3. Atomic States, Shells, and Configurations
A one-electron atomic state is defined by the quantum numbers
nlmlms or nljmj, 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
ml, ms, and mj represent
the projections of the corresponding angular momenta along a particular
direction; thus, for example,
ml = -l, -l + 1
... l and ms = � 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 Nn′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(2l + 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 3p6 configuration thus represents a full
subshell and
3s2 3p6 3d10
represents a full shell for n = 3.
The parity of a configuration is even or odd according to
whether Σili 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 nlj or by
nl 2LJ 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
2S + 1 = 2s + 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
1H 1s 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 2n - 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.
