Spherical Gaussian-Type Function

In molecular calculations it is convenient to express G-spinors in terms of Spherical Gaussian-type Functions (SGTF) [107]... 

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... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Double Zeta + Polarization functions Extended Hartree-Fock Electron Spectroscopy for Chemical Analysis Floating Ellipsoidal Gaussian Orbital Floating Spherical Gaussian Orbital Generalized Atomic Effective Potential Gaussian Type Orbital... 

Each component of generally contracted with Gaussian type spherical harmonics functions. Contraction coefficients of the basis sets are determined by four-component atomic calculations [5],... 

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... 

They applied their method to radicals such as HeH, LiH, H4, He2" as a p-type lobe function, constructed from two equivalent spherical Gaussians,... 

Fig. 8.10. Two spherically symmetric GTOs of the "Is-type C r 1,0) (a) These are used to form the difference orbital (b) G(r 1, —0.5/)—G(r 1, +0.5/), where / is the unit vector along the j -axis. For comparison (c), the Gaussian-type p, orbital is shown xG r 1.01. It can be seen that the spherical orbitals may indeed simulate the 2p ones. Similarly, they can model the spatial functions of arbitrary complexity.

Fig. 1 a, b. Positions of spherical Gaussians in molecular fragment FSGO basis sets for (a) the methyl radical, and (b) ethylene. Pairs of spherical Gaussians labelled and are combined as in Eq. (3) to form p -type basis functions... 

Work by Frost and co-workers in the mid 1960s abandoned the idea of AO-based functions to arrive at an even more compact basis set. They suggested the use of s-type Gaussians that were not fixed at the atomic centers, but could float in space so as to optimally represent each localized pair of electrons. Because only one function was needed for each pair of electrons, the basis sets used in floating spherical Gaussian (FSGO) scheme were often referred to as subminimal. Extensions of Frost s idea to ellipsoidal Gaussians of the form... 

---