Three-center integrals

In the case of hydrogen molecule, the term (ri l/ r + R ri2), which involves three centers, does not show up in the calculation. We will not discuss this integral in the present work. 

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. 

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... 

In common applications EP values must be calculated over several thousand points, therefore the efficiency of the computation technique is of crucial importance. Calculation of the first term on the right-hand side of Eq. (3) is trivial, but not of the second term because of the three-center one-electron potential integrals. 

The molecular electron density function needed for EP calculation can be obtained through ab initio as well as various semi-empirical methods. Since ab initio calculations are not economical for large molecules (several hundred atoms), the use of well-parameterized semi-empirical methods are still justified. When semi-empirical methods are used the three-center potential integrals usually disappear, and therefore the electronic contribution can be easily calculated by Slater-type orbitals. In ab initio methods (primitive or contracted) Gaussian-type orbitals are used for calculating the three-center integrals because their calculations are clumsy with Slater-type orbitals. 

Thus, the three-center potential integrals are not retained. Ferenczy et al. found that the AMI MEP maps are able to reproduce the main characteristics of the HF/STO-3G MEP maps for the water, formaldehyde, formamide and the cytosine molecules and for the cyanate ion [40]. However, the NDDO AMI MEP maps gave deeper minima which were closer to the molecules than those in the HF/STO-3G MEP maps. The contour plots of MEP for the cytosine molecule are displayed in Fig. 2. It can be seen that the NDDO AMI MEP map correctly predicts the N3 nitrogen atom as a primary protonation center instead of the 02 oxygen atom. This finding is in agreement with the HF/STO-3G MEP map [28] and the experimental as well as the theoretical proton affinities [29, 30]. Similar results were also obtained by Luque et al. based on the quasi ab initio MNDO MEP map [37], INDO/S, HF/4-31G and HF/6-31G calculations showed an opposite order of protonation [27, 31] as discussed earlier. 

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]. 

All inner shell electrons are presumed to be spherically symmetric and contribute nothing to the field-gradient, and all the two and three-center integrals in the above expression presumed to cancel exactly the field-gradient produced by the residual nuclear charges of the atoms B, C.., etc. Thus... 

For example, three center nuclear attraction integrals (AB C) will reduce to an expression involving two center integrals of (AA C) and (BB C) type. 

Because of the many center nature of the fourth integral case, a detailed analysis of three center nuclear attraction integral problem is given. Using the ideas developed in Sections 4 and 5, it is described how the three center integrals become expressible in terms of one and tv/o center ones. An example involving s-type WO-CETO functions is presented as a test of the developed theory of the preceding chapters. 

Variation of the Three Center Nuclear Attraction Integrals with L . 

Figure 6.1 Variation of the three center Nuclear Attraction integral <1Sa1Sb C> value.

Figure 6.1 shows the three center nuclear attraction integral variation using the same Eab covering of Figure 5.1, and the same... 

Table 7.1 shows how repulsion integrals over CETO functions can be constructed with reliable accuracy, as three center nuclear attraction integrals were computed. 

One of the authors (R.C.) wants to thank Prof. S. Fraga for the challenging proposal made to him in 1969, concerning computation of three center STO nuclear attraction integrals. The interest of Prof. S. Huzinaga on the possible use of STO s in the same manner as GTO s, and the test atomic calculations made by him in 1977 on the second and third rows of the periodic table [73] are also deeply acknowledged. 

Fig. 10. Results of simplified LCAO discussion of the three-center orbital (41). Orthogonality of the AO s and constancy of the Coulomb integrals have been assumed. The ratio of i onance integrals y/p = Q corresponds to an open three-center bond, that for y/p = 1 to a central three-center bond, and that for p/y = 0 to an ordinary electron-pair bond. The lowest energy states is E. ...

A two-electron integral is invariant against a common translation of its four centers. This property was first explicitly described by Komornicki et al. (1977). Thus the 12 derivatives with respect to the coordinates of the four centers can be expressed in terms of the nine derivatives with respect to the coordinates of three centers the gradient with respeet to the fourth center can be determined from the condition that the sum of the x components of the forces on the four centers should vanish. Much of the discussion of the general... 

There are roughly integrals to evaluate, assuming the same number of functions in the CDB as in the orbital basis. At most three centers are involved in the integrals. 

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