Nuclear critical charge for two-electron ion in Lagrange mesh method

TL;DR

This paper uses the Lagrange mesh method to precisely determine the nuclear critical charge for a two-electron ion, confirming previous results and providing highly accurate energy calculations near the critical charge.

Contribution

The study applies the Lagrange mesh method to accurately compute the critical charge and ground state energy of two-electron ions, improving precision over previous methods.

Findings

Confirmed the critical charge to 11 decimal digits.

Achieved 14-decimal accuracy for H$^-$ ground state energy.

Validated the Lagrange mesh method for near-critical charge calculations.

Abstract

The Schroedinger equation for two electrons in the field of a charged fixed center $Z$ is solved with the Lagrange mesh method for charges close to the critical charge $Z_{cr}$. We confirm the value of the nuclear critical charge $Z_{cr}$ recently calculated in Estienne et al. {\em Phys. Rev. Lett. \bf 112}, 173001 (2014) to 11 decimal digits using an inhomogeneous (non-uniform) three-dimensional lattice of size $70 \times 70 \times 20$. We show that the ground state energy for H$^-$ is accurate to 14 decimals on the lattice $50 \times 50 \times 40$ in comparison with the highly accurate result by Nakashima-Nakatsuji, {\it J. Chem. Phys. \bf 127}, 224104 (2007).