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Perspective
High-Fidelity Potential Energy Surfaces for Gas Phase andGas-Surface Scattering Processes from Machine Learning
Bin Jiang, Jun Li, and Hua GuoJ. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.0c00989 • Publication Date (Web): 09 Jun 2020
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Submitted to JPCL, 3/25/2020, revised 4/23/2020
High-Fidelity Potential Energy Surfaces for Gas Phase and Gas-Surface Scattering Processes from Machine Learning
Bin Jiang,1,* Jun Li,2,* and Hua Guo3,*
1Hefei National Laboratory for Physical Science at the Microscale, Key Laboratory of Surface
and Interface Chemistry and Energy Catalysis of Anhui Higher Education Institutes, Department
of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China
2School of Chemistry and Chemical Engineering and Chongqing Key Laboratory of Theoretical
and Computational Chemistry, Chongqing University, Chongqing 401331, China
3Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New
Mexico 87131, USA
*: corresponding authors, [email protected], [email protected], [email protected]
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Abstract
In this Perspective, we review recent advances in constructing high-fidelity potential
energy surfaces (PESs) from discrete ab initio points, using machine learning tools. Such PESs,
albeit with substantial initial investments, provide significantly higher efficiency than direct
dynamics methods and/or high accuracy at a level that is not affordable by on-the-fly approaches.
These PESs not only are a necessity for quantum dynamical studies due to delocalization of wave
packets, but also enable the study of low-probability and long-time events in (quasi-)classical
treatments. Our focus here is on inelastic and reactive scattering processes, which are more
challenging than bound systems because of the involvement of continua. Relevant applications
and developments for dynamical processes in both the gas phase and at gas-surface interfaces are
discussed.
TOC graphic
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The concept of a potential energy surface (PES), first introduced by Born and Oppenheimer
(BO) through the separation of electronic and nuclear motions in molecular systems,1, 2 plays a
central role in physical chemistry. The PES establishes a relationship between a nuclear
configuration and the forces acting on nuclei, which governs nuclear dynamics leading to
molecular spectroscopy, energy transfer, and chemical reactivity. (Although electronic
degeneracies cause complications, the construct of PESs remains relevant, as discussed below).
The BO PES is appealing to chemists because nuclear dynamics can be intuitively related to its
topography. While it is not a measurable property, the PES is nonetheless an entity that
experimentalists seek to determine.3
Apart from diatomic systems, accurate determination of a global PES from first principles has
been a challenge until very recently.4-6 The difficulties were partly due to insufficient accuracy and
high costs of electronic structure theory. Thanks to more efficient quantum chemistry algorithms
and the dramatic increase in computing power, it is now possible to run ab initio calculations at
tens to hundreds of thousands configurations for reasonably large systems.7 Correlated
wavefunction based ab initio theory for molecular systems, such as coupled cluster8 and
configuration interaction theories,9 can now achieve chemical accuracy (1 kcal/mol) effectively
with uniform reliability in the dynamically relevant configuration space. For gas-solid interfaces,
such high-level ab initio methods are generally not available, but one can still rely on reasonably
accurate and highly efficient density functional theory (DFT).10 The bottleneck now becomes the
development of a faithful representation of the PES from a large number of discrete data points,
whose dimensionality is the number of nuclear degrees of freedom (DOFs) of the system.
A popular alternative to constructing PESs is direct dynamics, in which forces acting on nuclei
are computed on the fly in a (quasi-)classical treatment of nuclear dynamics.11, 12 Besides
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avoidance of constructing the PES, this approach has many other advantages, particularly for
exploring the potential landscape of a complex system. However, it can be computationally
expensive and extremely inefficient for rare and/or long-time events. To mitigate the high
computational costs, direct dynamics is often not run at the highest possible level of electronic
structure theory, compromising the reliability of calculated dynamics. Furthermore, this method is
very difficult to apply to quantum dynamics as a wave packet tends to delocalize. Hence, for
extensive dynamical studies, a high-fidelity PES based on ab initio points can provide significant
numerical savings over the on-the-fly calculations, even when the initial investment can be quite
substantial.
Pull Qoute inserted here: For extensive dynamical studies, a high-fidelity PES based on ab
initio points can provide significant numerical savings over the on-the-fly calculations.
In recent years, there have been keen and fast developing interests in developing high-
dimensional PESs using modern machine-learning (ML) methods.13 To this end, ML allows
accurate predictions of properties, such as PESs with a large number of nuclear DOFs, based on a
limited set of first principles data, such that new ab initio calculations are not needed. This is
possible because BO PESs are typically smooth functions of nuclear coordinates. There has been
tremendous progress in constructing interaction potentials for condensed phase materials as well
as large molecules using ML methods,14 which has enabled simulations of many properties that
were impossible to imagine only a few years ago. For these many-body problems, the key for ML-
based approaches is to express the PES in terms of atomic contributions because it can be readily
scaled to large systems with thousands of atoms.15-17 In this Perspective, we will instead focus on
molecular systems that involve dissociation continua, relevant to scattering and reactions. Such
systems are challenging in a different respect because scattering is highly sensitive to details of
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the PES. As a result, the PES is required to give a high-fidelity description of not only strongly
interacting regions, but also asymptotes that define the initial and final states of scattering.18, 19
While the aforementioned atomistic representation is viable,20-22 it might not be the most effective
for such problems. Due to space limitations, we will mostly focus on new and significant
developments since our review on this topic four years ago.23
Pull Quote inserted here: ML allows accurate predictions of properties, such as PESs with a
large number of nuclear DOFs, based on a limited set of first principles data, such that new ab
initio calculations are not needed.
It should be noted that there is no strict boundary between modern ML methods and
traditional interpolation/fitting approaches to represent PESs,6, 24-26 as none of them assumes a
physically derived functional form.13 While there has been a long history of using the simplest ML
technique, namely linear regression, in representing PESs,4, 5 the introduction of permutationally
invariant polynomials (PIPs) in terms of all internuclear distances by the Bowman group27, 28 led
to a powerful means to represent high-dimensional PESs with high fidelity. Indeed, the use of all
internuclear distances, despite their redundancy for systems with more than four atoms, greatly
simplified the coordinate system designed for different molecular geometries, thus enabling a
rubust fit of the global PES. Furthermore, they also streamlined the adaptation of permutation
symmetry in the system, which is of utmost importance in polyatomic molecules that contain
identical nuclei. The PIP representation also allows analytical expression for derivatives, which
can in turn be explicitly used in the fitting to improve the fitting quality.29 The readers are referred
to excellent reviews on the PIP method,6, 30 and some recent applications in relatively large reactive
systems.31, 32 In this Perspective, we will be focusing on two modern ML methods, namely neural
networks (NNs)13, 23, 33-36 and kernel-based regression13, 24, 37, 38 in constructing reactive PESs.
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An artificial NN consists of layers of interconnected mathematic functions mimicking
biological neurons.39 Theoretically, it provides an ultra-flexible form that is capable of
representing any multi-dimensional real analytical function with arbitrary precision. In a typical
NN-based approach, parameters in an NN are non-linearly optimized to minimize a cost function,
which can be written as a root-mean-square difference between the training data and NN output.
A key issue in this approach is the selection of points for the training set. Ideally, the distribution
of points should cover the dynamically relevant configuration space, but the number of points
should be as few as possible. The former is often achieved by running trajectories on the fly or on
primitive PESs to explore the configuration space. The latter can be accomplished by using
distance metrics and energy cutoffs to eliminate unnecessary points. More recently, there has been
strong interest in automated schemes in selecting the points.40-42
Unlike NNs, kernel-based regression approaches, such as Gaussian process (GP) regression,43
are non-parametric ML methods with no specific functional form. Indeed, GP is a probabilistic
model as a collection of prior normally distributed random functions characterized by its mean and
covariance functions, where the covariance or kernel function k(x, x′) describes the correlation
between the Gaussian distributions at two different configurations x and x′. According to Bayes’
theorem, the posterior probability distribution is also a Gaussian, whose mean and variance
correspond to the prediction and uncertainty at an unknown point, via maximizing the likelihood
of the GP model with the training data.
While the NN and GP methods are formally equivalent in some special cases,37 they both offer
advantages and disadvantages.44 For instance, GP explicitly provides uncertainty of the prediction
at an unknown configuration based on Bayesian estimation, but suffers from poor scalability due
to its interpolation nature. NNs, on the other hand, are capable of representing a much larger data
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set, thus finer details of a PES, with relatively high efficiency, but they typically require more ab
initio calculations. They are both able to learn energy and gradients simultaneously, which
significantly improves the fitting quality.45-47 Consequently, they are amenable to analytical
derivatives. These ML methods are generally superior to traditional interpolation methods, in
accuracy and/or efficiency. There have been some investigations on the comparison of different
ML methods in terms of their performance and computational costs.22, 48-50 However, the efficacy
of a ML method depends on the problem at hand, thus a comparison of applicability and efficiency
in one system might not be extendable to another system. Typically, ML methods excel in
interpolation, but often do poorly in extrapolation. No further details are given here for either the
NN or GP machinery, as they can be found in standard literature.
Pull quote inserted here: GP explicitly provides uncertainty of the prediction at an unknown
configuration based on Bayesian estimation, but suffers from poor scalability due to its
interpolation nature. NNs, on the other hand, are capable of representing a much larger data set,
thus finer details of a PES, with relatively high efficiency, but they typically require more ab initio
calculations.
Comparing to condensed phase systems, molecules typically have much stronger directional
bonding. As a result, it is natural to use traditional structural descriptors, such as bond lengths and
angles, or internuclear distances, in constructing PESs for gas phase systems.4 They are also
preferred for ML representations of molecular PESs.6, 23, 30, 33, 35 For example, Zhang and coworkers
used internuclear distances in representing global NN PESs from high-level ab initio data to
achieve fitting errors within a few meV,51, 52 which is necessary for quantum scattering calculations
in polyatomic reactions.
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An important property of the BO PES is its invariance under the complete nuclear
permutation and inversion (CNPI) group.53 The PIPs provide a convenient way to adapt the
translation, inversion, as well as permutation symmetry in NN representation of PESs. In this PIP-
NN approach, permutation symmetry is enforced by using low-order PIPs in the input layer of an
NN.54 Different from the PIP approach in which symmetrized polynomials are used as the fitting
bases,6, 30 the PIP-NN method fits ab initio data using NNs in a space spanned by PIPs. As a result,
higher fidelity can be achieved with exact symmetry, but without modifications of NNs. It should
be noted that for systems with more than three atoms, PIPs with a higher order than the number of
internuclear distances are needed to avoid false permutations.55 This creates some redundancy, but
does not affect the fitting accuracy. Since our last review in 2016,23 this method has been
successfully applied to a wide range of reactive systems with up to seven atoms, including KRb +
KRb,56 Be+ + H2O,57 OH + SO,58 OH + O2,59 OH + H2O,60 O + C2H2,61 OH + HO2,62 N2 + HOC+,63
Cl + CH4,64 F + CH3OH,65 and Cl + CH3OH.66 The latter two cases involve fifteen DOFs, with
multiple reaction channels and rich dynamics. In Figure 1, the Cl + CH3OH PES is shown, which
contains two reaction channels to HCl + CH3O/CH2OH products.
Pull quote inserted here: The PIP-NN method fits ab initio data using NNs in a space spanned
by PIPs. As a result, higher fidelity can be achieved with exact symmetry, but without
modifications of NNs
In the PIP-NN approach, the number of PIPs increases quickly with the number of identical
nuclei in the system, thus lowering its efficiency. For instance, 1331 PIPs were used as the input
layer of an NN for fitting an OH + CH4 PES,67 which has five hydrogen atoms. As many of the
PIPs are redundant, it is advantageous to express PIPs in terms of non-redundant terms
corresponding to primary and secondary invariant polynomials.68 This can be achieved by
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decomposing any high-order PIPs into a product of two low-order polynomials with a remainder
invariant polynomial of the same order.28 Hence, one can use only non-redundant PIPs as the input
layer of NNs. For the H + H2S system, for example, only 9 non-redundant PIPs were used in the
PIP-NN PES for the three H nuclei, a significant reduction from 23 3th-order PIPs.69
When the system becomes more complicated, the decomposition may not be done completely.
Consequently, some redundancy remains in lower-order PIPs. It is known that all invariant
polynomials under a given symmetry group can be represented as polynomials in a finite
generating set of invariant polynomials.70 Although its form is not unique, there exists for a given
system a complete set of generating invariants, which can be determined using computer algebra
software like SINGULAR.71 Such an approach was first demonstrated by Opelka and Domcke,72
and investigated further by Zhang and coworkers.73 These generating invariants are denoted as
fundamental invariants (FIs) and the resulting NN fitting approach is called FI-NN.73 As the size
of the FIs is much smaller than that of PIPs, FI-NN is typically more efficient than PIP-NN,
resulting in a faster evaluation of the PES.36 The FI-NN method has been successfully applied to
several reactive systems.74-77
In some special cases, such as non-reactive scattering, full permutation symmetry is not
required. Without losing generality, a PES for a collection of weakly interacting subsystems can
be expressed as , where Vintra denotes the intramolecular PESs of subsystems and intra interV V V
Vinter represents the interaction among the constituent molecules. Because identical atoms among
different molecules are not exchanged in non-reactive events, there is little incentive to enforce the
permutation symmetry between them, as suggested by Truhlar, Bowman, and their coworkers.78,
79 Indeed, we have shown recently that a reduced set of PIPs can be used to adapt the permutation
symmetry in the intermolecular PIP-NN PESs.80 This reduced set of PIPs can be obtained by
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excluding all terms that are relevant only to the intramolecular subsystems, thus simplifying the
symmetry adaptation.
Even with all the improvements discussed above, it is still difficult to construct PESs for
systems with more than ten atoms and strong interactions, especially when there are many identical
atoms. To this end, the atomistic NN (AtNN) method of Behler and Parrinello15, 81 becomes more
attractive. As discussed in more detail below, AtNN expresses the PES as a sum of atomistic
contributions, each encoded by mapping functions that describe its local environment, which is
generally affected by other atoms within a specified cutoff radius. Each atomistic component of
the same element shares identical NN architecture and fitting parameters, naturally preserving
permutational invariance. Recent tests have demonstrated that the AtNN approach is capable of
describing global reactive PESs accurately, but it requires the determination of different NNs for
all atom types in the system. Furthermore, a large number of mapping functions and data points
are often needed to give an accurate description of the PES because of its strong anisotropy,
potentially slowing down the evaluation of the AtNN PES.20-22 In addition, long-range interactions,
especially for charged systems, are difficult to be described effectively by the AtNN approach.
In addition to NNs, kernel-based regression represents another class of ML methods. The use
of kernels in PES interpolation can be traced back to the reproducing kernel Hilbert spaces (RKHS)
method.24 Recently, GP regression has gained increasing popularity in developing PESs because
of several attractive features.82 This ML model was first introduced to construct force-fields83 and
high-dimensional atomistic PESs for condensed phase systems.16 It was also applied to represent
global PESs for calculating spectroscopic properties49, 50, 84-86 and reaction dynamics.87-92 In many
of these studies, GP regression has been shown to accurately and efficiently reproduce existing
PESs with often much fewer points than NNs,50, 87, 89, 92 which is now recognized as a distinct
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advantage of GP over other ML methods. Its prowess has also been demonstrated in constructing
accurate PESs from high level ab initio points for the S + H2 reaction88 and NaK + NaK collision.91
Another unique property of GP is that it provides the statistical uncertainty of a prediction, thus
offering an effective guide to add new data points. Known as “active learning” in the context of
ML, such algorithms based on the search of the maximum variance of the GP model have been
validated in a variety of systems.85, 89 Interestingly, Krems and coworkers proposed a GP model to
identify new data points by optimizing quantum dynamics results.92 This active learning strategy
is quite efficient, enabling GP PESs converged to the J = 0 reaction probability with only 30 points
for the H + H2 reaction and 290 points for the OH + H2 reaction. Another advantage is that
overfitting is generally not a concern in GP regression.50 A drawback of GP regression is its
relatively poor scaling of computational costs compared to NNs. Indeed, the numerical cost of
training and evaluating a GP model scales O(M3) and O(M), respectively, where M is the number
of points. It is therefore necessary to develop GP PESs with as few points as possible. Krems and
coworkers recently showed that both interpolation and extrapolation accuracy of a GP PES can be
significantly improved by increasing the complexity of kernels instead of increasing the number
of points.86 While most GP PESs discussed here are interpolated in terms of internal coordinates
of the system, the use of PIPs instead improves the description of permutation symmetry,49 in the
same spirit of the PIP-NN approach.
We note that there are significant efforts to apply ML methods to describe BO PESs for more
complex reactions.93-95 The resulting PESs, while extremely important, are typically not at the level
of accuracy as those discussed in this Perspective and thus not covered.
All discussion so far has been restricted to adiabatic PESs. When electronic degeneracies,
such as conical intersections (CIs),96, 97 are present, the adiabatic PESs contain cusps and the
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derivative coupling becomes singular at the CI seam. Under such circumstances, it is preferable to
represent the PESs and their couplings in a diabatic potential energy matrix (DPEM), in which the
diagonal potentials are coupled by the off-diagonal terms.98, 99 The elements of the DPEM are all
smooth functions of nuclear coordinates and thus amenable to ML representations. Furthermore,
it removes the intrinsic singularities at the degeneracy, rendering better behaved dynamics.100
Recently, NNs have been used to represent DPEMs for several systems.101-109 For example, Lenzen
et al. proposed an NN-based implementation for the X + CH4 HX + CH3 reactions using a
diabatization-by-ansatz approach. NNs were employed to represent both the diagonal and off-
diagonal elements of the DPEM, trained exclusively from adiabatic energies without derivative
couplings, where diabatization near a CI seam is determined from a symmetry-based ansatz.105
Permutation symmetry adds an extra layer of complexity in representing DPEMs. Depending
on the CNPI symmetry of each diabatic state, the diagonal and off-diagonal terms might belong to
different irreducible representations. For the 1,21A system of NH3, for example, the diagonal terms
are symmetric with respect to the permutation of H atoms, while the off-diagonal terms are
antisymmetric.110 These permutation symmetries can be enforced using an extension of the PIP-
NN method, in which the antisymmetric off-diagonal terms are represented as a PIP-NN multiplied
by an antisymmetric prefactor.103 A similar approach has been used in the NN-based simultaneous
diabatizing-fitting approach by Yarkony and coworkers,107-109 which generates the DPEM from
adiabatic energies, gradients and derivative couplings. The NN-based fits of DPEMs generally
have a higher fidelity than PIP-based ones. Very recently, Guan et al. demonstrated that this NN-
based approach can be extended to fitting of electronic properties such as dipole functions108 and
spin-orbit couplings,109 for which the diabatic representation is indispensable. Finally, we note in
passing that there is emerging interest in developing excited state PESs of more complicated
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systems from direct dynamics simulations.111-114 However, these efforts have so far been restricted
to semi-global adiabatic PESs at relatively low levels of theory without dissociation continua.
A gas-surface process differs from a gas-phase reaction in at least two aspects.115 First, the
molecule can interact with the surface at multiple sites. Second, the motion of surface atoms, and
sometimes electrons, allows energy exchange between the impinging molecule and the surface
during scattering and may significantly influence the reaction dynamics.116-119 As a result, PESs
for describing scattering at gas-surface interfaces are in general more complicated than their gas-
phase counterparts in terms of symmetry, dimensionality, and complexity. Furthermore, the
electronic structure characterization is almost exclusively given in DFT. Most non-NN-based first-
principles PESs used the corrugation reduction procedure (CRP),120 designed to reduce the strong
corrugation near the surface. They were however limited to describing diatomic scattering on static
surfaces with six molecular DOFs included, due to its interpolation nature. Higher-dimensional
PESs for gas-surface scattering/reaction from DFT calculations have emerged recently using NNs.
Interestingly, NNs have a long history in the development of molecule-surface PESs and
remain a formidable force in this arena. Indeed, the first ever NN-based PES was for CO adsorption
on frozen Ni(111).121 Many subsequent studies have focused on incorporating the periodicity of
the rigid surface into NNs. For example, Lorenz et al. first proposed a set of symmetry-adapted
coordinates as the input of the NN PES for H2 reactive scattering on rigid Pd(100),122 which were
later generalized by Reuter and coworkers using Fourier expansions of atomic Cartesian
coordinates on different crystal facets.123, 124 While the periodicity of rigid surfaces was taken into
account in these studies, the permutational invariance of the molecular DOFs was not rigorously
fulfilled. More recently, Zhang and coworkers developed high-fidelity NN PESs for diatomic and
polyatomic dissociation at rigid metal surfaces.125-130 These PESs were symmetrized by mapping
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the data points into a symmetry irreducible area on the surface surrounded by high-symmetry sites
and sorting the coordinates relevant to identical atoms. This approach could lead to discontinuous
derivatives at the boundary, which might affect energy conservation of classical trajectories.
We have extended the PIP-NN approach to gas-surface systems by incorporating both the
permutation symmetry of the molecule and surface periodicity.131 Differing from the gas-phase
counterpart, here the PIPs are crafted in terms of not only internuclear distances but also the
primitive Fourier expansions of atomic coordinates in the molecule on the periodic surface. These
PIPs were demonstrated to be essential in obtaining more accurate NN fits131 than the intuitive
symmetrization of the Fourier terms.123, 124 Since our review in 2016,23 the PIP-NN method has
been applied to several systems, including CO + Co(1120),132 NH3 + Ru(0001)133 and CH4 +
Ni(111),134 as well as a multi-channel reaction of CH3OH on Cu(111).135 We note in passing that
in gas-surface applications, the practical PIP-NN implementation is essentially the same as the FI-
NN method.73
These aforementioned methods represent the entire PES by a single NN and consider the
molecular DOFs only. They are difficult to include surface DOFs, which would increase the
dimensionality of the PES dramatically. As the condensed-phase characteristics of the substrate
becomes prominent in such problems, atomistic ML methods, which have revolutionized the
development of interaction potentials for condensed phase systems,13 are better suited. Taking the
example of AtNN, the PES is represented by a sum of energies of the constituent atoms in the
system. Each atomic type is represented by a distinct NN, which is trained to learn its local
environment in all possible configurations accessible by dynamics.15 Since all surface atoms with
the same elemental type share the same NN, this approach substantially reduces the complexity of
NNs and is naturally invariant under permutation of identical nuclei. The difficulties in
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representing a high-dimensional PES are thus converted to how to best encode the local
environmental information of each atom in the so-called mapping functions (or equivalently
descriptors136 or fingerprints137), being the input of NNs. The atomistic descriptors can be obtained
by, for example, the radial and angular atom centered symmetry functions (ACSFs) summing over
interactions between the central atom and neighboring atoms.15, 81 These ACSFs are relatively
short ranged so that only the local environment is encoded. Importantly, these descriptors are
invariant with respect to rotation, translation, and inversion, so that the corresponding AtNN PES
fulfills the required symmetry and/or periodicity of the systems across the gas and condensed
phases. For detailed implementation, the reader is referred to authoritative reviews by Behler.13, 14
The first application of the AtNN method to describe molecular scattering from a solid
surface was reported in 2017,138 trained from a data set generated from DFT direct dynamics. The
AtNN PESs for the HCl + Au(111) system, approximated with a periodic model that involves 60
DOFs, were shown to reproduce quantitatively not only the energy loss profile of scattered HCl
molecules,138 but also the dissociative sticking probabilities,139 obtained from direct dynamics
simulations, but with a ~105-fold speedup. Later, Zhang et al. extended the AtNN representation
to describe polyatomic (CO2) reactive scattering on Ni(100),47 trained with only ~10000 snapshots
selected from as few as 50 direct dynamics trajectories, taking advantage of simultaneous fitting
of both energies and forces. Likewise, Kroes and coworkers developed reactive AtNN PESs for
N2 on Ru(0001)140 and CH4 on Cu(111),141 which enabled efficient molecular dynamics
calculations of the dissociative sticking probability as low as ~10-5. In another example, an AtNN
PES for CO scattering from Au(111) was developed, which was instrumental for understanding
low-probability and long-time events such as trapping, as the molecule explores both
chemisorption and physisorption wells.142 More recently, AtNN PESs for the NO + Au(111)143
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and CO2 + Pt(111) (shown in Figure 2) systems144 have been developed, and they provided
unprecedented new physical insights into these important surface processes. These examples
demonstrate that the AtNN method is capable of constructing highly efficient and accurate high-
dimensional PES for scattering and reactions at metal surfaces including both molecular and
surface DOFs. A distinct advantage of analytical PESs is the ability to explore site specificity of
the reaction barrier, which offers a view of the “chemical shape” of the surface.144
Pull Qoute inserted here: The AtNN method is capable of constructing highly efficient and
accurate high-dimensional PES for scattering and reactions at metal surfaces including both
molecular and surface DOFs.
As discussed in Sec. III, the BO approximation breaks down when electronic states become
degenerate. Indeed, the coupling between moving nuclei and metallic electrons is ubiquitous
thanks to the vanishing band-gap in metallic systems,145 although its strength is system-dependent.
The continuous electronic states of a metal surface render the nonadiabatic dynamical calculations
much more challenging than those in the gas phase. Two approximate models have been
proposed.145, 146 One is the stochastic surface hopping model147 based on probabilistic electronic
transitions among multiple discretized states. Because of difficulties in computing excited state
PESs and derivative couplings in metallic systems, very few systems have been investigated using
this approach.148, 149 Even in these examples, very limited first-principles information on
nonadiabatic couplings is available, preventing the use of ML methods to represent the DPEM. An
alternative is the electronic friction model,150, 151 in which the effective nuclear-electron coupling
is ascribed to a friction force to capture the instantaneous response of slowly moved nuclei to fast
electronic motion,152 resulting in a generalized Langevin equation.150 In this model, an analytical
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representation of the electronic friction tensor (EFT), , becomes a prerequisite for efficient Λ𝑖𝑗
dynamics similations of nonadiabatic dissipation, reprsenting a relevant topic of this Perspective.
Within a simple local density friction approximation,153 the tensorial friction reduces to a
scalar property, i.e., atomic friction coefficients, which depend on the metal electron density at the
positions where the atoms are embedded. In practice, the electron density surface (EDS) of a metal
surface is formally equivalent to the PES of a single atom adsorbed on the same surface. As a
result, it is quite straightforward to construct the three-dimensional EDS when the surface is
frozen.153 The EDS depends on surface atom displacements as well, when the surface is moving.
To this end, Jiang and coworkers have applied the AtNN model to accurately represent the EDS
of a movable Au(111) surface.143 More recently, EFT has been calculated by a first-order time-
dependent perturbation theory (TDPT) based on Kohn-Sham orbitals, fully accounting for the
electronic structure of the molecule-surface system.154-156 In such a case, the full tensorial
symmetry of EFT has to be taken into account, which is much more intricate than scalar properties
like potential energy and electron density. Specifically, EFT is not invariant but covariant under a
specific symmetry operation. Different NN representations for EFT of a diatomic molecule
scattering from a rigid surface have been developed independently by the Meyer group157 and Jiang
group.158 The molecule is either reoriented to a local reference frame or mapped to the symmetry
irreducible region of the surface, allowing one-to-one correspondence between the nuclear
configuration and the EFT that can be represented by NNs. However, by construction, neither
method is able to account for the influence of substrate structure and lattice motion on EFT. More
recently, Zhang and Jiang proposed to reconstruct the EFT form by manipulating the first and
second derivative matrices of multiple virtual outputs of atomistic NNs with respect to atomic
Cartesian coordinates.159 This strategy rigorously preserves positive semidefiniteness, directional
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property, and correct covariant symmetry of EFT. While this symmetrization scheme can work
with any atomistic ML methods, it was implemented with a physically inspired embedded atom
neural network (EANN) model developed by the same group.160 This EANN model uses atomic
density based descriptors, whose computational cost scales linearly with the number of
neighboring atoms, thus more efficient than the conventional ACSFs. Including both molecular
and surface DOFs, this tensorial representation for EFT will enable investigating adiabatic and
nonadiabatic energy dissipation during scattering and reactions at metal surfaces in a unified
framework.
It is clear from the discussion above that the emergence of ML methods has in the past few
years had a tremendous impact on high-fidelity development of PESs from first-principles data.
These PESs, often with high dimensionality, enabled dynamical calculations with unprecedented
accuracy and extraordinary efficiency, leading to new insights and discoveries. We note however
that the accuracy of ML PESs is ultimately limited by the level of electronic structure theory used
to generate the training set. Progress in that arena, possibly with ML, is expected to further improve
our ability to predict experimental observations.
There are already encouraging developments in extending ML approaches to construct
PESs for coupled electronic states in gas phase systems. To this end, DPEMs are the key, as their
elements are smooth functions of nuclear coordinates and thus amenable to ML methods. This
characteristic extends to other electronic properties, such as dipole moments, which can also be
accurately predicted using ML methods.109 However, stumbling blocks remain. For example, it is
well established that exact diabatization is unattainable for systems with more than two atoms.161
As a result, all diabatization schemes are approximate, and some might lead to unwanted effects.162
Numerically, the inclusion of derivative coupling and energy gradients in addition to energies leads
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to a very large data set, which potentially limits ML applications to relatively small systems. More
efficient approaches need to be found.
For gas-surface systems, we can envisage many possible improvements. For example, it
should be possible to train a single PES to describe the interaction of a molecule on different metal
facets, which will greatly improve our ability to understand the influence of defects on reactivity.
The currently ML approaches can be readily extended to describe a varying number of adsorbates
and surface atoms so as to investigate the influence of coverage and surface phonons on dynamics.
It is also interesting to apply ML approaches to more complex, such as the Eley-Rideal type,
surface reactions. Because DFT is not accurate enough in some cases,139, 163 higher level electronic
structure calculations are necessary, but they are limited by the number of points affordable. GP
regression may be more promising to tackle these systems as it requires much fewer points than
NNs.
As demonstrated in the ML fit of friction tensors, ML methods are expected to play a more
prominent role in representing vectorial and tensorial properties of molecular systems, such as
dipoles and polarizabilities. These properties are much harder to represent than scalar properties
because of their complicated covariances. Apart from the NN-based work described above,159 GP
has also been successfully applied.164 We expect more advances in this direction.
An ultimate question is whether it is possible to derive the PES directly from scattering
experiments. This inversion problem is very difficult because scattering attributes, such as
differential cross sections, are sensitively dependent on the details of the PES. Some efforts in this
direction has already been made,92 and more progress can be expected in the near future.
Finally, we note that rapid advances in ML methodology will certainly help to stimulate
new applications in developing PESs. State-of-the-art deep NNs based methods have been
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advanced to represent PESs165, 166 and some of them have been benchmarked in gas phase reactive
systems. For example, the latest continuous-filter convolutional neural networks model, known as
SchNet,165 has been applied to accurately refit global PESs of the H/Cl + H2 reactions and the
OCHCO+ and H2CO systems.167 It is interesting to explore whether these new ML methods,
designed to represent molecular properties of many systems, have better representability than
regular NNs and GP based methods for more complex multichannel reactions and/or involving
multi-conformers. The use of ML methods to describe long-range interactions also represents a
challenge, but kernel based methods have been shown to be effective.168 On the other hand, optimal
data selection strategies can be developed in the concept of “active learning”, with the ultimate
goal of sampling as few as possible points to generate an as accurate as possible ML PES. Other
ML concepts, e.g., classification and clustering,169 are able to assist more efficient geometric
recognition and partition in constructing PESs. More investigations in this direction are expected.
We also note in passing that several open-source ML packages containing the essential machinery
are available.29, 40, 46 Overall, we are extremely optimistic about the future of ML assisted
development of PESs for inelastic and reactive scattering.
Biographies
Bin Jiang is Professor of Chemical Physics at University of Science and
Technology of China. He received his PhD degree from Nanjing University and
worked with Prof. Hua Guo at University of New Mexico as a postdoctoral fellow.
His research interests focus on machine learning method development and applications to potential
energy surfaces across gas phase and gas-surface interfaces, as well as quantum/classical dynamics
of gas-surface reactions.
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Jun Li is Professor of Chemistry at Chongqing University, China. He received his
PhD degree from Sichuan University and worked with Prof. Hua Guo at University
of New Mexico as a postdoctoral fellow. His research interests include potential
energy surfaces, reaction kinetics and dynamics, for gas phase systems.
Hua Guo is Distinguished Professor at University of New Mexico. He received his
D.Phil. from Sussex University (UK) with the late Prof. John Murrell and did a
postdoc with Prof. George Schatz at Northwestern University. His research interests cover
mechanisms, dynamics, and kinetics of gas phase reactions, photochemistry, and surface reactions.
Acknowledgements:
B. J. was supported by National Key R&D Program of China (2017YFA0303500),
National Natural Science Foundation of China (91645202 and 21722306), and Anhui Initiative in
Quantum Information Technologies (AHY090200). J. L. was supported by National Natural
Science Foundation of China (21973009 and 21573027), Chongqing Municipal Natural Science
Foundation (cstc2019jcyj-msxmX0087), and Alexander von Humboldt Foundation (Humboldt
Fellowship for Experienced Researchers). H. G. thanks for support from National Science
Foundation (Grant No. CHE-1462109), Department of Energy (Grant No. DE-SC0015997) and
Department of Defense (Grant Nos. FA9550-18-1-0413 and W911NF-19-1-0283), and is grateful
for a Humboldt Research Award from the Alexander von Humboldt Foundation.
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Figure 1. Contour plots of the 15 dimensional PIP-NN PES for the Cl + CH3OH → HCl + CH3O reaction (a) along RHCl and RHO, and for the Cl + CH3OH → HCl + CH2OH reaction (b) along RHCl and RCH. All other coordinates are optimized.
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Figure 2. Contour plots of the 90 dimensional AtNN PES for CO* + O* CO2* CO2 reaction on Pt(111) along the CO2 reaction path in the vicinity of the reactive and non-reactive transition states. The first layer of the Pt surface and the reminder of the CO2 coordinates were optimized. Reproduced from Ref. 144 with permission.
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