Example 15.5
What are the term symbols for the two higher-energy states in the Na atom’s
D lines? Use the solution to Example 15.4 for the values of the various angu-
lar momenta.Solution
In this case, the momenta of the single valence electron dictate the total an-
gular momenta; that is, (L,S,J) (,s,j). For the two upper electronic states,
the (L,S,J) quantum numbers are (1,^12 ,^12 ) and (1,^12 ,^32 ). (How did we know
that S^12 ? Because for a single electron, the vector sum of the single spin an-
gular momentum is ^12 .) Since L1, both term symbols are P states, and since
S^12 for both states, the multiplicities are both 2(^12 )
1 2. Therefore, the
two term symbols are(^2) P
1/2 and
(^2) P
3/2
These two term symbols, and others like them, are used to label not only
electronic states but also electronic transitions. For example, upon knowing
that the lowest-energy ground state of the Na atom has a^2 S1/2term symbol
(a fact that can be determined from the electron configuration of its valence
shell), the two transitions involved in the sodium D emission lines are labeled
(^2) P1/2→ (^2) S1/2 (^2) P3/2→ (^2) S1/2
Such labels make it easier to express the identities of the electronic states
involved in an electronic transition.
We are beginning to focus on quantum numbers Land Sinstead ofand
s(the spin for a single electron). This is because for multielectron atoms, the
quantum numbers and sare not “good” quantum numbers. The quantum
numbers and swere originally defined in terms of a single electron. Recall
that the concepts of electron shells and subshells were defined using the hy-
drogen atom and then applied as an approximation to larger atoms (“electron
configurations”). For multielectron systems, the eigenvalue equations involv-
ing and sare not strictly satisfied. Even though we presume to label electrons
as having a principal, orbital angular momentum (total and z-component) and
a spin angular momentum (again, total and z-component) using the aufbau
principle, such a labeling is an approximation. A better description of reality is
that an unfilled shell has a totalorbital angular momentum Land a totalspin
angular momentum S.Land S, and subsequently J, are the good quantum
numbers.
The situation is not as complicated as it might seem, because Land Sare
determined from the vector combinations of the individual and squantum
numbers from the electrons in the unfilled shell. Consider the simplest case,
two electrons in the outermost, unfilled subshell. (Remember that filled sub-
shells contribute no netorbital or spin angular momentum.) Two electrons
having individual orbital angular momenta 1 and 2 can couple so that the
net orbital angular momentum can have the possiblevalues
L 1
2 → 1 2 in integral steps (15.10)
where again the arrow means “through.” That is, the possible values ofL
range from the integers 1
2 through 1 2 in integral increments.
The absolute values imply that Lcan never be negative. For example, for two
15.5 Multiple Electrons: Term Symbols and Russell-Saunders Coupling 527