Degeneracy in molecules

The above results mainly apply to the Longuet-Higgins E x e problem, but this historical survey would be incomplete without reference to early work on the much more challenging problems posed by threefold or higher electronic degeneracies in molecules with tetrahedral or octahedral symmetry [3]. For example, tetrahedral species, with electronic symmetry T or T2, have at least five Jahn-Teller active vibrations belonging to the representations E and T with individual coordinates (Qa,Qb) and (Qx. Qx. Q ) say. The linear terms in the nine Hamiltonian matrix elements were shown in 1957 [3] to be... 

The discussion so far has focused on two-state conical intersections, which are the most common conical intersections, and which have been studied extensively. Three-state degeneracies imposed by symmetry have been studied in the context of the Jahn-Teller problem for many decades,but only minor attention had been given to accidental three-state degeneracies in molecules until recently. As most molecular systems in nature have low or no symmetry, these accidental intersections may have a great impact on the photophysics and photochemistry of those molecular systems, as has been found in accidental two-state intersections.Three-state degeneracies may provide a more efficient relaxation pathway when more than one interstate transition is needed. Moreover, they introduce more complicated geometric phase effects,and they can affect the system s dynamics and pathways available for radiationless transitions. 

Some values of rotation and vibration wavenumbers and temperatures of small molecules, and the degeneracy in parentheses, are listed in table 4.16. One can see... 

In a number of cases the molecule under consideration has zero-order degenerate localized states, e.g., two carbonyl groups in a diketone. Interactions between the two chromophores will, in general, lift the zero-order degeneracy. In such a molecule there are, say, two discrete levels coupled... 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

Correlation effects in molecules are normally partitioned into near-degeneracy effects (static correlation) and dynamic correlation. Qualitatively they differ in the way they separate the electrons. Static correlation leads to a large separation in space of the two electrons in a pair, for example on two different atoms in a dissociation process. Dynamic correlation on the other hand deals... 

However, so far all the applications of the JT effect theory were realized only for chemically bonded systems in their high-symmetry configuration and transition states of chemical reactions. We show here that this is an unnecessary restriction the JT type instability is inherent to all the cases of degeneracy or pseudodegeneracy in molecular systems and condensed matter including nonbonded states in molecule formation from atoms, intermolecular interaction, and chemical reactions. 

Originally, the Jahn-Teller (JT) theorem was formulated as a prediction of a symmetry break for any non-linear polyatomic molecule with an orbital degeneracy in its ground state. However, as we understand it now, the real meaning of the JT theorem is a non-zero slope of the adiabatic potential energy surface (APES) at the point of electron degeneracy. This does not necessarily result in a lowering of... 

If the system has an internal angular momentum (associated with rotational states of molecules) there will, in the absence of an external field, be degeneracies in the system that will be practical to display explicitly in the expression for microscopic reversibility in Eq. (B.22). For systems with angular momenta, time reversal of the quantum equations of motion reverses the signs of both the momenta and their projections on a given direction, just like in a classical system. To express this explicitly, Eq. (B.13) is written as... 

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