Contracted Gaussian orbitals

Fig. 7.1 Variation of orbital overlap (Sij) as a function of intemuclear distance (r,y) using contracted Gaussian orbitals corresponding to a minimal basis of Fe...

Atomic natural orbital (ANO) basis sets [44] are fonned by contracting Gaussian fiinctions so as to reproduce the natural orbitals obtained from correlated (usually using a configuration interaction with... 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

These primitive Gaussians form the actual basis functions which are called contracted Gaussians. The atomic orbitals are then expressed as ... 

Contemporary basis sets are usually formed from atomic-orbital basis functions T), of contracted Gaussian form,... 

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

The global basis x used in the calculations rejKuted here is again of simple contracted gaussian form it contains four contracted s functions and three sets of contracted p functions on the chlorine atom three contracted s functions and two sets of contracted p functions on the lithium. When the orbitals are written in the matrix form 4> = xT, the first 8 columns of T, namely Toor, define the doubly occupied core orbitals the 3 columns of Tv i define the sigma lone p>air and the two orbitals of the bond pair and the 11 columns of Toom provide the complementary (virtual) space. 

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. 

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. 

The basis sets described here in most detail are those developed by Pople3 and coworkers [40], which are probably the most popular now, but most general-purpose (those not used just on small molecules or on atoms) basis sets utilize some sort of contracted Gaussian functions to simulate Slater orbitals. A brief discussion of basis sets and references to many, including the widely-used Dunning... 

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