Tell Machine Learning Potentials What They Are Needed For: Simulation-Oriented Training Exemplified for Glycine.

Machine learning potentials (MLPs) are widely applied as an efficient alternative way to represent potential energy surfaces (PESs) in many chemical simulations. The MLPs are often evaluated with the root-mean-square errors on the test set drawn from the same distribution as the training data. Here, we systematically investigate the relationship between such test errors and the simulation accuracy with MLPs on an example of a full-dimensional, global PES for the glycine amino acid. Our results show that the errors in the test set do not unambiguously reflect the MLP performance in different simulation tasks, such as relative conformer energies, barriers, vibrational levels, and zero-point vibrational energies. We also offer an easily accessible solution for improving the MLP quality in a simulation-oriented manner, yielding the most precise relative conformer energies and barriers. This solution also passed the stringent test by diffusion Monte Carlo simulations.

No Headache for PIPs: A PIP Potential for Aspirin Runs Much Faster and with Similar Precision Than Other Machine-Learned Potentials.

Assessments of machine-learning (ML) potentials are an important aspect of the rapid development of this field. We recently reported an assessment of the linear-regression permutationally invariant polynomial (PIP) method for ethanol, using the widely used (revised) rMD17 data set. We demonstrated that the PIP approach outperformed numerous other methods, e.g., ANI, PhysNet, sGDML, and p-KRR, with respect to precision and notably with respect to speed [Houston et al., J. Chem. Phys. 2022, 156, 044120]. Here, we extend this assessment to the 21-atom aspirin molecule, using the rMD17 data set, with a focus on the speed of evaluation. Both energies and forces are used for training, and the precision of several PIPs is examined for both. Normal mode frequencies, the methyl torsional potential, and 1d vibrational energies for an OH stretch are presented. We show that the PIP approach achieves the level of precision obtained from other ML methods, e.g., atom-centered neural network methods, linear regression ACE, and kernel methods, as reported by Kovács et al. in J. Chem. Theory Comput. 2021, 17, 7696-7711. More significantly, we show that the PIP PESs run much faster than all other ML methods, whose timings were evaluated in that paper. We also show that the PIP PES extrapolates well enough to describe several internal motions of aspirin, including an OH stretch.

Formic Acid-Ammonia Heterodimer: A New Δ-Machine Learning CCSD(T)-Level Potential Energy Surface Allows Investigation of the Double Proton Transfer.

The formic acid-ammonia dimer is an important example of a hydrogen-bonded complex in which a double proton transfer can occur. Its microwave spectrum has recently been reported and rotational constants and quadrupole coupling constants were determined. Calculated estimates of the double-well barrier and the internal barriers to rotation were also reported. Here, we report a full-dimensional potential energy surface (PES) for this complex, using two closely related Δ-machine learning methods to bring it to the CCSD(T) level of accuracy. The PES dissociates smoothly and accurately. Using a 2d quantum model the ground vibrational-state tunneling splitting is estimated to be less than 10-4 cm-1. The dipole moment along the intrinsic reaction coordinate is calculated along with a Mullikan charge analysis and supports the mildly ionic character of the minimum and strongly ionic character at the double-well barrier.

A New A Priori Method to Avoid Calculation of Negligible Hamiltonian Matrix Elements in CI Calculation.

Hamiltonian matrices typically contain many elements that are negligibly small compared to the diagonal elements, even with methods to prune the underlying basis. Because for general potentials the calculation of H-matrix elements is a major part of the computational effort to obtain eigenvalues and eigenfunctions of the Hamiltonian, there is strong motivation to investigate locating these negligible elements without calculating them or at least avoid calculating them. We recently demonstrated an effective means to "learn" negligible elements using machine learning classification (J. Chem. Phys. 2023, 159, 071101). Here we present a simple, new method to avoid calculating them by using a cut-off value for the absolute difference in the quantum numbers for the bra and ket. This method is demonstrated for many of the same case studies as were used in the paper above, namely for realistic H-matrices of H2O, the vinyl radical, C2H3, and glycine, C2H5NO2. The new method is compared to the recently reported machine learning approach. In addition, we point out an important synergy between the two methods.

Ab Initio Potential Energy Surface for NaCl-H2 with Correct Long-Range Behavior.

We report a full dimensional ab initio potential energy surface for NaCl-H2 based on precise fitting of a large data set of CCSD(T)/aug-cc-pVTZ energies. A major goal of this fit is to describe the very long-range interaction accurately. This is done in this instance via the dipole-quadrupole interaction. The NaCl dipole and the H2 quadrupole are available through previous works over a large range of internuclear distances. We use these to obtain exact effect charges on each atom. Diffusion Monte Carlo calculations are done for the ground vibrational state using the new potential.

A Status Report on "Gold Standard" Machine-Learned Potentials for Water.

Owing to the central importance of water to life as well as its unusual properties, potentials for water have been the subject of extensive research over the past 50 years. Recently, five potentials based on different machine learning approaches have been reported that are at or near the "gold standard" CCSD(T) level of theory. The development of such high-level potentials enables efficient and accurate simulations of water systems using classical and quantum dynamical approaches. This Perspective serves as a status report of these potentials, focusing on their methodology and applications to water systems across different phases. Their performances on the energies of gas phase water clusters, as well as condensed phase structural and dynamical properties, are discussed.

Machine learning classification can significantly reduce the cost of calculating the Hamiltonian matrix in CI calculations.

Hamiltonian matrices in electronic and nuclear contexts are highly computation intensive to calculate, mainly due to the cost for the potential matrix. Typically, these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices of H2O, C2H3 (vinyl), and C2H5NO2 (glycine) using full-dimensional ab initio-based potential surfaces. We then show that many of these small elements can be replaced by zero with small errors of the resulting full set of eigenvalues, depending on the threshold value for this replacement. As a result of this empirical evidence, we investigate three machine learning approaches to predict the zero elements. This is shown to be successful for these H-matrices after training on a small set of calculated elements. For H-matrices of vinyl and glycine, of order 15 552 and 8828, respectively, training on a percent or so of elements is sufficient to obtain all eigenvalues with a mean absolute error of roughly 2 cm-1.

Interfacing q-AQUA with a Polarizable Force Field: The Best of Both Worlds.

Polarizable force fields are pervasive in the fields of computational chemistry and biochemistry; however, their empirical or semiempirical nature gives them both weaknesses and strengths. Here, we have developed a hybrid water potential, named q-AQUA-pol, by combining our recent ab initio q-AQUA potential with the TTM3-F water potential. The new potential demonstrates unprecedented accuracy ranging from gas-phase clusters, e.g., the eight low-lying isomers of the water hexamer, to the condensed phase, e.g., radial distribution functions, the self-diffusion coefficient, triplet OOO distribution, and the temperature dependence of the density. This represents a significant advancement in the field of polarizable machine learning potential and computational modeling.

Ring-Polymer Instanton Tunneling Splittings of Tropolone and Isotopomers using a Δ-Machine Learned CCSD(T) Potential: Theory and Experiment Shake Hands.

Tropolone, a 15-atom cyclic molecule, has received much interest both experimentally and theoretically due to its H-transfer tunneling dynamics. An accurate theoretical description is challenging owing to the need to develop a high-level potential energy surface (PES) and then to simulate quantum-mechanical tunneling on this PES in full dimensionality. Here, we tackle both aspects of this challenge and make detailed comparisons with experiments for numerous isotopomers. The PES, of near CCSD(T)-quality, is obtained using a Δ-machine learning approach starting from a pre-existing low-level DFT PES and corrected by a small number of approximate CCSD(T) energies obtained using the fragmentation-based molecular tailoring approach. The resulting PES is benchmarked against DF-FNO-CCSD(T) and CCSD(T)-F12 calculations. Ring-polymer instanton calculations of the splittings, obtained with the Δ-corrected PES are in good agreement with previously reported experiments and a significant improvement over those obtained using the low-level DFT PES. The instanton path includes heavy-atom tunneling effects and cuts the corner, thereby avoiding passing through the conventional saddle-point transition state. This is in contradistinction with typical approaches based on the minimum-energy reaction path. Finally, the subtle changes in the splittings for some of the heavy-atom isotopomers seen experimentally are reproduced and explained.

PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials.

We wish to describe a potential energy surface by using a basis of permutationally invariant polynomials whose coefficients will be determined by numerical regression so as to smoothly fit a dataset of electronic energies as well as, perhaps, gradients. The polynomials will be powers of transformed internuclear distances, usually either Morse variables, exp(-ri,j/λ), where λ is a constant range hyperparameter, or reciprocals of the distances, 1/ri,j. The question we address is how to create the most efficient basis, including (a) which polynomials to keep or discard, (b) how many polynomials will be needed, (c) how to make sure the polynomials correctly reproduce the zero interaction at a large distance, (d) how to ensure special symmetries, and (e) how to calculate gradients efficiently. This article discusses how these questions can be answered by using a set of programs to choose and manipulate the polynomials as well as to write efficient Fortran programs for the calculation of energies and gradients. A user-friendly interface for access to monomial symmetrization approach results is also described. The software for these programs is now publicly available.

Δ-Machine Learned Potential Energy Surfaces and Force Fields.

There has been great progress in developing machine-learned potential energy surfaces (PESs) for molecules and clusters with more than 10 atoms. Unfortunately, this number of atoms generally limits the level of electronic structure theory to less than the "gold standard" CCSD(T) level. Indeed, for the well-known MD17 dataset for molecules with 9-20 atoms, all of the energies and forces were obtained with DFT calculations (PBE). This Perspective is focused on a Δ-machine learning method that we recently proposed and applied to bring DFT-based PESs to close to CCSD(T) accuracy. This is demonstrated for hydronium, N-methylacetamide, acetyl acetone, and ethanol. For 15-atom tropolone, it appears that special approaches (e.g., molecular tailoring, local CCSD(T)) are needed to obtain the CCSD(T) energies. A new aspect of this approach is the extension of Δ-machine learning to force fields. The approach is based on many-body corrections to polarizable force field potentials. This is examined in detail using the TTM2.1 water potential. The corrections make use of our recent CCSD(T) datasets for 2-b, 3-b, and 4-b interactions for water. These datasets were used to develop a new fully ab initio potential for water, termed q-AQUA.

Semiclassical and VSCF/VCI Calculations of the Vibrational Energies of trans- and gauche-Ethanol Using a CCSD(T) Potential Energy Surface.

A recent full-dimensional Δ-Machine learning potential energy surface (PES) for ethanol is employed in semiclassical and vibrational self-consistent field (VSCF) and virtual-state configuration interaction (VCI) calculations, using MULTIMODE, to determine the anharmonic vibrational frequencies of vibration for both the trans and gauche conformers of ethanol. Both semiclassical and VSCF/VCI energies agree well with the experimental data. We find significant mixing between the VSCF basis states due to Fermi resonances between bending and stretching modes. The same effects are also accurately described by the full-dimensional semiclassical calculations. These are the first high-level anharmonic calculations using a PES, in particular a "gold-standard" CCSD(T) one.

A Machine Learning Approach for Rate Constants. III. Application to the Cl(2P) + CH4 → CH3 + HCl Reaction.

The temperature dependence of the thermal rate constant for the reaction Cl(3P) + CH4 → HCl + CH3 is calculated using a Gaussian Process machine learning (ML) approach to train on and predict thermal rate constants over a large temperature range. Following procedures developed in two previous reports, we use a training data set of approximately 40 reaction/potential surface combinations, each of which is used to calculate the corresponding database of rate constant at approximately eight temperatures. For the current application, we train on the entire data set and then predict the temperature dependence of the title reaction employing a "split" data set for correction at low and high temperatures to capture both tunneling and recrossing. The results are an improvement on recent RPMD calculations compared to accurate quantum ones, using the same high-level ab initio potential energy surface. Both tunneling at low temperatures and significant recrossing at high temperatures are observed to influence the rate constants. The recrossing effects, which are not described by TST and even sophisticated tunneling corrections, do appear in experiment at temperatures above around 600 K. The ML results describe these effects and in fact merge at 600 K with RPMD results (which can describe recrossing), and both are close to experiment at the highest experimental temperatures. These results are in accord with a recent high-level experiment-theory study of this reaction.

Quantum Calculations on a New CCSD(T) Machine-Learned Potential Energy Surface Reveal the Leaky Nature of Gas-Phase Trans and Gauche Ethanol Conformers.

Ethanol is a molecule of fundamental interest in combustion, astrochemistry, and condensed phase as a solvent. It is characterized by two methyl rotors and trans (anti) and gauche conformers, which are known to be very close in energy. Here we show that based on rigorous quantum calculations of the vibrational zero-point state, using a new ab initio potential energy surface (PES), the ground state resembles the trans conformer, but substantial delocalization to the gauche conformer is present. This explains experimental issues about identification and isolation of the two conformers. This "leak" effect is partially quenched when deuterating the OH group, which further demonstrates the need for a quantum mechanical approach. Diffusion Monte Carlo and full-dimensional semiclassical dynamics calculations are employed. The new PES is obtained by means of a Δ-machine learning approach starting from a pre-existing low level density functional theory surface. This surface is brought to the CCSD(T) level of theory using a relatively small number of ab initio CCSD(T) energies. Agreement between the corrected PES and direct ab initio results for standard tests is excellent. One- and two-dimensional discrete variable representation calculations focusing on the trans-gauche torsional motion are also reported, in reasonable agreement with experiment.

The MD17 datasets from the perspective of datasets for gas-phase "small" molecule potentials.

There has been great progress in developing methods for machine-learned potential energy surfaces. There have also been important assessments of these methods by comparing so-called learning curves on datasets of electronic energies and forces, notably the MD17 database. The dataset for each molecule in this database generally consists of tens of thousands of energies and forces obtained from DFT direct dynamics at 500 K. We contrast the datasets from this database for three "small" molecules, ethanol, malonaldehyde, and glycine, with datasets we have generated with specific targets for the potential energy surfaces (PESs) in mind: a rigorous calculation of the zero-point energy and wavefunction, the tunneling splitting in malonaldehyde, and, in the case of glycine, a description of all eight low-lying conformers. We found that the MD17 datasets are too limited for these targets. We also examine recent datasets for several PESs that describe small-molecule but complex chemical reactions. Finally, we introduce a new database, "QM-22," which contains datasets of molecules ranging from 4 to 15 atoms that extend to high energies and a large span of configurations.

q-AQUA: A Many-Body CCSD(T) Water Potential, Including Four-Body Interactions, Demonstrates the Quantum Nature of Water from Clusters to the Liquid Phase.

Many model potential energy surfaces (PESs) have been reported for water; however, none are strictly from "first-principles". Here we report such a potential, based on a many-body representation at the CCSD(T) level of theory up to the four-body interaction. The new PES is benchmarked for the isomers of the water hexamer for dissociation energies, harmonic frequencies, and unrestricted diffusion Monte Carlo (DMC) calculations of the zero-point energies of the Prism, Book, and Cage isomers. Dissociation energies of several isomers of the 20-mer agree well with recent benchmark energies. Exploratory DMC calculations on this cluster verify the robustness of the new PES for quantum simulations. The accuracy and speed of the new PES are demonstrated for standard condensed phase properties, i.e., the radial distribution function and the self-diffusion constant. Quantum effects are shown to be substantial for these observables and also needed to bring theory into excellent agreement with experiment.

Enabling Rapid and Accurate Construction of CCSD(T)-Level Potential Energy Surface of Large Molecules Using Molecular Tailoring Approach.

The construction of a potential energy surface (PES) of even a medium-sized molecule employing correlated theory, such as CCSD(T), is arduous due to the high computational cost involved. The present study reports the possibility of efficiently constructing such a PES of molecules containing up to 15 atoms and 550 basis functions by employing the fragment-based molecular tailoring approach (MTA) on off-the-shelf hardware. The MTA energies at the CCSD(T)/aug-cc-pVTZ level for several geometries of three test molecules, viz., acetylacetone, N-methylacetamide, and tropolone, are reported. These energies are in excellent agreement with their full calculation counterparts with a time advantage factor of 3-5. The energy barrier from the ground to transition state is also accurately captured. Further, we demonstrate the accuracy and efficiency of MTA for estimating the energy gradients at the CCSD(T) level. As a further application of our MTA methodology, the energies of acetylacetone at ∼430 geometries are computed at the CCSD(T)/aug-cc-pVTZ level and used for generating a Δ-machine learning (Δ-ML) PES. This leads to the H-transfer barrier of 3.02 kcal/mol, well in agreement with the benchmarked barrier of 3.19 kcal/mol. The fidelity of this Δ-ML PES is examined by geometry optimization and normal mode frequency calculations of global minima and saddle point geometries. We trust that the present work is a major development for the rapid and accurate construction of PES at the CCSD(T) level for molecules containing up to 20 atoms and 600 basis functions using off-the-shelf hardware.

Permutationally invariant polynomial regression for energies and gradients, using reverse differentiation, achieves orders of magnitude speed-up with high precision compared to other machine learning methods.

Permutationally invariant polynomial (PIP) regression has been used to obtain machine-learned potential energy surfaces, including analytical gradients, for many molecules and chemical reactions. Recently, the approach has been extended to moderate size molecules with up to 15 atoms. The algorithm, including "purification of the basis," is computationally efficient for energies; however, we found that the recent extension to obtain analytical gradients, despite being a remarkable advance over previous methods, could be further improved. Here, we report developments to further compact a purified basis and, more significantly, to use the reverse differentiation approach to greatly speed up gradient evaluation. We demonstrate this for our recent four-body water interaction potential. Comparisons of training and testing precision on the MD17 database of energies and gradients (forces) for ethanol against numerous machine-learning methods, which were recently assessed by Dral and co-workers, are given. The PIP fits are as precise as those using these methods, but the PIP computation time for energy and force evaluation is shown to be 10-1000 times faster. Finally, a new PIP potential energy surface (PES) is reported for ethanol based on a more extensive dataset of energies and gradients than in the MD17 database. Diffusion Monte Carlo calculations that fail on MD17-based PESs are successful using the new PES.

A CCSD(T)-Based 4-Body Potential for Water.

High-level, ab initio calculations find that the 4-body (4-b) interaction is needed to account for near-100% of the total interaction energy for water clusters as large as the 21-mer. Motivated by this, we report a permutationally invariant polynomial potential energy surface (PES) for the 4-body interaction. This machine-learned PES is a fit to 2119 symmetry-unique, CCSD(T)-F12a/haTZ 4-b interaction energies. Configurations for these come from tetramer direct-dynamics calculations, fragments from an MD water simulation at 300 K, and tetramer fragments in a variety of water clusters. The PIP basis is purified to ensure that the PES goes rigorously to zero in monomer+trimer and dimer+dimer dissociations. The 4-b energies of isomers of the hexamer calculated with the new PES are shown to be in better agreement with benchmark CCSD(T) results than those from the MB-pol potential. Tests on larger clusters further validate the high-fidelity of the PES. The PES is shown to be fast to evaluate, taking 2.4 s for 105 evaluations on a single core of 2.4 GHz Intel Xeon processor, and significantly faster using a parallel version of the PES.

MULTIMODE Calculations of Vibrational Spectroscopy and 1d Interconformer Tunneling Dynamics in Glycine Using a Full-Dimensional Potential Energy Surface.

A full-dimensional, permutationally invariant polynomial potential energy surface for glycine recently reported (R. Conte et al., J. Chem. Phys. 2020, 153, 244301) is used with the code MULTIMODE to determine the IR absorption spectra for Conformers I and II using a new separable dipole moment function. The calculated spectra agree well with the experimental ones. The full-dimensional nature of the potential allows us also to examine dynamical results, such as tunneling rates. Remarkably, using a one-dimensional path based on the potential energy surface to estimate the tunneling rate from Conformer VI to Conformer I, good agreement is found with the recent experimental measurement. Finally a brief comparison of our potential energy surface with a recently reported sGDML one is made.