Adiabatic representation generalization

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

As a result, the generalized crude adiabatic representation of the clamped-nucleus Hamiltonian satisfies ... 

The consideration that the velocity of electrons is much higher than that of the nuclei (a consequence of their much smaller masses) leads to the Born-Oppenheimer approximation, perhaps the better known example of the near separability of variables. We reconsider it in view of subsequent generalizations. Using the language of classical mechanics, we will speak of adiabatic separability, which can be shown to be related to a semiclassical expansion, i.e. to an asymptotic expansion in h. See references [11-14] where we also discuss a post-adiabatic representation. 

We refer to Chapter 4 for a detailed discussion on the definition and explicit construction of diabatic states. The diabatic representation is generally advantageous for the computational treatment of the nuclear dynamics if the adiabatic potential-energy surfaces exhibit degeneracies such as conical intersections. Moreover, the diabatic representation often reflects more clearly than the Born ppenheimer adiabatic representation the essential physics of curve crossing problems and is thus very useful for the construction of appropriate model Hamiltonians for polyatomic systems. 

It should be stressed that for multidimensional curve crossing problems the low-order Taylor expansions (8), (9) and (19) are justified only in the diabatic electronic representation. In the adiabatic representation, curve crossings generally lead to rapid variations of potential-energy functions and transition dipole moments, rendering a low-order Taylor expansion of these functions in terms of nuclear coordinates meaningless. 

The second transformation with generator S2 is equivalent to the nonadiabatic transformation from the adiabatic representation into the final one, which we shall call general , i.e. the representation that involves the adiabatic case as well as the nonadiabatic one. This representation is defined through new quasiparticles denoted simply without bar... 

If we proceed from the general to the adiabatic representation with zero c coefficients we obtain exactly the Born-Handy ansatz (28.14) from the first principle derivation ... 

Interaction potentials are introduced for general electronic representations and are discussed in the adiabatic representation to emphasize the different nature of long-range and short-range interactions, particularly for polyatomic systems. The size of a target, i.e., the number of atoms in it, is shown to influence how one approaches the collision dynamics. We also describe areas of recent progress in quantal, semiclassical, and classical dynamics. 

Delos, J.B. and Thorson, W.R. (1972) Studies of the potential-curve-crossing problem. II. General theory and a model for dose crossings. General theory and a model for dose crossings. Phys. Rev., A., 6, 728 Delos, J.B. and Thorson, W.R. (1975) Diabatic and adiabatic representations for atomic collision processes. J. Chem. Phys.,... 

H3 (and its isotopomers) and the alkali metal triiners (denoted generally for the homonuclears by X3, where X is an atom) are typical Jahn-Teller systems where the two lowest adiabatic potential energy surfaces conically intersect. Since such manifolds of electronic states have recently been discussed [60] in some detail, we review in this section only the diabatic representation of such surfaces and their major topographical details. The relevant 2x2 diabatic potential matrix W assumes the fomi... 

The general potential U (8) has not been used before 1999 [52] because its numerical matrix representation requires huge basis sets, incompatible with the common computers. In order to avoid this situation, an approximation has been undertaken in previous studies the adiabatic approximation [54,55], Following an idea of Stepanov [56], Marechal and Witkowski assumed that the fast mode follows adiabatically the slow intermonomer motions, just as the electrons are assumed to follow adiabatically the motions of the nuclei in a molecule. It has been shown [57] that the adiabatic approximation is only suitable for very weak hydrogen bonds, as discussed in the next section. 

A more general description of the effects of vibronic coupling can be made using the model Hamiltonian developed by Koppel, Domcke and Cederbaum [65], The basic idea is the same as that used in Section III.C, that is to assume a quasidiabatic representation, and to develop a Hamiltonian in this picture. It is a useful model, providing a simple yet accurate analytical expression for the coupled PES manifold, and identifying the modes essential for the non-adiabatic effects. As a result it can be used for comparing how well different dynamics methods perform for non-adiabatic systems. It has, for example, been used to perform benchmark full-dimensional (24-mode) quantum dynamics calculations... 

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