Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices

TL;DR

This paper introduces an iterative method for calculating vibrational spectra that drastically reduces memory requirements by using sum of product basis functions and exploiting the Hamiltonian's structure, enabling calculations on high-dimensional systems.

Contribution

The paper presents a novel iterative approach that avoids storing full-dimensional vectors, significantly lowering memory costs in vibrational spectra calculations using SOP basis functions.

Findings

Achieved accurate spectra for a 20-D model with minimal memory use.

Successfully computed spectra for CH₃CN with results matching previous studies.

Reduced memory requirement from 8×10^11 GB to 1 GB for high-dimensional problems.

Abstract

We propose an iterative method for computing vibrational spectra that significantly reduces the memory cost of calculations. It uses a direct product primitive basis, but does not require storing vectors with as many components as there are product basis functions. Wavefunctions are represented in a basis each of whose functions is a sum of products (SOP) and the factorizable structure of the Hamiltonian is exploited. If the factors of the SOP basis functions are properly chosen, wavefunctions are linear combinations of a small number of SOP basis functions. The SOP basis functions are generated using a shifted block power method. The factors are refined with a rank reduction algorithm to cap the number of terms in a SOP basis function. The ideas are tested on a 20-D model Hamiltonian and a realistic CH$_3$CN (12 dimensional) potential. For the 20-D problem, to use a standard direct product iterative approach one would need to store vectors with about $10^{20}$ components and would hence require about $8 \times 10^{11}$ GB. With the approach of this paper only 1 GB of memory is necessary. Results for CH$_3$CN agree well with those of a previous calculation on the same potential.