Four-centre integrals

When the number of atoms in the molecule increase above two, three- and four-centre integrals enter the expressions. Matcha and Kern115 have given expressions for these. However, there is also the problem of the number of integrals to be calculated with large molecules this will be extremely time-consuming. 

Xa F X/j) <—(Xa Jy X/j) y l(x a0 g (5.13) -leant compared to the three- and four-centre integrals, pseudospectral methods may i i nrovide energies at the same accuracy as fully analytical methods. For small to medium ... 

The main difficulty in all these methods is the evaluation of so-called four-centre integrals, e.g. 

It can immediately be seen that all three- and four-centre integrals are set to zero under the ZDO approximation. If the ZDO approximation is applied to all orbital pairs then the Roothaan-Hall equations for a closed-shell molecule (Equation (2.226)) simplify considerably to give the following for p = n ... 

These auxiliary basis functions b have the same functional form as the orbital expansion and the coefficients c are obtained by a least-squares fitting procedure. Substituting for the density in the four-centre integrals gives a computationally less demanding three-centre, two-electron integral ... 

The four-centre integrals, Vijj-i, are dominated by the diagonal terms ... 

The neglect of the off-diagonal four-centre integrals is either called the Complete Neglect of Differential Overlap (CNDO), or Zero Differential Overlap (ZDO). Models that retain some other four-centre integrals are termed Intermediate Neglect of Differential Overlap (INDO). 

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