The occurrence and properties of molecular vibrations with V ( x ) = ax 4

Abstract

It is shown that in certain modes of vibration of plane rings the potential energy for small displacements is proportional to the fourth power of the displacement, provided that there is free rotation about the bonds of the ring. This type of vibration is termed a ‘fourth-power vibration’. It is likely to occur in cyclobutane and its derivatives, in a number of halides having the formula X 2 Y 6 , and in the hydrides of group III elements. The energies and wave functions of the first four levels of a one-dimensional oscillator with V ( x ) = ax 4 have been derived by a method of successive approximations, and asymptotic formulae are given for the higher levels. The wave functions are qualitatively similar to those of a harmonic oscillator, but the energy levels differ considerably. A comparison is made between energy levels for oscillators with V ( x ) = a q | x q | and different values of q . The selection rule for dipole radiation from a fourth-pow er vibration is discussed. Overtones will be more numerous than in the spectrum of a harmonic oscillator. Estimates are made of the spectrum frequencies of fourth-power vibrations in actual molecules, with special reference to cyclobutane and diborane. For these two molecules there are observed infra-red frequencies of approximately the expected value. The isotope effect should provide a means of discriminating experimentally between harmonic and fourth-power vibrations. The contribution of a fourth-power vibration to any thermodynamic function will differ from that of a harmonic vibration with the same fundamental spectrum frequency. Figures are given for the specific heat, where the difference should be detectable experimentally. In the general case V ( x ) = a q | x q | the energy levels derived from the quantum theory lead to expressions for the thermodynamic functions which agree with the predictions of classical theory at high temperatures.