Orbital exponent, choice

Once the least-squares fits to Slater functions with orbital exponents 1.0 are available, fits to Slater functions with other orbital exponents can be obtained by simply multiplying the a s in the above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculations. The two possibilities may be to use the best atom expo-nents( = 1.0 for H, for example) or to optimize exponents in each calculation. The best atom exponents might be a rather poor choice for molecular environments, and optimization of nonlinear exponents is not practical for large molecules, where the dimension of the space to be searched is very large. Acompromise is to use a set of standard exponents where the average values of exponents are optimized for a set of small molecules. The recommended STO-3G exponents are... 

In order to determine the relations among orbital exponents in a basis which will follow these guidelines, we look at the matrix elements contributing to on(q). To that end we consider eq. 17 for the plane wave operator (eq. 21) which involves evaluation of terms of the sort (p e i /i). We wish to determine how these matrix elements behave as a function of orbital exponent and momentum transfer, and we then propose a scheme for choice of orbital exponents that will keep the BSR satisfied to as high momentum transfer as possible. 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

If we now consider the numerical results quoted in Table 1 for the optimum exponents, three conclusions follow immediately. Firstly, the 1 s orbital on the heavy atom is unchanged by molecule formation this is to be expected. Second, the sp3 orbitals involved in the X—H bond are all contracted with respect to their free-atom values. Finally, the sp3 orbitals containing Tone pairs of electrons are largely unchanged or expanded slightly on molecule formation. In fact, of course, the optimum separate atoms minimal basis functions do not have the same orbital exponent for the 2 s and 2 p AOs. To facilitate comparisons therefore in Table 1 the optimum n = 2 exponent is given for the atoms when such a constraint is imposed (the qualitative conclusions are, in any event, unchanged by use of these exponents for comparison or a notional exponent of 1/4 (fs + 3 fp) or any reasonable choice). 

The choice of orbital exponent (a) to use for a particular atomic Slater orbital has been the subject of several investigations. Originally, Slater (9) proposed a set of empirical rules for choosing exponents however, these are not used frequently in modern calculations. Hartree-Fock self-consistent-field (SCF)... 

The parabolic dependence of energy band on k expected by the free electron model is not observed in Fig. 10. Different choices of orbital exponent giving less diffuse atomic orbitals promote such a behavior, but more work is needed to determine the criteria for choosing parameters in the infinite-model EH theory. Other choices of orbital exponents also predict cubic geometry to be more stable than linear. 

Thus the orbital exponents of the Is functions range from 0.5 to 16.7 bohr-1, the 2p exponents range from 1.0 to 10.5 bohr-1, and the 3d exponents from 1.5 to 4.1 bohr-1. The energy is generally found to be more sensitive to the particular choice of ft than the choice of a. 

To conclude this section on SCF dipole properties it is instructive to inspect the variation of the DZP results in Table IV with orbital exponent. As mentioned above, energy-optimized exponents (ap = aj = 1.0) yield poor results, and very low exponents = 0.15, p = 0.08) yield reasonable polarizabilities, but poor moments. There exists, however, an intermediate choice (a = 0.25, ap = 0.15) for which and are simultaneously quite acceptable. The same property was demonstrated to hold for a number of other small molecules and this so-called moment-optimized DZP basis was therefore advocated as the smallest set yielding reasonable Ecui and F isp energies for polar dimers. 

Let us continue using N2 as an example for how one usually varies the box within which the anion is constrained. One uses a conventional atomic orbital basis set that likely includes s and p functions on each N atom, perhaps some polarization d functions and some conventional diffuse s and p orbitals on each N atom. These basis orbitals serve primarily to describe the motions of the electrons within the usual valence regions of space. To this basis, one appends an extra set of diffuse ir-symmetry orbitals. These orbitals could be p j (and maybe d ) functions centered on each nitrogen atom, or they could be orbitals centered at the midpoint of the N-N bond. Either choice can be used because one only needs a basis capable of describing the large-r L = 2 character of the metastable Ilg state s wave function. One usually would not add just one such function rather several such functions, each with an orbital exponent aj that characterizes its radial extent, would be used. Let us assume, for example, that K such additional diffuse tt functions have been used. 

Preliminary studies for Te4 indicate different orbital exponents, 2.12 or 2.45, depending on the choice of total overlap energy or total overlap population as the criterion of stationary property. Further calculations for Sc4 with a 3d STO produce a flatter curve similar to that of Figure 1. 

The data in Figure 4.22 represent the best results for the choice that the Slater 2s exponent be varied to minimize the energy of the 2s orbital. This choice was made because, as you will find, in all the calculations possible on this spreadsheet, the main defect is the modelling of the 2s orbital, even with the best choice of Slater exponent. For the choice of best Slater 2s exponent the calculated 2s orbital energy in the basis of six contracted primitives is returned as —0.1234611 Hartree, which is some 4 kJ/mole in error from the exact value of —0.1250 Hartree. 

Let us consider the STO basis set first. The essence of this basis choice is to place on each nucleus one or more STOs. The number of STOs on a nucleus and the orbital exponent of each STO remain to be chosen. Generally, the larger the number of STOs and/or the greater the care taken in selecting orbital exponents, the more accurate the final wavefunction and energy will be. 

We must now consider the choice of the orbital exponents. Again, some sufficient but not necessary conditions for completeness have been established [17]. Thus, completeness with respect... 

Basis sets taken from nonrelativistic calculations are a good first choice for most problems since they provide a good representation of the valence electron distribution. However, we know that the shape of the wavefunction is sensitive to the nuclear charge close to the nucleus, and it is therefore important to choose basis functions for the most penetrating s and p orbitals to improve the fit at short range. The simplest way to do this is to add some larger exponents. 

Table 1.6 lists the Hehre, Stewart and Pople sto-3g) Gaussian sets (33) for hydrogenic Is, 2s and 2p orbitals, the minimum basis sets for the atoms of the first short period when scaled with suitable choices for the Slater exponents. 

The Slater exponents are parameters in any calculations involving these Gaussians sets, which we can vary to achieve a particular result. In the present case, the desired result is a fit to the numerical radial data for lithium 2s. For the valence shell regions of the lithium 2s orbital the fit in Figure 3.7 is almost complete for the choices G = 2.7 and G.5 = 0.675. However, that, of course, does not mean that we have discovered a universal set of Slater exponents for lithium. All we have done in this analysis is set the criterion for the choice of Slater exponents, the match to the output of the Herman-Skillman program... 

Figure 4.8 Calculation of the hydrogen 2,1 energy using the Slater 2s orbital rendered orthogonal to the hydrogenic Is orbital with the best choice for the Slater exponent subject to two conditions. In the first diagram, the SOLVER based solution is determined by the requirement that the Virial theorem coefficient be 1.00. In the second diagram, the best values for the kinetic and potential energy terms have been determined for the choice of the Slater 2s exponent. Note, the substantial cancellation of mismatches of the r grad transformed Slater function compared to the variation of the transformed exact function.

Note, becau.se no normalization factor has been included, the normalization integral equals 0.25. rather than 1.00 for the choice of exponent describing the hydrogenic orbital function, compare Table 1.1. 

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