Variational quantum approaches for computing vibrational energies of polyatomic molecules

Comparison of curvilinear coordinates within vibrational structure calculations based on automatically generated potential energy surfaces

For floppy molecules showing internal rotations and/or large amplitude motions, curvilinear internal coordinates are known to be superior to rectilinear normal coordinates within vibrational structure calculations. Due to the myriad definitions of internal coordinates, automated and efficient potential energy surface generators necessitate a high degree of flexibility, supporting the properties arising from these coordinates. Within this work, an approach to deal with these challenges is presented, including key elements, such as the selection of appropriate fit functions, the exploitation of symmetry, the positioning of grid points, or elongation limits for different coordinates. These elements are tested for five definitions of curvilinear coordinates, with three of them being generated in an automated manner. Calculations for semi-rigid molecules, namely H2O, H2CO, CH2F2, and H2CNH, demonstrate the general functionality of the implemented algorithms. Additional calculations for the HOPO molecule highlight the benefits of these curvilinear coordinates for systems with large amplitude motions. This new implementation allowed us to compare the performance of these different coordinate systems with respect to the convergence of the underlying expansion of the potential energy surface and subsequent vibrational configuration interaction calculations.

Using nested tensor train contracted basis functions with group theoretical techniques to compute (ro)-vibrational spectra of molecules with non-Abelian groups

In this paper, we use nested tensor-train contractions to compute vibrational and ro-vibrational energy levels of molecules with five and six atoms. At each step, we fully exploit symmetry by using symmetry adapted basis functions obtained from an irreducible tensor method. Contracted basis functions are determined by diagonalizing reduced dimensional Hamiltonian matrices. The size of matrices of eigenvectors, used to account for coupling between groups of coordinates, is reduced by discarding rows and columns. The size of the matrices that must be diagonalized is thus substantially reduced, making it possible to use direct eigensolvers, even for molecules with five and six atoms. The symmetry-adapted contracted vibrational basis functions have been used to compute J = 0 energy levels of the CH3CN (C3v) and J > 0 levels of CH4.

Performance of Effective Harmonic Oscillator Approach for the Calculations of Vibrational Transition Energies of Large Molecules

The accuracy and performance of the effective harmonic oscillator approximation for the description of anharmonic vibrational structure calculations are tested for large molecular systems and compared with experimental values along with vibrational self-consistent field and second-order perturbation theories. The effective harmonic oscillator approach is an effective single-particle approximation where the variational parameters are the centroids and widths of the multidimensional Gaussian product functions posited as the vibrational wave functions. A comprehensive calculation for 849 transitions that include the fundamentals, two and three quanta overtone transitions, and several combination bands of three polyaromatic hydrocarbons and one DNA nucleobase with a total of 231 normal modes are assessed. A comparison of EHO results with the experimental values is done for the polyaromatic hydrocarbons, and a close agreement is found between the two results. It also offers anharmonic eigenstates and eigenfunctions that are nearly identical with vibrational self-consistent field theory. An extensive analysis on the resultant wave functions of the excited states is performed. The overall root-mean-square deviation (RMSD) between these two methods for 849 transitions understudy is only about 8.3 cm-1, suggesting the effective harmonic oscillator as a viable alternative for the reliable calculations of transition energies of large molecular systems.

Neural canonical transformations for vibrational spectra of molecules

The behavior of polyatomic molecules around their equilibrium positions can be regarded as that of quantum-coupled anharmonic oscillators. Solving the corresponding Schrödinger equations enables the interpretation or prediction of the experimental spectra of molecules. In this study, we developed a novel approach for solving the excited states of anharmonic vibrational systems. The normal coordinates of the molecules are transformed into new coordinates through a normalizing flow parameterized by a neural network. This facilitates the construction of a set of orthogonal many-body variational wavefunctions. This methodology has been validated on an exactly solvable 64-dimensional coupled harmonic oscillator, yielding numerical results with a relative error of 10-6. The neural canonical transformations are also applied to calculate the energy levels of two specific molecules, acetonitrile (CH3CN) and ethylene oxide (C2H4O). These molecules involve 12 and 15 vibrational modes, respectively. A key advantage of this approach is its flexibility concerning the potential energy surface, as it requires no specific form. Furthermore, this method can be readily implemented on large-scale distributed computing platforms, making it easy to extend to investigating complex vibrational structures.

Vibrational Self-Consistent Field (VSCF) and Post-VSCF Method Calculations Combined with the Reference Interaction Site Model Self-Consistent Field Method Coupled with the Constrained Spatial Electron Density Distribution: Applications to NaHCOO in Aqueous Phase

Investigating vibrational behavior in solution is crucial for understanding molecular dynamics within a solvent environment. Notably, the analysis of Raman spectra for molecules in solution is important owing to its ability to unveil intricate solute-solvent interactions. Previous studies have effectively employed frequency calculations utilizing the reference interaction site model self-consistent field method in conjunction with constrained spatial electron density distribution (RISM-SCF-cSED) to understand molecular vibrations in solution, primarily focusing on fundamental vibrational modes. However, the oversight of overtones and combination tones in these studies prompted us to combine the vibrational self-consistent field (VSCF) and vibrational second-order Mo̷ller-Plesset perturbation (VMP2) methods with RISM-SCF-cSED to address these aspects theoretically. Illustrating the efficacy of this integrated approach, we computed the Raman spectra of sodium formate (NaHCOO) in water, revealing the necessity of accounting for molecular anharmonicity in solution vibrational analysis. Our findings underscore the potency of VSCF and VMP2 in conjunction with RISM-SCF-cSED as a robust theoretical framework for such calculations.

VSCF/VCI theory based on the Podolsky Hamiltonian

While the vibrational spectra of semi-rigid molecules can be computed on approaches relying on the Watson Hamiltonian, floppy molecules or molecular clusters are better described by Hamiltonians, which are capable of dealing with any curvilinear coordinates. It is the kinetic energy operator (KEO) of these Hamiltonians, which render the correlated calculations relying on them rather costly. Novel implementation of vibrational self-consistent field theory and vibrational configuration interaction theory on the basis of the Podolsky Hamiltonian are reported, in which the inverse of the metric tensor, i.e., the G matrix, is represented by an n-mode expansion expressed in terms of polynomials. An analysis of the importance of the individual terms of the KEO with respect to the truncation orders of the n-mode expansion is provided. Benchmark calculations have been performed for the cis-HOPO and methanimine, H2CNH, molecules and are compared to experimental data and to calculations based on the Watson Hamiltonian and the internal coordinate path Hamiltonian.

A two-step quadrature-based variational calculation of ro-vibrational levels and wavefunctions of CO2 using a bisector-x molecule-fixed frame

In this paper, we propose a new two-step strategy for computing ro-vibrational energy levels and wavefunctions of a triatomic molecule and apply it to CO2. A two-step method [J. Tennyson and B. T. Sutcliffe, Mol. Phys., 1986, 58, 1067] uses a basis whose functions are products of K-dependent "vibrational" functions and symmetric top functions. K is the quantum number for the molecule-fixed z component of the angular momentum. For a linear molecule, a two-step method is efficient because the Hamiltonian used to compute the basis functions includes the largest coupling term. The most important distinguishing feature of the two-step method we propose is that it uses an associated Legendre basis and quadrature rather than a K-dependent discrete variable representation. This reduces the cost of the calculation and simplifies the method. We have computed ro-vibrational energy levels with J up to 100 for CO2, on an accurate available potential energy surface which is known as the AMES-2 PES and present a subset of those levels. We have converged most levels up to 20 000 cm-1 to 0.0001 cm-1.

Efficient vibrationally correlated calculations using n-mode expansion-based kinetic energy operators

Due to its efficiency and flexibility, the n-mode expansion is a frequently used tool for representing molecular potential energy surfaces in quantum chemical simulations. In this work, we investigate the performance of n-mode expansion-based models of kinetic energy operators in general polyspherical coordinate systems. In particular, we assess the operators with respect to accuracy in vibrationally correlated calculations and their effect on potential energy surface construction with the adaptive density guided approach. Our results show that the n-mode expansion-based operator variants are reliable and systematically improvable approximations of the full kinetic energy operator. Moreover, we introduce a workflow to generate the n-mode expanded kinetic energy operators on-the-fly within the adaptive density guided approach. This scheme can be applied in studies of species and coordinate systems, for which an analytical form of the kinetic energy operator is not available.

From the Automated Calculation of Potential Energy Surfaces to Accurate Infrared Spectra

Advances in the development of quantum chemical methods and progress in multicore architectures in computer science made the simulation of infrared spectra of isolated molecules competitive with respect to established experimental methods. Although it is mainly the multidimensional potential energy surface that controls the accuracy of these calculations, the subsequent vibrational structure calculations need to be carefully converged in order to yield accurate results. As both aspects need to be considered in a balanced way, we focus on approaches for molecules of up to 12-15 atoms with respect to both parts, which have been automated to some extent so that they can be employed in routine applications. Alternatives to machine learning will be discussed, which appear to be attractive, as long as local regions of the potential energy surface are sufficient. The automatization of these methods is still in its infancy, and the generalization to molecules with large amplitude motions or molecular clusters is far from trivial, but many systems relevant for astrophysical studies are already in reach.

Modeling the dynamics of quantum systems coupled to large-dimensional baths using effective energy states

The quantum dynamics of a low-dimensional system in contact with a large but finite harmonic bath is theoretically investigated by coarse-graining the bath into a reduced set of effective energy states. In this model, the couplings between the system and the bath are obtained from statistically averaging over the discrete, degenerate effective states. Our model is aimed at intermediate bath sizes in which non-Markovian processes and energy transfer between the bath and the main system are important. The method is applied to a model system of a Morse oscillator coupled to 40 harmonic modes. The results are found to be in excellent agreement with the direct quantum dynamics simulations presented in the work of Bouakline et al. [J. Phys. Chem. A 116, 11118-11127 (2012)], but at a much lower computational cost. Extension to larger baths is discussed in comparison to the time-convolutionless method. We also extend this study to the case of a microcanonical bath with finite initial internal energies. The computational efficiency and convergence properties of the effective bath states model with respect to relevant parameters are also discussed.

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.

A controlled study of the effect of deviations from symmetry of the potential energy surface (PES) on the accuracy of the vibrational spectrum computed with collocation

Symmetry, in particular permutational symmetry, of a potential energy surface (PES) is a useful property in quantum chemical calculations. It facilitates, in particular, state labelling and identification of degenerate states. In many practically important applications, however, these issues are unimportant. The imposition of exact symmetry and the perception that it is necessary create additional methodological requirements narrowing or complicating algorithmic choices that are thereby biased against methods and codes that by default do not incorporate symmetry, including most off-the-shelf machine learning methods that cannot be directly used if exact symmetry is demanded. By introducing symmetric and unsymmetric errors into the PES of H2CO in a controlled way and computing the vibrational spectrum with collocation using symmetric and nonsymmetric collocation point sets, we show that when the deviations from an ideal PES are random, imposition of exact symmetry does not bring any practical advantages. Moreover, a calculation ignoring symmetry may be more accurate. We also compare machine-learned PESs with and without symmetrization and demonstrate that there is no advantage of imposing exact symmetry for the accuracy of the vibrational spectrum.

Comprehensive quantum chemical analysis of the (ro)vibrational spectrum of thiirane and its deuterated isotopologue

The (ro)vibrational spectra of thiirane, c-C2H4S, and its fully deuterated isotopologue, c-C2D4S, have been studied by means of vibrational configuration interaction theory, VCI, its incremental variant, iVCI, and subsequent variational rovibrational calculations, RVCI, which rely on multidimensional potential energy surfaces of coupled-cluster quality including up to four-mode coupling terms. Accurate geometrical parameters, fundamental vibrational transitions and first overtones, rovibrational spectra and rotational spectroscopic constants have been determined from these calculations and were compared with experimental results whenever available. A number of tentative misassignments in the vibrational spectra could be resolved and most results for the deuterated thiirane are high-level predictions, which may guide experiments to come. Besides this, a new implementation of infrared intensities within the iVCI framework has been tested for the transitions of the title compounds and are compared with results obtained from standard VCI calculations.

Vibrational heat-bath configuration interaction with semistochastic perturbation theory using harmonic oscillator or VSCF modals

Vibrational heat-bath configuration interaction (VHCI)-a selected configuration interaction technique for vibrational structure theory-has recently been developed in two independent works [J. H. Fetherolf and T. C. Berkelbach, J. Chem. Phys. 154, 074104 (2021); A. U. Bhatty and K. R. Brorsen, Mol. Phys. 119, e1936250 (2021)], where it was shown to provide accuracy on par with the most accurate vibrational structure methods with a low computational cost. Here, we eliminate the memory bottleneck of the second-order perturbation theory correction using the same (semi)stochastic approach developed previously for electronic structure theory. This allows us to treat, in an unbiased manner, much larger perturbative spaces, which are necessary for high accuracy in large systems. Stochastic errors are easily controlled to be less than 1 cm-1. We also report two other developments: (i) we propose a new heat-bath criterion and an associated exact implicit sorting algorithm for potential energy surfaces expressible as a sum of products of one-dimensional potentials; (ii) we formulate VHCI to use a vibrational self-consistent field (VSCF) reference, as opposed to the harmonic oscillator reference configuration used in previous reports. Our tests are done with quartic and sextic force fields, for which we find that with VSCF, the minor improvements to accuracy are outweighed by the higher computational cost associated the matrix element evaluations. We expect VSCF-based VHCI to be important for more general potential representations, for which the harmonic oscillator basis function integrals are no longer analytic.

Quantum Chemical Investigation of the Cold Water Dimer Spectrum in the First OH-Stretching Overtone Region Provides a New Interpretation

Intramolecular vibrational transition wavenumbers and intensities were calculated in the fundamental HOH-bending, fundamental OH-stretching, first OH-stretching-HOH-bending combination, and first OH-stretching overtone (ΔvOH = 2) regions of the water dimer's spectrum. Furthermore, the rotational-vibrational spectrum was calculated in the ΔvOH = 2 region at 10 K, corresponding to the temperature of the existing jet-expansion experiments. The calculated spectrum was obtained by combining results from a full-dimensional (12D) vibrational and a reduced-dimensional vibrational-rotational-tunneling model. The ΔvOH = 2 spectral region is rich in features due to contributions from multiple vibrational-rotational-tunneling sub-bands. Origins of the experimental vibrational bands depend on the assignment of the observed sub-bands. Based on our calculations, we assign the observed sub-bands, and our reassignment leads to new values for the vibrational band origins of the free donor and antisymmetric acceptor OH-stretching first overtones of ∼7227 and ∼7238 cm-1, respectively. The observed bands with origins at 7192.34 and ∼7366 cm-1 are assigned to the symmetric acceptor OH-stretching first overtone and the OH-stretching combination of the donor, respectively.

Neural Network with Optimal Neuron Activation Functions Based on Additive Gaussian Process Regression

Feed-forward neural networks (NNs) are a staple machine learning method widely used in many areas of science and technology, including physical chemistry, computational chemistry, and materials informatics. While even a single-hidden-layer NN is a universal approximator, its expressive power is limited by the use of simple neuron activation functions (such as sigmoid functions) that are typically the same for all neurons. More flexible neuron activation functions would allow the use of fewer neurons and layers and thereby save computational cost and improve expressive power. We show that additive Gaussian process regression (GPR) can be used to construct optimal neuron activation functions that are individual to each neuron. An approach is also introduced that avoids nonlinear fitting of neural network parameters by defining them with rules. The resulting method combines the advantage of robustness of a linear regression with the higher expressive power of an NN. We demonstrate the approach by fitting the potential energy surfaces of the water molecule and formaldehyde. Without requiring any nonlinear optimization, the additive-GPR-based approach outperforms a conventional NN in the high-accuracy regime, where a conventional NN suffers more from overfitting.

Accuracy of Different Electronic Basis Set Families for Anharmonic Molecular Vibrations: A Comprehensive Benchmark Study

In this work, the accuracy and convergence of different electronic basis set families for the computation of anharmonic molecular vibrational spectroscopic calculations are benchmarked. A series of 39 different basis sets from different families following their hierarchy are assessed on VSCF and VSCF-PT2 algorithms with commonly used MP2 and DFT based B3LYP-D potentials for a set of molecular systems. Such an effort has been validated in a previous work ( J. Phys. Chem. A 2020, 124, 9203-9221) with split-valence basis sets for fundamentals and intensities. Here, fundamental transitions, vibrationally excited states, and intensities are compared with the experimental data to estimate the accuracy for a series of Jensen, Dunning, Calendar, Karlsruhe, and Sapporo basis set families. The convergence of basis sets are also compared with the large ANO basis set. Comprehensive statistical error analysis in terms of accuracy and precision was carried out to assess the performance of each basis set. It is observed that the improvement for the calculated harmonic and anharmonic values from the smaller basis sets to the medium (i.e., triple-ξ) is considerable. Beyond this, from medium to large basis sets, the convergence is slow and mostly posits nearly converged values. Basis sets with and without diffuse functions offer characteristically different accuracies and convergence patterns. Finally, recommendations are given on the choice of basis set chosen as black-box which can balance between accuracy and computational time, estimation of the errors, and their selections especially for large molecules.

Ab Initio Rovibrational Spectroscopy of the Acetylide Anion

In this work the rovibrational spectrum of the acetylide anion HCC- is investigated using high-level electronic structure methods and variational rovibrational calculations. Using a composite approach the potential energy surface and dipole surface is constructed from explicitly correlated coupled-cluster accounting for corrections due to core-valence correlation, scalar relativistic effects and higher-order excitation effects. Previous approaches for approximating the latter are critically evaluated. Employing the composite potential, accurate spectroscopic parameters determined from variational calculations are presented. In comparison to the few available reference data the present results show excellent agreement with ground state rotational constants within 0.005% of the experimental value. Intensities determined from the variational calculations suggest the bending fundamental transition ν2 around 510 cm-1 to be the best target for detection. The rather weak CD stretching fundamental ν1 in deuterated isotopologues show a second-order resonance with the (0,20,1) state and the consequences are discussed in some detail. The spectroscopic parameters and band intensities provided for a number of vibrational bands in isotopologues of the acetylide anion should facilitate future spectroscopic investigations.

Computational infrared and Raman spectra by hybrid QM/MM techniques: a study on molecular and catalytic material systems

Vibrational spectroscopy is one of the most well-established and important techniques for characterizing chemical systems. To aid the interpretation of experimental infrared and Raman spectra, we report on recent theoretical developments in the ChemShell computational chemistry environment for modelling vibrational signatures. The hybrid quantum mechanical and molecular mechanical approach is employed, using density functional theory for the electronic structure calculations and classical forcefields for the environment. Computational vibrational intensities at chemical active sites are reported using electrostatic and fully polarizable embedding environments to achieve more realistic vibrational signatures for materials and molecular systems, including solvated molecules, proteins, zeolites and metal oxide surfaces, providing useful insight into the effect of the chemical environment on the signatures obtained from experiment. This work has been enabled by the efficient task-farming parallelism implemented in ChemShell for high-performance computing platforms.  This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Calculating Vibrational Excited State Absorptions with Excited State Constrained Minimized Energy Surfaces

The modeling and interpretation of vibrational spectra are crucial for studying reaction dynamics using vibrational spectroscopy. Most prior theoretical developments focused on describing fundamental vibrational transitions while fewer developments focused on vibrational excited state absorptions. In this study, we present a new method that uses excited state constrained minimized energy surfaces (CMESs) to describe vibrational excited state absorptions. The excited state CMESs are obtained similarly to the previous ground state CMES development in our group but with additional wave function orthogonality constraints. Using a series of model systems, including the harmonic oscillator, Morse potential, double-well potential, quartic potential, and two-dimensional anharmonic potential, we demonstrate that this new procedure provides good estimations of the transition frequencies for vibrational excited state absorptions. These results are significantly better than those obtained from harmonic approximations using conventional potential energy surfaces, demonstrating the promise of excited state CMES-based methods for calculating vibrational excited state absorptions in real systems.

Computing excited OH stretch states of water dimer in 12D using contracted intermolecular and intramolecular basis functions

Due to the ubiquity and importance of water, water dimer has been intensively studied. Computing the (ro-)vibrational spectrum of water dimer is challenging. The potential has eight wells separated by low barriers, which makes harmonic approximations of limited utility. A variational approach is imperative, but difficult because there are 12 coupled vibrational coordinates. In this paper, we use a product contracted basis whose functions are products of intramolecular and intermolecular functions computed using an iterative eigensolver. An intermediate matrix F facilitates calculating matrix elements. Using F, it is possible to do calculations on a general potential without storing the potential on the full quadrature grid. We find that surprisingly many intermolecular functions are required. This is due to the importance of coupling between inter- and intra-molecular coordinates. The full G₁₆ symmetry of water dimer is exploited. We calculate, for the first time, monomer excited stretch states and compare P(1) transition frequencies with their experimental counterparts. We also compare with experimental vibrational shifts and tunneling splittings. Surprisingly, we find that the largest tunneling splitting, which does not involve the interchange of the two monomers, is smaller in the asymmetric stretch excited state than in the ground state. Differences between levels we compute and those obtained with a [6+6]D adiabatic approximation [Leforestier et al. J. Chem. Phys. 137 014305 (2012)] are ∼0.6 cm^-1 for states without monomer excitation, ∼4 cm^-1 for monomer excited bend states, and as large as ∼10 cm^-1 for monomer excited stretch states.

Model Chemistry Recommendations for Scaled Harmonic Frequency Calculations: A Benchmark Study

Despite the widespread popularity of scaled harmonic frequency calculations to predict experimental fundamental frequencies in chemistry, sparse benchmarking is available to guide users on the appropriate level of theory and basis set choices (model chemistry) or deep understanding of expected errors. An updated assessment of the best approach for scaling to minimize errors is also overdue. Here, we assess the performance of over 600 popular, contemporary, and robust model chemistries in the calculation of scaled harmonic frequencies, evaluating different scaling factor types and their implications in the scaled harmonic frequencies and model chemistry performance. We can summarize our results into three main findings: (1) Using model-chemistry-specific scaling factors optimized for three different frequency regions (low (<1,000 cm^-1), mid (1,000-2,000 cm^-1), and high (>2,000 cm^-1)) results in substantial improvements in the agreement between the scaled harmonic and experimental frequencies compared to other choices. (2) Larger basis sets and more robust levels of theory generally lead to superior performance; however, the particular model chemistry choice matters and poor choices lead to significantly reduced accuracies. (3) Outliers are expected in routine calculations regardless of the model chemistry choice. Our benchmarking results here do not consider the intensity of vibrational transitions; however, we draw upon previous benchmarking results for dipole moments that highlight the importance of diffuse functions (i.e., augmented basis sets) in high-quality intensity predictions. In terms of specific recommendations, overall, the highest accuracy model chemistries are double-hybrid density functional approximations with a non-Pople augmented triple-ζ basis set, which can produce median frequency errors down to 7.6 cm^-1 (DSD-PBEP86/def2-TZVPD), which is very close to the error in the harmonic approximation, i.e., the anharmonicity error. Double-ζ basis sets should not be used with double-hybrid functionals as there is no improvement compared to hybrid functional results (unlike for double-hybrid triple-ζ model chemistries). Note that 6-311G* and 6-311+G* basis sets perform like a double-ζ basis set for vibrational frequencies. After scaling, all studied hybrid functionals with non-Pople triple-ζ basis sets will produce median errors of less than 15 cm^-1, with the best result of 9.9 cm^-1 with B97-1/def2-TZVPD. Appropriate matching of double-ζ basis sets with hybrid functionals can produce high-quality results, but the precise choice of functional and basis set is more important. The B97-1, TPSS0-D3(BJ), or ωB97X-D hybrid density functionals with 6-31G*, pc-1, or pcseg-1 are recommended for fast routine calculations, all delivering median errors of 11-12 cm^-1. Note that dispersion corrections are not easily available for B97-1; given its strong performance here, we recommend these be added to major programs in coming updates.

Quantum chemical rovibrational analysis of aminoborane and its isotopologues

Aminoborane, H₂ NBH₂ and its isotopologues, H₂ N¹⁰ BH₂ , D₂ NBD₂ , and D₂ N¹⁰ BD₂ , have been studied by high-level ab initio methods. All calculations rely on multidimensional potential energy surfaces and dipole moment surfaces including high-order mode coupling terms, which have been obtained from electronic structure calculations at the level of explicitly correlated coupled-cluster theory, CCSD(T)-F12, or the distinguishable cluster approximation, DCSD. Subsequent vibrational structure calculations based on second-order vibrational perturbation theory, VPT2, and vibrational configuration interaction theory, VCI, were used to determine rotational constants, centrifugal distortion constants, vibrationally averaged geometrical parameters and (an)harmonic vibrational frequencies. The impact of core-correlation effects is discussed in detail. Rovibrational VCI calculations were used to simulate the gas phase spectra of these species and an in-depth analysis of the ν₇ band of aminoborane is provided. Color-coding is used to reveal the identity of the individual progressions of the rovibrational transitions for this particular mode.

The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces

Kernel-based methods, including Gaussian process regression (GPR) and generally kernel ridge regression, have been finding increasing use in computational chemistry, including the fitting of potential energy surfaces and density functionals in high-dimensional feature spaces. Kernels of the Matern family, such as Gaussian-like kernels (basis functions), are often used which allow imparting to them the meaning of covariance functions and formulating GPR as an estimator of the mean of a Gaussian distribution. The notion of locality of the kernel is critical for this interpretation. It is also critical to the formulation of multi-zeta type basis functions widely used in computational chemistry. We show, on the example of fitting of molecular potential energy surfaces of increasing dimensionality, the practical disappearance of the property of locality of a Gaussian-like kernel in high dimensionality. We also formulate a multi-zeta approach to the kernel and show that it significantly improves the quality of regression in low dimensionality but loses any advantage in high dimensionality, which is attributed to the loss of the property of locality.

On the method and results of calculation of vibrational tetrahedral sub-level energies and resonance interactions in vibrational spectra of high symmetry molecules: CH4 and GeH4 as an application of the XY4 (Td) molecule

We report here the analytical description of one of the important problems in the study of XY₄ (T_d) molecules, namely, description of vibrational tetrahedral sub-level structures and resonance interactions caused by the high symmetry of a molecule. The results obtained are applied to description of the vibrational energy spectrum of the CH₄ and GeH₄ molecules.

Convergence of series expansions in rovibrational configuration interaction (RVCI) calculations

Rotational and rovibrational spectra are a key in astrophysical studies, atmospheric science, pollution monitoring, and other fields of active research. The ab initio calculation of such spectra is fairly sensitive with respect to a multitude of parameters and all of them must be carefully monitored in order to yield reliable results. Besides the most obvious ones, i.e., the quality of the multidimensional potential energy surface and the vibrational wavefunctions, it is the representation of the μ-tensor within the Watson Hamiltonian, which has a significant impact on the desired line lists or simulated spectra. Within this work, we studied the dependence of high-resolution rovibrational spectra with respect to the truncation order of the μ-tensor within the rotational contribution and the Coriolis coupling operator of the Watson operator. Moreover, the dependence of the infrared intensities of the rovibrational transitions on an n-mode expansion of the dipole moment surface has been investigated as well. Benchmark calculations are provided for thioformaldehyde, which has already served as a test molecule in other studies and whose rovibrational spectrum was found to be fairly sensitive. All calculations rely on rovibrational configuration interaction theory and the discussed high-order terms of the μ-tensor are a newly implemented feature, whose theoretical basics are briefly discussed.

Multidimensional Quantum Dynamical Simulation of Infrared Spectra under Polaritonic Vibrational Strong Coupling

Recent experimental and theoretical studies demonstrate that the chemical reactivity of molecules can be modified inside an optical cavity. Here, we provide a theoretical framework for conducting multidimensional quantum simulations of the infrared (IR) spectra for molecules interacting with cavity modes. A single water molecule under polaritonic vibrational strong coupling serves as an illustrative example. Combined with accurate potential energy and dipole moment surfaces, our cavity vibrational self-consistent field/virtual state configuration interaction (cav-VSCF/VCI) approach can predict the IR spectra when the molecule is inside or outside the cavity. The spectral signatures of Rabi splittings and shifts of certain bands are found to be strongly dependent on the frequency and polarization direction of the cavity modes. Analyses of the simulated spectra show that polaritonic vibrational strong coupling can induce unconventional couplings among the molecule's vibrational modes, suggesting that intramolecular vibrational energy transfer can be significantly accelerated by the cavity.

On the vibrations of formic acid predicted from first principles

In this article, we review recent first principles, anharmonic studies on the molecular vibrations of gaseous formic acid in its monomer form. Transitions identified as fundamentals for both cis- and trans form reported in these studies are collected and supported by results from high-resolution experiments. Attention is given to the effect of coordinate coupling on the convergence of the computed vibrational states.

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.

Calculation of absolute Raman scattering cross-sections using vibrational self-consistent field/vibrational configuration interaction wave functions

In the present study, the differential scattering cross-sections, depolarization ratios and Raman shifts of small molecular systems are obtained from configuration iteration wave functions of vibrational self-consistent field (VSCF) states. The transition polarizabilities were modeled using the Placzek approximation, neglecting those contributions not arising from the electric dipole mechanism. This theoretical approach is considered a good approximation for samples that absorb in the UV range if the excitation radiation falls in the visible region, as is the case of the molecules selected for the present study, namely: water, methane, and acetylene. Potential energy and electronic polarizability surfaces are calculated by the CCSD(T) and CC3 methods with aug-cc-p(C)V(T,Q,5)Z basis sets. The vibrational Hamiltonian includes the vibrational angular momentum contribution of the Watson kinetic energy operator. As expected, due to the variational nature of the VSCF and vibrational configuration interaction (VCI) methods, the Raman transition wavenumbers are substantially improved over the harmonic predictions. Surprisingly, the scattering cross-sections obtained using the harmonic approximation or the VSCF method better agrees with the experimental values than those cross-sections predicted using VCI wave functions. The more significant deviations of the VCI results from the experimental reference may be related to the significant uncertainties of the measured cross-sections. Still, it may also indicate that the VCI Raman transition moments may require a more accurate description of the electronic polarizability surface. Finally, the depolarization ratios calculated for H₂ O and C₂ D₂ using harmonic and VCI wave functions have similar accuracy, whereas, for C₂ H₂ and C₂ HD, the VCI results are more accurate.

Quantitative molecular simulations

All-atom simulations can provide molecular-level insights into the dynamics of gas-phase, condensed-phase and surface processes. One important requirement is a sufficiently realistic and detailed description of the underlying intermolecular interactions. The present perspective provides an overview of the present status of quantitative atomistic simulations from colleagues' and our own efforts for gas- and solution-phase processes and for the dynamics on surfaces. Particular attention is paid to direct comparison with experiment. An outlook discusses present challenges and future extensions to bring such dynamics simulations even closer to reality.

Structure, energetics, and spectroscopy of the chromophores of HHe+n, H2He+n, and He+n clusters and their deuterated isotopologues

The linear molecular ions H₂He⁺, HHe+2, and He+3 are the central units (chromophores) of certain He-solvated complexes of the H₂He+n, HHe+n, and He+n families, respectively. These are complexes which do exist, according to mass-spectrometry studies, up to very high n values. Apparently, for some of the H₂He+n and He+n complexes, the linear symmetric tetratomic H₂He+2 and the diatomic He+2 cations, respectively, may also be the central units. In this study, definitive structures, relative energies, zero-point vibrational energies, and (an)harmonic vibrational fundamentals, and, in some cases, overtones and combination bands, are established mostly for the triatomic chromophores. The study is also extended to the deuterated isotopologues D₂He⁺, DHe+2, and D₂He+2. To facilitate and improve the electronic-structure computations performed, new atom-centered, fixed-exponent, Gaussian-type basis sets called MAX, with X = T(3), Q(4), P(5), and H(6), are designed for the H and He atoms. The focal-point-analysis (FPA) technique is employed to determine definitive relative energies with tight uncertainties for reactions involving the molecular ions. The FPA results determined include the 0 K proton and deuteron affinities of the ⁴He atom, 14 875(9) cm^-1 [177.95(11) kJ mol^-1] and 15 229(8) cm^-1 [182.18(10) kJ mol^-1], respectively, the dissociation energies of the He+2 → He⁺ + He, HHe+2 → HHe⁺ + He, and He+3 → He+2 + He reactions, 19 099(13) cm^-1 [228.48(16) kJ mol^-1], 3948(7) cm^-1 [47.23(8) kJ mol^-1], and 1401(12) cm^-1 [16.76(14) kJ mol^-1], respectively, the dissociation energy of the DHe+2 → DHe⁺ + He reaction, 4033(6) cm^-1 [48.25(7) kJ mol^-1], the isomerization energy between the two linear isomers of the [H, He, He]⁺ system, 3828(40) cm^-1 [45.79(48) kJ mol^-1], and the dissociation energies of the H₂He⁺ → H+2 + He and the H₂He+2 → H₂He⁺ + He reactions, 1789(4) cm^-1 [21.40(5) kJ mol^-1] and 435(6) cm^-1 [5.20(7) kJ mol^-1], respectively. The FPA estimates of the first dissociation energy of D₂He⁺ and D₂He+2 are 1986(4) cm^-1 [23.76(5) kJ mol^-1] and 474(5) cm^-1 [5.67(6) kJ mol^-1], respectively. Determining the vibrational fundamentals of the triatomic chromophores with second-order vibrational perturbation theory (VPT2) and vibrational configuration interaction (VCI) techniques, both built around the Eckart-Watson Hamiltonian, proved unusually challenging. For the species studied, VPT2 has difficulties yielding dependable results, in some cases even for the fundamentals of the H-containing molecular cations, while carefully executed VCI computations yield considerably improved spectroscopic results. In a few cases unusually large anharmonic corrections to the fundamentals, on the order of 15% of the harmonic value, have been observed.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

Comparison of body definitions for incremental vibrational configuration interaction theory (iVCI)

Within incremental vibrational configuration interaction theory (iVCI), the vibrational state energy is determined by means of a many-body expansion, i.e., it is a sum of terms of increasing order, which allow for an embarrassingly parallel evaluation. The convergence of this expansion depends strongly on the definition of the underlying bodies, which essentially decompose the correlation space into fragments. The different definitions considered here comprise mode-based bodies, excitation level-based bodies, and energy-based bodies. An analysis of the convergence behavior revealed that accounting for resonances within these definitions is mandatory and leads to a substantial improvement of the convergence, that is, the expansions can be truncated at lower orders. Benchmark calculations and systematic comparisons of the different body definitions for a small set of molecules, i.e., ketene, ethene, and diborane, have been conducted to study the overall performance of these iVCI implementations with respect to accuracy and central processing unit time.

Reduced-dimensional vibrational models of the water dimer

A model based on the finite-basis representation of a vibrational Hamiltonian expressed in internal coordinates is developed. The model relies on a many-mode, low-order expansion of both the kinetic energy operator and the potential energy surface (PES). Polyad truncations and energy ceilings are used to control the size of the vibrational basis to facilitate accurate computations of the OH stretch and HOH bend intramolecular transitions of the water dimer (H₂ ¹⁶O)₂. Advantages and potential pitfalls of the applied approximations are highlighted. The importance of choices related to the treatment of the kinetic energy operator in reduced-dimensional calculations and the accuracy of different water dimer PESs are discussed. A range of different reduced-dimensional computations are performed to investigate the wavenumber shifts in the intramolecular transitions caused by the coupling between the intra- and intermolecular modes. With the use of symmetry, full 12-dimensional vibrational energy levels of the water dimer are calculated, predicting accurately the experimentally observed intramolecular fundamentals. It is found that one can also predict accurate intramolecular transition wavenumbers for the water dimer by combining a set of computationally inexpensive reduced-dimensional calculations, thereby guiding future effective-Hamiltonian treatments.

Efficient and Automated Quantum Chemical Calculation of Rovibrational Nonresonant Raman Spectra

An outline of a newly developed program for the simulation of rovibrational nonresonant Raman spectra is presented. This program is an extension of our recently developed code for rovibrational infrared spectra [J. Chem Phys. 152 (2020) 244104] and relies on vibrational wavefunctions from variational configuration interaction theory to allow for an almost fully automated calculation of such spectra in pure ab initio fashion. Due to efficient contraction schemes this program requires modest computational resources and it can be controlled by only a few lines of input. As the required polarizability surfaces are also computed in an automated fashion, this implementation enables the routine application to small molecules. For demonstrating its capabilities, benchmark calculations for water H216O are compared to reference data and spectra for the beryllium dihydride dimer, Be2H4 (D2h), are predicted. The inversion symmetry of the D2h systems lead to complementary infrared and Raman spectra, which are needed both for a comprehensive investigation of this system.

Performance of Vibrational Self-Consistent Field Theory for Accurate Potential Energy Surfaces: Fundamentals, Excited States, and Intensities

The performance of vibrational structure calculations beyond harmonic approximation in the framework of the vibrational self-consistent field method with second-order perturbation corrections (VSCF-PT2) is investigated in conjunction with very accurate potential energy surfaces (PESs) given by various coupled-cluster electronic structure theories. The quality of anharmonic calculations depends on the accuracy of the underlying multidimensional PES obtained from its functional form, which is given by the level of electronic structure theory. Two such highest levels of typical coupled-cluster electronic structure methods, CCSD and the ″gold standard″ CCSD(T), along with their variants such as CCD, CR-CCL (completely renormalized CR-CC(2,3) approach), and CCSD(TQ) are tested for the construction of accurate anharmonic potentials without any fitting or ad hoc scaling and using cc-pVTZ basis sets. The accuracy of VSCF-PT2 theory in comparison to experimental values is tested for a series of 16 molecules with 135 fundamental bands, 64 overtones, and combination bands and also for 39 intensities. It is found that CCD and CCSD bind the potential tighter than CCSD(T) and the computed VSCF-PT2 transitions are more blue-shifted showing higher deviation from the experiment. In general, VSCF-PT2 results computed at the CCSD(T) potential offer a good cost/accuracy ratio, with the mean absolute deviation and the mean absolute percentage error with the experiment being ∼16 cm^-1 and 1.38, respectively, for fundamentals. Additionally, while the CR-CCL and CCSD(TQ) methods offer similar levels of accuracies as compared to CCSD(T), the former offers a better accuracy/cost ratio than the latter and is a suitable alternative to CCSD(T).

Easy representation of multivariate functions with low-dimensional terms via Gaussian process regression kernel design: applications to machine learning of potential energy surfaces and kinetic energy densities from sparse data

We show that Gaussian process regression (GPR) allows representing multivariate functions with low-dimensional terms via kernel design. When using a kernel built with high-dimensional model representation (HDMR), one obtains a similar type of representation as the previously proposed HDMR-GPR scheme while being faster and simpler to use. We tested the approach on cases where highly accurate machine learning is required from sparse data by fitting potential energy surfaces and kinetic energy densities.

Advances in vibrational configuration interaction theory - part 2: Fast screening of the correlation space

For larger molecules, the computational demands of configuration selective vibrational configuration interaction theory (cs-VCI) are usually dominated by the configuration selection process, which commonly is based on second order vibrational Møller-Plesset perturbation (VMP2) theory. Here we present two techniques, which lead to substantial accelerations of such calculations while retaining the desired high accuracy of the final results. The first one introduces the concept of configuration classes, which allows for a highly efficient exploitation of the analogs of the Slater-Condon rules in vibrational structure calculations with large correlation spaces. The second approach uses a VMP2 like vector for augmenting the targeted vibrational wavefunction within the selection of configurations and thus avoids any intermediate diagonalization steps. The underlying theory is outlined and benchmark calculations are provided for highly correlated vibrational states of several molecules.

Computational vibrational spectroscopy of molecule-surface interactions: what is still difficult and what can be done about it

Interactions of molecules with solid surfaces are responsible for key functionalities for a range of currently actively pursued technologies, including heterogeneous catalysis for synthesis or decomposition of molecules, sensitization, surface...

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].

Variational Energy Decomposition Analysis of Charge-Transfer Interactions between Metals and Ligands in Carbonyl Complexes

Variational energy decomposition analyses have been presented to quantify the σ-dative, ligand-to-metal forward charge transfer (CT) and the π-conjugative, metal-to-ligand backward charge delocalization on a series of isolelectronic transition-metal carbonyl complexes M(CO)₆, including M = Ti^2-, V^-, Cr, Mn⁺, and Fe²⁺. Although the qualitative features of these energy terms are understood, well-defined quantitative studies have been scarce. Consistent with early findings, electrostatic and Pauli exchange effects play a key role in σ-donation, resulting in blue shifts in ligand vibrational frequency in the complex geometries. Excluding chemical bonding interactions between the CO ligand and the metal fragments in the energy decomposition analysis, we found that loosely bound electrostatic complexes can be formed at a longer metal-to-ligand distance due to the exponential decay of Pauli exchange. In all complexes, the overall binding stabilization can be attributed to CT effects, with opposing trends between σ-donation and π-back bonding that follows an order of Ti^2- (4.4) > V^1- (2.6) > Cr (1.5) > Mn¹⁺ (1.1) > Fe²⁺ (0.5) in π-to-σ CT ratio. These electronic and energetic features are mirrored in the vibrational frequency shifts induced by different factors. The present investigation may help stimulate the use of energy decomposition techniques to understand the structure and activity of metallocatalysts using density functional theory.

Anharmonic Vibrational Calculations Based on Group-Localized Coordinates: Applications to Internal Water Molecules in Bacteriorhodopsin

An efficient anharmonic vibrational method is developed exploiting the locality of molecular vibration. Vibrational coordinates localized to a group of atoms are employed to divide the potential energy surface (PES) of a system into intra- and inter-group contributions. Then, the vibrational Schrödinger equation is solved based on a PES, in which the inter-group coupling is truncated at the harmonic level while accounting for the intra-group anharmonicity. The method is applied to a pentagonal hydrogen bond network (HBN) composed of internal water molecules and charged residues in a membrane protein, bacteriorhodopsin. The PES is calculated by the quantum mechanics/molecular mechanics (QM/MM) calculation at the level of B3LYP-D3/aug-cc-pVDZ. The infrared (IR) spectrum is computed using a set of coordinates localized to each water molecule and amino acid residue by second-order vibrational quasi-degenerate perturbation theory (VQDPT2). Benchmark calculations show that the proposed method yields the N-D/O-D stretching frequencies with an error of 7 cm-1 at the cost reduced by more than five times. In contrast, the harmonic approximation results in a severe error of 150 cm-1. Furthermore, the size of QM regions is carefully assessed to find that the QM regions should include not only the pentagonal HBN itself but also its HB partners. VQDPT2 calculations starting from transient structures obtained by molecular dynamics simulations have shown that the structural sampling has a significant impact on the calculated IR spectrum. The incorporation of anharmonicity, sufficiently large QM regions, and structural samplings are of essential importance to reproduce the experimental IR spectrum. The computational spectrum paves the way for decoding the IR signal of strong HBNs and helps elucidate their functional roles in biomolecules.

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.

Disentangling the Complex Vibrational Spectra of Hydrogen-Bonded Clusters of 2-Pyridone with Ab Initio Structural Search and Anharmonic Analysis

Vibrational spectra of 1:1 clusters of 2-pyridone (2PY) with water, ammonia, and other hydrogen bond-forming molecules have been measured by several experimental groups over the past two decades. Complex vibrational signatures associated with the N-H stretching fundamental at 3 μm are often observed. Several anharmonic coupling schemes have been proposed; however, the origin of these commonly seen complex features remains unclear. In this work, we present our theoretical analysis on the structure and vibrational spectra of these clusters using ab initio random search and ab initio anharmonic algorithms, respectively. Low-energy conformers were found to be hydrogen-bonded clusters and their vibrational spectra at 3 μm were simulated with ab initio anharmonic algorithms. We demonstrate that simple anharmonic mechanisms of Fermi resonance (FR), coupling between NH stretching modes, and overtone/combinations of skeleton modes of 2PY can lead to the complex vibrational signatures observed experimentally. Since this vibrational coupling scheme is inherent to the cis-amides with adjacent N-H and C═O groups when a hydrogen bond is formed with 2PY as the donor and acceptor, we believe that such a phenomenon is general to other hydrogen-bonding systems with the same functional group.