For the helium atom and its isoelectronic second-period ions (H-, He, Li+…, Ne8+), we applied the marginal-conditional exact factorization to a variationally optimized ground-state internal wavefunction, F(r1,r2,r12)=y(r2,r12)c(r1|r2,r12), where r1,r2 are the electron-nucleus distances, and r12 is the interelectronic distance. Applying the variation principle we derived coupled eigenvalue equations for the conditional, c(r1|r2,r12), and marginal, y(r2,r12), amplitudes. The first has as eigenvalue a nonadiabatic potential energy surface (NAPES), U(r2,r12), which plays the role of potential in the second equation, which has as eigenvalue the energy of the system. Such NAPES provides energetic and structural information about the atom or ion, analogously to a Born-Oppenheimer PES for a molecule. In particular, a classical-like interpretation of its topography implies that the most stable spatial configurations have r12&gt;r2. Moreover, the NAPES exhibits a stability basin that runs along r2, separated from the r12=r2 diagonal by a low barrier. For Z=2,…,10, the associated marginal distribution functions, Dm(r2,r12)=r2r12|y(r2,r12)|2, are slanted toward the r12&gt;r2 half-plane, as expected, but do not manifest the topological symmetry breaking associated with the presence of the basin, i.e., they are unimodal. On the other hand, for Z=1 (H-) the marginal distribution becomes bimodal, preserving such topological symmetry breaking. Calculations for model systems with 1&lt;Z&lt;2, show that the topological transition from unimodal to bimodal is continuous.

Topological symmetry breaking across the helium isoelectronic series

ACS Spring 2022

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