# Stochastic many-body perturbation theory for anharmonic molecular vibrations

## Abstract

A new quantum Monte Carlo (QMC) method for anharmonic vibrational zero-point energies and transition frequencies is developed, which combines the diagrammatic vibrational many-body perturbation theory based on the Dyson equation with Monte Carlo integration. The infinite sums of the diagrammatic and thus size-consistent first- and second-order anharmonic corrections to the energy and self-energy are expressed as sums of a few m- or 2m-dimensional integrals of wave functions and a potential energy surface (PES) (m is the vibrational degrees of freedom). Each of these integrals is computed as the integrand (including the value of the PES) divided by the value of a judiciously chosen weight function evaluated on demand at geometries distributed randomly but according to the weight function via the Metropolis algorithm. In this way, the method completely avoids cumbersome evaluation and storage of high-order force constants necessary in the original formulation of the vibrational perturbation theory; it furthermore allows even higher-order force constants essentially up to an infinite order to be taken into account in a scalable, memory-efficient algorithm. The diagrammatic contributions to the frequency-dependent self-energies that are stochastically evaluated at discrete frequencies can be reliably interpolated, allowing the self-consistent solutions to the Dyson equation to be obtained. Thismore »

- Authors:

- Department of Chemistry, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Urbana, Illinois 61801 (United States)
- (Japan)

- Publication Date:

- OSTI Identifier:
- 22419820

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 141; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; DEGREES OF FREEDOM; EVALUATION; FREQUENCY DEPENDENCE; MATHEMATICAL SOLUTIONS; MONTE CARLO METHOD; PERTURBATION THEORY; POTENTIAL ENERGY; SELF-ENERGY

### Citation Formats

```
Hermes, Matthew R., Hirata, So, E-mail: sohirata@illinois.edu, and CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012.
```*Stochastic many-body perturbation theory for anharmonic molecular vibrations*. United States: N. p., 2014.
Web. doi:10.1063/1.4892614.

```
Hermes, Matthew R., Hirata, So, E-mail: sohirata@illinois.edu, & CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012.
```*Stochastic many-body perturbation theory for anharmonic molecular vibrations*. United States. doi:10.1063/1.4892614.

```
Hermes, Matthew R., Hirata, So, E-mail: sohirata@illinois.edu, and CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012. Thu .
"Stochastic many-body perturbation theory for anharmonic molecular vibrations". United States.
doi:10.1063/1.4892614.
```

```
@article{osti_22419820,
```

title = {Stochastic many-body perturbation theory for anharmonic molecular vibrations},

author = {Hermes, Matthew R. and Hirata, So, E-mail: sohirata@illinois.edu and CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012},

abstractNote = {A new quantum Monte Carlo (QMC) method for anharmonic vibrational zero-point energies and transition frequencies is developed, which combines the diagrammatic vibrational many-body perturbation theory based on the Dyson equation with Monte Carlo integration. The infinite sums of the diagrammatic and thus size-consistent first- and second-order anharmonic corrections to the energy and self-energy are expressed as sums of a few m- or 2m-dimensional integrals of wave functions and a potential energy surface (PES) (m is the vibrational degrees of freedom). Each of these integrals is computed as the integrand (including the value of the PES) divided by the value of a judiciously chosen weight function evaluated on demand at geometries distributed randomly but according to the weight function via the Metropolis algorithm. In this way, the method completely avoids cumbersome evaluation and storage of high-order force constants necessary in the original formulation of the vibrational perturbation theory; it furthermore allows even higher-order force constants essentially up to an infinite order to be taken into account in a scalable, memory-efficient algorithm. The diagrammatic contributions to the frequency-dependent self-energies that are stochastically evaluated at discrete frequencies can be reliably interpolated, allowing the self-consistent solutions to the Dyson equation to be obtained. This method, therefore, can compute directly and stochastically the transition frequencies of fundamentals and overtones as well as their relative intensities as pole strengths, without fixed-node errors that plague some QMC. It is shown that, for an identical PES, the new method reproduces the correct deterministic values of the energies and frequencies within a few cm{sup −1} and pole strengths within a few thousandths. With the values of a PES evaluated on the fly at random geometries, the new method captures a noticeably greater proportion of anharmonic effects.},

doi = {10.1063/1.4892614},

journal = {Journal of Chemical Physics},

number = 8,

volume = 141,

place = {United States},

year = {Thu Aug 28 00:00:00 EDT 2014},

month = {Thu Aug 28 00:00:00 EDT 2014}

}