Computing vibrational energy levels by solving linear equations using a tensor method with an imposed rank

TL;DR

This paper introduces a tensor-based method using CP format to efficiently compute vibrational energy levels of high-dimensional molecules by solving linear equations without rank reduction or orthogonalization.

Contribution

It presents a novel CP tensor approach with fixed rank for solving linear equations to find vibrational energy levels, avoiding tensor rank growth and orthogonalization.

Findings

Successfully computed vibrational levels of a 64-D model Hamiltonian.

Accurately calculated energy levels of acetonitrile (12-D).

Method reduces memory requirements for high-dimensional vibrational calculations.

Abstract

Present day computers do not have enough memory to store the high-dimensional tensors required when using a direct product basis to compute vibrational energy levels of a polyatomic molecule with more than about 5 atoms. One way to deal with this problem is to represent tensors using a tensor format. In this paper, we use CP format. Energy levels are computed by building a basis from vectors obtained by solving linear equations. The method can be thought of as a CP realization of a block inverse iteration method with multiple shifts. The CP rank of the tensors is fixed and the linear equations are solved with an Alternating Least Squares method. There is no need for rank reduction, no need for orthogonalization, and tensors with rank larger than the fixed rank used to solve the linear equations are never generated. The ideas are tested by computing vibrational energy levels of a 64-D bilinearly coupled model Hamiltonian and of acetonitrile(12-D).