Quantum defects and Rydberg series

Spectra of atoms often exhibit phenomena associated with a Rydberg series, that is, states where one of the electrons is in an nl orbital, with n assuming a sequence of values. An example is the $1s^22snd\;^3\!D$ series in Be, $n=3,4,5,\ldots$. For such a series, a useful concept is that of a quantum defect parameter $\delta$. In hydrogen, the ionization energy (IE) is -1/(2n²) au. In complex neutral systems, the effective charge would be the same at large r. As n increases, the mean radius becomes larger and the probability of the electron being in the hydrogen-like potential increases. Thus one could define an effective quantum number $n^* = n - \delta$ such that

$$\mbox{IE}(nl) = \frac{1}{2 (n -\delta)^2}.$$ (44)

In general, when one is not dealing with neutral systems, the equation must be modified to

$$\mbox{IE}(nl) =\frac{1}{2}\left[\frac{(Z-N+1)}{(n -\delta)}\right]^2.$$

Often this parameter is defined with respect to observed data, but it can also be used to evaluate Hartree-Fock energies where $\mbox{IE}=\varepsilon_{nl,nl}/2$ so that $\varepsilon_{nl,nl} = [(Z-N+1)/((n -\delta)]^2$. table:qd shows the screening parameter, effective quantum number, and quantum defect for the Hartree-Fock 2snd 3 D/ and 1 D/ terms in Be as a function of n. For the triplet series, the screening is less than 3, so the $Z_{eff}= Z-\sigma(nl)$ is greater than one. At the same time, for the singlet series, the screening is slightly greater than 3, making the effective nuclear charge less than one. This is the effect of exchange and it is reflected also in the quantum defect parameter which is positive for the triplets and negative for the singlet series.

Table 3.6: The screening and quantum defect parameters of the 2sndRydberg series in Be

n	3 D/				1 D/
	$\sigma(nl)$	n*	$\delta(nl)$		$\sigma(nl)$	n*	$\delta(nl)$
3	2.960	2.968	0.032		3.012	3.014	-0.014
4	2.972	3.960	0.040		3.008	4.012	-0.012
5	2.979	4.957	0.043		3.006	5.013	-0.013
6	2.983	5.955	0.045		3.005	6.013	-0.013

 
2001-01-09