Assessing Permutationally Invariant Polynomial and Symmetric Gradient Domain Machine Learning Potential Energy Surfaces for H3O2.

The singly hydrated hydroxide anion OH-(H2O) is of central importance to a detailed molecular understanding of water; therefore, there is strong motivation to develop a highly accurate potential to describe this anion. While this is a small molecule, it is necessary to have an extensive data set of energies and, if possible, forces to span several important stationary points. Here, we assess two machine-learned potentials, one using the symmetric gradient domain machine learning (sGDML) method and one based on permutationally invariant polynomials (PIPs). These are successors to a PIP potential energy surface (PES) reported in 2004. We describe the details of both fitting methods and then compare the two PESs with respect to precision, properties, and speed of evaluation. While the precision of the potentials is similar, the PIP PES is much faster to evaluate for energies and energies plus gradient than the sGDML one. Diffusion Monte Carlo calculations of the ground vibrational state, using both potentials, produce similar large anharmonic downshift of the zero-point energy compared to the harmonic approximation of the PIP and sGDML potentials. The computational time for these calculations using the sGDML PES is roughly 300 times greater than using the PIP one.

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 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.

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.

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.

Electronic relaxation and dissociation dynamics in formaldehyde: pump wavelength dependence.

The effect of the incident UV pump wavelength on the subsequent excited state dynamics, electronic relaxation, and ultimate dissociation of formaldehyde is studied using first principles simulation and Coulomb explosion imaging (CEI) experiments. Transitions in a vibronic progression in the à ← X̃ absorption band are systematically prepared using a tunable UV source which generates pulses centered at 304, 314, 329, and 337 nm. We find, both via ab initio simulation and experimental results, that the rate of excited state decay and subsequent dissociation displays a prominent dependence on which vibronic transition in the absorption band is prepared by the pump. Our simulations predict that nonadiabatic transition rates and dissociation yields will increase by a factor of >100 as the pump wavelength is decreased from 337 to 304 nm. The experimental results and theoretical simulations are in broad agreement and both indicate that the dissociation rate plateaus rapidly after ≈2 ps following an ultrafast sub-ps rise.

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.

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.

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.

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.

Full-dimensional potential energy surface for acetylacetone and tunneling splittings.

We present a full-dimensional potential energy surface for acetylacetone (AcAc) using full and fragmented permutationally invariant polynomial approaches. Previously reported MP2/aVTZ energies and gradients are augmented by additional calculations at this level of theory for the fits. Numerous stationary points are reported as are the usual metrics to assess the precision of the fit. The electronic barrier height for the H-atom transfer is roughly 2.2 kcal mol-1. Diffusion Monte Carlo (DMC) calculations are used to calculate the ground state wavefunction and zero-point energy of acetylacetone. These together with fixed-node DMC calculations for the first excited-state provide the predicted tunneling splitting due to the barrier to H-transfer separating two equivalent wells. Simpler 1d calculations of this splitting are also reported for varying barrier heights including the CCSD(T) barrier height of 3.2 kcal mol-1. Based on those results the DMC splitting of 160 cm-1 with a statistical uncertainty of roughly 21 cm-1, calculated using the MP2-based PES, is estimated to decrease to 100 cm-1 for a barrier of 3.2 kcal mol-1. The fragmented surface is shown to be fast to evaluate.

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.

Δ-machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory.

"Δ-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients close to a coupled cluster (CC) level of accuracy. Here, we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation VLL→CC = VLL + ΔVCC-LL, and demonstrated for CH4, H3O+, and trans and cis-N-methyl acetamide (NMA), CH3CONHCH3. For these molecules, the LL PES, VLL, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients and ΔVCC-LL is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For CH4, these are new calculations adopting an aug-cc-pVDZ basis, for H3O+, previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA, new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA, training is done with 4696 CCSD(T) energies.

Presenting and sharing clinical data using the eTRIKS Standards Master Tree for tranSMART.

MOTIVATION: Standardization and semantic alignment have been considered one of the major challenges for data integration in clinical research. The inclusion of the CDISC SDTM clinical data standard into the tranSMART i2b2 via a guiding master ontology tree positively impacts and supports the efficacy of data sharing, visualization and exploration across datasets. RESULTS: We present here a schema for the organization of SDTM variables into the tranSMART i2b2 tree along with a script and test dataset to exemplify the mapping strategy. The eTRIKS master tree concept is demonstrated by making use of fictitious data generated for four patients, including 16 SDTM clinical domains. We describe how the usage of correct visit names and data labels can help to integrate multiple readouts per patient and avoid ETL crashes when running a tranSMART loading routine. AVAILABILITY AND IMPLEMENTATION: The eTRIKS Master Tree package and test datasets are publicly available at https://doi.org/10.5281/zenodo.1009098 and a functional demo installation at https://public.etriks.org/transmart/datasetExplorer/ under eTRIKS-Master Tree branch, where the discussed examples can be visualized.

A Machine Learning Approach for Rate Constants. II. Clustering, Training, and Predictions for the O(3P) + HCl → OH + Cl Reaction.

Following up on our recent paper, which reported a machine learning approach to train on and predict thermal rate constants over a large temperature range, we present new results by using clustering and new Gaussian process regression on each cluster. Each cluster is defined by the magnitude of the correction to the Eckart transmission coefficient. Instead of the usual protocol of training and testing, which is a challenge for present small database of exact rate constants, training is done on the full data set for each cluster. Testing is done by inputing hundreds of random values of the descriptors (within reasonable bounds). The new training strategy is applied to predict the rate constants of the O(3P) + HCl reaction on the 3A' and 3A″ potential energy surfaces. This reaction was recently focused on as a "stress test" for the ring polymer molecular dynamics method. Finally, this reaction is added to the databases and training is done with this addition. The freely available database and new Python software that evaluates the correction to the Eckart transmission coefficient for any reaction are briefly described.

Permutationally invariant polynomial potential energy surfaces for tropolone and H and D atom tunneling dynamics.

We report permutationally invariant polynomial (PIP) fits to energies and gradients for 15-atom tropolone. These include standard, augmented, and fragmented PIP bases. Approximately, 6600 energies and their associated gradients are obtained from direct-dynamics calculations using DFT/B3LYP/6-31+G(d) supplemented by grid calculations spanning an energy range up to roughly 35 000 cm-1. Three fragmentation schemes are investigated with respect to efficiency and fit precision. In addition, several fits are done with reduced weight for gradient data relative to energies. These do result in more precision for the H-transfer barrier height. The properties of the fits such as stationary points, harmonic frequencies, and the barrier to H-atom transfer are reported and compared to direct calculations. A previous 1D model is used to obtain the tunneling splitting for the ground vibrational state and qualitative predictions for excited vibrational states. This model is applied to numerous fits with different barrier heights and then used to extrapolate the H and D atom tunneling splittings to values at the CCSD(T)-F12 barrier. The extrapolated values are 2.3 and 0.14 cm-1, respectively for H and D. These are about a factor of two larger than experiment, but within the expected level of agreement with experiment for the 1D method used and the level of the electronic structure theory.

Full-dimensional, ab initio potential energy surface for glycine with characterization of stationary points and zero-point energy calculations by means of diffusion Monte Carlo and semiclassical dynamics.

A full-dimensional, permutationally invariant potential energy surface (PES) for the glycine amino acid is reported. A precise fit to energies and gradients calculated at the density functional theory (DFT)/B3LYP level of electronic-structure theory with Dunning's aug-cc-pVDZ basis set is performed involving 20 000 low-energy points and associated Cartesian gradients plus about 50 000 additional higher-energy points. The fact that newly calculated DFT/B3LYP energies for the main stationary points are close to the coupled-cluster single-double-triple [CCSD(T)] values, recently reported in the literature, provides reassurance about the accuracy of the constructed PES. Eight conformers and numerous saddle points are identified and characterized by describing geometries, relative stability, and harmonic frequencies. Stochastic and dynamical approaches are employed to study the vibrational ground state. Specifically, diffusion Monte Carlo simulations and approximate quantum dynamics, performed by means of the adiabatic switching semiclassical initial value representation technique, provide zero-point energies in excellent agreement with each other. The PES we report is sufficiently complete to permit spectroscopic and dynamical studies on glycine, which may be of interest to the biochemical and astrochemistry communities.

H atom Product Channels in the Ultraviolet Photodissociation of the 2-Propenyl Radical.

The H atom product channels in the ultraviolet photodissociation of 2-propenyl (CH2CCH3) radical were investigated in the wavelength region 224-248 nm using photofragment translational spectroscopy. The CH2CCH3 radicals were generated by 193 nm photodissociation of 2-chloropropene and 2-bromopropene precursors. The H atom photofragment yield spectra from both precursors revealed a broad feature peaking near 232 nm. The translational energy distributions of the H + C3H4 products peaked around 7-8 kcal/mol and extended close to the maximum excess energy. The fraction of the total available energy released as products' translation was nearly a constant (∼0.16 using the 2-chloropropene precursor and ∼0.18 using the 2-bromopropene precursor) in the wavelength range 224-248 nm. The angular distribution of the H atom product was isotropic. Quasi-classical trajectory (QCT) calculations were performed on the ground-state potential energy surface of CH2CCH3 for its decomposition at a 124 kcal/mol excitation energy (equivalent to 230 nm photolysis photon energy). The calculations yielded branching ratios for different dissociation product channels, 32% H + allene, 35% H + propyne, 0.5% H + cyclopropene, and 32% methyl + acetylene. The experimental and QCT translational energy distributions of the H atom loss channels qualitatively agreed, consistent with the main H atom product channels being the H + allene and H + propyne dissociations. The time scale of the 2-propenyl dissociation on the ground electronic state was calculated to be ∼2 ps, smaller compared to that of the overall UV photodissociation (≥10 ps, implied on the basis of the isotropic H atom product angular distribution). The mechanism of the UV photodissociation of 2-propenyl is consistent with unimolecular dissociation proceeding on the ground electronic state after internal conversion of the electronic excited states.

Teaching vibrational spectra to assign themselves.

A new paradigm for assigning vibrational spectra is described. Instead of proceeding from potential to Hamiltonian to eigenvalues/eigenvectors/intensities to spectrum, the new method goes "backwards" directly from spectrum to eigenvectors. The eigenvectors then "assign" the spectrum, in that they identify the basis states that contribute to each eigenvalue. To start, we demonstrate an algorithm that can obtain useful estimates of the eigenvectors connecting a real, symmetric Hamiltonian to its eigenvalues even if the only available information about the Hamiltonian is its diagonal elements. When this algorithm is augmented with information about transition intensities, it can be used to assign a complex vibrational spectrum using only information about (1) eigenvalues (the peak centers of the spectrum) and (2) a harmonic basis set (taken to be the diagonal elements of the Hamiltonian). Examples will be discussed, including application to the vibrationally complex spectral region of the formic acid dimer.