Four center integrals

In 1965, however, the computational resources needed for the full SCF approach were not yet available. Practical MO theories therefore still needed approximations. The main problem is the calculation and storage of the four-center integrals, denoted (fiv I Aa), needed to calculate the electron-electron interactions within the... 

In order to overcome these weaknesses, Pople and co-workers reverted to a more complete approach that they first proposed in 1965 [14], neglect of diatomic differential overlap (NDDO). In NDDO, all four-center integrals (pv are considered in which p and v are on one center, as are 2 and cr (but not necessarily on the same one as and v). Furthermore, integrals for which the two atomic centers are diEFer-ent are treated in an analogous way to the one-center integrals in INDO, resulting... 

The first three types of integrals involve one or two centers. The fourth type of integral involves up to four centers. 

All three- and four-center two-electron integrals, which are by far the most numerous of the two-electron integrals, are neglected. 

Craig, D. P., Proc. Roy. Soc. [London) A202, 498, Electronic levels in simple conjugated systems. I. Configuration interaction in cyclobutadiene. (ii) All the interelectron repulsion integrals, three- and four-centered atomic integrals, are included. 

Thus, we can alternatively express the Coulomb contribution in equation (7-12) solely in terms of the basis functions as the following four-center-two-electron integrals (since the four basis functions qfl, r v, q/ r a can be attached to a maximum of four different atoms)... 

If the basis set is restricted to one pn basis function on each sp2 carbon, if the two-electron integrals ignore all three-center or four-center ones, and if we exclude exchange components, one has the Pariser-Parr-Pople model. If, further, all two-electron integrals are set to zero except for the repulsion between opposite spins on the same site and the one-electron tunneling terms are restricted to nearest neighbors, the result is the Hubbard Hamiltonian... 

An alternative set of functions used to build up atomic orbitals are Gaussian functions that have a radical dependence expf-ftr2). A linear combination of these functions gives a reasonable representation of an atomic orbital. The functions are used for computational expediency in ab initio calculations, because four-center Gaussian integrals can be reduced to two-center integrals, which are relatively easy to calculate on a computer (21). 

Table 2 illustrates the overall effect of both these approximations, and other ones to be dealt with subsequently, on the number of one-, two-, three- and four-center integrals involved in a calculation on propane. 

Fig. 5.10 Schematic depictions of the physical meaning of some two-electron repulsion integrals (Section 5.2.3.6.5). Each basis function (j> is normally centered on an atomic nucleus. The integrals shown here are one-center and two-center two-electron repulsion integrals - they are centered on one and on two atomic nuclei, respectively. For molecules with three nuclei three-center integrals arise, and for molecules with four or more nuclei, four-center integrals arise...

The HF method tends to overestimate the barriers, making unstable molecules seem stabler than they really are. Geometries are discussed further in Section 5.5.1. Approximate versions of the MP2 method that speed up the process with little loss of accuracy are available in some program suites LMP2, localized MP2, and RI-MP2, resolution of identity MP2. LMP2 starts with a Slater determinant which has been altered so that its MOs are localized, corresponding to our ideas of bonds and lone pairs (Section 5.2.3.1), and permits only excitations into spatially nearby virtual orbitals [93]. RI-MP2 approximates four-center integrals (Section 5.3.2) by three-center ones [94]. 

Thus, four center integrals may be obtained as simple matrix multiplications of the three-index fields. This allows one to combine RI with most approximations without needing major modifications to existing algorithms and codes. 

Note that the added term, which involves the fit twice, is in that sense less accurate than Sambe and Felton s original term, which involves the fit only once. The point is, however, that if the fitted density is expressed as the exact density minus an error term, then the error in Eq. (5) is the Coulomb repulsion of the error with itself, which is always second order and positive [12]. This approach is readily extended to four-center Coulomb integrals [13],... 

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