Contracted multipole integrals

Here, RapyS is the distance vector between the primitive centers aap and byg. Using contracted multipole integrals instead of primitive integrals, we obtain the following expansion for contracted integrals ... 

Let us now consider an alternative procedure for the evaluation of the contracted multipole moments (9.13.36). First, we calculate - for each primitive overlap distribution contributing to the two-electron integral - local multipole moments of the type (9.13.20), each with origin at the centre of the primitive overlap distribution ... 

Here the q s are spherical multipole moments (monopole, dipole, quadrupole, etc.) of charge distributions and fig, respectively. Note that we used square brackets to denote an uncontracted two-electron integral. A suitable generalization for contracted integrals is described in the next section. 

In Section 9.13.1, we considered the evaluation of a primitive two-electron integral by a bipolar multipole expansion, with the origins of q" (P) and q (O) at the centres P and Q of the overlap distributions f2afc(ri) and f cdlrz). Let us now consider the evaluation of a two-electron integral over a contracted set of overlap distributions, which we write here in the form... 

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