5.3 Molecular symmetry and point groups
6 min read•july 30, 2024
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 axes and planes. This classification system is key to predicting molecular properties and behavior.
By assigning molecules to point groups, we can better understand their polarity, , 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
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- 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 (, , , etc.)
- Improper rotation axes () 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 () and horizontal ()
- 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 () 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 (, , , etc.)
- The identity operation () 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 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 , , and 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)
- (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 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 (meso compounds)
Spectroscopy and Selection Rules
- Symmetry-based 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)
Key Terms to Review (31)
Benzene: Benzene is a cyclic hydrocarbon composed of six carbon atoms connected by alternating single and double bonds, forming a planar, hexagonal structure. This unique arrangement allows benzene to exhibit exceptional stability and symmetry, making it an important compound in both organic chemistry and molecular physics. Its molecular symmetry leads to its classification within point groups, which helps in understanding its vibrational modes and chemical reactivity.
Burnside's Lemma: Burnside's Lemma is a theorem in group theory that provides a method for counting distinct objects under group actions, particularly useful in the study of molecular symmetry and point groups. It states that the number of distinct configurations (or orbits) can be calculated by averaging the number of points fixed by each group element over all elements of the group. This concept connects to how symmetry operations in molecules can lead to different arrangements without altering their fundamental structure.
C2: C2 refers to a specific type of molecular symmetry that indicates a two-fold rotational axis within a molecule. This means that if a molecule is rotated 180 degrees around this axis, the arrangement of atoms will remain unchanged. This symmetry element is crucial for understanding molecular structures and their properties, as it helps classify molecules into point groups, impacting their physical and chemical behavior.
C2v: The c2v point group is a classification of molecular symmetry that includes two vertical planes of symmetry and a two-fold rotational axis. This symmetry group helps in analyzing the molecular structure, predicting molecular vibrations, and determining the selection rules for spectroscopic transitions. The c2v point group is common in molecules with certain geometric arrangements, such as bent shapes, providing essential insights into their properties and behaviors.
C3: The term c3 refers to a specific type of rotational symmetry present in molecular structures, indicating that a molecule can be rotated by 120 degrees about an axis and appear unchanged. This symmetry is critical in classifying molecules within point groups, helping to identify the geometric arrangement of atoms and the resulting physical properties. Understanding c3 symmetry allows scientists to predict molecular behavior, interactions, and stability based on its symmetrical characteristics.
C4: C4 refers to a specific type of molecular symmetry characterized by a four-fold rotational axis. This means that when a molecule with C4 symmetry is rotated by 90 degrees about this axis, it appears indistinguishable from its original orientation. This symmetry is important for understanding molecular behavior, as it influences properties like vibrational modes and optical activity.
Character table: A character table is a mathematical tool used in group theory that summarizes the symmetry properties of a molecule by listing the characters of its irreducible representations for each symmetry operation in a given point group. It provides valuable information about the behavior of molecular vibrations, electronic states, and other physical properties under symmetry operations, making it essential for understanding molecular symmetry and point groups.
Chirality: Chirality refers to the property of a molecule that makes it non-superimposable on its mirror image, akin to how left and right hands are mirror images but cannot be perfectly aligned. This unique characteristic is crucial in molecular symmetry and point groups, as it influences how molecules interact with polarized light and biological systems, often resulting in different behaviors or functions based on their chirality.
Cs: In molecular symmetry, 'cs' refers to a specific point group that describes the symmetry of certain molecules, particularly those with a single plane of symmetry. This point group is denoted by the letter 'c' indicating a mirror plane, and 's' indicating that the molecule possesses a center of symmetry. The cs point group is significant as it helps categorize molecules based on their symmetrical properties and informs the study of molecular vibrations and electronic transitions.
D3h: The d3h point group describes a specific symmetry characteristic of certain molecules that have a trigonal planar geometry with an additional horizontal mirror plane. This point group is essential in molecular symmetry and helps identify the symmetry elements and their relations, including rotational axes and mirror planes, which are critical for understanding molecular behavior and properties.
Degeneracy: Degeneracy refers to the phenomenon where two or more quantum states share the same energy level. This concept is vital in molecular symmetry and point groups, as it influences how molecules behave under symmetry operations, affecting their vibrational modes and electronic structures. Understanding degeneracy helps explain the stability and reactivity of molecules based on their symmetrical properties.
Dnh: dnh refers to a specific type of molecular symmetry associated with point groups that have a certain arrangement of symmetry elements, including a principal rotation axis and horizontal mirror planes. This term is crucial for understanding how molecules can be classified based on their symmetrical properties, which in turn affects their physical and chemical behavior. The 'd' indicates the presence of a dihedral rotation axis, while 'n' signifies the number of rotational symmetries around that axis.
E: In the context of molecular symmetry and point groups, 'e' represents a specific irreducible representation of symmetry. This notation is commonly used in character tables to describe symmetric properties of molecules, especially diatomic and polyatomic structures. The 'e' representation typically indicates that the corresponding symmetry operations involve two degenerate states, which can be critical in understanding molecular vibrations and electronic states.
Eugene Wigner: Eugene Wigner was a Hungarian-American physicist and mathematician, known for his contributions to the field of quantum mechanics and nuclear physics. His work on molecular symmetry and point groups is foundational in understanding the symmetries that govern molecular structures and behaviors, particularly in how particles interact at a quantum level.
Group theory: Group theory is a branch of mathematics that studies algebraic structures known as groups, which are sets equipped with an operation that combines any two elements to form a third element while satisfying certain conditions. In the context of molecular symmetry, group theory is essential for understanding how molecules behave under symmetry operations and classifying them into point groups based on their geometric symmetries.
I: In the context of molecular symmetry and point groups, 'i' refers to the inversion center or center of inversion, which is a specific type of symmetry element. It represents a point in space such that for any point at coordinates (x, y, z), there is an equivalent point at (-x, -y, -z). This symmetry element plays a crucial role in determining the overall symmetry and classification of molecules within point groups.
Ih: The term 'ih' refers to the identity operation in the context of molecular symmetry and point groups. It signifies that a molecule remains unchanged when it undergoes this operation, meaning it has perfect symmetry. This concept is crucial for understanding how molecules can be classified into different point groups based on their symmetrical properties.
Infrared activity: Infrared activity refers to the ability of a molecule to absorb infrared radiation, which occurs due to the changes in its dipole moment during molecular vibrations. This phenomenon is crucial in understanding how molecules interact with infrared light, particularly in spectroscopy where it is used to identify and analyze molecular structures. Infrared activity is directly linked to molecular symmetry and point groups, as the symmetry of a molecule determines which vibrational modes are active in the infrared region of the electromagnetic spectrum.
Inversion center: An inversion center, also known as a center of inversion or inversion point, is a point in a molecule where, if you were to take any atom and invert it through this point, the resulting structure would be indistinguishable from the original. This symmetry element plays a crucial role in defining molecular symmetry and helps categorize molecules into different point groups based on their symmetrical properties.
Oh: 'oh' refers to a specific type of molecular symmetry associated with the octahedral point group, characterized by high symmetry elements including proper rotations and improper rotations. This symmetry is significant for understanding molecular geometries and how molecules interact with each other. Molecules with 'oh' symmetry often exhibit unique physical and chemical properties due to their symmetrical nature, influencing aspects like vibrational modes and electronic transitions.
Reflection: Reflection is a geometric operation where a figure is flipped over a specific line, called the line of reflection, creating a mirror image. In molecular symmetry, reflection is crucial as it helps in understanding how molecules can be superimposed on themselves through this operation, aiding in the classification of molecular shapes and structures into point groups.
Robert S. Mulliken: Robert S. Mulliken was an American physicist and chemist known for his contributions to molecular orbital theory and the understanding of molecular symmetry. His work laid the foundation for modern theories of bonding and molecular structure, emphasizing the importance of electron distribution in molecules. Mulliken's insights into how electrons occupy molecular orbitals have deep implications in the analysis of ionic bonding and electronegativity, as well as the classification of molecules based on their symmetry properties.
Rotation: Rotation refers to the movement of a molecule or its constituent parts around a specific axis. In the context of molecular symmetry, rotation plays a crucial role in defining how molecules can be superimposed on themselves and how they exhibit symmetry properties that can be categorized into point groups.
Selection rules: Selection rules are criteria that determine the allowed transitions between quantum states in a system, dictating which molecular vibrations or electronic transitions can occur under certain conditions. These rules arise from the conservation laws and the symmetries of the molecular system, influencing the intensity and probability of spectroscopic transitions. Understanding selection rules is essential for interpreting spectroscopic data, as they dictate which energy levels can interact during processes like Raman scattering or infrared absorption.
Sn: In molecular symmetry, 'sn' refers to a specific type of rotational axis known as an improper rotation axis. This concept combines both rotation and reflection, where a molecule is rotated around an axis and then reflected through a plane perpendicular to that axis. The 'n' indicates the number of times the operation can be performed in a full 360-degree rotation, which is essential in determining the symmetry properties of a molecule and its corresponding point group.
Symmetry breaking: Symmetry breaking refers to the phenomenon where a system that is initially symmetric becomes asymmetric due to certain interactions or conditions. This concept plays a vital role in understanding how molecular structures and behaviors can change, influencing various properties such as stability, reactivity, and phase transitions.
Td: The term 'td' refers to a specific type of molecular symmetry in the context of point groups. It is associated with the tetrahedral symmetry and describes molecules that have a highly symmetrical arrangement of atoms in three-dimensional space. The 'td' point group is characterized by four equivalent vertices located at the corners of a tetrahedron, which allows for specific rotational and reflectional symmetry operations.
Tetrahedral complex: A tetrahedral complex is a type of coordination compound where a central metal atom is surrounded by four ligands positioned at the corners of a tetrahedron. This arrangement is significant because it influences the complex's chemical and physical properties, including its symmetry and interactions with light. The tetrahedral geometry plays a crucial role in determining the complex's behavior in various chemical reactions and its ability to form bonds with different types of ligands.
Wigner's Theorem: Wigner's Theorem states that any symmetry transformation that preserves the inner product of quantum states can be represented by a unitary operator or an anti-unitary operator. This principle is crucial for understanding how symmetries are applied to quantum mechanical systems, linking the concepts of symmetry with physical observables and molecular structures.
σh: The term σh refers to a horizontal mirror plane in molecular symmetry that bisects a molecule horizontally, creating two symmetrical halves. This type of symmetry is crucial for identifying and classifying molecules within point groups, helping to determine their physical and chemical properties. Understanding σh is essential for analyzing molecular structures and predicting behavior during interactions and reactions.
σv: In molecular symmetry, σv represents a vertical mirror plane that bisects a molecule and is perpendicular to the principal axis of symmetry. This concept is crucial in understanding how molecules can exhibit certain symmetries, which in turn influences their physical and chemical properties. The presence of σv in a molecule's symmetry can help identify the point group it belongs to, which is essential for predicting vibrational modes and spectroscopic behavior.