Degeneracy lifting

In Eigure 9.9, we have presented another cartoon that attempts to show the potential energy surfaces in the space of the degeneracy lifting coordinates as one traverses a third coordinate X3. Of conrse, this is really a fonr-dimensional pictnre, which is not easy to assimilate. Nevertheless, it shonld be clear that the degeneracy... 

Figure 9.10. The conical intersection hyperline traced out by a coordinate X3 plotted in a space containing the coordinate X3 and one coordinate from the degeneracy-lifting space X1X2. See color insert.

Porphyrins, 21 14, 36, 135 -based manganese complexes, 46 400-402 as cobalt complex ligants, 44 284-290 compared to phthalocyanines, 7 75 complexes, 19 144, 145, 147 complex stability, 42 135-137 degeneracy lifting, 36 206 metalloporphyrins, DNA cleavage and, 45 271-283... 

We now introduce a pair of mass-weighted nuclear coordinates, gxi and 2 2 > (see Section 1) that describe mass-weighted rectilinear displacements along xi (Qo) and X2 (Qo), respectively. This basis set is the most convenient one for the present analysis because it simplifies the degeneracy-lifting terms of (17). The expansions for 8f and 8/i in this basis are ... 

Here we demonstrate the degeneracy lifting and appearance of the long-rage orbital order by the classical Monte-Carlo method in a flnite size cluster system [3,7]. We calculate the staggered orbital correlation function... 

The result is that all of the wrong-form case (b)-like terms in FRyT are cancelled by terms from —2f N 1 and the expected B[N+ (N+ +1) — A+2] form for FR T is obtained. However, the transformation from case (b) to (d) converts the diagonal and degeneracy-lifting case (b) Hel into non-diagonal form in case (d). This is illustrated by the unitary (orthogonal) transformation of a two-state Hel matrix... 

The optimization of a conical intersection is a constrained optimization in the space orthogonal to the two degeneracy-lifting coordinates, Xj and Xg. The optimization has an additional requirement, namely, the energy difference between the two states at the conical intersection is zero. Thus, the gradient used in our conical intersection optimization algorithm is the sum of two gradients ... 

Fig. 29. Term diagram of a Co(II) ion (i) in aTj environment with degeneracy lifting of the T, states by a dynamic Jahn-Teller effect indicated as a symmetric term splitting (left-hand side of the vertical dotted line) (ii) in an environment of lower symmetry (e.g., Q) with removal of the degeneracy (right-hand side of the vertical dotted Une). The best fit for Co(II)-ethylene is obtained for 64/6=6.6 (vertical dotted line). Reprinted from [32] with permission of Academic Press, Inc...

Fig. 21. Same plots as Fig. 20 with the accidential degeneracy lifted = 0.225 (left) and = 0.425 (right).

Fig.5.19a-c. Degenerate mode frequency spectrum of a confocal resonator (d = R) (a), degeneracy lifting in a near-confocal resonator (d = 1.1/ ) (b), and the spectrum of fundamental modes in a plane-mirror resonator (c)... 

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