Laguerre basis

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

The functions S(m are tesseral (i.e., real combinations of spherical) harmonics, Lf are Laguerre functions, and T(a) are gamma functions (Powell and Craseman, 1961) k is restricted to 0 k n and it must have the same parity as n. The constant A, in the case of a finite basis, can be used to optimize this basis. The matrix elements required in this basis can be easily computed from Eq. (14) and the relation... 

Figure 3 shows the eigenvalues for this (s-wave) problem In a basis of 50 Laguerre functions of the form... 

In this section we shall examine the competition between singularity and localisation effects in Rayleigh-Ritz variational calculations performed by John Loeser and Dudley Herschbach [22] on heliumlike ions for a wide range of D and Z, using a Pekeris-type basis of products of generalised Laguerre functions... 

Typically the Schulz-Flory weight function (Eq. 10.41) is used and discrete Laguerre polynomials (Eqs. 10.42, 10.43) are used as basis functions. 

Trial functions such as the Laguerre functions (4.4.6) may also be interpreted as arising from Gram-Schmidt orthogonalization of the basis set x" using the inner product... 

Obviously any basis set method is heavily reliant on the choice of appropriate expansion functions. Conventional vibrational basis set have usually been constructed from products of one-dimensional expansions of orthogonal polynomials. In particular Hermite or associated Laguerre... 

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

While expanding in a basis of orthogonal functions is fairly easily understood, care must be taken in choosing an appropriate set of basis functions. In this case the symmetry of the basis functions chosen must match that of the problem, as seen below. The importance of symmetry in the problem is beautifully presented by the choice of basis for the radial coordinate. Consider two choices of the basis for the radial coordinate - a Fourier basis and a Laguerre basis. That is, the radial functions can be expanded in a basis of Fourier functions (sines and cosines) or Laguerre functions. The collocation points can be loosely thought of as the nodes of the basis functions. 

The Laguerre basis suits the symmetry of the problem, because its collocation points (think of nodes of the Laguerre functions) lie between 0 and oo, and the collocation grid spacing is nonuniform. Of course, we have prior intuition that this basis is better suited because the analytic solutions of the hydrogen atom Schrodinger equation are Laguerre polynomials. 

Comparison botweon Mappad Fourier and Laguerre basis... 

Figure 4. Comparison between the use of a Laguerre basis and a mapped-Fourier basis for the calculation of energy eigenvectors and eigenvalues of the hydrogen atom.

The Laguerre functions (6.5.16) form a complete orthonormal set of functions that combine a fixed-exponent exponential decay with a polynomial in r. They therefore arise quite naturally when searching for a suitable set of one-electron basis functions for atomic applications. Because of their obvious similarity with the hydrogenic wave functions, we shall use for the Laguerre functions the same notation as for the hydrogenic functions, referring, for example, to X2m ... 

To illustrate the convergence properties of the Laguerre functions, we shall use these functions to expand the numerical Hartree—Fock orbitals of the ground state of the carbon atom. However, to see how such expansions are obtained, we shall first consider in general terms the expansion of a function /(x) in a set of basis functions Thus, we wish to determine an expansion in... 

Fig. 63. Lcast-squarcs expansions of the l.v. 2s and 2p orbitals of the carbon P ground stale in Laguerre functions with = 1 (atomic units). The radial distribution function.s ol the carbon orbitals are depicted using thick grey lines. The dashed, doited and full thin lines correspond to Laguerre expansions containing 2, 8 and 15 terms, respectively. Also plotted are the errors in the expansions against the number of basis functions employed.

The conclusions we may draw from this example are quite clear. We cannot hope to expand all the orbitals of an atomic system accurately in a small number of Laguerre functions with a single, fixed exponent. Instead, we must take a more pragmatic approach and introduce functions specifically designed to reproduce as closely as possible the different orbitals of each atom. In this way, we may hope to reproduce the electron distribution of each atom with a small number of terms. Indeed, our example suggests that it may be possible to obtain a crude but qualitatively correct description with no more than one basis function for each occupied AO - if we use basis functions with variable exponents. 

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