Hermite integration

These integrals can be solved by Hermite integration in which ... 

Huang, B. Du, X. 2006. A robust design method using variable transformation and Gauss-Hermite integration. Int. J. Numer. Methods Eng. 66 1841-1858. 

A numerical scheme such as the trapezoidal rule or Simpson s rule [109] can be used for the integral over cos0, with Gauss-Hermite integration used to integrate over r. 

First, consider the Hermite integrals with e = 0. From (9.4.12), we know that these Hermite integrals reduce to... 

Together with (9.5.32), this expression allows us to generate recursively the higher-order multipole moments from those of lower orders, noting that all Hermite integrals with f > e are zero ... 

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... 

The obvious way to generate the integrals (9.9.40) is first to calculate the expansion coefficients and Hermite integrals and then to carry out the summation. As the number of Cartesian orbitals in each shell scales as I , the number of Cartesian integrals scales as For each Cartesian integral, we carry out a simultaneous summation over three indices, each of which scales as L, so the total operation count scales as V p. ... 

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 ... 

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. 

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 ... 

Next, we invoke the recurrence relation for the Hermite integrals (9.9.18) ... 

Inserting the definition of the Boys function (9.8.1) in (9.11.27), we may write the Hermite integral in the following manner ... 

To complete the evaluation of the Hermite integrals by Rys quadrature, we must derive a scheme for the evaluation of the modified Hermite polynomials. 

In the two-electron case, we may calculate the Hermite integrals in the same manner... 

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