Optimization of exponents

For molecules the optimization of exponents is very important for small basis sets but it can never alone absorb the deficiency due to a lack of expansion functions. The number and secondly the kind of STO basis set functions (i.e. s, p, d,. .. type) are the most important considerations. The decreasing importance of the exponent optimization as the basis set grows is observable from Table 2.4,... 

The solution to this dilemma is to recognize that the nucleus has a finite size, and that this should be accounted for. Ishikawa and coworkers showed that the use of a finite nucleus instead of a point nucleus allowed for more compact basis sets [12] and also eliminated problems with basis set balance close to the nucleus [13]. Visser et al. [14] performed a full relativistic optimization of exponents for the one-electron atoms Sn and U with and without a finite nucleus, showing that the use of a finite nuclear radius significantly decreased the maximum exponent. 

Usually, contractions are determined from atomic SCFcalculations. In these calculations one uses a relatively large basis of uncontracted Gaussians, optimizes all exponents, and determines the SCF coefficients of each of the derived atomic orbitals. The optimized exponents and SCF coefficients can then be used to derive suitable contraction exponents and contraction coefficients for a smaller basis set to be used in subsequent molecular calculations. 

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... 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

Even-tempered basis sets have the same ratio between exponents over the whole range. From chemical considerations it is usually preferable to cover the valence region better than the core region. This may be achieved by well-tempered basis sets. The idea is similar to the even-tempered basis sets, tire exponents are generated by a suitable formula containing only a few parameters to be optimized. The exponents in a well-tempered basis of size M are generated as ... 

I Meanwhile others object to the suggestion that the optimization of basis sets are carried out by reference to experimental data. While accepting that the exponents and contraction coefficients are generally optimized in atomic calculations, they insist that these optimizations are in themselves ab initio. 

The Veillard basis set [23] (1 ls,9p) has been used for A1 and Si, and the (1 ls,6p) basis of the same author has been retained for Mg. However, three p orbitals have been added to this last basis set, their exponents beeing calculated by downward extrapolation. The basis sets for Al, Si and Mg have been contracted in a triple-zeta type. For the hydrogen atom, the Dunning [24] triple-zeta basis set has been used. We have extended these basis sets by mean of a s-type bond function. We have optimized the exponents a and locations d of these eccentric polarization functions, and the internuclear distance R of each of the studied molecules. These optimized parameters are given in Table 3. 

A main source of model bias lies in the choice of exponents in the single-exponential-type functions r exp (-ar) that are commonly used as the radial parts of the deformation functions this choice is often more of an art than a science [4]. Very little is known about the optimal values to be used for elements other than those of the first two rows. Selection of the best value for the exponents n is usually carried out by systematically varying exponents and monitoring the effects on the R indices and/or residual densities [8, 9]. The procedure can in some cases be unsatisfactory, as is the case when very diffuse functions centred on one atom are used to model most of the density in the bond, and even some of the density on neighbouring atoms [10]. 

Table 3. Optimized orbital exponents orbital (4), and total (/) mean excitation energies of Si for confinement radii R (a.u.) = 4,6, and o°... 

Figure 5.1. Verification of Eqs. (5.1) and (5.2) by means of Mulliken charges deduced from STO-3G calculations involving complete optimizations of geometry and orbital exponents. The net charge q is of (5.1) for the lower line and qa of Eq. (5.2) for R—H [44].

At a quite different level of approximation, this class of compounds was investigated by means of STO-3G calculations involving a detailed optimization of all the geometric and exponent parameters [42]. The Mulhken net atomic charges and the... 

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