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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 GF method ： ウィキペディア英語版 GF method The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by E. Bright Wilson to obtain certain ''internal coordinates'' for a vibrating semi-rigid molecule, the so-called ''normal coordinates'' ''Q''k. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to obtaining vibrational amplitudes of the atoms as a function of time. In Wilson's GF method it is assumed that the molecular kinetic energy consists only of harmonic vibrations of the atoms, ''i.e.,'' overall rotational and translational energy is ignored. Normal coordinates appear also in a quantum mechanical description of the vibrational motions of the molecule and the Coriolis coupling between rotations and vibrations. It follows from application of the Eckart conditions that the matrix G−1 gives the kinetic energy in terms of arbitrary linear internal coordinates, while F represents the (harmonic) potential energy in terms of these coordinates. The GF method gives the linear transformation from general internal coordinates to the special set of normal coordinates. == The GF method == A non-linear molecule consisting of ''N'' atoms has 3''N'' − 6 internal degrees of freedom, because positioning a molecule in three-dimensional space requires three degrees of freedom, and the description of its orientation in space requires another three degree of freedom. These degrees of freedom must be subtracted from the 3''N'' degrees of freedom of a system of ''N'' particles. The interaction among atoms in a molecule is described by a potential energy surface (PES), which is a function of 3''N'' − 6 coordinates. The internal degrees of freedom ''q''1, ..., ''q''3''N''−6 describing the PES in an optimal way are often non-linear; they are for instance ''valence coordinates'', such as bending and torsion angles and bond stretches. It is possible to write the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule. This is why Wilson linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate ''q''t is denoted by ''S''t. The PES ''V'' can be Taylor expanded around its minimum in terms of the ''S''t. The third term (the Hessian of ''V'') evaluated in the minimum is a force derivative matrix F. In the harmonic approximation the Taylor series is ended after this term. The second term, containing first derivatives, is zero because it is evaluated in the minimum of ''V''. The first term can be included in the zero of energy. Thus, :$2V\; \backslash approx\; \backslash sum\_^\; F\_\; S\_s\backslash ,\; S\_t.$ The classical vibrational kinetic energy has the form: :$2T\; =\; \backslash sum\_^\; g\_(\backslash mathbf)\; \backslash dot\_s\backslash dot\_t\; ,$ where ''g''''st'' is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Mixed terms $S\_s\backslash ,\; \backslash dot\_t$ generally present in curvilinear coordinates are not present here, because only linear coordinate transformations are used. Evaluation of the metric tensor g in the minimum q0 of ''V'' gives the positive definite and symmetric matrix G = g(q0)−1. One can solve the two matrix problems :$\backslash mathbf^\backslash mathrm\; \backslash mathbf\; \backslash mathbf\; =\backslash boldsymbol$ \quad \mathrm\quad \mathbf^\mathrm \mathbf^ \mathbf = \mathbf, simultaneously, since they are equivalent to the generalized eigenvalue problem :$$ \mathbf \mathbf \mathbf = \mathbf \boldsymbol, where $\backslash boldsymbol=\backslash operatorname(f\_1,\backslash ldots,\; f\_)$ where ''fi'' is equal to $4^\_i^$ ($\_i$ is the frequency of normal mode ''i''); $\backslash mathbf\backslash ,$ is the unit matrix. The matrix L−1 contains the ''normal coordinates'' ''Q''k in its rows: :$Q\_k\; =\; \backslash sum\_^\; (\backslash mathbf^)\_\; S\_t\; ,\; \backslash quad\; k=1,\backslash ldots,\; 3N-6.\; \backslash ,$ Because of the form of the generalized eigenvalue problem, the method is called the GF method, often with the name of its originator attached to it: Wilson's GF method. By matrix transposition in both sides of the equation and using the fact that both G and F are symmetric matrices, as are diagonal matrices, one can recast this equation into a very similar one for FG . This is why the method is also referred to as Wilson's FG method. We introduce the vectors :$\backslash mathbf\; =\; \backslash operatorname(S\_1,\backslash ldots,\; S\_)$ \quad\mathrm\quad \mathbf = \operatorname(Q_1,\ldots, Q_), which satisfy the relation :$$ \mathbf = \mathbf \mathbf. Upon use of the results of the generalized eigenvalue equation, the energy ''E'' = ''T '' + ''V'' (in the harmonic approximation) of the molecule becomes: :$$ 2E = \dot \mathbf^\dot^\mathrm\mathbf\mathbf ::$$ = \dot \; \left( \mathbf^\mathrm \mathbf^ \mathbf\right) \; \dot^\mathrm \left( \mathbf^\mathrm\mathbf\mathbf\right)\; \mathbf ::$$ = \dot\dot^\mathrm\boldsymbol\mathbf = \sum_^ \big( \dot_t^2 + f_t Q_t^2 \big). The Lagrangian ''L'' = ''T'' − ''V'' is :$$ L = \frac \sum_^ \big( \dot_t^2 - f_t Q_t^2 \big). The corresponding Lagrange equations are identical to the Newton equations :$$ \ddot_t + f_t \,Q_t = 0 for a set of uncoupled harmonic oscillators. These ordinary second-order differential equations are easily solved, yielding ''Q''''t'' as a function of time; see the article on harmonic oscillators. 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「GF 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.