Gaussian-type function contracted

In this equation, the original Gaussian-type functions are called primitive functions to distinguish them from the contracted ones. The primitive functions are specified only by the orbital exponent, a p, contracted coefficient, c p, and coordinate vector of the center of the function, Ra. By using these contracted Gaussian-type functions as basis functions, it was expected that highly accurate results could be obtained with far fewer basis functions. 

Minimal basis functions (e.g. STO-LG) contain only minimal required contracted functions for each atom. For example, since electrons occupy atomic orbitals up to the 2p orbital in the carbon atom, five contracted Gaussian-type functions corresponding to Is, 2s, 2p, 2py, and 2p orbitals are necessary at the minimum. The minimal basis functions approximating Slater-type orbitals (STO) corresponding to atomic orbitals with L prinoitive functions are called STO-LG basis functions. 

The efficiency of sets of Gaussian-type functions can be increased, particularly in the parametrization of core molecular orbitals, by employing contracted Gaussian-type functions instead of the primitive Gaussian-type functions discussed in the preceding subsectioa Contracted Gaussian-type functions are defined as... 

H impurity therefore contained four contracted Gaussian-type functions. At this point it should be emphasized that the purpose of this study is not to obtain quantitative numbers for the Li-H interaction where, of course, correlation becomes important and a much better basis set had to be used, but to investigate the effect of an impurity in an infinite system going beyond the simple, non-self-consistent ti t-binding approximation with one orbital per unit cell. However, it should also be pointed out that the difference in binding energies between LiH and L12 in the aforementioned basis and geometry has been found to be approximately 1.1 eV [1.12 eV-( — 0.005) eV] while the Hartree-Fock limit is approximately 1.3 eV (1.5 eV-0.2 eV) and the correlated value is also about 1.3 eV (2.3eV-1.0eV).< >... 

As remarked in the main text, basis AOs employed by a host ESS seldom have realistic resemblance to the physical orbitals of an atomic system. We may illustrate the differences by considering the fluorine Is, 2s, and 3s atomic orbitals previously exhibited in Fig. 2.4. For the standard basis set (6-311-1—l-G ) of contracted Gaussian-type functions that were employed to construct the atomic wavefimction, we consider the corresponding lowest three s-type basis AOs (often labeled Is, 2s, and 3s in ESS program output). 

The molecular electron density function needed for EP calculation can be obtained through ab initio as well as various semi-empirical methods. Since ab initio calculations are not economical for large molecules (several hundred atoms), the use of well-parameterized semi-empirical methods are still justified. When semi-empirical methods are used the three-center potential integrals usually disappear, and therefore the electronic contribution can be easily calculated by Slater-type orbitals. In ab initio methods (primitive or contracted) Gaussian-type orbitals are used for calculating the three-center integrals because their calculations are clumsy with Slater-type orbitals. 

In our four-component molecular approach, thus, we use spin-coupled, kinetically balanced, generally contracted Gaussian-type spinors (GTSs) as basis functions. The basis expansion is... 

A comprehensive study of a variety of contracted Gaussian-type basis sets was made by Archibald and Sabin [30]. They used a (7s,3p) set of dementi [31] and (9s,5p) and ( ls,7p) sets of Huzinaga [32], in each case varying the way in which the functions are contracted. Choosing an (85,5p) basis as a com-... 

Different from the contracted Gaussian-type basis functions mentioned later, primitive functions usually have a standardized form to save the effort of developing a different computational program for each basis function. The primitive functions corresponding to s, p, d, and f atomic orbitals are represented analogically to the... 

For contracted Gaussian-type basis functions, various types of functions have been suggested. The following are several major Gaussian-type basis functions (for further details, see, e.g., (Jensen 2006)) ... 

Fig. 3. Orbital amplitude plots for a set of even-tempered Is Gaussian-type functions. The function shown in the top left-hand comer is the most diffuse, that is it has the smallest exponent, whilst that shown in the bottom right-hand comer is the most contracted and has the largest exponent...

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