# Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

## TL;DR

This paper introduces a novel factorization of the two-electron atom's wavefunction, leading to a potential energy surface analogy that enhances understanding of Coulomb correlation effects in helium's ground state.

## Contribution

It develops an exact factorization approach for two-electron atoms, deriving pseudoeigenvalue equations that reveal a molecular-like potential energy surface for atomic structure analysis.

## Key findings

- Reveals a nonadiabatic potential energy surface for helium's ground state.
- Shows correlation-induced symmetry breaking and quantum phase transition.
- Provides a new interpretative framework for Coulomb correlation in atoms.

## Abstract

For the $S$ states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances $r_1$ and $r_2$, and a conditional amplitude, which depends functionally on the interelectronic distance $r_{12}$ and parametrically on $r_1$ and $r_2$. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point $(r_1,r_2)$, such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.