Slater type orbitals complex form

Once we have decided to let the molecular orbitals be linear combinations of atomic orbitals, we need to decide the mathematical form we will use for the atomic orbitals. One choice would be to simply use the hydrogenic wavefunctions adapted for other atoms. Such a function is called a Slater-type orbital (STO Figure 14.8 A). These wavefunctions have radial forms possessing terms such as e" " ( = Z/n). Although such functions are used in some kinds of semi-empirical calculations (see below), they generally are not used in ab initio quantum mechanics. The reason for this is that it is very difficult to evaluate the complex two-electron integrals (/ and K) with STOs. To address this issue, other types of basis sets have been developed. 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

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