University of Basra College of Science Master of Physics

Properties and physical applications

of
group theory and symmetry
Abdulhafiz Amer
 Index
Introduction to Symmetry and Group Theory 4
Symmetry 4
Symmetry Operations and Symmetry Elements: 5
Identity E: 5
Rotation axis 𝑪𝒏. 6
The principal axis 6
Reflection, σ 8
Inversion, i 9
n-fold Axis of Improper Rotation, 𝑺𝒏 10
Symmetry Element 12
Point Group 13
Space Group 19
Definition of space groups 19
Classification of space groups 20
Symmetry operations in space groups 21
Glide Planes 21
Screw Axes 23
Hermann-Mauguin notation for space groups 25
Relationship between point groups and space groups 25
Point groups as subgroups of space groups 25
Space groups as extensions of point groups 26
Applications of point groups and space groups 27
1.Symmetry in crystal structures 27
2.Symmetry in phonon dispersion relations 27
3.Symmetry in electronic band structures 27
4.Symmetry-based selection rules for transitions 28
Determination of point groups and space groups 31
Experimental methods for symmetry determination 31
X-ray diffraction and space group determination 31
Computational methods for symmetry analysis 31
Consequences of symmetry breaking 32
Ferroelectricity and noncentrosymmetric point groups 32
Piezoelectricity and polar point groups 32
Magnetism and time-reversal symmetry breaking 32
Symmetry and Physical Properties 33
Representation Symmetry Operations as A Matrices 34
The Identity Operation, E 34
The Reflection Operation, σ 34
The n-fold Rotation Operation, Cn 35
The Inversion operation, I 36
Represented Molecules by Reducible Representations: 37
How Do You Get a Reducible Representation for a Molecule? 40
1. Choose a Basis Set 40
2. Determine the Point Group 40
3. Apply Symmetry Operations to the Basis 40
4. Build the Reducible Representation Table 40
5. Reduce Γ into Irreducible Representations 40

2
Character Tables 41
Properties of Characters of Irreducible Representations in Point Groups 42
Mullikan Symbol of irreducible representations 42
Properties of Character table 44
Character table for point group C4h 47
Example : Water (C2v) 56
1 - The bending vibration transforms like A1: fully symmetric : 56
2 - The asymmetric stretch transforms like B2 : 56
3 - The 2px orbital on hydrogen transforms like B1 58
Refrence 60

3
Introduction to Symmetry and Group Theory
Symmetry can help resolve many chemistry and physics problems and usually the first
step is to determine the symmetry. If we know how to determine the symmetry of small
molecules, we can determine symmetry of other targets which we are interested in. Usually,
it is not only the symmetry of molecule but also the symmetries of some local atoms,
molecular orbitals, rotations and vibrations of bonds, etc. that are important.

For example, if the symmetries of molecular orbital wave functions are known, we can
find out information about the binding. Also, by the selection rules that are associated with
symmetries, we can explain whether the transition is forbidden or not and also we can predict
and interpret the bands we can observe in Infrared or Raman spectrum. The qualitative
properties of molecular orbitals can be obtained using symmetry from group theory (whereas
their precise energetics and ordering have to be determined by a quantum chemical method).

Group Theory is a branch of the mathematical field of algebra. In quantum chemistry,

group theory can applied to ab initio or semi-empirical calculations to significantly reduce
the computational cost. Symmetry operations and symmetry elements are two basic and
important concepts in group theory. When we perform an operation to a molecule, if we
cannot tell any difference before and after we do the operation, we call this operation a
symmetry operation. This means that the molecule seems unchanged before and after a
symmetry operation. As Cotton defines it in his book, when we do a symmetry operation to a
molecule, every points of the molecule will be in an equivalent position.

Symmetry
 It means similarity and indicates the regularity of things. Congruence is the highest
degree of symmetry.
 Symmetry is found in all aspects of daily life, as well as in various sciences.

Symmetry in molecules:

Symmetry in molecules is their identicalness. If we perform a symmetry operation

on a molecule and the result is a state indistinguishable from the original state, the
molecule is said to be symmetrical to this symmetry operation.

For example, a water molecule (H2O) when rotated 180 degrees returns to the same
original state. Therefore, it is symmetrical to the 180-degree rotation (symbolized by C2).

4
Symmetry Operations and Symmetry Elements:
A symmetry element is the tool used to perform symmetry operations. It can be a point,
an axis (a specific line), or a plane.

Figure 1 symmetry element

Symmetry Operation: is the operation that is applied to the system to obtain an identical
state.

There are five basic symmetry operations:

1- Identity
2- Rotation axis
3- Reflection
4- Inversion
5- Improper Rotation

Identity E:
The identity operator, (E), consists of doing nothing, and the corresponding symmetry
element is the entire molecule. Every molecule possesses at least this operation. For example,
the CHFClBr molecule in. The identify symmetry operation is not indicated since all
molecule exhibit this symmetry.

Figure 2 CHFClBr Molecule

5
Rotation axis 𝑪𝒏 .
The rotation operation (sometimes called proper rotation), Cn, rotates an object about an
axis by 2π/n radians or 360∘/n . Rotation by Cn leaves the molecule unchanged. The H2O
molecule has a C2 axis (Figure 3). Molecules can have more than one Cn axis, in which case
the one with the highest value of n is called the principal axis. In some high symmetry
systems, there may be more than one principal axis. Note that by convention, rotations are
counterclockwise about the axis. Cn rotations are indicated via vectors with labels as
indicated below.

Figure 3 Examples of n-fold Axis of Rotation: (left) The water molecule contains a C2
axis. (right) Ethane contains both C2 and C3 axes.

We always want to express rotations in their simplest equivalent fractions of m/n:

𝐶42 = 𝐶2 ; 𝐶64 = 𝐶32 ; 𝐶86 = 𝐶43

Rotating an object n times brings the object back to the original object and is equivalent to
the identity operation, E :

𝐶𝑛𝑛 = 𝐸

𝐶𝑛 ′ : Secondary rotation axis (order nnn)

𝐶’’𝑛 : Another secondary 𝐶𝑛 axis .

These are secondary 𝐶𝑛 axes — rotation axes of order n , but not the principal axis.

The prime (′) and double prime (″) notation distinguishes between different

The principal axis

The principal axis in molecular symmetry (or point group theory) is the highest-order
rotation axis (Cn) in a molecule — the axis around which the molecule can be rotated by
360°/n and look the same.

Definition:

 The principal axis is the main axis of symmetry for the molecule.
 It is the Cn axis with the highest value of n (i.e., the largest fold rotation).
6
  It is used as a reference to define the orientation of other symmetry elements (like
mirror planes, secondary axes, etc.).

For Example cyclopropane molecular (in D3h Point Group) has 3 rotating axis by n=2 ,
and 2 rotating axis by n=3

Figure 4

*Note: In the C3 diagram, the two axes of symmetry are exactly the same, meaning there
are two symmetrical rotations of 120 degrees, one to the right and the other to the left.

Here, the primary symmetry element is C3, and the secondary symmetry element is C2,
denoted Cn', because:

7
Reflection, σ
Reflection, σ, defines the bilateral symmetry about a plane (mirror plane / reflection
plane). Reflection in the plane leaves the molecule looking the same. In a molecule that also
has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane
and is labeled σv, while one perpendicular to the axis is called a horizontal mirror plane and
is labeled σℎ. A vertical mirror plane that bisects the angle between two C2 axes is called a
dihedral mirror plane, σd. If no principal axis exist, σℎ is defined as the plane of the
molecule. σ symmetry is indicated as a plane on molecules; since they often bisect atoms,
which should be clearly indicated.

Figure 5 Examples of reflection symmetry. (left) The ammonia molecule contains three
identical reflection planes. All are designated as vertical symmetry planes (σv) because
they contain the principle rotation axis.(middle) The water molecule contains two
different reflection planes. (right) benzene contains a total of seven reflection planes,
one horizontal plane (σh) and six vertical planes (σv and σd).

For any mirror plane, performing two successive reflections about the same plane brings
objects back to their original configuration:

σσ = σ2 = E

Figure 6 cyclopropane

Some sources use the symbol (‘) to indicate the presence of more than one plane of
reflection symmetry. For example:

σv’ indicates the presence of more than one plane of perpendicular reflection symmetry, as in
the cyclopropane molecule ( Figure 6 ), which belongs to the D3h group.

8
Inversion, i
Inversion, i, through the center of symmetry leaves the molecule unchanged. Inversion
consists of passing each point through the center of inversion and out to the same distance on
the other side of the molecule.

If inversion symmetry exists, a line drawn from any atom through the center will connect
with an equivalent atom at an equivalent distance from the center. Examples of molecules
with centers of inversion is shown in Figure 7 . Centers of inversion are indicated via a point,
which may or may not overlap with an atoms. The inversion center is always located at the
central point of the molecule and there can only be one inversion center in any system. The
centers of inversion in the examples below do not overlap with atoms.

Figure 7 Examples of Center of Inversion Symmetry. (left) Benzene and (right) staggard
ethane have centers of inversion (green balls).

𝑖𝑖 = 𝑖 2 = 𝐸

Molecules with no inversion symmetry are said to be centrosymmetric.

9
n-fold Axis of Improper Rotation, 𝑺𝒏
Improper rotations, Sn, are also called rotation-reflections. The rotation-reflection
operation consists of rotating by Cn about an axis, followed by reflecting in a plane
perpendicular to the same axis. Improper rotation symmetry is indicated with both an axis
and a plan as demonstrated in the examples in Figure 8.

Figure 8 Examples of Improper axis of rotation. (left) Staggered ethane contains

an S6 axis of improper rotation. (right) Methane contains three S4 axes of improper
rotation.

Note : S1 is the same as reflection and S2 is the same as inversion.

The lowest-order improper rotation that is not a simpler operation is S3. The pattern of
successive operations depends on if n is even or odd. The general relationships
for Sn operations are:

If n is even : 𝑺𝒏𝒏 = 𝑬 :

 Molecule returns to original orientation configuration after each full

rotation
 There are an even number of rotation operators and reflection operators

If n is odd : 𝑺𝒏𝒏 = 𝝈 𝒂𝒏𝒅 𝑺𝟐𝒏

𝒏 = 𝑬:

 First full rotation:

o Molecular does not return to original configuration after first rotation
o There are an odd number of rotation operators and reflection operators
 Second full rotation:
o Molecule returns to original orientation configuration
o There are an even number of rotation operators and reflection operators

When m is even, there is always a corresponding proper rotation (Cn):

𝐶𝑛𝑚 𝑚<𝑛
𝑆𝑛𝑚 ={
𝐶𝑛𝑚−𝑛 𝑚>𝑛

10
If 𝑆𝑛𝑛 with even n exist , 𝐶𝑛/2 exist .

If 𝑆𝑛𝑛 with odd n exist , both 𝐶𝑛 and and σ perpendicular to Cn exist .

11
Symmetry Element
The symmetry process can only occur in the presence of a symmetry element. For
example, for the water molecule H2O, there are symmetry processes accompanied by
symmetry elements, as follows:

1. Rotational symmetry C2, accompanied by a line of symmetry C2 (as shown in Figure a

below).
2. Reflection symmetry σyz, accompanied by a symmetry plane (mirror plane) σyz (as
shown in Figure b).
3. Reflection symmetry σxz, accompanied by a symmetry plane (mirror plane) σxz (as
shown in Figure c).

Figure 9

a b c

12
Point Group
 The set of symmetry operations that can be performed on a molecule is called a point
group.
 Molecules with the same symmetry operations belong to the same point group.
 This means that a molecule can have more than one symmetry operation, and each
symmetry operation is accompanied by a symmetry element.

For example, the compound [OsCl3(CO)3]-

It has four symmetry operations:

Symmetry operation Description Structure

The molecule remains the

Identity E
same.

180 degree rotation about

180 rotation ( C2 ) the line of symmetry passing
through the center of Os

13
 Reflection at the xz plane
Reflection on the xz plane
gives symmetry about the xz
σxz
symmetry plane.

Reflection at the yz plane

Reflection on the yz plane
gives symmetry about the yz
(σyz )
symmetry plane.

𝛔yz 𝛔xz C2

Together, these four symmetry operations represent a point group, and for this molecule, it
belongs to the point group C2v. We currently have 10 known major groups, and the following
table shows these points and the symmetry operations for each group.

14
 Table 1 Point Group and Symmetry Operation

Poi
nt
Symmetry Operation
Gro
up
1 C∞v E 2C∞Φ ∞σv
Linear
2 groups D∞h E 2C∞Φ ∞σi i 2S∞Φ ∞C2
3 C1
4 Nonaxial
Ci Ei
groups
5 Cs Eσ
6 C2 E C2
7 C3 E C3 (C3)2
8 C4 E , C4 , C2 , (C4)3
9 Cn groups
C5 E C5 (C5)2 (C5)3 (C5)4
10 C6 E C6 C3 C2 (C3)2 (C6)5
11 C7 E C7 (C7)2 (C7)3 (C7)4 (C7)5 (C7)6
12 C8 E C8 C4 (C8)3 C2 (C8)5 (C4)3 (C8)7
13 D2 E C2 (z) C2 (y) C2 (x)
14 D3 E 2C3 (z) 3C'2
15 D4 E 2C4 (z) C2 (z) 2C'2 2C''2
16 Dn groups D5 E 2C5 (z) 2(C5)2 5C'2
17 D6 E 2C6 (z) 2C3 (z) C2 (z) 3C'2 3C''2
18 D7 E 2C7 2(C7)2 2(C7)3 7C'2
19 D8 E 2C8 2C4 2(C8)3 C2 4C'2 4C''2
20 C2v E C2 σv(xz) σv'(yz)
21 C3v E 2C3 3σv
22 C4v E 2C4 C2 2σv 2σd
Cnv groups
23 C5v E 2C5 2C52 5σv
24 C6v E 2C6 (z) 2C3 (z) C2 (z) 3 v 3d
25 C7v E 2C7 2(C7)2 2(C7)3 7v

15
 Poi
nt
Symmetry Operation
Gro
up
26 C8v E 2C8 2C4 2(C8)3 C2 4v 4d
27 C2h E C2 i σh
28 C3h E C3 C32 σh S3 S35
29 Cnh groups C4h E C4(z) C2 (C4)3 i (S4)3 h S4
30 C5h E C5 (C5)2 (C5)3 (C5)4 h S5 (S5)7 (S5)3 (S5)9
31 E C6(z) C3 C2 (C3)2 (C6)5 i (S3)5 (S6)5 h
C6h
S6 S3
32 D2h E C2 (z) C2 (y) C2 (x) i (xy) (xz) (yz)
33 D3h E 2C3 (z) 3C'2 h (xy) 2S3 3v
34 E 2C4 (z) C2 2C'2 2C''2 i 2S4 h 2v 2
D4h
d
35 Dnh groups
D5h E 2C5 2(C5)2 5C'2 h 2S5 2(S5)3 5 v
36 E 2C6 (z) 2C3 C2 3C'2 3C''2 i 2S3 2S6
D6h
h (xy) 3 d 3v
37 D7h 2C7 2(C7)2 2(C7)3 7C'2 h 2S7 2(S7)5 2(S7)3 7 v
38 D8h
39 D2d E 2S4 C2 (z) 2C'2 2d
40 D3d E 2C3 3C'2 i 2S6 3d
41 D4d E 2S8 2C4 2(S8)3 C2 4C'2 4d
42 Dnd group D5d E 2C5 2(C5)2 5C'2 i 2(S10)3 2S10 5 d
43 D6d E 2S12 2C6 2S4 2C3 2(S12)5 C2 6C'2 6d
44 E 2C7 2(C7)2 2(C7)3 7C'2 i 2(S14)5 2(S14)3
D7d
2S14 7d
45 E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7
D8d
C2 8C'2 8 d
46 S2 It is identical Ci : E , i
47 S4 E S4 C2 (S4)3
48 S6 E C3(z) (C3)2 i (S6)5 S6
Sn groups
49 S8 E S8 C4 (z) (S8)3 C2 (S8)5 (C4)3 (S8)7
50 S10 E C5 (C5)2 (C5)3 (C5)4 i (S10)7 (S10)9 S10 (S10)3
51 S12 E S12 C6 S4 C3 (S12)5 C2 (S12)7 (C3)2 (S4)3

16
 Poi
nt
Symmetry Operation
Gro
up
(C6)5 (S12)11

52 T E 4C3 4(C3)2 3C2

53 Th E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 3h
54 Td E 8C3 3C2 6S4 6d
Cubic
55 O E 8C3 6C'2 6C4 3C2 =(C4)2
groups
56 E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6
Oh
d
57 I E 12C5 12(C5)2 20C3 15C2
58 E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3
Ih
20S6 15

Reference

Character tables for point groups : http://symmetry.jacobs-university.de/

For Examples with 3-D Figure : https://symotter.org/gallery

17
 To determine which point group the molecule belongs to, we follow the following
algorithm:

18
Space Group
 A space group extends the idea of point groups by adding translational symmetry.
 In crystals, atoms are arranged periodically in space, so besides rotation and
reflection, translations (shifting by a certain distance) are a fundamental part of
symmetry.
 Space groups describe how motifs (groups of atoms) are repeated throughout the
whole crystal.
 There are 230 distinct space groups in three-dimensional space.

Space groups include:

1. Pure translations (moving by a lattice vector),

2. Screw axes (rotation + translation along the axis),
3. Glide planes (reflection + translation along the plane),
4. As well as all the point group operations.

 Real crystals are not just single points — they are huge, repeating lattices of atoms
arranged through space.
 Space groups describe not only the rotations, reflections, and inversions (from point
groups) but also translations — movements through space that still land you on an
identical piece of the structure.

Space groups combine:

o Point group symmetries

o Translations (shifting by regular steps through the lattice)
o Plus new kinds of operations:
 Screw axis: rotate, then translate a bit along the axis
 Glide plane: reflect, then translate a bit along the plane

Example of space group symmetry:

After rotating and sliding along a screw axis, a crystal structure may look exactly the
same.

In a crystal like quartz, if you rotate 120° around an axis and then move slightly along that
axis, the crystal looks the same. That’s a screw axis — part of a space group operation.

Definition of space groups

 Space groups are mathematical groups that describe the symmetry of a crystal in
three dimensions, including both point group symmetry and translational
symmetry
 They consist of a combination of symmetry operations that leave the crystal
structure invariant
 Space groups are crucial for understanding the arrangement of atoms in a crystal
and its resulting properties

19
Classification of space groups
 Space groups are classified based on their symmetry elements and the relationships
between them
 There are 230 unique three-dimensional space groups, which are divided into 7
crystal systems and 14 Bravais lattices
 Each space group is characterized by a unique combination of symmetry
operations and a specific arrangement of atoms in the unit cell .
 In two dimensions, there are 17 unique space groups, also known as wallpaper
groups
 These 2D space groups describe the possible symmetries of patterns on a plane
 In three dimensions, there are 230 unique space groups, which describe the
symmetries of crystal structures
 The number of space groups in 3D is determined by the combination of the 32
crystallographic point groups and the 14 Bravais lattices

Figure 10 : lists the

14 Bravais lattices
and shows a unit
cell for each. In this
figure, red lattice
points are at the
corners of unit cells
and blue spheres
are the extra points
due to centering.
Symbols such as
1P, 2P, 2C, 222P,
etc., beneath each
figure describe
their symmetries
and whether they
are primitive (P) or
centered (I, F, A, B,
C). These are the
standard labels we
use for each of the
14 lattices.

20
Symmetry operations in space groups 1
In 3D, we assign atomic arrangements to different space groups that have different space
symmetries. Each space group is characterized by a combination of one of the 14 Bravais
lattices with a unit cell that has a particular symmetry. We call the different kinds of possible
symmetry operators, collectively, space group operators. Space symmetry includes the point
group symmetries that we discussed previously. And it also includes glide planes and screw
axes. These two are special kinds of space group operators that involve combinations of
point symmetry and translational symmetry, in much the same way that rotoinversion axes
involve rotation and inversion applied simultaneously.

Glide Planes
Glide planes differ from normal mirror planes
because they involve translation before reflection.
Figure 11 shows some examples. In drawing a, a
single atom is repeated by a horizontal glide plane
(dashed red line). The red arrows show how the
atom repeats by translation followed by reflection.
In drawing b, a three-atom motif repeats according
to a horizontal glide plane. And in drawing c, we see
a 2D pattern with glide plane symmetry. The pattern
contains no mirrors, rotation axes, and no inversion
center. Yet, it contains significant symmetry.

Space group operators and other symmetry

elements combine in many ways. For example,
Figure 11d shows a pattern that contains horizontal
glide planes (blue dotted lines) and vertical mirror
planes (solid red lines). As we have seen previously,
combination of symmetry elements often requires
other symmetry to be present. The combination of a
glide plane with a mirror means that this pattern
contains 2-fold rotation axes (red lens shapes). This
pattern has symmetry equivalent to plane group
(2mg) depicted in Figure 12.

Figure 11 : Examples of glide

planes

https://geo.libretexts.org/Bookshelves/Geology/Mineralogy_(Perkins_et_al.)/11%3A_Crystallography/11.05%3A_Symm
etry_of_Three_Dimensional_Atomic_Arrangements/11.5.01%3A_Space_Group_Operators

21
 Figure 12 The 17 possible plane symmetries

Figure 13 shows a glide plane in three dimensions; a

motif of 8 atoms repeats by vertical translation and
reflection. This figure involves a simple motif with lots of
space around it. The arrangements of atoms in most
minerals are generally more complicated and difficult to
draw in 3D in a way that shows their symmetries clearly. In
part, this is because a glide plane affects all atoms in a
structure, not just the ones adjacent to the plane .

Atomic arrangements may contain any of six symmetry

elements involving reflection, singly or in combination.
These include proper mirror planes and 5 different kinds of
glide planes. Glide planes may have any of the orientations
that mirror planes can have. So, they may be parallel to
the a-, b-, or c-axis, parallel to a face diagonal, or parallel to
a main (body) diagonal. For most glide planes, the
magnitude of the translation (t) is half the unit cell
dimension in the direction of translation. The direction of
gliding distinguishes different glides and allows us to
classify them. The table below summarizes the possibilities. Figure 13

22
 operator type of operation orientation of translation translation*

m proper mirror none none

1
a axial glide parallel to a /2 a
1
b axial glide parallel to b /2 b
1
c axial glide parallel to c /2 c
1
n diagonal glide par. to a face diagonal /2 t
1
d diamond glide par. to main diagonal /4 t

*t = the unit cell dimension in the direction of translation

Table 2 Space Symmetry Operators Involving Reflection

Screw Axes
Screw axes result from the simultaneous application of translation and rotation. We combine 2-, 3-,
4-, or 6-fold rotation operators with translation to produce these symmetry elements. Many
combinations are possible.

Figure 14 shows a few examples.

Figure 14 : Examples of screw axes

A screw axis has the appearance of a spiral staircase. We rotate a motif, translate it, and
get an additional motif. As with proper rotation axes (rotation axes not involving translation),
each n-fold screw operation involves rotation of 360o/n. After n repeats, the screw has come
full circle. The translation associated with a screw axis must be a rational fraction of the unit
cell dimension or the result will be an infinite number of atoms, all in different places in
different unit cells. We label screw axes using conventional symbols.

In Figure 14 , they are 61, 41, 31, and 21. In the labels, the large 6, 4, 3, or 2 signifies 6-
fold, 4-fold, 3-fold, or 2-fold rotation. The subscript tells the translation distance. A 61 screw
axis, for example, involves translation that is 1/6 of the unit cell dimension in the direction of
the screw axis. 41, 31, and 21 axes involve translations that are 1/4, 1/3, and 1/2 the unit cell

23
dimension. Similarly, 62, 63, 64, and 65 screw axes (not shown) would involve translations
of 2/6, 3/6, 4/6, and 5/6 of the cell dimension.

Figure 15 :Some 4-fold screw axes

Figure 16a shows application of a 42 operator to a single atom. The translation distance
is /2 the unit cell height, and we must go through four 90o rotations and two unit cells to get
1

another atom that is directly above the starting atom. All unit cells must be identical, but the
42 operation gives a bottom and top unit cell with atoms in different places. The only way
this operator can be made consistent is to add the extra atoms shown in Figure 17b. In other
words, the presence of a 42 axis requires the presence of a 2-fold axis of symmetry.

Figure 18c shows a 43 axis and Figure 19d shows a 41 axis. After four applications of
either operator the total rotation is 360°, bringing the fourth point directly above the first. For
the 41 axis, after four applications the total translation is equivalent to one unit cell length.
But for the 43 axis it is three unit cell lengths because the translation is 3/4 of the unit cell. The
only way the 43 operator can be made consistent is to add the extra atoms shown in Figure
20d. Note that the 43 and 41 axes produce patterns that are mirror images of each other. The
two axes are an enantiomorphic pair, sometimes called right-handed and left-handed screw
axes.

When all combinations are considered, we get the 21 possible rotation axes (either proper
rotation axes or screw axes) listed in the table below. As with proper rotational axes, some
screw axes are restricted to one or a few crystal systems. For example, 31 and 32, which are
an enantiomorphic pair, only exist in the rhombohedral system. Similarly, the 6n axes only
exist in the hexagonal system.

24
 operator type of operation rotation angle translation*
1 identity 360o none
o
1 inversion center 360 none
o
2 proper 2-fold 180 none
21 2-fold screw 180o 1/2 t
o
2 mirror 180 none
o
3 proper 3-fold 120 none
o
31 3-fold screw 120 1/3 t
32 3-fold screw 120o 1/3 t
o
3 3-fold rotoinversion 120 none
o
4 proper 4-fold 90 none
o
41 4-fold screw 90 1/4 t
42 4-fold screw 90o 2/4 = 1/2 t
o
43 4-fold screw 90 3/4 t
o
4 4-fold rotoinversion 90 none
6 proper 6-fold 60o none
o
61 6-fold screw 60 1/6 t
o
62 6-fold screw 60 2/6 = 1/3 t
o
63 6-fold screw 60 3/6 = 1/2 t
64 6-fold screw 60o 4/6 = 2/3 t
o
65 6-fold screw 60 5/6 t
o
6 6-fold rotoinversion 60 none
*t = the unit cell dimension in the direction of translation
Table 3 : Space Symmetry Operators Involving Rotation

Hermann-Mauguin notation for space groups

 Hermann-Mauguin notation is used to describe space groups using symbols
 The notation includes the symbols for the point group symmetry elements,
followed by the symbols for the translational symmetry elements
 Examples of Hermann-Mauguin notation for space groups include 𝑃21 /𝑐
(primitive monoclinic with a two-fold screw axis and a glide plane) and 𝐹𝑚3̅𝑚
(face-centered cubic with full octahedral symmetry)

Relationship between point groups and space groups

 Point groups and space groups are closely related, as space groups are built upon
the foundation of point group symmetry
 Point groups describe the local symmetry of a molecule or crystal, while space
groups describe the global symmetry of the entire crystal structure
 Understanding the relationship between point groups and space groups is essential
for a comprehensive analysis of crystal symmetry

Point groups as subgroups of space groups

 Every space group contains one or more point groups as subgroups
 The point group of a space group is obtained by removing all translational
symmetry elements from the space group
 For example, the space group P4mm has the point group C4v as a subgroup.

25
Space groups as extensions of point groups
 Space groups can be viewed as extensions of point groups, incorporating
translational symmetry elements
 The translational symmetry elements in a space group are combined with the point
group symmetry elements to create the full symmetry of the crystal structure
 For example, the point group D2h can be extended to the space group Pnma by
adding translational symmetry elements

The link below shows all the Space Groups :

https://spacegroups.symotter.org/structures.html

26
Applications of point groups and space groups
Point groups and space groups have numerous applications in solid state physics,
chemistry, and materials science. They provide a framework for understanding the
symmetry-dependent properties of molecules and crystals. Some key applications include the
analysis of crystal structures, selection rules for transitions, phonon dispersion relations, and
electronic band structures.

1. Symmetry in crystal structures

 Space groups are used to describe the symmetry of crystal structures
 The symmetry of a crystal determines the arrangement of atoms in the unit cell and
the overall lattice structure
 Examples of crystal structures with different space groups include diamond
(𝐹𝑑3̅𝑚), graphite (𝑃63 /𝑚𝑚𝑐), and perovskite (𝑃𝑚3̅𝑚)

Crystal symmetry heavily influences physical properties:

 Piezoelectricity (ability to generate electric charge under stress) occurs only in

crystals that lack a center of symmetry.
 Optical activity (rotation of light polarization) depends on the symmetry of the
structure.
 Ferroelectricity (spontaneous electric polarization) also requires specific
symmetry conditions.

2. Symmetry in phonon dispersion relations

 Space group symmetry influences the phonon dispersion relations in crystals
 Phonon modes at high-symmetry points in the Brillouin zone are classified
according to the irreducible representations of the point group of the wavevector
 The symmetry of the phonon modes determines their degeneracy and the shape of
the dispersion curves.
 Examples of high-symmetry points in the Brillouin zone include the Γ point
(center), X point (edge center), and L point (hexagonal face center)

3. Symmetry in electronic band structures

 Space group symmetry also affects the electronic band structure of crystals
 The symmetry of the crystal determines the degeneracy and connectivity of the
electronic bands
 High-symmetry points in the Brillouin zone are used to label the electronic states
and to analyze the band structure
 Examples of electronic band structures with different symmetries include the
direct bandgap in GaAs (zinc blende structure, 𝐹4̅3𝑚 ) and the indirect bandgap in
Si (diamond structure, 𝐹4̅3𝑚 )

27
 4. Symmetry-based selection rules for transitions
Symmetry-based selection rules determine whether certain physical transitions (such as
electronic, vibrational, or optical transitions) are allowed or forbidden based on the
symmetry properties of the initial and final states.These rules are crucial in fields like
spectroscopy, quantum mechanics, and solid-state physics.

Examples of symmetry-based selection rules include the Laporte rule (transitions between
states of opposite parity are forbidden in centrosymmetric molecules) and the spin selection
rule (transitions between states of different spin multiplicity are forbidden)

The matrix element for a transition typically looks like:

⟨𝜓𝑓𝑖𝑛𝑎𝑙 |𝑂̂|𝜓𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ⟩

where 𝑂̂ is an operator (e.g., the electric dipole operator for optical transitions).

A transition is allowed if this integral is invariant under all symmetry operations of the
point group — meaning it behaves like the totally symmetric representation. Otherwise, the
transition is forbidden.

( I ) Electric Dipole Selection Rules

For electric dipole transitions (important in optical absorption and emission):

 The transition is allowed if the direct product : Γ𝑖𝑛𝑖𝑡𝑎𝑙 × Γ𝑂̂ × Γ𝑓𝑖𝑛𝑎𝑙 contains the
totally symmetric representation.

Here:

 Γ𝑖𝑛𝑖𝑡𝑎𝑙 symmetry of the initial state.

 Γ𝑓𝑖𝑛𝑎𝑙 symmetry of the final state.
 Γ𝑂̂ symmetry of the operator (for electric dipole, transforms like coordinates x,y,z).

( II ) In a center of inversion (inversion symmetry) system:

 Electric dipole transitions can only occur between states of opposite parity (i.e.,
even g and odd u).
 Transitions g→g , u→u are forbidden.

A center of inversion (also called an inversion center) is a point in space such that
for every point (x,y,z), there is an equivalent point at (−x,−y,−z).

In simpler words:

If you draw a straight line from any atom through the center of inversion,
there is an identical atom the same distance on the opposite side.

This is a symmetry element in crystallography and molecular symmetry.

A molecule or crystal with a center of inversion is called centrosymmetric.

28
 For Example : CO2 (carbon dioxide molecule):

It is a straight line: O = C = O.

The carbon atom is at the center, and the two oxygen atoms are symmetrically placed
on either side.

Diamond crystal:

Each carbon atom has another directly opposite through a center of inversion.

How Inversion Affects Transitions

In systems with a center of inversion:

States (wavefunctions) have parity:

 Even parity (gerade, g): symmetric under inversion — wavefunction doesn't

change sign.
 Odd parity (ungerade, u): antisymmetric under inversion — wavefunction
changes sign.

For electric dipole transitions:

Transitions are allowed only between states of opposite parity:

 g→u are allowed.

 g→g are forbidden.

This is because the electric dipole operator (associated with position r⃗\vec{r}r) is
odd under inversion (it changes sign).

( III )Atomic hydrogen:

 1s→2p transition is allowed (different angular momentum quantum numbers and

opposite parity).
 1s→2s transition is forbidden (same parity).

Atomic hydrogen is a simple atom: one proton, one electron. It has energy levels like
1s,2s,2p,3s,3p,3d, etc.

Each orbital has a definite parity:

 s-orbitals (like 1s,2s,3s) have even parity (g).

 p-orbitals (like 2p,3p) have odd parity (u).
 d-orbitals (like 3d) have even parity (g).

Thus:

 1s→2p (even to odd) → Allowed transition.

 1s→2s (even to even) → Forbidden by dipole selection rules.

Why?

29
 Because under inversion:

 1s and 2s are both symmetric (even).

 The dipole operator is antisymmetric.
 So the matrix element vanishes → transition forbidden.

Molecular oxygen O2 has a center of inversion.

 In crystals like centrosymmetric salts (e.g., NaCl), electronic transitions are

strongly constrained by parity.
 In non-centrosymmetric systems (where there is no center of inversion), many
more transitions are allowed because parity is not a good quantum number.

( IV ) Vibrational (Infrared and Raman) Selection Rules

Infrared (IR) activity:

A vibration mode is IR active if it transforms like x,y, or z

Raman activity:

A vibration is Raman active if it transforms like quadratic functions

𝑥 2 , 𝑦 2 , 𝑧 2 , 𝑥𝑦, 𝑦𝑧, , 𝑧𝑥.

Thus, symmetry dictates whether a given vibrational mode shows up in IR or Raman

spectra.

We will discuss this topic in detail using the water molecule later.

30
Determination of point groups and space groups
 Determining the point group and space group of a molecule or crystal is essential
for understanding its symmetry and physical properties
 Several experimental and computational methods are used to determine the
symmetry of a system
 These methods include X-ray diffraction, spectroscopy, and computational
symmetry analysis

Experimental methods for symmetry determination

 Experimental techniques such as X-ray diffraction, neutron diffraction, and
electron diffraction are used to determine the crystal structure and symmetry
 Spectroscopic methods, such as infrared and Raman spectroscopy, can provide
information about the point group symmetry of molecules and crystals
 Polarized light microscopy can be used to identify the crystal system and point
group based on the optical properties of the sample

X-ray diffraction and space group determination

 X-ray diffraction is the most common method for determining the space group of a
crystal
 The diffraction pattern provides information about the lattice parameters, crystal
system, and the presence of certain symmetry elements
 Systematic absences in the diffraction pattern can be used to identify the space
group
 Examples of systematic absences include the absence of (0k0) reflections for a 21
screw axis along the b-axis and the absence of (h00) reflections for an a-glide
plane perpendicular to the a-axis

Computational methods for symmetry analysis

 Computational methods, such as group theory and first-principles calculations, are
used to analyze the symmetry of molecules and crystals
 Group theory can be used to determine the point group and space group of a
system based on its symmetry elements
 First-principles calculations, such as density functional theory (DFT), can provide
information about the electronic structure and symmetry-dependent properties of a
system
 Examples of computational symmetry analysis include the use of
the FINDSYM program to identify the space group of a crystal structure and the
use of the Bilbao Crystallographic Server to generate the irreducible
representations of a space group

31
Consequences of symmetry breaking
 Symmetry breaking occurs when a system undergoes a phase transition that lowers
its symmetry
 The breaking of symmetry can lead to the emergence of new physical properties
and phenomena
 Examples of symmetry breaking include ferroelectricity, piezoelectricity, and
magnetism

Ferroelectricity and noncentrosymmetric point groups

 Ferroelectricity is a property of certain materials that exhibit a spontaneous electric
polarization that can be reversed by an external electric field
 Ferroelectric materials belong to noncentrosymmetric point groups, which lack an
inversion center
 Examples of ferroelectric materials include barium titanate (BaTiO3, point group
4mm) and lead zirconate titanate (PZT, point group 4mm)

Piezoelectricity and polar point groups

 Piezoelectricity is the ability of certain materials to generate an electric charge in
response to applied mechanical stress
 Piezoelectric materials belong to polar point groups, which have a unique polar
axis
 Examples of piezoelectric materials include quartz (SiO2, point group 32) and zinc
oxide (ZnO, point group 6mm)

Magnetism and time-reversal symmetry breaking

 Magnetism arises from the breaking of time-reversal symmetry in a material
 Magnetic materials have a spontaneous magnetic moment that can be aligned by
an external magnetic field
 The magnetic symmetry of a material is described by its magnetic point group and
magnetic space group
 Examples of magnetic materials include iron (Fe), which belongs to the point
group m3m and the space group Im ̅ 3m, and nickel (Ni), which belongs to the
point group m3m and the space group 𝐹𝑚 ̅3𝑚.

32
Symmetry and Physical Properties
Carrying out a symmetry operation on a molecule must not change any of its physical
properties. It turns out that this has some interesting consequences, allowing us to predict
whether or not a molecule may be chiral or polar on the basis of its point group.

For a molecule to have a permanent dipole moment , it must have an asymmetric charge
distribution. The point group of the molecule not only determines whether the molecule may
have a dipole moment , but also in which direction(s) it may point.

If a molecule has a Cn axis with n>1, it cannot have a dipole moment perpendicular to
the axis of rotation (for example, a C2 rotation would interchange the ends of such a dipole
moment and reverse the polarity, which is not allowed – rotations with higher values
of n would also change the direction in which the dipole points). Any dipole must lie parallel
to a Cn axis.

Also, if the point group of the molecule contains any symmetry operation that would
interchange the two ends of the molecule, such as a σh mirror plane or a C2 rotation
perpendicular to the principal axis, then there cannot be a dipole moment along the axis. The
only groups compatible with a dipole moment are Cn, Cnv and Cs. In molecules belonging
to Cn or Cnv the dipole must lie along the axis of rotation.

One example of symmetry in chemistry that you will already have come across is found in
the isomeric pairs of molecules called enantiomers. Enantiomers are non-superimposable
mirror images of each other, and one consequence of this symmetrical relationship is that
they rotate the plane of polarized light passing through them in opposite directions. Such
molecules are said to be chiral, meaning that they cannot be superimposed on their mirror
image. Formally, the symmetry element that precludes a molecule from being chiral is a
rotation-reflection axis Sn. Such an axis is often implied by other symmetry elements present
in a group.

Figure 21

For example, a point group that has Cn and σh as elements will also have Sn. Similarly, a
center of inversion is equivalent to S2. As a rule of thumb, a molecule definitely cannot have
be chiral if it has a center of inversion or a mirror plane of any type (σh, σv or σd), but if these
symmetry elements are absent the molecule should be checked carefully for an Sn axis before
it is assumed to be chiral.

33
Representation Symmetry Operations as A Matrices
Matrices can be used to map one set of coordinates or functions onto another set. Matrices
used for this purpose are called transformation matrices. The symmetry operations in a group
may be represented by a set of transformation matrices Γ(g), one for each symmetry element
g.

Each individual matrix is called a representative of the corresponding symmetry

operation, and the complete set of matrices is called a matrix representation of the group. The
matrix representatives act on some chosen basis set of functions, and the actual matrices
making up a given representation will depend on the basis that has been chosen. The
representation is then said to span the chosen basis. The basis we will use are unit vectors
pointing in the x, y, and z directions.

The transformation matrix for any operation in a group has a form that is unique from the
matrices of the other members of the same group; however, the character of the
transformation matrix for a given operation is the same as that for any other operation in the
same class. Each symmetry operation below will operate on an arbitrary vector, u, where :

The Identity Operation, E

The first rule is that the group must include the identity operation E (the ‘do nothing’
operation). The matrix representative of the identity operation is simply the identity matrix
and leaves the vector unchanged:

Every matrix representation includes the appropriate identity matrix.

The Reflection Operation, σ

The reflection operation reflects the vector u over a plane. This can be the xy, xz,
or yz plane. The matrix is similar to the identity matrix, with the exception that there is a sign
change for the appropriate element. The reflect matrix in the xy plane is:

Notice that the element for the dimension being reflected is the on that is negative. In the
above case, since z is being reflected over the xy plane, the z element in the matrix is
34
negative. If we were to reflect over the xz plane instead, the y element would be the one that
is negative:

The n-fold Rotation Operation, Cn

The Cn operator rotates the molecule about an axis. The counterclockwise rotation of
vector u about the z axis is:

For clockwise rotation, the sign on the sinθ terms are reversed. This matrix simplifies
dramatically for the C2 rotation:

Rotation matrices operating about the

35
The Inversion operation, I
The inversion operation inverts every point:

These operations represented by matrices are usually denoted by the symbol 𝚪(g), where g
represents any of the symmetry operations:

𝟏 𝟎 𝟎 𝒙 𝒙
𝚪 (𝑬) ∶ (𝟎 𝟏 𝟎 ) ( 𝒚) = )
( 𝒚
𝟎 𝟎 𝟏 𝒛 𝒛

−𝟏 𝟎 𝟎 𝒙 −𝒙
𝚪(𝑪𝟐 ) ∶ (𝟎 −𝟏 𝟎) (𝒚) = (−𝒚)
𝟎 𝟎 𝟏 𝒛 𝒛

−𝟏 𝟎 𝟎 𝒙 −𝒙
𝚪(𝝈𝒗 ) ∶ (𝟎 −𝟏 𝟎 ) (𝒚) = (−𝒚)
𝟎 𝟎 −𝟏 𝒛 −𝒛

𝟎 𝟎 𝟏 𝒙 𝒙
𝚪(𝝈′𝒗 ) ∶ (𝟎 𝟏 𝟎) (𝒚) = (𝒚)
𝟏 𝟎 𝟎 𝒛 𝒛

36
Represented Molecules by Reducible Representations:
A reducible representation (often denoted Γ ) is a representation of a group (a set of
symmetry operations) where the corresponding matrices can be broken down (or reduced)
into simpler, smaller irreducible representations (irreps2).

In molecular symmetry, we use reducible representations to describe how vectors,

orbitals, or atomic displacements transform under the symmetry operations of a molecule’s
point group.

In addition to operators, we can define properties of molecules using a matrix

representation. Before making the matrix, we need to carefully choose a basis set that defines
the information we want to extract For example, let's say we want to know the symmetry of
the valence s orbitals in ammonia, NH3, which is in the C3v point group. We will select a
basis (sN,s1,s2,s3) that consists of the valence s orbitals on the nitrogen and the three
hydrogen atoms. We need to consider what happens to this basis when it is acted on by each
of the symmetry operations in the C3v point group, and determine the matrices that would be
required to produce the same effect. The basis set and the symmetry operations in
the C3v point group are summarized in the figure below.

sN : refer to the s-state in Nitrogen

s1 : refer to the s-state in first Hydrogen

s2 : refer to the s-state in 2nd Hydrogen

s3: refer to the s-state in 3rd Hydrogen

The effects of the symmetry operations on our chosen basis are as follows:

Figure 22 Ammonia NH3 Element Group

2
The term “irreps” is short for irreducible representations, which are the fundamental building blocks in group
theory — especially in the study of molecular symmetry.

37
By inspection, the matrices that carry out the same transformations are:

These six matrices therefore form a reducible representation for the C3v point group in
the (sN,s1,s2,s3) basis . that these matrices reduce down to the irreducible representations

38
found in the character tables. These reducible representations multiply together according to
the group multiplication table and satisfy all the requirements for a mathematical group.

We choose different basis sets to extract different properties of molecules. For example,
we could include representations of the valence p orbitals in N in our basis set to obtain the
structure and symmetry of the molecular orbitals for ammonia. To understand understand
the molecular motions of ammonia (translates, rotates, and vibrates), we could place a x, y,
and z unit vectors on each atom to represent the their motion, and then construct our
matrices.

39
How Do You Get a Reducible Representation for a Molecule?
1. Choose a Basis Set
Decide what you want to analyze:

 Atomic positions (for vibrations)

 Atomic orbitals (for bonding)
 Bond vectors (for IR/Raman activity)

Example: Use the x, y, z displacements of atoms in the molecule as your basis (for
vibrational modes).

2. Determine the Point Group

Identify the molecule’s point group using symmetry elements.

Example:

 H₂O → C₂v
 NH₃ → C₃v
 Benzene → D₆h

3. Apply Symmetry Operations to the Basis

For each symmetry operation in the point group:

 Apply it to each vector in the basis.

 Count how many vectors stay unchanged (mapped onto themselves).
 This count is the character (the trace of the transformation matrix) for that operation.

4. Build the Reducible Representation Table

Operation E C₂ 𝝈𝒗 (𝒙𝒛) 𝝈′𝒗 (𝒚𝒛)
Γ (e.g.displacements) # # # #

5. Reduce Γ into Irreducible Representations

Use the reduction formula:
𝟏
𝒂𝒊 = ∑ 𝝌𝚪 (𝑹) ∙ 𝝌𝐢 (𝑹) ∙ 𝒏𝑹
𝒉
𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏𝒔

Where:

 𝒂𝒊 = number of times irrep i appears

 h = order of the group (total number of symmetry operations)
 𝝌𝚪 (𝑹) = character of Γ for operation R
 𝝌𝐢 (𝑹) = character of irreducible representation i
 𝒏𝑹 = number of operations in that class .

40
Character Tables
A character table is a 2 dimensional chart associated with a point group that contains the
irreducible representations of each point group along with their corresponding matrix
characters. It also contains the Mulliken symbols used to describe the dimensions of the
irreducible representations, and the functions for symmetry symbols for the Cartesian
coordinates as well as rotations about the Cartesian coordinates. As shown in fig below for
C2v Point Group :

All operations in the character table are contained in the first row of the character table, in
this case E, C2, σv & σ’v, these are all of the operations that can be preformed on the
molecule that return the original structure. The first column contains the three irreducible
representations A1, A2 ,B1, B2 .

The character of the irreducible representation denotes what the operation does. A value of
1 represents no change, -1 opposite change and 0 is a combination of 1 & -1 (0’s are found in
degenerate molecules. The final two columns Rotation and Translation represented
by Rx,Ry, Rz & x, y, z respectively.

Each Rx, Ry, Rz & x, y, z term is the irreducible symmetry of a rotation or translation
operation. Like wise the final column the orbital symmetries relates the orbital wave function
to a irreducible representation.

These basis functions could be:

 Molecular orbitals
 Atomic orbitals (like px, dxy)
 Vibrational modes
 Electronic states
 Linear combinations of atomic orbitals (LCAOs)

41
 Each function belongs to a certain symmetry species (A, B, E...) depending on how it
behaves under the group’s symmetry operations.

Properties of Characters of Irreducible Representations in Point Groups

1. There is always a totally symmetric representation in which all the characters are 1.
e.g. In C2v, A1 is totally symmetric.
2. The order of the group (ℎ) is the total number of symmetry operations in the group.
e.g. In C2v, ℎ=4
3. Similar operations are listed as classes (R) and appear as columns in the table.
e.g. In C2v, there are four classes of operations, E, C2, σv(xz), and σ′v(yz)
4. The number of irreducible representations (rows) must equal the number of classes
(columns). This results in all character tables being square.
e.g. In C2v, there are four classes and four irreducible representations.
5. The sum of squares of all characters under E is equal to the order of the group: ℎ = ∑[𝜒𝑖]2
e.g. In C2v, h=12+12+12+12=4
6. For any irreducible representation (𝑖 ), the sum of squares of its characters multiplied by the
number of operations in the class is the order of the group: ℎ = ∑[𝜒𝑖(𝑅)]2
e.g. For A2 in C2v, h=(1×1)2+(1×1)2+(−1×1)2+(−1×1)2=4
7. Irreducible representations are orthogonal. For any two representations (i
and 𝑗): ∑[𝜒𝑖∗ (𝑅) 𝜒𝑗(𝑅)] = 0
e.g. For B1 and B2 of C2v, [1×1]+[−1×−1]+[1×−1]+[−1×1]=0

Mullikan Symbol of irreducible representations

𝜒(𝐶 ′ 2 ) 𝑜𝑟
Notation E 𝜒(𝐶𝑛 ) 𝜒(𝜎ℎ ) 𝜒(𝑖)
𝜒(𝜎𝑣 )
A 1 +
B 1 -
E 2
T 3
Subscript (1) +
Subscript (2) ‫ـــ‬
Superscript ( ́ ) +
Superscript ( ́ ́ ) ‫ـــ‬
Subscript (g) +
subscript (u) ‫ـــ‬
Table 4

42
Symbol Physical Meaning
(singly degenerate or one dimensional) symmetric with respect to
A rotation of the principle axis .

(singly degenerate or one dimensional) anti-symmetric with respect

B
to rotation of the principle axis
E (doubly degenerate or two dimensional)
T (thirdly degenerate or three dimensional )
symmetric with respect to the Cn principle axis, if no perpendicular
Subscript 1
axis, then it is with respect to σv
anti-symmetric with respect to the Cn principle axis, if no
Subscript 2
perpendicular axis, then it is with respect to σv
Subscript g symmetric with respect to the inverse
subscript u anti-symmetric with respect to the inverse
Prime ( ́ ) symmetric with respect to σh (reflection in horizontal plane)
double anti-symmetric with respect to σh ( opposite reflection in horizontal
prime ( ́ ́ ) plane)

The symbols ( g,u) only appears in point groups with an inversion center (i) , and its
mean

 g (gerade): symmetric under inversion (i)

 u (ungerade): antisymmetric under inversion (i)

Degener
Label Symmetry Behavior Meaning / Physical Behavior
acy
Totally symmetric vibrations (Raman
Ag non symmetric under inversion
active), s-orbitals
antisymmetric in some rotations
Out-of-phase symmetric modes,
Bg non or mirror planes, but symmetric
bending/d-orbitals like d_(x²–y²)
under inversion
Degenerate symmetric vibrations (e.g.,
Eg doubly symmetric under inversion
d_xz, d_yz)
Often “silent” or z-type translations
Au non antisymmetric under inversion
(possibly IR inactive)
Bu non antisymmetric under inversion x or y-direction motions, IR active
Degenerate asymmetric vibrations, x/y
Eu doubly antisymmetric under inversion
translations, IR active

43
Properties of Character table
Here we will discuss the properties of the character table, and we will deal with the group
C3v to prove these properties. I would like to point out that this is not a complete method for
finding the character table, but the purpose here is to identify its properties through this
group, and we will discuss another group later in a scientific manner:

1. Symmetry order:

It represents the number of symmetry operations in the group: it is a symmetry operation E

+ two rotations of 120 degrees (2C3) + three perpendicular reflections (3σv). Therefore, we
have:

h=1+2+3=6

2. The number of irreducible representations = the number of symmetry groups:

We have three symmetry groups: E, 2C3, and 3σv. Therefore, we must have three
irreducible representations, which we will denote by the symbols Γ1, Γ2, and Γ3.

C3v E 2C3 3σv

Γ1 X11 X12 X13
Γ2 X21 X22 X23
Γ3 X31 X32 X33

3. The elements of the first row always express perfect symmetry... that is, they are
all equal to 1.

C3v E 2C3 3σv

Γ1 1 1 1
Γ2 X21 X22 X23
Γ3 X31 X32 X33

4. The sum of the squares of the first column must be equal to the symmetry order
h.

We use this property to find the elements of the first column.

𝟏 + (𝑿𝟐𝟏 )𝟐 + (𝑿𝟑𝟏 )𝟐 = 𝒉 ⟹ 𝟏 + (𝑿𝟐𝟏 )𝟐 + (𝑿𝟑𝟏 )𝟐 = 𝟔

For this relationship to be true, X21 = 2 and X31 = 1.

C3v E 2C3 3σv

Γ1 1 1 1
Γ2 1 X22 X23
Γ3 2 X32 X33

44
 Physical Meaning (Typical
Irrep Symmetry Example Motion/Orbital Type
Examples)

Totally symmetric → symmetric Symmetric (z, x², y², Symmetric stretch, s

A₁
stretch, s orbital, z-axis motion z²) orbital

Asymmetric rotation (e.g.,

A2 Antisymmetric (Rz) Rotation around z
torsion), sometimes IR inactive
Antisymmetric in one plane
B1 Antisymmetric (x) Motion along x
(e.g., x-direction motion)
Antisymmetric in another plane
B2 Antisymmetric (y) Motion along y
(e.g., y-direction motion)

45
 5. Irreducible representations are orthogonal to each other.

We take advantage of this property to find X22 , X23 , X32 , X33 . The orthogonality
property is given by the formula:

〈Γ1〉. 〈Γ2〉 = 0

𝑤ℎ𝑒𝑟𝑒 ∶ 〈𝑥1 , 𝑥2 , 𝑥3 〉. 〈𝑦1 , 𝑦2 , 𝑦3 〉 = 𝑥1 𝑦1 + 𝑥2 𝑦2 + 𝑥3 𝑦3

First : From orthonormal of Γ1 & Γ2

〈 1 , 1 , 1 〉Γ1 ∗ 〈 1 , X 22 (𝟐) , X 23 (𝟑) 〉Γ2 = 1 + 𝟐X 22 + 𝟑X 23 = 0

Where (2) results from the fact that there are two rotational symmetry operations (2C3)

and (3) results from the fact that there are three reflection symmetry operations (3σv)

For this relationship to be true, 𝑋22 = 1, 𝑋23 = −1

C3v E 2C3 3σv

Γ1 1+ 1+ 1+
Γ2 +1 +1 -1
Γ3 +2 X32 X33

Second : By the same mechanism of perpendicularity as above, we find 𝑋33 and 𝑋32 from
the perpendicularity of Γ3 to Γ2 or Γ1:

〈 1 , 1 , 1 〉Γ1 ∗ 〈 2 , X 32 (𝟐) , X 33 (𝟑) 〉Γ3 = 2 + 𝟐X 32 + 𝟑X 33 = 0

For this relationship to be true, 𝑋32 = −1 𝑎𝑛𝑑 𝑋33 = 0

C3v E 2C3 3σv

Γ1 1 +1 +1
Γ2 1 +1 -1
Γ3 2 -1 0

From Millikan's table of representations, we find that Γ3 Γ2 Γ1 are represented by the

symbol: A1, B2, E

C3v E 2C3 3σv

A1 1 +1 +1
B2 1 +1 -1
E 2 -1 0

46
Character table for point group C4h
C4h has 8 symmetry groups, so it must have 8 irreducible representations.

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤1 𝑋11 𝑋12 𝑋13 𝑋14 𝑋15 𝑋16 𝑋17 𝑋18

𝛤2 𝑋21 𝑋22 𝑋23 𝑋24 𝑋25 𝑋26 𝑋27 𝑋28
𝛤3 𝑋31 𝑋32 𝑋33 𝑋34 𝑋35 𝑋36 𝑋37 𝑋38
𝛤4 𝑋41 𝑋42 𝑋43 𝑋44 𝑋45 𝑋46 𝑋47 𝑋48
𝛤5 𝑋51 𝑋52 𝑋53 𝑋54 𝑋55 𝑋56 𝑋57 𝑋58
𝛤6 𝑋61 𝑋62 𝑋63 𝑋64 𝑋65 𝑋66 𝑋67 𝑋68
𝛤7 𝑋71 𝑋72 𝑋73 𝑋74 𝑋75 𝑋76 𝑋77 𝑋78
𝛤8 𝑋81 𝑋82 𝑋83 𝑋84 𝑋85 𝑋86 𝑋87 𝑋88

From Note (1) the first row is symmetrical.

C4v 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4

𝛤1 1 1 1 1 1 1 1 1

47
 𝜞𝟐 : in z-axis

𝐸̂ : 𝑧 → 𝑧 ; 𝑧’ = [1] 𝑧 ; 𝑋21 = Tr(E) = 1

𝐶̂4 : 𝑧 → 𝑧 ; 𝑧’ = [1] 𝑧 ; 𝑋22 = Tr(𝐶̂4) = 1

𝐶̂2 : 𝑧 → 𝑧 ; 𝑧’ = [1] 𝑧 ; 𝑋23 = Tr(𝐶̂2) = 1

𝐶̂43 : 𝑧 → 𝑧 ; 𝑧’ = [1] 𝑧 ; 𝑋24 = Tr(𝐶̂43) = 1

𝜎̂ℎ : 𝑧 → −𝑧 ; 𝑧’ = [−1] 𝑧 ; 𝑋25 = Tr(𝐶̂43 ) = -1

𝑆̂43 : 𝑧 → −𝑧 ; 𝑆̂4 = 𝐶43 𝜎̂ℎ [𝑧] ; 𝑧’ = [−1] 𝑧 ; 𝑋26 = Tr(𝐶̂43) = -1

𝑖̂ : 𝑧 → −𝑧 ; 𝑖̂ = 𝑖[𝑧] = −𝑧 ; 𝑧’ = [−1] 𝑧 ; 𝑋26 = Tr(𝐶̂43) = -1

𝑆̂4 : 𝑧 → −𝑧 ; 𝑆̂4 = 𝐶4 𝜎̂ℎ [𝑧] ; 𝑧’ = [−1] 𝑧 ; 𝑋26 = Tr(𝐶̂43) = -1

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤2 1 1 1 1 -1 -1 -1 -1 z

The figure below shows the effect of symmetry operations on the z-axis, where (a)
represents E, C4, C2, C43, and (b) represents the remaining operations.

a b

Figure 23

𝑎 𝑏 𝑐
( )
𝑇𝑟 𝑚𝑎𝑡𝑟𝑖𝑥 = 𝑇𝑟 ( 𝑑 𝑒 𝑓) = 𝑎 + 𝑒 + 𝑖
𝑔 ℎ 𝑖

48
 𝜞𝟑 : in x-axis
Here the x-axis is connected to the y-axis by the reflection plane 𝜎̂ℎ (which is nothing but
the xy plane), so both x and y have similar representations.

operator Axis affected Matrix Form of Axis affected Tr ( operator matrix )

𝑥 → 𝑥 𝑥’ 1 0 𝑥
𝐸̂ : 𝑦 → 𝑦 [ ]=[ ][ ] 𝑋31 =2
𝑦′ 0 1 𝑦

𝑥 → 𝑦 𝑥’ 0 1 𝑥
𝐶̂4 : 𝑦 → −𝑥 [ ′]=[ ][ ] 𝑋32 = 0
𝑦 −1 0 𝑦

𝑥 → −𝑥 𝑥’ −1 0 𝑥
𝐶̂2 : 𝑦 → −𝑦 [
𝑦 ′]=[ ][ 𝑦 ] 𝑋33 = −2
0 −1

𝑥 → −𝑦 𝑥’ 0 −1 𝑥
𝐶̂43 : 𝑦 → 𝑥 [
𝑦 ′]=[ ][𝑦 ] 𝑋34 = 0
1 0

𝑥 → 𝑥 𝑥’ 1 0 𝑥
𝜎̂ℎ : [ ]=[ ][ ] 𝑋35 = 2
𝑦 → 𝑦 𝑦′ 0 1 𝑦

3 𝑥 → −𝑦 𝑥’ 0 −1 𝑥
𝑆̂4 = 𝐶̂ 4 𝜎̂ ℎ : 𝑦 → 𝑥 [
𝑦 ′]=[
1 0
][𝑦 ] 𝑋36 = 0

𝑥 → −𝑥 𝑥’ −1 0 𝑥
𝑖̂ : 𝑦 → −𝑦 [ ′]=[ ][ 𝑦 ] 𝑋37 = −2
𝑦 0 −1

𝑥 → −𝑦 𝑥’ 0 −1 𝑥
𝑆̂4 = 𝐶̂ 4 𝜎̂ ℎ : 𝑦 → 𝑥 [
𝑦′
]=[ ][𝑦 ] 𝑋38 = 0
1 0

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤3 2 0 -2 0 2 0 -2 0 x , y

The following figure shows the effect of symmetry operations on the components of the x
and y axes.

49
𝛤4 : in xz & yz - plane
It can be derived by multiplying 𝛤2 which is in the z-axis by 𝛤3 which is for the y,z axes.

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤2 1 1 1 1 -1 -1 -1 -1 z
𝛤3 2 0 -2 0 2 0 -2 0 x , y
𝛤4 2 0 -2 0 -2 0 2 0 xz , yz

50
𝛤5 : in xy - plane

operator Axis affected Matrix Form of Axis affected Tr ( operator matrix)

𝑥→ 𝑥
𝐸̂ : 𝑦→ 𝑦 x' y’ = [1] x y 𝑋51 = 1

𝑥→ 𝑦
𝐶̂4 : 𝑦 → −𝑥 x' y’ = [1] y (-x) = [-1] yx 𝑋52 = −1

𝑥→ −𝑥
𝐶̂2 : 𝑦→ −𝑦 x' y’ = [1] (-x) (-y) = [1] xy 𝑋53 = 1

𝑥 → −𝑦
𝐶̂43 : 𝑦→ 𝑥 x' y’ = [1] (-y) x =[-1] yx 𝑋54 = −1

𝜎̂ℎ : 𝑥𝑦 → 𝑦𝑥 x' y’ = [1] xy 𝑋55 = 1

𝑥 → −𝑦
𝑆̂43 = 𝐶̂43 𝜎̂ℎ : 𝑦→ 𝑥 x' y’ = [1] (-y) x =[-1] yx 𝑋56 = −1

𝑥→ −𝑥
𝑖̂ : 𝑦→ −𝑦 x' y’ = [1] (-x) (-y)=[1] xy 𝑋57 = 1

𝑥→ 𝑦
𝑆̂4 = 𝐶̂ 4 𝜎̂ ℎ : 𝑦 → −𝑥 x' y’ = [1] y (-x) = [-1] yx 𝑋52 = −1

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤5 1 -1 1 -1 1 -1 1 -1 xy

The figure below shows the effect of symmetry operations on the xy plane.

51
Figure 24

52
 𝛤5 :Rotation axis Rz

From the figure above, which shows the system rotation axes Rx, Ry, Rz, we see that

• The effect of symmetry operations on Rz is equivalent to the effect of perfect

symmetry (the system in its original state).
• The effect of symmetry operations on Rx is equivalent to their effect on the xz plane.
• The effect of symmetry operations on Ry is equivalent to their effect on the yz plane.

So Rx Ry Rz does not give new representations.

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤1 1 1 1 1 1 1 1 1
𝛤2 1 1 1 1 -1 -1 -1 -1 z
𝛤3 2 0 -2 0 2 0 -2 0 x , y
𝛤4 2 0 -2 0 -2 0 2 0 xy , xz
𝛤5 1 -1 1 -1 1 -1 1 -1 xy
𝛤6
𝛤7 𝑋71 𝑋72 𝑋73 𝑋74 𝑋75 𝑋76 𝑋77 𝑋78
𝛤8 𝑋81 𝑋82 𝑋83 𝑋84 𝑋85 𝑋86 𝑋87 𝑋88

We notice in the table above that both 𝛤2 and 𝛤3 have two degeneracies (E = 2) and they
represent new representations written in terms of complex functions.

53
 Now, we have 7 irreducible representations, and it remains to find the last one, which we
can find from the orthogonality condition,

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤1 1 1 1 1 1 1 1 1 Rz

𝛤2 1 1 1 1 -1 -1 -1 -1 z
𝛤3 2 0 -2 0 2 0 -2 0 x , y
𝛤4 2 0 -2 0 -2 0 2 0 xz , yz Rx , Ry
𝛤5 1 -1 1 -1 1 -1 1 -1 xy
𝛤6 𝑋61 𝑋62 𝑋63 𝑋64 𝑋65 𝑋66 𝑋67 𝑋68

First: From Note (1), we find 𝑋61 , where the symmetry order here is h = 8:

h = 8 Number of symmetry elements

n = 8 Number of irreducible representations
n = 6 Number of real irreducible representations

2
ℎ = 12 + 12 + (12 ) + (12 ) + 12 + 𝑋61 =8 → 𝑋61 = 1

Secondly, from the orthonormal of 𝛤1 with 𝛤6 :

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4

𝛤1 1 1 1 1 1 1 1 1
𝛤6 1 𝑋62 𝑋63 𝑋64 𝑋65 𝑋66 𝑋67 𝑋68

〈Γ1〉. 〈Γ6〉 = 0

〈1 , 1 ,1 ,1 , 1, 1 , 1 , 1〉. 〈1 , 𝑋62 , 𝑋63 , 𝑋64 , 𝑋65 , 𝑋66 , 𝑋67 , 𝑋68 〉 = 0

1(1) + (1)𝑋62 + (1)𝑋63 + (1)𝑋64 + (1)𝑋65 + (1)𝑋66 + (1)𝑋67 + (1)𝑋68 = 0

1 + 𝑋62 + 𝑋63 + 𝑋64 + 𝑋65 + 𝑋66 + 𝑋67 + 𝑋68 = 0

In order for this condition to be met, it must be

𝑋62 = −1 𝑋63 = 1 𝑋64 = −1 𝑋65 = −1 𝑋66 = 1 𝑋67 = −1 𝑋68 = 1

Then

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4

𝛤6 1 -1 1 -1 -1 1 -1 1

It only meets the conditions of orthogonality with the rest of the functions.

54
 C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤1 1 1 1 1 1 1 1 1 Rz

𝛤2 1 1 1 1 -1 -1 -1 -1 z
𝛤3 2 0 -2 0 2 0 -2 0 x , y
𝛤4 2 0 -2 0 -2 0 2 0 xy , xz Rx , Ry
𝛤5 1 -1 1 -1 1 -1 1 -1 xy
𝛤6 1 -1 1 -1 -1 1 -1 1
Table 5 C4h Charecter table

If we assume that we swapped the elements in 𝛤6 , here is another option for 𝛤6 :

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝛤1 1 1 1 1 1 1 1 1 Rz

𝛤2 1 1 1 1 -1 -1 -1 -1 z
𝛤3 2 0 -2 0 2 0 -2 0 x , y
𝛤4 2 0 -2 0 -2 0 2 0 xy , xz Rx , Ry
𝛤5 1 -1 1 -1 1 -1 1 -1 xy
𝛤6 1 -1 -1 1 -1 1 -1 1

The condition of perpendicularity between 𝛤6 and 𝛤5 will disappear:

(1*1)+ (-1*-1)+ (1*-1)+ (-1*1)+ (1*-1)+ (-1*1)+ (1*-1)+ (-1*1)

= 1+1-1-1-1-1-1-1= -6 ≠0

Any case other than that in Table 2 is false.

Now, using Millikan's symbol table, we write Table 5 in the form:

C4h 𝐸̂ 𝐶̂4 𝐶̂2 𝐶̂43 𝜎̂ℎ 𝑆̂43 𝑖̂ 𝑆̂4 Basis Components

𝐴𝑔 1 1 1 1 1 1 1 1 Rz

𝐴𝑢 1 1 1 1 -1 -1 -1 -1 z
𝐸𝑔∗ 2 0 -2 0 2 0 -2 0 x , y
𝐸𝑢∗ 2 0 -2 0 -2 0 2 0 xz , yz Rx , Ry
𝐵𝑔 1 -1 1 -1 1 -1 1 -1 xy
𝐵𝑢 1 -1 1 -1 -1 1 -1 1

55
 Example : Water (C2v)
In a molecule like H₂O, there are different types of vibrations:

 Symmetric stretch: both H atoms move in and out together

 Asymmetric stretch: one H moves in, one moves out
 Bend: the angle between H–O–H opens and closes

Look : https://www.chem.purdue.edu/jmol/vibs/h2o.html to show this 3 mods

The bend is when the two hydrogen atoms swing in and out, changing the bond angle, but
keeping bond lengths nearly constant.

Water has symmetry operations: E, C₂, σv(xz), σv′ (yz)

1 - The bending vibration transforms like A1: fully symmetric :

This means that when you apply all the symmetry operations of the molecule’s point
group (like rotation, reflection, inversion, etc.) to the bending vibration, its overall shape
and direction do not change — it’s symmetric under all operations .

2 - The asymmetric stretch transforms like B2 :

This means the motion of the atoms during the asymmetric stretching vibration
behaves, under symmetry operations, the same way as the B₂ irreducible representation in
the molecule’s point group (like C₂v for H₂O). It doesn’t stay fully symmetric, but follows
the specific transformation rules of the B₂ symmetry species.

In a molecule like H₂O, the asymmetric stretch is a type of vibrational mode where:

a- One hydrogen atom moves toward the oxygen,

b- The other hydrogen atom moves away from the oxygen
c- The angle between the O–H bonds stays about the same.

This is called "asymmetric" because the motion is not the same on both sides — it’s out
of sync.

What Does "Transforms Like B₂" Mean?

This tells you the symmetry behavior of the vibration.

When we say : The asymmetric stretch transforms like B₂

We mean: : If you apply all the symmetry operations of the molecule's point group (in
this case, C₂v for water), this vibration behaves according to the B₂ row in the character
table.

That row includes signs (characters) showing how a function changes under operations
like:

 E (identity): no change → character = +1

 C₂ (180° rotation): flips sign → character = –1

56
  σv(xz): reflection in the molecular plane → character = +1
 σv′(yz): reflection in the vertical plane perpendicular to the molecular plane →
character = –1

Visual Interpretation
When you apply a C₂ rotation (180° around the z-axis) to the asymmetric stretch:

 The two H atoms swap positions

 Because one is moving inward and the other outward, this flips the sign of the motion
→ This is why B₂ has a –1 under C₂.

The same logic applies to reflections: the asymmetric stretch responds differently under
each mirror plane — it matches the B₂ row in the character table.

Summary
Saying “the asymmetric stretch transforms like B₂” means:

 The motion of the atoms under symmetry operations matches the B₂ symmetry type
in the character table.
 It's antisymmetric under some operations (like rotation and reflection).
 This determines how it appears in vibrational spectroscopy, and whether it’s IR or
Raman active.

57
3 - The 2px orbital on hydrogen transforms like B1
What is the 2pₓ Orbital on Hydrogen?

 The 2pₓ orbital points left-right (along the x-axis).

 If a hydrogen atom has a pₓ orbital, it sticks sideways from the nucleus — aligned
along the x-direction.

What Does "Transforms Like B₁" Mean?

In group theory, every function (orbital, vibration, etc.) transforms according to a

symmetry species — also called an irreducible representation (irrep).

When we say: "The 2pₓ orbital on H transforms like B₁"

we’re saying: If you apply all the symmetry operations of C₂v to that orbital, the resulting
signs and shapes match the B₁ row of the character table.

In C₂v:

Operation What It Does Effect on 2pₓ (x-direction)

E (identity) do nothing stays the same → +1
C₂ (180° rot.) flips x → –x orbital flips sign → –1
σ_v(xz) mirror in xz-plane orbital is in-plane → +1
σ_v′(yz) mirror in yz-plane orbital flips across plane → –1

So:

 The 2pₓ orbital responds to these operations with the exact pattern of characters:
(+1, –1, +1, –1)
 This matches the B₁ irreducible representation in the C₂v character table

Why Is This Useful?

Knowing that the 2pₓ orbital transforms as B₁ lets us:

 Determine how it combines with other orbitals (LCAO theory)

 Assign symmetry labels in molecular orbital diagrams
 Predict selection rules for spectroscopy
 Simplify quantum chemical calculations using group theory

For example: If an MO (molecular orbital) has B₁ symmetry, it can only mix with other
orbitals of B₁ symmetry.

58
Summary
Saying “The 2pₓ orbital on hydrogen transforms like B₁” means:

 Its symmetry behavior under the molecule’s operations matches the B₁ row of the
character table.
 It flips or stays the same in specific ways depending on the operation.
 This is key for building MOs, analyzing bonding, and understanding spectra.

These labels tell you:

 Whether an orbital or motion is symmetric or antisymmetric
 Whether it will mix or stay separate from others
 Whether it’s IR or Raman active

59
Refrence

[1] https://symotter.org/gallery

[2] http://symmetry.jacobs-university.de/

[3] https://geo.libretexts.org/Bookshelves/Geology/Mineralogy_(Perkins_et_al.)/

[4] https://spacegroups.symotter.org/
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Text
[5]
book_Maps/Physical_Chemistry_(LibreTexts)/
[6] Molecular Symmetry and Group Theory - R. C. Maurya, J. M. Mir

[7] Group Theory With Applications in Chemical Physics - P. W. M. JACOBS

[8] Open AI – Chat gpt

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