In recent years, there has been increasing interest in combining machine learning with computational chemistry. Many machine learning applications in theoretical chemistry have focused on the development of potential energy surfaces (PES) based on ab initio data pointsManzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665 Another computational resource that has gained popularity in recent years is the use of Graphics Processing Units (GPUs) to accelerate calculations such as molecular dynamics simulations Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25 In this work, we take advantage of the knowledge gained from both of these areas, and apply these approaches to the evaluations of the potential energy for Diffusion Monte Carlo (DMC) calculations. Diffusion Monte Carlo is a stochastic simulation approach through which an ensemble of localized functions (walkers) converges to the ground state vibrational wave function at long propagation times. This is accomplished through the ensemble randomly sampling a potential energy surface. These calculations require on the order of 10⁸ to 10¹0 potential energy evaluations, making the call to a potential energy surface the most expensive part of these calculations. To this end, we have developed a general technique for “learning” already existing potential energy surfaces using neural networks, which can then easily be evaluated on GPUs using TensorFlow. The technique entails running a small-scale DMC simulation to collect training data, coordinates and energies, and using this data to fit a subspace of the potential energy surface that is relevant for the DMC simulations. We find that using the neural network potential energy surface maintains the fidelity of the wave function while also enabling the use of ensemble sizes orders of magnitude larger than previously used for the same amount of wall time. We test this method on small systems such as CH5+, H2O, and water dimer.

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Manzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665. Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25.

Diffusion Monte Carlo (DMC) is a technique for obtaining the ground-state solution to the vibrational time-independent Schrödinger equation based on a stochastic sampling of an electronic potential energy surface (PES). Ideally, the electronic energies used in DMC are calculated at the CCSD(T) level of theory. Recently, Tom Miller’s group at Caltech has developed a technique for calculating accurate CCSD(T) corrections to the electronic energy at the cost of a standard Hartree-Fock calculation using Gaussian process regression, a machine learning techniqueWelborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779Cheng, L., Kovachki, N. B., Welborn, M., Miller, T. M. JCTC. 2019 15 (12), 6668-6677. DMC is a well-suited technique to validate these PESs, referred to as MOB-ML surfaces, since DMC samples the entire region of the PES in which the vibrational ground state wave function has amplitude. Despite the speed of these MOB-ML PESs relative to an ab initio calculation, calculating the Hartree-Fock energy for every one of thousands of DMC configurations over tens of thousands of time-steps makes the use of these surfaces computationally challenging. To this end, we combined two techniques to bring these calculations into computational feasibility. First, we implemented a DMC algorithm to take advantage of massively parallel high performance computing environments. Second, we use this DMC code to collect training data for a neural network (NN). We use a generic algorithm to train the NN to learn the MOB-ML surface based on the points sampled by the DMC. Using TensorFlow, the evaluation of the NN can then easily be done using graphics processing units (GPUs). We have performed small-scale DMC calculations on H2O and CH5+ using these MOB-ML surfaces, and have also run large-scale NN+MOB-ML DMC simulations to obtain a high resolution ground state vibrational wave function and measure of the zero-point energy. Due to the generality of both the MOB-ML surface generation and the NN fitting workflow, we can easily extend this work to larger systems, such as large ion-water clusters or a Criegee intermediate, where running the DMC with traditional electronic structure would be intractable.

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Welborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779

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In ab initio calculations of diatomic spectroscopy, discrete points are typically computed along the potential energy curve. The points are fitted to a continuous function, such as a polynomial, a Morse function, or a spline, so that ro-vibrational energy levels can be computed and conventional constants extracted. The choice of fitting function is arbitrary and affects the results, which is undesirable in predictive work. Here we suggest using lower-level theory to create a dense, high-resolution grid, to be used as a guide. Instead of fitting the sparse, high-level data directly, the energy differences between the high-level points and the guiding potential are fitted. This simple strategy reduces the uncertainty from the choice of fitting function, yielding more precise predictions.

Based on the the Coriolis interaction conceived by Teller and Tisza (1932), the fundamental Wilson-Howard Hamiltonian (WHH), formulated in 1936Wilson, Jr., E. B. & Howard, J. B. 1936 J. Chem. Phys. 4, 260 includes everything about the vibration-rotation interaction. Enormous technical advances followed both in vibrational and rotational spectroscopy and have produced extremely rich data. Then God sent us the genius Jim Watson to deeply study the WHH and harvest all its fruits in the most sublime and direct manner. Out of the many jewels Watson left us I single out three works. (1) The theory of the centrifugal distortion of asymmetric-top molecules (1966)Watson, J. K. G. 1966 J. Chem. Phys. 45, 1360; 46, 1935; 1968 J. Chem. Phys. 48, 4517 To solve the indeterminacy of centrifugal distortion constants of non-planar asymmetric tops, Watson discovered an angular momentum operator P_xP_yP_z + P_zP_yP_x which commutes with the rotational Hamiltonian and produces an additional relation between centrifugal constants making the solutions possible. This is the most frequently quoted paper in the field of molecular spectroscopy. (2) Simplification of the WHH (1968)Watson, J. K. G. 1968 Mol. Phys. 15, 479 For the first term of rearranged WHH H = 1/2Σ(Π_α-π_α)μ_αβ(Π_β-π_β) Watson discovered commutation relation [π_α, μ_αβ]=0. using absolutely beautiful tensor algebra, thus simplifying the Hamiltonian. This is the most fundamental work in the field of molecular spectroscopy and represents the triumph of tensor algebra. (3) Forbidden rotational transitions (1971)Watson, J. K. G. 1971 J. Mol. Speectrosc. 40, 536 The ∆K=0 selection rule of a symmetric top molecule corresponds to cylindrical symmetry. Since the actual symmetry is C₃ rather than cylindrical for say NH₃, ∆K=3 forbidden transitions are weakly allowed. Watson's theory showed that non-polar molecules such as CH₄ and H₃⁺ are polar in some rotational levels and undergo forbidden rotational transitions. This theory has greatly influenced molecular astrophysics. e.g. Oka, T., Geballe, T. R., Goto, M., Usuda, T., McCall, B. J., & Indriolo, N. 2019 ApJ, 883, 54html:<hr /><h3>Footnotes: Wilson, Jr., E. B. & Howard, J. B. 1936 J. Chem. Phys. 4, 260, Watson, J. K. G. 1966 J. Chem. Phys. 45, 1360; 46, 1935; 1968 J. Chem. Phys. 48, 4517: Watson, J. K. G. 1968 Mol. Phys. 15, 479: Watson, J. K. G. 1971 J. Mol. Speectrosc. 40, 536: e.g. Oka, T., Geballe, T. R., Goto, M., Usuda, T., McCall, B. J., & Indriolo, N. 2019 ApJ, 883, 54

Vibrational perturbation theory is a commonly-used method for obtaining anharmonic corrections to harmonic zero-order wave functions and energies. The method comes in two main variants, canonical Van Vleck perturbation theory, which has been used extensively by Sibert and coworkersSibert, E. L. J. Chem. Phys. 88, 4378-4390 (1987).nd Rayleigh-Schrödinger perturbation theory, which was used by Nielsen to derive analytic corrections up through second-order.Nielsen, H. H. Phys. Rev. 60, 794-810 (1941).Nielsen, H. H. Rev. Mod. Phys. 23, 90-136 (1951). In this work, we introduce an adaption to the Rayleigh-Schrödinger formalism. This approach, which is a generalization of approaches found in the physics literature,Sakurai, J. J.; Liboff, R. L. Modern Quantum Mechanics. Am. J. Phys. (1986).rovides a simple route to obtain high-order corrections to the vibrational energies and wave functions for a system. The intensive symbolic algebra required in both the Van Vleck and Nielsen approaches is replaced by a sparse, numerical linear algebra for which efficient libraries already exist. This approach is formally equivalent to standard methods and admits a straight-forward approach to account for degeneracies. We apply this method to obtaining 4^th and higher order corrections to the vibrational energies of a series of water clusters making use of curvilinear internal coordinates.

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Sibert, E. L. J. Chem. Phys. 88, 4378-4390 (1987).a Nielsen, H. H. Phys. Rev. 60, 794-810 (1941).

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Sakurai, J. J.; Liboff, R. L. Modern Quantum Mechanics. Am. J. Phys. (1986).p

The fundamental Ritz combination principle [1] originally found for atoms has also been applied to molecules as a method to reconstruct the energy states from measured lines without relying on any model Hamiltonian. In 2006 Nesbitt and coworkers [2] proposed to apply it to protonated methane, CH₅⁺, which was first done in 2015 [3] and extended in 2017 [4] by our group. Currently, we are elaborating this method to a universal, easy to use tool which can be used for arbitrary spectra. Essentials of the program and examples, including the floppy He-H₃⁺ system, will be discussed. [1] W. Ritz, On a new law of series spectra, Astrophys. J. 28 (1908), p. 237. [2] C. Savage, F. Dong, D.J. Nesbitt, Toward a quantum-mechanical understanding of the high-resolution infrared spectrum of CH₅⁺, in: Contribution TA05, 61st International Symposium on Molecular Spectroscopy, Columbus, OH, USA, 2006. [3] Oskar Asvany, Koichi M. T. Yamada, Sandra Brünken, Alexey Potapov, Stephan Schlemmer. Experimental ground-state combination differences of CH₅⁺. Science, 347 (2015), pp. 1346-1349. [4] S. Brackertz, S. Schlemmer, O. Asvany, Searching for new symmetry species of CH₅⁺ – From lines to states without a model, J. Mol. Spectrosc. 342 (2017), p. 73-82.

A three dimensional discrete variable representation (DVR) is developed for the general case of a rare gas atom interacting with an asymmetric top molecule and applied to the case of argon-haloethylene heterodimers. The position of the rare gas is specified relative to the principal inertial axis system (in a I^r representation) by the spherical polar coordinates (r, θ, ϕ). While the DVR for r is straightforward, those for θ and ϕ presented particular challenges, which are discussed. In common with all DVR approaches, the present DVR is well suited for systems where the intermolecular potential is calculated on a grid of discrete points and provides easy access to the wave functions in the DVR coordinates. The method is applied to the well-characterized argon-cis-1,2-difluoroethylene system and used to predict the tunneling splitting and argon-molecule vibrational wave functions in argon-vinyl chloride for which the molecular structure and rotational spectrum remain a puzzle.

Variational rovibrational calculations for small molecules with up to 3 atoms are nowadays a routine task. Tennyson, J. Chem. Phys. 145, 120901 (2016).y employing accurate potential energy surfaces it is possible to achieve so called “spectroscopic accuracy” of 1 cm⁻¹ for small linear molecules. Schröder and Sebald, J. Chem. Phys. 144, 044307 (2016); Makhnev et al., J. Phys. Chem. A 122, 1326 (2018).owever, rovibrational calculations for linear molecules with more than 3 atoms are still rather scarce. Mladenovi\'c, J. Chem. Phys. 141, 224304 (2014); Chubb et al., J. Chem. Phys. 149, 14101 (2018). new variational method (C8v4) will be presented that is able to obtain rovibrational term energies and wave functions of tetraatomic linear molecules based on Watsons isomorphic Hamiltonian for linear molecules. Watson, Mol. Phys. 14, 465 (1970).he rovibrational wavefunction is expanded in (symmetrized) products of harmonic oscillator (1D and 2D) and rigid-rotor functions. The intricacies related to the vibrational angular momentum in linear molecules require a careful study of symmetry properties. Kinetic energy matrix elements in the chosen basis set can be evaluated in a fast mixed numerical/analytical fashion. The main computational bottlenecks are the integration of the potential energy matrix and the diagonalisation of the rovibrational Hamiltonian. An efficient hybrid MPI/OMP parallelization has been implemented, exploiting the block structure of the Hamiltonian, that yields a significant reduction of the computational time. The diagonalization employs a K-contraction scheme that reduces the number of basis functions in the final rovibrational Hamilonian by at least an order of magnitude. Example benchmark calculations for acetylene (HCCH) are presented. The converged (N_\textvib = 2785) results for HCCH up to J_\textmax = 2 accurately reproduce earlier variational calculations Bramley and Handy, J. Chem. Phys. 98, 1378 (1993).n both g and u symmetry.

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 Tennyson, J. Chem. Phys. 145, 120901 (2016).B  Schröder and Sebald, J. Chem. Phys. 144, 044307 (2016); Makhnev et al., J. Phys. Chem. A 122, 1326 (2018).H  Mladenovi\'c, J. Chem. Phys. 141, 224304 (2014); Chubb et al., J. Chem. Phys. 149, 14101 (2018).A  Watson, Mol. Phys. 14, 465 (1970).T  Bramley and Handy, J. Chem. Phys. 98, 1378 (1993).i

The linear propynylidynium l-C₃H⁺ (\textX¹Σ⁺) is part of the select group of interstellar cations. Its discovery was accompanied by a controversy in the astrophysical community. Pety et al., Astron. Astrophys. 548, 68 (2012);  Huang et al., Astrophys. J. Lett. 768, 25 (2013).ollowing its initial detection, theoretical calculations questioned the assignment based on 2 order vibrational perturbation theory (VPT2) calculations. The matter was ultimately resolved by laboratory rotational spectra. Brünken et al., Astrophys. J. Lett. 783, 36 (2014).he failure of VPT2 was subsequently attributed to a shallow CCC bending potential. Botschwina et al., Astrophys. J. 787, 72 (2014).iscrete Variable Representation (DVR) calculations Mladenovi\'c J. Chem. Phys. 141, 224304 (2014).ater confirmed the D₀/D_\texte ratio but resulted in a large sextic centrifugal distortion constant H₀ exceeding the astronomical value by an order of magnitude. Using a new variational method for tetraatomic linear molecules (C8v4; see also P5277), based on Watsons isomorphic Hamiltonian for linear molecules, theory and experiment are reconciled. The C8v4 calculations confirm the small size of H₀ obtained in previous experimental studies. A high-level composite ab initio potential energy function (PEF) has been developed, combining explicitly correlated coupled-cluster results with corrections for core-valence correlation, scalar relativistic effects and higher-order correlation as well as the diagonal Born-Oppenheimer correction. Large scale C8v4 calculations using this PEF show excellent agreement with the available experimental rotational and vibrational parameters. Brünken et al., J. Phys. Chem. A 123, 8053 (2019).he presented rotational spectroscopic parameters of excited vibrational states should facilitate forthcoming experimental spectroscopic studies on l-C₃H⁺.

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 Pety et al., Astron. Astrophys. 548, 68 (2012);  Huang et al., Astrophys. J. Lett. 768, 25 (2013).F  Brünken et al., Astrophys. J. Lett. 783, 36 (2014).T  Botschwina et al., Astrophys. J. 787, 72 (2014).D  Mladenovi\'c J. Chem. Phys. 141, 224304 (2014).l  Brünken et al., J. Phys. Chem. A 123, 8053 (2019).T

Chlorine oxides, namely chlorine monoxide (ClO) and its head-to-tail dimer (ClOOCl), are thought to be important participants in the catalytic destruction of ozone in the Antarctic polar vortex. Chlorine peroxide photolysis is the crucial step in the ozone-depleting mechanism, requiring accurate measurements of absorption cross sections to estimate the amount of ozone destroyed by this process in the polar stratosphere. However, there are large inconsistencies in previous experimental determinations of absorption cross sections in the longer wavelength tail region, where chlorine peroxide photolysis is atmospherically relevant. We have used a model based upon Condon's reflection principle to construct a simulated absorption spectrum for dissociative excited states, as is the case for chlorine peroxide. The simulated spectrum uses oscillator strengths and excitation energies generated from equation-of-motion coupled-cluster (EOM-CC) techniques that include effects of triple excitations and agrees well with experimental spectra. This method shows promise in elucidating semi-quantitative features of the absorption cross sections without encountering common experimental complications.

The importance of the breakdown of the Born-Oppenheimer (BO) approximation in the vicinity of a conical intersection has spurred the recent development of computational approches that involve vibronic coupling between electronic states. Computational chemistry has mainly focused on conical intersections along reaction paths. Jahn-Teller type conical intersections offer a spectroscopically accessible alternative that can be used for benchmarking these calculations. The presence of Jahn-Teller and pseudo-Jahn-Teller couplings not only leads to additional terms in the rotational Hamiltonian but also modify how well known terms like Coriolis coupling and spin-rotation coupling are computed. In this talk we build upon the general quantum chemical method pioneered by Köppel, Domcke and Cederbaum H. KöPPEL AND W. DOMCKE AND L. S. CEDERBAUM  Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 57, (1984), 59-246o tackle molecules with vibronic coupling wherein ab initio electronic structure calculations in the adiabatic limit are used as the starting point for calculating properties of molecules in the regime where the BO approximation breaks down. A model vibronic Hamiltonian is parameterized using derivatives of adiabatic potential energy surfaces. Vibronic eigenvectors are obtained from this model Hamiltonian that allow the calculation of parameters in the rotation-spin Hamiltonian that can be determined experimentally from high resolution spectroscopy. H. KöPPEL AND W. DOMCKE AND L. S. CEDERBAUM  Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 57, (1984), 59-246t

Effects due to vibronic and spin angular momentum have long been observed in rotationally resolved spectra of Jahn-Teller active molecules. The breakdown of the Born-Oppenheimer approximation in these molecules leads to complications in their vibronic spectra. The molecular picture is further complicated due to the coupling of spin angular momenta with rotational and vibronic angular momentum. Values of these coupling constants have been determined by many high resolution rotationally resolved spectroscopic experiments. However, the presence of a conical intersection in the adiabatic potential energy surfaces has made getting a reliable quantum chemistry calculation of these terms challenging. In this talk we present methods that have been developed to make first principles calculations of vibronic angular momenta and Coriolis coupling constants from quantum chemistry. Further methods have been developed to calculated spin related terms (spin-orbit quenching and spin rotation coupling), using ab initio methods and model vibronic hamiltonian. We report a comparision of calculations of these parameters with corresponding results determined from high resolution spectroscopy for molecules like methoxy radical and cyclopentadienyl radical.

The vibronic structures of the low-lying electronic states in linear polyatomic molecules, which are utilized to construct closed optical cycling, play crucial roles in the laser-cooling processes. The construction of a multi-state Köeppel-Domcke-Cederbaum (KDC) quasidiabatic Hamiltonian with spin-orbit coupling, linear vibronic coupling, and Renner-Teller effects taken into account is reported aiming to obtain accurate vibronic levels and wave functions for laser-coolable triatomic molecules. The parameters for this KDC Hamiltonian were obtained from relativistic equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) calculations. Discrete variable representation (DVR) calculations were then carried out to obtain the vibronic levels and wave functions. The accuracy of the present parametrization for the KDC Hamiltonian is demonstrated with calculations for vibronic levels of the X²Σ and A²Π states of the SrOH molecule.