Calculation of Mean Atomic Positions in Vibrating Polyatomic Molecules

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Abstract
A new method is described for calculating effects of anharmonic vibrations on bond lengths and angles in polyatomic molecules. It is shown that the quantum mechanical analog of Newton's second law of motion leads to simple equilibrium conditions from which it is possible to calculate mean displacements of atoms from positions of minimum potential energy. The method resembles the standard first‐order perturbation method insofar as it relates mean displacements to mean‐square displacements which can be computed from the zeroth‐order harmonic problem. It gives a much more direct calculation, however, circumventing the laborious nonlinear transformations between internal and normal coordinates encountered in the perturbation method. Advantages and limitations are discussed. Applications to several problems are outlined briefly, including isotope effects and effects of temperature on mean molecular structures. The approach, extended to crystal lattices, is shown to account for Grueneisen's relation which correlates thermal expansion and heat capacity. Numerical results of calculations are presented for H2O, D2O, NH3, and ND3, comparing the present method with other methods. Contrary to prevailing opinion, the mean apical angle of NH3 appears to be smaller than that of ND3.