The Laguerre functions

In an attempt to retain the exponential form of the hydrogenic wave function while avoiding the problems associated with the use of continuum states, we are led to the following class of functions  

Section 6.5.1. The functions and more generally are known as the Laguerre functions, a term that is also used for the generalization of L (x) to nonintegral complex indices n and or. 

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  

The essential difference between the hydrogenic and Laguerre functions is the absence of the inverse quantum number in the latter functions. The Laguerre functions are therefore considerably more compact than the hydrogenic ones for large n, as seen from the average value of r (see Exercise 6.5) 

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Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

Another example, of trivial nature, is the Stark effect, where inspection shows that there are only two nonvanishing terms remaining in (38), to yield of course the exact result, though i Ijas been thought that the set (21) involving the Laguerre functions... 

Moreover, with the help of the Laguerre functions of Table 3.1, the full normalized expression of the associate radial wave functions may be analytically formulated yet, the physical significance will be carried by the squared quantity in radial normalization framework, as mean-... 

These are the Laguerre functions of order 1/2 with eigenvalues... 

This equation does not depend on the m quantum number since the single electron is assumed to move in a central field. The radial functions are thus the same for orbitals, which are different only in m. A number of known sets of functions satisfy the mentioned boundary conditions. The Laguerre functions are known to be solutions of Equation 2.13. R(0) has to be finite at the nucleus and R(r) 0 exponentially when r oo, and this is, in fact, satisfied by Laguerre functions that are of the following type ... 

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... 

This chapter introduces a method for building Laplace transfer function models from noisy step response data. The algorithm is based on the Laguerre functions and exploits their orthonormal properties to produce a simple, yet effective approach. 

The parameter p is called the time scaling factor for the Laguerre functions. This parameter plays an important role in their practical application and will be discussed in detail in Section 2.3. (Note The set of Laguerre functions presented in Equations (2.4) differs by a factor of —1 for even values of i when compared with the set of Laguerre functions presented by Lee (1960). However, this does not affect the orthonormal properties of these functions.)... 

In parallel with the above time domain description, an approximation of the process transfer function using the Laguerre functions can also be developed. The Laplace transform of the impulse response h t) in Equation (2.5) leads to the continuous-time transfer function of the process... 

It is important to be able to efficiently generate values for the Laguerre functions. There are several ways to do so and each way requires a different amount of computational effort. 

Method A. For low model orders, the Laguerre functions can be generated using Equations (2.4) directly. 

We can now generate the Laguerre functions in Equations (2.4) by setting X = 2pt in the Laguerre polynomials... 

By examining Equation (2.16), we find that the Laguerre functions in Equar tions (2.4) satisfy the following set of differential equations... 

Using the orthonormal properties of the Laguerre functions, Equation (2.30) is equivalent to... 

Lemma 2.1 For some p > 0, the Laplace transforms of the Laguerre functions given in Equations (2.10) satisfy the following equality... 

An equivalent result to Equation (2.85) can be obtained by taking the Laplace transform of the derivatives of the Laguerre functions. It can be readily shown that, from the state space representation of the Laguerre network in Equation (2.19)... 

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. 

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... 

Use of the principal quantum number n for the HO system emphasizes the close relationship of the HO functions (6.6.3) with the Laguerre functions (6.5.17) but complicates the expression for the... 

As for the Laguerre functions in Section 6.5.4, we may adjust the exponents to improve the convergence. With an exponent of 11, the Is HO expansion yields an error less than 0.1 with 3 terms included and less than 0.01 with 21 terms. Long expansions are needed for the 2s and 2p functions as well. Again, the HO functions converge much more slowly than the corresponding Laguerre functions, even with optimal exponents. 

We here examine the radial extent of the hydrogenic radial functions R i and the Laguerre functions In terms of the associated Laguerre polynomials L (x), these functions may be expressed as... 

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