Rates of convergence of the partial-wave expansion beyond Kato's cusp condition II: evaluations for the prefactors on the ground state of the helium atom

TL;DR

This paper analyzes the convergence rates of partial-wave expansions for the helium atom's ground state, deriving prefactors and identifying cancellation effects that influence the energy convergence behavior.

Contribution

It provides explicit derivations of the prefactors for the partial-wave expansion and explains the origin of convergence cancellations, extending previous work on helium atom wavefunctions.

Findings

Partial-wave energy increments converge as L^{-2N-6}.

Cancellations significantly affect convergence rates.

Evidence suggests assumptions on wavefunction regularities can be relaxed.

Abstract

This article is a continuation of our previous work (Phys. Rev. A 88, 032511 (2013)). The prefactors for the partial-wave expansion of the helium atom are derived. Due to series of cancellations, the partial-wave increments of the energy converge as $L^{-2N-6}$. The origin of these cancellations is identified from alternative expressions of the partial-wave energies. There is some evidence that the assumptions of regularities for the exact wavefunction can be reduced.