This paper examines the vibronic energy levels of a symmetrical non-linear molecule in a spatially doubly degenerate electronic state which is split in first order by a doubly degenerate vibrational mode. The vibronic levels are classified by a quantum number, which in certain cases is formally related to the combined angular momentum of electronic and vibrational motion, and numerical values are obtained for the energies of these levels as functions of this quantum number and a dimensionless parameter measuring the magnitude of the electronic-vibrational coupling. It is shown that the selection rules for transitions from these vibronic levels to those of a non-degenerate electronic state allow changes in vibrational energy of any integral number of quanta, as though the Jahn-Teller effect were equivalent to a distortion which makes allowed vibrational transitions which would otherwise be forbidden. Numerical values are given for the oscillator strengths of vibronic absorption or emission bands involving transitions between a Jahn-Teller distorted state and an electronically non-degenerate state. It is found that in transitions from the latter to the former the vibrational structure of the electronic band exhibits two intensity maxima if the distortion is large.

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According to the Jahn-Teller theorem, symmetrical molecules with degenerate electronic states are unstable. Such molecules therefore take up a distorted shape. If there is near-degeneracy, the symmetrical shape may also be unstable. We have studied the distortion in some particular cases. The approach is to minimize the total electronic energy with respect to distortions of the nuclear framework, the latter being considered to be static. There are always several equivalent distortions of equal energy, so that a static distortion fails to remove the degeneracy. The discussion of dynamic effects is postponed to a subsequent paper. A linear molecule of formula BAB , for which two electronic states of opposite symmetry are sufficiently nearly degenerate, will be stable in a configuration with unequal A—B separations, and unstable when symmetrical. This example illustrates some of the main physical features of Jahn-Teller distortions in a simple manner. Although we know of no example where the symmetrical structure is actually unstable, there are examples where the tendency towards distortion noticeably reduces the force constant of the asymmetric vibration. Octahedral complexes AB 6 with degenerate electronic states occur in many situations (e. g. paramagnetic crystals, F-centres, luminescent centres and exciton states in cubic crystals). The orbital degeneracy may be threefold ( T 1 and T 2 ) or twofold. In the former case the stable distortion is found to be either of tetragonal symmetry about a [100] direction, or of trigonal symmetry about a [111] direction. The twofold degenerate situation leads to a more complicated situation. If one neglects anharmonic effects there appears to be an infinity of distortions minimizing the energy; a more detailed consideration of the anharmonic terms shows that the stable distortions are of elongated tetragonal character. This result has an important bearing on complexes involving the cupric ion.

The Newtonian definition of the mass-centre can be generalized to the restricted theory of relativity in several ways. Three in particular lead to fairly simple expressions in terms of instantaneous variables for quite general systems. Of these only one is independent of the frame in which it is defined. It suffers from the disadvantage that its components do not commute (in classical mechanics, do not have zero Poisson brackets), and are therefore unsuitable as generalized co-ordinates in mechanics. Of the other two, one is particularly . simply defined, and the other has commuting co-ordinates. The Poisson brackets can be derived from quite general considerations because the various mass-centres are expressible in terms of integrals of the energy-momentum tensor which are directly connected with the infinitesimal operators of the group of Lorentz transformations. The definitions are readily applicable to a single particle in theories, such as are current for elementary particles, where a co-ordinate observable does not exist, but an energy-momentum tensor does, and furnish the nearest approach possible to such observables. They are applied to electrons, particles of spin 0 and ℏ (scalar- and vector-meson theories), and to photons.

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