Boys function

This is a process that increases as the cube of the basis set size, and the optimization of the (Q )Boys function is therefore an MLis method. The Edmiston-Ruedenberg... 

The character s death is signified as a random victim of state-sanctioned terror. Significantly, whilst the Boy demonstrates practical concern and help towards Lear in life, it is in death that the Boy embodies a deeper reality. The ghost of the Gravedigger s Boy functions as a kind of Fool to Bond s Lear as was the Fool to Shakespeare s traumatised former king. In King Lear the Fool disappears without account or return. In Lear there is no cathartic transcending of... 

The integral appearing on the right-hand side of (9.7.13) is of central importance in the calculation of Coulomb-interaction integrals over Gaussian distributions. It belongs to a class of functions that we shall refer to as the Boys function... 

Since we have expressed the potential ftom the first chaige distribution in terms of the Boys function (9.7.15), this interaction becomes... 

In Section 9.7, we found that the electrostatics of spherical Gaussian charge distributions are determined by the Boys function F ix). Since this function plays such an important role in the evaluation of molecular integrals over Gaussian functions, we shall in this section discuss its properties and evaluation in more detail. 

We now consider the evaluation of the Boys function. Since this function is at the centre of integral evaluation, it is important that it is calculated efficiently. Different methods of evaluating the Boys function have been suggested in the literature [11]. Some of these approaches are discussed in the present subsection. 

These two approximations - (9.8.9) for large values of jc and (9.8.11) truncated at = 6 for small values - are illustrated for Fo(x) in Figure 9.6. Although we have a reasonable approximation to the Boys function by these two methods, they are inadequate for use in integral calculations, in which we would like errors of order 10" ° or smaller. Using the above two approximations, this accuracy is attained only in the regions x < 0.18 and x > 19.35. More accurate methods are therefore needed in practice. 

Fig. 9.6. The Boys function Fo(x) (thick grey line) as approximated by the long-range formula (9.8.9) for large x (dashed line) and by a sixth-order Taylw expansion (9.8.11) around jc = 0 for small x (dotted line).

A practicable alternative is to pretabulate the function at regular intervals x, for small arguments. During the calculation of the integrals, the Boys function at jc is then expanded around the nearest tabulated point x, = x — Ax ... 

The Boys functions of different orders are related by recursion. Integrating the Boys function by parts, we obtain for upward recursion (see Exercise 9.3)... 

We therefore only need to calculate the Boys function for the highest or the lowest order needed, obtaining the others by downward or upward recursion. For small x, upward recursion is unstable since it involves the difference between two almost equal numbers, making downward recursion the preferred strategy. 

The Boys function is related to the incomplete gamma function, one of the special functions of mathematical physics. From Section 6.5.1, we recall that the Euler gamma function is defined as... 

To establish the relation of the Boys function to the incomplete gamma function, we substitute... 

From this expression, we note the following close relationship between the Boys function and the incomplete gamma function ... 

Indeed, in electronic-structure theory, F (x) is often loosely referred to as the incomplete gamma function. In order to avoid any confusion with the special functions P n,x) and y(n,x), we have here adopted the term the Boys fimction for F (x). From (9.8.20), we may interpret the Boys function as the function V(n + weighted by the incomplete gamma function... 

We are here interested in the relationship of the Kummer function to the Boys function. From (9.8.35) we obtain, using (9.8.37),... 

Comparing with (9.8.11), we find that the Boys function is a (renormalized) special case of the Kummer function ... 

Therefore, the standard relations that have been established for the Kummer function hold for the Boys function as well. For example, for the Boys function, the general relations [9]... 

The integrals appearing in these expressions are the Coulomb potential and interaction integrals for spherical charge distributions, introduced in Section 9.7.1. Equations (9.7.23) and (9.7.24) may now be used to replace these integrals by the Boys function. Taking the normalization of the charge distributions (9.7.1) and (9.7.2) into account, we obtain... 

These derivatives may be further simplified since the Boys function depends only on the relative separation of the two centres, giving the following major simplifications ... 

These simplifications occur since we use Hermite (rather than Cartesian) Gaussians to describe the nonspherical distributions. Since the derivatives of the Boys function play such an important role, we introduce the Hermite Coulomb integrals... 

Hence, to calculate Coulomb integrals over Hermite Gaussians, we simply take the derivatives of the Boys function. We also see that field and field-gradient integrals may be calculated in the same way as Coulomb-potential integrals. It remains, however, to develop a method for calculating the derivatives of the Boys function. 

To develop a scheme for evaluating the Hermite Coulomb integrals (9.9.9), we note that the first derivative involves the first-order Boys function... 

In general, thaefore, higher derivatives are linear combinations of Boys functions of different orders and our task is now to develop a recursive scheme by which the Hermite integrals R,uv for... 

Using (9.9.14) and differentiating the nth order Boys function, we find... 

In this way, all Hermite Gaussians of order f -f- -h u < may be calculated from the Boys functions of order n < by recursion. The same recurrences may be used to evaluate the integrals (9.9.11) except that Xpq replaces Xpc and so on. 

We must also consider the cost of generating the expansion coefficients and the Hermite integrals in (9.9.40). The number of Hermite integrals scales as L p and their evaluation scales as L p. The L p dependence for the constmction arises from the use of the auxiliary four-index integrals R iuv where each index scales as L. The evaluation of the Boys function scales only as Lp. Finally, the number of expansion coefficients E / scales as L p and their evaluation also since no... 

Boys functions Hermite integrals Expansion coefficients Lrp V L p L p ... 

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