Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions

Compensation States Approach in the Hybrid Diabatization Scheme: Extension to Multidimensional Data and Properties

The diabatization of reactive systems for more than just a couple of states is a very demanding problem and generally requires advanced diabatization techniques. Especially for dissociative processes, the drastic changes in the adiabatic wave functions often would require large diabatic state bases, which quickly become impractical. Recently, we addressed this problem by the compensation states approach developed in the context of our hybrid diabatization scheme. This scheme utilizes wave function as well as energy data in combination with a diabatic potential model. In regions where the initial diabatic state basis becomes insufficient for an appropriate representation of the adiabatic states, new model states are generated. The new model states compensate for the state space not spanned by the initial diabatic basis. Such a compensation state is obtained by projecting the initial diabatic state space out of the adiabatic wave function. This yields a very efficient basis representation of the electronic Hamiltonian. The present work presents two new aspects. First, it is shown how other operators like the spin-orbit operator in the framework of the Effective Relativistic Coupling by Asymptotic Representation (ERCAR) can be evaluated in this compact model state space without losing the correct wave function information and accuracy. Second, the extension of the approach to multidimensional potential energy surface models is presented for methyl iodide including the C-I dissociation coordinate and the angular H3C-I bending coordinates.

The effective relativistic coupling by asymptotic representation approach for molecules with multiple relativistic atoms

The Effective Relativistic Coupling by Asymptotic Representation (ERCAR) approach is a method to generate fully coupled diabatic potential energy surfaces (PESs) including relativistic effects, especially spin-orbit coupling. The spin-orbit coupling of a full molecule is determined only by the atomic states of selected relativistically treated atoms. The full molecular coupling effect is obtained by a diabatization with respect to asymptotic states, resulting in the correct geometry dependence of the spin-orbit effect. The ERCAR approach has been developed over the last decade and initially only for molecules with a single relativistic atom. This work presents its extension to molecules with more than a single relativistic atom using the iodine molecule as a proof-of-principle example. The theory for the general multiple atomic ERCAR approach is given. In this case, the diabatic basis is defined at the asymptote where all relativistic atoms are separated from the remaining molecular fragment. The effective spin-orbit operator is then a sum of spin-orbit operators acting on isolated relativistic atoms. PESs for the iodine molecule are developed within the new approach and it is shown that the resulting fine structure states are in good agreement with spin-orbit ab initio calculations.

Using Collocation to Solve the Schrödinger Equation

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

Exact quantum dynamics developments for floppy molecular systems and complexes

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

High-dimensional neural network potentials for accurate vibrational frequencies: the formic acid dimer benchmark

In recent years, machine learning potentials (MLP) for atomistic simulations have attracted a lot of attention in chemistry and materials science. Many new approaches have been developed with the primary aim to transfer the accuracy of electronic structure calculations to large condensed systems containing thousands of atoms. In spite of these advances, the reliability of modern MLPs in reproducing the subtle details of the multi-dimensional potential-energy surface is still difficult to assess for such systems. On the other hand, moderately sized systems enabling the application of tools for thorough and systematic quality-control are nowadays rarely investigated. In this work we use benchmark-quality harmonic and anharmonic vibrational frequencies as a sensitive probe for the validation of high-dimensional neural network potentials. For the case of the formic acid dimer, a frequently studied model system for which stringent spectroscopic data became recently available, we show that high-quality frequencies can be obtained from state-of-the-art calculations in excellent agreement with coupled cluster theory and experimental data.

Neural Network Potentials: A Concise Overview of Methods

In the past two decades, machine learning potentials (MLPs) have reached a level of maturity that now enables applications to large-scale atomistic simulations of a wide range of systems in chemistry, physics, and materials science. Different machine learning algorithms have been used with great success in the construction of these MLPs. In this review, we discuss an important group of MLPs relying on artificial neural networks to establish a mapping from the atomic structure to the potential energy. In spite of this common feature, there are important conceptual differences among MLPs, which concern the dimensionality of the systems, the inclusion of long-range electrostatic interactions, global phenomena like nonlocal charge transfer, and the type of descriptor used to represent the atomic structure, which can be either predefined or learnable. A concise overview is given along with a discussion of the open challenges in the field.

Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

On the Cartesian Representation of the Molecular Polarizability Tensor Surface by Polynomial Fitting to Ab Initio Data

We describe an approach to constructing an analytic Cartesian representation of the molecular dipole polarizability tensor surface in terms of polynomials in interatomic distances with a training set of ab initio data points obtained from a molecular dynamics (MD) simulation or by any other available means. The proposed formulation is based on a perturbation treatment of the unmodified point dipole polarizability model of Applequist [ J. Am. Chem. Soc. 1972, 94, 2952] and is shown here to be, by construction (i) free of short-range or other singularities or discontinuities, (ii) symmetric and translationally invariant, and (iii) nonreliant on a body-fixed coordinate system. Permutational invariance of like nuclei is demonstrated to be readily applicable, making this approach useful for highly fluxional and reactive systems. Derivation of the method is described in detail, adding brief didactic numerical examples of H₂ and H₂O and concluding with an MD simulation of the Raman spectrum of H₅O₂⁺ at 300 K with the polarizability tensor fitted to CCSD(T)/aug-cc-pVTZ data obtained using the HBB-4B potential [ J. Chem. Phys. 2005, 122, 044308].

Machine Learning Force Fields

In recent years, the use of machine learning (ML) in computational chemistry has enabled numerous advances previously out of reach due to the computational complexity of traditional electronic-structure methods. One of the most promising applications is the construction of ML-based force fields (FFs), with the aim to narrow the gap between the accuracy of ab initio methods and the efficiency of classical FFs. The key idea is to learn the statistical relation between chemical structure and potential energy without relying on a preconceived notion of fixed chemical bonds or knowledge about the relevant interactions. Such universal ML approximations are in principle only limited by the quality and quantity of the reference data used to train them. This review gives an overview of applications of ML-FFs and the chemical insights that can be obtained from them. The core concepts underlying ML-FFs are described in detail, and a step-by-step guide for constructing and testing them from scratch is given. The text concludes with a discussion of the challenges that remain to be overcome by the next generation of ML-FFs.

Mean-field density matrix decompositions

We introduce new and robust decompositions of mean-field Hartree-Fock and Kohn-Sham density functional theory relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning one-electron reduced density matrices into either bond-wise or atomic contributions, are compared to alternatives from the literature with regard to both molecular energies and dipole moments. Besides commenting on possible applications as an interpretative tool in the rationalization of certain electronic phenomena, we demonstrate how decomposed mean-field theory makes it possible to expose and amplify compositional features in the context of machine-learned quantum chemistry. This is made possible by improving upon the granularity of the underlying data. On the basis of our preliminary proof-of-concept results, we conjecture that many of the structure-property inferences in existence today may be further refined by efficiently leveraging an increase in dataset complexity and richness.

Neural Network Potential Energy Surfaces for Small Molecules and Reactions

We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

Quantum Chemistry in the Age of Machine Learning

As the quantum chemistry (QC) community embraces machine learning (ML), the number of new methods and applications based on the combination of QC and ML is surging. In this Perspective, a view of the current state of affairs in this new and exciting research field is offered, challenges of using machine learning in quantum chemistry applications are described, and potential future developments are outlined. Specifically, examples of how machine learning is used to improve the accuracy and accelerate quantum chemical research are shown. Generalization and classification of existing techniques are provided to ease the navigation in the sea of literature and to guide researchers entering the field. The emphasis of this Perspective is on supervised machine learning.

A reactive, scalable, and transferable model for molecular energies from a neural network approach based on local information

Despite the ever-increasing computer power, accurate ab initio calculations for large systems (thousands to millions of atoms) remain infeasible. Instead, approximate empirical energy functions are used. Most current approaches are either transferable between different chemical systems, but not particularly accurate, or they are fine-tuned to a specific application. In this work, a data-driven method to construct a potential energy surface based on neural networks is presented. Since the total energy is decomposed into local atomic contributions, the evaluation is easily parallelizable and scales linearly with system size. With prediction errors below 0.5 kcal mol^-1 for both unknown molecules and configurations, the method is accurate across chemical and configurational space, which is demonstrated by applying it to datasets from nonreactive and reactive molecular dynamics simulations and a diverse database of equilibrium structures. The possibility to use small molecules as reference data to predict larger structures is also explored. Since the descriptor only uses local information, high-level ab initio methods, which are computationally too expensive for large molecules, become feasible for generating the necessary reference data used to train the neural network.

Fitting potential energy surfaces to sum-of-products form with neural networks using exponential neurons

In this paper, the use of the neural network (NN) method with exponential neurons for directly fitting ab initio data to generate potential energy surfaces (PESs) in sum-of-product form will be discussed. The utility of the approach will be highlighted using fits of CS2, HFCO, and HONO ground state PESs based upon high-level ab initio data. Using a generic interface between the neural network PES fitting, which is performed in MATLAB, and the Heidelberg multi-configuration time-dependent Hartree (MCTDH) software package, the PESs have been tested via comparison of vibrational energies to experimental measurements. The review demonstrates the potential of the PES fitting method, combined with MCTDH, to tackle high-dimensional quantum dynamics problems.

Two new methods for computing vibrational energy levels

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix–vector products.

Representing potential energy surfaces by high-dimensional neural network potentials

The development of interatomic potentials employing artificial neural networks has seen tremendous progress in recent years. While until recently the applicability of neural network potentials (NNPs) has been restricted to low-dimensional systems, this limitation has now been overcome and high-dimensional NNPs can be used in large-scale molecular dynamics simulations of thousands of atoms. NNPs are constructed by adjusting a set of parameters using data from electronic structure calculations, and in many cases energies and forces can be obtained with very high accuracy. Therefore, NNP-based simulation results are often very close to those gained by a direct application of first-principles methods. In this review, the basic methodology of high-dimensional NNPs will be presented with a special focus on the scope and the remaining limitations of this approach. The development of NNPs requires substantial computational effort as typically thousands of reference calculations are required. Still, if the problem to be studied involves very large systems or long simulation times this overhead is regained quickly. Further, the method is still limited to systems containing about three or four chemical elements due to the rapidly increasing complexity of the configuration space, although many atoms of each species can be present. Due to the ability of NNPs to describe even extremely complex atomic configurations with excellent accuracy irrespective of the nature of the atomic interactions, they represent a general and therefore widely applicable technique, e.g. for addressing problems in materials science, for investigating properties of interfaces, and for studying solvation processes.

Construction of high-dimensional neural network potentials using environment-dependent atom pairs

An accurate determination of the potential energy is the crucial step in computer simulations of chemical processes, but using electronic structure methods on-the-fly in molecular dynamics (MD) is computationally too demanding for many systems. Constructing more efficient interatomic potentials becomes intricate with increasing dimensionality of the potential-energy surface (PES), and for numerous systems the accuracy that can be achieved is still not satisfying and far from the reliability of first-principles calculations. Feed-forward neural networks (NNs) have a very flexible functional form, and in recent years they have been shown to be an accurate tool to construct efficient PESs. High-dimensional NN potentials based on environment-dependent atomic energy contributions have been presented for a number of materials. Still, these potentials may be improved by a more detailed structural description, e.g., in form of atom pairs, which directly reflect the atomic interactions and take the chemical environment into account. We present an implementation of an NN method based on atom pairs, and its accuracy and performance are compared to the atom-based NN approach using two very different systems, the methanol molecule and metallic copper. We find that both types of NN potentials provide an excellent description of both PESs, with the pair-based method yielding a slightly higher accuracy making it a competitive alternative for addressing complex systems in MD simulations.

Vibrations of H+(D+) in stoichiometric LiNbO3 single crystal

A first principles quantum mechanical calculation of the vibrational energy levels and transition frequencies associated with protons in stoichiometric LiNbO(3) single crystal has been carried out. The hydrogen contaminated crystal has been approximated by a model one obtains by translating a supercell, i.e., a cluster of LiNbO(3) unit cells containing a single H(+) and a Li(+) vacancy. Based on the supercell model an approximate Hamiltonian operator describing vibrations of the proton sublattice embedded in the host crystal has been derived. It is further simplified to a sum of uncoupled Hamiltonian operators corresponding to different wave vectors (ks) and each describing vibrations of a quasi-particle (quasi-proton). The three dimensional (3D) Hamiltonian operator of k=0 has been employed to calculate vibrational levels and transition frequencies. The potential energy surface (PES) entering this Hamiltonian operator has been calculated point wise on a large set of grid points by using density functional theory, and an analytical approximation to the PES has been constructed by non-parametric approximation. Then, the nuclear motion Schrödinger equation has been solved by employing the method of discrete variable representation. It has been found that the (quasi-)H(+) vibrates in a strongly anharmonic PES. Its vibrations can be described approximately as a stretching, and two orthogonal bending vibrations. The theoretically calculated transition frequencies agree within 1% with those experimentally determined, and they have allowed the assignment of one of the hitherto unassigned bands as a combination of the stretching and the bending of lower fundamental frequency.