Molecular symmetry and point groups are crucial concepts in understanding molecular geometry and bonding. They help us classify molecules based on their symmetry elements, like rotation axes and reflection planes. This classification system is key to predicting molecular properties and behavior.

By assigning molecules to point groups, we can better understand their polarity, chirality, and spectroscopic properties. This knowledge connects directly to molecular geometry and hybridization, helping us predict and explain molecular shapes and bonding patterns.

Molecular Symmetry Elements

Types of Symmetry Elements

• Symmetry elements are geometric features that describe the symmetry of a molecule
• Four main types of symmetry elements exist: rotation axes, reflection planes, inversion centers, and improper rotation axes
• Identifying symmetry elements is crucial for assigning molecules to their appropriate point groups and understanding their properties

Rotation and Improper Rotation Axes

• Rotation axes (Cn) are imaginary lines about which a molecule can be rotated by 360°/n to produce an identical configuration
• The order of the axis (n) represents the number of times the molecule can be rotated to produce an identical configuration (C2, C3, C4, etc.)
• Improper rotation axes (Sn) involve a rotation followed by a reflection through a plane perpendicular to the axis of rotation
• The order of the improper rotation axis (n) is the number of times the molecule can be rotated and reflected to produce an identical configuration (S4, S6, etc.)

Reflection Planes and Inversion Centers

• Reflection planes (σ) are imaginary planes that divide a molecule into two mirror images
• Two types of reflection planes exist: vertical (σv) and horizontal (σh)
• Vertical reflection planes (σv) contain the principal rotation axis and bisect the angle between two C2 axes
• Horizontal reflection planes (σh) are perpendicular to the principal rotation axis
• Inversion centers (i) are points in a molecule where all atoms can be reflected through the center to produce an identical configuration

Point Group Assignment

Point Group Classification System

• Point groups are a classification system for molecules based on their symmetry elements
• Molecules with the same set of symmetry elements belong to the same point group
• The Schoenflies notation is commonly used to denote point groups, with letters and subscripts representing the symmetry elements present in the molecule (Cs, C2v, D3h, etc.)
• The identity operation (E) is present in all point groups and represents the molecule in its original configuration

Common Point Groups and Their Symmetry Elements

• The Cn point groups have an n-fold rotation axis (C2, C3, C4, etc.)
• The Cnv point groups have an n-fold rotation axis and n vertical reflection planes (C2v, C3v, C4v, etc.)
• The Dnh point groups have an n-fold rotation axis, n vertical reflection planes, and a horizontal reflection plane (D2h, D3h, D4h, etc.)
• The Dn point groups have an n-fold rotation axis and n two-fold rotation axes perpendicular to the principal axis (D2, D3, D4, etc.)
• The Td, Oh, and Ih point groups are highly symmetric and represent tetrahedral, octahedral, and icosahedral symmetry, respectively
• Molecules with no symmetry elements other than the identity operation belong to the C1 point group (asymmetric molecules)

Examples of Molecular Point Groups

• Water (H2O) belongs to the C2v point group, with a C2 axis and two vertical reflection planes (σv and σv')
• Ammonia (NH3) belongs to the C3v point group, with a C3 axis and three vertical reflection planes (σv)
• Methane (CH4) belongs to the Td point group, with four C3 axes, three C2 axes, and six reflection planes (σd)
• Benzene (C6H6) belongs to the D6h point group, with a C6 axis, six C2 axes, a horizontal reflection plane (σh), and six vertical reflection planes (σv)

Symmetry Operations and Character Tables

Symmetry Operations

• Symmetry operations are the specific actions (rotations, reflections, inversions, and improper rotations) that can be performed on a molecule to produce an identical configuration
• Each point group has a unique set of symmetry operations that define its symmetry
• The number of symmetry operations in a point group is equal to the order of the point group
• Examples of symmetry operations include the identity operation (E), rotation about an axis (Cn), reflection through a plane (σ), inversion through a center (i), and improper rotation (Sn)

Character Tables and Irreducible Representations

• Character tables are matrices that provide information about the symmetry operations and irreducible representations of a point group
• The character of a symmetry operation is the trace (sum of diagonal elements) of the matrix representation of that operation
• The character indicates how the operation transforms the basis functions of the irreducible representation
• Irreducible representations (Γ) are the smallest sets of basis functions that transform according to the symmetry operations of the point group
• Irreducible representations are labeled as A, B, E, and T, with subscripts and superscripts denoting additional properties (A1, A2, B1, B2, E, T1, T2, etc.)

Examples of Character Tables

• The C2v point group, common in molecules like water and hydrogen peroxide, has four symmetry operations: E, C2, σv(xz), and σv'(yz)
• The C2v character table consists of four irreducible representations: A1, A2, B1, and B2
• The D3h point group, found in molecules like boron trifluoride and the carbonate ion, has 12 symmetry operations: E, 2C3, 3C2, σh, 2S3, 3σv
• The D3h character table has six irreducible representations: A1', A2', E', A1", A2", and E"

Symmetry and Molecular Properties

Polarity and Symmetry

• Molecular symmetry plays a crucial role in determining the polarity of molecules
• Polarity refers to the uneven distribution of electron density in a molecule, resulting in a net dipole moment
• Molecules with high symmetry, such as those belonging to the Td or Oh point groups, are typically non-polar due to the cancellation of individual bond dipole moments (methane, sulfur hexafluoride)
• Molecules with lower symmetry, such as those in the C2v or C3v point groups, may be polar if the individual bond dipole moments do not cancel out, resulting in a net dipole moment (water, ammonia)

Chirality and Symmetry

• Chirality is the property of a molecule being non-superimposable on its mirror image
• Chiral molecules lack an improper rotation axis (Sn) and are optically active, meaning they rotate plane-polarized light
• Molecules with an improper rotation axis (Sn) or a reflection plane (σ) are achiral and optically inactive (molecules belonging to the Cs, Ci, Dnd, and Dnh point groups)
• The presence of a chiral center (usually a carbon atom with four different substituents) is a common cause of chirality in organic molecules (amino acids, sugars)
• Some molecules with chiral centers may still be achiral if they possess additional symmetry elements, such as a reflection plane or an inversion center (meso compounds)

Spectroscopy and Selection Rules

• Symmetry-based selection rules can be used to predict the allowed transitions in vibrational and electronic spectroscopy
• The allowed transitions depend on the symmetry of the initial and final states and the symmetry of the transition moment operator
• For a transition to be allowed, the direct product of the irreducible representations of the initial state, the transition moment operator, and the final state must contain the totally symmetric irreducible representation (A1 or A1')
• Forbidden transitions, which do not satisfy the selection rules, may still occur with low intensity due to vibronic coupling or other perturbations (n → π transitions in carbonyl compounds)