Gaussian-type orbitals, computational

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. 

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. 

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. 

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... 

Alternatively, these one-center coulomb integrals can be computed from first principles using Slater or Gaussian type orbitals. 

The arrival of ab initio programs based on Gaussian-type orbitals and more powerful computers reduced all these developments to historical souvenirs. It should be emphasized, however, that Sandorfy s work constituted the first attempt to go beyond the 7t-electron approximation, and the first molecular orbital treatment of polyatomic molecules that took into account all valence electrons without using group orbitals. 

Two different basis sets were used (1) basis set I was medium size, and (2) split-valence basis set II taken from Huzinaga [24] (9s and 5p gaussian-type orbital contracted into [3s, lp] for the C,N,0 atoms). In basis set II, more flexibility has been allowed to the description of the valence shells by adapting a triple- contraction completed with one p type (for hydrogen) or one d type (for C,N,0) polarization function. Then two ab initio calculations were carried out (1) all atoms were described with basis set I, and (2) basis set II was used for all the atoms but those belonging to the methyl and phenyl substituents due to computer limitations. 

In this contribution we have reviewed the applicability, accuracy and computational efficiency of the local spin density functional approach to the chemistry of transition metal complexes and clusters using a linear combination of Gaussian-type orbital basis set for the calculation of electronic structures, ground state geometries and vibrational properties. 

Potential energy curves for singlet and triplet A j, B, and B j states of COF j have been computed using ab initio projected-unrestricted Hartree-Fock theory with a contracted Gaussian type orbital basis set [273]. However, symmetry was strictly maintained for these excited states, so the poor agreement between the predicted and experimental band onsets (which was readily acknowledged by Brewer et al. [273]) comes as little surprise. 

We determine the ground-state electronic structure of solids within Density Functional Theory (DFT) and the usual KS variational procedure, all implemented in the computational package gtoff [14]. The results of the all-electron, full-potential calculations are Bloch eigenfunctions p,k(r), expressed as linear combinations of Gaussian Type Orbitals (GTOs), and KS eigenvalues p k-... 

Despite the huge increase in computational effort, this direct symmetry-adapted LCAO method was used to study ozone [22], tetrahedral Ni4 [23], and D5h-symmetric ferrocene (Fe(C5H5)2) [24] using molecular orbital (MO) contraction coefficients in the linear-combination-of-Gaussian-type orbital (LCGTO) computer code of [25]. Obviously, symmetry-adapted calculations are important enough to pay an order-TV computational price. The reasons are first, and foremost, that the calculations converge, and second that the wavefunction and one-electron orbitals can be used to address experiment, which typically must first determine the symmetry of the molecule. 

H. Tatewaki and S. Huzinaga, J. Comput. Chem., 1, 205 (1980). A Systematic Preparation of New Contracted Gaussian-Type Orbital Sets. 111. Second-Row Atoms from Li Through Ne. 

At that time this was a formidable calculation. It was performed on an IBM 370/158, which was not a supercomputer, but nevertheless a respectable mainframe. A basis of 144 Gaussian type orbitals (GTOs) was used for restart purposes integrals were stored on tape, requiring two tapes of 170 Mb for one point on the PES. About 150 points were computed, so that 300 tapes were needed. A tape reel had a diameter of 10.5 in. and, including its case, was about 1 in. wide, so that a rack of about 7.5 m long and 30 X 30 cm wide had to be used to store the 50 Gb of information. One point on the surface took... 

For the calculations we used the Munich version of the linear combination of Gaussian-type orbital density functional (LCGTO-DF) code. ° The computationally economic local spin-density approximation (LSDA) to the exchange-correlation functional has been successfully used in chemical applications since the seventies. This functional (employed here in the parameterization suggested by Vosko, Wilk, and Nusair, has been shown to describe accurately impor-... 

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. 

Because of the latter computational difficulties, another choice of basis sets has been preferred, mainly by the ab initio quantum-chemistry community, and this basis set has actually given an important computer program its name. According to Boys [59], a Gaussian-type orbital (GTO) may also serve as a basis function, and the radial-dependent part scales as... 

The alternative to semi-empirical methods is the full self-consistent calculation, the so-called ab initio approach. It is based on computing all integrals in Eq. (52), with the atomic Slater-type orbitals (STO), exp(—or), replaced by Gaussian-type orbitals... 

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