Diabatic coupling elements

Fig-1 Adiabatic potentials (solid lines) and diabatic potentials (dashed lines) corresponding to the electronic states e and e. Kee-(Q) schematically represents the coordinate dependence of the non adiabatic coupling elements (adiabatic representation) whereas Uee (Q) schematically represents the coordinate dependence of the non diabatic coupling elements (diabatic representation)... 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

To Specify the off-diagonal elements 14 / of the diabatic potential matrix, we consider two cases of the diabatic coupling Vkk/ which are of particular interest in molecular physics ... 

As is apparent from Eq. (7), the energy of the electronic transition in CT complexes is determined as the sum of the two terms, the difference in the energy of the diabatic states (Ada) and the electronic coupling element (HDa). Depending on their relative values, there can be two limiting cases represented by weak complexes with Hda Ada and strong complexes with Ada HDA. 

Fig. 8. Typical profiles of the potential-energy surfaces for electron-transfer in the isergonic region (for 2 = 1.8 eV). The dotted lines represent the diabatic (non-interacting) states (HDa = 0). Adiabatic states (solid lines) are presented with the values of the electronic coupling element Hda...

Figure 3. Plot of the diabatic (Ga, Gb) and adiabatic (G, G2) free-energies of the reactants and products against the reaction coordinate for an electron transfer reaction with AG° = 0. i/ab is the electronic coupling element between the diabatic states of the reactants and products and X is the reorganization energy for the reaction.

A rigorous dynamical investigation should involve coupled motion on both PES s. However, that would require two global PES s and the non-adiabatic coupling between them. This information is not available at the present time. Mahapatra et al. [130] recently determined two PES s and calculated the coupling element in a diabatic representation. However, they restricted their interest to the calculation of the absorption spectrum and the internal conversion and therefore considered only relatively small O-NO distances, too small for dissociation calculations. Global PES s for... 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

Everything considered, it is much more convenient, for the two-electronic-state expansion in systems displaying conical intersections, to use the diabatic equation (108) rather than the adiabatic one (74). When doing so, however, the errors associated with the neglect of the W(,1)ad(R) and W(2)d(R) terms appearing in (106) should be estimated, and if necessary the effect of these terms can be introduced a posteriori by perturbation methods. The same is true for the errors in the coupling elements introduced when using the DIM approach. 

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