Vibration–rotation kinetic energy operators: A geometric algebra approach

Abstract

The elements of the reciprocal metric tensor g(qiqj), which appear in the exact 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. 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. In order to show that the rotational measuring vectors produced by the present method agree with the known results, the general formulas are applied to the triatomic bond-z, and to the triatomic angle bisector frame. All the rotational measuring vectors are also explicitly derived for a new triatomic body frame defined in terms of two Jacobi vectors. As a final application, all the rotational measuring vectors are presented for a new N-atomic frame defined in terms of N−1 Jacobi vectors, and for a simple N-atomic frame defined in terms of N nuclear position vectors (N=3,4,5,6,…).