Family Basis Set

One of the prime concerns in the use of finite basis expansions is to keep the expansion short, or more bluntly, to get the best description of the system with the smallest number of basis functions. As we have seen 

In addition to the growth of the small component p space, this also leads to another possible problem. If the large component s and d functions are optimized independently, they may well have exponents that are quite close in value. This is no problem for the large component, where symmetry makes these functions linearly independent. However, for the small component, these end up in the same symmetry, and one may in fact encounter serious linear dependencies. Again, this may be removed by projection or other methods, but from the point of view of basis sets size it is very unfortunate to fill the primitive space with functions that overlap strongly. 

While SCF basis sets are the starting point for molecular calculations, it is necessary to augment the basis sets that describe the atomic orbitals in order to describe polarization of the atomic orbitals in the molecule and to describe electron correlation. These two requirements have some overlap, and the functions used for electron correlation are often adequate for molecule formation. Both of these effects usually involve basis functions with higher angular momentum than in the atomic valence orbitals. This presents a challenge in relativistic calculations because the requirements of kinetic balance for the small component adds one further unit of angular momentum to the basis for any function added to the large component for polarization or correlation. 

Optimizing correlating and polarization functions presents issues that are similar to those of the SCF functions. If the spin-orbit components of a subshell are sufficiently different, it might be necessary to use j- or k-optimization for the correlating functions. The use of j-optimization raises a fundamental question what do we now mean by a polarization function A j = 1/2 shell contains both an si/2 function and a pi/2 function, but the charge distributions for both of these are spherical. Polarization by a uniform electric field introduces functions with the angular momentum incremented by one unit. Thus for a, j = 1/2 shell, the polarization functions would have to come from a j = 3/2 shell, which contains both a P3/2 and dy2 function. If the basis set already contains a j = 3/2 shell with appropriate radial distributions, it is not necessary to add another function, just as in the non-relativistic case it is not necessary to add p functions for polarization of an s set if the functions already exist in the basis set. 

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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... 

Also, apart from the compilation of dual family basis sets for the series potassium to element (118) [30], and the Universal Gaussian Basis of Malli et al. [26], there are few extensive series for the heavier elements which can be explored for systematic investigations. Among the tasks that need to be undertaken is a more complete derivation of higher quality basis sets, including polarization and correlation functions, and a systematic investigation of even-tempered basis sets. 

The discussion above applies to uncontracted basis sets. Contracted basis sets present a few further problems. To properly represent the spin-orbit splitting, the two spin-orbit components should be contracted separately. The contraction is now j -dependent, rather than f-dependent, and can only be represented directly in a 2-spinor basis. The problem is not now confined to the small component. If the large-component scalar basis set includes contractions for both spin-orbit components, the product of the contracted basis functions for each spin-orbit component with the spin functions generates a representation for both spin-orbit components. Thus there is a duplication of the basis set that is close to linearly dependent, and some kind of scheme to project out linearly dependent components, either numerically or by conversion to a 2-spinor basis, is mandatory. The same applies to the small component. For example, the contracted p sets for the large-component and d sets both span the same space, but because of the contraction the (i-generated set cannot be made a subset of the -generated set, even if a dual family basis set is used. 

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

A variety of compound methods have been developed in an attempt to accurately model the thermochemical quantities we have been considering. These method-s attempt to achieve high accuracy by combining the results of several different calculations as an approximation to a single, very high level computation which is much too expensive to be practical. We will consider two families of methods the Gaussian-n methods and the Complete Basis Set (CBS) methods. 

The Complete Basis Set (CBS) methods were developed by George Petersson and several collaborators. The family name reflects the fundamental observation underlying these methods the largest errors in ab initio thermochemical calculations result from basis set truncation. 

As we noted in Chapter 7, the CBS family of methods all include a component which extrapolates from calculations using a finite basis set to the estimated complete basis set limit. In this section, we very briefly introduce this procedure. 

It has been shown [14] for both types ofbasis sets (1.1) and (1.2) that a given set of dimension n can be regarded as a member B of a family of basis sets that in the limit n oo become complete both in the ordinary sense and with respect to a norm in the Sobolev space - which is the condition for the eigenvalues and eigenfunctions of a Hamiltonian to converge to the exact ones. However, as to the speed of convergence the two basis sets (1.1) and (1.2) differ fundamentally. 

In terms of the three-dimensional local coordinate transformations R leading to the local basis set transformations Tf k>, the entire macromolecular system is naturally covered with a family of local coordinate systems. These local coordinate systems are also pairwise compatible, since the actual transformation V< > between any two such local systems of some serial indices k and k can be given explicitly as... 

In Table 1 we have collected the Cauchy moments, S(k), calculated for the Ne atom using the CCSD model and the n-aug-cc-pVAZ basis-set family. As can be seen from the table, single augmentation is not sufficient for the calculation of the Cauchy moments. On the other hand, going beyond the double augmentation... 

Table 1. The basis-set convergence of the Cauchy moments S(k) [a.u.] for Ne calculated with CCSD model and the n-aug-cc-VXZ basis-set family (all electrons correlated)...

The relative energies of the three protonated species are well reproduced by all methods from the Gn family. This can largely be explained by (a) the fact that all these methods involved CCSD(T) or QCISD(T) steps (and apparently triple excitations are quite important here) (b) the relatively rapid basis set convergence noted above, which means that it is not really an issue that the CCSD(T) and QCISD(T) steps are carried out in relatively small basis sets. CBS-QB3 likewise reproduces the relative energetics quite well. 

In die Pople family of basis sets, the presence of diffuse functions is indicated by a + in die basis set name. Thus, 6-31- -G(d) indicates that heavy atoms have been augmented with an additional one s and one set of p functions having small exponents. A second plus indicates the presence of diffuse s functions on H, e.g., 6-311- -- -G(3df,2pd). For the Pople basis sets, die exponents for the diffuse functions were variationally optimized on the anionic one-heavy-atom hydrides, e.g., BH2 , and are die same for 3-21G, 6-3IG, and 6-3IIG. In the general case, a rough rule of thumb is diat diffuse functions should have an exponent about a factor of four smaller than the smallest valence exponent. Diffuse sp sets have also been defined for use in conjunction widi die MIDI and MIDIY basis sets, generating MIDIX+ and MIDIY-I-, respectively (Lynch and Truhlar 2004) the former basis set appears pardcularly efficient for the computation of accurate electron affinities. 

In die Dunning family of cc-pVnZ basis sets, diffuse functions on all atoms are indicated by prefixing with aug . Moreover, one set of diffuse functions is added for each angular momentum already present. Thus, aug-cc-pVTZ has diffuse f, d, p, and s functions on heavy atoms and diffuse d, p, and s functions on H and He. An identical prescription for diffuse functions has been used by Jensen (2002) in connection with the pc-n basis sets. 

Neither of these basis set families is satisfactory for accurate calculations without the addition of polarization functions. Various ad hoc rules have been developed over the years for polarization exponents. Since SCF polarization is less sensitive to exponent choice than correlation, it is reasonable to let the latter determine the exponents. By fitting the results of correlated calculations on closed-shell hydrides, Ahlrichs and Taylor [47] arrived at the following formulas for d exponents (ay)... 

Results and Discussion. The calculated aluminum shielding tensors for A1H, A1F, A1C1 and A1NC for a variety of basis sets are presented in Table II. This particular family of basis sets was chosen as it is the one most commonly used in shielding calculations, especially for large systems. The experimental magnetic shielding tensors are obtained from... 

G. A. Petersson, T. G. Tensfeldt, and J. A. Montgomery Jr.,/. Chem. Phys., 94,6091 (1991). A Complete Basis Set Model Chemistry. III. The Complete Basis Set-Quadratic Configuration Interaction Family of Methods. 

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