Slater and Gaussian Type Orbitals

There are two types of basis functions (also called Atomic Orbitals (AO), although they in general are not solutions to an atomic Schrodinger equation) commonly used 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

Gaussian type orbitals can be written in terms of polar or Cartesian coordinates as shown in eq. (5.2). 

The increase in the number of GTO basis functions, however, is more than compensated for by the ease of which the required integrals can be calculated. In terms of computational efficiency, GTOs are therefore preferred and are used almost universally as basis functions in electronic structure calculations. Furthermore, essentially all applications take the GTOs to be centred at the nuclei. For certain types of calculations the centre of a basis function may be taken not to coincide with a nucleus, for example being placed at the centre of a bond or between non-bonded atoms for improving the calculation of van der Waals interactions. 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

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A) SLATER-TYPE ORBITALS AND GAUSSIAN-TYPE ORBITALS... 

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 ... 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

The situation is quite similar in chemistry. Due to decades of experience with Hartree-Fock and Cl calculations much is known about the construction of basis functions that are suitable for molecules. Almost all of this continues to hold in DFT — a fact that has greatly contributed to the recent popularity of DFT in chemistry. Chemical basis functions are classified with respect to their behaviour as a function of the radial coordinate into Slater type orbitals (STOs), which decay exponentially far from the origin, and Gaussian type orbitals (GTOs), which have a gaussian behaviour. STOs more closely resemble the true behaviour of atomic wave functions [in particular the cusp condition of Eq. (19)], but GTOs are easier to handle numerically because the product of two GTOs located at different atoms is another GTO located in between, whereas the product of two STOs is not an STO. The so-called contracted basis functions , in which STO basis functions are reexpanded in... 

There are two types of basis functions (also called Atomic Orbitals, AO) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). In term of computational efficiency, GTOs are preferred, and used almost universally as basis functions in electronic structures calculations. Having decided on the type of function, the most important factor is the number of functions to be used. 

As mentioned above the radial function R r) for many-elcctron atoms needs to be approximated in some way. The atomic orbitals most frequently employed in molecular calculations are Slater type orbitals (STOs) and Gaussian type orbitals (GTOs). Their mathematical form m ikes them relatively easy to handle in machine calculations. An STO with principal quantum number n is written as... 

Orbital and density plots from linear combinations of Slater or Gaussian type orbitals. Available by anonymous ftp on infomeister.osc.edu (128.146.36.5) in directory /pub/chemistry/software/SOURCES/FORTRAN/moplot2. Silicon Graphics, Convex, and VAX. 

In molecular quantum chemistry two types of atomic-like basis sets are used Slater-type orbitals (STO) and Gaussian-type orbitals (GTO). In fact, it is not really correct to call them orbitals . They are better described as basis-set functions, since they are Slater-type or Gaussian-type functions used to approximate the shapes of the orbitals defined as one-electron wavefunctions. Using the acronyms accepted in sohd-state theory it would be possible even call LCAO methods for crystals to all-electron or full potential hnear combination of Slater (Gaussian)-type functions FP LS(G)TF method [458], compare with the acronym FP LAPW (full potential linear combination of angmented plane waves). 

The atomic and molecular wave functions are usually described by a linear combination of either Gaussian-type orbitals (GTO) or Slater-type orbitals (STO). These expressions need to be multiplied by a center dependent factor expf ip-A). Further the STOs in momentum space need to be multiplied by Yim(6p,p). Examining the expressions [4], one notices the Gaussian nature of the GTOs even after the FT. The STOs are significantly altered on FT. From the expressions in Table 5.1, STOs are seen to exhibit a decay which is the decay of the slowest Is... 

The molecular electron density function needed for EP calculation can be obtained through ab initio as well as various semi-empirical methods. Since ab initio calculations are not economical for large molecules (several hundred atoms), the use of well-parameterized semi-empirical methods are still justified. When semi-empirical methods are used the three-center potential integrals usually disappear, and therefore the electronic contribution can be easily calculated by Slater-type orbitals. In ab initio methods (primitive or contracted) Gaussian-type orbitals are used for calculating the three-center integrals because their calculations are clumsy with Slater-type orbitals. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

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