Key Concepts of Point Groups

Why This Matters

Point groups are the foundation of symmetry analysis in crystallography—and symmetry is everything when it comes to understanding molecular properties, crystal structures, and spectroscopic behavior. You're being tested on your ability to identify symmetry elements, classify molecules into the correct point group, and predict physical properties like optical activity, dipole moments, and vibrational modes based on that classification. The point group determines what's allowed and what's forbidden at the molecular level.

Don't just memorize the names and their symmetry elements. Focus on understanding what operations each group contains, how groups relate to each other hierarchically, and what molecular or crystallographic examples belong where. When you can look at a structure and systematically identify its symmetry elements, you'll nail both multiple-choice identification questions and FRQs asking you to predict properties from symmetry.

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Low-Symmetry Point Groups

These groups contain minimal symmetry operations and represent the starting point for understanding more complex arrangements. The fewer symmetry elements present, the more "general" the object's shape.

C1 (Identity)

• Contains only the identity operation (E)—every object has at least this symmetry, making C1 the "catch-all" for asymmetric structures
• No symmetry elements besides doing nothing; molecules here are completely asymmetric
• Chiral molecules with no symmetry belong here—critical for understanding optical activity

Ci (Inversion)

• Single inversion center (i) transforms each point $(x, y, z)$ to $(-x, -y, -z)$ through the origin
• Determines centrosymmetry—molecules with only an inversion center have no net dipole moment
• Key for chirality analysis: presence of i alone (without other elements) still allows chirality in some contexts

Cs (Mirror Plane)

• Single mirror plane (σ) reflects all points across a plane of symmetry
• Bilateral symmetry is the everyday term—think of a molecule with one "flat side"
• Molecules are achiral if they possess any mirror plane, making Cs important for stereochemistry

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Cyclic Rotation Groups (Cn)

Cyclic groups contain a single n-fold rotation axis as their primary symmetry element. Rotation by $360°/n$ brings the object into an equivalent position.

C2 (2-fold Rotation)

• 180° rotation (C2 axis) produces two equivalent orientations of the molecule
• Simplest rotational symmetry—common in bent molecules like water (though water has higher symmetry overall)
• Two operations total: E and C2, making this the minimal cyclic group

C3 (3-fold Rotation)

• 120° rotation (C3 axis) generates three equivalent positions around the axis
• Trigonal symmetry appears in molecules like triphenylphosphine when lacking mirror planes
• Operations include E, C3, and C3² (two successive 120° rotations = 240°)

C4 (4-fold Rotation)

• 90° rotation (C4 axis) creates four equivalent positions
• Contains C2 as a subgroup—a 180° rotation is automatically present (C4² = C2)
• Rare in isolated molecules but important in crystal structures with tetragonal symmetry

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Rotation Groups with Mirror Planes (Cnv)

These groups combine an n-fold rotation axis with vertical mirror planes containing that axis. The "v" indicates mirror planes are vertical—parallel to the principal axis.

C2v (2-fold Rotation with Vertical Mirrors)

• One C2 axis plus two σv planes—the mirror planes intersect along the rotation axis
• Water (H2O) is the classic example—bent geometry with the C2 axis bisecting the H-O-H angle
• Four operations total: E, C2, σv, σv'—important for vibrational spectroscopy analysis

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Dihedral Groups (Dn)

Dihedral groups contain an n-fold principal axis plus n perpendicular 2-fold axes. The "D" indicates the presence of these additional rotation axes, creating higher symmetry.

D2 (Three Perpendicular 2-fold Axes)

• Three mutually perpendicular C2 axes—no single "principal" axis, all equivalent
• Orthorhombic crystals and twisted biphenyls exhibit this symmetry
• Operations include E, C2(x), C2(y), C2(z)—four total, all rotations

D3 (3-fold with Three Perpendicular 2-folds)

• One C3 axis plus three C2 axes perpendicular to it, spaced 120° apart
• Trigonal prismatic geometry—common in certain coordination compounds
• Six operations total: E, 2C3, 3C2—forms the basis for D3h and D3d groups

D4 (4-fold with Four Perpendicular 2-folds)

• One C4 axis plus four C2 axes perpendicular to it (two sets: through vertices and through edges)
• Square antiprism geometry and certain metal complexes exhibit this symmetry
• Eight operations total—significantly higher symmetry than D2 or D3

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High-Symmetry Point Groups

These groups describe the most symmetric molecular and crystallographic arrangements, containing multiple rotation axes of different orders plus mirror planes. These are the "Platonic solid" symmetries.

Td (Tetrahedral)

• Four C3 axes through vertices, three C2 axes through edge midpoints, six σd planes—24 total operations
• Methane (CH4) is the textbook example—four equivalent C-H bonds arranged tetrahedrally
• No inversion center—Td molecules can be polar along certain directions and show specific spectroscopic selection rules

Oh (Octahedral)

• Three C4 axes, four C3 axes, six C2 axes, plus inversion and mirror planes—48 total operations
• SF6 and most metal hexacarbonyls M(CO)6 exhibit this symmetry
• Contains inversion center—all Oh molecules are nonpolar and centrosymmetric

Ih (Icosahedral)

• Six C5 axes, ten C3 axes, fifteen C2 axes, plus inversion and mirrors—120 total operations
• C60 (buckminsterfullerene) and viral capsids display this rare, high symmetry
• Highest point group symmetry possible—only appears in structures with 5-fold axes, forbidden in crystals

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Quick Reference Table

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Self-Check Questions

1. Which two low-symmetry point groups (C1, Ci, Cs) allow for molecular chirality, and which one guarantees a molecule is achiral?

2. A molecule has a 3-fold rotation axis but no mirror planes or other rotation axes. What point group does it belong to, and how many symmetry operations does that group contain?

3. Compare and contrast D2 and D4: what symmetry elements do they share, and what distinguishes their principal axes?

4. Why does Td symmetry allow for IR activity in all vibrational modes while Oh symmetry does not? What structural feature accounts for this difference?

5. If you're asked on an FRQ to identify a point group that appears in viral capsid structures but cannot exist in crystalline solids, which group would you name and why is it forbidden in crystals?