Two-electron integral calculations

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. 

To this point, the basic approximation is that the total wave function is a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ab initio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ab initio calculation. However, there are two main th ings to be con sidered in the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222), will require the fewest possible terms for an accurate representation of a molecular orbital. The second one is the speed of two-electron integral calculation. 

These new basis functions can easily be shown to be orthonormal. It also turns out that two-electron integrals calculated using these orthogonalized basis functions do indeed satisfy the ZDO approximation much more closely than the ordinart basis functions. 

In the field of quantum chemistry, many algorithms have been developed to improve the efficiency of the two-electron integral calculations over the years. One of the most frequently used algorithms is the screening based on the Schwartz (K.H.A. Schwartz) inequality (1888). That is, based on the inequality,... 

Results comparable in quality to those presented by Visser et al. could also be obtained by more approximate schemes of relativistic quantum chemistry. Visser et al. indeed state that their results could serve as a reference for more approximate methods. A considerable reduction in the computational effort could be obtained if a two-component all-electron scheme instead of the four-component DHFR approach would be applied 99.2% of the two-electron integrals calculated by Visser et al. result from the presence of the lower components in the one-particle functions. In addition the computational effort for the SCF iterations and the subsequent integral transformation also would be considerably... 

The present work describes a breakthrough in two-electron integral calculations, as a result of Coulomb operator resolutions. This is particularly significant in that it eliminates the arduous orbital translations which were necessary until now for exponential type orbitals. The bottleneck has been eliminated from evaluation of three- and four- center integrals over Slater type orbitals and related basis functions. 

Once the requisite one- and two-electron integrals are available in the MO basis, the multiconfigurational wavefunction and energy calculation can begin. Each of these methods has its own approach to describing tlie configurations d),. j included m the calculation and how the C,.] amplitudes and the total energy E are to be... 

For all calculations, the choice of AO basis set must be made carefully, keeping in mind the scaling of the two-electron integral evaluation step and the scaling of the two-electron integral transfonuation step. Of course, basis fiinctions that describe the essence of the states to be studied are essential (e.g. Rydberg or anion states require diffuse functions and strained rings require polarization fiinctions). 

Split valence basis sets generally give much better results than minimal ones, but at a cost. Remember that the number of two-electron integrals is proportional to kf , where W is the number of basis functions. Whereas STO-3G has only live ba.sis functions for carbon, 6-31G has nine, resulting in more than a tenfold increase in the size of the calculation,... 

The amount of computation for MP2 is determined by the partial tran si ormatioii of the two-electron integrals, what can be done in a time proportionally to m (m is the u umber of basis functions), which IS comparable to computations involved m one step of(iID (doubly-excitcil eon figuration interaction) calculation. fo save some computer time and space, the core orbitals are frequently omitted from MP calculations. For more details on perturbation theory please see A. S/abo and N. Ostlund, Modem Quantum (. hern-isir > Macmillan, Xew York, 198.5. 

III an SCF calculation. many iterations may beneetled to achieve SCr con vergeiice. In each iteration all the two-electron integrals are retrieved to form a Fock matrix. Fast algorith m s to retrieve the two-cicetron s integrals arc important. 

RalTenetti [R. C. RalTenetli. Chem. I hys. Lett. 20, iiiS.bfl 97iS ) proposed another way to store the two-electron integrals in ah iniiio calculations. RatTenetti rewrote (93) on page 2.31 to read... 

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

Thus, in a two-electron integral of the form p,v a), the product < (1)< (1) (where 0 and may be on different centres) can be replaced by a single Gaussian function that is centred at the appropriate point C. For Cartesian Gaussian functions the calculation is more complicated than for the example we have stated above, due to the presence of the Cartesian functions, but even so, efficient methods for performing the integrals have been devised. 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

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