Bipolar multipole expansion

Substituting the partial-wave expansion in the form (9.13.18) into (9.13.1), we arrive at the bipolar multipole expansion of the two-electron integral... 

Note that, for spherical overlap distributions centred at P and Q, (9.13.24) and (9.13.25) are the only nonzero multipole moments and the monopole expansion (9.13.27) then repcsents an exact expression for the two-electron integral (assuming disjoint charge distributions) - see also the discussion in Section 9.12.3. In the same mannra-, the multipole expansion (9.13.19) terminates exactly after a finite number of terms whenever the charge distributions of the electrons are one-centre functions, whose centres are chosen as origins of the multipole expansions. In general, however, the bipolar multipole expansion does not terminate and the expansion is then truncated when the remainder is sufficiently small as discussed in Section 9.13.2. 

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... 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

The electrostatic, induction, and dispersion terms can be expanded in a convergent series closely related to the multipole expansion, but fully accounting for the charge-overlap effects, the so-called bipolar expansion introduced by Buehler and Hirschfelder199,200. In the local coordinate systems with the origins located at the centers of masses of the monomers A and B, separated by the distance R, and with their x and y axes parallel and aligned along the z axes, the distance between two particles in space can be expressed as follows,... 

The expansion given above is exact, except for r12 = 0. If the terms resulting from the regions II-IV are neglected, one recovers the standard multipole expansion of the interaction operator. Substituting the bipolar expansion of 1 /r12 and analogous expansions for other terms of the operator V into the matrix elements (0 V J), the bipolar expansion of a given polarization correction is obtained. 

The term bipolar is used to describe this expansion since it involves two sets of multipole moments, with different origins. In matrix notation, the multipole expansion (9.13.19) may be written as... 

There are other approaches for calculations of the TERIs. The uniformly charged sphere model is used in LNDO/S, the multipole modeP is used in MNDO/d, AMl/d, and PM3(tm) the bipolar expansion of the Klopman-Ohno potential is used in PM3d. 

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