# A theory of phonon induced friction on molecular adsorbates

Ardavan Farahvash  Department of Chemistry, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Adam P. Willard
<EMAIL_ADDRESS>Department of Chemistry, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

In this paper we provide a theory for how surface phonons couple to molecular
adsorbates. Our theory maps the extended dynamics of a solid’s vibrational
motions to a generalized Langevin equation, and by doing so, encapsulates
these dynamics in a single quantity: the non-Markovian friction. The different
frequency components of this friction are the phonon modes of the solid
weighted by their coupling to the adsorbate degrees of freedom. Using this
formalism, we demonstrate that physisorbed species couple primarily to the
acoustic phonons while chemisorbed species couple to dispersionless local
vibrations. We subsequently derive equations for phonon-adjusted reaction
rates using transition state theory and demonstrate that these corrections
improve agreement with experimental results for CO desorption rates from
Pt(111).

##  1 Introduction

When a molecule binds to a solid surface, its properties and dynamics become
intertwined with those of the solid. This interaction affects chemical
reaction rates and mechanisms via changes to both the electron and nuclear
properties of the bound molecule. Recent advances in theoretical techniques
have given a powerful perspective into the salient effects of surface binding
on molecular electronic structure.1, 2, 3, 4 However, despite numerous
experimental and computational studies demonstrating the significant
influences surface vibrations sometimes have on reaction rates and
mechanisms,5, 6, 7, 8, 9 there remains comparatively little general insight
into the role of solid nuclear vibrations on surface chemical processes.

In this paper, we provide a general theory of how surface phonons couple to
molecular adsorbates. Our method is extends our previous method for utilizing
the generalized Langevin equation (GLE) for describing the influence of
surface vibrational modes on molecular adsorbates. 10 In our previous method,
the GLE was employed to describe the effective dynamics of a surface binding
site, while in the method we present here the surface is integrated over so
that the GLE can be employed to describe the effective dynamics of the
adsorbate itself. Using this technique, we demonstrate that phonons interact
in a very different manner with physisorbed and chemisorbed species.
Specifically, physisorbed species couple primarily to acoustic phonons across
a broad range of frequencies, while chemisorbed species couple primarily to
dispersionless local vibrations. By combining these observations with harmonic
transition state theory, we derive equations explicitly showing how phonons
alter reaction rates at solid surfaces and demonstrate that these phononic
corrections agree with experimental measurements of desorption rate constants.

Over the past four decades, a multiplicity of experimental and theoretical
studies have demonstrated how surface vibrations modulate reactions dynamics
at metal interfaces. For example, dissociation of both methane and
$\text{N}_{\text{2}}$ on both a wide variety metal catalysts has been shown to
be deeply intertwined with surface atom motion 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 5. Another example is the enhancement of many reactions (such
as the oxidation of CO or ethanol) by surface acoustic waves 23, 24, 25, 26,
27, 28, 29, 7. In these experiments, a piezoelectric driver is used to apply
surface waves at a catalytic interface at specific frequencies and
polarization. For certain choices of frequency and polarization, rate
enhancement is observed while for others no noticeable change is made. While
several explanations have been given for this effect, there remains no clear
consensus 7. Much of the lack of consensus arises from the inability of
current theory to connect the length and timescales of atomistic dynamics with
the mesoscopic scale of experimentally relevant surface acoustic waves 30.
Inspired greatly by these experiments, we have sought to develop a theory
which can capture how surface phonons couple to molecular adsorbates and
bridge the gap between mesoscopic and molecular scales.

In order to accomplish this task, we will integrate over the solid nuclear
coordinates to derive an effective equations of motion for the adsorbate. As
we demonstrate in Section 2, such a procedure naturally leads to a generalized
Langevin equation (GLE),

$\accentset{\mbox{\large..}}{\mathbf{x}}_{A}=-\frac{\partial\tilde{V}}{\partial\mathbf{x}_{A}}-\int_{0}^{t}\mathbf{K}(t-\tau)\accentset{\mbox{\large.}}{\mathbf{x}}_{A}(\tau)d\tau+\mathbf{R}(t),$
(1)

where ${\mathbf{x}}_{A}$ is a vector of the mass-weighted displacements of the
adsorbate degrees of freedom, $\tilde{V}$ is the effective potential energy
surface for the adsorbate, $\mathbf{K}$ is the non-Markovian friction or
memory kernel, and $\mathbf{R}$ is the conjugate random force. The friction
kernel, $\mathbf{K}(t)$, is central to determining the dynamics of the GLE and
thus the dynamics of the adsorbate degrees of freedom. It encodes all relevant
information about the solid’s phonon frequencies, displacements, and how they
couple to the adsorbate. Naturally, the friction kernel also plays a
significant role in determining how phonons modulate reaction rates of
adsorbed species.

Figure 1:  2D schematic illustrating the dominant phonon modes in terms of
their coupling to the adsorbate or contribution to the friction kernel.
$\omega_{\mathrm{as}}$ is the frequency of the adsorbate-surface bond and the
$\omega_{\mathrm{D}}$ is the solid’s Debye frequency.

The primary results of this paper are two-fold. First, using our GLE
formalism, we demonstrate a general pattern for how surface phonons interact
with adsorbates, wherein species that are weakly coupled to the surface
(physisorbed) interact primarily with acoustic phonons, and species that are
strongly coupled to the surface (chemisorbed) interact primarily with
dispersionless local modes. The key value that separates these regimes is the
ratio of the solid’s Debye frequency to the frequency of the adsorbate-surface
bond. Figure 1 provides a schematic of this result. Second, using harmonic
transition state theory, we derive a general formula for how phonons alter
desorption rates from solid surfaces,

$k_{d}=\frac{\sqrt{\frac{\mu}{m}\omega^{2}_{\mathrm{as}}-K(t=0)}}{\omega_{\mathrm{as}}}\times\frac{\tilde{\omega}_{\mathrm{D}}}{\omega_{\mathrm{D}}}\times
k_{d0}$ (2)

where $k_{d0}$ is the rate constant for a static surface,
$\omega_{\mathrm{as}}$ is the frequency of the adsorbate-surface bond,
$K(t=0)$ is the instantaneous friction along the desorption coordinate, $\mu$
is the reduced mass of the adsorbate and adsorption site, $m$ is the adsorbate
mass, $\omega_{\mathrm{D}}$ is the solid’s natural Debye frequency, and
$\tilde{\omega}_{\mathrm{D}}$ is the adjusted maximum phonon frequency due to
adsorption. We demonstrate the adjustments from Eq. 2 improvement agreement
with experimental results for CO desorption from Pt(111).

The remainder of the paper is organized as follows. In Section 2, we present
the formal theory behind the phonon-induced GLE and highlight the
characteristics of the friction kernel. We then analyze results for the
friction kernel across a range of values for $\omega_{\mathrm{as}}$ for a
variety of metal surfaces. In Section 3 we provide further insight into the
results presented in Section 2 by analyzing the role of dispersion plays in
the phonon-induced friction. Finally, in Section 4, we use transitions state
theory to examine how phonons affect reaction rates at solid surfaces.

##  2 Coupling surface vibrations to adsorbates

###  2.1 Theory

We seek to understand the effect of a solid’s surface vibrations on adsorbate
degrees of freedom through integrating out the solid’s nuclear degrees of
freedom to derive effective equations of motion for the adsorbate. In order to
do so we must make two critical assumptions. First, we assume that the nuclear
degrees of freedom of the adsorbate and solid can be described classically and
adiabatically. There are cases where adsorbate dynamics are non-adiabatic, and
in such cases a mixed quantum-classical approach has been shown to be
effective 31, 32, but such a theory is beyond the scope of this paper. Second,
we assume that the solid vibrations and surface-adsorbate interaction are both
harmonic. Clearly, this assumption is an oversimplification, and therefore its
consequences and limitations should be thoroughly tested. In this paper, we
will develop theory within this harmonic approximation, and intend to
subsequently publish results examining the role of anharmonicity.

To begin, let $\mathbf{x}_{A}$ and $\mathbf{x}_{S}$ refer to the the mass-
weighted displacements of the adsorbate and solid nuclei respectively from
their equilibrium positions. We decompose the total potential energy into
contributions from the adsorbate and solid,

$V(\mathbf{x}_{A},\mathbf{x}_{S})=V_{A}(\mathbf{x}_{A})+V_{AS}(\mathbf{x}_{A},\mathbf{x}_{S})+V_{S}(\mathbf{x}_{S}),\\\
$ (3)

where $V_{A}$ is the adsorbate potential energy containing all intramolecular
and intermolecular interactions, $V_{S}$ is the solid potential energy
surface, and $V_{AS}$ the adsorbate-solid interaction. By expanding $V_{AS}$
and $V_{S}$ to second order around the equilibrium displacements we may define
$\mathbf{G}_{AS}$, $\mathbf{G}_{A}$, $\mathbf{G}_{S}$, and $\mathbf{H}_{S}$
whose elements are given by,

$\displaystyle
G_{AS;ij}=\frac{\partial^{2}V_{AS}}{\partial{x}_{A,i}\partial{x}_{S,j}},$ (4)
$\displaystyle
G_{A;ij}=\frac{\partial^{2}V_{AS}}{\partial{x}^{2}_{A,i}}\delta_{ij},$ (5)
$\displaystyle
G_{S;ij}=\frac{\partial^{2}V_{AS}}{\partial{x}^{2}_{S,i}}\delta_{ij},$ (6)
$\displaystyle
H_{S;ij}=G_{S;ij}+\frac{\partial^{2}V_{S}}{\partial{x}_{S,i}\partial{x}_{S,j}},$
(7)

where $\delta_{ij}$ is the Kronecker delta, $\mathbf{H}_{S}$ is the solid’s
mass-weighted Hessian, $\mathbf{G}_{AS}$ is the coupling between adsorbate and
solid degrees of freedom, and $\mathbf{G}_{A}$ and $\mathbf{G}_{S}$ are
diagonal matrices of oscillation frequencies for the adsorbate and solid
degrees of freedom respectively. All derivatives in Eqns. 4-7 are evaluated at
zero, per the definition of $\mathbf{x}_{A}$ and $\mathbf{x}_{S}$.

Using this expansion, the equations of motion for adsorbate and solid atoms
are,

$\displaystyle\accentset{\mbox{\large..}}{\mathbf{x}}_{A}=-\frac{\partial
V_{A}}{\partial\mathbf{x}_{A}}-\mathbf{G}_{A}\mathbf{x}_{A}-\mathbf{G}_{AS}\mathbf{x}_{S},$
(8)
$\displaystyle\accentset{\mbox{\large..}}{\mathbf{x}}_{S}=-\mathbf{G}^{T}_{AS}\mathbf{x}_{A}-\mathbf{H}_{S}\mathbf{x}_{S},$
(9)

respectively. We now introduce solid phonon coordinates
$\mathbf{u}_{S}=\mathbf{U}^{T}\mathbf{x}_{S}$ where $\mathbf{U}$ is a matrix
with the eigenvectors of $\mathbf{H}_{S}$ as columns. In terms of these
coordinates the equations of motion may be expressed as,

$\displaystyle\ddot{\mathbf{x}}_{A}=-\frac{\partial
V_{A}}{\partial\mathbf{x}_{A}}-\mathbf{G}_{A}\mathbf{x}_{A}-\mathbf{C}\mathbf{u}_{S},$
(10)
$\displaystyle\ddot{\mathbf{u}}_{S}=-\mathbf{C}^{T}\mathbf{x}_{A}-\boldsymbol{\omega}^{2}\mathbf{u}_{S},$
(11)

where $\mathbf{C}=\mathbf{G}_{AS}\mathbf{U}$ is a matrix of couplings between
each adsorbate coordinate and solid phonon mode, and
$\boldsymbol{\omega}^{2}=\mathbf{U}^{T}\mathbf{H}_{S}\mathbf{U}$ is a diagonal
matrix of the square frequency of the solid phonons. Solving Eq. 11 in terms
of $\mathbf{x}_{A}$ and plugging back in to Eq. 10 yields a GLE for the
adsorbate degrees of freedom,

$\accentset{\mbox{\large..}}{\mathbf{x}}_{A}=-\frac{\partial
V_{A}}{\partial\mathbf{x}_{A}}-\left[\mathbf{G}_{A}-\mathbf{K}(t=0)\right]\mathbf{x}_{A}(t)-\int_{0}^{t}\mathbf{K}(t-\tau)\accentset{\mbox{\large.}}{\mathbf{x}}_{A}(\tau)d\tau+\mathbf{R}(t),$
(12)

where the friction kernel $\mathbf{K}(t)$ is given by,

$\mathbf{K}(t)=\mathbf{C}\frac{\cos(\boldsymbol{\omega}t)}{\boldsymbol{\omega}^{2}}\mathbf{C}^{T},$
(13)

and statistics of the random force $\mathbf{R}(t)$ are related to
$\mathbf{K}(t)$ by the second fluctuation-dissipation theorem,

$\frac{\left\langle\mathbf{R}(t)\mathbf{R}^{T}(0)\right\rangle}{k_{\mathrm{B}}T}=\mathbf{K}(t).$
(14)

The positive frequency components of the Fourier transform of the friction
kernel are (up to a multiplicative constant),

$\bar{K}_{ab}(\omega)=\sum_{j}\frac{C_{aj}C_{bj}}{\omega^{2}_{j}}\delta{(\omega-\omega_{j})},$
(15)

where $\omega_{i}$ are the solid’s phonon frequencies, which in the case of
the continuous spectrum of frequencies leads to the expression,

$\bar{K}_{ab}(\omega)=\frac{C_{a}(\omega)C_{b}(\omega)}{\omega^{2}}\rho(\omega).$
(16)

Eq. 12 through 16 contain a great deal of insight. Eq. 12 illustrates that, in
the harmonic approximation, the influence of phonon modes on adsorbates can be
entirely captured in terms of the friction $\mathbf{K}(t)$, as it determines
both the properties of $\mathbf{R}(t)$ via the fluctuation-dissipation theorem
and the deviation of the adsorbate potential from a fixed solid. Eq. 15 and
Eq. 16 reveal that the Fourier transform of this friction kernel is equivalent
to the phonon density of states of the solid
$\rho(\omega)=\sum_{j}\delta{(\omega-\omega_{j})}$ weighted by the adsorbate-
phonon coupling $\mathbf{C}$. The Fourier transform of the friction kernel is
often called the spectral density of the environment. For any given model of
an adsorbate and metal, such quantities may be readily calculated.

In many cases Eq. 13 and Eq. 15 may further be simplified by realizing that
adsorbates couple only to a handful of surface sites and thus
$\mathbf{G}_{AB}$ and $\mathbf{C}$ are quite sparse. Consider the limit where
a single atom per adsorbate couples only to a single surface atom with a force
constant, $\mu\omega_{\mathrm{as}}^{2}$, where $\mu$ is the reduced mass of
the adsorbate and surface atom. In such a case, Eq. 13 and 15 may be
simplified to,

$K(t)=\frac{\mu^{2}}{mM}\omega_{\mathrm{as}}^{4}\sum_{j}\frac{U^{2}_{sj}}{\omega^{2}_{j}}\cos(\omega_{j}t),$
(17)
$\bar{K}(\omega)=\frac{\mu^{2}}{mM}\omega_{\mathrm{as}}^{4}\sum_{j}\frac{U^{2}_{sj}}{\omega^{2}_{j}}\delta{(\omega-\omega_{j})},$
(18)

where $U_{sj}$ is the expansion coefficient of the surface displacement $s$ in
the $j$th normal mode, $m$ is the mass of the adsorbate atom, and $M$ is the
mass of the surface atom.

###  2.2 Results and discussion

We have computed the phonon induced friction for a simple model of CO on
Pt(111). The CO was assumed the interact with a single adsorption site, as
experimental structures show that CO adsorbs primarily atop Pt(111) sites at
low surface coverages 33, 34. Both theory and experiment show that the
frequency of the Pt-CO stretch, which we define as $\omega_{\mathrm{as}}$, is
around $480\ \mathrm{cm}^{-1}$ 34, 35, 36. In this section we will vary
$\omega_{\mathrm{as}}$, in order asses how to magnitude of the surface-
adsorbate coupling affects the phonon-induced friction kernel $K(t)$. The
metal potential energy surface $V_{S}$ was modeled using an effective medium
theory (EMT) forcefield developed by Norskov et al.37 More details about
calculations can be found in the Section 5.

Figure 2:  (A) Friction kernel, (B) spectral density, and (C) density of
states for Pt(111) and three values of $\omega_{\mathrm{as}}$. The density of
states in (C) was calculated using the 4x4x8 surface slab.

Figure 2. illustrates results for phonon friction kernel in the direction
normal to the surface plane $K(t)$, it’s Fourier transform $\bar{K}(\omega)$,
and the phonon density of states $\rho(\omega)$, for three values of
$\omega_{\mathrm{as}}$ using a 4x4x8 and 8x8x8 Pt(111) surface slab. These
three values were chosen because they represent three distinct physical limits
of the adsorbate-surface coupling. The leftmost column of Figure 2. —
$\omega_{\mathrm{as}}=100\ \mathrm{cm}^{-1}$ — represents a weak coupling
limit where the adsorbate-surface interaction frequency is less than
platinum’s Debye frequency $\omega_{\mathrm{D}}=156\mathrm{cm}^{-1}$. We
identify this limit with physisorption, as noble gases have been shown to have
surface interaction frequency of $100\mathrm{cm}^{-1}$ or less with Pt(111).
The rightmost column — $\omega_{\mathrm{as}}=500\ \mathrm{cm}^{-1}$ —
represents a strong coupling limit where the adsorbate-surface frequency is
much greater than the solid’s Debye frequency. We identify this limit with
chemisorption as it is near the aforementioned interaction frequency for CO.
The center column — $\omega_{\mathrm{as}}=300\ \mathrm{cm}^{-1}$ — represents
an intermediate case. Delta function peaks in $\bar{K}(\omega)$ and
$\rho(\omega)$ were broadened to thin Lorentzians of width $1\mathrm{cm}^{-1}$
for ease of visualization.

For all cases of $\omega_{\mathrm{as}}$, the adsorbate couples strongly to a
surface acoustic mode around $\omega=10\mathrm{cm}^{-1}$. The frequency of
this peak is observed to be highly dependent on the dimensions of the surface
slab, consistent with the behavior of an acoustic phonon. In Section 3 we will
give a more detailed theoretical and numerical analysis of the size effects
observed here, fully accounting for phonon dispersion. In the physisorbed
limit, this acoustic mode dominates over the others, resulting in highly non-
Markovian friction. The mode associated with this peak can be shown to arise
from the flexing of the lattice in the direction perpendicular the the
surface. Snapshots of the motion mode are provided Figure 2A. and entire
movies are provided in the electronic SI.

Figure 3: Illustration of the modes with the largest amplitude contributions
to $K(t)$ and $\bar{K}(\omega)$ in the (A) $\omega_{\mathrm{as}}=100\
\mathrm{cm}^{-1}$ (B) $\omega_{\mathrm{as}}=500\ \mathrm{cm}^{-1}$ cases.

In contrast to the physisorbed limit, in the chemisorbed limit, the adsorbate
couples most strongly to a high frequency mode around $\omega=210\
\mathrm{cm}^{-1}$. This mode arises from the local oscillations of the surface
site the adsorbate is bound to. The frequency of this mode is seen to be
independent of unit cell size, suggesting that it is dispersionless. The large
amplitude of this mode signifies that in the chemisorbed limit, the collective
motions of the bulk phonons of the solid do not contribute significantly to
the adsorbate’s motion. Instead, the surface site’s local oscillations (which
are shifted in frequency due to the presence of the adsorbate) dominate the
contribution to the forces on the adsorbate. Many published models for
reactive scattering on solid surfaces will describe solid vibrations using
only a single, harmonically bound surface atom 12, 38, 39, 40, 41. These
results explain why such a method is successful for strongly coupled species.

Despite this large shift in the spectral density when varying
$\omega_{\mathrm{as}}$, in Figure 1C. we demonstrate that the phonon density
of states is nearly identical to that of a bare surface. The only significant
change is the presence of the aforementioned surface site local mode present
at high values of $\omega_{\mathrm{as}}$. These results seem physically valid
given that the low frequency acoustic modes of a solid should be unaffected by
gaseous species especially at low pressures and surface coverages.

In addition to Pt(111) with EMT we have tested several different surfaces of
different crystal structures, facets, elemental compositions, and using
different forcefields. These results are presented in the Section S1 of the
SI. While the details of the friction kernel and spectral density certainly
vary (e.g. Ru has a much higher Debye frequency than Pt) the behavior in the
limits of physisorption ($\omega_{\mathrm{as}}<\omega_{\mathrm{D}}$) and
chemisorption ($\omega_{\mathrm{as}}>>\omega_{\mathrm{D}}$) remains the same
as in Figure 1. This independence with respect to the details of the atomistic
model is appealing and can be understood from a perturbative perspective.
Specifically, in the physisorbed limit, the motion of adsorption site(s) is a
small perturbation to the bulk phonon modes, while in the chemisorbed limit,
the bulk phonon modes are a small perturbation to the motion of the adsorption
sites. In the SI, we rigorously examine this statement by comparing
perturbative schemes to the exact results for $K(t)$ and $\bar{K}(\omega)$.

Finally, we note that in this section we have only studied the effects of
phonons for a clean surface, and presumably surface heterogeneity’s, such as
steps and defects, may add further richness and complexity to the picture we
have provided here. We also note that we have only studied elemental solids,
and the optical modes of polyatomic crystals could add another interesting
dimension. While we believe such studies may prove fruitful, for the purposes
of brevity we do not discuss such results in this paper.

##  3 Phonon dispersion

The size-dependence of the acoustic peak in Figure 2 is troubling. In many
studies, atomistic models are used as convenient substitutes when simulating
macroscopic solids, particularly in application to molecular adsorption.
Figure 2 demonstrates that such simulations suffer from finite size errors due
to the maximum wavelength allowed by the boundary conditions of the surface
unit cell. We emphasize that these finite-size artifacts are present even when
using periodic boundary conditions to evaluate interatomic
interactions/forces, as we did in the previous section, and that they can have
a significant impact when computing measurable quantities such as sticking
coefficients 10. In this section, we illustrate how to generalize the theory
presented in Section 2 in order to compute phonon friction kernels in the
limit of an infinite surface by integrating the frequencies and eigenmodes
across the first Brillouin zone.

In crystalline solids, displacement by a lattice vector returns the same
solid. This symmetry can be used to calculate the phonon frequencies and
displacements of the bulk solid using a spatial Fourier transform of the mass-
weighted Hessian. This result is known as Bloch’s theorem. Let $a$ and $b$ be
the indices of two primitive unit cells, and let $\mathbf{H}_{S}(a,b)$ be the
mass-weighted Hessian of the crystal
${H}_{S;ij}(a,b)=\frac{\partial^{2}V}{\partial{x}_{S,i}(a)\partial{x}_{S,j}(b)}$.
The Fourier transform of this matrix may be expressed as,

$\mathbf{D}(\mathbf{k})=\sum_{a,b}\mathbf{H}_{S}(a,b)e^{i\mathbf{k}\cdot(\mathbf{r}_{a}-\mathbf{r}_{b})},$
(19)

where $\mathbf{r}_{a}$ is the origin of the $a$th cell.
$\mathbf{D}(\mathbf{k})$ is known as the dynamical matrix satisfying,

$\mathbf{D}(\mathbf{k})\mathbf{U}_{j}(\mathbf{k})=-\omega_{j}^{2}(\mathbf{k})\mathbf{U}_{j}(\mathbf{k}).$
(20)

where $\omega_{j}$ is the $j$th phonon band frequency and $\mathbf{U}_{j}$ is
the corresponding polarization vector. For 3D monatomic crystals, the
primitive (Wigner-Seitz) cell consists of a single atom with three degrees of
freedom, thus producing three phonon bands. These bands are the acoustic
transverse and longitudinal modes, and characterized by linear dispersion at
low wavenumbers. They are illustrated for FCC platinum in Figure 4A.

In a surface slab, the presence of anisotropy breaks the symmetry of the 3D
crystal. This symmetry breaking leads to additional surface mode bands, which
can be acoustic (Rayleigh waves), but non-acoustic bands are also often
observed 42, 43, 44. In Figure 4B. and 4C. we demonstrate the phonon
dispersion of a Pt(111) surface calculated via EMT. These results were
calculated using a 4x4x8 primitive surface replicated in a 6x6 super-cell. We
verified convergence with respect to supercell size. Of course, the number of
surface phonon bands depends on the size of the surface unit cell used in
computing $\mathbf{D}(\mathbf{k})$, however we demonstrate in Figure S6
results are similar across different unit cell sizes.

Figure 4: Phonon dispersion curves. (Right) Bulk Pt dispersion curves
calculated using an EMT forcefield and a 10x10x10 atom supercell. (Middle)
Dispersion curves for a 4x4x8 atom surface slab replicated in a 6x6 surface
cell (Right) Same as middle but with an adsorbed CO molecule corresponding to
$\omega_{\mathrm{as}}=480\ \mathrm{cm}^{-1}$.

Figure 4B illustrates the phonon dispersion for a bare surface, while Figure
4C illustrates the dispersion for a surface with an adsorbed CO. The bands are
colored based on their mean slope between the $\Gamma$ and $K$ high-symmetry
points. The three lowest frequency bands are all surface acoustic modes. The
flexing mode depicted in Figure 3A. is precisely one of these bands,
confirming that the corresponding peak in the phonon spectral density in
Figure 1. arises from an acoustic phonon. The remaining modes are nearly
dispersionless — especially the highest frequency mode in Figure 4C, which
corresponds to the surface site local vibration depicted in Figure 3C. The
dispersionless nature of this mode supports why it was not seen to be
dependent on surface slab size in Figure 1.

Using these dispersion relations, we can average the friction kernel across
the first Brillouin-zone,

$K(t)=\frac{\mu^{2}}{mM}\omega_{\mathrm{as}}^{4}\sum_{j}\sum_{\mathbf{k}}\
\frac{\left|U_{sj}(\mathbf{k})\right|^{2}}{\omega_{j}(\mathbf{k})^{2}}\cos(\mathbf{\omega}_{j}(\mathbf{k})t),$
(21)
$\bar{K}(\omega)=\frac{\mu^{2}}{mM}\omega_{\mathrm{as}}^{4}\sum_{j}\sum_{\mathbf{k}}\
\frac{\left|U_{sj}(\mathbf{k})\right|^{2}}{\omega_{j}(\mathbf{k})^{2}}\delta(\omega-\mathbf{\omega}_{j}(\mathbf{k})),$
(22)

where the outer sum goes over all phonon bands, and the inner sum over the
first Brillouin zone. The outer sum and inner sum together sum over all phonon
modes. Eqns. 21 and 22 simply separate the sum in Eqns. 17 and 18 into two
parts: the outer sum varying intra-cell displacements and the inner sum
varying inter-cell displacements. In principle, this procedure eliminates the
dependence on boundary conditions and generalizes results from a surface unit
cell to an infinite surface. However, the depth of the surface (corresponding
to the non-periodic dimension) is still limited to the depth of the surface
unit cell. The results in previous section essentially correspond to only
taking the value of Eqns. 21 and 22 at $\mathbf{k}=0$.

Figure 5:  Spectral density for Pt(111) surface phonons. The black line is
equivalent to the 4x4x8 results presented in Figure 1. The blue dashed line
was calculated by averaging the results for the black-line across $k$-space
(Eqns. 21 and 22).

Figure 5 demonstrates the results of Eqns. 21 and 22 and compares to previous
results for a single unit cell slab. Naturally, the strong coupling of the
adsorbate to a single acoustic mode is broadened, resulting in a much flatter
spectral density in the low frequency ranges. Such a flat spectral density is
characteristic of Markovian (white) noise. The high frequency range of the
spectral density is largely unaltered, due to the dispersionless nature of the
high frequency modes. In the SI, we discuss how a flat spectral density for
acoustic phonons is consistent with predictions from continuum elastic theory.

##  4 Phonon effects on reaction rates

The most salient motivation for the study of the phononic friction is to
elucidate the role of phonons in reactions rates at catalytic interfaces. As
was noted first by Kramers45, and expanded on by many others46, 47, 48, 49,
the friction of the environment plays a critical role in determining reaction
rates, especially in determining rate prefactors. In this section, we use
transition state theory (TST) to demonstrate how surface phonons affect
reaction rates near thermal equilibrium, emphasizing the role of the phonon
friction kernel. We then specialize to the case of desorption rates and
compare the results of our model to experimental temperature dependent rate
constants for CO and Xe desorption from Pt(111). Many details of the formalism
given this section have been omitted for the purpose of providing a concise
presentation of the theory that focuses on physical ramifications. A thorough
presentation of the theory is provided in the SI for interested readers.

###  4.1 Transition State Theory

####  4.1.1 General theory

The Eyring TST equation for the rate constant may be expressed as,

$k=\frac{1}{\beta h}\frac{Q^{{\ddagger}}}{Q_{R}}e^{-\beta\Delta
E^{{\ddagger}}},$ (23)

where $h$ is Planck’s constant, $\Delta E^{{\ddagger}}$ is the potential
energy difference between the reactant state and the transition state,
$Q^{{\ddagger}}$ is the partition function at the transition state, and
$Q_{R}$ is the partition function in the reactant basin. By expanding said
partition functions to second order about the transition state and the
reactant basin, we can analytically evaluate $Q_{R}$ and $Q^{{\ddagger}}$
yielding,

$k=\frac{1}{2\pi}\times\frac{\prod\limits^{N-1}_{i=0}f_{i}}{\prod\limits^{N-1}_{i=1}f_{i}^{{\ddagger}}}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (24)

where $N$ is the total number of modes, $f_{i}$ are the vibrational
frequencies in the reactant basin, $f_{i}^{{\ddagger}}$ are the non-imaginary
vibrational frequencies at the transition state. Note that since the
transition state is defined to be a saddle point, it always has one imaginary
frequency mode and this mode corresponds the reaction coordinate. Pollak
illustrated that Eq. 24 is equivalent to Kramers and Grote-Hynes rate
theories50. In a gas phase molecular reaction, $f_{i}$ are the
eigenfrequencies of the molecular mass-weighted Hessian. However, in a
reaction at a solid interface, both phonon modes and molecular/adsorbate modes
will contribute to the rate constant,

$k=\frac{1}{2\pi}\times\prod\limits^{N_{S}-1}_{i=0}\frac{\omega_{i}}{\omega_{i}^{{\ddagger}}}\times\frac{\prod\limits^{N_{A}-1}_{i=0}\tilde{f}_{i}}{\prod\limits^{N_{A}-1}_{i=1}\tilde{f}^{{\ddagger}}_{i}}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (25)

where $N_{S}$ and $N_{A}$ are the number of solid and adsorbate degrees of
freedom respectively, $\omega_{i}$ and $\omega_{i}^{{\ddagger}}$ are the
phonon frequencies in the reactant and transition state respectively, and
$\tilde{f}_{i}$ and $\tilde{f}^{{\ddagger}}_{i}$ are effective molecular
frequencies in the reactant and transition state respectively. Let
$\mathbf{H}_{A}$ be the mass-weighted Hessian of the adsorbate degrees of
freedom, then $\tilde{f}_{i}$ and $\tilde{f}^{{\ddagger}}_{i}$ are the
eigenfrequencies of an effective Hessian,

$\tilde{\mathbf{H}}_{A}=\mathbf{H}_{A}-\mathbf{C}\boldsymbol{\omega}^{-2}\mathbf{C}^{T},$
(26)

where the shift term is equal to the instantaneous (zero-time) friction,
$\mathbf{K}(t=0)=\mathbf{C}\boldsymbol{\omega}^{-2}\mathbf{C}^{T}$.

Comparing Eqns. 24 and 25 reveals that phonons introduce two distinct
corrections to rate constant: the ratio of phonon frequencies between the
reactant and transition state, and the shift in adsorbate frequencies. The
ratio of phonon frequencies is a kinetic correction factor. It is largest when
the reactant molecules are strongly coupled to the surface and the transition
state is not. In Figure 2C we demonstrated that an adsorbate strongly coupled
to a surface affects only the highest frequency phonon mode, leaving the bulk
of the phonon density of states unchanged. Thus, we can approximate the ratio
of phonon frequencies as,

$\prod\limits^{N_{S}-1}_{i=0}\frac{\omega_{i}}{\omega_{i}^{{\ddagger}}}\approx\frac{\tilde{\omega}_{\mathrm{D}}}{\omega_{\mathrm{D}}},$
(27)

where $\omega_{max}$ is the highest frequency phonon mode when the reactants
are bound to the surface and $\omega_{\mathrm{D}}$ is the bare solid Debye
frequency.

In contrast, the shift in the adsorbate frequencies is a thermodynamic
correction arising from phonons shifting the free energy surface along the
reaction coordinate. Indeed, if we denote $\tilde{f}_{0}$ as the adsorbate
normal mode along the reaction coordinate Eq. 25 can be simplified to,

$k=\frac{\tilde{f}_{0}}{2\pi}\times\prod\limits^{N_{S}-1}_{i=0}\frac{\omega_{i}}{\omega_{i}^{{\ddagger}}}\times
e^{-\beta(\Delta E^{{\ddagger}}+T\Delta\tilde{S}^{\ddagger})},$ (28)

where $\Delta\tilde{S}^{\ddagger}$ is the effective barrier entropy,

$\Delta\tilde{S}^{\ddagger}=k_{\mathrm{B}}\sum^{N_{A}-1}_{i=0}\ln\left(\frac{\tilde{f}^{{\ddagger}}_{i}}{\tilde{f}_{i}}\right).$
(29)

In the harmonic approximation this barrier entropy is independent of
temperature, but in a more general context it can be shown to be temperature
dependent (see SI).

####  4.1.2 Surface desorption

In surface desorption, the details of the adsorbate degrees of freedom are
unimportant as the reaction coordinate can be defined to be the center of mass
of the adsorbate and it’s distance to the surface. Furthermore, for most
desorption processes the transition state is not coupled to the surface 51 and
we can use Eq.27. With these assumptions the desorption rate constant is,

$k_{d}=\frac{\tilde{\omega}_{\mathrm{as}}}{2\pi}\times\frac{\tilde{\omega}_{\mathrm{D}}}{\omega_{\mathrm{D}}}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (30)

where $\Delta E^{{\ddagger}}$ is the desorption barrier and
$\tilde{\omega}_{\mathrm{as}}$ is the effective adsorbate-surface interaction
frequency satisfying,

$\tilde{\omega}_{\mathrm{as}}=\sqrt{\frac{\mu}{m}\omega^{2}_{\mathrm{as}}-K(t=0)}.$
(31)

While transition state theory is fundamentally a classical theory 52, 53, 54,
55, studies have demonstrated improved agreement with experiment when
introducing quantum corrections, such as adding a correction for the
rotational motion of the molecule or using quantum harmonic oscillator
partition functions instead of classical oscillator partition functions 51,
56. In the subsequent section, we will compare 4 different models for the
desorption rate constant to experimental results: (1) a fixed-surface model
using classical harmonic oscillator partition functions and a rotation factor,

$k_{d1}=\frac{\omega_{\mathrm{as}}}{2\pi}\times\frac{2I}{\hslash^{2}\beta}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (32)

(2) a phonon-corrected model using classical harmonic oscillator partition
functions and a rotation factor,

$k_{d2}=\frac{\tilde{\omega}_{\mathrm{as}}}{2\pi}\times\frac{\tilde{\omega}_{\mathrm{D}}}{\omega_{\mathrm{D}}}\times\frac{2I}{\hslash^{2}\beta}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (33)

(3) a fixed-surface model using quantum harmonic oscillator partition
functions and a rotation factor,

$k_{d3}=\frac{1-e^{-\beta\hslash\omega_{\mathrm{as}}}}{2\pi\beta\hslash}\times\frac{2I}{\hslash^{2}\beta}\times
e^{-\beta\Delta E^{{\ddagger}}},$ (34)

and (4) a phonon-corrected model using quantum harmonic oscillator partition
functions and a rotation factor,

$k_{d4}=\frac{1-e^{-\beta\hslash\tilde{\omega}_{\mathrm{as}}}}{2\pi\beta\hslash}\times\frac{1-e^{-\beta\hslash\tilde{\omega}_{\mathrm{D}}}}{1-e^{-\beta\hslash\omega_{D}}}\times\frac{2I}{\hslash^{2}\beta}\times
e^{-\beta\Delta E^{{\ddagger}}}.$ (35)

In the equations above $I$ denotes the moment of inertia of the adsorbate.
Note that by "fixed-surface" we do not mean a surface at absolute zero, but
rather a surface that acts as an ideal, structureless, thermal environment.

###  4.2 Results and discussion

We have compared results from Eqns. 32 to 35 to experimental temperature
dependent desorption rate constants for CO and Xe from a Pt(111) surface.
Parameters used in computing Eqns. 32 to 35 are presented in Table 1. Note
that because Xe is an atomic species, its moment of inertia is 0, and thus we
ignore the rotational partition function factors for Xe calculations. The
phonon corrections in Eq. 33 and Eq. 35 were computed using a 4x4x8 EMT
surface slab and subsequently averaged across the first Brillouin zone as
described in Section 3. In the SI, we illustrate that the rate constant
corrections we present here are not sensitive to the size of the surface slab
used, or even whether or not one accounts for phonon dispersion.

Table 1: Parameters used for computing desorption rate constants shown in Figure 6. |  $E^{{\ddagger}}$ ($\mathrm{eV}$) |  $\omega_{\mathrm{as}}$ ($\mathrm{cm^{-1}}$) |  $\tilde{\omega}_{\mathrm{as}}$ ($\mathrm{cm^{-1}}$) |  $\tilde{\omega}_{\mathrm{D}}$ ($\mathrm{cm^{-1}}$)
---|---|---|---|---
CO |  1.4756 |  48034 |  164 |  203
Xe |  0.24557 |  2858 |  21 |  156

The CO desorption rate constants were taken from Ref. 56 and the Xe desorption
rates were taken from Ref. 57. In Ref. 56, desorption rate constants were
calculated by fitting the time-dependent flux from a beam scattering
experiment to two models: a single exponential model and a bi-exponential
model. The single exponential model fit the entire flux signal, mixing
contributions from terrace and steps. Meanwhile the biexponential model
separated the flux into a fast component, arising from terrace desorption, and
a slow component, arising from step to terrace diffusion followed by terrace
desorption. While our TST calculations do not include the role of steps, we
compare the results of our models to data both from the single exponential
model and the fast component of the biexponential for thoroughness and
transparency.

Figure 6:  Rate constants for desorption from a Pt(111) surface. (A) CO
desorption. Grey squares are experimental data which mixed contributions from
both steps and terraces. Black circles refer experimental data where the
kinetics of terrace desorption was isolated. (B) Xe desorption.

In Figure 6A, we see that the phonon corrected models (Eqns. 33 and 35) give
improved agreement with experimental results for CO desorption. In particular,
$k_{d4}$ the quantum, flexible surface model and the terrace desorption rate
constants yield the best agreement. The improved agreement when using Eqns. 33
to 35 versus Eqns. 32 to 34 arises from the reduced adsorbate-surface
frequency $\tilde{\omega}_{as}$. Physically, the flexible surface relaxes the
stiffness of the adsorbate-surface bond, leading to a lower frequency of
attempts over the barrier and a lower rate prefactor.

In Figure 6B, we demonstrate results for Xe. Here, all the TST models lie
essentially on top of each other and are lower in value than the experimental
rate constants, although by a small margin. The smaller phonon corrections for
Xe versus CO are a natural result of the weaker interactions with the surface.
Weaker coupling means a small $K(t=0)$, leading to
$\tilde{\omega}_{\mathrm{as}}\approx\omega_{\mathrm{as}}$. Furthermore, weak
coupling also results in a phonon density of states that is unchanged from a
bare lattice, implying $\tilde{\omega}_{\mathrm{D}}=\omega_{\mathrm{D}}$. In
general, the theory presented in this section suggests the phonon corrections
to the rate constant are much larger for chemisorbed species than physisorbed
species.

Of course, it is worth emphasizing that using a slightly different values for
the surface binding energy, $\Delta E^{{\ddagger}}$, can substantially shift
the quality of agreement of theoretical calculations with experiment. The
major impediment to the first principles calculation of chemical rates is
still the calculation of the barrier energy, and the phonon corrections to the
rate constants seem to be a comparatively minor factor, even for chemisorbed
species. Indeed, the purpose of this section was not to demonstrate that the
magnitude of phonon corrections to reaction rates is large, but rather to
illustrate that the theoretical models we developed in Sections 2 and 3, when
combined with transition state theory, produce physically interpretable
results which correspond well with existing experimental measurements.

##  5 Methods

All calculations were performed using the Atomic Simulation Environment (ASE)
python package 59, 60, 61, 62, 63, 64. Mass-weighted Hessians were computed
using a finite-difference scheme in the vibrations module of ASE. Dynamical
matrices were computed using the phonon module. The displacement size used in
finite difference calculations was 0.01. 100 points were used to uniformly
sample the first Brillouin zone.

When computing the friction kernels and spectral densities, all adsorbates
were treated as an effective adatom, with a given coupling between the center
of mass of the adsorbate and a surface atom. Friction kernels were computed
for the three Cartesian degrees of freedom of this adatom. Additionally, a
single atom in the bottom layer of each surface slab was constrained in order
to remove center of mass motion. Without a constraint on the solid, the center
of mass diffusion of the entire lattice will dominate the contribution to the
friction kernel. Results for friction kernels were computed using individual
surface atoms as adsorption sites and subsequently averaged.

##  6 Conclusions

In this manuscript we have developed a theory for how surface phonons couple
to molecular adsorbates based on the generalized Langevin equation. By
integrating out the solid degrees of freedom (assuming they could be described
harmonically) we derived a GLE for the adsorbate, wherein the friction is
merely a sum of the phonon frequency of the solid weighted by their coupling
to the adsorbate. We demonstrated that this friction kernel depends
sensitively on the frequency of the adsorbate-surface bond. When the frequency
of this bond is smaller than the Debye frequency of the solid, adsorbates
couple primarily to the acoustic phonons of solid. When the frequency of the
bond is much larger than the Debye frequency of the solid, adsorbates couple
primarily to the dispersionless local vibrations of the adsorption site.
Subsequently, we used harmonic transition state theory to derive phononic
corrections to reaction rate constants. We show that these corrections
improved agreement between theory and experiment for CO desorption rates from
Pt(111). In subsequent publications we intend on further elaborating on the
results presented in this paper by: (1) analyzing how anhamonicity affects the
phononic friction, (2) examining the interplay between phonon and solvent
fluctuations in solvated surface reactions, and (3) using the models derived
in this paper to examine how surface acoustic waves can activate chemical
reactions.

## Supplementary Material

See supplementary material for results for friction kernels for surfaces other
than Pt(111), a perturbative analysis of the physisorbed and chemisobed
limits, more details on convergence of the friction kernel with respect to
supercell size, a discussion of continuum elastic theory, and more details on
the derivation of rate constants using transition state theory.

## Data Availability

Data that support the findings of this study are available from the
corresponding author upon reasonable request. The code used to process memory
kernels is available on Github https://github.com/afarahva/glepy.

## Acknowledgements

AF and APW were supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-SC0019441. This research used resources of the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231. Ardavan Farahvash acknowledges
support from the National Science Foundation Graduate Research Fellowship
program.

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