Gaussian functions, in basis set

An important issue of the application of electronic structure theory to polyatomic systems is the selection of the appropriate basis set. As usual in quantum chemistry, a compromise between precision and computational cost has to be achieved. It is generally accepted that in order to obtain qualitatively correct theoretical results for valence excited states of polyatomic systems, a Gaussian basis set of at least double-zeta quality with polarization functions on all atoms (or at least on the heavy atoms) is necessary. For a correct description of Rydberg-type excited states, the basis set has to be augmented with additional diffuse Gaussian functions. Such basis sets were used in the calculations discussed below. 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

Molecular Orbital Calculations. The most sophisticated and theoretically rigorous of the molecular orbital methods are ab initio calculations. These are performed with a particular mathematical function describing the shape of the atomic orbitals which combine to produce molecular orbitals. These functions, or basis sets, may be chosen based on a convenient mathematical form, or their ability to reproduce chemical properties. Commonly used basis sets are a compromise between these two extremes, but strict ab initio calculations use only these mathematical functions to describe electronic motion. Representative of ab initio methods is the series of GAUSSIAN programs from Carnegie-Mellon University (11). In general, these calculations are computationally quite intensive, and require a large amount of computer time even for relatively small molecules. 

As said above it is possible to use the same Gaussian-type standard basis sets of ab initio theory for DFT calculations. Concerning the quality of the basis set which is necessary to obtain reliable results, it is advisable to use for Ge at least a split-valence basis set which should be augmented by a d-type polarization function such as 6-31G(d). Better basis sets of triple-zeta quality with more polarization functions up to 6-311G(3df) have been developed for Ge which belong to the standard basis sets in Gaussian 9831. Other basis sets for Ge are available, e.g., from the compilations of Huzinaga et al.32 and Poitier et al.33 and from the work of Aldrichs et al 34. 

All these calculations require a set of atomic orbitals from which MOs can be calculated (the basis set). The earliest to be used were Slater-type orbitals (STOs) but these are mathematically inconvenient, and the STO-3G minimal basis set, which uses gaussian functions to mimic Slater orbitals, is commonly used. More sophisticated gaussian basis sets, which lead to improved accuracy, carry labels such as 6-31G(d) and 6-31++G(dp). Successive increases in basis set size (STO-3G—>3-21G—>3-31G(d)—>6-311G(3df)) give improved bond-length accuracy. 

Two different basis sets were used (1) basis set I was medium size, and (2) split-valence basis set II taken from Huzinaga [24] (9s and 5p gaussian-type orbital contracted into [3s, lp] for the C,N,0 atoms). In basis set II, more flexibility has been allowed to the description of the valence shells by adapting a triple- contraction completed with one p type (for hydrogen) or one d type (for C,N,0) polarization function. Then two ab initio calculations were carried out (1) all atoms were described with basis set I, and (2) basis set II was used for all the atoms but those belonging to the methyl and phenyl substituents due to computer limitations. 

In this contribution we have reviewed the applicability, accuracy and computational efficiency of the local spin density functional approach to the chemistry of transition metal complexes and clusters using a linear combination of Gaussian-type orbital basis set for the calculation of electronic structures, ground state geometries and vibrational properties. 

The Gaussian lobe function method was introduced by Preuss and developed for routine calculations by Whitten. The contraction of Gaussian lobe function (GLF) basis sets is made along the same lines as with Cartesian GTF s. As regards the primitives, the s-type functions are expressed in the usual way. But the primitives of p, d, f,. .. types are expressed as linear combinations of s-Gaussians (lobe functions) placed at different points so as to retain the proper symmetry (see Fig. 2,4 ). Thus, a p-type function on nucleus A may be... 

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