Gaussian radial functions

The discussion in this section follows closely that for the exponential orbitals in Section 6.5. First, a complete set of orthonormal functions with Gaussian radial forms and a fixed exponent a is introduced. From these orbitals, we next arrive at a more flexible and useful set of nonorthogonal basis functions by simplifying the polynomial part and by introducing variable exponents. As for the exponential functions in Section 6.5, the performance of the Gaussian-based functions is illustrated by carrying out simple expansions of the ground-state orbitals of the carbon atom. We 

We consider an electron in a spAiraically symmetric harmonic potential [9]. In polar coordinates, the Hamiltonian of such a three-dimensional isotropic harmonic oscillator (HO) is given by 

The energy levels of the isotropic HO depend on the quantum numbers n and I in the following manner  

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 

according to (6.6.5), R fir) and have the same principal quantum number 

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This equation is formally similar to equation 3.41, and the problem is still how to find the most appropriate coefficients for the linear combinations, those that will produce a wavefunction that correctly reproduces total energies, ionization potentials, and spacings between electronic levels. The problem for isolated atoms is only formally different from the problem for molecules, because in both cases the calculation is eventually carried out by choosing a number of primitive gaussian radial functions, by mixing them in the appropriate Slater determinant, and by finding the best coefficients for the combination into the final atomic or molecular orbitals (the orbitals of an isolated atom are the molecular orbitals for a one-atom molecule) [4]. But do not confuse basis orbitals and complete orbitals for example, the 3s atomic orbital of a chlorine atom may be a combination of Is, 2s, and 3s gaussians. 

In mathematics there is a large number of complete sets of one-particle functions given, and many of those may be convenient for physical applications. With the development of the modern electronic computers, there has been a trend to use such sets as render particularly simple matrix elements HKL of the energy, and the accuracy desired has then been obtained by choosing the truncated set larger and larger. Here we would like to mention the use of Gaussian wave functions (Boys 1950, Meckler 1953) and the use of the exponential radial set (Boys 1955), i.e., respectively... 

L. Kiernan, J.D. Mason and K. Warwick, Robust initialisation of Gaussian radial basis function networks using partitioned k-means clustering. Electron. Lett., 32 (1996) 671-672. 

Of the several approaches that draw upon this general description, radial basis function networks (RBFNs) (Leonard and Kramer, 1991) are probably the best-known. RBFNs are similar in architecture to back propagation networks (BPNs) in that they consist of an input layer, a single hidden layer, and an output layer. The hidden layer makes use of Gaussian basis functions that result in inputs projected on a hypersphere instead of a hyperplane. RBFNs therefore generate spherical clusters in the input data space, as illustrated in Fig. 12. These clusters are generally referred to as receptive fields. 

Other types of radial functions have been applied, including Gaussian-type functions (Stewart 1980), and harmonic oscillator wave functions (Kurki-Suonio 1977b). 

A hydrogen atom is described by the Gaussian radial density function... 

A final point about basis functions concerns the way in which their radial parts are represented mathematically. The AOs, obtained from solutions of the Schrbdin-ger equation for one-electron atoms, fall-off exponentially with distance. Unfoitu-nately, if exponentials are used as basis functions, computing the integrals that are required for obtaining electron repulsion energies between electrons is mathematically very cumbersome. Perhaps the most important software development in wave function based calculations came from the realization by Frank Boys that it would be much easier and faster to compute electron repulsion integrals if Gaussian-type functions, rather than exponential functions, were used to represent AOs. 

Figure 2.2. Radial dependence of basis functions a) correct exponential decay (STO) (b) primitive Gaussian-type function (solid line) vs. an STO (dotted line) (c) least-squares expansion of the STO in terms of three Gaussian-type orbitals (STO-3G).

The basis set may be constructed from radial functions of either Slater (exponential) or gaussian form, multiplied by appropriate angular (0 and functions. This scheme is frequently referred to by the acronym LCAO-MO -SCF. 

Figure 4.3 Output from a Gaussian radial basis function for a single input value x.

One other network that has been used with supervised learning is the radial basis function (RBF) network.f Radial functions are relatively simple in form, and by definition must increase (or decrease) monotonically with the distance from a certain reference point. Gaussian functions are one example of radial functions. In a RBF network, the inputs are fed to a layer of RBFs, which in turn are weighted to produce an output from the network. If the RBFs are allowed to move or to change size, or if there is more than one hidden layer, then the RBF network is non-linear. An RBF network is shown schematically for the case of n inputs and m basis functions in Fig. 3. The generalized regression neural network, a special case of the RBF network, has been used infrequently especially in understanding in vitro-in vivo correlations. 

In order to overcome these problems, the core electrons are often excluded from the calculation (frozen-core approximation), and their effect on the valence electrons is parameterized in the form of a pseudo potential based on a relativistic atomic calculation [12]. In connection with GTO basis sets, the most common form of pseudo potential is the effective core potential (ECP) using Gaussian-type radial functions to describe the potential [13-16]. 

In nonlinearly separable cases, SVM maps the vectors into a higher dimensional feature space using a kernel function K(xh x ). The Gaussian radial basis... 

Provided the inverse of ( ) exists, the solution w of the interpolation problem can be explicitly calculated, and has the form w = < )-1 y. The most popular and widely used radial basis function is the Gaussian basis function... 

Quiney 1988). The small component s radial function has been fixed according to the kinetic balance condition (Stanton and Havriliak 1984), which has its origin in the coupled nature of Dirac s first-order differential equations and is introduced to keep the method variationally stable. The index A denotes the coordinates of the nucleus s centre RA of atom A, to which the basis function is attached, i.e. rA = r — RA. As an alternative, Cartesian Gaussians,... 

Unlike the basic Slater function the basic Gaussian function provides a very deficient approximation to a Is atomic radial function. Whereas the Slater function has a nonzero derivative at the nucleus and so can model the cusp of the radial function, the Gaussian function exhibits a zero derivative at the origin [compare the derivatives of... 

Table 1.5 The Gaussian basis sets proposed by Reeves to represent the hydrogenic radial functions. The table entries, for each basis set, are the exponents, a, of the primitive Gaussians and then in the second columns the complete normalized coefficients, d, of the linear combinations.

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