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Local approximation for the Hartree-Fock exchange potential: A deformation approach. (English) Zbl 0956.81097

The authors treat here a system of \(N\) coupled stationary “one-electron” Schrödinger equations \[ -\Delta \phi_j/ 2+ V_{\text{ext}} \phi_j+ V_H \phi_j+ V_{\text{ex}} \phi_j= \varepsilon_j \phi_j, \quad 1\leq j\leq N. \] Here \(V_{\text{ext}}\): an external potential, \(V_H(x):= \int \{\rho(y)/|x-y|\} dy\), \(x\in \mathbb{R}^3\) by \(\rho:= \sum_{j=1}^N|\phi_j|^2\), and the “exchange term” \((V_{\text{ex}} \phi_j)(x):= -\sum_{k=1}^N \{\int dx' \phi_j(x') \overline{\phi}_k(x')/|x-x'|\}\phi_k(x)\). As a result of \(V_{\text{ex}} \phi_j\) for \(N\to \infty\), they give a justification of the Slater-approximation \(V_s \phi_j(x):= -C\rho^{1/3}(x) \phi_j(x)\), \(C=1.4771\). They use the setting of \(\Omega= ]- L/2,L/2 [^3\) with periodic boundary condition, and a deformation \(f\) of the space such that \(\det (\partial f_i/\partial x_j)= \rho/\rho_0\), \(\rho_0= N/|\Omega|\).
Theorem: Suppose that \(|\varepsilon |_2= |(\rho- \rho_0)/ \rho_0|_2\to 0\) in \(\mathbb{C}^2(\Omega)\) as \(N\to \infty\) \[ \begin{split} V_{av}(x):= \int_{\mathbb{R}^3} |x-x'|^{-1} |\sum_{j=1}^N \phi_j(x) \overline{\phi}_j (x')|^2/ \rho(x) dx'\\ =- C\rho(x)^{1/3} \{1+ O(|\varepsilon |_2^2)+ O(\log (N)/ N^{2/3})+ ML/ (CN^{1/3})\}, \end{split} \] as \(N\to \infty\) holds for \(\forall x\in \Omega\), where \(C=1.7931\).

MSC:

81V70 Many-body theory; quantum Hall effect
81V10 Electromagnetic interaction; quantum electrodynamics
81V45 Atomic physics
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