Synopsis The anharmonic vibrational state theory based on quantum Monte Carlo method is one of the most accurate methods to solve a vibrational Schrödinger equation of molecular systems. In anharmonic vibrational state analyses, one should take account of the inter-mode couplings, which cannot be included by the conventional normal mode coordinates only. In this study, thus, we have developed a novel vibrational coordinate based on backflow transformation which enable us to take correlation effects in quantum many-body systems directly.
Synopsis The anharmonic vibrational state theory based on quantum Monte Carlo method is one of the most accurate methods to solve a vibrational Schrödinger equation of molecular systems. In anharmonic vibrational state analyses, one should take account of the inter-mode couplings, which cannot be included by the conventional normal mode coordinates only. In this study, thus, we have developed a novel vibrational coordinate based on backflow transformation which enable us to take correlation effects in quantum many-body systems directly.
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.
Neural quantum states (NQS) are a promising ansatz for solving many-body quantum problems due to their inherent expressiveness. Yet this expressiveness can only be harnessed efficiently for treating identical particles if the suitable physical knowledge is hardwired into the neural network itself. For electronic structure, NQS based on backflow determinants have been shown to be a powerful ansatz for capturing strong correlation. By contrast, the analogue for bosons, backflow permanents, is unpractical due to the steep cost of computing the matrix permanent and due to the lack of particle conservation in common bosonic problems. To circumvent these obstacles, we introduce a modal backflow (MBF) NQS design and demonstrate its efficacy by solving the anharmonic vibrational problem. To accommodate the demand of high accuracy in spectroscopic calculations, we implement a selected-configuration scheme for evaluating physical observables and gradients, replacing the standard stochastic approach based on Monte Carlo sampling. A vibrational self-consistent field calculation is conveniently carried out within the MBF network, which serves as a pretraining step to accelerate and stabilize the optimization. In applications to both artificial and ab initio Hamiltonians, we find that the MBF network is capable of delivering spectroscopically accurate zero-point energies and vibrational transitions in all anharmonic regimes.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.
One of the primary challenges prohibiting demonstrations of practical quantum advantages for near-term devices amounts to excessive measurement overheads for estimating relevant physical quantities such as ground state energies. However, with major differences between the electronic and vibrational structures of molecules, the question of how the resource requirements of computing anharmonic, vibrational states can be reduced remains relatively unexplored compared to its electronic counterpart. Importantly, bosonic commutation relations, distinguishable Hilbert spaces and vibrational coordinates allow manipulations of the vibrational system that can be exploited to minimize resource requirements. In this work, we investigate the impact of different coordinate systems and measurement schemes on the number of measurements needed to estimate anharmonic, vibrational states for a variety of three-mode (six-mode) molecules. We demonstrate an average of 3-fold (1.5-fold), with up to 7-fold (2.5-fold), reduction in the number of measurements required by employing appropriate coordinate transformations, based on an automized construction of qubit Hamiltonians from a conventional vibrational structure program.
Anharmonic vibrational many-body methods are developed for and applied to small molecules and extended systems in a bound potential energy surface (PES). Diagrammatically size-consistent and basis-set-free vibrational coupled-cluster (XVCC) theory for both zero-point energies and transition frequencies, the latter through the equation-of-motion (EOM) formalism, is defined for an nth-order Taylor-series PES. Quantum-field-theoretical tools (the rules of normal-ordered second quantization and Feynman--Goldstone diagrams) for deriving their working equations are established. The equations of XVCC and EOM-XVCC are derived and implemented with the aid of computer algebra. Algorithm optimizations known as strength reduction, intermediate reuse, and factorization are carried out before code generation,producing algorithms with optimal cost scaling. A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. An extended STEOM-XVCC (Ext-STEOM-XVCC) method is defined as an mth order configuration interaction method with the doubly similarity-transformed Hamiltonian including up to mth-order excitation operators. Because the doubly transformed Hamiltonian is dressed with higher-order anharmonic effects, the frequencies of overtones and combinations obtained are different and superior to the corresponding EOM-XVCC method. We compare and contrast the Ext-STEOM-XVCC method to its electronic counterpart. We apply the previously established second-order size-extensive vibrational many-body perturbation (XVMP2) method to the anharmonic phonon dispersion curves of a model two-mass system and the optical phonons of polyethylene. We obtain accurate results despite the presence of multiple Fermi-resonances in the crystalline systems.
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 CH 2 O molecule, the zero-point energies, and the vibrational energy redistribution dynamics of polyenes C 2 n H 2 n +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.