The Relativistic Schr\"odinger Equation through FFTW3: An Extension of quantumfdtd

TL;DR

This paper extends the quantumfdtd code to include the relativistic Schr"odinger equation and introduces optimized FFT-based kinetic energy calculations, enhancing the tool's capabilities for large-scale quantum simulations.

Contribution

The authors develop and release quantumfdtd v3, adding relativistic kinetic energy support, optimized FFT-based schemes, and enhanced features like external potentials and parity projection.

Findings

Successful implementation of relativistic kinetic energy term

Performance improvements with FFT-based schemes

Compatibility with previous non-relativistic code versions

Abstract

In order to solve the time-independent three-dimensional Schr\"odinger equation, one can transform the time-dependent Schr\"odinger equation to imaginary time and use a parallelized iterative method to obtain the full three-dimensional eigenstates and eigenvalues on very large lattices. In the case of the non-relativistic Schr\"odinger equation, there exists a publicly available code called quantumfdtd which implements this algorithm. In this paper, we (a) extend the quantumfdtd code to include the case of the relativistic Schr\"odinger equation and (b) add two optimized FFT-based kinetic energy terms for non-relativistic cases. The new kinetic energy terms (two non-relativistic and one relativistic) are computed using the parallelized Fast Fourier Transform (FFT) algorithm provided by the FFTW library. The resulting quantumfdtd v3 code, which is publicly released with this paper, is backwards compatible with version 2, supporting explicit finite differences schemes in addition to the new FFT-based schemes. Finally, the original code has been extended so that it supports arbitrary external file-based potentials and the option to project out distinct parity eigenstates from the solutions. Herein, we provide details of the quantumfdtd v3 implementation, comparisons and tests of the three new kinetic energy terms, and code documentation.