The Multiconfiguration Equations for Atoms and Molecules: Charge Quantization and Existence of Solutions
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We prove the existence of ground-state solutions for the multiconfiguration self-consistent field equations for atoms and molecules whenever the total nuclear charge Z exceeds N−1, where N is the number of electrons. Moreover, we show that for arbitrary values of Z and N the scattering charge, i.e., the asymptotic amount of charge lost by an energy-minimizing sequence, is integer-quantized. Our analysis applies to the MC equations of arbitrary rank. As special cases we recover, in a new and unified way, the existence theorems of Zhislin [Zh60] for the N-body Schrödinger equation (infinite rank MC) and of Lieb & Simon [LS77] for the Hartree-Fock equations (rank-N MC). Our approach is a direct study of an invariant, orbital-free formulation in N-body space of the underlying variational principle. Proofs involve (i) the geometric N-body localization methods for the linear Schrödinger equation first introduced by Enss [En77] (and developed in [Sim77, Sig82]), which can be adapted to become powerful tools in nonlinear many-body theory as well, (ii) weak convergence methods from the theory of nonlinear partial differential equations, (iii) careful analysis of the structure of the one- and two-body density matrices of the bound and scattering fragments delivered by geometric localization, which allows us to overcome the fact that the rank of the fragments is not reduced by localization.
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