Contracted basis function

Table I. The pattern of basis sets used in the present work. It is described as follows For orbital basis sets, C (521/41) means that the basis set for carbon contains 8 s basis functions, contracted to 3, (521), and 6 p basis functions, contracted to 2, (41). For auxiliary basis sets, C (622/522) means that, there are 6 s basis functions, 2 for p and 2 for d. The first part (622) concerns charge density, and the second part (522) the exchange-correlation. Exponents and coefficients will be sent upon request...

The previously introduced acronyms DZP, TZ2P etc., refer to the number of contracted basis functions. Contraction is especially useful for orbitals describing the inner (core) electrons, since they require a relatively large number of functions for representing the... 

The experimental ground-state geometry [138] was used with the molecule in the xy plane. ANO basis functions contracted as C,N/ Asl>pld, H/2sl/ , were used, supplemented with Islpld Rydberg functions in the cation charge centroid. The study of the electronic spectrum of imidazole involved tt— t7, 7t, and Rydberg states (in the gas... 

HyperChem offers an easy way to interactively add certain basis functions to a molecular system. The Extra Basis Function dialog box can be used to add an S, P, D, SP, or SPD shell to the selected atom(s). These extra basis functions are primitives with no contractions. Thus, the extra basis functions are uniquely defined by the shell type and the value of the exponent. 

HyperChem quantum mechanical calculations are ab initio and semi-empirical. Ab initio calculations use parameters (contracted basis functions) associated with shells, such as an s shell, sp shell, etc., or atomic numbers (atoms). Semi-empirical calculations use parameters associated with specific atomic numbers. The concept of atom types is not used in the conventional quantum mechanics methods. Semi-empirical quantum mechanics methods use a rigorous quantum mechanical formulation combined with the use of empirical parameters obtained from comparison with experiment. If parameters are available for the atoms of a given molecule, the ab initio and semi-empirical calculations have an a priori aspect when compared with a molecular mechanics calculation, letting... 

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 ... 

Gaussian Basis Functions for use in Molecular Calculations III Contraction of (10s, 6p) Atomic Basis Sets for the First-Row Atoms T. FI. Dunning, Jr... 

Take note of Dunning s notation. He writes the primitives (10s6p) and the contracted basis functions in square brackets [5s3p]. To give a detailed example, consider the oxygen atom set in Table 9.7. 

Consider now making the variational coefficients in front of the inner basis functions constant, i.e. they are no longer parameters to be determined by the variational principle. The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

Combining the full set of basis functions, known as the primitive GTOs (PGTOs), into a smaller set of functions by forming fixed linear combinations is known as basis set contraction, and the resulting functions are called contracted GTOs (CGTOs). 

The basis functions are normally the same as used in wave mechanics for expanding the HF orbitals, see Chapter 5 for details. Although there is no guarantee that the exponents and contraction coefficients determined by the variational procedure for wave functions are also optimum for DFT orbitals, the difference is presumably small since the electron densities derived by both methods are very similar. ... 

Huxley, H.E. Hanson, J. (1960). The molecular basis of contraction in cross-straited muscle. In Structure and Function of Muscle (Bourne, G.H., ed.), Vol. I, pp. 183-227. Academic Press, New York. 

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