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Lecture 4
Electronic Microstates & Term Symbols
Suggested reading: Shriver and Atkins, Chapter 20.3
or Douglas,
g , 1.4-1.5
Recap from last class: Quantum Numbers
F
Four
quantum numbers:
b
““n”” , “l”,
“l” “m
“ l”,
” and
d ““ms”
Or,, equivalently:
q
y “n” , “l”,, “j”,
j , and “mj”
Orbital angular
momentum:
Associated with
rotation off electron
l
cloud around
nucleus
Spin angular
momentum:
Total angular
momentum:
Associated with
rotation/spin of
electron around it’s
axis
Vector sum of
orbital and spin
angular
momentum
Recap from last class: Quantum Numbers
F
Four
quantum numbers:
b
““n”” , “l”,
“l” “m
“ l”,
” and
d ““ms”
Or,, equivalently:
q
y “n” , “l”,, “j”,
j , and “mj”
Orbital angular
momentum:
L  
   1 
Lz  m 
Spin angular
momentum:
Total angular
momentum:
S   ss  1
J   j  j  1
S z  ms
J z  m j
 = 0, 1, 2, ….n1
s = 1/2
j = -s,…,+s
m = -...,
 0,
0 … +
ms = ±1/2
mj = -j,…0,…+j
j 0 +j
Electronic Configurations and Microstates
• Electronic configurations tell us the number of electrons in each
orbital, but they don’t tell us how the electrons occupy the
orbitals.
i.e.,, don’t always
y want to draw:
•“Terms”: the energy levels and configurations of atoms and ions
(and molecules)
• Useful in understanding ionic and coordination compound
spectra
General Form of Terms
naTb
•n: principle quantum number
•a = 2S+1 “multiplicity”
p
y
(S is the total spin angular momentum of all electrons)
**a number**
For example, if two electrons are in two different
orbitals we might have antiparallel spins:
S=s1+s2
= ½ - ½ =0
Or parallel spins: S=s1+s2 = ½ + ½ = 1
General Form of Terms
naTb
•n: principle quantum number of valence electrons
•a = 2S+1 “multiplicity”
p
y
(S is the total spin angular momentum of all electrons)
**a number**
•T=L (total orbital angular momentum of all electrons; vector sum
of , the orbital momentum of individual electrons)
**a letter**
l
L=0, 1, 2, 3, 4  S, P, D, F, G
•b=J (total angular momentum)
**a number**
Examples of Term Symbols in the literature
Free Ions/ Atoms
Molecules
Sundararajan Dalton Trans
Sundararajan,
Trans., 2009
2009, 6021-6036
6021 6036
Wang, Nature Materials 10 (2011)
Co(PPH3)2Cl2 Used in palladium-catalyzed
coupling reactions, 2010 Nobel Prize
Hydrogen (1 electron atom)
Ground state: 1s
n=1; =0
L= =
 =0
S=s= ½
Only
l one term is possible
bl
(i.e., only one energy level
for the one microstate))
2S
½
(read “doublet S”)
J=L+S(vector sum)
=0+½
In a magnetic field
field, due to the Zeeman effect,
effect the 12S1/2 term yields two
closely-spaced energy levels (mj=½, -½)
Multiplicity terms
S= ½  2S+1=a=2
doublet (electron can be spin up or spin down)
S= 1  2S+1=a=3
triplet (three different spin configurations,
configurations with
spin wavefunction χ)
 
(both electrons spin up)
1

(  ) (electrons are indistinguishable)
2
 
(both electrons spin down)
Multiplicity terms
S= ½  2S+1=a=2
doublet (electron can be spin up or spin down)
S= 1  2S+1=a=3
triplet (three different spin configurations,
configurations with
spin wavefunction χ)
S=3/2
/  2S+1=a=4
quartet (four
(f
d ff
different
spin configurations)
f
)
 
1

(    )
3
1

(    )
3
 
Hydrogen (1 electron atom)
Excited state: 2p1
n=2; =1
L= = 1
S=s= ½
J=L+S
= 1+½ or 1-½
= ½ or 3/2
Two terms are possible
(i.e., two energy levels,
depending on the
orientations of the electron
spin with respect to the
orbital angular momentum
of the electron)
2P
1/2
or 2P3/2
In a magnetic field
field, due to the Zeeman effect,
effect the 22P1/2 term yields
two closely-spaced energy levels (mj=½, -½) while the 22P3/2 yields
four (mj=3/2, ½, -½, -3/2)
Helium (2 electron atom)
L  1   2
S  s1  s2
G
Ground
d state: 11s2
n=1; 1=2=0
L= 1+2= 0
S=s1+s2= ½-½ (Pauli) = 0
J L S( t sum)=0
J=L+S(vector
)0
1S
0
Helium (2 electron atom)
L  1   2
S  s1  s2
G
Ground
d state: 11s12p
2 1
1=0; 2=1
L= 1+2= 1
S=s1+s2= ½-½ or ½+½
= 0 or 1
J=L+S(vector sum)=0
1P
0
3P
0
“singlet P”
“triplet
p P”
Carbon (6 electron atom)
L  1   2   3   4   5   6
S  s1  s2  s3  s4  s5  s6
G
Ground
d state: 11s22s
2 22p
2 2
1=0; 2=0; 3=0; 4=0; 5=1; 6=1
L= 2, 1, 0  D, P, S
S= 0 or 1  2S+1 = 1, 3
J L S( t sum)=3,
J=L+S(vector
) 3 2,
2 1
The electrons in the 1s and
2s states will be spin
paired. The two electrons
in p can be either in px, py,
or pz orbitals, and either
paired or unpaired:
m=-1
m=0
m=+1
Microstates
Microstates: the different ways in which electrons
can occupy certain orbitals
Grouping together the microstates that have the same
energy when electron-electron repulsions are taken
( , the
into account,, yyields the terms (i.e.,
spectroscopically distinguishable energy levels)
Number of
microstates:
2 N 0!

(2 N 0  N e )! N e !
No=# degenerate orbitals (i.e., three degenerate p orbitals)
Ne=# electrons
Carbon, continued: [He]2p2
2 N 0!
2(3!)


 15
(2 N 0  N e )! N e ! (2  3  2)!2!
No=# degenerate orbitals for p=3
Ne=#
# electrons in p
p=22
What are the microstates,
microstates and which has the lowest
energy?
Carbon, continued: [He]2p2
ms=+1/2
ml
+1
0
ms=-1/2
‐1
+1
0
‐1
ML=ml
MS=ms
1
1
1
2
0
1
3
‐1
1
4
1
‐1
5
0
‐1
6
‐1
‐1
7
2
0
8
1
0
9
0
0
10
1
0
11
0
0
12
‐1
0
13
0
0
14
‐1
0
15
‐2
0
Carbon, continued: [He]2p2
MS=m
 s
ML=ml
+1
0
‐1
‐2
0
1
0
‐1
1
2
1
0
1
3
1
+1
1
2
1
+2
2
0
1
0
Note: the array is symmetric about the lines through ML=0
and MS=0, providing a check of the tabulation
ML=22, 1,
1 0
0, -11, - 2  L
L=22  a D term
The values in the array occur only for MS=0  S=0
**a 1D term!**
Subtracting out the 1D term:
MS=m
 s
+1
0
‐1
‐1
1
1
1
0
1
2
1
+1
1
1
1
‐2
ML=ml
+2
2
ML=1, 0, -1  L=1  a P term
The values in the array occur MS=1, 0, -1  S=1
term!**
**aa 3P term!
Subtracting out the 3P term:
MS=m
 s
+1
0
‐1
‐1
0
0
0
0
0
1
0
+1
0
0
0
‐2
ML=ml
+2
2
ML=0  L=0  a S term
MS=0  S=0
term!**
**aa 1S term!
The energies of the terms
1 Hund’s
1.
Hund s rule: For a given configuration,
configuration the term with
the greatest multiplicity lies lowest in energy
 the triplet term of a configuration (if one is
permitted) will have a lower energy than a singlet term
p
y, the term with
2. L_Rule: For a term of a ggiven multiplicity,
the greatest value of L lies lowest in energy
 if L is high, the electrons can effectively avoid each
other
3. J-Rule: if subshell is less than half filled, lowest J is
lowest energy.
If greater than half filled, highest J is lowest energy
q to half-filled,, onlyy one J possible
p
If equal
Energy Diagram for Carbon
Eneergy
Ground state of Carbon: 3P0 (“triplet P0”)
Beyond Russell-Saunders coupling
For light atoms and the 3d series (Z30),
(Z30) the energies
of the microstates are determined first by the electron
spin (S), then their orbital angular momentum (L).
i.e, as before, total momentum J is determined by
g
momenta,, then the
summingg first the orbital angular
spins, and then combining both: Russel-Saunders
coupling
For heavier atoms, must consider spin-orbit coupling:
“jj-coupling”.
l
For each
h electron,
l
find
f d j=+s, then
h sum j’s
of each electron to find total J of atom/ion
For this class, knowledge of Russell-Saunders
coupling is sufficient