Bokanowski, Olivier; Mauser, Norbert J. Local approximation for the Hartree-Fock exchange potential: A deformation approach. (English) Zbl 0956.81097 Math. Models Methods Appl. Sci. 9, No. 6, 941-961 (1999). 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 \(C=1.7931\). Reviewer: Hideo Yamagata (Osaka) Cited in 21 Documents MSC: 81V70 Many-body theory; quantum Hall effect 81V10 Electromagnetic interaction; quantum electrodynamics 81V45 Atomic physics Keywords:stationary one-electron Schrödinger equations; Slater-approximation PDFBibTeX XMLCite \textit{O. Bokanowski} and \textit{N. J. Mauser}, Math. Models Methods Appl. Sci. 9, No. 6, 941--961 (1999; Zbl 0956.81097) Full Text: DOI References: [1] DOI: 10.1007/BF02097241 · Zbl 0771.46038 · doi:10.1007/BF02097241 [2] DOI: 10.1007/BF02097395 · Zbl 0802.47061 · doi:10.1007/BF02097395 [3] Becke A. D., Phys. Rev. 38 (1988) · doi:10.1103/PhysRevA.38.3098 [4] DOI: 10.1142/S021820259600016X · Zbl 0867.35075 · doi:10.1142/S021820259600016X [5] DOI: 10.1063/1.531468 · Zbl 0889.92030 · doi:10.1063/1.531468 [6] DOI: 10.1002/(SICI)1097-461X(1998)68:4<221::AID-QUA1>3.0.CO;2-X · doi:10.1002/(SICI)1097-461X(1998)68:4<221::AID-QUA1>3.0.CO;2-X [7] DOI: 10.1007/BF01206884 · Zbl 0539.47028 · doi:10.1007/BF01206884 [8] Catto I., Acad. Sci. 322 pp 357– (1996) [9] DOI: 10.1017/S0305004100016108 · JFM 56.0751.04 · doi:10.1017/S0305004100016108 [10] DOI: 10.1007/BF03156228 · Zbl 0057.22805 · doi:10.1007/BF03156228 [11] DOI: 10.1103/PhysRev.106.364 · Zbl 0080.44505 · doi:10.1103/PhysRev.106.364 [12] DOI: 10.1103/PhysRevA.43.2179 · doi:10.1103/PhysRevA.43.2179 [13] Kohn W., Phys. Rev. 140 (1965) · doi:10.1103/PhysRev.140.A1133 [14] DOI: 10.1016/0001-8708(77)90108-6 · Zbl 0938.81568 · doi:10.1016/0001-8708(77)90108-6 [15] DOI: 10.1007/BF01609845 · doi:10.1007/BF01609845 [16] DOI: 10.1002/andp.19554520102 · Zbl 0068.40203 · doi:10.1002/andp.19554520102 [17] DOI: 10.1088/0370-1328/72/2/302 · Zbl 0085.21504 · doi:10.1088/0370-1328/72/2/302 [18] DOI: 10.1002/qua.560290113 · doi:10.1002/qua.560290113 [19] DOI: 10.1007/BF01205672 · Zbl 0618.35111 · doi:10.1007/BF01205672 [20] DOI: 10.1103/PhysRev.81.385 · Zbl 0042.23202 · doi:10.1103/PhysRev.81.385 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.