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.
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.
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.
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 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.
Thus, The classical vibrational kinetic energy has the form: where gst is an element of the metric tensor of the internal (curvilinear) coordinates.
Evaluation of the metric tensor g in the minimum s0 of V gives the positive definite and symmetric matrix G = g(s0)−1.
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 .
We introduce the vectors which satisfy the relation Upon use of the results of the generalized eigenvalue equation, the energy E = T + V (in the harmonic approximation) of the molecule becomes: The Lagrangian L = T − V is The corresponding Lagrange equations are identical to the Newton equations for a set of uncoupled harmonic oscillators.
These ordinary second-order differential equations are easily solved, yielding Qt as a function of time; see the article on harmonic oscillators.
Especially for a torsion angle, which involves 4 atoms, it requires tedious vector algebra to derive the corresponding values of the
Now, which can be inverted and put in summation language: 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).
There are several related coordinate systems commonly used in the GF matrix analysis.