Core integral

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

All calculations were performed on the Cray-2 computers at the Minnesota Supercomputer Center. In some cases the two-electron Integrals could be kept in the 256 megaword central memory of the Cray-2, and in these cases an "in-core" integral and SCF code(53) was used. The largest in-core calculations possible in... 

There are two versions of CNDO, CNDO/1 and an improved version, CNDO/2. First look at CNDO/1. Consider the core integrals Hcrwhere both orbitals are the same (i.e. the same orbital occurs twice in the integral (r(l) /7J °re ,.(l))) and are on the same atom A. Recall the example of an ab initio calculation on HHe+ (Section 5.2.36e). Consider, say, element (1,1) of that Hcore matrix. From Eq. 5.116 ... 

The core integrals with the same orbital [Pg.404]

The core integrals H A with different orbitals (fir and s, on the same atom A are not simply taken as being proportional to the overlap integral, as in CNDO (Eq. 6.12), but rather are also (like the case of both orbitals on the same atom) evaluated from Eq. 6.10, which in this case becomes... 

The core integrals /with different orbitals s, on different atoms A and B are taken, as in CNDO (cf. Eq. 6.12), to be proportional to the overlap integral between proportionality constant is the arithmetic mean of parameters for atoms A and B ... 

The overlap integral is calculated from the basis functions although the overlap matrix is taken as a unit matrix as far as the Roothaan-Hall equations go (Section 6.2.2). These core integrals are sometimes called core resonance integrals. 

Using a partition of the core Hamiltonian into atom-centered fractions, the one-center-one-orbital core integral may be written... 

In the component, the core integral term is assigned values derived from experimental atomic spectroscopy.57-58 All one-center two-electron terms are retained, and the two-center two-electron terms are obtained from the following ... 

The variations in the four different types of integral must now be derived. For the core integral we have... 

We shall not discuss at length further simplifications, known as next-neighbors interactions (for core integrals), zero-differential-overlap (for two-electron repulsion integrals) etc., which were introduced into the formalism of the n molecular orbital theory after the basic work of Goeppert-Mayer and Sklar. Detailed reviews on these topics have been published 38,39,40,41,42). Let us just show why zero-differential-overlap can be justified in terms of orthogonalized orbitals 43,22,44). 

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