Calculations Employing Valence Internal Coordinates

The well-known GF matrix technique of E. B. Wilson and his colleagues for calculating the harmonic frequencies of polyatomic molecules is based on the use of valence coordinates, also referred to as internal coordinates. What is presented here is merely a sketch of the method a fuller discussion would require extensive use of matrix algebra, which is beyond the scope of this book. The appendix on matrices in this chapter serves only as a very short introduction to such methods. For details reference should be made to the classical work of E. B. Wilson, J. C. Decius and P. C. Cross (WDC) in the reading list. 

The contribution of the potential energy to the Hamiltonian is given in the harmonic approximation by 

In Equation 3.43 F now is the matrix of the force constants in valence coordinates. Again one finds normal mode coordinates, Qi, corresponding to the normal mode 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28) 

The matrix to be diagonalized for finding the vibrational frequencies is the matrix product of the above G matrix for Cartesian coordinates and the corresponding F matrix for Cartesian displacement coordinates. It is noted in passing that the GF matrix is generally not symmetric, i.e. 

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