The replusive forces between isotopic molecules

Abstract

It is usually assumed that the forces of attraction and repulsion between two molecules depend only on their electronic structures and are independent of the nuclear masses. While this assumption is undoubtedly true to a first approximation it fails to take into account the zero-point energy associated with the nuclear vibrations, which will modify the charge distribution both of the nuclei and of the electronic shells. Since the zero-point energy depends upon the nuclear mass, this may lead to differences between the behaviour of a pair of isotopic molecules such as H 2 and D 2 . If the restoring force of the nuclear vibration is not directly proportional to the displacement of the nuclei from their mean position, then the mean internuclear distance will be different for the two isotopes. The magnitude of this anharmonic effect can be calculated from spectrum data, and it is found that for H 2 and D 2 the difference is less than 10 -11 cm., and hence negligible. However, the energy of interaction of two molecules will not be a linear function of the inter­ nuclear distance within the molecules, so that even for harmonic oscillations the observed interaction will depend upon the magnitude of the zero-point energy. Both the attractive and the repulsive intermolecular forces will be affected in this way, but it is difficult to treat the former owing to the absence of any exact treatment of exchange forces between molecules. The problem is more easily attacked in the case of the Coulomb forces (which have a net repulsive effect), and the present paper constitutes an attempt to estimate the isotope effect for this type of force. It will be necessary to neglect the mutual deformation of the charge distributions caused by the approach of the two molecules. This assumption is usually made in dealing with Coulomb forces, and it is unlikely to introduce serious error in calculating the isotopic difference. The total Coulomb interaction between two molecules can then be written as G = G n + G n c + G e (1) where G n is the interaction between the nuclei of the two molecules, G ne the interaction between the nuclei of one molecule and the electron shell of the other, and G e the interaction between the two electron shells. The principles involved are most easily seen by considering G n for two diatomic molecules.