Four-center, two-electron

All three- and four-center two-electron integrals, which are by far the most numerous of the two-electron integrals, are neglected. 

Thus, we can alternatively express the Coulomb contribution in equation (7-12) solely in terms of the basis functions as the following four-center-two-electron integrals (since the four basis functions qfl, r v, q/ r a can be attached to a maximum of four different atoms)... 

The novel four-center two-electron delocalized o-bishomoaromatic species 593, 594,599,601a, and 603 are representatives of a new class of 2jt-aromatic pericyclic systems. These may be considered as the transition state of the Woodward-Hoffmann allowed cycloaddition of ethylene to ethylene dication or dimerization of two ethylene radical cations985 (Figure 3.25, 604). Delocalization takes place among the orbitals in the plane of the conjugated system, which is in sharp contrast to cyclobutadiene dication 605 having a conventional p-type delocalized electron structure (Figure 3.25). 

Schleyer, P. v. R., Bremer, M., Schotz, K., Kausch, M. Schindler, M. 1987 Four-center two-electron bonding in a tetrahedral topology. Experimental realization of three-dimensional homoaromaticity in the l,3-dehydro-5,7-adamantanediyl dication. Angew. Chem. Int. Ed. Engl. 26, 761 763. 

Scheme 4.17 presents a couple of other strange-looking cationic species which were discovered in studies in a related field. In connection with the problem of dodecahedrane synthesis via the isomerization of pagodane 34 (cf. data in Schemes 4.10 and 4.11), Olah s and Prinzbach s groups engaged in studies of the behavior of pagodane derivatives under superacid conditions. Their hope was to force the cationic isomerization of 34 to 3. Despite all attempts, this route was unworkable. As a reward for these apparently futile efforts they were able to observe the unexpected formation of a very stable cationic species, the pagodane dication 55 (Scheme 4.17). The pattern of its NMR spectra combined with the nature of its quenching adduct 56, and the theoretical analysis of possible alternatives, enabled the authors to ascribe to this dication the unprecedented four-center/two-electron delocalized bis-homoaromatic structure.

The energy level diagram shown in Fig. 13 along with the six electrons from the Lewis structure shown earlier shows that the lower three BMO s are filled. These MO s are four-center two-electron bonds and are therefore delocalized over the whole molecule. 

A different approach to dynamical correlation in large systems is to approximate the four-center two-electron integrals in, for example, Eq. (53) by sums of products of three-center integrals, as done in the resolution-of-identity MP2 (RI-MP2) method [64]. As the number of three-center integrals is much smaller than the number of four-center integrals, this approach reduces the computational cost of MP2 calculations dramatically. To ensure high accuracy in the calculations, special auxiliary basis sets have been developed for the RI expansion. Although the RI expansion... 

Figure 5.10. Schematic depictions of the physical meaning of some two-electron repulsion integrals (section 5.2.3.6e). Each basis function (j> is normally centered on an atomic nucleus. The integrals shown here are one-center and two-center two-electron repulsion integrals - they involve one and two atoms, respectively. In calculations with more than two atoms three-center and four center two-electron integrals can arise.

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