### 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

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,

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

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.