Molecular Symmetry and Chirality

Molecular Symmetry and Chirality
Introduction and Overview
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The symmetry of a molecule describes how its different parts relate to one another geometrically. symmetry plays an important role in many areas of chemistry, with effects on:

physical properties: e.g. dipole moment, chirality
spectroscopic properties: transition intensities, geometric equivalence of groups or nuclei
bonding interactions: bonds require overlap of atomic orbitals of correct symmetry

Terms and Definitions

Symmetry Operation: a spatial manipulation performed on a molecule that leaves it in an configuration identical to and superimposable upon the original configuration.

Symmetry Element: an axis, plane, or point about which a symmetry operation is performed.

Point Group: a symbol that identifies all the symmetry elements present in a molecule.

Symmetry Elements and Operations

Element	Operation	Symbol	ANotes
none	identity (no operation)	E	a molecule is always unchanged if no operation is performed, so all molecules possess E E = C₁ i.e. rotation by 360° about any axis is just like doing nothing
proper rotation axis	rotate by 360º/n	C_n	axis with highest value of n is the principal axis C₂ axes perpendicular to the principal C_n are called C₂' axes
mirror plane	reflection	σ	σ parallel to and containing C_n is vertical: σ_v σ parallel to C_n and bisecting two C₂' axes is dihedral: σ_d σ perpendicular to C_n is horizontal: σ_h
inversion centre	inversion	i	all atoms are moved through inversion centre to an equal distance on opposite side i may or may not be an atomic centre
improper rotation axis	rotate by 360º/n, then reflect ^ to axis	S_n	combination rotation/reflection operation odd n value requires both C_n and σ_h (e.g. BF₃ has an S₃ axis, and also has both a C₃ and a σ_h) even n value may or may not have C_n and σ_h (e.g. CH₄ has an S₄ axis, but has neither a C₄ nor a σ perpendicular to the S₄) S₁ = σ, S₂ = i

Effects on Molecular Properties

Chirality:
To be chiral, a molecule must lack both i and σ.*
Molecules belonging to groups C_n (including E) and D_n are chiral.

* This is a small oversimplification. The most precise requirement for chirality is the lack of any S_n element, but because hardly anybody really knows what those look like, there are a series of increasingly precise shortcuts used to Spotting a Chiral Compound. Or, you can just skip to the summary at the end.

Polarity:
To be polar, a molecule must lack all of i, C₂' axes, and σ_h.
Molecules belonging to groups C_s, C_n (incl E) and C_nv (including C_∞v) may be polar.