Temperate orbitals

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... 

R. D. Bardo and K. Ruedenberg, /. Chem. Phys., 60, 918 (1974). Systematic Approach to Extended Even-tempered Orbital Bases for Atomic and Molecular Calculations. 

The p and d orbitals with m = 0, i.e., the z and z orbitals, respectively, which lie along the z axis, can conveniently be called the polar or axial orbitals. Analogously, the p and d orbitals with m = /, i.e., the x, y, xy, and x -y orbitals, which lie in the equatorial plane (z = 0) can conveniently be called the equatorial orbitals. The remaining d orbitals with m = 1, i.e., the xz and yz orbitals, can conveniently be called the temperate orbitals, since their lobes are oriented towards the temperate zones of an earth where the z axis passes through the north and south poles. 

D. F. Feller and K. Reudenberg, Systematic approach to extended even-tempered orbital bases for atomic and molecular calculations, Theoret. Chim. Acta 52 231 (1979) M. W. Schmidt, and K. Ruedenberg, Effective convergence to complete orbital bases and to the atomic Hartree-Fock limit through systematic sequences of Gaussian primitives, J. Chem. Phys. 71 395 (1979). 

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 ... 

It has been shown that Slater type orbitals restricted to Is, 2p, 3 /,... functions can lead to good results for total energies. Orbital exponents, C, may be chosen by the even-tempered formula93-99... 

Bardo, R. D. and Ruedenberg, K., Even-tempered atomic orbitals. VI. Optimal orbital exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and oxygen in molecules,. 1. Chem. Phys. 60, 918-931 (1974). 

---