Gaussian type basis functions

Evaluation of the integrals that arise in the calculations was for some time a primary problem in the field. A most important development in this context was the introduction of Gaussian-type basis functions by Boys [32], who showed that all of the integrals in SCF theory could be evaluated analytically if the radial parts of the basis functions were of the form P(x,y, z) exp(-r2). The first ten functions are listed by Hehre, Radom, Schleyer and Pople [33] and we repeat them here ... 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

The use of the Gaussian-type basis functions allows the possibility to evaluate analytically both Coulomb and exchange-correlation energies [28,29]. The energy gradient is written as... 

In the DKS framework, the DKH transformation is based on the one-particle potential V = Vejf In most applications, the potential is restricted to the nuclear potential V c for representations of the wave functions with Gaussian-type basis functions, all pertinent matrix elements can be evaluated analytically [16,18]. In this proximate second-order DKH approach, the electron-electron contributions to the DKS Hamiltonian, Vh and Vxc, are formally considered as a-posteriori add-ons they neither enter the definition of the DK transformation nor are they subjected to the relativistic transformation. We will refer to this model, where only the nuclear attraction field is used in the DKH transformation, as DKnuc. 

The set of all sudi functions with n, /, and m being integers but with having all possible positive values forms a complete set. The parameter is called the orbital exponent. To get a truly accurate representation of the Hartree-Fock orbitals, we would have to include an infinite number of Slater orbitals in the expansions. In practice, one can get very accurate results by using only a few judiciously chosen Slater orbitals. (Another possibility is to use Gaussian-type basis functions see Section 15.4.)... 

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)) ... 

It is worthwhile to note that although only Gaussian-type basis functions have so far been explained, they are not the only option in quantum chemistry caiculations. Actually, Slater-type basis functions, which reproduce the shapes of orbitais more accurately, as mentioned above, are used in, e.g., the Amsterdam Density Functional (ADF) program, in which numerical integral calculations are carried out with Slater-type functions. 

Basis set Slater type and Gaussian type basis functions. 

To solve the Kohn-Sham equation, a spinor basis that is composed of the direct product of the atomic basis functions (usually Gaussian-type basis functions) and the one-electron spin functions, aa), where a) is the atomic basis function and a) = la) and P) are the one-electron spin functions. The molecular orbitals can be expanded as linear combinations of the spinor basis functions,... 

Comparison of self-consistent field calculations using Gaussian-type basis functions with fully numerical studies for atomic systems. ... 

The reason for the introduction of Gaussian-type basis functions is the fact that two-electron integrals can be solved analytically. All relativistic one- and two-electron integrals can be evaluated using standard techniques that have been developed in nonrelativistic quantum chemistry. For a detailed discussion we may therefore refer to the book by Helgaker, Jorgensen and Olsen [284] and include here only a few general comments. 

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