Physical Chemistry , 1st ed.

number of states is the sum of these degeneracies, which is 45. There are 45

individual electronic states within the d^2 electron configuration.

These examples might require a rethinking of the idea of excited states.

Previously, we have considered an excited state as any state above the ground

state, and these excited states have usually been obvious changes in relatively

major quantum numbers for the system. In the case of the hydrogen atom, the

electronic states have a quantized energy dictated by the principal quantum

number n, and the electronic spectrum of hydrogen is due to changes in the n

quantum number (and more specifically to concurrent changes in the quan-

tum number, but this is not immediately apparent because of the degeneracy of

the hydrogen electronic states). Moreover, we have used a hydrogen-atom ap-

proximation for the labeling of electronic states for multiple-electron atoms, and

so we have used the labels 1s,2s,2p, and so on for the orbitals of larger atoms.

One might presume, then, that electronic spectra are due to changes in elec-

trons from one orbital to another, as with the hydrogen atom or even the

sodium atom discussed above (in that case we were treating the single valence

atom of sodium as a hydrogen-like system). However, for atoms with a multi-

electron valence subshell, it is more complicated. For such systems, excited

states occur withinthe lowest-energy electron configuration. Only one of the

term symbols represents the lowest-energy ground state of the atom. The other

term symbols are, by definition, excited states. This is despite the fact that all

of the states are part of the same electron configuration.

The next question is, then, which of the term symbols represents the ground

electronic state? In 1925–1927, after a detailed examination of spectra, Friedrich

Hund formulated some rules to determine the term symbol for the ground

state.Hund’s rulesare:

1. The term(s) having the higher multiplicity are lower in energy. If this
   unambiguously determines the term symbol for the ground state, then
   stop here.


2. Of the term symbols having the highest multiplicity, the higher the value
   ofL, the lower the energy.


3. If the valence subshell is less than half-filled, the lower the J, the lower the
   energy. If the valence subshell is more than half-filled, the higherthe J,
   the lower the energy. (Subshells that are exactly half-filled will always
   have an S term symbol as the highest-multiplicity state and will therefore
   have only one possible value for J.)
   According to these rules, the lowest-energy state for a carbon atom in the p^2
   configuration is predicted to be^3 P 0 , which is the case. The^3 P 1 electronic state
   is slightly higher in energy (16.4 cm^1 ), the^3 P 2 state is slightly higher still
   (43.5 cm^1 ). The^1 D 2 electronic state is much higher in energy (in fact, it is
   10,194 cm^1 above the ground state), and finally the^1 S 0 state is the highest-
   energy state (21,648 cm^1 above the ground state) in the manifold of elec-
   tronic states within the p^2 electron configuration. Figure 15.5 illustrates the dif-
   ferent states of this manifold. (Hund’s rules are also applicable to molecular
   electronic states, as we will discuss later in this chapter.)

Example 15.8

Determine the expected ground state of an atom of Ni, which has a d^8 va-

lence subshell configuration.

15.5 Multiple Electrons: Term Symbols and Russell-Saunders Coupling 531

Energy

21648.4 cm^1

16.4 cm^1

43.5 cm^1

10193.7 cm^1

(^1) S 0
(^1) D 2
(^3) P 2
(^3) P 1
(^3) P 0
Figure 15.5 Carbon atoms have five distinct
electronic energy levels within the 1s^22 s^22 p^2 elec-
tron configuration, only one of which is the
ground state.