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

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.

arXiv comment15 pages, 6 figures

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• Phase Space Approach to Solving the Time-independent Schrödinger Equation
• Constrained Adiabatic Trajectory Method (CATM): a global integrator for explicitly time-dependent Hamiltonians
• Quantum dynamics by the constrained adiabatic trajectory method

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