15.5 Multiple Electrons: Term Symbols
and Russell-Saunders Coupling
Understanding the momentum combinations of lone electrons in a subshell is
straightforward. But what about the majority of atomic systems, neutral or
ionized, that have more than one electron in the outermost subshell? How do
we understand all the possible ways the orbital and spin angular momenta of
multiple electrons can interact?
Two common systems are used to understand multiple-electron spin-orbit
coupling. They are called the Russell-Saunders coupling schemeand the j-j cou-
pling scheme.The Russell-Saunders (or RS) coupling scheme is valid for low-Z
(that is, low atomic number, typically 30 or lower) atoms where spin-orbit cou-
pling is relatively weak. It treats total orbital angular momentum of multiple
electrons separately from the total spin angular momentum. The j-jcoupling
scheme is used for high-Z(Z30) atoms in which spin-orbit coupling is so
large that a total angular momentum jfor each individual electron must be de-
termined first. We will consider the RS coupling scheme exclusively in this text.
Before we consider the coupling scheme, we need to introduce the idea of
the term symbol. We will be working with total orbital angular momenta and
total spin angular momenta, and being vector quantities the individual angu-
lar momenta of two or more electrons can combine. A term symbolis a short-
hand device for indicating the values of the orbital, spin, and total angular mo-
menta of an electronic state (and since the energy of the state depends on the
values of these angular momenta, the term symbol becomes a useful way to la-
bel an electronic state).
IfLis the quantum number indicating the vector sum of the orbital angu-
lar momentum of an electronic state, and Srepresents the vector sum of the
spin angular momentum of the electronic state, and Jrepresents the total an-
gular momentum of the electronic state, then the term symbol would be con-
structed as2S (^1) L
J (15.9)
Instead of using the numerical value for L, a letter label is used (just like the
s,p,d,f,...subshell designations stand for 0,1,2,3,...in atoms).The
following capital letters are used to indicate the value for L:
L Letter designation
0 S
1 P
2 D
3 F
..
..
..
Such letters are commonly used to refer to various electronic states; that
is, one hears of S states, P states, and so on. The left superscript is not Sbut
2 S 1 (for reasons that become clear shortly). The quantity 2S 1 is called
the multiplicityof the state. States that have a multiplicity of 1 are called singlet
states, and states that have multiplicities of 2 are doublet states. There are also
triplet, quartet, and so on, states. The right subscript,J, indicates the total
angular momentum and is determined by Land S, as before. The following
example shows how term symbols are constructed.
526 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure