Spherical Gaussian basis functions

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. 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Spherical harmonics, Cartesion Gaussian basis functions, 261... 

An interesting approach for comparison with XPS experiments is a generalization of the Floating Spherical Gaussian Orbital (FSGO) technique. In this method, each electron pair is represented by a single Gaussian basis function whose exponent and position are obtained by a variational procedure or by reference... 

The use of a separable potential of the form of Equation 8 in Equation 6 to obtain solutions of the form of Equation 9 can be shown to be equivalent to using the functions ct (r) in the Schwinger variational principle for collisions (13). At this stage the functions ai(r) can be chosen to be entirely discrete basis functions such as Cartesian Gaussian (lA) or spherical Gaussian (15) functions. We note that with discrete basis functions alone the approximate solution satisfies the scattering boundary condition. Such basis... 

Several common basis sets are built in GAUSSIAN 90, which is capable of handling both Cartesian (such as 6d) and spherical (such as 5d) Gaussian basis functions. Molecular geometries can be input in the form of Cartesian coordinates or the Z-matrix. Geometry optimization to both minima and transition states is possible. The HF analytical energy second derivatives needed for the vibrational frequencies calculations can be computed with either a standard or a direct CPHF program. 

The value of the ECP depends not only on an electron s coordinates, but also on the projection of the wave function of one electron (holding the other electron coordinates constant) onto the spherical harmonics, T/m. The local term, WL+i(r), depends only on the distance of the electron from the nucleus. The angular potentials wi(r) are determined so that, beyond some cutoff distance, the pseudo-orbitals obtained from an ECP calculation match those of an all-electron calculation, but are nodeless and smoothly go to zero within the cutoff radius [133], The w are then fit to a Gaussian expansion [134] so that the potential can be rapidly integrated over Gaussian basis functions. 

Use of these Cartesian Gaussian-type functions leads, for example, to six components of d symmetry instead of the true five components. This can be shown to be equivalent to the addition of a 3s function to the basis set and can lead to numerical problems associated with near-linear dependence if the s basis set is sufficiently large. The use of spherical Gaussian-type functions, which are often defined as... 

As an alternative to the spherical Gaussian basis sets introduced so far, Cartesian Gaussian functions. 

Some applications of perturbation theory to molecular problems would benefit from the simplicity of an extended floating spherical Gaussian basis. Adamowicz and Bartlett ° have developed a procedure for projecting large conventional basis set wave functions onto a floating spherical Gaussian basis. 

Normally, a spherical Gaussian basis set is made up of functions with minimum n-value, that is, Is, 2p, 3d, — We now get two different cases depending on whether k is positive or negative ... 

The quantities r Yij 9,) and Yi,m(9,)/r + appearing in equations (67) and (68) are known as solid spherical harmonics. Because of their central role in multipole approximations, it is important to have optimized procedures for their generation and manipulation. An equivalent and efficient reformulation of the solid spherical harmonics has been published recently, with the interesting property of having very simple derivatives with respect to Cartesian coordinates, which are required for the computation of forces, or for obtaining useful recurrence relations for the integrals of solid spherical harmonic with Gaussian basis functions. 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

These permutations on coordinates are equivalent to operations on the basis functions. We will use shifted spherical Gaussians for this example (these functions will be discussed in a detailed way below in this chapter) ... 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

An alternative solution to the gauge-independence problem in molecular calculations is to attach the complex phase factors directly to the atomic basis functions or atomic orbitals (AOs) rather than to the MOs. Thus, each basis function—which in modern calculations usually corresponds to a Gaussian-type orbital (GTO)—is equipped with a complex phase factor according to Eq. 87. A spherical-harmonic GTO may then be written in the from... 

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