VB theory

In summary, the major dilemma of VB theory in its classical form was that its beautifully simple concepts, which served many generations of chemists and were embodied in countless elementary textbooks on valency, could not be reconciled with the exigencies of actual calculation - at least with the means then available. Other methods, more easily adapted to ab initio development, began to overshadow, and eventually to totally eclipse, the area of quantum chemistry in which Pauling played such a major role. 

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Simple VB theory [75] uses for the basis set five low-lying structures that differ in their spin pairing characteristics, as shown in Figure 26. Similar to the case of the radical, the degenerate 2 f e lowest singlet state of... 

There is an equivalent way of generating solutions to the electronic Schrodinger equation which conceptually is much closer to the experimentalists language, known as Valence Bond (VB) theory. We will start by illustrating the concepts for the H2 molecule, and note how it differ from MO methods. 

The main features of the chemical bonding formed by electron pairs were captured in the early days of quantum mechanics by Heitler and London. Their model, which came to be known, as the valence bond (VB) model in its later versions, will serve as our basic tool for developing potential surfaces for molecules undergoing chemical reactions. Here we will review the basic concepts of VB theory and give examples of potential surfaces for bond-breaking processes. 

When we try to apply VB theory to methane we run into difficulties. A carbon atom has the configuration [HeJ2s22pvl2p l,1 with four valence electrons (34). However, two valence electrons are already paired and only the two half-filled 2/ -orbitals appear to be available for bonding. It looks as though a carbon atom should have a valence of 2 and form two perpendicular bonds, but in fact it almost always has a valence of 4 (it is commonly tetravalent ) and in CH4 has a tetrahedral arrangement of bonds. 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

The second approach is addressed to elaborate methods able to derive from accurate calculations the points of interest for the interpretation The strategy, in general, consists in the adoption of a simpler model (the mathematical aspects of the model are again concerned) and the task consists in reducing the information coming from the full in-depth calculation (not the the numerical values of observables and other statutory quantities alone) to the level of the simpler model. For example accurate calculations may be reduced at the level of a simple VB theory (Robb, Hiberty) or of a simple MO perturbation scheme (Bernardi) making more transparent the interpretation. 

Simple VB theory [75] uses for the basis set five low-lying structures that differ in their spin pairing characteristics, as shown in Figure 26. Similar to the case of the radical, the degenerate 1 li 2 state is the lowest singlet state of D h symmetry. It lies on the lowest singlet surface and is constructed by the combination of the five type-V structures (in fact, only four are linearly independent). These structures transform as Aj in C2v, and will be referred to as Ai(I) structures. 

Using Valence Bond (VB) theory, the central atoms of the molecules with formulas AB2U2 and AB3U should undergo sp3 hybridized with predicted bond angles of 109.5°. If no hybridization occurs, bonds would be formed by the use of p orbitals. Since the p orbitals are oriented at 90° from each other, the bond angles would be 90°. Note that hybridization is only invoked if the actual molecular geometry data indicate that it is necessary. 

The positive charge in 28 is stabilized by /j-cr-C-C-hyperconj ligation with the C-C-ring bonds of the two cyclopropyl moieties. In the parlance of VB theory this is described by resonance of 28 with non-bonding resonance limiting structures, the homoallenyl cation type structure 28a, the homopropargyl cation type structure 28b and the Dewar-type limiting resonance structure 28c. 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

VB and MO theories can be applied to simple molecular systems as follows. According to VB theory, if (p.A and

wave functions of independent systems a and b then the total wave function iff and total energy E are written as follows ... 

MO wave functions in the above form give equal importance to covalent and ionic structures, which is unrealistic in homonuclear diatomic molecules like H2. This should be contrasted with (/>Vb> which in its simple form neglects the ionic contributions. Both and i//MO are inadequate in their simplest forms while in the VB theory the electron correlation is overemphasized, simple MO theory totally neglects it giving equal importance to covalent and ionic structures. Therefore neither of them is able to predict binding energies closer to experiment. The MO theory could be... 

Modem valence bond (VB) theories such as Spin-Coupled theory, together with DFT and Molller-Plesset MO methods, and ab initio molecular dynamics, were employed to study structure/dynamics in representative carbonium ions. 

Quantum mechanics provide many approaches to the description of molecular structure, namely valence bond (VB) theory (8-10), molecular orbital (MO) theory (11,12), and density functional theory (DFT) (13). The former two theories were developed at about the same time, but diverged as competing methods for describing the electronic structure of chemical systems (14). The MO-based methods of calculation have enjoyed great popularity, mainly due to the availability of efficient computer codes. Together with geometry optimization routines for minima and transition states, the MO methods (DFT included) have become prevalent in applications to molecular structure and reactivity. 

We have performed Spin-Coupled calculations on a series of selected carbonium ions (55). The Spin-Coupled calculations allow the study of chemical structure of die molecule, since chemical structure and connectivity are central features of VB theory. Spin-Coupled calculations for CHS+ in CSI show the system as bonded by the intuitively proposed 3c2e bond, which connects the carbon atom to two hydrogens, and three ordinary 2c2e bonds between the carbon and the other hydrogens, commonly called the tripod (Figure 3). 

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