Basis Gaussian-type orbitals

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 ... 

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. 

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... 

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, 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. 

The notation STOnG means a minimal atomic basis of Slater Type Orbitals is used each of which is expanded into M-simple Gaussian Type Orbitals. 

The quantum mechanical polarizability is calculated using the DFT, with B3P86 (Becke s three-parameter functional [53] with the non-local correlation provided by Perdew [54]). The basis set used for the water molecules is 6-311 + +G. Because of the very diffuse nature of the anion F, the basis set used is the specially designed, and very extensive, fully uncontracted 14s 9p 6d 2f Gaussian-type orbitals [55]. All the QM calculations were made with the Gaussian98 program [56]. 

Large basis sets of Gaussian-type orbitals (up to 35s21p2ld9g6h4i) were used, taken from the universal basis set ofMalli etal. [55]. Many electrons were correlated (5 in B,... 

Dahle, P., Helgaker, T., Jonsson, D., Taylor, RR. Accurate quantum-chemical calculations using Gaussian-type geminal and Gaussian-type orbital basis sets applications to atoms and diatomics. Phys. Chem. Chem. Phys. 2007, 9, 3112-26. 

Figure 2.2. Radial dependence of basis functions a) correct exponential decay (STO) (b) primitive Gaussian-type function (solid line) vs. an STO (dotted line) (c) least-squares expansion of the STO in terms of three Gaussian-type orbitals (STO-3G).

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

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