Efficient Procedure for the Numerical Calculation of Harmonic Vibrational Frequencies Based on Internal Coordinates

We propose a general procedure for the numerical calculation of the harmonic vibrational frequencies that is based on internal coordinates and Wilson's GF methodology via double differentiation of the energy. The internal coordinates are defined as the geometrical parameters of a Z-matrix structure, thus avoiding issues related to their redundancy. Linear arrangements of atoms are described using a dummy atom of infinite mass. The procedure has been automated in FORTRAN90 and its main advantage lies in the nontrivial reduction of the number of single-point energy calculations needed for the construction of the Hessian matrix when compared to the corresponding number using double differentiation in Cartesian coordinates. For molecules of C-1 symmetry the computational savings in the energy calculations amount to 36N - 30, where N is the number of atoms, with additional savings when symmetry is present. Typical applications for small and medium size molecules in their minimum and transition state geometries as well as hydrogen bonded clusters (water dimer and timer) are presented. In all cases the frequencies based on internal coordinates differ on average by <1 cm(-1) from those obtained from Cartesian coordinates.