General formulation of the vibrational kinetic energy operator in internal bond-angle coordinates

Abstract

A general formulation of the vibrational kinetic energy operator expressed in internal bond-angle coordinates is presented. This formulation is based on Podolsky’s expression for the covariant form of the Laplace–Beltrami operator. When a valid set of internal bond-angle coordinates is employed, it is possible to adapt a systematic approach to solve for the Jacobian determinant governing the coordinate transformation from Cartesian coordinates. In the general case of an arbitrary N-atom system, this Jacobian always factorizes to a simple form. This allows one to evaluate all the terms that contribute to V̂′, the effective potential that arises from transforming the kinetic energy operator to internal coordinates. We discuss restrictions on the choice of internal vibrational coordinates that may be included in a valid set. We then provide tabular information from which the vibrational kinetic energy operator for any molecular system can be constructed directly with no matrix inversion or chain rule manipulation required.

Polyspherical Coordinate Systems on Orbit Spaces with Applications to Biomolecular Shape

“…We show that any non-collinear configuration of three or more atoms can be non-singularly described by some Zsystem internal coordinate system. Thus Z-systems provide the first mathematically rigorous and chemically adequate general theory of valence internal coordinates, and they are consistent with the guidelines stated in [31]. Despite assertions in that paper, there do exist valence internal coordinate systems (combinations of bond lengths, bond angles, and wedge angles) which are not Z-system coordinate systems and which fail to satisfy the guideline "bond angle coordinates must all subtend two of the bond length coordinates present in a valid set".…”

“…If |N | = N then there should be 3N − 6 independent internal coordinates assigned to each non-collinear shape. Internal coordinate systems have been studied by chemists and molecular physicists for a long time because of their importance for simplifying the study of molecular vibrations, scattering processes, and geometry optimization [6], [7], [16], [20], [28], [31], [32], [67], [70], [76], [77], [90]. It is possible to devise internal coordinate systems which use only distances as coordinates; this approach is related to distance geometry [18] and rigidity theory [36], [85].…”

“…For attempts in two dimensions to utilize both distances and angles as internal coordinates, see [88], [89]. Internal coordinate systems which are based on collections of bond lengths, bond angles, and torsion angles [70], [82], are loosely called valence coordinates by chemists, but until recently [31] there has been little effort to understand exactly which combinations of these valence coordinates form "good" internal coordinate systems. In fact, the recent appearance of [31] illustrates the fact that the term "valence coordinate system" has never been rigorously defined.…”

“…Internal coordinate systems which are based on collections of bond lengths, bond angles, and torsion angles [70], [82], are loosely called valence coordinates by chemists, but until recently [31] there has been little effort to understand exactly which combinations of these valence coordinates form "good" internal coordinate systems. In fact, the recent appearance of [31] illustrates the fact that the term "valence coordinate system" has never been rigorously defined. This problem is becoming more urgent now that some molecular dynamics simulations are being done in internal coordinates [67], [41], [56].…”

Some studies on generalized coordinate sets for polyatomic molecules

“…A well-known example of internal coordinates is the so called BAT coordinates that consist of bond length, angle, and torsional coordinates. 15,16 BAT coordinates are more natural than Cartesian coordinates for describing the dynamics of polyatomic molecules, as they have already taken into account the constraints imposed by the chemical bonding connectivity. Thus, it will be of broad interest to present some properties on the generalized forces of BAT coordinates.…”