Molecular integrals over STOs

For the basis set xi-> , X there are b different possibilities for each basis functionin (rj I tu), and use of the identities (ri I tu) = (sr tu) = [Eq. (14.47)] shows that there are about b /8 different electron-repulsion integrals to be evaluated. Accurate SCF molecular calculations on small- to medium-size molecules might use from 50 to 500 basis functions, producing from 700000 to 10 electron-repulsion integrals. Computer evaluation of three- and four-center integrals over STO basis functions is very time consuming. 

Integrals over CETO functions can be described over WO-CETO s and transformed afterwards to the working molecular axis framework as needed. A similar procedure is used when dealing with STO functions, see for example [34e]. 

In principle, since GTF s form a complete set, the exact molecular orbitals may be expressed in terms of them. Unfortunately the behavior of the Gausslans near the nucleus and far away is incorrect, so many more GTF s than STO s are needed to approximate the exact orbitals to the same degree of accuracy. This point will be discussed in the next sections. Here we note only once again that this disadvantage is overweighed by the ease with which the integrals over GTF s are computed. 

This completes our sketches of the derivations of the molecular integrals involving the GTF basis. Unlike the case of an STO basis, all the overlap and energy integrals can be obtained in a closed form giving the GTF basis an overwhelming practical superiority over the STO basis. 

The natural attractiveness of the STO basis is countered by the numerical difficulties associated with the computation of the various molecular integrals for polyatomic molecules of general geometry. An enormous amount of work has been done over the past thirty years to try to solve these computational problems but despite this effort it is still true to say that the use of the STO basis is only practical for atoms and linear molecules. Even here we arc being slightly optimistic it is only atoms, diatomics and linear triatomics which are well within reach for the STO basis. 

Despite the interest to obtain AO integral algorithms over cartesian exponential orbitals or STO fimctions [43] in a computational universe dominated by GTO basis sets [2], this research was started as a piece of a latter project related to Quantum Molecular Similarity [44], with the concurrent aim to have the chance to study big sized molecules in a SCF framework, say, without the need to manipulate a huge number of AO functions. 

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