The nonadiabatic coupling

If the only change in the molecular charge distribution between the states a and b is the position of the transferred electron (i.e. if we assume that the other electrons are not affected) then rab is the transfer distance, that is, the separation between the donor and acceptor centers. Eq. (16.97) is known as the Mulliken-Hush formula. 

Equation (16.97) is useful for estimating the nonadiabatic coupling associated with electron transfer reactions mirrored by an equivalent optical transition. An 

The derivation of this formula, outlined in Appendix 16A, is limited by the assumption that the electronic coupling can be described in a two-state framework. For a discussion of this point and extension to more general situations see the paper by Bixon and Jortner cited at the end of this chapter. 

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By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

The extended Bom-Oppenheimer approximation based on the nonadiabatic coupling terms was discussed on several occasions [23,25,26,55,56,133,134] and is also presented here by Adhikari and Billing (see Chapter 3). 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

Figure 6. Initial rovibrational state specified reaction probabilities. Solid line exact quantum mechanical numerical solution. Solid line with solid square generalized TSH with use of the nonadiabatic coupling vector. Solid line with open circle generalized TSH with use of Hessian. Sur= 1(2) means the ground (excited) potential energy surface. Taken from Ref. [51].

The C matrix, the columns ofwhich, Cj(, are the eigenvectors of H, is normally not too different from the matrix defined above. However, the QDPT treatment, applied either to an adiabatic or to a diabatic zeroth-order basis, is necessary in order to prevent serious artefacts, especially in the case of avoided crossings [27]. The preliminary diabatisation makes it easier to interpolate the matrix elements of the hamiltonian and of other operators as functions of the nuclear coordinates and to calculate the nonadiabatic coupling matrix elements ... 

Fig. 8 shows the nonadiabatic coupling functions between states, which are completely dominated by the double crossing feature. The coefficient mixing term is a very good approximation of the total matrix elements, thus confirming the considerations already put forward with regard to the singlets. 

In this section we give the relations between the nonadiabatic coupling matrix elements in the quasi-diabatic and adiabatic representations. We do not obtain simple... 

In the previous section, we discussed the calculation of the PESs needed in Eq. (2.16a) as well as the nonadiabatic coupling terms of Eqs. (2.16b) and (2.16c). We have noted that in the diabatic representation the off-diagonal elements of Eq. (2.16a) are responsible for the coupling between electronic states while Dp and Gp vanish. In the adiabatic representation the opposite is true The off-diagonal elements of Eq. (2.16a) vanish while Du and Gp do not. In this representation, our calculation of the nonadiabatic coupling is approximate because we assume that Gp is negligible and we make an approximation in the calculation of Dp. (See end of Section n.A for more details.)... 

CLASSICAL DESCRIPTION OF NON ADIABATIC QUANTUM DYNAMICS 253 is given in terms of the nonadiabatic coupling matrices of first and second order ... 

Let us first consider the population probability of the initially excited adiabatic state of Model 1 depicted in Fig. 17. Within the first 20 fs, the quantum-mechanical result is seen to decay almost completely to zero. The result of the QCL calculation matches the quantum data only for about 10 fs and is then found to oscillate around the quantum result. A closer analysis of the calculation shows that this flaw of the QCL method is mainly caused by large momentum shifts associated with the divergence of the nonadiabatic couplings F = We therefore chose to resort to a simpler approximation... 

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