Truncation overlap integrals

The exchange repulsion energy in EFP2 is derived as an expansion in the intermolecular overlap. When this overlap expansion is expressed in terms of frozen LMOs on each fragment, the expansion can reliably be truncated at the quadratic term [44], This term does require that each EFP carries a basis set, and the smallest recommended basis set is 6-31-1— -G(d,p) [45] for acceptable results. Since the basis set is used only to calculate overlap integrals, the computation is very fast and quite large basis sets are realistic. 

The approximations in the SHM are its peremptory treatment of the overlap integrals S (Section 4.3.4, discussion in connection with Eqs. 4.55), its drastic truncation of the possible values of the Fock matrix elements into just a, jl and 0 (Section 4.3.4, discussion in connection with Eqs. 4.61), its complete neglect of electron spin, and its glossing over (although not exactly ignoring) interelectronic repulsion by incorporating this into the a and jl parameters. 

Simplified versions of this expansion have been used with success, however, in which the expansion is truncated to only the first term, yielding a Mulliken-like approximation, but the coefficient C is not set to the overlap, but rather is forced to yield the correct integral, ... 

The Hubbard model can be derived from eqn.(2) by (i) making a zero differential overlap (ZDO) approximation and (ii) by assuming the range of Coulomb interactions to be truncated to just on-site repulsion by virtue of strong shielding of the interactions by the conduction electrons in metals. The ZDO approximation [25] restricts the nonzero electron repulsion integrals to those of the form... 

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. 

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