Obara-Saika scheme

The evaluation of the ERI in the Obara-Saika scheme can be summarized as follows (see Figure 6) ... 

Figure 6 A schematic outline of the Obara-Saika scheme...

Having discussed the Cartesian Gaussian functions and their overlap distributions, we are now ready to consider the evaluation of the simple one-electron integrals. By simple, we here mean the standard molecular integrals that do not involve the Coulomb interaction. In the present section, we thus discuss the evaluation of overlap integrals and multipole-moment integrals by the Obara-Saika scheme [5], based on the translational invariance of the integrals. We also... 

As the next example of the Obara-Saika scheme for simple one-electron integrals, we consider the integrals over the differential operators... 

Obara-Saika scheme [5] - in which we avoid the intermediate Hermite integrals altogether, obtaining the final Cartesian integrals by recursion directly from the Boys function. 

THE OBARA-SAIKA SCHEME FOR ONE-ELECTRON COULOMB INTEGRALS... 

We have now succeeded in setting up a set of recurrence relations by means of which the two-electron Cartesian integrals may be obtained from the Boys function. The resulting expressions are rather complicated, however, involving as many as eight distinct contributions. Unlike the McMurchie-Davidson scheme, the Obara-Saika scheme does not treat the two electrons separately since the recurrences (9.10.24) and (9.10.25), for example, affect the indices of all four orbitals. In Section 9.10.3, we shall see how the Obara-Saika recurrences may be simplified considerably when used in conjunction with two other types of recurrence relations the electron-transfer recurrences and the horizontal recurrences. 

In the original Obara-Saika scheme, the two-electron integrals are generated from the Boys function by means of rather unwieldy recurrence relations such as (9.10.24). It is possible, however, to break the generation of the two-electron integrals up into several smaller steps, each of which involves a simpler set of recurrences. Consider the following three-step path to the Cartesian integrals. 

In this way, we may build up the full set of Cartesian integrals by the use of three sets of recurrence relations, each of which is considerably simpler than the full Obara-Saika scheme. The horizontal recurrences (9.10.28) and (9.10.29) do not involve the orbital exponents and may therefore be applied cfier the transformation of the integrals to the contracted basis, as will be discussed later. In passing, we note that this procedure may also be used for the one-electron integrals, replacing the first step by the similar recurrence relation... 

From our discussion of the Obara-Saika scheme three techniques emerge, differing in the use of the electron-transfer and the horizontal recurrence relations see Table 9.5. In the OS4 scheme, no use is made of the transfer and horizontal recurrences - the primitive Cartesian integrals... 

Table 9.5 Cost and memray requirements of the two-electron Obara- -Saika scheme ...

In the previous sections, we considered two appoaches to the calculation of Coulomb integrals over Gaussian orbitals - the McMurchie-Davidson scheme and the Obara-Saika scheme - both of which employ Boys functions as intermediates. In the present section, a different approach is... 

In the McMurchie-Davidson and Obara-Saika schemes, the Coulomb integrals are generated as linear combinations of Boys functions f of different orders n. Referring back to the discussion of these schemes, we find that the two-electron integrals may be written in the form... 

In the following, we shall first show how Gaussian quadrature can be simplified for the special case of even polynomials and weight functions such as those in (9.11.3). The orthogonal polynomials needed for the calculation of Coulomb integrals (9.11.3) are then introduced, and finally we show how a Gaussian-quadrature scheme for the evaluation of Coulomb integrals can be developed based on the McMurchie-Davidson and Obara-Saika schemes. 

In the Rys-quadrature scheme presented above, the one- and two-dimensional integrals were calculated using the McMurchie-Davidson scheme. These integrals may also be obtained from the Obara-Saika scheme, as we shall now discuss. 

Comparing with the McMurchie-Davidson scheme in Section 9.9.6 and the Obara-Saika scheme in Section 9.10.4, we note that the cost scales in the same manner for the schemes that employ the electron-transfer relation - the MDl, OSl and R1 schemes, all of which are dominated by the electron-transfer step and by the horizontal step (L ). For high angular momentum,... 

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