Overlapping between orbitals

Finally, the matrix S from Equation 31 is the overlap matrix, indicating the overlap between orbitals. 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

Soo is exactly and Z7oo is roughly proportional to the square of overlap between individual molecular orbitals of both subsystems A and B. The dependence of 7oo on orbital overlap has been discussed in detail 75h The last term in Eq. (11), U ooSoo, is roughly proportional to the fourth power of overlap between orbitals and hence much smaller in magnitude at great enough intermolecular distances. Therefore it is usually added to the higher order contributions. 

The stereochemistry of the cyclobutene isomerizations and the reverse processes of this type, involving the formation of a bond between the ends of a linear system containing a number of 7i--electrons, has been discussed by Woodward and Hoffmann (1965). They term such processes electrocyclic and consider that their steric course is determined by the symmetry of the highest occupied molecular orbital of the open-chain isomer. In an open-chain system containing 4 7T-electrons (such as butadiene), the symmetry of the highest occupied ground-state orbital is such that bonding interaction between the ends of the chain must involve overlap between orbital envelopes on opposite faces of the system, and this can only occur in a conrotatory process ... 

Fio. 23. Diagrammatic representation of overlap between orbital of metal atom and... 

Since we neglected overlap between orbitals of subunits, overlap densities a(l)b(l) and a (2)b (2) are negligible. The matrix element //, for the singlet state can be approximated by the electrostatic interaction between transition densities aa on subunit A and bb on subunit B ... 

Group (c) satellites (dotted). These arise from electron correlation effects which are beyond the FISCI mechanism. The large spatial overlap between orbitals with the same principal quantum number n =v3 still favours electron correlation effects between 3s3p6 and 3s23p43d, but ISCI and FSCI are responsible for the presence of 3s23p43d 2S+1Le final ionic states with symmetries different from 2Se. Most of these states can be identified in the electron spectrum. 

That is, there is no overlap between orbitals on different atoms. Note that this drastic approximation violates the basic bonding principle To form a bond there must an overlap of orbitals. 

Detailed calculations on reactants visualized as oblate spheres have shown that Hel also depends on the shape and nodal character of overlapping orbitals [56]. Thus, Hel for planar n-conjugated systems is predicted to vary depending on the relative orientation of the molecules (Fig. 15 and 16). Orbital character has special importance in PET where overlap between orbitals of the quencher and the rapidly oscillating wavefunctions of the excited reactant determines the extent of electron coupling. 

There are a number of concerted reactions involving cyclic transition states which are characterized by the maintenance throughout of an overlap between orbitals of the correct symmetry. These reactions are known as pericyclic reactions and the rules that govern them are known as the Woodward-Hoffmann rules. A typical example of a reaction of this type is the Diels-Alder reaction of a diene and a dienophile. 

By contrast, in the [1,2] anionic H shift, the dominant FMO interaction is between the Is orbital of the one-atom component and t/q of the two-atom component. For there to be positive overlap between orbitals where bond-making and bond-breaking take place, the two-atom component must react antarafacially. Thus, for this shift to be a thermally allowed process, the H atom must have partial bonds to the top and bottom faces of the C2 unit simultaneously. Because this arrangement is geometrically impossible, [1,2] anionic H shifts are thermally disallowed reactions. 

This approach is relevant when the overlap between orbitals belonging to neighbouring sites is large (for instance in C - C covalent bonds). In case of orbital degeneracy, two low-lying states g,a) or u,a) are available for the electron pair which may then form a singlet or a triplet. Of course, the Hund s first rule tells us that the triplet is more stable, as often encountered. 

For those methods in which atomic orbitals are not orthogonal, the overlap between orbitals must be considered as in the Mulliken population analysis [Mulliken,1955a, 1955b]. In this case, the equation... 

In the standard (non-extended) Hnckd method, the overlap between orbitals (S) is neglected in the calcnlation of the normalization factor. As a result, the sum of the squares of the coefficients is 1. In what follows, when we describe symmetry-adapted orbitals as normalized, we shall always adopt that approximation. 

You have just seen that overlap between orbitals governs NMR coupling constants other situations where orbital overlap is important are ... 

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