pelectrons ( 1 2 1), the possible values for Lare 2, 1, and 0. These pos-
sible values ofLindicate the possible vector combinations of the mquantum
numbers of the two electrons.
For the Sof a multielectron, unfilled-subshell atom, a similar relation-
ship applies. For the simple case of two electrons, the possiblevalues ofSare
given by
Ss 1
s 2 →s 1 s 2 in integral steps (15.11)For electrons,s^12 , so for two electrons the possible values ofSare 1 and 0.
These possible values ofScorrespond to the possible vector combinations of
the msquantum numbers of the two electrons. The vector combinations of
multiple and svalues are similar to those depicted in Figures 15.2 and 15.3.
How do these rules help us in understanding electronic energy levels of
atoms? The first step is to recognize that an atom can have all combinations of
orbital and spin angular momentum—that is, all possible combinations ofL
and S—that are possible. The one immediate additional factor to consider is
the Pauli principle. For example, the carbon atom has a ground-state electron
configuration 1s^22 s^22 p^2 .Within this electron configuration,the atom can have
several possible combinations ofLand S, only one of which is the lowest-energy
ground state. This means that there are excited statesof the carbon atom that
still have the electron configuration 1s^22 s^22 p^2. Each of these states, ground
and excited, will have its own term symbol, so within this electron configura-
tion several possible term symbols label the individual energy levels. For a car-
bon atom, the possible values ofLare 2, 1, and 0 (satisfy yourself that this is
the case), and the possible values for Sare 1 and 0 (again, satisfy yourself that
this is true). All possible combinations ofLand Slead to the following possi-
ble term symbols,Jnot included:(^1) S, (^1) P, (^1) D, (^3) S, (^3) P, and (^3) D
Although these are all of the combinations, some of them are ruled out by
the Pauli principle. For example, the term symbol^3 D implies that for both
electrons,m
1 and that the spins are oriented in the same direction (that
is, the msfor both electrons is the same). This implies that both electrons have
the same set of four quantum numbers, which is forbidden by the Pauli prin-
ciple. Therefore, the^3 D term symbol cannot and does not existfor this electron
configuration. A similar argument can be made for the^3 S term: both electrons
could have m0 and msthe same, but this is forbidden by the Pauli princi-
ple. Therefore, the^3 S term symbol does not exist either.
The^1 P term symbol also does not exist, not because of Pauli principle
reasons, but because the remaining term symbols collectively define all the
possible ways the two pelectrons can couple their orbital and spin angular
momentum. A^1 P term symbol is redundant and therefore unnecessary. So, the
possible term symbols for the ground-state electron configurationare
(^1) S, (^1) D, and (^3) P
Again, note the distinction between electron configuration and term symbols.
All three of the above term symbols describe certain states of the two pelec-
trons in the ground-state electron configuration of the carbon atom (or, for
that matter, any atom that has a p^2 valence subshell). However, because they
represent different total orbital and spin angular momenta, they represent
states that have different total energies, even though they are for a carbon atom
having a 1s^22 s^22 p^2 electron configuration.
528 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure