Primitive Gaussian functions

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

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),... 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

Another interesting possibility, which will not be discussed here, may consist into expanding the radial factors e and re as a linear combination of primitive gaussian functions [69]. Studies about this simpler approximation is imder way in our Laboratory. 

Figure 3.8. Molecular orbitals for the oxygen atom, with indication of their quantum numbers (main, orbital angular momentum and projection along axis of quantisation). Shown is the oxygen nucleus and the electron density (where it has fallen to 0.0004 it is identical for each pair of two spin projections), but with two different shades used for positive and negative parts of the wavefunction. The calculation uses density functional theory (B3LYP) and a Gaussian basis of 9 functions formed out of 19 primitive Gaussian functions (see text for further discussion). The first four orbitals (on the left) are filled in the ground state, while the remaining ones are imoccupied.

Figure 3.9. Molecular orbitals (iso-density at 0.0004) for the diatomic oxygen molecule, calculated as in Fig. 3.8 using a Gaussian basis of 18 functions formed from 38 primitive Gaussian functions. The basis states numbered 1 to 8 are occupied in the Oj ground state.

The contraction coefficients dma are held fixed during a calculation and the functions 3m(r) are primitive Gaussian function characterized by the order of their monomial pre-factor and their exponent am... 

Exponents of a set of primitive Gaussian function have been optimized to yield the lowest pseudo atom energies for all first- and second-row elements with an atomic DFT code employing the appropriate GTH potential for each element. A family basis set scheme has been adopted using the same set of exponents for each angular momentum quantum number of the occupied valence... 

Orbital s—an SCF orbital contracted from 10 primitive Gaussian functions. 

There are two considerations in the light of these comparisons. In the first place, we do not know the accuracy of the Herman-Skillman result. Secondly, it may be possible to improve the fit to the boron data by suitable choice of a modified Slater exponent or the use of a larger linear combination of primitive Gaussian functions. 

Provide for up to six primitive Gaussian functions to model a Slater function, by entering either zero values or the exponents and coefficients listed in Table 1.6 in cells D 5 to I 6. In Figure 2.10, the cell values correspond to the jsto-6g 2s basis set. 

Form the projections of the normalized primitive Gaussian functions on the radial mesh in columns D 11 to I 3011 in the usual manner. Enter the basic formula in cell D 11, with... 

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