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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 Holstein–Herring method ： ウィキペディア英語版 Holstein–Herring method The Holstein–Herring method, also called the Surface Integral method, also called Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei (Pauli Exclusion Principle). ==Theory== The basic idea pioneered by Holstein and Herring approach can be illustrated for the hydrogen molecular ion or more generally, atom-ion systems or ''one-active electron'' systems, as follows. We consider states that are represented by even or odd functions with respect to behavior under space inversion. This is denoted with the suffixes g and u from the German ''gerade'' and ''ungerade'' and are standard practice for the designation of electronic states of diatomic molecules, whereas for atomic states the terms ''even'' and ''odd'' are used. The electronic Schrödinger equation can be written as: :$$ \left( -\frac \nabla^2 + V \right) \psi = E \psi~, where ''E'' is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function $\backslash psi=\backslash psi(\backslash mathbf)$ depending on the spatial coordinates of the electron and where $V$ is the electron-nuclear Coulomb potential energy function. For the hydrogen molecular ion, this is: :$$ V = - \frac \left( \frac + \frac \right) For any gerade (or even) state, the electronic Schrödinger wave equation can be written in atomic units ($\backslash hbar=m=e=4\; \backslash pi\; \backslash varepsilon\_0\; =1$) as: :$$ \left( -\frac \nabla^2 + V(\textbf) \right) \psi_ = E_ \psi_ For any ungerade (or odd) state, the corresponding wave equation can be written as: :$$ \left( -\frac \nabla^2 + V(\textbf) \right) \psi_ = E_ \psi_ For simplicity, we assume real functions (although the result can be generalized to the complex case). We then multiply the gerade wave equation by $\backslash psi\_$on the left and the ungerade wave equation on the left by $\backslash psi\_$and subtract to obtain: :$$ \psi_ \nabla^2 \psi_ - \psi_ \nabla^2 \psi_ = \; . where $\backslash Delta\; E\; =\; E\_\; -\; E\_$ is the ''exchange energy splitting''. Next, without loss of generality, we define orthogonal single-particle functions, $\backslash phi\_A^\; =\; \backslash frac\; +\; \backslash phi\_B^\; -\; \backslash phi\_B^$ are in general ''polarized'' i.e. they are not pure eigenfunctions of angular momentum with respect to their nuclear center, see also below). Note, however, that in the limit as $R\; \backslash rightarrow\; \backslash infty$, these localized functions $\backslash phi\_^$. We denote $M$ as the mid-plane located exactly between the two nuclei (see diagram for hydrogen molecular ion for more details), with $^3$ space is divided into left ($L$) and right ($R$) halves. By considerations of symmetry: :$$ \left. \psi_ \right|_M = \mathbf \cdot \left. \mathbf \psi_ \right|_M = 0 \; . This implies that: :$$ \left. \phi_^ \right|_M \; , \qquad \phi_^ \cdot \left. \mathbf \phi_^ \phi_B^2 ~dV and conversely. Integration of the above in the whole space left to the mid-plane yields: :$$ 2 \int_ \psi_ \psi_ ~ dV = \int_ ( \phi_A^2 - \phi_B^2 ) ~ dV = 1 - 2 \int_R \phi_A^2 ~ dV and :$$ \int_ ( \psi_ \nabla^2 \psi_ - \psi_ \nabla^2 \psi_ ) ~dV = \int_ ( \phi_^ - \phi_^ ) ~dV From a variation of the divergence theorem on the above, we finally obtain: :$$ \Delta E = \phi_A^ where $d$ and the ground state $1\; s\; \backslash sigma\_g$ (as expressed in molecular notation—see graph for energy curves), was found to be: :$$ \Delta E = E_ - E_ = \frac \, R \, e^ Previous calculations based on the LCAO of atomic orbitals had erroneously given a lead coefficient of $4/3$ instead of $4/e$. While it is true that for the Hydrogen molecular ion, the eigenenergies can be mathematically expressed in terms of a generalization of the Lambert W function, these asymptotic formulae are more useful in the long range and the Holstein–Herring method has a much wider range of applications than this particular molecule. 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「Holstein–Herring method」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.