Basis Slater-type orbitals

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. 

Hoffmann introdueed the so-ealled extended Hiiekel method. He found that a value for K= 1.75 gave optimal results when using Slater-type orbitals as a basis (and for ealeulating the Sg,v)- The diagonal h elements are given, as in the eonventional Hiiekel method, in terms of valenee-state IP s and EA s. Cusaehs later proposed a variant of this parameterization of the off-diagonal elements ... 

In quantum ehemistry it is quite eommon to use eombinations of more familiar and easy-to-handle "basis funetions" to approximate atomie orbitals. Two eommon types of basis funetions are the Slater type orbitals (STO s) and gaussian type orbitals (GTO s). STO s have the normalized form ... 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

Hoffmann introdueed the so-ealled extended Hiiekel method. He found that a value for K= 1.75 gave optimal results when using Slater-type orbitals as a basis (and for ealeulating the... 

The complete neglect of differential overlap (CNDO) method is the simplest of the neglect of differential overlap (NDO) methods. This method models valence orbitals only using a minimal basis set of Slater type orbitals. The CNDO method has proven useful for some hydrocarbon results but little else. CNDO is still sometimes used to generate the initial guess for ah initio calculations on hydrocarbons. 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact solutions of the Schrodinger equation for the hydrogen atom (or any one-electron atom, such as Li" ). Hyper-Chem uses Slater atomic orbitals to construct semi-empirical molecular orbitals. The complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and are not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

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... 

In Table 5, we show the calculated magnetic susceptibilities for AlH and SiH+, and the Lipseomb et al. [7] obtained values for AlH using an extended basis set of field independent Slater type orbitals. 

Representation of each molecular orbital as a linear combination of atomic orbitals (atomic basis sets). Atomic basis sets are usually represented as Slater type orbitals or as combinations of Gaussian functions. The latter is very popular, due to a very fast algorithm for the computation of bielectronic integrals. 

The corresponding estimate for the second eigenvalue (2s orbital energy) is —0.1789. These results are in good agreement with the actual HF/STO-3G ( Hartree-Fock method with a variational basis set of three-term Gaussians for each Slater-type orbital 10) eigenvalues eis = —2.3692 and e2s = —0.1801. 

Each CGTO can be considered as an approximation to a single Slater-type orbital (STO) with effective nuclear charge f (zeta). The composition of the basis set can therefore be described in terms of the number of such effective zeta values (or STOs) for each electron. A double-zeta (DZ) basis includes twice as many effective STOs per electron as a single-zeta minimal basis (MB) set, a triple-zeta (TZ) basis three times as many, and so forth. A popular choice, of so-called split-valence type, is to describe core electrons with a minimal set and valence electrons with a more flexible DZ (or higher) set. 

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