GF method

The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar 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, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar 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. 2V \approx \sum_{s,t=1}^{3N-6} F_{st} S_s\, S_t. The classical vibrational kinetic energy has the form: 2T = \sum_{s,t=1}^{3N-6} g_{st}(\mathbf{s}) \dot{S}_s\dot{S}_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\, \dot{S}_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 s 0 of V gives the positive definite and symmetric matrix G = g(s 0 ) −1 . One can solve the two matrix problems \mathbf{L}^\mathrm{T} \mathbf{F} \mathbf{L} =\boldsymbol{\Phi} \quad \mathrm{and}\quad \mathbf{L}^\mathrm{T} \mathbf{G}^{-1} \mathbf{L} = \mathbf{E}, simultaneously, since they are equivalent to the generalized eigenvalue problem \mathbf{G} \mathbf{F} \mathbf{L} = \mathbf{L} \boldsymbol{\Phi}, where \boldsymbol{\Phi}=\operatorname{diag}(f_1,\ldots, f_{3N-6}) where f i is equal to 4{\pi}^{2}{\nu}_i^{2} ( {\nu}_i is the frequency of normal mode i); \mathbf{E}\, is the unit matrix. The matrix L −1 contains the normal coordinates Q k in its rows: Q_k = \sum_{t=1}^{3N-6} (\mathbf{L}^{-1})_{kt} S_t , \quad k=1,\ldots, 3N-6. \, 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 \mathbf{s} = \operatorname{col}(S_1,\ldots, S_{3N-6}) \quad\mathrm{and}\quad \mathbf{Q} = \operatorname{col}(Q_1,\ldots, Q_{3N-6}), which satisfy the relation \mathbf{s} = \mathbf{L} \mathbf{Q}. 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{s}}^\mathrm{T} \mathbf{G}^{-1}\dot{\mathbf{s}}+ \mathbf{s}^\mathrm{T}\mathbf{F}\mathbf{s} = \dot{\mathbf{Q}}^\mathrm{T} \; \left( \mathbf{L}^\mathrm{T} \mathbf{G}^{-1} \mathbf{L}\right) \; \dot{\mathbf{Q}}+ \mathbf{Q}^\mathrm{T} \left( \mathbf{L}^\mathrm{T}\mathbf{F}\mathbf{L}\right)\; \mathbf{Q} = \dot{\mathbf{Q}}^\mathrm{T}\dot{\mathbf{Q}} + \mathbf{Q}^\mathrm{T}\boldsymbol{\Phi}\mathbf{Q} = \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 + f_t Q_t^2 \big). The Lagrangian L = T − V is L = \frac{1}{2} \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 - f_t Q_t^2 \big). The corresponding Lagrange equations are identical to the Newton equations \ddot{Q}_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. S_t =\sum_{A=1}^N \sum_{i=1}^3 s^t_{Ai} \, x_{Ai}= \sum_{A=1}^N \mathbf{s}^t_{A} \cdot \mathbf{x}_{A}, \quad \mathrm{for}\quad t = 1,\ldots,3N-6, where s A t is known as a Wilson s-vector. If we put the s^t_{Ai} into a (3N − 6) × 3N matrix B, this equation becomes in matrix language \mathbf{s} = \mathbf{B} \mathbf{x}. The actual form of the matrix elements of B can be fairly complicated. Especially for a torsion angle, which involves 4 atoms, it requires tedious vector algebra to derive the corresponding values of the s^t_{Ai} . See for more details on this method, known as the Wilson s-vector method, the book by Wilson et al., or molecular vibration. Now, \mathbf{s} = \mathbf{L} \mathbf{Q} = \mathbf{L} \mathbf{l}^{tr} \mathbf{q} = \mathbf{B} \mathbf{M}^{-1/2} \mathbf{q} \equiv \mathbf{D} \mathbf{q}, which can be inverted and put in summation language: Q_k = \sum_{A=1}^N \sum_{i=1}^3 D^k_{Ai}\, d_{Ai} \quad \mathrm{for}\quad k=1,\ldots, 3N-6. Here D is a (3N − 6) × 3N matrix, which is given by (i) the linearization of the internal coordinates s (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process). \mathbf{s} = \mathbf{B}\mathbf{x} , i.e. the matrix \mathbf{B} transforms the Cartesian coordinates to (linearized) internal coordinates. \mathbf{x} = \mathbf{M}^{-1/2}\mathbf{q}, i.e. the mass matrix \mathbf{M}^{1/2} transforms Cartesian coordinates to mass-weighted Cartesian coordinates. \mathbf{q} = \mathbf{l} \mathbf{Q}, i.e. the matrix \mathbf{l} transforms the normal coordinates to mass-weighted Cartesian coordinates. \mathbf{s} = \mathbf{L}\mathbf{Q}, i.e. the matrix \mathbf{L} transforms the normal coordinates to internal coordinates. Note the useful relationship: \mathbf{L}=\mathbf{B} \mathbf{M}^{-1/2} \mathbf{l}. These matrices allow one to construct the G matrix quite simply as \mathbf{G} = \mathbf{B} \mathbf{M}^{-1} \mathbf{B}^{\rm T}. \sum_{A=1}^N \mathbf{s}^t_{A} = 0\quad\mathrm{and}\quad \sum_{A=1}^N \mathbf{R}^0_A\times \mathbf{s}^t_A= 0, \quad t=1,\ldots,3N-6. These conditions follow from the Eckart conditions that hold for the displacement vectors, \sum_{A=1}^N M_A\; \mathbf{d}_{A} = 0 \quad\mathrm{and}\quad \sum_{A=1}^N M_A\; \mathbf{R}^0_{A} \times \mathbf{d}_{A} = 0.

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