The Salem-Klopman Equation

Using perturbation theory, Klopman and Salem14 derived an expression for the energy (AE) gained and lost when the orbitals of one reactant overlap with those of another. Their equation has the following form  

The derivation of this equation involves approximations and assumptions. It is valid only when S is small. The integral S for a C C bond being formed by p orbitals overlapping in a a sense reaches a maximum value of 0.27 at a distance of 1.74 A and then rapidly falls off. Thus, any reasonable estimate of the distance apart of the atoms in the transition structure cannot fail to make S small. The integral [3 is roughly proportional to S, so the third term of Equation 3.4 is a second-order term. With S always small, the higher-order terms are very small indeed, and we neglect them. 

As two molecules collide, three major forces operate. 

Having identified the causes for the ease of a reaction like this, we must next use the ideas behind Equation 3.4 to identify the sites of reactivity in each of the reacting species. To find the contribution of the Coulombic forces, we need the total electron population on each atom. For the allyl anion, the excess n charge (Fig. 1.29) is 

When we add the contribution from the frontier orbitals, the picture is even more striking. The HOMO of the anion has coefficients at C-l and C-3 of 0.707, and similarly the LUMO of the cation has coefficients at C-l and C-3 of 0.707. In both frontier orbitals the coefficient on C-2 is zero. Thus the frontier orbital term is overwhelmingly in favour of reaction of C-l (C-3) of the anion with C-l (C-3) of the cation. 

We have now seen how the attraction of charges and the interaction of frontier orbitals combine to make a reaction between two such species as the allyl anion and allyl cation both fast and highly regioselective. We should remind ourselves that this is not the whole story another reason for both observations is that the reaction is very exothermic when a bond is made between C-l and C-1 the energy of a full a bond is released with cancellation of charge, which we could not easily do if reaction were to take place at C-2 on either component. In other words, we find, as we often shall, that we are in the situation of Fig. 3.3a—the Coulombic forces and the frontier orbital interaction on one side, and the stability of the product on the other, combine to lower the energy of the transition structure. 

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We cannot, then, expect this approach to understanding chemical reactivity to explain everything. Most attempts to check the validity of frontier orbital theory computationally indicate that the sum of all the interactions of the filled with the unfilled orbitals swamp the contribution from the frontier orbitals alone. Even though the frontier orbitals make a weighted contribution to the third term of the Salem-Klopman equation, they do not account quantitatively for the many features of chemical reactions for which they seem to provide such an uncannily compelling explanation. Organic chemists, with a theory that they can handle easily, have fallen on frontier orbital theory with relief, and comfort themselves with the suspicion that something deep in the patterns of molecular orbitals must be reflected in the frontier orbitals in some disproportionate way. 

Another way of looking at the same problem uses the Salem-Klopman equation (Equation 3.4). Using only the HOMO of a nucleophile and the LUMO of an electrophile, Klopman simplified Equation 3.4 to Equation 4.2 ... 

Fig. 5.2 The Salem-Klopman equation applied to the Biirgi-Dunitz angle...

A fundament of the quantum chemical standpoint is that structure and reactivity are correlated. When using quantum chemical reactivity parameters for quantifying relationships between structure and reactivity one has the advantage of being able to describe the nature of the structural influences in a direct manner, without empirical assumptions. This is especially valid for the so-called Salem-Klopman equation. It allows the differentiation between the charge and the orbital controlled portions of the interaction between reactants. This was shown by the investigation of the interaction between the Lewis acid with complex counterions 18> (see part 4.4). 

The interactions of the occupied orbitals of one reactant with the unoccupied orbitals of the other are described by the third term of the Klopman-Salem-Fukui equation. This part is dominant and the most important for uncharged reaction partners. Taking into account that the denominator is minimized in case of a small energy gap between the interacting orbitals, it is clear that the most important interaction is the HOMO-LUMO overlap. With respect to the Diels-Alder reaction, one has to distinguish between two possibilities depending on which HOMO-LUMO pair is under consideration. The reaction can be controlled by the interaction of the HOMO of the electron-rich diene and the LUMO of the electron-poor dienophile (normal electron demand) or by the interaction of the LUMO of an electron-poor diene and the HOMO of an electron-rich dienophile (inverse electron demand cf Figure 1). 

Frontier Orbital theory supplies an additional assumption to this calculation. It considers only the interactions between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). These orbitals have the smallest energy separation, leading to a small denominator in the Klopman-Salem equation. The Frontier orbitals are generally diffuse, so the numerator in the equation has large terms. 

Molecular Orbital (HOMO) of one component and the Lowest Unoccupied Molecular Orbital (LUMO) of the other. Within the scope of FMO theory, the reactivity of two reaction partners towards each other is described quantitatively by the Klopman-Salem-Fukui relationship (equation i)60 62 ... 

The kinetic aspects of these reactions were inspected by the frontier molecular orbital (FMO) method for the 1,3-dipolar cycloaddition reactions of R Ns + R2NCO or R2N3 + R NCO affording the corresponding tetra-zolinone (97JHC113). Making use of the Klopman-Salem equation, from... 

This paradox has not yet been resolved. The work of Epiotis62 offers the following clue. The Klopman-Salem equation involves orbital energies, for which we may substitute ionization potentials and electron affinities, by virtue of Koopmans theorem. If we think, as we should, in terms of states rather than orbitals, the excitation energy of a state in which an electron is transferred from a donor D to an acceptor A is the difference In—A a between the ionization potential and the electron affinity only if the donor and the acceptor are infinitely far apart. When they are closer, a negative electrostatic term C must be added. The energies of the possible types of state take the form... 

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