Schmidt-Ruedenberg sets

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. 

Even-temp)ered basis sets can be systematically increased (or decreased) in size by following the empirical prescription of Schmidt and Ruedenberg[29] in which the parameters ant and Pnt depend on the number of functions, n, in the basis t. The following empirical formulae lead to a sequence of basis sets which become complete in the limit of large n... 

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]. 

The parameters employed in generating sequences of even-tempered basis sets for the neutral F and Ne atoms were taken from the work of Schmidt and Ruedenberg[29] -... 

In the scheme which we label (a), the same basis set is employed for both the ground and excited state. Therefore, the same integrals over basis functions are used for both states. The values of the even-tempered parameters a and /3 for M = 6 are those which were optimized for the ground state of the atom as reported by Schmidt and Ruedenberg [9]. These values are given in Table 1 of reference [9j. Larger basis sets were generated by means of the recursion formulae [10] ... 

In scheme (6) the basis set is optimized by invoking the variation principle for each state considered. For the ground state the optimized values of the even-tempered parameters ao and / o given by Schmidt and Ruedenberg [9] are used. We add the subscript 0 to distinguish ground state values. For the excited state optimal ai and / i values for a sequence of Mi values are determined. 

In our third scheme, which we label (c), we optimized the parameter a and /3 the smallest basis set considered, i.e. M = 6 and then determine values of these parameters for the basis sets of larger size by using the recursions (40) and (41) with the values of a and b taken from the work of Schmidt and Ruedenberg [9]. 

The use of a systematic sequence of basis sets can, of course, be usefully combined with the use of a universal basis set(52). In Figure 8, we display the results of Hartree-Fock calculations on the radial beryllium-like ions Li", B+, C2+, N +, Ol++, F5+, Ne6+, using the basis set given by Schmidt and Ruedenberg(56) specifi-ically for the beryllium atom. It can be seen that for the positive ions the basis sets give a uniform convergence rate. In... 

On the basis of atomic matrix Hartree-Fock calculations using even-tempered sets of Gaussian-type functions of various sizes in which a and f were optimized with respect to the energy, Ruedenberg and his co-workers proposed empirical functional forms for the dependence of a and P on N in particular, Schmidt and Ruedenberg158 put forward the forms... 

The basis sets employed in the present work contained 2N s-type functions and N p- functions. The, so-called regularized , coefficients a , bg, ag, bg, Up, bp, flp, bp used in eqns. (7) and (8) to generate the parameters am and Pg g for these even-tempered basis sets were taken from Schmidt and Ruedenberg[65] for each atom considered. 

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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