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.
Bibliographical noteFunding Information:
Support for this work was provided by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences [award number?DE-FG02-12ER46875]. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration [contract number DE-AC04-94-AL85000].
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- Green's function theory
- Potential energy surfaces
- anharmonic vibrations
- tensor decomposition