Gaussian-type orbitals

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

Cartesian Gaussian-type orbitals (GTOs) Jfa.i.f( ( characterized by the quantum numbers a, b and c, which detail the angular shape and direction of the orbital, and the exponent a which governs the radial size . 

Because th e calculation of m n Iti-ceiiter in tegrals that are in evitable for ah iniiio method is very difficult and time-con sum in g. Ilyper-Chem uses Gaussian Type Orbital (GTO) for ah initio methods. In truly reflecting a atomic orbital. STO may he better than GTO. so HyperC hem uses several GTOs to construct a STO. The number of GTOs depends on the basis sets. For example, in the minimum STO-3G basis set IlyperGhem uses three GTOs to construct a STO. 

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 ... 

Alternatively, these one-eenter eoulomb integrals ean be eomputed from first prineiples using Slater or Gaussian type orbitals. 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

FIGURE 10.1 Approximating a Slater-type orbital with several Gaussian-type orbitals. 

Gaussian theory (Gl, G2, G3) a method for extrapolating from ah initio results to an estimation of the exact energy Gaussian-type orbital (GTO) mathematical function for describing the wave function of an electron in an atom... 

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. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The breakthrough for molecular applications came with Boys s classic paper (1950) on the use of Gaussian-type orbitals (GTOs). These basis functions have an exponential dependence of exp (— (ar /al)) rather than exp(—( r/ao))-The quantity a is called the Gaussian exponent. Normalized Is and 2p GTOs are... 

Least squares representations of Slater-type atomic orbitals as a sum of Gaussian-type orbitals are presented. These have the special feature that... 

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... 

Gaussian type orbitals can be written in terms of polar or cartesian coordinates... 

Gaussian-type orbitals, the computational requirements grow, in the limit, with the fourth power in the number of basis functions on the SCF level and with even a higher power for methods including correlation. Both the conceptual and the computational aspects prevent the computational study of important problems such as the chemistry of transition metal surfaces, interfaces, bulk compounds, and large molecular systems. 

Haberlen, O.D. and Rdsch, N. (1992) A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method application to AuFl, AuCl and Au2. Chemical Physics Letters, 199, 491-496. 

Andzelm, J., Wimmer, E., 1992, Density Functional Gaussian-Type-Orbital Approach to Molecular Geometries, Vibrations, and Reaction Energies , J. Chem. Phys., 96, 1280. 

Gaussian probability, linear thermodynamics quadratic expansion, 12-13 regression theorem, 17-20 Gaussian-type orbitals (GTOs), 257-258 Gauss s law, diatomic molecules, internal electric field computations, 249-250... 

Gallant, R. T. and A. St-Amant. 1996. Linear scaling for the charge density fitting procedure of the linear combination of Gaussian-type orbitals density functional method. Chem. Phys. Lett. 256, 569. 

Robin et al.162 investigated the photoelectron spectrum of unsubstituted cyclo-propenone and interpreted its results with the aid of Gaussian-type orbital calculations of double-zeta quality for the electronic ground state using the experimentally established133 geometry of cyclopropenone. 

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... 

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