Diabatic energy

It also furnishes the following relation between the diagonal adiabatic energy matrix and the nondiagonal diabatic energy one... 

The —(/i /2p)W (Rx) matrix does not have poles at conical intersection geometries [as opposed to W (R )] and furthermore it only appears as an additive term to the diabatic energy matrix (q ) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

Elements of the matrix —are usually small in the vicinity of a conical intersection and can be added to to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) contains both the singular... 

Since the second-derivative coupling matrix is only an additive teiin in Eq. (87), we can merge it with the diabatic energy matrix and define a 2 x 2 diabatic matrix... 

The two diabatic energy profiles are expressed in terms of harmonic forms having a common force constant as ... 

Figure 16.3. The two lowest states during the dissociation of cyclopropane along the C2H4 relaxed path. The dashed lines, indicating the diabatic energies, were not computed but have been added merely to guide the eye.

Figure 13. Excitation energies (with respect to the vibrational ground state) of the second, third, and fourth excited vibrational states of XCO as functions of the mass m. The dashed lines schematically indicate diabatic energy curves. The inset shows the expectation values of the kinetic energies (measured in terms of the corresponding values in the ground vibrational state) of the fourth excited state [(1, 0, 0) for HCO], (Reprinted, with permission of the American Institute of Physics, from Ref. 17).

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). 

It is assumed that target states p are indexed for each value of q such that a smooth diabatic energy function Ep(q) is defined. This requires careful analysis of avoided crossings. The functions should be a complete set of vibrational functions for the target potential Vp = Ep, including functions that represent the vibrational continuum. All vibrational basis functions are truncated at q = qd, without restricting their boundary values. The radial functions fra should be complete for r < a. 

In Figure 3 the plot of the diabatic energies referring to the two main VB structures is shown against the LiF interatomic distance. It is quite evident looking at this diagram that at the equilibrium geometry (duF = 1.7A), both in vacuo and in solution, that the molecule is characterized by essentially an ionic description dominates in the expres-... 

Figure 3. Diabatic energy curves (hartree) for the dissociation of LiF in vacuo (solid line) and in aqueous solution (dashed line). The LiF distance is measured in A. Curves A refer to the covalent structure and curves B to the ionic structure corresponding to Li+F. ...

Note above that the GMH and adibatic formulations are equivalent in terms of building the CT free energy surfaces. The distinctions seen in Figure 15 may seem to contradict to this statement. The problem is resolved by noting that the requirement 0 imposed by the GMH formulation makes the diabatic energy gap nonzero for self-exchange transitions ... 

Since the TS is given in terms of the diabatic energy surfaces for the reactant and product, it is also clear that activation energies will be too high. For evaluating relative"... 

Some important free-energy relationships are presented in terms of the diabatic energy profiles G, and Gf in Figure 3. The vertical and horizontal shifts of the G/ profile relative to that for C, correspond, respectively, to the driving force of the ET process (—AG,y ) and the reorganization energy (z) of nuclear modes (shifts of equilibrium coordinate values). 

Evidently the average adiabatic energy (G2 + Gi)/2, like the average diabatic energy, is described by a parabola with force constant 22 centered at X = 1/2 with its minimum vertically displaced by 2/4 + AG°/2 relative to the diabatic minimum. 

As shown in Figure 3, the splitting at the intersection of the diabatic energy curves lowers the barrier by Ffab- Further, as //ab increases, the reactant and product minima of the adiabatic curves move closer together. The positions of the minima (reactant s and product s equilibrium configurations) are given by... 

From Eq. 3b the vertical difference between the diabatic energies at the equilibrium configuration (adiabatic minimum) of the reactants for AG° = 0 is given by... 

The diagonal elements of the EVB hamiltonian correspond to the diabatic energies of the valence bond states and are given by a regular force field expression... 

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