Elements

The point group symmetry of a molecule is defined by the presence or absence of 5 types of symmetry element.

• Symmetry axis: an axis around which a rotation by ${\displaystyle {\tfrac {360^{\circ }}{n}}}$ results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated C_n. Examples are the C₂ axis in water and the C₃ axis in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is aligned with the z-axis in a Cartesian coordinate system.
• Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is generated. This is also called a mirror plane and abbreviated σ (sigma = Greek "s", from the German 'Spiegel' meaning mirror).^[7] Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σ_v) and one perpendicular to it horizontal (σ_h). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σ_d). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).
• Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. In other words, a molecule has a center of symmetry when the points (x,y,z) and (−x,−y,−z) are identical. For example, if there is an oxygen atom in some point (x,y,z), then there is an oxygen atom in the point (−x,−y,−z). There may or may not be an atom at the inversion center itself. Examples are xenon tetrafluoride where the inversion center is at the Xe atom, and benzene (C
  ₆H
  ₆) where the inversion center is at the center of the ring.
• Rotation-reflection axis: an axis around which a rotation by ${\displaystyle {\tfrac {360^{\circ }}{n}}}$, followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated S_n. Examples are present in tetrahedral silicon tetrafluoride, with three S₄ axes, and the staggered conformation of ethane with one S₆ axis. An S₁ axis corresponds to a mirror plane σ and an S₂ axis is an inversion center i. A molecule which has no S_n axis for any value of n is a chiral molecule.
• Identity, abbreviated to E, from the German 'Einheit' meaning unity.^[8] This symmetry element simply consists of no change: every molecule has this symmetry element, which is equivalent to a C₁ proper rotation. It must be included in the list of symmetry elements so that they form a mathematical group, whose definition requires inclusion of the identity element. It is so called because it is analogous to multiplying by one (unity).^[9]

Point groups and their symmetry operations

Groups

The symmetry operations of a molecule (or other object) form a group. In mathematics, a group is a set with a binary operation that satisfies the four properties listed below.

In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C₄ rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C₄. By convention the order of operations is from right to left.

A symmetry group obeys the defining properties of any group.

1. closure property:
   For every pair of elements x and y in G, the product x*y is also in G.
   ( in symbols, for every two elements x, y ∈ G, x*y is also in G ).
   This means that the group is closed so that combining two elements produces no new elements. Symmetry operations have this property because a sequence of two operations will produce a third state indistinguishable from the second and therefore from the first, so that the net effect on the molecule is still a symmetry operation. This may be illustrated by means of a table. For example, with the point group C₃, there are three symmetry operations: rotation by 120°, C₃, rotation by 240°, C₃² and rotation by 360°, which is equivalent to identity, E.
   

   C2v point group multiplication tab

   Point group C₃ Multiplication table

   	E	C3	C32
   E	E	C3	C32
   C3	C3	C32	E
   C32	C32	E	C3

2. This table also illustrates the following properties
3. Associative property:
   For every x and y and z in G, both (x*y)*z and x*(y*z) result with the same element in G.
   ( in symbols, (x*y)*z = x*(y*z ) for every x, y, and z ∈ G)
4. existence of identity property:
   There must be an element ( say e ) in G such that product any element of G with e make no change to the element.
   ( in symbols, x*e = e*x = x for every x ∈ G )
5. existence of inverse element:
   For each element x in G, there must be an element y in G such that product of x and y is the identity element e.
   ( in symbols, for each x ∈ G there is a y ∈ G such that x*y = y*x = e for every x ∈ G )

The order of a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.

Point groups and permutation-inversion groups

Flowchart for determining the point group of a molecule

The successive application (or composition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v' symmetry operation: σ_v*C₂ = σ_v'. ("Operation A followed by B to form C" is written BA = C).^[9] Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, where S is the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.

This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarises all symmetry operations that all molecules in that category have.^[9] The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.

One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates^[10] and these operations commute with the vibronic Hamiltonian. They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins.^[11] Point groups describe the geometrical symmetry of a molecule whereas permutation-inversion groups describe the energy-invariant symmetry.

Examples of point groups

Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl₃, POF₃, XeO₃, and NH₃ all share identical symmetry operations.^[12] They all can undergo the identity operation E, two different C₃ rotation operations, and three different σ_v plane reflections without altering their identities, so they are placed in one point group, C_3v, with order 6.^[9] Similarly, water (H₂O) and hydrogen sulfide (H₂S) also share identical symmetry operations. They both undergo the identity operation E, one C₂ rotation, and two σ_v reflections without altering their identities, so they are both placed in one point group, C_2v, with order 4.^[13] This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.^[9] Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.

Common point groups

The following table lists many of the point groups applicable to molecules, labelled using the Schoenflies notation, which is common in chemistry and molecular spectroscopy. The descriptions include common shapes of molecules, which can be explained by the VSEPR model. In each row, the descriptions and examples have no higher symmetries, meaning that the named point group captures all of the point symmetries.

Point group	Symmetry operations[14]	Simple description of typical geometry	Example 1	Example 2	Example 3
C1	E	no symmetry, chiral	bromochlorofluoromethane (both enantiomers shown)	lysergic acid	L-leucine and most other α-amino acids except glycine
Cs	E σh	mirror plane	thionyl chloride	hypochlorous acid	chloroiodomethane
Ci	E i	inversion center	meso-tartaric acid	mucic acid (meso-galactaric acid)	1,2 dibromo 1,2 dichloroethane
C∞v	E 2C∞Φ ∞σv	linear	hydrogen fluoride (and all other heteronuclear diatomic molecules)	nitrous oxide (dinitrogen monoxide)	hydrocyanic acid (hydrogen cyanide)
D∞h	E 2C∞Φ ∞σi i 2S∞Φ ∞C2	linear with inversion center	oxygen (and all other homonuclear diatomic molecules)	carbon dioxide	acetylene (ethyne)
C2	E C2	"open book geometry", chiral	hydrogen peroxide	hydrazine	tetrahydrofuran (twist conformation)
C3	E C3 C32	propeller, chiral	triphenylphosphine	triethylamine	phosphoric acid
C2h	E C2 i σh	planar with inversion center, no vertical plane	trans-1,2-dichloroethylene	trans-dinitrogen difluoride	trans-azobenzene
C2v	E C2 σv(xz) σv'(yz)	angular (H2O) or see-saw (SF4)	water	sulfur tetrafluoride	Dichloromethane
C3h	E C3 C32 σh S3 S35	propeller	boric acid	phloroglucinol (1,3,5-trihydroxybenzene)	benzotrifuroxan
C3v	E 2C3 3σv	trigonal pyramidal	ammonia (if pyramidal inversion is neglected)	phosphorus oxychloride	cobalt tetracarbonyl hydride, HCo(CO)4
C4v	E 2C4 C2 2σv 2σd	square pyramidal	xenon oxytetrafluoride	pentaborane(9), B5H9	nitroprusside anion [Fe(CN)5(NO)]2−
C5	E 2C5 2C52	five-fold rotational symmetry	C-reactive protein	[Fe(Me5-Cp)(P5)]	Corannulene derivative
C5v	E 2C5 2C52 5σv	'milking stool' complex	Cyclopentadienyl nickel nitrosyl (CpNiNO)	corannulene	Pentamethylcyclopentadienyl nickel nitrosyl (Cp*NiNO)
D2	E C2(x) C2(y) C2(z)	twist, chiral	biphenyl (skew conformation)	twistane (C10H16)	(δ,δ)-trans-[Co(en)2Cl2]+
D3	E C3(z) 3C2	triple helix, chiral	Tris(ethylenediamine)cobalt(III) cation	tris(oxalato)iron(III) anion	tris(en)cobalt(III)
D2h	E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)	planar with inversion center, vertical plane	ethylene	pyrazine	diborane
D3h	E 2C3 3C2 σh 2S3 3σv	trigonal planar or trigonal bipyramidal	boron trifluoride	phosphorus pentachloride	cyclopropane
D4h	E 2C4 C2 2C2' 2C2" i 2S4 σh 2σv 2σd	square planar	xenon tetrafluoride	octachlorodimolybdate(II) anion	Trans-[CoIII(NH3)4Cl2]+ (excluding H atoms)
D5h	E 2C5 2C52 5C2 σh 2S5 2S53 5σv	pentagonal	cyclopentadienyl anion	ruthenocene	C70
D6h	E 2C6 2C3 C2 3C2' 3C2‘’ i 2S3 2S6 σh 3σd 3σv	hexagonal	benzene	bis(benzene)chromium	coronene (C24H12)
D7h	E C7 S7 7C2 σh 7σv	heptagonal	tropylium (C7H7+) cation
D8h	E C8 C4 C2 S8 i 8C2 σh 4σv 4σd	octagonal	cyclooctatetraenide (C8H82−) anion	uranocene	bis(cot)thorium(IV)
D2d	E 2S4 C2 2C2' 2σd	90° twist	allene	tetrasulfur tetranitride	diborane(4) (excited state)
D3d	E 2C3 3C2 i 2S6 3σd	60° twist	ethane (staggered rotamer)	dicobalt octacarbonyl (non-bridged isomer)	cyclohexane chair conformation
D4d	E 2S8 2C4 2S83 C2 4C2' 4σd	45° twist	sulfur (crown conformation of S8)	dimanganese decacarbonyl (staggered rotamer)	octafluoroxenate ion (idealized geometry)
D5d	E 2C5 2C52 5C2 i 2S103 2S10 5σd	36° twist	ferrocene (staggered rotamer)	Ruthenocene (stag)	fulleride ion
S4	E 2S4 C2		1,2,3,4-tetrafluorospiropentane (meso isomer)[15]	tetramethyl-cot	bis(dth)copper(I)
Td	E 8C3 3C2 6S4 6σd	tetrahedral	methane	phosphorus pentoxide	adamantane
Th	E 4C3 4C32 i 3C2 4S6 4S65 3σh	pyritohedron	[Fe(C5H5N)6]2+	[Th(NO3)6]2-	Fe(H2O)6
Oh	E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd	octahedral or cubic	sulfur hexafluoride	molybdenum hexacarbonyl	cubane
I	E 12C5 12C52 20C3 15C2	chiral icosahedral or dodecahedral	Rhinovirus	snub dodecahedron	human polio virus
Ih	E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ	icosahedral or dodecahedral	Buckminsterfullerene	dodecaborate anion	dodecahedrane

Representations

A set of matrices that multiply together in a way that mimics the multiplication table of the elements of a group is called a representation of the group. For example, for the C_2v point group, the following three matrices are part of a representation of the group:

${\displaystyle \underbrace {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\\\end{bmatrix}} _{C_{2}}\times \underbrace {\begin{bmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{bmatrix}} _{\sigma _{\text{v}}}=\underbrace {\begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}} _{\sigma '_{\text{v}}}}$

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are all that are needed as all other representations of the group can be described as a linear combination of the irreducible representations. Also, the irreducibile representations are those matrix representations in which the matrices are in their most diagonal form possible.

Transformation matrix symmetry operations