Hydrogen optimized exponents

A similar type basis set was developed by Spackman, " who added to a standard 6-3IG basis set a set of diffuse s and d Gaussian functions for first-row atoms and a set of diffuse s and p functions for hydrogen. The exponents of the s functions were fixed at 0.25 times the most diffuse s exponent in the 6-31G basis, and the exponents of the additional p and d functions were optimized to maximize the polarizabilities for a series of AH compounds. This basis, referred to as 6-31G(+sd-l-sp), was reported to give MP2 polarizabilities accurately to 2% of experiment. Like the 6-31G-I-PD basis, because of its relatively small size, this basis has also been used for hyperpolarizability calculations. 

One set of diffuse d-fiinctions with a momentum-optimized exponent of 0.25 provides an already reasonable value for the dispersion stabilization [55]. Two sets of d-polarization functions with standard exponents of 1.6 and 0.4 do not provide better description. The addition of a diffuse sp shell to the standard d-polarization functions (6-31+G basis set) is not sufficient. Computational studies conclude that p-polarization functions on hydrogen atoms are not very important, and the f-polarization functions on second-row elements bring only a small improvement in stabilization [35,36]. 

The second part of this paper concerns the choice of the atomic basis set and especially the polarization functions for the calculation of the polarizability, o , and the hyperpo-larizabiliy, 7. We propose field-induced polarization functions (6) constructed from the first- and second-order perturbed hydrogenic wavefunctions respectively for a and 7. In these polarization functions the exponent ( is determined by optimization with the maximum polarizability criterion. These functions have been successfully applied to the calculation of the polarizabilities, a and 7, for the He, Be and Ne atoms and the molecule. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

The Veillard basis set [23] (1 ls,9p) has been used for A1 and Si, and the (1 ls,6p) basis of the same author has been retained for Mg. However, three p orbitals have been added to this last basis set, their exponents beeing calculated by downward extrapolation. The basis sets for Al, Si and Mg have been contracted in a triple-zeta type. For the hydrogen atom, the Dunning [24] triple-zeta basis set has been used. We have extended these basis sets by mean of a s-type bond function. We have optimized the exponents a and locations d of these eccentric polarization functions, and the internuclear distance R of each of the studied molecules. These optimized parameters are given in Table 3. 

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). 

Figure 4.11 Calculation of the 2s orbital energy in hydrogen with the sto-3g ls> basis set, Table 1.6, for the 2s Slater function rendered orthogonal to the sto-3g ls> function. The initial calculation returns a poor estimate of the energy terms and 2 for the minimization condition on the least-squares integral of Chapter 3. Optimization based on the minimization of the energy, using SOLVER on the Slater exponent, returns closer agreement with the exact results.

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