Basis Functions Slater-Type

Irrespective of whether we use Gaussian functions, Slater type exponentials or numerical sets, certain categories of functions that can help to characterize the quality of a basis set... 

The spatial part of each ip], is expanded in terms of the fundamental set of basis functions (Slater- or Gauss-type functions) gm... 

Basis set Slater type and Gaussian type basis functions. 

By proper choice of the contraction parameters one can thus use basis functions that are approximate atomic Hartree-Fock functions. Slater-type functions, etc., while still evaluating integrals only with primitive Gaussian functions. A procedure that has come into wide use is to fit a Slater-type orbital (STO) to a linear combination of L = 1,2, 3,... primitive Gaussian functions. This is the STO-LG procedure (the procedure is commonly referred to as STO-NG, but since N represents the number of electrons everywhere in this book, we prefer an alternative symbol). In particular, STO-3G basis sets are often used in polyatomic calculations, in preference to evaluating integrals with Slater functions. We will use STO-3G basis sets for our model calculations on H2 and HeH. We need to explicitly consider the form that the contraction (3.212) takes if is to approximate a s Slater-type function. 

Plane Waves and Atomic-like Basis Sets. Slater-type Functions... 

The DKH approach allows relativistic all-electron DFT calculations to be performed using the traditional PDA and GGA approximations, as well as meta-GGA and hybrid density functionals. Slater-type basis sets were used in the relativistic DFT calculations, based on the ZORA Hamiltonian and applied for heavy metals and their surfaces [562,563]. 

Several functional forms have been investigated for the basis functions Given the vast experience of using Gaussian functions in Hartree-Fock theory it will come as no surprise to learn that such functions have also been employed in density functional theory. However, these are not the only possibility Slater type orbitals are also used, as are numerical... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

In honour of J. C. Slater, we refer to such basis functions as Slater-type orbitals (STOs). Slater orbital exponents ( = (Z — s)/n ) for atoms through neon are given in Table 9.2. 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

The first step in reducing the computational problem is to consider only the valence electrons explicitly, the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third rows of the periodic table have four basis functions (one s- and one set of p-orbitals, pj, , Pj, and Pj). The large majority of semi-empirical methods to date use only s- and p-functions, and the basis functions are taken to be Slater type orbitals (see Chapter 5), i.e. exponential functions. 

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... 

If one includes functions with n - / even in (1.1) (i.e. one uses set b) the basis is formally overcomplete. However the error decreases exponentially with the size of the basis [2,16]. Unfortunately for this type ofbasis the evaluation of the integrals is practically as difficult as for Slater type basis functions, such that basis sets of type (b) have not been used in practice. 

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