Atomic orbital integrals

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

In usual MO calculations with the ZORA Hamiltonian, the atomic orbital integrals derived from the ZORA Hamiltonian are simple and are evaluated numerically in direct space. In our study, however, we use the resolution of identity (RI) approximation with finite basis functions to evaluate them. To this end we use the relation. 

MO The atomic orbital integrals are generated once and stored externally, then transformed to the molecular orbital basis. The transformed (MO) integrals are also stored externally. The four index transformation, eq 11, requires work and disk storage (see the Integral Transformation section). This is the approach used until recently for all correlated energy methods. 

In-Core The atomic orbital integrals are generated once and stored in canonical order in main memory (i.e., including zeroes). This requires large amounts of memory - N /8... 

Direct The atomic orbital integrals are recomputed as needed. This does not require 0(N" ) internal or external storage but does involve additional computational effort. For large molecules, substantial savings are possible which compensate for this additional effort. However, direct methods are the only choice when memory and disk are exhausted and consequently are inevitably used for the largest calculations. 

Escf is the self-consistent field energy, f is the electric field, a is a nuclear coordinate, is the one-electron atomic orbital integral, If is related to the derivative of the molecular orbital coefficients with respect to a by... 

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