Molecular symmetry
In chemistry, molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as whether or not it has a dipole moment, as well as its allowed spectroscopic transitions. To do this it is necessary to use group theory. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the WoodwardHoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.
There are many techniques for determining the symmetry of a given molecule, including Xray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.
Point group symmetry concepts
Rotational axis (C_{n}) 
Improper rotational elements (S_{n})  

Chiral no S_{n} 
Achiral mirror plane S_{1} = σ 
Achiral inversion centre S_{2} = i  
C_{1}  
C_{2} 
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 results in a molecule indistinguishable from the original. This is also called an nfold rotational axis and abbreviated C_{n}. Examples are the C_{2} axis in water and the C_{3} 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 zaxis 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). 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 2fold 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^{}
_{6}H^{}
_{6}) where the inversion center is at the center of the ring.  Rotationreflection axis: an axis around which a rotation by , followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an nfold improper rotation axis, it is abbreviated S_{n}. Examples are present in tetrahedral silicon tetrafluoride, with three S_{4} axes, and the staggered conformation of ethane with one S_{6} axis. An S_{1} axis corresponds to a mirror plane σ and an S_{2} 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. This symmetry element simply consists of no change: every molecule has this symmetry element, which is equivalent to a C_{1} 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).
Operations
The five symmetry elements have associated with them five types of symmetry operation, which leave the geometry of the molecule indistinguishable from the starting geometry. They are sometimes distinguished from symmetry elements by a caret or circumflex. Thus, Ĉ_{n} is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, the C_{4} axis of the square xenon tetrafluoride (XeF_{4}) molecule is associated with two Ĉ_{4} rotations in opposite directions (90° and 270°), a Ĉ_{2} rotation (180°) and Ĉ_{1} (0° or 360°). Because Ĉ_{1} is equivalent to Ê, Ŝ_{1} to σ and Ŝ_{2} to î, all symmetry operations can be classified as either proper or improper rotations.
For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any angle Φ is a symmetry operation.
Symmetry groups
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 C_{4} rotation about the zaxis and a reflection in the xyplane, denoted σ(xy)C_{4}. By convention the order of operations is from right to left.
A symmetry group obeys the defining properties of any group.
 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 C_{3}, there are three symmetry operations: rotation by 120°, C_{3}, rotation by 240°, C_{3}^{2} and rotation by 360°, which is equivalent to identity, E.
Point group C_{3} Multiplication table E C_{3} C_{3}^{2} E E C_{3} C_{3}^{2} C_{3} C_{3} C_{3}^{2} E C_{3}^{2} C_{3}^{2} E C_{3}
 This table also illustrates the following properties
 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)
 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 )
 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 permutationinversion groups
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 C_{2} rotation followed by a σ_{v} reflection is seen to be a σ_{v}' symmetry operation: σ_{v}*C_{2} = σ_{v}'. ("Operation A followed by B to form C" is written BA = C). 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. 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 (vibrationelectronic) coordinates 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 (rotationvibrationelectronic) Hamiltonian, requires the use of the appropriate permutationinversion group as introduced by LonguetHiggins. Point groups describe the geometrical symmetry of a molecule whereas permutationinversion groups describe the energyinvariant symmetry.
Examples of point groups
Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl_{3}, POF_{3}, XeO_{3}, and NH_{3} all share identical symmetry operations. They all can undergo the identity operation E, two different C_{3} 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. Similarly, water (H_{2}O) and hydrogen sulfide (H_{2}S) also share identical symmetry operations. They both undergo the identity operation E, one C_{2} rotation, and two σ_{v} reflections without altering their identities, so they are both placed in one point group, C_{2v}, with order 4. 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. 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  Simple description of typical geometry  Example 1  Example 2  Example 3 

C_{1}  E  no symmetry, chiral  bromochlorofluoromethane (both enantiomers shown) 
lysergic acid 
Lleucine and most other αamino acids except glycine 
C_{s}  E σ_{h}  mirror plane  thionyl chloride 
hypochlorous acid 
chloroiodomethane 
C_{i}  E i  inversion center  mesotartaric acid 
mucic acid (mesogalactaric acid) 

C_{∞v}  E 2C_{∞}^{Φ} ∞σ_{v}  linear  hydrogen fluoride (and all other heteronuclear diatomic molecules) 
nitrous oxide (dinitrogen monoxide) 
hydrocyanic acid (hydrogen cyanide) 
D_{∞h}  E 2C_{∞}^{Φ} ∞σ_{i} i 2S_{∞}^{Φ} ∞C_{2}  linear with inversion center  oxygen (and all other homonuclear diatomic molecules) 
carbon dioxide 
acetylene (ethyne) 
C_{2}  E C_{2}  "open book geometry", chiral  hydrogen peroxide 
hydrazine 
tetrahydrofuran (twist conformation) 
C_{3}  E C_{3} C_{3}^{2}  propeller, chiral  triphenylphosphine 
triethylamine 
phosphoric acid 
C_{2h}  E C_{2} i σ_{h}  planar with inversion center, no vertical plane  trans1,2dichloroethylene 
transdinitrogen difluoride 
transazobenzene 
C_{2v}  E C_{2} σ_{v}(xz) σ_{v}'(yz)  angular (H_{2}O) or seesaw (SF_{4})  water 
sulfur tetrafluoride 
Dichloromethane 
C_{3h}  E C_{3} C_{3}^{2} σ_{h} S_{3} S_{3}^{5}  propeller  boric acid 
phloroglucinol (1,3,5trihydroxybenzene) 

C_{3v}  E 2C_{3} 3σ_{v}  trigonal pyramidal  ammonia (if pyramidal inversion is neglected) 
phosphorus oxychloride 
cobalt tetracarbonyl hydride, HCo(CO)_{4} 
C_{4v}  E 2C_{4} C_{2} 2σ_{v} 2σ_{d}  square pyramidal  xenon oxytetrafluoride 
pentaborane(9), B_{5}H_{9} 
nitroprusside anion [Fe(CN)_{5}(NO)]^{2−} 
C_{5}  E 2C_{5} 2C_{5}^{2}  fivefold rotational symmetry  Creactive protein 

C_{5v}  E 2C_{5} 2C_{5}^{2} 5σ_{v}  'milking stool' complex  Cyclopentadienyl nickel nitrosyl (CpNiNO) 
corannulene 

D_{2}  E C_{2}(x) C_{2}(y) C_{2}(z)  twist, chiral  biphenyl (skew conformation) 
twistane (C_{10}H_{16}) 

D_{3}  E C_{3}(z) 3C_{2}  triple helix, chiral  Tris(ethylenediamine)cobalt(III) cation 
tris(oxalato)iron(III) anion 

D_{2h}  E C_{2}(z) C_{2}(y) C_{2}(x) i σ(xy) σ(xz) σ(yz)  planar with inversion center, vertical plane  ethylene 
pyrazine 
diborane 
D_{3h}  E 2C_{3} 3C_{2} σ_{h} 2S_{3} 3σ_{v}  trigonal planar or trigonal bipyramidal  boron trifluoride 
phosphorus pentachloride 
cyclopropane 
D_{4h}  E 2C_{4} C_{2} 2C_{2}' 2C_{2}" i 2S_{4} σ_{h} 2σ_{v} 2σ_{d}  square planar  xenon tetrafluoride 
octachlorodimolybdate(II) anion 
Trans[Co^{III}(NH_{3})_{4}Cl_{2}]^{+} (excluding H atoms) 
D_{5h}  E 2C_{5} 2C_{5}^{2} 5C_{2} σ_{h} 2S_{5} 2S_{5}^{3} 5σ_{v}  pentagonal  cyclopentadienyl anion 
ruthenocene 
C_{70} 
D_{6h}  E 2C_{6} 2C_{3} C_{2} 3C_{2}' 3C_{2}‘’ i 2S_{3} 2S_{6} σ_{h} 3σ_{d} 3σ_{v}  hexagonal  benzene 
bis(benzene)chromium 
coronene (C_{24}H_{12}) 
D_{7h}  E C_{7} S_{7} 7C_{2} σ_{h} 7σ_{v}  heptagonal  tropylium (C_{7}H_{7}^{+}) cation 

D_{8h}  E C_{8} C_{4} C_{2} S_{8} i 8C_{2} σ_{h} 4σ_{v} 4σ_{d}  octagonal  cyclooctatetraenide (C_{8}H_{8}^{2−}) anion 
uranocene 

D_{2d}  E 2S_{4} C_{2} 2C_{2}' 2σ_{d}  90° twist  allene 
tetrasulfur tetranitride 
diborane(4) (excited state) 
D_{3d}  E 2C_{3} 3C_{2} i 2S_{6} 3σ_{d}  60° twist  ethane (staggered rotamer) 
dicobalt octacarbonyl (nonbridged isomer) 
cyclohexane chair conformation 
D_{4d}  E 2S_{8} 2C_{4} 2S_{8}^{3} C_{2} 4C_{2}' 4σ_{d}  45° twist  sulfur (crown conformation of S_{8}) 
dimanganese decacarbonyl (staggered rotamer) 
octafluoroxenate ion (idealized geometry) 
D_{5d}  E 2C_{5} 2C_{5}^{2} 5C_{2} i 2S_{10}^{3} 2S_{10} 5σ_{d}  36° twist  ferrocene (staggered rotamer) 

S_{4}  E 2S_{4} C_{2}  1,2,3,4tetrafluorospiropentane (meso isomer) 

T_{d}  E 8C_{3} 3C_{2} 6S_{4} 6σ_{d}  tetrahedral  methane 
phosphorus pentoxide 
adamantane 
T_{h}  E 4C_{3} 4C_{3}^{2} i 3C_{2} 4S_{6} 4S_{6}^{5} 3σ_{h}  pyritohedron  
O_{h}  E 8C_{3} 6C_{2} 6C_{4} 3C_{2} i 6S_{4} 8S_{6} 3σ_{h} 6σ_{d}  octahedral or cubic  sulfur hexafluoride 
molybdenum hexacarbonyl 
cubane 
I  E 12C_{5} 12C_{5}^{2} 20C_{3} 15C_{2}  chiral icosahedral or dodecahedral  Rhinovirus 

I_{h}  E 12C_{5} 12C_{5}^{2} 20C_{3} 15C_{2} i 12S_{10} 12S_{10}^{3} 20S_{6} 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:
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.
Character tables
For any group, its character table gives a tabulation (for the classes of the group) of the characters (the sum of the diagonal elements) of the matrices of all the irreducible representations of the group. As the number of irreducible representations equals the number of classes, the character table is square.
The representations are labeled according to a set of conventions:
 A, when rotation around the principal axis is symmetrical
 B, when rotation around the principal axis is asymmetrical
 E and T are doubly and triply degenerate representations, respectively
 when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion.
 with point groups C_{∞v} and D_{∞h} the symbols are borrowed from angular momentum description: Σ, Π, Δ.
The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the righthand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.
The character table for the C_{2v} symmetry point group is given below:
C_{2v}  E  C_{2}  σ_{v}(xz)  σ_{v}'(yz)  

A_{1}  1  1  1  1  z  x^{2}, y^{2}, z^{2} 
A_{2}  1  1  −1  −1  R_{z}  xy 
B_{1}  1  −1  1  −1  x, R_{y}  xz 
B_{2}  1  −1  −1  1  y, R_{x}  yz 
Consider the example of water (H_{2}O), which has the C_{2v} symmetry described above. The 2p_{x} orbital of oxygen has B_{1} symmetry as in the fourth row of the character table above, with x in the sixth column). It is oriented perpendicular to the plane of the molecule and switches sign with a C_{2} and a σ_{v}'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B_{1} irreducible representation. Likewise, the 2p_{z} orbital is seen to have the symmetry of the A_{1} irreducible representation (i.e.: none of the symmetry operations change it), 2p_{y} B_{2}, and the 3d_{xy} orbital A_{2}. These assignments and others are noted in the rightmost two columns of the table.
Historical background
Hans Bethe used characters of point group operations in his study of ligand field theory in 1929, and Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy. The first character tables were compiled by László Tisza (1933), in connection to vibrational spectra. Robert Mulliken was the first to publish character tables in English (1933), and E. Bright Wilson used them in 1934 to predict the symmetry of vibrational normal modes. The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.
Molecular rotation and molecular nonrigidity
As discussed above in the section Point groups and permutationinversion groups, point groups are useful for classifying the vibrational and electronic states of rigid molecules (sometimes called semirigid molecules) which undergo only small oscillations about a single equilibrium geometry. LonguetHiggins introduced a more general type of symmetry group suitable not only for classifying the vibrational and electronic states of rigid molecules but also for classifying their rotational and nuclear spin states. Further, such groups can be used to classify the states of nonrigid (or fluxional) molecules that tunnel between equivalent geometries (called versions) and to allow for the distorting effects of molecular rotation. These groups are known as permutationinversion groups, because the symmetry operations in them are energetically feasible permutations of identical nuclei, or inversion with respect to the center of mass (the parity operation), or a combination of the two.
For example, ethane (C_{2}H_{6}) has three equivalent staggered conformations. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C_{3} axis. Although each conformation has D_{3d} symmetry, as in the table above, description of the internal rotation and associated quantum states and energy levels requires the more complete permutationinversion group G_{36}.
Similarly, ammonia (NH_{3}) has two equivalent pyramidal (C_{3v}) conformations which are interconverted by the process known as nitrogen inversion. This is not the point group inversion operation i used for centrosymmetric rigid molecules (i.e., the inversion of vibrational displacements and electronic coordinates in the nuclear center of mass) since NH_{3} has no inversion center and is not centrosymmetric. Rather, it is the inversion of the nuclear and electronic coordinates in the molecular center of mass (sometimes called the parity operation), which happens to be energetically feasible for this molecule. The appropriate permutationinversion group to be used in this situation is D_{3h}(M) which is isomorphic with the point group D_{3h}.
Additionally, as examples, the methane (CH_{4}) and H_{3}^{+} molecules have highly symmetric equilibrium structures with T_{d} and D_{3h} point group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational centrifugal distortion. The permutationinversion groups required for the complete study of CH_{4} and H_{3}^{+} are T_{d}(M) and D_{3h}(M), respectively.
In its ground (N) electronic state the ethylene molecule C_{2}H_{4} has D_{2h} point group symmetry whereas in the excited (V) state it has D_{2d} symmetry. To treat these two states together it is necessary to allow torsion and to use the double group of the permutationinversion group G_{16}.
A second and less general approach to the symmetry of nonrigid molecules is due to Altmann. In this approach the symmetry groups are known as Schrödinger supergroups and consist of two types of operations (and their combinations): (1) the geometric symmetry operations (rotations, reflections, inversions) of rigid molecules, and (2) isodynamic operations, which take a nonrigid molecule into an energetically equivalent form by a physically reasonable process such as rotation about a single bond (as in ethane) or a molecular inversion (as in ammonia).
See also
 Parity (physics) § Molecules
 Irreducible representation § Applications in theoretical physics and chemistry
 WoodwardHoffmann rules § Correlation diagrams
 Hapticity § Hapticity and fluxionality
 Character table
 Crystallographic point group
 Point groups in three dimensions
 Symmetry of diatomic molecules
 Symmetry in quantum mechanics