Solving the Schr\"odinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients

TL;DR

This paper develops and analyzes advanced splitting methods with complex coefficients for solving the Schrödinger eigenvalue problem via imaginary time propagation, achieving higher accuracy and efficiency.

Contribution

It introduces new splitting schemes with complex coefficients, including sixth-order methods, that outperform existing techniques for Schrödinger eigenvalue problems.

Findings

New high-order splitting methods with complex coefficients are more efficient.

Proposed methods outperform existing schemes in numerical tests.

Variable order algorithms enhance computational efficiency.

Abstract

The Schr\"odinger eigenvalue problem is solved with the imaginary time propagation technique. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. High order fractional time steps of order greater than two necessarily have negative steps and can not be used for this class of diffusive problems. However, there exist methods which use fractional complex time steps with positive real parts which can be used with only a moderate increase in the computational cost. We analyze the performance of this class of schemes and propose new methods which outperform the existing ones in most cases. On the other hand, if the gradient of the potential is available, methods up to fourth order with real and positive coefficients exist. We also explore this case and propose new methods as well as sixth-order methods with complex coefficients. In particular, highly optimized sixth-order schemes for near integrable systems using positive real part complex coefficients with and without modified potentials are presented. A time-stepping variable order algorithm is proposed and numerical results show the enhanced efficiency of the new methods.