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For the $S$ states of two-electron atoms, we introduce an exact and uniquefactorization 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 interelectronicdistance $r_{12}$ and parametrically on $r_1$ and $r_2$. Applying thevariational principle, we derive pseudoeigenvalue equations for these twoamplitudes, which cast the internal Schr\"odinger equation in a form akin tothe Born-Oppenheimer separation of nuclear and electronic degrees of freedom inmolecules. The marginal equation involves an effective radial Hamiltonian,which contains a nonadiabatic potential energy surface that takes into accountall interparticle correlations in an averaged way, and whose unique eigenvalueis 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 stateof He as prototype, we show that the nonadiabatic potential energy surfaceaffords a molecularlike interpretation of the structure of the atom, and aidsin the analysis of energetic and spatial aspects of the Coulomb correlation, inparticular correlation-induced symmetry breaking and quantum phase transition.

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