Nuclear attraction integrals

It remains to specify the elements of the one-electron core Hamiltonian, Hj y, containing the kinetic energy and nuclear attraction integrals. 

When expressing the nuclear attraction integrals in the FSGO basis, one has explicitly ... 

Use of overlap integrals and electrostatic potentials, essentially nuclear attraction integrals, when dealing with two electron repulsion integrals. 

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. 

Nuclear Attraction integrals are also studied at this section end, providing the first application example of the technique outlined in section 5 above. 

Nuclear attraction integrals being the first touchstones of the present procedure, are discussed here, at this section end, opening the way to two electron integrals problem which will be discussed next in section 7. 

There are four kinds of nuclear attraction integrals. Three of them do not need the expansion (5.2) and are presented first. 

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. 

One encounters an expression similar to the one appearing in the Nuclear Attraction Integrals of the same kind as these discussed in section 6.2, as shown in equation (6.21). 

Some Computational Results on Nuclear Attraction Integrals... 

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... 

In this sense, electronic spin-spin contact integrals can be computed as simple bilinear functions of overlap integrals, and electron repulsion integrals are expressible as bilinear fimctions of some sort of two electron nuclear attraction integrals. 

It is straightforward to see how integration with respect to the electron 2, for example, is equivalent to compute a nuclear attraction integral like ... 

The integration over the coordinates of the electron number 2 will be done over a prolate spheiical coordinate system rB2,rv2,rBi), and the analytic expression of the nuclear attraction integral shown in equation (7.5) obtained. This integration method ensures that the variable rjj will not produce any mathematical problem as it has been observed in section 6.5. Note that the constant parameter, which is the usual intercenter distance, when dealing with this kind of coordinate system, will act here as a variable and it has to be integrated when considering the electron (1) coordinates. 

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. 

Pairs of shell-pairs are required for most PRISM calculations but, for example, if nuclear-attraction integrals are being calculated as outlined in Section 3.2.4, the "shell-quartets" are constructed by pairing the shell-pairs with the infinite-exponent Gaussian at each of the nuclei. 

From (130), it can also be seen that the Shibyya-Wulfman integrals can be interpreted as a species of nuclear attraction integrals, as Koga has pointed out [33, 34]. To see this, we note that ydx) obeys the Schrodinger equation... 

Koga T, Matsuhashi T (1987) Sum rules for nuclear attraction integrals over hydrogenic orbitals. J Chem Phys 87(8) 4696-4699... 

The application of a modified electron-nucleus potential together with analytical basis functions requires the evaluation of appropriate matrix elements (nuclear attraction integrals) ... 

This notation covers both the non-relativistic and relativistic cases (scalar orbitals and 4-component spinors, respectively), the indices i and j carry information to identify the basis functions imambiguously. The integrals in Eq. (122) may involve only a single centre A = B — C [ atomic integrals ]), two centres, or three centres A, B, C all different). The difficulty of their evaluation increases with the number of centres. In addition, every type of basis functions requires its own implementation of nuclear attraction integrals. This task has been accomplished, at least to some paxt, for various potentials and Slater-type or Gauss-type basis functions. For technical reasons (ease of evaluation of multi-centre integrals) the latter type is usually preferred. 

Nuclear attraction integrals for the combination of the homogeneous finite nucleus model (see Sect. 4.3) with Gauss-type basis functions were implemented by Ishikawa et al. [101], by Matsuoka et al. [102,103], and by dementi et al. [104-106]. 

O. Matsuoka, Nuclear attraction integrals in the homogeneously charged sphere model of the atomic nucleus, Chem. Phys. Lett. 140 (1987) 362-366. 

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