Even-tempered orbital basis

Let us comment at this point on the use of the even-tempered method for generating orbital exponents. A universal basis set clearly need not be an even-tempered one. However, even-tempered basis functions do span the one-electron space in a fairly uniform fashion. Unlike a set of exponents determined by energy optimization for a particular system, we expect that the even-tempered functions will be more suitable for transferring between systems. Even-tempered orbital exponents are defined through the recursion... 

Even-tempered basis sets [40] consist of GTOs in which tlie orbital exponents belonging to series of... 

Even-tempered basis sets (M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3961 (1979)) consist of GTOs in which the orbital exponents ak belonging to series of orbitals consist of geometrical progressions ak = a, where a and P characterize the particular set of GTOs. 

The contracted basis in Figure 28.3 is called a minimal basis set because there is one contraction per occupied orbital. The valence region, and thus chemical bonding, could be described better if an additional primitive were added to each of the valence orbitals. This is almost always done using the even-tempered method. This method comes from the observation that energy-optimized exponents tend to nearly follow an exponential pattern given by... 

Sets of orbital exponents r (/, k) have been proposed mainly by Huzinaga [3], van Duijneveldt [4], Pople et al. [5]. A systematic construction of basis sets of arbitrary dimension is possible in terms of the even tempered concept of Ruedenberg et al. [6,7 ], or of some more sophisticated generalizations [8,9,10]. For a recent comprehensive review on basis sets see Feller and Davidson [11]. 

The Hartree-Fock ground state of the F anion is described by orbitals of s Emd of p symmetry. In the first part of this study, attention was restricted to the convergence of the second order many-body perturbation theory component of the correlation energy for stematically constructed even-tempered basis sets of primitive Gaussian-typ>e functions of s and p symmetry. 

TABLE 1. Self-Consistent Field energies (in hartree) of the He atom for some excited states as a function of the size, M, of the even-tempered basis set used to parametrize the orbitals. In the column headed (a) the same even-tempered basis set optimized for the ground state, is used for aU states ... 

In this paper, the systematic truncation of a distributed universal even-tempered basis set capable of supporting high precision in both mauix Haitree-Fock and second order many-body perturbation theory calculations is explored using the ground state of the boron fluoride molecule as a prototype. The truncation procedure adopted is based on the magnitude of the orbital expansion coefficients associated with a given basis function in each of the occupied orbitals. 

In our previous work [15-22], we have used a universal sequence of distributed even-tempered basis set in which the exponents for each symmetry type form a geometric series according to the formula tk = on each expansion centre. Different parameters a and fi were used for the functions of x-type in order to provide the additional flexihility required to obtain a satisfactory description of any cusp at the point upon which the functions were centred. Common values of a and /3 were used for p-type functions and functions of higher symmetry. The orbitals, both occupied and unoccupied, are then approximated by an expansion of the form... 

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