Hydrogen wavefunction

Because the hydrogen wavefunctions are mutually orthogonal, this sum of integrals simplifies to... 

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. 

In treating complex atoms and molecules, these hydrogenlike wavefunctions are still too complicated in most applications. In treating low-energy (up to a few eV) problems, the behavior of the wavefunctions near the nuclei of the atoms is not important. Therefore, in the hydrogen wavefunctions, only the term with the highest power of r is significant. Based on this observation, Zener (1930) and Slater (1930) proposed a simplified form of the atomic wavefunctions ... 

In this subsection, we show that by evaluating the modified Bardeen integral, Eq. (7.14), with the distortion of the hydrogen wavefunction from another proton considered, as shown by Holstein (1955), an accurate analytic expression for the exact potential of the hydrogen molecular ion is obtained. 

The hydrogenic requirement that the wavefunction be finite at the origin has been replaced by the requirement that atr2r0 the wavefunction be shifted in phase from the hydrogenic wavefunction by r. This new boundary condition necessitates g functions as well as the regular / functions in the coulomb r > r0... 

Values of (r-4) and (r-6) for hydrogenic wavefunctions have been calculated analytically, and the expressions are given in Table 2.3. Examining the forms of Table 2.3, it is apparent that both (r-4) and (r-6) exhibit n-3 scalings, due to the normalization of the radial wavefunction. However, they exhibit different i dependences (r-4) scales as 5, and (r-6) scales as 8. As a result of the very different i scalings the contributions of the dipole and quadrupole polarizabilities to the quantum defect are easily separated by measurements of the intervals between several i series. Furthermore, for high t, (r -4> , and as a result, for high ,... 

Here

zero order representation of the Ba atom as long as the inner turning point of the outer n(. electron is at a larger radius than the outer turning point of the Ba+ n C electron. 

The wavefunctions Auger electron are of interest. 

Figure 7.6 Radial wavefunctions Pls(r) = rRls(r) of helium. HYDR is the hydrogenic wavefunction with Z = 2 HF is the Hartree-Fock wavefunction. From [BJ066].

Using hydrogenic wavefunctions for the radial functions R /r) for simplicity, one gets... 

Using hydrogenic wavefunctions with Z = 2, one obtains the following results for the lowest expansion coefficients F( ri, r2) [GMM54] ... 

Use the hydrogen wavefunctions to find the value of z where the wavefunction i/sP+ = (fas + Pip,)/ fl is most positive, and the position where it is most negative. 

The hydrogenic wavefunctions have the general analytical form ... 

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