Two-electron integrals format

So only the two-electron integrals wilh p. > v. and I>aand [p.v > 7.a need to he computed and stored. Dp.v.la on ly appears m Gpv, and Gvp, w hereas ih e original two-electron integrals con tribute to other matrix elemen is as well. So it is m iich easier to form ih e Fock matrix by using the siipermairix D and modified density matrix P th an the regular format of the tw O-electron in tegrals and stan dard den sity m atrix. 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

Indeed, it is this convenient fact which enables all the SCF methods outlined so far to be implemented compactly the underlying two-electron integral transformation (four-index multiplications) has been contained into the formation of J and K matrices and some one-electron (two-index matrix multiplications) transformations. However, if we use any method which demands the existence of the MO-based repulsion integrals with either i j or k or both)... 

In the remainder of this chapter we will discuss different ways to parallelize the computation of the two-electron integrals with focus on how to achieve load balance. Two-electron integral computation in a quantum chemistry code is often performed in the context of a procedure such as Fock matrix formation or two-electron integral transformation, but we here consider the integral computation separately to specifically concentrate on strategies for distributing the work. Examples of parallel two-electron integral computation in quantum chemical methods are discussed in chapters 8,9, and 10. 

An alternative strategy was to develop methods wherein the two-electron integrals are parameterized to reproduce experimental heats of formation. As such, these are semi-empirical molecular orbital methods—they make use of experimental data. Beginning first with modified INDO (MINDO/1, MlNDO/2, and MINDO/3, early methods that are now little used), the methodological development moved on to modified neglect of diatomic differential overlap (MNDO). A second MNDO parameterization was created by Dewar and termed Austin method 1 (AMI), and finally, an "optimized" parametrization termed PM3 (for MNDO, parametric method 3) was formulated. These methods include very efficient and fairly accurate geometry optimization. The results they produce are in many respects comparable to low-level ab initio calculations (such as HF and STO-3G), but the calculations are much less expensive. 

The formation of the b vector therefore reduces to a contraction of the one- and two-electron integrals with the one- and two-particle density matrices. 

Our first task in the parallelization of HF theory is the parallelization of the Fock matrix formation. The computation of the Fock matrix is dominated by the two-electron integral part G = F — A wide variety of approaches to the computation of G has been implemented. Here we will illustrate several possible schemes. Some of these will require that at least one processor is able to store all elements of the matrices involved, whereas in others storage of matrix elements is fully distributed. 

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