Molecular Symmetry, Point Groups and an Introduction to Group Theory

Symmetry is not decoration — it is the single most economical language chemistry has for predicting which vibrations absorb infrared radiation, which electronic transitions are allowed, how many NMR-distinct environments a molecule has, and how atomic orbitals combine into molecular orbitals. This chapter builds that language from the ground up: the symmetry elements and operations a molecule can possess, the systematic assignment of a molecule to a point group, and the elementary group theory — representations, characters and the reduction formula — that turns a point group from a label into a working tool. The treatment follows Housecroft & Sharpe's integration of group theory with structural inorganic chemistry.

You should have to hand: a model-building habit (sketch or build the molecule before classifying it), the trigonometric values at 0°, 60°, 90°, 120° and 180°, and comfort with $3\times3$ matrices, since every symmetry operation is ultimately a linear transformation of a point in space.

1. Symmetry Elements and Symmetry Operations

A symmetry operation is a movement of a molecule — a rotation, reflection, or inversion — that produces a configuration indistinguishable from the original: every atom is carried onto a position occupied by an identical atom, and the overall shape is unchanged. A symmetry element is the geometric entity (a point, line or plane) with respect to which the operation is carried out. The five types of operation that exhaust all possibilities for an isolated molecule are:

• $E$, the identity. Do nothing. Every molecule, however irregular, possesses $E$; it is required for the group axioms below.
• $C_n$, proper rotation. Rotation by $360^\circ/n$ about a $n$-fold rotation axis. The axis of highest $n$ found in a molecule is its principal axis. A molecule may have several $C_n$ axes of different orders; $C_n^n = E$.
• $\sigma$, reflection. Reflection through a mirror plane. Three kinds are distinguished by their relation to the principal axis: $\sigma_h$ (horizontal, perpendicular to the principal axis), $\sigma_v$ (vertical, containing the principal axis), and $\sigma_d$ (dihedral, a vertical plane that bisects the angle between two $C_2$ axes perpendicular to the principal axis).
• $i$, inversion. Every point $(x,y,z)$ is carried to $(-x,-y,-z)$ through a single centre of inversion, which need not coincide with an atom (it does not in benzene's centre, but it does in $\mathrm{XeF_4}$'s central Xe).
• $S_n$, improper rotation. Rotation by $360^\circ/n$ about an axis followed by reflection in the plane perpendicular to that axis, performed as one combined operation. Note that $S_1 = \sigma$ and $S_2 = i$, so these are not independent operations but special cases of $S_n$.

Key distinction. A symmetry element is where you perform the operation (a line, a plane, a point); the symmetry operation is the thing you actually do. A single $C_4$ axis generates four distinct operations: $C_4$, $C_4^2 (= C_2)$, $C_4^3$, and $C_4^4 (= E)$.

2. Worked Example: Finding the Elements of $\mathrm{NH_3}$ and $\mathrm{BF_3}$

Ammonia, $\mathrm{NH_3}$ is a trigonal pyramid. Its principal axis is the $C_3$ axis through N and the centroid of the three H atoms, generating operations $C_3$ and $C_3^2$. Each N–H bond lies in a mirror plane that also contains the $C_3$ axis and bisects the H–N–H angle opposite it: three such $\sigma_v$ planes exist. There is no $\sigma_h$ (the H$_3$ plane is not a symmetry plane, since N sits above it), no $i$, and no $S_n$. The full operation list is $\{E, C_3, C_3^2, \sigma_v, \sigma_v', \sigma_v''\}$ — six operations, point group $C_{3v}$.

Boron trifluoride, $\mathrm{BF_3}$ is trigonal planar. It shares the $C_3$ axis through B perpendicular to the molecular plane, but now that very molecular plane is a mirror plane perpendicular to $C_3$, so it is $\sigma_h$. In addition, each B–F bond defines a $C_2$ axis in the plane, giving three $C_2$ axes perpendicular to $C_3$; three $\sigma_v$ planes, each containing $C_3$ and one $C_2$, also appear (these count as $\sigma_d$ by some conventions since they bisect adjacent $C_2$ axes, but the standard labelling for $D_{3h}$ calls them $\sigma_v$). There is also an $S_3$ axis coincident with $C_3$. The planarity that adds $\sigma_h$ and the three $C_2$'s is exactly what distinguishes $D_{3h}$ ($\mathrm{BF_3}$) from $C_{3v}$ ($\mathrm{NH_3}$): a nonplanar pyramid cannot have $\sigma_h$ or the perpendicular $C_2$ axes.

3. Assigning Point Groups: A Systematic Procedure

A point group is the complete set of symmetry operations of a molecule, so named because every operation leaves at least one point (the centre of mass) fixed. Rather than guessing, Housecroft & Sharpe's flow-chart logic proceeds in order:

1. Is the molecule linear? If so, it is $C_{\infty v}$ (no centre of symmetry, e.g. HCN) or $D_{\infty h}$ (has a centre of symmetry, e.g. $\mathrm{CO_2}$, $\mathrm{H_2}$).
2. Does it have two or more $C_n$ ($n \geq 3$) axes? Then it belongs to one of the special high-symmetry groups: $T_d$ (tetrahedral, e.g. $\mathrm{CH_4}$, $\mathrm{SiF_4}$), $O_h$ (octahedral, e.g. $\mathrm{SF_6}$, $[\mathrm{PtCl_6}]^{2-}$), or $I_h$ (icosahedral, e.g. $\mathrm{B_{12}H_{12}^{2-}}$, $\mathrm{C_{60}}$).
3. Otherwise, find the principal axis $C_n$ (the one of highest order; if none exists other than $C_1$, go to step 5).
4. Are there $n$ $C_2$ axes perpendicular to $C_n$? If yes, the molecule is in the $D$ family; if no, it is in the $C$ or $S$ family.
   • $D$ family: then ask — is there $\sigma_h$? → $D_{nh}$. No $\sigma_h$ but $n$ $\sigma_d$ planes? → $D_{nd}$. Neither? → $D_n$.
   • $C$/$S$ family: is there $\sigma_h$? → $C_{nh}$. Are there $n$ $\sigma_v$ planes containing $C_n$? → $C_{nv}$. Is there an $S_{2n}$ axis coincident with $C_n$ and nothing else? → $S_{2n}$. None of these? → $C_n$.
5. If no element survives step 3 except possibly $\sigma$ or $i$: a single mirror plane alone gives $C_s$; a centre of inversion alone gives $C_i$; neither gives the trivial group $C_1$ (asymmetric molecules, e.g. CHFClBr).

Why order matters. Checking for linear and for the cubic/icosahedral groups before hunting for a principal axis prevents the common error of trying to classify $\mathrm{CH_4}$ by looking for "the" $C_n$ axis — it has four equivalent $C_3$ axes and three $C_2$ axes simultaneously, which is precisely the signature of $T_d$, not of any single-axis family.

4. Chirality and the Decisive Role of $S_n$

A molecule is chiral (non-superimposable on its mirror image) if and only if it possesses no improper rotation axis of any order — remembering that $\sigma = S_1$ and $i = S_2$ count as improper rotations. This is a sharper test than "no mirror plane and no centre of symmetry": a molecule in point group $S_4$ possesses an improper axis yet neither a standalone $\sigma$ nor $i$, and it is still achiral. Tetraphenylmethane with four identically twisted (propeller) phenyl rings realises this $S_4$ case. In practice, point groups $C_n$ and $D_n$ (rotations only, no $\sigma$, no $i$, no $S_n$) are the chiral ones routinely encountered, e.g. $\Delta$/$\Lambda$ tris-chelate complexes such as $[\mathrm{Co(en)_3}]^{3+}$, which belongs to $D_3$.

5. Groups, Multiplication Tables and Classes

Mathematically, a group is a set of elements (here, symmetry operations) together with a combination rule (successive application) satisfying four axioms: closure (the product of any two operations is itself an operation of the set), associativity, existence of an identity $E$, and existence of an inverse for every element. The order $h$ of the group is the number of operations (for $C_{3v}$, $h=6$). Multiplying every pair of operations generates the group's multiplication table; this table is not generally commutative ($AB \neq BA$ in general; a point group is non-Abelian exactly when some class contains more than one operation, e.g. $2C_3$ in $C_{3v}$, because a rotation and a reflection need not commute).

Operations fall into classes: $C_3$ and $C_3^2$ are related by a similarity transformation ($X^{-1} C_3 X = C_3^2$ for a suitable $X$ in the group) and so belong to the same class, written $2C_3$. Members of a class are geometrically equivalent — they share the same character in every representation (Section 6) — which is why character tables list classes, not individual operations, as columns.

6. Representations and Character Tables

Each symmetry operation can be represented by a matrix describing how it transforms a chosen basis (e.g. the $x,y,z$ axes, or the three N–H bond vectors of ammonia). The set of these matrices, one per operation, forms a representation of the group, and the character $\chi(R)$ of operation $R$ is the trace (sum of diagonal elements) of its matrix — chosen because the trace is invariant under the change of basis that leaves physical conclusions unchanged.

A representation is reducible if it can be block-diagonalised into (equivalently, written as a sum of) smaller representations; it is irreducible if it cannot be reduced further. The irreducible representations (irreps) of a point group are tabulated once and for all in its character table: rows are irreps (labelled $A_1, A_2, B_1, B_2, E, T_1, \dots$ by convention — $A/B$ for one-dimensional, $E$ for two-dimensional, $T$ for three-dimensional irreps, not to be confused with the identity operation $E$), columns are classes of operations, and entries are characters. The final columns show which irreps transform like $x,y,z$ (used for IR activity and bonding) and like the quadratic functions $x^2, xy,$ etc. (used for Raman activity).

The $C_{2v}$ character table, needed repeatedly below, is:

| $C_{2v}$ | $E$ | $C_2$ | $\sigma_v(xz)$ | $\sigma_v'(yz)$ | linear | |---|---|---|---|---|---| | $A_1$ | 1 | 1 | 1 | 1 | $z$ | | $A_2$ | 1 | 1 | $-1$ | $-1$ | — | | $B_1$ | 1 | $-1$ | 1 | $-1$ | $x$ | | $B_2$ | 1 | $-1$ | $-1$ | 1 | $y$ |

Every irrep has $\chi(E)$ equal to its dimension; here all four are one-dimensional, consistent with $h=4=\sum_i \ell_i^2$ (the dimension theorem, where $\ell_i$ is the dimension of irrep $i$).

7. The Reduction Formula and Molecular Vibrations

Given a reducible representation (for instance, the transformation properties of all $3N$ Cartesian displacement coordinates of an $N$-atom molecule), the number of times $n_i$ that irrep $i$ occurs is extracted by the reduction formula:

where the sum runs over classes, $g_R$ is the number of operations in class $R$, $\chi(R)$ is the character of the reducible representation for that class, and $\chi_i(R)$ is the corresponding character of irrep $i$ from the table.

Worked example — the vibrations of $\mathrm{H_2O}$. Water is $C_{2v}$. Placing a Cartesian axis set on each atom, the reducible representation $\Gamma_{3N}$ for all $9$ displacement coordinates has characters $\chi(E)=9$, $\chi(C_2)=-1$, $\chi(\sigma_v)=1$, $\chi(\sigma_v')=3$ (computed from how many atoms are unmoved by each operation, multiplied by the contribution per unmoved atom: $+3$ for $E$ per atom unmoved, $-1$ for $C_2$, $+1$ for a reflection, per atom lying in the mirror). Applying the reduction formula with $h=4$ gives $\Gamma_{3N} = 3A_1 + A_2 + 2B_1 + 3B_2$. Removing the three translations ($A_1+B_1+B_2$, transforming as $x,y,z$) and three rotations ($A_2 + B_1 + B_2$, transforming as rotations about $x,y,z$) leaves the vibrational representation:

exactly the $3N-6 = 3$ vibrational modes expected for this non-linear triatomic: a symmetric stretch and a bend (both $A_1$) and an antisymmetric stretch ($B_2$). (For a linear molecule the count is $3N-5$, since rotation about the molecular axis is not a physical degree of freedom.)

8. IR and Raman Activity from Symmetry

A vibrational mode is infrared-active if and only if its irrep is the same as one of the irreps spanning $x, y, z$ (the "linear" row of the character table) — because the transition-moment integral is non-zero only then. It is Raman-active if its irrep matches one of the quadratic functions ($x^2, y^2, z^2, xy, xz, yz$). For $C_{2v}$, all of $A_1, B_1, B_2$ transform as some combination of $x,y,z$, so all three modes of water ($2A_1+B_2$) are IR-active; the same three irreps also appear among the quadratic functions, so all three are Raman-active too. The general mutual exclusion rule — a mode active in one technique is inactive in the other — holds strictly only for molecules possessing a centre of inversion $i$ (point groups containing $i$, e.g. $C_i$, $C_{2h}$, $D_{2h}$, $D_{3d}$, $D_{\infty h}$, $O_h$; note $C_{3h}$ and $D_{3h}$ have no $i$), because $i$-containing groups split cleanly into modes symmetric under $i$ (gerade, Raman-active, IR-silent) and antisymmetric under $i$ (ungerade, IR-active, Raman-silent). This is the experimental fingerprint used to confirm whether a molecule such as $\mathrm{N_2O_4}$ is centrosymmetric.

Common Exam Pitfalls

• Classifying by eye instead of by the flow chart. Students often decide "it looks like $C_{3v}$" from a mental picture; always locate the actual highest-order axis and systematically test for perpendicular $C_2$'s and for $\sigma_h$ before assigning the group, especially for molecules close in appearance to $D_{3h}$ versus $C_{3v}$ (planar vs. pyramidal).
• Forgetting that $\sigma = S_1$ and $i = S_2$. This causes students to wrongly call an $S_4$-only molecule "achiral because it has no mirror plane and no inversion centre" — check instead for any $S_n$, which $S_4$ itself already is.
• Confusing symmetry elements with symmetry operations. A single $C_3$ axis is one element but generates two non-trivial operations, $C_3$ and $C_3^2$; the order of the group counts operations, not elements.
• Missing perpendicular $C_2$ axes that are "hidden" in the bonds. In $D_{3h}$ $\mathrm{BF_3}$, each B–F bond is itself a $C_2$ axis; failing to find these leads to the wrong (too low-symmetry) point group $C_{3v}$.
• Applying the reduction formula with the wrong $h$ or omitting a class multiplicity $g_R$. Every term in the sum must be weighted by how many operations are in that class, not just evaluated once per distinct symmetry type.
• Assuming IR and Raman activity are always mutually exclusive. The exclusion rule is a consequence of a centre of inversion; for non-centrosymmetric molecules like $\mathrm{H_2O}$, the same mode can be active in both spectroscopies simultaneously.

Summary

• A symmetry operation ($E, C_n, \sigma, i, S_n$) maps a molecule onto an indistinguishable configuration; the point group is the complete set of such operations.
• Assign point groups systematically: linear → cubic/icosahedral special cases → locate the principal $C_n$ → test for perpendicular $C_2$'s ($D$ vs. $C$/$S$ family) → test for $\sigma_h$, $\sigma_v$/$\sigma_d$, or $S_{2n}$.
• A molecule is chiral if and only if it has no improper axis $S_n$ of any order (remembering $\sigma=S_1$, $i=S_2$).
• A group satisfies closure, associativity, an identity, and inverses; operations related by a similarity transformation form a class.
• Characters $\chi(R)$ are matrix traces, invariant under basis change; irreducible representations are tabulated in the character table, with $h = \sum_i \ell_i^2$.
• The reduction formula $n_i = \frac{1}{h}\sum_R g_R\chi(R)\chi_i(R)$ decomposes any reducible representation, e.g. the $3N$ Cartesian displacements of a molecule, giving $\Gamma_{\text{vib}}$ after removing translations and rotations ($3N-6$ modes, or $3N-5$ if linear).
• A vibration is IR-active if its irrep spans $x,y,z$; it is Raman-active if its irrep spans a quadratic function; the two are mutually exclusive only when the molecule has a centre of inversion.