Integral-direct Fock matrix evaluation

A simple iterative DHF procedure for solving the generalized eigenvalue problem FC = ESC takes the form [113] 

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Evaluate the integrals. In a conventional algorithm, they are stored on disk and read in for each iteration. In a direct algorithm, integrals are computed a few at a time as the Fock matrix is formed. 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

Large-scale SCF treatments have been made possible with the direct SCF method developed by Almlof and co-workers. This procedure is based on the evaluation of two-election integrals on die fly whenever they are needed in the construction of the Fock operator (in the matrix representation required in SCF treatments). The efficiency of direct SCF method.s results from the fact that many integrals give negligible contributions and can be safely neglected. This requires reliable estimates for the integral values, TURBOMOLE employs the mathematically best separable bounds for this purpose. ... 

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