One-Centre Integrals

In a closed-shell system, P = P) = P and the Fock matrix elements can be obtained by making this substitution. If a basis set containing s, p orbitals is used, then many of the one-centre integrals nominally included in INDO are equal to zero, as are the core elements Specifically, only the following one-centre, two-electron integrals are non-zero (/x/x /x/x), (pit w) and (fti/lfM/). The elements of the Fock matrix that are affected can then be written a." Uxllow s ... 

Apart from the normalizing factor, the two-centre overlap integral then reduces to a simple one-centre integral of the form... 

The Fermi contact contribution to a coupling constant, V(A-B), is proportional to the product of the 5-electron densities at the coupled nuclei, sA(0)sB(0). (11-17) Both the orbital and dipolar contributions are proportional to the product of the one-centre integrals (r 3)A and (r 3)B, where (r 3)A and (r 3)B are the expectation values of r for valence shell p orbitals on atoms A and B respectively. (14-17) Thus one can write the total coupling constant... 

Ref. 118. Sum-over-state perturbation method. The values of the one-centre integrals. s3(0) and (r 3)( are quoted from those summarized by Morton. (120)... 

Include one-centre integrals of all classes (these are so cheap to make there is little point in excluding them) together with the (LL LL) integrals which would be needed for the equivalent nonrelativistic Hartree-Fock calculation. 

Coupling constants are usually analysed at present in terms of a MO theory developed by Pople and Santry. They showed that coupling constants involving directly bonded atoms arise almost entirely from the Fermi contact interaction between nuclear moments and electron spins in 5 orbitals. Using the LCAO approximation and retaining only one-centre integrals, they derived the following expression for Ja-b—... 

One-centre integrals both basis functions and the attracting centre all share a common origin. These integrals are atomic and are trivial to evaluate in the local polar coordinates. 

One-centre integrals both charge distributions are on one centre and this centre is common. These are atomic integrals and their evaluation is straightforward. 

One-centre integrals the attracting nucleus coincides with the point in space on which the product of the two GTFs generates a GTF function. 

Two-electron integrals are approximated as in eq. (3.87), except that one-centre integrals (jUa aIVaCTa) are preserved. 

By means of an independent electron model and a minimum basis set of valence shell atomic orbitals, Pople and Santry (6) developed the MO expressions appropriate to SOS calculations of spin-spin coupling. A truncated set of excited states is produced by promoting a single electron at a time from occupied to unoccupied MO s. In keeping with the nature of the independent electron model, only one-centre integrals are included in the calculations. 

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