Exchange operator Hartree-Fock calculations

In the more general case, i.e., when the fockian contains a nonlocal exchange operator, like in hybrid DFT and in Hartree-Fock calculations, the relation seen Eq. 73 does not hold any more and the commutator of the position operator with the fockian contains an exchange contribution [76, 77], which gives rise to an additional term ... 

By combining the separation into Coulomb and exchange terms with the multipole technique of Section 9.14, it is possible to arrive at an operation count that scales linearly with the size of the system, allowing us to cany out direct Hartree-Fock calculations for very large systems. However,... 

In many calculations beyond the Hartree-Fock level a first step is the transformation of at least some integrals. For the simplest such calculation, second-order perturbation theory, integrals with two indices transformed into the occupied MO basis axe required. Such integrals appear in many situations, including the MO basis formulation of coupled-perturbed Hartree-Fock theory. We can represent the first phase of this transformation as obtaining Coulomb and exchange operators ... 

One guesses at an initial set of wave functions, , and constructs the Hartree-Fock Hamilton S which depends on the through the definitions of the Coulomb and exchange operators, (/ and One then calculates the new set of , and compares it (or the energy or the density matrix) to the input set (or to the energy or density matrix computed from the input set). This procedure is continued until the appropriate self-consistency is obtained. 

Harrison, et al. have reported an efficient, accurate multiresolution solver for the Kohn-Sham and Hartree-Fock self-consistent field methods for general polyatomic molecules. The Hartree-Fock exchange is a nonlocal operator, whose evaluation has been a computational bottleneck for electronic structure calculations, scaling as for small molecules... 

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