Approximate Separability and Choice of Coordinates for Excited Vibrations of Polyatomic Molecules and Clusters

Abstract

To simplify the dynamics of coupled anharmonic vibrations in polyatomic systems, methods such as the self-consistent field (SCF) approximation and the adiabatic approximation assume that motions in different modes are separable. Success of such methods thus depends considerably on a good choice of the coordinates that are being mutually separated.

This study examines the physical considerations that can be used in making adequate choices of coordinate systems in SCF calculations of small molecules and van der Waals clusters. Both Cartesian and curvilinear coordinate systems are discussed. Part of this article reviews results of previous studies on this topic, e.g. the optimization of Cartesian coordinates by coordinate rotation, and the introduction of ellipsoidal coordinates for the bending-stretching spectrum of HCN. New results are presented for Xe(He)₂, a prototype of the “Three Balls” problem; and for I₂He, a prototype of the “Stick and Ball” systems. Comparative SCF calculations using hyperspherical, ellipsoidal and Jacobi coordinates are made, in a full 3D framework. Hyperspherical modes are found optimal for the “Three Balls”, ellipsoidal coordinates prove optimal for the “Stick and Ball”. Physical explanation for these findings is offered, and related insight is obtained into the vibrational motions involved. Suggestions are made of possible extensions of some of the above themes for other systems.