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# Symmetry Elements

Symmetry elements of the molecule are geometric entities: an imaginary point, axis or plane in space, which symmetry operations: rotation, reflection or inversion, are performed. , Their recognition leads to the application of symmetry to molecular properties and can also be used to predict or explain many of a molecule’s chemical properties. Symmetry elements and symmetry operations are two fundamental concepts in group theory, which is the mathematical description of symmetry properties that describe the structure, bonding, and spectroscopy of molecules.

Contents

  1. Point of symmetry operations
1.1. Identity, E
1.2. Proper Rotation, Cn
1.3. Reflection, σ
1.4. Inversion, i
1.5. Improper Rotation, Sn
2. Point groups
3. Example: symmetry of benzene


1. Point symmetry operations
Point symmetry of a molecule results when there exists at least one point in space that remains indistinguishable from the original molecule after any symmetry operation is applied. In other words, a rotation, reflection or inversion operations are called symmetry operations if, and only if, the newly arranged molecule is indistinguishable from the original arrangement. There are five kinds of point of symmetry elements that a molecule can possess, thus, there are also five kinds of point of symmetry operations. All symmetry elements in a molecule must share at least one point in common and this point occurs at the center of the molecule.

1.1. Identity, E
Identity operation comes from the German ‘Einheit’ meaning unity. This symmetry element means no change. All molecules have this element.

1.2. Proper Rotation, Cn
Proper rotation operates with respect to an axis called a symmetry axis (also known as n-fold rotational axis). An axis around which a rotation by 360°/n (or 2π/n) results in an identical molecule before and after the rotation. The axis with the highest n is called the principal axis.

In general, a molecule contains nCn operations such that {Cn1, Cn2, Cn3,…, Cnn-1, Cnn} where Cnn = E. For example, if there exists C5 axis then there exists 5C5 (2C51, 2C52, C55) operations:

• C51 = C54 since the operation results indistinguishable molecule in clockwise and counter-clockwise, respectively.
• C52 = C53 since the operation results indistinguishable molecule in clockwise and counter-clockwise, respectively.
• C55 = E

1.3. Reflection, σ
Reflection operates with respect to a plane called a plane of symmetry (also known as a mirror plane). There exist three types of mirror planes:

• σh – a horizontal mirror plane of a molecule is perpendicular to the primary axis of a molecule.
• σv – a vertical mirror plane of a molecule includes the primary axis of a molecule and passes through the bonds (atoms).
• σd – a dihedral (also known as a diagonal mirror plane) of a molecule includes the primary axis of a molecule while bisecting the angle between two C2 axes that are perpendicular to it. Therefore, ơd does not pass through the bonds (atoms).

NOTE: σd is a special case of a σv.

1.4. Inversion, i
Inversion operates with respect to a point called a center of symmetry (also known as an inversion center). It gives the same result to rotating a molecule around C2 axis then reflecting it with respect to a mirror plane that is perpendicular to C2. For example, SF6 in figure 2 has an inversion point at the center S.

1.5. Improper Rotation, Sn
Improper rotation operates with respect to an axis called rotation-reflection axis. In other words, it is a combination operation of a rotation about an axis by 360°/n (or 2π/n) followed by reflection in a plane perpendicular to the rotation axis.

2. Point groups
The complete collection of symmetry elements of a molecule forms the basis of a mathematical group, and the collection of symmetry operations that are interrelated to each other via certain kind of rules is known as a point group:

• Closure – if any two symmetry operations are in the same group then their product, resulting in another operation, will also be in the same group:
                                    If A∈G and B∈G, then (A∩B)∈G

• Associativity – the law of associativity applies to all symmetry operations:
                                          (AB)C=A(BC)

• Identity – there exists an operation that commutes with other operations (identity, E) and leaves them unchanged:
                                      If A∈G and E∈G, then AE=EA=A

• Inverses – for every symmetry operation in the group, there exists an inverse operation that their product results identity:
                       If A∈G, then there exists A-1∈G such that AA-1=A-1A=E


• Molecular Point Group (1.2)
• Matrices (1.3)
• Representations (1.4)

3. Example: symmetry of benzene
Benzene is one of the molecules that possess various symmetry elements and symmetry operations.

4. References
 Pfenning, Brian W. (2015). Principles of Inorganic Chemistry. Hoboken: John Wiley & Sons, Inc.. pp. 195.
 https://www-e.openu.ac.il/symmetry/symmetry-tutorial.html

# Molecular Point Groups

A Point Group describes all the symmetry operations that can be performed on a molecule that result in a conformation indistinguishable from the original. Point groups are used in Group Theory, the mathematical analysis of groups, to determine properties such as a molecule's molecular orbitals.

## Assigning Point Groups

While a point group contains all of the symmetry operations that can be performed on a given molecule, it is not necessary to identify all of these operations to determine the molecule's overall point group. Instead, a molecule's point group can be determined by following a set of steps which analyze the presence (or absence) of particular symmetry elements.

1. Determine if the molecule is of high or low symmetry.
2. If not, find the highest order rotation axis, Cn.
3. Determine if the molecule has any C2 axes perpendicular to the principal Cn axis. If so, then there are n such C2 axes, and the molecule is in the D set of point groups. If not, it is in either the C or S set of point groups.
4. Determine if the molecule has a horizontal mirror plane (σh) perpendicular to the principal Cn axis. If so, the molecule is either in the Cnh or Dnh set of point groups.
5. Determine if the molecule has a vertical mirror plane (σv) containing the principal Cn axis. If so, the molecule is either in the Cnv or Dnd set of point groups. If not, and if the molecule has n perpendicular C2 axes, then it is part of the Dn set of point groups.
6. Determine if there is an improper rotation axis, S2n, collinear with the principal Cn axis. If so, the molecule is in the S2n point group. If not, the molecule is in the Cn point group.

The steps for determining a molecule's overall point group are shown in the included flowchart.

### Example: Finding the point group of benzene (C6H6)

1. Benzene is neither high or low symmetry
2. Highest order rotation axis: C6
3. There are 6 C2 axes perpendicular to the principal axis
4. There is a horizontal mirror plane (σh)

Benzene is in the D6h point group.

## Low Symmetry Point Groups

Low symmetry point groups include the C1, Cs, and Ci groups.

Low Symmetry Groups
Group Description Example
C1 only the identity operation (E) CHFClBr
Cs only the identity operation (E) and one mirror plane C2H2ClBr
Ci only the identity operation (E) and a center of inversion (i) C2H2Cl2Br

## High Symmetry Point Groups

High symmetry point groups include the Td, Oh, Ih, C∞v, and D∞h groups. The table below describes their characteristic symmetry operations. The full set of symmetry operations included in the point group is described in the corresponding character table.

High Symmetry Groups
Group Description Example
C∞v linear molecule with an infinite number of rotation axes and vertical mirror planes (σv) HBr
h linear molecule with an infinite number of rotation axes, vertical mirror planes (σv), perpendicular C2 axes, a horizontal mirror plane (σh), and an inversion center (i) CO2
Td typically have tetrahedral geometry, with 4 C4 axes, 3 C2 axes, 3 S4 axes, and 6 dihedral mirror planes (σd) CH4
Oh typically have octahedral geometry, with 3 C4 axes, 4 C3 axes, and an inversion center (i) as characteristic symmetry operations SF6
Ih typically have an icosahedral structure, with 6 C5 axes as characteristic symmetry operations B12H122-

## D Groups

The D set of point groups are classified as Dnh, Dnd, or Dn, where n refers to the principal axis of rotation. Overall, the D groups are characterized by the presence of n C2 axes perpendicular to the principal Cn axis. Further classification of a molecule in the D groups depends on the presence of horizontal or vertical/dihedral mirror planes.

D Groups
Group Description Example
Dnh n perpendicular C2 axes, and a horizontal mirror plane (σh) benzene, C6H6 is D6h
Dnd n perpendicular C2 axes, and a vertical mirror plane (σv) propadiene, C3H4 is D2d
Dn n perpendicular C2 axes, no mirror planes [Co(en)3]3+ is D3

## C Groups

The C set of point groups are classified as Cnh, Cnv, or Cn, where n refers to the principal axis of rotation. The C set of groups are characterized by the absence of n C2 axes perpendicular to the principal Cn axis. Further classification of a molecule in the C groups depends on the presence of horizontal or vertical/dihedral mirror planes.

C Groups
Group Description Example
Cnh horizontal mirror plane (σh) perpendicular to the principal Cn axis boric acid, H3BO3 is C3h
Cnv vertical mirror plane (σv) containing the principal Cn axis ammonia, NH3 is C3v
Cn no mirror planes P(C6H5)3 is C3

## S Groups

The S set of point groups are classified as S2n, where n refers to the principal axis of rotation. The S set of groups are characterized by the absence of n C2 axes perpendicular to the principal Cn axis, as well as the absence of horizontal and vertical/dihedral mirror planes. However, there is an improper rotation (or a rotation-reflection) axis collinear with the principal Cn axis.

S Groups
Group Description Example
S2n improper rotation (or a rotation-reflection) axis collinear with the principal Cn axis 12-crown-4 is S4

# Matrix (1.3)

A matrix is a rectangular array of quantities or expressions in rows (m) and columns (n) that is treated as a single entity and manipulated according to particular rules. The dimension of a matrix is denoted by m × n. In inorganic chemistry, molecular symmetry can be modeled by mathematics by using group theory. The internal coordinate system of a molecule may be used to generate a set of matrices, known as a representation, that corresponds to particular symmetry operations. Matrix modeling thus allows for symmetry operations performed on the molecule to be represented in an identical fashion mathematically.

## Matrix Operations

The sum of two matrices, A and B, is carried out by adding or subtracting the element of one matrix with the corresponding element of another matrix. These operations may only be performed on matrices of identical dimension.

A + B = $\sum _{i=1}^{m}(\sum _{j=1}^{n}A_{ij}+B_{ij})$ where i refers to a particular row and j to a particular column.

Example:

${\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{pmatrix}}+$ ${\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\end{pmatrix}}=$ ${\begin{pmatrix}{a_{11}+b_{11}}&{a_{12}+b_{12}}&{a_{13}+b_{13}}\\{a_{21}+b_{21}}&{a_{22}+b_{22}}&{a_{23}+b_{23}}\end{pmatrix}}$ ### Scalar Multiplication

Multiplication of a matrix by a scalar, c, multiplies every element within the matrix by the scalar.

cA = c · Ai,j

Example:

c $\cdot$ ${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}c\cdot a_{11}&c\cdot a_{12}\\c\cdot a_{21}&c\cdot a_{22}\end{pmatrix}}$ ### Matrix Multiplication

Matrix multiplication entails computing the dot product of the row of one matrix, A, with the column of another matrix, B. Matrix multiplication is only defined if the number if columns of A, denoted by n, is equal to the number of rows of B, denoted by m. Their product is then the m × n matrix, C. Matrix multiplication entails some mathematical properties. First, it is associative; in other words, (A × B) × C = A × (B × C). Furthermore, matrix multiplication is not commutative; in other words, A × B =/= B × A

Cm×n = Am×c · Bc×n = $\sum _{k=1}^{c}A_{i,k}B_{k,j}$ Example:

${\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{pmatrix}}\times {\begin{pmatrix}b_{11}\\b_{21}\\b_{31}\end{pmatrix}}={\begin{pmatrix}{a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}}\\{a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31}}\end{pmatrix}}$ ### Row Operations

There are three kinds of elementary row operations that are used to transform a matrix:

Type Definition Operation
Row Switching The swapping of one row with that of another row $\mathrm {R} _{i}\leftrightarrow \mathrm {R} _{j}$ Row Addition The addition of a multiple of one row to another row $\mathrm {R} _{i}+k\mathrm {R} _{j}\rightarrow \mathrm {R} _{i}$ Row Multiplication Multiplication of a row by a scalar, c, with c ≠ 0 c$\mathrm {R} _{i}\rightarrow \mathrm {R} _{i}$ ## Square Matrices

Square matrices are matrices where the number of rows and number of columns are equal, resulting in an n × n matrix.

### Identity Matrix

The identity matrix, In, is a diagonal matrix which has all elements along the main diagonal equal to 1 and all other elements equal to 0. Multiplication of another matrix by the identity matrix leaves the first unchanged. Moreover, multiplication with the identity matrix is commutative; in other words, A × I = I × A.

Example:

A·I3 = ${\begin{pmatrix}a&b&c\\d&e&f\end{pmatrix}}\cdot {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}={\begin{pmatrix}a&b&c\\d&e&f\end{pmatrix}}$ ### Trace

Only applicable to square matrices, the trace or character, $\chi$ , of a matrix is the sum of its diagonal entries along the main diagonal.

### Determinant

The determinant of a matrix, denoted det(A), is a real number computed from a square matrix. A non-zero determinant implies matrix invertibility, which further implies that the set of linear equations comprising the matrix has exactly one solution.

For a 2 × 2 matrix, the determinant is computed as follows:

det(A) = ${\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc$ For a 3 × 3 matrix, the determinant is computed as follows:

det(A) = ${\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}$ Higher order determinants may be calculated by using Cramer's Rule.