The Optimization of Basis Sets

One possibility is to resort to even-tempered basis sets. These are sets where the exponents form a geometric series such that the exponent of 

---

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 optimization of basis set non-linear parameters, appearing in equation (5.2), constitute one of the main steps in the preliminary work before many center integral evaluation. There will be described only a step by step procedure in order to optimize non-linear parameters of the involved fimctions one by one. 

The nature of basis sets suitable for 4-component relativistic calculations is described. The solutions to the Dirac equation for the hydrogen atom yield the fundamental properties that such basis functions must satisfy. One requirement is that the basis sets for the large and small component be kinetically balanced, and the consequences of this are discussed. Schemes for the optimization of basis sets and choice of symmetry and shell structure is discussed, as well as the advantages offered the use of family sets for scalar basis sets. Special considerations are also required for the description of correlation and polarization in these calculations. Finally the applicability of finite basis sets in actual applications is discussed... 

FIGURE 34.4 Fukui functions determined using Equation 34.18. Long-dashed lines are determined using conventional calculations (no potential wall) with the aug-cc-pVTZ basis set dotted lines use conventional calculations with the optimally compact basis sets identified in Table 34.4 solid lines use the potential wall approach of Section 34.5, with the aug-cc-pVTZ basis. (Reprinted from Tozer, D J. and De Proft, F., J. Chem. Phys., 127, 034108, 2007. With permission.)... 

Br2 and HBr.—We have already observed the phenomenon of basis-set dependence in pseudopotential calculations i.e. since some pseudopotentials are parameterized with respect to a particular VE basis unpredictable behaviour results when they are used in conjunction with other (perhaps more flexible) basis sets for which they are not optimal. Now we give details of calculations which illustrate the associated problem of pseudopotential parameterization dependence , i.e. the effect of changing the number of terms used to fit the desired functional form for the pseudopotential. 

Fig. 8 The effect of basis set on the specific rotation of (S)-methyloxirane calculated with the B3LYP functional at 589 nm. The key labels the method of geometry optimization. Basis set numbers correspond to the following 1 = 6-31G(d), 2 = 6-31+-l-G(d,p), 3 = 6-31++G(2d,2p), 4 = cc-pVDZ, 5 = cc-pVTZ, 6 = aug-cc-pVDZ, 7 = d-aug-cc-pVDZ, 8 = mixed-cc-PVTZ (aug-cc-pVTZ(C,0) and aug-cc-pVDZ(H)), 9 = aug-cc-pVTZ, 10 = Sadlej-pVTZ. Data to prepare the plot were taken from [145]...

Optimization of basis sets is not something the average user need to worry about. Optimized basis sets of many different sizes and qualities are available either in the forms of tables, or built into the computer programs. The user merely has to select a suitable basis set. However, if the interest is in specialized properties the basis set may need to be tailored to meet the specific needs. For example if the property of interest is an accurate value for the electron densitv at the nucleus Ifor example for determining The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

The optimization of basis function exponents is an example of a highly non-lmear optimization (Chapter 14). When the basis set becomes large, the optimization problem is no longer easy. The basis functions start to become linearly dependent (the basis approaches completeness) and the energy becomes a very flat function of the exponents. Furthermore, the multiple local minima problem is encountered. An analysis of basis... 

Optimization of basis sets is not something the average user need to worry about. Optimized basis sets of many different sizes and qualities are available either in the forms of tables, or built into the computer programs. The user merely has to select a suitable basis set. However, if the interest is in specialized properties the basis set may-... 

The electron affinities calculated with the extended, optimized basis sets are shown in Table 13. First of all, the results clearly show that at the Hartree-Fock level (ROHF or UHF) the SF6 molecule is destabilized after the addition of an electron. All calculated values of electron affinity decrease systematically with the expansion of basis set. Furthermore, the combination of the most recent functionals (DF4) gives a value of EA =... 

---