Gaussian function exponents

The solution to this problem is to use more than one basis function of each type some of them compact and others diffuse, Linear combinations of basis Functions of the same type can then produce MOs with spatial extents between the limits set by the most compact and the most diffuse basis functions. Such basis sets arc known as double is the usual symbol for the exponent of the basis function, which determines its spatial extent) if all orbitals arc split into two components, or split ualence if only the valence orbitals arc split. A typical early split valence basis set was known as 6-31G 124], This nomenclature means that the core (non-valence) orbitals are represented by six Gaussian functions and the valence AOs by two sets of three (compact) and one (more diffuse) Gaussian functions. 

A Gaussian expansion contains two parameters the coefficient and the exponent. The most flexible way to use Gaussian functions in ab initio molecular orbital calculations permits both of these parameters to vary during the calculation. Such a calculation is said to use... 

Functions 10 through 13 comprise the diffuse s function (note the small value for the exponent a, which will fall off to zero at a much greater distance than the earlier gaussian functions). Functions 14 through 19 are d functions. This basis set uses six-component d functions d 2, dy2, dj2, d, d, d. They are constructed using the exponent and D-COEF coefficient from tne final section of the preceding table. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

The MCSCF and the subsequent perturbation calculations were done using a 6-31+G basis set expanded by a set of spd Rydberg functions. Exponents of this additional gaussians were 0.032 and 0.028 for the s and p shells for the oxygen atom, and 0.023 and 0.021 for the carbon atom. For the d functions, a common value of 0.015 was chosen for both heavy atoms. 

The calculation performed for the metastable N (ls2s) + He system has necessitated somewhat larger Cl spaces (200-250 determinants) in order to reach the same perturbation threshold ri = 0.01, the la molecular orbital being not frozen for this calculation.The basis of atomic orbitals has been also expanded to a 10s6p3d basis of gaussian functions for nitrogen reoptimized on N (ls ) for the s exponents and on N (ls 2p) for the p exponents and added of one s and one p diffuse functions [22]. For such excited states,... 

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),... 

The two most popular basis sets consist of either Slater-type orbitals8 (STO s) or Gaussian functions. When using STO s one or more are placed on each nucleus - the more the better. The so-called minimal basis set consists of only those STO s which correspond to the occupied a.o. s in the seperated atom limit. Instead of using Slater s rules to determine orbital exponents they may be varied in order to minimize the energy. Once this optimization has been done for a small molecule the values so established can be used in bigger problems. The basis can be improved by adding additional STO s for various nuclei, e.g. with different orbital exponents. If every minimal basis a.o. is represented by two such STO s a "double Q" set is obtained. The only restriction on the number and type of STO that can be added, seems to be computer time. 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

The aim of this paper is ascertain whether it is possible to determine the ground state second-order correlation energy of the hydrogen molecule to sub-millihartree accuracy using a basis set containing only s-type Gaussian functions with exponents and distribution determined by an empirical, but physically motivated, procedure. 

The ultimate goal of quantum mechanical calculations as applied in molecular modeling is the a priori compulation of properties of molecules with the highest possible accuracy (rivaling experiment), hut utilizing the fewest approximations in the description of the wave-function. Al> initio. or from first principles, calculations represent the current state of the an ill this domain. Ah i/tirio calculations utilize experimental data on atomic systems to facilitate the adjustment of parameters such as the exponents ol the Gaussian functions used to describe orbitals within the formalism. 

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