Integral two-center

The ZDO approximation is made only for two-center integrals one-center coulomb Za,b = and exchange... 

There are some boundary conditions which can be used to fix parameters and Ag. For example, when the distance between nucleus A and nucleus B approaches zero, i.e., R g = 0.0, the value of the two-electron two-center integral should approach that of the corresponding monocentric integral. The MNDO nomenclature for these monocentric integrals is. 

The main difference between CNDO, INDO and NDDO is the treatment of the two-electron integrals. While CNDO and INDO reduce these to just two parameters (7AA 7ab), all the one- and two-center integrals are kept in the NDDO approximation. Within an sp-basis, however, there are only 27 different types of one- and two-center integrals, while the number rises to over 500 for a basis containing s-, p- and d-functions. 

Roothan CJ (1951) A study of two-center integrals useful in calculations on molecular structure. J ChemPhys 19 1445... 

For small distances R(k) (S2 A) between the nuclei N and k the two-center integrals in (ADD)z, can be approximated by expanding (5.1a) in powers of r(k)/R(k), where r(k) denotes the distance of the electron from the nucleus k125). For R(k) S 2.5 A, the unpaired electron can be considered to be concentrated at the nuclei k, so that the distant contribution may be approximated by the classical electron-nuclear point-dipole formula... 

Of the matrix elements (im V/ /W), only the two-center integrals between first or second nearest neighbors are retained. The coefficients aim then... 

The SDCI calculations are somewhat more involved in calculations of atomic real-space core-valence partitioning models because of the two-center integrals (2.10) and (2.11) that require definite integration limits to cover the appropriate core and valence subspaces. Foitunately, these calculations are greatly aided by most efficient standard techniques. 

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). 

Pople et al. 25> pointed out that while the results obtained for two-center integral evaluation in a full Roothaan S.C.F.M.O. treatment are independent of the choice of axis, the same is not true in simplified versions. Such integrals are sensitive to the choice of coordinate system and the hybridization of the orbitals. 

The implementation of such a model mostly depends on the choice of the atomic orbitals. Linear combinations of Slater Type Orbitals arc natural and moreover allow a good description of one-center matrix elements even at large intemuclear distances. However, a complete analytical calculation of the two-center integrals cannot be performed due to the ETF, and time consuming numerical integrations [6, 7] are required (demanding typically 90% of the total CPU time). 

We use an alternative to this method, that enables a fast and accurate evaluation of the two-center integrals. Analytical integration is possible when linear combinations of Gaussian Type Orbitals are used to describe the atomic states [1,9, 10]. Imperfect behavior of such gaussian functions at large distances does not affect the results, since the two-center matrix-elements (7) have an exponential decay for increasing intemuclear distances. For example, for integrals and expressed in cartesian coordinates, one has to evaluate expressions such as... 

The 0 e Renner-Teller vibronic system describes an orbital doublet ( ) interacting with a two-dimensional vibrational mode of s symmetry. In Section 2 we determine the general formal structure of the electron-phonon interaction matrices with orbital electronic functions of different symmetry (p-like, d-like, /-like, etc.), exploiting their intuitive relation with the Slater-Koster matrices of the two-center integrals. A direct connection with the form obtained through the molecule symmetry is discussed in Section 3. 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

For an orbital doublet of/-like functions we proceed in the same way, using the two-center integrals recently obtained by Doni [20]. Following the patterns previously traced, we arrive at ... 

The SLG-MINDO/3 and SCF-MINDO/3 methods have approximately the same accuracy while calculating the heats of formation of organic compounds. Significant deviations from the experiment (for both wave functions) are observed for the branched compounds. The heats of formation are significantly overestimated for both types of wave functions. It is clear that the intrabond correlation has not much to do with this defect. The NDDO parametrization partially rectifies this by a more detailed account of two-center integrals. 

The two-electron integrals pq kl] are < p(l)0fc(2) e2/ri2 0,(l)0j(2) > and may involve as many as four orbitals. The models of interest are restricted to one and two-center terms. Two electrons in the same orbital, [pp pp], is 7 in Pariser-Parr-Pople (PPP) theory[4] or U in Hubbard models[5], while pp qq are the two-center integrals kept in PPP. The zero-differential-overlap (ZDO) approximation[3] can be invoked to rationalize such simplification. In modern applications, however, and especially in the solid state, models are introduced phenomenologically. Particularly successful models are apt to be derived subsequently and their parameters computed separately. 

The exact evaluation of the two-center integrals may be carried out using the expressions of Roothaan (1951b) for Slater orbitals, with the orbital exponents as determined from Eq. (19). 

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. 

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