Diabatic curve

The vibronic interaction between the level vi of the diabatic potential curve Vf(R) and the level v2 of another diabatic curve V2 R) is reduced to 

In the diabatic model, the electronic part of the matrix element HiiVli2lV2 is often assumed to be independent of R. Then the nuclear and electronic coordinates can be separated in the integration of Eq. (3.3.4). By integration over the electronic coordinates, one obtains 

Indeed, as for any electronic quantity, the value of He actually depends on R. However, this dependence is usually weak. Equation (3.3.5) holds even if the independence of He is a linear function of the fi-centroid (Halevi, 1965) defined by 

The significance of the. R-centroid is illustrated as follows. The electronic matrix element, He(R), may be expanded in a power series about an arbitrarily chosen internuclear distance, R (most usefully, R = Rc, the internuclear distance at which the two potential curves cross), 

Then the vibrational matrix elements of He(R) are expressed in terms of Rn centroids, 

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Figure B3.4.17. When a wavepacket comes to a crossing point, it will split into two parts (schematic Gaussians). One will remain on the same adiabat (difFerent diabat) and the other will hop to the other adiabat (same diabat). The adiabatic curves are shown by fidl lines and denoted by ground and excited die diabatic curves are shown by dashed lines and denoted 1, 2.

An ab-initio study of the lowest states of BH. Quasi-diabatic curves and vibronic couplings M. Persico, R. Cimiraglia andF. Spiegelmann... 

Fig. 1. Schematic potential energy curves for a neutral transition metal atom (M) inserting into the H-R bond of a hydrocarbon. Diabatic curves are shown as dashed lines, adiabatic curve shown as a solid line.

Figure 5. Energy diagram for charge separation resolved into reactant-like and product-like diabatic surfaces. The two diabatic curves do not intersect, but interact, to give an avoided crossing, whose energy gap is twice the electronic coupling, Vei, for the interaction. Also depicted is the Marcus-Hush classical rate expression for nonadiabatic ET.

Charge Localized vs. Delocalized Wavefunctions. In the spirit of the Condon approximation discussed above, we do not include the full dependence of the purely electronic matrix elements on q n, but rather evaluate them where the diabatic curves... 

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

Figure 1. Diabatic potential energy curves for Nal with an expanded view of the adiabatic potential curves, e, and Ej, near the diabatic curve crossing.

Figure 3. ELF isocontours (0.78) for the four states AA, BB, AB and Triplet at points in the quasi-diabatic curves a= 0, nil and ti. ELF basins involving hydrogen atoms are not displayed for clarity.

As an example we can take the excited states of NO. It has been shown that there are two excited states of the same symmetry ( 11) whose vibrational levels are best interpreted on the basis of diabatic curves which cross as in Fig. 1 (75-7 7). One of these states (B) arises from the electron excitation to an antibonding valence molecular orbital and the other (C) from excitation to a Rydberg orbital. The Born-Oppenheimer adiabatic curves cannot cross (by virtue of the non-crossing rule which is to be discussed in a later section) and must fullow the dashed curves shown in the figure. 

Historically the first application of symmetry to potential energy surfaces was to prove the so-called non-crossing rule. In its simplest form this may be stated as potential energy curves for states of diatomic molecules of the same symmetry do not cross . We have already seen in section 2 that this should be qualified to apply to adiabatic curves, as in some situations it may be convenient to define diabatic curves wdiich do cross. 

The process may then be described by classical trajectories and by transition probabilities for changing the potential surfaces when these trajectories come close to a crossing. The transition probability for changing the diabatic curves at a crossing line R, is given by a Landau-Zener... 

Figure 3.45 Schematic free energy curves in the solvent coordinate z for the discussion of the equilibrium solvation location of the Cl seam in Figure 3.42 Solid curves are the adiabatic curves for very small but finite electronic coupling, while the dashed curves are diabatic curves for zero coupling, (a) The symmetric case, where the filled circle represents the location of the minimum free energy in the upper adiabatic state in the presence of finite electronic coupling, while the open circle represents a free energy minimum when the electronic coupling vanishes exactly (6 = 90°). (b) An asymmetric case where the two surfaces intersect for z > 1 and the equilibrium location of the Cl seam fails.

One of key developments of modern VB methods is the ability to compute barriers for elementary reactions (6,7), sometimes with high accuracy (6). Equally important is the current capability to analyze these barriers using the VB diagrams, VBSCD or VBCMD, described in Chapter 6, and to compute reactivity quantities like the promotion gap, G, and the resonance energy of the TS, B. These are multi-layered calculations in which both the adiabatic and diabatic curves are calculated variationally. 

The second caveat concerns the limitation of the quasi-variational procedure to produce an entire diabatic curve when there are low lying excited states that cut through it. For example, in the case of H-abstraction by an electronegative atom, or in SN2 reactions (7), the ionic structure of the bond lies below the R and P image states in the VBSCD. Therefore, past the crossing point of the... 

Formulated in another way, the metal (M)-level is suddenly pulled down from well above to well below the ligand (L) valence levels a distance larger than the exchange splitting 2H]2 as a consequence, the system cannot follow adiabatically but undergoes a diabatic (curve crossing) transition to an excited state of the ionic system. 

Very often, as in the case of the Woodward-Hoffmann rules, the crossing between the diabatic curves can be predicted without resorting to complicated calculations. A simple and interesting review on VB correlation diagrams has been recently given by Shaik and Shurki [7] (see also other useful Chapters in this volume). 

The energy curves are plotted in Fig. 5. For this case, the diabatic curve corresponding to the ionic structure is the higher at all the distances considered. For this reason, the polarization of the solvent is less pronounced than for LiF because it depends on a wavefunction which is dominated by covalent character. 

The solvation contributions to the free energy functional of Eq. (4) cannot be distinguished on the scale of Fig. 5 for the lower two curves, which are respectively, the total free energy and the covalent structure diabatic curve. Instead in the region of the minimum of the upper curve, the solvent lowers the free energy by about 15 kcal/mol, but this effect is not sufficient to product a particularly significant variation in the total wavefunction. 

The systematics found in the alkali hydride potentials suggest that it might be possible to model a simple ionic potential to reflect the behavior of the whole series of alkali hydrides. Here we construct a "practical" diabatic curve which reflects the physical properties, e.g. the dipole moment function.. We expect our diabatic curve to follow the ionic part of the adiabatic potential and to begin to deviate in the avoided crossing region. 

Based on our definition for the "practical" diabatic curves, we require that the ionic curve agree with the inner wall of the adiabatic curve. The fitted parameters A and p are listed in Table V. 

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