Finding symmetry elements
A practical workflow, with three molecules you can “practice on mentally.”
A symmetry operation is an action (rotate, reflect, invert, etc.) that moves a molecule into a configuration that is indistinguishable from the original. The corresponding geometric object (axis, plane, point) is called a symmetry element.
Common symmetry elements (what to look for)
- Identity \(E\): “do nothing” (every molecule has this).
- Proper rotation axis \(C_n\): rotation by \(360^\circ/n\) gives an indistinguishable molecule.
- Mirror plane \(\sigma\): reflection through a plane. Common special cases: \(\sigma_h\) (perpendicular to principal axis), \(\sigma_v\) (contains principal axis), \(\sigma_d\) (contains principal axis and bisects angles).
- Center of inversion \(i\): every point \((x,y,z)\) maps to \((-x,-y,-z)\).
- Improper rotation axis \(S_n\): rotate by \(360^\circ/n\) then reflect through a plane perpendicular to that axis (very common in highly symmetric molecules).
A quick workflow (works on almost any molecule)
- Find the best rotation axis (largest \(n\)): that’s the principal axis.
- Look for mirror planes (especially planes that contain the principal axis).
- Check for inversion (\(i\)) through the molecular “center.”
- Decide what else must be present (e.g., extra \(C_2\) axes, \(S_n\) axes) based on the pattern you see.
- Name the point group once you know the set of elements.
Molecular symmetry is commonly described using the Schönflies notation, a systematic labeling scheme that classifies molecules according to the symmetry elements they possess. Each Schönflies symbol is a compact summary of a molecule’s symmetry.
Basic structure of Schönflies symbols
A Schönflies symbol consists of a capital letter that identifies the overall symmetry family, followed by subscripts and modifiers that specify the key symmetry elements.
- \(C\) groups: Molecules with a single principal rotation axis \(C_n\). These are the most common symmetry groups for small molecules.
- \(D\) groups: Molecules with a principal axis \(C_n\) and multiple \(C_2\) axes perpendicular to it. These are typically higher-symmetry molecules.
- \(T\), \(O\), \(I\) groups: Molecules with tetrahedral, octahedral, or icosahedral symmetry (very high symmetry; less common in simple organic molecules).
Subscripts and modifiers
Additional letters describe the presence of mirror planes and inversion:
- \(v\) (vertical): mirror planes that contain the principal axis (\(C_{nv}\), e.g., formaldehyde).
- \(h\) (horizontal): a mirror plane perpendicular to the principal axis (\(C_{nh}\), \(D_{nh}\)).
- \(d\) (dihedral): mirror planes that contain the principal axis and bisect angles between perpendicular \(C_2\) axes (\(D_{nd}\)).
- No modifier: the group contains only rotation axes (e.g., \(C_n\), \(D_n\)).
Flowchart: How to assign a point group
Use the following step-by-step procedure to determine the point group of a molecule. Always start with the highest possible symmetry and work downward.
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Is the molecule linear?
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If yes:
- If it has a center of inversion → \(D_{\infty h}\) (e.g., \(N_2\), \(CO_2\))
- If it does not → \(C_{\infty v}\) (e.g., \(HCl\))
- If no, continue.
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If yes:
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Find the highest-order rotation axis
(\(C_n\)). This is the principal axis.
- If no \(C_n\) axis exists → go to Step 6.
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Are there multiple \(C_2\) axes perpendicular to the principal axis?
- If yes → the molecule belongs to a \(D_n\) family.
- If no → the molecule belongs to a \(C_n\) family.
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Is there a horizontal mirror plane
(\(\sigma_h\), perpendicular to the principal axis)?
- If yes → append \(h\) (\(C_{nh}\) or \(D_{nh}\))
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If no \(\sigma_h\), are there vertical mirror planes?
- If planes contain the principal axis → append \(v\) (\(C_{nv}\))
- If planes bisect angles between perpendicular \(C_2\) axes → append \(d\) (\(D_{nd}\))
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If no rotation axis exists:
- If the molecule has one mirror plane → \(C_s\)
- If the molecule has a center of inversion → \(C_i\)
- If neither → \(C_1\)
In General: rotation axes define the “letter,” mirror planes and inversion define the subscripts.
Special cases
- \(C_1\): only the identity operation (\(E\)); no symmetry beyond “existence.”
- \(C_s\): a single mirror plane and no rotation axis.
- \(C_i\): a center of inversion but no rotation axis.
- \(D_{\infty h}\): linear molecules with a center of inversion (e.g., \(N_2\)).
- \(C_{\infty v}\): linear molecules without inversion (e.g., \(HCl\)).
Big idea: the Schönflies symbol encodes the most important symmetry elements of a molecule in a compact, standardized way. Learning to recognize these patterns makes point-group assignment faster and more reliable.
Example 1: Formaldehyde, H\(_2\)CO (point group \(C_{2v}\))
Formaldehyde is planar, with the carbonyl group defining a natural “molecular plane.” If we place the molecule in the page, the key elements are:
- Principal axis \(C_2\): an axis through C and O that lies in the molecular plane. A 180° rotation swaps the two H atoms and leaves C and O in place.
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Two vertical mirror planes \(\sigma_v\):
- One is the molecular plane itself (reflection keeps all atoms in the plane).
- The other plane is perpendicular to the molecular plane but still contains the C=O axis; it bisects the H–C–H angle and swaps the two H atoms.
- No inversion center (\(i\) absent) and no \(\sigma_h\) relative to the principal axis in the way required for the \(C_{2h}\) family.
Summary: \( \{E, C_2, \sigma_v, \sigma_v' \} \Rightarrow C_{2v} \).
Example 2: trans-1,2-dichloroethene (point group \(C_{2h}\))
trans-dichloroethene is planar. The trans arrangement is what introduces an inversion center. The key elements are:
- Principal axis \(C_2\): an axis perpendicular to the molecular plane passing through the midpoint of the C=C bond. A 180° rotation swaps left ↔ right, mapping each Cl to the other Cl and each H to the other H.
- Horizontal mirror plane \(\sigma_h\): the molecular plane is perpendicular to the \(C_2\) axis.
- Center of inversion \(i\): located at the midpoint of the C=C bond; each atom maps to an identical atom across the center (Cl ↔ Cl, H ↔ H).
- No vertical mirror planes containing the principal axis (the trans substitution breaks them).
Summary: \( \{E, C_2, i, \sigma_h\} \Rightarrow C_{2h} \).
Example 3: Benzene, C\(_6\)H\(_6\) (point group \(D_{6h}\))
Benzene is a classic “high symmetry” molecule: a planar regular hexagon with identical substituents. It has many symmetry elements; here are the most important ones to recognize quickly.
- Principal axis \(C_6\): through the center of the ring, perpendicular to the plane. Rotations by 60° leave it unchanged. (This also implies \(C_3\) and \(C_2\) along the same axis.)
- Six additional \(C_2\) axes in the ring plane: three pass through opposite C atoms; three pass through opposite C–C bonds. This collection of multiple \(C_2\) axes together with the principal axis is characteristic of the \(D\) families.
- Horizontal mirror plane \(\sigma_h\): the molecular plane (the ring plane).
- Vertical mirror planes \(\sigma_v\) and/or \(\sigma_d\): planes that contain the principal axis and slice the ring in symmetry-related ways.
- Center of inversion \(i\): at the center of the ring.
- Improper axes \(S_6\) (and others): high-symmetry planar molecules with a principal \(C_n\) and a \(\sigma_h\) typically have related \(S_n\) operations as well.
Summary: benzene has a \(C_6\) principal axis + many \(C_2\) axes (the “\(D_6\)” part), plus \(\sigma_h\) and \(i\) (the “\(h\)” part), giving \(D_{6h}\).
Big idea: identifying symmetry elements is largely pattern recognition—start with the best rotation axis, then ask “what planes/center must also be present?” The point group name is a compact summary of those symmetry elements.