The A Matrix in Molecular Vibration–Rotation Theory

• Published in: Journal of molecular spectroscopy Vol. 205; no. 2; pp. 227 - 231
• Main Authors: Winnewisser, Brenda P., Watson, James K.G.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01-02-2001
• Subjects: A matrix; ab initio force fields; Cartesian coordinates; force constants; internal coordinates; A matrix; internal coordinates; force constants; ab initio force fields; Cartesian coordinates
• ISSN: 0022-2852; 1096-083X
• DOI: 10.1006/jmsp.2000.8268

Crawford's A matrix in the theory of molecular vibrations is, in a sense, the inverse of Wilson's B matrix, but is not unique because B is rectangular. We consider the general form of A and then use the Eckart conditions to obtain the solution A = M−1BTG−1, which has been widely used. Although the internal-coordinate harmonic force constants f = ATFXA, where FX are the Cartesian force constants, are superficially isotope-dependent, we show that this dependence vanishes. More generally, solutions of the form A = WBT(BWBT)−1, where W is an arbitrary nonsingular square matrix, are shown to give an f matrix that is independent of W.