The elements of the reciprocal metric tensor, which appears in the exact (non-relativistic) internal kinetic energy operators of polyatomic molecules, can, in principle, be written as the mass-weighted sum of the inner products of measuring vectors associated to the nuclei of the molecule. In the case of vibrational degrees of freedom, the measuring vectors are simply the gradients of the vibrational coordinates with respect to the position of the nucleus in question. They can be calculated either by the direct vectorial differentiation, or from the variation of the appropriate coordinate along the path of the particle. It is more difficult to find these vectors for the rotational degrees of freedom, because the components of the total angular momentum operator are not conjugated to any rotational coordinates. However, by the methods of geometric algebra, the rotational measuring vectors are easily calculated for any geometrically defined body-frame, without any restrictions to the number of particles in the system. The kinetic energy operators produced by the present approach are in perfect agreement with the previously published results. The methods of geometric algebra have been recently applied (with good success) to the description of the large amplitude inversion vibration of ammonia.

janne.pesonen@helsinki.fi