Slater integrals

The //yj matrices are, in practice, evaluated in temis of one- and two-electron integrals over the MOs using the Slater-Condon mles [M] or their equivalent. Prior to fomiing the Ffjj matrix elements, the one-and two-electron integrals. 

Expanding the Slater determinants and integrating out the spin part and collecting terms that are the same under exchange of electeon indices, we have... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

The one-eenter exchange integrals that INDO adds to the CNDO schcmccan be related to th e-Slater-Condon param eters h", O. and F used to describe atomic spectra. In particular, for a set of s, p,. p,.. t, atom ie orbitals, all the on e-ecn ter in tegrals are given as ... 

The algorithms in Z[ DO/S are almost the same as those in ZlNDO/1, except of the one-center two-electron integral, b . ZINDO/S uses em pirical value of in stead of ii sin g ah initio vaine in terms of the Slater orbitals. 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

The coefficients and the exponents are found by least-squares fitting, in which the overlap between the Slater type function and the Gaussian expansion is maximised. Thus, for the Is Slater type orbital we seek to maximise the following integral ... 

There is considerable variation in the values assigned to the election repulsion integrals in Exercise 8.9.1. Salem (1966) points out that calculation using Slater orbitals leads to... 

Two Slater type orbitals, i andj, centered on the same point results in the following overlap integrals ... 

In this form, it is elear that E is a quadratie funetion of the Cl amplitudes Cj it is a quartie funetional of the spin-orbitals beeause the Slater-Condon rules express eaeh <

Cl matrix element in terms of one- and two-eleetron integrals < > and... 

These density matriees are themselves quadratie funetions of the CI eoeffieients and they refleet all of the permutational symmetry of the determinental funetions used in eonstrueting F they are a eompaet representation of all of the Slater-Condon rules as applied to the partieular CSFs whieh appear in F. They eontain all information about the spin-orbital oeeupaney of the CSFs in F. The one- and two- eleetron integrals < I f I > and < (l)i(l)j I g I (l)ic(l)i > eontain all of the information about the magnitudes of the kinetie and Coulombie interaetion energies. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

Alternatively, these one-eenter eoulomb integrals ean be eomputed from first prineiples using Slater or Gaussian type orbitals. 

Here, p2 and G represent the well known Slater-Condon integrals in terms of whieh the eoulomb and exehange integrals ean be expressed ... 

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