Cartesian Hermite polynomials

Here is a normalization factor, // ( ) is a Hermite polynomial, and m is a dimensionless positional coordinate obtained by dividing the Cartesian coordinate... 

The connection between the Cartesian Hermite Gaussian and the Hermite polynomials is established by finding the expansion coefficients of... 

The Hermite polynomials of degree n, H x), that appear in the HO functions in Cartesian form fulfil the orthogonality relation [11,12]... 

The three-dimensional HO functions (6.6.24) possess a rather complicated nodal structure. If we retain only the highest-order terms in the Hermite polynomials, we arrive at the following set of nodele.ss Cartesian GTOs ... 

Let us consider the expansion of two-centre Cartesian overlap distributions in Hermite Gaussians. Since the overlap distribution is a polynomial of degree i -p j in xp (9.2.24), it may be expanded exactly in the Hermite polynomials of degree t < i + j. We therefore write... 

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). 

The Hermite Gaussians differ from the Cartesian Gaussians (9.2.3) only in the polynomial factors, which for the Hermite Gaussians are generated by differentiation. 

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