On the kinetic energy operator and the general triatomic molecule

Abstract

In this paper we present the quantum mechanical kinetic energy operator (KEO) of a triatomic molecule in a form which exhibits the separation of the vibrational-rotational motion and their coupling, and which is independent of the masses. First, with the help of matrix methods, we separate the translational and rotational motion for the general N-body system. In this derivation the origin of the “body-fixed” axes is at the center of mass but we do not specify any conditions defining the orientation of these axes with respect to the molecule. Next, we obtain simple expressions for the triatomic molecule. These result from setting the body-fixed frame of reference coincident with the instantaneous principal axes of inertia, and from choosing suitable sets of generalized internal coordinates. In the two sets considered, one can group the terms of the KEO in a way that clearly indicates how pure vibration around equilibrium plus rigid rotation are the predominant contribution to energy. The purely vibrational part of the KEO has a concise explicit form. In both sets one of the coordinates, which describes the antisymmetric mode in the case of a symmetric bent XY 2 molecule, appears only through its associated momentum in the KEO. Moreover, one of the two sets puts the purely vibrational part in a separable form. All these features suggest that the above choice would be useful to study large molecular deformations. Finally turning to the potential, we express the internuclear distances in terms of the generalized coordinates. Simple combinations of the latter are linear functions of the former squared, the coefficients containing the masses only. Thus, a matrix of constants relates two isotopic varieties of a molecular species. The symmetric XY 2 molecule illustrates further simplifications in the harmonic approximation.