Symmetry
and
Spectroscopy
P. R. Bunker and Per Jensen:
Molecular Symmetry and Spectroscopy, 2nd Edition,
2nd Printing, NRC Research Press, Ottawa, 2006
(ISBN 0-660-19628-X). $49.95 for 747 pages.
paperback. BJ1
P. R. Bunker and Per Jensen:
Fundamentals of Molecular Symmetry,
IOP Publishing, Bristol, 2004
(ISBN 0-7503-0941-5). $57.95
paperback. BJ2
Examples of point group symmetry
H2O
C2v
C3H4
D2d
CH3F
C3v
C60
Ih
Examples of point group symmetry
H2O
C2v
C3H4
D2d
CH3F
C3v
C60
Ih
Point group symmetry of H2O
y
(-x)
z
The point group C2v consists of
the four operations E, C2y, yz, and xy
The word ´group´ is loaded. To see how
we do two operations in succession
Point groups: Number of rotation axes
and reflection planes.
y
y
σyz
(-x)
1
(-x)
z
2
z
2
1
C2
y
(-x)
z
σxy = C2 σyz
y
σyz
(-x)
1
y
(-x)
z
2
z
2
1
C2
y
σxy
(-x)
z
Multiplication Table for H2O Point group
y
C2v = {E, C2, yz, xy }
(-x)
z
Multiplication table (=Rrow Rcolumn, in succession)
E
C2
yz
xy
E
E
C2
yz
xy
C2
C2
E
xy
yz
yz
yz
xy
xy
xy
yz
E
C2
C2
E
σxy = C2 σyz
Use multiplication table to prove that it is a “group.”
{E, C2, yz, xy } forms a “group“ if it obeys
the following GROUP AXIOMS :
•All possible products RS = T belong to the group
•Group contains identity E (which does nothing)
•The inverse of each operation R1 (R1R =RR1 =E ) is in
the group
•Associative law (AB )C = A(BC ) holds
C2v
E
C2
yz
xy
E
E
C2
yz
xy
C2
C2
E
xy
yz
yz
yz
xy
xy
xy
yz
E
C2
C2
E
{E, C2, yz, xy } forms a “group“ if it obeys
the following GROUP AXIOMS :
•All possible products RS = T belong to the group
•Group contains identity E (which does nothing)
•The inverse of each operation R1 (R1R =RR1 =E ) is in
the group
•Associative law (AB )C = A(BC ) holds
Fermi:
C2v
‘‘Group theory CE
2
is just a bunch yz
of definitions‘‘ xy
E
E
C2
yz
xy
C2
C2
E
xy
yz
yz
yz
xy
xy
xy
yz
E
C2
C2
E
•All possible products RS = T belong to the group
•Group contains identity E (which does nothing)
•The inverse of each operation R1 (R1R =RR1 =E ) is in
the group
•Associative law (AB )C = A(BC ) holds
Not a GROUP
E
C2
yz
xy
E
E
C2
yz
xy
C2
C2
E
xy
yz
yz
yz
xy
xy
xy
yz
E
C2
C2
E
•All possible products RS = T belong to the group
•Group contains identity E (which does nothing)
•The inverse of each operation R1 (R1R =RR1 =E ) is in
the group
•Associative law (AB )C = A(BC ) holds
Is a GROUP
(subgroup of C2v)
Rotational
subgroup
E
C2
yz
xy
E
E
C2
yz
xy
C2
C2
E
xy
yz
yz
yz
xy
xy
xy
yz
E
C2
C2
E
PH3 at equilibrium
Symmetry elements:
C3, 1, 2, 3
C3 Rotation axis
k Reflection plane
Symmetry operations:
C3v = {E, C3, C32, 1, 2, 3 }
Multiplying C3v symmetry operations
Reflection
Rotation
2
C3
=  2 1
Reflection
Multiplication table for C3v
C32 = σ2σ1
Multiplication table for C3v
C32 = σ2σ1
Note that C3 = σ1σ2
Multiplication table for C3v
Rotational
subgroup
Multiplication table for C3v
3 classes
A matrix group
´M1 =
´M2 =
´M3 =
1
0
0
1
1
3

2 2
1
3


2
2
1
3

2
2
1
3

2
2
1
M4 =
´ M5 =

0 1
1

2
3
2
3
2
1
2
1
3

2
2
1
3

2
2

´ M6 =
0
Multiplication table for the matrix group
M1
M2
M3
M4
M5
M6
M1
M1
M2
M3
M4
M5
M6
M2
M2
M3
M1
M6
M4
M5
M3
M3
M1
M2
M5
M6
M4
M4
M4
M5
M6
M1
M2
M3
M5
M5
M6
M4
M3
M1
M2
M6
M6
M4
M5
M2
M3
M1
Products are Mrow Mcolumn
E C3 C32 σ1 σ2 σ3
Multiplication tables
have the ‘same shape’
This matrix group forms a “representation” of the C3v group
These two groups are isomorphic.
Irreducible Representations
The matrix group we have just introduced
is an irreducible representation of the C3v
point group.
The sum of the diagonal elements (character)
of each matrix in an irreducible representation
is tabulated in the character table of the
point group.
The characters of this irreducible rep.
C3v
´M1 =
´M2 =
´M3 =
1
0
0
1
1
3

2 2
1
3


2
2
1
3

2
2
1
3

2
2

E
C3
C32
C3v
2
-1
1
M4 =
´ M5 =
0
0 1
1

2
3
2
3
2
1
2
1
3

2
2
1
3

2
2
1
0
2
0
3
0

-1
´ M6 =
The characters of this irreducible rep.
E C3 σ1
C32 (12)
σ2
E (123)
1
2
3σ3
3 classes
A1
1
1
1
A2
1
1
1
E
2
1
0
The 2D representation M = {M1, M2, M3, ....., M6}
of C3v is the irreducible representation E. In this
table we give the characters of the matrices.
Elements in the same class have the same characters
Character Table for the point group C3v
E C3 σ1
C32 (12)
σ2
E (123)
1
2
3σ3
Two 1D
irreducible
representations
of the C3v group
A1
1
1
1
A2
1
1
1
E
2
1
0
The 2D representation M = {M1, M2, M3, ....., M6}
of C3v is the irreducible representation E. In this
table we give the characters of the matrices.
Elements in the same class have the same characters
The matrices of the E irreducible rep.
C3v
´M1 =
´M2 =
´M3 =
1
0
0
1
1
3

2 2
1
3


2
2
1
3

2
2
1
3

2
2

E
C3
C32
C3v
1
M4 =
´ M5 =
0
0 1
1

2
3
2
3
2
1
2
1
3

2
2
1
3

2
2
1
2

´ M6 =
3
The matrices of the A1 + E reducible rep.
C3v
´M1‘ =
´M2‘ =
100
010
001
1 0 0
0 1 3
0
´M3‘ =
E
2 2
1
3


2
2
1 0
0 1
0
C3
C3v
M4‘ =
´ M5‘ =
1 0 0
1 3
0 2 2
0
0
3

2
2
1
3

2
2
10 0
01 0
0 0 1
C32
´ M6‘ =
1 0
0 1
0
1
2
3
2
2
3

2
1
2
0

3
2
1
2
3
The matrices of the A2 + E reducible rep.
C3v
´M1‘‘ =
´M2‘‘ =
´M3‘‘ =
100
010
001
E
1 0 0
0 1 3
2 2
3 ‘ 1

0 2 2
1 0
0 1
0
C3
C3v
M4‘‘ =
´ M5‘‘ =
-1 0 0
1 3
0 2 2
0
0
3

2
2
1
3

2
2
-1 0 0
01 0
0 0 1
C32
´ M6‘‘ =
3
2
1
2
-1 0 0
0 1  3
0
2
3

2
2
1
2
1
2
3
Character table for the point group C2v
E
C2 E*
σyz (12)*
σxy
E (12)
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1
Irreducible representations are
“symmetry labels”
Some of Fermi’s definitions
•
•
•
•
•
•
•
Group
Subgroup
Multiplication table of group operations
Classes
Representations
Irreducible and reducible representations
Character table
See, for example, pp 14-15 and Chapter 5 of BJ1
Some of Fermi’s definitions
•
•
•
•
•
•
•
Group
Subgroup
Multiplication table of group operations
Classes
Representations
Irreducible and reducible representations
Character table
See, for example, pp 14-15 and Chapter 5 of BJ1
Irreducible representations
The elements of irrep matrices satisfy the
„Grand Orthogonality Theorem“ (GOT).
We do not discuss the GOT here, but we list three
consequences of it:
• Number of irreps = Number of classes in the group.
• Dimensions of the irreps, l1, l2, l3 … satisfy
l12 + l22 + l32 + … = h,
where h is the number of elements in the group.
• Orthogonality relation
• Irreducible and reducible representations
These are used as ‘‘symmetry labels‘‘
on energy levels.
Which energy levels can ‘‘interact‘‘
and which transitions can occur.
Can also determine whether certain
terms are in the Hamiltonian.
BUT
IN SOME CIRCUMSTANCES
THERE ARE PROBLEMS IF WE
TRY TO USE POINT GROUP
SYMMETRY TO DO THESE THINGS
How do we use point group symmetry if
the molecule rotates and distorts?
H3
+
D3h
C2v
Or if tunnels?
NH3
3
1
2
C3v
2
1
3
D3h
What are the symmetries
of B(CH3)3, CH3.CC.CH3,
(CO)2, (NH3)2,…?
Nonrigid molecules (i.e. molecules
that tunnel) are a problem
if we try to use a point group.
Also
What should we do if
we study transitions (or
interactions) between
electronic states that have
different point group
symmetries at equilibrium?
Point groups used for classifying:
The electronic states for any molecule
at a fixed nuclear geometry
(see BJ2 Chapter 10), and
The vibrational states for molecules,
called “rigid” molecules, undergoing
infinitesimal vibrations about a
unique equilibrium structure
(see BJ2 Pages 230-238).
To understand how we use symmetry
labels and where the point group
goes wrong we must study
what we mean by “symmetry”
Rotations and
reflections
Permutations
and the inversion
J.T.Hougen, JCP 37, 1422 (1962); ibid, 39, 358 (1963)
H.C.Longuet-Higgins, Mol. Phys., 6, 445 (1963)
P.R.B. and Per Jensen, JMS 228, 640 (2004) [historical introduction]
See also BJ1 and BJ2
Symmetry not from geometry
since molecules are dynamic
• Centrifugal distortion
eg. H3+ or CH4 dipole
moment
• Nonrigid molecules:
eg. ethane, ammonia,
(H2O)2, (CO)2,…
• Breakdown of BOA:
eg. HCCH* - H2CC
Also symmetry applies
to atoms, nuclei and
fundamental particles.
Geometrical point group
symmetry is not possible
for them.
We need a more general
definition of symmetry
Symmetry Based on Energy Invariance
Symmetry operations are operations
that leave the energy of the system
(a molecule in our case) unchanged.
Using quantum mechanics we define a symmetry operation
as follows:
A symmetry operation is an operation that
commutes with the Hamiltonian:
(RH – HR)n = [R,H]n = 0
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation
Isotropic Space----------Rotation
Identical electrons------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q,s)
(-p,-q,s)
Reversal symmetry-----Time reversal (p,q,s) (-p,q,-s)
Ch. conj. Symmetry-----Particle
antiparticle
P(E*)
T
C
Symmetry Operations (energy invariance)
• Uniform Space ----------Translation
Separate translation…
Translational momentum
Ψtot = Ψtrans Ψint
int = rot-vib-elec.orb-elec.spin-nuc.spin
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation K(spatial) group,
J, mJ or F,mF labels
Isotropic Space----------Rotation
Identical electrons-------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q)
(-p,-q)
P(E*)
Reversal symmetry-----Time reversal (p,s)
(-p,-s) T
Ch. conj. Symmetry-----Particle
antiparticle
C
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation Symmetric group Sn
Isotropic Space----------Rotation
Identical electrons-------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q)
(-p,-q)
P(E*)
Reversal symmetry-----Time reversal (p,s)
(-p,-s) T
Ch. conj. Symmetry-----Particle
antiparticle
C
For the BeH molecule (5 electrons)
Ψorb-spin transforms as D(0) of S5
PEP
Slater determinant ensures antisymmetry
so do not need S5
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation
Isotropic Space----------Rotation
Identical electrons-------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q)
(-p,-q)
Reversal symmetry-----Time reversal (p,s)
(-p,-s)
Ch. conj. Symmetry-----Particle
antiparticle
P(E*)
T
C
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation
Isotropic Space----------Rotation
Identical electrons-------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q)
(-p,-q)
Reversal symmetry-----Time reversal (p,s)
(-p,-s)
Ch. conj. Symmetry-----Particle
antiparticle
P(E*)
T
C
CNPI group = Complete Nuclear Permutation Inversion Group
Symmetry Operations (energy invariance)
•
•
•
•
•
•
•
Uniform Space ----------Translation
Isotropic Space----------Rotation
Identical electrons-------Permute electrons
Identical nuclei-----------Permute identical nuclei
Parity conservation-----Inversion (p,q)
(-p,-q)
Reversal symmetry-----Time reversal (p,s)
(-p,-s)
Ch. conj. Symmetry-----Particle
antiparticle
P(E*)
T
C
CNPI group = Complete Nuclear Permutation Inversion Group
EXAMPLE:
The CNPI group for H2O is C2v(M) = {E, (12), E*, (12)*}
The CNPI Group for the Water Molecule
The Complete Nuclear Permutation Inversion (CNPI) group
for the water molecule is C2v(M) = {E, (12), E*, (12)*}
H1
+
e
H2
O
(12)
H2
+
e
H1
(12)*
E*
H2
H1
-
e
O
O
We compare C2v and this CNPI group
Multiplication table (Rrow Rcolumn)
C2v
CNPI
E
C2
E
E
C2
yz
xy
yz
xy
E
(12)
E*
(12)*
E
E
(12)
E*
(12)*
C2
C2
E
xy
yz
(12)
(12)
E
(12)*
E*
yz
yz
xy
xy
xy
yz
E
C2
C2
E
E*
E*
(12)*
E
(12)
(12)*
(12)*
E*
(12)
E
C2v and CNPI are isomorphic!
We compare C2v and this CNPI group
Rotational
subgroup
Permutation
subgroup
E
C2
E
E
C2
yz
xy
yz
xy
E
(12)
E*
(12)*
E
E
(12)
E*
(12)*
C2
C2
E
xy
yz
(12)
(12)
E
(12)*
E*
yz
yz
xy
xy
xy
yz
E
C2
C2
E
E*
E*
(12)*
E
(12)
(12)*
(12)*
E*
(12)
E
CNPI group of water: Character table
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
This group is called C2v(M)
1
Why RH = HR used to Define Symmetry?
For the water molecule (  nondegenerate and R2 = E for all R) :
H = E
RH = RE
HR = ER
Thus R = c since E is nondegenerate.
However R2 = E, so R(RΨ) = Ψ, but R(RΨ) = c2Ψ. Thus c2 = 1,
c = ±1 and R = ±
Symmetry of H restricts symmetry of eigenfunctions Ψ
+ Parity
R=E*
Ψ1+(x)
- Parity
Ψ2-(x)
x
x
Ψ-(-x) = -Ψ-(x)
Ψ3+(x)
x
Ψ+(-x) = Ψ+(x)
Eigenfunctions of H
must satisfy
E*Ψ = ±Ψ
”Why RH = HR used to Define Symmetry?”
continued……
For the water molecule (with nondegenerate states):
RH=HR
implies
H = E
RH = RE
HR = ER
Thus R = c. However R2 = E, so c = ±1 and R = ±
Symmetry of H restricts symmetry of eigenfunctions Ψ
Allows us to SYMMETRY LABEL the energy
levels using the irreps of the symmetry group
There are four symmetry types of H2O wavefunction
R = ±
A1
A2
B1
B2
E
1
1
1
1
(12) E*
1
1
1
-1
-1
-1
-1
1
(12)*
1
-1
1
-1
Possible labels would be (1,1), (1,-1), (-1,-1), and (-1,1).
However.
systematic
∫ΨaHΨbdτMore
= 0 ifgenerally
symmetries
of Ψa andare
Ψb the
are irreducible
different.
representation labels (or symmetry labels) from the
symmetry
∫Ψ μΨ dτ group.
= 0 if symmetry of product is not A
a
b
1
A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1
The Symmetry Labels of the C2v(M) Group of H2O
R = ±
A1
A2
B1
B2
E
1
1
1
1
(12) E*
1
1
1
-1
-1
-1
-1
1
(12)*
1
-1
1
-1
∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.
∫ΨaμΨbdτ = 0 if symmetry of product is not A1
A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1
The Symmetry Labels of the C2v(M) Group of H2O
R = ±
A1
A2
B1
B2
E
1
1
1
1
(12) E*
1
1
1
-1
-1
-1
-1
1
(12)*
1
-1
1
-1
Labeling is not just bureaucracy…it is useful
because of:
The vanishing integral theorem
BJ1 pp114-117, BJ2 pp 136-139
The Symmetry Labels of the C2v(M) Group of H2O
R = ±
A1
A2
B1
B2
E
1
1
1
1
(12) E*
1
1
1
-1
-1
-1
-1
1
(12)*
1
-1
1
-1
Labeling is not just bureaucracy…it is useful
because of:
The vanishing integral theorem
BJ1 pp114-117, BJ2 pp 136-139
But first let’s look at three things we overlooked:
Rn=E with n>2, degenerate states, symmetry of a product
n
Suppose R = E where n > 2.
n
We still have RΨ = cΨ for nondegenerate Ψ, but now R Ψ = Ψ.
n
Thus c = 1 and c =
n
√1
If n = 3 we introduce
 = ei2/3
and c = 1,ε or
ε2 (=ε*)
C3(M) E (123)
C3 (132)
C 32
1
1
1
A
1
1
1
Ea
1

*
Eb
1
*

iπ
e
= -1
ei2π = 1
n
Suppose R = E where n > 2.
n
We still have RΨ = cΨ for nondegenerate Ψ, but now R Ψ = Ψ.
n
Thus c = 1 and c =
n
√1
If n = 3 we introduce
 = ei2/3
C3(M) E (123)
C3 (132)
C 32
1
1
1
A
1
1
1
Ea
1

*
Eb
1
*

and c = 1,ε or
ε2 (=ε*)
A pair of
separably
degenerate
irreps. Degenerate
because of T
For nondegenerate states we had
this as the effect of a symmetry
operation on an eigenfunction:
For the water molecule (  nondegenerate) :
H = E
RH = RE
HR = ER
Thus R = c since E is nondegenerate.
What about degenerate states?
ℓ-fold degenerate energy level with energy En
R Ψnk = D[R ]k1Ψn1 + D[R ]k2Ψn2 + D[R ]k3Ψn3 +…+ D[R ]kℓΨnℓ
For each relevant symmetry operation R, the constants
D[R ]kp form the elements of an ℓℓ matrix D[R ].
ForT = RS it is straightforward to show that
D[T ] = D[R ] D[S ]
The matrices D[T ], D[R ], D[S ] ….. form an ℓ-dimensional
representation that is generated by the ℓ functions Ψnk
The ℓ functions Ψnk transform according to this
representation
Symmetry of a product:C2v(M) example
The symmetry of the product of two nondegenerate states is easy:
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1
B1 x B2, A1 x A2, B1 x A2, B2 x A2, B1 x B1,…
A2
A2
B2
B1
A1
Symmetry of a product. Example: C3v
A1  A1 = A1
A1  A2 = A2
A2  A2 = A1
A1  E = E
A2  E = E
E  E = A1  A2  E
E  E:
4
1
0
Reducible representation
Characters of the product representation are the products
of the characters of the representations being multiplied.
See pp 109-114 in BJ1
Symmetry of a product. Example: C3v
A1  A1 = A1
A1  A2 = A2
A2  A2 = A1
A1  E = E
A2  E = E
E  E = A1  A2  E
We say that E x E
E  E:
4
1
0
A1
Reducible representation
Characters of the product representation are the products
of the characters of the representations being multiplied.
See pp 109-114 in BJ1
Back to the vanishing
integral theorem
+ Parity
- Parity
Ψ+(x)
Ψ-(x)
x
x
Ψ-(-x) = -Ψ-(x)
Ψ+(x)
x
Ψ+(-x) = Ψ+(x)
∫Ψ+Ψ-Ψ+dx = 0
- parity
+ Parity
- Parity
Ψ+(x)
Ψ-(x)
The vanishing integral
theorem
x
x
Ψ
(-x)
=
-Ψ
(x)
+
Ψ0(x)
∫f(τ)dτ =
if symmetry of f(τ) does not contain A1
x
Ψ+(-x) = Ψ+(x)
∫Ψ+Ψ-Ψ+dx = 0
- parity
Diagonalizing the molecular
Hamiltonian
Schrödinger equation
Eigenvalues and eigenfunctions are found by
diagonalization of a matrix with elements
To apply the vanishing integral rule we look at symmetry of
Diagonalizing the molecular
Hamiltonian
= 0
if Γ(integrand) does not contain Γ(s)
Hmn vanishes if Γ(
) and Γ(
) are different
The Hamiltonian matrix
factorizes, for example for H2O
Integrated absorption intensity for a line is:
~ ~
I(f ← i) = ∫ ε(ν)dν
line
8π3 Na
______
=
(4πε0
)3hc2
νif F(Ei )S(f ← i) Rstim(f→i)
Frequency factor
F(Ei ) = [ gie-Ei/kT ] / ∑ gje-Ej/kT Boltzmann factor
j
S(f ← i) =
∑ | ∫ Φf* μA Φi dτ |2 Line strength
A=X,Y,Z
Rstim(f→i) = 1 – exp (-hνif /kT ) Stimulated emission
Selection rules for transitions
The intensity of a transition is proportional to the square of
Z
μZ =
Σi qi Zi
For the integral to be non-vanishing, the integrand must
have a totally symmetric component.
Product of symmetries of Φs must contain that of μZ
Symmetry of Z
Z has symmetry *
What is
* ???
Symmetry of Z
Z has symmetry *
* has character +1 under all permutations P
1 under all permutation-inversions P*
Symmetry of Z for H2O
* = A2
The Symmetry Labels of the C2v(M) Group of H2O
Γ(H) = Γ(s)
R = ±
A1
A2
B1
B2
E
1
1
1
1
(12) E*
1
1
1
-1
-1
-1
-1
1
(12)* Symmetry of H
1
-1
Symmetry of μZ
1
-1
Γ(μZ) = Γ*
Using symmetry labels and the vanishing
integral theorem we deduce that:
∫Ψa*HΨbdτ = 0 if symmetries of Ψa and Ψb are different.
0
0
0
0
As a result the Hamiltonian matrix is block diagonal.
∫Ψa*μΨbdτ = 0 if symmetry of product ΨaΨb is not Γ*
Determining symmetry and
reducing a representation
Example of using the symmetry operation (12):
(12)

r 1´
r2´
´
H2
We have (12) (r1, r2, ) = (r1´, r2´, ´)
We see that (r1´, r2´, ´) = (r2, r1, )
H1
3
E
r1

3
r2
r 1´
2
1
´
1
3
(12)
r1

3
r2
r 2´
2
1
r 2´
´
2
r1 ´
r2 ´
´
1
r1 ´
r2 ´
´
r 1´
2
=
r1
r2

=
r2
r1

=
r1
r2

=
r2
r1

3
E*
r1

r2
2
1
´
2
r2´
1
r 1´
r1 ´
r2 ´
´
3
3
(12)*
r1
1

1
r2
2
2
´ r ´
2
r 1´
3
r1 ´
r2 ´
´
R
a =
E
r1
r2

(12)
r1
r2

E*
r1
r2

(12)*
r1
r2

a´
=
r1 ´
r2 ´
´
=
r1 ´
r2 ´
´
=
r1 ´
r2 ´
´
=
r1 ´
r2 ´
´
= D[R] a
=
r1
r2

=
r2
r1

=
r1
r2

=
r2
r1

=
1
0
0
0
1
0
0
0
1
r1
r2

=3
=
0
1
0
1
0
0
0
0
1
r1
r2

=1
=
1
0
0
0
1
0
0
0
1
r1
r2

=
0
1
0
1
0
0
0
0
1
r1
r2

=3
=1
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1

3
1
3
1
aA1 =
1
4
Γ = Σ aiΓi
i
A reducible representation
( 13 + 11 + 13 + 11) = 2
( 13 + 11  13  11) = 0
aB1 =
1
4
1
4
aB2 =
1
4
( 13  11 + 13  11) = 1
aA2 =
( 13  11  13 + 11) = 0
 = 2 A1  B2
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1

3
1
3
1
aA1 =
1
4
Γ = Σ aiΓi
i
i
A reducible representation
( 13 + 11 + 13 + 11) = 2
( 13 + 11  13  11) = 0
aB1 =
1
4
1
4
aB2 =
1
4
( 13  11 + 13  11) = 1
aA2 =
( 13  11  13 + 11) = 0
 = 2 A1  B2
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1

3
1
3
1
aA1 =
1
4
Γ = Σ aiΓi
i
i
A reducible representation
( 13 + 11 + 13 + 11) = 2
( 13 + 11  13  11) = 0
aB1 =
1
4
1
4
aB2 =
1
4
( 13  11 + 13  11) = 1
aA2 =
( 13  11  13 + 11) = 0
 = 2 A1  B2
E (12) E* (12)*
A1
1
1
1
1
A2
1
1
1
1
B1
1
1 1
1
B2
1
1
1
1

3
1
3
1
aA1 =
1
4
Γ = Σ aiΓi
i
i
A reducible representation
( 13 + 11 + 13 + 11) = 2
( 13 + 11  13  11) = 0
aB1 =
1
4
1
4
aB2 =
1
4
( 13  11 + 13  11) = 1
aA2 =
( 13  11  13 + 11) = 0
 = 2 A1  B2
We know now that r1, r2, and  generate the
representation 2 A1  B2
Consequently, we can generate from r1, r2, and
 three „symmetrized“ coordinates:
S1 with A1 symmetry
S2 with A1 symmetry
S3 with B2 symmetry
For this, we need projection operators
Pages 102-109 of BJ1
Projection operators:
General for li-dimensional irrep i
Diagonal element of representation matrix
Symmetry operation
Simpler for 1-dimensional irrep i
1
Character
Projection operators:
General for li-dimensional irrep i
Simpler for 1-dimensional irrep i
Character
Diagonal element of representation matrix
1
Symmetry operation
Projection operators:
General for li-dimensional irrep i
Simpler for 1-dimensional irrep i
Character
Diagonal element of representation matrix
1
A1
E
1
(12)
1
E*
1
Symmetry operation
(12)*
1
PA1 = (1/4) [ E + (12) + E* + (12)* ]
Projection operators:
General for li-dimensional irrep i
Simpler for 1-dimensional irrep i
Character
Diagonal element of representation matrix
1
A1
B2
E
1
1
(12)
1
-1
E*
1
1
Symmetry operation
(12)*
1
-1
PA1 = (1/4) [ E + (12) + E* + (12)* ]
PB2 = (1/4) [ E – (12) + E* – (12)* ]
Projection operator for A1 acting on r1
S1 = PP11A1A1r1 =
1
4
[ E + (12) + E* + (12)* ]r1
=
1
4
[ r1 + r2 + r1 + r2 ] =
S2 = PP11A1A1 =
1
4
[ E + (12) + E* + (12)* ]
=
1
4
[ + + +] =
S3 = P11 r1 =
1
4
[ E  (12) + E*  (12)*] r1
=
1
4
[ r1  r2 + r1  r2 ] =
2  =
PP11BB2
1
4
[ E  (12) + E*  (12)* ]
=
1
4
[   +    ] =
2
PBB2
1
2
[ r1 + r2 ]

1
2
[ r1  r2 ]
0
 Is „annihilated“ by PP11BB2
2
Projection operators for A1 and B2
S1 = PP11A1A1r1 =
1
4
[ E + (12) + E* + (12)* ]r1
=
1
4
[ r1 + r2 + r1 + r2 ] =
S2 = PP11A1A1 =
1
4
[ E + (12) + E* + (12)* ]
=
1
4
[ + + +] =
S3 = P11 r1 =
1
4
[ E  (12) + E*  (12)*] r1
=
1
4
[ r1  r2 + r1  r2 ] =
2  =
PP11BB2
1
4
[ E  (12) + E*  (12)* ]
=
1
4
[   +    ] =
2
PBB2
1
2
[ r1 + r2 ]

1
2
[ r1  r2 ]
0
 Is „annihilated“ by PP11BB2
2
Projection operators for A1 and B2
S1 = PP11A1A1r1 =
1
4
[ E + (12) + E* + (12)* ]r1
=
1
4
[ r1 + r2 + r1 + r2 ] =
S2 = PP11A1A1 =
1
4
[ E + (12) + E* + (12)* ]
=
1
4
[ + + +] =
S3 = P11 r1 =
1
4
[ E  (12) + E*  (12)*] r1
=
1
4
[ r1  r2 + r1  r2 ] =
2
PBB2
1
1
2
[ r1 + r2 ]

1
2
[ r1  r2 ]
Aside: S1, S2 and S3 have
the symmetry and form of the
4
normal coordinates. See
pp 269-277 in BJ1, and 232-233 in BJ2
1
4
[H,R] Symmetry and conservation laws
(see chapter 14 of BJ2)
iħ ∂Ψ/∂t = HΨ
where Ψ is a function of q and t
Does symmetry change with time?
∂<Ψ|R|Ψ>/∂t = <∂Ψ/∂t|R|Ψ> + <Ψ|∂(RΨ)/∂t>
= <∂Ψ/∂t|R|Ψ> + <Ψ|R|∂Ψ/∂t>
i
__
=
[<HΨ|R|Ψ> - <Ψ|R|HΨ>]
ħ
i <Ψ|[H,R]|Ψ> (H is Hermitian)
= __
ħ
=0
So Far:
• Point group (geometrical) symmetry
• H2O and PH3 point groups used as examples
• Group theory definitions: Irreducible reps and Ch. Tables
• Reducible representations and projection operators
• Problems using point groups: Rotation, tunneling,…
• Use [H,R]=0 to define R as a symmetry operation
• Introduce the CNPI group
• Explain why [H,R]=0 used to define symmetry
• Can label energy levels (the Ψ generate a representation.)
• Vanishing integral theorem
• Forbidden interactions and forbidden transitions
• Conservation of symmetry
Where are we going?
HΨn = EnΨn
H = H0 + H’
where H0Ψn0 = En0Ψn0
En0 is ℓ-fold degenerate: Eigenfunctions are Ψn10,Ψn20,…,Ψnℓ0
We want to symmetry label the energy levels using the
irreducible representations of a symmetry group.
We do this because it helps us to do many things:
Which En0 can be mixed by H’: Block diagonalize H-matrix.
Selection rules for transitions: Only if connected by Γ*.
Nuclear spin statistics and intensity ratios
Tunneling splittings, Stark effect, Zeeman effect,
Breakdown of Born-Oppenheimer approximation…
The basis of what we do using symmetry is that:
The ℓ-fold degenerate eigenfunctions Ψn10,Ψn20,…,Ψnℓ0
GENERATE an ℓ-fold irreducible representation of the
symmetry group (this labels the energy level En0).
R
Ψn10
Ψn20
.
.
Ψnℓ0
=
D(R)
Ψn10
Ψn20
.
.
Ψnℓ0
The matrices D(R) form an irreducible representation
The above follows from the fact that [H,R] = 0.
To obtain the matrices D(R), and hence the irreducible
rep. label, we need to know the Ψni0 and to know how the
symmetry ops transform the coordinates in the Ψni0.
But first of all we must decide
on the symmetry group that
we are going to use.
It could be the CNPI group
BUT…
There are problems with the CNPI Group
Number of elements in the CNPI groups of various
molecules
C6H6, for example, has a 1036800-element CNPI group,
but a 24-element point group at equilibrium, D6h
Huge groups and (as we shall see) multiple symmetry labels