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 Qk. 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 3N − 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 3N 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 3N − 6 coordinates. The internal degrees of freedom s1, ..., s3N−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 st is denoted by St. The PES V can be Taylor expanded around its minimum in terms of the St. 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,
2 V ≈ ∑ s , t = 1 3 N − 6 F s t S s S t . {\displaystyle 2V\approx \sum _{s,t=1}^{3N-6}F_{st}S_{s}\,S_{t}.}
The classical vibrational kinetic energy has the form:
2 T = ∑ s , t = 1 3 N − 6 g s t ( s ) S ˙ s S ˙ t , {\displaystyle 2T=\sum _{s,t=1}^{3N-6}g_{st}(\mathbf {s} ){\dot {S}}_{s}{\dot {S}}_{t},}
where gst is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Mixed terms S s S ˙ t {\displaystyle 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 s0 of V gives the positive definite and symmetric matrix G = g(s0)−1. One can solve the two matrix problems
L T F L = Φ a n d L T G − 1 L = E , {\displaystyle \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
G F L = L Φ , {\displaystyle \mathbf {G} \mathbf {F} \mathbf {L} =\mathbf {L} {\boldsymbol {\Phi }},}
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