pp 3D 1D 3P 1P
3S 1S
m
1 0
l
-1

m
1
l

m
1 1 1 0 0 0 0 0 0 0 0 0 -1 -1 -1
s


↑
↑
-1 0 0 -1 2
↑

1
↓
↓↑

0 1 0 1 0
↑

↓
↓
Atomic terms symbols for two equivalent electrons
12
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
l - l coupling
Summary
s-s coupling
L-S coupling
Magnetic field
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L - Scoupling: the orbital momenta for all electrons is coupled to obtain a total orbital angular momentum L and then the individual spin angular momenta is coupled to obtain the total spin angular momentum S. The total angular momentum J is then obtained by vector addition of L and S.
Sz M S M S S , S 1, , S
J LS J
m
s
Total M om e ntum
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J J 1
J L S , L S 1, , L S
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Chemistry Department of Fudan University
11
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
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Chemistry Department of Fudan University
20112011-9-30 Chemistry Department of Fudan University
7
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Atomic terms symbols for two non-equivalent electrons
M S ms
1 2 0 1 1 1 0 0 1 2 3 2 1 -1 0 1 1 1 0
Slater method
ML
1 0 -1 -2
m
l
L2 S 0
M S ms
1 2 0 1 1 1 0 0 0 1 2 1 0 -1 0 1 1 1 0
1
D
ML
1 0 -1 -2
m
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
1. The term with maximum multiplicity lies lowest in energy. 2. For a given multiplicity, the term with the largest value of L lies lowest in energy. 3. For atoms with less than halffilled shells, the level with the lowest value of J lies lowest in energy. If the subshell is more than half filled, the state with maximum J is lowest in energy.
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Chemistry Department of Fsical Chemistry I Quantum Numbers of Atom
Electron
Chapter II Atomic Structure and Spectrum
A tom
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Physical Chemistry I
Chapter II Atomic Structure and Spectrum
L=0, L=2,
S=0 S=1
1S 3D
singlet triplet
p2
↓
-1 0 -2 -1 -1 0 -2 1
↑
↓
↑
↓↑

0 -1
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8
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Number of possible States of the configuration f7
C
m ( number N ( possible
electrons ) spin orbitals )
max S max L
C
7 14
Count the total ms Count the total ml
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Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Quantum Numbers of Atom
Electron l Atom L Relation L l
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Physical Chemistry
Chapter II Atomic Structure & Spectrum
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Chemistry Department of Fudan University
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Spin multiplicity
Atomic Term Symbol
Atomic Levels
2 S 1
L
2 S 1
LJ
If the Pauli principle does not have to be considered, we can write all the term states directly.
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l
Chemistry Department of Fudan University
9
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
M S ms
1 2 0 1 1 1 0
0 0 1 2 1 0
j - jcoupling: the orbital angular momentum l
and spin angular momentum s of each electron are coupled to obtain a total angular momentum j. The all the individual j are coupled to obtain the total angular momentum J of an atom.
ˆ ' 0 dt E E H
Angular Momentum
l l l 1
lz
L LL 1
LZ
LZ M L
L l1 l2 ,..., l1 l2 Lz lz
Orbital
component
lz ml
M L ml
ml l , l 1,,l M L L, L 1,, L
R elation
s
A ngular M om entum
S S
S
s

s
s s 1
S S 1 S s1 s 2 , , s1 s 2 Sz Sz MS
s 1/ 2
Spin
sz
com ponent
s
z
s z m s m s s , s
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Atomic terms for various configuration
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ˆ H ˆ0H ˆ' H
ˆ 0 0 E 0 0 H