Basis functions general considerations

The generalized eigenvalue problem is unfortunately considerably more complicated than its regular counterpart when S = I. There are possibilities for accidental cases when basis functions apparently should mix, but they do not. We can give a simple example of this for a 2 x 2 system. Assume we have the pair of matrices... 

On the basis of thermodynamic considerations, some of the lattice sites in the crystal are vacant, and the number of vacant lattice sites generally is a function of temperature. The movement of a lattice atom into an adjacent vacant site is called vacancy diffusion. In addition to occupying lattice sites, atoms can reside in interstitial sites, the spaces between the lattice sites. These interstitial atoms can readily move to adjacent interstitial sites without displacing the lattice atoms. This process is called interstitial diffusion. The interstitial atoms may be impurity atoms or atoms of the host lattice, but in either case, interstitial atoms are generally present only in very dilute amounts. However, these atoms can be highly mobile, and in certain cases, interstitial diffusion is the dominant diffusion mechanism. 

The second problem is the much more realistic one of the effect of a limited basis set expansion. This is clearly a more serious problem because only for linear molecules or those with a few first-row atoms can the Hartree-Fock limit be reached at present. For many of the molecules with which theoretical chemists deal, wave-functions of such accuracy are not available but it may be some comfort to know that even if they were they need not give very good answers It should be mentioned that Brillouin s theorem applies to any SCF wavefunction, but unless the wavefunction is near the Hartree-Fock limit the electron distribution cannot be expected to be a close representation of the true one. No general treatment of this problem has been given neither does one seem possible since it would depend on the ways in which the basis set under consideration was weak, and these may be many. 

Extra considerations are required to construct suitable sets of polynomials, which provide basis functions for the irreducible subspaces of the cubic and icosahedral point groups. Clearly, such a set of central functions is invariant under the point group G. Eor such a function, f, then (fgi, fg2,. .., fgn> is a subspace of the central functions invariant under G. But it is not, in general, an irreducible subspace, i.e. it may contain further subspaces that transform according to different irreducible representations. 

Now consider the basis functions used. Generally, each MO is written as a linear combination of one-electron functions (orbitals) centered on each atom. For diatomic molecules, one can use Slater functions [Eq. (11.14)] for the AOs. To have a complete set of AO basis functions, an infinite number of Slater orbitals are needed, but the true molecular Hartree-Fock wave function can be closely approximated with a reasonably small number of carefully chosen Slater orbitals. A minimal basis set for a molecular SCF calculation consists of a single basis function for each mner-shell AO and each valence-shell AO of each atom. An extended basis set is a set that is larger than a minimal set. Minimal-basis-set S(2F calculations are easier than extended-basis-set calculations, but the latter are considerably more acciurate. 

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