A variational approach for linearly dependent moving bases in quantum dynamics: application to Gaussian functions

TL;DR

This paper introduces a variational method to handle linear dependence in non-orthogonal, time-dependent basis sets for quantum dynamics, ensuring accurate and unitary evolution, demonstrated on a double-well potential with Gaussian functions.

Contribution

It develops a variational approach that manages linear dependence and changes in basis space dimensionality during quantum evolution, improving accuracy and unitarity.

Findings

Method converges to exact quantum dynamics.

Ensures unitarity of the evolution.

Effective for Gaussian basis functions in complex potentials.

Abstract

In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schr\"odinger equation. The method is based on: i) the definition of a linearly independent working space, and ii) a variational construction of the propagator over finite time-steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are done on a quartic double-well potential with Gaussian basis function whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.