Gaussian functions, electronic structure calculation

There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schrodinger equation) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals have die functional form... 

The electronic structure calculations were carried out using the hybrid density functional method B3LYP [15] as implemented in the GAUSSIAN-94 package [16], in conjunction with the Stevens-Basch-Krauss (SBK) [17] effective core potential (ECP) (a relativistic ECP for Zr atom) and the standard 4-31G, CEP-31 and (8s8p6d/4s4p3d) basis sets for the H, (C, P and N), and Zr atoms, respectively. 

The Gaussian orbitals are very important in practical applications. In spite of their wrong asymptotic behaviour at both r —> 0 and r —> °o, nearly all molecular electronic structure calculation programs have been constructed using Gaussian sets of one-electron functions. In this example the Gaussian basis has been selected as... 

Almost all contemporary ab initio molecular electronic structure calculations employ basis sets of Gaussian-type functions in a pragmatic approach in which no error bounds are determined but the accuracy of a calculation is assessed by comparison with quantities derived from experiment[l] [2]. In this quasi-empirical[3] approach each basis set is calibrated [4] for the treatment of a particular range of atoms, for a particular range of properties, and for a particular range of methods. Molecular basis sets are almost invariably constructed from atomic basis sets. In 1960, Nesbet[5] pointed out that molecular basis sets containing only basis sets necessary to reach to atomic Hartree-Fock limit, the isotropic basis set, cannot possibly account for polarization in molecular interactions. Two approaches to the problem of constructing molecular basis sets can be identified ... 

For the two-electron integrals, the four basis functions may be located on 1, 2, 3 or 4 different aTotmc ceiiteis. has already—been mentioned that exponential type basis functions (x exp(—ar)) fundamentally are better suited for electronic structure calculations, however, it turns out that the calculation of especially 3- and 4-centre two-elpc TQn integrals is very rime-consuming for exponential functions. Gaussian functions... 

Gaussian functions are appropriate functions for electronic structure calculations not only because of the widely recognized fact that they lead to molecular integrals which can be evaluated efficiently and accurately but also because such functions do not introduce a cusp into the approximation for the wave function at a physically inappropriate point when off-nuclei functions are employed. Furthermore, Gaussian functions are suitable for the description of wave functions in the vicinity of nuclei once the point nucleus model is abandoned in favour of a more realistic finite nucleus model. 

Although the first proposals(l) (2) to use Gaussian basis functions in molecular electronic structure calculations were made in the late 1940s and early 1950s, in a recent historical review Shavitt(3) records how the first applications met with very limited success because they used rather small numbers of Gaussians, and little additional work was reported in the fifties but then interest in Gaussian basis sets and exploration of variants and extensions increased substantially in the early sixties . In 1960, Nesbet(4) identified two approaches to the design of molecular basis sets ... 

Gaussian basis functions in molecular electronic structure calculations was first suggested by McWeeny115,116 and independently by Boys117 in 1950. However, the supposedly more physical Slater (exponential) Type Orbitals (STO) remained the basis function of choice for many years because they correctly describe not only the cusp associated with the nucleus on which they are centred but also afford a suitable representation of the long range behaviour. Shavitt118 records that... 

Gaussian-type functions are the most widely used basis functions in molecular electronic structure calculations. Two forms of Gaussian-type functions are in common usage. Cartesian Gaussian-type functions have the form... 

The above equations are sufficient to evaluate the gradients of all integrals that are met in electronic structure calculations. In actual practice, however, many tricks are used to reduce the computational work. The evaluation of the gradients when Slater-type functions are used (see Sections 3.5 and 3.6) is more difficult, but could proceed along the same lines as for Gaussian functions, at least for the two-center integrals that usually appear in semiempirical models. 

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