Double-zeta Slater functions/orbitals

Figure 3.8 The two worksheets in the spreadsheet, fig3-8.xls, for the calculation of the orthonormal double-zeta Slater radial function of the lithium 2s orbital. The comparison graphs in this figure show the possible starting situation, with both C coefficients equal 1 and the C2 equal 0. In both cases, changing the value of cj in cell SCSI 1 leads to the best fit results shown in the next diagram.

Many calculations for atoms have led to the development of a number of recipes for deciding the best values of and n. A further important issue is the size of the basis set. A minimal basis set of STOs for an atom would include one function for each SCF occupied orbital with different n and / quantum numbers in equation (6.56) for the chlorine atom, therefore, the minimal basis set would include s, 2s, 2p, 3s and 3p functions, each with an optimised Slater orbital exponent . A higher order of approximation would be to double the number of STOs (the double zeta basis set), with orbital exponents optimised ultimately the Hartree-Fock limit is reached, as it has been for all atoms from He to Xe [13]. 

In the 1960s, this last statement was not acceptable and, indeed, it was to minimize such discrepancies that prompted Clementi to propose his optimized double-zeta sets. That better results can be obtained using this approach is shown in Figure 1.12, wherein the comparisons are made between the optimized Clementi double-zeta function and the numerical results. The detail of this calculation requires the involvement of the variation principle to determine the relative weightings of the two components of the double-zeta basis by minimizing the calculated 2s orbital energy. This requires that the 2s function be rendered orthonormal [Chapters] to the Is function in lithium. Thus, all four Slater functions in the Clementi double-zeta basis, the two for the 1 s and the two for the 2s, contribute to... 

The pre-exponential factors in the equation 1.12 normalize the Slater approximations to the radial components of atomic orbitals. Normality is not an inherent property of linear combinations of Slater orbitals, for example, as in Table 1.3, and it is important to check any published coefficients to determine whether normalization is included. In addition, the Slater orbitals for a set of atomic orbitals in an atom are not mutually orthogonal. The results of atomic structure calculations using Slater orbitals, either as single functions or in linear combinations, as in double-zeta sets, of course, are mutually orthogonal, since this property of the eigenfunctions, is mirrored in the final linear combinations returned by the calculations for the eigenvalues. 

Figure 3.9 Best choices for the coefficients of the linear combinations forming the double-zeta representation of the Is and 2s Slater orbitals for lithium in the formation of the orthonormal function.

That the rendering of the Is and 2s functions orthonormal leads to linear combinations ranging over all four of the Slater functions in the double-zeta basis for the two orbitals... 

In order to evidence the effects of the crystal field, the central ion must be described by a wavefunction built up with a sufficiently flexible basis set. We have selected the double-zeta set (DZ) of Slater type orbitals proposed by Clementi [3]. The wave-function for the isolated NO J ion with this basis set had been already published by us some years ago [4]. 

An accurate analysis on this point was made by Richardson for the Ng molecule and leads to very interesting conclusions. This author calculated the field gradient by using the Ransil s molecular orbitals in the approximation (i) (zeta constant), (iii) (zeta varied in the molecule), and a new type of molecular orbital [approximation (iv)] with a more extended basis, in which each atomic orbital 2s and 2p of the nitrogen atom is composed of two Slater-type functions with different zeta parameters (double zeta approximation). 

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