Physical Chemistry , 1st ed.

angular momentum and spin angular momentum, that determines the total

electronic eigenvalue energy and thus dictates the changes in energy recorded

in an electronic spectrum. It is important, then, to understand how the orbital

and spin angular momenta interact. This interaction is called spin-orbit cou-

pling.Spin-orbit coupling acts to make individual electronic energies slightly

different from the equation above, depending on how the spin angular mo-

mentum is interacting with the orbital angular momentum. The overall effect

is to splitthe energy levels into a larger number of discrete energy levels. The

net result is that the electronic spectrum of a multielectron atom is more

complicated.

Experiments have indicated that the total angular momentum and the zcom-

ponent of the total angular momentum for an electron are quantized. (This

situation is very similar to the rotations of molecules.) As such, the allowed

values for total angular momenta quantum numbers are similar to those for

orbital or spin angular momenta. We will adopt the convention of using the

quantum numbers and mto refer to the orbital angular momenta of an

electron,sand msto label the spin angular momenta, and introduce the quan-

tum numbers jand mjto refer to the total angular momentum and the zcom-

ponent of the total angular momentum for a single electron. As with all angu-

lar momenta,mjcan have 2j 1 possible values, ranging from jto j. We also

adopt the convention of using capital letters for the various quantum numbers

for the total angular momenta of several electrons. We will use ,m, and so

on for a single electron, but L,ML,S,MS,J, and MJfor the various combined

momenta of more than one electron.

Orbital and spin angular momenta combine (that is, couple) in vector fash-

ion. Consider the electron having 0 (that is, an electron in the ssubshell)

as shown in Figure 15.2. The spin angular momentum sis always ^12 for an elec-

tron, but it can be oriented in two different directions (corresponding to the

quantum number mshaving values of ^12 or ^12 ). The total angular momen-

tum, labeled by the quantum number j, is determined from the combination

of the and svalues, or simply ^12 . However, the jvector can have two possible

orientations with respect to the z-axis, corresponding to two different possible

values ofmj, as shown in Figure 15.2.

For a single p-subshell electron,jcan have two possible values, correspond-

ing to the two possible vector combinations of the vector (which has mag-

nitude 1) and the svector (which has magnitude ^12 ). This is illustrated in Figure

15.3. For the pelectron:

j s or  s (15.6)

 1 

1

2

 or 1 

1

2



j

3

2

 or 

1

2



Generally, the possible values ofjare

j s→ s in integer steps (15.7)

where the arrow means “through.” For j^12 ,mjcan be either ^12 or ^12 , just as

for the single selectron. However, for j^32 ,mjcan have values of ^32 , ^12 ,^12 ,or

^32 . Thus, the single pelectron has more possible values for its total angular mo-

mentum, and more possible zcomponents of its total angular momentum,

than a single selectron.

15.4 Angular Momenta: Orbital and Spin 523



1

mj  2

1

 0 s  2



1

mj   2

1

 0 s  2



3

j  2

1

 1 s  2



1

j  2

1

 1 s  2

Figure 15.3 The combination of the orbital
angular momentum of a p-subshell electron (rep-
resented by a vector) with the spin angular mo-
mentum can also lead to two possible total angu-
lar momenta. Compare with Figure 15.2.

Figure 15.2 The combination of the orbital
angular momentum of an s-subshell electron
(represented by a single dot) with the spin angu-
lar momentum yields a total angular momentum
jof^12 that can have two possible zorientations,
corresponding to mj^12 and mj ^12 , as shown.