Even-tempered Gaussians

The chief initial advantage to be gained from the even-tempered approximation is the reduction in the number of parameters to be optimized. For example, all exponents in an (s,p) basis set on carbon can be specified in terms of only four parameters (a p, ap, and pp) regardless of the number of primitives. 

It was subsequently discovered that the optimized a(N) and P(N), where N is the number of primitives, can be parameterized as functions of N. This 

The ease of generating arbitrarily large even-tempered sets has meant that some of the lowest energy SCF and Cl wave functions for first-row atoms and many small molecules have been produced with these basis sets. Tables 4 and 5 illustrate the sorts of results that may be achieved. 

Wilson and co-workers have proposed using a large enough even-tempered primitive set that the same set of exponents can be used for all first row elements.Of course, this requires very large basis sets if neon is to be treated with the same accuracy as lithium. 

The concepts of even-tempered exponents and universal basis sets were combined along with six constraints to produce geometrical basis sets for all atoms from H to Sr. The same set of exponents (but differing numbers of s, p, and d functions) appears. Errors in the atomic energies range from 0.0002 hartree for the 6s hydrogen basis to 0.0005 hartree for the (I3s,8p) carbon basis to 0.0555 hartree for the (20s,13p,8d) Sr basis. 

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The key feature is - both for the expansion of 1/r or e " in terms of even-tempered Gaussians - that, for large n, the cut-off error goes as exp(-anh) with h the step size and that the discretization errors goes as exp(—6/h), with a and b constants. While - for fixed n a small h is good for the discretization error, it is bad for the cut-off error and vice versa. The best compromise is that h 1/%/ni which implies that the overall error goes as exp(—Cy )-... 

We were able to show analytically - in an unexpectedly tricky way (the mathematical ingredients of which are in the appendix) - that the error of an expansion of the function 1/r in terms of an even-tempered Gaussian basis of dimension n goes as exp(-ci/n) provided that the two parameters of the even-tempered basis are optimized. 

Schmidt and Ruedenberg[29] optimized the constants at, a t,bt and hj, for individual atomic species. However, for the large and flmdble even-tempered basis sets required for calculations of high precision it has been recognized that the precise values of these constants is not critical and, building on the concept of a universal basis set[30]-[36], a universal systematic sequence of even-tempered Gaussian basis sets has been investigated[37] [38]. 

Figure 1 Comparison of the convergence behaviour of the second order correlation energy component for the F anion with sequences of even-tempered Gaussian basis sets containing s- and p-type functions designed for the neutral F and Ne atoms with the behaviour of this component for the Ne atom. The cmrves are labelled as follows - (a) AE2 Ne [A e]) (b) AE2 F -, [iVe]) (c) AEtiF--, [F]).

A revival of the idea of choosing exponents by means of geometric progression, though in a somewhat different form, is met with even--tempered Gaussian basis setsl. ... 

An overview of the development of the finite difference Hartree-Fock method for diatomic molecules has been presented. Given the power of the currently available workstations the exact solutions of the HF equations for systems containing 20-25 electrons can be obtained routinely. The approach has been successfully used to calibrate even-tempered gaussian basis sets the discrepancies between the finite difference and the finite basis set results for a number of diatomic molecules (including a 40-electron system GaF) could be narrowed to a couple of iHartree. 

Convergence of the ground state energy of the hydrogen molecular ion using a systematic sequence of distributed even-tempered Gaussian basis sets. 

Systematic Sequences of Even-Tempered Gaussian Primitives for Diatomic Molecules in Solution A Preliminary Study using Continuum Solvation Models... 

Systematic sequences of even-tempered Gaussian primitive functions... 

Bohr using a systematic sequence of even-tempered Gaussian primitive functions in the gas phase and in three... 

Estimated limits based on large uncontracted even-tempered Gaussian calculations. The Cl results for oxygen are based on the estimated full Cl energy derived from large multireference single- and double-excitation calculations. 

Table 5 Examples of Extended Even-Tempered Gaussian Basis Set C Calculations on Small Molecules" ...

R. C. Raffenetti, Int. J. Quantum. Chem. Symp., 9, 289 (1975). Even-Tempered Gaussian Atomic Orbitals First-Row Atoms. 

S. Wilson, Theor. Chim. Acta, 58, 31 (1980). Universal Systematic Sequences of Even-Tempered Gaussian Primitive Functions in Electronic Correlation Studies. 

For all the aluminumlike systems studied, even-tempered basis sets (Schmidt and Ruedenberg (1979)] of 26s22p20d Gaussian-type were used for MCDF SCF. In basis sets of even-tempered Gaussians [Schmidt and Ruedenberg (1979)], the exponents, C/ci are given in terms of the parame-... 

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