Elements of symmetry refer to specific geometric features or operations that describe the spatial arrangement of molecules or crystals. These features allow a molecule or crystal to be manipulated in space (rotated, reflected, or inverted) and still retain its original orientation or appearance. Symmetry plays a crucial role in understanding the properties of molecules and crystalline solids, influencing their physical properties, spectroscopic characteristics, and reactivity patterns.
There are several elements of symmetry that are key in the classification of molecules and crystals:
1. Center of Symmetry (Inversion Center)
2. Axis of Symmetry (Rotation Axis)
3. Plane of Symmetry (Mirror Plane)
4. Improper Axis of Symmetry (Rotation-Reflection Axis)
Each of these elements corresponds to a symmetry operation, an action that can be performed on a molecule that leaves the molecule indistinguishable from its original configuration.
1. Center of Symmetry (Inversion Center)
The center of symmetry, also called the inversion center (denoted as \(i\)), is a point in space located in the middle of a molecule. If every atom in the molecule can be mapped onto an equivalent atom by inverting through this central point, the molecule is said to possess a center of symmetry.
Operation:
In an inversion operation, every point on the molecule is reflected through the center of symmetry to the opposite side, at an equal distance from the center.
Examples:
Benzene (C6H6): In benzene, the center of symmetry lies at the center of the ring. Each atom on one side of the ring has an equivalent atom on the opposite side.
Ethylene (C2H4): The center of symmetry is located between the two carbon atoms in the ethylene molecule, where each atom has a mirror-image counterpart on the opposite side of the inversion center.
Applications:
The presence or absence of a center of symmetry helps in determining whether a molecule will show dipole moments or certain types of spectroscopic transitions (e.g., in infrared spectroscopy, molecules without a center of symmetry can be IR active).
 2. Axis of Symmetry (Rotation Axis)
The axis of symmetry (denoted as Cn) refers to an imaginary line about which the molecule can be rotated by a certain angle, and the molecule appears indistinguishable from its original position. This is a key element in understanding the geometry of many molecules, especially those that possess cyclic or repeating structures.
Rotation Axis:
A rotation by (360°/n) around the axis leaves the molecule unchanged.
The number (n) in (Cn) indicates the order of rotation. For example, C2 means a 180° rotation, C3 means a 120° rotation, and so on.
Examples:
Water (H2O): Water has a C2 axis of symmetry. A 180° rotation about this axis (which bisects the H-O-H bond angle) leaves the molecule unchanged.
Methane (CH4): Methane has multiple rotational axes of symmetry, including a C3 axis along each of the four tetrahedral directions. It also has C2 axes along lines passing through pairs of hydrogen atoms.
Benzene (C6H6): Benzene possesses a C6 axis, as a 60° rotation about the axis perpendicular to the plane of the ring leaves the molecule unchanged.
Types of Rotation Axes:
Principal Axis of Symmetry: The highest-order rotation axis in a molecule is called the principal axis of symmetry. It is the axis with the highest value of (n).
Applications:
The presence of rotational axes plays a critical role in group theory and molecular orbital theory, where the symmetry of a molecule helps predict molecular orbitals, energy levels, and reactivity.
3. Plane of Symmetry (Mirror Plane)
A plane of symmetry (denoted as σ) is an imaginary plane that divides a molecule into two halves, such that one half of the molecule is the mirror image of the other half.
Types of Planes of Symmetry:
Horizontal Plane (σh): A plane perpendicular to the principal axis of symmetry.
Vertical Plane (σ_v): A plane that includes the principal axis of symmetry.
Diagonal Plane (σ_d): A vertical plane that bisects the angle between two adjacent rotational axes.
Operation:
If a molecule can be reflected across a plane of symmetry and remains indistinguishable from its original orientation, the molecule has a mirror plane symmetry.
Examples:
Ammonia (NH3): Ammonia has three σ_v planes of symmetry, each passing through the nitrogen atom and one of the hydrogen atoms.
Ethylene (C2H4): Ethylene has a σ_h plane that lies in the plane of the molecule, dividing it into two mirror-image halves.
Water (H2O): Water has a vertical mirror plane (\(σ_v\)) that cuts through the oxygen atom and the bond axis of the two hydrogen atoms.
Applications:
Molecules with mirror planes can often show optical inactivity if the reflection results in superimposability, which is common in meso compounds.
4. Improper Axis of Symmetry (Rotation-Reflection Axis)
The improper axis of symmetry (denoted as Sn) combines two symmetry operations: a rotation followed by a reflection in a plane perpendicular to the rotation axis.
Operation:
The (Sn) operation consists of a rotation by 360°/n about an axis, followed by a reflection through a plane perpendicular to that axis.
Examples:
Tetrafluoromethane (CF4): CF4 has an improper axis of symmetry, specifically \(S_4\), due to its tetrahedral structure.
Applications:
Improper axes of symmetry are important in understanding the chirality of molecules. If a molecule possesses an improper axis, it is achiral. This is essential in stereochemistry, where improper axes help determine if a molecule has the potential to rotate plane-polarized light.
Summary of Elements of Symmetry
| Element of Symmetry | Symbol | Definition | Examples |
| Center of Symmetry | i | A point through which every atom has a corresponding atom on the opposite side | Benzene, Ethylene |
| Axis of Symmetry | Cn​ | An axis around which the molecule can be rotated by 360∘/n360°/n360∘/n and remain unchanged | Water (C2), Benzene (C6​) |
| Plane of Symmetry | σ | A plane that divides the molecule into two mirror-image halves | Water (σv​), Ethylene (σh​) |
| Improper Axis | Sn​ | A combination of a rotation and reflection | CF4 (tetrafluoromethane) |
Applications of Symmetry in Chemistry
1. Molecular Orbital Theory: Symmetry elements help in determining the molecular orbitals in molecules. The symmetry of atomic orbitals defines the allowed combinations for bonding and anti-bonding molecular orbitals.
2. Spectroscopy: Symmetry elements influence the selection rules in spectroscopic techniques, including infrared (IR) and Raman spectroscopy. For example, molecules with a center of symmetry typically show no IR absorption but may be Raman active (mutual exclusion rule).
3. Crystal Structures: Symmetry elements are fundamental in the classification of crystal structures. Crystals are classified into different crystal systems based on their symmetry elements. For instance, the cubic system has the highest degree of symmetry.
4. Stereochemistry: Symmetry is crucial in stereochemistry, particularly in determining whether molecules are chiral or achiral. Molecules with symmetry elements like a plane of symmetry or a center of symmetry are generally achiral, while those without such elements are often chiral.
5. Reactivity: Symmetry influences the reactivity of molecules. Molecules with high symmetry may be less reactive in certain reactions, as symmetry can stabilize molecular orbitals and reduce the likelihood of interaction with reagents.
Conclusion
The study of elements of symmetry is fundamental in understanding molecular structure, physical properties, and chemical behavior. Whether in molecular orbital theory, spectroscopy, crystal structure classification, or stereochemistry, the recognition of symmetry elements provides insights into how molecules and crystals interact with their environment and react in various conditions. Symmetry serves as a powerful tool in predicting and explaining the behaviors of molecules in both theoretical and practical contexts.