The VW transformation -- A Simple Alternative to the Wilson GF Method

TL;DR

The paper introduces the VW transformation as a simpler, computationally efficient alternative to the Wilson GF method for calculating molecular vibration frequencies and normal modes, avoiding complex internal coordinate transformations.

Contribution

The VW transformation replaces the force constant matrix with a non-singular matrix, simplifying calculations while preserving all vibrational properties, and eliminates the need for internal coordinate transformations.

Findings

VW transformation yields identical eigenmodes and frequencies as the GF method.

It simplifies computations by working directly in Cartesian coordinates.

The new matrix W is non-singular, enabling conventional matrix techniques.

Abstract

An alternative, the VW transformation, is proposed to replace the Wilson GF method for calculating molecular vibration frequencies and normal modes. The VW transformation yields precisely the same eigenmodes and and eigenfrequencies that are found with the GF method. The transformation proceeds entirely in the mass-normalized Cartesian coordinates of the individual atoms, with no transformations to internal coordinates. The VW transformation thus offers an enormous computational simplification over the GF method, namely the mathematical apparatus needed to transform to internal coordinates is eliminated. All need for new researchers to understand the complex matrix transformations underlying the GF method is thus also removed. The VW transformation is not a projection method; the internal vibrations remain dispersed over all 3N atomic coordinates. In the VW method, the 3N x 3N force constant matrix V is replaced with a new 3N x 3N matrix W. The replacement is physically transparent. The matrix W has the same internal normal modes and vibration frequencies that V does. However, unlike V, W is not singular, so the normal modes and normal mode frequencies can be obtained using conventional matrix techniques.