Contracted Gaussians

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... 

McLean, A.D. Chandler, G.S. Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z=ll-18 J. Chem. Phys. 72 5639-5648, 1980. 

The contracted Gaussian functions are a linear combination of the primitive Gaussian functions. That is,... 

Linear combinations of primitive gaussians like these are used to form the actual basis functions the latter are called contracted gaussians and have the form ... 

Ref. 29. MO-SCF calculations using medium-size contracted Gaussian basis set. 

Schaefer, A. Huher, C. Ahlrichs, R. Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr. J. Chem. Phys. 1994, 100, 5829-5835. 

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 basis sets consist of 34 (6-311+G(d,p)), 92 (aug-cc-pVTZ), 172 (aug-cc-pVQZ), and finally 287 (aug-cc-pV5Z) contracted Gaussian functions and include polarization functions up to d- (6-311+G(d,p)), f- (aug-cc-pVTZ), g- (aug-cc-pVQZ), and even h-character (aug-cc-pV5Z). 

Schafer, A., Horn, H., Ahlrichs, R., 1992, Fully Optimized Contracted Gaussian Basis Sets for Atoms Li to Kr , J. Chem. Phys., 97, 2571. 

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 . 

Example 1. Dissociation of H O by symmetric stretch Reference calculations are first performed at equilibrium geometry, using a simple basis of contracted gaussian functions [8]. The molecule is then dissociated by symmetric stretch, energies being calculated at bond length intervals of AR = Q.2Re up to i = 6.0J e (where dissociation is effectively complete). 

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. 

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