OVERLAP DETERMINANT METHOD

In order to eliminate the problems with the invariance, we proposed some time ago a topological approximation based on the so-called overlap determinant method [43]. This approximation is based on the transformation matrix T that describes the mutual phase relations of atomic orbitals centred on molecules R and P, and thus plays in this approach the same role as the so-called assigning tables in the overlap determinant method (Eq. 4)... 

Robert Ponec, Overlap Determinant Method in the Theory of Pericyclic Reactions, Springer, Berlin, 1995. 

Primes with the AO basis of the product are used to denote the fact that the corresponding atomic orbitals x can differ from the AO basis of the reactant, for example because of different spatial orientation. This distinction between the AO bases of the reactant and the product is very important since it is just precisely from here that the possibility arises to exploit the formalism for the discrimination between the forbidden and the allowed mechanisms, i.e., in our case, between the conrotation and the disrotation. The basis of this discrimination are the so-called assigning tables, the physical meaning of which is just in providing the detailed specification of the mutual transformation between bases % and x. which is the necessary prerequisite for the calculation of the overlap integral S p. The same problem was encountered also by Trindle [33], and his mapping analysis failed to find broader use only because of considerable numerical complexity. On the other hand, the overlap determinant method solves this problem much more simply and its use is really a matter of seconds using only pen and paper. 

Another important type of pericyclic reactions are cycloadditions. As an example of the use of the overlap determinant method for this class of reactions, let us analyze... 

This reaction is known to be allowed by a supra-antara mechanism, while the alternative supra-supra process is forbidden. The overlap determinant method also leads to the same results. The first step of the analysis consists again in the selection of the irreducible core. This core is in our case formed by the set of two disappearing ethene 7t bonds and by two newly created a bonds of cyclobutane. On the basis of this irreducible core, the structure of the reacting molecules is described by the approximate wave functions (30),... 

Extension of the original formalism of the overlap determinant method to photochemical reactions [56] can again be best demonstrated by reactions the course of which is governed by the Woodward-Hofimaim rules. In this case the most important result is the exact reproduction of the reversal of the stereochemical course of the reaction in comparison with analogous thermal processes (see Table 1). As an example let us analyze first the photochemical isomerization of 1,3 butadiene to cyclobutene. The most important modification enters into the formalism at the level of the construction of the irreducible core, where it is necessary to respect the fact that the reactant does not enter into the reaction in the ground, but in the excited state. This circumstance finds its reflection in that one of the n bonds entering into the irreducible core from the part of the butadiene is to be replaced by the... 

As a further example demonstrating the application of the overlap determinant method to cycloadditions, let us analyze the 2 + 2 dimerization of ethene. As has already been stated above, the irreducible core of this reaction for the ground state process is formed by the set of two disappearing ethene 7t bonds and the set of two newly created cyclobutane ct bonds. The fact that we are interested here in the photochemical process finds its reflection in the formal replacement of one of the ethene n bonds by the corresponding virtual bond. Let it be for example bond 7134. The bonds forming the irreducible core of photoreaction are then again described by the usual linear combinations (57) ... 

This approach, based on the exploitation of quantum chemical quantities equivalent to the classical concept of valence was recently subject of a number of stupes [77-78], of which the most inspiring for our purposes was the approach by Jug [78], Incorporating his definition of the valence Wj of atom i (66) into the framework of the generalized overlap determinant method it is possible to introduce the so-called topological valencies w(cp) (67)... 

Introduction of the generalized overlap determinant method and of closely related... 

The usefulness of this index, especially for the applications to chemical reactivity, can be best demonstrated by formally incorporating the eq.(70) into the framework of the generalized overlap determinant method, where structures A and B are identified... 

In order the overlap integral of the functions Or and Op could be calculated, these functions are to be transformed into the common AO basis. Let it be, e.g., the AO basis X of the reactant. Within the framework of the overlap determinant method this transformation is described by the so-called assigning tables (120), the aim of which is to express the mutual phase relations of corresponding AO bases. 

Similarly as with the overlap determinant method the primes are used to denote the fact that the AO bases x and % serving to describe the molecular orbitals of the reactant and the product are generally different. Using the known expressions for the first order spinless density matrices (136), the original definition expression for the similarity index (137) can be rewritten in the form (138). 

In order to eliminate this unconvenient feature various procedures were proposed of which the simplest is probably the topological approximation arising from the philosophy of the overlap determinant method with its assigning tables. Rewriting these tables in the alternative matrix form (140), the problem of the multicenter integrals can be substantially simplified. 

On the basis of trivial identities (144) resulting fi-om the incorporation of the above formalism into the fiamework of the generalized overlap determinant method and the idempotency relations (145), the equation (142) can be rewritten in the final form (139) presented in the chapter 6. 

---