Two-electron repulsion integral

The calculation of the two-electron repulsion integrals in ab initio method is inevitable and time-consuming. The computational time is mainly dominated by the performance of the two-electron integral calculation. The following items can control the performance of the two-electron integrals. 

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Set this threshold to a small positive constant (the default value is 10" ° Hartree). This threshold is used by HyperChem to ignore all two-electron repulsion integrals with an absolute value less than this value. This option controls the performance of the SCF iterations and the accuracy of the wave function and energies since it can decrease the number of calculated two-electron integrals. 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

The term ( iv X.o) in Equation 32 signifies the two-electron repulsion integrals. Under the Hartree-Fock treatment, each electron sees all of the other electrons as an average distribution there is no instantaneous electron-electron interaction included. Higher level methods attempt to remedy this neglect of electron correlation in various ways, as we shall see. 

In this latter formula, the two electron repulsion integral is written following Mulliken convention and the one electron integrals are grouped in the matrix e. In this way, the one-electron terms of the Hamiltonian are grouped together with the two electron ones into a two electron matrix. Here, the matrix is used only in order to render a more compact formalism. 

Here the symbol e represents the one-electron integral matrix and iaja lKia ) is the usual two-electron repulsion integral in the Condon and Shortley notation. 

As above, we have changed the subscripts to indicate that the summations run over the orbitals rather than the electrons. The two-electron repulsion integrals Jab and K /, are formally defined as... 

The Fock integrals first encountered in equation (A.45) are constructed from kinetic energy integrals, nuclear-electron attraction integrals, and two-electron repulsion integrals, as follows, continuing from equation (A.49) ... 

The supermatrix G, which contains the two-electron repulsion integrals, has elements defined by... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

Taking this into account, there are only six unique two-electron repulsion integrals, whose values are ... 

Fig. 5.10 Schematic depictions of the physical meaning of some two-electron repulsion integrals (Section 5.2.3.6.5). Each basis function (j> is normally centered on an atomic nucleus. The integrals shown here are one-center and two-center two-electron repulsion integrals - they are centered on one and on two atomic nuclei, respectively. For molecules with three nuclei three-center integrals arise, and for molecules with four or more nuclei, four-center integrals arise...

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

Note that the name coulomb integral has a different meaning in HMO theory (where it refers to the energy of the orbital Xr in the field of the nuclei) to Hartree-Fock theory discussed below (where it refers to a two-electron repulsion integral). 

Use of overlap integrals and electrostatic potentials, essentially nuclear attraction integrals, when dealing with two electron repulsion integrals. 

As an example, we describe the parallel computation of two-electron repulsion integrals over Gaussian basis functions. These integrals are usually computed over shell blocks of integrals to make optimal reuse of intermediate quantities, where the computation of each shell block is totally independent from that of other shell blocks.2 -34 jhis is an example of an embarrassingly parallel application. The four nested loops that exist in this algorithm are ... 

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