Orbitals Gaussian

Atoms are special, because of their high symmetry. How do we proceed to molecules The orbital model dominates chemistry, and at the heart of the orbital model is the HF-LCAO procedure. The main problem is integral evaluation. Even in simple HF-LCAO calculations we have to evaluate a large number of integrals in order to construct the HF Hamiltonian matrix, especially the notorious two-electron integrals 

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 

In the case of d-type orbitals, there are six Cartesian GTOs with pre-exponential factors of x, xy, y, xz, yz and z - Only five are linearly independent, e combi nation 

We see from the figure that the position vector of the electron relative to the centre of is r — Fa, so we need a factor of 

Apart from the normalizing factor, the two-centre overlap integral then reduces to a simple one-centre integral of the form 

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Functions with higher / values and with sizes like those of lower-/ valence orbitals are also used to introduce additional angular correlation by pemiitting angularly polarized orbital pairs to be fomied. Optunal polarization functions for first- and second-row atoms have been tabulated and are included in the PNNL Gaussian orbital web site data base [45]. 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

The symbol ao is the first Bohr radius, approximately 52.9 pm, and to is the permittivity of free space.) As we will see in later chapters, Gaussian orbitals are... 

The next step was to represent each Slater atomic orbital as a fixed linear combination of Gaussian orbitals so a Slater-type orbital with exponent f is written as a sum of GTOs with exponents a, q 2, and so on. For example, in the case of three primitive GTOs we might write... 

This calculation has been made here using the 4s basis set (which includes no polarisation p gaussian orbitals). The energy obtained in this way is very good it reproduces the energy obtained by diagonalisation viz. -0.59088 H cf the Table 1) with an error equal to 0.02 eV. 

Scuseria, G. E., 1999, Linear Scaling Density Functional Calculations With Gaussian Orbitals , J. Phys. Chem. A, 103, 4782. 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

This consideration excludes the free-electron model for example, and the Floating Spherical Gaussian Orbital approach in its simplest form. 

The Gaussian orbitals are very important in practical applications. In spite of their wrong asymptotic behaviour at both r —> 0 and r —> °o, nearly all molecular electronic structure calculation programs have been constructed using Gaussian sets of one-electron functions. In this example the Gaussian basis has been selected as... 

SCF calculations for the hypothetical compound H3PO were carried out by Marsmann et al.. The authors studied the effect of adding either a c/ or another p orbital to a phosphorus atom depicted in terms of seven s and three p Gaussian orbitals. An approximate model of H3PO based on the valence bond method has been published by Mitchell. ... 

In recent years a compromise has been found which presently dominates polyatomic calculations. Each function fj is expanded as a linear combination of gaussian orbitals (f is then called a contracted gaussian function). Since this is basically a numerical fitting procedure, various choices have been suggested for the contraction scheme. The most popular choices are presently Pople s approximations (15) to Slater orbitals and Dunning s approximations (16) to free atom Hartree-Fock orbitals. 

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