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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kohn–Sham equations ： ウィキペディア英語版 Kohn–Sham equations In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the Schrödinger equation of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.〔 〕〔 〕 The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as ''vs''(r) or ''v''eff(r), called the Kohn–Sham potential. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to :$\backslash left(-\backslash frac\backslash nabla^2+v\_(\backslash mathbf\; r)\backslash right)\backslash phi\_(\backslash mathbf\; r)=\backslash varepsilon\_\backslash phi\_(\backslash mathbf\; r)$ This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here, ''εi'' is the orbital energy of the corresponding Kohn–Sham orbital, ''φi'', and the density for an ''N''-particle system is :$\backslash rho(\backslash mathbf\; r)=\backslash sum\_i^N\; |\backslash phi\_(\backslash mathbf\; r)|^2.$ The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham (沈呂九), who introduced the concept at the University of California, San Diego in 1965. ==Kohn–Sham potential== In Kohn-Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as :$E()\; =\; T\_s()\; +\; \backslash int\; d\backslash mathbf\; r\backslash \; v\_(\backslash mathbf\; r)\backslash rho(\backslash mathbf\; r)\; +\; E\_()\; +\; E\_()$ where ''Ts'' is the Kohn–Sham kinetic energy which is expressed in terms of the Kohn–Sham orbitals as :$T\_s()=\backslash sum\_^N\backslash int\; d\backslash mathbf\; r\backslash \; \backslash phi\_i^$ *(\mathbf r)\left(-\frac\nabla^2\right)\phi_i(\mathbf r), ''v''ext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), ''EH'' is the Hartree (or Coulomb) energy, :$E\_=\backslash int\; d\backslash mathbf\; r\backslash int\; d\backslash mathbf\text{'}\backslash \; .$ and ''E''xc is the exchange-correlation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals to yield the Kohn–Sham potential as :$v\_(\backslash mathbf\; r)\; =\; v\_(\backslash mathbf)\; +\; e^2\backslash int\; d\backslash mathbf\text{'}\; +\; .$ where the last term :$v\_(\backslash mathbf\; r)\backslash equiv$ is the exchange-correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory. The Kohn–Sham orbital energies ''εi'', in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as :$E\; =\; \backslash sum\_^N\; \backslash varepsilon\_i\; -\; V\_()\; +\; E\_()\; -\; \backslash int\; \backslash rho(\backslash mathbf)\; d\backslash mathbf$ Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kohn–Sham equations」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.