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Chapter 4
Group theory
Group theory will be presented in this chapter as a tool for spectroscopy. Indeed group theory
makes it possible to
- construct and classify molecular orbitals,
- classify electronic, vibrational, rotational and nuclear spin wave functions,
- predict which states are allowed,
- predict physical properties (existence of electric dipole moment, optical activity etc.),
- predict selection rules (electric dipole transitions, configuration interaction. etc.)
However, group theory does not make any quantitative predictions. The interest of group
theory lies in simplifying some problems like those mentioned above that arise in molecular
spectroscopy.
4.1 Symmetry operations
4.1.1 Definition of a group
A group G is a set of elements A, B, C, ... connected by a combination rule (written as a
product, for example A · B) which has the following properties:
1. the closure: for all elements A and B of the group G, A · B = C is also an element of
the group G.
2. the associativity: the combination rule must be associative, i. e. A·(B·C) = (A·B)·C.
3. the identity: there must be an element, the identity E (also called unit), such that
E · R = R · E = R for all elements R of the group.
4. the inverses: each element R must have an inverse R
− 1
which is also a group element
such that R · R
− 1
= R
− 1
· R = E.
102 CHAPTER 4. GROUP THEORY
In group theory, the elements considered are symmetry operations. For a given molecular
system described by the Hamiltonian
H, there is a set of symmetry operations
Oi which
commute with
H:
[
Oˆ
i,^
Hˆ
]
H and
Oi thus have a common set of eigenfunctions and the eigenvalues of
Oi can be used
as labels for the eigenfunctions (see Lecture Physical Chemistry III). This set of operations
defines (with the multiplication operation) a symmetry group. In molecular physics and
molecular spectroscopy two types of groups are particularly important, the point groups and
the permutation-inversion groups.
4.1.2 Point group operations and point group symmetry
The point groups adequately describe molecules that can be considered as rigid on the
timescale of the spectroscopic experiment, which means molecules that have a unique equi-
librium configuration with no observable tunneling between two or more equivalent configu-
rations.
The symmetry operations of the point groups are:
- the identity E which leaves all coordinates unchanged.
- the proper rotation Cn by an angle of 2π/n in the positive trigonometric sense (i. e.
counter-clockwise). The symmetry axis with highest n is chosen as principal axis. If
a molecule has a unique Cn axis with highest n, the molecule has a permanent dipole
moment that lies along this axis (e. g. H 2 O, NCl 3 in Figure 4.1). If a molecule has
several Cn axes with highest n, the molecule has no permanent dipole moment (e. g.
CH 4 ).
O
H H
C 2
N
Cl
Cl
C 3
Cl
Figure 4.1: C 2 rotation of H 2 O and C 3 rotation of NCl 3.
- the reflection through a plane σ; the reflections are classified into two categories:
104 CHAPTER 4. GROUP THEORY
been developed, originally by Christopher Longuet-Higgins and Jon T. Hougen (see Bunker
and Jensen, Molecular Symmetry and Spectroscopy, 1998). Their concept relies on the fact
that the symmetry operations, i. e. the permutation-inversion operations leave
H unchanged.
The symmetry operations of the CNPI groups are:
- the permuation (ij) of the coordinates of two identical nuclei i and j which denotes
the exchange of the nucleus i with the nucleus j (see Figure 4.4 for examples),
- the cyclic permutation (ijk) of the coordinates of three identical nuclei i, j, and k,
i. e. the nucleus i will be replaced by the nucleus j, j by k and k by i (see Figure 4.
for example),
O
H (^) (1) H (^) (2)
(12) O
H (^) (2) H (^) (1)
N
H (^) (2)
H (1)
(123)
H (^) (3)
N
H (^) (3)
H H (^) (1) (2)
F(2) H (^) (2) F(1)
H (^) (1) (12)
F(1) H (^) (1) F(2)
H (^) (2)
Figure 4.4: Examples of (i j) and (i j k) permutations.
- all possible circular permutations of n identical nuclei (for example, the (1 2 3 4 5 6)
permutation in benzene),
∗
of all coordinates of all particles through the center of the lab-fixed
frame,
- the permutation followed by an inversion (ij)
∗
= E
∗
· (ij) of all coordinates of all
particles
- the cyclic permutation followed by an inversion (ijk)
∗
of all coordinates of all
particles,
- all possible circular permutations followed by an inversion of all coordinates of n iden-
tical nuclei.
4.2. IMPORTANT CONCEPTS IN A GROUP 105
The permutation operations only affect identical nuclei, therefore the molecular Hamiltonian
is left unchanged upon these operations. Moreover the molecular Hamiltonian depends on
distances rather than positions, hence the inversion operation also leaves
H unchanged.
The CNPI groups represent a more general description that can also be applied to rigid
molecules. Indeed each point group is isomorphous to a CNPI group although the symmetry
operations are not identical (for example, the inversion i of a point group symmetry is not
the same as the inversion of a permutation inversion group E
∗
Example: the point group C3v is isomorphous to S 3 = {E, (1 2 3), (1 3 2), (1 2), (1 3), (2 3)}, which
means that there is a one to one correspondence between the two sets of operations.
However, one disadvantage of the CNPI groups is their size which can become very large. For
example, the CNPI for CH 4 contains 48 symmetry operations, and that of benzene 1036800!
In the case of non-rigid systems, this problem is usually solved by using a subgroup, i. e. a
subset of the group which forms a group under the same combination rule. These subgroups
are called molecular symmetry (MS) groups.
In the case of rigid molecules, most of the time the point groups are used. In the following,
we will consider rigid molecules only and restrict ourselves to point group symmetry, but all
concepts can be extended to the CNPI and MS groups.
4.2 Important concepts in a group
4.2.1 Order, conjugated elements and classes
The order of a group is equal to the number of elements in the group. The discrete
(or finite) groups have a finite order (for example C2v is a group of fourth order), while
continuous groups have infinite orders (C∞v for example).
Let us consider two operations Oˆi and Oˆj that leave Hˆ unchanged when applied individually.
Hence, they must also leave
H unchanged when applied in succession. The notation
Oi ·
Oj
means that
Oj acts first, and
Oi second. In other words,
Oi ·
Oj must be a symmetry operation
Oˆ
k if^
Oˆ
i and^
Oˆ
j are symmetry operations, which is a corollary of the closure property of a
group. Very often it is useful to build the so-called multiplication table which summarizes
all possible
Oi ·
Oj combinations.
4.2. IMPORTANT CONCEPTS IN A GROUP 107
For example, the C2v group is abelian (see Table 4.1). However, not all groups are Abelian.
An example of a non-abelian group is the point group C3v.
Example: CH 3 Cl in the C3v group
The symmetry operations are E, C 3 , C 2 3 , σ a , σ b , σ c , hence the group is of order 6. With the help of
Figure 4.6, one can derive the multiplication table of the C3v point group. One sees that the group is
not Abelian because not all operations commute (e. g., C 3 · σ
a = σ
c and σ
a · C 3 = σ
b ). Moreover, not
all operations are their own inverse (e. g., C 3 · C 3 = E).
Figure 4.6: The operations of the C3v point group with the example of the CH 3 Cl molecule
represented as a Newmann projection (adapted from F. Merkt and M. Quack in Handbook of
high-resolution spectroscopy, 2011).
108 CHAPTER 4. GROUP THEORY
st
operation (right)
C3v E C 3 C
2 3
a
b
c
nd
operation (left)
E E C 3 C
2 3
a
b
c
C 3 C 3 C
2 3
E σ
c
a
b
C
2 3
C
2 3
E C 3 σ
b
c
a
a
a
b
c
E C 3 C
2 3
b
b
c
a
C
2 3
E C 3
c
c
a
b
C 3 C
2 3
E
Table 4.2: Multiplication table of the C3v point group.
If
Oi,
Oj , and
Ok are all group elements and if they obey the relation
Ok ·
Oi ·
O
− 1 k
Oj , (4.3)
then Oˆi and Oˆj are called conjugated elements. All conjugated elements in a group form
a class.
Example: Elements of the point group C3v that belong to the same class as C 3
We consider C 3 =
Oi and apply each operation
Ok of C3v according to Equation (4.3) with the help
of the multiplication table in order to find the conjugated elements of C 3.
Oi
Ok
O
− 1 k
Ok ·
Oi ·
O
− 1 k
Oj
Oi
Ok
O
− 1 k
Ok ·
Oi ·
O
− 1 k
Oj
C 3 E E E · C 3 · E = C 3 C
2 3
E E E · C
2 3
· E = C
2 3
C 3 C 3 C
2 3 C^3 ·^ C^3 ·^ C
2 3 =^ C^3 C
2 3 C^3 C
2 3 C^3 ·^ C
2 3 ·^ C
2 3 =^ C
2 3
C 3 C
2 3 C^3 C
2 3 ·^ C^3 ·^ C^3 =^ C^3 C
2 3 C
2 3 C^3 C
2 3 ·^ C
2 3 ·^ C^3 =^ C
2 3
C 3 σ
a σ
a σ
a · C 3 · σ
a
︸ ︷︷ ︸ σc
= C
2 3 C
2 3 σ
a σ
a σ
a · C
2 3 ·^ σ
a
︸ ︷︷ ︸
σb
= C 3
C 3 σ
b σ
b σ
b · C 3 · σ
b ︸ ︷︷ ︸ σa
= C
2 3 C
2 3 σ
b σ
b σ
b · C
2 3 ·^ σ
b
︸ ︷︷ ︸ σc
= C
2 3
C 3 σ c σ c σ c · C 3 · σ
c
︸ ︷︷ ︸
σb
= C
2 3
C
2 3 σ c σ c σ c · C
2 3 · σ
c
︸ ︷︷ ︸ σa
= C
2 3
C 3 and C 2 3 are conjugated; they are elements of the same class of order 2.
Similarly, one can show that σ
a , σ
b and σ
c form a class of order 3.
The order k of an element Oˆi is the smallest integer k ≥ 1 with Oˆ
k i
= E. This property
110 CHAPTER 4. GROUP THEORY
Therefore x, y, and z correspond to the following representations designated by Γ:
C2v E C z 2 σ xz σ yz
(z) 1 1 1 1
(x) 1 − 1 1 − 1
(y) 1 − 1 − 1 1
Case 2: instead of using x, y, or z, more complicated functions can be used to generate a one-
dimensional representation, such as x
2 , y
2 , z
2 , xy, xz or yz as listed in the fourth column of the
character table. For example, one takes the functions Ψ 4 = x
2 and Ψ 5 = xy:
Ψ 4 = x 2 : E x 2 = (E x)(E x) = x 2 Ψ 5 = xy : E xy = (E x)(E y) = xy
C
z 2 x
2 = (C
z 2 x)(C
z 2 x) =^ x
2 C
z 2 xy^ = (C
z 2 x)(C
z 2 y) =^ xy
σ xz x 2 = (σ xz x)(σ xz x) = x 2 σ xz xy = (σ xz x)(σ xz y) = −xy
σ
yz x
2 = (σ
yz x)(σ
yz x) = x
2 σ
yz xy = (σ
yz x)(σ
yz y) = −xy
It is easy to verify that χ (xy) = χ (x) × χ (y) , a result that can be written as a direct product. To
evaluate a direct product, one multiplies the characters of each class of elements pairwise and
obtains as direct product a representation of the group:
(xy) = Γ (x) ⊗ Γ (y) = (1 -1 1 -1) ⊗ (1 -1 -1 1) = (1 1 -1 -1).
Case 3: one can also look at the transformation properties of rotations and for example take Ψ 6 = Rz
as illustrated in Figure 4.7.
Figure 4.7: The Rz rotation of water.
Ψ 6 = Rz : E Rz = Rz
C
z 2 Rz^ =^ Rz
σ xz Rz = −Rz Direction of rotation reversed.
σ
yz Rz = −Rz Direction of rotation reversed.
Rz transforms as follows:
C2v E C z 2 σ xz σ yz
(Rz ) 1 1 − 1 − 1
4.2. IMPORTANT CONCEPTS IN A GROUP 111
Representations of higher dimensionality can be obtained by looking at the transformation
properties of two or more functions. Indeed, to construct an n-dimensional representation of
a group, one takes n linear independent functions or vectors Ψi, i = 1, ..., n spanning a given
n-dimensional space. Applying the group operations on Ψi leads to a transformed function
which is a linear combination of the original functions:
OΨi =
n
j=
bji(
O)Ψj. (4.7)
Example: two dimensional representation
x y
of the C2v group
E
x
y
x
y
x
y
⎠ ,^ with^ χ
x y) E
C
z 2
x
y
−x
−y
x
y
⎠ ,^ with^ χ
x y) Cz 2
σ xz
x
y
x
−y
x
y
⎠ ,^ with^ χ
x y) σxz^
σ yz
x
y
−x
y
x
y
⎠ ,^ with^ χ
x y) σyz^
The two-dimensional representation of
x y
has thus the following characters:
C2v E C
z 2 σ
xz σ
yz
2 × 2 Matrix
x y)^2 − 2 0 0
If the matrices of all elements of a representation of a group can be simultaneously brought
into block-diagonal form by a given coordinate transformation, the representation is said to
be reducible, if not, it is irreducible.
The character table of a group lists all irreducible representations and gives for each rep-
resentation the character of each class of elements.
4.2. IMPORTANT CONCEPTS IN A GROUP 113
4.2.3 Reduction of reducible representations
There is a systematic mathematical procedure to perform the reduction of representations.
All representations in a character table form a set of orthogonal vectors that span the complete
space:
O^ ˆ
(i)
O) × χ
(j)
O) = hδij , (4.8)
where h represents the order of the group and
O runs over all the elements of the group. (Note
that some classes of non-Abelian groups contain more than one element!). Any reducible
representation can thus be expressed as a linear combination of irreducible representations
red
k
c
red k
(k)
where Γ
(k)
represents an irreducible representation. The expansion coefficients c
red k
can be
determined using the reduction formula Equation (4.10):
c
red k
h
O^ ˆ
red
O) × χ
k
O) (4.10)
Example : two-dimensional representation spanned by 1s atomic orbitals
1s(1) 1s(2)
centred on the H
atoms of a water molecule H 2 O in the C2v group (see Figure 4.8)
O
H (2) H (1)
Figure 4.8: 1s atomic orbitals on the H atoms of H 2 O.
E
1s(1)
1s(2)
1s(1)
1s(2)
1s(1)
1s(2)
⎠ thus^ χE = 2
C
z 2
1s(1)
1s(2)
1s(2)
1s(1)
1s(1)
1s(2)
⎠ thus^ χCz 2
σ xz
1s(1)
1s(2)
1s(2)
1s(1)
1s(1)
1s(2)
⎠ thus^ χσxz^ = 0
σ yz
1s(1)
1s(2)
1s(1)
1s(2)
1s(1)
1s(2)
⎠ thus^ χσyz^ = 2
114 CHAPTER 4. GROUP THEORY
C2v E C
z 2 σ
xz σ
yz
A 1 1 1 1 1
A 2 1 1 − 1 − 1
B 1 1 − 1 1 − 1
B 2 1 − 1 − 1 1
2 × 2 Matrix
(1s) 2 0 0 2
Reduction of Γ
(1s) = (2 0 0 2) :
c
(1s) A 1
(2 × 1 + 0 × 1 + 0 × 1 + 2 × 1) = 1
c
(1s) A 2
(2 × 1 + 0 × 1 + 0 × (−1) + 2 × (−1)) = 0
c
(1s) B 1
(2 × 1 + 0 × (−1) + 0 × 1 + 2 × (−1)) = 0
c
(1s) B 2
(2 × 1 + 0 × (−1) + 0 × (−1) + 2 × 1) = 1
(1s) = A 1 ⊕ B 2.
This means that one can therefore construct one linear combination of the two 1s(H) orbitals of H 2 O
with A 1 symmetry (totally symmetric) and one with B 2 symmetry as will be shown in the following.
4.3 Useful applications of group theory
4.3.1 Determination of symmetrized linear combinations of atomic orbitals
To find the symmetrized linear combination of atomic orbitals (LCAO), one uses so-called
projectors
P. The projector associated with the irreducible representation Γ is defined by
P
Γ
h
O^ ˆ
(Γ)
O) ×
O. (4.11)
The application of
P
Γ
onto the atomic orbitals provides a LCAO of symmetry Γ.
116 CHAPTER 4. GROUP THEORY
orbitals. The five symmetrized orbitals listed above can be used to form five molecular orbitals
according to the following diagram (Figure 4.11) which does not take the 1s and 2s orbitals on the
oxygen into account because only valence electrons are considered for the formation of chemical bonds.
H 1s
O 2p
p x p y
B 1 B 2
p z
A 1
b (^1)
B 2 A 1
a 1
b (^2)
a 1
b 2
Figure 4.11: Valence molecular orbitals of H 2 O built from symmetrized H(1s) “ligand” or-
bitals and the 2p atomic orbitals of O. The labels of the molecular orbitals refer to their
symmetry in lower case letters.
From the electronic configuration of each atom, there are six valence electrons (O ... (2p)
4 , H (1s)
1 )
to place in the Molecular Orbitals (MOs) following Pauli’s Aufbau-principle gives the ground state
configuration: ...(b 2 )
2 (a 1 )
2 (b 1 )
2 with an overall symmetry A 1. Because four of the six electrons are
in bonding orbitals and two in a non bonding px orbital, one expects two chemical bonds in H 2 O.
The energetical ordering of the two bonding MO of B 2 and A 1 symmetry depends on the HOH angle
α defined in Figure 4.12. Whereas the a 1 orbital becomes nonbonding at α = 180 ◦ , the b 2 orbital
remains bonding at α = 180
◦ but becomes antibonding at small angles.
Figure 4.12: Bond angle α.
4.3. USEFUL APPLICATIONS OF GROUP THEORY 117
4.3.2 Symmetry of normal modes
We consider the 3N -dimensional reducible representation Γ 3 N spanned by the set of 3N
Cartesian coordinates of the N atoms in a molecule and reduce it into irreducible repre-
sentations of the corresponding group. The molecule can also be characterized by its 3N
displacement coordinates i. e. the translations (t), rotations (r) and vibrations (v). There-
fore:
Γ 3 N = Γt ⊕ Γr ⊕ Γv (4.12)
The representation of the vibrational modes Γv can be deduced from Γ 3 N subtracting the
representations Γt and Γr as indicated in the character table.
Example: The vibrational modes of H 2 O
The total representation is 3 × 3 = 9-dimensional. All irreducible representations of C2v are one-
dimensional, and only three vibrational modes (3N -6) exist in H 2 O. The symmetry of these modes will
be obtained by eliminating the six irreducible representations corresponding to the three translational
and the three rotational degrees of freedom of the molecule.
Figure 4.13: Coordinates used to derive the Γ 9 representation of H 2 O in the C2v group.
In the basis set (or representation) Γ 9 = {x 1 , y 1 , ..., z 3 }, the C2v symmetry operations are represented
by 9 × 9 matrices.
4.3. USEFUL APPLICATIONS OF GROUP THEORY 119
The reducible 9-dimensional representation is therefore:
C2v E C
z 2 σ
xz σ
yz
A 1 1 1 1 1 = Γ
z
A 2 1 1 -1 -1 = Γ
Rz
B 1 1 -1 1 -1 = Γ
x = Γ
Ry
B 2 1 -1 -1 1 = Γ
y = Γ
Rx
The Γ 9 representation can then be reduced using the reduction formula of Equation (4.10)):
cA 1 =
cA 2 =
cB 1
cB 2
Γ 9 = 3A 1 ⊕ A 2 ⊕ 2B 1 ⊕ 3B 2.
From these nine irreducible representations, three correspond to translations (Γ
x = B 1 , Γ
y = B 2 ,
z = A 1 ) and three correspond to rotations (Γ
Rx = B 2 , Γ
Ry = B 1 , Γ
Rz = A 2 ). The remaining three,
namely 2A 1 ⊕ B 2 , correspond to the three vibrational modes of H 2 O (3N − 6 = 3, because H 2 O is a
nonlinear molecule). To determine these modes one can use the projection formula of Equation (4.11).
Let us consider the vibrational mode of symmetry B 2 as an example. In practice it is convenient to
first treat the x, y and z displacements separately and then to combine the x, y, and z motions.
For the x-dimension:
Pˆ B^2 x 1 =^
(1Ex 1 − 1 C
z 2 x 1 − 1 σ
xz x 1 + 1σ
yz x 1 )
(x 1 + x 2 − x 2 − x 1 ) = 0.
The B 2 mode does not involve x-coordinates.
For the y- and z-dimensions:
Pˆ B^2 y 1 =^
(1Ey 1 − 1 C
z 2 y 1 − 1 σ
xz y 1 + 1σ
yz y 1 )
(y 1 + y 2 + y 2 + y 1 ) =
(y 1 + y 2 ).
P^ ˆ B^2 z 1 =^
(1Ez 1 − 1 C
z 2 z 1 − 1 σ
xz z 1 + 1σ
yz z 1 )
(z 1 − z 2 − z 2 + z 1 ) =
(z 1 − z 2 ).
The B 2 mode involves both y and z coordinates. Drawing the displacement vectors one obtains a
vectorial representation of the motion of the H atoms in the B 2 mode. The motion of the O atom
120 CHAPTER 4. GROUP THEORY
can be estimated in a same way or reconstructed by ensuring that the center of mass of the molecule
remains stationary.
Figure 4.14: Determination of the nuclear motion of the B 2 mode of water.
The mode can be easily identified as the asymmetric stretching mode.
4.3.3 Symmetry of vibrational levels
The nomenclature to label the vibrational states of a polyatomic molecule is
v 1 1
v 2 2
v 3 N − 6 3 N − 6
where νi designate the mode and vi the corresponding vibrational quantum number. Usually
only the modes νi for which vi = 0 are indicated. The notation
(v 1 , v 2 , · · · , v 3 N − 6 ) (4.14)
is also often used. For the ordering of the modes, the totally symmetric modes come first
in order of descending frequency, then the modes corresponding to the second irreducible
representation in the character table in order of descending frequency, etc.
To find the overall symmetry of the vibrational wavefunction one must build the direct
product
Γvib = (Γν
1
v 1
2
v 2
3 N − 6
v 3 N − 6
Example: The three vibrational modes of H 2 O ν 1 is the O-H symmetric stretching mode (˜ν 1 =3585 cm
− 1 )
of symmetry A 1 , ν 2 is the H-O-H bending mode (˜ν 2 =1885 cm
− 1 ) of symmetry A 1 and ν 3 is the O-H
asymmetric stretching mode (˜ν 3 =3506 cm
− 1 ).
We consider the state with v 1 = 2, v 2 = 1, v 3 = 3. In the first notation, this will correspond to :
