Hermite Coulomb integrals

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... 

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... 

Having obtained the Hermite Coulomb integrals, we are now in a position to calculate the Cartesian Coulomb integrals... 

The overlap and Hamiltonian matrix elements over Hermite Gaussians must then be evaluated. The one-electron operators present no special problems and need not be discussed further. The Coulomb integrals are identical to those in Hartree-Fock theory, i.e. 

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 Section 9.7, we discussed the evaluation of Coulomb integrals over spherical Gaussians. We now go one step further and consider nonspherical electron distributions as described by Hermite Gaussians. The one-electron Coulomb integral can then be expressed as... 

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. 

NexL at each abscissa, we calculate the modihed Hermite polynomials (9.11.30) using the recurrence relations (9.11.36). The resulting polynomial values are then eontraeted with the Hermite-to-Cartesian expansion coefficients, yielding the one-dimensional d artesian integrals (9.11.43)-(9.11.45). The expansion coefficients, which may be obtained from the two-term recurrence relations (9.5.15)-(9.5.17), are the same for all the abscissae. The final Cartesian one-electron Coulomb integral is obtained by carrying out the summation (9.11.42). 

We shall derive the recurrence relations for the Coulomb-potential integrals from the recurrence relations for the expansion coefficients in Section 9.5.1 and for the Hermite integrals in Section 9.9.2. Consider the integral (9.10.2) with i incremented by 1 ... 

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