### Abstract

In this review, we summarise recent developments in our laboratory in the field of many-body quantum-mechanical calculations of the anharmonic vibrational structure of molecules. Our size-extensive vibrational self-consistent field (XVSCF) and size-extensive second-order many-body perturbation (XVMP2) methods are, unlike their parent methods (VSCF and VMP2), defined in diagrammatic formulations of the energies and Dyson self-energies, leading to manifestly size-consistent expressions for zero-point energies and anharmonic vibrational frequencies calculable with much greater efficiency. The effective one-mode potentials of XVSCF are quadratic and hence the Schrödinger equation for each mode can be solved analytically, unlike VSCF, where a basis-set expansion of wave functions on more complex one-mode potentials need to be performed; VSCF potentials and their minima (anharmonic geometry) are shown to reduce to the quadratic potentials and their minima (also given analytically) of XVSCF in the thermodynamic limit. By self-consistently solving the Dyson equation with frequency-dependent self-energies, XVMP2 has the ability to calculate anharmonic frequencies of fundamentals as well as combinations and overtones in the presence of strong anharmonic resonance without a multireference or quasi-degenerate formulation, which tends to be non-size-consistent.To eliminate the computational bottleneck of XVSCF and XVMP2, which is the high-rank force-constant evaluation, we have developed alternative algorithms in which the diagrammatic equations are recast as a small number of high-dimensional integrals and then evaluated stochastically using a Metropolis Monte Carlo (MC) method. These MC-XVSCF and MC-XVMP2 methods not only remove the need for force-constant evaluation or storage, but also take into account force constants of up to infinite order according to their importance. They are a new branch of quantum Monte Carlo which can calculate frequencies (excitation energies) directly without fixed-node errors.

Original language | English (US) |
---|---|

Pages (from-to) | 71-97 |

Number of pages | 27 |

Journal | International Reviews in Physical Chemistry |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

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### Keywords

- Anharmonic vibrations
- Diagrams
- Dyson equation
- Monte Carlo

### Cite this

*International Reviews in Physical Chemistry*,

*34*(1), 71-97. https://doi.org/10.1080/0144235X.2014.1001220

**Diagrammatic theories of anharmonic molecular vibrations.** / Hermes, Matthew R; Hirata, So.

Research output: Contribution to journal › Article

*International Reviews in Physical Chemistry*, vol. 34, no. 1, pp. 71-97. https://doi.org/10.1080/0144235X.2014.1001220

}

TY - JOUR

T1 - Diagrammatic theories of anharmonic molecular vibrations

AU - Hermes, Matthew R

AU - Hirata, So

PY - 2015/1/1

Y1 - 2015/1/1

N2 - In this review, we summarise recent developments in our laboratory in the field of many-body quantum-mechanical calculations of the anharmonic vibrational structure of molecules. Our size-extensive vibrational self-consistent field (XVSCF) and size-extensive second-order many-body perturbation (XVMP2) methods are, unlike their parent methods (VSCF and VMP2), defined in diagrammatic formulations of the energies and Dyson self-energies, leading to manifestly size-consistent expressions for zero-point energies and anharmonic vibrational frequencies calculable with much greater efficiency. The effective one-mode potentials of XVSCF are quadratic and hence the Schrödinger equation for each mode can be solved analytically, unlike VSCF, where a basis-set expansion of wave functions on more complex one-mode potentials need to be performed; VSCF potentials and their minima (anharmonic geometry) are shown to reduce to the quadratic potentials and their minima (also given analytically) of XVSCF in the thermodynamic limit. By self-consistently solving the Dyson equation with frequency-dependent self-energies, XVMP2 has the ability to calculate anharmonic frequencies of fundamentals as well as combinations and overtones in the presence of strong anharmonic resonance without a multireference or quasi-degenerate formulation, which tends to be non-size-consistent.To eliminate the computational bottleneck of XVSCF and XVMP2, which is the high-rank force-constant evaluation, we have developed alternative algorithms in which the diagrammatic equations are recast as a small number of high-dimensional integrals and then evaluated stochastically using a Metropolis Monte Carlo (MC) method. These MC-XVSCF and MC-XVMP2 methods not only remove the need for force-constant evaluation or storage, but also take into account force constants of up to infinite order according to their importance. They are a new branch of quantum Monte Carlo which can calculate frequencies (excitation energies) directly without fixed-node errors.

AB - In this review, we summarise recent developments in our laboratory in the field of many-body quantum-mechanical calculations of the anharmonic vibrational structure of molecules. Our size-extensive vibrational self-consistent field (XVSCF) and size-extensive second-order many-body perturbation (XVMP2) methods are, unlike their parent methods (VSCF and VMP2), defined in diagrammatic formulations of the energies and Dyson self-energies, leading to manifestly size-consistent expressions for zero-point energies and anharmonic vibrational frequencies calculable with much greater efficiency. The effective one-mode potentials of XVSCF are quadratic and hence the Schrödinger equation for each mode can be solved analytically, unlike VSCF, where a basis-set expansion of wave functions on more complex one-mode potentials need to be performed; VSCF potentials and their minima (anharmonic geometry) are shown to reduce to the quadratic potentials and their minima (also given analytically) of XVSCF in the thermodynamic limit. By self-consistently solving the Dyson equation with frequency-dependent self-energies, XVMP2 has the ability to calculate anharmonic frequencies of fundamentals as well as combinations and overtones in the presence of strong anharmonic resonance without a multireference or quasi-degenerate formulation, which tends to be non-size-consistent.To eliminate the computational bottleneck of XVSCF and XVMP2, which is the high-rank force-constant evaluation, we have developed alternative algorithms in which the diagrammatic equations are recast as a small number of high-dimensional integrals and then evaluated stochastically using a Metropolis Monte Carlo (MC) method. These MC-XVSCF and MC-XVMP2 methods not only remove the need for force-constant evaluation or storage, but also take into account force constants of up to infinite order according to their importance. They are a new branch of quantum Monte Carlo which can calculate frequencies (excitation energies) directly without fixed-node errors.

KW - Anharmonic vibrations

KW - Diagrams

KW - Dyson equation

KW - Monte Carlo

UR - http://www.scopus.com/inward/record.url?scp=84926150846&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84926150846&partnerID=8YFLogxK

U2 - 10.1080/0144235X.2014.1001220

DO - 10.1080/0144235X.2014.1001220

M3 - Article

AN - SCOPUS:84926150846

VL - 34

SP - 71

EP - 97

JO - International Reviews in Physical Chemistry

JF - International Reviews in Physical Chemistry

SN - 0144-235X

IS - 1

ER -