# The Pulay matrix scaling method

### Application of quantum chemical methods

The approach of Pulay to the calculation of force constants from ab initio quantum mechanics can be used to estimate the relative magnitude of interaction force constants. However, it is still not feasible to calculate the fundamental frequencies of vibration directly from an ab initio quantum chemical approach. For this reason Pulay developed the scaled internal coordinate method. The internal coordinates are calculated from by quantum chemical method (e.g. Hartree-Fock or density functional). The matrix is then scaled by a set of parameters according to the prescription

Fij scaled = (CiCj)1/2 Fij internal

The method employed to obtain the estimate of the force constants is to geometry-optimize the molecule. Once a stable minimum in the potential energy surface has been reached the forces on each atom vanishes:

The first derivative is used to step the geometry towards the minimum.