Point degenerate

III) the point x = 0 is a structurally unstable critical point (degenerate critical point) ... 

In the analysis of autonomous non-gradient systems the methods of gradient system theory have proved useful. The notions such as a stationary (critical) point, degenerate stationary point, structural stability (instability), morsification, phase portrait can be directly transferred to autonomous systems. A qualitative description of dynamical autonomous systems is constructed analogously with the description of gradient systems. 

For a given pair of valence and conduction bands, there must be at least one and one critical points and at least tluee and tluee critical points. However, it is possible for the saddle critical points to be degenerate. In the simplest possible configuration of critical points, the joint density of states appears as m figure Al.3.19. 

The constant of integration is zero at zero temperature all the modes go to the unique non-degenerate ground state corresponding to the zero point energy. For this state S log(g) = log(l) = 0, a confmnation of the Third Law of Thennodynamics for the photon gas. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

Equations (16)-(20) show that the real adiabatic eigenstates are everywhere smooth and continuously differentiable functions of Q, except at degenerate points, such that E (Q) — E, [Q) = 0, where, of com se, the x ) are undefined. There is, however, no requirement that the x ) should be teal, even for a real Hamiltonian, because the solutions of Eq. fl4) contain an arbitrary Q dependent phase term, gay. Second, as we shall now see, the choice that x ) is real raises a different type of problem. Consider, for example, the model Hamiltonian in Eq. (8), with / = 0 ... 

The sum excludes m = n, because the derivation involves the vector product of (n Vq H n) with itself, which vanishes. The advantage of Eq. (43) over Eq. (31) is that the numerator is independent of arbiriary phase factors in n) or m) neither need be single valued. On the other hand, Eq. (43) is inapplicable, for the reasons given above if the degenerate point lies on the surface 5. 

The first of these questions is deferred to Section VI. The second is addressed by considering tJie degeneracy condition VV r,o) = W (/..O). One solution lies at r = 0, and there ai e three other s at r = k/l and o = 7t, —rt/3 [30.3 IJ.. A circuit of with r < k/i therefore encloses a single degenerate point, wliich accounts for the Tiormar sign change. 1. whereas as circuit with... 

When two electronie states are degenerate at a particular point in configuration space, the elements of the diabatie potential energy matiix can be modeled as a linear function of the coordinates in the following fonn ... 

To obtain potential surfaces for two electronic states that will be degenerate at these points, we write a Hamiltonian as a 2 x 2 matrix in a diabatic representation in the following 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. 

Setting the diabatic basis equal to the adiabatic basis at the degenerate point, Ro, the expansions can be written in vector notation as... 

The major features of the PES aiound the degenerate point can now be easily analysed if we write the vector Q in the basis of where the... 

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]... 

Figure 5. A cut across the ground state (GS) and the excited state (ES) potential surfaces of the H4 system. The parameter Qp is the phase preserving nuclear coordinate connecting the H(lll) with the transition state between H(I) and H(1I) (Fig, 4). Keeping the phase of the electronic wave function constant, this coordinate leads from the ground to the excited state. At a certain point, the two surfaces must touch. At the crossing point, the wave function is degenerate.

Figure 20, The potential surface near the degeneracy point of a degenerate E state that distorts along two coordinates and Q. The parameter is the stabilization energy of the ground state (the depth of the moat ), [Adapted from [70]].

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