Physical Chemistry 20130605 week 10 Wednesday June 5, 2013 page 1
1
1
3
√2
1
P
IS2P
{
P
(1S(1)2Px (2)+1S(2)2Px (1))
√2
(1S(1)2Px (2)-1S(2)2Px (1))
∫ ψ*k Ĥψk dτ = Ek
Instead of energy levels, we call them terms or term symbols. 1S and 3P are terms. In multielectron
atoms, 2P is higher energy state than 2S.
Spin functions for 3P are:
α(1)α(2)
or
(α(1)β(2)+α(2)β(1))
1
or β(1)β(2)
√2
There are each symmetric and 3P spatial function is antisymmetric so the wave spin function is overall
antisymmetric (like it should be for a fermion such as an electron). That’s 9 way degenerate (3 for x, 3
for y, 3 for z). Likewise, 1P is 3 way degenerate.
Total orbital and spin angular momenta (Russell-Saunders coupling)
An electron in an atom has two motions: orbital and spin. Both are angular momentum.
l1 = orbital angular momentum of electron #1
s1 = spin angular momentum of electron #1
l2 = orbital angular momentum of electron #2
s2 = spin angular momentum of electron #2
L = total orbital angular momentum
S = total spin angular momentum
⃑ =l 1 +l
L
2
S⃑= l 1 +l
2
S= ∑ si
i
value of L
0
1 2 3
symbol (code), term symbols S(not spin) P D F
triplet ess = 3S is from that code (not spin)
If S = 0 then 2S+1 = 1 and we get 1P. For 2S+1L the 2S+1 is multiplicity.
If S = ½ + ½ = 1 then 2S+1=3 (or if both are negative).
4 5
G H
If S = ½ - ½ = 0 then 2S+1 = 1.
For 1S2P l=0 since l=0 for S and l=1 for P so L=0+1 and we get P. Then either S = ½+ ½ =1 and 2S+1=3
so we have 3P or S = -½ + ½ = 0 so 2S+1 so we have 1P.
For S=0
MS=0 and there’s one possible spin function:
1
√2
For S=1
For MS=-1
(α(1)β(2)-β(1)α(2)) The vectors cancel so S=0
MS= -1, 0, or 1 (three possible spin functions):
β(1)β(2)
-½-½=-1
For MS=0
(α(1)β(2)+β(1)α(2))/√2
For MS=1
α(1)α(2)
½+½=1
symmetric
½-½ =0
symmetric
symmetric
Hund’s Rules:
1. Of all the Terms (arising from a configuration), the one with the highest spin angular momentum will
have the lowest energy. Thus triplet states are lower than singlet states.
2. Among the Terms of a configuration with the same spin angular momentum the term with the
highest orbital angular momentum will have the lowest energy. Therefore E(3P) < E(1D) < E(1S)
Many-electron atoms and the periodic table
Lithium has 3 protons and 3 electrons so its Hamiltonian has 6 terms for potential energy. Three of
these are attractions between the nucleus and the electrons. The other three are repulsions between
electrons. r12 is the distance from electron 1 to electron 2. r13 is the distance from electron 1 to electron
3. r23 is the distance from electron 2 to electron 3. r1 is the distance from the nucleus to electron 1. r2 is
the distance from the nucleus to electron 2. r3 is the distance from the nucleus to electron 3.
e'2 e'2 e'2 Ze'2 Ze'2 Ze'2 ℏ
+
+
+
(∇2 +∇2 +∇2 )
r12 r13 r23 r1
r2
r3 2m 1 2 3
-
ℏ
(∇2 +∇2 +∇2 ) is the kinetic energy of the 3 electrons
2m 1 2 3