Even- and Well-tempered Basis Sets

yand 5 parameters are optimized for each atom. The exponents are the same for all types of angular momentum functions, and s-, p- and d-functions (and higher angular momentum) consequently have the same radial part. 

A well-tempered basis set has four parameters, compared with two for an even-tempered one, and is consequently capable of giving a better result for the same number of functions. Petersson et alJ have proposed a somewhat more general parameterization based on expanding the logarithmic exponents in a polynomial of order K in the basis function number. 

Optimization of basis sets is not something the common user needs to worry about. Optimized basis sets of many different sizes and qualities are available either in the forms of tables, websites or stored internally in 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 interested is an accurate value for the electron density at the nucleus (for example for determining the Fermi contact contribution to spin-spin coupling (see Section 10.7.6)) then basis functions with very large exponents are required. 

Alternatively, for calculating hyperpolarizabilites, very diffuse functions are required. In such cases, the basis function optimization is in terms of the property of interest, and not in terms of energy, i.e. basis functions are added until the change upon addition of one extra function is less than a given threshold. 

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 

The constants a,a b and b depend only on the atom type and the type of function. 

Even-tempered basis sets have the same ratio between exponents over the whole 

The constants a,a b and b depend only on the atom type and the type of function. Even-tempered basis sets have the advantage that it is easy to generate a sequence of basis sets which are guaranteed to converge towards a complete basis. This is useful if 

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  

---

When developing the large basis sets which are required for the reliable calculation of van der Waals interactions, it is important to ensure that the basis set be constructed and extended in a systematic fashion. In recent work, Wells and Wilson have employed systematic sequences of even-tempered basis sets of Gaussian-type functions in conjunction with many-body perturbation theory calculations to study van der Waals interaction potentials, thereby ensuring basis set superposition errors and size-inconsistency problems are controlled. They used the Boys-Bernardi procedure as a test for the magnitude of the basis set superposition error rather than as a correction. 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

---