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Archive for Rational Mechanics and Analysis

, Volume 169, Issue 1, pp 35–71 | Cite as

The Multiconfiguration Equations for Atoms and Molecules: Charge Quantization and Existence of Solutions

  • G. Friesecke
Article

Abstract

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.

Keywords

Variational Principle Nuclear Charge Nonlinear Partial Differential Equation Direct Study Geometric Localization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • G. Friesecke
    • 1
  1. 1.Mathematics Institute University of Warwick CoventryUK

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