Paper Title: An iterative Monte Carlo method to solve nonlinear second-order differential equations.



The dynamical α-Rényi entropies of local Hamiltonians grow at most linearly in time
    Sougato Bose 0000-0001-8726-0566
    August 5, 2024
===================================================================================



§ ABSTRACT 
The Monte Carlo method is a thriving and mathematically beautiful numerical technique used extensively, nowadays, to deal with many demanding problems in diverse fields. Here, we present an iterative Monte Carlo algorithm to work out very general nonlinear second-order differential equations, with Dirichlet boundary conditions. An example of its usage is, also, reported.  




§ INTRODUCTION

Since the conception of the Monte Carlo method (MCM) in its modern form <cit.>, this tantalising mathematical idea has become a powerful and successful route to solve extremely difficult and even “intractable" <cit.> problems in many areas of knowledge <cit.>. In particular, the usage of the Monte Carlo probabilistic approach for the numerical solution of differential equations has a long history <cit.> and represents one of its most useful applications in science and technology. In this regard, the amazing relation discovered between Brownian motion and potential theory <cit.> has prompted the employment of the random walk-based MCM to solve the very fundamental Poisson equation <cit.>, i.e., ∇^2 ϕ( r⃗ ) =-g. Despite the importance of this subject, to the best of our knowledge, there exists no general Monte Carlo algorithm to solve the Poisson equation when g also depends on ϕ( r⃗ ) and/or on its derivatives, apart from an explicit dependence on r⃗. In this context, below, we introduce and exemplify an iterative Monte Carlo method that allows to treat numerically ordinary differential equations of the form d^2 y( x ) dx^2=F(  x, y( x ), d y( x ) dx), with Dirichlet boundary conditions. Notably, this reported probabilistic procedure can be readily expanded to deal with partial differential equations.   



§ THE METHOD

A second-order ordinary differential equation (ODE) for y(x), with Dirichlet boundary conditions, is generally stated as: 


    E(x, y( x ), d y( x ) dx, d^2 y( x ) dx^2)=0,
  

such as y(a)=y_a and y(b)=y_b, ∋  a<b.

If Eq. (<ref>) can be put in the form 


    d^2 y( x ) dx^2=F(  x, y( x ), d y( x ) dx),


the Iterative Monte Carlo Method (IMCM), discussed in the following, can be utilised to solve Eq. (<ref>), for y(a)=y_a and y(b)=y_b. As a preliminary, let us, then, consider a simpler case of such equation, i.e., 


    d^2 y( x ) dx^2=F( x )


(notice that the RHS term of the previous equation only contains an explicit dependence on the variable x). The usual random walk-based Monte Carlo algorithm to solve Eq. (<ref>) comprises the next steps <cit.>:  



STEP 1: Given a uniform mesh {  x_0=a, x_1, x_2,..., x_N-1, x_N=b }, for each free point x_i, such that i=1,...,N-1, generate a large number, K, of random walks, which start at x_i and end when hitting an absorbing boundary site, either x_0 or x_N.



STEP 2: If the j-th walk arrives at the boundary after m^(i,j) steps and has visited the sequence of locations {  P_0^(i,j)=x_i, P_1^(i,j),P_2^(i,j),..., P_m^(i,j)^(i,j)=x_end^(i,j)}, calculate the Monte Carlo estimator for y(x_i) from 


    y(x_i)=1K∑_j=1^K  y(x_end^(i,j))-h^22K∑_j=1^K{∑_l=0^μ^(i,j) F( P_l^(i,j))  } ,

 
where μ^(i,j)=m^(i,j)-1 and y(x_end^(i,j)) being the value, y_a or y_b, of the function y(x) at the particular absorbing boundary point, x_0=a or x_N=b, reached by the j-th random walk initiated at x_i. Observe that the approximate values of y(x) for all the points in the mesh can be obtained for any arbitrary sequence of the indices in the set 1,...,N-1, and, besides, that the computer implementation of the procedure can be easily parallelised.   



Now, regarding the more general ODE stated by Eq. (<ref>), in theory, the typical random walk methodology reviewed above can not be applied to work out such ODE, since the RHS of Eq. (<ref>) depends, precisely, on the unknown functions y( x ) and y'(x)=d y( x ) dx. However, and given that, at the end, these two sought functions depend on x, the first version of our Iterative Monte Carlo Method (IMCM) proposes the use of an initial guess function, y^[0](x), with a well-known and simple dependence on x in the interval [x_0, x_N] and that fulfills the associated boundary conditions y^[0](x_0)=y_0=y_a and y^[0](x_N)=y_N=y_b; this trial function will allow us to evaluate the last term in the RHS of Eq. (<ref>) and, consequently, to set up the ensuing practicable estimator for the value of the next guess function y^[1](x) at x_i:


    y^[1](x_i)=1K∑_j=1^K  y(x_end^(i,j)) - 
    h^22K∑_j=1^K{∑_l=0^μ^(i,j) F( P_l^(i,j), y^[0](P_l^(i,j)), y^[0]'(P_l^(i,j)))  } ,


for i=1,...,N-1, with y^[0]'(x_i) being the numerical derivative of y^[0](x) at each node (which can be calculated through any finite difference formula). After determining the values of y^[1](x) for all the free nodes, the corresponding numerical derivative has to be got to proceed with successive approximations of y(x), obtained via the iteration prescription 


    y^[s+1](x_i)=1K∑_j=1^K  y(x_end^(i,j)) - 
    h^22K∑_j=1^K{∑_l=0^μ^(i,j) F( P_l^(i,j), y^[s](P_l^(i,j)), y^[s]'(P_l^(i,j)))  } ,


complemented with a finite difference approximation of its derivative. In a similar way to the scheme associated to Eq. (<ref>), the prior iteration expression can be used to get y^[s+1](x_i) for all the free points following any order of the mesh indices. Additionally, this first version of the IMCM, given by Eq. (<ref>), can be likewise parallelised. 

On the other hand, based on the idea of successive displacement we can try now to improve our IMCM as follows: 



STEP 1: Advance an initial guess function, y^[0](x), in [x_0, x_N], that satisfies the boundary conditions of the problem and, then, initialise ỹ(x_i)=y^[0](x_i), for i=0,...,N. 



STEP 2: For each free node, perform K absorbing random walks starting at x_i to obtain and update immediately ỹ(x_i) employing the formula 


    ỹ(x_i)=1K∑_j=1^K  y(x_end^(i,j)) - 
    h^22K∑_j=1^K{∑_l=0^μ^(i,j) F( P_l^(i,j), ỹ(P_l^(i,j)), ỹ'(P_l^(i,j)))  } .




STEP 3: Right after calculating ỹ(x_i) in STEP 2, differentiate it numerically to produce ỹ'(x_i).



STEP 4: Repeat entirely the previous steps 2 and 3 to complete a cycle of T iterations. 



This last stochastic and successive displacement process represents a refined, and seemingly hastened, version of the IMCM introduced here (the pseudocode of the corresponding algorithm can be consulted in Fig. <ref>).   







§ AN EXAMPLE OF APPLICATION

Let us consider the following nonlinear second-order ODE of the type d^2 y( x ) dx^2=F(  x, y( x ) ) <cit.>: 


    d^2 y( x ) dx^2=π^2 sin(π x ) +e^sin(π x )+x-e^-y( x ) ,


with y( 0)=0 and y( 1 ) =-1. Eq. (<ref>) has the exact solution 


    y^exact( x ) = -sin(π x )-x,


which it will be utilised below to asses the performance of our approximate numerical IMCM results. 

Coding the enhanced IMCM procedure described at the end of Section <ref> (see Fig. <ref>), such that  


    F( x, y( x ) ) = π^2 sin(π x ) +e^sin(π x )+x-e^-y( x ) ,


we obtained the data included in Fig. <ref>. Therein, we show the exact solution given by Eq. (<ref>) (red open symbols), the linear initial guess (green interrupted line), the IMCM results after a first iteration with Eq. (<ref>) (blue continuous line), and the IMCM prediction after ten iterations with Eq. (<ref>) (black continuous line). In each Monte Carlo iteration we have employed a fixed grid in [ 0,1 ], with N=100 equally-sized partitions, combined with K=10^5 random walks, to estimate the value of y(x) at each non-boundary node of the grid via Eq. (<ref>). The error in the s-th iteration, E^( s ) is calculated from 


    E^( s )= √(∑_i=0^N[ y^exact( x_i ) - ỹ^( s ) ( x_i ) ] ^2) ,
 

with ỹ^( s )( x ) being the s-th IMCM approximation. The corresponding errors for the IMCM results for iterations 1 and 10, plotted in Fig. <ref>, are 1.001× 10^0 and 2.787× 10^-2, respectively.

  



§ CONCLUSIONS
 
In this paper, we have proposed an Iterative Monte Carlo Method to solve second-order ODEs of the general form d^2 y( x ) dx^2=F(  x, y( x ), d y( x ) dx), with Dirichlet boundary conditions. Our algorithm is based on the classic random walk approach, but it is enriched with an iterative process starting from a trial or guess function, which allows the evaluation of the RHS of the ODE. To illustrate its performance, we have also included an example of application. The presented IMCM can be extended straightforwardly to second-order partial differential equations. 

100
	Ulam1987 N. G. Cooper, N. G., Los Alamos Science Number 15: Special Issue on Stanislaw Ulam 1909-1984 (1987). doi:10.2172/1054744
	
	Traub1994 J. F. Traub and H. Woźniakowski, Breaking Intractability, Scientific American 270, Issue 1, 102 (1994). doi:10.1038/scientificamerican0194-102
	 
	Hammersley1964 J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Methuen & Co. Ltd., 1964).
	
	Courant1928 R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100, 32 (1928). doi.org/10.1007/BF01448839 (English translation in IBM J. Res. Develop. 11, 215 (1967). doi.org/10.1147/rd.112.0215).
	
	Hersh1969 R. Hersh and R. J. Griego, Brownian Motion and Potential Theory, Scientific American 220, Issue 3, 66 (1969). doi:10.1038/scientificamerican0369-66
	
	Sadiku2015 M. N. O. Sadiku, R. C. Garcia, S. M. Musa, and S. R. Nelatury, Markov Chain Monte Carlo Solution of Poisson's Equation, International Journal on Recent and Innovation Trends in Computing and Communication 3, 106 (2015). doi:10.17762/ijritcc.v3i1.3770 
	
	Yin2014 P. Yin, Y. Huang, and H. Liu, An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation, Communications in Computational Physics 16, 491 (2014). doi:10.4208/cicp.270713.280214a  
	
	  
	                




 Paper Title: Describe, Transform, Machine Learning: Feature Engineering for Grain Boundaries and Other Variable-Sized Atom Clusters.

























































[cor1]Corresponding author

BYU_CS]C. Braxton Owens
LANL1]Nithin Mathew
BYU_Phys]Tyce W. Olaveson
NASA]Jacob P. Tavenner
LANL1]Edward M. Kober
Baylor]Garritt J. Tucker
BYU_Phys]Gus L. W. Hartcor1
gus.hart@gmail.com
BYU_ME]Eric R. Homercor1
eric.homer@byu.edu

[BYU_CS]organization=Department of Computer Science Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            


[LANL1]organization=Group T-1, Theoretical Division, Los Alamos National Laboratory,
            
            city=Los Alamos,
            postcode=87544, 
            state=NM,
            country=USA
            
            
[LANL2]organization=Center for Non-Linear Studies, Theoretical Division, Los Alamos National Laboratory,
            
            city=Los Alamos,
            postcode=87544, 
            state=NM,
            country=USA
            
            
[NASA]organization=KBR, Inc., Intelligent Systems Division, NASA Ames Research Center,
            
            city=Moffett Field,
            postcode=94035, 
            state=CA,
            country=USA
            
            
[Baylor]organization=Department of Physics, Baylor University,
            
            city=Waco,
            postcode=76798, 
            state=TX,
            country=USA
            

[BYU_Phys]organization=Department of Physics and Astronomy Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            

[BYU_ME]organization=Department of Mechanical Engineering Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            

§ ABSTRACT 

Obtaining microscopic structure-property relationships for grain boundaries are challenging because of the complex atomic structures that underlie their behavior. This has led to recent efforts to obtain these relationships with machine learning, but representing a grain boundary structure in a manner suitable for machine learning is not a trivial task. There are three key steps common to property prediction in grain boundaries and other variable-sized atom clustered structures. These are: (1) describe the atomic structure as a feature matrix, (2) transform the variable-sized feature matrices of different structures to a fixed length common to all structures, and (3) apply machine learning to predict properties from the transformed feature matrices. We examine these feature engineering steps to understand how they impact the accuracy of grain boundary energy predictions. A database of over 7000 grain boundaries serves to evaluate the different feature engineering combinations. We also examine how these combination of engineered features provide interpretability, or the ability to extract insightful physics from the obtained structure-property relationships.






Grain Boundaries Atomic Structure Structure Descriptor Machine learning Feature Engineering Structure-Property Relationships














§ INTRODUCTION


Due to the impact of grain boundaries (GBs) on material properties <cit.>, there is a need to better understand the relationship between the structure of a GB and its corresponding properties. With expanding computing power, increasingly large amounts of data, and advances in data-driven approaches, there has been a push for suitable representations of GBs in order to predict their properties <cit.>. However, accurate property prediction is not the only measure of success. Models and representations that provide insight into structure-property relationships are key to advance our understanding.
 
Representing a GB starts with defining its structure, since it has both macroscopic and microscopic characteristics. Macroscopically, five degrees of freedom are used to define the GB character: Three to define a misorientation between two crystals, (often given by a normalized rotation axis [uvw] and angle θ), and two to define a boundary plane (hkl). Microscopically, the positions of the atoms result in 3n degrees of freedom. Additionally, a GB can assume various metastable configurations under any given set of macroscopic constraints.<cit.>. 

While there have been impressive developments in macroscopic representations for better understanding GBs <cit.>, the atomic structure is what defines a GB's properties. This article concentrates on the microscopic structure-property relationships since the macroscopic structure acts as a constraint on the microscopic structure.   

One microscopic method for defining a GB is the structural unit model <cit.>. This model describes the atomic structure of a quasi two-dimensional GB as a series of repeating atomic “structural units" that are characteristic of the boundary's local atomic arrangement. This simplifies analysis of its geometry and properties as long as it is quasi two-dimensional. In recent years, the structural unit model has been modified to more accurately represent a GB by considering the effects of metastable structures <cit.>. Other early methods for defining local atomic environments for characterizing GBs include: the centrosymmetry parameter (CSP) <cit.>, Voronoi index <cit.>, excess volume <cit.>, common neighbor analysis (CNA) <cit.>, the Polyhedral Unit Model <cit.>, and local entropy <cit.>. 





Although the early methods have been used mainly to classify GB atoms <cit.> they have also been used for machine learning predictions of atomic level properties <cit.>. Advanced methods, some of which are described below, have also been used to predict atomic level properties in GBs <cit.>. 


However, when using machine learning to predict properties of a GB as a whole, the descriptor must be transformed in some way to achieve a consistent feature size. This notion of transformation is an important one for variable-sized atomic structures such as GBs because different GBs will have different numbers of atoms (features) in their structure. Transformation is one step in a three step feature engineering process common to almost all machine learning predictions for variable-sized atom structures, such as GBs. 
These steps are illustrated in Figure <ref> and are described as follows:
1. Describe the atomic structure with an encoding algorithm, descriptor, or fingerprint of some kind, which is often represented as a matrix or vector. 2. Transform the mapping of the variable length descriptor for each structure to a fixed length descriptor common across all structures in a dataset. 3. Apply Machine Learning models or regression algorithms, to learn and then predict the property of a given atomic structure. 

We illustrate the consistency of these steps, occasionally combined in different orders or sequences, in numerous applications of machine learning to predict properties based on microscopic GB structure in the paragraphs below. These examples will also serve to highlight the variety of descriptors, transforms, and machine learning algorithms employed by the community.

Snow et al. <cit.> utilized the graphs underlying CNA to describe the atoms in each GB, categorizing them into 2205 distinct environments. They then performed principal component analysis to reduce these environments into 84 principal components, which constitutes a second description step. To standardize the representation size, a transform was applied, representing each GB by the proportions of the 84 components present. This transformed representation was used as input to a linear regression machine learning model to predict GB energy.

Guziewiski et al. <cit.> also explore this concept of proportions, utilizing the diamond-structure identification and the polyhedral template matching algorithms to describe GB atoms. This was transformed into a fixed-length density metric for each GB by counting the number of atoms within each polyhedral template class and normalizing this by the GB area. Random forest machine learning models were then used to predict both the GB energy and the tensile strength of the GB. 

Gomberg et al. <cit.> utilized a specialized pair correlation function <cit.> to describe their GB structures. This function is unique due to its use of a probability distribution function, allowing equal sampling for each GB and simultaneously transforming the descriptor into a fixed length. This representation was further refined to the first two principal components, constituting a second describe step, which is then used as input into a regression machine learning model. Dang and Yu extended Gomberg's method by incorporating the standard deviation of the pair correlation function through a weight parameter <cit.>.

More recently, the GB community has used atomic structure descriptors developed by the machine learned interatomic potential community. These descriptors are attractive because they are inspired by the symmetries and physical response inherent to the atoms, as described by Musil et al.<cit.>.
Rosenbrock et al. <cit.> implemented one of these physics inspired descriptors, called the smooth overlap of atomic positions (SOAP), to describe the GB and then transformed the SOAP descriptor into a fixed-length vector by averaging over the atom environments. These are then used to predict GB energy using a support vector machine learning model. Later, Fujii et al.<cit.> used SOAP to calculate a local distortion factor, describing how similar a GB atom's environment is to a bulk atom's environment. This was then transformed into a fixed-length using complete-linkage clustering. The GB thermal conductivity was then predicted by a ridge regression machine learning algorithm.

 
In this paper, we examine the impact of using different descriptors, transforms, and machine learning models for feature engineering for predicting GB energy, as illustrated in Figure <ref>. The exact descriptors, transforms, and machine learning algorithms we employ are listed in Figure <ref> and described in detail in the Methods Section. The feature engineering is tested on a dataset comprising 7304 aluminum GBs, which provides comprehensive coverage of the 5-dimensional macroscopic space of crystallographic character <cit.>. The interplay of various descriptors, transforms, and machine learning algorithms are analyzed for their effect on the accuracy of their predictions. In addition, we examine how feature reduction impacts the results for select cases. Finally, we assess the interpretative ability of some key descriptors to establish meaningful connections to the inherent structure of the GB. Prioritizing interpretability is imperative to increase our understanding of atomic GB structure-property relationships.






§ RESULTS





 §.§ Accuracy of Predictions


As illustrated in Figure <ref>, there are essentially 3 methods or “knobs” that can be adjusted to improve accuracy of predictions: descriptor, transform, and machine learning algorithm. As noted above, the descriptors, transforms, and machine learning algorithms we employ are listed in Figure <ref> and described in detail in the Methods Section. 



We start by examining Figure <ref>, which shows parity plots comparing machine learning predictions of GB energy against the GB energy values calculated in LAMMPS, as reported in <cit.>. There is one plot for each of the 7 descriptors examined in this work: The Atomic Cluster Expansion (ACE), SOAP, Atom Centered Symmetry Functions (ACSF), Strain Functional (SF) descriptors, a Graph description, CSP, and CNA. For each of these 7 descriptors, the results are reported for the combination of transform and machine learning algorithm that resulted in the highest overall accuracy predicted for GB energy using this dataset. The accuracy of predictions is measured by the mean absolute error (MAE) and R-squared (R^2) values. These metrics are used in tandem to best illustrate the performance of a given model. Finally, these 7 descriptors are accompanied by a parity plot labeled as “Random SOAP” where the SOAP descriptor is used as input, but with the GB energy values shuffled so that every SOAP vector points to a random GB energy value in the set. This serves as a worst-case reference value since a shuffled dataset would be expected to have no correlations.

In examining Figure <ref>, the SOAP descriptor combined with Linear Regression achieves the highest accuracy, with a low MAE of 3.89 mJ/m2 and a high R^2 of 0.99, indicating near-perfect correlation between predicted and actual values. In contrast, the Random SOAP model, where GB energies are shuffled, has a high MAE of 46.96 mJ/m2 and a negative R^2 of -0.23, confirming no predictive capability. While ACE and SF descriptors also achieve high accuracy, ACSF exhibit intermediate performance and descriptors like graph (graph2vec), CNA, and CSP exhibit significantly higher MAE and lower R^2, indicating poorer predictive performance. This suggests that the higher complexity descriptors do indeed capture more relevent information for predicting grain boundary energy.

Figure <ref> illustrates key aspects of feature engineering used to predict GB energy. First, it can be seen that due to the nature of this dataset, with many GBs concentrated about the mean GB energy value of 497 mJ/m2, it is possible to obtain a relatively low MAE, even in the case of the “Random SOAP” model. It is for this reason that we report both the MAE and the R^2 values. Caution must be exercised in assuming a model is good just because the MAE is low. One must also see a high R^2 value to show that the model results in correlated predictions; a negative value for the R^2 metric indicates that it would have been better to simply predict the mean.

Second, one can see that the `average' transform is selected as the transform providing the most accurate predictions in four of seven cases. Third, in three of seven cases, the machine learning algorithm that provides the highest accuracy is linear regression. In the other four of seven cases, MLPRegression is the most accurate. But the three cases with linear regression have much better predictions than those with MLPRegression. Fourth, one can see that the combination of MAE and R^2 values provide a nice summary of the accuracy that can be visibly seen in the parity plots. Fifth, the stark contrasts of the MAE and R^2 values between the SOAP and “Random SOAP” models illustrates that there is valuable information in the features of the averaged SOAP that is correlated with the GB energy of a given structure. This is strictly true when comparing SOAP and “Random SOAP” and likely true when comparing “Random SOAP” with the other descriptors which categorize the atomic information of the GBs in distinct manners.





While important insights can be gained from these comparisons in Figure <ref>, great care must be exercised because they are not direct comparisons; the models are different. Better comparisons can be made by holding as many variables constant between the models as possible. The three key steps (describe, transform, and machine learning) represent a 3-dimensional space of combinations, as illustrated in Figure <ref>. However, since not all combinations were evaluated, we choose to examine 2-dimensional subsets, holding constant one of the three methods, as illustrated by the yellow bands in Figure <ref>. The methods that are held constant are chosen because they typically preform better than their counterparts for the GB energy predictions examined in this work. 

The first subset we analyze compares different descriptors and machine learning algorithms, keeping the averaging transform constant since it performes well for 4 of 7 descriptors in Figure <ref>. The accuracy comparison for this subset is given in Figure <ref>a. The ACE, SOAP, and SF descriptors performed exceptionally well across multiple machine learning algorithms. The R^2 values were consistently above 0.95 in the majority of cases, indicating strong predictive capabilities, while the MAE values were comparatively low, <10 mJ/m2 in the majority of cases, reflecting accuracy in the energy predictions. ACSF had higher errors (MAE just less than 20 mJ/m2 and R^2 around 0.8). CSP, CNA, and the graph descriptor had errors around 25 mJ/m2 and R^2 < 0.67.


One might be tempted to make quick judgements about the quality of the different descriptors, but caution must be exercised because the number of features for each descriptor vary drastically, as shown by Table <ref> in the Methods section. Each descriptor forms a unique basis to represent an environment. For the SOAP, ACE, and SF techniques, the user specifies the desired order of radial basis functions, the degree of spherical harmonics, or the polynomial order to be included in the representation. Similarly, for the ASCFs, the user selects sets of 2- and 3-body functions. The graph used in combination with graph2vec creates a nearest-neighbor connected graph where the user defines a cutoff distance for the nearest neighbors. In contrast, the CNA and CSP descriptors are both singular-valued quantities for each atom, though it is noted that CNA is built on top of local graphs that could be used instead of the integer classification, as in <cit.>.


In the cases where the user picks the level of expansion, the user can determine what cutoff can be used to obtain a desired level of accuracy; this trades computation time and memory required for a larger basis that hopefully captures more of the physics. For example, when we increased the number of ACSF features from 8 to 37, the R^2 value increased from 0.65 to 0.81 but it required significantly more computation time.
One could also compare the fact that ACSFs and SFs have approximately the same number of terms, 37 and 36 respectively, but the ACSFs perform worse than SFs.  This suggests that more or different interaction pairs are probably required to detect the structural features that determine GB energy with ACSF. In fact, with a greater understanding of the ACSF implementation, it is possible that one could obtain higher accuracy even with the same number of terms. 
Although we see that this method of “covering your bases” by increasing the number of features does quite well in terms of accuracy, there is power in a pre-training choice of basis based on physical beliefs about the material, which ideas are discussed in the interpretability section of the results.

What is remarkable about the two singular-valued descriptors, CSP and CNA, is that, despite being less accurate than the SOAP, ACE, SF, and ACSF descriptors, they still achieve respectable accuracy (c.f. Figure <ref>a).  
In fact, it is simple enough that we show the equation for the LR model, 
γ = 280.26 × x_CSP + 173.14, 
where γ is the GB energy and x_CSP is the average of the scalar CSP values for a given boundary. Similarly for CNA, the single coefficient linear function is 
γ = 505.88 × x_CNA -370.71, 
where x_CNA is the average of the integer CNA values that refer to different structure types.

Finally, in Figure <ref>a, we include the graph descriptor, despite its use of a different transform, graph2vec. This descriptor performs worse than the CSP and CNA descriptors, even though it encodes more information than the singular-valued descriptors. However, as with the other descriptors, several parameters could be optimized for better predictions, including the cutoff distance of the graph, the weighting of the graph, and the selection of subgraphs in the graph2vec transform.

In summary, the effect of descriptor on accuracy in Figure <ref>a shows that in general, more features is better. The density-based descriptors, many of which are created for machine learned interatomic potentials, appear to be better at capturing the complex and intricate nature of the local atomic environments. 

In examining the role of machine learning algorithms for a given descriptor, one can see that the linear models (LR, LASSO, RidgeCV) generally outperform other types of machine learning models for nearly all descriptors when using the average transform (c.f. Figure <ref>a). The SVM (SVR) and Nearest Neighbor (KNN) models frequently rank next in performance, followed closely by Neural Network (MLPRegression) and ensemble (AdaBoostRegressor) methods. All these models perform better than the "Random SOAP" input, demonstrating their ability to identify meaningful correlations between the features and GB energy. 

The second subset we examine in the 3-D space holds the SOAP descriptor constant and varies the transform and machine learning methods, as illustrated in Figure <ref>. The results for this subset are illustrated in Figure <ref>b, where it should be noted that the average column in Figure <ref>b is the same as the SOAP column in Figure <ref>a due to the intersection of the two 2-D cross-sections. This analysis shows that in all cases in Figure <ref>b, the averaging transform significantly outperforms the other methods; MAE is less than 10 mJ/m2 for the average transform in most cases and greater than 29 mJ/m2 for the other transforms in all cases. 

All the transforms assume some prior on the important features to transform. The average transform assumes that an average environment is the most important information to preserve. KMeans clustering assumes that the clustering in the dataset and the locations of those clusters is the most important information to preserve. Largest simplex assumes that it is important to represent the data with subsections of the data that are far apart and maximize the simplex volume. CUR assumes that specific subsets, actual rows and columns of the original data, are critical and obtains these through matrix decomposition.

Given that GB energy is calculated from the sum of the energy of all the atoms divided by the area of the boundary, it is not surprising that the best tranform is an average of the atomic environments. In other words, the assumption behind the average transform aligns closely with the calculation method for GB energy, making it a suitable representation. Conversely, the assumptions underlying other transforms do not as accurately capture the relationship between the atomic collection and GB energy. However, there may be other cases where a different transform better matches with the property of interest. For example, in some cases extreme values of a distribution control the behavior, such as in fracture, and a different transform may better capture that relationship. Therefore, we hypothesize that the best transform for accurate predictions is one that preserves the relationship between the way a collection of atoms relates to the property of interest.

However, it is also possible that the choices made in this work about the transform hyperparameters resulted in poor predictions we did not seek to optimize these hyperparameters. For example, the target rank used in the CUR transform is 20 and the largest simplex transform employed 10 dimensions; it is not clear that these values are sufficient or insufficient. Similarly, the KMeans transform employed 100 clusters and it is not clear that this is representative of the number of clusters in any of the descriptors. 

It is also worth noting that the SOAP descriptor appears to fall victim to the curse of high dimensionality when transformed with KMeans clustering. On average, the KMeans clustering transform performs worse when applied to SOAP. This is likely because the high dimensionality of SOAP descriptors leads to a phenomenon known as “distance uniformity". In high-dimensional spaces, feature values tend to be equidistant from each other, making it difficult for clustering algorithms to distinguish between similar and dissimilar data points. 

The third and fourth 2-D subsets 
we examine from the 3-D space illustrated in Figure <ref>
hold a different machine learning algorithm constant. Specifically, the linear regression model was picked because of its high accuracy and MLPRegression was also picked because of the popularity of deep learning models. The results for these subsets are presented in Figures <ref>c and <ref>d where the effect of different descriptors and transforms can be seen. First, in comparing the two tables, the average transform is better with linear regression in all but the case of ACSF. The higher accuracy between linear regression and MLPRegression is evenly divided for the largest simplex transform. MLPRegression is better in two of three cases for the CUR transform and in one of four cases for the KMeans transform. However, in many of these cases, the accuracy values approach or exceed predictions by “Random SOAP”, making it difficult to judge the value of the improvements. Furthermore, these all perform worse than the average transform.  

In the case of CSP and CNA, the GaussianKDE transform performs better than the average transform for both linear regression and MLPRegression, with the exception of CNA by linear regression. Also for these, MLPRegression performs better than linear regression for three of four cases considered. In these singular-valued descriptors, the more sophisticated GaussianKDE transform and MLPRegression allow it to obtain slightly better predictions. 

This examination of the effect of all three key steps (describe, transform, machine learning) shows that the descriptor plays an outsize role in the quality of the predictions. However, the transform of the features also plays an important role and some important information can be lost at this step if care is not exercised. Finally, the machine learning algorithm appears to play more of a secondary role; if the features are correlated with the property of interest, multiple algorithms can often extract the relationship (though some methods appear to perform better than others depending on the circumstance).

   



 §.§ Feature Selection


Although SOAP achieves the highest accuracy when predicting GB energy, in our implementation SOAP also uses the most features of any of the descriptors (c.f. Table <ref>). ACE and SF achieve comparable accuracy but only use 121 and 36 terms, respectively. In fact, these two occasionally outperform SOAP. It should be noted that any of these could be adjusted to use more or fewer terms to achieve higher or lower levels of accuracy. One can also use feature selection methods to remove redundant or irrelevant information in the machine learned structure-property models. Feature selection is an important step towards interpretable machine learning models because of the challenge of interpreting the meaning of high-dimensional descriptors <cit.>.

To identify lower dimensional representations we implement a feature selection method that uses the least absolute shrinkage and selection operator (LASSO) to identify what terms are most important for retaining high accuracy. LASSO is formulated as a minimization of a least squares error plus an L1 norm regularization term scaled by the parameter λ which controls the trade-off between fitting the model accurately to the training data and keeping the model coefficients (parameters) small and sparse. This LASSO model is defined by


    min_β{1/Ny - Xβ^2_2 + λβ_1 }


For this feature selection analysis, we use SOAP due to its prominence in GB predictions and SF for its accuracy and interpretability, as detailed in the following section. Figure <ref> plots the number of non-zero terms(blue) and the model accuracy(red) of LASSO as the size of λ increases for both SOAP (Figure <ref>a) and SF (Figure <ref>b). The sparsity of the parameters increases with increased λ values. This model is trained using the average transform and illustrates just how many parameters can be neglected while maintaining a high R^2 value. The `elbow' of the R^2 curve marks where the model begins to significantly drop in accuracy.


Figure <ref> illustrates that at low λ values SF retains all 36 terms and achieves an R^2 value of 0.95. SOAP starts with more terms but can be reduced to 209 terms with a comparable R^2 value. At the high end of λ values, SOAP and SF achieve an R^2 value of 0.43 using 4 and 2 terms, respectively,. 

This illustrates that although the accuracy decreases, both of these descriptors can be reduced to a very small representation space while preserving the most important information. It is noteworthy that averaged values of both CSP and CNA achieve higher accuracy predictions of GB energy with a single scalar value with R^2 values of 0.67 for both. Thus, these two singular-valued descriptors are quite expressive and are better than just a few terms of the other descriptors. But, perhaps this is not surprising since these descriptors were designed to easily identify defects and other changes in structure with a single value, while the other descriptors were created to provide a more nuanced description of an environment with a much larger number of terms.  Consequently, while feature selection can be helpful in removing redundant and irrelevant information, the descriptors selected have a big impact on identifying important features that contribute to the machine-learned structure-property model.





 §.§ Interpretability




The goal in this work is to obtain structure-property models that are both accurate and interpretable. Interpretability is of critical interest for the advancement of science since machine learning models could easily become black box models that we can't understand. We have examined the impacts that the descriptor, transform, and machine learning model have on accuracy and methods to select the features that have the biggest impact on the models. We now focus on extracting interpretable information from these models. 



  
Descriptors
ACE, SOAP, SF, ACSF, CSP, and CNA all represent the local environment around each atom. These representations often involve sums over neighboring atoms, where each term in the sum depends on the distance and/or angular relationship between the central atom and its neighbors. ACE, SOAP, and SF all represent the angular relationships in terms of spherical harmonic expansions, while ACSF uses a more explicit n body expansion of the angular terms. SOAP utilizes a truncated Gaussian function for the radial distribution and then recasts its basis function in a bispectrum approach. This results in the mixing of angular and radial nodes in the invariant basis functions. ACE utilizes multiple radial functions within a specified cutoff radius to capture the radial environments. In SF, the radial and angular nodes of the expansion are kept separate resulting in descriptors that are analogous to an atomic orbital expansion. SF then represents this information in the minimal set of invariants required to characterise deformations up to the 4th order (i.e., second derivatives or curvature of the strain). ACSF uses multiple Gaussian radial terms which are then convolved with the angular terms. All of the methods carry out their expansions to a level of accuracy that can be defined by the user.

In contrast to the other methods, the graph description (used in the graph2vec transform) characterizes the GB in a periodic, weighted graph. In this framework, the nodes of the graph correspond to the spatial positions of atoms and the edges are weighted by the distance between neighboring atoms within a cutoff distance. Thus, this method captures the complex arrangements of the atoms in a GB as a whole.

All the descriptors transform the 3D spatial coordinates of atoms into a high-dimensional feature space. This transformation is designed to capture complex interactions and symmetries, but it also means that the resulting descriptors are often far removed from the intuitive, three-dimensional space in which atoms actually exist. The use of physical descriptors (such as bond bending terms or spherical harmonics) leaves open the possibility that terms could be interpretable. However, the derivation of the terms often renders this difficult, and most approaches do not offer any interpretations of their descriptors. This is not a criticism of these descriptors as they were not designed for the purpose of interpretability. 

Many of these descriptors were designed for machine learned potentials or for classification of atom neighborhoods. In other words, many of these descriptors were meant for forward modeling only, but to extract interpretable information, we need to go backwards through forward models. Therefore, pointing out weaknesses in backwards applications to extract specific terms is not a completely fair criticism. Nevertheless, because we wish to understand which descriptors provide this backwards path to interpretability we examine them anyway.

In a backwards application, ACE and SOAP are less than ideal because they do not retain knowledge of how their rotationally invariant descriptors are oriented in space. In contrast, the SF approach explicitly retains such terms. Additionally, for ACE and SOAP, one can know which degree of the spherical harmonics are important as illustrated in Supplemental Figure S2, 
but finer structural detail within that degree is lost. It may be possible to preserve some of that information if interpretability is desired. In a backwards application of ACSF, it is not clear to the authors how one would interpret the values of the 2- and 3-body functions. At a minimum the summation over neighbors makes it difficult to connect with specific atomic structures. However, it is possible that improved application of ACSF descriptors could be useful in extracting the local symmetries and deviations therefrom that could result in interpretability. 

Graph2vec's interpretability suffers due to its abstract representation of graph embeddings. In its current implementation, there is no method by which to go backwards and make use of the subgraphs that were extracted, but this could be added for improved interepretability. CSP and CNA are non-unique descriptors where multiple atomic configurations can produce identical values, making it challenging to pinpoint specific structural features responsible for observed properties.


As a result, the inherent complexity of these descriptors makes it challenging to reverse-engineer the key features of the structure-property relationships. Most of these descriptors do not support easy interpretability because they were not designed for this purpose. However, the SF approach is distinct in characterizing the invariants of physical deformations up to the 4th order. This descriptor is examined in additional detail in the case study below.




  
Transforms
Just like descriptors, the ability to go backwards through transforms is crucial for interpretability. Averaging, the best transform for the most accurate descriptors, can not be applied backwards. One can only make conclusions about the average atomic environments. As explained in the Accuracy subsection, there are assumptions inherent to each of the transforms. Averaging best matches the physics of how GB energy is calculated. 

On the other hand, the largest simplex and CUR methods actually preserve specific features from the input matrix. By selecting the most important rows, these techniques effectively fix the size of the matrix that describes the GB. In other words, they choose the atomic environments that, even before training on data, are likely to preserve valuable information based on the assumptions of these transforms listed earlier. This deliberate selection ensures that the matrix representation of each GB remains concise and meaningful. Therefore, both CUR and largest simplex methods identify original atomic environments that can be analyzed for their interpretability and impact on the predictions.

The KMeans clustering transform, while fixing the matrix representation, does not preserve the original rows of the matrix but rather a number of cluster centers. Thus it is like averaging where information is lost in the transform application.  In the case of KMeans clustering the original rows can be identified by finding the nearest neighbors to the cluster centers. Although this can be difficult in higher dimensions. 

Finally, it is important to note however that other properties of interest might have different relationships with the atoms involved and therefore a different transform might work best, as discussed in the Accuracy subsection above. 



  
Machine Learning Models
Finally, the ability to extract interpretable information from the machine learning models depends heavily on the model used. When linear models are used, one can readily identify the features that have the greatest impact on the predictions based on values of the coefficients. The relationships are easily defined and understood. One can even reduce the features using a feature selection or regularization approach <cit.>, as discussed above, to more easily identify the important features. However, in non-linear models identifying the most important features is not as simple. 

If a KNN performs well, that may be indicative of clustering in the input features, which would suggest that the clustering is relevant to the property of interest. Ensemble methods like AdaBoostRegressor fit the data multiple times, focusing more on difficult cases with additional iterations. While decision trees provide inherent interpretability, ensembles of them make that more difficult. However, these algorithms can export importance scores to learn about which features are of greatest interest.

In our Support Vector Regression (SVR) model, we utilized a linear kernel to fit the data with a hyperplane. The choice of a linear kernel allowed us to maintain a straightforward relationship between the features and the output. This approach ensured that the model remained easily interpretable, as each feature's effect on the output could be independently assessed through the corresponding coefficients. By employing a linear kernel, we avoided the complexities associated with high-dimensional transformations, which are common in nonlinear kernel methods.

A neural network model uses a series of layers whose connectedness and construction can be highly variable. In the neural network each layer typically has a non-linear activation function to learn non-linear relationships. While this may improve predictions, the overall structure of the neural network makes it difficult to extract interpretable information.

However, in any of these cases, one can use additional tools, such as those that fit into the category of Explainable AI <cit.>, to extract the features of greatest impact or importance.

Having just discussed how the method selected for each step in the process of predicting structure-property relationships impacts interpretability, the discussion has remained theoretical. In the following subsection, we examine a case study where we can be more specific about the ability to extract meaningful information from machine learned models.




 §.§ Case Study of Interpretability





    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

Here we examine how a combination of one descriptor, one transform, and features selection in two different machine learning models provide interpretability. We employ the SF descriptor because, as noted above, it was defined with the express purpose of retaining a physical meaning. We employ the average transform because, as noted above, it retains a connection to how our property of interest, GB energy, is calculated. Finally, we examine feature selection in two different machine learning methods to illustrate the differences related to interpretability. 


As discussed above, regularized linear models allow easy identification of the most important features in a model. We revisit the results of the LASSO application to the SF illustrated in Figure <ref> and described in the Feature Selection subsection above. As the model complexity is reduced, the terms that remain can be considered the most important for GB energy prediction and interpretability. The last five SF terms to be removed by LASSO, and therefore the top five terms for predicting GB energy, are listed in Table <ref>. Next to each term in Table <ref> is the sign of the correlation of that term in the model with its effect on the GB energy prediction.


To understand the impact of a non-linear model, we employ Extra Trees regression. For interpretability, this is used in conjunction with SHapley Additive exPlanations (SHAP) analysis <cit.>. SHAP values explain the prediction of an instance by computing the contribution of each feature to the prediction. This method is based on game theory and helps in attributing the prediction output to individual features, offering a fair and consistent way to understand the model behavior.





Figure <ref> plots the SHAP analysis of an Extra Trees regression, showcasing the top 10 SF features sorted by their average SHAP value. 
A SHAP value indicates how much a given feature changes the output of the model compared to the baseline prediction; positive and negative SHAP values correspond to a positive and negative effect on the predicted property, respectively. The colors represent the values of the features for each data point; with red and blue values corresponding to high and low values of a particular feature, respectively. For example, if a dot is blue and located on the right side (positive SHAP value), the low value of that feature increases the value predicted by the model. 
The top five features from the SHAP analysis are also listed in Table <ref>, along with the sign of the correlation between the term and its effect on the predicted GB energy in that model.




There is significant alignment between the LASSO and SHAP analysis, as illustrated in Table <ref>. Four of the top five SF terms are the same in both analyses. Furthermore, the sign on the correlation of these four terms is the same. These four terms are P4I8, P1I0, P2I0, and P4I9, and are accompanied by the P3I4 and O2I0 terms that only appear in one model. 

To further confirm the expected correlation of these top terms from the LASSO and SHAP analysis, we plot the average SF values for each GB against both excess volume and energy in Figure <ref>. Excess volume is included as it is known to have a positive correlation with GB energy <cit.>. It is noted that correlation plots for all SF terms are plotted in Supplemental Figures S8-S10. 
The top 5 features from these supplmental plots with the highest correlation with GB energy are listed in Table <ref> along with the sign of the correlation. It is worth noting that five of the six correlations plotted in Figure <ref> have the same sign as that of the models listed in Table <ref>. The exception is the O2I0 term, which has an opposite sign in the model but only has a weak correlation with energy.

As noted earlier, SF comes with an added interpretability benefit since each SF descriptor classifies something unique about the deformation. Each SF term can be classified into one of five categories: density, deformation magnitude, deformation type, and internal and external orientation[Supplemental Figures S8-S10 
identify the categories for all 36 terms. Also, note that some of the categories from <cit.> have been renamed: deformation magnitude was net deformation, deformation type was directionality, and the two orientation types were not listed as a categories.] <cit.>.

The P4I8 term is one of three density metrics, and is an r^4 average for all the atoms in the neighborhood of a given atom. Figure <ref> shows that there are reasonable correlations with excess volume and GB energy, which haveR^2 values of 0.53 and 0.72, respectively.

The P1I0, P2I0, and P3I4 terms are all categorized as deformation magnitude.
P1I0 characterizes the gradient in the density. 
P2I0 characterizes the net deviatoric strain, akin to the von Mises strain invariant of the neighborhood. There should be substantial deformations of the fcc structures in the lattice neighboring the GB and shearing will be one of the primary means for this. The P3I4 term measures the extensional contribution of the strain gradient. The importance of this term can be attributed to the presence of strain gradients at GBs, especially in the case of GBs that can be represented as dislocation arrays. It is similar to the P1I0 term, where it is a gradient term, but it is weighted by r^3 rather than r^1. As illustrated in Figure <ref>, the P1I0 and P3I4 have strong correlations with excess volume and energy, while P2I0's correlations are weak.


The P4I9 metric is an internal orientation metric that defines the orientation between the local lattice and the P4I6 measure of the shear. The latter is similar to the net shear metric P2I0, except that it is weighted by r^4 rather than r^2.  
Finally, the O2I0 is an external orientation metric that is also highlighted by the LASSO method; this terms defines the orientation of the lattice shearing (measured by P2I0) with respect to the normal of the GB. Thus, it has some similarity to P4I9. Given that aluminum is not isotropic (albeit with a relatively small Zener ratio), it is not surprising that the amount of shear necessary to accommodate the mismatch at a GB will be related to the direction of that shear i.e., the crystal orientation. 

Interestingly, the P4I9 and O2I0 terms both have little to no correlation with excess volume and energy on their own. But in conjunction with the other terms in the models, they are deemed more important for the prediction of GB energy than other terms that have strong correlation with energy. For example, the P2I2 term (a density metric based on r^2 weighting) has much stronger correlation with energy than the P4I9 term, but the models are not predicting based on any single term alone, but the combined effect of multiple terms. Thus, some of these terms may provide more of a secondary effect that can distinguish nuanced variations of the GB energy and such effects may not be apparent in 2-D cross-sections examining single-value correlations.

The consistency of the top features and their identifiable correlations with energy in all but one case illustrate that this combination of SF descriptor, average transform, and both linear and non-linear machine learning models is capturing useful trends. The average transform tells about general trends of the atoms in the GB but not about specific local atomic environments. The linear model provides detailed insight into the influence of each parameter. The non-linear model provides this insight through the SHAP analysis. Most importantly, the SF descriptor connects the features with physical, interpretable attributes of the GB structure-property relationship.

However, the analysis also uncovers the fact that different models will extract different features. While there are only minor differences in the top five features listed in Table <ref>, the difference becomes more dramatic as more terms are compared. In addition, other models will extract entirely different features; a Bayesian linear model applied to the data showed correlations with many of the SF metrics listed above, but sometime with the same effect and sometimes with an opposite effect on the GB energy. However, the Bayesian analysis, which is described in the supplemental material, required an additional step in describing the data (  subtracting the mean and dividing by the standard deviation). The model input is slightly different and leads to different correlations as a result. Care must be taken since the machine learning is only finding correlations in the data that has been through feature engineering, it is not finding the causal relationships for the predictions.




§ DISCUSSION

Prediction in variable-sized atom-clustered structures consistently require three steps: describe, transform, machine learning. Each of these steps, illustrated in Figure <ref>, play a role in the resulting model accuracy and interpretation. Such feature engineering is frequently employed to provide more accurate predictions or better interpretation of the results. In the fast growing environment of machine learning and artificial intelligence models, the diversity in steps taken by different groups makes it challenging to know which methods lead to improved accuracy and interpretability. We have examined this challenge in GBs, which, like all variable-sized atom clusters, require a transform to obtain consistent feature sizes. By attempting to standardize the various steps of the feature engineering process, we have aimed to understand how each step in the process affects the accuracy and interpretability of the resulting model predictions.


Descriptors play an essential role in taking atom structure information, most often represented by Cartesian coordinates, and mapping that to a feature vector that encodes the most critical information.
Our findings underscore the robustness of physics-inspired structural representations <cit.> in capturing the intricate behaviors of GBs. Notably, descriptors such as the SOAP, ACE, and SF demonstrated superior predictive accuracy, underscoring their potential in advancing computational materials science. SF stands out in this group of accurate predictors because it has a low feature count and each feature has a physical meaning in terms of the strain in the neighborhood of each atom. Perhaps one of the interesting conclusions is that higher order deformations (i.e., strain gradients and higher) should be considered for accurate predictions of GB energy. This is likely the main reason behind higher predictive capability of SOAP and ACE compared to SF, as the current implementation of SF considers only up to 4th order terms, whereas spherical harmonics up to 12th order were considered for both SOAP and ACE. This is also corroborated by LASSO analysis for SOAP descriptors (Supplemental Figures S1 and S2), 
which shows persistence of l=8,10 terms. 


In short, it appears that better accuracy can always be achieved with additional descriptor information. For example, there are numerous cases where concatenation of one or more descriptor improves the learning <cit.>. But, longer feature vectors complicate interpretability. Additionally, some descriptors that can encode a lot of information cannot be processed in reverse to provide physical insight from model predictions. Therefore, those descriptors that encode physics that can be readily extracted from a model prediction are likely to provide the greatest insight into the resulting structure-property models. 


The transforms applied to obtain consistent feature sizes from the variable-sized input data impact both the accuracy and interpretability of the resulting model. As hypothesized in this work, the average transform provides the best accuracy because it is the most similar to the procedure used to calculate GB energy from the atomic structure; the excess energy, relative to the bulk energy, for all the atoms is summed and divided by the area of the boundary. We further hypothesize that other properties of interest may benefit from descriptors that capture the important features of that property. For example, properties controlled by extreme values in a distribution may benefit from a transform that captures the structural aspects of those extreme values. However, while some transforms might provide more accurate predictions, they may make interpretation difficult. The average transform is one of these and it can not be processed in reverse to tell us how the distribution of values that were averaged might be critical to a certain structure-property relationship.


The machine learning models and algorithms also play a significant role in both accuracy and interpretability. As discussed above, simpler models are preferred to complex models and the linear models provide high accuracy in many cases. The linear models are easier to interpret because the contribution of any given feature can be easily discerned. However, non-linear models can make use of Explainable AI tools <cit.>, such as SHAP analysis discussed above, to extract interpretable meaning from the resulting model. Models can sometimes be overly complex; we suggest opting for simpler linear models whenever feasible.

From an accuracy standpoint, we recognize that we have not considered all combinations of descriptors, transforms, and machine learning models. Neither have we done an exhaustive search of the hyperparameters for each of the descriptors, transforms and machine learning models beyond a simple gridsearch of hyperparameters for the machine learning models. We cannot guarantee that higher accuracy couldn't be achieved with adjustments to the models we examined. But, by providing a systematic approach, and a consistent dataset across all the models evaluated in this work, we provide a framework and benchmark against which new and improved models can be tested, like the MNIST dataset has served for benchmarking machine learning efforts in optical character recognition <cit.>. Furthermore, by providing standard steps and language for the comparison of models along with a deliberate attempt to employ principles of the machine learning community, we hope the grain boundary community can identify the best methods to obtain structure-property relationships that will drive innovation.

 
Feature selection is a tool that can be used in conjunction with machine learning models to reduce the feature vector to those items that are the most critical for the accuracy in the model. The assumption from an interpretability standpoint is that these selected features are the most important for the model and can therefore be used to obtain insight into the structure-property models.


The final case study illustrated how the SF descriptor provides insight into the density and deformations that correlate with GB energy. Four of the top five features were shared between a linear and non-linear model.
This nuanced view, where some features are consistently highlighted across methods while others are unique to specific approaches, offers a richer understanding of the predictive landscape. It suggests that while some attributes of GBs are universally recognized by various predictive models, others may be more method-dependent, possibly due to underlying assumptions or mathematical formulations inherent to each technique. This insight not only enriches our understanding of feature selection dynamics but also guides further investigation into the specific roles these features play in material structure-property relationships.

The implications of these results offer a pathway towards more precise and efficient predictive models that can be instrumental in materials design and engineering. By enhancing our ability to predict GB properties, these findings could facilitate the development of materials with optimized mechanical properties, thereby having a profound impact on various industrial applications.







§ METHODS


In the application of machine learning grain boundaries (GBs), feature engineering <cit.> is crucial for enhancing model performance by tailoring input data to more accurately reflect the underlying problem and prepare for predictive modeling. Figure <ref> illustrates the three key steps taken for predicting GB properties: employing descriptors to encode or describe the structure (step 1), transforms to standardize the data size across different GBs (step 2), and  machine learning or prediction algorithms to predict a GB property from the input data (step 3). The methods and techniques employed in this work for the three key steps are described below.



 §.§ Descriptors


The descriptors employed in this work are inspired by a phylogenetic tree of structural representations shown in Figure <ref>. The tree is reproduced and adapted from a paper titled “Physics-Inspired Structural Representations for Molecules and Materials,” by Musil et al.  <cit.>. They emphasized the requirements for developing descriptors that map the atomic positions in Cartesian coordinates to a new metric space commonly called the feature space, the representation <cit.>, descriptor <cit.>, or fingerprint <cit.>. This mapping is required for predictive modeling because atom locations represented as Cartesian coordinates cannot uniquely characterise materials <cit.>. Specifically, atom locations do not preserve symmetries and any translation or rotation of a material will change the Cartesian coordinates.
Thus, a descriptor should address requirements of completeness, symmetry, smoothness, and additivity. These requirements and the general process is discussed in additional detail in the supplemental materials. 





In this work, we considered descriptors from all the branches. A more complete description of all the methods considered is provided in the supplemental materials. In the end, we selected the following methods listed in the tree by Musil et al.  <cit.>: smooth overlap of atomic positions (SOAP), atomic cluster expansion (ACE) <cit.>, and Behler-Parinello's atom centered symmetry functions (ACSF) <cit.>. These three methods are boxed in green in Figure <ref>. We also include several other methods. First, we include a recent Strain Functional (SF) Descriptor, which define the minimal set of invariants required to characterise deformations up to the 4th order (i.e., second derivatives or curvature of the strain) <cit.>. This method is similar to SOAP in having smoothly declining neighborhood function, compared to the sharper neighborhood cutoffs of ACE, MTP and SNAP. Second, we include two methods commonly used to identify GB atoms: Centrosymmetry Parameter (CSP) and Common Neighbor Analysis (CNA). Finally, we include a periodic graph description of a boundary where every atom in the GB is defined by a node, which is connected to edges within a certain cutoff distance. Each of these additional methods is listed in Figure <ref> in green under the branch we identified as most relevant. Techniques considered but not selected from other branches are discussed in the supplemental materials, with reasons being that in some cases they were not appropriate for characterization of single species large GB structures and in other cases, code was not readily available for implementation.

 
 



 
The SOAP, ACE, and ACSF implementations all required user defined parameters that impact how many terms is used to represent an atom environment. SOAP and ASCF are implemented using the `DScribe' python library <cit.>, a powerful tool designed for creating descriptors in atomistic systems. For SOAP, all default parameters were used excluding , , , and . ACSF are unique in the sense that the user defines a list of interactions i.e. pairs, triplets for the various functions used by ACSF. Since these lists can not be described briefly here, they are included in the supplemental materials. The implementation of the ACE algorithm was achieved using the `ace.jl' package within the Julia programming environment <cit.>. A basis was created using the following parameters: , , , , , and . The SF descriptors are currently 4th order, but can be extended to higher orders. SF used the following parameters:  and . The graph description has the nodes of the periodic graph listed as their spatial coordinates. The edges are weighted by the distance to their neighbors that are within  Å of the atom (node) of interest. CNA <cit.> and CSP <cit.> both represent each atom's environment with a single value, which is calculated during the creation of the dataset using LAMMPS <cit.>. Table <ref> shows the feature lengths of each of our descriptors denoted by M.






 §.§ Transforms

Since GBs and other variable-sized atom-clustered structures can have variable numbers of features, the feature size must be standardized through some sort of transform. This will transform the N_i × M feature representation from the descriptor to a P × M feature representation that is identical for all atomic structures. 


In this work we examine 6 possible transforms :  average, CUR or skeleton matrices <cit.>, KMeans clustering <cit.>, the largest simplex <cit.>, kernel density estimation (GaussianKDE) <cit.>, and graph2vec <cit.>.  The motivation behind this set of transforms, along with a more detailed explanation of each is provided in the supplemental materials. 


The Average, largest simplex, CUR and Kmeans transforms are used with almost all descriptors. GaussianKDE is used only with CSP and CNA and graph2vec is used only with the graph descriptor. 
Table <ref> lists the theoretical input and output size of each transform along with the P values used in this work. It is noted that no attempt was made to find the optimal P value for each implementation. Explanations for why each P value was chosen is given below along with the descriptions of the transforms.




 §.§ Machine Learning

Machine learning is oftentimes yet another mapping to a different feature space. To understand the impact of this step, the research employed a comprehensive and systematic approach to compare a diverse set of machine learning algorithms. This set includes three linear models: LinearRegession, Lasso, and RidgeCV; one support vector machine: SVR; two ensemble methods: Extra Trees and AdaBoost; one nearest neighbor method: KNN; and one neural network (deep learning) model: MLPRegresssion. All of these methods are available for implementation via the sklearn python library <cit.>. This set of algorithms were selected using the sklearn documentation where various supervised methods are grouped by methodology. 


Central to this methodology was the implementation of grid search for hyperparameter optimization across all algorithms. This was implemented in an effort to provide a uniform and equitable basis for comparison between models. 

We disclose that beyond the grid search, which required user input, little additional effort was taken for the optimization of the hyperparameter values. Typically convergence was costly from a time perspective for the non-averaging transforms due to the increased complexity, so the range of hyperparameters defined in the grid search was customised according to convergence time and not accuracy. While we recognize that this may limit the accuracy of these predictions, our data consisted of only a train and test set, so this also minimized the chances of overfitting.

A 5-fold cross-validation strategy was applied to all models.  
This approach assesses the models' performance on unseen data, providing a dependable estimate of their generalization capabilities. For performance assessment, both the Mean Absolute Error (MAE) and the Coefficient of Determination (R^2) were recorded. The MAE provided an understanding of the average absolute error made by the models, while the R^2 offered insights into the proportion of variance in the dependent variable that could be explained by the independent variables, acting as another method for measuring the accuracy of the model.

Comprehensive documentation of all aspects of the model training process, including hyperparameter values, cross-validation details, and performance metrics, was maintained for the reproducibility of the research. A code base with a sample dataset is included to ensure that the experiments can be replicated and validated by others in the scientific community.



 §.§ Dataset

To compare the diverse descriptors, transforms, and machine learning techniques, we employ a recently published atomistic dataset of aluminum GBs  <cit.>. This dataset provides a comprehensive sampling of the five degrees of crystallographic character. The datasest includes 7304 pure Aluminum GB structures and their corresponding energies. The construction and process of obtaining the minimum energy structure for each GB is described in detail <cit.> and the structures are available for download <cit.>.
These GB structures were created with the emperical embedded atom model (EAM) potential created by Mishin et al. <cit.>. Thus, the physics inherent to these structures is limited to the accuracy of this emperical potential and the methods used to construct the GBs.
For purposes of limiting data storage, each GB includes ± 15 Å of atoms relative to the expected location of the GB. Because the size of some of the GBs in the original dataset were too large for some of the descriptor implementations, we use a subset of 7174 GBs, which are those GB structures that contain less than 35,000 atoms. It is expected that this dataset contains sufficient diversity both in crystallographic character and atomic structure to serve as a robust basis for the comparisons and interpretations provided in this work.




§ ACKNOWLEDGEMENTS

This work was primarily supported by the U.S. National Science Foundation (NSF) under Award #DMR-1817321. EMK and NM were supported by internal LANL LDRD funding (XX9A, XXG0).



§ COMPETING INTERESTS


All authors declare no financial or non-financial competing interests. 



§ DATA AVAILABILITY


The datasets generated and/or analysed during the current study are available in the Mendeley Data repository, https://doi.org/10.17632/4ykjz4ngw <cit.>.



§ CODE AVAILABILITY


The underlying code for this study is available at 

https://github.com/braxtonowens/gbcomparehttps://github.com/braxtonowens/gbcompare. 



§ CREDIT AUTHOR STATEMENT


C. Braxton Owens: Conceptualization, Methodology, Software, Formal analysis, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization;
Gus L. W. Hart, Eric R. Homer - Conceptualization, Methodology, Formal analysis, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Funding acquisition
Tyce W. Olaveson: Conceptualization, Software, Formal analysis;
Nithin Mathew, Jacob P. Tavenner, Edward M. Kober, Garritt J. Tucker - all performed the following related to the Strain Functional analysis: Formal analysis, Writing - Review & Editing;


 elsarticle-num 
 



































 Paper Title: Two-stage assembly of patchy ellipses: From bent-core particlesto liquid crystal analogs.

 





 
 



^1Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India.
^2School of Chemistry, University of Birmingham, Birmingham B15 2TT, UK.
^3Institut für Theoretische Physik, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
^4CNR-ISC, Uos Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy.





§ ABSTRACT 
We investigate the two-dimensional behavior of colloidal patchy ellipsoids specifically designed to follow a two-step assembly process from the monomer state to mesoscopic liquid-crystal phases, via the formation of so-called bent-core units at the intermediate stage. Our model comprises a binary mixture of ellipses interacting via the Gay-Berne potential and decorated by surface patches, with the binary components being mirror-image variants of each other - referred to as left-handed and right-handed ellipses according to the position of their patches. The surface patches are designed so as in the first stage of the assembly the monomers form bent-cores units, i.e. V-shaped dimers with a specific bent angle. The Gay-Berne interactions, which act between the ellipses, drive the dimers to subsequently form the characteristic phase observed in bent-core liquid crystals. We numerically investigate – by means of both Molecular Dynamics and Monte Carlo simulations – the described two-step process: we first optimize a target bent-core unit and we then fully characterize its state diagram in temperature and density, defining the regions where the different liquid crystalline phases dominate.


Two-stage assembly of patchy ellipses: From bent-core particles to liquid crystal analogs
    Varsha Banerjee^1
    August 5, 2024
=========================================================================================







§ INTRODUCTION



During the past years bent-core liquid crystals (BCLCs) have attracted a rapidly increasing interest both in experimental and theoretical investigations <cit.>. The constituent units of these systems are bent-core (BC) colloidal entities (typically ranging in size from nm to μm) that have a (sharp or smooth) bent shape. This characteristic shape can be realized in different ways, for instance by merging two ellipsoids (or sphero-cylinders) via their tips at a (possibly flexible) angle <cit.> or by creating bead-resolved, bent strings of spherical particles (often referred to as “banana-shaped” colloids) <cit.>. The reason for the considerable interest in such BC particles is the large variety of (ordered) phases that these entities can form: fascinating chiral nematic, smectic, splay, and twist-bend phases (despite the achiral nature of the BC constituents) are just a few examples. 


Here we propose a model system where the BC units result from the spontaneous assembly of suitably designed particles characterized by shape and bond anisotropy, namely we design anisotropic, patchy colloids such that they form V-shaped dimers. To introduce shape anisotropy, we choose – in our two-dimensional setup – elliptic particles, which interact via the well-established Gay-Berne (GB) potential <cit.>. The relevant parameters that characterize this interaction are the aspect ratio of the ellipses and their interaction range. The other ingredient to our model is the patchiness of the colloids <cit.>: patchy colloids have surfaces decorated with well defined regions (both in terms of position and of spatial extent) that support a different interaction behaviour with respect to the bare surface. Thus, depending on their mutual interaction properties these regions can either foster or suppress patch-patch directional bonding. Patchiness has become a widely used standard tool (both in experiments and in theoretical/simulation-based investigations) to trigger in bottom-up processes the self-assembly of decorated particles into highly complex bonding patterns, ranging from liquid-crystalline mesophases, over complex ordered structures and quasi-crystalline lattices to structural glasses <cit.>.

In the present contribution we describe in detail the process that starts from the formation of BCs via the self-assembly of elliptic patchy colloids and ends up in the self-organization of these entities into BCLCs. Our design of the elliptic patchy particles envisages a binary mixture of mirror-symmetric units: while both species have the same elliptic shape, the two types of particles are mirror symmetric with respect to their patch decoration (leading thus to left- and right-handed particles). In fact, the two patches (per particle) are located close to the tip of the colloid. The form of the patch-patch interaction is based on the work of Russo et al. <cit.>, while the choice of the position, interaction range and interaction strength of the patches influences the features of the BC units.
In a first step we demonstrate that our particles are indeed able to self-assemble via the related patch-patch bonding mechanism into stable BC entities; we demonstrate that our model is characterized by a high versatility, as the bonding angle can be tailored via the patch positions and/or the patch-patch interaction strength. The impact of all these parameters on the actual shape, stability and yield of these units is studied in detail. In a second step we consider an ensemble of such BCs and study their self-assembly into mesophases. Focusing for the moment on a particular set of model parameters we are able to trace out in a qualitative manner a diagram of states (in the temperature-density plane), which shows the emergence of isotropic and smectic phases.  All investigations have been carried out via extensive computer-simulations: we have applied both molecular dynamics (MD) and – in a complementary manner – Monte Carlo (MC) simulations, using different types of thermodynamic ensembles. As the proposed BC model can be easily extended to support BCs with different sets of features, it reveals itself as a powerful tool to gain a deeper insight into this fascinating class of functional materials with potential applications in optics, photonics, and sensing. The rest of our paper is organized as follows. Sec. <ref> provides the theoretical framework which includes the model and the methodology to characterize the bent-cores. The computational details of the MD and MC procedures are provided in Sec. <ref>. The results and discussion follow in Sec. <ref>. Finally, Sec. <ref> provides the summary and conclusion.  




§ THEORETICAL FRAMEWORK




 §.§ Model


We want to design a simple model in which colloidal monomers form a bent-core (BC) entity first, which then self-assemble into mesoscopic liquid crystalline phases. Such a targeted design process would allow for exploring a large number of phases which emerge by changing the shape as well as the surface chemistry of the colloidal monomers. 

To simulate such a system we use a simple Gay-Berne (GB) potential <cit.> for the elliptic colloidal monomers which are decorated on their surface by suitably placed patches (index `P'). The form of the patch potential is adopted from the work of Russo et al. <cit.>. Thus the interaction potential between a pair of particles i and j  is given by (where we have suppressed for the moment the arguments)


    V_ij = V_ GB + V_ P,


where V_ GB is the Gay-Berne potential acting between the centers of mass of the two particles, and V_ P is the interaction potential due to all pairs of patches located on the surfaces of particles i and j.

The GB potential V_ GB characterizes the shape-anisotropy of the elliptical particles and is defined as follows:

    V_GB(û_i, û_j, r⃗_ij) = 4ϵ(û_i, û_j,r̂_ij)[(σ_0/r_ij - σ(û_i, û_j, r̂_ij)+σ_0)^12. 
    . - (σ_0/r_ij - σ(û_i, û_j, r̂_ij)+σ_0)^6],


where σ_0 is the unit distance of the interactions.  r⃗_ij is the vector connecting the centers of mass of the particles and û_i and û_j are the unit vectors representing the orientations of the particles as shown in Fig. <ref>(a). ϵ = ϵ_0 ϵ_1^μϵ_2^ν is the orientation dependent interaction strength, where ϵ_0 represents the energy unit of the interactions; the parameters μ and ν modify the well depth of the potential; σ is the anisotropic contact distance. Further ϵ_1, ϵ_2 and σ are defined as:

    ϵ_1(û_i, û_j, r̂_ij)    = 1 - χ'/2{(r̂_ij.û_i + r̂_ij.û_j)^2/1+χ'û_i . û_j+(r̂_ij.û_i - r̂_ij.û_j)^2/1 - χ'û_i . û_j}, 
    ϵ_2(û_i, û_j)    = [1-χ^2(û_i . û_j)^2]^-1/2,
    σ(û_i, û_j, r̂_ij)    = σ_0[1-χ/2{(r̂_ij.û_i + r̂_ij.û_j)^2/1+χû_i . û_j+(r̂_ij.û_i - r̂_ij.û_j)^2/1 - χû_i . û_j}]^-1/2.

The energy anisotropy parameter χ' and the shape anisotropy parameter χ are defined as:

    χ'    = (κ'^1/μ-1) / (κ'^1/μ + 1), 
    χ   = (κ^2-1) / (κ^2 + 1),


where κ' = ϵ_s / ϵ_e is the energy anisotropy with ϵ_s and ϵ_e being the well depths of the side-to-side and end-to-end configurations. κ = σ_∥ / σ_⊥ is the aspect ratio of the elliptic particles with σ_∥ and σ_⊥ being the major and minor axes of the particles. The GB potential therefore contains four key parameters to describe interacting elliptic particles: κ, κ', μ, and ν. There exist a number of GB homologues in the context of nematic LCs depending on the values chosen for the four parameters  <cit.>. We will discuss the appropriate choice for obtaining the BCs and their LC-like self-assembly shortly.

The patch-patch interactions V_ P are described by a sum of inverted Gaussian potentials <cit.>:

    V_ P(d) = - ∑_k ∈ P_i∑_l ∈ P_jϵ_klexp[-1/2(d_kl/α)^2],


where ϵ_kl is the strength of the interaction between patches k and l, P_i and P_j are the set of patches on the particles i and j, respectively, and d_kl is the distance between the centers of patches k and l, which decorate the surface of the elliptical particles, as shown in the schematic of Fig. <ref>(a). The width of the inverted Gaussian, α, defines the range of the patch-patch interactions. In addition to being able to compute gradients required for molecular dynamics (MD) simulations, this choice of the patch potential also allows for oblique patch placements, i.e., at an angle from the tip of the ellipses. Such placements promote the assembly of stable BCs. 

Let us consider the schematic in Fig. <ref>(a) that shows ellipsoidal particles with two patches placed obliquely; the reason for a two-patch model will be clear soon. In our model there are two kinds of patch placements on the ellipsoids: one to the “left” (red particles) and the other to the “right” (green particles). The green particles have patches G_1 and G_2 at angles -θ_1 and -θ_2 from the major axis. Similarly, the red particles have patches R_1 and R_2 at angles θ_1 and θ_2 from the major axis. The interaction strengths ϵ_G_1 R_1 and  ϵ_G_2 R_2 are set equal to ϵ_p, while all other interactions are set to 0: these include notably ϵ_G_1G_1, ϵ_G_1G_2, ϵ_R_1R_1, ϵ_R_1R_2, ϵ_G_1 R_2 and ϵ_G_2 R_1. We consider a pair of left-handed and right-handed particles to be a BC unit if the patch interaction strength V_p is 
sufficiently large to prevent dissociation of this arrangement by thermal fluctuations. In our simulations, we find that a choice of V_p < -3k_ BT  (T being the temperature and k_ B being the Boltzmann constant) ensures that the BCs once formed remain bonded.  

Fig. <ref>(b) shows a schematic of a BC colloid corresponding to the above model description. This entity is characterized by the director n̂, the polarization p̂, and the bend angle γ (as indicated). In our investigations we have found that single patch particles are characterized by a broad distribution of the bend angles, as shown in Fig. <ref>(c). Reducing the width of the  angle distribution would require lowering the value of α, which, as we shall see later, negatively affects the kinetics of BC self-assembly. Further, multiple single-patch ellipses can adhere in clusters (or asters). 
Having two patches on the other hand results in smaller magnitudes of the bend angle and lesser flexibility.  
Thus the two-patch ellipsoids represent an optimal choice to obtain BCs at a high yield. 
The model parameters θ_i, ϵ_p, and α have a significant impact on the BC yield and phases. We will study their influence in the rest of the paper.




 §.§ Methods





  §.§.§ Bent-core characteristics


Let us refer to the schematic in Fig. <ref> that depicts a pair of left-handed and right-handed patchy ellipsoids. Let their center of mass, after the formation of the BC, be represented by r⃗_ cm^ l and r⃗_ cm^ r, respectively. The center of mass of the BC unit is given by the vector average:


    R⃗_ cm = r⃗_ cm^ l + r⃗_ cm^ r/2.


Further, the local director n̂ and the polarization p̂ can be expressed in terms of the orientations u⃗_l and u⃗_r as follows:


    n̂= u⃗_ l-u⃗_ r/|u⃗_ l-u⃗_ r|,   p̂ =u⃗_ l+u⃗_ r/|u⃗_ l+u⃗_ r|,


Similarly, the bend angle γ can also be defined in terms of the orientation vectors as follows:


    γ=cos^-1( u⃗_ l·u⃗_ r/|u⃗_ l||u⃗_ r|).


As discussed in the context of Fig. <ref>(b), the BC unit is characterized by n̂, p̂, and γ. We will use Eqs. (<ref>) and (<ref>) for the evaluation of these quantities in all our simulations. 



  §.§.§ Q-Tensor



To characterize the global orientational order of the mesoscopic phases formed by the BC units, we compute the Q-tensor as an appropriate order parameter for quantifying both the nematic and the polar order <cit.>. Its components are defined as follows:


    Q_αβ = 1/N_ b∑_i=1^N_ b1/2(3 n_i,αn_i,β - δ_αβ),


where N_ b is the number of BCs in the ensemble, α, β∈{x,y} represent the Cartesian components of n̂_i corresponding to the i^th BC. The largest eigenvalue of Q gives the nematic order parameter 𝒮 and the corresponding eigenvector represents the global director n. The nematic order parameter can also be interpreted as 𝒮=P_2(n̂_i·𝐧), where P_2(x) = (3x^2-1)/2 is the Legendre polynomial of degree two and the over-bar indicates an average over all the molecules. The isotropic phase corresponds to S = 0 and a fully aligned nematic phase has S = 1. 
The polar order can be computed as 𝒫 = |p̂_i·𝐩|, where 𝐩 is normal to 𝐧. A polarization 𝒫 = 1 implies that the polar vectors p̂_i of the BCs are all pointing in the same direction as 𝐩. In this case, aggregates of BCs exhibit nematic as well as polar order. 𝒫 = 0 indicates random or staggered (e.g., anti-ferroelectric) orientation of p̂_i of the BCs. Note that in this case, the aggregate could still exhibit nematic order.



  §.§.§ Radial distribution function



The radial distribution function g(r) (RDF) which measures correlations between pairs of particles can provide information about the presence of density modulations in the system and is a useful tool for identifying the structure of the aggregates or domains. This function is defined as <cit.>


    g(r) = 1/ρ_ b N_ b⟨∑_i,ji≠ j^N_ bδ(r-r_ij) ⟩,


where r_ij = |r⃗_ij| is the separation distance between the centers of mass of BCs i and j, N_ b is the number of BCs, and  ρ_ b = N_ b/L^2 is their number density. In practice, the δ-function is replaced by a counting function which is normalized appropriately by the area of the finite but small region in which the counting is done. The division by ρ_ b ensures that g(r) = 1 when the particles are completely uncorrelated, as this is, for instance realized in an ideal gas. Sharp peaks indicate the presence of structure and oscillatory behavior marks the presence of translational order in the system. We can also define a characteristic lengthscale (or correlation length) over which the particle positions are correlated. Typically, it is the distance r over which the peaks in g(r) decay to – say – 0.2 times the maximum value of the RDF <cit.>.

In order to identify layered phases such as the smectic one, we resolve the RDF along axes in the local frame of reference of the BC units and average the results. The RDF resolved parallel to the local director n̂_i of the BCs, g_∥(r_∥), is defined as follows <cit.>:


    g_∥(r_∥)=1/ρ_b N_b⟨∑_i,ji≠ j^N_bδ(r_∥ - r_ij, ∥)Θ(h/2 - r_ij,  ⊥)/h⟩,


where r_ij, ∥= |r⃗_⃗i⃗j⃗·n̂_i| and r_ij,⊥=|r⃗_⃗i⃗j⃗×n̂_i| are the parallel and perpendicular components of the separation r_ij with respect to n̂_i; further, h is the average height of the BCs (i.e., average extension of the BCs in the direction perpendicular to n̂_i, see Fig. <ref>b) and Θ (x) is the Heaviside step function, which ensures that only BCs within the same layer as the reference BC (with index i) are included in the computation. Similarly, g_⊥(r_⊥) which resolves RDF in the direction perpendicular to the local director is defined as <cit.>


    g_⊥(r_⊥)=1/ρ_ b N_ b⟨∑_i,ji≠ j^N_ bδ(r_⊥ - r_ij, ⊥)Θ(w/2 - r_ij,  ∥)/w⟩,


where w is the average width of the BCs (see Fig. <ref>b). 



§ COMPUTATIONAL DETAILS



Our starting point to obtain BC colloids and their phases is a collection of an equal number of right-handed and left-handed two-patch ellipsoids, and specified patch-patch interactions. We study the self-assembly using extensive molecular dynamics (MD) and Monte Carlo (MC) simulations. As discussed in detail in Sec. <ref>, the ellipsoids interact via the GB potential which is characterized by four essential parameters: κ, κ^', μ and ν. There are a large variety of GB homologues which differ from each other in terms of the values chosen for the four parameters. The primary focus in selecting their values has been (i) to obtain the nematic and smectic LC phases and (ii) to obtain convergence with experimental data <cit.>. The choice of κ=3.0 is elementary as for real LC systems, the length-to-breadth ratio of the constituent molecules is typically equal to or greater than 3:1. 
A phase diagram of the GB model in the κ^'-T indicates that the ordered phase as κ^'→ 1 is primarily nematic <cit.>. A value of κ^' > 1 introduces energy anisotropy, favoring side-to-side alignment of ellipses. These values promote nematic as well as smectic order characteristics of LCs. The parameters μ and ν modify the well depths of the potential, and hence their impact on the nematicity and smecticity is very subtle <cit.>. A popular choice of parameters is κ=3, κ^'=5, μ=1 and ν=3 due to Berardi et al. <cit.>. They ensure the formation of LC  phases and agreement with experimental data in many real systems <cit.>. In the present work, our focus is to first create BC units, and then study their assembly in ordered phases. Consequently, we choose κ^'=1 to promote patch-patch aggregation for the formation of BCs in the first stage, which then assemble to form ordered phases due to the GB interactions in the second stage. 


 


 §.§ Details of MD simulations

MD simulations are performed in NVT and NPT ensembles using the LAMMPS software <cit.>, employing the Nosé-Hoover thermostat and barostat for temperature and pressure coupling, as needed <cit.>. Periodic boundaries with minimum image convention have been used for the simulations. All calculations were done in reduced LJ units: T^*= k_BT/ϵ_0, Δ t^*= Δ t/ √(m σ_0^2/ ϵ_0), where k_B is the Boltzmann constant. The star is dropped in subsequent discussions. The equations of motion are integrated using the velocity-Verlet algorithm with simulation time step Δ t = 0.001 <cit.>. 

We initialize the system with N=4000 patchy ellipsoids in a square box of length L depending on the desired density ρ=(πσ_∥σ_⊥)N/L^2, where σ_∥  and σ_⊥ are the major and minor axes of the ellipsoids. For all studied densities, we initialize the particles on a regular lattice and allow the system to equilibrate at T=3 for 10^5 MD steps. The resulting homogeneous state is then quenched to the desired temperature. Simulations were performed up to 10^7 steps and relevant quantities measured at intervals of 5× 10^3 steps. All presented data have been averaged over 10 independent runs.



 §.§ Details of MC simulations



In a complementary fashion we have used Monte Carlo (MC) <cit.> simulations to investigate the structural properties of our BC entities. To be more specific we have used so-called “virtual-move” MC (VMMC) simulations <cit.>, an algorithm that proposes simultaneous translational and rotational moves of clusters of particles according to gradients of interaction energies of the particles. In short, a seed particle is subjected to a “virtual” trial move; based on the emerging bond energies before and after the trial move, neighbouring particles are added to a virtual cluster. This process is iterated for each particle in the cluster and the positions of all particles involved are updated simultaneously (for more details we refer to Refs. <cit.>). 

The VMMC simulations were performed in the NVT ensemble, applying standard periodic boundary conditions. Due to the fact that the degree of parallelization of the VMMC algorithm is strongly limited, we had to restrict ourselves to considerably smaller ensembles (as compared to the MD simulations): a typical ensemble size of N = 900 particles was used. Starting in a square box (whose size is imposed by the desired density ρ) half of the particles are left- and right-handed, respectively. At a given temperature the system is equilibrated during 4 × 10^7 VMMC steps, while the production part of the simulations extended over 2 × 10^7 VMMC steps. For each state point we considered ten independent MC runs, each of them starting from different initial conditions. For the highest densities we started from ordered arrays of parallel unbounded ellipses, while for the lowest densities we started from random configurations.  All quantities were averaged over 4 × 10^3 independent configurations per single run, adding to  4 × 10^4 per state point. 














§ RESULTS AND DISCUSSION





 §.§ First step assembly stage

Before studying the mesoscopic phases of the system, it is useful to understand the effect of the model parameters on the formation of BCs. The presented patchy colloidal analog of BCLCs has three model parameters and two system parameters to control the properties of the BCs: patch interaction strength ϵ_p (defined in units of ϵ_0), patch interaction range α, patch positions θ_i, density ρ, and temperature T. We perform simulations in the NVT ensemble with density ρ=0.4 and temperature T=1.2, corresponding to an equilibrium isotropic fluid state of the underlying Gay-Berne particles. Unless stated otherwise, the patch positions are fixed at θ_1 = 5 and θ_2 = 10. 

We start by demonstrating the effect of patch-patch interaction strength ϵ_p on the formation of BCs, keeping the patch range constant at α = 0.1. The important metrics to characterize the BCs are: (i) the fraction of particles f that have self-assembled into BCs, (ii) the average bend angle γ̅ of the BCs and (iii) the flexibility of the BCs. Figs. <ref>(a)-(c) show prototypical equilibrated snapshots at time t=10^4 for ϵ_p = 7.5, 15 and 26.25. The red and green patchy ellipsoids which yield a BC unit are coloured purple. The asters, formed when more than two ellipsoidal particles adhere, are coloured cyan. (See insets in Figs. <ref>(a)-(c).)  The BC yield f for different values of ϵ_p is shown in Fig. <ref>(d). The inset shows the patch-patch interaction V_p(d) (inverted Gaussian) for the different values of ϵ_p. The primary effect of varying ϵ_p is to vary the well-depth. Fig. <ref>(e) shows that the corresponding distribution of bend angles while Fig. <ref>(f) shows f and γ̅ as a function of ϵ_p.
For ϵ_p = 7.5, f remains quite low even after long simulation times. This is because V_p in this case is weaker than thermal energy k_BT and is not sufficient to keep the left-handed and right-handed particles bonded, see the morphology in Fig. <ref>(a). Fig. <ref>(e) shows that the corresponding distribution of bend angles is broad.
For an intermediate value of ϵ_p = 15, V_p is sufficiently strong with respect to k_BT, thus keeping the individual building blocks bonded, as shown in Fig. <ref>(b). As a result, we achieve a BC yield close to 1.
For ϵ_p=30, V_p is even stronger, making single bonds also stable against thermal fluctuations. In addition to the BCs, this leads to the creation of asters formed due to the patch at θ_1=5. Single bonds with more than one monomer of the opposite handedness are also stable in this case as seen in the morphology of Fig. <ref>(c). Consequently, the BC yield is also reduced, as observed in Fig. <ref>(d). The corresponding bend angle distribution shown in Fig. <ref>(e), which excludes the asters, is narrower than that for ϵ_p = 7.5, indicating the reduced flexibility of the BCs, as is expected with stronger patch-patch interactions. Interestingly, for the range of ϵ_p values over which the BC formation is efficient, we do not notice a significant variation in the flexibility of the bend angle as revealed by Fig. <ref>(f).



The other crucial parameter for controlling the self-assembly of BCs is the patch interaction range α. Graphically, it represents the well width of inverted Gaussian potential. 
Guided by the observations in Fig. <ref>, we set the patch strength ϵ_p=15 to explore the effect of α. Figs. <ref>(a)-(c) show the snapshots for α = 0.03, 0.1 and 0.15.
For the small value of 0.03, the effective region within which two complementary patches can make a stable bond is significantly reduced. This not only requires the bonding monomers to be close to each other but also that they are in near-perfect alignment in order to make a double bond. As a result, the probability of opposite-handed particles forming a BC is quite small. This negatively affects the kinetics of the self-assembly leading to a low rate of BC production, as seen in Fig. <ref>(d). The resultant BCs are also less flexible and prone to dissociate into monomers upon the smallest fluctuations. 
As α increases to 0.1 in Fig. <ref>(b), the monomers have a higher probability of making bonds and the rate of BC self-assembly increases. The flexibility of BCs also increases as the increased patch range allows for a larger window of angles over which the monomers can remain bonded, as shown in Fig <ref>(e) and (f). Both these factors contribute to a larger BC yield. For larger patch ranges α≥ 0.15 in Figs. <ref>(f), the rate of self-assembly of BCs is high. However, the yield is significantly reduced due to the formation of asters. They coexist with the BCs as shown in Fig <ref>(c), and the BCs display a large fluctuation in their bend angle for larger values of α. 

From Figs. <ref>(f) and <ref>(f), it can be seen that for fixed values of the patch angles, there is a sizable region in the (ϵ_p-α) parameter space which yields an efficient formation of BCs. The distribution of the bend angle γ is predominantly dictated by the patch range α and to a lesser extent by the patch strength ϵ_p. 

It is also educative to understand the role of the patch placements [θ_1, θ_2] on the BCs. Choosing the ϵ_p = 15 and α = 0.1, Fig. <ref> shows typical morphologies for four different choices of patch positions: (a) [0, 2]; (b) [0, 5]; (c) [3, 6]; (d) [5, 10]. The corresponding bend angle distributions P(γ) vs. γ and the BC yield f are also shown in Figs. <ref>(e) and (f). Their choice affects the bend angle. For example [0, 5] yields γ̅≃ 145, but shifted placements [5, 10] results in γ̅≃ 82. Note that this angle is dependent on the shape anisotropy κ as well. For instance, the mean bend angle would be larger for less eccentric ellipses, i.e., for κ < 3. BC yields are also affected as in cases such as [0, 2], there is a larger possibility of non-BC aggregations such as the asters. Further, we can also expect distinct phases and their textures as the patch positions are varied: [0, 2] nudges the arrangement in a nematic phase while [0, 5] introduces a splay in the texture. Our preferred choice has been [5, 10] as it yields BC formation ≃ 1. The interplay of α, θ_1 and θ_2 can be expected to yield a range of bend angles and novel phases. This is an exhaustive investigation that deserves attention. However, first it is essential to understand the methodology to create model BCs from patchy ellipsoids and subsequently study their self-assembly into states with long-range liquid crystalline order. We prioritize establishing these protocols in the present work and will address the α-θ_1-θ_2 state diagram in an ensuing study.



 §.§ Second step assembly stage

We now turn our attention to the ordering assemblies from the initial monomeric state. Figs. <ref>(a)-(d) show the time evolution of the morphology. The corresponding BC yields f are shown in Fig. <ref>(e).  
At time t=0, when the system is quenched from T=2 to 1.2, the system is in a homogeneous isotropic state where the monomers have not formed any BCs [Fig. <ref>(a)]. In the absence of any patches, this would remain an isotropic fluid, given our choice of the GB parameters. Due to the presence of attractive patches, opposite-handed monomers start adhering. The BC yield f rapidly approaches 1 as time increases. An unusual observation from Figs. <ref>(c) and (d) is that the evolving ordered state has anti-ferroelectric smectic order. 
We also evaluate the nematic and polar order parameters (𝒮 and 𝒫) for the evolving morphology in Fig. <ref>(f). Note that the polar direction p̂ of the neighbouring BCs in the ordered phase are antiparallel (or staggered). Consequently, we evaluate the polarization by considering only the magnitude of p̂ along the direction of polarization p⃗. In other words, we take the absolute value of (p̂_i·𝐩) before averaging over all BCs. As seen in Fig. <ref>(f), 𝒮 is close to zero at early times as the system is in an isotropic state. As time evolves, there is the emergence of BCs and subsequent liquid crystalline order due to the GB interactions between the BC units. The BCs start forming small locally aligned regions, which then coalesce into larger domains with liquid crystalline order [Fig. <ref>(c) and (d)]. Consequently, there is an increase in the nematic and polar order parameters, as seen in Fig. <ref>(f). These observations, along with Figs. <ref>(e), emphasize the two-stage self-assembly in our model: the BC formation followed by long-range liquid crystalline order in the smectic anti-ferroelectric phase.


The full set of BC states (phases) for a specific choice of model parameters (ϵ_p = 15, α = 0.1) in the optimal window is shown in the ρ - T state diagram of Fig. <ref>(a). The different symbols indicate the different states observed in our simulations: isotropic state (indigo circles), anti-ferroelectric smectic (magenta triangles), arrested or frozen state (green squares) and disordered state (blue diamonds). The background color represents the BC yield. We see that there is a sizeable region in the ρ-T space where the BC yield is greater than 0.9.
Panels (b) to (e) show typical morphologies obtained from MD (left) and MC (right) simulations for the four states obtained at indicated parameter values of the corresponding filled symbols in the state diagram. In (b),  there is formation of BCs, but the system is in an isotropic state as the Gay-Berne attraction between them is not sufficient to overcome the thermal fluctuations. 
The region between the dashed lines is where we obtain the smectic anti-ferroelectric phase shown in the snapshot (c). For these parameter values, the BCs are fluid enough to self-assemble into large ordered domains. At low temperatures (T < 0.9), there are locally aligned regions of BCs, but there is a lack of long-range order as these clusters are frozen. Such a kinetically arrested or frozen state due to low temperatures is shown in panel (d). At very high densities ρ > 0.6, the particles lack the free volume to move and reorient, leading to a dense disordered BC state, as shown in panel (e). Similar morphologies are also observed in the NPT ensemble with pressure-tuned to achieve corresponding densities, see Fig. <ref>.

In order to characterize the structure of the phases, we evaluate the directional radial distribution functions (RDF) resolved along axes in the local frame of reference of the BCs as described in section <ref>. This is schematically illustrated in Fig. <ref>(a) where for each BC i, we compute the RDF with other BCs along parallel or perpendicular directions to its local alignment n̂_i. Only those BCs that are within a finite region perpendicular to the direction of the correlation evaluation are included. The directional RDFs at ρ=0.4 and T=1.5, corresponding to the isotropic phase, are shown in Fig. <ref>(b). We observe only a few peaks in g_⊥(r_⊥) and hardly any in g_∥(r_∥), indicating that at most, only two or three BCs stack along their short axis. In the case of the smectic anti-ferroelectric phase (ρ=0.4 and T=1.2), we observe multiple equidistant peaks along either direction, indicating the arrangement of BCs into multiple layers, as shown in Fig. <ref>(c). Interestingly, the correlations decay much more slowly in the direction of the long axis of the BCs (g_⊥(r_⊥)). In the frozen state at ρ=0.4 and T=0.9 [Fig. <ref>(e)], we find equidistant peaks that decay rapidly, corresponding to short-range correlations along either direction. This is due to the presence of multiple small domains of locally aligned BCs. In the dense disordered state (ρ=0.7 and T=1.2), we only find short-range correlations along the short axis of the BCs, as seen from Fig. <ref>(f). In all cases, our MD and MC results are in agreement with each other. The variation in the correlation lengths along the two directions (l_n and l_p) with temperature T is shown in Fig. <ref>(d), for two densities (ρ = 0.4 and 0.7). At the higher density, the system is in a dense disordered state, and hence the correlation lengths do not vary with temperature. At the lower density, we see a gradual increase in the correlation lengths with a peak centered around T = 1.2, corresponding to the increased smectic anti-ferroelectric alignment, beyond which the correlations decay rapidly as the system enters the high-temperature isotropic phase. The plots also show that the size of the LC domains along the long axis of the BCs is larger than that in the orthogonal direction. The current investigation of parameters belonging to patch-patch interactions provides only isotropic and smectic anti-ferroelectric phases. This is because we have not fully explored the effect of patch positions on the formation of mesoscopic phases. Our preliminary explorations of the α-θ_1-θ_2 space reveal a variety of phases, e.g., nematic and splay-bends, through appropriate choice of the model parameters.







§ SUMMARY AND CONCLUSION



We have developed a patchy ellipsoid model capable of generating mesoscopic liquid crystalline phases in a hierarchical two-stage self-assembly process, via the formation of BC dimers in the first stage. This is achieved through a binary mixture of ellipses having a pair of complementary and obliquely placed patches with specific combination rules to guide the formation of BCs. The patches are modeled as points on the surface of the ellipses interacting via an inverted Gaussian potential. The second stage of the assembly into ordered liquid phases is governed by the Gay-Berne interactions acting between the ellipses themselves. Our model has three crucial parameters: (i) the patch interaction strength ϵ_p, (ii) the patch interaction range α and (iii) the patch positions θ_1 and θ_2, that control the formation of the BCs, their mean bend angle and flexibility. 

We perform extensive MD and MC simulations to elucidate the model and the effect of the model parameters. Our main observations are summarized below:

(a) For fixed patch positions θ_1 and θ_2 and patch range α, we observe a window of ϵ_p values for which the BC yield f is close to 1. For small values of ϵ_p, the particles do not make stable bonds, whereas for large values of ϵ_p, assemblies of more than two complementary monomers (asters) are formed due to increased stability of single bonds.

(b) Similarly, with all other parameters fixed, we obtain high BC yields for a wide range of values of α. Although smaller values of α also can produce BCs, we observe low yields in our simulations due to slow kinetics. Larger values of α promote the formation of asters as the monomers can make erroneous bonds due to increased range.

(c) For a choice of parameters in the optimal window, we study the evolution of the BC yield and the nematic and polar orders, which show that the assembly into ordered phases occurs in two-stages with the BC formation preceding the long-range ordering.

(d) We obtain the state diagram of the model in the ρ-T plane, which exhibits isotropic, smectic anti-ferroelectric and disordered arrested states (due to low temperatures or/and high densities) for the given choice of model parameters. We compute correlation functions and order parameters of these states and show that there is consistency between MD and MC simulations. Further, there is consistency in the morphologies between NVT and NPT ensembles as well, at appropriately chosen parameter values.

(e) The patch positions θ_1 and θ_2 can also be varied to generate BCs with different configurations leading to a myriad of interesting liquid crystalline phases, which we shall explore in subsequent works.

The assembly scenarios of these systems are very rich and, so far, the available models for BC entities from patches have not been able to predict all the observed BCLC phases. Our work sets the stage for these future investigations. In contrast to the molecular BCs often studied in experiments, our model provides a colloidal route to create BCLCs. Generating them at the colloidal scale in experiments is challenging but could have huge potential applications. We believe that our study offers one possible solution for the experimental realization of these interesting fluids.





§ ACKNOWLEDGEMENTS

AKS gratefully acknowledges the HPC computing facility of IIT Delhi. JT, EB and GK gratefully acknowledge the generous share of CPU time, offered by the Vienna Scientific Cluster (VSC) under project number fs71898. JT acknowledges financial support from the EU-funded doctorate college ENROL. EB acknowledges support from the Austrian Science Fund (FWF) under Proj. No. Y-1163-N27. VB acknowledges SERB (India) for research grants CRG/2022/004871 and CRG/2019/001836. AB acknowledges partial financial support from CRG/2019/001836 during a visit to    IIT Delhi that initiated this work.
















































45
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

[Lubensky and Radzihovsky(2002)]Lubensky2002
  author author T. Lubensky and author L. Radzihovsky, @noop  journal journal
  Phys. Rev. E volume 66, pages 031704
  (year 2002)NoStop
[Ros et al.(2005)Ros,
  Serrano, de La Fuente, and Folcia]Ros2005
  author author M. B. Ros, author J. L. Serrano,
  author M. R. de La Fuente,   and author C. L. Folcia,  @noop  journal journal J. Mater. Chem.  volume 15, pages 5093 (year
  2005)NoStop
[Takezoe and Takanishi(2006)]Takezoe2006
  author author H. Takezoe and author Y. Takanishi, @noop  journal journal
  Jpn. J. Appl. Phys. volume 45, pages
  597 (year 2006)NoStop
[Jákli et al.(2018)Jákli, Lavrentovich, and Selinger]Jakli2018
  author author A. Jákli, author O. D. Lavrentovich,  and author J. V.  Selinger, @noop  journal journal Rev. Mod. Phys. volume 90, pages 045004 (year 2018)NoStop
[Johnston et al.(2002)Johnston, Low, and Neal]Johnston2002
  author author S. J. Johnston, author R. J. Low,   and author M. P. Neal, 10.1103/PhysRevE.65.051706 journal journal Phys. Rev. E volume 65, pages 051706 (year 2002)NoStop
[Bates and Luckhurst(2005)]Bates2005
  author author M. A. Bates and author G. R. Luckhurst, 10.1103/PhysRevE.72.051702 journal journal Phys. Rev. E volume
  72, pages 051702 (year 2005)NoStop
[Bates(2006)]Bates2006
  author author M. A. Bates, @noop  journal journal Phys.
  Rev. E volume 74, pages 061702
  (year 2006)NoStop
[Dewar and Camp(2004)]Dewar2004
  author author A. Dewar and author P. J. Camp, 10.1103/PhysRevE.70.011704 journal
  journal Phys. Rev. E volume 70,  pages 011704 (year 2004)NoStop
[Dewar and Camp(2005)]Dewar2005
  author author A. Dewar and author P. J. Camp, 10.1063/1.2062007 journal journal J. Chem. Phys. volume 123, pages 174907 (year 2005)NoStop
[Chiappini et al.(2019)Chiappini, Drwenski, van Roij, and  Dijkstra]Chiappini2019
  author author M. Chiappini, author T. Drwenski,
  author R. van Roij,  and  author M. Dijkstra, 10.1103/PhysRevLett.123.068001 journal journal Phys. Rev. Lett. volume 123, pages 068001 (year 2019)NoStop
[Berardi et al.(1995)Berardi, Fava, and Zannoni]Berardi1995
  author author R. Berardi, author C. Fava,   and author C. Zannoni,  @noop  journal journal Chem. Phys.
  Lett. volume 236, pages 462 (year 1995)NoStop
[Pawar and Kretzschmar(2010)]Kretzschmar_2010
  author author A. Pawar and author I. Kretzschmar, @noop  journal journal
  Macromol. Rapid Commun. volume 31, pages 150 (year 2010)NoStop
[Bianchi et al.(2011)Bianchi, Blaak, and Likos]Bianchi_pccp_2011
  author author E. Bianchi, author R. Blaak,   and author C. N. Likos,  @noop  journal journal Phys. Chem. Chem.
  Phys. volume 13, pages 6397 (year 2011)NoStop
[Ravaine and Duguet(2017)]ravaine2017synthesis
  author author S. Ravaine and author E. Duguet, @noop  journal journal
  Current opinion in colloid & interface science volume 30, pages 45 (year 2017)NoStop
[Bianchi et al.(2017)Bianchi, Capone, Coluzza, Rovigatti, and van Oostrum]Bianchi2017Limiting
  author author E. Bianchi, author B. Capone,
  author I. Coluzza, author L. Rovigatti,  and author P. D. J. van Oostrum, 10.1039/C7CP03149A journal journal
  Phys. Chem. Chem. Phys. volume 19, pages 19847 (year 2017)NoStop
[Smallenburg et al.(2013)Smallenburg, Leibler, and Sciortino]SmallenburgLeibler_2013
  author author F. Smallenburg, author L. Leibler,  and author F. Sciortino, @noop  journal journal
  Phys. Rev. Lett. volume 111, pages
  188002 (year 2013)NoStop
[Chen et al.(2011)Chen,
  Diesel, Whitmer, Bae,
  Luijten, and Granick]Granick_2011
  author author Q. Chen, author E. Diesel,
  author J. K. Whitmer, author S. C. Bae, author
  E. Luijten,  and author
  S. Granick, @noop  journal journal J. Am. Chem. Soc. volume 133, pages 7725 (year
  2011)NoStop
[Wang et al.(2012)Wang,
  Wang, Breed, Manoharan,
  Feng, Hollingsworth, Weck,  and Pine]Pine_2012
  author author Y. Wang, author Y. Wang, author D. R. Breed, author
  V. N. Manoharan, author
  L. Feng, author A. D. Hollingsworth, author M. Weck,  and author D. J.  Pine, @noop  journal journal Nature volume 491, pages
  51 (year 2012)NoStop
[Morphew et al.(2018)Morphew, Shaw, Avins, and Chakrabarti]Chakrabarti_2018
  author author D. Morphew, author J. Shaw,
  author C. Avins,  and author D. Chakrabarti, @noop 
  journal journal ACS Nano volume 12, pages 2355 (year
  2018)NoStop
[He et al.(2020)He,
  Gales, Ducrot, Gong,
  Yi, Sacanna, and Pine]Pine_2020
  author author M. He, author J. P. Gales,
  author É. Ducrot, author Z. Gong, author
  G.-R. Yi, author S. Sacanna,  and author D. J. Pine, @noop  journal
  journal Nature volume 585, pages 524 (year 2020)NoStop
[Posnjak et al.(2024)Posnjak, Yin, Butler, Bienek, Dass, Sharp, and Liedl]Liedl_2024
  author author G. Posnjak, author X. Yin,
  author P. Butler, author O. Bienek, author
  M. Dass, author I. D. Sharp,  and author T. Liedl, @noop  journal journal Science volume 384, pages
  781 (year 2024)NoStop
[Liu et al.(2024)Liu,
  Matthies, Russo, Rovigatti,
  Narayanan, Diep, McKeen,
  Gang, Stephanopoulos, Sciortino, Yan, Romano, and Šulc]Sulc_2024
  author author H. Liu, author M. Matthies,
  author J. Russo, author L. Rovigatti, author
  R. P. Narayanan, author
  T. Diep, author D. McKeen, author O. Gang, author N. Stephanopoulos, author F. Sciortino, author H. Yan,
  author F. Romano,  and author P. Šulc, @noop 
  journal journal Science volume 384, pages 776 (year
  2024)NoStop
[Karner and Bianchi(2024)]karner2024anisotropic
  author author C. Karner and author E. Bianchi, @noop  journal journal
  Nanoscale Advances volume 6, pages
  443 (year 2024)NoStop
[Russo et al.(2009)Russo,
  Tartaglia, and Sciortino]Russo2009
  author author J. Russo, author P. Tartaglia,   and author F. Sciortino,  @noop  journal journal J. Chem. Phys.  volume 131 (year 2009)NoStop
[Gay and Berne(1981)]Gay1981
  author author J. Gay and author B. Berne,  @noop  journal journal J. Chem. Phys.  volume 74, pages 3316 (year
  1981)NoStop
[Berardi and RJ(1993)]Berardi1993
  author author R. Berardi and author A. RJ,  @noop  journal journal J. Chem. Soc.
  Farady Trans volume 89, pages 4069
  (year 1993)NoStop
[Bates and Luckhurst(1999)]Bates1999
  author author M. Bates and author G. Luckhurst, @noop  journal journal J.
  Chem. Phys. volume 110, pages 7087
  (year 1999)NoStop
[Zannoni(2001)]Zannoni2001
  author author C. Zannoni, @noop  journal journal J.
  Mater. Chem. volume 11, pages 2637
  (year 2001)NoStop
[De Gennes and Prost(1993)]De1993physics
  author author P.-G. De Gennes and author J. Prost, @noop  title The physics of liquid
  crystals, number 83 (publisher Oxford
  university press, year 1993)NoStop
[Wei and Patey(1992)]Wei1992
  author author D. Wei and author G. Patey,  @noop  journal journal Phys. Rev. Lett.  volume 68, pages 2043 (year
  1992)NoStop
[Allen and Tildesley(2017)]Allen2017
  author author M. P. Allen and author D. J. Tildesley, @noop  title Computer simulation
  of liquids (publisher Oxford university press, year 2017)NoStop
[Frenkel and Smit(2023)]Frenkel2023
  author author D. Frenkel and author B. Smit,  @noop  title Understanding molecular simulation:
  from algorithms to applications (publisher Elsevier,  year 2023)NoStop
[Billeter and Pelcovits(1998)]Jeffrey1998
  author author J. Billeter and author R. Pelcovits, 10.1063/1.168726 journal
  journal Computer in Physics volume
  12, pages 440 (year 1998)NoStop
[Puri(2004)]Puri2004
  author author S. Puri, https://doi.org/10.1080/01411590410001672648 journal journal Phase Transitions volume 77, pages 407 (year 2004)NoStop
[Weis and Levesque(1993)]Weis1993
  author author J. J. Weis and author D. Levesque, @noop  journal journal
  Phys. Rev. E volume 48, pages 3728
  (year 1993)NoStop
[Cañeda-Guzmán et al.(2014)Cañeda-Guzmán, Moreno-Razo, Díaz-Herrera, and Sambriski]Caneda2014
  author author E. Cañeda-Guzmán, author J. Moreno-Razo, author E. Díaz-Herrera,  and author E. Sambriski, @noop  journal
  journal Mol. Phys. volume 112,  pages 1149 (year 2014)NoStop
[Birdi et al.(2022)Birdi,
  Underwood, Wilding, Puri,  and Banerjee]Birdi2022
  author author N. Birdi, author T. L. Underwood, author N. B. Wilding, author S. Puri,   and author V. Banerjee,  @noop  journal journal Phys. Rev. E  volume 105, pages 024706 (year 2022)NoStop
[LAM()]LAMMPS
  @noop  howpublished
  <https://lammps.sandia.gov>NoStop
[Plimpton(1995)]Plimpton1995
  author author S. Plimpton, @noop  journal journal J.
  Comput. Phys. volume 117, pages 1
  (year 1995)NoStop
[Hoover(1985)]Hoover1985
  author author W. G. Hoover, https://doi.org/10.1103/PhysRevA.31.1695 journal journal Phys. Rev. A volume
  31, pages 1695 (year 1985)NoStop
[Hoover(1999)]Hoover1999
  author author W. G. Hoover, @noop  title Time Reversibility,
  Computer Simulation, and Chaos (publisher World Scientific,
  Singapore, year 1999)NoStop
[Andersen(1983)]Andersen1983
  author author H. C. Andersen, @noop  journal journal J.
  Comput. Phys. volume 52, pages 24
  (year 1983)NoStop
[Whitelam and Geissler(2007)]VMMC_1
  author author S. Whitelam and author P. L. Geissler, http://dx.doi.org/10.1063/1.2790421
  journal journal J. Chem. Phys. volume 127, eid 154101 (year
  2007)NoStop
[Whitelam et al.(2009)Whitelam, Feng, Hagan, and Geissler]VMMC_2
  author author S. Whitelam, author E. H. Feng,
  author M. F. Hagan,  and  author P. L. Geissler, 10.1039/B810031D journal journal Soft
  Matter volume 5, pages 1251 (year 2009)NoStop
[Hedges()]VMMC_3
  author author L. O. Hedges, http://vmmc.xyz title Libvmmc,  NoStop




 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: Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory.


Modeling stochastic eye tracking data: A comparison of quantum generative adversarial networks and Markov models
    Pedro Lind
    August 5, 2024
================================================================================================================



[remember picture,overlay]
	[anchor=north east,inner sep=20pt] at (current page.north east)
	
    < g r a p h i c s >
;


§ ABSTRACT 
We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. 
The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. 
By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. 
The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.


0.9
 AMS classification: 65M99, 81-08




Keywords: absorbing boundary conditions, variable potential, Schrödinger equation,
Titchmarsh-Weyl m-function, unbounded domain






§ INTRODUCTION

In this paper we consider the linear Schrödinger problem of the form

    u_t+∂_x^2u   =V(x)u,   (x,t)∈×(0,T],
    
       u(x,0)   =u_0(x),   x∈,

where T denotes the finite evolution time, and u_0 is an initial wave packet supported in a finite interval Ω_ int=[x_-,x_+] with x_-<x_+. 
It is well known that under mild conditions the Cauchy problem (<ref>) has a unique solution
u∈ C(^+,L^2()), cf. <cit.>, e.g.:

Let u_0∈ L^2() and real-valued potential V∈ L^∞().
Then the problem (<ref>) has a unique solution u∈ C(^+,L^2()).
Moreover, the “energy” is preserved, i.e.
    u(.,t)_L^2()=u_0_L^2(),  ∀ t ≥ 0.



The Schrödinger problem (<ref>) is defined on an unbounded domain x∈.
To numerically simulate its solution, it is common practice to truncate the domain to a bounded one,
for example, Ω_ int,T=Ω_ int×(0,T].
Absorbing boundary conditions (ABCs) are thus necessary for well-posedness at the two artificially introduced boundaries, Σ_±,T={x_±}×(0,T].

Numerical simulation of the linear Schrödinger equation on
unbounded domains with an external potential has been a hot
research area for nearly thirty years, cf. the concise review article <cit.>.
An ABC is called exact if the solution of the truncated domain problem remains the same
as that of the original unbounded domain problem.
The exact ABC is guaranteed to exist due to the well-posedness
of the linear Schrödinger problem (<ref>),
but it can only be formulated analytically for some special potentials,
such as constant potential <cit.>, linear potential <cit.>, symmetric periodic potential <cit.>,
isotropic free particle potential,
Morse potential, harmonic potential, and Bargeman potential,
cf. e.g. <cit.>.

In the more general case, i.e., for general variable potential problems
one is led to design approximate analytical ABCs for a given frequency regime with respect to some a priori criterion.
Methods in this category include the pseudo-differential calculus method <cit.>, the perfectly matched layer (PML) method <cit.>, 
and the operator splitting method <cit.>.
To the authors' knowledge, all of them are essentially based on the high frequency approximations.
For low-frequency problems, the ABCs would be less accurate by these methods.

This paper proposes a new approach to the design of ABCs for the Schrödinger problem.
Inspired by the work of Alpert, Greengard, and Hag­strom <cit.> on the fast evaluation of nonreflecting boundary kernels for time-domain wave propagation,
we approximate the Titchmarsh-Weyl m-function (equivalently, the exact DtN operator) in the frequency domain by a rational function with respect to an appropriate spectral parameter.
In the time domain, the nonreflecting boundary kernels are thus approximated by a sum of exponentials, which makes the approximate ABCs easy to implement.

The rationality of the above treatment is due to the analyticity property and the asymptotic behavior of the m-function.
Since our approximation is performed in the whole frequency regime, the proposed ABCs are expected to be more versatile and accurate, 
especially in the low-frequency regime, thus overcoming the typical high-frequency restriction.
Note that the Titchmarsh-Weyl m-function is nothing else but the so-called total symbol in microdifferential calculus,
which is treated by an asymptotic expansion to obtain a hierarchy of ABCs, cf. <cit.>.
Note also that Titchmarsh-Weyl theory is used in the analysis of initial value problems for Schrödinger equations operator-valued potentials <cit.> or strongly singular potentials <cit.>. It is also used in practical applications in quantum mechanics <cit.> and in option pricing in mathematical finance <cit.>.

This paper is organized as follows.
 In Section <ref> we review the basic facts of the Titchmarsh-Weyl theory for Schrödinger operators in one dimension for ease of later reference.
Then, in Section <ref> we discuss the Titchmarsh-Weyl m-function (i.e. the exact Dirichlet-to-Neumann operator) and explain the algorithm used to compute the m-function numerically.
Thus, at least from a numerical point of view, the exact ABC is explicitly known, see Section <ref>. 
However, when simulating the Schrödinger equation (<ref>), the difficulty does not lie in the computation of the m-function in a frequency domain method, 
which is presented in Section <ref>, but in its inverse Laplace transformation, which is too expensive.
For this reason, in Section <ref> we introduce a rational approximation of the m-function in the frequency domain
to obtain an approximate ABC that can be computed efficiently using a fast evaluation technique <cit.> in the time domain.
We discuss practical implementation issues and finally in Section <ref> we conclude with numerical results
illustrating that our new approach leads to an efficient and reliable algorithm for the time-dependent Schrödinger equation with a general variable potential.






§ THE TITCHMARSH-WEYL THEORY

We will review here, for convenience of later work, the essentials of
Titchmarsh-Weyl (TW) theory for Schrödinger operators in one dimension.
The interested reader may consult <cit.> or <cit.> for a more detailed presentation.

First, consider the Schrödinger operator L on the real line given by

    L=-∂_x^2+V(x),     x∈,

with a real-valued, locally integrable potential V∈ L^1_ loc().
Let x_0∈ be an arbitrarily chosen point, called reference point.
In the following we will study how the solutions depend on this parameter x_0.

To do this, we consider θ(x;x_0,λ) and ϕ(x;x_0,λ) to be the fundamental solutions of the Schrödinger eigenvalue problem.

    -u_xx+V(x)u=λ u,     x∈,  λ∈,

with the following initial conditions at the reference point x_0:
 

    θ(x_0;x_0,λ)   =1,    θ_x(x_0;x_0,λ)=0,
    ϕ(x_0;x_0,λ)   =0,    ϕ_x(x_0;x_0,λ)=1.


It can be shown that under these assumptions
θ(x;x_0,λ) and ϕ(x;x_0,λ) exist on the whole real axis,
and they are entire functions of λ and real for λ∈.
Now, as a basic fact of TW theory, the equation (<ref>) has at least one solution ψ_±, called Weyl's solution with


    ψ_±(x_0;x_0,λ)=1,

and


    ψ_±(x;x_0,λ)∈ L^2(^x_0_±)

for any λ∈_+.

Here ^x_0_± stands for the interval [x_0,±∞) and _+ denotes the upper half of the complex plane, i.e. _+={z∈  | z>0}.
A potential V(x) is said to be in the limit-point case at ±∞ if and only if there exists only one Weyl's solution in the corresponding L^2 space.
The reader will immediately realize that the assumption of V(x) in the limit-point case is necessary for the well-posedness of the Schrödinger problem (<ref>) in a more general setting.
At positive infinity, a standard sufficient condition for the limit-point case is given by <cit.>:


Let V(x) be a continuous real-valued function on (x_0,∞) and suppose that there
exists a positive differentiable function M(x) such that

  
  (i) V(x)≥-M(x) if x>x_0;
  
  (ii) ∫_x_1^∞(M(x))^-1/2 dx=∞ for any x_1>x_0;
  
  (iii) M'(x)/(M(x))^3/2 is bounded near ∞.

Then V(x) is in the limit-point case at  ∞.

An analogous result can be given at negative infinity point.

According to this theorem, a potential V(x) is in the limit-point case
provided that V(x)≥-k x^2 for some constant k and for all sufficiently large x.
This implies that the restriction on the potential for the limit-point case is very weak:
 It only excludes some especially strange potentials, which may not be physically relevant at all.
Roughly speaking, the limit case does not admit potentials that tend too fast
(faster than quadratically) to -∞ for x→±∞.

Due to the boundary conditions (<ref>) we can write

    ψ_±(x;x_0,λ)= θ(x;x_0,λ)+m_±(x_0,λ) ϕ(x;x_0,λ),

with some uniquely determined coefficient,
the Titchmarsh-Weyl m-function m_±(x_0,λ).
This function plays a fundamental role in the spectral theory of
the Schrödinger operator (<ref>) on the half-line ^x_0_±.

We will now summarize some of the most important properties of the
Titchmarsh-Weyl m-function. First,

    m_±(x_0,λ)  is analytic with respect to λ to \ and  m_±_+→_+

and is therefore called a Herglotz function (or Nevanlinna or Pick function),
cf. <cit.>.
It is easy to show that this Herglotz-property is directly related to the positive type of the DtN-map in the sense of memory equations,
cf. <cit.> for the corresponding case of constant external potential.
Thus it is an essential ingredient of the stability w.r.t. the L^2 norm.

We also have the symmetry property

    m_±(x_0,λ)=m_±(x_0,λ̅)

and the local singularities of m

are real and and most of them are first order, i.e.

    lim_ϵ→0+(-ϵ) m_±(x_0,λ+ϵ)   ≥0,  λ∈,


cf. <cit.>.

Another important property is given by the Borg-Marchenko theorem <cit.>, 
which states that the Titchmarsh-Weyl m-function m_±(x_0,λ) 
uniquely determines the potential V(x) at x>x_0 (or x<x_0).
Moreover, since ψ_±(x;x_0,λ) changes with a simple multiplication 
when the reference point x_0 changes, one has

    m_±(x,λ)= ∂_xψ_±(x;x_0,λ)/ψ_±(x;x_0,λ).

Thus, it is easy to verify that the m function satisfies the following Riccati equation:

    ∂_xm_±(x,λ)=-m_±^2(x,λ)+V(x)-λ.







§ THE EXACT ABC BY TITCHMARSH-WEYL THEORY

We apply the Laplace transform

    (x,s) = ℒ(u(x,t))(s) = ∫_0^+∞ u(x,t)  e^-st dt,    s>0,

to the Schrödinger equation (<ref>) on the right exterior domain Ω_+={x∈| x>x_+} and on the left exterior domain Ω_-={x∈| x<x_-}.
In the frequency domain, the Schrödinger equation is a second order homogeneous ODE

    -_xx+V(x)=λ,   x∈Ω_±,

with λ= s∈_+. The exact absorbing boundary condition of the DtN form in the frequency domain is thus

    û_x(x_±,λ) = m_±(x_±,λ)û(x_±,λ).



Only in some special cases, the m-function has a closed analytic form <cit.>.
For example, in the case of a constant potential V≡ V_0 one gets

    m_+(x_+,λ)=-√(-λ+V_0).

If the potential represents a harmonic oscillator, i.e. V(x)=x^2 on the interval [0,∞),
one obtains a meromorphic m-function given by the ratio of two gamma functions:

    m_+(0,λ)=-2Γ(3/4-1/4λ)/Γ(1/4-1/4λ).










Finally, for the Bargmann potential

    V(x)=-8β^2 β-γ/β+γe^-2β x/(1+β-γ/β+γe^-2β x)^2,  β>0,  γ≥ 0,

one obtains the m-function

    m_+(0,λ)=-√(-λ)-γ^2-β^2/√(-λ)+γ.


In the general case, however, numerical methods must be considered.
This problem has been studied in many papers, e.g. <cit.>.
In this paper we simply compute the m-function by evolving the Riccati equation (<ref>) 
with the classical fourth-order Runge-Kutta scheme and setting an initial data m_±(x_±,λ,λ)=∓√(-λ) 
at a sufficiently distant point x_±,λ=± 200. 
This treatment is reasonable since the potentials in our numerical tests actually decay to zero for x→∞.







§ THE FREQUENCY-DOMAIN METHOD

The solution of the time-dependent Schrödinger equation could then be computed 
using the following frequency-domain method:


  * Fix σ>0.
For each s=σ+μ with μ∈, solve the

Laplace-transformed Schrödinger equation in the bounded interval 
[x_-,x_+]:

    -_xx+V(x)   =  s  - u_0(x),   x∈[x_-,x_+],
    
    
      _x(x_-)   =m_-(x_-, s) (x_-),
    _x(x_+)   =m_-(x_+, s) (x_+).


  * Perform the inverse Laplace transformation

    u(x,t)    = ℒ^-1(û(s,t))(x)
       =1/2π∫_σ-∞^σ+∞e^st(s,t) ds
    =
    e^σ t/2π∫_-∞^∞e^ ftu(x,σ+ f) df,


to derive the wave function u(x,t) for any t∈(0,T].


In the numerical implementation, some parameters have to be tuned.
The function û is smoother for larger damping factor σ, but the evolution time span is then limited
because an exponential term is involved in the inverse Laplace transformation. 
As a common practice, we set σ=1/T, where T is a prescribed evolution time. 
The integration domain is unbounded in (<ref>) and must be truncated. 
We introduce a cut-off frequency f_c and confine the integration to the interval [-f_c,f_c]. 
In addition, to get rid of high frequency oscillations in the inverse-transformed function, we should introduce another filtering function χ,
which remains 1 over a sufficiently large frequency band with zero frequency as its center, and vanishes smoothly near the endpoints of [-f_c,f_c]. 
A good candidate (empirically) is

    χ=exp(-(1.2f/f_c)^20).

After these treatments, we then derive an approximate inverse transformation as

    e^σ t/2π∫_-∞^∞e^ ftu(x,σ+ f) df≈e^σ t/2π∫_-f_c^f_cχ(f) e^ ftu(x,σ+ f) df.

The right side is computed with an appropriate quadrature scheme.




§ THE TIME-DOMAIN METHOD

It follows from Section <ref> that the exact ABC we are looking for is now explicitly known, at least from a numerical point of view.
But this is not the whole story for simulating the solution of the time-dependent Schrödinger equation.
The difficulty lies not in computing the m-function itself, but in computing its inverse Laplace transformation.
Of course, a numerical inverse transformation is possible, but it would be too expensive.

Therefore, in this section we design an approximate ABC based on the rational approximation of the m-function.
The kernel functions are of exponential type with respect to the half-order time derivative operator, 
so the fast evaluation technique proposed in <cit.> (cf. Appendix) is applicable.
For some alternative fast evaluation methods, we refer the reader to <cit.> and the references therein.
The rational approximation is realized by solving a least squares problem, 
an analogous technique to that used in <cit.> for fast evaluation of the boundary kernel functions of the hyperbolic wave equation.

In the time domain, the truncated Schrödinger problem reads

    u_t+∂_x^2u    = V(x)u,   (x,t)∈[x_-,x_+]×(0,T],
    
      u(x,0)   =u_0(x),   x∈[x_-,x_+],
    
      u_x(x_±,t)    = ℒ^-1(m_±(x_±, s) û(x_±,s))(t),   t∈ (0,T].

To simplify the notation, we will focus on the right boundary at x=x_+.
Let us recall that the DtN map in the frequency domain is

    û_x(x_+,s) = m_+(x_+, s) û(x_+,s).

Returning to the time domain we have to consider the convolution

    u_x(x_+,t) = K(t)*u(x_+,t),
        with  K(t) = ℒ^-1(m_+(x_+, s))(t).

There are two major difficulties here.
First, it is generally hard to compute K(t)
and second, the convolution involved naturally leads to a nonlocal-in-time DtN map.

To get an idea, let us first consider two specific simple examples.
In the case of the free Schrödinger (V≡ 0), cf.(<ref>), we have

    m_+( s)=-√(- s)   and thus   ℒ^-1(m_+( s))=-e^-π/4 ∂_t^1/2,

with the half-order time derivative defined as

    ∂_t^1/2 v(t) = 1/√(π)  d/ d t∫_0^t v(τ)/√(t-τ)  dτ,

which can be efficiently evaluated with some existing methods,
e.g. <cit.>.
Second, we consider the Bargmann potential (<ref>), and in this case
the m-function is

    m_+(0,is)=-√(- s)-γ^2-β^2/√(- s)+γ,

thus we have

    ℒ^-1(m_+(0, s)) = -e^-π/4 ∂_t^1/2
        -(γ^2-β^2)(e^-π/4 ∂_t^1/2+γ)^-1.

This operator can then be evaluated efficiently by introducing an unknown function and using the fast methods for ∂_t^1/2.

Inspired by these two examples it naturally leads us to think about the possibility of
approximating the m-function with a rational function with respect to a new spectral parameter k=√(- s) (NOT s), i.e.,

    m_+(x_+, s)≈m̃_+(x_+, s)
      =-√(- s)+∑_n=1^dα_n/√(- s)+β_n.

Once this is done, we can then replace the exact m-function with the approximate alternative m̃,
which leads to the approximate kernel function

    ℒ^-1(m̃_+(x_+, s)) 
       = -e^-π/4 ∂_t^1/2+∑_n=1^dα_n(e^-π/4 ∂_t^1/2+β_n)^-1.

The analogous idea for the Bargmann potential (<ref>) can then be used to handle this kernel function.

The answer for the possibility is affirmative considering that the asymptotic expansion has been given in <cit.> as

    m_+(x_+,λ)=-√(-λ)+(1/√(r)),   r→∞,

where λ=μ r, r∈ and the convergence is uniform for μ in any compact subset of _+, cf. <cit.>.

Now putting

    _+(λ)=m_+(x_+,λ)+√(-λ),

in view of (<ref>) we know that _+(λ) is analytic in _+ with respect to λ and
it tends to zero for λ→∞.
We then use the method of Alpert, Greengard and Hagstrom  <cit.> to approximate _+(λ) with a rational function with respect to √(-λ) (NOT λ).
In terms of (<ref>) we consider the following nonlinear least squares problem

    ϵ=min_P,Q∫_-∞+σ^∞+σ|P(√(-λ))/Q(√(-λ))-_+(λ)|^2 |d√(-λ)|,

where P, Q are polynomials with (P)+1=(Q)=d, and d is determined by making ϵ≤ϵ_0, where ϵ_0 is a prescribed tolerance number.
This nonlinear problem (<ref>) is then solved using the technique of
 linearization and orthogonalization <cit.>.
Finally, by expressing P/Q as a sum of poles, we arrive at

    m_+(x_+,λ)≈m̃_+(x_+,λ)=-√(-λ)+∑_n=1^dα_n/√(-λ)+β_n.

Note that the coefficients α_n and β_n should appear as conjugate pairs due to the symmetry property (<ref>).
Unfortunately, it is not clear whether the rational approximation m̃_+(x_+,λ) in (<ref>) still has the important Herglotz-property of the m-function.
Moreover, the Herglotz-property cannot be checked by some conditions on the poles due to the leading square root in (<ref>).



Applying the same idea to the m-function m_-(x_-,λ) and we get
the approximate boundary condition in the frequency domain

    _x(x_±,s) = (∓√(- s)+∑_n=1^d_±α_n,±/√(- s)+β_n,±)(x_±,s).

If we introduce new unknowns ŵ_n,± as

    ŵ_n,± = (x_±,s)/√(- s)+β_n,±,

then we can rewrite (<ref>) as


    _x(x_±,s)±√(- s)    = ∑_n=1^d_±α_n,±ŵ_n,±,
    √(- s) ŵ_n,±+β_n,±ŵ_n,±   = (x_±,s),     n=1,…,d_±.


In the time domain, the approximate boundary condition is


    u_x(x_±,t)± e^-π/4 ∂_t^1/2u(x_±,t)
         = ∑_n=1^d_±α_n,±w_n,±(t),
    
      e^-π/4 ∂_t^1/2w_n,±(t)+β_n,±w_n,±(t)   =u(x_±,t),     n=1,…,d_±.






The final approximate truncated time-domain problem is formulated as

    u_t+∂_x^2u   =V(x)u,   (x,t)∈Ω_ int×(0,T],
    
      u(x,0)   =u_0(x),   x∈Ω_ int,
    
      u_x(x_±,t)   ± e^-π/4 ∂_t^1/2u(x_±,t)
      = ∑_n=1^d_±α_n,±w_n,±(t),
    
      e^-π/4 ∂_t^1/2w_n,±(t)   +β_n,±w_n,±(t)
    =u(x_±,t),     n=1,…,d_±.






§ NUMERICAL RESULTS

In this section, we present some numerical results to test the accuracy of the proposed methods.
In each example, the standard Crank-Nicolson scheme for time discretization is used.
The fast evaluation of the half-order time derivative operator (<ref>) is performed with the method of Zheng <cit.>.
The computational domain is chosen to be Ω_ int=[x_-,x_+]=[-5.5] and the initial data is a Gaussian beam: u_0(x)=e^-x^2+4 x.
We use an 8^ th-order FEM method with 1024 elements for the spatial discretization and a uniform time step of size Δ t=10^-4.





 §.§ The Free Schrödinger Equation

The exact solution of the free Schrödinger equation (V(x)≡0) is

    u_ exa(x,t) = √(/-4t+)exp(- x^2-4x+16t/-4t+).

We set the cut-off frequency f_c in (<ref>) to be f_c=256 and used a filtering function χ(f)=exp(-(1.2f/f_c)^20).
The following Table <ref> shows the relative L^2-errors to the exact solution at certain time points when using 8097 quadrature points with Simpson's rule.

Using this simple example, we can see that the frequency method with truncation and filtering works quite well: the magnitude of the relative L^2-errors is at most on the order of 10^-7.





 §.§ The Coulomb-like Potential

In the second example we test the time-domain method with a Coulomb-like potential

    V(x)=1/√(1+x^2).

We set σ for Step 1 to σ=1 and the tolerance number for the nonlinear least squares problem (<ref>) to ϵ_0=10^-8. 
Here we get 4 poles.
Figure <ref> shows the time evolution in a colored contour plot. 
One can see how the initial beam spreads out as time increases.

Figure <ref> shows how the relative L^2-error evolves in time.
In this example with a varying external potential the magnitude
of the relative L^2-errors remains below 10^-5.






 §.§ The Gaussian Barrier

Next, we change the potential to a Gaussian barrier

    V(x)=30e^-36(x-8)^2

with a height of 30, located in the exterior domain x>5 and centered at x=8. 
We set σ=1 and ϵ_0=10^-4 and get 21 poles with the nonlinear least squares algorithm. 
Figure <ref> shows the time evolution of the solution. 
One can clearly see how the initial beam propagates, spreads, and is (partially) reflected by the Gaussian barrier (<ref>). 
The time evolution of the corresponding relative L^2-error is shown in Figure <ref>.
The relative L^2 errors remain below 5× 10^-4.

Unfortunately, the nonlinear least squares algorithm used in this paper failed to produce a rational approximation within an error tolerance much smaller than ϵ_0. 
A more efficient algorithm is still desired, and this problem is currently under investigation.












§ CONCLUSION AND OUTLOOK

In this work we presented a new approach to simulate the solution to the Schrödinger equation with a general space-dependent potential in unbounded domains. 
Both frequency-domain and time-domain methods have been developed.

Future work will consist of implementing a more sophisticated algorithm for computing the m-function.
Instead of solving the Riccati equation (<ref>), we will consider computing the Weyl circles <cit.>
or the boundary control approach of Avdonin, Mikhaylov and Rybkin <cit.>.
We will also look for a more stable algorithm for its rational approximation.
It will also be clarified how this rational approximation can be made to preserve the essential 
Herglotz-property of the analytic m-function.
This study will allow us to perform a rigorous stability analysis of this new approach.
Finally, as a future goal, we want to extend our approach to the multi-dimensional Schrödinger problem, following the idea of Amrein and Pearson <cit.>.






§ APPENDIX: FAST EVALUATION METHOD

Here we present a short description of the method in <cit.> for evaluating the
half-order time derivative ∂_t^1/2. 
For any smooth function v=v(t) with v(0)=v'(0)=0, 
it is known that the semi-discrete half-order derivative

    D_t^1/2v(t_n):=√(2/Δ t)∑_m=0^nα_m  v(t_n-m)

with

    α_m=β_k=(2k)!/2^2k(k!)^2   , m=2k,
    
    -β_k   , m=2k+1,

gives a second-order approximation of ∂_t^1/2v(t_n) (see <cit.>).
Suppose there exists a sum of decaying exponentials satisfying

    β̃_k = ∑_j=1^Lw_j e^-s_jk,   s_j>0,   |β_k-β̃_k|≤ϵ, k=0,1,…,[N/2].

Here N is the total number of time steps. 
If ϵ is small enough, it is reasonable to approximate (<ref>) with

    D̃_t^1/2v(t_n) := √(2/Δ t)(v(t_n)-v(t_n-1)) + √(2/Δ t)∑_m=2^nα̃_mv(t_n-m),

where

    α̃_m=
    β̃_k   , m=2k,
    
     -β̃_k   , m=2k+1.

Set v_k=v(t_k), 𝐯=(v_0,v_1,…), and define

    ℱ_ odd(w,s;𝐯,k)
    
    :=∑_m=1^kwe^-smv_2k+1-2m

and

    ℱ_ even(w,s;𝐯,k)
    
    :=∑_m=1^kwe^-smv_2k-2m.

Thus,
ℱ_ odd(w,s;𝐯,0)=ℱ_ even(w,s;𝐯,0)=0.
In addition, we have the following recursions

    ℱ_ odd(w,s;𝐯,k)
       =   e^-s[wv_2k-1+ℱ_ odd(w,s;𝐯,k-1)],
    ℱ_ even(w,s;𝐯,k)
       =   e^-s[wv_2k-2+ℱ_ even(w,s;𝐯,k-1)].

The summation (<ref>) is then computed within O(L) operations
as

    ∑_m=2^nα̃_m v_n-m=
    ∑_j=1^Lℱ_ even(w_j,s_j;𝐯,k)-∑_j=1^Lℱ_ odd(w_j,s_j;𝐯,k-1)   , n=2k,
    ∑_j=1^Lℱ_ odd(w_j,s_j;𝐯,k)-∑_j=1^Lℱ_ even(w_j,s_j;𝐯,k)   ,
    n=2k+1.

In <cit.> for N=1,000,000, the authors found a sum of 81
decaying exponentials that approximates β_k with an error of less than 5.0× 10^-11.



00

akto
T. Aktosun,
Mathematical Studies in Nonlinear Wave Propagation,
Contemporary Mathematics 379,
 American Mathematical Society, Providence, RI, 2005.


AlpertGreengardHagstrom
B. Alpert, L. Greengard and T. Hagstrom,
Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation,
SIAM J. Numer. Anal. 37 (2000), 1138–1164.

AmPe04
W.O. Amrein and D.B. Pearson,
M operators: a generalisation of Weyl-Titchmarsh theory,
J. Comput. Appl. Math. 171 (2004), 1–26.








AntoineBesse2003
X. Antoine and C. Besse, Unconditionally stable discretization
schemes of non-reflecting boundary conditions for the
one-dimensional Schrödinger equation,
 J. Comput. Phys. 181 (2003), 157–175.

AnBeMo04
X. Antoine, C. Besse and V. Mouysset,
Numerical Schemes for the simulation of the two-dimensional
Schrödinger equation using non-reflecting boundary conditions,
Math. Comp. 73 (2004), 1779–1799.

AnBeDe06
X. Antoine, C. Besse and S. Descombes,
Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations,
SIAM J. Numer. Anal. 43 (2006), 2272–2293.

TBC
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle,
A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations,
Commun. Comput. Phys. 4 (2008), 729–796. 

AnBeSz09
X. Antoine, C. Besse and J. Szeftel,
Towards accurate artificial boundary conditions for nonlinear PDEs through examples,
CUBO 11 (2009), 29–48.

ArEhSo03
A. Arnold, M. Ehrhardt and I. Sofronov,
Discrete transparent boundary conditions for the Schrödinger
equation: fast calculation, approximation, and stability,
Comm. Math. Sci. 1 (2003), 501–556.












 


AMR07
S. Avdonin, V. Mikhaylov and A. Rybkin,
The boundary control approach to the Titchmarsh-Weyl m-function, 
Comm. Math. Phys. 275 (2007),  791–803.

Benn80
C. Bennewitz and W.N. Everitt,
Some remarks on the Titchmarsh-Weyl m-coefficient.
Proc. Pleijel Conference, Uppsala, Sweden, 1979, pp. 49–108.






Benn20
C. Bennewitz, R. Weikard and M. Brown,
Spectral and scattering theory for ordinary differential equations, Springer, 2020.






Borg
G. Borg,
Uniqueness theorems in the spectral theory of y”+(λ-q(x))y=0,
Proc. 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo,
1952, pp. 276–287.

Brown
B.M. Brown, V.G. Kirby and J.D. Pryce,
A numerical method for the determination of the Titchmarsh-Weyl m-coefficient,
Proc. R. Soc. Lond. A  435 (1991),  535–549.









 











Chiba90
Y. Chiba and S. Ohnishi,
Resonance-state calculation applying the Weyl-Titchmarsh theory: 
Application for the quantum-confined Stark effects on excitons in a GaAs–Al_xGa_1-xAs quantum well,
Phys. Rev. B 41 (1990), 6065–6068











Codd55
E.A. Coddington and N. Levinson,
Theory of Ordinary Differential Equations,
McGraw-Hill, New York, 1955.









Danielyan
A.A. Danielyan and B.M. Levitan,
On the asymptotic behavior of the Weyl-Titchmarsh m-function,
Math. USSR Izvestiya  36 (1991), 487–496.






EhAr01
M. Ehrhardt and A. Arnold,
Discrete transparent boundary conditions for the Schrödinger equation,
Riv. Mat. Univ. Parma  6 (2001), 57–108.

EhMi04
M. Ehrhardt and R.E. Mickens,
Solutions to the discrete Airy equation: Application to parabolic equation calculations,
J. Comput. Appl. Math. 172 (2004), 183–206.

Ehr07
M. Ehrhardt,
Discrete transparent boundary conditions for Schrödinger-type equations
for non-compactly supported initial data,
Appl. Numer. Math. 58 (2008), 660–673.

EZ
M. Ehrhardt and C. Zheng,
Exact artificial boundary conditions for problems with periodic structures,
J. Comput. Phys. 227 (2008), 6877–6894.

Zheng
M. Ehrhardt, H. Han and C. Zheng,
Numerical simulation of waves in periodic structures,
 Commun. Comput. Phys. 5 (2009), 849–870.











Simon
F. Gesztesy and B. Simon,
A new approach to inverse spectral theory, II. General real potentials
and the connection to the spectral measure,
Ann. Math.  152 (2000),  593–643.

GZ
F. Gesztesy and M. Zinchenko,
On spectral theory for Schrödinger operators with strongly singular potentials,
Math. Nachr. 279 (2006), 1041–1082.

GZ13
F. Gesztesy, R. Weikard and M. Zinchenko,
Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials,
Oper. Matrices 7(2) (2013), 241–283.










GDB98
G.N. Gibson, G. Dunne and K.J. Bergquist,
Tunneling Ionization Rates from Arbitrary Potential Wells,
Phys. Rev. Lett. 81 (1998), 2663–2666.

GKW00
N. Gordon, J.P. Killingbeck and M.R.M. Witwit,
Numerical Determination of the Titchmarsh-Weyl m-coefficient,
Comput. Math. & Math. Phys. 40, (2000), 108–117.











JiGr04
S. Jiang and L. Greengard,
Fast evaluation of nonreflecting boundary conditions for the Schrödinger
equation in one dimension,
Comp. Math. Appl. 47 (2004), 955–966.






Kirby
V.G. Kirby,
A numerical method for determining the Titchmarsh-Weyl m-coefficient and its application
to certain integro-differential inequalities,
Ph.D. Thesis, University of Wales, College of Cardiff, UK, 1990.

AK12
A. Kostenko, A. Sakhnovich and G. Teschl, 
Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, 
Int. Math. Res. Not. 8 (2012), 1699–1747.







LiZhang
Y. Li and J.E Zhang,
Option pricing with Weyl-Titchmarsh theory,
Quant. Finance 4 (2004), 457–464.

March
V.A. Marchenko,
Certain problems in the theory of second-order differential operators,
Doklady Akad. Nauk SSSR 72 (1950), 457–460. (Russian).












Pazy
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations,
Springer, New York 1983.

Pol02
A.D. Polyanin,
Handbook of Linear Partial Differential Equations for Engineers and Scientists,
Chapman & Hall/CRC, 2002.

RS2
M. Reed and B. Simon,
Methods of modern mathematical physics II: Fourier analysis, self-adjointness,
Academic Press, San Diego 1975.












Sims07
R. Sims,
Reflectionless Sturm-Liouville equations,
J. Comput. Appl.  Math.  208 (2007), 207–225.

Titchmarsh
E.C. Titchmarsh,
Eigenfunction Expansions Associated with Second-Order Differential Equations,
Oxford University Press, London, 1946, Volumes 1 and 2.






 

Weyl
H. Weyl,
Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen,
Math. Ann. 68 (1910), 220–269.

WGK99
M.R.M. Witwit, N. Gordon and J.P. Killingbeck,
Numerical computation and analysis of the Titchmarsh-Weyl m(l) function for some simple potentials,
J. Comput. Appl. Math. 106 (1999), 131–143.

ZXW08
J. Zhang, Z. Xu and X. Wu,
Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations,
Phys. Rev. E 78 (2008), 026709.

ZhengFast
C. Zheng,
Approximation, stability and fast evaluation of exact artificial boundary condition for
the one-dimensional heat equation,
J. Comput. Math. 25 (2007), 730–745.

ZhengPML07
C. Zheng,
A perfectly matched layer approach to the nonlinear Schrödinger wave equations,
J. Comput. Phys. 227 (2007), 537–556.




 Paper Title: Beyond full statevector simulation with Qibo.



 TIF-UNIMI-2024-7







^1Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE
^2TIF Lab, Dipartimento di Fisica, Università degli Studi di Milano
^3School of Computing, Australian National University, Canberra, ACT, Australia
^4Instituto de Física, Universidade Federal do Rio de Janeiro, P.O. Box 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil

^5European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland


andrea.pasquale@unimi.it

§ ABSTRACT 
    In this proceedings, we present two new quantum circuit simulation protocols recently added as optional backends to , an open-source
    framework for quantum simulation, hardware control and calibration.
    We describe the current status of the framework as for version .
    In detail, the two new backends for Clifford and tensor networks simulation are presented and benchmarked against the state-of-the-art.




§ INTRODUCTION

With the significant achievements reached by quantum technologies <cit.>,
the interest towards quantum computing is consistently growing. However, the noise
affecting these early devices is preventing large-scale applications of quantum algorithms.
Waiting for quantum computers to reach a level of reliability that allows for
full-scale execution of quantum computing routines without limitations on the problem size,
it is necessary to continuously improve the simulation tools we have at our disposal.
In particular, the development of classical simulation techniques is crucial to
overcome the intrinsic representation limit in full statevector simulation, where
the memory required to fully represent a qubit system explodes exponentially as
the number of qubits increases.

Several open-source software packages for classical simulation of quantum systems
are available. While some of them, including  <cit.> and
 <cit.>,  <cit.>,
 <cit.> and  <cit.>, provide general tools for quantum
simulations, other libraries such as  <cit.>
mostly focus on specific applications like quantum machine learning.

In this context, for the last 4 years we have been developing  <cit.>,
an open-source software
framework for quantum computing. At its early stages, was mainly dedicated
to full statevector (including density matrix) simulation <cit.>,
and managed to achieve performance competitive with state-of-the-art simulators,
thanks, also, to a Just-In-Time compilation approach <cit.>.
The current layout of the framework, as of version , is shown in
Fig. <ref>. provides a language Application Programming Interface
(API) to deploy quantum algorithms using either a circuit-based or a quantum
annealing-based approach. Moreover, we provide general-purpose tools that are useful in quantum information theory,
including calculation of distances among quantum states and quantum channels.

's modular structure enables the deployment of any component of the language
API on different software and hardware platforms, which we refer to
as backends.
We are now going to briefly present all the backends available in version .
The package <cit.> upon installation is equipped with a backend based on
 <cit.>, which provides adequate performances for simulating circuits with
a relatively low number of qubits n (i.e. n ≤ 20). We also provide a more efficient
general purpose simulator, called ,
which supports hardware acceleration. More specifically, allows multi-threaded CPU execution
using  <cit.>, while GPU support is enabled both through  <cit.>
and through compatibility with NVIDIA's  <cit.> library.
To improve the performance, defines custom operators that
exploit the sparsity of the matrix representation of several quantum
gates that are often used in quantum computing.
Such improvements are also reflected in the quantum annealing approach when the Trotter decomposition is used.
We also provide backends supporting automatic differentiation specifically designed for quantum machine learning
applications, including one based on  <cit.>
primitives, and a recently added one based on the popular machine learning framework  <cit.>.
The possibility of integrating automatic differentiation tools within
such a modular environment can be exploited to develop and test both pure quantum
and hybrid classical-quantum machine learning algorithms <cit.>.

As a middleware software package, also provides a backend for quantum hardware execution.
This backend is called , which defines a dedicated API to perform instrument
control, driver operations, as well as compilation
of circuits into customizable native gates sets.
Alongside , we provide a dedicated package to characterize and calibrate
self-hosted quantum devices:  <cit.>.
Besides simulation and hardware backends, there is an ongoing effort to provide cloud access, giving
users the possibility to manage their self-hosted quantum hardware by using 's hardware control tools,
or to deploy quantum circuits on well-known quantum cloud providers, including IBM Quantum <cit.> and IonQ <cit.>.

This was a brief overview to showcase the wide modularity and diversity that provides. As a demonstration of the
ongoing effort to continuously support new platforms and extend 's compatibility with state-of-the-art quantum
computing software stacks we recently added a backend that interfaces with  <cit.>.

Below, we focus on two simulation backends recently added to the framework, which are dedicated to tensor network simulation of quantum circuits and fast simulation of Clifford circuits.



§ SIMULATION OF CLIFFORD CIRCUITS

With full statevector simulation, it is only possible to simulate circuits up to
a limited number of qubits: John Preskill describes this limitation in Ref. <cit.>
introducing the approximate yet explanatory term of 50-qubit barrier.
The main limitations are imposed by  huge memory requirements and very long computation times <cit.>.
Over the last decades, researchers have studied extensively Clifford circuits, a class of circuits that can be simulated in polynomial times <cit.>.
The quantum states generated by Clifford circuits are called stabilizers states <cit.>,
and have many applications in quantum information theory <cit.>.
As current quantum hardware gets closer to the requirements needed for quantum error correction<cit.>,
such circuits have started to receive more attention, leading to the development of techniques to classically simulate them.



Based on this, a new backend has been developed in which focuses on fast and efficient simulation of Clifford circuits.
The implementation in is based on the phase-space formalism introduced in Ref. <cit.>.
From the perspective of code design, the implementation makes full use of 's modularity, allowing for alternative backends to be easily plugged in.
The basic implementation proposed is based on  primitives. This enables the single-threaded simulation of Clifford circuits on CPUs.
By taking advantage of , we also provide implementations based on  and .
The integration with the  backend for multi-threaded computations on CPUs is done via custom kernels that are compiled just-in-time.
For GPU integration, we make use of custom CUDA C kernels through .

To evaluate our implementation, we compare our Clifford simulator with Clifford simulators available in  and .
The results are shown in Fig. <ref>.
We tested all the simulators on the same dataset consisting of 100 random Clifford circuits for each system syze n.
These circuits were sampled uniformly following Ref. <cit.>.
We ran the CPU benchmarks on an AMD EPYC 7773X processor and the GPU benchmarks on a NVIDIA RTX Quadro a6000.
Our  and  backends asymptotically approach performance similar to ,
whereas our -based implementation displays an advantage over the whole range of qubits considered.
For instance, for n = 1000, our  backend is up to one order of magnitude faster than , and almost two order of magnitudes faster than .

Here, we make a remark about our GPU implementation.
As expected, the overhead of copying the data from the host to the device is dominating the results for a small number of qubits.
However, an improvement due to the huge parallelization capabilities of GPUs is expected to appear at n ≳ 2^11.
Since we benchmarked average runtimes up to 1000 qubits, this crossing in performance was not observed yet, making further investigation of bigger systems a necessity.




§ TENSORNETWORK SIMULATION USING 


After showing that for specific type of quantum circuits we can reduce the computational time,
we now introduce a second popular approach to simulate large quantum circuits:
classical approximation methods.
A popular method for approximating quantum circuits are tensor networks (TN) <cit.>,
which represents states or operators as network of smaller tensors reducing both memory and computational requirements.
They are successfully used, for instance, to solve one-dimensional strongly-correlated quantum systems <cit.>.
On the other hand, the effectiveness of representing large-scale systems
comes at the cost of introducing truncation errors, which make these techniques less
effective when there is the need to know the quantum state more exhaustively.

Within the framework, we have recently developed , a subpackage which enables to execute quantum circuits using tensor network like computations,
allowing to support large-scale simulation of quantum circuits in .
interfaces with state-of-the-art quantum TN simulation libraries such
as  <cit.> from NVIDIA and  <cit.>.
Both Matrix Product States and generic TN are supported. is designed to support
High Performance Computing configurations including single node GPUs, as well as multi-node multi-GPU configuration using
Message Passing Interface or the NVIDIA Collective Communication Library (NCCL) from .

To showcase the capabilities of the library we perform comparison between and performances in Fig.<ref>.
We execute a variational quantum circuit on a  A100 GPU for several number of qubits.
As expected, using full statevector simulation with it possible to run only up to 40,
while using we show how the curve flattens and we observe that we are able to simulate a variational circuit with up to 400 qubits.
Moreover, given the slow rise of the curve we expect to increase the number of qubits with appropriate memory requirements.



§ OUTLOOK


In this proceedings, we have described the latest updates available in .
After a brief overview on all modules currently available in the framework, we have put the focus
on two new simulation methodologies: Clifford simulation and Tensor Networks (TN). We have shown that our Clifford simulator
is competitive with state-of-the-art libraries. We further demonstrated that, despite its early development stage,
our TN implementation is able to simulate efficiently circuits with up to 400 qubits.
Future developments of the framework include having a dedicated module to perform QML algorithms,
which we refer to as , and we are looking forward to expand to support also quantum chemistry, as well as Quantum optimization problems.
Finally, although has been developed in , we are looking to separate core elements to take advantage of the better performance offered by other languages,
e.g.  and .



§ ACKNOWLEDGEMENTS

This project is supported by ’s Quantum Research Center.
The authors thank all contributors for helpful discussion and Liwei Yang and Andy Tan Kai Yong
for their support in developing .
M.R. is supported by ’s Quantum Technology Initiative () through the Doctoral Student Program.

unsrtnat





 Paper Title: Fast and scalable finite-element based approach for density functional theory calculations using projector-augmented wave method.












Department of Computational and Data Sciences, Indian Institute of Science Bengaluru, India




Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA


phanim@iisc.ac.in

Department of Computational and Data Sciences, Indian Institute of Science Bengaluru, India




             

§ ABSTRACT 









Gaining ab initio insights into complex  material systems at nanometer length-scale demand enormous computational resources due to the strict accuracy requirements of density functional theory (DFT) calculations compounded with the cubic scaling complexity of DFT with number of atoms. 
In this work, we present a computationally efficient methodology that utilizes a local real-space formulation of the  projector augmented wave (PAW) method  discretized with finite-element (FE) basis to enable accurate and large-scale electronic structure calculations. This is achieved through the synergy of the PAW method's ability to transform the underlying all-electron DFT problem into a form that involves smoother electronic fields and the potential of systematically convergent higher-order finite-element basis to achieve significant computational gains. In particular, we employ higher-order quadrature rules to evaluate integrals involving PAW-atomic data accurately, allowing the use of coarser FE meshes for electronic fields. 
Further, we have developed efficient computational strategies for solving the underlying FE discretized PAW generalized eigenproblem by employing the Chebyshev filtered subspace iteration approach to compute the desired eigenspectrum in each self consistent field iteration. These strategies leverage the low-rank perturbation of FE basis-overlap to invert FE discretized PAW overlap matrix in conjunction with exploiting the FE-cell level structure of both the local and nonlocal parts of the discretized PAW Hamiltonian and overlap matrices. Using the proposed approach we benchmark the accuracy and performance on various representative examples involving periodic and non-periodic systems with state-of-the-art plane wave based PAW implementations.  Furthermore, we also demonstrate a significant computational advantage over state-of-the-art plane-wave basis for large-scale systems comprising of around 35,000 electrons. Our approach () significantly reduces the degrees of freedom required to achieve desired accuracy when compared to norm-conserving pseudopotentials, thereby enabling large-scale DFT simulations at an order of magnitude lower computational cost.





                              
Fast and scalable finite-element based approach for density functional theory calculations using projector-augmented wave method
    Phani Motamarri
    August 5, 2024
================================================================================================================================







§ INTRODUCTION 


Electronic structure calculations based on Kohn-Sham density functional theory (DFT) have significantly advanced our understanding of material properties. The Kohn-Sham approach to DFT <cit.> reformulates the many-body problem of interacting electrons into an equivalent computationally tractable problem of non-interacting electrons within an effective mean field governed by electron density.  This reformulation has made it possible to predict a wide range of material properties, including mechanical, transport, electronic, magnetic and optical properties, over the past six decades. Pseudopotential approximations in DFT <cit.> have been crucial in making these calculations computationally efficient, as they focus only on valence electrons that participate in chemical bonding. To this end, these approximations transform an all-electron DFT calculation into an equivalent problem involving smooth pseudo-wave-functions for valence electrons, using an effective external potential (pseudopotential) that lumps the effect of core-electrons and the singular Coulomb nuclear potential. In a numerical setting, pseudopotential DFT calculations are computationally efficient since they only compute valence electronic wavefunctions, allowing for the use of coarser grids or lower plane-wave energy cutoffs to resolve these smooth pseudo electronic fields. Over the years, pseudopotentials have evolved from norm-conserving <cit.> to ultrasoft potentials <cit.>, and now to the state-of-the-art projector augmented wave (PAW) <cit.> method, enabling accurate predictions of material properties across diverse material systems without the need for expensive all-electron DFT calculations. However, the stringent accuracy required for DFT calculations to compute meaningful properties, combined with the cubic-scaling computational complexity with the number of electrons, demands significant computational resources even for pseudopotential DFT calculations, typically limiting their application to material systems with at most few thousands of electrons. Overcoming the current limits on ab-initio calculations has been a decades-long pursuit by researchers around the world, and efforts over the past three decades have focussed on developing accurate and computationally efficient methods for solving the Kohn-Sham DFT problem. These include efficient discretization methods for solving DFT equations, reduced-scaling algorithms for lowering the computational complexity of the DFT problem as well as efforts to improve the parallel scalability of the widely used codes. Among the various discretization methods, plane wave (PW) basis is the most popular method of choice  for metallic and heterogenous material systems due to the computational efficiency afforded by these basis sets owing to its spectral convergence. Despite the popularity of PW basis for DFT calculations, the limitations of these Fourier methods stemming from the poor scalability on parallel computing architectures, the inefficiency in treating non-periodic systems, and the inability to handle generic boundary conditions led to the development of systematically improvable, efficient, and scalable real-space discretization techniques based on finite-difference <cit.>, finite-elements <cit.>, wavelets <cit.> and other reduced-order basis techniques <cit.>. 

The finite-element (FE) basis, a compactly supported piece-wise polynomial basis, is a relatively new approach in real-space methods for DFT, offering several advantages over other commonly used basis sets.  FE basis sets are systematically convergent for any material system, can accommodate generic boundary conditions including fully periodic, semi-periodic, non-periodic. Their locality allows one to exploit fine-grained parallelism on modern heterogenous architectures while providing excellent parallel scalability on distributed systems<cit.>. As demonstrated in previous works <cit.>, finite-element-based methods have significantly outperformed plane-wave basis in the case of norm-conserving pseudopotential DFT calculations, as the system size increases for a given accuracy of ground-state energy and forces. Furthermore, adaptive resolution offered by FE basis sets enables all-electron calculations <cit.> but mixed all-electron and pseudopotential calculations <cit.>. The open-source code DFT-FE <cit.>, the workhorse behind the ACM Gordon Bell Prize 2023, inherits these features while incorporating scalable and efficient solvers for the solution of Kohn-Sham equations. The DFT-FE code has shown excellent scalability massively parallel many-core CPU and hybrid CPU-GPU architectures (tested up to  200,000 cores on many-core CPUs and  40,000 GPUs on hybrid CPU-GPU architectures), while simulating material systems as large as 600,000 electrons <cit.>. Furthermore, DFT-FE has been recently employed in a range of scientific studies involving large-scale DFT calculations on material systems with tens of thousands of electrons – computing the electronic structure of large DNA molecules <cit.>, predicting energetics of extended defects in crystalline materials <cit.>, investigating phase transformations in doped nanofilms <cit.>, evaluating spin Hamiltonian parameters of systems with defects <cit.> and solving inverse DFT problem <cit.> 



We note that the aforementioned finite-element-based DFT calculations are all limited to the use of norm-conserving pseudopotentials. These pseudopotentials can become very hard for material systems with elements involving d or f electrons and these DFT calculations can be computationally demanding. Further, these often require semi-core states to be treated as valence states for better transferability, resulting in harder pseudopotentials with more valence electrons, particularly for alkali, alkaline earth metals, or early transition metals. Previous works <cit.> have shown that these hard norm-conserving pseudopotentials require a large number of degrees of freedom (30,000 to 50,000 finite-element basis functions per atom) to achieve chemical accuracy, leading to high computational costs for solving the Kohn-Sham DFT eigenvalue problem. While ultrasoft pseudopotentials relax the norm conservation condition <cit.>, resulting in softer potentials, their construction is complex and requires extensive testing to ensure accuracy and transferability. To address these issues, Blöchl <cit.> developed the projector augmented wave (PAW) method, formally an all-electron approach that generalizes the LAPW method and the pseudopotential approach. The PAW method recovers all-electron wavefunctions by introducing a linear transformation operator (𝒯) acting on smoothly varying pseudo wavefunctions. Within the frozen core approximation, it treats core states as frozen and retrieves the oscillatory behavior of all-electron valence wavefunctions near atomic centers through the action of 𝒯 on the smooth pseudo-valence wavefunctions. This allows one to reformulate the all-electron Kohn-Sham DFT eigenproblem into an equivalent PAW eigenvalue problem involving only smooth pseudo-valence wavefunctions, striking a balance between accuracy and computational efficiency. As a result, the PAW approach within the frozen core approximation has become widely adopted in popular plane-wave based DFT codes <cit.>.




The real-space discretization techniques have also been employed in the context of PAW implementations and are primarily based on finite-difference approximations <cit.>, wavelet based methods <cit.> and psinc basis <cit.>. However, most of these PAW based DFT calculations deal with material systems involving few thousands of electrons excepting the linear scaling DFT approach <cit.> using the PAW method applied to material systems with a band-gap. The computational efficiency and scalability of these real-space implementations compared to state-of-the-art plane-wave-based PAW implementations remains unknown. This work introduces the local real-space formulation of the PAW method amenable for finite-element (FE) discretization, proposing efficient strategies to solve the underlying FE-discretized PAW generalized eigenvalue problem using a self-consistent field iteration approach. Our methodology significantly reduces FE degrees of freedom required to achieve chemical accuracy using the PAW method, outperforming the optimized norm-conserving (ONCV) pseudopotential based DFT calculations employed in previous FE based approaches for a given system size. Additionally, we demonstrate that our method can handle fully periodic, non-periodic, and semi-periodic boundary conditions for generic material systems as large as 50,000 electrons. Moreover, our method offers substantial computational efficiencies over state-of-the-art plane-wave-based approaches for systems with over 5,000 electrons.  Notably, to the best of our knowledge this represents the first development of a fast and scalable approach for the PAW method within the framework of finite-element discretization.

We adopt the original Blöchl's <cit.> PAW method in our current work and do not employ the Kresse's modifications <cit.> that include the compensation charge contribution for computing the exchange-correlation term, which increases computational costs and introduces numerical instabilities <cit.>. The key aspects of the proposed finite-element (FE) approach for the PAW method include: (i) local reformulation of the extended PAW electrostatics involving smooth pseudo-electron density and the compensation charge making it amenable for FE discretization; (ii) deducing the discretised governing equations by evaluating the variational derivative of the locally reformulated all-electron DFT energy functional in the PAW method; (iii) employing higher-order quadrature rules to efficiently and accurately evaluate integrals involving atom-centered functions, allowing the use of coarser FE grids for electronic fields and yet capture the relevant PAW atomic data; (iv) implementation of a self-consistent field (SCF) iteration approach based on Anderson mixing of total charge density using an efficient real-space strategy to minimize the weighted total charge density residual norm; (v) an efficient scheme to evaluate the inverse of FE discretized PAW overlap matrix  using Woodbury's formula in conjunction with Gauss-Lobatto-Legendre quadrature rules and a block-wise matrix inversion for the overlap of atomic projectors in real-space; and (vi) leveraging this efficient computation of ^-1 to design a Chebyshev polynomial based filter <cit.> in each SCF iteration to compute the space rich in occupied eigenstates, and solving the PAW generalized eigenvalue problem by projecting onto this filtered space. The proposed formulation has been implemented in a distributed setting on CPUs using the message passing interface (MPI) for communication across multiple nodes to enable large-scale PAW based DFT calculations. We benchmark the accuracy and performance of our method against plane-wave-based PAW implementations on various representative non-periodic, semi-periodic, and fully periodic systems. Compared to reference data from plane-wave calculations, our results show excellent agreement in ground-state energies, energy variation with bond length and lattice parameters, formation energies, surface energies, and band structures. Relative to widely used plane-wave based PAW implementations, our finite-element based approach demonstrates significant computational advantage both in terms of CPU node-hrs and minimum wall time with increasing systems sizes reaching up to 5× and 10×, respectively, for non-periodic systems (∼18,000 electrons) and  3× and 6×, respectively, for periodic systems (∼34,000 electrons). Our implementation also exhibits excellent parallel scaling efficiency (∼70%) at 1200 degrees of freedom per MPI rank for material systems, achieving nearly a 10-fold speed-up over the state-of-the-art DFT-FE code using norm-conserving ONCV pseudopotentials for system sizes close to 50,000 electrons, with a significant reduction in computational resource requirements.


 
 The remaining sections of the manuscript is organised as follows. Section <ref> briefly introduces the Blöchl PAW method followed by a discussion on the local real-space formulation of the PAW method and the governing equations to be solved. Subsequently, Sec. <ref> introduces the finite-element discretization for the PAW governing differential equations and Sec. <ref> details the proposed computational methodology in achieving an efficient and scalable solution procedure for solving the FE discretized PAW generalised eigenvalue problem using a self-consistent field iteration approach. Section <ref> presents  comprehensively accuracy and performance benchmarks of our implementation comparing with state of the art PAW based plane-wave(PW) codes on representative benchmark systems. Additionally, this section presents a comparative study of the current work and the norm-conserving pseudopotential (ONCV) in the DFT-FE code in terms of parallel scalability and time to solution. Finally, we conclude with a short discussion summarizing the key findings and outline future prospects arising from this work in Sec. <ref>




§ REAL-SPACE FORMULATION OF PROJECTOR AUGMENTED WAVE METHOD









 
The ground-state properties of a material system comprising of N_a nuclei and N_e electrons is governed by the following all-electron Kohn-Sham DFT energy functional <cit.>



    E[{_i }, {^a}] = T_s[{_i }] + E_xc[n()] + J[ρ(),{^a} ]

where {_i } and {^a} denote the set of single-electron orthonormal wavefunctions and the set of nuclear position vectors respectively. We restrict our discussion here to spin-unpolarized case, while the extension to spin-polarized systems is straightforward and not elaborated here. The term T_s[{_i }] in Eqn. (<ref>) denotes the kinetic energy of the non-interacting electrons where

    T_s[{_i }] = 2∑_i^Nf_i∫_i^*()(-1/2∇^2)_i()d,

with f_i denoting the orbital occupancy function and the integer N ≥ N_e/2 denotes the number of single-electron wavefunctions. Further, E_xc[n()] in Eqn. (<ref>) is the exchange-correlation energy that accounts for the quantum-mechanical many-body effects. Adopting the generalized gradient approximation (GGA) <cit.> for the exchange-correlation functional description in this work, we have

    E_xc[n()] = ∫ϵ_xc[n(),∇ n()]d.

where n()= 2∑_i^Nf_i|_i()|^2 represents the all-electron density. The total electrostatic energy in Eqn. (<ref>) is denoted by J[ρ(),{^a}] = E_el[ρ(),{^a}] - E_self[ {^a}]. Here, E_el is the electrostatics energy associated with the total charge ρ()=n()+ ∑_ab^a(), where b^a()=-𝒵^aδ( - ^a) denotes the nuclear charge density with 𝒵^a being the atomic number of  a^th atom.  This electrostatics energy E_el involves a convolution in real-space given by:

    E_el[ρ,{^a} ] = 1/2∫∫ρ()ρ()/|-|dd

where the domain of integration for  in the above Eqn. (<ref>) is ℝ^3, while the domain of integration for  in the case of an infinite-periodic system is over the unit-cell of interest, whereas in the case of a finite system, it is over ℝ^3. Further, for a periodic system the index a refers to atoms at all lattice sites with the compact support of b^a() lying in the unit cell of interest, while for a finite system, it refers to all the atoms in the system. Finally, the self-interaction energy corresponding to the nuclear charge density b^a() is denoted as E^a_self[^a] and hence the total self-interaction energy is given by E_self[ {^a}] = ∑_aE^a_self[^a].



Within the frozen-core approximation in PAW method, the valence all-electron Kohn-Sham (KS) wavefunctions({_i}) that are orthogonal to the core wavefunctions are recovered from pseudo-smooth(PS) wavefunctions ({_i}) using a linear transformation operator(𝒯)  <cit.> given by:

    |_i⟩ = 𝒯|_i⟩ = (ℐ + ∑_a 𝒯^a) |_i⟩

The index i = {1,2...,N} in Eqn. (<ref>) runs over the valence states with  N ≥ N_v/2 where N_v being the number of valence electrons. Furthermore, the linear transformation operator (𝒯) in Eqn. (<ref>) is defined such that it differs from the Identity operator (ℐ) by a set of atom-specific transformations 𝒯^a that has an effect only inside a certain atom-centered ball Ω_a with radius | - _a| < r_c^a referred to as augmentation sphere. Consequently, the all-electron valence Kohn-Sham wavefunctions can be expressed as

    _i() = _i()  + 
       ∑_a∑_α(^a_α()-^a_α())⟨^a_α|_i|⟩

where {^a_α} are the all-electron partial waves, {^a_α} are the pseudo-smooth partial waves that are used to expand {_i()} and {_i()} inside the augmentation sphere. Further, {^a_α} in Eqn. (<ref>) are the projector functions that are orthogonal to the pseudo smooth partial waves  i.e., ⟨^a_α|^a_β|=⟩δ_αβ. In the above Eqn. (<ref>) and subsequently in the manuscript, the atom-centered fields are denoted with superscript `a', implicitly denoting their functional dependence on ( - ^a). Furthermore, the atom-centered functions {^a_α}, {^a_α}, {^a_α} in Eqn. (<ref>) are of the separable form ϑ_α(r)S_lm  (θ), with ϑ_α(r)  being a radial function and S_lm(θ) is the real-spherical harmonic function that denotes the angular part of these atom-centered functions.
We note that r=|-^a| and  θ∈ℝ^2 represents the azimuthal and polar components of (-^a). 
Moreover, we note that the compact support of the functions ^a_α() and (^a_α()-^a_α()) lies in the augmentation sphere Ω_a, and consequently the decomposition in Eqn. (<ref>) transforms the orthonormality requirement of all-electron Kohn-Sham wavefunctions to 𝒮-orthonormality (𝒮=𝒯^†𝒯) condition on the pseudo-smooth(PS) wavefunctions {_i} i.e. ⟨_i|𝒮|_j⟩=δ_ij. 
 Eqn. (<ref>), further allows the decomposition of the all-electron density n() into a pseudo-smooth (PS) electron density () and an atom-centered correction which replaces the PS electron-density inside the augmentation sphere with the true all-electron density and is given by

    n()    = () + ∑_a(n^a() - ^a())
    with,  ()    = 2∑_i^Nf_i|_i()|^2 + _c() 
    while,  n^a()    = n^a_c()+ ∑_αβ^a_α()^a_β()D^a_αβ
    and  ^a()    =^a_c()+ ∑_αβ^a_α()^a_β()D^a_αβ

In the above equations, n^a() represents the atom-centered all-electron density and ^a(), the atom-centered PS electron density that matches with n^a() outside the augmentation sphere. Further, the matrix-components of the atom-dependent Hermitian matrix ^a (often referred to as channel occupancy matrix <cit.> or spherical density matrix <cit.>) are given by

    D^a_αβ = 2∑_if_i ⟨_i|^a_α|⟨%s|%s⟩⟩^a_β|_i

Using the decompositions in Eqns. (<ref>) and (<ref>), the all-electron kinetic energy T_s[{_i}] and the exchange correlation energy E_xc[n()] can be split into a smooth contribution  and an atom-centered correction term as follows:

    T_s[{_i }]    = T_s[{_i}] + ∑_aT^a_core + ∑_a∑_αβΔ T^a_αβD^a_αβ
     
    E_xc[n()]    = E_xc[()] +
     ∑_aΔ E^a_xc

where T^a_core is the kinetic energy contribution from the core states while the terms Δ T^a_αβ, Δ E^a_xc are atomic corrections computed as

    Δ T^a_αβ =  -1/2∫_Ω_a( ^a_α()∇^2^a_β() - ^a_α()∇^2^a_β()) d
       Δ E^a_xc = ∫_Ω_a(ϵ_xc[n^a,∇ n^a]-ϵ_xc[^a,∇^a])d


Finally, the all-electron electrostatics energy is decomposed into a smooth contribution and an atom-centered correction term by introducing a compensation charge ^a() compactly supported in Ω_a so that the multipole moments of (n^a() - ^a() + b^a() -^a()) vanish  <cit.> inside Ω_a. To this end, the compensation charge ^a() in terms of atom-centered spherical shape functions ^a_lm() is given by

    ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm()

The atom-centered shape functions ^a_lm() in the above Eqn. (<ref>) are compactly supported in Ω_a and are usually constructed in the form of Gaussians, sinc-squared or Bessel functions <cit.> to satisfy the condition ∫_Ω_a r^lS_lm(θ)^a_l'm'()d = δ_ll'δ_mm'. Furthermore κ^a and Δ ^a_lmαβ in Eqn. (<ref>) are given by

    κ^a    = ∫_Ω^a(n^a_c(r)-^a_c(r))r^2dr-𝒵^a/√(4π)
    Δ ^a_lmαβ   = ∫_Ω_ar^l(^a_α()^a_β() - ^a_α()^a_β() )S_lm(θ̂)d


Now, the all-electron electrostatic energy can be computed as

    J[ρ(),{^a}] =  E_el[(),{^a} ] + ∑_aΔ E^a_el

where the smooth total density is given by () = () + (), with ()=∑_a^a(). The electrostatic energy involving () is given by

    E_el[(),{^a} ] = 1/2∫∫()()/|-|dd

and the atom-centered correction term is given by

    Δ E^a_el = 
    E^a_el[ρ^a()] - E^a_el[^a()]  - E^a_self[^a]   where
       E^a_el[ρ^a()] = 1/2∫_Ω_a∫_Ω_aρ^a()ρ^a()/|-|dd
       E^a_el[^a()]   = 1/2∫_Ω_a∫_Ω_a^a()^a()/|-|dd

with ρ^a() = n^a()+b^a() and ^a() = ^a()+^a(). The choice of the compensation charge in Eqn. (<ref>) allows one to evaluate  Δ E^a_el only within the augmentation sphere Ω_a.

Combining the various atom-centered correction terms from different energy contributions into Δ E^a, the all-electron energy in Eqn. (<ref>) in PAW method is reformulated as follows:

    E[{_i }, {^a}] = E[{_i}, {^a}] +∑_aΔ E^a[{D^a_αβ}]

 where we define E[{_i}, {^a}] and Δ E^a[{D^a_αβ}] as,

    E[{_i}, {^a}]    =  T_s[{_i } ] + E_xc[()] + E_el[(),{^a} ]  
    Δ E^a[{D^a_αβ}]    = Δ E^a_xc + Δ E^a_el + ∑_αβΔ T^a_αβD^a_αβ + T^a_core

and the details of evaluating Δ E^a[{D^a_αβ}] are mentioned in Appendix-<ref>.
At this juncture, we note from Eqn.(<ref>) that the all-electron Kohn-Sham energy functional of {_i} has been decomposed into an energy functional of smooth wavefunctions {_i} and a term that includes atomic corrections. The validity of this decomposition is reliant on two key assumptions. First, it is assumed that the augmentation spheres are non-overlapping and to achieve this, specialised PAW data sets are generated based on the application of interest <cit.>. Second assumption is that the set of partial waves ({_α} and {_α}) forms a complete basis within the augmentation sphere. However, in practice, only a finite number of partial waves are used. The error arising from this incompleteness is mitigated by introducing an atom-specific zero potential v̅^a()  with a compact support only inside augmentation sphere <cit.>. 

The electrostatic interaction energy involving the smooth total density E_el[(),{^a}] in Eqn. (<ref>) is extended in real-space and can be reformulated as a local variational problem in the spirit of <cit.>. Subsequently, the variational problem involved in computing this electrostatic energy contribution is given by:

    E_el[,{^a}] = -max_φ∈γ{1/8π∫|∇φ()|^2 d - ∫()φ()d}

where φ(,{^a}) denotes the trial function corresponding to electrostatic potential due to smooth electron-density (()) and the sum of atom-centered compensation charges (()). Finally, the problem of determining the all-electron ground state energy and the all-electron density for a given position of nuclei is expressed as the following variational problem involving the smooth electronic fields:



    E_GS = min_{_i}∈𝒳max_φ∈γ{ T_s[{_i}] - 1/8π∫|∇φ()|^2 d + ∫()φ()d + E_xc[()] + ∑_aΔ E^a[{D^a_αβ}] }


where the function space 𝒳 = {{_i} | <_p,_q >_𝕏 = δ_pq} with <_p,_q> denoting the 𝒮-inner product ⟨_p|𝒮|_q|$⟩ defined on𝕏. The space𝕏denotes a suitable function space that guarantees the existence of the minimisers. The domains used for numerical computations that are non-periodic corresponds to a large enough domain such that the wavefunctions{_i}decays to 0 and in periodic calculations, these correspond to supercells with periodic boundary conditions. Denoting such an appropriate domain byΩ, the appropriate function spaces for𝕏andγare𝕏=γ=H^1_0(Ω)in the case of non-periodic problems and𝕏 = γ= H^1_per(Ω)in the case of periodic problems.



 §.§ Governing Equations

The stationarity condition with respect to smooth fields{_i}corresponding to the PAW variational problem in Eqn. (<ref>) gives rise to nonlinear generalized Hermitian eigenvalue problem (GHEP) of the following form:

    ℋ_i = ε_i𝒮_i

whereℋis a Hermitian operator with

    ℋ = -1/2∇^2 + V_eff(,{^a}) + ℋ_nloc
       ℋ_nloc_i :=  ∑_a∑_αβ^a_α()Δ h^a_αβ∫^a_β() _i()d

and𝒮is a positive-definite Hermitian operator with

    𝒮 = ℐ + 𝒮_nloc
       𝒮_nloc_i :=  ∑_a∑_αβ^a_α()Δ s^a_αβ∫^a_β() _i()d

The set({ε_i},{ _i})are the eigenvalue and eigenfunction pairs of the generalised eigenvalue problem (<ref>) and theNsmallest eigenvalue and eigenvector pairs are used to compute the PS density()and the matrix^aemploying the Eqns. (<ref>) and Eqn. (<ref>) respectively. Further, the effective potential (V_eff) comprising of the total electrostatic potential, exchange-correlation potential and zero potential is computed as

    V_eff  = V_el + V_xc + V̅ =  δ E_el/δ + δ E_xc/δ + ∑_av̅^a

 where the electrostatic potentialδE_el/δ=φ(,{^a})is computed as the solution of the Poisson equation associated with the smooth total density()as given by eq.(<ref>).
Furthermore, the atom centered contributionsΔh^a_αβandΔs^a_αβinℋ_nlocand𝒮_nlocare expressed as   
    
    Δ h^a_αβ = δΔ E^a/δ D^a_αβ + ∑_lmΔ^a_lmαβ∫^a_lm()φ()d
        - ∫_Ω_av̅^a()^a_α()^a_β()d

    Δ s^a_αβ  = ∫_Ω_a(^a_α()^a_β()-^a_α()^a_β())d = √(4π)Δ^a_00αβ

Finally, the set of governing differential equations to be solved in the PAW method are:

    (-1/2∇^2 + V_eff(,{^a} )+ℋ_nloc)_i =ε_i(ℐ+𝒮_nloc)_i
      
        2∑_if(ε_i,μ) = N_v,   f(ε_i,μ) = 1/1+ exp(ϵ_i-μ/σ)
       () = 2∑_if(ε_i,μ)|_i()|^2, () = ∑_a^a()
       ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm()
       Û^a_α i = ∫_α^a()_i()d, D^a_αβ = 2∑_if(ϵ_i,μ)Û^a^*_α iÛ^a_iβ
       () = () + (),  -1/4π∇^2φ(,{^a}) = ()

whereÛ^a^*_αiis the complex conjugate ofÛ^a_αi,μis the Fermi-energy andf_i:=f(ϵ_i,μ)is the orbital occupancy function lying in the interval[0,1]which is computed -via- the Fermi-Dirac distribution withσ= k_BTwhereTis the smearing temperature andk_Bis the Boltzmann constant.

When dealing with periodic crystals, 2D periodic slabs, or surfaces, we invoke the Bloch theorem <cit.> along the periodic axes. Instead of solving the DFT problem on a large periodic supercell, we solve the reduced problem on smaller unit cells with periodic boundary conditions. By invoking Bloch theorem, the eigenfunctions of the PAW generalised eigenvalue problem can be expressed as

    _i,() = e^i._i,()

wherei = √(-1)and_i,()is a unit-cell periodic complex valued function satisfying_i,(+_r) = _i,()for all lattice vectors_rwithdenoting a point in the first Brillouin zone(BZ) of the reciprocal lattice. The set of governing equations for the PAW formalism with Bloch wavefunctions_i,()can be recast as:

    (-1/2∇^2 - i·∇ + 1/2||^2 + V_eff(,{^a} )+ℋ^_nloc)_i,    = ε_i,(ℐ+𝒮^_nloc)_i,  on  Ω_p,  ∀∈ BZ 
    ℋ^_nloc_i, :=  ∑_a∑_αβ∑_re^-i· (-_r) ^a_α(-(^a+_r))Δ h^a_αβU^a_β i, 
    𝒮^_nloc_i, :=  ∑_a∑_αβ∑_re^-i· (-_r) ^a_α(-(^a+_r))Δ s^a_αβU^a_β i
    
    2∑_i_BZ2f(ε_i,,μ) d = N_v,   f(ε_i,,μ)  = 1/1+ exp(ϵ_i,-μ/σ), () = ∑_i_BZ2f(ϵ_i,,μ)|_i,()|^2 d, () = ∑_a^a()
    U^a,_α i := ∫_Ω_p∑_r'e^i· (-_r')p_α^a( - (^a+L_r'))_i,()d,  D^a_αβ = 2∑_i_BZf(ϵ_i^,μ)U^a,*_α iU^a,_i βd
    ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm(), () = ()+(), -1/4π∇^2φ(,{^a}) = ()

where_BZdenotes the volume average of the integral over the first Brillouin zone (BZ) corresponding to the periodic unit cellΩ_p. We further note that the equations corresponding to non-periodic case in Eqn.(<ref>) can be easily recovered from the above Eqn. (<ref>) by considering (= 0,_r= 0) and replacing the periodic unit-cell domainΩ_pwith a non-periodic simulation domainΩ. We note that semi-periodic boundary conditions can be handled in a similar way depending on the number of periodic directions. Furthermore, we note that the set of equations in Eqn.(<ref>) or Eqn.(<ref>) involves a nonlinear generalized Hermitian eigenvalue problem (GHEP) that needs to be solved self-consistently in conjunction with the Poisson equation. In particular, we seek to solve the fixed point iteration[(),{ ^a}] = F([(),{ ^a}])using a self-consistent field (SCF) iteration procedure where the evaluation of the mapF([(),{ ^a}])involves the solution of the GHEP and the Poisson equation in Eqn.(<ref>) or Eqn.(<ref>) at every SCF iteration.









§ PAW-FE: FINITE ELEMENT DISCRETIZATION FOR PAW METHOD









Finite-element method involves decomposing the spatial domain of interest into non-overlapping subdomains called finite-elements (or cells) using a finite-element (FE) mesh. The underlying FE basis functions spanning a finite-element subspace are piecewise polynomials that exhibit systematically convergent behaviour <cit.>. Furthermore, these basis functions are strictly local, having a compact support only in the finite-element cells sharing a finite-element node, making them well-suited for massive parallelization. In contrast to conventional finite-element basis typically constructed from a tensor product of Lagrange polynomials interpolated through equidistant nodal points in a given finite-element, in this work, we specifically employ spectral finite-element basis that areC^0continuous Lagrange polynomials generated using Gauss-Lobatto-Legendre(GLL) nodal points that are the roots of derivatives of Legendre polynomials. We refer to previous works <cit.> about the advantages provided by higher-order adaptive spectral finite-element-based methods to solve the Kohn-Sham DFT problem in the context of norm-conserving pseudopotentials and all-electron DFT calculations. A crucial aspect of the computational methodology described below involves the use of these higher-order spectral finite-elements in conjunction with the reduced order Gauss-Lobatto Legendre (GLL) quadrature rules combined with the non-overlapping nature of augmentation spheres that enables the efficient inversion of the PAW overlap matrix. We discuss here the finite-element discretization of the governing equations deducing the algebraic generalized eigenvalue problem to be solved in the PAW method.


As seen from Eqn.(<ref>), the real-space formulation of the PAW method presented in this work results in a min-max problem involving the smooth pseudo-wavefunctions and the electrostatic potential. Employing a single FE discretization for both the wavefunctions and the electrostatic potential  can result in non-variational behaviour of the electronic ground-state energy. Therefore, we seek a solution for the electrostatics problem -via- the Poisson equation with a more stringent accuracy than the eigenvalue problem solution employing different orders of FE interpolating polynomial basis for the electrostatic potential and the wavefunctions. To this end, we represent the electronic fields in Eqn.(<ref>) in the FE basis and are given by 

    ^h_i,() = ∑_J=1^MN^h,p_J()^J_i,, φ^h,p_el() = ∑_J=1^M_elN^h,p_el_J()φ^J

where^J_i,,φ^Jdenote the FE discretized fields, whileN^h,p_J:1 ≤J ≤MandN^h,p_el_J:1 ≤J ≤M_eldenote the strictly local Lagrange polynomial basis functions of degreepandp_el, respectively spanning the finite-element subspace, generated using the nodes of the FE triangulation𝒯^hwith the characteristic mesh size denoted byhandp_el > pas discussed. To this end, the finite-element discretization of the Kohn-Sham generalized eigenvalue problem in Eqn.(<ref>) results in an algebraic generalized Hermitian eigenvalue problem^_i, = ε^h_i,^_i,, where^and^denote the discretized Hamiltonian matrix and the PAW overlap matrices of sizeM ×Mwithε^h_i,denoting theith eigenvalue corresponding to the discrete eigenvector_i,( see Appendix-<ref> for derivation of the FE discretized algebraic EVP). As discussed in  Appendix-<ref>, the finite-element discretized Hamiltonian and PAW overlap matrices has both local and non-local contributions and hence can be decomposed as^=^,loc+ ^,nlocand^=+ ^,nlocwhere the matrix entries of the local contributions^,locandare given by

    H_IJ^,loc = _Ω_p{1/2∇ N^h,p_I()·∇ N^h,p_J()   
       + V^h_eff() N_I^h,p()N_J^h,p()
       + 1/2||^2N_I^h,p()N_J^h,p() - i· N^h,p_I()∇ N^h,p_J()
       +^h_GGA() ·(N_I^h,p()∇ N^h,p_J()+N^h,p_J()∇ N^h,p_I() )} d
       M_IJ = ∫_Ω_pN_I^h,p()N_J^h,p()d

with

    V^h_eff =  (φ^h,p_el()+ V̅() +∂ϵ_xc(, ∇)/∂|_ = ^h) 
       ^h_GGA() = ∂ϵ_xc(, ∇)/∂∇|_ = ^h

and the discrete electrostatic potentialφ^h,p_el()inV^h_effabove is computed by solving the finite-element discretized Poisson equation= wheredenotes the Laplace operator discretized using the finite-element basisN^h,p_el_J:1 ≤J ≤M_elwhiledenotes the electrostatic potential vector whose entries are the expansion coefficientsφ^Jin Eqn.(<ref>) anddenotes the forcing vector corresponding to, the right-hand side of the Poisson equation in Eqn.(<ref>). To this end, the entries ofandare given by

    L_IJ   = 1/4π∫_Ω_p∇ N^h,p_el_I()·∇ N^h,p_el_J()d,
    
    c_I   = ∫_Ω_p^h()N^h,p_el_I()d

Finally, the matrix entries corresponding to the non-local contributions in^and^are given by

    ^,nloc = ∑_a^a,Δ^a_^a,†, ^,nloc = ∑_a^a,Δ^a_^a,†
       C^a,_J α = ∫_Ω_p∑_re^-i·(-_r)^a_α(-(^a+_r))N^h,p_J()d

where the atom-dependent coupling matricesΔ^a_andΔ^a_are of sizen^a_pjn^a_pjwithn^a_pjdenoting the number of projectors for atom indexaand the corresponding matrix entries denoted asΔh^a,h_αβandΔs^a_αβrespectively are given by


    Δ h^a,h_αβ = δΔ E^a,h/δD^a_αβ - ∫_Ω_av̅^a()^a_α()^a_β()d
        + ∑_lmΔ^a_lmαβ∫_Ω_p∑_r^a_lm(-(^a+_r))φ^h_el()d
       Δ s^a_αβ  = √(4π)Δ^a_00αβ

    
Since the FE basis functions are localised in real-space, the matrices^,locand^,locare sparse. Since the projectors^a(-^a)have a compact support within the augmentation sphereΩ_a, we note that theM ×n^a_pjmatrix^a,is also a sparse matrix for every atoma. Consequently, we can now rewrite the discretized Hamiltonian matrix and the overlap matrix within the PAW formalism as 
 
    ^ = ^_loc + ^Δ_^^†, ^ =  + ^Δ_^^†

where theM ×n_projmatrix^ = [^1,... ^a,...^N_a, ], withn_proj = ∑_a n^a_pjdenoting the total number of atomic projectors in the system. In the above Eqn.(<ref>),Δ_andΔ_are the block diagonal matrices of sizen_proj ×n_projwith each block corresponding to the atom-dependent coupling matricesΔ^a_andΔ^a_respectively as introduced in Eqn.(<ref>).



    
     
       
    
    
 Finally, the discretized governing equations corresponding to Eqn. (<ref>) that need to be solved in PAW-FE are given by:

    ^_i, = ε^h_i,^_i,
    
     2∑_n^N_BZf(ε^h_i,,μ)d = N_v,  f(ε^h_i,,μ) = 1/1+ exp(ε^h_i,-μ/σ)
    ^h() = 2∑_i^N_BZf(ε^h_i,,μ)|^h_i,()|^2d, 𝖴^a,_i = ^a,†_i,
    
    D^a,h_αβ = 2∑_i^N_BZf(ε^h_i,,μ)𝖴^a,^*_α i𝖴^a,_β i d
    ^h() = ∑_a^a,h(), ^h() = ^h()+^h(), =


We note that the discretized generalized Hermitian eigenvalue problem (GHEP) in the above Eqn.(<ref>) is a nonlinear eigenproblem and is solved forNsmallest eigenvalue-eigenvector pairs, which in turn are used to evaluate the smooth pseudo density^h()and the atom-centred density matrix{ D^a,h_α,β}. This nonlinear GHEP is self-consistently solved
in conjunction with the discretized Poisson equation subject to the applied boundary conditions in order to obtain the ground-state electronic fields and the energy. As noted before, replacing the periodic unit-cell domainΩ_pwith a non-periodic simulation domainΩand considering=0,_r=0, we recover the finite-element discretized equations corresponding to non-periodic boundary conditions from Eqn.(<ref>). Subsequently, the total all-electron energy in the PAW method,E^hin terms of the finite-element discretized ground-state solution fields(ε^h_n,,^h(), D^a,h_αβ, φ^h,p_el )is given by
    
    E^h =    2∑_i=1^N_BZf(ε^h_i,,μ)ε^h_i, d - ∑_a∑_αβΔ h^a_αβD^a,h_αβ
       - ∫_Ω_p[ ^h()V^h_eff() + ^h_eff,GGA() ·∇^h() ] d
       + 1/2∫_Ω_p^h()φ^h,p_el()d + E_xc[^h()] 
       + ∑_aΔ E^a[{D^a,h_αβ}]

The various computational aspects involved in efficiently solving the nonlinear generalized eigenvalue problem (Eqn. (<ref>)) using a self-consistent field (SCF) iteration procedure are discussed in the subsequent section. In the following section, we choose to drop theindex in the FE discretized electronic fields, vectors and matrices introduced in Eqn. (<ref>) for brevity, and we refer to them asũ_i^h(),_i,,andwithout theindex.  







§ SELF CONSISTENT FIELD ITERATION

In this section, we begin by describing the self-consistent field iteration (SCF) procedure adopted in our work to solve the nonlinear PAW generalized eigenvalue problem, highlighting the real-space mixing strategy employed in our work suited towards solving the Kohn-Sham fixed point problem inherent in the PAW method. Subsequently, we describe the Chebyshev filtering subspace iteration (ChFSI) approach and the related modifications to adapt ChFSI for efficiently solving the underlying generalized eigenvalue problem within a given SCF iteration. Additionally, we elaborate on the key computational strategies that leverage the underlying sparse finite-element structure and the compact support of the PAW projectors in real-space to facilitate fast and scalable calculations of computationally intensive steps in the ChFSI approach.




The finite-element discretized governing equations (<ref>) in the PAW method involves the solution of a nonlinear generalized eigenvalue problem that needs to be solved self-consistently as alluded to before. Since the total charge density is the sum of the smooth pseudo density () and the compensation charge () which is a function of the spherical density matrix^a, we note that the Kohn-Sham fixed point iteration map in the PAW method is written in terms of total charge density i.e_out^h() = F(_in^h()). To this end, we start with an input guess for the total charge density_in = _in^h + ^h(^a_in)and the mapF(_in^h())involves evaluating the effective potential (V^h_eff(),^h_GGA()), the non-local Hamiltonian coupling matrix (Δ^a_) using^h_into compute the eigenpairs (ε^h_i,_i) of the PAW generalized eigenvalue problem which are in turn used to  output total charge density_out = _out^h + ^h(^a_out). The above procedure is summarised in steps 2-5 of the Algorithm(<ref>) that describes the SCF iteration and these steps are continued until||^h_out() - ^h_in()||is lesser than the required tolerance in some  appropriate  norm. 
The computationally intensive step involved in the SCF iteration is solving the discretized generalised eigenvalue problem (step-4 of algo.(<ref>)). This step asymptotically scales cubically with the system size, making it crucial to employ efficient numerical strategies to delay the onset of cubic scaling and also reduce the number of SCF iterations. 

The following subsection describes the mixing strategy employed in our work to accelerate the convergence of the SCF procedure.


 §.§ Total charge density mixing

In contrast to DFT calculations involving norm-conserving pseudopotentials and all-electron methods, where pseudo-valence electron density or the all-electron density is mixed during the SCF iteration, in the PAW method, the total charge density (() = () + ()) needs to be updated self consistently. This is because the compensation charge()depends on{^a}which in turn depends on pseudo-smooth wavefunctions (see Eqn.(<ref>)) and hence needs to be mixed at every SCF iteration.

We employ an-stage Anderson <cit.> mixing scheme to accelerate the SCF iteration in the current work, with appropriate modifications relevant to the real-space PAW framework as discussed in this section. Denoting input total charge densities and output total charge densities at then^thSCF iteration as^(n)_in()and^(n)_out()respectively, the input total charge density for(n+1)^thSCF iteration^(n+1)_in()is computed as

    ^(n+1)_in = γ_mix_out + (1-γ_mix) _in

where_in(out)is expressed as a linear combination of input and output total charge densities fromnprevious SCF iterations. To this end, we have

    _in(out) = ^n_in(out) + ∑_k=1^n-1 c_k (^(n-k)_in(out) - ^(n)_in(out))

where we determine the linear combination coefficientsc_kby minimizing the norm square of the residualR = ||_res||^2_W = ||_out() - _in()||^2_Wwith the weighted norm defined as 

    ||_res||^2_W := ⟨_res|ℐ+κ_c|_res|⟩eq:weightnorm

 whereκ_cinbasis is the Coulomb kernel1/|- '|andℐis the Identity operator.κ_cin this weighted norm definition penalizes the term⟨_res|_res|$⟩ during the Anderson minimization step at longer wavelength changes of total charge density (i.e. at smaller magnitude of wavevectors, || → 0 in Fourier representation) in order to suppress charge sloshing in material systems with vanishing bandgaps as  explored in prior works <cit.>. Since the objective function  (<ref>) in Anderson mixing is non-local in real-space, we now provide a prescription to recast this equation into a local-form using an auxiliary potential corresponding to residual charge density _res() -via- the solution of the Poisson problem. To this end, we first note that  

    ⟨_res|κ_c|_res|   ⟩=  ∫∫_res()_res(')/|-'|dd'
       = ∫_res() φ_res() d

where φ_res() = ∫_res(')/|-'|d' denotes the auxiliary potential and is computed by solving the Poisson problem -1/4π∇^2 φ_res() = _res().  

 
 

Finally, from this Poisson equation and Eqn. (<ref>), we have the following

    ⟨_res|κ_c|_res|=⟩∫() φ_res() d =  -1/4π∫∇^2 φ_res() φ_res() d, 
       = -1/4π∫(∇.(φ_res∇φ_res) - |∇φ_res()|^2] ) d, 
       = 1/4π∫|∇φ_res()|^2 d,

where we used the divergence theorem for zeroing out the boundary term to get the last equality in the above equation.

Employing the finite-element discretization for the underlying electronic fields, we solve the following minimization problem in the Anderson mixing scheme.

    ||^h_res()||^2_W  :=  1/4π∫|∇φ^h_res()|^2d + ∫|_res^h()|^2d

where, φ^h_res() is the solution of the FE discretized Poisson equation _res=_res with  denoting the discretized Laplacian operator as given in Eqn. (<ref>) and
_res is the forcing vector given by ∫^h_resN_I^h,p_el d while the entries of the solution vector _res denotes the linear combination coefficients of the expansion of φ^h_res() in FE basis. Finally, the Anderson mixing coefficients c_k in Eqn. (<ref>) obtained by minimizing the weighted norm defined in Eqn.(<ref>) are employed to determine the input total charge density _in^h,(n+1) using Eqn. (<ref>). In other words, the linear combination coefficients c_k are used to determine ^h_in() and {^a,h_in} for the subsequent (n+1)^th SCF iteration. 

Using ^h(), the Hamiltonian() is computed and subsequently we solve the FE discretized generalized eigenvalue problem _i=ε_i_i. The following subsections describes the details of the efficient solution strategies employed in the current work.






 











 §.§ Subspace iteration procedure for PAW generalised eigenvalue problem

We employ the Chebyshev filtered subspace iteration approach(ChFSI) <cit.> to solve the FE discretized generalized non-linear eigenvalue problem in Eqn.(<ref>) for the smallest N eigenvector and eigenvalue pairs. ChFSI belongs to a class of iterative orthogonal projection methods and has been successfully employed in conducting large-scale real space DFT calculations over the past few years <cit.>. This approach relies on a Chebyshev filtering procedure that constructs a subspace rich in the wanted eigenvectors exploiting the fast growth property of Chebyshev polynomials outside the interval [-1,1], followed by a Rayleigh-Ritz step. The ChFSI approach can be viewed as a nonlinear subspace iteration procedure in the context of self-consistent field (SCF) iteration employed in solving the DFT nonlinear eigenvalue problem, allowing the reuse of the filtered subspace V spanned by the eigenvectors of a given SCF iteration to a subsequent iteration progressively improving it. To this end, the eigenspace of interest is refined at every SCF iteration, leading to a close approximation of the desired eigenspace of the non-linear eigenvalue problem as the SCF iterations proceeds towards convergence. Most importantly, the ChFSI method is naturally amenable for efficient use of high-performance computing architectures since the construction of the filtered subspace does not involve any interaction amongst the vectors, allowing for matrix-multivector multiplications to be done blockwise or in parallel over a few blocks, reducing the peak memory requirement <cit.>. 






Unlike most of the prior works <cit.> that employed ChFSI for standard eigenvalue problems, we need to solve here a large sparse generalized Hermitian eigenvalue problem (GHEP) as given in Eqn. (<ref>).  In the following, we describe the ChFSI approach adopted in the current work to solve this GHEP arising out of the finite-element (FE) discretization of the PAW method. 



  
Construction of Chebyshev filtered subspace: To solve the GHEP in Eqn. (<ref>), we consider the matrix 
^-1 that has the same eigenpairs as =ϵ.  To amplify the eigenspace of the interest corresponding to the N smallest eigenvectors (wanted spectrum) and dampen the unwanted spectrum, we seek to construct a Chebyshev polynomial filter involving  ^-1. This requires us to efficiently evaluate the inverse of , a large FE discretized sparse matrix. We now recall from Eqn.(<ref>) that  is of the form =  + Δ_^†, a rank-n_proj perturbation of .  We now use the Woodbury formula <cit.> to compute the inverse of , that can be expressed as

    ^-1   = M^-1 - ^-1Δ __inv^†^-1
    where  Δ__inv   = (Δ_^-1 + ^†^-1)^-1

We note that the above Equation (<ref>) requires the computation of inverse of the large M × M sparse FE basis overlap-matrix (), and the inverse of a smaller matrix (Δ_^-1 + ^†^-1) of size n_proj n_proj that scales with number of atoms. To evaluate ^-1, Gauss-Lobatto-Legendre (GLL) quadrature rules coincident with the finite-element nodes of spectral finite-elements are employed in this work. These GLL points are reduced-order quadrature rules and are used only to construct  in Eqn. (<ref>), rendering the matrix  diagonal denoted by _D. This leads to a trivial evaluation  <cit.> of ^-1 by inverting the diagonal entries of _D to be denoted as _D^-1 in the subsequent text. The other term in ^-1 which needs careful treatment, is the inverse of  (Δ_^-1 + ^†^-1). Recall from Eqns. (<ref>) and (<ref>), 
Δ_ is block diagonal with each block corresponding to each atom `a' and hence Δ_^-1 can be evaluated as the inverse of individual blocks Δ_^a. Furthermore, using the fact that the PAW augmentation spheres are localized in real-space and seldom overlap, the matrix ^†_D^-1 can be  approximated  as a block diagonal matrix with each block corresponding to an atom in the given material system. Hence, the block diagonal form of the Δ^a __inv matrix in real-space allows us to compute its inverse atom-by-atom. To this end, the approximate ^-1 can be recast as follows:

    ^-1   = _D^-1 - ∑_a_D^-1^aΔ^a __inv^a†_D^-1
    where  Δ^a __inv   = (Δ_^a^-1 + ^a^†_D^-1^a)^-1

In this work, we seek to construct a subspace rich in the desired eigenvectors of the PAW generalized eigenvalue problem =ϵ in Eqn. (<ref>)  by building a Chebyshev polynomial filter of =^-1 a close approximation to ^-1. To this end, the filtering procedure first relies on mapping the unwanted spectrum of  to [-1,1] and the desired spectrum to (-∞,-1). This is accomplished by computing a shifted and scaled matrix  through the affine transformation given by = 2/b-a -b+a/b-aI where a and b denote the upper bounds of wanted and unwanted  spectrum respectively. The upper bound b of the unwanted spectrum is obtained inexpensively by using a few steps of generalized Lanczos iteration <cit.> as discussed in the Algorithm (<ref>) which employs = _D + ∑_a^aΔ^a_^a^† and its inverse ^-1 as discussed in Eqn. (<ref>). A good approximation to the upper bound a of the wanted spectrum is obtained as the highest generalized Rayleigh quotient of  in the Chebyshev filtered subspace of the previous SCF iteration.

We now construct the filtered subspace by computing the action of a degree-m Chebyshev polynomial filter T_m() on an input subspace _in where the subspace _in is usually the eigenspace from the previous SCF iteration and is a matrix of size M × N. This action is accomplished -via- the three-term recurrence relation of Chebyshev polynomials  as given below:

    _f = T_m()_in where  T_m+1() = 2 T_m() - T_m-1()

The rapid growth property of Chebyshev polynomials outside [-1,1] amplifies the components in _in along the direction of the desired eigenvectors (occupied states) while dampening the components along the unwanted eigenvectors (unoccupied states), leading to the filtered subspace _f to be a close approximation of the eigenspace of interest of the PAW eigenproblem =ϵ in a given SCF iteration.
 



  
Rayleigh-Ritz step: Upon computing the filtered subspace _f in a given SCF iteration, we project the Hamiltonian() and PAW-overlap() onto the filtered subspace _f to construct projected matrices ^p and ^p of size N N as shown in step 5 of algo.(<ref>). The Rayleigh-Ritz step now involves solving a subspace eigenvalue problem for N eigenpairs and is of the form

    ^p = ^pΛ

where  and Λ  denote the eigenvector matrix and the diagonal eigenvalue matrix of the smaller subspace eigenvalue problem Eqn.(<ref>) respectively. Finally, a subspace rotation step (see step 7 of Algo.(<ref>)) is carried out to obtain the -orthonormal eigenvectors  in the finite-element subspace which is in turn used to compute the output total charge density ρ^h() = n^h + b().

A good approximation to the eigenspace of interest in the ChFSI approach employed in this work relies on the choice of the degree m of the Chebyshev polynomial filter. This choice depends on the separation of eigenvalues within the desired region of eigenspectrum and also, the magnitude of the largest eigenvalue (λ_max) of the matrix ^-1 that determines the ratio of spectral widths of the wanted to the unwanted regions of the eigenspectrum. We note that this λ_max is dictated by the fineness of the mesh size of the underlying finite-element discretization, i.e., it increases with decreasing mesh size. In the PAW method within the frozen core approximation, the electronic fields {ψ̃_i} (or equivalently {ũ^_i} in the periodic case) and φ vary smoothly compared to the electronic fields in the case of ONCV pseudopotentials for most atomic species. This allows for the use of coarser finite-element meshes in the case of PAW formalism, leading to the choice of Chebyshev polynomial degree between 10 and 30 for accurately approximating the desired eigenspace. The reduced degrees of freedom and the lower Chebyshev polynomial degree required to reach the desired chemical accuracies in the proposed finite-element framework for the PAW approach gives four to five fold computational advantage compared to current state-of-the-art finite-element code for DFT using ONCV pseudopotentials (DFT-FE) as will be demonstrated in the subsequent sections.

The ChFSI approach described here involves computational intensive operations notably in two key steps: first, constructing a subspace by computing , where  represents any one of the matrices  , , ^-1, and second, performing a Rayleigh-Ritz step by computing matrix-matrix products of the form ^†. We will now elucidate efficient strategies that exploit the compact support nature of the finite-element basis to evaluate these operations in a computationally efficient manner.











 §.§ Efficient computational methodologies

In the proposed ChFSI approach in Algorithm (<ref>), the efficient computation of matrix-matrix products of the form  and ^† is critical for enabling fast, accurate and scalable PAW-FE calculations on parallel computing architectures. We first consider the case of evaluating the matrix multivector products of the form  arising in the construction of Chebyshev filtered subspace, where the matrix  can be one of discretized Hamiltonian matrix(), PAW overlap() matrix or its inverse. We note here that the matrix  is a large sparse matrix of dimension M × M with M ≈10^3 - 10^8 while  is a large dense matrix of dimension M × N  with N ≈10 - 10^4.  A naive strategy often adopted in the finite-element literature to compute  would be to construct the FE discretized global sparse matrix  using sparse-matrix storage data-structures and then compute the sparse matrix-dense matrix product . This strategy would be highly inefficient on modern parallel computing systems involving multi-threaded and multi-node architectures owing to the expensive data movement costs involved in the underlying sparse-matrix algorithms. To this end, we employ a cell-matrix approach <cit.> that avoids the construction of global sparse-dense matrix-products and instead employs dense-dense matrix-products at FE cell level to reduce memory requirements, enhance cache locality, and improve arithmetic intensity of computation. We describe below the key mathematical aspects used in this implementation strategy to efficiently evaluate   during the ChFSI procedure in solving the PAW generalized eigenvalue problem.




Denoting the simulation domain by Ω_p, we begin by partitioning Ω_p into subdomains Ω^(t) ∀ t = 1,2,...,n_t, where n_t is the number of subdomains with each subdomain Ω^(t) assigned to an MPI task t. Further, let E_t be the number of finite-element (FE) cells and m_t be the number of basis functions (number of finite-element nodes) in each subdomain Ω^(t) such that Ω^(t) = ∪^E_t_e=1Ω^(e,t), where the notation (e,t) refers to the index of the FE cell 'e' in an MPI task 't'. This allows us to decompose the integrals over the domain Ω_p in Eqns. (<ref>), (<ref>) to a series of integrals over smaller domains Ω^(e,t) as

    ∫_Ω_pf()d = ∑_t∑_e∫_Ω^(e,t)f()d

Finally, the integrals over each domain Ω^(e,t) are computed using a suitable quadrature rule as:

    ∫_Ω^(e,t)f()d = ∑^N_q_qf(_q)w_q

where N_q is the total number of quadrature points in the domain Ω^(e,t) while w_q, _q are the quadrature weights and sampling points respectively. 3D Gauss Legendre quadrature rules that are tensor products of 1D quadrature rules <cit.> are employed in this work, and the order of these rules employed are determined such that errors due to the quadrature rule are of higher order than the discretization error. The flexibility in the choice of quadrature rules for evaluating various integrals involving PAW atomic data in  Eqns. (<ref>),(<ref>) allows the use of coarser finite-element grids to represent electronic fields and yet capture the relevant atom-centered data, a notable advantage compared to finite-difference approach, a popular real-space discretization DFT method, where special schemes need to be devised to retain the use of coarser grid resolution for smoother electronic fields without egg-box effects <cit.>. 

We note that the operation  can be decomposed as = _loc + _nloc (where := , , ^-1 and _loc:= _loc, , _D^-1 respectively). To this end, the mathematical expressions for  relevant to our implementation strategy within the finite-element framework can be written as 
 

    = [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T(^(e,t)_loc + ∑_a^(a,e,t)Δ^a _(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) )^(e,t)^(t)^(t)]


    = [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T(^(e,t) + ∑_a^(a,e,t)Δ^a _(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) )^(e,t)^(t)^(t)]


    ^-1 = ^-1_D - [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T  ^-1_D^(e,t)∑_a^(a,e,t)Δ^a __inv(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) ^(e,t)^(t)^(t)]^-1_D
      

In the Eqns.(<ref>)-(<ref>), the Boolean sparse matrix ^(t) represents the partitioner matrix that acts on  (or ^-1_D) to provide the subdomain level matrix ^(t) while preserving the continuity of the nodal field across the subdomain boundaries. The m_t  m_t sparse matrix ^(t) represents the constraint matrix that constrains the values of the m_t × N matrix ^(t) at the specific FE nodes. These constraints either satisfy the necessary boundary conditions (periodic, non-periodic, or semi-periodic) imposed on the discretized electronic wavefunctions or address constraints from non-conforming meshes <cit.>. Furthermore, the n_p^3× m_t Boolean matrix ^(e,t) denotes the sub-domain to FE cell map associated with the sub-domain Ω^t whose action on ^(t)^(t) results in the extraction of the finite-element cell-level entries from ^(t)^(t) while ensuring the continuity condition of the discretized electronic wavefunction across the finite-element cells within the partitioned sub-domain Ω^(e,t). Efficient computation of  illustrated in Eqns.(<ref>)-(<ref>) can be summarized as a sequence of the following steps:

    
  * Precompute the entries of the cell-level matrices ^(e,t)_loc, ^(e,t)_loc ^(a,e) and Δ^a_, the atom dependent coupling matrices (Δ^a_:= Δ^a_, Δ^a_, Δ^a__inv) using the quadrature rules as discussed in Eqns.(<ref>) and (<ref>))
    
  * Extract the finite-element cell-level matrix:
 ^(e,t)= ^(e,t)^(t)^(t),∀ (e,t)
    
  * Evaluate the partial non-local operator action at the FE cell level: 
    
Δ^a_  = ∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at^(e,t) where δ_e^a e indicates that this computation is done in only those FE cells e^a lying in the augmentation sphere (Ω_a) centered around an atom `a'. Furthermore, δ_t^a t indicates that this computation is done only on those MPI tasks t^a which are associated with Ω_a.
    
  * Evaluate the local operator action along with the impending non-local operator action:

    ^(e,t) = ^(e)_loc^(e,t) + ∑_a^(a,e,t)Δ^a _Δ^a_
    
  * Assembly of the global vector =:

    = ∑_t^n_t^(t)^T ^t^T∑_e^E_t^(e,t)^T^(e,t)

We remark that the finite-element cell-level matrices ^e,t_loc and the coupling matrix Δ^a_ are computed at every SCF iteration while ^e,t_loc is computed for a given finite-element mesh. Further Δ^a _ is computed only once while Δ^a_^-1 is evaluated
only once per ground-state calculation.

Following the evaluation of =, the key computationally intensive operation in the construction of Chebyshev filtered subspace, the subsequent step is the Rayleigh-Ritz step that requires the evaluation of matrix-matrix products of the form ^† (see Algorithm 3). To this end, the matrices , are partitioned row wise into _p,_p across various MPI tasks. To this, the local _p^†_p is evaluated to obtain the smaller matrix N N in each MPI tasks and finally, the local contribution is summed up across all MPI tasks to obtain ^† = ∑_p_p^†_p. 

So far, in this section we discussed the various computational methodologies employed in the proposed PAW-FE approach that can enable accurate large-scale DFT calculations on material systems up to tens of thousands of electrons in a computationally efficient manner. The following section discusses the accuracy and performance results of the proposed computational approach while benchmarking with state-of-the-art plane Wave(PW) codes Abinit<cit.> and Quantum Espresso<cit.>.











§ RESULTS AND DISCUSSION






In this section, we demonstrate the accuracy, performance and parallel scalability of the proposed method  on representative benchmark problems involving periodic, semi-periodic and non-periodic boundary conditions. We first discuss the accuracy of our implementation on non-periodic systems involving molecular systems and small metallic nano-clusters by comparing energetics with plane-wave-based PAW implementations. We then demonstrate the performance of  in terms of both computational node-hours and minimum wall time with respect to plane-wave calculations for increasingly large sizes of metallic Cu nanoclusters containing up to 17537 electrons (923 atoms). Subsequently, we demonstrate the accuracy of our method compared to plane-wave calculations on semi-periodic and fully periodic systems involving unit-cells, large super-cells, surfaces. We further discuss the performance of our method in terms of computational node-hrs and minimum wall time for increasingly large sizes of Cu_3Pt periodic supercells containing up to 34,304 electrons (2048 atoms) compared to plane-wave DFT calculations using PAW. Finally, we showcase the computational efficiency and parallel scalability of  compared to norm-conserving ONCV pseudopotential DFT calculations using  on large-scale material systems involving ∼50,000 electrons.

All the DFT calculations reported in this work employ GGA <cit.> exchange-correlation functional of the PBE <cit.> form. Furthermore, we employ the PAW data-sets from the pseudo-dojo database <cit.>. In all our simulations, we use Fermi-Dirac smearing with T=500 K.  Additionally, for the mixing of the total charge density involving (()) and {^a}, we employ the n-stage Anderson mixing <cit.> as discussed in the Section <ref>.

For all accuracy validation studies in this work, we employ refined finite element (FE) meshes with FE interpolating polynomial (p) of degree 7. These meshes are constructed such that the differences in the all-electron ground-state energies (Eqn. (<ref>)) between successively sub-divided meshes reach 10^-6 Ha/atom. In addition, the choice of quadrature integration rules and FE interpolating polynomial for electrostatics p_el is chosen such that the energy variation with respect to these parameters is an order of magnitude lower than this discretization error. In the case of plane-wave calculations we use [ version  is used for the accuracy validation] code and the cut-off energy for wavefunctions E^wfc_cut is chosen so that the ground-state energy is converged up to 10^-6 Ha/atom while simultaneously ensuring that energy change with respect to the cut-off energy for total charge density E^rho_cut is an order of magnitude lower.

The comparative study involving the performance of our  implementation with that of plane-wave calculations has been conducted by solving a given DFT problem to the same level of accuracy. To this end, we choose the discretization parameters such that the discretization error with respect to a very refined calculation in each case to be around ∼2 × 10^-4 Ha/atom for our comparisons. We employ () code[ version  with ELPA is used for the performance benchmarking study. The authors observed  was computationally efficient than  on few benchmark examples.] for conducting our performance studies using plane-wave DFT calculations. Furthermore, we use two metrics to compare the performance of PAW-FE and the plane-wave calculations: (i) computational cost (η_c) in node-hrs, and (ii) minimum wall time (τ^min_c) in secs. The computational cost η_c is obtained by multiplying the minimum number of compute nodes required to fit a given problem with the average wall-time per SCF iteration[Average wall-time per SCF is obtained by dividing the wall time taken for the self-consistent field iteration procedure with the number of SCF iterations taken to reach a default convergence threshold for self-consistency]. Further, the minimum wall time (τ^min_c) is obtained by computing average wall time per SCF by increasing the number of compute nodes till a parallel scaling efficiency of around 35% is reached. All the simulations involved in the accuracy and performance benchmarking studies were performed on high-memory CPU nodes of PARAM Pravega[PARAM Pravega is one of India's fastest supercomputers stationed at Indian Institute of Science comprising of 156 High Memory Intel Xeon Cascade-Lake based CPU nodes (7,488 Cores) where each node consists of 48 Intel Xeon 6248 processor, 768 GB memory and Mellanox HDR Infiniband interconnect between all the nodes for fast MPI communication.]. 



 §.§ Non-periodic systems

In this subsection, we consider the case of non-periodic systems for our accuracy and performance benchmarking. In particular, for accuracy validation studies, we consider isolated molecules such as CO, CH4, H2O and also metallic nano-clusters such as Cu13, Al13. We benchmark the accuracy of   with  by comparing the energy vs bond length curves and the ground state energies computed at the minimum energy configuration. Following the accuracy validation study, we assess the performance of our implementation by comparing the computational cost (η_c) and minimum wall time(τ^min_c) of   with Quantum Espresso(), a widely used code employing plane-wave basis. In this performance benchmarking study, we consider three-dimensional icosahedron<cit.> shaped Cu nanoparticles of various sizes ranging from 55 atoms (1045 electrons) to 923 atoms (17537 electrons). Homogeneous boundary conditions have been imposed on the electrostatic potential (φ^h()) for these calculations using   while in plane-wave codes, periodic boundary conditions (PBCs) are only admissible and hence PBCs are employed using a suitable vacuum to minimise image-image interactions. 



  §.§.§ Accuracy benchmarking

 In all the accuracy benchmarking studies reported in this subsection, we choose refined finite-element (FE) meshes with FE interpolating polynomial degree p to be 7 and the mesh size h to be around 1.0 Bohr while in the case of [. We use the =0 in  to enforce Blöchl formulation of exchange-correlation(XC) contribution with =0 to avoid any approximation in the XC correction term computation.], we use a plane-wave cut-off energy E^wfc_cut=60 Ha. Figure (<ref>) reports the energy as a function of inter-atomic distance obtained from  and   for the molecular systems O2,  H2O, NH3 and CH4. Since the scale of ground-state energies is different in the two implementations compared here, the energies in the plots of Fig. (<ref>) are measured relative to the minimum ground-state energy obtained in this study using  and  for each of the above molecular systems. We now describe the details of the simulations conducted for our comparative study: (i) Oxygen: The energy as a function of O-O bond length is plotted in Fig. (<ref>)(a) and the bond length for the minimum energy configuration is found to be 2.30 Bohr in   which is in excellent agreement with , (ii) Ammonia: The trigonal pyramidal geometry of ammonia is considered where the N-H bond lengths are varied such that the H-N-H bond angle is maintained to be 106.2^∘. In the Fig. (<ref>)(b), we plot the energy variation as a function of this N-H bond length and we observe the N-H bond length at the minimum energy configuration to be around 1.93 Bohr using   which matches very closely to the value obtained with , (iii) Water: The O-H bond lengths in the water molecule are varied while ensuring that the planar H-O-H bond angle of 104.46^∘ is maintained. Energy of the molecule is plotted as a function of this O-H bond length in Fig. (<ref>)(c) and we observe from our   calculations that the O-H bond length at the minimum energy configuration is 1.83 Bohr which is in close agreement with , (iv) Methane: The tetrahedral geometry of methane is considered where the C-H bond lengths are varied such that H-C-H angle is maintained at 109.5^∘. Fig. (<ref>)(d) shows the comparison between  and  calculations. These results indicate that the C-H bond length at the minimum energy configuration is 2.07 Bohr in both cases.
 
 To summarize the observations from Fig. (<ref>), we find an excellent agreement in all the energy vs bond-length curves between  and  for the representative molecules considered in this work. Furthermore, the bond length at the minimum energy configuration obtained from our implementation shows a very close match with that of plane-wave calculations as summarized in Table <ref>. We now compare the ground-state energies of a few representative systems obtained using   with that of  in Table<ref>. To this end, we consider the above four molecular systems at the minimum energy configuration obtained from Fig. <ref>. Additionally, we consider the 13-atom icosahedron nanoclusters Cu_13 and Al_13 with 4.8 Bohr and 5.3 Bohr as the nearest neighbour distance, respectively <cit.>. To have a one-on-one comparison of ground-state energies with  for these non-periodic systems considered in this work, we compute pseudo valence energy in  using the expression in Eqn.(<ref>) as opposed to all-electron energy in Eqn.(<ref>). We note that Table (<ref>) shows an excellent match between the ground-state energies obtained from   and , and the differences in the energies are O(10^-5)Ha/atom. Having established the agreement of  with a plane-wave code in terms of energetics, we now focus on performance benchmarking of  against the plane-wave implementations.
 
  
 






  §.§.§ Performance benchmarking

We benchmark the performance of  with Quantum espresso <cit.>, a widely used open-source plane-wave (PW) code, by comparing the average CPU time per SCF in terms of computational node-hrs(η_c) and minimum wall time(τ^min_c) with increasing sizes of Cu nanoclusters<cit.> solved to similar level of accuracy between the two approaches. These icosahedron-shaped nanoclusters considered in this study are constructed with 4.8 Bohr nearest neighbour distance<cit.>, and the nanocluster size is increased by varying the number of shells. To this end, non-periodic DFT calculations using the PAW method are conducted on Cu3-shell (2793 electrons), Cu4-shell (5871 electrons), Cu5-shell (10,659 electrons), Cu6-shell (17,537 electrons). The two metrics (η_c,τ^min_c) used to benchmark the performance results are discussed at the beginning of this section. In , the degree of the FE interpolating polynomial p is chosen to be 5, and a mesh size h=1.2 Bohr is employed with a Chebyshev polynomial degree of 20 in the ChFSI step. While in the case of  , the plane-wave cut-off energy E^wfc_cut=25 Ha is used with Γ-point for k-point sampling and default settings are used for the eigensolver. These discretization parameters are chosen so that the discretization error in the ground-state energies obtained in both   and   is ∼ 2 × 10^-4Ha/atom. Furthermore, we used a vacuum of around 20 Bohr and 15 Bohr from the farthest atom of the nanocluster in  and  respectively, ensuring that ground-state energies are converged up to O(10^-5)Ha/atom with vacuum size.

Table (<ref>) reports the average CPU time per SCF iteration in terms of computational node-hrs (η_c) and the number of basis functions in  and  for various sizes of Cu nanoclusters studied here. From this table, we find that for system sizes greater than ∼ 1000 electrons,  becomes more efficient than . We observe the efficiency gains for  increases with system size over , achieving gains of around 3.1×, 4.3× and 5.1× for Cu_4-shell, Cu_5-shell and Cu_6-shell respectively, the benchmark systems considered for CPU-time comparison.  
 This increase in computational gains is attributed to the need for using more processors to satisfy the peak memory requirement, where the efficient parallel scalability of  provides the necessary advantage. Further, in this case of non-periodic systems, the spatial adaptivity of finite-element basis allows the FE mesh to be coarse-grained into the vacuum in contrast to a uniform spatial resolution of the plane-wave basis in , which is also a key factor for the superior gains observed in . Additionally, we estimate the computational complexity of  from Table(<ref>) and is found to be of N_e^1.8 while for  we observe it to be of N_e^2.6. Hence, we expect to see further gains of  over  with increasing system size.
 
 Finally, the Table <ref> compares the average time per SCF in terms of minimum wall time(τ^min_c) obtained in both  and . We note that the minimum wall time of  is measured at ∼ 30%-40% scaling efficiency while a scaling efficiency of ∼ 50%- 55% is reached for [  could be scaled further on more nodes, however, the maximum available 140 high-memory nodes on PARAM-Pravega limited the further reduction of minimum wall time]. Even for a small system involving Cu3-shell comprising of 2793 electrons, we observe a gain of around 3.2 in the minimum wall time for  compared to , and we see almost a staggering speed up of 10 for Cu6-shell comprising of 17,537 electrons, the largest system considered in this study. The increased speedup with system sizes is attributed to the superior scaling of  which requires only nearest neighbour communication and the various efficient computational methodologies employed in solving the PAW eigenvalue problem using the Chebyshev filtered subspace iteration approach (ChFSI) in  (see Section <ref>).












  


 §.§ Periodic/Semi-periodic systems

In this subsection, we consider the case of periodic and semi-periodic systems for our accuracy and performance benchmarking. In particular, we benchmark the accuracy of  with  by comparing energy versus lattice parameters of representative bulk systems involving Li2O, Cu, Cu3Pt and Li unit cells. Further, we compute and benchmark with  the formation energy of Li2O and the surface energies of the semi-periodic systems – Zr(001) and La(011) terminated Lithium Lanthanum Zirconium Oxide (LLZO), an electrolyte material. Finally, we compare the band structure of C diamond primitive cell obtained from  and .  Following the accuracy benchmarking study, we assess the performance of  with   on two metrics (η_c, τ^min_c) on increasing sizes of Cu3Pt supercells.






  §.§.§ Accuracy benchmarking

In all the accuracy benchmarking studies of periodic/semi-periodic systems reported here, we compute the all-electron PAW ground state energy using Eqn. (<ref>) in . Further, we choose refined finite-element (FE) meshes with FE interpolating polynomial degree to be 7 and the mesh size h to be around 1.0 Bohr while in the case of  , we use a plane-wave cut-off energy E^wfc_cut=60 Ha.



  
Energy vs lattice parameter: Figure (<ref>) shows the variation of ground-state energy as a function of lattice parameter for the case Li2O(cubic), Cu(fcc), Cu3Pt(fcc) and Li(bcc) unitcells. We use orthogonal unit-cells in each case and employ shifted 888 Monkhorst-Pack k-point grid for sampling the Brillouin zone <cit.>. In Fig.(<ref>), we plot the energies relative to the minimum ground-state energy obtained in this study using  and  for each of the above material systems. This is done to account for different energy scales in the two implementations considered here. From Fig.(<ref>), we observe an excellent correspondence between  and . Furthermore, we also observe from Table. <ref> that the lattice parameters at the minimum energy configurations show an excellent match for all the representative periodic systems considered here.


 


  
Formation energy: Using the ground-state energies at the minimum energy configuration obtained from Fig. <ref>, we compute the formation energy Δ E^Li2O_f of Li2O as 
Hartree-per-formula unit(Ha/f.u.) using the following expression:

    Δ E^Li2O_f = 1/4E_Li2O - E_Li - 1/2E_O2

To this end, Δ E^Li2O_f is found to be -0.223095 Ha/f.u. in , while in the case of , we obtain a value of -0.223126 Ha/f.u. This results in a difference of ∼ 3.210^-5 Ha/f.u., establishing a good agreement in the formation energy computed between the two codes




  
Surface energies: Next, we compute and validate the surface energies of LLZO (Lithium Lanthanum Zirconium Oxide), a material of interest for solid electrolytes, focusing on La(011) and Zr(001) terminations. We obtain the coordinates from a study of LLZO particle morphology<cit.>. The slab is placed at the center of the simulation domain with 25 Bohr vacuum on either side. In  , semi-periodic boundary conditions are employed for the electronic fields, with homogeneous Dirichlet boundary conditions for φ^h() on the boundary parallel to the surface. While in , periodic boundary conditions are employed. The surface energy(γ) is computed using the following expression in the units of J/m^2 

    γ = 1/2A_surface( E_surface - E_bulk)

where A_surface is the surface area, E_surface is the ground-state (GS) energy of the material systems with the surface of interest and E_bulk is the GS energy of the corresponding bulk system and the values obtained from  and  are shown in Table <ref>. From this table, we observe a close match in surface energies computed from both codes. The difference in surface energies are ∼ 210^-3 J/m^2. Additionally, it is observed that the Zr(001) surface has higher surface energy in both codes, consistent with observations from previous studies<cit.>.





  
Band structure comparisons: We now compute the band-structure of the diamond primitive cell of Carbon with lattice constant 3.373 Bohr. The Brioullin Zone(BZ) and the k-point path are shown in Fig.(<ref>) and were obtained using seeK-path<cit.>. A shifted 888 Monkhorst-Pack k-point grid was used for the initial SCF calculation for sampling the Brillouin zone, following which a non-self-consistent field (NSCF) calculation was performed using the k-points specified along the high symmetry points of Fig.(<ref>) to extract the band structure. Figure (<ref>) shows the overlay of the band structure computed using  and .  We observe an excellent match in the eigenvalues and, hence, the band structure between the two codes.  
    

The comprehensive benchmarking studies discussed here establish the accuracy of our  computational methodology and implementation for periodic/semi-periodic systems. We now proceed to discuss the performance aspects of  for periodic systems with increasing system sizes.






















  §.§.§ Performance benchmarking

We benchmark here the performance of  with  similar to the non-periodic systems discussed previously and compare the computational efficiency and minimum wall time on increasingly large supercells of Cu3Pt crystal solved to similar levels of accuracy between the two approaches. The supercells are constructed by stacking of Cu3Pt unit cells along the three lattice dimensions and to this end, we consider supercells of sizes 4× 4 × 4 (4288 electrons), 5 × 5 × 5 (8375 electrons), 6 × 6 × 6 (14472 electrons), 7 × 7 × 7 (22981 electrons), 8 × 8 × 8 (34304 electrons). We perform a Γ-point calculation with periodic boundary conditions imposed on all electronic fields in both  and . Further, the discretization parameters are chosen such the discretization error in the ground-state energies obtained in both cases is 210^-4Ha/atom. In particular, in  , the degree of the FE interpolating polynomial p is chosen to be 6, and a mesh size of h=1.8 Bohr is employed with Chebyshev polynomial degree of 20 in the ChFSI step. While in the case of  , the plane-wave cutoff energy E^wfc_cut = 25 Ha is used with default settings for the eigensolver.

Table(<ref>) and (<ref>) presents the computational cost[The computing system used for conducting the performance benchmarking studies are high memory nodes ( 768 GB per node) which favors   as the minimum number of nodes required to fit peak memory requirement is lesser.] and the minimum wall time comparison between  and . We recall that the computational cost (η_c) is computed as the product of minimum number of nodes required to fit the problem and the average wall time per SCF iteration. From the Table.(<ref>), we observe that the number of basis functions required by  is a factor of 6 higher than  and we begin to see computational gains in  for systems greater than 8000 electrons. The efficiency gains we achieve with an increase in system sizes are around 1.7×, 2.3× and 3.2× for 6 × 6 × 6, 7 × 7 × 7, 8 × 8 × 8 Cu3Pt supercells respectively, the benchmark systems considered for CPU-time comparison in this work. We note that the computational gains obtained in the periodic case are lesser than the case of non-periodic systems because of the degrees of freedom advantage that plane-wave (PW) codes provide for bulk systems (∼ 500 PW basis functions per atom) compared to non-periodic systems (∼ 5000 PW basis functions per atom) to reach the desired accuracy.  However, on estimating the computational complexity of  from Table (<ref>) we find that  scales as N_e^2.2 while  scales as N_e^2.8 indicating that we will obtain an increase in computational gains for  compared to  as the system size.

Finally, the Table (<ref>) compares the average time per SCF in terms of minimum wall time (τ^min_c) between  and . To this end, we observe a gain of around 2.3× for Cu3Pt 5 × 5 × 5 supercell with 8375 electrons and the computational gains increase to almost 7.7× for system size comprising of 34,304 electrons. Even for fully periodic calculations, these increased gains with system size underscore the importance of fast and scalable computational methods developed in this work.


 
 





 




 §.§ Scalability and parallel efficiency

In this subsection, we compare the computational cost and parallel scaling efficiency of the proposed computational methodology  relative to the norm-conserving pseudopotential (ONCV) calculations <cit.> using the  code solved to a similar level of accuracy. We remark here that the transferability of ONCV pseudopotential DFT calculations have been shown to be on par with PAW approaches <cit.> for most material systems in contrast to the early norm-conserving Troullier-Martins (TM) pseudopotentials <cit.> and have become one of the popular method of choices in most DFT implementations including . However, as alluded to before, these ONCV pseudopotentials can become very hard, requiring a large number of degrees of freedom to reach the desired accuracy for most material systems involving elements with d or f electrons. Towards this study, we consider 999 Cu3Pt supercell and an icosahedral-shaped Pt8-shell nanocluster <cit.>, with the nearest neighbour distance being 5.27 Bohr, as representative systems for periodic and non-periodic calculations. We obtain the PAW data set and ONCV pseudopotential data from pseudodojo <cit.> repository for our comparative study. The finite-element mesh parameters in both cases are determined such that the discretization error is around 210^-4 Ha/atom. Table (<ref>) shows the details of number of electrons, finite-element (FE) mesh parameters, number of FE basis functions between the ONCV based  and  calculations and the Chebyshev polynomial degree employed during the ChFSI step. The simulations for this study were conducted on the OLCF supercomputer Frontier [Frontier is the world's fastest exascale computer. Each compute node of Frontier has  64-core AMD Epyc 7713 processor with 512 GB memory and multiple slingshot NICs for communication between nodes.] CPUs. 

To assess the strong scaling behaviour between  using ONCV pseudopotentials and the  calculations, we plot the average wall time per SCF iteration τ_c against the number of compute nodes in Fig. (<ref>).
We observe that  is almost an order of magnitude (∼ 10× to 12×) faster than the  calculations using the ONCV pseudopotentials in terms of computational node-hrs for the large-scale representative systems considered here. Furthermore, we observe similar or better parallel scaling efficiency, while requiring 4× to 7× less number of compute nodes to fit the given problem in the case of . The efficiency gains obtained in  can be attributed to (i) coarser finite-element mesh size leading to much lesser degrees of freedom compared to the ONCV pseudopotential-based  calculations to reach the desired chemical accuracy, (ii) reduced Chebyshev polynomial degree during the ChFSI step due to the coarser mesh size in  calculations, (iii) lesser number of valence electrons within the frozen-core approximation of the PAW method in the case of Pt leading to better parallel scaling efficiency of ChFSI due to reduced communication costs in the Rayleigh-Ritz(RR) step. We further note that the computational gains of  compared to   using ONCV pseudopotential calculations may vary depending on the pseudopotential repository used, and the current comparisons use the data sets obtained from pseudo-dojo<cit.> repository, as mentioned before.








    










§ SUMMARY

In this work, we present a computationally efficient real-space methodology for large-scale density functional theory (DFT) calculations using finite-element (FE) discretization of the projector-augmented wave (PAW) method. To the best of our knowledge, this represents the first finite-element-based approach for DFT calculations, combining the efficiency of PAW formalism involving smooth electronic fields with the ability of systematically convergent higher-order finite-element basis to achieve significant computational efficiencies on modern computing systems. The methodology exploits the compactly supported nature of FE basis functions and further takes advantage of the adaptive resolution offered by the FE basis to handle generic boundary conditions efficiently. Furthermore, this work demonstrates a substantial computational advantage over the current state-of-the-art plane-wave-based PAW implementations for medium to large-scale material systems, thereby pushing the boundaries of length-scales that can be realised from ab initio calculations than possible today. 

The development of the proposed approach is based on the amalgamation of the following key ideas. First, we deduced the finite-element (FE) discretized governing equations corresponding to Blöchl's PAW method by making use of a locally reformulated electrostatics functional involving the smooth electron density and compensation charge. Further, higher-order quadrature rules were utilized to accurately evaluate the underlying integrals involving atom-centered functions in these governing equations, thereby allowing the use of coarser FE grids for the various electronic fields. Secondly, we employed a self-consistent field (SCF) iteration approach based on Anderson mixing of smooth electron density and atom-dependent spherical density matrix. This is accomplished by minimizing the weighted norm of the total charge density residual, which is locally reformulated using the gradients of the electrostatic potential corresponding to this residual. Next, we developed an efficient strategy to compute the inverse of the FE-discretized PAW overlap matrix  which is essential for solving the underlying generalized eigenvalue problem. Leveraging this strategy for computing ^-1, we designed a Chebyshev polynomial-based filter within each SCF iteration to compute the desired eigenspace. The FE-cell-level structure of both the discretized Hamiltonian () and the inverse PAW overlap matrix (^-1) is exploited to devise computationally efficient strategies that  employ cache-friendly matrix-matrix products to evaluate the action of the Chebyshev filter on the trial subspace during the course of Chebyshev filtered subspace iteration procedure.

The accuracy of the proposed method was validated against a plane-wave implementation of PAW across various non-periodic, semi-periodic, and fully periodic representative benchmark systems. Compared to the plane-wave calculations, our proposed methodology() showed excellent agreement in ground-state energies, energy variation with bond length and lattice parameters, formation energies, surface energies, and band structures. Furthermore, compared to the state-of-the-art plane-wave implementation, the efficient computational strategies employed in  enabled a 5 reduction in computational cost in non-periodic systems(∼18000 electrons) and a 3 in periodic systems (∼34000 electrons) with more significant gains observed as system size increases. Additionally, in terms of minimum wall time we observe substantial speed-ups for both periodic and non-periodic systems (∼ 7×-10×) using . Additionally, the proposed computational strategies exhibit excellent parallel scaling efficiency (∼70%) even at 1200 degrees of freedom per MPI task and achieves a substantial ten-fold reduction in computational cost over current state-of-the-art  code using norm-conserving pseudopotentials for system sizes close to 50,000 electrons.

The proposed work, , offers a systematically convergent basis capable of handling generic boundary conditions while substantially reducing the number of basis functions required to achieve desired chemical accuracy compared to the current state-of-the-art finite element-based DFT methods using norm-conserving pseudopotentials. Furthermore, the computational methodologies developed are well-suited for modern GPU architectures, which can significantly accelerate  calculations and reduce computational time—a focus of future investigation. This work constitutes a significant step towards enabling ab initio calculations of complex systems at larger length scales, thereby aiding in tackling a diverse set of key scientific and technological problems. Additionally, it paves the way for longer time-scale molecular dynamics simulations with ab initio accuracy by providing a more comprehensive energy landscape for robust statistical and machine learning-based computational models.



§ ACKNOWLEDGEMENTS

The authors would like to thank Prof. Vikram Gavini for many helpful discussions and his valuable suggestions. The authors gratefully acknowledge the seed grant from Indian Institute of Science and SERB Startup Research Grant from the Department of Science and Technology India (Grant Number: SRG/2020/002194). The research used the resources of PARAM Pravega at Indian Institute of Science, supported by National Supercomputing Mission (NSM) R&D for exa-scale grant (DST/NSM/R&D Exascale/2021/14.02). This research also used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. K.R. would like to acknowledge Prime Minister Research Fellowship(PMRF) from the Ministry of Education India for financial support. S.D acknowledges the support of Toyota Research Institute. P.M. acknowledges Google India Research Award 2023 and Indo-Korean Science and Technology center Bengaluru for financial support during the course of this work.  




§ ENERGY EVALUATION

The reformulated all-electron energy functional Eqn.(<ref>) in the PAW method is expressed in terms of E[{_i}, {^a}] and Δ E^a[{D^a_αβ}].  To this end, the atom-centered correction term is expressed as

    Δ E^a[{D^a_αβ}] = Δ E^a_xc + Δ E^a_el + ∑_αβΔ T^a_αβD^a_αβ + T^a_core

  In this section, we discuss the evaluation of various terms involved in computing Δ E^a[{D^a_αβ}] for completeness which is inline with prior works<cit.>. 


  
Kinetic Energy correction: 


    


    Δ T^a_αβ =    (1/2∑_l_α l_β∑_m_α m_β(∫dϕ_α(r)/drdϕ_β(r)/drr^2drδ_l_α l_βδ_m_α m_β + ∫_Ω_aϕ_α(r)ϕ_β(r)dr(l_α+1)(l_α)δ_l_α l_βδ_m_α m_β). 
       -. 1/2∑_l_α l_β∑_m_α m_β(∫_Ω_ad_α(r)/drd_β(r)/drr^2drδ_l_α l_βδ_m_α m_β + ∫_Ω_a_α(r)_β(r)dr(l_α+1)(l_α)δ_l_α l_βδ_m_α m_β))



  
Electrostatics energy correction


    


    Δ E^a_el = 
    E^a_el[ρ^a()] - E^a_el[^a()]  - E^a_self[^a] = Δ𝒞^a + ∑_αβΔ𝒞^a_αβD^a_αβ + ∑_αβα'β'D^a*_αβΔ𝒞^a_αβα'β'D^a_α'β'


    where,  Δ𝒞^a =    1/2( ∫_Ω_a∫_Ω_an_c^a()n_c^a()/|-|dd - ∫_Ω_a∫_Ω_añ_c^a()ñ_c^a()/|-|dd - (κ^a)^2∫_Ω_a∫_Ω_a_00^a()_00^a()/|-|dd)  
        - κ^a ∫_Ω_a∫_Ω_añ_c^a()_00^a()/|-|dd -√(4π)𝒵^a∫_Ω_an^a_c(r)/rdr


    Δ𝒞^a_αβ =   ∫_Ω_a∫_Ω_aϕ^a_i()ϕ^a_j()n^a_c()/|-|dd - ∫_Ω_a∫_Ω_a^a_i()^a_j()ñ^a_c()/|-|dd  -𝒵^a∫_Ω_aϕ^a_i()ϕ^a_j()/||d
        -κ^a ∫_Ω_a∫_Ω_a^a_i()^a_j()g_00^a()/|-|dd - Δ^a_00αβ(κ^a ∫_Ω_a∫_Ω_a_00^a()_00^a()/|-|dd + ∫_Ω_a∫_Ω_añ^a_c()_00^a()/|-|dd)


    and  Δ𝒞^a_αβα'β' =   1/2[ ∫_Ω_a∫_Ω_aϕ^a_α()ϕ^a_β()ϕ^a_α'()ϕ^a_β'()/|-|dd - ∫_Ω_a∫_Ω_a^a_α()^a_β()^a_α'()^a_β'()/|-|dd]  
        -∑_L[ 1/2Δ^a_lmα'β'∫_Ω_a∫_Ω_a^a_α()^a_β()_lm^a()/|-|dd + 1/2Δ^a_lmαβ∫_Ω_a∫_Ω_a^a_α'()^a_β'()_lm^a()/|-|dd]
       -∑_lm1/2Δ^a_lmαβ∫_Ω_a∫_Ω_a_lm^a()_lm^a()/|-|ddΔ^a_lmα'β'

where, r = |-^a| and κ^a,Δ^a_lmαβ are defined in Eqn.(<ref>). We note that 1/|-| in the above equations can be expanded in terms of real spherical harmonics as

    1/|-| =
    ∑_ℓ∑_m=-ℓ^ℓ4π/2ℓ+1min(r,r')^ℓ/max(r,r')^ℓ+1
    S_ℓ,m(θ)S_ℓ,m(θ')

where r = |-^a| and θ∈ℝ^2 represents the azhimuthal and polar coordinates of  in spherical coordinates, with r' = |-^a| and θ' represents the azhimuthal and polar coordinates of  in spherical coordinates. Using the above Eqn.(<ref>), the integrals involved in Eqns. (<ref>), (<ref>), (<ref>) can be
evaluated in spherical coordinate system as discussed in detail in the Appendix of <cit.>.




  
Exchange Correlation Energy correction:


    

The evaluation of atom centred exchange-correlation is evaluated on a radial grid using the Lebedev quadrature rule<cit.> for the angular integration and Simpson's rule for the radial integration.

    Δ E^a_xc = ∫_Ω_a(ϵ_xc[n^a,∇ n^a]-ϵ_xc[^a,∇^a])d

where

    n^a() = n^a_c(r)S_00(θ) + ∑_αβϕ^a_α(r)ϕ^a_β(r)S_l_α m_α(θ)S_l_β m_β(θ)D^a_αβ


    ∇ n^a() = [          ∂ n^a_c(r)/∂ rS_00(θ) + ∑_αβ(∂ϕ^a_α(r)/∂ rϕ^a_β(r) + ∂ϕ^a_β(r)/∂ rϕ^a_α(r) )S_l_α m_α(θ)S_l_β m_β(θ)D^a_αβ;        ∑_αβ1/r(ϕ^a_α(r)∂ S_l_α m_α(θ)/∂ϑϕ^a_β(r)S_l_β m_β(θ) + ϕ^a_β(r)∂ S_l_β m_β(θ)/∂ϑϕ^a_α(r)S_l_α m_β(θ))D^a_αβ; ∑_αβ1/r sin(ϑ)(ϕ^a_α(r)∂ S_l_α m_α(θ)/∂ςϕ^a_β(r)S_l_β m_β(θ) + ϕ^a_β(r)∂ S_l_β m_β(θ)/∂ςϕ^a_α(r)S_l_α m_β(θ))D^a_αβ ]

where θ = [ϑ,ς] ∈ℝ^2 represents the polar and azhimuthal coordinates in spherical coordinate system. Further Eqn.(<ref>) can be simplified  using the below expressions

    ∂ S_lm(θ)/∂ϑ = - √(2l+1)/√(2l+3)√((l+1)^2 - m^2)S_l+1,m(θ)/sin(ϑ)


    ∂ S_lm(θ)/∂ς = -|m|S_lm̅(θ),  where,  m̅ = -m

The expression for ^a() can be derived following the above steps in a similar fashion. Finally, the exchange-correlation energy correction term is evaluated as

    Δ E^a_xc = ∑_qw_q∫(ϵ_xc[n^a(r,θ_q),∇ n^a(r,θ_q)]-ϵ_xc[^a(r,θ_q),∇^a(r,θ_q)]) r^2dr

Further, the atom-centered exchange correlation potential correction (Δ V^a, αβ_xc) is expressed as

    Δ V^a,αβ_xc = ∑_qw_q(∫δ E^a_xc[n^a,∇ n^a]/δ D^a_αβϕ^a_α(r)ϕ^a_β(r) r^2dr - ∫δ E^a_xc[^a,∇^a]/δ D^a_αβ^a_α(r)^a_β(r) r^2dr)S_l_α m_α(θ_q)S_l_β m_β(θ_q)

where, q refers to the Lebedev quadrature points used to sample θ and w_q are the corresponding quadrature weights while the above radial integrals are evaluated until augmentation radius (r^a_c) using the adaptive Simpsons integration rule. We note that the Eqns. (<ref>), (<ref>)  are efficiently evaluated by recasting the integrands into a dense matrix-matrix multiplication in our  implementation.




  
PAW Valence Energy for non-periodic systems:
Recall from Eqn.(<ref>), the FE discretized all-electron PAW energy for a non-periodic system corresponding to a simulation domain Ω can be computed using the following expression

    E^h =    2∑_i=1^Nf(ε^h_i,μ)ε^h_i - ∑_a∑_αβΔ h^a_αβD^a,h_αβ 
            - ∫_Ω[ ^h()V^h_eff() + ^h_eff,GGA() ·∇^h() ] d
       + 1/2∫_Ω^h()φ^h,p_el()d + E_xc[^h()]+ ∑_aΔ E^a[{^a,h}]  where  ^h() = ^h() + ^h()

    We note that the above expression for E^h includes the energy contribution of the frozen core, and in order to exclude the frozen core contribution in E^h and compare the ground-state energies one-on-one with  <cit.> in our benchmarking studies, we introduce the PAW valence energy E^h_v for non-periodic systems as follows
    
    E^h_v = E^h - ∑_aE_xc[n^a_c() ] -∑_aC^a_corr,  where,  E_xc[n^a_c() ] = ∫_Ω_aϵ_xc[n^a_c,∇ n^a_c]d
       and 𝒞^a_corr = ∫_Ω∫_Ω^a_c()^a_c()/|-|dd - (E^a_el[n^a_c()+b^a()] - E^a_el[^a_c()]), ^a_c() = ^a_c() + κ^a_0^a()



§ VARIATIONAL FORMULATION OF THE PAW ENERGY FUNCTIONAL AND FINITE-ELEMENT DISCRETIZATION

In this section, we present the variational formulation of the PAW energy functional and subsequently derive the finite-element discretized equations to be solved in the PAW method. For brevity, the expressions derived below assume the case of a non-periodic isolated system or a periodic system involving a large supercell with Γ-point sampling. To this end, we make the assumption that the wavefunctions are real; however, the derivation is trivially extendable for unit-cell calculations with multiple -point sampling of the Brillouin zone involving complex wavefunctions. We begin with the all-electron Kohn-Sham energy functional in Eqn.(<ref>) to be minimized subject to 𝒮-orthonormality constraints. Consequently, we introduce the Lagrange multiplier matrix (Λ) for enforcing these constraints and the functional to be minimized to determine the ground-state energy for given positions of nuclei {^a} in the PAW method can be written as follows:

    F[{_i },φ,Λ] = E[{_i },φ] - ∑_ijλ_ij(∫_i()𝒮_j()d-δ_ij)

Now, taking the variational derivative of the above functional with respect to _n, we have

    2 f_n∫1/2∇_n()·∇δ_n()d + 2 f_n∫(φ()+V̅)_nδ_n d + ∑_a∑_αβΔ^a_lmαβ∫δ D^a_αβ/δ_n()δ_n()d∫^a_lm()φ()d
        + ∫δ E_xc/δ()(∫δ()/δ_n()δ_n()d)d + ∫δ E_xc/δ∇()·[∫(δ∇()/δ_n()δ_n() + δ∇()/δ∇_n()δ∇_n() )d] d
        +∑_a∑_αβδΔ E^a/δ D^a_αβ∫δ D^a_αβ/δ_n()δ_n()d = 2∑_j(λ_nj+ λ_jn)/2∫δ_n()𝒮_j()d

where,

    δΔ E^a[{D^a_αβ}]/δ D^a_αβ = (Δ T^a_αβ + Δ E^a_el,αβ + 2∑_α'β'Δ E^a_el,αβα'β'D^a_α'β' + Δ V^a, αβ_xc - ∫_Ω_av̅^̅a̅(r)S_00(θ)^a_α()^a_β()d) =  Δ h^a_αβ

with

    ∫δ D^a_αβ/δ_n()δ_n()d = 2 f_n∫_α^a() δ_n()d∫_β^a() _n()d

Finally, we have

    f_n∫1/2(∇δ_n()·∇_n() +(δ E_xc/δ() + φ())δ_n()_n())d + f_n∫δ E_xc/δ∇() ·( _n()δ_n() + δ_n()_n() ) d
       + f_n∑_a∑_αβ∫^a_α()δ_n()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫_lm^a()φ()d)∫^a_β()_n()d= ∑_jλ^S_nj∫δ_n()𝒮_j()d

where the entries λ^S_nj = λ_nj+λ_jn/2 are the entries of a symmetric matrix Λ^S = 1/2(Λ + Λ^T).
Similarly, taking the variational derivative of Eqn. (<ref>) with respect to φ(), we get

    1/4π∫∇φ()·∇δφ()d = ∫()δφ()d
 
where the simulation domain for Eqns.(<ref>) and (<ref>) is represented by Ω. The variatonal quantities are such that _n() ∈𝕏 and φ()∈γ with 𝕏=γ = H^1_0(Ω) in the case of non-periodic problems and 𝕏=γ=H^1_per(Ω) in the case of periodic problems.
 Now, we let 𝕍^h ∈𝕏, 𝕍^h_el∈γ denote the finite-dimensional subspaces of dimensions M and M_el respectively, and in the discrete setting, we seek the approximate solution _n^h()∈ 𝕍^h, φ^h()∈𝕍^h_el  such that 

    f_n∫_Ω1/2(∇δ_n()·∇^h_n() +(δ E_xc/δ() + φ^h())δ_n()^h_n())d + f_n∫δ E_xc/δ∇() ·( ^h_n()δ_n() + δ_n()^h_n() ) d
       + f_n∑_a∑_αβ∫^a_α()δ_n()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫^a_lm()φ^h()d)∫^a_β()^h_n()d= ∑_jλ^S_nj∫δ_n()𝒮^h_j()d
       ∀ δ_n() ∈𝕍^h

and 

    1/4π∫∇φ^h()·∇δφ()d = ∫()δφ()d   ∀ δφ∈𝕍^h_el

Further, let the finite dimensional subspace 𝕍^h be spanned by the finite-element basis { N^h,p_I() }  for   1 ≤ I ≤  M and the subspace 𝕍^h_el be spanned by the FE basis {N^h,p_el_I()}, for  1 ≤ I ≤   M_el. This allows us to express the approximate solution ^h_n() uniquely as ^h_n() = ∑_JN^h,p_J()u_n^J and, subsequently φ^h() as φ^h() = ∑_JN_J^h,p_el()φ^J.
 Moreover, since Eqn.(<ref>) is valid for all δ_n() ∈𝕍^h, choosing δ_n() = N^h,p_I() successively for every I, we now obtain the FE discretized equations to be solved in the form of the following matrix equations 

    f_n∑_J∫1/2∇ N^h,p_I()·∇ N^h,p_J()u_n^J d + f_n∑_J∫V^h_eff()N_I^h,p()N_J^h,p()u_n^J d
     +     f_n∑_J∫V^h_GGA·(N^h,p_I()∇ N_J^h,p() + ∇ N_I^h,p()N^h,p_J())u_n^J d
       + f_n∑_J∑_a,αβ∫N^h,p_I()^a_β()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫^a_lm()φ^h()d)∫_α^a()N_J^h,p()u_n^Jd
       = ∑_jλ^S_nj∑_J(∫N^h,p_I()N^h,p_J()u_j^Jd + ∑_a,αβ∫N^h,p_I()^a_β()d√(4π)Δ ^a_00αβ∫_α^a()N_J^h,p()u_j^Jd)

Rewriting the above expression in the matrix-form we have

    𝖴 = 𝖴Λ^S

where, the matrix 𝖴 = [𝗎_1... 𝗎_n ...𝗎_N] with the entries of the vector 𝗎_n given by u^J_n for 1 ≤ J ≤  M which are the linear combination coefficients of ^h_n() when expressed in FE basis. Now, we note that the matrix Λ^S in the Eqn. (<ref>) is a symmetric matrix which is orthogonally diagonalizable to be of the form Λ^S = ^T
 and let ϵ^h_nf_n be the entries of the diagonal matrix . Furthermore, introducing the unitary transformation of the discretized wavefunction matrix 𝖴 using the orthogonal matrix  diagonalizing Λ^S, we have = 𝖴 which can then be used to transform the matrix equation Eqn. (<ref>) to obtain the following generalised eigenvalue problem in the PAW method

    _n = ϵ^h_n_n

where the entries of the FE-discretized Hamiltonian () and the PAW overlap matrix are given by Eqns.(<ref>) to (<ref>) in Section (<ref>). Additionally, since Eqn.(<ref>) is satisfied ∀ δφ∈𝕍^h_el, we choose δφ() = N_I^h,p_el() successively for all I to finally obtain the following FE discretized Poisson's equation

    1/4π∑_J∫∇ N_I^h,p_el()· N_J^h,p_el()φ^Jd = ∫^h()N_I^h,p_el()d
 




§ CHEBYSHEV FILTERED SUBSPACE ITERATION

As discussed in Section (<ref>), we use a self-consistent field iteration (SCF) procedure to solve the nonlinear PAW generalised eigenvalue problem. This involves the solution of a linearized generalised eigenvalue problem at each SCF iteration for which we employ the Chebyshev filtered subspace iteration(ChFSI) approach. As outlined in Algo.(<ref>), ChFSI requires upper and lower bounds of the eigenspectrum corresponding to the FE discretized eigenvalue problem = ϵ^h_n for which we use the generalized Lanczos procedure discussed in Algo.(<ref>).
     

   









 Paper Title: Comparative analysis of structural, elastic, electronic, phonon, thermal and optical properties of two $\text{Na}_6\text{Ge}_2\text{Se}_6$ phases from first principles calculations.






Department of Physics, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

Department of Chemistry, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

Department of Physics, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

§ ABSTRACT 

The demand for new alkali metal chalcogenide materials is continuously increasing due to their potential applications across various technological fields. Recently, a new compound, Na6Ge2Se6, was computationally predicted, representing a new phase distinct from the experimentally observed Na6Ge2Se6 reported in 1985. Notably, this newly predicted phase displays a lower total energy compared to the previously known experimental phase, as determined by first-principles calculations. In this study, we undertake a thorough comparative analysis of the structural, elastic, electronic, phonon, thermal, and optical properties of these two Na6Ge2Se6 phases. Our results show that both phases meet mechanical and dynamical stability criteria. The electronic band structure analysis confirms the semiconducting nature of both materials, with a 2.97 eV indirect band gap for the predicted phase and a 2.93 eV direct band gap for the observed phase. Optically, both phases exhibit strong absorption in the ultraviolet region. Thermal properties analysis reveals that the predicted phase is more thermodynamically stable below 907 K, while the observed phase shows greater thermodynamic stability above this temperature.




Comparative analysis of structural, elastic, electronic, phonon, thermal and optical properties of two Na6Ge2Se6 phases from first principles calculations
    Aleksandr Chernatynskiy
    August 5, 2024
==========================================================================================================================================================








§ INTRODUCTION


Ternary alkali metal-based chalcogenides are characterized by the general formula ABZ where A signifies an alkali metal (Li, Na, K), B includes main group elements (e.g., Ge, Bi) or transition metals (e.g., Fe, Cu), while Z represents chalcogen elements (e.g., S, Se). These compounds have gained considerable interest recently for their potential applications in energy conversion/storage, optoelectronics, and thermoelectrics. These areas are vital for supporting the advancement of current technologies and industries, especially in addressing the global energy challenge as society aims to transition from fossil fuels to renewable, efficient, and eco-friendly energy solutions. In response to these demands, numerous ABZ compounds have been synthesized and explored.

In the quest for improved energy storage solutions, the focus on ion-battery anodes is crucial due to their significant impact on battery performance. ABZ materials like LiLnSe2 and NaFeS2 have shown high storage capacities and efficient energy cycling in recent studies <cit.>, with NaFeS2 showcasing excellent cycling stability as a Li-ion battery anode, suggesting potential for enhanced battery lifespan and reliability. Optoelectronically, ABZ compounds such as NaSbS2 offer strong absorption in the visible spectrum, and their affordability, abundance, and non-toxicity make them attractive for solar energy applications<cit.>. Furthermore, in thermoelectric applications, ABZ materials like CsAg5Te3 exhibit promising mid-temperature performance with high ZT values<cit.>, thanks to their intrinsically low thermal conductivity<cit.>. Ternary chalcogenides have also played a crucial role in enabling the synthesis of more complex multinary chalcogenides in a rational manner<cit.>. Many of these compounds have demonstrated significant technological potential. For instance, outstanding electric thermal properties have been observed in materials like Na_0.95Pb_19SbTe_22, achieving a ZT value greater than 1 across one of the broadest temperature ranges (475 to 650 K) reported for any single material<cit.>. Additionally, a study on Na_1-xK_xAsQ_2 (Q=S,Se) highlighted its potential applications in signal processing and data transmission, attributed to its nonlinear optical properties<cit.>.

Although many ABZ compounds have been synthesised, the existing collection of these ternary ABZ compounds still lacks many possible combinations. The process of discovering new materials through purely experimental means is often prolonged, as the growth and characterization of their properties can be time-consuming. With the increase of  computational power, simulations are increasingly employed in material discovery, for both screening of the properties of already known materials as well as prediction of the new compounds. Recently, a new structure of the chalcogenide, Na6Ge2Se6 was computationally predicted using simulated annealing and first-principles methods<cit.>. This compound represents a new phase distinct from the experimentally observed compound reported in 1985 with the same composition (Na6Ge2Se6)<cit.>. The latter has been utilized recently in synthesizing a potential nonlinear optical material, Na8Mn2(Ge2Se6)2<cit.>. Notably, the newly predicted Na6Ge2Se6 phase shows a lower total energy compared to the observed phase from first-principles calculations, suggesting it as a potential meta-stable state of Na6Ge2Se6 that has not been reported before. Furthermore, the observed Na6Ge2Se6 phase has received little attention, with no comprehensive analysis conducted to the best of our knowledge. This gap in research has motivated us to perform a comparative analysis of the structural, elastic, electronic, phonon, thermal, and optical properties of these two compounds, which could serve as a valuable reference for future studies.




















§ COMPUTATIONAL METHODOLOGY


In this study, we examined the ground state properties of both predicted and observed Na6Ge2Se6 phases using density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)<cit.>. This involved solving the Kohn-Sham equations<cit.> to determine the ground state energies of a crystalline system. Given the close proximity of the ground state energies between the predicted and observed phases, calculations were performed using two distinct approaches for the exchange-correlation potential: the Generalized Gradient Approximation (GGA) within the Perdew-Burke-Ernzerhof framework (GGA-PBE)<cit.> and the hybrid functionals from Heyd-Scuseria-Ernzerhof (HSE06)<cit.>. In both Na6Ge2Se6 structures,  Integration over the Brillouin zone was executed via the tetrahedron method with Gaussian smearing, employing a 5×5×5 Monkhorst–Pack k-point mesh<cit.>. The energy cutoff for the plane wave basis set was set to 600 eV. The Ge 3d^104s^34p^2, Se 4s^2p^4, Na 2p^63s^1 electrons are treated as valence electrons. convergence thresholds were 1e-8 eV for the electronic self-consistent loop and 1e-7 eV for structure optimization.

Further, we explored the elastic and dynamical properties of optimized structures using first-principle calculations with the GGA-PBE functional within VASP. This analysis allowed for the calculation of elastic constants C_ij and several elastic properties, including the bulk modulus B, Young's modulus E and shear modulus G, along with optical properties such as the dielectric function ϵ(ω), absorption coefficient α(ω) and conductivity σ(ω) using the VASPKIT<cit.> tool. The Phonopy package<cit.> facilitated the calculation of phonon dispersion relations, phonon density of states, and related thermal properties. Considering the tendency of GGA-PBE to underestimate band gaps in semiconductors and insulators, the HSE06 functional<cit.> was applied to investigate electronic and optical properties. The electronic band structures and density of states for both phases were visualized using the open-source Python library Pymatgen<cit.>.





§ CRYSTAL STRUCTURE

We analyzed the Na6Ge2Se6 structures using two different exchange-correlation energy functionals: GGA-PBE and the HSE06 hybrid functional. The calculated energies per atom are presented in Table <ref>. Both PBE and HSE06 functionals indicated lower energies for the predicted Na6Ge2Se6 phase by 16 meV/atom. To confirm the convergence of our DFT calculations, we conducted two additional structure relaxations with the GGA-PBE functional: one with a 400 eV energy cutoff and a 3×3×3 k-point mesh, and another with a 500 eV energy cutoff and a 4×4×4 k-point mesh. For the predicted phase, both of these settings produced energy per atom within 0.006% of the most accurate result listed in the Table <ref>. For the observed phase, the difference is even smaller, within 0.002%. The energy difference between the predicted and observed phases listed above is an order of magnitude larger than 0.06%, thus numerical accuracy is sufficient to make conclusions about the energy order of two structures.

This leads us to conclude that the predicted Na6Ge2Se6 phase is energetically more favorable than the observed one at 0 K.

The predicted Na6Ge2Se6 crystallizes in the space group R3 (No.148), whereas the observed phase is found in the space group P2_1/c (No.14). Both structures exhibit identical ethane-like selenide [Ge2Se6] units, featuring a Ge-Ge bond length of 2.49 Å for the predicted phase and 2.47 Å for the observed phase, and an average Ge-Se bond length of 2.39 Å  as detailed in Table <ref>. These bond lengths are slightly greater than the originally observed Na6Ge2Se6 data<cit.>, which recorded a Ge-Ge bond length of 2.43 Å  and a Ge-Se bond length of 2.33 Å. The [Ge2Se6] ethane-like dimer configuration bears similarity to the [P2Se6] in Mg2P2Se6<cit.>, and [Si2Se6] in Na4MgSi2Se6<cit.>. For both Na6Ge2Se6 phases, the formed [Ge2Se6] dimers are isolated from each other, similar to the cases in K6Ge2Se6<cit.> and Cs6Ge2Se6<cit.>. The predicted Na6Ge2Se6 phase exhibits cell parameters of a=11.349 Å, b=11.349 Å, c=10.900 Å, α=90, β=90 and γ=120, while the observed Na6Ge2Se6 has a=8.445 Å, b=12.038 Å, c=8.316 Å, α=90, β=118.877 and γ=90 with a smaller unit cell volume. The predicted Na6Ge2Se6 features only one crystallographically unique Na atom, one Ge atom, and one Se atom, contrasting with the observed Na6Ge2Se6, which has three unique Na atoms, one Ge atom, and three Se atoms. In the predicted Na6Ge2Se6, each Na atom coordinates with five Se atoms to form a [NaSe5] pyramid. This pyramid shares corners with its neighboring [NaSe5] pyramids, whereas in the observed phase, each Na atom bonds with six Se atoms to form a [NaSe6] slightly distorted octahedron. These octahedra are connected by face-sharing with their neighboring [NaSe6] octahedron, as illustrated in Fig. <ref>(c) and Fig. <ref>(d). The electron localization function (ELF) provides a visual representation of valence electron distribution<cit.>, highlighting features that correspond to chemical bonding characteristics. Fig. <ref> shows that areas near the center of Se atoms and between Ge/Ge atoms exhibit ELF values close to 1.







§ ELASTIC PROPERTIES




Analyzing the elastic properties offers insights into the mechanical behavior of compounds, including their stability, ductility, and brittleness. Such information is especially valuable in industrial and device manufacturing sectors, enhancing our understanding of the forces acting in solids. To calculate elastic constants, we employed the energy-strain method<cit.>, as facilitated by VASPKIT<cit.> based on DFT calculations with GGA-PBE functional.

For the predicted Na6Ge2Se6 phase, which crystallizes in the trigonal crystal system, There are five independent
elastic constants: C_11, C_33, C_44, C_12, C_13 with the added relation

    C_66 = (C_11 - C_12)/2

This contrasts with its observed counterpart, which crystallizes in monoclinic crystal system with lower symmetry that has thirteen independent elastic constants: C_11, C_12, C_13, C_15, C_22, C_23, C_25, C_33, C_35 , C_44, C_46 ,C_55 and C_66. The configurations of these constants constitute the elastic matrices for each respective phase as follows:

    C_predicted =
    [ 32.06 10.86 13.31     ·     ·     ·;     · 32.06 13.31     ·     ·     ·;     ·     · 21.06     ·     ·     ·;     ·     ·     · 12.09     ·     ·;     ·     ·     ·     · 12.09     ·;     ·     ·     ·     ·     · 10.60 ]


    C_observed =
    [ 28.28 10.82 15.82     ·  2.45     ·;     · 32.08 15.21     ·  1.55     ·;     ·     · 40.55     ·  4.76     ·;     ·     ·     · 10.43     ·  2.81;     ·     ·     ·     · 10.67     ·;     ·     ·     ·     ·     · 10.75 ]






These elastic constants are pivotal for assessing other elastic properties. A crystal must satisfy the Born-Huang criteria<cit.> to be deemed mechanically stable. For the predicted structure, the necessary and sufficient Born-Huang criteria include:

    C_11 > |C_12|, C_44 > 0, C_66 > 0


    2C_13^2 < C_33(C_11 + C_12)
























Due to the intricacy of these criteria for the monoclinic system, their detailed equations are not listed here. Crucially, the calculated elastic constants for both Na6Ge2Se6 phases fulfill their respective stability criteria, indicating mechanical stability for both the predicted and observed phases. 

Constants C_11, C_22 and C_33 quantify the response to uniaxial strain along the three principal axes. The differences between these constants measure the extent of anisotropy in linear compressibility. From matrix C_predicted and C_observed, we can see that for predicted phase, C_11=C_22>C_33 which means linear compressibility is isotropic in the x- and y-directions and greater than in the z-direction. 
For the observed phase, C_33>C_22>C_11, indicating that the observed Na6Ge2Se6 phase is characterized by anisotropic linear compressibility. While both phases have similar resistance to deformation along the x- and y-directions, the observed phase is significantly stiffer in the z-direction.

On the other hand, C_44, C_55 and C_66 assess the material’s resistance to shear deformation about these axes. The elastic constant matrix for the predicted Na6Ge2Se6 phase exhibits C_44=C_55 and slightly smaller C_66, indicating its shear modulus is weakly anisotropic. For this phase, the resistance to shear deformation is isotropic about the x- and y-directions and greater than that in the z-direction. Comparing both phases, the observed phase shows slightly higher resistance to shear deformation about the principal axes.

The off-diagonal terms indicate coupling between different deformation modes. For instance, C_12 correlates the stress applied in the x-direction to the resultant strain in the y-direction. Analysis of the matrices suggests that both phases have a similar response to deformation in the y-direction when stress is applied in the x-direction. However, the predicted phase shows greater shear strain in the z-direction induced by stress in both the x- and y-directions.





Utilizing the independent elastic constants, we can derive key mechanical properties such as the bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) through the following relations, with the calculated values presented in Table <ref>:

    B=B_V+B_R/2


    G=G_V+G_R/2


    E=9BG/3B+G


    ν=3B-2G/2(3B+G)

Here, B_V, B_R and G_V, G_R represent the Voigt<cit.> and Reuss<cit.> values of the bulk modulus and shear modulus respectively. Furthermore, we evaluated Kleinman's parameter (ζ) using VASPKIT<cit.>. 

The bulk and shear moduli offer insights into a crystal's mechanical behavior, indicating its resistance to volumetric and shape deformations, respectively. The data in the Table <ref> reveal that the observed Na6Ge2Se6 phase possesses a greater resistance to deformation. This is expected as the unit-cell volume of the predicted phase is greater than that of the observed phase. Pugh's ratio (B/G) serves as a gauge for a material's ductility or brittleness, with a threshold value of 1.75 distinguishing between the two behaviors where the material is claimed to be brittle, if B/G<1.75 and classified to be ductile if B/G>1.75<cit.>. Our findings suggest that both phases exhibit ductility. The stiffness and thermal shock resistance of these materials are inferred from Young's modulus (E)<cit.>, indicating superior stiffness in the observed phase. Conversely, a lower Young's modulus hints at enhanced thermal shock resistance<cit.> for the predicted phase. Poisson's ratio (ν) sheds light on the nature of bonding forces within a crystal and aids in evaluating its mechanical properties, including stability against shear<cit.>. Values for the predicted and observed phases are 0.268 and 0.283, respectively, falling within the central force solids' expected range of 0.25 to 0.50<cit.>. This metric also predicts the material's ductility or brittleness, classifying those with ν>0.26 as ductile and those below 0.26 as brittle<cit.>. Thus, our analysis corroborates both phases' ductility, aligning with conclusions drawn from Pugh's ratio. Lastly, the Kleinman parameter (ζ), varying between 0 and 1, indicates the dominance of bond stretching or bending in resisting external stress<cit.> where the lower value of the Kleinman parameter ζ indicates a minimal role of bond bending in resisting external stress, whereas a higher value suggests a negligible contribution of bond stretching or contracting to resist externally applied stress. For both Na6Ge2Se6 phases, a ζ>0.5 indicates a major role for bond bending in its mechanical strength.

Fig. <ref> shows the 3D and 2D representations of Young’s modulus and linear compressibility for both Na6Ge2Se6 phases. For the predicted phase, the Young’s modulus and linear compressibility in the xy-plane form a circle, as shown in Figures <ref>(e) and (f), suggesting both properties are isotropic within the xy-plane. However, both properties show significant deviation from a circular form within the yz- and xz-planes, indicating that Young’s modulus and linear compressibility are anisotropic overall for the predicted Na6Ge2Se6. For the observed phase, Figures <ref>(g) and (h) show that both properties are anisotropic, with the 2D representation of Young’s modulus in the xy- and yz-planes and linear compressibility in the xy-plane more closely resembling a circle. This suggests relative isotropy for both properties within the corresponding planes.












§ ELECTRONIC PROPERTIES

We present the electronic band structure along with the partial and total density of states (DOS) in Fig. <ref>. These calculations were performed using optimized lattice parameters along high-symmetry directions in the first Brillouin zone. The partial density of states (PDOS) on orbitals and elements are depicted in Fig. <ref>. The band structures and DOS for both Na6Ge2Se6 phases were determined using the HSE06 hybrid exchange-correlation functional<cit.> to ensure enhanced accuracy<cit.>. The energy reference point is set at the top of the valence band. The band structure analysis reveals that the predicted Na6Ge2Se6 phase possesses an indirect band gap, with the conduction band minimum (CBM) located at the X-point and the valence band maximum (VBM) at the Z-point, resulting in a band gap of 2.972 eV. In contrast, the observed Na6Ge2Se6 phase exhibits a direct band gap of 2.932 eV, with both the CBM and VBM situated at the Y_2-point. 

From Fig. <ref>, we observe that both phases share similar orbital and element contributions to the total DOS. Regions close to the Fermi level are predominantly influenced by contributions from Se-p and Ge-s orbitals, while Na’s involvement near the Fermi level is minimal. This observation suggests that the band gaps in both Na6Ge2Se6 phases are largely attributable to the [Ge2Se6] dimers. Within the valence band, ranging from -4 to 0 eV, the primary contributions come from Se-p and Ge-p orbitals in both phases. This trend extends to the lower regions of the conduction band, from 4 to 8 eV, where Na-p displays a slight contribution to the total DOS in this region.







§ PHONON AND THERMAL PROPERTIES


Investigating phonon properties is fundamental for understanding crystalline materials, as it provides insights into structural stability, phase transitions, and how vibrations influence their thermal behavior. In our study of the phonon dispersion and phonon band structures for both Na6Ge2Se6 phases, we employed the Phonopy package<cit.> to create a series of 2×2×2 supercell structures with various displacements. We then conducted force calculations using VASP, employing the GGA-PBE exchange-correlation functional. The phonon frequencies and eigenvectors were determined from the dynamical matrices, which were calculated based on the force constants derived using Phonopy.

The phonon dispersion relations along high-symmetry paths for both phases are depicted in Fig. <ref>. The unit cell of the predicted phase comprises 14 atoms, resulting in 42 phonon branches, while the observed phase has 28 atoms, resulting in 82 branches, which include three acoustic and the rest optical branches. The frequency spectrum of these modes spans from 0 to 10 THz, without any discernible gap between the acoustic and optical modes for either phase.
In the lower frequency domain, below 7 THz, Na and Se atoms predominantly contribute to the optical branches. In the higher frequency range, the contributions mainly stem from Ge and Se. This distribution underscores the strong bonding present within the [Ge2Se6] dimers, reflecting their structural integrity across both phases of Na6Ge2Se6. The lack of negative frequency branches within the dispersion plots affirms the dynamic stability of these phases at zero pressure.





The Debye temperature (θ_D) represents the temperature corresponding to the highest energy vibrational mode in a solid, derived from the equation<cit.>:

    θ_D=h/k_B[3n/4πV_0]^1/3v_m

Here, h is the Planck’s constant, k_B is the Boltzmann’s constant, n is the number of atoms in the unit cell, V_0 is the unit cell's equilibrium volume, and v_m is the material's average speed of sound. The average speed of sound v_m can be determined from the material's mass density (ρ), along with its bulk (B) and shear modulus (G), through the equations for longitudinal (v_l) and transverse (v_t) sound speeds:

    v_m=[1/3(3/v_t^3 +1/v_l^3)]^1/3


    v_t=√(G/ρ)


    v_l=√(3B+4G/3ρ)

The calculated values for v_t, v_l, v_m and θ_D are provided in Table <ref>. These values indicate that the predicted Na6Ge2Se6 phase exhibits a slower longitudinal speed of sound but a higher transverse one than observed phase, although the values differ only slightly. The Debye temperature of the predicted phase is approximately 0.89% lower than that of its counterpart.

Phonon calculations offer insights into additional thermal properties such as entropy (S), constant-volume heat capacity (C_v), and Helmholtz free energy (F), which are depicted in Fig. <ref>. The heat capacity curves, as one can see from Fig. <ref>(b), are almost indistinguishable as one expects from similar speeds of sound and Debye temperatures. They show standard Debye theory behavior with the increasing temperature: quantum T^3 at low temperature, saturating to the constant classical Dulong-Petit limit<cit.> above Debye temperature.

The calculation of the Helmholtz free energy (F) within the harmonic approximation<cit.> facilitates an evaluation of the thermal contributions to the relative stability of two phases, as described by the following equation:

    F=   E_total + 1/2∑_qνħω(qν) 
       +k_BT∑_qνln[1-exp(-ħω(qν)/k_BT]

Here, E_total denotes the total energy of the crystal, available in Table <ref>. The summed terms represent the Helmholtz free energy attributable to phonons<cit.>, with the initial sum reflecting the zero-point energy (ZPE) that is independent of temperature. The predicted phase's ZPE is approximately 1 meV/atom higher than that of the observed phase. The subsequent sum accounts for the temperature-dependent term referring to the thermally induced occupation of phonon modes. The Helmholtz free energies, plotted in Fig. <ref>(d) for temperatures ranging from 0 to 1000 K, with a closer look at the range of 905 to 912 K in Fig. <ref>(c), illustrate that the predicted phase possesses lower free energy from 0 to approximately 907 K. Since its rate of decrease with temperature is more gradual than that of the observed phase, the free energy of the observed phase becomes lower when the temperature exceeds 907 K. This suggests that the predicted phase is more thermodynamically stable than the observed phase in the temperature range from 0 to 907 K, indicating that this predicted phase is very likely to be experimentally observable. 













§ OPTICAL PROPERTIES

A material's optical behavior is characterized by several energy/frequency-dependent parameters, such as the dielectric function ϵ(ω), absorption coefficient α(ω), conductivity σ(ω), energy-loss function L(ω), reflectivity R(ω) and refractive index n(ω). The outcomes of these calculated properties are depicted in Fig. <ref>, based on formulas integrated within VASPKIT<cit.>.

    ϵ(ω)=ϵ_1(ω)+iϵ_2(ω)


    α(ω)=√(2)ω/c[√(ϵ_1^2+ϵ_2^2)-ϵ_1]^1/2


    L(ω)=ϵ_2/ϵ_1^2+ϵ_2^2


    n(ω)=[√(ϵ_1^2+ϵ_2^2)+ϵ_1/2]^1/2


    R(ω)=(n-1)^2+k^2/(n+1)^2+k^2



To examine the anisotropic characteristics, we calculated the optical parameters along the three principal axes and presented the optical spectra for incident photon energies up to 25 eV. For the predicted phase, these parameters are isotropic in the x- and y-directions. Therefore, Fig. <ref> displays the optical parameters and the corresponding derived optical properties plotted only in the y- and z-directions for the predicted phase.

The complex dielectric function is essential for describing a material's optical properties, as it serves as the foundation from which other energy-dependent optical constants are derived. We illustrate the dielectric functions for both predicted and observed Na6Ge2Se6 phases in Fig. <ref>(c) and (i), respectively. The real part of the dielectric function (ϵ_1) initially increases, reaches a peak, then sharply decreases, eventually dropping below zero. The imaginary part (ϵ_2), indicative of dielectric loss, closely aligns with the optical absorption coefficient. From the figure we can see ϵ_1 and ϵ_2 peak in different directions. For the predicted phase, ϵ_1 peaks in z-direction, and ϵ_2 peaks in y-direction. This contrasts with the observed phase, where both ϵ_1 and ϵ_2 peak in the same y-direction. The material's electronic characteristics can be inferred from the absorption coefficient (α(ω)), depicted in Fig. <ref>(a) and (g). Both phases begin to absorb at photon energies of approximately 3.1 eV and 2.9 eV respectively, indicating their optical band gaps. The absorption coefficients in three different directions follow a similar trend, with peaks at 7.4 eV for the predicted phase and 9.5 eV for the observed phase, both within the ultraviolet spectrum. The material's reflection and absorption properties, related to its loss function, are illustrated in Fig. <ref>(e)(k) and (d)(j). The edge trailing of the material's reflection and absorption spectra matches the peak in the loss function, showing a sharp increase around 12 eV for both phases. This spike indicates enhanced absorption and reflection of electromagnetic waves in the ultraviolet region, with the predicted phase exhibiting much notable anisotropy along y- and z-directions with the maximum appearing in y-direction for both phase. Optical conductivity which represents electrical conductivity over specific photon energy ranges is shown in Fig. <ref>(b) and (h). Conductivity sharply increases from zero at photon energies of 3.1 eV and 2.9 eV for the predicted and observed Na6Ge2Se6 phases respectively, mirroring the optical band gap observed in the absorption curves in Fig. <ref>(a)(g). In summary, the predicted and observed phases display similar optical properties, with the predicted phase exhibiting a higher degree of optical anisotropy in the y- and z-directions, while being isotropic in the x- and y-directions.







§ CONCLUSION

A comprehensive comparison was conducted between two Na6Ge2Se6 phases: a computationally predicted phase and an experimentally observed phase. This comparison spans their structural, elastic, electronic, phonon, thermal, and optical properties, employing first-principles methods.

The predicted Na6Ge2Se6 phase is the energetically favored structure at zero pressure and temperature based on first-principles calculations. Both phases feature the identical [Ge2Se6] dimer within their crystal structures, with significant differences observed in the bonding environment between Se and Na atoms (five Na atoms bond with one Se atom in the predicted phase, whereas six Na atoms bond with one Se atom in the observed phase).

Regarding elastic properties, both phases meet the criteria for mechanical stability and share similar characteristics. These include their resistance to volumetric and shape deformations, ductility, nature of bonding forces (central force), and bond bending as a major role in resisting external stress. The differences arise from their anisotropic characteristics. While the predicted phase is isotropic in the x- and y-directions, it exhibits greater anisotropy in the y- and z-directions compared to the observed phase.

Electronic structure analyses reveal a closely matched energy band gap between the two phases, primarily contributed by the [Ge2Se6] dimers, with only a 1.3% difference. However, the electronic band structure indicates that the predicted phase exhibits an indirect band gap, while the observed phase has a direct band gap.

The optical behavior between the phases is similarly aligned, with both exhibiting a comparable optical band gap (3.1 eV for the predicted phase and 2.9 eV for the observed phase) and enhanced absorption and reflection of electromagnetic waves in the ultraviolet region. Although the predicted phase is isotropic in the x- and y-directions, it displays a greater degree of optical anisotropy in the y- and z-directions, aligning with the anisotropic nature identified in the elastic properties.

Phonon dispersion analyses affirm the dynamic stability of both phases. Further calculations of thermal properties indicate that the predicted phase is more thermodynamically stable in the temperature range from 0 to 907 K, suggesting a high likelihood of experimental observation.



§ ACKNOWLEDGEMENTS

The authors thank NSF, USA (grant No. DMR1809128) for the funding of this project. The authors also acknowledge the usage of the HPC cluster "Foundry" at Missouri S&T funded by NSF award OAC-1919789.



§ DATA AVAIABILITY STATEMENT

The data that support the findings of this study are available upon request.









































 Paper Title: Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Metasurfaces using a Static Mode Representation.







Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy
 

Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy


Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy



Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy
 

[]carlo.forestiere@unina.it
Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy


§ ABSTRACT 
Metasurfaces, comprising large arrays of interacting subwavelength scatterers, pose significant challenges for general-purpose computational methods due to their large electric dimensions and multiscale nature. This paper introduces an efficient boundary element method specifically tailored for metasurfaces, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Our method combines the Multilevel Fast Multipole Algorithm (MLFMA) with a representation of surface current density unknowns using static modes, a set of entire domain basis functions dependent only on object shape. The reduction in the number of unknowns by the static mode expansion (SME), combined with the 𝒪(N log N) complexity of MLFMA matrix-vector products, significantly reduces CPU time and memory requirements for solving scattering problems from metasurfaces compared to classical MLFMA with RWG basis functions. The MLFMA-SME method offers substantial benefits for the analysis and optimization of metalenses and metasurfaces.


Multilevel Fast Multipole Algorithm 
 for Electromagnetic Scattering by Large Metasurfaces 
 using a Static Mode Representation
    Carlo Forestiere
    August 5, 2024
===============================================================================================================================






§ INTRODUCTION


Metasurfaces are collections of interacting subwavelength scatterers, typically operating near resonance. By carefully designing the geometry and arrangement of these scatterers, metasurfaces can engineer effective constitutive relations.  They are also versatile in shaping and manipulating electromagnetic waves for various applications, including imaging and analog computing <cit.>. Metalenses are one common class of metasurfaces, in which the unit cells are designed to impart a specific, position-dependent phase shift to transmitted light, resulting in a global focusing effect. Accurate simulation of metasurfaces and metalenses, with efficient use of time and memory, is crucial for understanding their fundamental properties and optimizing their performance. 

Metasurfaces design often relies on the “unit cell” approximation <cit.>. This approach constructs a library of the electromagnetic response of meta-atoms, such as disks, pillars, or nanofins, as a function of a design parameter, like the disk radius, pillar height, or the nanofin orientation, assuming the structure to be locally periodic.  Unfortunately, the “unit-cell” approach has several limitations: the most critical limitation is the assumption that the interaction between adjacent meta-atoms is similar to that in periodic arrays, which necessitates a slowly changing structure. This constraint limits the design space, resulting in narrow-bandwidth operation, low efficiencies and many types of non-ideality.

General-purpose electromagnetic simulation tools, based on either integral or differential formulation, face significant challenges. Differential formulations are easy to implement and result in sparse matrices but require truncation of the computational domain through the introduction of convenient boundary conditions.   Among them, the  Finite-Difference Time-Domain (FDTD) is widely used to model metasurfaces. However, the finite resolution leads to inaccuracies in wave speed, resulting in phase accumulation errors as the size of the scattering region increases <cit.>. Hence, even though the complexity of FDTD simulations nominally increases linearly with the size of the array, maintaining a fixed error requires increasing the resolution. This makes the approach prohibitive when applied using traditional computing platforms. 

Integral formulations are appealing because they define the unknowns only within the objects' volumes or on their boundaries if the objects are spatially piecewise homogeneous, and naturally satisfy radiation conditions at infinity. However, they usually involve dense matrices, which are computationally demanding to invert, even with acceleration techniques like the fast multipole method <cit.>.

A brute-force strategy for scaling full-wave simulations involves adapting general-purpose computational electromagnetic algorithms to massive parallelization for CPU-based <cit.> or GPU-based computing platforms <cit.>. However, it is more efficient to tailor existing methods to the specific characteristics of the problem of electromagnetic scattering from metasurfaces, which generally consist of repeated particle shapes arranged in aperiodic positions and arbitrary orientations, typically subwavelength.

 In the present work, we propose a fast boundary integral method specifically tailored for large arrays of subwavelength scatterers, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation <cit.>. We combine the Multilevel Fast Multiple Method with the boundary integral equation method employing an expansion of the unknowns of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation in terms of a particular class of entire-domain basis functions, defined on the individual meta-atom, and denoted as static modes. As demonstrated in Ref. <cit.>, static modes drastically reduce the total number of unknowns in scattering problems involving arrays of penetrable particles compared to discretization based on sub-domain basis functions, such as RWG.


This work is organized as follows: In Section <ref>, we recall the PMCHWT formulation and its discretization. In Section <ref>, we briefly review the static mode expansion <cit.>, and in Section <ref> we combine the the fast multipole method with this expansion. In Section <ref>, we first validate the proposed method by comparing, for arrays of spheres, the results with those obtained using the multiparticle Mie theory <cit.>. We then assess the computational burden in terms of CPU time and memory requirements by comparing the static mode expansion to a more traditional approach using RWG basis functions. Finally, we present results for three large particle arrays: a golden angle spiral array, a two-layer Moiré superlattice, and a 100 λ× 100 λ canonical metalens.



§ PMCHWT SURFACE INTEGRAL EQUATION

 
	Let us consider a system composed of  particles illuminated by a time-harmonic electromagnetic
	field Re{E⃗_inc e^jω t}. Each particle is a linear, homogeneous, isotropic material that occupies the three-dimensional domain Ω_, with = 1 ,...,. The material has permittivity
	ε^+(ω), permeability μ^+(ω) and it is surrounded by a background
	medium with permittivity ε^-(ω) and permeability
	μ^-(ω). The equivalent electric j⃗_e^
	and magnetic j⃗_m^ surface current densities, defined on ∂Ω_, are
	solutions of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) <cit.> surface integral problem
 
 
 
 
    [ 𝒵^11    ⋯  𝒵^1;    ⋮    ⋱    ⋮; 𝒵^ 1    ⋯   𝒵^ ][ j⃗^1;    ⋮;  j⃗^ ] = [ v⃗^1;    ⋮;  v⃗^ ] ,

	
with

    𝒵^ = [   ζ^- 𝒯_-^ + ζ^+ 𝒯_+^δ_           𝒦_-^ + 𝒦_+^δ_;       -(𝒦_-^ + 𝒦_+^δ_^) 1/ζ^-𝒯_-^ + 1/ζ^+𝒯_+^δ_ ],

where δ_ is the Kronecker delta, and

    j⃗^ = [ j⃗_e^; j⃗_m^ ], v⃗^ = [ e⃗_0^; h⃗_0^ ],

	
    e⃗_0^ = - n_×n_×E⃗_inc |_∂Ω_, h⃗_0^ = - n_×n_×H⃗_inc |_∂Ω_.


The operators 𝒯_±^ and 𝒦_±^ are, respectively, the EFIE and MFIE integral operators, defined as

    𝒯_±^{u⃗} (r⃗) =
       j k^±n_×n_×∫_∂Ω_ g^± (r⃗-r⃗') u⃗(r⃗') ' + 
       1/jk^±n_×n_×∫_∂Ω_∇ g^± (r⃗ - r⃗') ∇'_S ·u⃗(r⃗') ',


    𝒦_±^{u⃗}(r⃗) = n_×n_×∫_∂Ω_u⃗(r⃗') ×∇' g^±(r⃗ - r⃗') ',

where ∇_S' · denotes the surface divergence and g^± is the homogeneous space Green's function of the region Ω_±, i.e.

    g^± (r⃗ - r⃗') = e^-jk^±r⃗ - r⃗'/4πr⃗ - r⃗',
 
with k^± = ω√(μ^±ε^±) and ζ^± = √(μ^± /ε^±).

Apart from a multiplicative factor, the operator 𝒯_±^{u⃗} (r⃗) takes a surface electric current density u⃗ defined on the ν-th particle and outputs the tangential component of the electric field it generates on the μ-th particle. Similarly, the operator 𝒦_±^{u⃗} (r⃗) takes a surface magnetic current density u⃗ defined on the ν-th particle and outputs the tangential component of the electric field it generates on the μ-th particle.



 §.§ Galerkin equations

Aiming at the solution of the PMCWHT eq. (<ref>), we represent the equivalent electric and magnetic surface currents on the particle Ω_
as a combination of a set of basis function ℬ_φ^ = {φ^_n(r⃗)}_n = 1^N, so:

    j⃗_e^(r⃗) = ∑_n = 1^Nα^,e_n φ^_n (r⃗) 
    j⃗_m^(r⃗) = ∑_n = 1^Nα^,m_n φ_n^ (r⃗).

We find the finite-dimensional approximation of the PMCHWT by substituting eq. (<ref>) in eq. (<ref>) and by projecting along the same set of basis functions, according to a Galerkin projection scheme:

    [ Z^11    ⋯  Z^1;    ⋮    ⋱    ⋮; Z^ 1    ⋯   Z^ ][ J⃗^1;    ⋮;  J⃗^ ] = [ V⃗^1;    ⋮;  V⃗^ ] ,

where

    Z^ = [     ζ^- T_-^ + ζ^+ T_+^δ_             K_-^ + K_+^δ_;          -(K_-^ + K_+^δ_) 1/ζ^- T_-^ + 1/ζ^+ T_+^δ_ ],



    [T_±^]_mn = ⟨φ_m^ | 𝒯_±^ | φ_n^⟩,  	[K_±^]_mn = ⟨φ_m^ | 𝒦_±^ | φ_n^⟩,



    J⃗^= [ J⃗_e^; J⃗_m^ ], V⃗^ = [ E⃗_0^; H⃗_0^ ],


    J⃗^_e = [α_1^,e, … ,α_N^,e]^T, J⃗_m^ = [α_1^,m, … ,α_N^,m]^T,



    [E⃗_0^]_n = ⟨φ_n^ | e⃗_0^⟩,  [H⃗_0^]_n = ⟨φ^_n | h⃗_0^⟩,


    ⟨u⃗ | v⃗⟩ = ∫_∂Ωu⃗^*(r⃗) ·v⃗(r⃗) ,

If the particles are identical, each block Z^ is a square matrix of dimension 2N. Therefore, the finite-dimensional system (<ref>) has 2N degrees of freedom, making the resolution of problems with a large number of particles prohibitive when using sub-domain basis function as the RWG <cit.> (where N is the number of edges on the surface mesh of the particle), even when acceleration techniques like fast multipole algorithms are implemented.
In the following, we denoted as φ the sub-domain basis function, such as the Rao-Wilton-Glisson <cit.>.

Let us introduce another set of basis functions ℬ_ψ^ = {ψ^_n(r⃗)}_n=1^N. These functions are related to the basis ℬ_φ^ through the linear transformation

    ψ_i^(r⃗) = ∑_j=1^N m_ijφ^_j(r⃗),

where M = [m_ij] is the transformation matrix. To find the representations of the operators T_±^ and K_±^ in the new basis ℬ_ψ, we can transform them from the basis ℬ_φ representation using

    {A_±^}_ψ = M {A_±^}_φ M^T,

where {A_±^}_φ denotes the representation of the operators A_±^ = T_±^, K_±^ in the basis ℬ_φ.




§ STATIC SURFACE CURRENT MODES



Guided by the fact that the particles constituting the building blocks of a metasurface, i.e., the meta-atoms, are typically smaller than the wavelength, we seek a set of entire-domain basis functions defined on the surface of an individual meta-atom that can efficiently describe the electric and magnetic surface current densities in this regime. Mathematically, we aim for basis functions that diagonalize some of the operators involved in the PMCHWT formulation in the static limit. This set of entire-domain basis functions, defined on the surface of an individual meta-atom and termed static surface current density modes, has recently been introduced and proven effective in representing surface current density fields  <cit.>. These modes consist of two distinct subsets: longitudinal modes, characterized by a zero surface curl, and transverse modes, with zero surface divergence. These subsets are constructed by introducing two separate auxiliary eigenvalue problems. These two eigenvalue problems are associated with two operators, 𝒯_0^⊥ and 𝒯_0^∥, which are obtained from the static-limit of the first and second operators in Eq. (<ref>), respectively.


Let Ω be one of the meta-atom {Ω_}_ = 1^ in the array, whose boundary ∂Ω is “sufficiently regular" <cit.>; n is the normal to ∂Ω pointing outward. The longitudinal static surface current modes are nontrivial solutions of the eigenvalue problem:

    𝒯_0^{j⃗^_k }( r⃗)  =  γ_k^ j⃗^_k     on ∂Ω,

where

    𝒯_0^{w⃗}( r⃗) = n×n×∇∮_∂Ω g_0 ( r⃗  - r⃗') ∇' _S  · w( r⃗' ) ',

∇_S· denotes the surface divergence, and g_0 is the homogeneous space static Green's function

    g_0 ( r⃗ - r⃗' ) = 1/4 π 1/r⃗ - r⃗' .

Eq. (<ref>) can be numerically solved by using as expansion and test functions the star basis functions <cit.>.
Longitudinal static modes constitute a basis for the square-integrable functions defined on ∂Ω and having zero surface curl, and diagonalize the first operator in Eq. (<ref>) in the small-particle limit.


The transverse static surface current modes are nontrivial solutions of the eigenvalue problem:

    𝒯_0^⊥{j⃗^⊥_k }( r⃗)  = γ_k^⊥j⃗^⊥_k     on ∂Ω,

with 

    𝒯_0^⊥{w⃗}( r⃗) =  -n×n×∮_∂Ω g_0 ( r⃗ - r⃗' ) w⃗( r⃗' ) '.
 

This equation can be numerically solved by using as expansion and test function the loop functions <cit.>.
Trasverse static modes constitute a basis for the square integrable functions defined on ∂Ω having zero surface divergence, and diagonalize the second operator in Eq. (<ref>) in the small-particle limit.
Several characteristics make the use of static modes appealing: i) The retarded Green's function, serving as the kernel for integral operators in surface integral formulations, can be decomposed into the static Green's function, which possesses an integrable singularity, and a distinct term that is a regular function. The integral operators that include the static Green's function are diagonalized by the static modes, thus regularizing the overall problem. ii) Employing the expansion of static modes, along with suitable rescaling and rearrangement of the unknowns, renders the surface integral formulation resistant to the low-frequency breakdown issue. iii) Static modes are solely dependent on the object's shape, allowing for their universal application across diverse operating frequencies and material properties of the object. This enables a consistent, frequency-independent representation of any scattering scenario involving objects of a particular shape, surpassing other basis sets, such as characteristic modes <cit.>, which vary with frequency and material properties.

As demonstrated in Ref. <cit.>, the transverse static modes {j⃗_n^⊥}_n=1^N^⊥, associated with the first N^⊥ eigenvalues γ_n^⊥ (sorted in descending order), and the longitudinal static modes {j⃗_m^}_m = 1^N^,
associated with the first N^ eigenvalues γ_m^ (sorted in ascending order), can be used as basis function to represent the equivalent electric and magnetic surface currents on Ω_, namely ℬ_ψ^ = {ψ^_i}_i=1^N^⊥ + N^ =  {j⃗_1^⊥,...,j⃗_N^⊥^⊥,j⃗_1^,...,j⃗_N^^}. The transformation (<ref>) can be used to express the (<ref>) in the static modes basis starting from the RWG representation. For each couple (,) of particles we have

    T̂^ = [ T̂^_⊥⊥  T̂^_⊥;  T̂^_⊥   T̂^_;        ], K̂^ = [ K̂^_⊥⊥  K̂^_⊥;  K̂^_⊥   K̂^_;        ],


    T̂^_α, β = M^α T^ M^βT, K̂^_α, β = M^α K^ M^βT,


    T̂^ = MT^ M^T, K̂^ = M K^ M^T,

where  α, β = ⊥, and

    M = [ M^⊥;  M^ ].




As the static surface modes are a very effective description of the surface current density field on objects whose dimensions are small compared to the wavelength, the dimension of the operators T̂ and K̂ is typically smaller than that of T and K, the transformation (<ref>) effectively acts as a compression. Consequently, the system (<ref>) now has 2(N^⊥ + N^) degrees of freedom, which is typically drastically smaller than 2n_e derived from the RWG basis, where n_e is the number of edges in the single-particle mesh. In the following, we denoted as ψ the basis function in the static modes basis.
 


§ FAST MULTIPOLE METHOD



We apply the Multilevel Fast Multipole Algorithm (MLFMA) <cit.> to solve the PMCHWT system (<ref>) using a basis of entire-domain static modes to represent electric and magnetic surface currents. MLFMA accelerates the matrix-vector product by enabling groups of basis functions to interact collectively rather than through pairwise interactions between every possible pair of basis functions.

Initially, basis functions are grouped.

Following this, the impedance matrix Z is decomposed as the sum of two matrices: the near-field and the far-field matrices. The near-field matrix is associated with interactions among basis functions within adjacent groups, while the far-field matrix is associated with interactions between non-adjacent basis functions. 

Traditionally, grouping basis functions involves a hierarchical tree structure, such as a Quadtree or Octree <cit.>. The bounding box, i.e., the smallest box containing the entire object, is first identified. This box is then recursively subdivided until, at the finest level, the size of the box is a fraction of the wavelength, typically λ/4. 


At the finest level, standard Quadtree/Octree approaches might divide a meta-atom into different groups, making the grouping of entire-domain basis functions ambiguous. Leveraging the fact that we deal with meta-atoms whose dimensions are smaller than the wavelength, we propose grouping basis functions by considering each particle as a finest-level group, as shown in Fig. <ref>(a). At this level, all static modes ψ^_i are physically associated with the center of the particle Ω_.  For coarser levels, we hierarchically group the basis functions using a standard Quadtree structure (Fig. <ref>b), which is particularly suitable for structures where the lateral dimensions significantly exceed the vertical ones, as in the case of metasurfaces and metalenses. 

Following the fast multipole prescriptions, we decompose the impedance matrix Z as the sum of its block-diagonal part (near part) and the off-diagonal part (far part). For identical particles, the self-particle blocks Z^ are identical, i.e. Z^ = Z_s for = 1,...,. Therefore, Z_s can be evaluated and stored once, leading to significant savings in memory requirements and CPU time. To address the singularities in Green's function when r⃗  = r⃗', the numerical integration of shape function times Green's function or its gradient are evaluated using the techniques introduced in Refs. <cit.> and <cit.>.

According to (<ref>), the far-field part of Z is 


    Z^ = [   ζ^- T_-^       K_-^;      -K_-^ 1/ζ^- T_-^ ] ≠,

with , = 1,...,. Since only k^- appears when ≠, in the following we simplify the notations by omitting the "-" symbol, thus k = k^-.

To evaluate the operators T^ and K^ when Ω_ is far from Ω_ (i.e. ≠), we rewrite the equations (<ref>) and (<ref>) as

    T_mn^= -jk ∫_∂Ω_ψ_m^(r⃗) ·∫_∂Ω_G(r⃗ , r⃗')ψ_n^(r⃗'⃗)  '  ,


    K_mn^=  ∫_∂Ω_ψ_m^(r⃗) ·∫_∂Ω_ψ_n^(r⃗'⃗)  ×∇ ' g(r⃗ - r⃗')   '  ,

where G is the dyadic Green's function

    G(r⃗,r⃗')  = ( I - 1/k^2∇∇') g(r⃗- r⃗'),

and I is the unit dyad. Then, using the addition theorem <cit.>, Green's function can be expressed as

    e^-jkr⃗ - r⃗'/4πr⃗-r⃗' = 
    ∫_𝕊^2 e^-jkk·(r⃗ - r⃗_) T_L(k, r⃗_)  e^jkk·(r⃗' - r⃗_)dk,

with r⃗∈Ω_, r⃗' ∈Ω_, r⃗_ = r⃗_ - r⃗_, where r⃗_ and r⃗_ are, respectively, the centers of the particles Ω_ and Ω_, 𝕊^2 = {k∈ℝ^3 : k_2 = 1} is the unit sphere,  T_L is the translation function <cit.> 

    T_L( k, r⃗_) =  
    -jk/(4π)^2∑_ℓ = 0^L (-j)^ℓ (2ℓ +1) h_ℓ^(2) (kr⃗_) P_ℓ (k·r_),

where h_ℓ^(2) is the second kind spherical Hankel function of order ℓ, P_ℓ is the Legendre Polynomial of order ℓ. 

Using the expansion (<ref>) in Eqs. (<ref>) and (<ref>), we obtain


    T^_mn =  -jk∫_𝕊^2ψ̂^_m(k)· T_L(k, r⃗_)(I - kk) ·ψ̂^*_n(k)  dk,



    K^_mn = -jk∫_𝕊^2k×ψ̂^_m(k)· T_L(k, r⃗_)  ψ̂^*_n(k) dk,

where and ψ̂^_m is the k-space representation of the basis function

    ψ̂^_m(k) = ∫_∂Ω_ψ_m^(r⃗) e^-jkk· (r⃗ - r⃗_).


It is well known that truncating the series (<ref>) at a finite order L requires careful handling due to numerical instabilities caused by the nature of spherical Hankel functions <cit.>. Ideally, one would choose a high order to minimize error, but these functions are highly oscillatory at higher order, meaning that increasing L does not necessarily enhance accuracy <cit.>. Furthermore, these functions diverge at small arguments, that is, when the electrical distance between two centers is too small, leading to a low-frequency breakdown <cit.>. However, in this field of application, such instabilities are not observed at optical frequencies, because the particles are not too close together.



 



§ RESULTS AND DISCUSSION




The execution of the numerical algorithm can be subdivided into five stages:

	
  i) Assembly of the self-interaction impedance matrix Z_s of an individual meta-atom and the right-hand side vector V⃗ using RWG basis functions. Since the particles are identical, Z_s only needs to be evaluated and stored once. The computational complexity and memory requirement for assembling and storing Z_s, in terms of the number of particles, is (2n_e)^2𝒪(1).
	
  ii) Generation of the static modes of an isolated object. Specifically, the longitudinal static modes are obtained by solving the eigenvalue problem (<ref>) using star functions as both expansion and testing functions; the transverse static modes are obtained by solving the eigenvalue problem (<ref>) using loop functions as both expansion and testing functions. In Fig. <ref>, we present the calculated first ten longitudinal and ten transverse static modes for a nanodisk.

	
  iii) Evaluation of the MLFMA operators, consisting of the translation function T_L, the radiation and receive operators ψ̂ and ψ̂^†. The time complexity and memory requirements for assembling T_L are on the order of 𝒪(log). Since the operators ψ̂ and ψ̂^† are evaluated for each particle, their time and memory complexities are on the order of 𝒪(). These complexities can be reduced to 𝒪(1) if the particles have the same spatial orientation (i.e., there is no rotation between particles).
	
  iv) Compression stage, i.e. transitioning from the representation of the matrices Z_s, ψ̂ and ψ̂^† in terms of the RWG basis to the representation in terms of the static modes basis. This stage can be integrated into steps i) and iii), as there is no need to directly store the RWG representation of these operators.
	
  v) Solution of the PMCHWT system using an iterative method accelerated by the Multilevel Fast Multipole Algorithm (MLFMA), which reduces the computational complexity of the matrix-vector product from 𝒪(^2) to 𝒪(log). This stage is the most time-consuming, as the entire system is solved with a complexity of 𝒪(I log), where I is the number of iterations required. To accelerate the convergence of the Generalized Minimal Residual Method (GMRES) <cit.>, we employ a block-diagonal preconditioner, where each block is the single-particle self-interaction impedance matrix. We choose a relative tolerance of 10^-4. It effectively reduces I to 𝒪(^α), with α <1.

In the following simulations, the truncation number L of the translation function (<ref>), is initially set to 10 at the finest level. This value is subsequently doubled at each successive level <cit.>. The numerical evaluation of the integrals over the unit sphere is performed using K = 2L^2 point <cit.>.
For interpolation, we employ a 4× 4 Lagrange interpolation scheme.
The FORTRAN code used to implement this algorithm operates on a single CPU (Intel Xeon-Gold 6140 M 2.3 GHz).



 §.§ Validation

We validate the MLFMA-SME method by investigating the scattering from a particle array composed of 100 identical gold nano-spheres, each of radius R = 100 nm, and arranged according to a golden angle spiral pattern <cit.>. The i-th particle is centered at (x_i,y_i), according to 

    x_i = r_i cos(i  α)
    
    y_i = r_i  sin(i  α),

where i  = 1,...,; r_i = c√(i), c is a constant scaling factor, α = 2π/ϕ^2 is the golden angle, where ϕ = (1+√(5))/2 is the golden ratio. We choose the scaling factor as c = √(3)R.

The gold permittivity is obtained by interpolating experimental data <cit.>. The array is illuminated by a linearly polarized plane, propagating along the axis orthogonal to the array's plane, linearly polarized along the x-axis, with varying wavelength λ in the interval [300,700]nm. Each particle's surface is discretized by a triangular mesh having 500 nodes, 996 triangles, and 1494 edges. The scattering cross section, C_sca, is plotted in Fig. <ref> a as function of λ. The solution, computed by the MLFMA-SME, utilizing N^ = N^⊥ = 10 modes per particle, corresponding to a total number of 4000 unknowns, is compared against the multiparticle Mie theory <cit.>, where the field expansions are truncated at the order M = 15. The  relative error on the scattering cross section is defined as 

    ϵ{C_sca} = |C_sca - C̅_sca|/C̅_sca,

where C̅_sca  is the reference solution obtained by the multiparticle Mie theory. 
The achieved error is lower than 0.02 all over the investigated spectral range.

For the same array, in Fig. <ref>b we show the normalized scattered far field for a fixed wavelength λ = 600 nm. The reference Mie solution, obtained for M = 30, is included for comparison.  An excellent agreement is found.


 §.§ Computational costs


We conducted a complexity study on metasurfaces consisting of a varying number up to 5· 10^4 of identical gold nano-spheres, each of radius R = 100 nm. The particles are arranged according to a golden-angle spiral pattern.  This aperiodic pattern is particularly convenient for our study as it is deterministic, and allows us to increase the dimension of the array one particle at a time without appreciably changing the array's density. The array is excited by a linearly polarized plane wave of wavelength λ = 600 nm, propagating along the axis orthogonal to the array's plane. Each sphere has been discretized by a surface mesh, shown in the inset of Fig. <ref> (a), with 100 nodes, 196 triangles, 294 edges, which results in an average edge length approximately equal to λ/15.
In Fig <ref>, we present a comparison between the CPU time and memory requirements using the MLFMA with RWG basis function and static mode expansion with N^ = N^⊥ = 10.  Simulation results show that the use of MLFMA, in both cases RWG and SME, reduces the computational complexity of the matrix-vector product from 𝒪(^2) to 𝒪(log), as expected. The number of degrees of freedom, and the computational time of each case study reported in Fig. <ref>  are listed in Tab. <ref>.
In particular, with the use of the SME, this simulation is approximately 3 ÷ 5 times faster. 



However, the speed-up depends on various factors, such as the number of points used for interpolation/anterpolation between different levels of the quadtree, or the number of points used to sample the transfer function T_L on the unit sphere. Above all, mesh density on each particle plays a very important role. Indeed, the advantages introduced by the MLFMA-SME become much more apparent with denser meshes, necessary for discretizing irregular objects, such as nanofins and nanorods typically used in metalens. The reason why the speed-up is less than the number of unknowns' compression ratio is attributable to the k-space representation of the MLFMA operators, which means that the computational complexity is not only linked to the number of unknowns but also to the electrical size of the array.



Another crucial advantage introduced by SME in the MLFMA concerns memory requirements. Although the translation function T_L is the same in both RWG and SME, as it doesn't depend on the basis function, the matrices representing the radiation and receive operators are κ times smaller, and the self impedance matrix Z_s is κ^2 times smaller, where κ = n_e/(N^∥ + N^⊥) is the total number of unknowns' compression ratio. Additionally, after m iterations, computing the GMRES solution requires storing all the basis vectors of the Krylov subspace 𝒦_m, leading to a memory requirement of 𝒪(mN), where N is the total number of unknowns. As N increases, the storage cost may become prohibitive, even for a small number of iterations m. As previously mentioned, SME can drastically reduce N, allowing for the simulation of larger structures that were impracticable before.


 



 §.§ Electrically large particle arrays



The use of the MLFMA-SME finds its natural application in the numerical solution of the electromagnetic scattering problem by large metasurfaces, where an array, whose dimension can be much larger than the incident wavelength, is made of  objects with identical shape but with different orientation and size. We show three exemplificative case studies of the application of this method to metasurfaces and metalenses: a golden angle spiral of gold bricks, and a Moirè lattice of gold disks and a canonical metalens of TiO_2 nano-fins.

In all three cases, we model the shape of the meta-atoms as a superellipsoid, whose boundary has the implicit equation 

    [ (x/a_x)^2/m + (y/a_y)^2/m]^m/n + (z/a_z)^2/n - 1 = 0,

We used the public domain code developed by Per-Olof Persson and Gilbert Strang <cit.> to generate the surface mesh. 

We first consider a finite-size golden angle spiral array having linear dimension of 115 λ. The array is made of 40 · 10^3 gold bricks of dimension 200 × 200 × 100 nm. We model this shape as a superellipsoid, using Eq. (<ref>) with m = n = 0.25, a_x = a/2, a_y = a_z = a. Each particle is described by a surface mesh having 436 nodes, 868 triangles, and 1302 edges. The i-th particle is rotated with respect to x by an angle θ_i = 2π (i-1)/. The array is excited by a plane wave, linearly polarized in the plane of the array, along the 𝐱̂-axis, and propagating in the direction orthogonal to the array's plane with wavelength λ = 600 nm. The array size is about 70 μm. In Fig. <ref>a, we show the electric field magnitude on the particles' surface. The solution is computed by using the MLFMA-SME as detailed in this work using the static mode expansion with N^ = N^⊥ = 10, i.e. 40 unknowns per particle. The total computational time is about 20.4 hours on a single CPU. 








Next, we now consider an example of 2.5D metastructures, which are stacked layers of interacting metasurface layers. They provide sufficient degrees of freedom to implement efficient multifunctional devices. Specifically, we consider a two-layer Moiré superlattice array of 20160 gold nanodisks of radius R = 100 nm. We model this shape as a superellipsoid, whose boundary has the implicit equation (<ref>), with m = 1, n = 0.25, a_x = a_y = R, a_z = R/2.  Each particle is described by a surface mesh having 588 nodes, 1172 triangles, and 1758 edges. The centers of the particles are arranged at the vertices of two layers, obtained by periodically replicating a hexagon with a side length of 400 nm. The second layer is located at a height h = 300 nm above the first and is rotated by an angle θ = 5° with respect to it. The array is excited by a plane wave, linearly polarized in the plane of the array, along the x-axis, and propagating in the direction orthogonal to the array's plane with wavelength λ = 600 nm. The array size is about 46 × 53  μm, which is roughly (76× 87)λ. In Fig. <ref>b we show the electric field magnitude on the particles' surface, while in Fig. <ref> we show the longitudinal and transverse modes. The solution is computed using the MLFMA-SME with N^ = N^⊥ = 10. The total computational time is less than 6 hours on a single CPU.

Eventually, we consider an example of a canonical metalens <cit.>, composed of 10^4 TiO_2 nanofins of dimension 410× 85× 600 nm. We model this shape as a superellipsoid, whose boundary has the implicit equation (<ref>), with m = n = 0.25. The total computational time is about 15 hours on a single CPU. Each particle is described by a surface mesh having 198 nodes, 392 triangles, and 588 edges.  To achieve the desired phase profile, the i-th particle, centered at (x_i,y_i), is rotated with respect to x
by an angle <cit.>

    θ_i = π/λ(f - √(f^2 + x_i^2 + y_i^2)),
 where f is the focal length. The metalens has a desired numerical aperture (NA) of 0.8, resulting in a focal length f = 37.5λ. The array is excited by a plane wave, circularly polarized in the plane of the array and propagating in the direction orthogonal to the array's plane with wavelength λ = 660 nm. The array size is 66 × 66  μm, which is  (100× 100)λ. In Fig. <ref>c we show the electric field magnitude on the particles' surface. The solution is computed using the MLFMA-SME with N^ = N^⊥ = 10. In Fig. <ref>d we show the normalized intensity E⃗(x,z)^2 on the x-y cross-sectional plane. The achieved focal length is consistent with the prescribed value. 




§ CONCLUSION

In this work, we proposed a fast integral equation solver suitable for large particle arrays, even those hundreds of wavelengths in diameter, comprising numerous arbitrarily shaped objects with varying orientations and sizes. These scenarios are commonly encountered in the electromagnetic modeling of metasurfaces and metalenses. Our approach combines the Multilevel Fast Multipole Algorithm (MLFMA) with the PMCHWT formulation, utilizing a particular class of entire domain basis function defined on the isolated particle, termed Static Mode <cit.>.

To validate the accuracy of the proposed method, we analyze the scattering from finite aperiodic arrays of nano-spheres and compare the results with those obtained using the multiparticle Mie theory <cit.>. We report the CPU-time and memory requirements as functions of the number of particles, comparing the static mode expansion to a more traditional approach using RWG basis functions. Our findings indicate that the MLFMA-SME is 3 to 5 times faster and requires about 5 times less memory than the traditional MLFMA employing RWG basis functions. Furthermore, this speed and memory advantages significantly improve as the density of the surface mesh increases.

Additionally, we demonstrate the capabilities of the MLFMA-SME method on different arrays: a golden-angle spiral array with a diameter of 115 λ, consisting of 40,000 gold bricks; a two-layer Moiré superlattice made up of 20,160 gold nanodisks, with overall dimensions of 76λ× 87λ; a canonical metalens composed of 10,000 TiO_2 nanofins, covering an area of 100λ× 100λ and having focal length f=37.5λ.

Our results show that SME drastically reduces the total number of unknowns, leading to significant savings in CPU time and memory requirements for the numerical solution of scattering problems from arrays of identical particles compared to the classical MLFMA implementation with sub-domain basis functions. This advancement enables the simulation of large structures that were previously impracticable.



This work was supported by the Italian Ministry of University and Research under the PRIN-2022, Grant Number 2022Y53F3X “Inverse Design of High-Performance Large-Scale Metalenses".


ieeetr

10

silva_performing_2014
A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta,
  “Performing mathematical operations with metamaterials,” Science,
  vol. 343, no. 6167, pp. 160–163, 2014.

yu_flat_2014
N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature
  Materials, vol. 13, pp. 139–150, Feb. 2014.
Bandiera_abtest: a Cg_type: Nature Research Journals Number: 2
  Primary_atype: Reviews Publisher: Nature Publishing Group Subject_term:
  Metamaterials Subject_term_id: metamaterials.

aieta_multiwavelength_2015
F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic
  metasurfaces by dispersive phase compensation,” Science, vol. 347,
  pp. 1342–1345, Mar. 2015.
Publisher: American Association for the Advancement of Science.

xue_fullwave_2023
W. Xue, H. Zhang, A. Gopal, V. Rokhlin, and O. D. Miller, “Fullwave design of
  cm-scale cylindrical metasurfaces via fast direct solvers,” Aug. 2023.
arXiv:2308.08569 [physics].

greengard_fast_1987
L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,”
  Journal of Computational Physics, vol. 73, pp. 325–348, Dec. 1987.

chew_fast_2000
W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song, eds., Fast and
  Efficient Algorithms in Computational Electromagnetics.
Boston: Artech House, July 2000.

fostier_asynchronous_2008
J. Fostier and F. Olyslager, “An Asynchronous Parallel MLFMA for
  Scattering at Multiple Dielectric Objects,” IEEE Transactions
  on Antennas and Propagation, vol. 56, pp. 2346–2355, Aug. 2008.
Conference Name: IEEE Transactions on Antennas and Propagation.

donepudi_higher_2001
K. Donepudi, J.-M. Jin, S. Velamparambil, J. Song, and W. C. Chew, “A higher
  order parallelized multilevel fast multipole algorithm for 3-D
  scattering,” IEEE Transactions on Antennas and Propagation, vol. 49,
  pp. 1069–1078, July 2001.
Conference Name: IEEE Transactions on Antennas and Propagation.

velamparambil_analysis_2005
S. Velamparambil and W. C. Chew, “Analysis and performance of a distributed
  memory multilevel fast multipole algorithm,” IEEE Transactions on
  Antennas and Propagation, vol. 53, pp. 2719–2727, Aug. 2005.
Conference Name: IEEE Transactions on Antennas and Propagation.

ergul_fast_2013
O. Ergül and L. Gürel, “Fast and accurate analysis of optical metamaterials using surface integral equations and the parallel multilevel fast multipole
  algorithm,” in 2013 International Conference on Electromagnetics in Advanced Applications (ICEAA), pp. 1309–1312, Sept. 2013.

waltz_massively_2007
C. Waltz, K. Sertel, M. A. Carr, B. C. Usner, and J. L. Volakis, “Massively
  Parallel Fast Multipole Method Solutions of Large
  Electromagnetic Scattering Problems,” IEEE Transactions on
  Antennas and Propagation, vol. 55, pp. 1810–1816, June 2007.
Conference Name: IEEE Transactions on Antennas and Propagation.

fijany_massively_1995
A. Fijany, M. Jensen, Y. Rahmat-Samii, and J. Barhen, “A massively parallel
  computation strategy for FDTD: time and space parallelism applied to
  electromagnetics problems,” IEEE Transactions on Antennas and
  Propagation, vol. 43, pp. 1441–1449, Dec. 1995.
Conference Name: IEEE Transactions on Antennas and Propagation.

hughes_perspective_2021
T. W. Hughes, M. Minkov, V. Liu, Z. Yu, and S. Fan, “A perspective on the
  pathway toward full wave simulation of large area metalenses,” Applied
  Physics Letters, vol. 119, p. 150502, Oct. 2021.

chang_surface_1977
Y. Chang and R. Harrington, “A surface formulation for characteristic modes of
  material bodies,” IEEE Transactions on Antennas and Propagation,
  vol. 25, pp. 789–795, Nov. 1977.
Conference Name: IEEE Transactions on Antennas and Propagation.

wu_scattering_1977
T.-K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric
  bodies of revolution,” Radio Science, vol. 12, no. 5, pp. 709–718,
  1977.

poggio_chapter_1973
A. J. Poggio and E. K. Miller, “CHAPTER 4 - Integral Equation
  Solutions of Three-dimensional Scattering Problems,” in Computer Techniques for Electromagnetics (R. Mittra, ed.), International
  Series of Monographs in Electrical Engineering, pp. 159–264,
  Pergamon, Jan. 1973.

harrington_field_1993
R. F. Harrington, Field computation by moment methods.
Wiley-IEEE Press, 1993.

forestiere_static_2023
C. Forestiere, G. Gravina, G. Miano, G. Rubinacci, and A. Tamburrino, “Static
  Surface Mode Expansion for the Electromagnetic Scattering From
  Penetrable Objects,” IEEE Transactions on Antennas and
  Propagation, vol. 71, pp. 6779–6793, Aug. 2023.
Conference Name: IEEE Transactions on Antennas and Propagation.

xu_electromagnetic_1995
Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl.
  Opt., vol. 34, no. 21, pp. 4573–4588, 1995.

rao_electromagnetic_1982
S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of
  arbitrary shape,” IEEE Transactions on Antennas and Propagation,
  vol. 30, pp. 409–418, May 1982.
Conference Name: IEEE Transactions on Antennas and Propagation.

forestiere_electromagnetic_2019
C. Forestiere, G. Gravina, G. Miano, M. Pascale, and R. Tricarico,
  “Electromagnetic modes and resonances of two-dimensional bodies,” Physical Review B, vol. 99, p. 155423, Apr. 2019.
Publisher: American Physical Society.

monk_finite_2003
P. Monk, Finite Element Methods for Maxwell's Equations.
Clarendon Press, Apr. 2003.

wilton_novel_1993
D. Wilton, J. Lim, and S. Rao, “A novel technique to calculate the
  electromagnetic scattering by surfaces of arbitrary shape,” URSI Radio
  Science Meeting, p. 322, 1993.

burton_study_1995
M. Burton and S. Kashyap, “A study of a recent, moment-method algorithm that
  is accurate to very low frequencies,” Applied Computational
  Electromagnetics Society Journal, vol. 10, pp. 58–68, 1995.

bucci_use_1995
O. M. Bucci and G. D. Massa, “Use of characteristic modes in
  multiple-scattering problems,” Journal of Physics D: Applied Physics,
  vol. 28, pp. 2235–2244, Nov. 1995.

garbacz_modal_1965
R. Garbacz, “Modal expansions for resonance scattering phenomena,” Proceedings of the IEEE, vol. 53, pp. 856–864, Aug. 1965.
Conference Name: Proceedings of the IEEE.

harrington_characteristic_1972
R. Harrington, J. Mautz, and Yu Chang, “Characteristic modes for dielectric
  and magnetic bodies,” IEEE Transactions on Antennas and Propagation,
  vol. 20, pp. 194–198, Mar. 1972.

chen_characteristic_2015
Y. Chen and C.-F. Wang, Characteristic Modes: Theory and
  Applications in Antenna Engineering.
John Wiley & Sons, June 2015.

engheta_fast_1992
N. Engheta, W. Murphy, V. Rokhlin, and M. Vassiliou, “The fast multipole
  method (FMM) for electromagnetic scattering problems,” IEEE
  Transactions on Antennas and Propagation, vol. 40, pp. 634–641, June 1992.

coifman_fast_1993
R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the
  wave equation: a pedestrian prescription,” IEEE Antennas and
  Propagation Magazine, vol. 35, pp. 7–12, June 1993.
Conference Name: IEEE Antennas and Propagation Magazine.

song_multilevel_1997
J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for
  electromagnetic scattering by large complex objects,” IEEE Transactions
  on Antennas and Propagation, vol. 45, pp. 1488–1493, Oct. 1997.
Conference Name: IEEE Transactions on Antennas and Propagation.

graglia_numerical_1993
R. Graglia, “On the numerical integration of the linear shape functions times
  the 3-D Green's function or its gradient on a plane triangle,” Antennas and Propagation, IEEE Transactions on, vol. 41, pp. 1448 –1455,
  Oct. 1993.

hodges_evaluation_1997
R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the
  use of vector triangle basis functions,” Microwave and Optical
  Technology Letters, vol. 14, no. 1, pp. 9–14, 1997.

song_error_2001
J. Song and W. C. Chew, “Error analysis for the truncation of multipole
  expansion of vector Green's functions [EM scattering],” IEEE
  Microwave and Wireless Components Letters, vol. 11, pp. 311–313, July 2001.
Conference Name: IEEE Microwave and Wireless Components Letters.

hastriter_error_2003
M. L. Hastriter, S. Ohnuki, and W. C. Chew, “Error control of the translation
  operator in 3D MLFMA,” Microwave and Optical Technology Letters,
  vol. 37, no. 3, pp. 184–188, 2003.

abramowitz_handbook_1964
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
  with Formulas, Graphs, and Mathematical Tables.
New York: Dover, ninth dover printing ed., 1964.

koc_error_1998
S. Koc, J. Song, and W. Chew, “Error analysis for the multilevel fast
  multipole algorithm,” in IEEE Antennas and Propagation Society
  International Symposium. 1998 Digest. Antennas: Gateways to the
  Global Network. Held in conjunction with: USNC/URSI National
  Radio Science Meeting (Cat. No.98CH36, vol. 3, pp. 1758–1761
  vol.3, June 1998.

dembart_accuracy_1998
B. Dembart and E. Yip, “The accuracy of fast multipole methods for Maxwell's
  equations,” IEEE Computational Science and Engineering, vol. 5,
  pp. 48–56, July 1998.
Conference Name: IEEE Computational Science and Engineering.

darve_efficient_2004
E. Darve and P. Havé, “Efficient fast multipole method for low-frequency
  scattering,” Journal of Computational Physics, vol. 197, pp. 341–363,
  June 2004.

saad_gmres_1986
Y. Saad and M. H. Schultz, “GMRES: A Generalized Minimal Residual
  Algorithm for Solving Nonsymmetric Linear Systems,” SIAM
  Journal on Scientific and Statistical Computing, vol. 7, pp. 856–869, July
  1986.
Publisher: Society for Industrial and Applied Mathematics.

trevino_circularly_2011
J. Trevino, H. Cao, and L. Dal Negro, “Circularly Symmetric Light
  Scattering from Nanoplasmonic Spirals,” Nano Letters, vol. 11,
  pp. 2008–2016, May 2011.
Publisher: American Chemical Society.

johnson_optical_1972
P. B. Johnson and R. W. Christy, “Optical Constants of the Noble
  Metals,” Physical Review B, vol. 6, pp. 4370–4379, Dec. 1972.

persson_simple_2004
P.-O. Persson and G. Strang, “A Simple Mesh Generator in Matlab,”
  SIAM Review, vol. 46, no. 2, pp. 329–345, 2004.

khorasaninejad_metalenses_2016
M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso,
  “Metalenses at visible wavelengths: Diffraction-limited focusing and
  subwavelength resolution imaging,” Science, vol. 352, no. 6290,
  pp. 1190–1194, 2016.






