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

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 electronnucleus distances r1 and r2, and a conditional amplitude, which depends functionally on the interelectronic distance r12 and parametrically on r1 and r2. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schrödinger 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 (r1, r2), 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.

Characterization of the continuous transition from atomic to molecular shape in the three-body Coulomb system

“…The procedure is analogous to the one presented in ref. [29], making the correspondences r 1 → r 13 , r 2 → r 23 , r 12 → r 12 , and using the Hamiltonian (10) instead of the fixednucleus Hamiltonian. Hence, here we present only the most relevant equations and interpretations, and refer the reader to Ref.…”

“…Hence, here we present only the most relevant equations and interpretations, and refer the reader to Ref. [29] for further details. Following Hunter [31], we introduce the MCEF of the spatial eigenfunction [29] Φ(r 12 , r 13 , r 23 ) = ψ(r 13 , r 23 )χ(r 12 |r 13 , r 23 ), (12) by defining marginal and conditional amplitudes ψ(r 13 , r 23 ) := e iθ(r13,r23)…”

“…where, from here onwards, angular brackets express integrals over r 12 with the Jacobian r 12 . The phase exp[iθ(r 13 , r 23 )], with θ real, is arbitrary and can be chosen to set the symmetries of ψ and χ with respect to the exchange r 13 ↔ r 23 [29,34]. Now we can write…”

“…Then, in Subsec. III A we apply our view to the threeparticle system, exploiting the marginal-conditional exact factorization (MCEF) of the eigenfunction introduced earlier [29].…”

“…In fact, in Ref. [29] we showed that an exact nonadiabatic PES (NAPES) [31][32][33][34][35] can be rigorously constructed for the electrons in the He atom, and that it has interpretive power, in particular for making connections with classical structure ideas. In Subsec.…”