Two-state degeneracy

The Two-State Model Time-Independent Pertirrbation Two States- Degeneracy... 

We shall refer to three special cases, namely, the two-state degeneracy, the tri-state degeneracy and the tetra-state degeneracy. The first two cases were already discussed before and will be mentioned here just for the sake of completeness, but we shall elaborate on the tetra-surface case (N=4) which is presented here for the first time. 

E10.ll The electronic energies and degeneracies for the first two states of atomic fluorine are given below. All other states are at such high energies that their contribution is negligible. 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

Luttinger-Tisza method is burdened by independent minimization variables, while analysis of the values of the Fourier components F k) makes it possible to immediately exclude no less than half of the variable set and to obtain a result much more quickly. Degeneracy of the ground state occurs either due to coincidence of minimal values of Vt (k) at two boundary points of the first Brillouin zone k = b]/2 and k = b2/2, or as a result of the equality Fj (k) = F2 (k) at the same point k = h/2. The natural consequence of the ground state degeneracy is the presence of a Goldstone mode in the spectrum of orientational vibrations.53... 

The NRT description of TS complexes is closely related to the general two-state valence bond model of Shaik and Pross.93 This model emphasizes the coupled changes in two adiabatic states that evolve from distinct diabatic valence-bond (VB) configurations r and states cross in energy at s = s, ... 

If there are n uncoupled oscillators (all equivalent), states have degeneracies that can be simply seen from Eq. (6.11). They are depicted in Figure 6.2. It is convenient to represent the wave functions using the notation of Halonen and Child (1983). 12,0,0,0,0,0 > denotes a state with two quanta in the same oscillator, while II, 1,0,0,0,0 > denotes two states with one quantum in one oscillator and another in another oscillator. 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

To determine the various state degeneracies, the gkim, we, of course, have to know something about the states themselves. Sometimes symmetry considerations are sufficient to determine the gklm, even when the various state energies are totally unknown. We will consider only two simple examples here. 

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