Hermite basis functions

The Hermite basis functions (p, t) have the following form ... 

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

Tab. 1 Selected vibrational eigenstates of the potential energy surface of figure 5. The first twelve are the lowest energy ones. 1 iy> and 1 , /> indicate Hermite basis functions localized on the left and right well, respectively the subscripts refer to the quantum numbers on the r, (i and k) and rj (/ and /) coordinates.

To understand the criteria for basis set choice, then, we need consider only the behavior of the primitive integrals. The primitive integrals over the basis functions can be expressed in terms of Hermite polynomials... 

We also introduce abstract basis functions, / >, whose v-space matrix elements are related to Hermite polynomials H/(v)... 

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. 

Up to now the basis functions Ni x) are still arbitrary and not restricted to a finite element approach. In the finite element frame a suitable approximation is given by Lagrange- or Hermite-interpolation polynomials. 

In order to achieve continuity of the first derivatives of the approximate solution we use hermite cubics [8] and thus, have to consider two types of basis functions. By tensor product we construct d-dimensional basis functions (2 types). Now, we restrict ourselves on the two dimensional case and get four types of hermite bicubic basis functions as shown in Fig.3.1. Furthermore, we choose the hierarchical approach [2, 9]. Then, on every hierarchy level k = ki- -k2 subspaces Ski,k2 re spanned by the basis functions with supports as indicated in Fig.3.2. Here, ki denotes the hierarchy level in direction x,. Notice that one rectangle depicts the above mentioned four different types of basis functions. Now, the usual full grid space Vs is spanned by the whole set of subspaces that are shown in Fig.3.2. However, the sparse grid space Vs is constructed only by the subspaces below the dotted line in Fig.3.2, i.e. Vn is given by the direct sum... 

Fig. 3.1. The four types of hermite bicubic basis functions.

Quantum dynamics simulations of the UV absorption spectrum and of the electronic state population dynamics of the molecule excited by a short laser pulse resonant with the transition to the bright B2u t t ) state, based on the models described in Sect. 5.3 were performed using the MCTDH method in the multi-set formalism (see Sect. 4.2.5 in Chap. 4). For the representation of the Hamiltonian and the wave function, a Hermite polynomial DVR scheme [60] was used for all the degrees of freedom. The number of SPF and primitive basis functions used in the calculations are listed in Table 5.4. Test calculations with both larger primitive and SPF bases have been... 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

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. 

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