The Mean Amplitudes of Thermal Vibrations in Polyatomic Molecules. II. An Approximate Method for Calculating Mean Square Amplitudes

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Abstract
A simple method of the calculation of the mean amplitudes of both directly bonded atom pairs and nonbonded ones is proposed by approximating the function cothx by 1/x+x/4. The enumeration of the L matrix, the most time‐consuming procedure, is replaced by the calculation of the inverse matrix F−1 from the potential energy matrix. The formulae obtained are applicable with reasonable accuracy (4 percent) to almost all the cases except the atom pairs having higher frequencies than 1200 cm−1. For the latter cases modified formulas are presented by using hyperbolic cotangent function. It is also revealed that, if some allowance be made for the errors of the mean amplitudes, a definite value can be assigned to the mean amplitude of a given pair of atoms, without referring to the structure of the other part of the molecule. Reliable values of the mean amplitudes are computed for some typical atom pairs, by the use of the force constants of Urey‐Bradley type obtained by the analysis of the vibrational spectra.