S-type Gaussian

The ASA density for this atom is expressed in terms of the s-type Gaussians as... 

The aim of this paper is ascertain whether it is possible to determine the ground state second-order correlation energy of the hydrogen molecule to sub-millihartree accuracy using a basis set containing only s-type Gaussian functions with exponents and distribution determined by an empirical, but physically motivated, procedure. 

The Hartree-Fock limit for the ground state of the hydrogen molecule can be achieved in principle by using a basis set of s-type Gaussian functions distributed along the internuclear axis. Such a basis set will be termed an axis basis set. The atom-centred basis functions, designated (ac), are located on the points... 

However, it should be emphasized that when these recursions are applied to a set of s-type Gaussian fimctions centred on some point, they generate a complete set of fimctions of s-symmetry, which should be supplemented by complete sets of functions of higher symmetry centred on the same point and/or complete sets of functions centred on other points. 

H Fourier space Flartree-Fock (FS-RFIF) with using distributed basis set of s-type Gaussian function (DSGF)... 

I. Flamant et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF. The method briefly is to use RHF-Bloch states (p (k,r), which are doubly occupied up to the Fermi energy Ep and orthonormalized. k and n the wave number and the band index, respectively. 

In this present calculation, therefore, the two extra s-type Gaussian primitives are added to simulate the effect of polarization by constructing sp-hybrid functions [0a( ) and < (f)] at each atomic site, with for the bond length R... 

Another, even more drastic approximation was proposed by Wilhite and Euwema. In this method, the entire charge distribution function (which is the product of two contracted Gaussians) is replaced by a few s-type Gaussian functions. To minimize the error, coefficients and exponents of the replacement functions are calculated by equating the magnitudes of several multipole moments of the original and approximate charge distributions. 

In most applications, this number is set equal to the atomic number of the element involved, Z. In some applications, is set to the number of electrons attributed to the atom according to the results of some type of population analysis. The isolated atom electron densities are optimal linear combinations of s-type Gaussians. That is. 

The tVi are the expansion coefficients for the M s-type Gaussians, and we can see immediately the link between Eq. [42] and the wave function quadrature. So, for the calculation of ASA-based promolecular electron densities, we first need to develop a scheme for the fitting of the atomic densities. The exponents of the Gaussians may be chosen from, e.g., a well-tempered series.The coefficients may then be fitted against the true atomic ab initio electron density. Once these exponents and coefficients are set, these Gaussian exponents and coefficients are universally applicable. Promolecular densities p (r) can then be obtained quickly from Eq. [41]. 

This constraint automatically makes any promolecular density for some molecule, calculated with Eq. [41], fulfill the normalization constraint for that molecule. Such a constraint is most easily handled by a Lagrange multiplier. In ASA, normalized s-type Gaussians are applied, so the following constraint is introduced ... 

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