Configurational Instability of Degenerate Electronic States

Abstract

The implications of the Jahn-Teller theorem are discussed with special reference to the case where the forces tending to lower the symmetry of electronically degenerate molecular states are of the same order as the restoring forces encountered during typical vibrations. It is shown that the resulting dynamical situation may be described as a particular kind of coupling between low-frequency electronic motions and nuclear modes. I N 1937, Jahn and Teller 1 demonstrated that electronically degenerate states of nonlinear molecules are unstable with respect to certain asymmetric displacements of their nuclei. If the nuclei are of infinite mass, two possibilities may be envisaged. On the one hand, the molecule may dissociate since it possesses no stable nuclear configurations. On the other, it may take one of several new shapes having lower symmetry. In the present note, we shall investigate this latter possibility and consider particularly the effect of finite nuclear masses. For, if the stability attained by assuming an asymmetric nuclear configuration is no more than the zero-point energy of a typical vibrational mode-or if the concomitant displacement is no larger than a zero-point amplitude-it is clear that a special coupling between electronic and nuclear motions will arise. ILLUSTRATIVE EXAMPLEConsider a molecule with six identical nuclei, whose initial configuration is that of a regular hexagon (Deh). The bond distance is fixed so that the molecule is stable at least with respect to totally symmetric displacements. The molecule is supposed to be in a doubly-degenerate electronic state of symmetry E\ u . To be definite, the electronic eigenfunctions ^°, ^JS°=^J.°* are chosen so that they simply acquire factors 00, co b =Q)~~1 respectively on rotating the nuclear framework through 2ir/6 radians [«=exp(2iri/6)].Excepting the translational and rotational degrees of freedom, all possible nuclear displacements may be described by linearly combining symmetry coordinates of species a\ g , /5i M , fag, Pzu, €i«, 2€2^, t 2u . These may be chosen to be eigenfunctions of the sixfold rotation operator also, so that they are in general complex. When used collectively, we call them s u (u=l, • • •, 12); more specifically, however, they are si w (ai g ), s 2 (3) (ftw)> s^(M, s^ifcu), *5 (±1) (ei0, *6 (±2) (* 2ff ), *7 (±2) (e 2 "), Ss (±2) ( e 2u)y where s r (n) acquires the factor cc n under CQ and s r {~n) is its complex conjugate. The Hamiltonian for the molecule as a whole consists of two parts: T, the kinetic energy operator for the nuclei and "0, which is called the electronic Hamiltonian. So defined, V contains the kinetic energy operator for 1 H. A. Jahn and E. Teller, Proc. Roy. Soc. (London) A164, 117 (1937). the electrons, their potential energy in the field of the nuclei, their mutual repulsions and the mutual repulsion of the nuclei; it depends on the nuclear coordinates parametrically only. We shall suppose that it is possible to develop V as a Taylor's series in the symmetry coordinates about the hexagonal ref...

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