Iterative active-space selection for vibrational configuration interaction calculations using a reduced-coupling VSCF basis

Kylin-V: An open-source package calculating the dynamic and spectroscopic properties of large systems

Quantum dynamics simulation and computational spectroscopy serve as indispensable tools for the theoretical understanding of various fundamental physical and chemical processes, ranging from charge transfer to photochemical reactions. When simulating realistic systems, the primary challenge stems from the overwhelming number of degrees of freedom and the pronounced many-body correlations. Here, we present Kylin-V, an innovative quantum dynamics package designed for accurate and efficient simulations of dynamics and spectroscopic properties of vibronic Hamiltonians for molecular systems and their aggregates. Kylin-V supports various quantum dynamics and computational spectroscopy methods, such as time-dependent density matrix renormalization group and our recently proposed single-site and hierarchical mapping approaches, as well as vibrational heat-bath configuration interaction. In this paper, we introduce the methodologies implemented in Kylin-V and illustrate their performances through a diverse collection of numerical examples.

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.

Potential energy surfaces and dynamic properties via ab initio composite and density functional approaches

Vibrational spectroscopy enables critical insight into the structural and dynamic properties of molecules. Presently, the majority of theoretical approaches to spectroscopy employ wavefunction-based ab initio or density functional methods that rely on the harmonic approximation. This approximation breaks down for large molecules with strongly anharmonic bonds or for molecules with large internuclear separations. An alternative to these methods involves generating molecular anharmonic potential energy surfaces (potentials) and using them to extrapolate the vibrational frequencies. This study examines the efficacy of density functional theory (DFT) and the correlation consistent Composite Approach (ccCA) in generating anharmonic frequencies from potentials of small main group molecules. Vibrational self-consistent field Theory (VSCF) and post-VSCF methods were used to calculate the fundamental frequencies of these molecules from their potentials. Functional choice, basis set selection, and mode-coupling are also examined as factors in influencing accuracy. The absolute deviations for the calculated frequencies using potentials at the ccCA level of theory were lower than the potentials at the DFT level. With DFT resulting in bending modes that are better described than those of ccCA, a multilevel DFT:ccCA approach where DFT potentials are used for single vibrational mode potentials and ccCA is used for vibrational mode-mode couplings can be utilized for larger polyatomic systems. The frequencies obtained with this multilevel approach using VCIPSI-PT2 were closer to experimental frequencies than the scaled harmonic frequencies, indicating the success of utilizing post-VSCF methods to generate more accurate representations of computed infrared spectra.

Theoretical Insights into the Vibrational Structure of Carbon Dioxide Rare-Gas Complexes

Two new flexible-monomer two-body ab initio potential energy surfaces (PESs) for the neon and krypton van der Waals complexes with carbon dioxide were developed, extending our previous work on the Ar-CO2 molecule. The accuracy of the PESs was validated by their agreement with the vibrational spectrum of the rare-gas complexes. The intermolecular and intramolecular vibrational excitation energies were computed at the vibrational self-consistent field and vibrational configuration interaction levels of theory. Overall, the agreement between theory and experiment is excellent throughout the vibrational spectra. The observed slight splitting of the bending modes, resulting from their nondegeneracy in the complexes, is confirmed by our computations, and the results qualitatively agree with the experiment. The splitting increases with increasing polarizability of the rare-gas atom. Additionally, we explain a discrepancy in the mode assignment in the intermolecular region of the neon complex with our VCI character assignment.

Matrix Product State Formulation of the MCTDH Theory in Local Mode Representations for Anharmonic Potentials

The matrix product state formulation of the multiconfiguration time-dependent Hartree theory, MPS-MCTDH, reported previously [Kurashige, J. Chem. Phys. 2018, 19, 194114] is extended to realistic anharmonic potentials with n-mode representations beyond the linear vibronic coupling model. For realistic vibrational potentials, the local mode representation should give a more compact representation of the potentials, i.e., lowering the dimensionality of the entanglements, than the normal coordinates, and the MPS-MCTDH formulation should work more efficiently and maintain the accuracy with a small bond dimension of the MPS ansatz. In fact, it was confirmed that the use of the local coordinates made the interaction matrices diagonal dominant and the number of terms in the n-body expansion of the potentials was significantly reduced. The method was applied to the IR spectrum of the CH₂O molecule, the zero-point energies, and the vibrational energy redistribution dynamics of polyenes C_2nH_2n+2. The results showed that the efficiency of the MPS-MCTDH method is significantly accelerated by the use of local coordinates even if the long-range interactions are included in the potential.

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.

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.

Multicomponent heat-bath configuration interaction with the perturbative correction for the calculation of protonic excited states

In this study, we extend the multicomponent heat-bath configuration interaction (HCI) method to excited states. Previous multicomponent HCI studies have been performed using only the variational stage of the HCI algorithm as they have largely focused on the calculation of protonic densities. Because this study focuses on energetic quantities, a second-order perturbative correction after the variational stage is essential. Therefore, this study implements the second-order Epstein-Nesbet correction to the variational stage of multicomponent HCI for the first time. Additionally, this study introduces a new procedure for calculating reference excitation energies for multicomponent methods using the Fourier-grid Hamiltonian (FGH) method, which should allow the one-particle electronic basis set errors to be better isolated from errors arising from an incomplete description of electron-proton correlation. The excited-state multicomponent HCI method is benchmarked by computing protonic excitations of the HCN and FHF^- molecules and is shown to be of similar accuracy to previous excited-state multicomponent methods such as the multicomponent time-dependent density-functional theory and equation-of-motion coupled-cluster theory relative to the new FGH reference values.

Advances in vibrational configuration interaction theory - part 1: Efficient calculation of vibrational angular momentum terms

Finite basis vibrational configuration interaction theory (VCI) is a highly accurate method for the variational calculation of state energies and related properties, but suffers from fast growing computational costs in dependence of the size of the correlation space. In this series of papers, concepts and techniques will be presented, which diminish the computational demands and thus broaden the applicability of this method to larger molecules or more complex situations. This first part focuses on a highly efficient implementation of the vibrational angular momentum (VAM) terms as occurring in the Watson Hamiltonian and the prediagonalization of initial subspaces within an iterative configuration selective VCI implementation. Working equations and benchmark calculations are provided, the latter demonstrating the increased performance of the new algorithm.

Reduced Two-Electron Interactions in Anharmonic Molecular Vibrational Calculations Involving Localized Normal Coordinates

Spatially localized vibrational normal mode coordinates are shown to reduce the importance of calculating the full set of two-electron terms in the molecular electronic Schrödinger equation. Electron correlation and dispersion interactions become less significant in (E,E)-1,3,5,7-octatetraene vibrational self-consistent field calculations when displacing remote atoms along multiple coordinates. Electron correlation interactions between spatially remote modes are also found to be less important compared to their corresponding uncorrelated interaction terms. Attenuation of the Coulomb operator indicates that the two-electron terms between remote electrons become less important for accurately describing the strongly contributing mode-coupling terms between sets of localized vibrational modes.

Near-Exact CASSCF-Level Geometry Optimization with a Large Active Space using Adaptive Sampling Configuration Interaction Self-Consistent Field Corrected with Second-Order Perturbation Theory (ASCI-SCF-PT2)

An accurate description of electron correlation is one of the most challenging problems in quantum chemistry. The exact electron correlation can be obtained by means of full configuration interaction (FCI). A simple strategy for approximating FCI at a reduced computational cost is selected CI (SCI), which diagonalizes the Hamiltonian within only the chosen configuration space. Recovery of the contributions of the remaining configurations is possible with second-order perturbation theory. Here, we apply adaptive sampling configuration interaction (ASCI) combined with molecular orbital optimizations (ASCI-SCF) corrected with second-order perturbation theory (ASCI-SCF-PT2) for geometry optimization by implementing the analytical nuclear gradient algorithm for ASCI-PT2 with the Z-vector (Lagrangian) formalism. We demonstrate that for phenalenyl radicals and anthracene, optimized geometries and the number of unpaired electrons can be obtained at nearly the CASSCF accuracy by incorporating PT2 corrections and extrapolating them. We demonstrate the current algorithm's utility for optimizing the equilibrium geometries and electronic structures of six-ring-fused polycyclic aromatic hydrocarbons and 4-periacene.

Exploring the anharmonic vibrational structure of carbon dioxide trimers

Our previously developed mbCO2 potential [O. Sode and J. N. Cherry, J. Comput. Chem. 38, 2763 (2017)] is used to describe the vibrational structure of the intermolecular motions of the CO₂ trimers: barrel-shaped and cyclic trimers. Anharmonic corrections are accounted for using the vibrational self-consistent field theory, vibrational second-order Møller-Plesset perturbation (VMP2) theory, and vibrational configuration interaction (VCI) methods and compared with experimental observations. For the cyclic structure, we revise the assignments of two previously observed experimental peaks based on our VCI and VMP2 results. We note that the experimental band observed near 13 cm^-1 is the out-of-phase out-of-plane degenerate motion with E″ symmetry, while the peak observed at 18 cm^-1 likely corresponds to the symmetric out-of-plane torsion A″ vibration. Since the VCI treatment of the vibrational motions accounts for vibrational mixing and delocalization, overtones and combination bands were also observed and quantified in the intermolecular regions of the two trimer isomers.

Vibrational heat-bath configuration interaction

We introduce vibrational heat-bath configuration interaction (VHCI) as an accurate and efficient method for calculating vibrational eigenstates of anharmonic systems. Inspired by its origin in electronic structure theory, VHCI is a selected CI approach that uses a simple criterion to identify important basis states with a pre-sorted list of anharmonic force constants. Screened second-order perturbation theory and simple extrapolation techniques provide significant improvements to variational energy estimates. We benchmark VHCI on four molecules with 12-48 degrees of freedom and use anharmonic potential energy surfaces truncated at fourth and sixth orders. When compared to other methods using the same truncated potentials, VHCI produces vibrational spectra of tens or hundreds of states with sub-wavenumber accuracy at low computational cost.

Theoretical Studies of Two Key Low-Lying Carbenes of C5H2 Missing in the Laboratory

The equilibrium geometries and spectroscopic properties of two key singlet carbenes, buta-1,3-diynylcarbene (6) and 2-methylenebicyclo[1.1.0]but-1(3)-en-4-ylidene (9), which have not been experimentally observed to date, are investigated using high-level coupled-cluster (CC) methods. The current theoretical study necessitates new experimental data on C₅H₂ isomers considering the relevance of these molecules to interstellar chemistry. Bent-pentadiynylidene (4) has been missing in the laboratory and the prime focus of our earlier theoretical work. The present theoretical study indicates that isomers 6 and 9 are also viable experimental targets. Apart from ethynylcyclopropenylidene (2), pentatetraenylidene (3), ethynylpropadienylidene (5), and 3-(didehydrovinylidene)cyclopropene (8), which are identified by Fourier transform microwave spectroscopy, the dipole moments of elusive 4, 6, and 9 are also nonzero (μ ≠ 0). The relative energies of these isomers, calculated at the CCSDT(Q)/CBS level of theory, with respect to linear triplet pentadiynylidene (1) reveal that they all lie within 25.1 kcal mol^-1. Therefore, geometric, energetic, aromatic, and spectroscopic parameters are reported here, which may assist the efforts of molecular spectroscopists in the future. Anharmonic vibrational calculations on isomers 6 and 9 indicate that the former is loosely bound and would be challenging to be detected experimentally. Among the undetected carbenes, 9 may be considered as a potential target molecule considering its higher polarity and aromatic nature.

Understanding the anharmonic vibrational structure of the carbon dioxide dimer

Understanding the vibrational structure of the CO₂ system is important to confirm the potential energy surface and interactions in such van der Waals complexes. In this work, we use our previously developed mbCO2 potential function to explore the vibrational structure of the CO₂ monomer and dimer. The potential function has been trained to reproduce the potential energies at the CCSD(T)-F12b/aug-cc-pVTZ level of electronic structure theory. The harmonic approximation, as well as anharmonic corrections using vibrational structure theories such as vibrational self-consistent field, vibrational second-order Møller-Plesset perturbation, and vibrational configuration interaction (VCI), is applied to address the vibrational motions. We compare the vibrational results using the mbCO2 potential function with traditional electronic structure theory results and to experimental frequencies. The anharmonic results for the monomer most closely match the experimental data to within 3 cm^-1, including the Fermi dyad frequencies. The intermolecular and intramolecular dimer frequencies were treated separately and show good agreement with the most recent theoretical and experimental results from the literature. The VCI treatment of the dimer vibrational motions accounts for vibrational mixing and delocalization, such that we observe the dimer Fermi resonance phenomena, both in the intramolecular and intermolecular regions.

A-VCI: A flexible method to efficiently compute vibrational spectra

The adaptive vibrational configuration interaction algorithm has been introduced as a new method to efficiently reduce the dimension of the set of basis functions used in a vibrational configuration interaction process. It is based on the construction of nested bases for the discretization of the Hamiltonian operator according to a theoretical criterion that ensures the convergence of the method. In the present work, the Hamiltonian is written as a sum of products of operators. The purpose of this paper is to study the properties and outline the performance details of the main steps of the algorithm. New parameters have been incorporated to increase flexibility, and their influence has been thoroughly investigated. The robustness and reliability of the method are demonstrated for the computation of the vibrational spectrum up to 3000 cm^-1 of a widely studied 6-atom molecule (acetonitrile). Our results are compared to the most accurate up to date computation; we also give a new reference calculation for future work on this system. The algorithm has also been applied to a more challenging 7-atom molecule (ethylene oxide). The computed spectrum up to 3200 cm^-1 is the most accurate computation that exists today on such systems.

Reduced Basis Set Dependence in Anharmonic Frequency Calculations Involving Localized Coordinates

Localized normal coordinates are known to be effective in speeding up anharmonic frequency calculations by reducing the complexity of the nuclear Hamiltonian and wave function. Displacing atoms in localized coordinates can also cause relatively small changes in the electronic structure, which can be exploited for further computational efficiency improvements during ab initio electronic structure calculations of the potential energy surface by reducing the electronic basis set dependence. Three different schemes for reducing the basis set dependence have been investigated in this work. These include combining localized coordinate schemes with general mixed basis sets, distance based force-field reductions, and using coordinate specific basis sets. The importance of accurately describing electronic interactions is found to diminish both for multicoordinate terms involving the displacement of remote atoms and when describing the interactions between more remote atoms within specific coordinates.

Discerning Inter- and Intramolecular Vibrations of Sulfur Polyaromatic Compounds

Thiophenes are an important class of molecules in fields as diverse as petrochemistry, molecular electronics, and optoelectronics. Thiophenic submolecular motifs are thought to play a role in molecular association and nanoaggregation phenomena in both pure materials and natural and synthetic mixtures. Vibrational (infrared and Raman) spectroscopy provides the means to characterize these species. In this work far-infrared photoacoustic and low-frequency Raman spectra of a series of polycyclic aromatic hydrocarbons containing sulfur have been measured and interpreted using DFT calculations based on a perturbational-variational method coupled with potential truncation. The approach and outcomes illustrate how inter- and intramolecular vibrations for thiophenic systems in single and multicomponent mixtures can be discriminated. This work offers the perspective to search the inter- and intramolecular signatures of the main submolecular motifs and heteroelements postulated as being present in the asphaltenes.

Intermediate vibrational coordinate localization with harmonic coupling constraints

Optimized normal coordinates can significantly improve the speed and accuracy of vibrational frequency calculations. However, over-localization can occur when using unconstrained spatial localization techniques. The unintuitive mixtures of stretching and bending coordinates that result can make interpreting spectra more difficult and also cause artificial increases in mode-coupling during anharmonic calculations. Combining spatial localization with a constraint on the coupling between modes can be used to generate coordinates with properties in-between the normal and fully localized schemes. These modes preserve the diagonal nature of the mass-weighted Hessian matrix to within a specified tolerance and are found to prevent contamination between the stretching and bending vibrations of the molecules studied without a priori classification of the different types of vibration present. Relaxing the constraint can also be used to identify which normal modes form specific groups of localized modes. The new coordinates are found to center on more spatially delocalized functional groups than their fully localized counterparts and can be used to tune the degree of vibrational correlation energy during anharmonic calculations.

On the benefits of localized modes in anharmonic vibrational calculations for small molecules

Anharmonic vibrational calculations can already be computationally demanding for relatively small molecules. The main bottlenecks lie in the construction of the potential energy surface and in the size of the excitation space in the vibrational configuration interaction (VCI) calculations. To address these challenges, we use localized-mode coordinates to construct potential energy surfaces and perform vibrational self-consistent field and L-VCI calculations [P. T. Panek and C. R. Jacob, ChemPhysChem 15, 3365 (2014)] for all vibrational modes of two prototypical test cases, the ethene and furan molecules. We find that the mutual coupling between modes is reduced when switching from normal-mode coordinates to localized-mode coordinates. When using such localized-mode coordinates, we observe a faster convergence of the n-mode expansion of the potential energy surface. This makes it possible to neglect higher-order contributions in the n-mode expansion of the potential energy surface or to approximate higher-order contributions in hybrid potential energy surfaces, which reduced the computational effort for the construction of the anharmonic potential energy surface significantly. Moreover, we find that when using localized-mode coordinates, the convergence with respect to the VCI excitation space proceeds more smoothly and that the error at low orders is reduced significantly. This makes it possible to devise low-cost models for obtaining a first approximation of anharmonic corrections. This demonstrates that the use of localized-mode coordinates can be beneficial already in anharmonic vibrational calculations of small molecules and provides a possible avenue for enabling such accurate calculations also for larger molecules.

Examining the impact of harmonic correlation on vibrational frequencies calculated in localized coordinates

Carefully choosing a set of optimized coordinates for performing vibrational frequency calculations can significantly reduce the anharmonic correlation energy from the self-consistent field treatment of molecular vibrations. However, moving away from normal coordinates also introduces an additional source of correlation energy arising from mode-coupling at the harmonic level. The impact of this new component of the vibrational energy is examined for a range of molecules, and a method is proposed for correcting the resulting self-consistent field frequencies by adding the full coupling energy from connected pairs of harmonic and pseudoharmonic modes, termed vibrational self-consistent field (harmonic correlation). This approach is found to lift the vibrational degeneracies arising from coordinate optimization and provides better agreement with experimental and benchmark frequencies than uncorrected vibrational self-consistent field theory without relying on traditional correlated methods.

Vibrational Signature of a Single Water Molecule Adsorbed on Pt(111): Toward a Reliable Anharmonic Description

In this study, we present a thorough benchmarking of our direct anharmonic vibrational variation-perturbation approach for adsorbed molecules on surfaces. We then use our method to describe the vibrational structure of a water molecule adsorbed on a Pt(111) surface and compare our results with the available experimental data. By using an explicitly correlated hybrid method to describe the molecule-surface interaction, we improve on the initial periodic PBE/DZP potential energy landscape and obtain vibrational frequencies that are of near-experimental accuracy. We introduce an implementation of anharmonic z-polarized IR intensity calculation and explain the absence of antisymmetric O-H stretch in the experimental data for the adsorbed water molecule, while the symmetric O-H stretch is predicted to be visible.

Pruning the Hamiltonian Matrix in MULTIMODE: Test for C2H4 and Application to CH3NO2 Using a New Ab Initio Potential Energy Surface

We report vibrational self-consistent field/virtual state configuration interaction energies of nitromethane using the code MULTIMODE and a new full-dimensional potential energy surface (PES). The PES is a precise, permutationally invariant linear least-squares fit to 17,049 electronic energies, using the CCSD(T)-F12b method with HaDZ basis (cc-pVDZ basis for H atoms, and aug-cc-pVDZ basis for C, O, N atoms). Nitromethane has 15 vibrational degrees of freedom, including one that is a nearly free internal methyl torsion, which is accurately described by the PES. This torsional mode makes vibrational calculations very challenging and here we present the results of calculations without it. Nevertheless, 14-mode calculations are still challenging and can lead to very large Hamiltonian matrices. To address this issue, we apply a pruning scheme, suggested previously by Handy and Carter, that reduces the size of matrix without sacrificing accuracy in the eigenvalues. The method is briefly described here in the context of partitioning theory. A new and more efficient implementation of it, coded in the latest version of MULTIMODE program, is described. The accuracy and efficiency are demonstrated for 12-mode C2H4 and then applied to CH3NO2. Agreement with available experimental values of the CH3NO2 14 fundamentals is very good. Diffusion Monte Carlo calculations in full dimensionality are done for the zero-point energy and wavefuction. These indicate that the torisonal motion is nearly a free-rotor in this state.

Pushing the limits in accurate vibrational structure calculations: anharmonic frequencies of lithium fluoride clusters (LiF)n, n = 2-10

The vibrational spectra of a series of small lithium fluoride clusters, i.e. (LiF)n, n = 2-10, were studied by vibrational configuration interaction (VCI) calculations relying on potential energy surfaces including three-mode coupling terms and being obtained from explicitly correlated local coupled cluster calculations. Due to the account for anharmonicity effects, the simulated spectra allow for a direct comparison with experimental data and may thus help to identify clusters in the experiments. Even structurally closely related clusters can clearly be distinguished by infrared spectroscopy. The largest system in this study required more than 1000 basis functions in the electronic structure calculations and more than 10(7) configurations in the vibrational structure calculations and became computationally feasible only due to a combination of different approximations and highly parallelized algorithms.

Efficient anharmonic vibrational spectroscopy for large molecules using local-mode coordinates

This article presents a general computational approach for efficient simulations of anharmonic vibrational spectra in chemical systems. An automated local-mode vibrational approach is presented, which borrows techniques from localized molecular orbitals in electronic structure theory. This approach generates spatially localized vibrational modes, in contrast to the delocalization exhibited by canonical normal modes. The method is rigorously tested across a series of chemical systems, ranging from small molecules to large water clusters and a protonated dipeptide. It is interfaced with exact, grid-based approaches, as well as vibrational self-consistent field methods. Most significantly, this new set of reference coordinates exhibits a well-behaved spatial decay of mode couplings, which allows for a systematic, a priori truncation of mode couplings and increased computational efficiency. Convergence can typically be reached by including modes within only about 4 Å. The local nature of this truncation suggests particular promise for the ab initio simulation of anharmonic vibrational motion in large systems, where connection to experimental spectra is currently most challenging.