Introduction:

The symmetry relationships in the molecular structure provide the basis for a mathematical theory, called group theory. The mathematics of group theory is predominantly algebra. Since all molecules are certain geometrical entities, the group theory dealing with such molecules is also called the ‘algebra of geometry’.

Symmetry Element:

A symmetry element is a geometrical entity such as a point, a line or a plane about which an inversion, a rotation or a reflection is carried out in order to obtain an equivalent orientation.

Symmetry Operation:

A symmetry operation is a movement such as an inversion about a point, a rotation about a line or a reflection about a plane in order to get an equivalent orientation.

The various symmetry elements and symmetry operations are listed in the table below.

Symmetry element	Symmetry operation	Schoenflies symbol	Hermann-Mauguin symbol
Centre of Symmetry or Inversion centre	Inversion	I	Ä®
Plane of symmetry	Reflection θ	σ	m
Axis of symmetry	Rotation through	Cn	n
Improper axis	Rotation followed by reflection in a plane perpendicular to axis	Sn	ñ
Identity element	Identity Operation	E

Centre of Symmetry:

A point in the molecule from which lines drawn to opposite directions will meet similar points at exactly the same distance. Some of the molecules, which have a centre of symmetry, are:

• N₂F₂
• PtCl₄
• C₂H₆

Plane of Symmetry:

A plane which divides the molecule into two equal halves such that one half is the exact mirror image of the other half. The molecules, which have plane of symmetry, are:

• H₂O
• N₂F₂
• C₂H₄

The broken line in the σ-plane: If we look from the left side (A) into the mirror plane, H_A appears to have gone on the other side and its image appears exactly at H_B. Similarly, viewing the structure of the H₂O molecule from the right side (B), the reflection of H_B appears at H_A. Configuration II is the result of this reflection operation and is equivalent to I. Another round of this operation on the molecule (configuration II) yields configuration III, which is identical to configuration I.

Proper Axis of Symmetry:

An axis passing through the molecule about which, when the molecule is rotated 360°/n, an equivalent orientation is produced. This is an axis of n-fold symmetry or an axis of order, as shown below.

• H₂O
• N₂H₃
• BF₃

Initially, the H₂O molecule is in configuration I, lying flat on the plane of the paper. After rotating it through an angle θ = 180° about an axis passing through the O atom (Z-axis) and along the HOH angle, configuration II is obtained. Configuration II is equivalent to configuration I, but not identical. A second, similar rotation about the Z-axis on configuration II results in configuration III. Configuration III is identical to the initial configuration I.

Principal Axis:

If there is more than one axis of symmetry, in many cases one of the axes is identified as the principal axis. This axis is selected in the following order:

1. The only axis
2. The highest order axis
3. The axis passing through the maximum number of atoms
4. The axis perpendicular to the plane of the molecule

The principal axis is taken as the vertical axis in the z-direction. The subsidiary axis is perpendicular to the principal axis and hence lies in the horizontal direction.

Molecule	Principal Axis	Subsidiary Axis
Water	C2	Nil
Ammonia	C3	Nil
BF3	C3	3C2
XeF4	C4	4C2
Cyclopentadienyl anion	C5	5C2
Benzene	C6	6C2
H2O2	C2	Nil
XeOF4	C4	Nil

(A C_n axis can combine with only n C₂ axes perpendicular to it or with no subsidiary axis.)

The plane of symmetry is also classified based on the principal axis. Planes that include or involve the principal axis are called vertical planes (σ_v), and planes perpendicular to the principal axis are called horizontal planes (σ_h).

Molecule	Principal Axis	Vertical Planes	Horizontal Planes
Water	C2	Nil	Nil
Ammonia	C3	Nil	One
BF3	C3	2	Nil
XeF4	C4	3	Nil
Cyclopentadienyl anion	C5	3	One
Benzene	C6	4	One
H2O2	C2	6	One
XeOF4	C4	4	Nil

Improper Axis of Symmetry:

An axis passing through the object about which, when the object is rotated through 360°/n followed by reflection in a plane perpendicular to the axis, produces an equivalent orientation.

For example: Ethane molecule (staggered form).

Configuration I and II are not equivalent, i.e., θ = 60°, and therefore the C₆ rotational operation is not a valid symmetry operation by itself. Similarly, configurations II and III are not equivalent, indicating that the σ operation perpendicular to the so-called C₆ rotational axis is also not a genuine symmetry operation. However, configurations I and III are equivalent, so that a C₆ operation followed by a σ operation perpendicular to C₆ constitutes a valid combined operation. This product operation results in an element called the S₆ axis.

Identity Element:

This element is obtained by an operation called the identity operation. After this operation, the molecule remains unchanged. This situation can be visualized in two ways:

1. We do not perform any operation on the molecule, or
2. We rotate the molecule by 360°.

Every molecule has this element of symmetry, and it co-exists with the identity of the molecule; hence the name identity element.

Point Group:

The symmetry elements can combine only in a limited number of ways, and these combinations are called point groups.

Nomenclature of the Point Group:

• There are certain conventions developed by two schools of thought for naming these point groups.
• The Schoenflies nomenclature is more popularly used for molecular point groups than that of Hermann-Mauguin.
• Crystal and space groups are named using Hermann-Mauguin symbolism.
• For example, H₂O and pyridine are assigned the point group symbol C_2v, which means the molecules contain a C₂ axis and 2 σ_v planes.

Identification of Molecular Point Groups:

The whole set of molecules can be divided into three broad categories:

1. Molecules of Low Symmetry (MLS)
2. Molecules of High Symmetry (MHS)
3. Molecules of Special Symmetry (MSS)

Molecules of Low Symmetry (MLS):

The starting point could be molecules containing no symmetry elements other than E. Such molecules are unsymmetrically substituted and belong to the C₁ point group.

For example: TeCl₂Br₂ in its gaseous phase, and tetrahedral carbon and silicon compounds with the formula AHFClBr (A = C, Si), belong to the C₁ point group.

Molecules of High Symmetry (MHS):

This category includes all molecules containing a C_n axis, either alone or with additional symmetry elements. The three main types of point groups are: C_n, D_n, and S_n.

C_n Type Point Group:

Molecules that contain only one C_n (proper) axis fall under the C_n point group. The presence of a C_n axis implies the presence of (n–1) distinct symmetry elements. Whether n is even or odd, a C_n operation generates a set of n symmetry elements including the identity E.

Thus, the order of the C_n group is n (h = n), and such molecules are designated as belonging to C_n point groups.

C_nv Point Groups:

This group contains a C_n axis and n σ_v planes of symmetry.

• When n is odd, all the planes are of the σ_v type.
• When n is even, there are n⁄2 planes of σ_v type and n⁄2 planes of σ_v′ type.

C_nh Point Groups:

This set of point groups is formed by adding a horizontal plane (σ_h) to a proper rotational axis, C_n.

The group has a total of 2n elements — n elements from the C_n operations, and the other n elements generated by combining C_n with σ_h, resulting in corresponding S_n axes.

When n is even, molecules belonging to the C_nh point group necessarily contain a centre of inversion (i).

D_n Point Groups:

These are purely rotational groups, meaning they contain only rotational axes of symmetry.

When a molecule contains only one C_n axis, it is classified under the C_n point group. However, if in addition to a primary C_n axis, there exists a set of n C₂ axes perpendicular to the C_n axis, the molecule belongs to the D_n point group.

The order of this rotational group is 2n, since the C_n operations generate (n – 1) + E elements, and the n C₂ axes contribute n additional operations.

For example, the gauche (skew) form of ethane belongs to the D₃ point group.

D_nh Point Groups:

This point group is formed by adding a horizontal mirror plane (σ_h) to the symmetry elements of a D_n group.

The order of the D_nh group is 4n.

• When n is even, the elements generated by the addition of σ_h are distinct and different from those already present in the D_n group.
• When n is odd, a set of n elements based on S_n axes is obtained.

An example of a molecule in this point group is B₂H₆, which belongs to the D_2h point group.

D_nd Point Groups:

This point group is obtained by adding a set of dihedral planes (nσ_d) to the elements of a D_n group.

To qualify for this point group, a molecule must possess:

• A C_n proper rotational axis
• n C₂ axes perpendicular to the C_n axis
• n σ_d planes (dihedral mirror planes)

This results in a total of 3n symmetry elements at minimum.

An example is cyclohexane in chair conformation, which belongs to the D_3d point group.

S_n Point Groups:

The S_n axis (improper rotation axis) is the sole group generator for the S_n point group, where n is even.

For n odd, the presence of an S_n axis implies the existence of 2n elements, which include a mirror plane (σ) that appears independently.

The presence of additional symmetry elements alongside the S_n axis leads to other point groups:

• If a plane perpendicular to the C_n or S_n axis is present, the molecule may belong to point groups like C_nh, D_nh, or D_nd.
• If n is even and there is no horizontal mirror plane (σ_h), but other elements exist along with S_n, the molecule most likely belongs to the D_nd point group.

An example of a molecule belonging to this group is SiO₄(CH₃)₄, which exhibits S₄ symmetry.

Point Groups and Their Detailed List of Symmetry Elements

Point group	Order of group, h	Type of symmetry elements
C1	1	E (=C1)
Ci	2	E, I (=S2)
Cs	2	E, σ
Cn - groups: (h = n)
C2	2	E, C2
C3	3	E, C31, C32
C4	4	E, C41, C42 (=C2), C43
C5	5	E, C41, C42, C43, C44
Cnv - groups: (h = 2n)
C2v	4	E, C2, σ
C3v	6	E, C31, C32, 3σv
C4v	8	E, C41, C42(=C2), C43, 2σv, 2σv'
Cnh - groups: (h = 2n)
C2h	4	E, C2, i=(S2), σh
C3h	6	E, C31, C32, S31, S35, σh
C4h	8	E, C41, C42(=C2), C43, S41, S43, σh, i=(S2)
Dn - groups: (h = 2n)
D2	4	E, C2, C2'
D3	6	E, C31, C32, 3C2
D4	8	E, 2C4, C2, 4C2
Dnh - groups: (h = 4n)
D2h	8	E, C2, 2C2', i=(S2), σh, 2σv
D3h	12	E, 2C3, 3C2, σh, 3σv, 2S3, (S31, S35)
D4h	16	E, 2C4, (C41, C42), C2=(C42), 2C2', 2C2", σh, 2σv, 3σd, i, 2S4 (S41, S43)
Dnd - groups: (h = 4n)
D2d	8	E, C2, 2C2', 2σd, 2S4
D3d	12	E, 2C3 (C31, C32), 3C2, i, 3σd, 2S6(S61, S63)
D4d	16	E, 2C4, (C41, C43), C2=(C42), 4C2', 4σd, 4S8(S81, S83, S85, S87)
Sn (n=even) - groups: (h = n)
S4	4	E, S41, S43, C2
S6	6	E, S61, S45, C31, C32, i
S8	8	E, S8, S81, S83, S85, S87, C41, C43, C2=(C42)
Infinite-point group (h=∞)
C∞v	∞	E, ∞, C∞, ∞σv
D∞v	∞	E, ∞, C∞, ∞σv, σh, i

D∞_v Point Group

This group is an extension of the D_nh group (∞). This group of molecules contains a C_∞ axis, an infinite number of C₂ axes perpendicular to the C_∞ axis, and a σ_h plane. Consequently, the molecule also possesses an infinite number of σ_v planes and a center of inversion (i). Thus, all centrosymmetric linear molecules belong to this point group.

Typical examples include homonuclear diatomic molecules such as N₂, O₂, H₂, F₂, and Cl₂.

Molecules Containing Multiple Higher-Order Axes

This is a special class of molecules which contain more than one type of rotational axes (n ≥ 2) that are neither perpendicular to the principal C_n axis (n-highest), as in D_n and related point groups, nor bear any perpendicular relationship. These high-symmetry molecules have shapes corresponding to the five Platonic solids: tetrahedral, octahedral, cube, dodecahedral, and icosahedral.

Tetrahedral Point Groups

The highest-fold axis in these point groups is the C₃ axis, which occurs in multiples. Molecules with only C₃ axes and additionally only C₂ axes belong to T, a pure rotational point group, since they contain only proper rotational axes. All other types of symmetry elements (σ_v, i, S_n) are absent in this group.

T: 8C₃ (4C₃₁, 4C₃₂), 3C₂, E

When σ_d, S₄ (collinear with C₂ axes) elements are added to the T group elements, we get a full group called T_d. The order of this group is 24.

T_d: 8C₃ (4C₃₁, 4C₃₂), 3C₂, E, 6S₄ (S₄₁, S₄₃), 6 σ_d

There is another uncommon point group, T_h, which can be obtained by adding three planes of symmetry (σ_h) to the T group. The order of this group is 24.

T_h: 8C₃ (4C₃₁, 4C₃₂), 3C₂, i, 3σ_h, 8S₆ (4S₆₁, 4S₆₅)

Example: Co(NO₂)₆³⁻

Octahedral Point Groups

This is another class of cubic groups. Additionally, octahedral point groups have multiple C₄ axes when compared to that of tetrahedral groups.

When the group contains only rotational axes, it is labelled as the O group. The order (h) of this group is 24.

O: E, 6C₄ (3C₄₁, 3C₄₂), 8C₃ (4C₃₁, 4C₃₂), 6C₂, 3C₂′ = 3C₄₂

To the O group elements, if 3σ_h and 6σ_d planes are added, a group of higher symmetry can be generated. The order of this group is 48.

O_h: E, 6C₄ (3C₄₁, 3C₄₂), 3C₂′ = 3C₄₂, 6C₂, 8C₃ (4C₃₁, 4C₃₂), i, 3σ_h, 6σ_d, 6S₄ (S₄₁, S₄₃), 8S₆ (4S₆₁, 4S₆₅)

Cubane

Icosahedral Groups:

This group contains molecules with either icosahedral or pentagonal dodecahedral shapes and belongs to I_h point groups. The molecules containing only the rotational elements are said to belong to the I point group. The order of this point group is 60, whereas the full group is 120.

I: E, 24C₅ (6C₅₁, 6C₅₂, 6C₅₃, 6C₅₄), 20C₃ (10C₃₁, 10C₃₂), 15C₂

I_h: E, 24C₅, 20C₃, 15C₂, 24S₁₀ (6S₁₀₁, 6S₁₀₃, 6S₁₀₇, 6S₁₀₉), 20S₆ (10S₆₁, 10S₆₅), i, 15σ

Fullerene

Great Orthogonality Theorem:

The matrices of the different Irreducible Representations (IR) possess certain well-defined interrelationships and properties. Orthogonality theorem is concerned with the elements of the matrices which constitute the IR of a group.

The mathematical statement of this theorem is,

$$\sum_{R}^{}[\Gamma_i(R)_{mn}][\Gamma_j(R)'m'n']^*=\frac{h}{\sqrt{l_il_j}}\delta_{ij}\delta_{mm'}\delta_{nn'}$$

Where,

• i, j – Irreducible Representations
• li, lj – Its dimensions
• h – Order of a group
• Γ_i(R)_mn – Element of m^th row, n^th column of the i^th representation
• Γ_j(R)'_m'n' – Element of m′^th row, n′^th column of the j′^th representation
• δ_ij, δ_mm′, δ_nn′ – Kronecker delta

Kronecker delta can have values 0 and 1. Depending on that, the main theorem can be expressed as three similar equations:

i.e.,

1. When Γ_i ≠ Γ_j and j ≠ i, then δ_ij = 0
   Therefore,
   Σ_R [ Γ_i(R)_mn ] [ Γ_j(R)'_m'n' ]^* = 0
2. When Γ_i = Γ_j and j = i, then δ_ij = 1
   Therefore,
   Σ_R [ Γ_i(R)_mn ] [ Γ_i(R)'_m'n' ]^* = 0
   
   From these two equations, we can state the Orthogonality Theorem as: “The sum of the product of the irreducible representations is equal to zero.”
3. When i = j, m = m′, n = n′
   Then,
   Σ_R [ Γ_i(R)_mn ] [ Γ_i(R)_mn ]^* = $\frac{h}{l_i}$

From the above equations some important rules of the irreducible representations of a group and there character were obtained.

Five Rules Obtained:

1. The sum of the squares of the dimensions of the representations equals the order (h) of the group.
   
   i.e., Σl_i² = l₁² + l₂² + l₃² + …… + l_n² = h
   
   Γ_i(E) – the character of the identity operation E in the i^th irreducible representation, which equals the dimension of the representation.
   
   i.e., Σ_i [Γ_i(E)]² = h
2. The sum of the squares of the characters in any irreducible representation is equal to h.
   
   i.e., Σ_R [Γ_i(R)]² = h
3. The vectors whose components are the characters of two different irreducible representations are orthogonal.
   
   i.e., Σ_R Γ_i(R) · Γ_j(R) = 0     when i ≠ j
4. In a given representation (reducible or irreducible), the characters of all matrices corresponding to operations in the same class are identical.
   
   Example: In the C_3v point group, the operations E, 2C₃, and 3σ_v belong to different classes. The characters of operations in the same class are identical for a particular irreducible representation.
5. The number of irreducible representations in a group equals the number of classes in the group.

Applications:

Applying these 5 rules we can develop the character table for various point groups. For most chemical applications, it is sufficient to know only the characters of the each of the symmetry classes of a group.

Steps for the Construction of a Character Table:

1. Write down all the symmetry operations of the point group and group them into classes.
2. Note that the number of irreducible representations (IRs) is determined using the group theory theorem.
3. Carefully follow the interrelationships of various group operations.
4. Use the orthogonality and normality theorems to fix the characters.
5. Generate a representation using basic vectors. Try using basis vectors such as X, Y, Z, R_x, R_y, R_z, etc., and check their transformation properties.

Character Table for C_2v Point Group:

1. For the C_2v point group, there are 4 symmetry operations:

• E (Identity)
• C_2z (180° rotation about the z-axis)
• σ_xz (reflection in the xz-plane)
• σ_yz (reflection in the yz-plane)

These operations form 4 classes. The corresponding irreducible representations are denoted as Γ₁, Γ₂, Γ₃, and Γ₄.

The character under the identity operation E for each representation is represented as l₁, l₂, l₃, l₄.

C2v	E	C2z	σxz	σyz
Γ1	l1
Γ2	l2
Γ3	l3
Γ4	l4

2. The sum of the squares of the dimensions of the symmetry operations = 4.

i.e., l₁² + l₂² + l₃² + l₄² = h = 4.

This can only be satisfied by four one dimensional representations.

C2v	E	C2z	σxz	σyz
Γ1	1
Γ2	1
Γ3	1
Γ4	1

The unknowns for Γ₁ is a₁, b₁, c₁, for Γ₂ is a₂, b₂, c₂.

C2v	E	C2z	σxz	σyz
Γ1	1	a1	b1	c1
Γ2	1	a2	b2	c2
Γ3	1	a3	b3	c3
Γ4	1	a4	b4	c4

3. Sum of the squares of the characters of any IR is equal to the order of the group.

i.e., 1² + a₁² + b₁² + c₁² = 4.

C2v	E	C2z	σxz	σyz
Γ1	1	1	1	1
Γ2	1	a2	b2	c2
Γ3	1	a3	b3	c3
Γ4	1	a4	b4	c4

4. The orthogonality theorem must be satisfied by all the symmetry operations.

i.e., Σ_R Γ_i(R) Γ_j(R) = 0

i.e., for Γ₁ · Γ₂

i.e., 1·1 + a₁·1 + b₂·1 + c₂·1 = 0

Let a₂ = 1, b₂ = -1, and c₂ = -1

Then Γ₁ · Γ₂ = 0

C2v	E	C2z	σxz	σyz
Γ1	1	1	1	1
Γ2	1	1	-1	-1
Γ3	1	a3	b3	c3
Γ4	1	a4	b4	c4

For Γ₃ · Γ₁

i.e., 1·1 + a₃·1 + b₃·1 + c₃·1 = 0

Let a₃ = -1, b₃ = 1, and c₃ = -1

Then Γ₁ · Γ₃ = 0

C2v	E	C2z	σxz	σyz
Γ1	1	1	1	1
Γ2	1	1	-1	-1
Γ3	1	-1	1	-1
Γ4	1	a4	b4	c4

For Γ₄ · Γ₁

i.e., 1·1 + a₄·1 + b₄·1 + c₄·1 = 0

Let a₄ = -1, b₄ = -1, and c₄ = 1

Then Γ₁ · Γ₄ = 0

C2v	E	C2z	σxz	σyz
Γ1	1	1	1	1
Γ2	1	1	-1	-1
Γ3	1	-1	1	-1
Γ4	1	-1	-1	1

Rules For Assigning Mullicon Symbols:

1. If the IR is unidimensional term A or B is used.
   If it is two dimensional E is used.
   If it is three dimensional T is used.
2. If one dimensional IR is symmetric with respect to the principle axis C_n, i.e., character of C_n is +1, the term A is used. If it is -1, the term B is used.
3. If IR is symmetric with respect to subsidiary axes then subscript 1 is given and is antisymmetric then subscript 2 is given.
4. Prime (′) and double prime (″) marks are used for indicating symmetric or antisymmetric with respect to horizontal plane.
5. ‘g’ and ‘u’ subscripts are given for those which are symmetric and antisymmetric respectively with respect to centre of symmetry.

C2v	E	C2z	σxz	σyz
A1	1	1	1	1
A2	1	1	-1	-1
B3	1	-1	1	-1
B4	1	-1	1	-1

In any character table there are 4 different areas.

• Area I – Characters of symmetry operations
• Area II – Mullicon Symbols
• Area III – Cartesian coordinates of rotation axes
• Area IV – Binary Products

Area III:

In order to assign the Cartesian coordinates, different operations are performed on each of the axes. Here we find the symbols X, Y, Z represent coordinates and rotations R_x, R_y, and R_z.

Consider a vector along the Z axis. The identity operation does not change the direction of the head of the vector. On performing C₂, σ_xz, and σ_yz operations, no change occurs. Hence, its characters are 1 1 1 1. Therefore, the vector 'Z' transforms under A₁.

The characters are 1 -1 1 -1 corresponding to B₁. And with respect to vector Y, the characters are 1 -1 -1 1 and therefore correspond to B₂.

A similar arrangement can be made for the rotation axes R_x, R_y, and R_z, representing rotation about the X, Y, and Z axes, respectively.

To determine how they transform, a curved arrow should be considered around the respective axes. If the direction of the head of the curved arrow does not change due to the symmetry operation, the character is +1; otherwise, it is -1.

The characters are 1 1 -1 -1. Therefore, it corresponds to A₂ and it becomes R_z.

The characters are 1 -1 1 -1. Therefore it will be B₁ and it becomes R_x. Similarly B₂ becomes R_y.

Therefore,

C_2v Character Table

C2v	E	C2z	σxz	σyz	Linear Functions, Rotations
A1	1	1	1	1	Z
A2	1	1	-1	-1	Rz
B1	1	-1	1	-1	X, Ry
B2	1	-1	-1	1	Y, Rx

Area IV:

Which represents the squares and binary products.

          A1   = Z    = 1  1  1  1
          A1²  = Z²   = 1  1  1  1   = A1
          B1   = X    = 1 -1  1 -1
          B1²  = X²   = 1  1  1  1   = A1
          B2   = Y    = 1 -1 -1  1
          B2²  = Y²   = 1  1  1  1   = A1
          
          XY   = B1 · B2  = 1  1 -1 -1  = A2
          XZ   = B1 · A1  = 1 -1  1 -1  = B1
          YZ   = B2 · A1  = 1 -1 -1  1  = B2
          

Therefore the actual character table for C_2v point group will be:

C_2v Character Table

C2v	E	C2z	σxz	σyz	Linear Functions, Rotations	Quadratic
A1	1	1	1	1	Z	X², Y², Z²
A2	1	1	-1	-1	Rz	XY
B1	1	-1	1	-1	X, Ry	XZ
B2	1	-1	-1	1	Y, Rx	YZ

Character Table for C_3v Point Group:

1. For C_3v point group, there are 6 symmetry operations and 3 classes, i.e., Γ₁, Γ₂, Γ₃.
2. The sum of the squares of the dimensions of the symmetry operations = 6.

i.e., l₁² + l₂² + l₃² = h = 6.

This can only be satisfied by, 2 one-dimensional and 1 two-dimensional representations.

C_3v Character Table

C3v	E	2C3	3σv
Γ1	1	a₁	b₁
Γ2	1	a₂	b₂
Γ3	2	a₃	b₃

3. The sum of the dimensions of Γ₁ is also 6.

Therefore, its characters are (1 1 1).

C_3v Character Table

C3v	E	2C3	3σv
Γ1	1	1	1
Γ2	1	a₂	b₂
Γ3	2	a₃	b₃

4. All operations must satisfy the orthogonality condition,

Σ_R Γ_i(R) · Γ_j(R) = 0

i.e., for Γ₁ · Γ₂:

1·1 + 2·a₂·1 + 3·b₂·1 = 0

Let a₂ = 1 and b₂ = −1

Then Γ₁ · Γ₂ = 0

C_3v Character Table

C3v	E	2C3	3σv
Γ1	1	1	1
Γ2	1	1	−1
Γ3	2	a₃	b₃

i.e., For Γ₃ · Γ₂:

2·1 + 2·a₃·1 − 3·b₃·1 = 0

Let a₃ = −1 and b₃ = 0

Then Γ₃ · Γ₂ = 0

C_3v Character Table

C3v	E	2C3	3σv
Γ1	1	1	1
Γ2	1	1	−1
Γ3	2	−1	0

For any character table, there are four areas.

For Area I:

Assign the Mulliken symbols.

C_3v Character Table

C3v	E	2C3	3σv
A1	1	1	1
A2	1	1	−1
E	2	−1	0

For Area III:

In order to assign the Cartesian coordinates, different operations are performed on each of the axes. Here we were finding the symbols X, Y, Z represent coordinates and rotations R_x, R_y and R_z.

Consider,

The characters are 1 1 1 corresponding to A₁.

The characters are 1 −1 1, the character corresponding to C₃ will be −1. Therefore, it will be E.

Similarly, for vector Y, we get 1 −1 1 and this is also E.

Similar arrangements could be made for the rotation axes R_x, R_y, and R_z.

The characters are 1 −1 1. Therefore, it corresponds to E and it will become R_x.

The characters are 1 1 −1. Therefore, it corresponds to A₂ and it will become R_z.

Similarly, for E the characters are 2 −1 0 and it will become R_y.

C_3v Character Table

C3v	E	2C3	3σv	Linear Functions, Rotations
A1	1	1	1	Z
A2	1	1	−1	Rz
E	2	−1	0	(X, Y), (Rx, Ry)

Area IV:

Which represents the squares and binary products.

A₁ = Z = 1 1 1

A₁² = Z² = 1 1 1 = A₁

XY = E = 2 −1 0 = E

XZ = E · A₁ = 2 −1 0 = E

YZ = E · A₁ = 2 −1 0 = E

Therefore the actual character table for C_3v point group will be,

C_3v Character Table

C3v	E	2C3	3σv	Linear Functions, Rotations	Quadratic
A1	1	1	1	Z	Z2
A2	1	1	−1	Rz
E	2	−1	0	(X, Y), (Rx, Ry)	(XY), (XZ), (YZ)

Some Important Character Tables for Molecular Point Groups:

1. Character Table for Non-Axial Point Groups
2. Character Table for C_n Point Groups
3. Character Table for C_nv Point Groups
4. Character Table for C_nh Point Groups
5. Character Table for D_n Point Groups
6. Character Table for D_nh Point Groups
7. Character Table for D_nd Point Groups
8. Character Table for S_n Point Groups
9. Character Tables for Higher Point Groups
10. Character Tables for Linear Point Groups