Stationary degenerate

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

The unperturbed Hamiltonian 3 is the same for all systems and is time-independent. The time-dependent perturbation G(t), different for each system, is considered as a stationary stochastic variable. We may, without loss of generality, suppose that the mean value of G(t) over the ensemble is equal to zero. We denote by a,p,y,. . . the eigenstates of supposed to be non-degenerate, and by fix, the corresponding energies. 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Figure 8.9 shows the various degenerate stationary-state boundaries from Fig. 6.18 as broken curves, and also (solid curve) a locus corresponding to Po k2 values for which the system can have a stationary-state solution with two eigenvalues simultaneously and exactly equal to zero. Mathematically the conditions for this are... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

The system of integral equations [Eq. (66)] is eventually discretized and solved with numerical linear algebra procedures. At each energy, the system (66) must be solved for each of the open channels. A complete set of linearly independent degenerate real (i.e., stationary) continuum solutions if"E is thus obtained. The stationary scattering states xjr E are not orthogonal it can be shown that their superposition is given by... 

Different combinations of degenerate orbitals simplify different problems. For example, the normal representations px, py, pz of the / = 1 orbitals are actually mixtures of different // / values. The stationary states with a single value of mi 0 have real and imaginary parts, and are difficult to visualize. 

In a hydrogen atom, the 2x and 2p orbitals are degenerate. Therefore, as discussed in the last section, any other combination of these orbitals would also be a stationary state with the same energy. The only reason for writing these orbitals as we did in Equa-... 

For degenerate states more than one state with eigenvalue E take the form (9). Both are stationary, but a complex superposition, although again a stationary function, does not have the form (9) and could describe particles in motion [48]. 

Figure 17. Example of almost degenerate quasi-species. For system of binary sequences with chain length v = 50 relative stationary concentrations of 51 mutant classes (, ) are shown in region 0.972 > q> 0.952. Notice extremely sharp transition at = 0.9638 when new master sequence /iso, and its neighbors become dominant. At = 0.9546 usual error threshold observed. Selective values used in this example are A,o, = 10, 4,50, = 9.9,4,4,1 = 2, and all other 4,j = 1 for 1 = 1,. 48 cf. figure 5).

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