Hartree-Fock critical nuclear charge in two-electron atoms.

Hugh G A Burton

doi:10.1063/5.0043105

Abstract

Electron correlation effects play a key role in stabilizing two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. We investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts an inner and outer electron. We also identify closed-shell and unrestricted critical nuclear charges where the highest occupied orbital energy becomes zero and the electron density detaches from the nucleus. Finally, we identify the importance of fractional spin errors and static correlation for small nuclear charges.

Highlights

How much positive charge is required to bind two electrons to a nucleus? This simple question has been subject to intense research and debate ever since the early 1930s.1–9 High-precision calculations have only recently converged on a critical nuclear charge for binding two electrons of Zc = 0.911 028 224 077 255 73(4).7–10 For Z > Zc, the two-electron atom (Z e e) is bound and stable, with an energy lower than the ionized system (Z e + e)
We identify the nuclear charge for HF symmetry-breaking ZsUbHF using a bisection method to locate the point where the lowest orbital Hessian eigenvalue of the restricted HF (RHF) solution vanishes
ZsUbHF > 1, making our result consistent with previous observations of unrestricted HF (UHF) symmetry breaking in the hydride anion

Summary

INTRODUCTION

These sudden qualitative changes in the HF wave function can be further probed using the average radial electronic positions, providing an indicator for ionization that does not rely on energetic properties. Previous studies on the two-electron atom using HF theory have primarily focused on the large Z, or “high-density,” limit (see Ref. 24) In this limit, the closed-shell RHF wave function provides a good approximation to the exact result, creating a model for understanding dynamic electron correlation.. The small-Z limit provides a model for understanding how to predict strong static correlation, which remains a major computational challenge In this contribution, we investigate the properties of the RHF and UHF ground-state wave functions in the small Z limit. RHF theory predicts a closed-shell critical point where half the electron density becomes ionized, leading to strong static correlation

TWO-ELECTRON ATOMIC HAMILTONIAN

COMPUTATIONAL DETAILS

Spin-symmetry breaking critical point

Closed-shell critical point

Fractional spin error

CONCLUDING REMARKS