Adiabatic terms

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Fig. 23. Diabatic terms of initial and final states (solid lines) and adiabatic terms (dashed lines). Adiabatic splitting is 2V. ...

At > 1, Bo 1 aiid the transition occurs along the lower adiabatic term F. At 1, Bo =... 

In the nonadiabatic regime A is proportional to the adiabatic splitting 2 Fd. The instanton trajectory crosses the barrier twice, each time bringing the factor A/A a associated with the probability to cross the nonadiabaticity region remaining on the same adiabatic term (and thus... 

In this section, the adiabatic picture will be extended to include the non-adiabatic terms that couple the states. After this has been done, a diabatic picture will be developed that enables the basic topology of the coupled surfaces to be investigated. Of particular interest are the intersection regions, which may form what are called conical intersections. These are a multimode phenomena, that is, they do not occur in ID systems, and the name comes from their shape— in a special 2D space it has the form of a double cone. Finally, a model Hamiltonian will be introduced that can describe the coupled surfaces. This enables a global description of the surfaces, and gives both insight and predictive power to the formation of conical intersections. More detailed review on conical intersections and their properties can be found in [1,14,65,176-178]. 

Two - particle energy correction correction to electron - electron correlation energy due to the phonon field. This non-adiabatic term represents full attractive contribution, and can be compared to the reduced form of Frohlich effective Hamiltonian which maximizes attractive contribution of electron - electron interaction and that can be either attractive or repulsive (interaction term of the BCS theory). For superconducting state transition at the non-adiabatic conditions, the two-particle correction is unimportant - see [2],... 

Fig. 2.3. The adiabatic terms of a system as functions of the generalized coordinate R. The dotted lines denote the crossing diabatic terms in the region of strong influence of the inter-term interaction. The indexes i and f denote the regions of the initial and final states.

Adiabatic Term describing an operation that occurs without the addition or removal of heat. 

The adiabatic term HAdiab is the same as (48) in the situation without Fermi... 

The right-hand side of (11.109) contains terms off-diagonal in the electronic state (i.e. the non-adiabatic terms). If the right-hand side is set equal to zero, we obtain the adiabatic eigenvalue problem,... 

Figure 2.1. Variation in the energy of the system along the reaction coordinate a diabatic terms of the reactant (1) and products (2) b adiabatic terms of the ground state (I) and excited state (II) V is the resonance integral.

As discussed earlier, matrix elements Ckm(t) of are supposed to be essentially different from zero near t = tp and to vanish as f — tp - oo. Let rkm be the characteristic time of changing Ckm near and htokm be the difference in adiabatic terms at tp. Then the N matrix will be quite different for the two limiting cases rkmiokm 1 and rkmwkm 1. If either of these conditions is fulfilled, then general methods can be suggested for the calculation of the matrix of nonadiabatic transitions. 

Here s + is" is the complex phase difference for two adiabatic terms U R) and U2(R) calculated from the complex branch point Rc to corresponding turning points, and dj and <52 are adiabatic phases for potentials Ux and t/2. Equation (13) is valid when two coupling regions do not overlap, that is, when 1. If, however, s 1, two (or more) branching points... 

Adiabatic terms in equation (38) are degenerate in quantum numbers S and w that yield the same value of the factor (— l)sw. After diagonalization of the matrix M., with and fP taken into account,... 

Thus, in the region where Ae or Kdlp Ae the adiabatic terms are... 

In the approximation of the dipole-dipole interaction, the adiabatic terms of a system A + B are... 

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