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The A Matrix in Molecular Vibration-Rotation Theory
Brenda P. Winnewisser1, James K. G. Watson
1Physikalisch-Chemisches Institut der Justus-Liebig-Universität, Heinrich-Buff-Ring 58, Giessen, D-35392, Germany
Journal of Molecular Spectroscopy
|February 13, 2001
Summary
Crawford's A matrix in molecular vibration theory is explored. We demonstrate that internal-coordinate force constants are independent of the specific A matrix used, resolving isotope-dependency concerns.
Area of Science:
- Physical Chemistry
- Theoretical Chemistry
- Computational Chemistry
Background:
- The molecular vibration theory relies on the relationship between Cartesian and internal coordinates.
- Crawford's A matrix is related to Wilson's B matrix but lacks uniqueness due to B's rectangular nature.
Purpose of the Study:
- To explore the general form of Crawford's A matrix.
- To derive a unique and widely used form of the A matrix using Eckart conditions.
- To investigate the isotope-dependency of internal-coordinate harmonic force constants.
Main Methods:
- Analysis of the general form of Crawford's A matrix.
- Application of Eckart conditions to derive a specific A matrix solution.
- Mathematical derivation to demonstrate the independence of force constants from the choice of A matrix.
Main Results:
- The commonly used A matrix solution A = M(-1)B(T)G(-1) is derived.
- Superficial isotope-dependence of internal-coordinate harmonic force constants is shown to vanish.
- Force constants are proven independent of an arbitrary nonsingular matrix W in the general A matrix form A = WB(T)(BWB(T))(-1).
Conclusions:
- The choice of A matrix does not affect the physically meaningful internal-coordinate harmonic force constants.
- This resolves potential ambiguities arising from the non-uniqueness of the A matrix in molecular vibration analysis.