Exact separation of radial and angular correlation energies in two-electron atoms

TL;DR

This paper presents an exact method to separate radial and angular correlation energies in two-electron atoms, revealing that 60.2% of helium's correlation energy is purely radial, challenging previous assumptions.

Contribution

The authors derive analytic matrix elements within the Hylleraas basis to precisely uncouple radial and angular correlation energies, improving understanding of electron correlation.

Findings

60.2% of helium's correlation energy is from purely radial interactions

Excluding radial-angular coupling reduces the radial correlation energy to 41.3%

The method agrees with numerical cubature for ground and excited states

Abstract

Partitioning of helium atom's correlation energy into radial and angular contributions, although of fundamental interest, has eluded critical scrutiny. Conventionally, radial and angular correlation energies of helium atom are defined for its ground state as deviations, from Hartree--Fock and exact values, of the energy obtained using a purely radial wavefunction devoid of any explicit dependence on the interelectronic distance. Here, we show this rationale to associate the contribution from radial-angular coupling entirely to the angular part underestimating the radial one, thereby also incorrectly predict non-vanishing residual radial probability densities. We derive analytic matrix elements for the high-precision Hylleraas basis set framework to seamlessly uncouple the angular correlation energy from its radial counterpart. The resulting formula agrees with numerical cubature yielding precise purely angular correlation energies for the ground as well as excited states. Our calculations indicate 60.2% of helium's correlation energy to arise from strictly radial interactions; when excluding the contribution from the radial-angular coupling, this value drops to 41.3%.