The first term can be included in the zero of energy. Thus, The classical vibrational kinetic energy has the form: where gst is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Mixed terms 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 simultaneously, since they are equivalent to the generalized eigenvalue problem where where fi is equal to ( is the frequency of normal mode i); is the unit matrix. The matrix L−1 contains the normal coordinates Qk in its rows: 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 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. Normal coordinates in terms of Cartesian displacement coordinates Often the normal coordinates are expressed as linear combinations of Cartesian displacement coordinates. Let RA be the position vector of nucleus A and RA0 the corresponding equilibrium position. Then is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of the internal curvilinear coordinates qt expresses the coordinate St in terms of the displacement coordinates where sAt is known as a Wilson s-vector. If we put the into a (3N − 6) × 3N matrix B, this equation becomes in matrix language The actual form of the matrix elements of B can be fairly complicated. Especially for a torsion angle, which involves 4 atoms, it requires