Doubly degenerate

Mulliken symbols The designators, arising from group theory, of the electronic states of an ion in a crystal field. A and B are singly degenerate, E doubly degenerate, T triply degenerate states. Thus a D state of a free ion shows E and Tj states in an octahedral field. 

The electrochemical features of the next higher fullerene, namely, [70]fullerene, resemble the prediction of a doubly degenerate LUMO and a LUMO + 1 which are separated by a small energy gap. Specifically, six reversible one-electron reduction steps are noticed with, however, a larger splitting between the fourth and fifth reduction waves. It is important to note that the first reduction potential is less negative than that of [60]fullerene [31]. 

The mode coordinates, tiansforming, respectively, as the — 1 and 1 components of the E (doubly degenerate) modes, have the form ... 

The Couplitig-Coefficierits lJ ABC abc) for the Complex Form of a Doubly Degenerate Representation in the Octahedral Group, Following G. F. Koster et al.. Properties of tke Thirt i-Two Point Groups, MIT Press, MA, 1963, pp, 8, 52. 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

An example that is closely related to organic photochemishy is the x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic tenn in Eq, (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat, The polar coordinates p and <(>, shown in Figure 20, can be used to write an expression for the energy ... 

PERMUTATIONAL SYMMETRY AND THE ROLE OF NUCLEAR SPIN IN THE VIBRATIONAL SPECTRA OF MOLECULES IN DOUBLY DEGENERATE ELECTRONIC STATES ... 

Notiee that the eigenvalue )ik = 3 appears twiee we say that the eigenvalue )ik = 3 is doubly degenerate. )ik = 2 is a non-degenerate eigenvalue. 

The second difference is the appearance of a doubly degenerate E symmetry species whose characters are not always either the - - 1 or — 1 that we have encountered in nondegenerate point groups. 

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