2D Nondirect Product Discrete Variable Representation for Schr\"odinger Equation with Nonseparable Angular Variables

TL;DR

This paper introduces a new nondirect product discrete variable representation (npDVR) method using 2D quadratures on the sphere, significantly improving convergence and efficiency for quantum problems with nonseparable angular variables.

Contribution

The paper presents a novel npDVR approach based on Lebedev and Popov 2D quadratures, enhancing computational convergence for nonseparable angular quantum problems.

Findings

npDVR accelerates convergence compared to traditional methods

Popov quadratures yield the fastest convergence and highest efficiency

Method effectively computes hydrogen atom spectra in arbitrary fields

Abstract

We develop a nondirect product discrete variable representation (npDVR) for treating quantum dynamical problems which involve nonseparable angular variables. The npDVR basis is constructed on spherical functions orthogonalized on the grids of the Lebedev or Popov 2D quadratures for the unit sphere instead of the direct product of 1D quadrature rules. We compare our computational scheme with the old one that used the product of 1D Gaussian quadratures in terms of their convergence and efficiency by calculating, as an example, the spectrum of a hydrogen atom in the magnetic and electric fields arbitrarily oriented to one another. The use of the npDVR based on the Lebedev or Popov 2D quadratures substantially accelerates the convergence of the computational scheme. Moreover, we get the fastest convergence with the npDVR based on the Popov quadratures, which has the largest efficiency coefficient.