第2章原子结构和原子光谱[3]
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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
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.
20112011-9-30
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
ˆ H ˆ0H ˆ' H
ˆ 0 0 E 0 0 H
The zeroth-order Schrodinger equation can be solved exactly. It is then assumed that the perturbation is sufficiently small that its ˆ ' 0 dt E E 0 E ' presence does not appreciably alter 0H the eigen-functions for the system.
What is the coupling of the angular Momentum?
20112011-9-30 Chemistry Department of Fudan University 2
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
20112011-9-30 Chemistry Department of Fudan University 3
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Quantum Numbers of Atom
Electron l Atom L Relation L l
The state of the multi-electron atom
The energy of a state for multi-electron atoms is related to the interactions between electrons outside the nucleus, namely, the coupling of the angular momentum of electrons
Sz M S M S S , S 1, , S
J LS J
m
s
Total M om e ntum
20112011-9-30
J J 1
J L S , L S 1, , L S
Chemistry Department of Fudan University 5
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
20112011-9-30
Chemistry Department of Fudan University
1
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Atomic Term Symbols
The arrangement of electrons in an atom is known as its electronic configuration.
-1 0 1 1 1 0
ML
1 0 -1 -2
m
l
L 1
M S ms
1 2 0 0 0 0 0 0 0 0 1 0 0
S 1
-1 0 0 0 0 0
3
P
1
ML
1 0 -1 -2
ML 0
ml
20112011-9-30
MS 0
S
10
Chemistry Department of Fudan University
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
14
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Perturbation Theory
ˆ E H
0
ˆ0H ˆ ) 0 E 0 (H
The Hamiltonian is partitioned into two parts: one is the zeroth-order Hamiltonian, and the other is the perturbation.
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
p2
↓
-1 0 -2 -1 -1 0 -2 1
↑
↓
↑
↓↑
0 -1
20112011-9-30
Chemistry Department of Fudan University
8
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
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.
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
20112011-9-30 Chemistry Department of Fudan University 6
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
L=0, L=2,
S=0 S=1
1S 3D
singlet triplet
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
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Physical Chemistry
Chapter II Atomic Structure & Spectrum
20112011-9-30
Chemistry Department of Fudan University
20112011-9-30
Chemistry Department of Fudan University
4
Physical Chemistry I Quantum Numbers of Atom
Electron
Chapter II Atomic Structure and Spectrum
A tom
12
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
l - l coupling
Summary
s-s coupling
L-S coupling
Magnetic field
20112011-9-30 Chemistry Department of Fudan University 13
M S ms
1 2 0 1 1 1 0 0 ห้องสมุดไป่ตู้ 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
20112011-9-30
Chemistry Department of Fudan University
11
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
20112011-9-30
Chemistry Department of Fudan University
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.
7
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Atomic terms symbols for two non-equivalent electrons
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.
20112011-9-30
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
ˆ H ˆ0H ˆ' H
ˆ 0 0 E 0 0 H
The zeroth-order Schrodinger equation can be solved exactly. It is then assumed that the perturbation is sufficiently small that its ˆ ' 0 dt E E 0 E ' presence does not appreciably alter 0H the eigen-functions for the system.
What is the coupling of the angular Momentum?
20112011-9-30 Chemistry Department of Fudan University 2
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
20112011-9-30 Chemistry Department of Fudan University 3
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Quantum Numbers of Atom
Electron l Atom L Relation L l
The state of the multi-electron atom
The energy of a state for multi-electron atoms is related to the interactions between electrons outside the nucleus, namely, the coupling of the angular momentum of electrons
Sz M S M S S , S 1, , S
J LS J
m
s
Total M om e ntum
20112011-9-30
J J 1
J L S , L S 1, , L S
Chemistry Department of Fudan University 5
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
20112011-9-30
Chemistry Department of Fudan University
1
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Atomic Term Symbols
The arrangement of electrons in an atom is known as its electronic configuration.
-1 0 1 1 1 0
ML
1 0 -1 -2
m
l
L 1
M S ms
1 2 0 0 0 0 0 0 0 0 1 0 0
S 1
-1 0 0 0 0 0
3
P
1
ML
1 0 -1 -2
ML 0
ml
20112011-9-30
MS 0
S
10
Chemistry Department of Fudan University
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
14
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Perturbation Theory
ˆ E H
0
ˆ0H ˆ ) 0 E 0 (H
The Hamiltonian is partitioned into two parts: one is the zeroth-order Hamiltonian, and the other is the perturbation.
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
p2
↓
-1 0 -2 -1 -1 0 -2 1
↑
↓
↑
↓↑
0 -1
20112011-9-30
Chemistry Department of Fudan University
8
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
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.
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
20112011-9-30 Chemistry Department of Fudan University 6
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
L=0, L=2,
S=0 S=1
1S 3D
singlet triplet
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
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
Physical Chemistry
Chapter II Atomic Structure & Spectrum
20112011-9-30
Chemistry Department of Fudan University
20112011-9-30
Chemistry Department of Fudan University
4
Physical Chemistry I Quantum Numbers of Atom
Electron
Chapter II Atomic Structure and Spectrum
A tom
12
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
l - l coupling
Summary
s-s coupling
L-S coupling
Magnetic field
20112011-9-30 Chemistry Department of Fudan University 13
M S ms
1 2 0 1 1 1 0 0 ห้องสมุดไป่ตู้ 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
20112011-9-30
Chemistry Department of Fudan University
11
Physical Chemistry I
Chapter II Atomic Structure and Spectrum
20112011-9-30
Chemistry Department of Fudan University
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.