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Lecture 02
Chemical Applications of Symmetry and Group Theory
Overview
This lecture introduces the fundamental concepts of symmetry elements and operations in chemistry, building upon the previous session's discussion of molecular symmetry. It details five primary symmetry elements: identity (E), proper axis of rotation (Cn), mirror planes (σ), center of symmetry (i), and improper axis of rotation (Sn). The lecture elaborates on identity as the 'do nothing' operation, present in all objects. It then explains proper axes of rotation, defining 'n' as the order of rotation (360°/angle) and introducing notation like C2, C3, C4, and C∞ for linear molecules. Examples like benzene and carbon dioxide illustrate these concepts. The discussion also touches upon molecules having multiple rotational axes and the potential for these axes to coincide or be distinct. Finally, the lecture introduces mirror planes (σ), including horizontal (σh), vertical (σv), and dihedral (σd) types, explaining their relationship to the principal axis of rotation and setting the stage for further exploration in the next session.
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Chapters
- •Recap of previous lecture on molecular symmetry.
- •Introduction to symmetry elements and operations.
- •Five main symmetry elements: Identity (E), Proper axis of rotation (Cn), Mirror planes (σ), Center of symmetry (i), Improper axis of rotation (Sn).
- •Focus on understanding these elements in detail.
- •Identity operation is denoted by E.
- •It is the 'do nothing' operation, leaving the object unchanged.
- •All objects possess at least one symmetry element: the identity element.
- •Results in an identical, not just indistinguishable, structure.
- •A Cn axis allows rotation by 360°/n to achieve an indistinguishable structure.
- •The 'n' in Cn represents the order of rotation.
- •Examples: C2 for 180° rotation, C3 for 120° rotation, C4 for 90° rotation.
- •C1 is equivalent to the identity operation (E).
- •C∞ exists for linear molecules like CO2, where any rotation yields an indistinguishable structure.
- •H2O molecule has a C2 axis bisecting the H-O-H angle.
- •BF3 molecule has a C3 axis perpendicular to the molecular plane.
- •Benzene molecule has a C6 axis passing through its center.
- •Molecules can have multiple rotational axes (e.g., BF3 has C3 and three C2 axes).
- •Rotational axes can coincide (e.g., Benzene's C6, C3, and C2 axes).
- •A mirror plane (σ) operation reflects an object, resulting in an indistinguishable structure.
- •Reflection transforms coordinates (x, y, z) to (-x, -y, -z) relative to the plane.
- •Types of mirror planes:
- •σh: Perpendicular to the principal axis of rotation.
- •σv: Contains the principal axis of rotation.
- •σd: Bisects the dihedral angle between the principal axis and adjacent C2 axes.
- •Summary of identity, rotation, and mirror plane symmetry elements.
- •Brief mention of center of symmetry (i) and improper rotation (Sn) to be covered later.
- •Preview of further discussion on mirror planes and examples in the next lecture.
- •Importance of these concepts for understanding molecular properties.
Key Takeaways
- 1Symmetry operations are movements that leave a molecule indistinguishable from its original state.
- 2The identity operation (E) is the simplest symmetry operation, present in all molecules.
- 3Proper axes of rotation (Cn) involve rotating a molecule by 360°/n.
- 4The order 'n' of a rotational axis is determined by the smallest angle of rotation that yields an indistinguishable structure.
- 5Linear molecules possess a C∞ axis, allowing indistinguishable structures for any rotation angle.
- 6Mirror planes (σ) involve reflection, creating an indistinguishable structure across the plane.
- 7Different types of mirror planes (σh, σv, σd) are defined relative to the principal axis of rotation.
- 8Molecules can possess multiple symmetry elements, including various rotational axes and mirror planes.