Degenerate States

By the Jahn-Teller theorem, systems with orbital degeneracy will distort so as to remove the degeneracy. However, many dynamic Jahn-Teller systems are known in which the time-average geometry is of the higher point-group. These systems are the kind that will be addressed here.  

The analytical RHF configuration interaction first derivative calculation developed by Liotard [43] has been modified to allow systems with degenerate states to be run.

Each of the degenerate states is a linear combination of microstates. Each microstate can be described by a Slater determinant [66,67], which represents a specific pattern of occupancy of molecular orbitals. Each M.O. is a linear combination of Slater atomic orbitals.

The whole state is best described by an equal mixture of the degenerate states of which it is composed. Note that this is NOT a combination of states, rather it is a mixture. An example of a combination of states is a state function, composed of a linear combination of microstates. In such a combination the phase-factor between microstates is significant, thus state(1)= (1/ 2)1/2(Microstate(a) + Microstate(b)) is different from state(2) (1/ 2)1/2(Microstate(a) - Microstate(b)). An example of a mixture of states is the 2T2g state of TiF63-, a d1 system, in which the best description of the state is an equal mixture of the three degenerate space components of T2g, and an equal mixture of the two spin components of the Kramer's doublet. The overall state is thus 1/6($\alpha$(T2g(x)+T2g(y)+T2g(z))+ $\beta$(T2g(x)+T2g(y)+T2g(z)).

If equimixtures are not used, then the Jahn-Teller theorem applies, and the system would immediately distort so as to remove the degeneracy. In the case of TiF63-, this would result in distortion from Oh to D4hsymmetry.