The VB Model

This model is very familiar to chemists in its semi-intuitive qualitative extension known as resonance theory, which was first proposed by Linus Pauling in his work 

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We often refer to Heitler and London s method as the valence bond (VB) model. A comparison between the experimental and the valence bond potential energy curves shows excellent agreement at large 7 ab but poor quantitative agreement in the valence region (Table 4.3). The cause of this lies in the method itself the VB model starts from atomic wavefunctions and adds as a perturbation the fact that the electron clouds of the atoms are polarized when the molecule is formed. 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

We can show the relationships among the state components pictorially (see Figure 16.17). Because the VBS is a spec of its implementation, we should be able to replicate the VBS model in the implementation and show how its links are all redundant. The invariants that relate the implementation and specification models are called retrievals. 

The frequency exaltation of the Kekule mode is mirrored by the structural manifestations in the twin states, discussed with reference to Figures 16 and 17. Thus, the repulsive jr-curve in the ground state softens the potential and thereby enables the ground-state molecule to distort along the Kekule mode when angular strain is exerted. In contrast, the attractive jr-curve in the twin excited state stiffens the potential and restores the local Deh symmetry of the benzene nucleus. The two physical effects are in perfect harmony and find a natural reflection in the VB model. 

Alkynyl substituents stabilize a radical center by the same 12 kcal/mol that on average is achieved by alkenyl and aryl substituents. From the point of view of the VB model this is due to the fact that propargyl radicals exhibit the same type of resonance stabilization as formulated for allyl and benzyl radicals in the right column of Table 1.1. In the MO model, the stability of propargyl radicals rests on the overlap between the one correctly oriented n system of the C=C triple bond and the 2 pz AO of the radical center, just as outlined for allyl and benzyl radicals in Figure 1.5 (the other 7t system of the C=C triple bond is orthogonal to the 2pz AO of the radical center, thus excluding an overlap that is associated with stabilization). 

The origin of the VB model the Heitler-London treatment of the hydrogen molecule... 

The VB model (17) is the basis of nearly all semi-empirical applications of VB theory to polyatomic molecules. 

It is well known that the properties of conjugated molecules are principally determined by their rc-electrons. Furthermore, planar conjugated molecules are prototypical in that their n-electrons could be separately treated from the remaining o -electrons. Hence, semiempirical theoretical models developed mainly for conjugated molecules treated only the jr-electrons explicitly but incorporated the effects of the o -electrons and the nuclei into some adjustable parameters featuring these models. The VB model (17) could be such a model, which may be further simplified as... 

Now we are concerned with how the VB model (18) can be solved for a given conjugated system. In fact, the model Hamiltonian (18) actually acts on the space of pure spin functions, either of the Weyl-Rumer (WR) form [34] or the simple product of one-electron spin functions. The matrix element between any two WR functions can be obtained by using Pauling s graphical rules [4], while the matrix element between two simple spin products is easily available using the following expression... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

S subspace is known to be fsN = (2S + l)Nl/(N/2 + S + l) (N/2-S)l [34], while the number of neutral Slater determinants with quantum number M, n(M), equals the binomial coefficient n(M)=C%l2 M. It is clear that in either case the dimension of the Hamiltonian matrix is exponentially proportional to the size of the system. In this chapter, the Slater determinants are chosen to be the N-electron basis functions, in which the VB model (20) is solved for various Sz spaces respectively. 

To illustrate the VB method, we consider several small conjugated molecules in the following. For larger conjugated molecules, the solution of the VB model (20) needs efficient computational techniques, which will be described in the next subsection. 

As done in MO theory, REPE values can be computed within the VB model if an appropriate reference structure and its energy are determined [22], Then... 

As stated above, a new scale is desirable to describe quantitatively the global aromaticity in the VB model. Since the global aromaticity measures the average benzene character of a conjugated system, resonance energy per hexagon (REPH), as defined by Eq. (48), may be a natural choice for BHs [22],... 

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