Definition of Diabatic States

The concept of a diabatic state has different definitions. Strictly speaking, a basis of diabatic states (ft, If...) should be such that Equation 10.9 is satisfied for any variation 3Q of the geometrical coordinates (Q). 

Having defined a diabatic state as a unique VB structure, or more generally as a linear combination of a subset of VB structures leading to a specific bonding scheme, the question is now How do we calculate such a state in a meaningful way  

The Nonvariational Method (Method I) An initial possibility is to keep the same orbitals that optimize the adiabatic state for the diabatic state something that seems simple and appealing. In practice, this would be done as follows  

The main problem with this diabatization procedure is that this does not guarantee the best possible orbitals for the diabatic states, except for cases where one uses a minimal basis set. Indeed, the BOVB orbitals are optimized so as to minimize the energy of the multistructure ground state, and are therefore the best compromise between the need to lower the energies of the individual VB structures and to maximize the resonance energy between all the VB structures. This latter requirement implies that the final orbitals are not the best possible orbitals to minimize each of the individual VB structures taken separately. It follows that the diabatic states calculated in this way will very often possess very high energy and are not recommended. 

The Quasi-variational method (Method II) An alternative approach, which we recommend, consists of optimizing each diabatic state separately, in an independent calculation. Consequently, the resulting orbitals of the diabatic states are different from those of the adiabatic states, and each diabatic state possesses its best possible set of orbitals. The diabatic energies are obviously lower compared with those obtained by the previous method, and are therefore quasi-variational. The diabatic energies of the covalent and ionic structures of F2, calculated with Methods I and II in the L-BOVB framework, are shown in Table 10.4. It is seen that the ionic structures have much lower energies in the quasi-variational procedure, and as such, the procedure can serve a basis for deriving quantities such as resonance energies (see below). 

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Closely related to the above merit of VB methods, the unique definition of diabatic states also allows us to derive the energy profiles for diabatic states. Since for many reactions the whole process can be described with very few resonance structures, the comparison between the diabatic and adiabatic state energy profiles can yield insight into the nature governing the reactions [22-24]. In fact, even for complicated enzymatic reactions, simple VB ideas have shown unparalleled value [25, 26]. However, the further utilization of the VB ideas at the empirical and semi-empirical levels should be carefully verified by benchmark ab initio VB... 

Equations (23) and (24) presume a knowledge of the relevant electronic states, either the charge-locahzed diabatic states A and (ps or the adiabatic states hi and F2 (eqnation 8). Typically, diabatic states are natural for studying electron transfer, whereas the adiabatic states are nsed for optical transitions. The generalized Mulhken-Hnsh approximation adopts the definition of diabatic states that are diagonal with respect to the component of the dipole moment operator along... 

In all calculations, standard Gaussian basis functions are used to construct the wave function for each specific diabatic state. For comparisons purposes, basis sets ranging from 3 -21G to aug-cc-p VTZ have been used. Specific details on the choice and definition of diabatic states are given below for each individual case. 

Hendekovic, J. (1982) Novel variational definition of diabatic states. Chem. Phys. Lett., 90, 193. 

However, this strict (and useless) definition can be applied only in the case of a complete set of diabatic states. When the study is restricted to a finite number of states of interest, this rule does not apply and the diabatic states must depend on the nuclear coordinates. They can be obtained from the corresponding adiabatic states through a imitary transformation ... 

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. 

For reproducing as closely as possible diabatic conditions, we have fixed the Cl—Cl bondlength at its neutral equilibrium value. This way, the system depends on two parameters as shown in Figure 1. Previous experimental and theoretical studies on similar systems, [1,18] have shown that electron jump from Li to the acceptor molecule CI2, which has, once relaxed, a positive vertical electron affinity (see Table 1), is likely to take place at a distance d, (see the definition of this parameter in Figure 1) which is superior to the LiCl equilibrium distance (MP2 value 2.0425 A). The description of this phenomenon in terms of MO and states will be briefly recalled in the next section. 

The use of the energy-gap reaction coordinate allows us to calculate solvent reorganization energies in a way analogous to that in the Marcus theory for electron transfer reactions.19 The major difference here is that the diabatic states for electron transfer reactions are well-defined, whereas for chemical reactions, the definition of the effective diabatic states is not straightforward. The Marcus theory predicts that... 

There are several fundamental reasons why the GMH and adiabatic formulations are to be preferred over the traditionally employed diabatic formulation. The definition of the diabatic basis set is straightforward for intermolecular ET reactions when the donor and acceptor units are separated before the reaction and form a donor-acceptor complex in the course of diffusion in a liquid solvent. The diabatic states are then defined as those of separate donor and acceptor units. The current trend in experimental design of donor-acceptor systems, however, has focused more attention on intramolecular reactions where the donor and acceptor units are coupled in one molecule by a bridge.The direct donor-acceptor overlap and the mixing to bridge states both lead to electronic delocalization, with the result that the centers of electronic localization and localized diabatic states are ill-defined. It is then more appropriate to use either the GMH or adiabatic formulation. 

Marcus model and diabatic states make more immediate (even if not strictly necessary) the introduction of the dynamical solvent coordinate, of which in Section 5.2 we have given a definition based on parameters of the continuum model, but other definitions are possible. Actually, the Russian school used this concept without giving formal definitions, at the best of our knowledge (several papers have been published in relatively minor Russian journals, with limited circulation in western countries in those years), and basing it on a description of the solvent as a continuum or a set of os-... 

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