Even and Well Tempered Basis Sets

The optimization of basis function exponents is an example of a highly non-linear optimization problem (Chapter 12). When the basis set becomes large, the optimization problem is no longer easy. The basis functions start to become linearly dependent (the basis set approaches completeness) and the energy becomes a very flat function of the exponents. Analyses of basis sets that have been optimized by variational methods reveal that the ratio between two successive exponents is approximately constant. Taking this ratio to be constant reduces the optimization problem to only two parameters for each type of basis function, independent of the size of the basis. Such basis sets have been labelled even-tempered basis sets, with the ith exponent given as Z = ab', where a and b are fixed constants for a given type of function and nuclear charge. It was later discovered that the optimum a and b constants to a good approximation can be written as functions of the size of the basis set, M.5

Zi = ab'; i = 1,2,...,M ln(ln b ) = b ln M + b' (5.3)

The constants a, a', b and b' depend only on the atom type and the type of function (s or p). Even-tempered basis sets have the advantage that it is easy to generate a sequence of basis sets that are guaranteed to converge towards a complete basis. This is useful if the attempt is to extrapolate a given property to the basis set limit. The disadvantage is that the convergence is somewhat slow, and an explicitly optimized basis set of a given size will usually give a better answer than an even-tempered basis of the same size.

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.6 The idea is similar to the even-tempered basis sets, with the exponents being 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 according to eq. (5.4).

The a, b, g and d 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.

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