McMurchie-Davidson scheme

The evaluation of relativistic Fock matrix elements depends on an efficient way to evaluate interaction integrals following the McMurchie-Davidson scheme [3]. The key step is the recognition that the product of two Gaussian functions on different nuclear centres A, Ajy is proportional to a third Gaussian on some intermediate centre A jy. It is thus easy to see that the product of two SGTF (10) can be put in the form... 

Alternatively, the McMurchie-Davidson scheme [133] can be used for the computation of the spatial part of the matrix elements of Hpy (see [106]). The central concept of this method is to expand the product of two Gaussians (the so-called overlap distribution) in terms of Hermite functions according to... 

Figure 4 A schematic outline of the McMurchie-Davidson scheme. Note that this scheme omits the transfer equation utilized by the example in the text...

THE McMURCHIE-DAVIDSON SCHEME FOR MULTIPOLE-MOMENT INTEGRALS... 

In the present section, we consider the evaluation of Cartesian Coulomb integrals by the McMurchie-Davidson scheme. FirsL we show how Coulomb integrals over Hermite Gaussians... 

In the McMurchie-Davidson scheme, the one-electron Coulomb integrals may be written in the following manner... 

Having considered the evaluation of the one-electron Coulomb integrals, let us turn our attention to the two-electron repulsion integrals within the McMurchie-Davidson scheme. Clearly, many of the considerations for the one-electron integrals apply also to the two-electron integrals. In addition, some new considerations arise because of the higher complexity of the two-electron integrals. 

Table 9.2 Cost and memory requirements of the vertical-horizontal McMurchie-Davidson scheme for one-electron Coulomb integrals...

The separate treatment of the two electrons in the McMurchie-Davidson scheme may be very useful in some circumstances. Consider, for example, the calculation of the Coulomb contribution to the Fock matrix, which, in the primitive Cartesian basis, may be written in the manner... 

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. 

Comparing the Obara-Saika and McMurchie-Davidson schemes, we find that MD4 scales more favourably than OS4, with a highest cost of L p (or L p ) rather than However,... 

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 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,... 

Obara-Saika or McMurchie-Davidson schemes. For more general charge distributions, representing, for example, a large part of a molecular system, it may be better to carry out the integration numerically, using Gaussian quadrature or some other numerical scheme. 

Using the McMurchie-Davidson scheme, verify that the general expression (9E.1.2) holds for the special cases px Px) and d dfi). 

According to (9.5.41), these integrals may, in the McMurchie-Davidson scheme, be written in the form... 

---