flips the sign around ϕ = 90^∘.
Most excited states are derived as the excitation of one electron in the HOMO in the ground state.
The excitation ratio of the HOMO is roughly 90%
except for 2a and 2b states around ϕ = 90^∘ for both molecules,
where the ratio is 20-70%.
The energies of these excited states were reported in Ref. <cit.>.
For both molecules, 
1a, 1b, and 2a (ϕ < 90^∘) or 1a, 1b, and 2b (ϕ > 90^∘)
are in the same approximate triplet state.
The rest 2a (2b) state is the singlet state for ϕ > 105^∘ (ϕ<75^∘),
and forms approximate triplet state with 3a and 3b for 90^∘ < ϕ < 105^∘ (75^∘ < ϕ < 90^∘).

The most stable structure of these excited states of H_2Te_2 and H_2Se_2
are achiral (ϕ = 0^∘ or 180^∘).
Actually, the MEC for the optimized structure in these excited states
is zero, indicating no enhancement for these molecules in these excited states.
This study of H_2Te_2 and H_2Se_2 is
not for the search for molecules with large MEC in excited states
but for the confirmation of the enhancement in electronic excited states.
Moreover, the optimized structure of electronic excited states 3a, 3b, and 4a 
have ϕ = 90^∘ (3a and 3b) or 105^∘ (4a) <cit.>.
Hence, some higher electronic excited states have chiral structures.




As explained in the previous section,
the usage of the adequate value of λ
is important for the evaluation of the MEC with the FFPT method.
In our work, the value of λ is 
determined by the comparison with the value of the MEC
in the CCSD computation using the Z-vector method.
The dependence of the value of the MEC on λ in the FFPT computations
is shown in 
Tables <ref> and <ref>.
For these computations, the dihedral angle is chosen to be ϕ = 45^∘,
since the integrated chirality is almost zero in optimized structure (ϕ = 90^∘)
and large at ϕ = 45^∘.
The deviation is the smallest at λ = 0.1,
and this value is adopted for both molecules.



















This single value of λ was used for all excited states and all angles of the same molecule
in the computation of Fig. <ref>.
Even for excited states, 
the small change of λ does not affect our results for excited states 
as seen in Table <ref>.
The adequate value of λ is 
determined by the ratio of the perturbation energy ∫ d^3x ⟨γ_5 (x) ⟩ to the molecular one.
The change of the molecular energy is small for the excitation and the change of the dihedral angle
compared to the increase of ∫ d^3x ⟨γ_5 (x) ⟩ by the excitation.
The increase of ∫ d^3x ⟨γ_5 (x) ⟩ from the ground state at ϕ = 45^∘
is several to ten times from Fig. <ref>.
In Tables <ref> and <ref>,
the deviation is less than 3% (H_2Te_2) and 5% (H_2Se_2) for λ = 1.0
and hence
for ten times larger perturbation Hamiltonian λ∫ d^3x ⟨γ_5 (x) ⟩
in the ground state at ϕ = 45^∘ 
the deviation is confirmed to be acceptably small.
Hence, 
the usage of the same λ in excited states is considered to be appropriate.





To understand the dependence of the MEC on the dihedral angle,
contributions from single molecular orbitals are studied.
In Fig. <ref>,
contributions from HOMO, HOMO-1, and HOMO-2
are shown as a function of the dihedral angle.
For both molecules,
contributions from HOMO, HOMO-1, and HOMO-2
are much larger than the total electron chirality summed up all the occupied orbitals.
These three contributions are almost canceled out.
This is the key to the idea of the enhancement of the electron chirality
and is consistent with the previous report <cit.>.
The pattern of the HOMO contribution is opposite to that of the MEC of 1a and 1b excited states.
This is consistent with the CBE hypothesis.




In our previous paper <cit.>,
it is reported that 
the value of the PVED in the first excited state of some molecules
can be predicted as the opposite sign of the HOMO contribution to the PVED
in the CBE hypothesis.
In this hypothesis,
when in the ground state the PVED contribution from the HOMO is larger than the total value of occupied orbitals,
the PVED in the first excited state is much larger than that in the ground state
by breaking the cancellation between contributions from the HOMO and the other orbitals.
Moreover, the PVED contribution from the HOMO in the ground state
dominates over any contribution from other occupied orbitals,
the PVED in the first excited state can be estimated from only the HOMO contribution.
Since the MEC of H_2Te_2 and H_2Se_2 is also derived as
a result of the cancellation between contributions from the HOMO and others orbitals <cit.>,
the conditions of the CBE hypothesis are satisfied for the MEC in H_2Te_2 and H_2Se_2.

In Fig. <ref>,
the estimate of the MEC in the CBE hypothesis 
is shown as well as the improved estimate as introduced in Sec. <ref>.

The predictions by the CBE hypothesis well realize 
the patterns of the MEC in 1a and 1b states.
Particularly, 
all predictions for ϕ > 90^∘
are perfectly consistent with computational results of the MEC.
For ϕ < 90^∘
the prediction in the 1a state of H_2Te_2
realizes precisely the MEC curve,
while  prediction curves for other states
have a few hundred percents deviations.
This inconsistency is considered to arise from 
the modification of occupied orbitals from the ground state
and the difference between HF and CCSD computations.
Nevertheless, 
the patterns of curves are consistent with the computational results.
The CBE hypothesis is considered to apply the MEC.
At most points, the simple CBE estimate does not deviate from the prediction by IE.
This is because the electron in the HOMO is dominantly excited at most points.
The prediction by the improved estimate seems to be worse
than the simple CBE estimate around ϕ = 90^∘.
This is because there are few data for IE points.
If there were additional data around ϕ = 90^∘,
the IE prediction was better than Fig. <ref>.










Finally, the dependence of the MEC on the dihedral angle is discussed.
It is known that the dependence of the MEC and PVED on the dihedral angle 
is almost the same in the ground state of H_2Te_2 and H_2Se_2 <cit.>.
For excited states, 
MEC curves have different patterns compared to PVED ones.
The pattern of MEC curves is like the teeth of a saw as shown in Fig. <ref>,
while PVED curves have a narrow peak <cit.>.
The difference in the dependence on the dihedral angle between the MEC and the PVED in the first excited states 
can be elucidated by the CBE hypothesis,
and the dependence is almost explained by the HOMO contribution dependence on the dihedral angle
as seen from Eqs. (<ref>) and (<ref>).
Therefore, 
the difference in the dependence is the difference in the HOMO contribution.
It is known that the PVED can be divided into contributions from each atom <cit.>.
Hence, to understand the difference in the HOMO contribution more deeply,
the HOMO contribution to the MEC is divided into atomic contributions,
i.e. heavier (Te or Se) and lighter (H) atomic contributions.
In Fig. <ref>,
atomic contributions in the HOMO to the MEC
is shown as a function of the dihedral angle.
For lines of HOMO(Te, Se) and HOMO(H),
the contribution from atomic orbitals of only a single atom (Te, Se, or H)
is taken into consideration.
These contributions are calculated with
QEDynamics (module for Dirac) <cit.>.
As seen from Fig. <ref>,
the HOMO contribution to the MEC 
is not the simple sum of atomic contributions
and the mixing by the superposition between atomic orbitals of heavier and lighter elements
is inevitably important.
This feature is a salient contrast to the PVED.
In Ref. <cit.>,
it was implied that the MEC as well as the PVED is also the very atomic nature,
since the electron chirality is derived as the product of the small and large components
of the four-component wave function.
Our result shows that the MEC is not so simple and not atomic nature.
This is also the reason why
computations of the MEC for molecules with only light elements such as H_2O_2 
requires additional diffuse functions for basis sets,
which reported in previous work <cit.>.





§ SUMMARY



For many molecules in the ground state,
the small value of the PVED and MEC
is derived as a result of the cancellation among larger contributions of valence electrons.
Hence, it was predicted that the PVED and MEC are enhanced 
for ionized or electronic excited states
by the cancellation breaking <cit.>.
Our previous papers <cit.>
showed that 
the PVED of H_2X_2 (X= O, S, Se, Te), CHFClBr, CHFClI, and CHFBrI
are enhanced in electronic excited states.
In this work, it has been shown that 
electronic excitation really enhances 
the electron chirality of the entire molecule, MEC,
for H_2Te_2 and H_2Se_2 as well as the PVED.
In our previous papers <cit.>,
the PVED in the first excited state 
is explained by the HOMO contribution to the PVED
if the PVED contribution from the HOMO in the ground state
dominates over any other contribution.
In this paper, 
it has been confirmed that 
this explanation can also be applicable to the MEC.
Then, the difference in the dependence on the dihedral angle between the MEC and PVED in excited states
has been discussed.
The dependence is determined by the dependence of HOMO contributions to the MEC and PVED.
While the PVED is known to be able to be divided into contributions from each atom,
it has been shown that the MEC is not explained with separate contributions from each atom.








This work was supported by Grants-in-Aid for Scientific Research (21H00072, and 22K12060).
The work of N. K. was supported by Grant-in-Aid for JSPS Fellows (23KJ1188)
and JST SPRING, Grant Number JPMJSP2110.
In this research work we used the supercomputer of ACCMS, Kyoto University.
The authors are grateful to Ayaki Sunaga for the contribution at the early stage of this work.


99 

zetaforce

A. Tachibana,
J. Mol. Model. 11, 301 (2005);


J. Mol. Struct. :(THEOCHEM) 943, 138 (2010).

Tachibana:2017
A. Tachibana,
New Aspects of Quantum Electrodynamics (Springer, 2017).

confirmation
M. Senami, J. Nishikawa, T. Hara, and A. Tachibana,
J. Phys. Soc. Jpn. 79, 084302 (2010);
T. Hara, M. Senami, and A. Tachibana, 
Phys. Lett. A 376, 1434 (2012);
M. Fukuda, M. Senami, and A. Tachibana, 
Prog. Theor. Chem. Phys. 27, 131 (2013);
M. Fukuda, K. Soga, M. Senami, and A. Tachibana,
Int. J. Quantum Chem. 116, 920 (2016).

CHFClBr
Ch. Daussy, T. Marrel, A. Amy-Klein, C. T. Nguyen, Ch. J. Bordé, Ch. Chardonnet,
Phys. Rev. Lett. 83, 1554 (1999);
M. Ziskind, C. Daussy, T. Marrel, and Ch. Chardonnet,
Eur. Phys. J. D 20, 219 (2002).

Darquie:2010
B. Darquié, C. Stoeffler, S. Zrig, J. Crassous, P. Soulard, P. Asselin, T. R. Huet,
L. Guy, R. Bast, T. Saue, P. Schwerdtfeger, A. Shelkovnikov, C. Daussy, A. Amy-Klein, and C. Chardonnet, 
Chirality 22, 870 (2010),
and references therein.

Senami:2019
M. Senami and K. Ito,
Phys. Rev. A 99, 012509 (2019).

Kuroda:2022
N. Kuroda, T. Oho, M. Senami, and A. Sunaga,
Phys. Rev. A 105, 012820 (2022).

Kuroda:2023
N. Kuroda, A. Sunaga, and M. Senami,
J. Phys. Soc. Jpn. 92 124301 (2023).

Bast:2011
R. Bast, A. Koers, A. S. P. Gomes, M. Iliaš, L. Visscher, P. Schwerdtfeger, T. Saue,
Phys. Chem. Chem. Phys. 13, 864 (2011).

Senami:2018
M. Senami, K. Inada, K. Soga, M. Fukuda, and A. Tachibana,
Prog. Theor. Chem. Phys. 31, 95 (2018).

Senami:2020
M. Senami and T. Shimizu,
Phys. Lett. A 384, 126796 (2020).

text:homochirality
For example, 
see U. Meierhenrich, 
“Amino Acids and the Asymmetry of Life",
(Springer-Verlag, Berlin, 2008).

CISS
R. Naaman,
J. Phys. Chem. Lett. 3, 2178 (2012);
R. Naaman, D. H. Waldeck,
Annu. Rev. Phys. Chem. 66, 263 (2015).

Senami:2021
M. Senami and T. Shimada (unpublished).

Berger:2003
R. Berger,
Phys. Chem. Chem. Phys. 5, 12 (2003).

Wisenfeld:1988
L. Wiesenfeld, Mol. Phys. 64, 739 (1988).

Laerdahl:1999
J. K. Laerdahl and P. Schwerdtfeger,
Phys. Rev. A 60, 4439 (1999).

Schwerdtfeger:2005
P. Schwerdtfeger, T. Saue, J. N. P. van Stralen, and L. Visscher,
Phys. Rev. A 71, 012103 (2005).

Eduardus
A. Landau, Eduardus, D. Behar, E. R. Wallach, L. F. Pašteka, S. Faraji, A. Borschevsky, and Y. Shagam,
J. Chem. Phys. 159, 114307 (2023);
Eduardus, Y. Shagam, A. Landau, S. Faraji, P. Schwerdtfeger, A. Borschevsky, and L. F. Pašteka,
Chem. Commun. 59, 14579 (2023).

textbook1
For example,
see J. D. Bjorken and S. D. Drell,
Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

DIRAC19
DIRAC, a relativistic ab initio electronic structure program, Release DIRAC19 (2019), written by A. S. P. Gomes, T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from I. A. Aucar, V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, L. Halbert, E. D. Hedegård, T. Helgaker, B. Helmich-Paris, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, J. K. Lærdahl, C. V. Larsen, Y. S. Lee, H. S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R. Di Remigio, K. Ruud, P. Sałek, B. Schimmelpfennig, B. Senjean, A. Shee, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, M. L. Vidal, S. Villaume, O. Visser, T. Winther, and S. Yamamoto (available at 
<http://dx.doi.org/10.5281/zenodo.3572669>,
see also <http://www.diracprogram.org>).

DIRAC:2020
T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall, E. Eliav, E. Fasshauer, T. Fleig, L. Halbert, E. D. Hedegård, B. Helmich-Paris, M. Iliaš, C. R. Jacob, S. Knecht, J. K. Lærdahl, M. L. Vidal, M. K. Nayak, M. Olejniczak, J. M. H. Olsen, M. Pernpointner, B. Senjean, A. Shee, A. Sunaga, and J. N. P. van Stralen,
J. Chem. Phys. 152, 204104 (2020).

ae2z
K.G. Dyall,
Theor. Chem. Acc. 99, 366 (1998);
Theor. Chem. Acc. 108, 365 (2002).

Dyall:2006
K.G. Dyall,
Theor. Chem. Acc. 115, 441 (2006).

Pople:1968
J. A. Pople, J. W. McIver, and N. S. Ostlund,
J. Chem. Phys. 49, 2960 (1968).

Pawlowski:2015
F. Pawlowski, J. Olsen, and P. Jørgensen,
J. Chem. Phys. 142, 114109 (2015).

Visscher:1995
L. Visscher, K. G. Dyall, and T. J. Lee, Int. J. Quantum Chem. 56, 411 (1995);
L. Visscher, T. J. Lee, and K. G. Dyall, J. Chem. Phys. 105, 8769 (1996).

Shee:2018
A. Shee, T. Saue, L. Visscher, and A. S. P. Gomes,
J. Chem. Phys. 149, (2018).

Sikkema:2009
J. Sikkema, L. Visscher, T. Saue, and M. Iliaš, J. Chem. Phys. 131, 124116 (2009).

Visscher_Gauss:1997
L. Visscher and K. G. Dyall, At. Data Nucl. Data Tables 67, 207 (1997).

acvXz
K.G. Dyall,
Theor. Chem. Acc. 108, 335 (2002);

J. Phys. Chem. A 113, 12638 (2009);


Theor. Chem. Acc. 135, 128 (2016).


QEDynamics 







QEDynamics (module for Dirac) (2023), written by M. Senami, K. Ichikawa, A. Tachibana
with contributions from M. Fukuda and K. Soga,
(https://github.com/mfukudaQED/QEDynamics_for_Dirac).










 Paper Title: Accurate Tunneling Splittings for Ever-Larger Molecules from Transfer-Learned, CCSD(T) Quality Energy Functions.




§ ABSTRACT 
This work combines state-of-the-art machine learning techniques with
highest-level electronic structure calculations and full-dimensional
quantum tunneling calculations to obtain a quantitative
characterization of tunneling splittings for system sizes that are
currently out of reach using traditional approaches. For
intramolecular hydrogen transfer in tropolone, the best computed
splitting including perturbative corrections in the ring-polymer
instanton calculations is 0.94 cm^-1 and compares with
0.974 cm^-1 from experiments. On the other hand, for
intermolecular double hydrogen transfer in the (propiolic
acid)–(formic acid) dimer, the computations yield 0.0147 cm^-1
which is larger by 40 % compared with experiment (0.0097 cm^-1)
but still in much better agreement than previous attempts (0.63
cm^-1). The strategy pursued in the present work is applicable to
yet larger systems and other properties of interest and provides a
rational route for highest-accuracy energy functions for prediction
and benchmarking electronic structure methods vis-a-vis experiments.




§ INTRODUCTION

Quantum mechanical tunneling is a central concept in chemistry and
physics.<cit.> One of the experimentally observable hallmarks
of tunneling processes is the splitting of energy levels. These
tunneling splittings are measurable and can be probed, e.g.,
using microwave<cit.> or infrared
spectroscopy.<cit.> The splittings are exquisitely
sensitive to the energetics and shape of the underlying potential
energy surface (PES) and provide valuable benchmarks to validate
state-of-the-art theoretical methodology.


Accurate PESs allow to carry out quantitative molecular simulations
that link directly to experiments. One of the most successful recent
approaches to devise such PESs uses a combination of electronic
structure calculations and machine learning techniques to give a
machine learning-based PES. Coupled cluster-based techniques such as
CCSD(T) are among the most accurate methods for closed-shell
systems. However, due to the unfavourable scaling of such calculations
(seventh power of the number of basis functions), the necessary number
of reference calculations required for training a full-dimensional PES
(∼ 10^4 or more) at this level of theory is only feasible for
small molecules with ≲ 5 heavy atoms. However, if a small
number of calculations can be afforded at such high levels, a
considerable boost in performance can be achieved using transfer
learning (TL)
techniques.<cit.>
The essence of TL is that the overall topographies of molecular PESs
at different levels of theory are often comparable which can be
exploited by retraining a low-level (LL) PES with small amounts of
data computed at a high level. Such an approach is explored in the
present work for two systems that have so far been intractable at the
CCSD(T) level of theory.


Computing accurate tunneling splittings for multidimensional systems
from fully quantum-mechanical methods is a formidable task. The
ring-polymer instanton (RPI) approach provides a semiclassical
approximation for tunneling calculations which scales well with system
size.<cit.> As instanton theory makes a
local harmonic approximation around the optimal tunneling pathway, it
typically deviates from quantum-mechanical benchmarks by up to 
20%.<cit.> This error can, however, be significantly
reduced using a new procedure to perturbatively correct the results
using information about third and fourth derivatives along the
path.<cit.>




The present work reports tunneling splittings for 15-atom tropolone
and 12-atom (propiolic acid)–(formic acid) dimer (PFD) (see
Figure <ref>) following the TL+RPI
approach.<cit.> These molecules are related
to malonaldehyde
<cit.>
and the formic acid dimer
(FAD)<cit.>,
which have been studied experimentally and theoretically in great
detail. Both molecules have been treated with RPI, which was
particularly successful for malonaldehyde. Using perturbatively
corrected RPI on two different malonaldehyde
PESs<cit.> gave a tunneling
splitting for hydrogen (H-)transfer of 22.1 cm^-1 whereas the
experimental value is 21.6 cm^-1. Compared with malonaldehyde and
FAD, far less theoretical and experimental work has been devoted to
their “larger siblings”, tropolone and PFD, mostly due to the
difficulty of obtaining CCSD(T) quality PESs for these systems of many
electrons.


The experimentally reported ground vibrational state (H/D) tunneling
splittings for tropolone are
(0.974/0.051) cm^-1.<cit.>
Theoretical works include a permutationally invariant polynomial PES
based on 6604 B3LYP reference energies and
forces.<cit.> This was recently improved to
near-CCSD(T) quality using the molecular tailoring approach
(MTA)<cit.> to obtain approximate CCSD(T) energies
and Δ-machine learning based on 2044
energies.<cit.> The (H/D) tunneling splitting as
evaluated on the Δ-machine learned PES was
(0.68/0.031) cm^-1, which was subsequently improved by scaling the
action using further MTA calculations to give
(0.92/0.042) cm^-1.


For PFD, the experimental energy splittings are
(0.009721/0.0001117) cm^-1 for H-H and D-D exchange,
respectively.<cit.> Theoretically, PFD has, for
example, been investigated using second-order vibrational perturbation
theory<cit.> and a one-dimensional model has been employed
to estimate tunneling splittings<cit.>, which,
however, were not in quantitative agreement with experiment. For
instance, the (H/D) tunneling splittings based on a rectilinear path
determined from a MP2/aug-cc-pVDZ intrinsic reaction coordinate scan
were (0.63/0.0245) cm^-1, and thus required a dramatic empirical
scaling in a subsequent step.





§ RESULTS AND DISCUSSION

For both tropolone and PFD, a sufficiently large number of
CCSD(T)/aug-cc-pVTZ calculations is currently out of reach
(e.g., for tropolone: “Ideally, we would learn the
difference between the DFT and the “gold standard” CCSD(T)
level. However, this is not feasible (for us) owing to N^7 scaling
...”<cit.>). In an attempt to reach CCSD(T) quality,
the present work employs TL strategies for tropolone and PFD. First,
LL reactive PESs were determined, see Supporting
Information. Schematics of the two PESs at the MP2 (low-)level
including molecular structures and the accuracy of the energy on
stationary points are shown in Figures <ref> and
<ref>. Comparing the energies of the stationary
points of the PESs with direct ab initio calculations yield
maximum deviations of 0.05 and 0.03 kcal/mol for tropolone and PFD,
respectively. Additional validations on a hold-out (test) set, see
Table <ref> and Figure <ref>, feature mean absolute
errors on energies (MAE(E)) well below 0.1 kcal/mol.




The barriers for H-transfer from the LL-PESs agree to within 0.001 and
0.01 kcal/mol with those from direct ab initio calculations at
the MP2 level which are 4.29 and 6.47 kcal/mol for tropolone and PFD,
respectively. Harmonic frequencies constitute an additional measure
for the quality of a PES around a stationary point. For the global
minimum and the transition state (TS) for H-transfer, the MAEs are
∼ 1 cm^-1 and ∼ 0.4 cm^-1 for tropolone and PFD,
respectively.  Tables <ref> and
<ref> compare the individual frequencies for
tropolone and PFD. This validates the quality of the LL-PESs compared
with reference electronic structure calculations at the MP2 level of
theory.


For the present work, the main target observable is the tunneling
splitting Δ^ H_ RPI for H-transfer. Despite the
remarkable accuracy of the two LL-PESs for tropolone and PFD when
compared with the reference MP2 data, the computed tunneling
splittings overestimate the experimental results by an order of
magnitude: The H-transfer tunneling splittings on the LL-PESs are
10.94 and 0.11 cm^-1, whereas experimental
measurements<cit.> give 0.974 and
0.0097211 cm^-1 for tropolone and PFD, respectively. This is a
clear indication that MP2 is not sufficient for quantitative
calculations of tunneling splittings. Hence, transfer learning to the
CCSD(T) level was explored next based on comparatively small numbers
of additional high-level calculations. The iterative data selection
procedure, which was identical for both systems, is described in
detail in the Supporting Information, with only a concise summary
provided here.




Based on an initial set of 25 judiciously selected structures, the
data set was iteratively extended by selecting n_i structures
along the refined minimum energy path (MEP) and instanton path (IP),
and by selecting n_i, pool structures from the LL pool of
structures (see Figure <ref>) using
active learning. Figure <ref> reports sets TL_0
to TL_2 for tropolone together with the MEPs (dot-dashed) and IPs
(dashed) on the LL- (black) and a representative TL-PESs (purple).
All steps involving the LL- and TL-PESs are rapid whereas the
high-level ab initio calculations are computationally demanding
(∼ 50 h for tropolone and ∼ 100 h for PFD). However, the
high-level ab initio calculations for multiple structures can
be run in parallel. Due to the rather small high-level data set sizes,
TL was repeated 10 times on different splits of the data.


The H-transfer barrier heights for tropolone and PFD from all TLs are
shown in Figure <ref>. For tropolone
(Figure <ref>A), the barrier height at the
MP2 level is 4.28 kcal/mol, which increases to 6.62 kcal/mol for
TL_2, i.e., after TL with 100 high-level data
points. Remarkably, the result with only 25 data points (TL_0)
appears converged as further addition of data points reduces the
uncertainty (the error bars) but not the average barrier
height. Similarly, for PFD the MP2 energy barrier of 6.46 kcal/mol
increases to 7.76 kcal/mol on the TL_3 PES. Again, going from TL_0
to TL_3 reduces the uncertainty and adjustments of the energy
barrier progressively decrease. It should be noted that obtaining
optimized geometries and harmonic frequencies from brute-force
electronic structure calculations at the CCSD(T)/aug-cc-pVTZ level is
not feasible for systems of the sizes studied here, but
straightforward when using the TL-PES.<cit.> From
the two TL-PESs the best estimate for the barrier heights are 6.62
kcal/mol and 7.76 kcal/mol which compare with 6.63 kcal/mol and 7.76
kcal/mol from single-point CCSD(T) calculations using the optimized
and TS structures obtained from the TL_2 and TL_3 PESs for
tropolone and PFD, respectively.





Previous work reported barrier heights of 6.69 kcal/mol for
tropolone<cit.> from a single-point
CCSD(T)/aug-cc-pVTZ calculation using a MP2/aug-cc-pVTZ optimized
structure. Alternatively, using Δ-machine learning to the
MTA-CCSD(T)/aug-cc-pVTZ level yields a barrier height of 7.18
kcal/mol.<cit.> For PFD, preliminary calculations at the
CCSD(T)-F12b/aug-cc-pVDZ level found E_ B∼ 8.0 kcal/mol for
double H-transfer in PFD.<cit.>




Ring polymer instanton simulations do not involve random numbers or statistical
errors which clearly distinguishes them from path-integral molecular
dynamics (PIMD)
methods<cit.> or
Diffusion Monte Carlo (DMC)
simulations.<cit.> Hence, the instanton
(once it is converged with τ→∞ and
N/τ→∞) is in principle uniquely determined by a
given PES. As shown in Figure <ref> the IP
differs from the MEP.  Representative IPs on the LL- and TL-PESs for
both systems are shown in Figure <ref> and as
a movie in the Supporting Information. For tropolone
(Figure <ref>A), the instanton is mainly
characterized by a hydrogen motion and slight adjustments in the CC
and CO bond lengths due to single-/double-bond migration. For PFD
(Figure <ref>B), more pronounced displacements
of the oxygen atoms, the monomer separation and wagging of the CCH
tail occur during double H-transfer.





The H- and D-transfer tunneling splittings as obtained from the
different TL-PESs for tropolone and PFD are shown in
Figure <ref>. For tropolone, the instanton
result on the LL-PES of 10.94 cm^-1 is brought much closer to the
experimental result of 0.974 cm^-1 using the TL-PESs. In
particular, the experimental result is within the standard deviation
of the calculations. Note that the excellent agreement of TL_0 and
its rather limited fluctuation around the mean probably appears by
chance. However, it does demonstrate that even a very small number of
judiciously chosen structures (25 for TL_0) suffices to
significantly improve the PES and enable accurate results to be
obtained. For TL_2, which is arguably the most accurate PES for
tropolone, Δ_ RPI^ H = 1.01 cm^-1 and
Δ_ RPI^ D = 0.055 cm^-1, which both overestimate
experimental values by ∼ 5%, and indicate that further
improvements should be possible which is addressed further below.


For PFD, gradually increasing the data set size from TL_0 to TL_3
led to smaller uncertainty and decreasing changes in Δ_
  RPI. The final transfer learning, TL_3, was performed to capture
the potential along the dissociation of the dimer (see
Figure <ref>). As anticipated this only marginally
affected the tunneling splitting. The values of Δ_ RPI^
  H = 0.014 cm^-1 and Δ_ RPI^ D = 0.00024
cm^-1 overestimate the experimentally reported splittings by ∼
50% and a factor of 2, respectively. Nevertheless, they are
considerably improved over the existing computational treatments that
reported 0.63 cm^-1.<cit.>






The tunneling splittings for the two systems considered here as
determined on the largest available TL data sets are summarized in
Table <ref>. For tropolone, the tunneling
splitting of the symmetric ^13C isotopologue has been determined
experimentally to be 0.97 cm^-1, compared with Δ_
  RPI^ SP = 1.00 cm^-1 from the present simulations. For
the ^18OH^18O isotopologue only an experimentally estimated
value Δ_ Expt.^ ^18O∼ 0.83 cm^-1 is
available so
far.<cit.> Hence,
the splitting was computed and yields Δ_ RPI^ SP,
  ^18O = 0.95 cm^-1.  Due to well-understood error
cancellations, instanton theory is normally particularly reliable in
predicting isotope effects.<cit.> Thus, the ratio
of calculated results can be used to obtain a new estimate of the true
tunneling splitting as Δ_ Expt.^ ^18O∼Δ_ Expt.^ H·Δ_ RPI^ SP, ^18O /
Δ_ RPI^ SP, H = 0.974·0.95/1.01 =
0.916 cm^-1. This value is considered an improved estimate for
Δ_ Expt.^ ^18O over Δ_ Expt.^
  ^18O∼ 0.83 cm^-1 and is a testable prediction for future
experiments from the present simulations. In all cases, the computed
splittings are larger than those reported experimentally. This is at
least partly due to inherent approximations of instanton theory.


Recently, a correction scheme for RPI-based tunneling splittings using
third- and fourth-order derivatives was
proposed.<cit.> For cases where the higher-order
derivatives are carried out numerically, it was shown that numerically
stable applications of this correction scheme require double-precision
arithmetics for training and inference of a neural network based
PES.<cit.> This goes beyond standard procedures
because machine learning models typically operate at 32-bit accuracy
(single-precision) to reduce the computational cost for training and
inference, and to lower memory requirements.<cit.> It
was found that single-precision can have detrimental effects
particularly if higher-order derivatives of the energy are
required.<cit.> For that reason, TL was repeated
using double-precision for the largest TL data set sizes and
Δ_ RPI^ DP was determined together with the
perturbatively corrected value Δ_PC^ DP. Due to the
higher computational burden of the double-precision
training/inference, the iterative data selection procedure was
performed using single-precision.


The averaged double-precision tunneling splittings Δ^ DP
(averaged over 10 independent TLs) for tropolone and PFD are presented
in Table <ref>. It is interesting to note that
even the uncorrected tunneling splittings Δ_ RPI^ SP
and Δ_ RPI^ DP differ slightly between single- and
double-precision arithmetics. These differences are probably because
single-precision does not give the Hessians to sufficient accuracy,
leading to small errors in the fluctuation factors
Φ. Comprehensive results to support this, including energy
barriers, actions S, fluctuation factors Φ, uncorrected
tunneling splittings and correction factors c_ PC for the
individual TLs are given in
Tables <ref>,
<ref>,
<ref> and
<ref>.  


For tropolone, the uncorrected (Δ_ RPI^ DP) and
perturbatively corrected (Δ_PC^ DP) tunneling
splittings are in excellent agreement with experiment and are the most
accurate results reported so far. Empirically, standard RPI theory for
Δ_ RPI was found to give splittings within ∼ 20% of
fully quantum-mechanical calculations using the same PES for typical
molecular systems, as long as the barrier height is significantly
higher than the zero-point energy along the tunneling
mode.<cit.> For
malonaldehyde<cit.> and tropolone, the
deviations from experiment are 16% and 10%, respectively, which is
within the expected range. If the results are perturbatively corrected
(Δ_PC^ DP), they are within 2% to 3% for
malonaldehyde<cit.> and
tropolone, respectively. As such, contrary to earlier
calculations<cit.> that do not include perturbative
corrections, the present calculations yield the “right answer for the
right reason”.


For PFD, however, the calculated splitting for H-transfer is roughly
50% larger than that found in the experiments, even after the
perturbative correction is applied. This is reminiscent of results for
FAD<cit.> for which Δ^ H_ RPI =
0.014 cm^-1 was found. The calculation compares with measured
splittings between 0.011 and 0.017
cm^-1<cit.>
(with 0.011 cm^-1 from microwave spectroscopy<cit.>
probably being the most reliable value), which also differs by more
than 20%.  In search of possible explanations, additional
calculations were carried out to probe (i) limitations in the quantum
chemical methods used (ii) PES representation errors and (iii)
limitations inherent in RPI theory and the perturbative correction.


Item (i) can be partly assessed by rescaling the action S with a
more accurate estimation of the instanton barrier height according to
S_ new = S ·√(E_ inst^ new / E_
    inst). The CCSD(T)/aug-cc-pVQZ instanton barrier for PFD
optimized on a representative TL-PES is E_ B^ new =
12.609 kcal/mol and compares to a barrier of 12.222 kcal/mol on the
TL-PES. Rescaling yields Δ_ RPI (Δ_ PC) of
0.0132 (0.0117) cm^-1 if c_ PC = 0.89 from ⟨
  TL_3⟩ (see Table <ref>) is
used. Although the results are closer to the measurements, they still
differ by more than 20% in particular for the leading order
estimation. To put this into context, S-scaling was also applied to
malonaldehyde. Using a representative
PES<cit.> (E_ inst=5.398 kcal/mol,
Δ_ RPI = 24.2 cm^-1, Δ_ PC=22.0 cm^-1)
and rescaling the action using E_ inst^ new =
5.608 kcal/mol from CCSD(T)/aug-cc-pVQZ calculations yields
Δ_ RPI (Δ_ PC) of 21.9
(19.8) cm^-1. Comparing the results with experimental
measurements<cit.> of 21.583 cm^-1 this is a
clear deterioration for Δ_ PC and shows that this simple
scaling approach can only be considered as a rough estimate of the
error. While inaccuracies in the level of theory cannot be ruled out
completely, the errors for PFD remain significantly higher than for,
e.g., malonaldehyde.


For point (ii) - deficiencies in representing the PES - it is noted
that PhysNet was used with the same hyperparameters for both tropolone
and PFD. Training on MP2 reference data yields comparable test set
errors MAE(E)/MAE(F) of 0.067/0.069 kcal/mol(/Å) for tropolone
and 0.034/0.133 kcal/mol(/Å) for PFD (see
Table <ref>). Hence, it appears unlikely that representation
errors lead to larger deviations in the tunneling splittings only for
PFD.  The quality of the PES is also influenced by the choice of
structures included.  Although the structures for both tropolone and
PFD were chosen following the same procedure and although Δ_
  RPI is converged, it cannot be ruled out that due to the soft
intermolecular modes present in PFD, additional sampling is
required. To assess this, scans of representative soft modes and their
comparison with ab initio CCSD(T) energies are given in
Figures <ref> and <ref> for
PFD, which show good agreement. 


Finally, computed tunneling splittings can be influenced by the
semiclassical approximations underlying RPI theory, see point
(iii). Given the excellent agreement between computations and
experiment for malonaldehyde and tropolone and the inferior
performance for the hydrogen-bonded dimer PFD, supplementary
calculations were performed for FAD. Using a CCSD(T)/aug-cc-pVTZ
level of theory PES for FAD<cit.> and supplementing it
with 667 additional structures along and around the instanton path
yields Δ_ RPI^ DP = 0.0158 cm^-1 and
Δ_PC^ DP = 0.0186 cm^-1. Again and similar to
PFD, this is consistently higher than the measured value of 0.011
cm^-1. Given that the soft modes are well captured by the PES (a
comparison of the PhysNet energies with their ab initio
CCSD(T) energies is given in Figure <ref>) it
seems likely that RPI theory faces challenges when applied to H-bonded
dimers, for example, due to strongly anharmonic out-of-plane
modes. This hypothesis can be tested in future work using
non-perturbative approaches such as PIMD to calculate the
tunneling splittings for a given PES.<cit.> 





§ CONCLUSION

The present work demonstrates that combining state-of-the-art
electronic structure and machine learning techniques with RPI theory
yields H- and D-transfer tunneling splittings in very good agreement
with experiments. The calculations for tropolone are the most accurate
to date due to the high-quality PES and because they include a
perturbative correction, leading to the “right answer for the right
reason.” In contrast, the best previous calculation of
0.92 cm^-1, which was carried out without these corrections,
probably benefited somewhat from cancellation of errors (i) incurred
by the instanton approximation, and (ii) arising from the
fragmentation scheme for obtaining a near-CCSD(T) quality
PES.<cit.> Key to the success in the present work is the
combination of transfer learning<cit.> and
including the perturbative correction to
RPI.<cit.> Importantly, brute-force calculation of
the required number of reference data (energies and forces) at the
CCSD(T) level of theory for training a machine learning PES is
unfeasible at present and for the foreseeable future. Likewise,
the Gaussian Process Regression instanton
approach<cit.> would probably also not be possible
as it requires computation of a few Hessians at the CCSD(T) level.  


Given the excellent performance of TL combined with RPI theory and
perturbative corrections for tunneling splittings of intramolecular
H-transfer (malonaldehyde and tropolone) it is anticipated that the
methods used in the present work can also be used to quantitatively predict tunneling splittings for similar
systems. This is particularly relevant because experimental
information is only available for a handful of
systems<cit.>
and reliable predictions facilitate exploration of as of now
uncharacterized systems. In a yet broader context, together with
complementary experimental information such as vibrational
spectroscopic data, tunneling splittings provide valuable information
to characterize the PES around and away from the minimum energy
structure for further refinements, for example by using “morphing”
techniques.<cit.>


The strategy followed here is generic and applicable to molecules for
which a few dozen very expensive ab initio calculations for
judiciously chosen geometries can be carried out. Application to yet
larger molecules necessitates advancements in either hardware
capabilities or electronic structure theory. Concerning the latter,
considerable effort is put into developing linear scaling CCSD(T)
methods, which are becoming competitive alternatives for accurate
computations of molecules exceeding 30
atoms.<cit.> Alternatively,
similarly to the GEMS approach<cit.>, machine
learning PESs can be built from molecular fragments that allow the
generation of ab initio reference data at higher levels of
theory. Since these approximate methods are inherently difficult to
test, the framework and predictions for, e.g., tropolone
provided here serve as a valuable benchmark for future
applications. Finally, the present approach - with suitable
adaptations in the data selection procedure - can also be applied to
other observables that are computationally expensive to determine
(e.g., quantum bound states of molecules or scattering cross
sections) or be combined with other quantum dynamics methods including
wavepacket propagation or PIMD simulations.




§ CONFLICT OF INTEREST

The authors declare no conflict of interest.



§ ACKNOWLEDGMENT

The authors gratefully acknowledge financial support from the Swiss
National Science Foundation through grants 200020_219779 (MM),
200021_215088 (MM), the NCCR-MUST (MM and JOR), the AFOSR (MM), and
the University of Basel (MM).


 

























§ SUPPORTING INFORMATION: ACCURATE TUNNELING SPLITTINGS FOR EVER-LARGER MOLECULES FROM TRANSFER-LEARNED, CCSD(T) QUALITY ENERGY FUNCTIONS







§ COMPUTATIONAL METHODS

In the following, the RPI method to calculate tunneling splittings is
introduced, followed by a detailed overview of the construction of the
low-level (LL) and high-level (HL) PESs using PhysNet.<cit.>



 §.§ Ring-Polymer Instanton Theory

The RPI method offers a semiclassical approximation for computing
tunneling splittings in molecular
systems.<cit.> In a
one-dimensional model, instanton theory is strongly related to the WKB
approximation.<cit.> Its main advantage, however, is that
it can also be applied to multidimensional systems, in which it
locates the optimal tunneling pathway.<cit.> This pathway,
known as the instanton, is defined as an imaginary-time
τ→∞ path connecting two degenerate wells which
minimizes the action, S. In computations, the path is constructed by
using an efficient ring-polymer optimization based on discretizing the
path into N ring-polymer beads and taking the limit
N→∞ (typically N ∼ 1000 is sufficient for
convergence). The potential U_N of a ring-polymer is given by

    U_N(x;β) = ∑_i=1^N V(x_i) +
        1/2(β_Nħ)^2∑_i=1^N (x_i+1 - x_i)^2 ≡S( x)/β_N ħ

with x = ( x_1, …, x_N) being the
mass-scaled coordinates of the beads, β = 1/k_b T and
β_N = β/N. The first term in
Equation <ref> corresponds to the sum over
all single bead potentials and the second term represents the harmonic
springs with frequency 1/(β_Nħ) that connect the different
beads. As the potential U_N and the action S are related by
division with (β_Nħ), S is determined by the distance
between neighbouring beads as well as the potential energy of each
bead. Therefore, the action S contains information
along the instanton path (IP). A detailed description of the
method is provided, e.g., in References  and
.


Once the IP has been located, fluctuations around the path are
computed to second order and the information is combined into the term
Φ, i.e., this is based on information around the
IP. This requires Hessians at each of the beads. The prediction
for the leading order tunneling splitting (in a double-well
system) is

    Δ_ RPI = 2ħ/Φ√(%s/%s)S2πħ e^-S/ħ.

Because S appears in the exponent it is particularly important to
determine this quantity with high accuracy. A limitation of the RPI
method for determining tunneling splittings is that fluctuations
around the instanton are treated harmonically (i.e., the
Hessians enter via Φ). To a good approximation, it is expected
that this captures the dominant tunneling contribution, except for
cases in which anharmonic effects perpendicular to the instanton are
significant or where the barrier is low. For this reason, Richardson
and coworkers recently developed a perturbatively corrected RPI
theory<cit.>, that accounts for additional
anharmonicity by including information from the third and fourth
derivatives of the potential along the instanton. The tunneling
splitting obtained from perturbatively corrected RPI theory is denoted
as Δ_ PC in the following, which, in practice is obtained
from scaling Δ_ RPI with a correction factor c_ PC
following Δ_ PC = c_ PC·Δ_ RPI.


All leading order splitting calculations in the present work were
carried out with N=4096 beads and at an effective temperature of
T_ eff = ħ/k_ Bτ = 25 K. The numerical convergence
of the leading order tunneling splittings is reported in
Tables <ref>(a) and
<ref>(a).  The perturbatively corrected
instanton calculations were carried out at T_ eff = 50 K and
with N=512 beads. The convergence sets are
reported in Tables <ref>(b) and
<ref>(b) and show that c_ PC converges
significantly more rapidly than the leading order term.




 §.§ Low-Level PESs

The data generation to construct PESs for tropolone and PFD (including
minima and transition states (TS) for hydrogen transfer) followed
similar strategies. While for tropolone sampling two conformers and
the corresponding TSs was sufficient, the PES for PFD involves
multiple conformational substates and dissociation products including
cis- and trans-formic acid, cis- and
trans-propiolic acid. The conformations and their relative
energies for both systems are illustrated in
Figures <ref> and <ref>. Including
monomers and dissociation products for PFD is not anticipated to
improve the tunneling splitting accuracy. However, accounting for
these states is essential for a global PES and desirable
future investigations.


First, normal mode vectors and force constants for all vibrations were
obtained at the MP2/aug-cc-pVDZ level of theory using
Gaussian.<cit.> This information was used to carry out
normal mode sampling<cit.> at temperatures T = [100,
  300, 500, 100, 1500, 2000] K to yield 13000 and 15000 structures
for tropolone and PFD, respectively. For tropolone 22 additional
structures were obtained along the CH-stretching mode to give a total
of 13022 structures. For PFD additional structures were generated by
running NVT MD simulations at 1000 K for cis- and
trans-formic acid, cis- and trans-propiolic
acid and PFD (9000 structures in total) using the semiempirical
GFN2-xTB method<cit.> as available in the atomic
simulation environment.<cit.> The region around the
TS as well as the non-hydrogen bonded complex of the monomers (van der
Waals complex) constituting PFD were sampled using a biasing harmonic
potential to control structural sampling (7000 structures in
total). Thus, the initial data set contained 31000 structures for PFD
and corresponding monomers. For all these structures, energies, forces
and dipole moments were determined at the MP2/cc-pVTZ level of theory
using MOLPRO.<cit.> For both systems two independent PhysNet
models were trained.


The PES pairs were used to carry out adaptive sampling
simulations.<cit.> For tropolone these were two rounds
of (NVT) simulations at 1000 and 2000 K, respectively. For PFD
adaptive sampling was run at 1000 K for the six stationary points on
the PES (see Figure <ref>) and the van der Waals
complex of the (trans-) formic acid and propiolic acid
monomers. Structures were saved if the predictions of the two PESs
differed by more than 0.5 kcal/mol. All adaptive sampling MD
simulations were carried out using the Langevin scheme with a time
step of 0.1 fs as implemented in ASE.<cit.> This
resulted in an additional 5619 structures for tropolone and 9013
structures for PFD (most of which cover the van der Waals complex,
i.e., 6300), respectively.


For both systems (tropolone and PFD with substates) the
configurational space of the PES was further sampled using diffusion
Monte Carlo simulations (DMC),<cit.> from
which 1356 structures were extracted for tropolone whereas for PFD 277
additional structures were obtained. All DMC simulations employ a GPU
implementation of the DMC algorithm for PhysNet and were run with 300
walkers, a step size of 5 a.u. and for a total of 60000
steps. Unphysical structures were detected and added to the training
dataset if the predicted energy was lower than the energy of the
minimum energy structure.


Finally, for both systems two independent PESs were retrained based
on a total of 19997 structures for tropolone and 40290 structures for
PFD. Using the two PES pairs another round of adaptive sampling and DMC
simulations was carried out. However, no further deficient structures
were detected.





 §.§ High-Level PESs / Data Selection Procedure




It has been recently demonstrated that transfer learning a LL-PES to a
higher level of electronic structure theory can be accomplished
efficiently by judiciously choosing reference structures for which
computationally expensive HL energies, forces and dipole moments are
determined.<cit.> This was shown for
malonaldehyde and it was found that transfer learning using 25 to 100
algorithmically selected structures and corresponding CCSD(T) data
performs on par with a PES that was transfer-learned using ∼ 900
structures. Following the road-map towards high-level of theory PESs
presented in Reference ,
high-level information was chosen for tropolone and
PFD. Figure <ref> illustrates the data
selection procedure for tropolone, which, however, is representative
for both systems. Note that it is not necessary to sample both
potential wells since PhysNet handles molecular symmetry by
construction.



    
  * The MEP (black dash-dotted) and the IP (black dashed) are
      determined on the LL-PES. From these, a total of six
      structures (brown rectangles), including the minimum structure,
      the TS structure, the structure of the IP that has the highest
      potential energy (i.e., at q=0) and three additional
      structures that are evenly distributed are manually selected.
    
  * Without additional a priori information it is
      advantageous to generate an initial pool of structures, that
      sample all relevant regions of the PES, to select from (note
      that at this stage no ab initio calculations are
      performed - the structures are sampled using MD, normal mode
      sampling on the low-level PES or are available as part of the
      LL-PES reference data). The coverage of the "structure pool"
      from which appropriate structures are selected is shown in
      Figure <ref>. Structures are
      selected from the pool such that their structural RMSD is
      maximized with respect to the existing set of six
      structures. These 19 structures are illustrated as salmon
      rectangles in Figure <ref>.
    
  * The 25 structures selected above (rectangles) form the data
      set for the first TL, which is referred to as TL_0 in the
      following. For these, CCSD(T) energies, forces and dipole
      moments are calculated using MOLPRO, which are used to
      transfer-learn the NN.
    
  * Next, the MEP (purple dash-dotted) and the IP (purple
      dashed) are determined on the TL_0 PES. From these, a total of
      five structures including the new minimum structure, the new TS
      structure and the new structure of the IP that has the highest
      potential energy and two evenly distributed structures are
      manually selected.
    
  * The five structures above are augmented with 20 structures
      that are chosen with active learning (i.e., two
      independent TL PESs are used to predict the pool of structures
      and the structures with the highest Δ E are selected).
    
  * For the 25 structures selected above (green open circles),
      CCSD(T) energies, forces and dipole moments are calculated using
      MOLPRO. They, together with the 25 structures from TL_0 form
      the data set for TL_1. The LL-PES is transfer-learned with the
      50 structures selected above (circles and rectangles).
    
  * The MEP and the IP are determined on the TL_1 PES (not
      shown in Figure <ref> because they are the
      same as for the TL_0 PES). If the minimum structure, the TS
      structure and the IP structure with the highest potential energy
      differ from the structures selected in the previous iteration,
      then they are selected again. This was not found to be necessary
      here.
    
  * Active learning is again employed to select the N (here
      N=50) structures with the highest Δ E form the pool.
    
  * For the N structures selected above (yellow ×
      symbols), CCSD(T) energies, forces and dipole moments are
      calculated using MOLPRO. They, together with the 50 structures
      from TL_1, form the data set for TL_2.
    
  * Repeat steps 7. to 9. until the observable(s) of interest
      converge. This closes the loop of the recommended procedure to
      obtain spectroscopically accurate PESs.






The procedure outlined above proved to be effective but may need to be
adapted for other systems, e.g. selecting a smaller or larger
number of structures in each iteration. The TL data sets for tropolone
contained 25/50/100 structures for TL_0/TL_1/TL_2, while for PFD
a fourth TL was carried out to correct the PES in the dissociation
limit of the two monomers. Thus, the TL data sets for PFD contained
25/50/100/(106 PFD + 300 monomer structures) for
TL_0/TL_1/TL_2/TL_3. Due to the rather limited data set sizes,
TL was repeated ten times using different data splits for each system
and TL data set size.



In the present case and on the computers used, the determination of
ab initio energy/force/dipole moment for a single structure
takes roughly 50 h for tropolone and 100 h for PFD. The high-level
ab initio calculations were performed at CCSD(T) level with
aug-cc-pVTZ basis set for PFD while for tropolone a slightly smaller
basis set was used (aug-cc-pVTZ basis on the atoms that are involved
in the hydrogen transfer reaction (i.e., transferring H-atom
and both O-atoms), while cc-pVTZ basis sets represent the rest of the
molecule.) This was necessary because of the enormous memory
requirements of such calculations.




§ RESULTS



 §.§ Low-level PES























 §.§ High-Level PES










§ CONVERGENCE OF THE RPI CALCULATIONS

The section contains the convergence of the numerical results for
tropolone and PFD, respectively.



 §.§ Tropolone












 §.§ PFD















 §.§ FAD









 Paper Title: Temporal Subspace Clustering for Molecular Dynamics Data.






Both authors contributed equally to this research.

  University of Vienna
  Vienna
  Austria
anna.beer@univie.ac.at


[1]

  LMU Munich
  Munich
  Germany

heinrigs-martin@t-online.de



  University of Vienna
  Vienna
  Austria
plant@univie.ac.at




  Aarhus University
  Aarhus
  Denmark
ira@cs.au.dk



§ ABSTRACT 
We introduce MOSCITO (MOlecular Dynamics Subspace Clustering with Temporal Observance), a subspace clustering for molecular dynamics data. 
MOSCITO groups those timesteps of a molecular dynamics trajectory together into clusters in which the molecule has similar conformations.
In contrast to state-of-the-art methods, MOSCITO takes advantage of sequential relationships found in time series data. Unlike existing work, MOSCITO does not need a two-step procedure with tedious post-processing, but directly models essential properties of the data. 
Interpreting clusters as Markov states allows us to evaluate the clustering performance based on the resulting Markov state models. 
In experiments on 60 trajectories and 4 different proteins, we show that the performance of MOSCITO achieves state-of-the-art performance in a novel single-step method.
Moreover, by modeling temporal aspects, MOSCITO obtains better segmentation of trajectories, especially for small numbers of clusters.





Temporal Subspace Clustering for Molecular Dynamics Data
    Ira Assent
    August 5, 2024
========================================================





§ INTRODUCTION

Proteins play an important role in every living organism.
Analyzing the transition between different shapes of a protein, also known as folding, is important for understanding the structure and function of proteins, e.g., for drug design. 
Many diseases like Alzheimer, Parkinson, and different types of cancer are connected to misfolding of certain proteins <cit.>. Thus, understanding such folding processes may help to prevent or cure diseases. 

Molecular dynamics data contains trajectories of proteins that change their shape over time. It is usually very high-dimensional, with thousands of time steps and hundreds of atoms for a single protein. 
Analyzing molecular dynamics data with traditional clustering is typically not meaningful due to the curse of dimensionality. To mitigate issues with high dimensionality, subspace clustering methods detect clusters in lower-dimensional subspaces <cit.>.
However, existing subspace clustering methods do not cater for the characteristics of molecular dynamics data.

In this paper, we introduce MOSCITO (Molecular Dynamics Subspace Clustering with Temporal Observance), a subspace clustering algorithm that uses temporal regularization to exploit sequential relationships found in molecular dynamics trajectories that describe folding activities of single molecules. MOSCITO is the first one-step approach to handle this complex data type by identifying the best subspace projection. It finds features tailored specifically to molecular dynamics data. Leveraging methods from video analytics, we show that MOSCITO finds temporally contiguous groups of of protein conformations.  

In trajectories, data points that are near in time are usually more closely related than those that are further apart. 
MOSCITO allows users to choose different sizes of time windows and different weighting methods for temporal regularization: binary, Gaussian, exponential, or logarithmic. 
Interestingly, the resulting clusters can be interpreted as states in a Markov State Model (MSM), which we use for comparing MOSCITO to approaches like SSC, PCA + k-Means, and TICA + k-Means.
As we demonstrate in thorough empirical evaluation, MOSCITO successfully clusters molecular dynamics data in a model that directly captures the essential features and temporal relationships.
Our main contributions are as follows:

    
  * We introduce MOSCITO, a subspace clustering method for molecular dynamics data that makes use of temporal relations between time steps that are close
    
  * MOSCITO finds temporally contiguous groups of protein conformations that are similar 
    
  * In contrast to currently used methods that require several steps, the clusters found by MOSCITO can directly be used as states of a Markov State Model describing the trajectory, yielding state-of-the-art quality with a one-in-all holistic approach



§ CLUSTERING TRAJECTORIES OF MOLECULAR DYNAMICS DATA
 

In this paper, we study the trajectories of a single molecule over time. A trajectory is given by the three-dimensional Cartesian coordinates of all atoms of the molecule for a number of time steps with fixed intervals.
Where most traditional clustering methods target numerical data in ℝ^d, clustering MD data brings some unique challenges as we outline in the following. 

Due to the size of MD data, automated methods for analyzing them are required. These often involve clustering methods to automatically group similar groups of atoms or timesteps.

In this paper, we focus on simulations of very long molecular dynamics trajectories that show folding processes of single molecules <cit.>. 
Based on the starting positions of a molecule's atoms, physical forces on all other atoms can be calculated, yielding simulated trajectories with a temporal resolution that usually lies in the femtosecond range.
While the overall shape of the molecules is decisive for its function, several features are shown to be relevant, e.g., the torsions between C^α atoms that form the backbone of the protein, distances between specific atoms, or the solvent accessible surface area.


There are different use cases for which a clustering of MD trajectories is required. 
E.g., clusters of atoms that move similarly over time can be used to find dynamic domains, i.e., fragments of a protein that are internally rigid <cit.>. 
Clustering time steps in which the conformation of the protein, i.e., the arrangement of its atoms in space, is similar is used for detecting relevant states of the protein. 
Those states can be linked to (local) valleys in the energy landscape of the protein and are e.g. used for developing Markov models that describe a protein trajectory. 
Thus, one goal is to separate metastable states of a protein and the transitions between those states. 





§ MARKOV STATE MODELING
 

Markov state models (MSMs) can be used to describe the dynamics of a MD trajectory by using a n × n transition matrix. To generate an MSM from an MD trajectory X={x_1,...,x_t}, 
discrete states of the trajectory need to be identified. 
State-of-the-art methods use a two-step approach in which they first reduce the dimensionality of the data by using PCA or TICA and then clustering it with k-Means into n discrete states S={S_1,...S_n}<cit.>. These states are used to calculate the transition probability matrix for a given lag time τ by counting the transitions from time step t to t+τ: 


    p_ij(τ)=ℙ(x_t+τ∈ S_j|x_t∈ S_i)


The resulting MSM can be used for further analysis of the trajectory. In this paper, we analyze the clustering performance of our method in comparison to state-of-the-art methods for clustering MD trajectories based on their suitability as Markov states.


The variational approach for Markov models (VAMP)<cit.> is a generalization of the Variational approach to conformational dynamics (VAC) <cit.>, <cit.> and provides a tool to evaluate MSMs. The VAC allows the approximation of eigenvalues and eigenfunctions with statistical observables <cit.>. It also defines a family of scoring functions, called the VAMP-r, which can be calculated from data. This score can be used to compare the choices of hyper-parameters, like the featurization of the data or the used clustering, used for constructing the MSM. The score is calculated from the singular value decomposition of the Koopman operator. The Koopman operator is explained below. It is important to note that the VAMP score can only be compared for MSMs constructed with the same lag time τ and cannot be used to find the optimal lag time τ itself.

The Koopman operator gives the conditional expected value of an arbitrary observable g at time t+τ for a given x_t. The Koopman operator K_τ of a Markov process is defined as follows <cit.>:


    K_τ g(x)=𝔼[g(x_t+τ|x_t=x]


When choosing the Dirac delta function δ_y centered at y, applying the Koopman operator evaluates the transition density of the dynamics <cit.>.


    K(τ)=C_00^-1/2C_01C_11^-1/2≈ U_mΣ_mV_m^T,

where Σ_m=diag(σ_1,...,σ_m) are the first m singular values. 



  
*VAMP-r Score
The VAMP-score <cit.> defines a function that can be used to score the quality of MSMs for a set lag-time τ.  This score can be used to compare the methods used to create the discrete trajectories from the same featured trajectory <cit.>.

With Σ_m as defined above (see <Ref>), the VAMP-r score can be written as:

    VAMP-r=∑_i=1^m σ_i^r


The VAMP-2-score is based on the sum of squared eigenvalues of the transition matrix and can be interpreted as kinetic content <cit.>. In this paper, we use the VAMP-2 score to score the performance of different discretization methods for MD trajectories since there usually is no given ground truth to compare to. The most commonly used scores are the VAMP-1, which is analogous to the Rayleigh trace, and the VAMP-2, which is analogous to the kinetic variance <cit.>. 




§ SUBSPACE CLUSTERING
 
In high-dimensional datasets where the curse of dimensionality renders traditional clustering infeasible, subspace clustering finds clusters in a union of lower dimensional subspaces <cit.>. For a dataset X=[x_1, x_2, ..., x_n]∈ℝ^d it is often assumed that vectors x_j are drawn from a union of k subspaces {S_i}_i=1^k of unknown dimensionality <cit.>. As MD data is typically very high dimensional, subspace clustering is a suitable approach.
For details on subspace clustering, see <cit.>.
In this paper, we focus on spectral-based subspace clustering.



The advantages of spectral subspace clustering methods include a relatively simple implementation and robustness regarding data corruption and initialization <cit.>. Spectral subspace clustering methods usually perform two steps. First, a subspace representation of the original data is learned and used to construct an affinity matrix. Then spectral clustering is used on the affinity matrix to obtain the final clustering. Many different spectral algorithms like sparse subspace clustering (SSC) <cit.> or Low-Rank Representation (LRR) <cit.> have been proposed, for an overview see <cit.>.

While spectral clustering has been successfully applied to numeric data, it lacks the ability to handle temporal data. We, thus, turn our attention to methods that have been developed with a different application goal in mind, but, as we argue, with similarities that make them attractive for the design of our MD subspace clustering method.

Temporal subspace clustering (TSC) <cit.> is a subspace clustering algorithm designed to segment video sequences into separate human motions. 
Video sequences are high-dimensional time series, with one frame of the video at each timestep. 
Significant changes in a video sequence usually are not immediate, but take place over many frames. 
Neighboring time frames are more correlated to each other than more distant frames. 
In that sense, molecular dynamics data is similar to video data. 
Unlike classical subspace clustering methods, which do not take any relations between the dimensions into account, TSC is designed to take advantage of the sequential relationship in time series data. 
TSC uses a dictionary learning approach together with a temporal Laplacian regularization function to encode those relationships in the dataset. Dictionary learning is a method that learns a set of basis vectors, a so-called dictionary <cit.>. The original data can then be represented as linear combinations of the basis vectors. TSC combines the dictionary learning approach with a temporal regularization function to encode the sequential relationships in the data. 
A non-negative dictionary and a corresponding coding are learned from the given data creating a robust affinity graph.



§ MOSCITO: TEMPORAL SUBSPACE CLUSTERING FOR MD DATA
  

We are now ready to describe our MOSCITO (MOlecular dynamics Subspace Clustering with Temporal Observance) approach, the first subspace clustering method for molecular dynamics data that captures temporal relations. 
Existing methods <cit.> often rely on a two- or three-step approach consisting of 1) dimensionality reduction, 2) clustering, and 3) finding macrostates by combining several clusters. As these steps are handled in isolation, their separate results may not be optimal overall. To avoid such undesirable side effects,  MOSCITO takes a holistic approach: 
Firstly, in subspace clustering, it inherently combines clustering with dimensionality reduction. 
Secondly, MOSCITO incorporates temporal relation by weighting sequential neighbors in a time window. In contrast, existing subspace clustering algorithms cannot capture e.g., correlation between consecutive time steps.

For our new method MOSCITO, we exploit the similarity between MD data and video data (see <Ref>) as well as the advances in the field of data mining on videos: TSC <cit.> clusters video data while incorporating the temporal relations between time steps as well as allowing clusters in subspaces of the data. Thus, we base MOSCITO on TSC. However, TSC is not able to handle MD data, because despite some similarities between the videos and MD data (e.g., smooth temporal transitions between time steps and potential clusters in subspaces), they also have strong differences: for every time step , MD data describes a chain of atoms with its 3d coordinates, while video data contains colors for every pixel, where pixels are in a 2d grid. 
Thus, we need to change several steps and include knowledge from the field of MD as shown in Figure <ref>, which provides an overview of MOSCITO. 
The blue parts are equivalent to those of TSC <cit.>, see <Ref>. 
The green parts are adapted or changed completely. 
At first, essential MD features 
are extracted from the MD trajectory. 
From the features and a temporal regularization function, we learn a dictionary and the corresponding representation coding matrix. 
We then construct an affinity graph, and generate temporal segments by clustering the affinity graph.






  
*Feature extraction 
From the input MD trajectors, we extract features that capture essential properties. 

Cartesian coordinates: 
The C^α atoms determine the shape and function of a protein, thus we use the common approach to focus solely on their 3d coordinates, rather than considering all atoms. 
For each frame,  coordinates are centered and aligned to the first frame of the trajectory.

Backbone torsions: 
The backbone torsions determine the main structure of a protein. The torsion angles ϕ and ψ describe the dihedral angles between atoms of the backbone. Backbone torsions better describe the structure of a protein than absolute coordinates <cit.> as internal angles are independent of the global rotation of the protein, removing the need to align all frames to a fixed rotation.  

Distance based: Heavy atoms are all atoms that are not hydrogen. We extract the minimal distance between heavy atoms <cit.> (we exclude the pairs at indices(i, i+1) and (i, i+2), as their distance is largely determined by their chemical bonds). Following <cit.>, distances d are transformed to the negative exponential function e^-d  

Flexible torsions: The flexible torsion feature consists of the χ_1 to χ_5 side chain torsions. The side chains of a protein are chains of atoms, connected to the backbone. The angles are transformed into sine and cosine angles.

Solvent accessible surface area: For each frame of the trajectory, the solvent accessible surface area (SASA) of each residue is calculated using the Shrake-Rupley algorithm <cit.>. The SASA of a residue is calculated by summing up the SASA of each of the residues atoms

3D shape histogram: Based on <cit.>, a combined 3D shape histogram for each frame of the trajectory is created. The state space is separated into five equally spaced shells around the origin of the coordinate system. The shells are split up further by combining them with 12 sectors into 60 cells. Histograms are created by counting the number of atoms in each cell. Unlike <cit.>, all atoms instead of only the surface atoms were used as we are not only interested in the shape and function of the protein at a certain time step, but also in its internal folded structure. Final histograms are normalized between 0 and 1.


  
*Temporal regularization
For trajectories, data points at neighboring time steps are usually more closely related than those at more distant time steps. Let the i-th column of the coding matrix Z a representation of vector x_i, then the encodings of its sequential neighbors, e.g., z_i-1, z_i+1 should be similar to z_i. To achieve this, we use a temporal Laplacian regularization function f(Z). 


    f(Z)=1/2∑_i=1^n ∑_j=1^n w_ij ||z_i - z_j||_2^2 = tr(ZL_T Z^T)

L_T is the temporal Laplacian matrix defined as L_T = D̃ - W. W is the weight matrix, capturing the sequential relationship in X, and D̃_ii = ∑_j=1^nw_ij is a diagonal matrix with the sum of the rows of W on its diagonal. 


The temporal regularization in MOSCITO weighs the sequential neighbors using a weight matrix W∈ℝ^n× n for a trajectory with n timesteps. 
Row i of W represents the weights of x_i's neighbors. 
In the base case, binary weights are used. 
This weighs all s sequential neighbors of x_i the same, independent of the actual temporal distance between the data points. 
In contrast to TSC, MOSCITO offers several 
Naturally, points in close temporal proximity are more strongly correlated than points further apart in time. To model the decrease in correlation, we introduce different decaying weightings: 

Binary:  weights of 1 for points in the neighborhood and 0 outside

Gaussian: Gaussian curve fitted to the neighborhood

Logarithmic: Logarithmic decay with a value of 1 for the direct neighbors dropping logarithmically to 0 for x_i+s and x_i-s

Exponential: Exponential decay w.r.t. the temporal distance 



  
*Dictionary learning Similar to TSC <cit.>, MOSCITO learns to represent dataset X by a dictionary D and a coding matrix Z, such that X≈ DZ <cit.>. 

Adopting a least square regression approach to dictionary learning, we use the following objective function <cit.>

    min_Z,D||X-DZ||_F^2+λ_1||Z||_F^2
    s.t.  Z ≥ 0, D ≥ 0, ||d_i||_2^2 ≤ 1,

where ||Z||_F^2 denotes the Frobenius norm, and we use λ_1 for balancing both terms. 

Importantly, MOSCITO integrates temporal regularization to capture relationships between nearby time steps. We achieve this by adding temporal regularization function f(Z) such that our new objective function is the same as in TSC <cit.>

    min_Z,D||X-DZ||_F^2+λ_1||Z||_F^2 + λ_2f(Z) 
    s.t.  Z ≥ 0, D ≥ 0, ||d_i||_2^2 ≤ 1,

with λ_2 tradeoff parameter <cit.>.

To solve the dictionary learning problem efficiently, we follow the alternating direction method of multipliers (ADMM) <cit.>. It decomposes the problem into subproblems that are more easy to solve. Using auxiliary variables U, V, Eq. <ref> is equivalent to

    min_Z,D,U,V||X-UV||_F^2+λ_1||V||_F^2 + λ_2f(V) 
    s.t.  U=D, V=Z, Z ≥ 0, D ≥ 0, ||d_i||_2^2 ≤ 1.

The augmented Lagrangian of Eq. <ref> is:

    ℒ    = 1/2||X-UV||_F^2 + λ_1||V||_F^2 + λ_2 tr(VL_TV^T) 
        + ⟨Λ, U_D⟩ + ⟨Π, V-Z⟩ + α/2||U-D||_F^2 + β/2||V-Z||_F^2,

with Λ and Π Lagrangian multipliers <cit.>, which can now be solved by alternatingly minimizing ℒ respective to V, U, Z and D. 

Updating V: To find the minimum of ℒ, its derivative with respect to V is set to zero, resulting in the following equation <cit.>:

    [I⊗ (U^TU+(λ_1+α)I) + λ_2L_T⊗ I] vec(V)=vec(U^TX-Π+α Z)


Updating U: As for V, set derivative of ℒ wrt. U to zero <cit.>:

    U=(XV^T-Λ+α D)(VV^T+α I)^-1


Updating Z and D: The update rules for Z and D are <cit.>: 

    Z=F_+(V+Π/β)
    ,     
        D=F_+(U+Λ/α)

where (F_+(A))_ij=max{A_ij, 0}, non-negativity constraint. Additionally, each column of D is normalized to unit length  ||d_i||_2^2≤1.



  
*Clustering
Using the learned coding matrix Z, an affinity graph is constructed. To capture sequential relationships, the following similarity measure is used <cit.>: 

    G(i, j)=z_i^T z_j/||z_i||_2||z_j||_2

Spectral clustering on graph G yields the final clusters.





§ EXPERIMENTS
 



  
Implementation 
We extract features of MD trajectories using libraries PyEMMA <cit.> and MDTraj[<https://www.mdtraj.org>].
Matrices are represented as NUMPY arrays and sparse matrices by scipy sparse matrices. 
We efficiently solve linear equations of the form Ax=b using the PyPardiso 0.4.2[<https://github.com/haasad/PyPardisoProject>] (version 2021.4) wrapper for the Intel oneMKL PARDISO library. 
For spectral clustering, we use the scikit-learn implementation. 
Our Python code for MOSCITO is https://github.com/Mhae98/MOSCITOpublicly available [<https://github.com/Mhae98/MOSCITO>]. 
For PCA, TICA and k-Means we use the implementations in PyEMMA.
The implementation of SSC is based on <cit.> and a Python translation of SSC [<https://github.com/abhinav4192/sparse-subspace-clustering-python>] 
All experiments run on a virtual machine with 64 CPUs and 512GB of RAM running Ubuntu 22.04. 





  
Datasets and parameter settings
We use MD trajectories of four proteins: 2F4K, Ace, 2WXC, and Savinase, which differ in the number of atoms | A|, number of Cα atoms | Cα|, and number of residues | R|, as summarized in Table <ref>, together with number of  trajectories and timesteps in each trajectory.

If not specified otherwise, parameter default values are 
 d= 60, s= 3, λ_1= 0.01, λ_2= 15, a= 0.1, b= 0.1.





 §.§ MOSCITO settings and features
 
In this section, we analyze the parameter sensitivity and impact of feature extraction for MOSCITO. 
For the following experiments, 10 trajectories of the 2WAV-protein are used. The Markov state models are constructed for k∈{10, 50, 100, 200} clusters with a lag time τ∈{1, 10, 100}. The average VAMP2-score of the 10 trajectories is calculated for each choice of clusters k and lag time τ. The VAMP2-score is calculated for five eigenvalues.
It is important to keep in mind that the VAMP2-score for different MSM lag times is not directly comparable, only for different numbers of clusters.




  §.§.§ Dictionary size
  d defines the dimensionality of the learned dictionary and thus the number of subspaces. As such, it is crucial for the clustering and runtime performance of MOSCITO, where larger dictionary size d provides more subspaces to represent the data.
We study d∈{10, 20, ..., 100}, and additionally s∈{2, 3, 4} number of sequential neighbors. Figure <ref> shows VAMP2-scores for different numbers of clusters (rows) and MSM lag times (columns): clustering performance improves for larger dictionary size d until it converges. For smaller MSM lag times, the VAMP2-score converges faster than for larger lag times, independent of the number of clusters. With an increasing number of sequential neighbors s, the overall shape of the curves is highly similar, with a slight preference for more neighbors that capture more of the temporal relationships.






  §.§.§ Sequential neighbors
 
Setting the number of sequential neighbors s in the temporal regularization to 0, the temporal aspect is not considered, with growing s longer temporal relationships are captured. The scores for s∈{0,...,12} are shown in Figure <ref> for different MSM lags (columns) and number of clusters (rows). 
The number of sequential neighbors s clearly affects clustering performance, with too few or too neighbors reducing performance, but with a broad stable range where temporal relationships are successfully captured.

Figure <ref> shows the clustering of a 2F4K protein trajectory with a long folded state into 5 clusters represented by different colors. With no (top) or few (center) sequential neighbors, time intervals comprised by a cluster can be very short.
Increasing the number of sequential neighbors (bottom) leads to continuous clusters with mostly uninterrupted blocks.  





  §.§.§ λ parameters
 
We study settings of the tradeoff parameters  λ_1 ∈{0.01, 0.05, …, 15} for sparsity  of the coding matrix Z, λ_2 ∈{0.01, 0.05, …, 35} for the temporal regularization function.
Figure <ref> shows that best performance is achieved for low values of λ_1  and higher values of λ_2 (light areas in the heatmap). This shows that temporal regularization is the more important factor for optimal clustering performance. 








  §.§.§ Learning rates
 
α, β∈{0.01, 0.05, …, 15} 
define how much Lagrangian multipliers Π and Λ change in each iteration of the algorithm.
Heat maps in Figure <ref> show that the results are stable across parameter settings, with VAMP2-scores in each heatmap relatively close, meaning our default values work well.





  §.§.§ Weighting mode
 
All weighting modes for the Laplacian regularization function: binary, Gaussian, logarithmic, and exponential, are tested for different numbers of sequential neighbors.
In Figure <ref> we see that for few sequential neighbors, the performance for all weighting modes is similar. For many sequential neighbors, performance starts to decrease. In most cases, the performance when using the binary weighting mode decreases earlier than for the other methods, but all reliably reach the same maximal score.






  §.§.§ Feature comparison
 
We analyze the above described features and their combinations via simple concatenation. 
Figure <ref> 
shows the average VAMP2-score for 10 trajectories of the 2WAX-protein, using the five largest eigenvalues for scoring. 






Best results for individual features are achieved with backbone torsions. Overall, combinations of backbone torsions, SASA, and shape histograms yield the best results. Figure <ref> shows the importance that MOSCITO gives to each of the features in its dictionary when clustering the full-dimensional feature space. While, e.g., the feature "flex" (flexible torsions) yields the worst VAMP2 score (left), it still has a high influence on the results (right), explaining the suboptimal VAMP2 score when using the full feature set.




 §.§ Clustering performance
 
In this section, the clustering performance of MOSCITO is compared to that of state-of-the-art approaches that are currently used in the field. We compare to PCA with k-Means and to TICA with k-Means because of their widespread use <cit.>. Additionally, we compare against the sparse spectral clustering algorithm <cit.>, a subspace clustering approach for molecular dynamics data.
For PCA and TICA, the default parameter settings of PyEmma <cit.> were used. SSC is a subspace clustering algorithm, thus it does not need any dimensionality reduction in the preprocessing. 

A general challenge in this kind of evaluation is the fact that no ground truth is given. As such, clustering performance is compared by constructing a MSM and scoring it using the VAMP2-score, in order to obtain quantitative insights. Still, visual inspection plays an important role in assessing the clustering performance, in particular, the temporal connectivity of clusters.



  
*Quantitative analysis
Figure <ref> shows the average VAMP2-score for 20 trajectories of the 2F4K protein in the first row. For MSM lag time 1, the performance of all methods is comparable with the exception of 10 clusters. There, MOSCITO and TICA + k-Means deliver the best performances. For longer MSM lag times, the scores separate with SSC performing better.




In the second row of Figure <ref>, we see the average VAMP2-score for 10 trajectories of the 2WXC protein. For the MSM lag times 1 and 10, the performance for all methods is relatively similar for large cluster sizes. For small cluster sizes, SSC performs poorly, while the other methods already reach close to their maximal score for only 10 clusters. For MSM lag time of 100, PCA + k-Means drastically drops in performance, while the other methods maintain comparable scores to each other.


The third row of Figure <ref> shows the average VAMP2-score for 28 trajectories of the Ace protein. Similar to the results of 2WXC, the performance for a MSM lag time of 1 and 10 is comparable between all methods for a large number of clusters. For few clusters, SSC again performs significantly worse. For MSM lag time of 100, the gap between the methods grows, with SSC performing best.

The last row of Figure <ref> shows the average VAMP2-score for 2 trajectories of Savinase. For all MSM lag times, SSC performs best while TICA + k-Means performs worst. Unlike for the other proteins, all methods require a large number of clusters to reach the maximal VAMP2-score. This might be due to the large size of Savinase and shorter trajectories compared to the other proteins.



  
*Qualitative analysis
The purpose of modeling temporal aspects is to obtain clusters with larger continuous segments, ideally retrieving macrostates directly.  Figure <ref> visually compares the segmentation of clustering methods, either by directly clustering the trajectory into five clusters or, alternatively by coarse-graining 50 clusters into five clusters using PCCA+ <cit.> (see <Ref>). We regard 
a 2F4K trajectory with a known long folded segment.


Directly clustering, i.e. without PCCA+, PCA + k-Means and SSC struggle to find the large folded state as one large block. MOSCITO and TICA+k-Means find similar clusterings and succeed to uncover the large folded state. When applying PCCA+, PCA + k-Means, TICA + k-Means, and MOSCITO generate similar clusterings, comparable to the ones of MOSCITO and TICA + k-Means without PCCA+. In this specific case, SSC fails to find the large segment, which is reflected in the respective VAMP2-score for this specific trajectory which is unusually low.



 §.§ Clustering runtime
 
In this section, the runtime of MOSCITO is studied and compared to the runtime of PCA + k-Means, TICA + k-Means, and SSC. The runtimes for MOSCITO are measured for 20 iterations. The following runtimes are the average of 10 2WXC-trajectories clustered into ten clusters, using all atom coordinates as features. 
The main factors influencing the runtime of MOSCITO, besides the size of the input data, are the dictionary size d and the number of sequential neighbors s. The effect of different dictionary sizes [10, ..., 100] on the runtime is shown left in Figure <ref>. The effect of different numbers of sequential neighbors [0, ..., 25] is on the right. 




The runtime of MOSCITO grows with increasing dictionary sizes. For optimal cluster performance, a large enough dictionary size has to be selected. Large values do not affect clustering performance negatively, but directly impact runtime. A dictionary size between 60 and 80 provides a good tradeoff.
The runtime generally grows with increasing number of regarded sequential neighbors. 
The overall run-to-run variance for the different trajectories is close to zero, indicating a stable runtime for a fixed number of iterations. 

For the comparison of the different clustering methods, backbone torsions were used. The comparison between the different clustering algorithms is shown in Figure <ref>.
PCA + k-Means and TICA + k-Means are both much faster than the subspace clustering methods.  Compared to SSC, MOSCITO is substantially faster. 



 §.§ Discussion
 
We conclude that MOSCITO provides at least comparable performance to state-of-the-art clustering methods for MD data. We evaluated the clustering performance by the clusters' suitability as Markov states using the VAMP2-score. The visual analysis of MOSCITOs clustering results shows a better clustering for higher numbers of sequential neighbors. Thus, MOSCITO is a very promising approach for creating MSMs in a one-step-approach.

MOSCITO's parameter robustness shows that there are good tradeoffs in dictionary size between runtime and clustering performance. The number of sequential neighbors matters, and our experiments indicate a good default value is between 3 and 5 to capture nearby temporal relations. 
In terms of feature extraction, MOSCITO on backbone torsions leads to best results, followed by Cα coordinates and SASA. Combining features does not improve the overall clustering performance. 
While simpler non-subspace methods are even faster, MOSCITO provides a faster runtime than SSC, and state-of-the-art quantitative results, with qualitative analysis showing the promise of the approach.




§ RELATED WORK
 
Distance-based clustering algorithms suffer from the curse of dimensionality when it comes to high-dimensional data like MD data.
Thus, the dimensionality is often reduced in a preprocessing step, e.g., with principal component analysis (PCA) or time-lagged independent component analysis (TICA) <cit.>. Both are widely used for analyzing MD data <cit.>. 
PCA transforms the data onto the axes with the highest variance, also known as principal components. 
PCA is a commonly used for MD data, e.g., in PyEMMA <cit.>, MSMBuilder [<http://www.msmbuilder.org>], or MDAnalysis [<https://www.mdanalysis.org/>].
TICA (Time-lagged independent component analysis) <cit.> linearly projects the data to independent, uncorrelated components (IC). 
The ICs are computed by maximizing the autocovariance for a fixed lagtime τ, and are frequently used for MD data (<cit.>, <cit.>).


State-of-the-art methods often follow a two-step approach, where the trajectories are discretized into a large number of microstates, and subsequently combined to the relevant macrostates.
Since the transitions between the metastable states are taking place over multiple timesteps
it is hard to assign time steps that are part of a transition to a metastable state clearly. 
Perron-Cluster Cluster Analysis (PCCA) (<cit.>,<cit.>) and a more robust version PCCA+ (<cit.>, <cit.>) fuzzily assign the microstates to their corresponding macrostates based on spectral methods for transition matrices:


The transition matrix P of a decomposable MSM with n_c macrostates can be reordered to a block-diagonal matrix P̃ with n_c blocks. 
For the matrix X containing the n_c dominant eigenvectors of P̃, the rows belonging to states in the same block are identical <cit.>. 
These rows represent the n_c corners of a (n_c-1) dimensional simplex.

For MSMs with transition states, i.e., nearly decomposable MSMs, the structure of the simplex is only slightly perturbed. To assign the microstates to the macrostates, they are fuzzily assigned to the n_c corners of the simplex based on their position in the simplex. The goal is to find a linear transformation A∈ℝ^n_c× n_c so that χ=XA where χ_i are the membership vectors. Those memberships all have to be positive and sum up to 1. 
The resulting macrostates generally cannot be considered to be a Markov chain <cit.>.


Time series data can be found in many fields, like traffic analysis, geology, computer vision, or molecular dynamics data. Many existing subspace clustering algorithms do not take the sequential properties of time series data into account <cit.>. One of the first subspace clustering methods taking advantage of sequential relationships in data was SpatSC introduced by <cit.>. A penalty term used to preserve sequential relationships in the learned affinity graph was added to the subspace clustering problem. Based on the same idea, OSC, a more robust subspace clustering method for sequential data was introduced by <cit.>. 
TSC <cit.> further improved on the idea by defining a Laplacian regularization to encode the temporal information. 
Our method MOSCITO builds upon TSC. 

To the best of our knowledge, only one subspace clustering method has been used on molecular dynamics data, namely SSC <cit.>, which was first introduced in the field of image processing <cit.>. We include SSC in our experiments as comparative method, thus we introduce it in detail in the following. 
Note that SSC does not take advantage of any properties that distinguish time series from other high-dimensional data, leading to suboptimal use of the available information in MD data.
The main idea behind SSC is to take advantage of the self-expressiveness of the data, i.e., assuming that every point y_i∈ Y can be rewritten as a linear combination of other points in the dataset. Each datapoint y∈∪_i=1^nS_i can be written as

    y_i=Yc_i,   c_ii=0,

where Y=[y_1,...,y_N] and c_i:=[c_i1  c_i2  ...  c_iN]^T with the constraint c_ii=0 eliminating the trivial solution of representing the point as itself. 
Since each subspace usually has more data points than dimensions the self-expressive dictionary Y is generally not unique. 
Among all those possible solutions for Equation <ref> there is a sparse solution so that the non-zero entries of c_i correspond to datapoints in the same subspace as y_i. This solution can be found by minimizing the l_1-norm of c_i, which can be solved efficiently and is known to prefer sparse solutions <cit.>.

These sparse representations are transferred to an undirected graph where points that are part of another point's sparse representation are connected. Based on this graph, spectral clustering detects the desired subspace clusters. 




§ CONCLUSION
 
We introduced MOSCITO, a subspace clustering method for high-dimensional molecular dynamics data. It uses temporal regularization to capture the relationship of sequential neighbors in time-series data.
While traditional and state-of-the-art methods rely on coarse-graining using PCCA+ in a two-step approach, MOSCITO achieves similar overall performance in a single clustering step.

Especially for small numbers of clusters, the temporal regularization in MOSCITO yields clusters consisting of connected time steps, fitting the real world more closely than alternative methods without relying on any postprocessing steps. 
Compared to current subspace clustering methods on MD data, MOSCITO provides a better runtime.
In future work, the integration of a multi-view approach into MOSCITO might take advantage of the additional information provided by using multiple features.


ACM-Reference-Format




 Paper Title: Efficient and Scalable Wave Function Compression Using Corner Hierarchical Matrices.






Contributed equally
Current address: Department of Chemistry, Brown University, Providence, RI, USA
Institute for Advanced Computational Science, Stony Brook University, Stony Brook, NY, USA 11794
Department of Chemistry, Stony Brook University, Stony Brook, NY, USA 11794


Contributed equally
Institute for Advanced Computational Science, Stony Brook University, Stony Brook, NY, USA 11794
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, USA 11794


Current Address: Intel Corporation RA2, Hillsboro, OR USA  97124
Institute for Advanced Computational Science, Stony Brook University, Stony Brook, NY, USA 11794
Department of Chemistry, Stony Brook University, Stony Brook, NY, USA 11794


[Correspondence email address: ]xiangmin.jiao@stonybrook.edu
Institute for Advanced Computational Science, Stony Brook University, Stony Brook, NY, USA 11794
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, USA 11794


[Correspondence email address: ]ben.levine@stonybrook.edu
Institute for Advanced Computational Science, Stony Brook University, Stony Brook, NY, USA 11794
Department of Chemistry, Stony Brook University, Stony Brook, NY, USA 11794

§ ABSTRACT 
The exponential scaling of complete active space (CAS) and full configuration interaction (CI) calculations limits the ability of quantum chemists to simulate the electronic structures of strongly correlated systems. Herein, we present corner hierarchically approximated CI (CHACI), an approach to wave function compression based on corner hierarchical matrices (CH-matrices)—a new variant of hierarchical matrices based on a block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to a truncated global singular value decomposition. The compression ratio is shown to improve with increasing active space size. By comparison of several alternative schemes, we demonstrate that superior compression is achieved by a) using a blocking approach that emphasizes the upper-left corner of the CI vector, b) sorting the CI vector prior to compression, and c) optimizing the rank of each block to maximize information density.



Efficient and Scalable Wave Function Compression Using Corner Hierarchical Matrices
    Benjamin G. Levine
    August 5, 2024
===================================================================================





§ INTRODUCTION



Complete active space (CAS) configuration interaction (CI) and its self-consistent field counterpart (CASSCF) remain the quantum chemist's workhorse method for describing multi-reference systems.<cit.> The most frequent targets for multi-reference calculations include strongly correlated systems, dissociated molecules, transition states, conical intersections involving the electronic ground state, and excited electronic states with multiple excited characters. CAS methods are viable when the multi-reference effects in question can be described in an active orbital space of fewer than ∼20 electrons. Unfortunately, the factorial scaling of the CAS wave function with active space size renders it impractical for anything larger.

As such, strategies for using compressed representations of the CASCI wave function have been widely explored in recent years, exploiting the large number of zero or near-zero elements of the CI coefficient vector (“configurational deadwood”<cit.>) to reduce compute time and storage. For example, restricted, generalized, and other active space methods use different truncations of the CAS expansion to reduce storage and cost.<cit.> Similarly, selected CI approaches use various physical criteria to develop a sparse representation of the CI vector on-the-fly.<cit.> Seniority CI achieves an efficient sparse representation of strongly correlated systems via a truncation of the CI space based on seniority number.<cit.> Full configuration interaction quantum Monte Carlo (FCI-QMC) methods can describe very large active spaces without explicit representation of the entire CI vector, allowing a fixed number of walkers to explore the most important regions of the CI vector stochastically.<cit.> An algorithm for choosing orbitals to maximize CI vector sparsity in exchange-coupled systems has also recently been reported.<cit.>

Other methods take advantage of data sparsity rather than sparsity. Whereas a sparse matrix has many zero elements, a data-sparse matrix may contain few zero elements but is compressible because it is low-rank or has other redundancies that may be exploited. For example, a highly efficient representation of the strongly correlated wave function of one-dimensional and quasi-one-dimensional systems can be achieved via the density matrix renormalization group (DMRG) approach.<cit.> The DMRG method can be thought of as leveraging the data sparsity of the CI vector. Rank-reduced configuration interaction methods also leverage data sparsity through a low-rank approximation of the CI vector,<cit.> and in a similar spirit, ideas from compressive sensing have been adapted to wave function compression.<cit.> Going one step further, it has been argued that the wave function itself is not an essential descriptor of electronic structure and can be replaced by the electron density<cit.> or reduced density matrices.<cit.> The search for an efficient and broadly applicable compressed wave function representation remains an active and important research direction, and recently there have been efforts to systematize the benchmarking of methods for compression of strongly correlated wave functions.<cit.>

In this paper, we present a proof-of-concept demonstration of a novel wave function compression strategy based on corner hierarchical matrices (CH-matrices), a new variant of hierarchical matrices. CH-matrices are motivated by H-matrices and H^2-matrices,<cit.> which use a block-wise low-rank approximation to a non-sparse matrix, leveraging data sparsity to minimize both storage and the computational cost of matrix operations. For N × N matrices, the H-matrix representation allows storage and matrix multiplication with quasi-linear scaling (𝒪(Nlog N) and 𝒪(Nlog^2 N), respectively). Open-source software is available to manipulate various flavors of hierarchical matrices, and given their block-wise nature, hierarchical matrices are well-suited for massively parallel implementation.

The effectiveness of H- and H^2-matrices relies on some degree of diagonal dominance of the matrices. This is because the matrix is subdivided into blocks, with smaller blocks on the diagonal and increasing block size as you move away from the diagonal, as illustrated in Figure <ref>. The blocks closest to the diagonal, which are shaded gray, are represented as dense matrices with no loss of accuracy. The off-diagonal blocks (shown in white) are approximated as low-rank. Typically, the rank is held constant despite the fact that the blocks are of different sizes, resulting in greater compression of the data farthest from the diagonal. These hierarchical-matrix techniques are beginning to see use in quantum chemistry. Chow and coworkers have leveraged hierarchical matrices to compress the electron repulsion integrals (ERI).<cit.> In this context, the hierarchical-matrix approach can be thought of as an algebraic generalization of the fast multipole method.<cit.> Using the proxy point method, the hierarchically compressed ERI tensor may be computed for an arbitrary contracted Gaussian basis, in a similar spirit. Unfortunately, the CASCI wave function is not diagonally dominant, so these hierarchical-matrix approaches are not very effective in our experience. Instead, we observe that the CASCI wave function is dominated by the upper-left corner of the CI vector. Based on this observation, we have developed a new variant of hierarchical matrices as illustrated in Figure <ref>, which we call Corner Hierarchical matrices (CH-matrices), to compress matrices with such patterns.

















Herein, we introduce our CH-matrix approach to compress the CASCI vectors of strongly correlated systems. In section <ref>, we define the corner hierarchically approximated CI (CHACI) representation of the wave function, along with several other candidate compression strategies. In section <ref>, we compare the performance of CHACI to several other approaches, including a truncated singular value decomposition approach that represents an optimal global low-rank approximation, akin to rank-reduced full CI (RRFCI). Finally, in section <ref>, we present conclusions and discuss the prospects for the direct optimization of CHACI wave functions. Some details and analysis of the CHACI algorithm are presented in the appendix.



§ THEORY AND METHODS


Ultimately we seek to directly optimize the electronic wave function in a compressed format. However, in the present paper, we do not solve for the wave function in a compressed format. The purpose of this paper is to demonstrate that hierarchical matrices provide a viable strategy for wave function compression. To this end, we will solve for the full CASCI wave function via traditional approaches, and then compress the resulting wave function into one of several forms. We can then decompress, and apply several metrics to quantify the accuracy of the compressed wave function.

Throughout this work, we will define the set of Slater determinants, {| αβ⟩}, in the CASCI wave function in terms of their composite α and β strings. The α (β) strings index the possible ways that N_α (N_β) spin-up (spin-down) electrons may occupy N_MO active spatial molecular orbitals. Therefore, the CASCI wave function may be represented by

    | C ⟩ = ∑_αβ C_αβ| αβ⟩.

The set of expansion coefficients, {C_αβ}, are typically known as the “CI vector.”   As such, it is typically thought of as an M_α M_β vector, but in this work, we will treat it as an M_α× M_β matrix, 𝐂. Here M_α and M_β represent the number of α and β strings, respectively,

    M_α = N_MO!/N_α!(N_MO - N_α)!

and

    M_β = N_MO!/N_β!(N_MO - N_β)!.

Where not otherwise noted, the strings will be ordered according to the scheme of Duch.<cit.>

Below, we define several strategies for wave function compression. CHACI compression, described in subsection <ref>, is the most efficient strategy we discovered in this work. Several other compression schemes, which we present in order to analyze the necessity of the various features of the CHACI algorithm, are also defined in subsection <ref>. In subsections <ref> we describe the model problems that we choose to test our algorithm and present computational details. In subsection <ref>, the performance metrics that we use to quantify the accuracy of the compressed wave function are defined.



 §.§ CHACI Wave Function Compression



The CHACI wave function compression scheme is based on what we call a corner hierarchical (CH) compression. In contrast to the H-matrix scheme illustrated in Figure <ref>, which was conceived to compress diagonally-dominant matrices, the CH scheme (Figure <ref>) is designed to compress matrices that are dominated by the upper-left corner of the matrix. Given the CI vector stored as an M× M matrix, the CH compression algorithm proceeds as follows. The total matrix (blue) is subdivided into four approximately equal-sized subblocks. The upper-right, lower-left, and lower-right subblocks (red in Figure <ref>) are compressed adaptively using either a low-rank or dense approximation. The upper left subblock is further subdivided, yielding four smaller subblocks. In a similar fashion, three (green) subblocks are again compressed. The number of levels is approximately logarithmic in M and determined a priori. Given the number of levels, we then determine whether to store each block as sparse or dense, as we describe in Algorithm <ref> and analyze its near optimality in Appendix <ref>.


Throughout this work, we will employ truncated singular value decomposition (TSVD) to compress individual blocks. The SVD of an m × n block, 𝐁, is

    𝐁 = 𝐔Σ𝐕^†,

where 𝐔 and 𝐕 are, respectively, m × m and n × n unitary matrices comprising the left and right singular vectors, and Σ is a diagonal matrix, with diagonal elements equal to the singular values. In the context of this work, m and n are typically equal, but they may differ in general for rectangular matrices. The SVD itself is an exact representation of the matrix 𝐁, and does not provide any compression. However, truncation by elimination of the smallest singular values and corresponding singular vectors provides an efficient low-rank approximation to 𝐁,

    𝐁≈𝐁̃ = a𝐁̃' = a𝐔_T Σ_T 𝐕_T^†,

where a is a scalar normalization factor, defined below. Upon compression 𝐔_T and 𝐕_T^† are now m × k and k × n, where k is the compression rank. Efficient compression can be achieved when an accurate representation of 𝐁 is achieved for k ≪min(n, m). TSVD compression decreases the Frobenius norm of the block, and therefore the overall wave function. To compensate for this, we rescale each block to maintain its original norm after compression, according to

    a = 𝐁_F/𝐔_T Σ_T 𝐕_T^†_F.

This block-wise renormalization prevents the shifting of population from sparser to denser blocks, which would be an artifact of global renormalization. Note that truncated SVD, without subsequent renormalization, is an optimal low-rank approximation to the block, in that the Frobenius norm of the difference between 𝐁̃' and 𝐁,

    𝐁̃' - 𝐁_F = ∑_i, j(B̃'_ij - B_ij)^2,

is minimized, for a given rank. However, this condition does not guarantee variational optimality (minimal energy).

In CHACI, CH compression of 𝐂 is supplemented by two additional strategies aimed at allowing more efficient compression. First, prior to compression, the rows (columns) of 𝐂 are sorted such that the rows (columns) are arranged in descending order according to their L^2 norms. In this way, the large elements are concentrated in the upper left corner of 𝐂, to the extent possible. 

Secondly, in the CHACI scheme, we do not employ the same value of the compression rank for all blocks. Instead, the SVD of each block is truncated to a different rank, such that the overall Frobenius norm of the compressed wave function, 𝐂̃_F, is optimized.  A detailed derivation and algorithm are presented in the appendix, but here we summarize the primary features.

The information density, ρ_i, associated with each singular vector pair is computed according to

    ρ_i = σ_i^2/n_row+n_col+1,

where σ_i is the singular value, and n_row and n_col are the lengths of the left and right singular vectors, respectively. Typically, n_row and n_col are equal in our algorithms, but they may differ in general for rectangular blocks. The information density can be thought of as the total contribution to 𝐂̃_F per unit of storage.

Only the singular vector pair whose information density is above a user-defined threshold will be stored.  At a given storage value, this algorithm approximately maximizes the Frobenius norm of the wave function, 𝐂̃_F.  Equivalently, this can be thought of as discarding the least amount of information.  However, this algorithm does not necessarily optimize the energy. However, when higher accuracy is required and the threshold for the Frobenius norm error is set very low, several of the upper-left corner blocks may become so dense that the storage cost of TSVD exceeds that of the dense format. To address this, we add another test after the TSVD: If the memory cost of TSVD is no less than the dense format, we directly store the block in dense format. For completeness, we present the pseudocode and its detailed analysis in the Appendix.



Taken together, we refer to CH compression of the reordered 𝐂 matrix with optimal rank as CHACI compression, without any additional modifier. To analyze the necessity of different features of the CHACI algorithm, we will present results for several other compression schemes, summarized in Table <ref>. The static rank (SR-) CHACI scheme is identical to CHACI, except that the rank of the compressed blocks is held constant rather than optimized per block. We introduce SR-CHACI in order to quantify the utility of dynamically increasing the rank. The unsorted (U-) CHACI scheme is identical to CHACI, except that the rows/columns remain ordered according to the original Duch scheme, rather than sorted by norm. This scheme is introduced to quantify the impact of sorting.

In addition, we will test H-matrix compression<cit.> of the wave function. The H-matrix representation is an existing hierarchical representation that is not based on CH blocking. Instead, the blocking structure of the matrix is optimized to represent diagonally dominant matrices, as pictured in Figure <ref>. In our implementation, the sparse blocks are compressed by the same TSVD approach that we used for CHACI. The rows/columns are not sorted, and the rank is held constant for all blocks.

Finally, as an important point of comparison, we include a non-hierarchical compression scheme: TSVD of the entire 𝐂 matrix. This form of the wave function is akin to that in RRFCI.<cit.> To maintain normalization, we rescale as in Eq. (<ref>). Given that implementation of a CI solver based on a hierarchically compressed wave function will be challenging, CHACI must provide a significant improvement in performance over TSVD to warrant further consideration.



 §.§ Model Problems and Computational Details






As a test case, we have chosen to compute the electronic structure of dodecacene (12-acene; Figure <ref>). Naively, one might think that longer acenes like 12-acene would have a simple closed-shell, aromatic electronic structure, but they are actually strongly correlated. These systems have significant poly-radical character, so much so that dodecacene is unstable under ambient conditions and has only recently been synthesized under ultrahigh vacuum.<cit.> The singlet ground state geometry of 12-acene was optimized at the CAM-B3LYP<cit.> level of theory. Floating occupation molecular orbital<cit.> (FOMO) CASCI calculations were then performed with
active spaces of 10-10, 12-12, 14-14, and 16-16, where active spaces are abbreviated ⟨number of active electrons⟩-⟨number of active orbitals⟩. Table <ref> presents the total storage required for the CI vector of the exact singlet wave function for each active space, measured in thousands of double-precision floating point values (kdoubles). The storage requirement for triplet wave functions is less by a factor of 0.7-0.8. The STO-3G minimal basis was used for all calculations. In all cases, both the singlet and triplet ground state wave functions are computed at the singlet ground state geometry. All calculations were performed in the TeraChem electronic structure software package.<cit.>



 §.§ Performance Metrics


We use several metrics to evaluate the performance of our various compression schemes. All are derived by comparing the properties of the compressed wave function, | C̃⟩, with those of the uncompressed wave function, | C ⟩. Hereafter we will refer to the latter as the “exact” wave function, understanding that it is only exact within the FOMO-CASCI approximation, with a given active space and basis. In all cases, the properties of the compressed wave function are generated by a) outputting the CI vector of the uncompressed wave function from TeraChem, b) applying one of the lossy compression schemes described above, implemented in an external code, c) decompressing the wave function back to its full dimensionality, and d) feeding the resulting CI vector back into TeraChem to compute the relevant property.

To evaluate how well our compressed wave function reproduces relative energies, we compute the vertical singlet-triplet gaps of the exact and compressed wave functions, respectively

    Δ E_S-T = ⟨ C_T | Ĥ | C_T ⟩ - ⟨ C_S | Ĥ | C_S ⟩

and

    ΔẼ_S-T = ⟨C̃_T | Ĥ | C̃_T ⟩ - ⟨C̃_S | Ĥ | C̃_S ⟩.

Here the subscripted T and S denote the triplet and singlet wave functions, respectively. We report the absolute difference between these values as the error in the gap,

    ΔΔ E_S-T = |ΔẼ_S-T - Δ E_S-T|.


To quantify the accuracy of the absolute energies, we also report signed errors in the singlet and triplet energies, respectively

    Δ E_S = ⟨C̃_S | Ĥ | C̃_S ⟩ - ⟨ C_S | Ĥ | C_S ⟩

and

    Δ E_T = ⟨C̃_T | Ĥ | C̃_T ⟩ - ⟨ C_T | Ĥ | C_T ⟩.

Per the variational principle, these errors are non-negative. Note that throughout this work, we compare the energies of the compressed wave functions to the energies of the exact wave functions using the same active space. Thus, our computed errors only include those associated with the compression algorithm, not with the size of the active space.

Lastly, as a separate measure of the accuracy of the wave function, we compute the absolute error in the total spin angular momentum squared (spin contamination) of the compressed singlet and triplet wave functions, respectively

    Δ⟨ S^2 ⟩ = ⟨C̃_S | Ŝ^2 | C̃_S ⟩

and

    Δ⟨ S^2 ⟩ = |⟨C̃_T | Ŝ^2 | C̃_T ⟩ - 2|.

The total spin expectation value is computed using the direct algorithm described in Ref. .

In many of the graphs below, the data is plotted as a function of the total number of double-precision variables stored in the compressed representation. The total storage is varied indirectly through the user-chosen parameters of the compression, as described in subsection <ref> and in the Appendix. This provides an apples-to-apples comparison of the accuracy of the wave function at a given level of compression across different compression schemes.



§ RESULTS AND DISCUSSION


Below, we analyze the performance of CHACI. In subsection <ref>, we compare the accuracy of CHACI compression to that of TSVD compression, as a function of the total storage. In subsection <ref>, we analyze the prospects for extending CHACI to larger active spaces. In subsections <ref> and <ref>, we analyze the necessity of optimal rank and sorting, respectively. Finally, in subsection <ref>, we compare H-matrix compression to TSVD compression.



 §.§ Accuracy of CHACI Compression Versus TSVD




We start by comparing the performance of CHACI compression to TSVD compression for the 14-14 active space. Figure <ref> presents the error in the singlet-triplet gap as a function of total storage. Clearly, CHACI outperforms TSVD in this case. The CHACI error is always less than 0.07 eV, even when only 28 kdoubles are stored (compared to 11,778 kdoubles for dense storage). Truncated SVD also achieves substantial compression, but errors of 0.2 eV remain even with 220 kdoubles stored.



Figures <ref>a and b present errors in the singlet and triplet absolute energies, respectively. Again, the performance of CHACI is far superior to TSVD, especially in the small-storage regime, where the TSVD errors are a substantial fraction of an eV. Figure <ref>c and d present spin error. Here CHACI again outperforms TSVD. Even with very modest (28 kdoubles) storage, the error in ⟨ S^2⟩ is 0.02, and the error drops rapidly toward zero with additional storage. In contrast, TSVD spin errors for the singlet case do not drop below 0.02 until nearly 400 kdoubles are stored.





Figures <ref> and <ref> demonstrate that the difference in performance between CHACI and TSVD increases when the active space size is increased from 14-14 to 16-16. In Figure <ref>, it can be seen that errors in the singlet-triplet gap are at or below 0.1 eV for all cases, when CHACI is employed, even when only 59 kdoubles are stored (compared to 165,637 kdoubles for dense storage). Errors decrease rapidly with additional storage. In contrast, TSVD errors are above 0.2 eV for all cases. Similarly large differences in performance are observed for errors in absolute energy and spin in Figure <ref>. As in the 14-14 case, CHACI does a much better job of maintaining the spin symmetry of the wave function than TSVD.



 §.§ Extrapolation of Performance to Larger Active Spaces




Ultimately, our goal is not to compute dense wave functions for subsequent compression. Our goal is to solve for large CI wave functions using a hierarchically compressed basis. Thus, in this section, we consider the behavior of CHACI compression as a function of active space size. In Figure <ref>, we consider several active spaces, comparing the convergence of several measures of the accuracy as a function of the percentage of dense storage used (the compression ratio). We find that as the size of the active space increases, the accuracies of both absolute energy and spin converge faster with increasing compression ratio.



To quantify this convergence behavior, we plot the compression ratio at which absolute energies of 0.2 eV accuracy are achieved as a function of the number of active orbitals/electrons in Figure <ref>. Both the singlet and triplet compression ratios converge quickly with increasing active space. Of the two, the triplet energy converges more slowly, thus we fit it to an exponential in order to extrapolate to larger active spaces. We find that the required compression ratio decays proportional to

    f(N_MO) ∝ e^-0.561N_MO.

Extrapolating to larger active spaces, we estimate that a 24-24 active space could converge to 0.2 eV accuracy at a storage cost of 77,370 kdoubles, which is less than the cost of the dense storage of a 16-16 active space (165,637 kdoubles). Though the convergence behavior is likely to be system-dependent, this result certainly encourages further study.



 §.§ Effect of Using Optimal Rank on Compression




In order to assess the necessity of the optimal rank procedure, we compare SR-CHACI (which uses a static rank) to CHACI and TSVD. Figure <ref> presents the error in the singlet-triplet gap as a function of the total storage for the 14-14 active space. Excepting one fortunate point at 160 kdoubles, the accuracy of SR-CHACI is significantly worse than that of CHACI, but still better than TSVD. However, considering the error in the absolute energies of the singlet and triplet states separately (Figure 11a and b, respectively), it is clear that this is due to error cancelation. Errors in the absolute energy of the singlet state derived from the SR-CHACI wave function are similar to those of TSVD, and much greater than those of CHACI. Further, errors in the SR-CHACI triplet state are slightly larger than TSVD.



Analysis of spin errors tells a similar story. CHACI is much superior to SR-CHACI at reproducing the spin of the original wave function, and SR-CHACI has similar (and sometimes larger) spin errors compared to TSVD. Taken together, we conclude that optimal rank is an essential component of the CHACI algorithm.



 §.§ Effect of Sorting on Compression




Here we assess the necessity of another feature of the CHACI compression algorithm: the sorting of rows/columns of the 𝐂 matrix prior to compression. To this end, we compare U-CHACI, in which the rows/columns remain unsorted, to CHACI and TSVD. Figure <ref> presents a heat-map of the uncompressed 𝐂 matrix of the singlet (panels a and c) and triplet (b and d) wave functions before (a and b) and after (c and d) sorting. Note that sorting concentrates larger elements into the upper-left corner of the matrix.



Figure <ref> compares the U-CHACI singlet-triplet errors to those of CHACI and TSVD. Though U-CHACI appears to be more accurate for predicting the relative energy than TSVD, it remains inferior to CHACI. Considering the errors in the absolute singlet and triplet energies (Figure <ref>a and b), we see that U-CHACI errors are on the order of the same size as those of TSVD, and considerably larger than those of CHACI. That being said, U-CHACI is solidly between CHACI and TSVD in its ability to accurately describe the total spin angular momentum (Figure <ref>c and d).



Taking this data together, we conclude that sorting is an essential component of the CHACI algorithm. However, our ultimate goal is not to compute the full wave function and subsequently compress it, and the type of a posteriori sorting that we use in CHACI would not be possible if we were to directly solve for the wave function in compressed form. But given that the Duch ordering of spin strings does not allow for efficient compression, the determination of an a priori scheme by which strings may be ordered for efficient compression remains an important open question.



 §.§ Effect of Upper Quadrant vs H-matrix Compression




Lastly, we consider compression of 𝐂 into the H-matrix format,<cit.> which is designed to leverage the diagonally dominant nature of the matrix, in contrast to the CH blocking scheme used in CHACI. Figure <ref> shows the error in the singlet-triplet gap, singlet absolute energy, and triplet absolute energy of the 14-14 wave function compressed into H-matrix format. H-matrix compression is inferior to TSVD, thus we conclude that the CH blocking scheme is an essential component of the CHACI scheme.



§ CONCLUSIONS


Here we have presented an exploratory study in which we employ hierarchical matrices to perform lossy compression of CASCI wave functions of a strongly correlated system. We have demonstrated superior compression compared to a global low-rank approximation of the wave function, and we have demonstrated that CH blocking, optimal rank, and row/column sorting are essential features of the algorithm.

These promising results suggest several avenues for future study. The development of an algorithm to directly solve for the wave function in the CHACI representation would allow the accurate solution of strongly correlated wave functions with active spaces well beyond those that are accessible via brute-force CASCI implementations. That hierarchical matrices are well suited for implementation on large-scale distributed computing systems is a noteworthy advantage of this approach. Yet a direct solution will only be possible after an effective a priori row/column sorting algorithm is developed, and the development of such an algorithm will require further analysis of the uncompressed wave function. 



§ ACKNOWLEDGEMENTS


This work was supported by a seed grant from the Institute for Advanced Computational Science. KOB, AT, and BGL are grateful for support from the National Science Foundation under grant CHE-1954519 and for start-up funding from Stony Brook University.





§ NEAR-OPTIMAL CHACI COMPRESSION


As described above, the CHACI algorithm introduces a strategy for
performing recursive partitioning. Its key idea is to approximate the CI vector via truncated SVD (TSVD) of a collection of small block matrices constructed hierarchically.  The algorithm
leverages the fact that many CI vectors, even of relatively strongly correlated systems, have data that predominantly reside in the matrix's upper-left corner.
Here we investigate this technique and propose an algorithm
that can achieve near-optimal storage efficiency within this framework.
For quantifying approximation accuracy, in this section, we employ the
squared Frobenius norm. Specifically, if 𝐀 represents
the original CI vector and 𝐀' symbolizes its CHACI
approximation, then the error is quantified by
𝐀 - 𝐀'_F^2.
Above, the numerical results consider the
energy and spin errors as additional metrics for error evaluation.



 §.§ Information Density

To derive the algorithm, we first introduce the concept of
information density, which quantifies the amount of information
contained in a singular value relative to the storage cost. Consider a matrix 𝐀∈ℝ^m × n with its singular value decomposition (SVD) given by

    𝐀 = 𝐔Σ𝐕^† = [𝐮_1, …, 𝐮_m]Σ
    [𝐯_1, …, 𝐯_n]^T,

where 𝐮_i ∈ℝ^m, 𝐯_i ∈ℝ^n, and Σ is an m× n diagonal matrix composed of σ_i along its diagonal. Without loss of generality, let us assume m≥ n. Here, 𝐀_F^2 = ∑_i=1^nσ_i^2 represents the total information contained in 𝐀. 
Storing the first k singular-value triplets would capture ∑_i=1^k σ_i^2 information and lose the remaining ∑_i=k+1^n σ_i^2. This is the best approximation of 𝐀 in Frobenius norm, as given by the TSVD. To quantify the efficiency of storing each singular-value triplet (𝐮_i, σ_i, 𝐯_i), we define the information density ρ_i as follows:

The information density ρ_i of the singular-value triplet (𝐮_i, σ_i, 𝐯_i) is the ratio of the information content σ_i^2 to the storage cost m+n+1, i.e.,

    ρ_i = σ_i^2/m+n+1.



This measure is particularly useful when comparing matrices of different sizes. For instance, a 4 × 4 matrix with a singular value of 5 has a higher information density of 25/9 compared to a 8 × 8 matrix with a singular value of 6, which has an information density of 36/17, less than the 4× 4 matrix. This indicates why smaller matrices with higher information density are often preferred for storage when aiming to minimize memory usage while maximizing retained information.









 §.§ Near-Optimal CHACI Algorithm


Based on the concept of information density, we propose our CHACI algorithm, which strategically chooses between dense and sparse formats for each block. The “near-optimal” descriptor emphasizes that among all CHACI configurations, this algorithm seeks to minimize memory usage within a small margin of the optimal solution that achieves the same accuracy. 
Algorithm <ref> outlines this near-optimal CHACI algorithm. This algorithm takes the number of levels p as an input, which should be approximately logarithmic in M. Based on our experiments, we choose p to be the smallest integer that satisfies k_0 2^p≥ M, where k_0=6, which can be computed recursively. The algorithm includes a mechanism to switch to dense storage for small blocks. It also takes a threshold for the information density as input, and varying this threshold would result in different storage requirement, as we did in our numerical tests in Section <ref>. It is possible to change the input to be a memory budget instead, and in this setting CHACI possesses some optimality properties, as we analyze next.







 §.§ Analysis of Near Optimality




In Section <ref>, we showed numerically that CHACI uses less memory than TSVD. To help understand why this is the case, we present a theoretical analysis of the CHACI algorithm. For simplicity and clarity, we make some simplifying assumptions in our analysis. First, we assume that each block is square, which is the case if the input matrix is a square matrix and its number of rows (and of columns) is a power of two. Second, we change the for and while loops in the algorithm to be in descending order of the information density among all the blocks. This modification requires maintaining a priority queue of the singular triplets of all the blocks, but it does not significantly affect the accuracy and performance compared to using the for-while loops in Algorithm <ref>.





Consider a CHACI-partitioned matrix with p levels, resulting in 3p + 1 blocks, each represented by an SVD. Denote these blocks as 𝐁_1 to 𝐁_3p+1, with each block 𝐁_i having a size m_i. Each singular value σ_i,j from block 𝐁_i contributes to the overall matrix's Frobenius norm squared, and hence

    𝐀_F^2 = ∑_i=1^3p+1∑_j=1^m_iσ_i,j^2.

Storing all singular-value triplets provides complete information, but practical constraints often require storage optimization. Therefore, we compute the information density for each block and prioritize those with the highest values to minimize storage while maintaining accuracy. Let
s_i represent the storage of the singular-value triplets and ρ_i the information density. The following lemma motivates us to minimize storage by sorting the singular-value triplets based on ρ_i:


Given a collection of positive-number pairs {(s_i, ρ_i)| 1≤ i≤ n}, where ρ_i is sorted in descending order, for an arbitrary selection of k pairs (s_j_1, ρ_j_1), …, (s_j_k, ρ_j_k), let r be a number between 1 and n such that

    ∑_i=1^r-1 s_i < ∑_i=1^k s_j_i≤∑_i=1^r s_i,

it then follows that

    ∑_i=1^r s_i ρ_i ≥∑_i=1^k s_j_iρ_j_i.



Let S=∑_i=1^r(s_i). Define a piecewise function f(x) over 0 < x ≤ S as

    f(x) = ρ_1    if  0 < x ≤ s_1, 
    ρ_2    if  s_1 < x ≤ s_1 + s_2, 
    ⋮
    ρ_r    if ∑_i=1^r-1 s_i < x ≤∑_i=1^r s_i, 
    
    0    otherwise.

Then,

    ∑_i=1^r s_i ρ_i = ∫_0^S f(x)   dx.

Sorting the ρ_j_i in descending order, or equivalently sorting j_i in ascending order, we redefine the pairs as (s_J_i, ρ_J_i), where J_i is the ith smallest element in the set {j_i}.
Define f̂(x) similarly as f(x) but with ρ_J_i in place of ρ_i, i.e.,

    f̂(x) = ρ_J_q   if ∑_i=1^q-1 s_J_i < x ≤∑_i=1^q s_J_i for  1≤ q ≤ m,
    
    0    otherwise.

Since ρ_i is sorted in descending order, we have f(x) ≥f̂(x) for all x ≥ 0. Therefore,

    ∑_i=1^r s_i ρ_i - ∑_i=1^m s_j_iρ_j_i = ∫_0^S (f(x) - f̂(x))   dx ≥ 0.



Lemma <ref> implies that by sorting the singular-value triplets based on their information densities, we can maximize the information retained for a given storage cost. This result suggests a greedy algorithm that selects the singular values with the highest information densities first, thereby minimizing storage while maintaining the desired accuracy. However, such a greedy strategy may not always yield the optimal solution for a given accuracy requirement. The following theorem provides a bound on the storage difference between the greedy algorithm and the optimal solution.



Given a collection of positive-number pairs {(s_i, ρ_i)| 1≤ i≤ n} as in Lemma <ref> and t∈(0, ∑_i^n s_i], for an arbitrary selection of k≤ n pairs (s_j_1, ρ_j_1), …, (s_j_k, ρ_j_k) such that ∑_i=1^k s_j_iρ_j_i≥ t, let r be the number of triplets such that

    ∑_i=1^r-1 s_iρ_i < t ≤∑_i=1^r s_iρ_i,

it then follows that

    ∑_i=1^r s_i - ∑_i=1^k s_j_i < N+2,

where N is the number of rows in the matrix. In other words, the storage difference between the greedy algorithm and the optimal CHACI solution is bounded by a number no more than N+2.


Assume that the selected pairs (s_j_i, ρ_j_i) satisfy that the storage

    ∑_i s_j_i≤∑_i=1^r-1 s_i.

Then by Lemma <ref>,

    ∑_i s_j_iρ_j_i≤∑_i=1^r-1 s_i ρ_i < t,

which conflicts with the fact that this selection satisfies the requirement

    ∑_i s_j_i·ρ_j_i≥ t.

Thus we have

    ∑_i s_j_i > ∑_i=1^r-1 s_i,

so

    ∑_i=1^r s_i - ∑_i s_j_i < ∑_i=1^r s_i - ∑_i=1^r-1 s_i = s_r ≤max_i=1^r{s_i}.

Since the largest block is of size ⌈ N/2⌉×⌈ N/2⌉, we have maxs_i≤ N+2.


In practice, the last block s_r is even much smaller than N+2. Hence, the storage difference between the greedy algorithm and the optimal solution is bounded by a number typically far smaller than N. Therefore, CHACI is nearly optimal in its storage requirement. In the following, we further show that CHACI typically requires less memory and is more efficient than applying the TSVD to the global matrix.









 §.§ Analysis of Complexity












The CHACI algorithms assume that the input CI vector is fully represented, requiring N^2 storage. The extra intermediate storage of the algorithm is about the same as the input size. Hence, we focus on the output's storage requirement, a common approach for compression techniques.

For each block i of size m_i × m_i, if stored in TSVD format with rank k_i, the storage requirement includes 2k_im_i for the vectors (each vector has m_i entries) and k_i for the singular values, totaling 2k_im_i + k_i. If a block is stored in a dense format, its requirement is m_i^2. Thus, the total storage for the output is

    Total output size = ∑_TSVD block  i (2k_im_i + k_i) + ∑_dense block  i m_i^2.

It is worth noting that the dense blocks are typically for the smaller blocks, and the total storage is dominated by the TSVD blocks. Note that

    ∑_i=1^3p+1 m_i ≤∑_j=1^p3 N/2^j +⌈N/2^p⌉≤ 3N+1,

and

    ∑_i=1^3p+1 (2m_i + 1) ≤ 6N+6p+3=6N+o(N).

For simplicity, assume all the blocks are stored in TSVD format. The output can be bounded by

    Total output size≲ 6Nmax{k_i}

for sufficiently large N. This bound is tight if the ranks {k_i} are uniform, but it is pessimistic if the ranks are highly non-uniform, as they are in practice for CI vectors.

Now we compare the storage requirement to that of global TSVD.
A global TSVD for the CI vector traditionally stores K singular values and vectors, resulting in an output storage of 2KN + K.
Hence, if the input matrix were a uniform low-rank matrix, CHACI would require more storage than the global TSVD.[H-matrices would also require more storage than the global TSVD if the input matrix is a uniform low-rank matrix.] However, CHACI is designed to exploit the structure of CI vectors, which are not uniformly low-rank. In our numerical experimentation, the
output size of near-optimal CHACI is typically about 2% of that of a global TSVD of equal accuracy.



The time complexity of the near-optimal CHACI algorithm involves computing TSVDs for various blocks. 
For each block i of size m_i × m_i, a rank-k_i TSVD can be approximately computed in 𝒪(k_im_i^2) using iterative methods.

An important detail worth emphasizing is Line <ref> of Algorithm <ref>. In this step, we would double the TSVD rank if the desired accuracy is not met. Such a doubling process results in a geometric progression of the TSVD computation cost, calculated as follows:

    ∑_j=1^⌈log_2 k_i ⌉ 2^j m_i^2 ≤ 4k_i m_i^2.

Consequently, the total time complexity for all blocks in the CHACI algorithm is

    Total time complexity = ∑_i=1^3p+1𝒪(k_i m_i^2),

where max{m_i}=N/2. If the rank increases were linear when recomputing the TSVD, the worst-case time complexity could potentially reach 𝒪(k_i^2 m_i^2) for each block, potentially increasing the overall time complexity.

Now we compare the time complexity to that of global TSVD.
Performing a global TSVD of an N × N matrix with a truncation rank K incurs a time complexity of 𝒪(KN^2). Since

    ∑_i=1^3p+1 m_i^2 ≤∑_j=1^⌈log N ⌉ 3N^2/2^2j≤ N^2.

As long as k_i < K, the near-optimal CHACI algorithm has a lower computational cost than a global TSVD. In practice, k_i ≪ K in the near-optimal CHACI algorithm, as we will demonstrate in our numerical experimentation. Hence, we can expect the near-optimal CHACI algorithm to be significantly faster than a global TSVD for moderately large N.







 Paper Title: AtmoSpec -- A Tool to Calculate Photoabsorption Cross-Sections for Atmospheric Volatile Organic Compounds.













    
    < g r a p h i c s >









§ ABSTRACT 
Characterizing the photolysis processes undergone by transient volatile organic compounds in the troposphere requires the knowledge of their photoabsorption cross-section – quantities often challenging to determine experimentally, particularly due to the reactivity of these molecules. We present a computational tool coined , which can predict a quantitative photoabsorption cross-section for volatile organic compounds by using computational photochemistry. The user enters the molecule of interest as a SMILES code and, after selecting a level of theory for the electronic structure (and waiting for the calculations to take place), is presented with a photoabsorption cross-section for the low-energy conformers and an estimate of the photolysis rate coefficient for different standardized actinic fluxes. More specifically,  is an automated workflow for the nuclear ensemble approach, an efficient technique to approximate the absolute intensities and excitation wavelengths of a photoabsorption cross-section for a molecule in the gas phase. This work provides background information on the nuclear ensemble approach, a guided example of a typical  calculation, details about the architecture of the code, and the current limitations and future developments of this tool.







§ INTRODUCTION


Understanding and predicting the photochemical reactivities and lifetimes of atmospheric volatile organic compounds (VOCs) upon sunlight absorption is of prime importance in informing chemical models used to simulate the molecular composition of our atmosphere.<cit.> However, the potential for photolysis of a large number of VOCs, in particular transient VOCs, remains unknown to date due to their short lifetime and instability, preventing their isolation or synthesis and rendering spectroscopic measurements highly challenging. 

The photolysis of a compound A following sunlight absorption, A ->[hν] B + C, is characterized by the photolysis rate -d[A]/dt=J[A], where J is the photolysis rate coefficient. This photolysis rate coefficient is defined as


    J=∫_λ_min^λ_maxϕ(λ)σ(λ) F(λ)dλ.


Equation (<ref>) provides the story of a photolysis process: the actinic flux (F(λ)) informs on the number of photons coming from a source (e.g., the sun) at a given wavelength per second and per unit area, and the likelihood of molecule A to absorb one of these photons is given by its photoabsorption cross-section σ(λ). If the molecule has absorbed light at wavelength λ, the quantum yield ϕ(λ) gives the probability that the photoexcited molecule will undergo the photolysis process described above and form B + C. Integrating the product ϕ(λ)σ(λ) F(λ) over a range of wavelength characteristic of the actinic region (280 - 400 nm) leads to the overall photolysis rate coefficient J. (We note at this point that J often depends on the temperature, pressure, and the solar zenith angle.)

Equation (<ref>) makes it clear that a key quantity to consider when studying the potential for photolysis of a VOC is its photoabsorption cross-section σ(λ). Photolysis data for a limited number of key VOCs can be found in databases like MPI-Mainz UV/Vis Spectral Atlas,<cit.> IUPAC,<cit.> NASA (JPL),<cit.> or atmospheric models such as TUV.<cit.> Information about the potential photolysis of VOCs is crucial to inform atmospheric models like the Master Chemical Mechanism used to simulate the chemical composition of the atmosphere.<cit.> As stated above, obtaining such quantities experimentally is often hampered by the instability of the molecule (or the difficulty in isolating it), and structure-activity relationships (SARs) – regularly used for ground-state properties of VOCs – often run out of steam when extended to photochemical processes, notably due to the possible delocalized nature of molecular excited states and the multichromophoric nature of some VOCs.<cit.>  

Recently, we proposed to use computational photochemistry to obtain an estimate of the photoabsorption cross-section of transient VOCs.<cit.> The computational strategy, based on the nuclear ensemble approach (NEA, see Sec. <ref> below for details),<cit.> offers an adequate description of the absolute cross-section and the position of the maxima in the photoabsorption cross-section for VOCs. While this technique does not describe vibronic progressions,<cit.> the width of each band is captured qualitatively well, which is a prime factor when considering the use of calculated photoabsorption cross-sections in Equation (<ref>).

Hence, the NEA could act as a doorway to predict the photoabsorption cross-section of transient VOCs and, hopefully, determine more robust photochemical SARs rules for compounds of the same family. However, the practical execution of the NEA workflow is rather tedious (yet not complex), which hinders its adoption outside of the computational photochemistry community. Here, we introduce  – an automated workflow and graphical interface to predict the photoabsorption cross-sections of VOCs based on the NEA and requiring only a limited number of operations from the user, making the theoretical prediction of cross-sections more accessible to researchers outside of the computational photochemistry community.

In the following, we briefly summarize the main conceptual ideas of the NEA, the different steps required for a typical calculation, and the limitations of the methods (Sec. <ref>).  We then walk the reader through an example calculation in  to determine the photoabsorption cross-section of glycolaldehyde (Sec. <ref>). We present the design and architecture of  and the details of the automatized NEA workflow implemented using the  infrastructure (Sec. <ref>). We finally stress the current limitations of  and highlight its future developments (Sec. <ref>). 



§ A BRIEF SURVEY OF THE NUCLEAR ENSEMBLE APPROACH



Different theoretical strategies can be used to predict the photoabsorption of gas-phase molecules like VOCs (see Ref.  for an overview), but here we focus exclusively on the NEA which represents a good tradeoff between computational efficiency and qualitatively robust results. Crucially, unlike other more complicated approaches, the NEA workflow is amenable to automation. The NEA, which is a numerical realization of the reflection principle,<cit.> proposes to project the ground-state nuclear density onto the excited electronic state(s) of interest.<cit.> In practice, the NEA consists of (i) approximating the ground-state probability density of the molecule of interest, (ii) sampling from this distribution a set of N_p molecular geometries (N_p ≈ 500-10000), and (iii) calculating, for each molecular geometry j (𝐑_j, where 𝐑 is a collective variable for all the nuclear coordinates of the molecule of interest), the corresponding vertical excitation energies (Δ E_0I(𝐑_j)) and transition dipole moments (μ_0I(𝐑_j)) to a subset of N_s excited electronic states of interest (labeled by I). For each geometry, each transition energy is broadened by a Gaussian (or Lorentzian) with a set width δ. The photoabsorption cross-section is obtained by averaging over all the N_s broadened electronic transitions for all the N_p geometries. Hence, the photoabsorption cross-section within the NEA for a given conformer is obtained as 

    σ(E) = π/3ħϵ_0 c1/N_p δ√(2π)∑_j=1^N_p∑_I=1^N_sΔ E_0I(𝐑_j) |μ_0I(𝐑_j)|^2 exp(-(E-Δ E_0I(𝐑_j))^2/2δ^2)  ,


with E being the photon energy. 

If the molecule of interest exists as multiple conformers at a given temperature, the protocol described above should be repeated for each conformer, and the total photoabsorption cross-section is obtained by summing the Boltzmann-weighted photoabsorption cross-section for each conformer.

The central idea of the NEA – to project a ground-state probability density vertically onto excited electronic state(s) – is justified for dissociative excited electronic states for which the corresponding photoabsorption cross-section does not exhibit vibronic progression.<cit.> In other words, the NEA cannot reproduce vibronic progressions as it does not consider the vibrational states of the excited electronic states.<cit.> For dissociative excited states (nσ^∗, πσ^∗, σσ^∗), which are of central interest for photolysis processes related to atmospheric photochemistry, the NEA can provide an accurate description of the photoabsorption cross-sections (absolute intensities and excitation wavelengths), given that an adequate electronic-structure method was employed for excitation energies and transition dipole moments – see Refs. for examples. However, the NEA was shown to provide a qualitatively correct description of the transition bands for photoabsorption cross-sections involving bound excited electronic states (like nπ^∗), capturing the proper envelope of each band as well as its location in energy and absolute intensity. The key ingredient for the success of the NEA in this later case lies in its ability to capture non-Condon effects by sampling molecular geometries away from the equilibrium geometry (i.e., the transition dipole moments in Eq. (<ref>) depends on the nuclear configuration of the sampled geometry, μ_0I(𝐑_j)). Hence, the NEA was successfully used in the context of atmospheric chemistry to describe the low-energy nπ^∗ band of the photoabsorption cross-section for carbonyl-containing VOCs.<cit.> Another potential limitation of the NEA is its inaccurate description of the tail of the photoabsorption cross-section, which should be stressed in the context of atmospheric photochemistry as a photolysis rate coefficient can be highly sensitive to a low-energy tail of the photoabsorption cross-section if it enters artificially deeper in the actinic region.

The NEA relies on the determination of an (approximate) ground-state nuclear probability density. The commonly employed approach is to use a harmonic approximation for the different vibrational modes of the molecule of interest, as one can determine analytically from this approximation a so-called Wigner distribution for both nuclear positions and momenta.<cit.> The key advantage of this approach is that a harmonic Wigner distribution only requires, for each conformer, an equilibrium geometry in the ground electronic state and its corresponding vibrational frequencies – two quantities readily available from any standard quantum-chemical software. Sampling molecular geometries from a harmonic Wigner distribution (used as a proxy for the ground-state nuclear probability density) is often called ‘Wigner sampling’. Wigner sampling is implemented in different codes to perform excited-state molecular dynamics, as the nuclear positions and momenta it provides can be used as initial conditions for excited-state dynamics. Using a harmonic Wigner distribution to provide the molecular geometries (nuclear positions) required by the NEA leads to accurate photoabsorption cross-sections for VOCs exhibiting rather harmonic modes (for example, acrolein<cit.>). However, care should be taken for flexible molecules, typically possessing low-frequency (anharmonic) normal modes, as they may lead to artifacts in the photoabsorption cross-section when the low-frequency mode is photoactive.<cit.> More advanced strategies can be used to obtain ground-state distributions for flexible molecules, such as ground-state ab initio molecular dynamics with a quantum (or colored-noise) thermostat<cit.> (QT) or path-integral molecular dynamics.<cit.> A detailed discussion of the NEA applied to VOC and its performances can be found in Ref. .

From a practical perspective, calculating the photoabsorption cross-section for a given VOC requires the following steps. First, one needs to find all molecular conformers that are meaningfully present at a given temperature (this step can be done manually or involve dedicated software for conformer exploration). Once the conformers are located, an electronic-structure method should be chosen for the subsequent ground- and excited-state calculations, ensuring that we have the best compromise between computational efficiency and accuracy (this step requires a quantum-chemical package). Once the electronic-structure method is selected, the following steps should be executed for each conformer: (i) geometry optimization and frequency calculation, (ii) sampling of N_p molecular geometries from the constructed harmonic Wigner distribution (this step often requires an external software), (iii) calculation of excitation energies and transition dipole moments for each sampled geometry, (iv) construction of the photoabsorption cross-section for the conformer (this step requires a bespoke code, sometimes the same as in step (ii)). The final photoabsorption cross-section is then calculated by weighting the photoabsorption cross-section of each conformer by its corresponding Boltzmann factor. The steps described above hopefully make it clear that determining the photoabsorption cross-section for a given VOC is not per se difficult, but it implies a rather tedious and repetitive set of operations, involving the use of different software packages. In the next Sections, we discuss how these steps have been turned into an automated workflow within . 



§ A GUIDED TOUR OF 



The goal of  is to transform the steps of an NEA calculation (described in the last paragraph of Sec. <ref>) into an automated workflow with the minimum effort for the user. The overall workflow of  is depicted in Fig. <ref> and this Section endeavors to describe each step of an NEA calculation with  from a user perspective. As an example, we propose to calculate the photoabsorption of glycolaldehyde, HCOCH2OH.






 §.§ Step 1 – Initial molecular structure and conformer search




 implements its graphical interface via a web application that can be accessed using any modern web browser. The user connects to  via its address on the machine where the code was installed (detailed installation information is provided on the GitHub page of <cit.>)
After launching , the user is first requested to provide the molecular structure for the VOC of interest, here glycolaldehyde (Figure <ref>a). The simplest way to provide this information is to use a SMILES (simplified molecular-input line-entry system) code – a simple ASCII string encoding the structure of a molecule.<cit.>
A SMILES code is easily obtained from a Lewis structure created with a molecular editor (for example, the SMILES code of glycolaldehyde is ). 
Once the SMILES code is provided, the user can click on the 'Generate molecule' button. 
 will proceed with a fast search of the possible conformers of the molecule using the ETKDG algorithm<cit.> implemented in the RDKit package,<cit.> followed by geometry optimization using the MMFF94 force field.<cit.> 
 will display all the conformers of the molecule in a ball-and-stick representation within a window for the user to confirm that this is indeed the molecule of interest (Figure <ref>b).  also displays the MMFF94 energies that serve as an initial estimate of their Boltzmann population. Both the conformer geometries and energetics are further refined in subsequent steps.
As an alternative to the input via SMILES, the user can also upload a file containing the molecular structure of interest, for example in an XYZ format. In this case,  does not perform a conformer search.

The user is then invited to provide a description and label for the calculation to generate an entry in  database (Figure <ref>c). The use of a database means that the results of this calculation can be recovered at any time in the future by selecting the entry via its label.




 §.§ Step 2 – Computational details for the NEA calculation




The second step is critical as it defines the computational details for the calculation of the photoabsorption cross-section. The user will be invited to provide information about the molecule (charge, multiplicity) and the level of electronic-structure theory required. The ground-state electronic structure information will be used to optimize the different conformers identified in Step 1 and calculate the vibrational frequencies required to build the harmonic Wigner distribution for the molecule. The excited-state electronic structure method defines the level of theory for the calculation of vertical excitation energies and transition dipole moments that will be required for building the photoabsorption cross-section based on Eq. (<ref>). In its current version,  is interfaced with the quantum-chemical software package ORCA,<cit.> which offers a broad range of capabilities for excited electronic states and is free for academic use.

While default parameters are provided for the calculation of a photoabsorption cross-section with a compromise between efficiency and accuracy (based on earlier experience with the NEA and VOCs),  offers different options to the users to refine their calculation and, perhaps more importantly, benchmark the level of electronic-structure theory. Hence, if the 'Number of Wigner samples' is set to 0, the user can perform single-point calculations on top of the optimized geometry of each conformer to test the sensitivity of vertical transitions to the quantum-chemical method.  gives access to LR-TDDFT (with and without the Tamm-Dancoff approximation), but allows the user to perform calculations with the wavefunction-based methods ADC(2) and EOM-CCSD. While these two electronic-structure methods are likely to be quite expensive for a full NEA calculation (i.e., with multiple Wigner samples), they offer an adequate point of comparison for single-point calculations. We note that implicit solvent can be included too, motivated by recent experiments where the photoabsorption cross-section of VOCs was deduced from their extinction coefficients in cyclohexane.<cit.>

Once all the computational details are provided, the user can define where the calculations should be launched (locally or remotely, for example on an HPC cluster that the user has access to) as well as the number of cores that the electronic-structure software can use. By clicking on the 'Submit' button, the user will launch the fully-automatized NEA calculation.



 §.§ Step 3 – Status of the calculation and outputs




Once the user clicks on the 'Submit' button in Step 2,  distributes the required calculations on the available resources (either in a serial or parallel mode). The user sees a status bar appearing (Figure <ref>) that summarizes the progress of the NEA workflow. At this stage, the user can close the  tab from their web browser, as the details of the molecule and its NEA workflow are saved in the  database under the descriptor provided in Step 1. The user can simply reopen  and search for the descriptor name in the  database to reload the current status of the calculation. 

For the user interested in the details of the NEA workflow, Step 3 gives direct access to all the output files provided by the quantum-chemical software in use (in the present example, ORCA). The button 'Show workflow details' (Figure <ref>) opens the organization tree of the workflow – geometry optimization and frequency calculation for each conformer, followed by the NEA calculation for each conformer – and all the respective output files can be found at the bottom of this tree. At a higher level of this tree,  provides a report of the key quantities extracted from the output files. 



 §.§ Step 4 – Results





Once all the calculations are done,  summarizes the results in the 'Step 4' and displays first the calculated photoabsorption cross-section (Figure <ref>).  provides a summary of the computational details for the NEA workflow (Figure <ref>a). In this particular case, we calculated the photoabsorption cross-section of glycolaldehyde with LR-TDDFT/TDA, using the ωB97X-D4 functional and the aug-cc-pVDZ basis set. 500 geometries were sampled from the harmonic Wigner distribution produced for each conformer (two conformers considered in total). 

The calculated photoabsorption cross-section is provided in an interactive window, together with any experimental photoabsorption cross-section that can be imported by the user (Figure <ref>b). In this particular example, the overall shape and width of the photoabsorption cross-section are properly captured by the NEA, while the intensity is slightly overestimated with the level of electronic-structure theory selected. Different tools are available to interact with the photoabsorption cross-section, and the user can save the results either as a CSV text file or a picture. A menu is also provided for tuning more advanced parameters related to the NEA (Figure <ref>c), such as the function used to broaden each vertical excitation within the NEA, its width (δ), or the overall energy unit used to represent the photoabsorption cross-section. A side window depicts the low-energy conformers considered to build the photoabsorption cross-section (Figure <ref>d) and their relative energy, calculated at the level of theory defined in Step 2.



The analysis of the conformers can be pushed further by clicking on the 'Show conformers' button (Figure <ref>a). In this mode,  shows the contribution of each conformer to the full photoabsorption cross-section. The user can select a conformer from the molecular-representation window, and its photoabsorption cross-section – weighted by the Boltzmann population of the conformer – is highlighted in red in the photoabsorption cross-section (Figure <ref>). In the specific case of glycolaldehyde presented here, a low-energy conformer with a Boltzmann population of 0.6% at 298 K shows a contribution to the photoabsorption cross-section that is shifted to longer wavelength (red line in Figure <ref>) in comparison to the dominant conformer exhibiting an intramolecular hydrogen bond (blue line in Figure <ref>). 



A last window gives access to an estimation of the photolysis rate constant J using Eq. (<ref>) with the calculated photolysis rate constant, a user-defined value for the photolysis quantum yield, and three options for the actinic fluxes (Figure <ref>). The user can select one of the three standardized actinic fluxes (Figure <ref>a) – 'high flux' (solar zenith angle = 0^∘, overhead ozone column = 200 DU), 'medium flux' (solar zenith angle = 60^∘, overhead ozone column = 350 DU), and 'low flux' (solar zenith angle = 90^∘, overhead ozone column = 500 DU) for a ground elevation of 0 km above sea level. An interactive window depicts the integrand of Eq. (<ref>), namely the product of the calculated photoabsorption cross-section, the quantum yield, and the chosen actinic flux (Figure <ref>c). An approximate wavelength-independent quantum yield can be provided by the user (Figure <ref>b) and its effect on the calculated J value is adapted in real-time so that the user can determine the upper and lower bounds for the photolysis rate constant based on the photoabsorption cross-section predicted by . 




§ SOFTWARE ARCHITECTURE AND IMPLEMENTATION



In this Section, we provide a brief description of the overall architecture of  as well as an overview of the software stack and libraries used to build this tool.



 §.§ Design goals

The overarching goal of  was to make the calculation of photoabsorption cross-section with the NEA accessible to non-experts in computational photochemistry. With that in mind, the following criteria were key for the development of the architecture and implementation of : (a) simple installation, (b) graphical interface that is easy to navigate, (c) input and output data should be stored, and easily searchable, and (d) the calculations should be fully reproducible.

A trivial way to address point (a) is to develop a software that does not necessarily need to be installed on the user's computer. This can be achieved by making the graphical interface run in a web browser, while the application itself may run on a remote computer. This strategy means that the application itself can be deployed by an expert user or IT service, while the user only needs to navigate to the correct URL in the web browser to interact with its graphical interface. As a secondary goal, we also aimed to make the installation of the application approachable and reproducible. We achieved that by delivering  as a Linux container (using Docker<cit.>), which bundles all the necessary dependencies together.
To achieve points (c) and (d), we utilized the workflow manager  (see below). 
We note that the points highlighted above align with the FAIR principles (findable, accessible, interoperable, reusable) for scientific data.<cit.> To further enhance the reusability of , the different parts of the software (such as the harmonic Wigner sampling module) will be made available as standalone Python packages, usable independently from the full  tool.



 §.§ Architecture and data flow


The overall architecture of  is depicted in Fig. <ref>. We have decided to utilize the existing ecosystem around the workflow manager .<cit.> With this choice, the implementation of the NEA workflow is fairly generic:  handles the tasks of submitting individual ab initio calculations to the quantum-chemical program (here, ORCA), parsing the resulting outputs and storing all the critical information in a relational database (PostgreSQL) and a custom file storage. This strategy gives us the flexibility to combine  with other quantum-chemical programs in the future. Moreover,  benefits from broad support for handling automatic job submission to remote HPC clusters (key to install  locally but utilize a remote HPC resource for the quantum-chemical calculations).

The other central part of  is its graphical interface (front-end). We again employed , which was specifically created to make quantum-chemical calculations and complex scientific workflows available through a user-friendly browser interface.<cit.> In , the front-end is built on top of a Jupyter notebook interface. While this choice is somewhat unconventional for a web application, it has the advantage that the graphical part of the code is written in the same programming language (Python) as the rest of the program. This strategy makes  (and other -based applications) more approachable for potential developers, in a scientific environment where Python is nowadays perceived as a de facto lingua franca.



The AtmoSpec code is fully open source, as is most of the software stack that it builds upon, with the major exception of ORCA, which is however available for free for academic use under a custom EULA license. (The link to the  GitHub repository can be found in Ref. .)



§ CAVEATS AND FUTURE DEVELOPMENTS



 provides a simple interface to calculate an approximate photoabsorption cross-section, but the user needs to keep in mind that this simplicity leads to a series of strict limitations. We list here the main current limitations, and we discuss in the following paragraph the future developments that may address some of these issues. (1)  only offers a rather modest choice of (single-reference) electronic-structure methods and, as such, is not equipped to treat molecules exhibiting a challenging electronic structure (e.g., radicals or metal complexes). (2)  relies on the NEA, meaning that the produced photoabsorption cross-sections do not exhibit any vibronic structure. Perhaps more importantly in the atmospheric context, the (low-energy) tail of the photoabsorption cross-sections is likely to be too high in intensity, even for purely dissociative states. (3)  uses a harmonic Wigner distribution to sample the geometries used in the NEA. This strategy may lead to artifacts when used for flexible VOCs, that is, molecules exhibiting low-frequency vibrational modes (< 200 cm^-1), as discussed in Ref. . Such modes can be removed from the sampling by applying a low-frequency cutoff in Step 2. The use of ab initio molecular dynamics combined with a quantum thermostat is desirable for sampling the ground-state probability density of such molecules, but harder to automatize. (4)  does not offer any speed-up for the calculation of photoabsorption cross-sections within the NEA. For VOCs, these calculations can take from a few hours to a few days, depending on the level of theory and the number of sampled geometries for the NEA. 

A series of future developments are planned for , some of them trying to address the limitations listed above. (a) To reduce the computational cost of a typical photoabsorption cross-section calculation, we plan to interface  with the GPU-accelerated electronic structure package TeraChem. In addition to this speed-up of the electronic-structure calculations, we will incorporate the idea of optimal sampling within .<cit.> The idea of the optimal sampling is to use a cheap level of electronic-structure theory (for example, ZIndo/S) to calculate the photoabsorption cross-section of the molecule of interest within the NEA and a normal number of sampled geometries (around 500). From this result, the optimal sampling can select a subset (around 50) of optimal geometries that best represent the overall photoabsorption cross-sections. The final photoabsorption cross-section can be obtained by performing the NEA on this subset of geometries using a higher level of electronic structure theory. (b) As part of the automation in , we plan to devise a strategy to pre-select an optimal level of theory based on the input molecular structure and automatically determine the number of excited electronic states important to describe the absorption of the molecule in the actinic region. (c)  could be generalized to other types of spectroscopies within the NEA framework, such as photoelectron spectroscopy<cit.> or X-ray absorption spectroscopy.<cit.> 





§ SUMMARY


In summary, this work introduced , a computational tool devised to predict the photoabsorption cross-section and other photolysis properties of transient volatile organic compounds for atmospheric applications.  offers an automated workflow for the nuclear ensemble approach, only requiring a minimal input from the (non-expert) user using a web browser interface. This work describes a typical calculation with , going through the details of each step of the workflow. We also highlighted the main limitations of the current implementation and its future developments.  is open source and freely available for download – the GitHub link can be found in Ref. . 








The authors would like to thank Fay Abu Al Timen, Marco Barnfield, Kirstin Gerrand, Will Hobson, Kon Nomerotski, and Emily Wright – undergraduate students at the University of Bristol – for their contribution to the development of the photolysis rate constant calculation widget. DH would like to thank the  and  developers that comprise an incredibly welcoming open source community, in particular Aliaksander Yakutovich, Jusong Yu, Xing Wang, Marnik Bercx, Sebastiaan Huber, Carl Simon Adorf, Carlo Pignedoli, and Giovanni Pizzi. DH would also like to thank the many users of AtmoSpec within the group in Bristol for their feedback and bug reports. The authors also acknowledge Sasha Madronich for providing the standardized actinic fluxes implemented in .  This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 803718, project SINDAM) and the EPSRC Grants EP/V026690/1, EP/X026973/1, and EP/Y01930X/1. This article is also based upon work from COST Action CA18212 - Molecular Dynamics in the GAS phase (MD-GAS), supported by COST (European Cooperation in Science and Technology).
















@ifundefinedendmcitethebibliography
  
45








f
subitem(mcitesubitemcount)

  
  

[Ervens et al.(2024)Ervens, Rickard, Aumont, Carter, McGillen,
  Mellouki, Orlando, Picquet-Varrault, Seakins, Stockwell, Vereecken, and
  Wallington]egusphere-2024-1316
Ervens, B.; Rickard, A.; Aumont, B.; Carter, W. P. L.; McGillen, M.;
  Mellouki, A.; Orlando, J.; Picquet-Varrault, B.; Seakins, P.; Stockwell, W.
  et al.  Opinion: Challenges and needs of tropospheric chemical
  mechanism development. EGUsphere 2024, 2024,
  1–35
[Curchod and Orr-Ewing(2024)Curchod, and
  Orr-Ewing]curchodorrewing2024
Curchod, B. F. E.; Orr-Ewing, A. J. Perspective on Theoretical and Experimental
  Advances in Atmospheric Photochemistry. The Journal of Physical
  Chemistry A 2024, 
[Keller-Rudek et al.(2013)Keller-Rudek, Moortgat, Sander, and
  Sörensen]keller2013mpi
Keller-Rudek, H.; Moortgat, G.; Sander, R.; Sörensen, R. The MPI-Mainz
  UV/VIS spectral atlas of gaseous molecules of atmospheric interest.
  Earth System Science Data 2013, 5, 365–373
[ACP(2021 (accessed July 2024))]ACPcopernicusWebsite
ACP copernicus website.
  https://acp.copernicus.org/articles/special_issue8.html, 2021 (accessed July
  2024)
[Burkholder et al.(2020)Burkholder, Sander, Abbatt, Barker,
  Cappa, Crounse, Dibble, Huie, Kolb, Kurylo, et al.
  others]burkholder2020chemical
Burkholder, J. B.; Sander, S. P.; Abbatt, J. P. D.; Barker, J. R.; Cappa, C.;
  Crounse, J. D.; Dibble, T. S.; Huie, R. E.; Kolb, C. E.; Kurylo, M. J.
  et al.  Chemical kinetics and photochemical data for use in
  atmospheric studies; evaluation number 19; 2020
[TUV(2021 (accessed July 2024))]TUVwebsite
TUV website.
  https://www2.acom.ucar.edu/modeling/tropospheric-ultraviolet-and-visible-tuv-radiation-model,
  2021 (accessed July 2024)
[Jenkin et al.(1997)Jenkin, Saunders, and
  Pilling]jenkin1997tropospheric
Jenkin, M. E.; Saunders, S. M.; Pilling, M. J. The tropospheric degradation of
  volatile organic compounds: a protocol for mechanism development.
  Atmospheric Environment 1997, 31, 81–104
[Saunders et al.(2003)Saunders, Jenkin, Derwent, and
  Pilling]saunders2003
Saunders, S. M.; Jenkin, M. E.; Derwent, R. G.; Pilling, M. J. Protocol for the
  development of the Master Chemical Mechanism, MCM v3 (Part A): tropospheric
  degradation of non-aromatic volatile organic compounds. Atmospheric
  Chemistry and Physics 2003, 3, 161–180
[Jenkin et al.(2003)Jenkin, Saunders, Wagner, and
  Pilling]jenkin2003
Jenkin, M. E.; Saunders, S. M.; Wagner, V.; Pilling, M. J. Protocol for the
  development of the Master Chemical Mechanism, MCM v3 (Part B): tropospheric
  degradation of aromatic volatile organic compounds. Atmospheric
  Chemistry and Physics 2003, 3, 181–193
[MCM(2021 (accessed July 2024))]MCMwebsite
Master Chemical Mechanism, MCM v3.3.1. http://mcm.york.ac.uk, 2021 (accessed
  July 2024)
[Prlj et al.(2021)Prlj, Marsili, Hutton, Hollas, Shchepanovska,
  Glowacki, Slavíček, and Curchod]prlj2021calculating
Prlj, A.; Marsili, E.; Hutton, L.; Hollas, D.; Shchepanovska, D.;
  Glowacki, D. R.; Slavíček, P.; Curchod, B. F. E. Calculating
  Photoabsorption Cross-Sections for Atmospheric Volatile Organic Compounds.
  ACS Earth and Space Chemistry 2021, 6, 207–217
[Prlj et al.(2020)Prlj, Ibele, Marsili, and
  Curchod]prlj2020theoretical
Prlj, A.; Ibele, L. M.; Marsili, E.; Curchod, B. F. E. On the theoretical
  determination of photolysis properties for atmospheric volatile organic
  compounds. Journal of Physical Chemistry Letters 2020,
  11, 5418–5425
[Crespo-Otero and Barbatti(2012)Crespo-Otero, and
  Barbatti]crespo2014spectrum
Crespo-Otero, R.; Barbatti, M. Spectrum simulation and decomposition with
  nuclear ensemble: formal derivation and application to benzene, furan and
  2-phenylfuran. Theoretical Chemistry Accounts 2012,
  131, 1237
[Schinke(1995)]schinkebook
Schinke, R. Photodissociation Dynamics: Spectroscopy and Fragmentation of
  Small Polyatomic Molecules; Cambridge University Press, Cambridge,
  1995
[Crespo-Otero and Barbatti(2018)Crespo-Otero, and
  Barbatti]crespo2018recent
Crespo-Otero, R.; Barbatti, M. Recent Advances and Perspectives on Nonadiabatic
  Mixed Quantum–Classical Dynamics. Chemical Reviews 2018,
  118, 7026–7068
[McGillen et al.(2017)McGillen, Curchod, Chhantyal-Pun, Beames,
  Watson, Khan, McMahon, Shallcross, and Orr-Ewing]mcgillen2017criegee
McGillen, M. R.; Curchod, B. F. E.; Chhantyal-Pun, R.; Beames, J. M.;
  Watson, N.; Khan, M. A. H.; McMahon, L.; Shallcross, D. E.; Orr-Ewing, A. J.
  Criegee intermediate–alcohol reactions, a potential source of functionalized
  hydroperoxides in the atmosphere. ACS Earth and Space Chemistry
  2017, 1, 664–672
[Sršeň et al.(2018)Sršeň, Hollas, and
  Slavíček]srvsevn2018uv
Sršeň, Š.; Hollas, D.; Slavíček, P. UV absorption
  of Criegee intermediates: quantitative cross sections from high-level ab
  initio theory. Physical Chemistry Chemical Physics 2018,
  20, 6421–6430
[McCoy et al.(2021)McCoy, Marchetti, Thodika, and
  Karsili]mccoy2021simple
McCoy, J. C.; Marchetti, B.; Thodika, M.; Karsili, T. N. A simple and efficient
  method for simulating the electronic absorption spectra of Criegee
  intermediates: Benchmarking on CH_2OO and CH_3CHOO. The Journal
  of Physical Chemistry A 2021, 125, 4089–4097
[Clarke et al.(2022)Clarke, Gibbard, Hutton, Verlet, and
  Curchod]clarke2022photochemistry
Clarke, C. J.; Gibbard, J. A.; Hutton, L.; Verlet, J. R. R.; Curchod, B. F. E.
  Photochemistry of the pyruvate anion produces CO_2, CO, CH_3^-, CH_3,
  and a low energy electron. Nature Communications 2022,
  13, 937
[Prlj et al.(2023)Prlj, Hollas, and
  Curchod]prlj2023deciphering
Prlj, A.; Hollas, D.; Curchod, B. F. E. Deciphering the Influence of
  Ground-State Distributions on the Calculation of Photolysis Observables.
  The Journal of Physical Chemistry A 2023, 127,
  7400–7409
[Hutton and Curchod(2022)Hutton, and Curchod]hutton2022
Hutton, L.; Curchod, B. F. E. Photodynamics of Gas-Phase Pyruvic Acid Following
  Light Absorption in the Actinic Region. ChemPhotoChem 2022,
  6, e202200151
[Marsili et al.(2022)Marsili, Prlj, and
  Curchod]marsili2022theoretical
Marsili, E.; Prlj, A.; Curchod, B. F. E. A theoretical perspective on the
  actinic photochemistry of 2-hydroperoxypropanal. The Journal of
  Physical Chemistry A 2022, 126, 5420–5433
[Persico and Granucci(2014)Persico, and Granucci]persico2014overview
Persico, M.; Granucci, G. An overview of nonadiabatic dynamics simulations
  methods, with focus on the direct approach versus the fitting of potential
  energy surfaces. Theoretical Chemistry Accounts 2014,
  133, 1–28
[Suchan et al.(2018)Suchan, Hollas, Curchod, and
  Slavíček]suchan2018importance
Suchan, J.; Hollas, D.; Curchod, B. F. E.; Slavíček, P. On the Importance of
  Initial Conditions for Excited-State Dynamics. Faraday Discussions
  2018, 
[Ceriotti et al.(2009)Ceriotti, Bussi, and
  Parrinello]ceriotti2009nuclear
Ceriotti, M.; Bussi, G.; Parrinello, M. Nuclear quantum effects in solids using
  a colored-noise thermostat. Physical Review Letters 2009,
  103, 030603
[Ceriotti et al.(2010)Ceriotti, Bussi, and
  Parrinello]ceriotti2010colored
Ceriotti, M.; Bussi, G.; Parrinello, M. Colored-noise thermostats à la
  carte. Journal of Chemical Theory and Computations 2010,
  6, 1170–1180
[Huppert et al.(2022)Huppert, Plé, Bonella, Depondt, and
  Finocchi]Finocchi2022
Huppert, S.; Plé, T.; Bonella, S.; Depondt, P.; Finocchi, F. Simulation of
  Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths.
  Applied Sciences 2022, 12
[Markland and Ceriotti(2018)Markland, and
  Ceriotti]markland2018nuclear
Markland, T. E.; Ceriotti, M. Nuclear quantum effects enter the mainstream.
  Nature Reviews Chemistry 2018, 2, 0109
[Hollas and Curchod(2024 (accessed July 2024))Hollas, and
  Curchod]daniel_hollas_2024_11075300
Hollas, D.; Curchod, B. F. E. AtmoSpec. 2024 (accessed July 2024);
  <https://doi.org/10.5281/zenodo.11075300>
[Weininger(1988)]smilescode
Weininger, D. SMILES, a chemical language and information system. 1.
  Introduction to methodology and encoding rules. Journal of Chemical
  Information and Computer Sciences 1988, 28, 31–36
[Riniker and Landrum(2015)Riniker, and Landrum]rdkit:etkdg
Riniker, S.; Landrum, G. A. Better Informed Distance Geometry: Using What We
  Know To Improve Conformation Generation. Journal of Chemical
  Information and Modeling 2015, 55, 2562–2574
[Wang et al.(2020)Wang, Witek, Landrum, and
  Riniker]rdkit:etkdgv3
Wang, S.; Witek, J.; Landrum, G. A.; Riniker, S. Improving Conformer Generation
  for Small Rings and Macrocycles Based on Distance Geometry and Experimental
  Torsional-Angle Preferences. Journal of Chemical Information and
  Modeling 2020, 60, 2044–2058
[rdk()]rdkit
RDKit: Open-source cheminformatics. https://www.rdkit.org
[Halgren and Nachbar(1996)Halgren, and Nachbar]MMFF94-IV
Halgren, T. A.; Nachbar, R. B. Merck molecular force field. IV. conformational
  energies and geometries for MMFF94. Journal of Computational Chemistry
  1996, 17, 587–615
[Neese et al.(2020)Neese, Wennmohs, Becker, and
  Riplinger]orcarev
Neese, F.; Wennmohs, F.; Becker, U.; Riplinger, C. The ORCA quantum chemistry
  program package. The Journal of Chemical Physics 2020,
  152, 224108
[Mapelli et al.(2023)Mapelli, Donnelly, Hogan, Rickard,
  Robinson, Byrne, McElroy, Curchod, Hollas, and Dillon]acp-23-7767-2023
Mapelli, C.; Donnelly, J. K.; Hogan, U. E.; Rickard, A. R.; Robinson, A. T.;
  Byrne, F.; McElroy, C. R.; Curchod, B. F. E.; Hollas, D.; Dillon, T. J.
  Atmospheric oxidation of new “green” solvents – Part 2: methyl pivalate
  and pinacolone. Atmospheric Chemistry and Physics 2023,
  23, 7767–7779
[Merkel(2014)]merkel2014docker
Merkel, D. Docker: lightweight linux containers for consistent development and
  deployment. Linux journal 2014, 2014, 2
[Wilkinson et al.(2016)Wilkinson, Dumontier, Aalbersberg,
  Appleton, Axton, Baak, Blomberg, Boiten, da Silva Santos, Bourne, Bouwman,
  Brookes, Clark, Crosas, Dillo, Dumon, Edmunds, Evelo, Finkers,
  Gonzalez-Beltran, Gray, Groth, Goble, Grethe, Heringa, 't Hoen, Hooft, Kuhn,
  Kok, Kok, Lusher, Martone, Mons, Packer, Persson, Rocca-Serra, Roos, van
  Schaik, Sansone, Schultes, Sengstag, Slater, Strawn, Swertz, Thompson,
  van der Lei, van Mulligen, Velterop, Waagmeester, Wittenburg, Wolstencroft,
  Zhao, and Mons]Wilkinson:2016aa
Wilkinson, M. D.; Dumontier, M.; Aalbersberg, I. J.; Appleton, G.; Axton, M.;
  Baak, A.; Blomberg, N.; Boiten, J.-W.; da Silva Santos, L. B.; Bourne, P. E.
  et al.  The FAIR Guiding Principles for scientific data management
  and stewardship. Scientific Data 2016, 3, 160018
[Uhrin et al.(2021)Uhrin, Huber, Yu, Marzari, and
  Pizzi]UHRIN2021110086
Uhrin, M.; Huber, S. P.; Yu, J.; Marzari, N.; Pizzi, G. Workflows in AiiDA:
  Engineering a high-throughput, event-based engine for robust and modular
  computational workflows. Computational Materials Science
  2021, 187, 110086
[Huber et al.(2020)Huber, Zoupanos, Uhrin, Talirz, Kahle,
  Häuselmann, Gresch, Müller, Yakutovich, Andersen, Ramirez, Adorf,
  Gargiulo, Kumbhar, Passaro, Johnston, Merkys, Cepellotti, Mounet, Marzari,
  Kozinsky, and Pizzi]Huber:2020aa
Huber, S. P.; Zoupanos, S.; Uhrin, M.; Talirz, L.; Kahle, L.;
  Häuselmann, R.; Gresch, D.; Müller, T.; Yakutovich, A. V.;
  Andersen, C. W. et al.  AiiDA 1.0, a scalable computational
  infrastructure for automated reproducible workflows and data provenance.
  Scientific Data 2020, 7, 300
[Yakutovich et al.(2021)Yakutovich, Eimre, Schütt, Talirz,
  Adorf, Andersen, Ditler, Du, Passerone, Smit, Marzari, Pizzi, and
  Pignedoli]aiidalab2021
Yakutovich, A. V.; Eimre, K.; Schütt, O.; Talirz, L.; Adorf, C. S.;
  Andersen, C. W.; Ditler, E.; Du, D.; Passerone, D.; Smit, B. et al.
  AiiDAlab – an ecosystem for developing, executing, and sharing scientific
  workflows. Computational Materials Science 2021, 188,
  110165
[Sršeň and Slavíček(2021)Sršeň, and
  Slavíček]optimalsampling2021
Sršeň, Š.; Slavíček, P. Optimal Representation of the
  Nuclear Ensemble: Application to Electronic Spectroscopy. Journal of
  Chemical Theory and Computation 2021, 17, 6395–6404
[Arbelo-González et al.(2016)Arbelo-González,
  Crespo-Otero, and Barbatti]doi:10.1021/acs.jctc.6b00704
Arbelo-González, W.; Crespo-Otero, R.; Barbatti, M. Steady and
  Time-Resolved Photoelectron Spectra Based on Nuclear Ensembles. Journal
  of Chemical Theory and Computation 2016, 12,
  5037–5049
[Muchova et al.(2023)Muchova, Hollas, Holland, Bacellar, Leroy,
  Barillot, Longetti, Coreno, de Simone, Grazioli, Chergui, and
  Ingle]D2CP05299G
Muchova, E.; Hollas, D.; Holland, D. M. P.; Bacellar, C.; Leroy, L.;
  Barillot, T. R.; Longetti, L.; Coreno, M.; de Simone, M.; Grazioli, C.
  et al.  Jahn–Teller effects in initial and final states:
  high-resolution X-ray absorption, photoelectron and Auger spectroscopy of
  allene. Physical Chemistry Chemical Physics 2023, 25,
  6733–6745





 Paper Title: Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension.




Line tension from dual-geometry sessile droplet measurements]Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension


d.tatyanenko@spbu.ru



Department of Statistical Physics, Faculty of Physics, Saint Petersburg State University
  7–9 Universitetskaya nab., St. Petersburg, 199034, Russia



§ ABSTRACT 
We perform a thermodynamic analysis of various contributions to the size dependence of the contact angle cosine for axisymmetric and cylindrical sessile droplets. This shows that a widely used method to determine the line tension from the slope of the contact angle cosine dependence on the three-phase contact line curvature (for axisymmetric droplets) provides a certain combination of the line tension, adsorptions at the three interfaces, and the macroscopic contact angle. To extract the leading-order contact-line-related contribution and determine the line tension, we propose a simple technique using contact angle size dependences for axisymmetric and cylindrical droplets at the same conditions.




[
    Konstantin D. Apitsin
    August 5, 2024
=========================






§ INTRODUCTION


The contact angle of a large sessile droplet can be found from the well-known classical Young equation <cit.>

    cos =  -

with σ the surface tension of a corresponding interface,  the macroscopic contact angle; α, β, and γ mark the liquid, gaseous and solid phases, respectively, and double Greek superscripts mark the corresponding interfaces. The surface tensions here are thermodynamic (defined as the surface excesses of the grand potential per unit area of the interface). For small droplets, the thermodynamic line tension κ (defined as the line excess of the grand potential per unit length of the three-phase contact line <cit.>) affects the contact angle <cit.>:

    cos =  -  - κ/.

Here θ is the contact angle, r is the radius of the three-phase contact line, subscript “a” refers to axisymmetric droplet (see Fig. <ref>(a)); the line tension κ is implied to be constant. Eq. (<ref>) is usually called the modified Young equation.

Subtracting the modified Young equation (<ref>) from the classical  one (<ref>) and dividing the both parts of the resulting equation by , one usually arrives at <cit.>

    cos - cos = κ/.

This equation is widely used for contact-angle-based measurements of the line tension in laboratory experiments (see, e.g., <cit.>) and computer modeling (e.g., molecular dynamics simulations <cit.>). According to Eq. (<ref>), the slope of the graph cos vs 1/ must be equal to - κ/. The surface tension  is typically measured or calculated separately.

However, some authors have noted that Eq. (<ref>) is generally not correct <cit.>, since it assumes the corresponding surface tensions in Eqs. (<ref>) and  (<ref>) to be equal, whereas they correspond to different equilibrium sessile droplet sizes, which implies different values of the chemical potential(s). There are arguments that adsorption at the solid–liquid interface and its effect on the surface tension  affect cos in the same manner and with a similar magnitude as it is measured in experiments <cit.>, so the line tension itself is not necessary to explain the dependence of cos on 1/. Consideration with use of an interface displacement  model shows <cit.> that the contributions of the line tension and the adsorption at the solid–gas interface (and its effect on the surface tension ) to the  dependence of cos on 1/ can be comparable in the presence of a precursor film. Some authors <cit.> suppose only  to depend on the droplet size and Eq. (<ref>) to be correct, but the slope of the graph cos vs 1/ to give a quantity not equal to - κ/.  Sometimes the resulting “line tension” is called the apparent line tension <cit.>. General thermodynamic analysis shows <cit.> that the value of this slope has contributions both from the line tension and the adsorptions at all three interfaces (see also <cit.> for similar arguments).
             

How to “extract” the line-tension contribution and determine the value of the line tension from the measurements or calculations of cosθ vs 1/r? Thermodynamic analysis shows <cit.> that, for cylindrical droplets [long liquid “channels”, or “filaments”, or “ridges” attached to the substrate surface, see Fig. <ref>(b)], there is no κ/r term in the equation corresponding to (<ref>). It means that dependence of the contact angle  on the size of a cylindrical droplet (, its half-width  on the substrate) is governed by dependences of the surface tensions and the droplet size on the chemical potential(s). This suggests the idea to compare or combine data of size dependence of cosθ for axisymmetric and cylindrical droplets at the same conditions to exclude the surface-tension-related contributions and determine the line tension.

In section <ref>, we present a thermodynamic analysis of various contributions to droplet size dependence of the contact angle for both axisymmetric and cylindrical droplets in a system with a single-component fluid and an ideal smooth homogeneous solid substrate in absence of external fields. Based on it, in section <ref> we propose a simple technique that allows to determine the line tension from a combination of contact angle size dependences for axisymmetric and cylindrical droplets in a manner similar to use of Eq. (<ref>).

Cylindrical droplets are often used in modeling, e.g., in molecular dynamics (MD) simulations <cit.> due to convenient use of periodic boundary conditions along their longitudinal axes and due to expected independence of the contact angle of (or a weak dependence on) the droplet width. However, they are hardly available for experimental measurements due to Plateau–Rayleigh-type instability <cit.>. Thus, the technique we propose is mainly oriented to modeling, especially with methods not allowing for direct calculation of the line excess of the grand potential (e.g., MD simulations). The dimensions of such simulated axisymmetric and cylindrical droplets, i.e., R, r, and θ, can be “measured”, and these data can be used to determine the line tension.





§ THERMODYNAMICS AND CONTACT ANGLES





 §.§ Equilibrium conditions for sessile droplets

For thermodynamic description of a small sessile droplet on a partially wettable substrate, we consider a materially open system of a fixed volume including the droplet itself, the substrate, and the surrounding gas (vapor). The liquid and gaseous phases are supposed to be a single-component fluid at a given values of chemical potential μ and temperature T (that supposed to be constant).  The grand thermodynamic potential of the system decomposes into bulk, surface, and line parts:

    Ω =    - p^α - p^β + ω^γ V^γ
        +   +  +  + κ L

with V the volume, A the area of the interface, L the length of the three-phase contact line, p the pressure in the bulk phase, ω the bulk density of the grand thermodynamic potential (in fluid phases equals -p). Other quantities as well as notations of phases and interfaces has been explained above.

Let us consider two types of droplet geometry: (1) axisymmetric droplet [shaped as a spherical segment in our “macroscopic” thermodynamic description, see Fig. <ref>(a)] and (2) a fixed-length piece of a cylindrical droplet [shaped as a segment of a circular cylinder in our “macroscopic” thermodynamic description, see Fig. <ref>(b)]. In both geometries, the cross-section of the droplet in our “macroscopic” thermodynamic description is a circular segment, therefore,

    r = R sinθ.

Thus, the shape and size of the droplet can be described by two independent variables, e.g., (R, r), (R, θ) or (r, θ).

In both geometries, choosing the  dividing surface to be the surface of tension, one can obtain equilibrium conditions for sessile droplets by expressing all the volumes, areas and the length of the three-phase contact line in (<ref>), and solving one of the following sets of two equations


    ∂Ω(R,r,T,μ)/∂ R = 0     and     ∂Ω(R,r,T,μ)/∂ r = 0,
       ∂Ω(R,θ,T,μ)/∂ R = 0    and     ∂Ω(R,θ,T,μ)/∂θ = 0, 
       ∂Ω(r,θ,T,μ)/∂ r = 0    and     ∂Ω(r,θ,T,μ)/∂θ = 0,


and then taking into account the geometric relation (<ref>). Since T and μ are constant, the pressures and the surface tensions in (<ref>) will be constant at such variations (for , this also requires / R = 0 which is true namely for the surface of tension chosen as the  dividing surface).

The droplet-size-dependent surface tensions  and  for spherical and cylindrical interfaces can generally differ, though their difference is expected to be very small at the same value of μ (see subsection <ref>).


Let us chose R and θ as independent variables describing the sessile droplet. Introducing non-variable quantities for the total volume occupied by the fluid ≡ V^α + V^β and the total area of the substrate ≡ + and using relation (<ref>), one can express the volumes, areas, and the contact line length for an axisymmetric droplet as

    = π^3/3 (2 + cos)(1 - cos)^2,
       = 2πsin, 
     = 2π^2 (1 - cos),
       = π^2 sin^2 ,
     =  - ,
       =  - .


Decomposition (<ref>) of the grand potential then takes form

    (,) = - ( p^α - p^β) π^3/3 (2 + cos)(1 - cos)^2 
        + 2 π^2 (1 - cos) + ( - ) π^2 sin^2 
        + 2πsin - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.5cm]
    - π^2 ( p^α - p^β) (2 + cos) (1 - cos)^2 
     + 4 π (1 - cos) + 2 π( - ) sin^2 
     + 2 πsin + 2 π∂/∂sin^2  = 0,
    
          - π^3  ( p^α - p^β) sin^3  + 2 π^2 sin + 2 πcos
        + 2 π^2 (  - ) sincos + 2 π^2 ∂/∂sincos = 0.

To exclude the pressure difference p^α - p^β, we multiply the first equation by (2 π)^-1sin^2   (1 - cos)^-2 and then subtract the second equation multiplied by (2 + cos)(2 π^2 sin)^-1 <cit.>. After some trigonometric simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an axisymmetric sessile droplet <cit.>

    p^α - p^β = 2/, 
    cos =  -  - / - /


For a cylindrical droplet,

    = ^2/4 (2  - sin 2),
     = ,
       = sin,
     =  - ,
       =  - ,

and the length  of the three-phase contact line does not depend on  and  [see Fig. <ref>(b)] and supposed to be constant. Contact angle  here is in radians. Decomposition (<ref>) of the grand potential then  takes form

    (,)    = - ( p^α - p^β) ^2/4  (2  - sin 2) + 
        + ( - ) sin +  - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.1cm]
    - ( p^α - p^β)  ( - sincos) + 
     + ( - ) sin + ∂/∂sin = 0,
    
       [t][8.1cm]
    - ( p^α - p^β) ^2 sin^2  + 
      +  ( - ) cos + ∂/∂cos = 0.

Here we have employed trigonometric equalities  sin 2 = 2 sincos and 1 - cos 2 = 2 sin^2. To obtain a simpler equation for p^α - p^β, we multiply the first equation by cos and subtract the second equation multiplied by sin. After some simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an cylindrical sessile droplet, slightly different from Eqs. (<ref>) and (<ref>):

    p^α - p^β = /, 
    cos =  -  - /.


The absence of the coefficient 2 in the Laplace equation  (<ref>) for the cylindrical droplet can be explained by the fact that its  interface is curved only along one direction, whereas the spherical droplet has equal curvatures along two directions. The line-tension terms in the generalized Young equations (<ref>) and (<ref>) originate from variations of the line term κ L in (<ref>). For an axisymmetric droplet, = 2π and ()/ = 2π + 2π / giving the two terms / + / in Eq. (<ref>). For a cylindrical droplet,  does not depend on  and ()/ =  / giving the only term / in Eq. (<ref>).

For both geometries, we suppose the line tension to depend not only on T and μ, but also on the droplet size. This dependence is usually overlooked and, in this case, Eq. (<ref>) transforms into the modified Young equation (<ref>), whereas Eq. (<ref>) transforms into formally classical Young equation (<ref>) (with surface tensions, however, depending on the chemical potential, i.e. on the equilibrium droplet size, that makes the contact angle  different from , see subsection <ref>).

We suppose the contact line radius (half-width) r to be the only appropriate size variable the line tension can be considered to depend on. It is not clear for the derivation we used in this paper, and some authors suppose the line tension to depend also on the contact angle θ <cit.>, corresponding “notional” derivatives also appear in the work <cit.>. However, a variational derivation of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), which considers all three variables R, r, and θ to be independent in some sense, gives an argument in favor of r as the mathematically appropriate size variable for κ in both geometries <cit.>.

Simple dropping the size-dependent term κ/ r in the generalized Young equations (<ref>) and (<ref>) make them inexact (even if κ is supposed to depend on the droplet size). It can be seen in a model system if the line tension is calculated directly <cit.>. Also, the generalized Young equation is the simplest way to calculate κ/ r if θ and r are found and the surface and line tensions (the latter for axisymmetric droplet only) are calculated directly <cit.>.




 §.§ Size dependence of contact angle and relations between pressures, surface tensions, droplet size and chemical potential


Comparing the generalized Young equation (<ref>) or (<ref>) with the classical Young equation (<ref>), one should take into account that these two equations are written for droplets corresponding to different values of the chemical potential μ. The classical Young equation corresponds to a macroscopically large (strictly speaking, infinite) drop, i.e. liquid–gas bulk phase coexistence at the binodal. Let us mark the values of the quantities corresponding to the binodal with subscript “∞”. The surface tensions at the binodal then should be denoted as , , and , and the classical Young equation (<ref>) should be rewritten as

    cos =  - .


Subtracting the generalized Young equations (<ref>) and
(<ref>) from  Eq. (<ref>) we obtain, for an axisymmetric droplet,

    cos - cos = δ + / + / r
    and a similar equation for a cylindrical droplet:cos - cos = δ + /.

Here δ≡ - with ≡ - and ≡ -. To make these equations more similar to Eq. (<ref>), let us introduce δ≡ -. Then, Eqs. (<ref>) and (<ref>) can be rewritten as

    ( cos - cos ) = δ + δcos + / + /, 
        ( cos - cos) = δ + δcos + /.


Now we see, that, in the exact counterpart of Eq. (<ref>) for the difference cos - cosθ, we have 4 terms for an axisymmetric droplet (instead of 1) and 3 terms for a cylindrical droplet (instead of 0). How do they depend on the droplet size [the radius (half-width) of the droplet on the substrate]?

For the  liquid–gas interface, let us write down the isothermal Gibbs adsorption equations:

    = - μ,

where  is the adsorption (the surface excess of matter per unit surface area) at the  interface for corresponding geometry. Since the surface of tension is chosen as the dividing surface in both geometries, these equations do not contain (/ R)  R terms.

For each of the solid–fluid interfaces, similar equations can be written. Generally, they are more complicated (see, e.g., Sec. 6.3 in <cit.> and Sec. 3.4.3 in <cit.>), but in some simple cases (e.g., for non-deformable substrate and such a choice of the solid–fluid dividing surfaces that the amounts of matter of the immobile component of solid do not change in the solid phase at any change of μ) they take <cit.> the same simple form as (<ref>), so we can write down

    ≡(  - ) = (  - ) μ.

Integrating Eqs. (<ref>) and (<ref>) from the value  at the binodal to the current value of μ, we obtain, introducing δμ≡μ -,

    δ = - ∫_^μ(μ') μ' ≃ - δμ,  
    δ = ∫_^μ(  - ) μ' ≃(  - ) δμ.

Since () = () =, the asymptotic expression in (<ref>) is the same for spherical and cylindrical surfaces at the same value of μ, but the exact integral is generally not if (μ) ≠(μ). Thus,  and  generally differ, but at least in the second order in δμ:

    -  = δ - δ = ((δμ)^2).


The last asymptotic expressions in Eqs. (<ref>) and (<ref>) are valid for small values of δμ. How is this quantity related to the droplet size? This can be estimated using the Laplace equation (<ref>) or (<ref>) for the axisymmetric or cylindrical droplet, respectively. Applying the isothermal Gibbs–Duhem equations p = n μ to liquid and gaseous phases, respectively, we obtain p^α - p^β≃δμ with  the number density of molecules in α liquid phase. Here we have neglected the compressibility of the liquid (≃) and the number density of molecules  in the gas (≪). With use of the Laplace equations (<ref>) and (<ref>), and geometric relation (<ref>), we obtain

    2/ = 2sin/≃δμ, 
    / = sin/≃δμ.

We now see that δμ∝ 1/R = (1/r), i.e., small values of δμ correspond to large droplets, and δμ→ 0 expectedly corresponds to R →∞ and r →∞.

Thus, for the correction terms in Eqs. (<ref>) and (<ref>) we have

    δ≃2sin/(  - )  = (δμ) = (1/r), 
    δ≃ - 2sin/ = (δμ) = (1/r), 
    /≃/ = (δμ) = (1/r), where = at μ = (→∞) is the line tension of the macroscopic (infinite) droplet, andδ≃sin/(  - ) = (δμ) = (1/r), 
    δ≃ - sin/ = (δμ) = (1/r).


As for the terms κ/ r, it can be shown <cit.> that they are ( (δμ)^2 ) = ( 1/r^2 ). For sufficiently large droplets, sinθ can be approximated with sin, with a relative error (1/r), that makes Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) valid with sin replaced by sin. The same applies to cos in the correction terms δcosθ in Eqs. (<ref>) and (<ref>).

Taking this into account, we can now write down the linear in 1/r approximations for Eq. (<ref>) and (<ref>):

    cos - cos≃/, 
    cos - cos≃/,




where

    ≡ + 2sin/(  -  - cos) 
    and   ≡sin/(  -  - cos)

are the quantities sometimes called the apparent line tensions <cit.> of axisymmetric and cylindrical droplets, respectively (see below).

Eqs. (<ref>) and (<ref>) show that the dependence of cos on 1/, even in the first order in 1/, is determined not only by the line tension, but also by the adsorptions at the three interfaces. In fact, Eq. (<ref>) should be treated as Eq. (<ref>) often is. This means that the slope of the graph cos vs 1/ at 1/→ 0 does not give the line tension itself but rather a combination of the line tension, the surface tension of the droplet free surface, adsorptions at the three interfaces, and the macroscopic contact angle. Some authors use the term “apparent line tension” meaning the value of the line tension calculated from the dependence of cosθ on 1/r (denoted further as cosθ (1/r)) at 1/r → 0 as if Eq. (<ref>) were the correct linear approximation. Then Eq. (<ref>) gives the value of the apparent line tension of axisymmetric droplets. More interesting, Eqs. (<ref>) and (<ref>) show that, for cylindrical droplets, cos also depends on 1/ in the first order in 1/. This has been found in simulations <cit.> and attributed to size dependence of the surface tension  expressed using the Tolman length <cit.> ≈/.  As it is seen from Eq. (<ref>), dependences of other surface tensions on the droplet size via the chemical potential could also made their contributions to the apparent line tension of cylindrical droplets.




§ LINE TENSION FROM DUAL-GEOMETRY CONTACT ANGLE DEPENDENCES ON DROPLET SIZE


One can easily notice that the apparent line tension (<ref>) of cylindrical droplets, proportional to the slope of the cos(1/)  graph at 1/→ 0, is exactly half of the second, adsorption-related, term in the apparent line tension (<ref>) of axisymmetric droplets, proportional to the slope of the cos(1/) graph at 1/→ 0. Thus, combining data for dependences cos(1/) and cos(1/), one can determine the line tension of a macroscopic droplet as

    =  - 2.


Using this relation, we have calculated the line tension  of a macroscopic droplet from the apparent line tensions  and  of sessile droplets simulated by Kanduč et al. <cit.> using data from Fig. 2(c) in <cit.>. As one can see in Fig. <ref>, the values of  calculated with Eq. (<ref>) are in good agreement with ones obtained in <cit.> by taking into account Tolman length <cit.> and “line tension stiffness” <cit.> corrections. It is clearly seen, that the values of  and  are comparable in these simulations that makes  visibly different from . In contrast, simulations of Isaiev et al. <cit.> show very weak dependence of cos(1/) comparing to cos(1/) (see Fig. 4 therein). This corresponds to ≪ and, thus, ≈.

             

Eqs. (<ref>)–(<ref>) suggest another idea. With dual-geometry sessile droplet measurements, it is not necessary to use the classical Young equation at all. Subtracting the generalized Young equation for axisymmetric droplets (<ref>)  from the one for cylindrical droplets (<ref>) we obtain

    cos - cos = / + / - /.

Taking into account Eqs. (<ref>), (<ref>) with subsequent comments and κ/ r = ( 1/^2 ), the first-order approximation to the combined equation (<ref>) will be

    cos - cos≃/.

Here dependences of  on μ and  and of  on μ are no longer taken into account since they give corrections ( 1/^2 ).

All the quantities in Eqs. (<ref>) and (<ref>) must be taken at the same values of the chemical potential and temperature. The chemical potential is often unknown in simulations or measurements. But, according to Eqs. (<ref>) and (<ref>) and taking into account Eq. (<ref>), the same value of δμ corresponds to the relation ≃/2 or, in the same order in δμ, to ≃/2. The former relation is expectedly more accurate, since it relies only on the Laplace equations (<ref>) and (<ref>), whereas the latter also uses approximation sin≃sin.

Thus, to determine the line tension from the measured (in the experiment or modeling) dependences of the cosθ on 1/r for both axisymmetric and cylindrical droplets at the same conditions, it is sufficient to calculate the difference

    cos|_ = /2 - cos|_  or  cos|_ = /2 - cos|_

and take a linear fit of it vs 1/ at sufficiently large values of . The slope of this graph will give /:

    cos|_ = /2 - cos|_≃/, 
    cos|_ = /2 - cos|_≃/.


An additional advantage of this technique is that the graph of the difference cos - cos vs 1/ must start from a known point (0 at 1/ = 0), thus the slope of the linear graph remains the only fitting parameter. When plotting cos(1/) and cos(1/) independently, each of them must be approximated by a straight line starting from an unknown point cos(0), which must be the same for axisymmetric and cylindrical droplets (and this, in principle, should be somehow taken into account to reduce the number of fitting parameters from 4 to 3). Unlike experimental measurements, droplet simulations require more computational resources as the droplet size increases, so approaching droplet sizes whose contact angles is macroscopic can be resource demanding. However, macroscopic contact angle , can be found, e.g., using the classical Young equation (<ref>) and calculating  and  in three simulations of  planar interfaces <cit.>.


Experiments with nanosized droplets suggest another possible advantage of using dual-geometry techniques to determine the line tension. For nanosized droplets, the graph cos vs 1/ can be fairly linear but this linear fit may work well only for small droplets and give a false value of cos at extrapolation down to 1/→ 0 (see Fig. 5 in <cit.>) and the slope that does not correspond to asymptotic Eq. (<ref>). In many simulations, the droplets are also nanosized, so the same problem can be with simulated droplets, both axisymmetric and cylindrical, giving false values of  and . If Eq. (<ref>) is then used, this would ultimately give a wrong value of . However, not only the slope would be wrong in such situation, but, most probably, the extrapolated values of cos would be different for cos vs 1/ and cos vs 1/ indicating an inconsistency. With use of the technique we proposed, such a situation would appear as a set of points of the quantity (<ref>) vs 1/ that do not fit a straight line starting from (0,0). Both such indicators would simply mean that the droplets are not large enough for the asymptotic Eqs. (<ref>), (<ref>), and (<ref>) to work and, thus, cannot be used to determine the contact angle  and the line tension  of macroscopic droplets.




§ OVERVIEW OF RESULTS AND DISCUSSION


Thermodynamic analysis we have performed shows that droplet size corrections to the contact angle cosine for axisymmetric sessile droplets attributed often to the line tension only, are determined generally by the line tension , its dependence on the droplet size (directly on the droplet radius  on the surface and via the chemical potential) and also by dependences of all three surface tensions on the droplet size (via the chemical potential). The same applies to corrections to the contact angle cosine for cylindrical sessile droplets, except for the line tension term / that is absent in the generalized Young equation (<ref>) for cylindrical droplets.

In the first order in 1/ (<ref>), this gives the slope of cos vs 1/ to be proportional not to the line tension  of macroscopic droplets [as it is widely believed according to Eq. (<ref>)] but rather to the so called “apparent line tension” , a sum of this line tension  and a combination of adsorptions at the three interfaces, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>).

For cylindrical droplets, the contact angle does also depend on the droplet half-width  on the surface already in the first order in 1/ (<ref>), giving the apparent line tension . It is determined by the same combination of the adsorptions, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>) but lacks the numerical coefficient 2 compared to the axisymmetric case (<ref>).

This gives a possibility to determine the line tension  from  and  found in dual-geometry measurements (e.g., for droplets simulated at the same conditions, i.e. at the same temperature with the same interaction potentials, etc.) as (<ref>). It was illustrated with calculations of  from  and  of MD-simulated droplets <cit.> (see Fig. <ref>).

We have also proposed a technique for  dual-geometry measurements based on calculation of the slope of the difference cos - cos at the same value of the chemical potential vs 1/. It can be implemented as a calculation of this difference (<ref>) at = /2 or = /2. The slope will equal / with  the surface tension of the planar liquid–gas interface. This technique has some advantages discussed in section <ref>.

The Laplace and the generalized Young equations in both geometries, as well as the Gibbs adsorption equation (<ref>) for the liquid–gas interface, were obtained and considered for the surface of tension as the liquid–gas dividing surface. At an arbitrary choice of this dividing surface, all these equations are more complicated <cit.>. Even though most researchers use these equations in their simple forms, the real choice of the dividing surface is often different or even not specified explicitly. In simulations, it can be done, e.g., by a boundary detection algorithm, and different algorithms can give <cit.> visibly different dependences of cosθ vs 1/r. This can be considered as an additional source of errors that need to be estimated.

As for the solid–liquid dividing surfaces, we have used the analogue of the Gibbs adsorption equation (<ref>) for the difference ≡ - written under some conditions that are not necessarily met at an arbitrary choice of these dividing surfaces. However, if the same choice is made in both geometries, these quantities will still cancel, both with use of Eq. (<ref>) and in the proposed technique where Eq. (<ref>) will remain valid. In contrast, the line tensions  and  (including their macroscopic value ) do generally depend on the choice of the solid–fluid dividing surfaces. For any specific choice, the dual-geometry approach should work, giving the value corresponding to chosen dividing surfaces.

Finally, let us discuss if the proposed technique can be easily adapted to determine the size-dependent line tension  and/or  (not only their limit  at r →∞). Eq. (<ref>) does not contain solid–fluid surface tensions, but it contains the surface tensions . They can be calculated separately for free spherical/cylindrical droplets of the same radius  of the liquid–gas surface as for the sessile droplet of the same type of geometry (thus, at the same value of chemical potential). However, since (μ,) and (μ,) are different functions, the expression / - / on the right hand side of Eq. (<ref>) is still ( (δμ)^2 ) = ( 1/^2 ) and, therefore, cannot be neglected even in the second order in 1/. Thus, Eq. (<ref>) can hardly be used to obtain more detailed information on the line tension from dual-geometry measurements of cosθ(1/r).



The authors thank  Volkov and  Polovinkin for valuable discussions and comments on the manuscript. The research has been carried out with financial support of the Russian Science Foundation, grant no. 22-13-00151, https://rscf.ru/en/project/22-13-00151/https://rscf.ru/en/project/22-13-00151/






 Paper Title: Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers.

 





 ./figures/ 

















 
















custompmatrix
  ([ )


 




















theoremTheorem
lemma[theorem]Lemma
claimClaim
corollary[theorem]Corollary
definitionDefinition
lemma1Lemma
exampleExample
conditionCondition


APS/123-QED 
]Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India

Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Graduate School of Science and Technology, Keio University, 7-1 Shinkawasaki, Saiwai-ku, Kawasaki, Kanagawa 212-0032, Japan
Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

Institute for Advancing Intelligence, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Academy of Scientific and Innovative Research (AcSIR), Ghaziabad- 201002, India



srinivasaprasannaa@gmail.com
Centre for Quantum Engineering, Research and Education, TCG CREST, Sector V, Salt Lake, Kolkata 700091, India
Academy of Scientific and Innovative Research (AcSIR), Ghaziabad- 201002, India


§ ABSTRACT 
Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form Ax⃗=b⃗ on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number (κ) scaling of the A matrix. In this work, we address the issue by introducing the post-selection-improved HHL (Psi-HHL) approach that involves a simple yet effective modification of the HHL algorithm to extract a feature of |x⟩, and which leads to achieving optimal behaviour in κ (linear scaling) for large condition number situations. This has  the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. We carry out two sets of simulations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large κ matrices via: (a) a set of toy matrices, for which we go up to size 64 × 64 and κ values of up to ≈ 1 million, and (b) a deep-dive into quantum chemistry, where we consider matrices up to size 256 × 256 that reach κ of about 466. The molecular systems that we consider are Li_2, RbH, and CsH. Although the feature of |x⟩ considered in our examples is an overlap between the input and output states of the HHL algorithm, our approach is general and can be applied in principle to any transition matrix element involving |x⟩. 


[
    V. S. Prasannaa
    August 5, 2024
===================







§ INTRODUCTION


The Harrow-Hassidim-Lloyd (HHL) algorithm was proposed to solve equations of the form Ax⃗=b⃗, where A is a known and sparse 2^N × 2^N matrix and b⃗ is 
a 2^N × 1 vector, on quantum computers with a time complexity that is exponentially lower in system size than those of the best known classical methods <cit.>. Further, via amplitude encoding, b⃗, which is a 2^N × 1 vector, is encoded into a log_22^N-qubit state (for brevity, we shall hereafter denote 2^N as n). Thus, when the HHL approach is applied to `killer applications' of quantum computing such as quantum chemistry, there is a natural exponential suppression in qubit number relative to the other known approaches such as the quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) algorithms <cit.>, thus opening new avenues to apply HHL for calculating properties of large molecules. 


Despite possessing the aforementioned advantages, the HHL algorithm relies upon the probability of successfully post-selecting 1 on the HHL ancilla qubit, that is, P(1) <cit.>. It is known that there is a substantial decrease in P(1) with increase in the condition number of the problem matrix A, which we shall denote by κ. In other words, the fraction of usable shots in the HHL algorithm becomes very low when large κ input matrices are involved. Unfortunately, it is not uncommon to encounter the issue while employing HHL for some interesting problems (for example, see Ref. <cit.> for application of the algorithm to hydrological systems), including quantum chemistry, which we shall discuss in detail in our Results section. Several works in literature have addressed the issue of handling large κ matrices either directly (where κ of the A matrix itself is altered via approaches such as preconditioning) or indirectly <cit.>. For a thorough survey, we direct the reader to Section 1.2 of Ref. <cit.>. 
 

In this work, we propose a new method to address the issue of handling A matrices with large condition numbers, by introducing the post-selection-improved HHL (Psi-HHL) algorithm, which relies upon the intuition of extracting the correct signal by subtracting a wrong signal from a mixed signal (that contains both the correct and wrong signals). A signal, in the context of our work, refers to a scalar that is a feature of |x⟩. For the purposes of this work, we pick the overlap between the input and output states of HHL, which in the case of molecules corresponds to their correlation energies (up to some normalization constants), as our signal, although it is to be noted that Psi-HHL itself is applicable to features as general as transition matrix elements. The wrong signal is obtained by successfully post-selecting outcome bit `0' instead of `1' (we call this approach HHL_1). The mixed signal is generated in an HHL quantum circuit by introducing an appropriately chosen 2 × 2 unitary just before measurement of the HHL ancilla (we name this procedure HHL_2). Figure <ref> presents schematic diagrams outlining the workings of the HHL and the Psi-HHL algorithms. 
 

This simple modification involving execution of two HHL routines enables one to carry out calculations for treating systems of linear equations involving large-κ matrices without having to contend with a large number of shots. In fact, we argue that the condition number scales linearly with system size with our approach for large κ situations, while still staying in the realm of a typical HHL computation. This compares favourably with several other notable works in literature, including but not limited to the original HHL work: 𝒪(κ^3) and 𝒪(κ^2) without and with amplitude amplification respectively <cit.>, 𝒪(κlog ^3κ) with variable time amplitude amplification <cit.>, and works by Childs et al: 𝒪(κpolylog (κ)) <cit.>, Wossnig et al: 𝒪(κ^2), and Costa et al's discrete quantum adiabatic approach based quantum linear solver algorithm <cit.>: 𝒪(κ). In particular, the last approach also achieves linear scaling in condition number, but we do so without invoking any additional algorithmic machinery  beyond the scope of the traditional HHL algorithm. In order to provide a very qualitative perspective, we note that for condition number of about 1 million, κ^3 : κ^2 : κlog^3κ : κ is ∼ 10^18: 10^12: 7.9 × 10^9: 10^6. For completeness, we add that the best known classical approach, conjugate gradient, scales as 𝒪(√(κ)) <cit.>, but its scaling with respect to system size is, of course, exponentially more expensive than that for HHL. 
 

The paper is organized as follows: Section <ref> presents the theory and methodology involved in our work by introducing the Psi-HHL approach, including a discussion on the complexity theoretic aspects (in κ) of Psi-HHL. Section <ref> presents our results for Psi-HHL applied to (a) suitably chosen sets of 4 × 4, 8 × 8, 16 × 16, 32 × 32, and 64 × 64 toy matrices, as well as for (b) molecular calculations (64 × 64 and 256 × 256 A matrix sizes) in the linearized coupled cluster (LCC) HHL and Psi-HHL frameworks. We conclude in Section <ref>. 
 

 



§ THEORY AND METHODOLOGY




 §.§ The HHL algorithm


We briefly sketch the relevant details of the HHL algorithm (we refer the readers to Refs. <cit.> for detailed descriptions of the algorithm) in Figs. <ref>(a) and <ref>(b), where an amplitude-encoded input b⟩ = ∑_i b_i ν_i ⟩, such that ∑_i |b_i|^2 = 1 where {ν_i ⟩} refers to the eigenbasis of A, is fed into the HHL algorithm along with n_r number of clock register qubits and one HHL ancillary qubit. The HHL algorithm itself involves QPE, followed by a non-trivial controlled-rotation module, and then a QPE^† module. At the end of the circuit, we have the following state vector,


    ∑_i b_i (√(1-C^2/λ̃^2_i)|0⟩ + C/λ̃_i|1⟩) ⊗|0^n_r⟩⊗|ν_i⟩.
 

In the above expression, {λ̃_i} refers to the estimates of the actual eigenvalues of A (1 ≥λ̃_i ≥ 1/κ) obtained via the QPE module of HHL. Typically, C is chosen to be the minimum eigenvalue of A, but it is worth noting that incurring such a large classical overhead can always be circumvented via approaches such as AdaptHHL <cit.>. When one post-selects the outcome `1' on the HHL ancilla qubit, one can think of it as choosing the `correct' signal. 
 

The solution vector |x⟩∝ A^-1 |b ⟩, and the input vector |b⟩ can be written as,
  
    |x⟩=|x⟩_ un/|x⟩_ un and |b⟩=|b⟩_ un/|b⟩_ un
 
respectively, where subscript `un' refers to unnormalized states— |b⟩_ un is the unnormalized input state after amplitude encoding and |x⟩_ un=∑_i b_i C/λ̃_i|ν_i ⟩, with ∑_i |b_i|^2 = 1. The probability of successfully post-selecting outcome `1' is given as P(1)=|x⟩_ un^2.
Finally, when one wants to predict a feature of the solution vector, say, the correlation energy of a molecular system (proportional to the overlap between |x⟩ and |b⟩), it can be done by appending an additional circuit (the Hong-Ou-Mandel module <cit.>), abbreviated as HOM in Fig. <ref>(b) that inputs the output, x ⟩, of the HHL algorithm and b ⟩, to output the overlap between the two quantities. The correlation energy is then calculated as 


    E_c^ HHL   =    - |b⟩_ un^2  |x⟩_ un  |⟨ b | x ⟩|  
       =    - |b⟩_un^2 ∑_i |b_i|^2 C/λ̃_i.
 



For the sake of simplicity, we set this quantity hereafter as the feature of |x⟩ that is of interest to us, for the rest of this work, although Psi-HHL can be applied to more general features, such as transition matrix elements involving |x⟩. Therefore, even for non-quantum chemistry examples that we shall later consider, we will use the term `correlation energy' to refer to the quantity from Eq. <ref>. The number of trials needed to successfully post-select outcome `1' is 𝒪(κ^2). Thus, in the event of large κ, the success probability falls quickly, making it practically difficult to carry computations using large κ matrices. In this work, we demonstrate a new approach— `Psi-HHL', towards handling large κ matrices by leveraging the probability of successfully post-selecting the `wrong' outcome `0', P(0), in HHL.   



 §.§ The Psi-HHL approach


In this section, we discuss the two essential ingredients— HHL_1 and HHL_2, which make up Psi-HHL as illustrated in Fig. <ref>(c). The HHL_1 module depicts the case where the `wrong' signal is post-selected from the traditional HHL given in Fig. <ref>(a) and Fig. <ref>(b). The HHL_2 entails  traditional HHL module appended with a mixing 2 × 2 unitary, given as R_Y(2α) and post-selecting 1 from the mixed signal. The role of this unitary is to mix the coefficients of |0⟩ and |1⟩ in the traditional HHL (see Eq. <ref>), thereby outputting a `mixed' signal. The core idea behind Psi-HHL is to subtract the correlation energies associated with HHL_1 and HHL_2 thus leaving us with the correlation energy corresponding to the `correct' signal. The gain in taking this route is the efficiency that we achieve in the number of shots expended. We summarize this in the following proposition: 

*proposition*Proposition

Psi-HHL yields the same correlation energy as traditional HHL through two implementations of HHL, but is more efficient in the number of shots than HHL, especially when dealing with A matrices with large condition numbers. 
 

We expound on the first part of the proposition in the following paragraphs, and the second in sub-section <ref>. 



  §.§.§ HHL_1: Computing the wrong signal


The first HHL module, HHL_1, serves to post-select the `wrong' signal of the HHL ancilla qubit, that is, the state attached to |0⟩ in Eq.<ref>. 
We re-write the state at the output of all the registers |ψ_ out⟩ except the clock qubits register, as follows, 
    |ψ_ out⟩   =   ∑_i b_i (√(1-C^2/λ̃^2_i)|0⟩⊗|ν_i⟩ + C/λ̃_i|1⟩⊗|ν_i⟩)
       =   ( |0⟩⊗  |x_ w⟩+ |1⟩⊗ |x_r⟩ ).
 
Note that here |x_w⟩ and |x_r⟩ are the unnormalised vectors in the superposition. The subscripts w and r refer to `wrong' and `correct' respectively, where for the latter, we use r since c is used already to denote `correlation'. The normalised solution vector |x_0⟩ which appears at the output register after successfully post-selecting outcome `0' is given by,

    |x_ 0⟩   =   |x_w⟩/|x_w⟩
       =   ∑_i b_i √(1 - C^2/λ̃_i^2)|ν_i⟩/ |x_w⟩,

where,

    |x_ w⟩ = √(P(0))= (∑_i |b_i√(1- C^2/λ̃_i^2)|^2)^1/2.
 Here, P(0) is the probability of successfully post-selecting outcome `0'. The associated correlation energy is then given as,

    E_ c^ w = -|b⟩_ un^2  |x_ w⟩  |⟨ b|x_0⟩| .

Here, |b⟩_ un is the unnormalized input vector after amplitude encoding, and

    |⟨ b|x_0⟩|= ∑_i  |b_i|^2 √(1- C^2/λ̃_i^2).
 



  §.§.§ HHL_2: Computing the mixed signal


The second HHL execution involves attaching the rotation gate R_Y(2α) on the HHL ancilla before the measurement as illustrated in the lower panel of Fig. <ref>(c). This seemingly trivial addition to the algorithm introduces a new degree of freedom via the angle α, enabling one to boost the probability of successful matrix inversion in the case of large κ. We recall that R_Y(2α) is a rotation about the Y axis on the Bloch sphere as given below

    R_Y(2α) = [  cos(α) -sin(α);  sin(α)  cos(α) ].
 
The state vector at the output of HHL_2 module |ψ'_ out⟩, ignoring the clock qubits register, after applying the unitary  in Eq. <ref> is given below,


    |ψ'_ out⟩   =   ∑_i b_i (√(1-C^2/λ̃_i^2)cos(α) - C/λ̃_isin(α) ) |0⟩⊗|ν_i⟩
       +     b_i ( √(1-C^2/λ̃_i^2)sin(α) + C/λ̃_icos(α)) |1⟩⊗|ν_i⟩
       =   (|0⟩⊗|x_ m⟩+|1⟩⊗|x'_ m⟩ ).


Note that in Eq. <ref>, |x_m⟩ and |x'_m⟩ are unnormalized vectors. The subscript `m' denotes `mixed', and while the unprimed vector accompanies |0⟩, the primed one accompanies |1⟩. The solution vector at the output after successfully post-selecting outcome `1' is given below


    |x'⟩   =   |x'_ m⟩/|x'_ m⟩
       =   ∑_i b_i ( √(1-C^2/λ̃_i^2)sin(α) + C/λ̃_icos(α)) |ν_i⟩/ |x'_ m⟩,


where 


    |x'_ m⟩   =   √(P'(1))
       =   (∑_i | b_i (√(1-C^2/λ̃_i^2)sin(α) +  C/λ̃_icos(α)) |^2)^1/2.


Here, P'(1) is the probability of successfully obtaining outcome `1' in HHL_2. The correlation energy is then obtained as 

    E_c^ m = -|b⟩_ un^2   |x_ m'⟩ |⟨ b|x'⟩|,

where

    |⟨ b|x'⟩| = ∑_i |b_i|^2 ( √(1- C^2/λ̃_i^2)sin (α) +C/λ̃_icos(α) ).
 
We choose α such that sin(α) and cos(α) are  positive and P'(1)≫ P'(0). P'(0) refers to the probability of post-selecting outcome `0' in HHL_2. 



  §.§.§ Psi-HHL: Subtracting signals
 

In order to prove this statement, we subtract Eq. <ref> and Eq. <ref>, thus obtaining 


    E_c^Psi-HHLcot(α)   =   E_c^m/sin (α) - E_c^w
       =     - |b⟩_ un^2 ∑_i |b_i|^2 C/λ̃_i  cot(α).

By comparing the right hand sides of Eq. <ref> and Eq. <ref>, we see that the expressions for E_c^ HHL and E_c^Psi-HHL differ only by a factor of cot(α) as shown below.


    E_c^Psi-HHL  =   E_c^m/ sin(α) - E_c^w/ cot(α) = E_c^ HHL.
 

Although theoretically, we obtain the same correlation energy from Psi-HHL, we observe that this technique is superior to traditional HHL in the way it handles large κ matrices, due to the optimal probability of success that one achieves while post-selecting, as we explain in the subsequent section.
Therefore, E_c^Psi-HHL can predict the true correlation energy, E_c, to a reasonable level of precision with fewer shots even for values of κ where HHL procedure fails to do so, which we illustrate in the Results section with examples. We find through our examples that for up to fairly large values of κ that are well beyond those that HHL can handle, Psi-HHL predicts the correlation energy with fewer shots. It is also important to note that throughout the derivation discussed above, we can replace |b⟩ with O|y⟩, where O is a suitably block-encoded observable <cit.>, and |y⟩ an arbitrary state that can be an output of another algorithm, and our results from Psi-HHL still hold. We also add that in problems where it is not immediately obvious a priori that the condition number is large, one cannot determine whether to employ HHL or Psi-HHL. However, this is not a problem as Psi-HHL intrinsically involves the regular HHL procedure, and hence if P(1) is significant, one can choose not to opt for the Psi-HHL procedure. 



 §.§ Complexity aspects
 

We now demonstrate the complexity (in κ) of the Psi-HHL algorithm. Since for the purposes of deriving condition number behaviour at the measurement-post-selection part of HHL, the distinction between the estimated eigenvalues of A, that is, λ̃_i, and the  actual eigenvalues, λ_i, does not matter, we denote all estimated eigenvalues in this section as simply λ_i for the sake of convenience. 
 

In order to elaborate on the second part of our Proposition, we invoke the probability with which one post selects the desired outcome successfully in terms of κ. 
 

a) Success probability of the post-selection process in HHL_1:  The probability of successfully post-selecting the outcome `0', P(0), in HHL_1 as given in Eq. <ref> can be expanded as <cit.> 


    P(0)   =   ∑_i |b_i√(1- C^2/λ_i^2)|^2 
       ≈   √((𝕀-C^2 A^-2))|b⟩^2,

where 𝕀 is the identity operator written in the eigenbasis of operator A, constant C= O(1/κ), where the condition number κ is taken as λ_ max/λ_ min  and |b⟩ is the input to the HHL circuit. The set of values {√(1-C^2/λ_i^2)} are the eigenvalues of the operator √(𝕀-C^2 A^-2), with the largest eigenvalue being √(1-C^2/λ_ max^2), and therefore 


    P(0)   ≤   √((𝕀-C^2 A^-2))_ op^2|b⟩ ^2 
    
    P(0)   ≤    1-C^2/λ_ max^2
       ≤    1-1/κ^2.

 
The first line uses the property of sub-multiplicativity of matrix norms, that is, A B≤A B. In  Eq. <ref>, we recall that we set C=λ_ min, ._ op is the operator norm, and |b⟩=1. Note that the smallest eigenvalue of this operator is √(1-C^2/λ_ min^2)=0, and corresponds to P(0)=0, which leads to expending a large number of shots in HHL_1. One could circumvent this issue by appropriately choosing a scaled C = γλ_min, where γ∈ 
( 0,1), leading to P(0)≥ 1-γ^2. This idea has been illustrated via an example as shown in Fig. <ref>, where we choose the scaling γ=0.5. 
 



b) Success probability of the post-selection process in HHL_2: A similar expression for the probability of successful post-selection in HHL_2,  P'(1), can be obtained by further expanding Eq. <ref> as follows, 


    P'(1)   ≈    [√((𝕀-C^2A^-2)) sin(α) 
       +    (C A^-1) cos(α) ]|b⟩^2.

The operator B √(𝕀-C^2A^-2)sin(α) + (C A^-1) cos(α) in its eigenbasis {|ν_i⟩} could be given in the diagonal form as 


    diag( ⋯, √(1-C^2/λ^2_max)sin(α)+(C/λ_max)cos(α), ⋯, 
    √(1- C^2/λ^2_min)sin(α)+(C/λ_min)cos(α),  
    ⋯ ),
 

where the entries that occur in the parentheses after `diag' are the diagonal elements of the matrix. Note that the largest and smallest eigenvalue of the matrix given in Eq. <ref> depend on the choice of α. When α=π/2, the component of that eigenvalue attached to sin(α) remains, and we do not extract the desired solution. However, when α is too small, say 0, the component attached to cos(α) remains and we get the same complexity as the traditional HHL approach. Therefore, α is chosen to satisfy the following properties: 


  * The success probability, P'(1) is large enough.

  * The solution component that matters to us does not vanish in the superpositions that occur along the diagonal entries in Eq. <ref>.


We demonstrate the case where α is close to π/2 and C=λ_min, in which case the operator in Eq. <ref> could be written in the form where the largest eigenvalue is the first entry and the smallest eigenvalue is the last entry, as given below,


    diag(⋯, √(1-1/κ^2)sin(α)+(1/κ)cos(α), ⋯, 
    cos(α)).


Using once again the sub-multiplicativity of matrix norms in Eq. <ref>, we can write 


    P'(1)   ≤    [√(𝕀-C^2A^-2) sin(α) 
       +    CA^-1cos(α) ] ^2_ op |b⟩^2 ,


where ._ op is the operator norm yielding the largest eigenvalue of the operator, and |b⟩=1. Then P'(1) is bounded the following way, 


    cos^2(α)≤ P'(1)   ≤   |[(√(1- 1/κ^2)) sin(α) + 1/κ cos(α) ]  |^2 
    
       =   (1- 1/κ^2) sin^2(α) + 1/κ^2 cos^2(α) 
       +    2(√(1- 1/κ^2))1/κsin(α)cos(α) 
       ≈    sin^2(α) + 1/κ sin(2α) +  1/κ^2cos(2α) 
       -   1/κ^3sin(α)cos(α).


In this bound, the left hand side is the square of the smallest eigenvalue of the matrix given in Eq. <ref>, while the right hand side is the square of the  largest eigenvalue of the matrix in Eq. <ref>. 

For the choice of C=γλ_min, where γ∈ ( 0,1), one can push the lower bound for the choice of given α. This implies that fewer measurements are required to invert matrices with large condition numbers, as compared to the traditional HHL technique illustrated in Fig. <ref>(a). We show this in Fig. <ref>, where for a given α=60^∘, we obtain 


    |[(√(1- γ^2/κ^2)) sin(α) + γ/κ cos(α) ]  |^2   ≤    P'(1) 
       ≤   |[(√(1- γ^2)) sin(α) 
       +   γcos(α) ]  |^2 .


While the A  and B matrices share the same basis {|ν_i⟩}, an eigenstate of B matrix that achieves the lower bound for P'(1) may not often correspond to the largest/smallest eigenvalue  of A matrix, for a given choice of angle α. 

Note that for an arbitrary input |b⟩ which is a nontrivial superposition of all the eigenvectors {|ν_i⟩} of A, we have the following condition,

    P'(1)   =   (∑_i | b_i (√(1-C^2/λ_i^2)sin(α) +  C/λ_icos(α)) |^2) 
       =   (∑_i | b_i|^2λ_i'^2).
 

Here, {λ_i'} are the eigenvalues of the operator √(𝕀-C^2A^-2) sin(α) + CA^-1cos(α). Eq. <ref> is a convex combination of the square of eigenvalues, λ_i'^2, therefore, an arbitrary input |b⟩ will yield a P'(1) satisfying the condition given in Eq. <ref>. We augment our findings with the simulations in the subsequent Sec. <ref>. 
 


§ RESULTS AND DISCUSSIONS
 

In this section, we consider two sets of examples to compare the performance of Psi-HHL with that of HHL: 



  * Toy A matrices of sizes 4 × 4, 8 × 8, 16 × 16, 32 × 32, and 64 × 64: For practical reasons associated with computational resources, we do not choose matrices whose κ grows with the A matrix size. Instead, we generate a sequence of A matrices with systematically increasing condition numbers for every system size. For example, for 4 × 4 matrix size, we begin with a sparse and diagonally dominant matrix (the latter is not a requirement but simplifies matters for our illustrations) such that its smallest diagonal entry is 2^-(n_r-1) and the largest 1. Recall that n_r is the number of QPE clock register qubits. We generate the sequence by varying n_r from 2 to 20. In such a sequence, κ grows exponentially with n_r, thus enabling us to reach the failure point of HHL rather rapidly and with existing computational resources. We also note that changing the position of the minimum diagonal element of A along the diagonal does not affect our analysis. 

  * Quantum chemistry: Since quantum chemistry is one of the main applications of interest with quantum computers, it is worth exploring this domain for our analysis <cit.>. However, since we no longer are considering toy examples, we begin by qualitatively assessing the applicability of HHL to the LCC problem by: 
 

  * Checking how the A matrices grow with system size (and along the way, defining the system size), thereby providing a handle on the classical hardness of the problem, and 

  * Checking how κ itself grows as a function of what we set to be the system size. Only after this qualitative analysis do we move to our main results for this problem domain. 
 

  * We carry out the following analyses: 


  * For the toy matrices, we compare HHL and Psi-HHL, and demonstrate the ability of the latter in predicting correlation energies across a range of condition number values. As we shall explain later in this section, we find that since κ grows slowly for the quantum chemistry case, it is not computationally feasible to carry out the analysis. We quantify the quality of our results for the correlation energies through percentage fraction difference, abbreviated as PFD, given by E_c-E_c^HHL/E_c× 100 for HHL and E_c-E_c^Psi-HHL/E_c× 100 for Psi-HHL. We denote the correlation energy obtained on a traditional computer by E_c, whereas E_c^HHL and E_c^Psi-HHL are the values for the quantity obtained from HHL and Psi-HHL respectively. 

  * For both the toy matrices as well as the quantum chemistry matrices, we study the PFDs of HHL and Psi-HHL approaches versus the number of shots. 

 

 



 §.§ Toy matrices: 4 × 4 through 64 × 64
 

We begin by discussing the details of our A matrices and b⃗ vectors. We pick our 4 × 4 matrix to be non-diagonal, and is given by  


    A=[  2^-(n_r-1)        0.00        0.00        0.00;        0.00        0.75 0.10× 10^-3        0.00;        0.00  0.10×10^-3        0.50        0.00;        0.00        0.00        0.00        1.00 ],

and b⃗=[ 0.10; 0.01; 0.20; 1.00 ]. The sparsity, s, of A is defined as the number of non-zero entries in the row with the most number of non-zero elements and is set to 2. Our choices for the other matrices, 8 × 8 through 64 × 64 can be found in Figure <ref> of the Appendix. We have set the following values for sparsity: 3 for 8 × 8, 4 for 16 × 16, 17 for 32× 32, and 7 for 64 × 64 matrices. Throughout, we have ensured that the eigenvalues of the A matrix lies between 1/κ and 1. We provide more examples for the 4 × 4 matrix size in Section <ref> of the Appendix. 
 

Figures <ref>(a) through <ref>(e) provides our findings for 4 × 4 through the 64 × 64 examples considered. Each data point in the sub-figures is the average result obtained over 10 repetitions, with each repetition involving 1 million shots. We pick α = 60^∘ for the Psi-HHL calculations. The corresponding data is given in Table <ref> of the Appendix. Our results obtained using other choices of angles, 70^∘ and 80^∘, are provided in Tables <ref> and <ref> respectively. We now return to Figure <ref>, where each sub-figure provides two sets of data: one which compares the probabilities, P(0), P(1), P'(0), and P'(1), obtained by executing HHL and Psi-HHL, and the other providing the PFDs for the two methods. We choose condition numbers all the way till about 1 million. Figures <ref> and <ref> of the Appendix shows these results for α of 60^∘, 70^∘, and 80^∘, with 10^5 and 10^6 shots respectively. The figures show that our data fits for HHL align with the expected behaviour of P(1) ∼𝒪(κ^-2). We also find that P'(1) scales as ∼𝒪(1) for large condition numbers (with the fit parameter a=0.75 matching with the Y-axis intercept for large condition number values). For completeness, we verify that the fit probabilities add to one (see Table <ref> of the Appendix). 
 

 

We now comment on the PFDs. We immediately see that HHL fails at some value of κ for the chosen number of shots (indicated by `nan' values in Tables <ref>, <ref>, and <ref> of the Appendix). This is due to the fact that as P(1) becomes very small, the fraction of the total number of shots that yield 1 upon measurement tend to very small values. Therefore, the number of shots available for the HOM module become too small to give any reasonable result beyond a certain value of κ. The bottom panel of each sub-figure of Figure <ref> shows that the error bars, in which the upper and lower points of the bars reflect the largest and the smallest PFD values obtained using HHL, increases as the condition number increases, till it reaches a critical value of κ, beyond which HHL fails. In contrast, we see that for Psi-HHL, even at κ ≈ 1 million, the length of the error bar is about 17 percent. 
 

We now verify that HHL does not fail upon increasing the number of shots, but the 
number requires to be about 1 billion in order to get a comparable range of PFDs as Psi-HHL, while Psi-HHL only incurs 10 million shots, as Figure <ref> shows. In particular, HHL gives a maximum PFD of 1.51 percent and a minimum of -2.98 percent with a billion shots, whereas Psi-HHL gives a maximum and minimum PFD of 2.33 and -2.97 percent respectively with only 10 million shots. 
 

We finally comment on our results from Figure <ref> on the larger size matrices (8 × 8 through 64 × 64), where for Psi-HHL, we consider α=60^∘. Figures <ref> through <ref> provides the data for 70^∘ and 80^∘ cases, with Figure <ref> also giving the information for 10^5 shots. We observe that broadly, the trends are similar to what we find for the smaller 4 × 4 cases. Figure <ref> of the Appendix summarizes the situations where HHL fails for all of the toy matrices considered, as a heat map. We find that for 1 million shots, HHL starts returning `nan' values between κ of 2^8 and 2^10. In view of the computational cost involved, we did not go all the way up to a condition number of 2^20 for matrices above 4 × 4 size, and we stopped much earlier for the 64 × 64 case. 
 



 §.§ Quantum chemistry matrices
 

Quantum chemistry is regarded as one of the killer applications of quantum computing <cit.>. Therefore, it is interesting to consider the performance of Psi-HHL in this scenario. In particular, we consider quantum chemistry problems via the LCC approach <cit.>, for this purpose. Quantum chemistry on quantum computers can involve different approaches to selecting systems of interest depending on the application domain, including but not limited to performing calculations on numerous candidate molecules of progressively increasing size, picking one molecule and systematically improving the precision in one's calculations via suitable active space choices (as well as better quality single particle bases, incorporating more physical effects in the molecular non-relativistic Hamiltonian, etc), etc. For the purposes of this work, since we only want to compare performance of the HHL and Psi-HHL approaches, we opt for the substantially simpler second approach. 
 

 



  §.§.§ The A matrix dimension


Since we are now going beyond toy examples, we begin with some preliminary qualitative remarks. We begin with the growth of the LCC A_mol matrix. We make a distinction between the dimension of the LCC matrix, A_mol, which is m × m, and the dimension of the associated A matrix, n × n, where we recall that n=2^N. We go from A_mol to A via padding, which we explain in Section <ref> of the Appendix. 

 

  * The number of rows (or columns), m, of the A_mol matrix, which we shall hereafter think of as the system size, is the number of particle-hole excitations that one considers for a calculation, which in turn is decided by choice of single particle basis and active space size. We denote the number of occupied orbitals as n_o and virtuals by n_v. It is common to employ the LCC approach by only considering single and double excitations (termed LCCSD), in which case A_mol matrix grows as m ∼ n_o^2n_v^2 for fixed values of n_o and n_v. As we include more excitations systematically, such as triples, quadruples, etc, while keeping the number of orbitals fixed, we observe that the A_mol matrix grows as ∼ n_o^ℰn_v^ℰ, which is exponential. ℰ refers to the excitation rank. 

  * For a given excitation rank, ℰ, as we increase the number of orbitals, the system size, m, grows further polynomially, for example, see Figure <ref> in the Appendix. For the purposes of this work, we restrict ourselves to LCCSD approximation (ℰ=2), and study the size of A_mol with increase in number of unoccupied orbitals, n_v, in a molecule, while keeping n_o fixed at a reasonable number. 
 

For our analyses, we do not attempt to consider a molecule in a given active space size to study the variation of m with choice of single particle basis, and defer the study for a future work. We choose three examples: the light Li_2, the moderately heavy RbH, and the heavy CsH neutral diatomic molecules, all in their equilibrium bond lengths (Li_2: 5.06 Bohr, RbH: 4.4726 Bohr, and CsH: 4.7135 Bohr)  <cit.>. The adjective `light' and `heavy' refer to the atomic number of the heavier atom in a diatomic molecule. We pick the Sapporo double zeta basis sets for RbH and CsH (Rb: Sapporo-DZP, Cs: Sapporo-DKH3-DZP-2012, and H: Sapporo-DZP) <cit.>, while for Li_2, we choose the STO-6G basis <cit.>. We employ the C_2v point group symmetry for RbH and CsH computations, while for the case of Li_2, we pick the D_2h symmetry. We fix n_o to 3, 13 , and 18 for Li_2, RbH, and CsH respectively. The technical details involved in the generation of A matrix as well as the b⃗ vector are discussed in Section [SM: chemistry-matrices-A]A2.A of the Appendix. 
 



  §.§.§ Condition number of the A matrix
 

Although HHL offers an exponential advantage in terms of system size, it is worth checking if for the LCCSD problem, the condition number itself scales favourably with system size, m, for the systems that we considered. We point out towards the end of Section [SM: chemistry-matrices-A]A2.A of the Appendix that it is sufficient to analyze κ versus m; this curve upper bounds κ versus n. We increase the active space, and thus m, in a systematic manner by carrying out an orbital character analysis, whose details we explain in Section [SM: chemistry-matrices-B]A2.B of the Appendix. Furthermore, for this analysis, we only consider RbH and CsH, and not Li_2 as the system is too small (in the STO-6G basis) to carry out condition number versus system size analysis. For RbH, we consider n_o=13 for our calculations, and vary the number of virtuals from 1 to 21, which corresponds to A_mol matrix size of 40 × 40 to 9065 × 9065 respectively. For CsH, n_o=18 and virtuals are varied from 1 to 16, while the A_mol matrix size goes from 67 × 67 to 11058 × 11058. 
 





Our results shown in Figure <ref> of the Appendix indicate that for RbH as well as for CsH, as we go from m of 40 to 9065 and 67 to 11059 respectively, κ grows polylogarithmically (fourth and sixth order respectively). This behaviour is also consistent with what one may expect from chemistry intuition: we begin by referring the reader to Table <ref> of the Appendix, where we list the largest and smallest eigenvalues of the A matrix for each value of m, for both RbH and CsH. The data shows that the increase in κ as we add virtual orbitals is driven by the decrease of the lowest eigenvalue of the A matrix. We note that every matrix element of the A matrix can be expressed as ⟨Φ_p|H|Φ_q ⟩, where H is the Hamiltonian and |Φ_p⟩ and |Φ_q⟩ are the excited determinants built from particle-hole excitations arising from the Hartree-Fock configuration. The smallest eigenvalue of the sparse and diagonally dominant A matrix, which we saw as being important in determining the growth of κ, 
can be qualitatively thought of as corresponding to the energy of the first excited state (assuming that the wave function of the first excited state has small contribution from the HF configuration). Thus, we expect that the decrease in the lowest eigenvalue of A saturates as we keep adding virtual orbitals, and eventually, the lowest eigenvalue of the A matrix will converge to a certain value. Thus, the increase of κ saturates too, thus leading one to expect that the condition number would vary as a polylogarithmic function of system size. We add at this juncture that while considering only two molecules does not allow any quantitative inference on the behaviour of condition number with system size, it gives a qualitative indication that quantum chemistry using HHL-LCCSD may a reasonable direction to consider. However, further analysis would be required to assess what one would expect for κ(m) in general for an HHL-LCC computation. We also note that we carried out a similar analysis for sparsity, and found that it too grows polylogarithmically for RbH and CsH (see Figure <ref> of the Appendix). 
 



  §.§.§ HHL and Psi-HHL results
 

We now discuss our results for the performance of the HHL and Psi-HHL algorithms for the molecules considered in this work.  
 

We begin by conducting a study similar to the one shown in Figure <ref> for the toy matrices. It is not practical to carry out an analysis similar to that in Figure <ref>, since unlike the toy matrices case, the condition number grows much slower and the computations become too expensive too soon. For the purposes of our analysis in Figure <ref>, we vary the number of shots and compare the performance of HHL and Psi-HHL in predicting the PFD, as well as the range across which the PFDs are spread, across 5 repetitions. For this analysis, we consider Li_2, RbH and CsH. The details are: 

 

  * Li_2: 21-qubit circuit with an A matrix size of 64 × 64 (built out of single and double excitations arising from 3 occupied and 8 virtual orbitals) in the STO-6G basis, with κ=64.07. 

  * RbH: 25-qubit circuit with an A matrix size of 256 × 256 (built out of single and double excitations arising from 13 occupied and 3 virtual orbitals) in the Sapporo double zeta basis, with κ=153.17. 

  * CsH: 26-qubit calculations with an A matrix size of 256 × 256 (built out of single and double excitations arising from 18 occupied and 2 virtual orbitals) in the Sapporo double zeta basis, with κ=466.59. Although RbH and CsH have the same A matrix size, the latter involves a 26-qubit calculation because we set an additional ancilla for CsH due to its larger atomic number. 


We note that for RbH, the same calculation would have incurred more qubits for VQE-UCCSD (32-qubit circuit) and QPE-CASCI (with n_r=8, 40-qubit circuit) computations, while we carry out a 25-qubit circuit evaluation in the case of HHL-LCC. Similarly, for CsH, VQE-UCCSD: 40 qubits, QPE-CASCI: 48 qubits, and HHL-LCC: 26 qubits. 
 

Our results are presented in Figure <ref>, with the accompanying data given in Table <ref> of the Appendix. For Li_2, the condition number is very small, and hence we only study the performance of HHL and Psi-HHL between 500 and 10000 shots. The data indicates that for Li_2, the Psi-HHL approach gives better average PFDs, especially in the regime of fewer number of shots (500 and 1000 shots). In the same regime, the spread in PFDs is also lesser for Psi-HHL. For the cases of RbH and CsH, as we increase the number of shots, the error bars for both HHL and Psi-HHL are seen to become much smaller. 
 


We now move to our next set of results. From a practical point of view, it is computationally not feasible to pick the larger system sizes from Figure <ref>. For potential energy curves using both HHL and Psi-HHL approaches, we consider: 



  * Li_2: 21-qubit circuit with an A matrix size of 64 × 64 (3 occupied and 7 virtual orbitals) in the STO-6G basis. 

  * RbH: 21-qubit circuit with an A matrix size of 64 × 64 (4 occupied and 4 virtual orbitals) in the Sapporo double zeta bases. 

  * CsH: 26-qubit calculations with an A matrix size of 256 × 256 (built out of single and double excitations arising from 18 occupied and 2 virtual orbitals) in the Sapporo double zeta bases. 
 

Furthermore, for each molecule, we also consider geometries in the neighbourhood of the equilibrium bond length (±1 Bohr in steps of 0.25 Bohr). With such a choice, the correlation energies required to be captured ranges from ∼ 10s of mHa (milli-Hartree) for Li_2 to RbH's ∼ 0.1-1 mHa to CsH's ∼ 10s of μHa, thus allowing us to check the performance of HHL and Psi-HHL across a range of correlation energy values. Calculating energies across many bond lengths further allows us to check across a (limited) range of condition numbers the behaviour of HHL and HHL. We note that the range of condition numbers corresponding to the considered geometries range from about 59 to 121 for Li_2, 14 to 19 for RbH, and 455 to 505 for CsH. Each calculation involves a million shots and our reported results are averaged over 5 repetitions. Figure <ref> shows the results when we plot Δ E, which is the difference between HHL (Psi-HHL) and the LCC correlation energy evaluated on a traditional computer, against the geometry. We find that Psi-HHL mostly performs better than HHL in predicting correlation energies. However, ... . This is not immediately obvious for the case of CsH, since the differences occur typically at μHa level. 
 




§ CONCLUSION
 

We propose the Psi-HHL algorithm to efficiently handle problems involving A matrices with large condition numbers. The approach involves post-selecting 0 in HHL (the `wrong signal'), followed by post-selecting 1 from an HHL-like circuit that contains an additional mixing 2 × 2 unitary (the `mixed signal'), and taking the difference between the wrong and the mixed signals to extract the correct signal. 
  

We present arguments to show that the complexity of Psi-HHL is optimal in condition number for large κ regimes; the complexity of the algorithm scales as κ. This is an improvement over earlier approaches in literature, such as HHL with amplitude amplification (scales quadratically, and incurs several additional gates due to its iterative nature) and HHL with variable time amplitude amplification (scales as κlog^3κ). It is also important to note that while the optimal κ behaviour can be recovered using a discrete quantum adiabatic approach based quantum linear solver algorithm <cit.>, we achieve the same result using Psi-HHL, which is a simple modification to the HHL algorithm. 
 

We apply our method to different toy matrices, where we systematically vary the condition number all the way to about 1 million while keeping the number of shots fixed to 1 million, and show that while HHL fails beyond a certain value, Psi-HHL continues to predict good results. We also demonstrate the performance of our proposed Psi-HHL approach for quantum chemistry examples, where we once again find that Psi-HHL yields better quality results in much fewer number of shots. En route, we also carry out a qualitative analysis to check suitability of treating LCC equations using the HHL algorithm. 
 

We anticipate that our work will pave the way for development of non-trivial Psi-HHL variants that would allow it to be integrated as an algorithmic primitive in larger algorithmic frameworks that use traditional HHL as a subroutine. 
 


We acknowledge National Supercomputing Mission (NSM) for providing computing resources of `PARAM Siddhi-AI’, under National PARAM Supercomputing Facility (NPSF), C-DAC, Pune and supported by the Ministry of Electronics and Information Technology (MeitY) and Department of Science and Technology (DST), Government of India. VSP thanks Prof. Aram Harrow for a very fruitful discussion on HHL as well as HHL-LCC. VSP and AJ thank Dr. Shibdas Roy for nice discussions on complexity. VSP and PBT thank Prof. Debashis Mukherjee for useful discussions on the quantum chemical sections of the work. KS acknowledges support from Quantum Leap Fragship Program (Grant No. JPMXS0120319794) from Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; Center of Innovations for Sustainable Quantum AI (JPMJPF2221) from Japan Science and Technology Agency (JST), Japan; and Grants-in-Aid for Scientific Research C (21K03407) and for Transformative Research Area B (23H03819) from Japan Society for the Promotion of Sciences (JSPS), Japan. 










§ APPENDIX







 
 
 




§ DETAILS ON CALCULATIONS INVOLVING TOY MATRICES FROM SIZE 4 × 4 THROUGH 64 × 64
  

 

In our Results section in the main text, we discussed the case of A not diagonal and the amplitude encoded state corresponding to b⃗ in an unequal superposition. We also provide our 4 × 4 through 64 × 64 matrices that we considered in the main text in Figure <ref>. 

We now provide additional data on two more choices for our 4 × 4 A matrices: 



  * A is diagonal and the amplitude encoded state corresponding to b⃗ is in an equal superposition. In particular, 	A=[ 2^-(n_r-1)       0.00       0.00       0.00;       0.00       0.75       0.00       0.00;       0.00       0.00       0.50       0.00;       0.00       0.00       0.00       1.00 ], b⃗=[ 1.00; 1.00; 1.00; 1.00 ]. The sparsity, s, of A is set to 1. 

  * A is diagonal and the amplitude encoded state associated with b⃗ is in an unequal superposition. In particular, A is the same as in case 1 given above, and b⃗=[ 0.10; 0.01; 0.20; 1.00 ]. 
 

We also present these results for α set to 60, 70, and 80 degrees (the main text only discussed the first case). Furthermore, while the main text considers 1 million shots, we also perform the same analyses for 10^5 shots. All of these results are presented in Figures <ref> and <ref> for the cases considering 10^5 and 10^6 shots respectively. The accompanying data is provided in Tables <ref>, <ref>, and <ref>, for 60, 70, and 80 degrees examples respectively. 









 



 

 

 

 

 













§ DETAILS ON CALCULATIONS INVOLVING QUANTUM CHEMISTRY MATRICES
 



 §.§ A matrix and b⃗ vector generation
 

We begin by explaining the procedure for generating the A matrix of size n. We use the configuration state function-based Graphical Unitary Group Approach-Configuration Interaction Singles and Doubles (GUGA-CISD) method available in the GAMESS program <cit.> to construct the Hamiltonian for a given molecule in a specified single particle basis. GUGA-CI ensures that the Hamiltonian is constructed in an orthogonal basis; hence, it is Hermitian. Also, because it is a spin-adapted CI method, it avoids spin contamination of the wave function, thereby giving accurate results. We remove the first row and the first column of the GUGA-CISD Hamiltonian matrix and subtract the diagonal entries of the resulting matrix with the energy obtained from Møller-Plesset theory to second order in perturbation (also called MP2 in literature). The latter step is carried out for the purpose of normal ordering, which in turn ensures that HHL would output the correlation part of the ground state energy directly instead of the total ground state energy (which can be written as a sum of the Hartree-Fock (HF) energy and the correlation energy, with the latter being much lesser than the former). This is desirable when one executes HHL-LCC calculations on NISQ era quantum hardware, as the error in the total ground state energy can be much greater than the correlation energy itself. We also note that unlike the traditional approach where the HF energy is subtracted from the diagonal entries, we take away the MP2 energy, in order to make sure that A is positive definite. In a sense, subtracting MP2 energy could be thought of as a level shift over and beyond subtracting the HF energy. 
 

We now focus our attention on the elements of the A matrix. The size of an A matrix, m, generated from a quantum chemical calculation, which we can denote as  A_mol, is not necessarily of the dimension n × n. In such cases, we embed the A_mol matrix of dimension m× m as a diagonal block in a larger matrix of dimension n × n, with the other diagonal block being an identity matrix of size (n-m) × (n-m), denoted as ℐ. In such a case, we solve a system of linear equations given by 


    A = [ A_mol     0;     0     ℐ ][ b⃗;  0 ] = [ x⃗;  0 ].


This approach ensures that the eigenvalues of A_mol remain unaffected, as long as its maximum eigenvalue is ≈ 1. We choose our matrices carefully so that this condition is satisfied. In the event that the maximum diagonal entry of A is less than 1, it is recommended to pad with the largest (or the smallest) diagonal entry of A so that the number of shots expended is reduced. As an example, we picked a 3 × 3 sparse matrix such that the largest diagonal entry is 0.70: [ 0.25 0.10 0.00; 0.10 0.45 0.20; 0.00 0.20 0.70 ], and considered two cases: padding with 1 and with 0.70. We found that kappa is larger in the latter (5.16) than in the former (4.21). 
 

Since for the problems we consider, κ increases with system size monotonically, and further, going from an unpadded to a padded matrix always right shifts a data point, it is sufficient to carry out κ versus m analysis for the unpadded case, as it upper bounds the complexity of the padded scenario of κ versus n. 

  



 §.§ Orbital character analysis
  

We now explain the procedure that we adopted for selecting orbitals, based on orbital character. We begin with RbH, and then proceed to explain our procedure for CsH (with accompanying data provided in Table <ref>). 
 

We fix the number of core orbitals to 13. We now describe the method adopted in choosing virtual molecular orbitals (MOs). Instead of taking the approach of ordering the virtual MOs by energy in ascending order and then imposing an energy cut-off to define our active space, we pick those MOs that are dominated by important atomic orbitals (AOs). We select the important AOs, and for the purposes of this work, we pick those AOs in the order of the principal quantum numbers. For example, in the case of RbH, we can decide the important AOs from the set that arise from our choice of basis, namely the Sapporo double zeta basis: 5s through 7p. For the purposes of our analysis on condition number versus system size, we pick all the AOs that the basis allows in order to obtain more data points for fitting (details given in Table <ref>). We now move to some specifics for RbH: for the smallest system size considered (1 virtual MO), we pick the first virtual from our sorted MO list, which is the MO dominated by the Rb 5s-5p_z hybrid, and thus our active space that is built out of 13 occupied MOs and 1 virtual one has a system size, m, of 40 × 40, and subsequently upon padding, an A matrix size, n, of 64 × 64. For the next system size, we add the next MO in the sorted list, which is dominated by Rb 5p_y, leading to a 119 × 119 matrix, which, after padding, becomes a matrix of size 128 × 128. We continue to construct A matrices in this fashion until we reach m=9065, for which κ is about 348. 
 

For the case of CsH molecule, we fix the number of occupied orbitals to 18 while progressively increasing the number of valence orbitals. The order in which we add virtual MOs as well as each such virtual MO's details are mentioned in Table <ref>. ]

 Paper Title: Exploiting Non-Abelian Point-Group Symmetry to Estimate the Exact Ground-State Correlation Energy of Benzene in a Polarized Split-Valence Triple-Zeta Basis Set.



§ ABSTRACT 

Local electronic-structure methods in quantum chemistry operate on the ability to compress electron correlations more efficiently in a basis of spatially localized molecular orbitals than in a parent set of canonical orbitals. However, many typical choices of localized orbitals tend to be related by selected, near-exact symmetry operations whenever a molecule belongs to a point group, a feature which remains largely unexploited in most local correlation methods. The present Letter demonstrates how to leverage a recent unitary protocol for enforcing symmetry properties among localized orbitals to yield a high-accuracy estimate of the exact ground-state correlation energy of benzene (D_6h) in correlation-consistent polarized basis sets of both double- and triple-ζ quality. Through an initial application to many-body expanded full configuration interaction (MBE-FCI) theory, we show how molecular point-group symmetry can lead to computational savings that are inversely proportional to the order of the point group in a manner generally applicable to the acceleration of modern local correlation methods.









Symmetry is a ubiquitous concept. Manifesting itself all around us—for instance, in the two separate halves of the human body, in leaves, snowflakes, and other objects of nature, but also in architecture, arts, and maths—symmetry may be identified everywhere and on all scales, from within the grandest entities of the universe to the smallest molecular constituents of matter. Beyond symmetry as simple mirror images, point groups generalize symmetry elements and operations, such as, reflections, inversions, and proper as well as improper rotations, and point-group arguments may even be used to predict and rationalize molecular properties and rule if certain spectroscopic transitions are allowed or not. Across the vastness of chemical compound space, the clear majority of all species will not be symmetrical in any way, but many small and modest-sized molecules of great chemical interest will be—ranging from diatomics of infinite point groups to buckminsterfullerene, the quintessential example of the icosahedral (I_h) point group with the largest number of possible symmetry elements.


The history of the exploitation of molecular point-group symmetries in quantum-chemical simulations is long and acclaimed <cit.>. The construction of a symmetry-adapted atomic orbital (AO) basis has traditionally been invoked to improve substantially upon the computational efficacy of mean-field methods. Regardless of any such symmetry adaptation of the AO basis, the resulting set of molecular orbitals (MOs) will transform as irreducible representations (irreps) of the full molecular point group, unless a symmetry-broken solution is converged upon. Consequently, correlation methods that operate in a basis of these MOs can be improved by exploiting the fact that various classes of integrals over totally symmetric operators will vanish whenever the direct products of the irreps themselves fail to contain a totally symmetric component. However, in the case of spatially localized molecular orbitals (LMOs)—obtained from a unitary rotation of a parent set of canonical molecular orbitals (CMOs) and favoured for both their ease of interpretation as well as their representational compactness—these are usually never optimized subject to the same sort of strict symmetry constraints due to the risk of suppressing their overall spatial locality as a direct and unwelcome consequence.

Be that as it may, we have recently proposed an algorithm capable of enforcing specific symmetry properties among LMOs of molecules belonging to both Abelian and higher-order point groups in a numerically exact fashion <cit.>. Specifically, given how a set of symmetry-equivalent MOs will require only a single symmetry-unique orbital to span all other members of a particular subspace through the application of a befitting set of symmetry operations, this feature holds the potential to facilitate equivalences of specific integrals and related quantities across many routine quantum-chemical methods. In the present Letter, we will demonstrate how to computationally exploit such LMO redundancies—within the context of an incremental approximation to exact electronic-structure theory in a fixed AO basis—to estimate the frozen-core full configuration interaction (FCI) energy of benzene to within unprecedented accuracy in polarized split-valence basis sets of both double- and triple-ζ quality.


In many-body expanded full configuration interaction (MBE-FCI) theory, exact properties within a given Hilbert space are approximated without recourse to an explicit sampling of the wave function by instead performing a multitude of complete active space configuration interaction (CASCI) calculations on restricted tuples of MOs.<cit.> As was recently demonstrated in Ref. , MBE-FCI calculations can now be performed in an entirely unbiased manner, unlike in previous versions of the method where FCI correlation properties were decomposed and solved for by enforcing a strict partitioning of a complete set of MOs into two complementary reference and expansion spaces. Instead, an adaptive algorithm automatically identifies an optimal reference space based on quantum fidelities between individual CASCI wave functions and the Hartree-Fock (HF) determinant, upon which the residual correlation in the remaining expansion space is recovered by means of an MBE in a set of individual spatially localized MOs or suitable clusters of these. In addition, the current state of MBE-FCI operates by truncating the latter space throughout the MBE subject to an a priori error tolerance, enabling more rapid and robust convergence of expanded properties onto the corresponding FCI solutions as well as resulting error bars associated with these <cit.>.

The present Letter is concerned with the exploitation of symmetry-equivalent sets of LMOs and the fact that tuples of these will give rise to identical correlation energies. For this purpose, we will make use of the petite list method <cit.>—a technique originally intended to accelerate the evaluation of integrals—to recover symmetry equivalences among MBE increments. By looping over all orbital combinations (the grande list) and applying the symmetry operations of the molecular point group to determine whether a given orbital tuple is lexicographically greater than any other symmetry-equivalent orbital tuple, it is possible to single out which tuple of MOs for a set of symmetry-equivalent active spaces will need to be explicitly considered. These symmetry-unique orbital combinations will then form the petite list. It is important to emphasize how our proposed use of symmetrized LMOs (vide infra) is generally applicable to a manifold of local electron-correlation schemes whenever these can formally express the correlation energy as a sum over contributions from individual MOs.


Our generalization of petite lists to arbitrary orbital tensors in an MBE requires that these objects must be subject to numerically exact symmetries. Due to the fact that cost functions of different orbital localization schemes possess numerous, densely clustered local minima, subject only to constraints on spatial locality, LMOs must be explicitly symmetrized as in Ref. . This procedure is of particular importance in the context of applications to orbital-based MBEs, given the recursive nature of such methods and the fact that any errors related to near-equivalences will propagate and become amplified throughout an expansion <cit.>.


Our implementation of symmetry within MBE-FCI theory differentiates between symmetry equivalences of increments and active spaces. A set of symmetry-equivalent active spaces all produce the same CASCI property but will only produce symmetry-equivalent increments in case this equivalence holds true for all subtuple active spaces. Thus, while a parent tuple may be symmetry-equivalent, its individual MOs may risk being neither equivalent nor invariant with respect to a given symmetry operation. The selected Pipek-Mezey <cit.> (PM) LMOs of benzene in Fig. <ref> provides an illustrative example of exactly this situation.




The upper panel of Fig. <ref> depicts two orbital tuples and how their CASCI correlation energies ({E_ia,E_ib}) and increments ({ϵ_ia,ϵ_ib}) are different. Both tuples involve the same occupied π-orbital, ϕ_i, and two different virtual orbitals, ϕ_a and ϕ_b, from a set of 12 symmetry-equivalent orbitals. While these two virtual orbitals can be transformed into each other by applying a vertical mirror plane, ϕ_i, on the other hand, is neither symmetry-invariant nor equivalent with any other π-orbital in the symmetry-equivalent set of cardinality 3 when applying said symmetry operation. On the contrary, the bottom panel of Fig. <ref> depicts active spaces formed from the complete set of occupied π-orbitals and the same two virtual orbitals, {ϕ_a,ϕ_b}. While the CASCI correlation energies are equal due to the symmetry-invariance of the π-orbital space with respect to all symmetry operations, the corresponding recursive increments will not be equal. In our implementation, whenever orbital tuples fail to produce symmetry-equivalent increments, the CASCI calculation in question is performed only once before the required increments get calculated from appropriate contributions only. The supporting information (SI) provides further details on our parallel implementation <cit.>.




Our proposed symmetry exploitation is next evaluated by performing MBE-FCI calculations on water (C_2v), ammonia (C_3v), and methane (T_d) in the standard cc-pVDZ basis set <cit.>. The results in Fig. <ref>—based on Foster-Boys <cit.> (FB) orbitals—show how both the total and redundant number of increment calculations increase exponentially, ensuring that the relative number of increment calculations to skip stays reasonably constant as the MBE order rises. The actual percentage is highly dependent on both the chosen point group of the molecular system and the nature of the LMOs themselves, as was previously touched upon in Ref. . The upper bound for symmetry exploitation is inversely proportional to the order of the molecular point group, h, an asymptotic limit which will only be reached when all MOs form symmetry-equivalent sets whose cardinality is equal to h. Given the limited size of all three molecular examples in Fig. <ref>, these upper bounds are never formally met since a few orbitals will always stay fully localized around the central (O, N, or C) atom.




By repeating the calculations of Fig. <ref> in the larger cc-pVTZ and cc-pVQZ basis sets, the results in Fig. <ref> demonstrate how the relative number of redundant calculations stays largely invariant under a change of the one-electron basis. However, in the case of extended basis sets, the localization cost function for the virtual orbital space may suffer from an abundantly large number of local minima <cit.>, and the symmetry properties of the converged minimum can adversely affect the ability to exploit point-group symmetry, as can be seen for NH_3 in the cc-pVTZ basis and CH_4 in the cc-pVQZ basis. On the other hand, the opposite may also hold true, as is clearly evident in the results for H_2O and NH_3 in the cc-pVQZ basis.




A popular symbol of the chemical sciences, the benzene molecule was recently the subject of a large, international blind challenge devoted to determining its ground-state (frozen-core) electronic energy in a modest correlation-consistent polarized double-ζ basis (cc-pVDZ) using a wide range of contemporary, high-accuracy approximations to FCI <cit.>. Following its publication, additional (post-blind) results have been reported. Among the entries in Ref. , the MBE-FCI method was able to converge the correlation energy of the benzene system to -863.0  mE_h by employing a reference space of 6 localized PM π-orbitals and a corresponding single-orbital expansion space, an estimated value situated close to the most dense part of the energy spectrum predicted by the suite of approximate FCI methods, cf. Fig. <ref>.


Alas, MBE-FCI was also by far the most expensive method benchmarked in Ref. , demanding a staggering 1.7 million core hours Computer architecture: Intel Xeon E5-2697v4 (Broadwell) hardware (36 cores @ 2.3 GHz, 3.56 GB/core).. One key reason for the large discrepancy in computational cost compared to other methods is the fact that MBE-FCI inherently involves a substantial amount of redundant increment calculations when applied to molecules of high symmetry. As an integral part of the present Letter, we will demonstrate how to lower the computational cost associated with running MBE-FCI calculations, not just to bring the method to a level on par with other modern FCI approximations, but importantly also to be able to enlarge the one-electron basis to one of polarized split-valence triple-ζ quality (cc-pVTZ). Beyond the application of improved screening protocols, tailored reference spaces, and clustered expansion spaces <cit.>, the proper exploitation of point-group symmetry in an optimized LMO basis is of paramount importance for achieving this objective. In addition, to further improve upon the accuracy of MBE-FCI, we will opt to target the gap in energy between FCI and a so-called base model (cf. Ref. ) rather than the full correlation energy <cit.>.


The SI contains a detailed discussion on how optimal settings behind our MBE-FCI calculations were chosen. In brief summary, a combination of a reference space consisting of the two sets of degenerate π-orbitals in the CMO basis and an expansion space of PM LMOs was found to allow for correlation to be optimally accounted for in the former space, while still permitting tight convergence of an MBE in the latter space upon moving to high expansion orders. Likewise, MO pairs rather than single MOs as the expansion objects were found to be beneficial in terms of reductions to the number of increment calculations that need to be evaluated. Finally, coupled cluster models with a full account of triple <cit.> (CCSDT) and quadruple <cit.> (CCSDTQ) excitations were employed as so-called base models, given how both are expected to constitute favourable and orbital-invariant approximations to FCI, unlike, for instance, the perturbative CCSD(T) <cit.> and CCSDT(Q) <cit.> models, cf. Fig. <ref>.


The results of MBE-FCI calculations on benzene in the cc-pVDZ basis—in full D_6h point-group symmetry and employing both choices of base models—are compared to other state-of-the-art methods in Fig. <ref>. While most other post-blind results for the correlation energy fall below the average result of Ref. , both energies predicted in the present Letter are instead shifted upwards, in perfect agreement with the CCSDTQ reference result, the residual error of which is expected to be marginal with respect to FCI. While the overall interpretation of both accuracy as well as reliability of all the individual entries in Fig. <ref> is left to the reader, our updated MBE-FCI results undoubtedly constitute a significant improvement upon the original result in Ref. , not merely in terms of accuracy (on account of our use of base models), but importantly also in terms of time-to-solution, due to orbital clustering and, foremost, the efficient exploitation of point-group symmetry. The MBE-FCI/CCSDT and MBE-FCI/CCSDTQ calculations required no more than 1,700 and 2,600 core hours, respectively Computer architecture used throughout the present study: Two Intel Gold 6130 per node (32 cores @ 2.1 GHz, 5.53 GB/core). The initial base model calculation on the full system has not been included in all reported timings as this preliminary step is not amenable to embarrassingly parallel computing in the same way as MBE-FCI is.; in comparison to the MBE-FCI calculation reported in Ref. , the computational effort has thus been reduced by a massive three orders of magnitude. In accumulative figures, a total of 89% of the required CASCI calculations were omitted due to our handling of point-group symmetries among the LMOs (2.6M redundant calculations).




Having reassessed the performance of the revamped MBE-FCI method in the modest cc-pVDZ basis, we will now extend the description of the benzene system to the larger cc-pVTZ basis set. While a CCSDTQ base model calculation is infeasible for a system with N = 254 orbitals to correlate, given its formal 𝒪(N^10) scaling, CCSDT can still be employed as a base model. Fig. <ref> compares the CCSDT reference result to alternative CC methods as well as a quantum Monte Carlo result from Ref. . Although not variational, it is arguably fair to assume that the CCSDT result is too high in energy for this system, also considering the comparison between CCSDT and CCSDTQ in the smaller cc-pVDZ basis, cf. Fig. <ref>. The perturbative CCSDT(Q) model, on the other hand, likely yields an energy slightly below the exact answer, again drawing conclusions on the basis of Fig. <ref>, but also its known tendency to overshoot correlation energies of the full CCSDTQ method <cit.>. The MBE-FCI/CCSDT entry in Fig. <ref> yields a correlation energy of -1028.7  mE_h, observed to fall right between the CCSDT and CCSDT(Q) result. While this energy will likely decrease slightly at potential later orders of the MBE, comparisons with the more accurate MBE-FCI result employing the CCSDTQ base model in the cc-pVDZ basis (Fig. <ref>) indicate that the resulting error will not exceed a conservative estimate of 1 kcal/mol. On account of our utilization of the full D_6h point group, a massive 238M CASCI calculations (90%) could be skipped due to symmetry equivalences, and the calculation completed in only a total of 39,700 core hours.


In summary, we have introduced a novel method for exploiting (non-)Abelian point-group symmetries in electronic-structure methods that operate in a basis of spatially localized molecular orbitals. On account of an exact symmetrization of said orbital basis, equivalences of tensor elements or incremental energy contributions, as in the present application to MBE-FCI theory, will lead to significant computational savings, which we have verified through numerical tests using various basis sets for a number of systems belonging to different molecular point groups. In combination with an efficient clustering of orbitals, the developments reported here have been demonstrated to considerably improve the computational efficacy of MBE-FCI theory. In further conjunction with correlated base models, we have applied MBE-FCI to the calculation of ground-state correlation energies of benzene in polarized split-valence basis sets of both double- and triple-ζ quality. Leveraging the full D_6h point group of the benzene system, our earlier cc-pVDZ result from Ref.  has been substantially refined at a significantly lower cost, and results in the larger cc-pVTZ basis have been made feasible in convincing agreement with high-level coupled cluster results. On account of the efficient exploitation of point-group symmetries reported herein, we believe MBE-FCI can help pave the way towards near-exact calculations on molecules and intermolecular complexes of unprecedented sizes. In addition, the integrated use of symmetry within incremental formulations of complete active space self-consistent field (CASSCF) methods holds promise of extending the scope of systems amenable to such electronic-structure treatments <cit.>.



§ ACKNOWLEDGMENTS


This work was supported by two research grants awarded to JJE, no. 37411 from VILLUM FONDEN (a part of THE VELUX FOUNDATIONS) and no. 10.46540/2064-00007B from the Independent Research Fund Denmark. The authors gratefully acknowledge the computing time granted on the Mogon II supercomputer by the Johannes Gutenberg-Universität Mainz (<hpc.uni-mainz.de>).



§ SUPPORTING INFORMATION


The supporting information (SI) contains a list of all acronyms of Figs. <ref> and <ref> (Table S1), further details on our implementation of symmetry within MBE-FCI and the  code, as well as details on the calibration of the MBE-FCI calculations behind Figs. <ref> and <ref>.



§ DATA AVAILABILITY


Data in support of the findings of this study are available within the article and its SI.







@ifundefinedendmcitethebibliography
  
44








f
subitem(mcitesubitemcount)

  
  

[Pitzer(1973)]Pitzer1973
Pitzer, R. M. Electron Repulsion Integrals and Symmetry Adapted Charge
  Distributions. J. Chem. Phys. 1973, 59, 3308
[Davidson(1975)]Davidson1975
Davidson, E. R. Use of Double Cosets in Constructing Integrals Over Symmetry
  Orbitals. J. Chem. Phys. 1975, 62, 400
[Dupuis and King(1977)Dupuis, and King]Dupuis1977
Dupuis, M.; King, H. F. Molecular Symmetry and Closed-Shell SCF Calculations.
  I. Int. J. Quantum Chem. 1977, 11, 613
[Takada et al.(1983)Takada, Dupuis, and King]Takada1983
Takada, T.; Dupuis, M.; King, H. F. Molecular Symmetry. IV. The Coupled
  Perturbed Hartree–Fock Method. J. Comput. Chem. 1983,
  4, 234
[Taylor(1985)]Taylor1985
Taylor, P. R. Symmetrization of Operator Matrix Elements. Int. J. Quantum
  Chem. 1985, 27, 89
[Taylor(1986)]Taylor1986
Taylor, P. R. Symmetry-Adapted Integral Derivatives. Theor. Chim. Acta
  1986, 69, 447
[Čársky et al.(1987)Čársky, Schaad, Hess, Urban, and
  Noga]Carsky1987
Čársky, P.; Schaad, L. J.; Hess, B. A.; Urban, M.; Noga, J. Use of Molecular
  Symmetry in Coupled-Cluster Theory. J. Chem. Phys. 1987,
  87, 411
[Häser et al.(1991)Häser, Almlöf, and
  Feyereisen]Haeser1991
Häser, M.; Almlöf, J.; Feyereisen, M. W. Exploiting Non-Abelian Point Group
  Symmetry in Direct Two-Electron Integral Transformations. Theor. Chim.
  Acta 1991, 79, 115
[Stanton et al.(1991)Stanton, Gauss, Watts, and
  Bartlett]Stanton1991
Stanton, J. F.; Gauss, J.; Watts, J. D.; Bartlett, R. J. A Direct Product
  Decomposition Approach for Symmetry Exploitation in Many-Body Methods. I.
  Energy Calculations. J. Chem. Phys. 1991, 94,
  4334
[Greiner and Eriksen(2023)Greiner, and Eriksen]Greiner2023
Greiner, J.; Eriksen, J. J. Symmetrization of Localized Molecular Orbitals.
  J. Phys. Chem. A 2023, 127, 3535
[Eriksen et al.(2017)Eriksen, Lipparini, and
  Gauss]Eriksen2017
Eriksen, J. J.; Lipparini, F.; Gauss, J. Virtual Orbital Many-Body Expansions:
  A Possible Route Towards the Full Configuration Interaction Limit. J.
  Phys. Chem. Lett. 2017, 8, 4633
[Eriksen and Gauss(2018)Eriksen, and Gauss]Eriksen2018
Eriksen, J. J.; Gauss, J. Many-Body Expanded Full Configuration Interaction. I.
  Weakly Correlated Regime. J. Chem. Theory Comput. 2018,
  14, 5180
[Eriksen and Gauss(2019)Eriksen, and Gauss]Eriksen2019
Eriksen, J. J.; Gauss, J. Many-Body Expanded Full Configuration Interaction.
  II. Strongly Correlated Regime. J. Chem. Theory Comput.
  2019, 15, 4873
[Eriksen and Gauss(2019)Eriksen, and Gauss]Eriksen2019a
Eriksen, J. J.; Gauss, J. Generalized Many-Body Expanded Full Configuration
  Interaction Theory. J. Phys. Chem. Lett. 2019, 10,
  7910
[Eriksen and Gauss(2020)Eriksen, and Gauss]Eriksen2020
Eriksen, J. J.; Gauss, J. Ground and Excited State First-Order Properties in
  Many-Body Expanded Full Configuration Interaction Theory. J. Chem.
  Phys. 2020, 153, 154107
[Eriksen and Gauss(2021)Eriksen, and Gauss]Eriksen2021
Eriksen, J. J.; Gauss, J. Incremental Treatments of the Full Configuration
  Interaction Problem. Wiley Interdiscip. Rev.: Comput. Mol. Sci.
  2021, 11, e1525
[Greiner et al.(2024)Greiner, Gauss, and Eriksen]Greiner2024a
Greiner, J.; Gauss, J.; Eriksen, J. J. Error Control and Automatic Detection
  of Reference Active Spaces in Many-Body Expanded Full Configuration
  Interaction. 2024; arXiv:2406.11343
[Friedrich et al.(2008)Friedrich, Hanrath, and
  Dolg]Friedrich2008
Friedrich, J.; Hanrath, M.; Dolg, M. Using Symmetry in the Framework of the
  Incremental Scheme: Molecular Applications. Chem. Phys. 2008,
  346, 266
[Pipek and Mezey(1989)Pipek, and Mezey]Pipek1989
Pipek, J.; Mezey, P. G. A Fast Intrinsic Localization Procedure Applicable for
  Ab Initio and Semiempirical Linear Combination of Atomic Orbital
  Wave Functions. J. Chem. Phys. 1989, 90, 4916
[Greiner and Eriksen()Greiner, and Eriksen]pymbe
Greiner, J.; Eriksen, J. J. : A Many-Body Expanded Correlation
  Code. See: <https://gitlab.com/januseriksen/pymbe>
[Dunning(1989)]Dunning1989
Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular
  Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem.
  Phys. 1989, 90, 1007
[Foster and Boys(1960)Foster, and Boys]Foster1960
Foster, J. M.; Boys, S. F. Canonical Configurational Interaction Procedure.
  Rev. Mod. Phys 1960, 32, 300
[Subotnik et al.(2004)Subotnik, Shao, Liang, and
  Head-Gordon]Subotnik2004
Subotnik, J. E.; Shao, Y.; Liang, W.; Head-Gordon, M. An Efficient Method for
  Calculating Maxima of Homogeneous Functions of Orthogonal Matrices:
  Applications to Localized Occupied Orbitals. J. Chem. Phys.
  2004, 121, 9220
[Subotnik et al.(2005)Subotnik, Dutoi, and
  Head-Gordon]Subotnik2005
Subotnik, J. E.; Dutoi, A. D.; Head-Gordon, M. Fast Localized Orthonormal
  Virtual Orbitals Which Depend Smoothly on Nuclear Coordinates. J. Chem.
  Phys. 2005, 123, 114108
[Blunt et al.(2019)Blunt, Thom, and Scott]Blunt2019
Blunt, N. S.; Thom, A. J. W.; Scott, C. J. C. Preconditioning and Perturbative
  Estimators in Full Configuration Interaction Quantum Monte Carlo. J.
  Chem. Theory Comput. 2019, 15, 3537
[Eriksen et al.(2020)Eriksen, Anderson, Deustua, Ghanem, Hait,
  Hoffmann, Lee, Levine, Magoulas, Shen, Tubman, Whaley, Xu, Yao, Zhang, Alavi,
  Chan, Head-Gordon, Liu, Piecuch, Sharma, Ten-no, Umrigar, and
  Gauss]Eriksen2020a
Eriksen, J. J.; Anderson, T. A.; Deustua, J. E.; Ghanem, K.; Hait, D.;
  Hoffmann, M. R.; Lee, S.; Levine, D. S.; Magoulas, I.; Shen, J.;
  Tubman, N. M.; Whaley, K. B.; Xu, E.; Yao, Y.; Zhang, N.; Alavi, A.; Chan, G.
  K.-L.; Head-Gordon, M.; Liu, W.; Piecuch, P.; Sharma, S.; Ten-no, S. L.;
  Umrigar, C. J.; Gauss, J. The Ground State Electronic Energy of Benzene.
  J. Phys. Chem. Lett. 2020, 11, 8922
[Lee et al.(2020)Lee, Malone, and Reichman]Lee2020
Lee, J.; Malone, F. D.; Reichman, D. R. The Performance of Phaseless
  Auxiliary-Field Quantum Monte Carlo on the Ground State Electronic Energy of
  Benzene. J. Chem. Phys. 2020, 153, 126101
[Loos et al.(2020)Loos, Damour, and Scemama]Loos2020
Loos, P.-F.; Damour, Y.; Scemama, A. The Performance of CIPSI on the Ground
  State Electronic Energy of Benzene. J. Chem. Phys. 2020,
  153, 176101
[Rask and Zimmerman(2021)Rask, and Zimmerman]Rask2021
Rask, A. E.; Zimmerman, P. M. Toward Full Configuration Interaction for
  Transition-Metal Complexes. J. Phys. Chem. A 2021,
  125, 1598
[Not()]Note-1
Computer architecture: Intel Xeon E5-2697v4 (Broadwell) hardware (36 cores @
  2.3 GHz, 3.56 GB/core).
[Noga and Bartlett(1987)Noga, and Bartlett]ccsdt_paper_1_jcp_1987
Noga, J.; Bartlett, R. J. The Full CCSDT Model for Molecular Electronic
  Structure. J. Chem. Phys. 1987, 86, 7041
[Noga and Bartlett(1988)Noga, and
  Bartlett]ccsdt_paper_1_jcp_1988_erratum
Noga, J.; Bartlett, R. J. Erratum: The Full CCSDT Model for Molecular
  Electronic Structure. J. Chem. Phys. 1988, 89,
  3401
[Scuseria and Schaefer, III(1988)Scuseria, and Schaefer,
  III]ccsdt_paper_2_cpl_1988
Scuseria, G. E.; Schaefer, III, H. F. A New Implementation of the Full CCSDT
  Model for Molecular Electronic Structure. Chem. Phys. Lett.
  1988, 152, 382
[Oliphant and Adamowicz(1991)Oliphant, and
  Adamowicz]ccsdtq_paper_1_jcp_1991
Oliphant, N.; Adamowicz, L. Coupled-Cluster Method Truncated at Quadruples.
  J. Chem. Phys. 1991, 95, 6645
[Kucharski and Bartlett(1992)Kucharski, and
  Bartlett]ccsdtq_paper_2_jcp_1992
Kucharski, S. A.; Bartlett, R. J. The Coupled-Cluster Single, Double, Triple,
  and Quadruple Excitation Method. J. Chem. Phys. 1992,
  97, 4282
[Raghavachari et al.(1989)Raghavachari, Trucks, Pople, and
  Head-Gordon]original_ccsdpt_paper
Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A Fifth-Order
  Perturbation Comparison of Electron Correlation Theories. Chem.
  Phys. Lett. 1989, 157, 479
[Bomble et al.(2005)Bomble, Stanton, Kállay, and
  Gauss]bomble_ccsdt_pq_jcp_2005
Bomble, Y. J.; Stanton, J. F.; Kállay, M.; Gauss, J. Coupled-Cluster
  Methods Including Noniterative Corrections for Quadruple Excitations.
  J. Chem. Phys. 2005, 123, 054101
[Not()]Note-2
Computer architecture used throughout the present study: Two Intel Gold 6130
  per node (32 cores @ 2.1 GHz, 5.53 GB/core). The initial base model
  calculation on the full system has not been included in all reported timings
  as this preliminary step is not amenable to embarrassingly parallel computing
  in the same way as MBE-FCI is.
[Eriksen et al.(2014)Eriksen, Kristensen, Kjærgaard,
  Jørgensen, and Gauss]eriksen_ccsd_pert_theory_jcp_2014
Eriksen, J. J.; Kristensen, K.; Kjærgaard, T.; Jørgensen, P.; Gauss, J.
  A Lagrangian Framework for Deriving Triples and Quadruples Corrections to
  the CCSD Energy. J. Chem. Phys. 2014, 140,
  064108
[Eriksen et al.(2016)Eriksen, Kristensen, Matthews,
  Jørgensen, and Olsen]eriksen_convergence_ccpt_jcp_2016
Eriksen, J. J.; Kristensen, K.; Matthews, D. A.; Jørgensen, P.; Olsen, J.
  Convergence of Coupled Cluster Perturbation Theory. J. Chem.
  Phys. 2016, 145, 224104
[Eriksen et al.(2015)Eriksen, Matthews, Jørgensen, and
  Gauss]eriksen_quadruples_pert_theory_jcp_2015
Eriksen, J. J.; Matthews, D. A.; Jørgensen, P.; Gauss, J. Communication:
  The Performance of Non-Iterative Coupled Cluster Quadruples Models.
  J. Chem. Phys. 2015, 143, 041101
[Eriksen et al.(2016)Eriksen, Matthews, Jørgensen, and
  Gauss]eriksen_open_shell_quadruples_jcp_2016
Eriksen, J. J.; Matthews, D. A.; Jørgensen, P.; Gauss, J. Assessment of the
  Accuracy of Coupled Cluster Perturbation Theory for Open-Shell Systems. II.
  Quadruples Expansions. J. Chem. Phys. 2016,
  144, 194103
[Greiner et al.(2024)Greiner, Gianni, Nottoli, Lipparini,
  Eriksen, and Gauss]Greiner2024
Greiner, J.; Gianni, I.; Nottoli, T.; Lipparini, F.; Eriksen, J. J.; Gauss, J.
  MBE-CASSCF Approach for the Accurate Treatment of Large Active Spaces.
  J. Chem. Theory Comput. 2024, 20, 4663




