### 13. Term Series, Quantum Defects, and Spectral-line Series

The Bohr energy levels for hydrogen or for a hydrogenic (one-electron) ion are given by

$$E_n = -\frac{Z^2}{n^2}\quad,$$

in units of *R*_{M}, the Rydberg for the appropriate nuclear mass. For a multielectron atom, the deviations of a series of (core) *nl* levels from hydrogenic *E*_{n} values may be due mainly to core penetration by the *nl* electron (low *l*-value series), or core polarization by the *nl* electron (high *l*-value series), or a combination of the two effects. In either case it can be shown that these deviations can be approximately represented by a constant quantum defect *δ*_{l} in the Rydberg formula,

$$E_{nl} = - ~ \frac{Z^2_c}{(n-\delta_1)^2} = - ~ \frac{Z^2_c}{(n^*)^2} ,$$

where *Z*_{c} is the charge of the core and *n* = n - δ* is the effective principal quantum number. If the core includes only closed subshells, the *E*_{nl} values are with respect to a value of zero for the (core)^{1}S_{0} level, i.e., the ^{1}S_{0} level is the limit of the (core)*nl* series. If the quantities in Eq. (11) are taken as positive, they represent term values or ionization energies; the term value of the ground level of an atom or ion with respect to the ground level of the next higher ion is thus the principal ionization energy.

If the core has one or more open subshells, the series limit may be the baricenter of the entire core configuration, or any appropriate sub-structure of the core, down to and including a single level. The *E*_{nl} values refer to the series of corresponding (core)*nl* structures built on the particular limit structure. The value of the quantum defect depends to some extent on which (core)*nl* structures are represented by the series formula.

The quantum defect in general also has an energy dependence that must be taken into account if lower members of a series are to be accurately represented by Eq. (11). For an unperturbed series, this dependence can be expressed by the extended Ritz formula

$$\begin{eqnarray*}

\delta&=& n - n^*\\ &~& ~\\

&=& \delta_0 + \frac{a}{(n-\delta_0)^2} + \frac{b}{(n-\delta_0)^4} +

\ldots \quad ,\end{eqnarray*}$$

with *δ*_{0},*a,b*... constants for the series (*δ*_{0} being the limit value of the quantum defect for high series members) [17]. The value of *a* is usually positive for core-penetration series and negative for core-polarization series. A discussion of the foundations of the Ritz expansion and application to high precision calculations in helium is given in Ref. [18].

A spectral-line series results from either emission or absorption transitions involving a common lower level and a series of successive (core)*nl* upper levels differing only in their *n* values. The principal series of Na I, 3*s*^{ 2}S_{1/2} - *np*^{ 2}P°_{1/2,3/2} (*n* ≥ 3) is an example. The regularity of successive upper term values with increasing *n* [Eqs. (11), (12)] is of course observed in line series; the intervals between successive lines decrease in a regular manner towards higher wavenumbers, and the series of increasing wavenumbers converges towards the term value of the lower level as a limit.