MO LCAO coefficients

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

The Pimentel-Rundle 3c/4e MO model can readily be generalized to hybrid orbitals (rather than pure AOs) and more general LCAO-MO mixing coefficients... 

Pople-Pariser-Parr PPP (CNDO) method. MO s from LCAO SCF, i.e. electron correlation taken into account. Equations to derive the LCAO coefficients by the variational procedure were proposed by Roothaan... 

The present article is an attempt to review those studies of pyridinelike heterocycles (mono-azines) and, to a lesser extent, their analogues and derivatives that have interpreted the behavior and estimated various physico-chemical properties of the compounds by the use of data calculated by the simplest version of the MO LCAO (molecular orbital, linear combination of atomic orbitals) method (both molecular orbital energies and expansion coefficients). In this review, attention is focused upon the use of the simple method because it has been applied to quite extensive sets of compounds and to the calculation of the most diverse properties. On the other hand, many fewer compounds and physico-chemical properties have been investigated by the more sophisticated methods. Such studies are referred to without being discussed in detail. In a couple of years, we believe, the extent of the applications of such methods will also be wide enough to warrant a detailed review. 

Obviously, an orbital boundary surface defines an interior and an exterior. Outside the boundary, the function cp has very small values because its square, summed over all space from the boundary wall to infinity, has a value of only 0.1. Recognizing this fact allows the LCAO approximation to be interpreted in physical terms. When we say that a molecular orbital is a linear combination of AOs, we imply that it is almost indistinguishable from cpk in the neighbourhood of atom k. This is because we are then inside the boundary of cpk and outside the boundary oi(pfl = k), so that cpk has finite values and contributions from [Pg.24]

If a correlation diagram is constructed by connecting reactant and product orbitals of equal symmetry that are localized in the same spatial region of the molecule and that show the same sign relations for the LCAO coefficients, crossings of correlation lines of equal symmetry may occur. The natural orbital correlations (Devaquet et al., 1978) obtained in this way are very similar to the intended correlations described earlier. The main difference is that in the first step the orbitals of the product are now not taken to be the AOs of the bond that is being broken, but rather MOs constructed from them. 

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