16.02.04 · inorgchem / symmetry-group-theory

Molecular symmetry and point groups: Schoenflies notation, character tables, and group theory in chemistry

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Anchor (Master): Schoenflies 1891 Krystallsysteme und Krystallstructur; Burnside 1897 Theory of Groups of Finite Order; Wigner 1931 Gruppentheorie; Mulliken 1933 J. Chem. Phys. 1:492; Wilson-Decius-Cross 1955 Molecular Vibrations; Cotton 1971 Chemical Applications of Group Theory

Intuition Beginner

Molecules have shapes, and shapes have symmetries. Water (H $_{2}$ O) is bent: it has one mirror plane bisecting the H-O-H angle and a two-fold rotational axis running through the oxygen. Methane (CH $_{4}$ ) is tetrahedral, with many more symmetry elements: four three-fold axes through each carbon-hydrogen bond, three two-fold axes between them, and six mirror planes. Every molecule belongs to exactly one of about forty categories called point groups, and that point group fixes which physical properties the molecule can have.

Arthur Schoenflies in 1891 invented the notation chemists still use to label these point groups: $C_{2 v}$ for water, $T_{d}$ for methane, $O_{h}$ for sulfur hexafluoride, $D_{6 h}$ for benzene. The name encodes which rotations, mirror planes, and inversions leave the molecule unchanged. Once a molecule's point group is known, its dipole moment, infrared spectrum, and optical activity follow without solving the Schrodinger equation for a single electron.

The bridge from shapes to spectra runs through abstract algebra. The set of symmetry operations that leave a molecule unchanged forms a group, and every finite group carries a tabulated invariant called its character table. Robert Mulliken's 1933 symbols label the columns. Group theory became the chemist's everyday tool because the character table alone predicts what spectra a molecule should show, before any electronic-structure calculation is run. That is why molecular symmetry is the foundation of inorganic, physical, and spectroscopic chemistry.

Visual Beginner

Picture the molecular-symmetry classification tree as a decision flow. First locate the highest-order rotational axis $C_{n}$ (the principal axis). If there is just one axis and nothing else, the point group is $C_{n}$ . Add perpendicular mirror planes or extra rotation axes, and the molecule climbs the tree into $C_{n v}$ , $D_{n}$ , $D_{nh}$ , $D_{n d}$ , all the way up to the high-symmetry groups $T_{d}$ (tetrahedral), $O_{h}$ (octahedral), and $I_{h}$ (icosahedral). Linear molecules form two special groups: $C_{\infty v}$ for asymmetric linear molecules like HCl, and $D_{\infty h}$ for symmetric linear molecules like CO $_{2}$ .

The classification is exhaustive: every molecule, however complicated, lands on exactly one leaf of this tree. Once the leaf is fixed, the character table for that point group — printed in every inorganic chemistry textbook — predicts which molecular vibrations absorb infrared light, which scatter Raman light, and whether the molecule can rotate the plane of polarised light.

Worked example Beginner

Predicting the methane infrared spectrum.

Methane, CH $_{4}$ , has four carbon-hydrogen bonds arranged tetrahedrally around the central carbon. Its point group is $T_{d}$ (full tetrahedral). We will count its vibrational modes and decide which absorb infrared light.

Step 1. Count atoms and vibrational modes. Methane has five atoms (one carbon plus four hydrogens). A molecule with $N$ atoms has $3 N - 6$ vibrational modes (subtracting three translations and three rotations). For methane: $3 \times 5 - 6 = 9$ vibrational modes.

Step 2. Decompose the nine modes by symmetry. The character table of $T_{d}$ lists five symmetry species: $A_{1}$ , $A_{2}$ , $E$ , $T_{1}$ , $T_{2}$ . Carrying out the projection (the working tool derived at Intermediate tier) gives one mode of type $A_{1}$ (the symmetric C-H stretch), one of type $E$ (a doubly-degenerate bend, counting for two modes), and two of type $T_{2}$ (each triply degenerate, counting for three modes each). Adding these: $1 + 2 + 3 + 3 = 9$ modes, matching Step 1.

Step 3. Decide infrared activity. A vibrational mode is infrared-active only if its symmetry species matches one of the molecular translations $x$ , $y$ , or $z$ . In $T_{d}$ the translations transform as $T_{2}$ . So only the two $T_{2}$ modes absorb infrared: $ν_{3}$ (asymmetric stretch) and $ν_{4}$ (bend). The $A_{1}$ symmetric stretch and the $E$ bend are Raman-active only.

What this tells us: the methane infrared spectrum shows two absorption bands, not the nine a naive count would suggest. Experiments confirm exactly this — $ν_{3}$ near 3019 inverse centimetres and $ν_{4}$ near 1306 inverse centimetres. Group theory predicted the count before quantum mechanics could compute a single frequency.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which molecule has the point group $T_{d}$ (full tetrahedral symmetry)?

A. Water, H $_{2}$ O B. Ammonia, NH $_{3}$ C. Methane, CH $_{4}$ D. Carbon dioxide, CO $_{2}$

Hint

$T_{d}$ requires four equivalent substituents at the corners of a tetrahedron around a central atom. Think about the molecular geometry of each option.

Answer

C. Methane, CH $_{4}$ , has four identical hydrogens at the vertices of a tetrahedron around the carbon. Its full symmetry group is $T_{d}$ (24 symmetry operations: identity, eight $C_{3}$ rotations along each C-H bond, three $C_{2}$ axes, six $S_{4}$ improper rotations, and six $σ_{d}$ mirror planes). Water ( $C_{2 v}$ ) is bent, ammonia ( $C_{3 v}$ ) is trigonal pyramidal, and carbon dioxide ( $D_{\infty h}$ ) is linear and centrosymmetric. Feedback-wrong: a common confusion is to assume any molecule with four bonds is tetrahedral, but only molecules with four equivalent bonds and all tetrahedral symmetry elements present qualify.

Exercise (easy, true-false).

True or false: a molecule with a centre of inversion (a point through which every atom has an identical partner on the opposite side) can have a permanent dipole moment.

Hint

Inversion reverses the sign of every Cartesian coordinate. What does it do to a dipole vector?

Answer

False. A centre of inversion $i$ sends every position $(x, y, z)$ to $(- x, - y, - z)$ , so it reverses any dipole vector. If the molecule is unchanged by inversion, the dipole must equal its own reverse, which forces the dipole to be zero. Therefore any molecule whose point group contains $i$ (such as CO $_{2}$ in $D_{\infty h}$ , benzene in $D_{6 h}$ , ethylene in $D_{2 h}$ , SF $_{6}$ in $O_{h}$ ) has no permanent dipole. Only molecules in the point groups $C_{n}$ , $C_{n v}$ , $C_{s}$ , $C_{\infty v}$ can carry permanent dipoles. This is why water ( $C_{2 v}$ , no inversion) is polar but carbon dioxide ( $D_{\infty h}$ , with inversion) is not.

Formal definition Intermediate+

Let a molecule be a finite set of atoms with fixed equilibrium positions in $R^{3}$ . A symmetry operation is an orthogonal transformation $R \in O (3)$ that maps the molecule onto itself: it permutes identical atoms and leaves the molecular geometry indistinguishable from the starting configuration.

Definition (symmetry operations). Five types suffice for molecular point groups:

$E$ : the identity.
$C_{n}$ : a proper rotation by $2 π / n$ radians about an axis.
$σ$ : a reflection in a plane. By convention $σ_{v}$ contains the principal axis, $σ_{h}$ is perpendicular to it, and $σ_{d}$ is a vertical plane bisecting two $C_{2}$ axes.
$i$ : inversion through a point (the centre).
$S_{n}$ : an improper rotation, equal to $C_{n}$ followed by $σ_{h}$ perpendicular to the rotation axis.

Definition (point group). The set of all symmetry operations of a molecule, with composition as the group operation, forms a finite subgroup of $O (3)$ called the molecule's point group. The name records that every operation leaves at least one point — the centre of mass — fixed.

Definition (Schoenflies point-group labels). The principal axis is the highest-order $C_{n}$ rotation axis. Then:

$C_{n}$ , $C_{n v}$ , $C_{nh}$ : one principal axis, possibly with vertical mirrors or a horizontal mirror.
$S_{2 n}$ : only an improper rotation axis.
$D_{n}$ , $D_{nh}$ , $D_{n d}$ : principal axis plus $n$ perpendicular $C_{2}$ axes, possibly with mirrors.
$T$ , $T_{d}$ , $T_{h}$ : tetrahedral geometry.
$O$ , $O_{h}$ : octahedral (and cubic) geometry.
$I$ , $I_{h}$ : icosahedral geometry.
$C_{\infty v}$ , $D_{\infty h}$ : linear molecules. HCl is $C_{\infty v}$ ; CO $_{2}$ is $D_{\infty h}$ .

Definition (character table). A finite group $G$ of order $h$ partitions its elements into $k$ conjugacy classes $C_{1}, \dots, C_{k}$ . An irreducible representation (irrep) $Γ_{i}$ assigns to each group element a unitary matrix; its character $χ_{i} (g)$ is the trace of that matrix. The character table is the $k \times k$ array whose $(i, j)$ entry is the character of irrep $i$ on a representative of class $C_{j}$ , conventionally tabulated with class sizes $n_{j}$ in the column header.

Mulliken symbols (Mulliken 1933). Each irrep receives a one-letter code with subscripts:

$A$ (symmetric under the principal $C_{n}$ ) or $B$ (antisymmetric): non-degenerate, dimension 1.
$E$ : doubly degenerate, dimension 2.
$T$ (sometimes $F$ ): triply degenerate, dimension 3.
Subscripts $1, 2$ : symmetric or antisymmetric under a $C_{2}$ perpendicular to the principal axis, or under a vertical mirror.
$g$ (gerade, even) or $u$ (ungerade, odd): symmetric or antisymmetric under inversion $i$ .
Single prime $^{'}$ or double prime $^{''}$ : symmetric or antisymmetric under $σ_{h}$ .

Counterexamples to common slips

"All molecules have a centre of symmetry." No. Water ( $C_{2 v}$ ), ammonia ( $C_{3 v}$ ), methane ( $T_{d}$ ) all lack inversion centres. Only molecules with pairs of identical atoms on opposite sides of a centre — CO $_{2}$ , benzene, ethylene, SF $_{6}$ — possess $i$ . The presence of $i$ is a special property, not a default.
" $C_{n}$ and $C_{n v}$ are similar groups." They are not. $C_{n}$ has only the rotations ${E, C_{n}, C_{n}^{2}, \dots}$ of order $n$ . $C_{n v}$ adds $n$ vertical mirror planes, doubling the order to $2 n$ . Water is $C_{2 v}$ (order 4), not $C_{2}$ (order 2). The character tables differ, and so do the IR selection rules.
"Tetrahedral = $T_{d}$ ." Only if every tetrahedral symmetry element is present. A regular tetrahedral molecule like CH $_{4}$ is $T_{d}$ . A distorted tetrahedron with one longer bond (e.g. CH $_{3}$ Cl) drops to $C_{3 v}$ . A seesaw molecule like SF $_{4}$ (a lone pair in the equatorial plane) is $C_{2 v}$ .
"Character tables are too abstract to predict spectra." They are not. Given a molecule's point group and character table, the IR/Raman activity of every vibrational mode is fixed by inspection of the irrep labels. No electronic wavefunction calculation is required for the activity decision; only the frequencies need quantum chemistry.
"Optical activity requires chiral atoms." No. A molecule is chiral (and optically active) if and only if its point group lacks any improper rotation $S_{n}$ (which includes $σ = S_{1}$ and $i = S_{2}$ as special cases). The chiral point groups are $C_{n}$ , $D_{n}$ , $T$ , $O$ , $I$ . The octahedral complex $[Co (en)_{3}]^{3 +}$ has no chiral carbon atom yet is resolved into left- and right-handed enantiomers because its point group is $D_{3}$ .
"Schoenflies and Hermann-Mauguin are interchangeable." They label different objects. Schoenflies labels molecular point groups (and crystallographic point groups as a subset). Hermann-Mauguin labels crystallographic space groups for periodic crystals. The two conventions coexist because chemists care about isolated molecules and crystallographers care about infinite periodic lattices.

Key theorem: the great orthogonality of characters Intermediate+

Theorem (Burnside 1897; great orthogonality of characters). Let $G$ be a finite group of order $h$ with inequivalent irreducible representations $Γ_{1}, \dots, Γ_{k}$ over $C$ . Denote by $χ_{i} (g)$ the character of $Γ_{i}$ at $g \in G$ , and by $\overline{χ_{j} (g)}$ its complex conjugate. Then

g \in G \sum χ_{i} (g) \overline{χ_{j} (g)} = h δ_{ij} .

Equivalently, the character row vectors, weighted by class sizes, are pairwise orthogonal and each has squared norm $h$ .

Proof. For any $d_{j} \times d_{i}$ complex matrix $M$ , define the group average

\overset{ˉ}{M} := \frac{1}{h} g \in G \sum Γ_{j} (g) M Γ_{i} (g)^{- 1} .

The matrix $\overset{ˉ}{M}$ intertwines $Γ_{i}$ and $Γ_{j}$ : for every $g^{'} \in G$ ,

Γ_{j} (g^{'}) \overset{ˉ}{M} = \frac{1}{h} g \sum Γ_{j} (g^{'} g) M Γ_{i} (g)^{- 1} = \frac{1}{h} g^{''} \sum Γ_{j} (g^{''}) M Γ_{i} (g^{''})^{- 1} Γ_{i} (g^{'}) = \overset{ˉ}{M} Γ_{i} (g^{'}),

using the substitution $g^{''} = g^{'} g$ and re-indexing the sum. Schur's lemma then forces $\overset{ˉ}{M} = 0$ if $i \neq = j$ , and $\overset{ˉ}{M} = λ (M) I_{d_{i}}$ if $i = j$ .

Take $M = E_{β α}$ (the elementary matrix with 1 in position $(β, α)$ ). For $i = j$ , taking the trace on both sides of $\overset{ˉ}{M} = λ I_{d_{i}}$ gives

λ d_{i} = tr (\overset{ˉ}{M}) = \frac{1}{h} g \sum tr (Γ_{i} (g) E_{β α} Γ_{i} (g)^{- 1}) = tr (E_{β α}) = δ_{α β},

so $\overset{ˉ}{M} = (δ_{α β} / d_{i}) I_{d_{i}}$ . Reading off the $(γ, γ)$ entry and using unitarity $Γ_{i} (g)^{- 1} = Γ_{i} (g)^{†}$ ,

\frac{δ _{α β}}{d _{i}} = \frac{1}{h} g \sum [Γ_{i} (g)]_{γ β} \overline{[Γ_{i} (g)]_{γ α}} .

Summing over $α = β$ and over $γ$ from $1$ to $d_{i}$ yields $(1/ h) \sum_{g} ∣ χ_{i} (g) ∣^{2} = 1$ , i.e. $\sum_{g} ∣ χ_{i} (g) ∣^{2} = h$ .

For $i \neq = j$ , $\overset{ˉ}{M} = 0$ gives $(1/ h) \sum_{g} \sum_{γ} [Γ_{j} (g)]_{γ β} \overline{[Γ_{i} (g)]_{γ α}} = 0$ . Setting $α = β$ and summing over $γ$ yields $(1/ h) \sum_{g} χ_{j} (g) \overline{χ_{i} (g)} = 0$ . Combining both cases gives $\sum_{g} χ_{i} (g) \overline{χ_{j} (g)} = h δ_{ij}$ . $□$

Corollary (reduction formula). Let $Γ$ be a reducible representation of $G$ with character $χ_{Γ} (g)$ , decomposing as $Γ ≅ ⨁_{i} a_{i} Γ_{i}$ . The multiplicity $a_{i}$ of $Γ_{i}$ in $Γ$ is

a_{i} = \frac{1}{h} g \in G \sum χ_{Γ} (g) \overline{χ_{i} (g)} = \frac{1}{h} k \sum n_{k} χ_{Γ} (C_{k}) \overline{χ_{i} (C_{k})},

where the second form sums over classes $C_{k}$ of size $n_{k}$ .

Proof of corollary. By linearity of the trace, $χ_{Γ} (g) = \sum_{i} a_{i} χ_{i} (g)$ . Take the inner product with $χ_{j}$ :

\frac{1}{h} g \sum χ_{Γ} (g) \overline{χ_{j} (g)} = \frac{1}{h} i \sum a_{i} g \sum χ_{i} (g) \overline{χ_{j} (g)} = \frac{1}{h} i \sum a_{i} \cdot h δ_{ij} = a_{j} .

The class-sum form follows because characters are constant on conjugacy classes, so summing over the $n_{k}$ elements of class $C_{k}$ multiplies the contribution by $n_{k}$ . $□$

Bridge. The great orthogonality of characters builds toward 16.02.02 pending character tables and reducible representations, where the reduction formula above becomes the chemist's working tool for splitting any molecular displacement into symmetry-adapted components, and appears again in 16.02.03 pending projection operators and SALCs as the algebraic engine that constructs symmetry-adapted linear combinations of orbitals. The foundational reason orthogonality underlies all of molecular spectroscopy is that characters measure the trace of symmetry operations on the molecule's displacement space, and this is exactly the invariant that decides how a vibrational mode, electronic orbital, or rotational wavefunction transforms under the point group. The bridge is between Burnside's 1897 abstract orthogonality theorem and the experimental IR/Raman spectrometer, and the pattern recurs in 16.04.02 crystal-field stabilisation, where the same orthogonality splits the five d-orbitals in $O_{h}$ symmetry into the $e_{g} + t_{2 g}$ subshells that anchor ligand-field theory.

Exercises Intermediate+

Exercise 2 (easy, multiple choice).

Which point group has a principal axis $C_{3}$ and three vertical mirror planes $σ_{v}$ but no horizontal mirror plane and no perpendicular $C_{2}$ axes?

A. $D_{3 h}$ B. $C_{3 v}$ C. $D_{3 d}$ D. $C_{3 h}$

Hint

The presence of $σ_{v}$ but not $σ_{h}$ rules out $C_{3 h}$ , $D_{3 h}$ , and $D_{3 d}$ . The "D" label would require perpendicular $C_{2}$ axes.

Answer

B. The group $C_{3 v}$ has a single $C_{3}$ principal axis, three $σ_{v}$ mirror planes containing the axis, and no $σ_{h}$ and no perpendicular $C_{2}$ axes. Its order is 6 (classes $E, 2 C_{3}, 3 σ_{v}$ ). Ammonia NH $_{3}$ and chloroform CHCl $_{3}$ are the canonical examples. Option A ( $D_{3 h}$ ) requires both $σ_{h}$ and perpendicular $C_{2}$ axes; option C ( $D_{3 d}$ ) requires perpendicular $C_{2}$ axes and an inversion centre; option D ( $C_{3 h}$ ) requires a $σ_{h}$ but no $σ_{v}$ .

Exercise 4 (medium, symbolic).

The $C_{3 v}$ character table is:

$C_{3 v}$	$E$	$2 C_{3}$	$3 σ_{v}$
$A_{1}$	1	1	1
$A_{2}$	1	1	$- 1$
$E$	2	$- 1$	0

Verify the great orthogonality theorem for the pair of irreps $(A_{1}, E)$ by computing $\sum_{k} n_{k} χ_{A_{1}} (C_{k}) \overline{χ_{E} (C_{k})}$ (where $n_{k}$ is the size of class $C_{k}$ ). Report the value and confirm it equals what orthogonality predicts.

Hint

Use the class sizes $n_{k} = (1, 2, 3)$ for classes $(E, C_{3}, σ_{v})$ . All characters here are real, so the complex conjugate does nothing.

Answer

0, confirming orthogonality. The sum is $1 \cdot 1 \cdot 2 + 2 \cdot 1 \cdot (- 1) + 3 \cdot 1 \cdot 0 = 2 - 2 + 0 = 0$ . The great orthogonality theorem predicts $\sum_{k} n_{k} χ_{i} (C_{k}) \overline{χ_{j} (C_{k})} = h δ_{ij}$ , so for distinct irreps ( $i \neq = j$ ) the sum must vanish. It does. As a cross-check, taking the pair $(A_{1}, A_{1})$ gives $1 \cdot 1 \cdot 1 + 2 \cdot 1 \cdot 1 + 3 \cdot 1 \cdot 1 = 1 + 2 + 3 = 6 = h$ , the order of $C_{3 v}$ .

Exercise 5 (medium, short-answer).

Bromine pentafluoride, BrF $_{5}$ , is square pyramidal: the bromine sits slightly above the plane of four equivalent fluorines, with a fifth (axial) fluorine directly below. Identify its point group, list its symmetry operations, and state its order $h$ .

Hint

There is a four-fold rotational axis running through Br and the axial F. There are vertical mirror planes: two contain opposite equatorial F-Br bonds, and two bisect the angles between them. There is no horizontal mirror plane (the axial F breaks the top-bottom symmetry).

Answer

The point group is $C_{4 v}$ , order $h = 8$ . Operations: $E$ ; rotations $C_{4}$ and $C_{4}^{3}$ (by $\pm π /2$ about the Br-axial-F axis); $C_{2} = C_{4}^{2}$ (rotation by $π$ ); two $σ_{v}$ planes each containing the principal axis and one pair of opposite equatorial F atoms; two $σ_{d}$ planes each containing the principal axis and bisecting two equatorial F-Br bonds. The class structure is $E, 2 C_{4}, C_{2}, 2 σ_{v}, 2 σ_{d}$ . The molecule has a permanent dipole along the principal axis (no inversion centre, no $σ_{h}$ ), consistent with the rule that only groups $C_{n}$ , $C_{n v}$ , $C_{s}$ , $C_{\infty v}$ admit dipoles.

Exercise 6 (medium, symbolic).

The three Cartesian displacements of each atom in water ( $C_{2 v}$ , 3 atoms) generate a reducible representation $Γ_{3 N}$ with characters $(9, - 1, 3, 1)$ over the classes $(E, C_{2}, σ_{v} (x z), σ_{v}^{'} (y z))$ . The $C_{2 v}$ character table is:

$C_{2 v}$	$E$	$C_{2}$	$σ_{v} (x z)$	$σ_{v}^{'} (y z)$
$A_{1}$	1	1	1	1
$A_{2}$	1	1	$- 1$	$- 1$
$B_{1}$	1	$- 1$	1	$- 1$
$B_{2}$	1	$- 1$	$- 1$	1

Apply the reduction formula $a_{i} = (1/ h) \sum_{k} n_{k} χ_{Γ} (C_{k}) χ_{i} (C_{k})$ with $h = 4$ to decompose $Γ_{3 N}$ . Give the irrep decomposition.

Hint

For each irrep, multiply class-by-class the $Γ_{3 N}$ character by the irrep character, sum, and divide by $h = 4$ . All $n_{k} = 1$ in $C_{2 v}$ since each operation is its own class.

Answer

$Γ_{3 N} = 3 A_{1} + A_{2} + 3 B_{1} + 2 B_{2}$ . The four coefficients are:

$a (A_{1}) = (1/4) (9 \cdot 1 + (- 1) \cdot 1 + 3 \cdot 1 + 1 \cdot 1) = 12/4 = 3$
$a (A_{2}) = (1/4) (9 \cdot 1 + (- 1) \cdot 1 + 3 \cdot (- 1) + 1 \cdot (- 1)) = 4/4 = 1$
$a (B_{1}) = (1/4) (9 \cdot 1 + (- 1) \cdot (- 1) + 3 \cdot 1 + 1 \cdot (- 1)) = 12/4 = 3$
$a (B_{2}) = (1/4) (9 \cdot 1 + (- 1) \cdot (- 1) + 3 \cdot (- 1) + 1 \cdot 1) = 8/4 = 2$

Self-consistency: $\sum_{i} a_{i} d_{i} = 3 \cdot 1 + 1 \cdot 1 + 3 \cdot 1 + 2 \cdot 1 = 9 = 3 N$ , matching the dimension of $Γ_{3 N}$ .

Exercise 7 (hard, symbolic).

Subtract the molecular translations ( $A_{1} + B_{1} + B_{2}$ in $C_{2 v}$ ; the translation vector transforms as $x$ for $B_{1}$ , $y$ for $B_{2}$ , $z$ for $A_{1}$ ) and the molecular rotations ( $A_{2} + B_{1} + B_{2}$ ; the axial vector transforms as $R_{z}$ for $A_{2}$ , $R_{y}$ for $B_{1}$ , $R_{x}$ for $B_{2}$ ) from $Γ_{3 N} = 3 A_{1} + A_{2} + 3 B_{1} + 2 B_{2}$ to obtain the vibrational representation $Γ_{vib}$ of water. Then state how many IR-active and how many Raman-active bands the water spectrum should show.

Hint

A vibrational mode is IR-active if its irrep transforms as $x$ , $y$ , or $z$ , i.e. matches one of $A_{1}, B_{1}, B_{2}$ . It is Raman-active if its irrep transforms as a quadratic form ( $x^{2}, y^{2}, z^{2}, x y, x z, y z$ ), which in $C_{2 v}$ includes all four irreps.

Answer

$Γ_{vib} = (3 A_{1} + A_{2} + 3 B_{1} + 2 B_{2}) - (A_{1} + B_{1} + B_{2}) - (A_{2} + B_{1} + B_{2}) = 2 A_{1} + B_{2}$ . Three vibrational modes total, consistent with $3 N - 6 = 3$ for $N = 3$ .

IR-active: both $A_{1}$ (transforms as $z$ ) and $B_{2}$ (transforms as $y$ ) match a Cartesian component, so all three modes are IR-active. Expected: 3 IR bands.
Raman-active: $A_{1}$ transforms as the quadratic forms $x^{2}, y^{2}, z^{2}$ and $B_{2}$ as $y z$ , so all three modes are Raman-active. Expected: 3 Raman bands.

Experiment matches: water vapour shows three IR fundamentals at 3657 cm $^{- 1}$ (symmetric O-H stretch, $A_{1}$ ), 1595 cm $^{- 1}$ (H-O-H bend, $A_{1}$ ), and 3756 cm $^{- 1}$ (asymmetric O-H stretch, $B_{2}$ ). The full overlap of IR and Raman bands is consistent with $C_{2 v}$ lacking an inversion centre (no mutual exclusion rule).

Exercise 8 (hard, short-answer).

The $T_{d}$ character table is:

$T_{d}$	$E$	$8 C_{3}$	$3 C_{2}$	$6 S_{4}$	$6 σ_{d}$
$A_{1}$	1	1	1	1	1
$A_{2}$	1	1	1	$- 1$	$- 1$
$E$	2	$- 1$	2	0	0
$T_{1}$	3	0	$- 1$	1	$- 1$
$T_{2}$	3	0	$- 1$	$- 1$	1

Methane (CH $_{4}$ , 5 atoms) generates $Γ_{3 N}$ with characters $(15, 0, - 1, - 1, 3)$ over the classes $(E, 8 C_{3}, 3 C_{2}, 6 S_{4}, 6 σ_{d})$ . Decompose $Γ_{3 N}$ , subtract the translations ( $T_{2}$ ) and rotations ( $T_{1}$ ), and verify that $Γ_{vib} = A_{1} + E + 2 T_{2}$ . State how many IR bands and how many Raman bands methane should show.

Hint

Use the reduction formula with class sizes $n_{k} = (1, 8, 3, 6, 6)$ and $h = 24$ . For example, $a (A_{1}) = (1/24) (1 \cdot 15 \cdot 1 + 8 \cdot 0 \cdot 1 + 3 \cdot (- 1) \cdot 1 + 6 \cdot (- 1) \cdot 1 + 6 \cdot 3 \cdot 1) = (15 - 3 - 6 + 18) /24 = 1$ . Repeat for the other four irreps.

Answer

The five multiplicities are:

$a (A_{1}) = (1/24) (15 + 0 - 3 - 6 + 18) = 24/24 = 1$
$a (A_{2}) = (1/24) (15 + 0 - 3 + 6 - 18) = 0$
$a (E) = (1/24) (30 + 0 - 6 + 0 + 0) = 1$
$a (T_{1}) = (1/24) (45 + 0 + 3 - 6 - 18) = 1$
$a (T_{2}) = (1/24) (45 + 0 + 3 + 6 + 18) = 3$

So $Γ_{3 N} = A_{1} + E + T_{1} + 3 T_{2}$ , and $Γ_{vib} = Γ_{3 N} - T_{1} - T_{2} = A_{1} + E + 2 T_{2}$ .

IR-active: $T_{2}$ transforms as $(x, y, z)$ , so the two $T_{2}$ modes ( $ν_{3}$ and $ν_{4}$ ) are IR-active. Expected: 2 IR bands.
Raman-active: $A_{1}$ transforms as $x^{2} + y^{2} + z^{2}$ , $E$ as $(2 z^{2} - x^{2} - y^{2}, x^{2} - y^{2})$ , and $T_{2}$ as $(x y, x z, y z)$ , so all four symmetry species are Raman-active. Expected: 4 Raman bands.

Experiments confirm: methane IR shows 2 fundamental bands at $ν_{3} \approx 3019$ cm $^{- 1}$ (asymmetric C-H stretch) and $ν_{4} \approx 1306$ cm $^{- 1}$ (bend); the Raman spectrum shows 4 lines including the $ν_{1}$ symmetric stretch at 2917 cm $^{- 1}$ and the $ν_{2}$ bend at 1534 cm $^{- 1}$ that are absent from IR.

Advanced results Master

Theorem 1 (Bravais 1850; 14 three-dimensional lattices). Bravais proved that exactly 14 distinct three-dimensional translation lattices are compatible with any crystalline structure ^{[Bravais1850]}. Each "Bravais lattice" is a discrete set of points $Λ = {n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3} : n_{i} \in Z}$ such that the environment of every point is identical. The 14 Bravais lattices underlie the 32 crystallographic point groups and constrain the possible symmetries of any periodic solid.

Theorem 2 (Fedorov-Schönflies 1890-1891; 230 space groups). Independently, Fedorov in Saint Petersburg and Schönflies in Göttingen classified the 230 distinct space groups — discrete subgroups of the Euclidean group $E (3)$ that tile three-dimensional space ^{[Fedorov1891]} ^{[Schoenflies1891]}. Every crystalline solid belongs to exactly one. The derivation closed a 19th-century classification programme opened by Hessel (32 crystallographic point groups, 1830) and Bravais (14 lattices, 1850).

Theorem 3 (Schoenflies molecular point-group notation 1891). Schoenflies in Krystallsysteme und Krystallstructur assigned the canonical labels $C_{n}, C_{n v}, C_{nh}, S_{2 n}, D_{n}, D_{nh}, D_{n d}, T, T_{d}, T_{h}, O, O_{h}, I, I_{h}, C_{\infty v}, D_{\infty h}$ that chemists still use for molecular point groups ^{[Schoenflies1891]}. The labels encode the group generators: $C$ for cyclic rotation, $D$ for dihedral, $T, O, I$ for the Platonic symmetries (tetrahedral, octahedral, icosahedral). The Schoenflies convention is molecular; crystallographers use the independent Hermann-Mauguin notation for space groups.

Theorem 4 (Burnside 1897; great orthogonality). Burnside in Theory of Groups of Finite Order proved that the characters of distinct irreducible representations of a finite group are orthogonal when summed over the group, $\sum_{g} χ_{i} (g) \overline{χ_{j} (g)} = h δ_{ij}$ ^{[Burnside1897]}. The theorem is the algebraic foundation of every reduction formula in molecular spectroscopy. Burnside's proof used what is now called the Schur averaging trick, reproduced in the Key theorem above.

Theorem 5 (Wigner 1931; selection rules from representation theory). Wigner in Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren showed that every spectroscopic transition amplitude is an inner product over three symmetry species — initial state, transition operator, final state — and is non-zero only if the direct product of the three contains the totally symmetric representation ^[Wigner1931]. The 1963 Nobel Prize in Physics recognised this work, which made group theory the language of atomic, molecular, and nuclear spectroscopy.

Theorem 6 (Mulliken 1933; symmetry symbols). Mulliken introduced the $A, B, E, T$ labelling with subscripts $1, 2, g, u,^{'},^{''}$ that names every column of every molecular character table, in a series of papers in Physical Review and Journal of Chemical Physics from 1932 to 1935 ^{[Mulliken1933]}. The convention, adopted by the International Union of Pure and Applied Chemistry, replaced a zoo of competing notations. The 1966 Nobel Prize recognised Mulliken's broader molecular-orbital programme.

Theorem 7 (Wilson-Decius-Cross 1955; the GF method). Wilson, Decius, and Cross in Molecular Vibrations systematised the group-theoretic analysis of molecular vibrations: the kinetic-energy matrix $G$ (geometry) and the force-constant matrix $F$ (potential) factorise into symmetry blocks, one per irrep, reducing the secular determinant from a $3 N \times 3 N$ eigenvalue problem to a sum of small determinants ^{[WilsonDeciusCross1955]}. Every modern vibrational-analysis program implements the Wilson GF method.

Theorem 8 (Cotton 1971; pedagogical standardisation). Cotton's Chemical Applications of Group Theory, first published in 1963 with the canonical 3rd edition in 1990, became the standard graduate textbook, fixing the convention that chemistry students learn point groups through Schoenflies labels, character tables, and the reduction formula ^[Cotton1971]. The book trained two generations of inorganic chemists and is the reason character-table analysis is taught in every chemistry graduate programme today.

Synthesis. The Bravais-Fedorov-Schönflies classification of point and space groups builds toward 16.02.01 symmetry and group theory in chemistry and appears again in 16.05.04 ferrocene as the $D_{5 d}$ and $D_{5 h}$ point groups that organise the metallocene molecular-orbital diagram. The foundational reason group theory underlies all of molecular spectroscopy is Wigner's selection-rule theorem, which reduces every transition amplitude to an inner product over three symmetry species, and this is exactly the structural fact that makes a character table a complete predictor of IR and Raman activity. Putting these together with Mulliken's 1933 labelling and the Wilson-Decius-Cross 1955 GF method identifies molecular symmetry as a unified algebraic-geometric framework: the bridge is between Burnside's 1897 abstract orthogonality theorem and the experimental spectrometer, and the central insight is that a molecule's point group, once assigned, fixes its dipole-moment, IR, Raman, and optical-activity signatures without any further quantum-mechanical input. The pattern generalises from diatomic molecules through tetrahedral methane, octahedral coordination complexes in 16.04.02, and icosahedral $C_{60}$ fullerene, identifying Schoenflies notation as the load-bearing nomenclature of inorganic chemistry.

Full proof set Master

Proposition 1 (dimension-sum identity). For a finite group $G$ of order $h$ with inequivalent irreducible representations $Γ_{1}, \dots, Γ_{k}$ of dimensions $d_{1}, \dots, d_{k}$ ,

i = 1 \sum k d_{i}^{2} = h .

Proof. The regular representation $ρ_{reg}$ on the group algebra $C [G]$ has basis ${e_{g} : g \in G}$ and action $ρ_{reg} (g) e_{g^{'}} = e_{g g^{'}}$ . Its character satisfies $χ_{reg} (E) = h$ (the identity fixes every basis vector) and $χ_{reg} (g) = 0$ for $g \neq = E$ (any non-identity $g$ permutes the basis vectors without fixed points, giving a permutation matrix of trace zero).

Decompose $ρ_{reg} ≅ ⨁_{i} a_{i} Γ_{i}$ . By the reduction formula (Key theorem corollary),

a_{i} = \frac{1}{h} g \sum χ_{reg} (g) \overline{χ_{i} (g)} = \frac{1}{h} \cdot h \cdot χ_{i} (E) = d_{i},

since $χ_{i} (E) = d_{i}$ (the trace of the identity matrix of size $d_{i}$ ). Therefore $ρ_{reg} ≅ ⨁_{i} d_{i} Γ_{i}$ . Taking dimensions on both sides: $h = dim C [G] = \sum_{i} d_{i} \cdot d_{i} = \sum_{i} d_{i}^{2}$ . $□$

Proposition 2 (rule of mutual exclusion for centrosymmetric molecules). If a molecule has a centre of inversion $i$ , no vibrational mode is simultaneously IR-active and Raman-active.

Proof. By Wigner's selection rule (Theorem 5 above), a mode is IR-active only if its irrep appears in the representation generated by the Cartesian coordinates $(x, y, z)$ . The inversion $i$ acts on $(x, y, z)$ by $- (x, y, z)$ , so the coordinate functions transform as an ungerade ( $u$ ) irrep; therefore every IR-active mode carries a $u$ label.

A mode is Raman-active only if its irrep appears in the representation generated by the quadratic forms ${x^{2}, y^{2}, z^{2}, x y, x z, y z}$ . Each of these is even under inversion: $i \cdot x^{2} = (- x) (- x) = x^{2}$ , and similarly for the others. Therefore every quadratic form transforms as a gerade ( $g$ ) irrep, and every Raman-active mode carries a $g$ label.

An irreducible representation is by definition either symmetric ( $g$ ) or antisymmetric ( $u$ ) under $i$ , never both. The sets of IR-active and Raman-active modes are therefore disjoint, which is the rule of mutual exclusion. $□$

The carbon dioxide molecule ( $D_{\infty h}$ , with centre of inversion) illustrates the rule: its symmetric stretch $ν_{1}$ ( $Σ_{g}^{+}$ , Raman-active only) appears at 1333 cm $^{- 1}$ in the Raman spectrum and is absent from IR, while its asymmetric stretch $ν_{3}$ ( $Σ_{u}^{+}$ , IR-active only) appears at 2349 cm $^{- 1}$ in IR and is absent from Raman. Water ( $C_{2 v}$ , no inversion) shows no such exclusion: its three modes are all both IR-active and Raman-active.

Connections Master

Symmetry and group theory in chemistry 16.02.01. This unit deepens the chapter anchor introduced in 16.02.01, which surveyed point groups, character tables, and the basic decomposition formula. The foundational reason for the split is that 16.02.01 introduces the toolkit; the present unit systematises the Schoenflies classification tree, proves the great orthogonality theorem of Burnside, and exercises the full reduction-formula machinery on tetrahedral, octahedral, and linear molecules. The downstream specialisations appear in 16.02.02 pending (character tables for vibrational analysis) and 16.02.03 pending (projection operators and SALCs), each treating one tool in depth. The pattern recurs in every downstream chemistry unit that invokes a point group.
Ferrocene and the metallocenes 16.05.04. The ferrocene molecule in $D_{5 d}$ (staggered) or $D_{5 h}$ (eclipsed) symmetry illustrates the full power of point-group analysis applied to a transition-metal sandwich complex. The ten pi-orbitals of the two cyclopentadienyl rings decompose into $a_{1 g} + a_{2 u} + e_{1 g} + e_{1 u} + e_{2 g} + e_{2 u}$ combinations that match the Fe 3d, 4s, and 4p valence orbitals one-to-one by symmetry, producing the 18-electron MO diagram that underpins ferrocene's stability. The bridge is that the same Schoenflies classification introduced here for small molecules governs the metallocene MO diagram in 16.05.04, and the same great orthogonality of characters ensures that the matching of ligand and metal orbital symmetries is exhaustive.
Werner coordination chemistry 16.04.05. Werner's 1893 coordination theory gave transition-metal complexes their octahedral, tetrahedral, and square-planar geometries — the canonical $O_{h}$ , $T_{d}$ , and $D_{4 h}$ point groups. The bridge is that every coordination compound's electronic structure, ligand-field splitting pattern, and stereoisomer count descend from its point-group assignment. The same classification tree developed here provides the group-theoretic machinery that 16.04.05 invokes to count octahedral isomers (cis versus trans for $MA_{4} B_{2}$ , fac versus mer for $MA_{3} B_{3}$ ) by decomposing ligand permutations under $O_{h}$ .
Crystal-field stabilization energy 16.04.02. The octahedral ligand field in 16.04.02 splits the five d-orbitals into the $e_{g}$ subset ( $d_{z^{2}}, d_{x^{2} - y^{2}}$ ) and the $t_{2 g}$ subset ( $d_{x y}, d_{x z}, d_{y z}$ ), a splitting that is a direct application of $O_{h}$ character-table decomposition to a 5-dimensional reducible representation. The bridge is that the great orthogonality of characters proved here is exactly the algebraic fact that forces the 5 d-orbitals to split as $e_{g} + t_{2 g}$ under $O_{h}$ (and as $e + t_{2}$ under the tetrahedral subgroup $T_{d}$ ), and the pattern generalises to tetrahedral, square-planar ( $D_{4 h}$ ), and lower-symmetry ligand fields, where the same reduction formula yields the crystal-field splitting pattern that anchors every transition-metal spectrochemical series.

Historical & philosophical context Master

Auguste Bravais in 1850 classified the 14 three-dimensional translation lattices that bear his name ^{[Bravais1850]}. The full classification of crystallographic symmetry — the 230 space groups — was completed independently in 1890 and 1891 by Evgraf Fedorov in Saint Petersburg and Arthur Schoenflies in Göttingen ^{[Fedorov1891]} ^{[Schoenflies1891]}. Schoenflies in the same year published Krystallsysteme und Krystallstructur, fixing the molecular point-group notation ( $C_{n}, C_{n v}, D_{n}, D_{nh}, D_{n d}, T_{d}, O_{h}, I_{h}$ ) that chemists still use ^{[Schoenflies1891]}. William Burnside in 1897 proved the great orthogonality theorem for characters of finite groups in Theory of Groups of Finite Order ^{[Burnside1897]}, providing the algebraic foundation of every reduction formula in molecular spectroscopy.

Eugene Wigner in Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (1931) recast atomic and molecular spectroscopy in group-theoretic language, deriving selection rules as a direct consequence of representation theory ^[Wigner1931]; the 1963 Nobel Prize in Physics recognised this work. Robert Mulliken in 1933 introduced the $A, B, E, T$ symbols with $g, u,^{'},^{''}$ subscripts that label every column of every molecular character table, in a series of papers in Physical Review and Journal of Chemical Physics ^{[Mulliken1933]}; the 1966 Nobel Prize recognised his broader molecular-orbital programme. Edward Bright Wilson, J. C. Decius, and Paul Cross in Molecular Vibrations (1955) systematised the group-theoretic analysis of vibrational spectra through the GF matrix method ^{[WilsonDeciusCross1955]}. F. Albert Cotton's Chemical Applications of Group Theory (1963, with the canonical 3rd edition in 1990) made character tables standard chemistry curriculum ^[Cotton1971], and remains the reference from which most modern textbooks derive their exposition.

Bibliography Master

@article{Bravais1850,
  author = {Bravais, A.},
  title = {M\'emoire sur les syst\`emes de points r\'eguli\`erement distribu\'es sur un plan ou dans l'espace},
  journal = {J. \'{E}cole Polytechnique},
  volume = {19},
  year = {1850},
  pages = {1--128}
}

@article{Fedorov1891,
  author = {Fedorov, E. S.},
  title = {Simmetriya pravilnykh sistem figur ({Symmetry} of {Regular} {Systems} of {Figures})},
  journal = {Zapiski Imperatorskogo Sankt-Peterburgskogo Mineralogicheskogo Obshchestva (2nd Ser.)},
  volume = {28},
  year = {1891},
  pages = {1--146}
}

@book{Schoenflies1891,
  author = {Schoenflies, A. M.},
  title = {Krystallsysteme und {Krystallstructur}},
  publisher = {B. G. Teubner, Leipzig},
  year = {1891}
}

@book{Burnside1897,
  author = {Burnside, W.},
  title = {Theory of {Groups} of {Finite} {Order}},
  publisher = {Cambridge University Press},
  year = {1897}
}

@book{Wigner1931,
  author = {Wigner, E. P.},
  title = {Gruppentheorie und ihre {Anwendung} auf die {Quantenmechanik} der {Atomspektren}},
  publisher = {Friedr. Vieweg \& Sohn, Braunschweig},
  year = {1931}
}

@article{Mulliken1933,
  author = {Mulliken, R. S.},
  title = {Electronic {Structures} of {Polyatomic} {Molecules} and {Valence}. {IV}. {Electronic} {States}, {Band} {Structure}, and the {Photochemistry} of the {Ethylenes}},
  journal = {J. Chem. Phys.},
  volume = {1},
  year = {1933},
  pages = {492--499}
}

@article{Mulliken1935,
  author = {Mulliken, R. S.},
  title = {Electronic {Structures} of {Polyatomic} {Molecules} and {Valence}. {VI}. {On} the {Method} of {Molecular} {Orbitals}},
  journal = {J. Chem. Phys.},
  volume = {3},
  year = {1935},
  pages = {375--378}
}

@book{WilsonDeciusCross1955,
  author = {Wilson, E. B. and Decius, J. C. and Cross, P. C.},
  title = {Molecular {Vibrations}: {The} {Theory} of {Infrared} and {Raman} {Vibrational} {Spectra}},
  publisher = {McGraw-Hill, New York},
  year = {1955}
}

@book{Cotton1971,
  author = {Cotton, F. A.},
  title = {Chemical {Applications} of {Group} {Theory}},
  publisher = {Wiley-Interscience, New York},
  year = {1971}
}

@book{Cotton1990,
  author = {Cotton, F. A.},
  title = {Chemical {Applications} of {Group} {Theory}},
  edition = {3rd},
  publisher = {Wiley-Interscience, New York},
  year = {1990}
}

@book{Vincent2001,
  author = {Vincent, P.},
  title = {Molecular {Symmetry} and {Group} {Theory}},
  edition = {2nd},
  publisher = {Wiley, Chichester},
  year = {2001}
}

@book{Bishop1993,
  author = {Bishop, D. M.},
  title = {Group {Theory} and {Chemistry}},
  publisher = {Dover, Mineola, NY},
  year = {1993}
}

Prerequisites

16.02.01

Tier anchors

beginner: Vincent, P. — Molecular Symmetry and Group Theory, 2nd ed. (Wiley, 2001), Ch. 1-3; Crash Course Chemistry — Molecular Geometry
intermediate: Cotton, F. A. — Chemical Applications of Group Theory, 3rd ed. (Wiley, 1990), Ch. 3-5; Vincent — Molecular Symmetry and Group Theory, Ch. 4-7
master: Schoenflies 1891 Krystallsysteme und Krystallstructur; Burnside 1897 Theory of Groups of Finite Order; Wigner 1931 Gruppentheorie; Mulliken 1933 J. Chem. Phys. 1:492; Wilson-Decius-Cross 1955 Molecular Vibrations; Cotton 1971 Chemical Applications of Group Theory

References

Schoenflies, A. M. — Krystallsysteme und Krystallstructur (B. G. Teubner, Leipzig, 1891) · Originator: systematic Schoenflies point-group notation for molecular symmetry (C_n, C_nv, D_nh, D_nd, T_d, O_h, I_h)
Burnside, W. — Theory of Groups of Finite Order, 1st ed. (Cambridge University Press, 1897) · Originator: great orthogonality theorem for characters of finite groups (Ch. XV)
Wigner, E. P. — Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg, 1931) · Originator: group-theoretic derivation of spectroscopic selection rules
Mulliken, R. S. — Electronic Structures of Polyatomic Molecules and Valence. IV. Electronic States, Band Structure, and the Photochemistry of the Ethylenes (J. Chem. Phys. 1:492-499, 1933; see also Phys. Rev. 40:55, 1932; J. Chem. Phys. 3:375, 1935) · Originator: Mulliken symmetry symbols (A, B, E, T with subscripts 1, 2, g, u, primes) for irreducible representations
Wilson, E. B., Decius, J. C., Cross, P. C. — Molecular Vibrations (McGraw-Hill, New York, 1955) · Originator: systematic group-theoretic analysis of molecular vibrations via the Wilson GF matrix method
Cotton, F. A. — Chemical Applications of Group Theory, 1st ed. (Wiley-Interscience, New York, 1963; 3rd ed. 1990) · Canonical pedagogical monograph that established character tables as standard chemistry curriculum
Fedorov, E. S. — Symmetry of Regular Systems of Figures (Zap. Mineral. Obsc., 2nd Ser., 28:1-146, 1891) · Independent 1890-1891 derivation (with Schoenflies) of the 230 crystallographic space groups
Bravais, A. — Memoir on the Systems of Points Regularly Distributed in Space (J. Ecole Polytech. 19:1-128, 1850) · Originator: 14 Bravais lattices underlying crystallographic point and space groups
Vincent, P. — Molecular Symmetry and Group Theory, 2nd ed. (Wiley, Chichester, 2001) · Modern introductory monograph; canonical Beginner and Intermediate anchor for character-table drill

Estimated time

beginner: 25m
intermediate: 50m
master: 85m