### Abstract

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm^{−1} or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

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

Pages (from-to) | 2120-2134 |

Number of pages | 15 |

Journal | Molecular Physics |

Volume | 115 |

Issue number | 17-18 |

DOIs | |

State | Published - Sep 17 2017 |

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### Keywords

- Green's function theory
- Potential energy surfaces
- anharmonic vibrations
- tensor decomposition

### Cite this

*Molecular Physics*,

*115*(17-18), 2120-2134. https://doi.org/10.1080/00268976.2017.1288937

**Low-rank canonical-tensor decomposition of potential energy surfaces : application to grid-based diagrammatic vibrational Green’s function theory.** / Rai, Prashant; Sargsyan, Khachik; Najm, Habib; Hermes, Matthew R.; Hirata, So.

Research output: Contribution to journal › Article

*Molecular Physics*, vol. 115, no. 17-18, pp. 2120-2134. https://doi.org/10.1080/00268976.2017.1288937

}

TY - JOUR

T1 - Low-rank canonical-tensor decomposition of potential energy surfaces

T2 - application to grid-based diagrammatic vibrational Green’s function theory

AU - Rai, Prashant

AU - Sargsyan, Khachik

AU - Najm, Habib

AU - Hermes, Matthew R.

AU - Hirata, So

PY - 2017/9/17

Y1 - 2017/9/17

N2 - A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm−1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

AB - A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm−1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

KW - Green's function theory

KW - Potential energy surfaces

KW - anharmonic vibrations

KW - tensor decomposition

UR - http://www.scopus.com/inward/record.url?scp=85014522541&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014522541&partnerID=8YFLogxK

U2 - 10.1080/00268976.2017.1288937

DO - 10.1080/00268976.2017.1288937

M3 - Article

AN - SCOPUS:85014522541

VL - 115

SP - 2120

EP - 2134

JO - Molecular Physics

JF - Molecular Physics

SN - 0026-8976

IS - 17-18

ER -