Hermite-Gaussian functions

We shall also require the Hermite Gaussian functions, defined by... 

The normal modes associated with a square-law medium, using the first two terms of Eq. (35), are known to be Hermite-Gaussian functions, as shown in Fig. 18. 

The analytical formulas for the integrals of generalized Hermite Gaussian functions were presented by Katriel [33] and Katriel and Adam [34]. They also investigated the effects of these basis functions for H2 and He test systems. A serious drawback of their approach is its coordinate dependence. 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

A different expansion relies on using Gram-Charlier polynomials, which are the products of Hermite polynomials and a Gaussian function [41] These polynomials are particularly suitable for describing near-Gaussian functions. Even and odd terms of the expansion describe symmetric and asymmetric deformations of the Gaussian, respectively. To ensure that P0(AU) remains positive for all values of AU, we take... 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

Herman-Kluk method, direct molecular dynamics, Gaussian wavepacket propagation, 380—381 Hermite basis functions ... 

A. Koster, Hermite gaussian auxiliary functions for the variational fitting of the Coulomb potential in density functional methods. J. Chem. Phys. 118, 9943-9951 (2003)... 

This is the important property of the Gaussian function, namely that its transform has the identical functional form as the original. This is called the property of self-reciprocity. All its derivatives also share the same property. The derivatives of the Gaussian function produce the well known Hermite functions. 

The computer codes of Sambe and Felton (SF) and Dunlap et al. (DCS) are based on the choice of a Hermite-Gaussian expansion set. Applying the variational theorem with the trial function of Eq. (37) and the LSD Hamiltonian of Eq. (36) leads to the usual matrix pseudo-eigenvalue problem ... 

Integrals involving the exchange-correlation potential r,c or the exchange-correlation energy density cannot be evaluated analytically so that further sets of auxiliary functions are introduced. (In practice and 6,<, behave similarly so that a common set is used to fit both functions.) The exchange-correlation basis (XCB) also consists of Hermite Gaussians... 

Such a possible feature can be found, as an example, within a typical set of solutions of the Schrodinger equation. The harmonic oscillator provides an obvious particular case of such an EH space. It is well known that harmonic oscillator solutions constitute the set of Hermite polynomials [73], weighted by a gaussian function [65]. These polynomials can be considered related to the GTO basis functions most widely used in contemporary Quantum Chemistry. First derivatives of Hermite polynomials are always well defined, producing another polynomial of the same kind. 

Together with the Rodrigues formula (see e.g. Abramowitz and Stegun [1]) for Hermite polynomials the derivatives of the Gaussian function are given by... 

---