Obara-Saika method integrals

As demonstrated by the Pople-Hehre method it is possible to achieve considerable reduction in the computational expense of contracted ERIs if large parts of the integral manipulation are performed after the contraction step. The McMurchie-Davidson and the Obara-Saika methods utilization of the transfer equation (17) to minimize the operation count has been shown. This idea can, however, be employed to the extent that all manipulations are performed on fully or partially contracted integrals. Recently a number of methods have been presented along those lines.The method of Gill and Pople will be used as an example of the approach because it is currently one of the most commonly used integral methods. Note the concept of early contraction, however, applies to any of the methods presented in the chapter. 

The most recent integral algorithms evolved with, and were influenced by, the advent of supercomputer technologies - if a new method cannot be "vectorized" and/or "parallelized", it faces a cool reception these days - and, of these, the Obara-Saika-Schlegel (OS) [53, 54], Head-Gordon-Pople (HGP) [55] and PRISM [61] algorithms are the most significant. 

The different scaling behaviour of the classical and nonclassical two-electron integrals has important ramifications. Thus, whereas the nonclassical integrals must be evaluated by the standard techniques such the McMurchie-Davidson, Obara-Saika and Rys schemes, the classical integrals may be evaluated, to an accuracy of 10 , more simply by the multipole method developed in Section 9.13. Moreover, to calculate the total Coulomb contribution to the Fock operator or to the energy in large systems, there is no need to evaluate the individual integrals explicitly. Rather, as discussed in Section 9.14, their contribution may be calculated much more efficiently by the fast multipole method, at a cost that scales only linearly with the size of the system. 

The methods of calculation of the one electron integrals on the atomic basis set, appearing in Eq. (2) have been greatly improved since 1969. The most recent method which we know has applications to the calculation of MEP values is by Obara and Saika [43] this paper also reports a concise discussion of preceding methods. 

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