Truncation exchange 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. 

A straightforward method bypasses the introduction of the auxiliary exchange-correlation fitting basis and evaluates those matrix elements directly by three dimensional numerical integration. This not only spares a computational step, it also avoids the limitations of the fitting basis set. There still arises a truncation error due to the numerical integration scheme. 

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

Further terms are not necessary in such a case the functions on both sides of the integral would differ with four and more spin-orbital exchanges (and Hamiltonian is still a sum of one-and two-electron operators). Similarly, the expansion in the integral ( Eo ff o) will also be truncated on the second term ... 

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