Evaluation of the two-electron interaction integral

Because evaluation of the two-electron interaction energies is very difficult, the development of SCF methods has evolved into two branches, one of which, the ab initio methods, attempt the rigorous and nonempirical evaluation of these terms. The other branch, semiempirical methods, avoids even attempting the evaluation of the integrals involved instead they are replaced with approximations. 

The MNDO/d formalism for the two-electron integrals has several advantages Conceptually, it employs the same approach as the established MNDO-type methods [19-22], including an effective reduction of the Coulomb interactions to allow for dynamic electron correlation. Moreover, it is rotationally invariant and computationally efficient, and analytical first and second derivatives of the integrals are available [121]. An alternative formalism has been implemented more recently in the SAMI approach [23,122], which apparently involves the analytical evaluation of the two-electron integrals in an spd basis and subsequent scaling (details still unpublished). 

Despite these modifications there remain a number of well-documented problems with the AM1/PM3 core-repulsion function [37] which has resulted in further refinements. For example, Jorgensen and co-workers have developed the PDDG (pair-wise distance directed Gaussian) PM3 and MNDO methods which display improved accuracy over standard NDDO parameterisations [38], However, for methods which include d-orbitals (e.g. MNDO/d [23,24], AMl/d [25] and AMI [39,40]) it has been found that to obtain the correct balance between attractive and repulsive Coulomb interactions requires an additional adjustable parameter p (previously evaluated using the one-centre two-electron integral Gss, Eq. 5-7), which is used in the evaluation of the two-centre two-electron integrals (Eq. 5-8). 

The two-electron integrals (Equation 6.32) are determined from atomic experimental data in the one-center case, and are evaluated from a semiempirical multipole model in the two-center case that ensures correct classical behavior at large distances and convergence to the correct one-center limit. Interestingly, this parameterization results in damped effective electron-electron interactions at small and intermediate distances, which reflects a (however less regular) implicit partial inclusion of electron correlation (Thiel, 1998). In this respect, semiempirical methods go beyond the HF level, and may accordingly be superior to HF ab initio treatments for certain properties that have a direct or indirect connection to the parameterization procedure. 

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