Any order imaginary time propagation method for solving the Schrodinger equation

TL;DR

This paper introduces a general class of high-order imaginary time propagation algorithms for solving the Schrödinger equation, enabling efficient computation of eigenstates with improved accuracy and parallelization potential.

Contribution

It develops a multi-product splitting method for imaginary time propagation of any even order, extending beyond the traditional fourth-order algorithms.

Findings

Algorithms up to 12th order demonstrate high-precision eigenstate computations.

Effective for parallel computing architectures.

Applicable to complex molecules like C60.

Abstract

The eigenvalue-function pair of the 3D Schr\"odinger equation can be efficiently computed by use of high order, imaginary time propagators. Due to the diffusion character of the kinetic energy operator in imaginary time, algorithms developed so far are at most fourth-order. In this work, we show that for a grid based algorithm, imaginary time propagation of any even order can be devised on the basis of multi-product splitting. The effectiveness of these algorithms, up to the 12$^{\rm th}$ order, is demonstrated by computing all 120 eigenstates of a model C$_{60}$ molecule to very high precisions. The algorithms are particularly useful when implemented on parallel computer architectures.