Isoenergetic point

Various theories have been proposed for horizontal transfer at the isoenergetic point. Gouterman considered a condensed system and tried to explain it in the same way as the radiative mechanism. In the radiative transfer, the two energy states are coupled by the photon or the radiation field. In the nonradiative transfer, the coupling is brought about by the phonon field of the crystalline matrix. But this theory is inconsistent with the observation that internal conversion occurs also in individual polyatomic molecules such as benzene. In such cases the medium does not actively participate except as a heat sink. This was taken into consideration in theories proposed by Robinson and Frosch, and Siebrand and has been further improved by Bixon and Jortner for isolated molecules, but the subject is still imperfectly understood. 

The theory of radiationless transition considers the transition to occur in two steps (i) horizontal transition from one energy state to the other at the isoenergetic point, for the two combining states and (ii) vibrational relaxation of the lower energy state. The step (i) is the rate determining step. The rate constant NR is given by the theory of Robinson and Frosch as,... 

As already described in part 2.3.2 for a set of pairs (R, a) (Fig. 3 a) the remaining geometric variables were optimized in keeping with a symmetry restriction (Fig. 3 b). The energy function E = f(R, a) obtained here in the gas phase is presented in Fig. 4 as a graphic representation of isoenergetic lines above the R-a-plane. Tire points marked by Roman numerals are minimas (educts I products III, IV, V) and the saddle... 

An interesting point can arise with Re(I) complexes (e.g., fac-Re+L(CO)3X where X is a halide, pyridine, nitrile, or isonitrile) where the n-n and MLCT states can be very nearly isoenergetic. This near-degeneracy allows the lowest state to be rather easily switched back and forth between n-n and MLCT by suitable choices of ligands and in some cases merely by altering the temperature 22 23 ... 

The free energy cost of introducing a kink is and is the difference between the free energy of a one kink path and that of a path of equivalent P that is confined to one well, E (since the wells are isoenergetic at the crossing point it does not matter which one [81]). The path sum can then be written as an expansion in kinks. The contribution of the set of paths with N kinks to the overall partition function consists of a factor from each kink of Qt = exp(- PFJ, and is... 

As has been pointed out in the past (e.g. concerning the linear-cyclic equilibrium in Ceand Cio carbon clusters (40)), Hartree-Fock underestimates the resonance stabilization of aromatic relative to non-aromatic systems (in the case at hand, between the N- and / -protonated isomers) and MP2 overcorrects. The structures are found to be nearly isoenergetic at the CCSD level inclusion of connected triple excitations favors the N-protonated ion. The direction of the effect of connected quadruples is somewhat unclear, and a CCSD(TQ) or CCSDT(Q) calculation impossible on systems this size, but the contribution will anyhow be much smaller in absolute magnitude than that of connected triple excitations, particularly for systems like these which are dominated by a single reference determinant. We may therefore infer that at the full Cl limit, the N-protonated species will be slightly more stable than its / -protonated counterpart. 

Recently, ab initio calculations of the relative energies of 12 and 13 have been reported.9 The results indicate that 12 and 13 are nearly isoenergetic, and an energy of only 4.4 kcal mol 1 is required to break the highly twisted 7t-bond in 13. The results also point to a possible singlet ground state for 12.9... 

Figure 2.18. Profile of isoenergetic surfaces of the excitonic dispersion in the vicinity of the bottom of the band in the model (2.139) of an orthorhombic crystal. We note the lengthening of the surfaces along c . Also, the wave vectors tend to orient perpendicular to d (or the b axis) in the vicinity of the point K = 0.

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