Joel M. Bowman

Vibrational second-order perturbation theory is a widely used method to perform postharmonic calculations of vibrational energies and infrared (IR) and Raman spectroscopy. We recently reported an application of this approach for vibrational energies using Fortran and Python software with machine-learned potentials (MLPs), with an application to 21-atom aspirin. Here, the software reported previously is extended to perform calculations of IR and Raman spectroscopy. Applications are made to six molecules, with up to 31 atoms, considered in a recent paper by Kapil and co-workers using MACE-based software for the MLP (MACE-OFF) and dipole moment/polarizability (MACE-MDP) to perform harmonic calculations of IR and Raman spectra with a linear approximation for the dipole and polarizability. Large differences are seen between those and the vibrational second-order perturbation theory (VPT2) spectra for XH-stretch fundamentals, and comparisons with experimental IR spectra for three of these molecules plus cholesterol show significant improvement of the VPT2 spectra over the harmonic ones. The extendability of the new software is demonstrated for 74-atom cholesterol using MACE software. The weakness of the VPT2 approach is shown for the IR spectrum of the formic acid dimer, using our PIP potential and dipole moment surface.

There has been a veritable explosion of methods and software to perform machine-learned regression on datasets of electronic energies and forces to develop high-dimensional machine-learned potential energy surfaces (ML-PESs). A major, but not deeply studied aspect is how well different ML-PESs represent the same dataset on which they were trained, beyond the standard statistical precision measures. Here, this is examined using several “stress tests,” for two widely used machine-learned potential approaches. One is based on permutationally invariant polynomial linear least-square regression and the other is the message-passing neural network PhysNet approach. The energies from the two PESs are directly compared as are the IR spectra determined from vibrational self-consistent field/vibrational configuration interaction calculations using fitted dipole moment surfaces. In addition, tunneling splittings for the hydrogen transfer between two equivalent structures are reported using three different methods: ring polymer instanton theory, diffusion Monte Carlo simulations, and the Q i m -path method. These calculations require the evaluation of on the order of one billion energies that are widely dispersed in the 15-dimensional configurational space. The two PESs yield results for these quantities in excellent agreement with each other.

Vibrational second order perturbation theory is a widely used method to perform post-harmonic calculations of vibrational energies and infrared (IR) and Raman spectroscopy. We recently reported an application of this approach for vibrational energies using Fortran and Python software with machine-learned potentials, with an application to 21-atom aspirin. Here we report extensions and applications to perform the calculations of IR and Raman spectroscopy. Applications are made to six molecules, with up to 31 atoms, considered in a recent paper by Kapil and co-workers using MACE-based software for the MLP (MACE-OFF) and dipole moment/polarizability (MACE-MDP) to perform double-harmonic calculations of IR and Raman spectra. Large differences are seen between those and the VPT2 spectra for XH-stretch fundamentals, and comparisons with experimental IR spectra show significant improvement of the VPT2 spectra over the double-harmonic calculations. The extendability of the new software is demonstrated for 74-atom cholesterol using MACE software. The strength and weakness of the VPT2 approach is shown for the IR spectrum of the formic acid dimer, using our PIP potential and dipole moment surface.

The rigid-rotor/harmonic-oscillator model is often insufficient for unbiased comparison with high-resolution microwave data and infrared measurements of medium-sized molecules, but direct high-level anharmonic force fields are rarely affordable for targets with many tens of normal modes. Here, we examine a pragmatic quartic force field/second-order vibrational perturbation theory (QFF/VPT2) route, in which different electronic structure levels are used for different layers of the calculation. In the L1//L0 implementation, a low level (L0) supplies the cubic and semidiagonal quartic force constants, whereas a high level (L1) supplies the equilibrium geometry and either the complete or a mode-selected harmonic reference. In this protocol, an already available machine-learning (ML)-derived QFF can be upgraded selectively with new high-level harmonic information, and the lower-level anharmonic layer does not need to be rebuilt or retrained. Thus, all harmonic terms, or only selected ones, can be promoted to L1, provided that the correspondence between the L0 and L1 coordinates is controlled. The approach is tested on benzoic acid and on the two observed conformers of aspirin, where the molecules are large enough that analytical high-order derivatives and full high-level QFFs are not routine options. Microwave and infrared (IR) observables are used together because they test different parts of the model: rotational constants probe the equilibrium geometry and vibrational averaging, whereas IR spectra probe the anharmonic force field and intensity patterns. Benzoic acid shows that L1//L0 and L1//L1 are equivalent within about 10 reciprocal centimeters for the present spectroscopic purpose and identifies modes with large OH stretching content as the relevant targets for L1 harmonic repair. Aspirin preserves the same pattern, despite its larger size and conformational flexibility. For the diagnostic OH stretch of aspirin I, direct finite differences and a preexisting permutationally invariant polynomial machine-learned potential give identical L0//L0 and L1//L0 VPT2 fundamentals. The combined microwave/IR analysis therefore supports a controlled, mode-selective high-level correction of low-level or machine-learned anharmonic layers while also clarifying the limits of the strategy when a high-quality machine-learned potential is not already available.

The determination of quartic force fields for use in vibrational second-order perturbation (VPT2) calculations, currently available in numerous electronic structure packages, becomes very expensive as the size of the molecule increases, especially if high-level coupled-cluster theory is used. Machine-learned potentials (MLPs) for large molecules and clusters offer a viable alternative to obtaining the quartic force field (QFF). Here, we report Fortran and Python software to determine the QFF and perform VPT2 calculations of energies from the MLPs. We describe this software and then apply it to H₂O and protonated oxalate as the test cases. The Fortran software is applied to 21-atom aspirin using a fast MLP reported by us. Despite the fact that there are 32,509 unique cubic force constants for aspirin, the computer time to calculate them using this MLP is trivial, i.e., around 1 min. The new software provides an efficient way to calculate quantum anharmonic energies, using the established VPT2 methodology, for machine learned potentials of large molecules.

Full-dimensional simulations reveal roaming as a general, nontraditional pathway. Its distinctive dynamics and energy redistribution significantly affect product branching and reaction rates, beyond the conventional minimum-energy-path paradigm.

Computational vibrational spectroscopy beyond the harmonic approximation relies on the molecular potential and ideally dipole and possibly higher moments of charge distributions. In the past decade, there has been a paradigm shift in generating highly accurate Machine-Learned potentials (MLPs). These are precise fits to thousands of electronic energies, using modern methods of regression. With such MLPs, it is possible to combine these with a variety of post-harmonic quantum methods ranging from perturbation theory to full variational calculations. After a short review of these methods, we focus on vibrational self-consistent field and configuration interaction (VSCF + VCI) calculations, as implemented in the code MULTIMODE. Two applications of this software to complex parts of the infrared spectra of formic acid dimer and the protonated oxalate anion are presented. Two new interfaces to MULTIMODE are then given. One is a Python-based GUI to enable user-friendly input to MULTIMODE. The second interface, PyFort, which is written in Fortran, uses MLPs written in Python in MULTIMODE via a C wrapper. Demonstrations of this are given for a PhysNet potential of Meuwly and co-workers for protonated oxalate anion (C2O4H−) and for the “universal” force field MACE-OFF of Csányi and co-workers. MULTIMODE VSCF + VCI vibrational energies of C2O4H− using the PhysNet MLP agree well with those using a permutationally invariant potential, trained on the datasets used to train the PhysNet MLP. A test of the MACE-OFF interface is done for H2CO. The PyFort software for both these examples is provided in the supplementary material.

Reduced dimensional models of tunneling have been a focus of theoretical chemistry since the birth of this field. The simplest model is the "reaction path" that connects a saddle point to the associated minima describing the reactant and the product. The widely used intrinsic reaction coordinate is a curved path that, perforce, contains a complex kinetic energy operator. In this paper, we examine a rectilinear reaction path, denoted the Q_im-path, and the associated 1d Hamiltonian. In the simplest version, the generally small Coriolis coupling terms are neglected. This Hamiltonian was introduced and shown to produce tunneling splittings of H- and D-transfer of malonaldehyde with good accuracy by Wang and Bowman in 2008. Here, we present further tests of this approach for tunneling splittings of H- and D-transfer for a number of molecules with symmetric double wells. These are tropolone, acetylacetone, malonaldehyde, and protonated oxalate anion, for which high-quality machine-learned potentials have been reported. Tests make use of the available benchmark results, including those from diffusion Monte Carlo calculations and, for tropolone, ring polymer instanton results. Extensions of the method for asymmetric double wells due to asymmetric isotope substitution and potential asymmetry, as well as to more than one degree of freedom, are reported.

We report and analyze vibrational self-consistent field and configuration interaction calculations of the infrared spectrum of protonated oxalate using precise machine-learned fits to electronic energies and dipole moment surfaces. For these fits, we use recently reported data by Meuwly and co-workers. Δ-machine learning is employed to elevate the MP2-level potential to the CCSD(T)/aug-cc-pVTZ level. In contrast to vibrational second-order perturbation theory and molecular dynamics simulations of this well-studied anion, the highly dispersed band in the expected OH-stretch region of the experimental spectrum is captured both in breadth and position. Calculations are also done for deuterated and tritiated isotopologs. The analysis of the eigenstates suggests that this band may exhibit “quantum chaos.”

Given the central importance of the protonated water dimer to the study of the hydrated proton, we report new fits to the previous CCSD(T) data set of Huang, Braams, and Bowman (HBB) that are more precise and, unlike HBB, provide fast gradients. The new fits, like the HBB one, are based on linear regression with permutationally invariant polynomials (PIPs). The fast gradients are provided via reverse differentiation. They cost roughly just three times the cost for an energy call and are roughly 20 times faster than the HBB numerical gradients. The two new PESs are fits to the original HBB data sets up to roughly 60,000 and to 110,000 cm^-1. Comparisons to the CCSD(T) benchmarks and to the HBB results for stationary points and Diffusion Monte Carlo ZPEs are reported and show good agreement.