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.