One-Electron Atoms with Cores

One-Electron Atoms with Cores

One-electron atom with core of net charge Z.

Question: Consider now an atom with one valence electron of principal quantum number $n$ outside a core (comprising the nucleus and filled shells of electrons) of net charge $Z$. Which of the following forms describes the lowest order correction to the hydrogenic energy $E_{n}={\frac {Z^{2}R}{n^{2}}}$?

• A. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{Z^2R}{n^2}+\delta}
• B. ${\frac {Z^{2}R}{n^{2}}}+{\frac {\delta }{n^{2}}}$
• C. ${\frac {Z^{2}R}{n^{2}}}+{\frac {\delta }{n^{3}}}$
• D. ${\frac {Z^{2}R}{(n-\delta )^{2}}}$

Both C and D are correct answers, equivalent to lowest order in $\delta$. This scaling can be derived via perturbation theory. In particular, we can model the atom with a core of net charge $Z$ as a hydrogenic atom with nuclear charge $Z$---described by Hamiltonian $H_{0}$---and add a perturbing Hamiltonian $H'$ to account for the penetration of the valence electron into the core. As we showed before, $\psi _{n\ell }(r){\overrightarrow {_{r\rightarrow 0}}}r^{\ell }n^{-{\frac {3}{2}}}$. Thus, for the Hamiltonian

$H=H_{0}+H',$

where the penetration correction $H'$ is localized around $r=0$, the deviation from hydrogenic energy levels $E_{n}$ is

$\Delta E_{n}=\langle \psi _{n\ell }|H'|\psi _{n\ell }\rangle \propto {\frac {1}{n^{3}}},$

where the proportionality constant depends on $\ell$. In particular, we can define the quantum defect $\delta _{\ell }$ such that

$\Delta E_{n}={\frac {\delta _{\ell }}{n^{3}}}2R.$

The energy levels then take the form

$-E_{n}={\frac {R}{n^{2}}}+{\frac {2R}{n^{3}}}\times \delta _{\ell }\simeq {\frac {R}{(n-\delta _{\ell })^{2}}}.$

This accounts for phenomenological formula for term values in the Balmer-Ritz formula,

$T_{n}={\frac {Z^{2}R}{n*^{2}}}.$

This formula, in which the $n*$ are non-integers but are spaced by integer differences, had been deduced spectroscopically long before it could be understood quantum mechanically.

It is possible to derive the quantum defect based on solving the radial Schrodinger equation in the JWKB approximation, which sheds light on the  relation of the quantum defect to the phase shift of an electron scattering off a modified Coulomb potential; and for a sample calculation of the quantum defect for a model atom.