7.2 Electron configurations and terms
7.2.1 Equivalent and non-equivalent electrons
e.g. for one p electron and one s-electron: l1= 1, l2
=0
(excited state of helium)
S = |s1 - s2|.. s1+s2 = 0, 1
L = | l1 - l2|.. l1+ l2 = 1
J = L + S … |L – S| =
L=1, S=1 → J = 2, 1, 0
→ 3P2, 3P1, 3P0
L=1, S=0 → J = 1
→ 1P1
e.g. for two non-equivalent p electrons: l1= l2
=1
(excited state of carbon)
S = |s1 - s2|.. s1+s2 = 0, 1
L = | l1 - l2|.. l1+ l2 = 2,1,0
J = L + S … |L – S| =
L=2, S=1 → J = 3, 2, 1
L=2, S=0 → J = 2
→ 3D3, 3D2, 3D1
→ 1D2
L=1, S=1 → J = 2, 1, 0
L=1, S=0 → J = 1
→ 3P2, 3P1, 3P0
→ 1P1
L=0, S=1 → J = 1, 0
L=0, S=0 → J = 0
→ 3S1 *
→ 1S0
* note that 3S1 does not exist since L+S = |L-S|=1.
e.g. for two equivalent* p electrons: l1= l2
=1
(carbon atom)
S = |s1 - s2|.. s1+s2 = 0, 1
L = | l1 - l2|.. l1+ l2 = 2,1,0
J = L + S … |L – S| =
•
L=2, S=0 → J = 2
→ 1D2
L=1, S=1 → J = 2, 1, 0
→ 3P2, 3P1, 3P0
L=0, S=0 → J = 0
→ 1S0
Two or more electrons are referred to as equivalent if they have the same value of n and l.
They must then differ in ml or ms
In order to determine the
number and type of allowed
terms use table of “microstates”
→ CHEM 442
http://physics.nist.gov/cgi-bin/AtData/display.ksh?XXE0qCqIXXT2XXI
7.2.2 Order of energy levels (Pauli and Hund’s Rules)
• Electron configurations are ordered according to the “Aufbau-Principle”
o Pauli-principle: no two electrons in an atom are allowed to share the same
quantum numbers n, l, ml, ms
o Electrons tend to occupy the orbital with the lowest energy available:
1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<…
• For a given configuration the order of the terms is given by Hund’s rules
o Hund’s rule #1: the term with the highest multiplicity is lowest in energy (e.g.
for the carbon 2s2 2p2 configuration 3P is lower than 1S and 1D)
o Hund’s rule #2: If the multiplicity is the same than the term with highest L
quantum number is lowest (e.g. for carbon 1D is lower than 1S)
o Hund’s rule #3: If the subshell is less than half full, than for a given 2S+1L term
the spin-orbit state with the lowest J quantum number is lowest in energy
(e.g. for carbon only two of the possible 6 electrons occupy the 2p subshell:
3
P0< 3P1<3P2; for oxygen’s 2s2 2p4 configuration: 3P0 > 3P1 > 3P2)
Wolfgang Pauli
(1900-1958)
German physicist who, in 1925, proposed the
Pauli exclusion principle, which states that no
two fermions may possess the same energy
(occupy the same quantum state) in a given atom.
He made fundamental contributions to quantum
mechanics.
His ability to make experiments self destruct simply by being in the same
room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991,
p. 48; Gamow 1985).
7.3 Magnitude of Spin-Orbit splitting
7.3.1 ...for one-electron atom
We had this before (Section 5, eq. 5.22)
For one-electron atom (“hydrogen-like”)
The correction due to spin orbit coupling is
! !
hc0
ESO = 2 ζ n , l l ⋅ s
"
hc
= 0 ζ n , l ( j ( j + 1) − # ( # + 1) − s ( s + 1) )
2
where ζn,l is the spin-orbit coupling constant (in cm-1)
The energy of the split level is shifted
ETotal = E(0) + ESO
Therefore, for s = ½ → j= l + ½ and j = l - ½
hc
1 
3
1  1 

ESO = 0 ζ n , l   # +  # +  − # ( # + 1) −  + 1 
2
2 
2
2  2 

for j= l + ½
hc
Etotal = E (0) + ζ n , j #
2
for j= l - ½
hc
Etotal = E (0) − ζ n , j ( # + 1)
2
(7.1)
(7.2)
(7.3)
7.3.2 ...for many-electron atoms
The correction due to spin orbit coupling is similar for the one-electron case
hc
ESO = 0 ζ n , l ( J ( J + 1) − L ( L + 1) − S ( S + 1) )
2
and the energy of the split level is
ETotal = E (0) + ESO
(7.4)
(7.5)
The intervals between neighboring levels (same L and S) are then given by
∆E = EJ +1 − EJ =
=
hc0
ζ n, l
2
hc0
ζ n , l ( J + 1)
2
( ( J + 1)( J + 2 ) − J ( J + 1) )
(7.6)
7.3.3 Ground state electron configuration and levels
Element
Configuration
Terms
Ground state level
Li
1s22s
2
S
2
S1/2
Be
1s22s2
1
S
1
S0
B
1s22s22p
2
P
2
P1/2
C
1s22s22p2
1
D,3P,1S
3
P0
N
1s22s22p3
4
S,2D,2P
4
S3/2
O
1s22s22p4
1
D,3Pi,1S
3
P2
F
1s22s22p5
2
Pi
2
P3/2
Ne
1s22s22p6
1
S
1
S0
7.3.4 Energy levels of the Helium atom
Singlet States
1
1
S0
Triplet States
1
P1
3
D2
3
S1
P2,1,0
3
D3,2,1
1s4p
1s3s
1s3p
1s3
1s3s
1s3p
1s3d
1s2p
1s2p
1s2s
1s2s
If net spin is S=0 → singlet states
→ not split
1s2
If net spin is S=1 → triplet states
→ split by SO coupling
7.3.5 New Selection Rules (difficult to derive)
∆S = 0
change of multiplicity is “spin-forbidden”
∆L = 0, ± 1
for the multi-electron atom but
∆l = ± 1
for each electron’s transition (within the orbital approximation)
∆J = 0, ± 1
except J=0 →J=0 is forbidden
e.g.
• transitions between singlet and triplet states are forbidden
1
S0 → 1P1
allowed
1
3
S0 → P1
forbidden
• transitions between, e.g. S and D states are forbidden
• transitions from s-orbitals to s-orbitals (or p-AO’s to p-AO’s, etc) are forbidden
1
S0 → 1P1
allowed
1
1
S0 → S0
forbidden
(Note: distinguish between S- states and s-electrons)
• transitions that involve ∆J=2 or larger are forbidden
3
P0 → 3D2
forbidden
3
P1 → 3D2
allowed
The helium atom
Triplet
Singlet
1
1
S0
P1
1
3
D2
S1
3
P2,1,0
3
D3,2,1
1s4p
1s3s
1s3p
1s3d
1s3s
1s3d
1s3p
1s2p
1s2s
1s2p
1s2s
1s2
7.4 More angular momentum coupling
7.4.1 The Zeeman effect: Atoms in magnetic fields
Usually the MJ levels are all degenerate: BUT
• the total angular momentum is associated with a “rotating and spinning” charge
• a magnetic moment results from the sum of spin and orbital angular momentum
• An external magnetic field will interact with the magnetic moment…
• …and lift the degeneracy: Some MJ states can align favourably with respect to the
B-field!
MJ
+2
!
B
+1
0
-1
-2
J =2
!
| J | = J (J +1) " = 6 "
Oxygen 2s22p4
Energy
3
3
3
MJ
P0
0
P1
+1
0
-1
+2
+1
0
-1
-2
P2
B (Tesla)
• Another new selection rule
∆M J = 0, ± 1
MJ
3/2
1/2
-1/2
-3/2
2
P3/2
2
P1/2
1/2
-1/2
1/2
-1/2
2
S1/2
B=0
B≠0
For the sodium atom
[Ne] 3s1 : L=0, S=1/2, J=1/2
[Ne] 3p1 : L=1, S=1/2, J=1/2
L=1, S=1/2, J=3/2
7.4.2 Hyperfine splitting: Interaction with the magnetic moment of the nucleus
→ electrons have total orbital angular momentum (given by L)
→ electrons have total spin (given by S)
→ protons and neutrons also have spin! (always s= ½ )
any nucleus will have spin: I = N(protons)/2 + N(neutrons)/2
Nuclear Spin Quantum Number
Isotope
I
Isotope
I
1
1/2
1
3/2
0
1/2
1
15
1/2
0
5/2
1/2
H
2
H
7
Li
12
C
13
C
14
N
N
16,18
O
17
O
F
23
Na
199
Hg
19
3/2
1/2
1. For an even number of protons and even number of neutrons, I = 0;
2. For an even number of neutrons and odd number of protons, I = integer/2;
3. For odd number of both protons and neutrons, I = integer.
The “most total” angular momentum
!
Nuclear spin is added to the total electron angular momentum J (same as before)
S = |s1 - s2|.. s1+s2
L = | l1 - l2|.. l1+ l2
! ! !
J = L+S
J = L + S … |L – S|
MS = -S, (1-S) … (S-1), S
ML = -L, (1-L) … (L-1), L
MJ = J ≤ MJ ≤ -J
Now in addition
! ! !
F =J +I
F = J + I … |J – I|
MF = F ≤ MF ≤ -F
Hyperfine splitting of the 2s and 3p H-atom levels
For the hydrogen atom I=1/2
2
P3/2
n=3
2
P1/2
F
2 = J+I
1 = J-I
1= J+I
0= J-I
∆F = 0, ± 1
n=2
2
S1/2
1= J+I
0= J-I
7.4.3 The final selection rules
Principal quantum number
∆n = any
Angular momentum quantum number
∆L = 0, ± 1
Within orbital approximation
∆l = ± 1
Spin angular momentum quantum number
∆S = 0
Spin-orbital coupling
∆J = 0, ± 1
Hyperfine coupling
∆F = 0, ± 1
With magnetic field
∆MF = 0, ± 1
7.4.4 Example for selection rules
consider all allowed transitions from
[2s2 2p2] 1D level
→ lower state has
J = L + S … |L – S| = 2+0 .. 2-0 = 2
• ∆S = 0: transitions to triplet or
quintet states are forbidden
• ∆L = 0, ± 1: transitions to Sand G- states are forbidden
leaves: only 1P1, 1D2, 1F3
• ∆l = ± 1:
[2s2 2p2] →[2s2 2p 3p]
transitions are forbidden
Leaves only:
[2s2 2p 3s] 1P1
[2s2 2p 3d] 1D2
[2s2 2p 3d] 1P1, 1F3
[2s2 2p 4s] 1P1
A table of lines from the 2s2 2p2 configuration is given at
http://physlab2.nist.gov/cgi-bin/AtData/display.ksh?XXT1XXO1q01q0q00j10001j0q0q0q0q2s2.2p2q0j0q0q1q_q0j0q0q0q0q_jXXRwq0qCqIXXI
---------------------------------------------------------------------------------------------------------------------------------------------------Wavelength | Rel. | Aki
| Acc |
Ei
Ek
| Configurations
| Terms | Ji - Jk | gi - gk | Type |
TP | Line
Vacuum
| Int. | (10^8s-1) |
| (cm-1)
(cm-1) |
|
|
|
|
|
Refs | Refs
(A)
|
|
|
|
|
|
|
|
|
|
|
---------------------------------------------------------------------------------------------------------------------------------------------------|
|
|
|
|
|
|
|
|
|
|
1357.134 |
| 1.35e-01 | C
| 10192.63 - 83877.31 | 2s2.2p2-2s2.2p(2P*)5s | 1D-1P* |
2 - 1
| 5 - 3 |
|
2 |
1357.659 |
| 1.08e-02 | C
| 10192.63 - 83848.83 | 2s2.2p2-2s2.2p(2P*)4d | 1D-3D* |
2 - 3
| 5 - 7 |
|
2 |
1359.275 |
| 2.16e-02 | C
| 10192.63 - 83761.26 | 2s2.2p2-2s2.2p(2P*)4d | 1D-3F* |
2 - 3
| 5 - 7 |
|
2 |
1359.439 |
| 9.74e-03 | C
| 10192.63 - 83752.41 | 2s2.2p2-2s2.2p(2P*)5s | 1D-3P* |
2 - 1
| 5 - 3 |
|
2 |
1364.164 | 120 | 1.57e-01 | C
| 10192.63 - 83497.62 | 2s2.2p2-2s2.2p(2P*)4d | 1D-1D* |
2 - 2
| 5 - 5 |
|
2 | 211
|
|
|
|
|
|
|
|
|
|
|
1459.031 | 100 | 4.76e-01 | B
| 10192.63 - 78731.27 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1P* |
2 - 1
| 5 - 3 |
|
1,2,3,6 | 211
1463.336 | 200 | 1.88e+00 | B
| 10192.63 - 78529.62 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1F* |
2 - 3
| 5 - 7 |
|
1,2,3,5 | 211
1467.402 | 120 | 5.49e-01 | B+
| 10192.63 - 78340.28 | 2s2.2p2-2s2.2p(2P*)4s | 1D-1P* |
2 - 1
| 5 - 3 |
|
1,2,3,6 | 211
1467.877 |
| 1.30e-02 | C
| 10192.63 - 78318.25 | 2s2.2p2-2s2.2p(2P*)3d | 1D-3D* |
2 - 3
| 5 - 7 |
|
6 |
1468.410 |
| 3.90e-02 | C
| 10192.63 - 78293.49 | 2s2.2p2-2s2.2p(2P*)3d | 1D-3D* |
2 - 1
| 5 - 3 |
|
6 |
|
|
|
|
|
|
|
|
|
|
|
1470.094 |
| 1.37e-02 | C+
| 10192.63 - 78215.51 | 2s2.2p2-2s2.2p(2P*)3d | 1D-3F* |
2 - 3
| 5 - 7 |
|
2,6 |
1472.231 |
| 8.01e-03 | C
| 10192.63 - 78116.74 | 2s2.2p2-2s2.2p(2P*)4s | 1D-3P* |
2 - 1
| 5 - 3 |
|
2 |
1481.763 | 150 | 3.92e-01 | B+
| 10192.63 - 77679.82 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1D* |
2 - 2
| 5 - 5 |
|
1,2,3,6 | 211
1542.177 |
| 2.22e-01 | C+
| 21648.01 - 86491.41 | 2s2.2p2-2s2.2p(2P*)5d | 1S-1P* |
0 - 1
| 1 - 3 |
|
1 |
1560.309 | 150 | 6.57e-01 | A
|
0.00 - 64089.85 | 2s2.2p2-2s.2p3
| 3P-3D* |
0 - 1
| 1 - 3 |
|
2n | 211
|
|
|
|
|
|
|
|
|
|
|
1560.682 | 400 | 8.86e-01 | A
|
16.40 - 64090.95 | 2s2.2p2-2s.2p3
| 3P-3D* |
1 - 2
| 3 - 5 |
|
2n | 211
1560.709 | 400 | 4.92e-01 | A
|
16.40 - 64089.85 | 2s2.2p2-2s.2p3
| 3P-3D* |
1 - 1
| 3 - 3 |
|
2n | 211
1561.340 | 100 | 2.94e-01 | A
|
43.40 - 64090.95 | 2s2.2p2-2s.2p3
| 3P-3D* |
2 - 2
| 5 - 5 |
|
2n | 211
1561.367 |
| 3.26e-02 | A
|
43.40 - 64089.85 | 2s2.2p2-2s.2p3
| 3P-3D* |
2 - 1
| 5 - 3 |
|
2n |
1561.438 | 400 | 1.18e+00 | A
|
43.40 - 64086.92 | 2s2.2p2-2s.2p3
| 3P-3D* |
2 - 3
| 5 - 7 |
|
2n | 211
|
|
|
|
|
|
|
|
|
|
|
1600.818 |
| 3.57e-03 | C
| 21648.01 - 84116.09 | 2s2.2p2-2s2.2p(2P*)4d | 1S-3P* |
0 - 1
| 1 - 3 |
|
2 |
1602.972 |
| 4.51e-01 | B
| 21648.01 - 84032.15 | 2s2.2p2-2s2.2p(2P*)4d | 1S-1P* |
0 - 1
| 1 - 3 |
|
1,2 |
1606.960 |
| 5.30e-02 | C
| 21648.01 - 83877.31 | 2s2.2p2-2s2.2p(2P*)5s | 1S-1P* |
0 - 1
| 1 - 3 |
|
2 |
1608.438 |
| 1.74e-02 | C
| 21648.01 - 83820.13 | 2s2.2p2-2s2.2p(2P*)4d | 1S-3D* |
0 - 1
| 1 - 3 |
|
2 |
1613.376 |
| 3.01e-04 | C+
|
0.00 - 61981.82 | 2s2.2p2-2s2.2p(2P*)3s | 3P-1P* |
0 - 1
| 1 - 3 |
|
2,4 |
|
|
|
|
|
|
|
|
|
|
|
1613.803 |
| 2.21e-04 | C+
|
16.40 - 61981.82 | 2s2.2p2-2s2.2p(2P*)3s | 3P-1P* |
1 - 1
| 3 - 3 |
|
2,4 |
1614.507 |
| 2.64e-04 | C+
|
43.40 - 61981.82 | 2s2.2p2-2s2.2p(2P*)3s | 3P-1P* |
2 - 1
| 5 - 3 |
|
2,4 |
1656.267 | 150 | 8.58e-01 | A
|
16.40 - 60393.14 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
1 - 2
| 3 - 5 |
| 2n,3n,4n,5n | 211
1656.929 | 120 | 1.13e+00 | A
|
0.00 - 60352.63 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
0 - 1
| 1 - 3 |
| 2n,3n,4n,5n | 211
1657.008 | 300 | 2.52e+00 | A
|
43.40 - 60393.14 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
2 - 2
| 5 - 5 |
| 2n,3n,4n,5n | 211
|
|
|
|
|
|
|
|
|
|
|
1657.379 | 120 | 8.64e-01 | A
|
16.40 - 60352.63 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
1 - 1
| 3 - 3 |
| 2n,3n,4n,5n | 211
1657.907 | 120 | 3.43e+00 | A
|
16.40 - 60333.43 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
1 - 0
| 3 - 1 |
| 2n,3n,4n,5n | 211
1658.121 | 150 | 1.44e+00 | A
|
43.40 - 60352.63 | 2s2.2p2-2s2.2p(2P*)3s | 3P-3P* |
2 - 1
| 5 - 3 |
| 2n,3n,4n,5n | 211
1751.827 | 500 | 9.07e-01 | B
| 21648.01 - 78731.27 | 2s2.2p2-2s2.2p(2P*)3d | 1S-1P* |
0 - 1
| 1 - 3 |
|
1,2,3,6 | 211
1763.909 |
| 3.59e-02 | C
| 21648.01 - 78340.28 | 2s2.2p2-2s2.2p(2P*)4s | 1S-1P* |
0 - 1
| 1 - 3 |
|
2 |
|
|
|
|
|
|
|
|
|
|
|
1765.366 |
| 1.04e-02 | C
| 21648.01 - 78293.49 | 2s2.2p2-2s2.2p(2P*)3d | 1S-3D* |
0 - 1
| 1 - 3 |
|
2 |
1930.905 | 1000 | 3.51e+00 | B+
| 10192.63 - 61981.82 | 2s2.2p2-2s2.2p(2P*)3s | 1D-1P* |
2 - 1
| 5 - 3 |
|
1,2,3,4,6 | 211
1992.012 |
| 4.01e-06 | D
| 10192.63 - 60393.14 | 2s2.2p2-2s2.2p(2P*)3s | 1D-3P* |
2 - 2
| 5 - 5 |
|
4 |
1993.620 |
| 7.82e-04 | C
| 10192.63 - 60352.63 | 2s2.2p2-2s2.2p(2P*)3s | 1D-3P* |
2 - 1
| 5 - 3 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
----------------------------------------------------------------------------------------------------------------------------------------------------
A table of lines from the 2s2 2p2 configuration is given at
http://physlab2.nist.gov/cgi-bin/AtData/display.ksh?XXT1XXO1q01q0q00j10001j0q0q0q0q2s2.2p2q0j0q0q1q_q0j0q0q0q0q_jXXRwq0qCqIXXI
Wavelength | Rel. | Aki
| Acc |
Ei
Ek
| Configurations
| Terms | Ji - Jk
Vacuum
| Int. | (10^8s-1) |
| (cm-1)
(cm-1) |
|
|
(A)
|
|
|
|
|
|
|
----------------------------------------------------------------------------------------------------------1459.031 | 100 | 4.76e-01 | B
| 10192.63 - 78731.27 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1P* |
2 - 1
1463.336 | 200 | 1.88e+00 | B
| 10192.63 - 78529.62 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1F* |
2 - 3
1481.763 | 150 | 3.92e-01 | B+
| 10192.63 - 77679.82 | 2s2.2p2-2s2.2p(2P*)3d | 1D-1D* |
2 - 2
1765.366 |
| 1.04e-02 | C
| 21648.01 - 78293.49 | 2s2.2p2-2s2.2p(2P*)3d | 1S-3D* |
0 - 1
1930.905 | 1000 | 3.51e+00 | B+
| 10192.63 - 61981.82 | 2s2.2p2-2s2.2p(2P*)3s | 1D-1P* |
2 - 1
1992.012 |
| 4.01e-06 | D
| 10192.63 - 60393.14 | 2s2.2p2-2s2.2p(2P*)3s | 1D-3P* |
2 - 2
1993.620 |
| 7.82e-04 | C
| 10192.63 - 60352.63 | 2s2.2p2-2s2.2p(2P*)3s | 1D-3P* |
2 - 1
-----------------------------------------------------------------------------------------------------------