### 3. Atomic States, Shells, and Configurations

A one-electron atomic *state* is defined by the quantum numbers
*nlm*_{l}m_{s} or *nljm*_{j}, with
*n* and *l* representing the principal quantum number and the orbital
angular momentum quantum number, respectively. The allowed values of *n*
are the positive integers, and *l* = 0, 1, ...,
*n* - 1. The quantum number *j* represents the angular
momentum obtained by coupling the orbital and spin angular momenta of an
electron, i.e., **j** = **l** + **s**, so
that *j* = *l* ± ^{1}/_{2}. The magnetic quantum numbers
*m*_{l}, *m*_{s}, and *m*_{j} represent
the projections of the corresponding angular momenta along a particular
direction; thus, for example,
*m*_{l} = -*l*, -*l* + 1
... *l* and *m*_{s} = ± ^{1}/_{2}.
The central field approximation for a many-electron atom leads to wave
functions expressed in terms of products of such one-electron states
[2,3]. Those electrons having the
same principal quantum number *n* belong to the *shell* for that
number. Electrons having both the same *n* value and *l* value belong
to a *subshell*, all electrons in a particular subshell being
*equivalent*. The notation for a *configuration* of *N*
equivalent electrons is *nl*^{ N}, the superscript usually
being omitted for *N* = 1. A configuration of several subshells
is written as
*nl*^{ N}n′*l*′^{ M} ... .
The numerical values of *l* are replaced by letters in writing a
configuration, according to the code *s*, *p*, *d* for
*l* = 0, 1, 2 and *f*, *g*, *h* ... for
*l* = 3, 4, 5 ..., the letter *j* being omitted.

The Pauli exclusion principle prohibits atomic states having two electrons with
all four quantum numbers the same. Thus the maximum number of equivalent
electrons is 2(2*l* + 1). A subshell having this number of
electrons is *full*, *complete*, or *closed*, and a subshell
having a smaller number of electrons is *unfilled*, *incomplete*, or
*open*. The 3*p*^{6} configuration thus represents a full
subshell and
3*s*^{2} 3*p*^{6} 3*d*^{10}
represents a full shell for *n* = 3.

The *parity* of a configuration is *even* or *odd* according to
whether Σ_{i}l_{i} is even or odd, the sum being
taken over all electrons (in practice only those in open subshells need be
considered).

### 4. Hydrogen and Hydrogen-like Ions

The quantum numbers *n*, *l*, and *j* are appropriate
[4]. A particular *level* is
denoted either by *nl*_{j} or by
*nl* ^{2}*L*_{J} with
*L* = *l* and *J* = *j*. The latter
notation is somewhat redundant for one-electron spectra, but is useful for
consistency with more complex structures. The *L* values are written with
the same letter code used for *l* values, but with roman capital letters. The
*multiplicity* of the *L term* is equal to
2*S* + 1 = 2*s* + 1 = 2.
Written as a superscript, this number expresses the *doublet* character of
the structure: each term for *L* ≥ 1 has two levels, with
*J = L* ± ^{1}/_{2}, respectively.
The Coulomb interaction between the nucleus and the single electron is
dominant, so that the largest energy separations are associated with levels
having different *n*. The hyperfine splitting of the
^{1}H 1*s* ground level
[1420.405 751 766 7(10) MHz] results from the interaction
of the proton and electron magnetic moments and gives rise to the famous
21 cm line. The separations of the 2*n* - 1 excited levels
having the same *n* are largely determined by relativistic contributions,
including the spin-orbit interaction, with the result that each of the
*n* - 1 pairs of levels having the same *j* value is
almost degenerate; the separation of the two levels in each pair is
mainly due to relatively small Lamb shifts.

### 5. Alkalis and Alkali-like Spectra

In the central field approximation there exists no angular-momentum coupling
between a closed subshell and an electron outside the subshell, since the net
spin and orbital angular momenta of the subshell are both zero. The *nlj*
quantum numbers are, then, again appropriate for a single electron outside
closed subshells. However, the electrostatic interactions of this electron with
the core electrons and with the nucleus yield a strong *l*-dependence of
the energy levels [5]. The differing
extent of "core penetration"" for *ns* and *np* electrons
can in some cases, for example, give an energy difference comparable to or
exceeding the difference between the *np* and
(*n* + 1)*p* levels. The spin-orbit fine-structure
separation between the *nl* (*l* > 0) levels
having *j* = *l* - ^{1}/_{2} and *l* + ^{1}/_{2}, respectively, is relatively small.