# 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 Woodward-Hoffmann 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 X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.

## Point group symmetry concepts

Examples of the relationship between chirality and symmetry :
| Rotational axis (C$_n$) | Improper rotational elements (S$_n$) | Improper rotational elements (S$_n$) | 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 ${\tfrac {360^{\circ }}{n}}$ results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated C$_n$. Examples are the 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 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). 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) of the molecule always look identical. For example, whenever there is an oxygen atom in some point (x,y,z), then there also has to be an oxygen atom in the point (−x,−y,−z). There may or may not be an atom at the inversion center itself. An inversion center is a special case of having a rotation-reflection axis about an angle of 180° through the center. Examples are xenon tetrafluoride (a square planar molecule), 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.
* Rotation-reflection axis: an axis around which a rotation by ${\tfrac {360^{\circ }}{n}}$, 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 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 z-axis and a reflection in the xy-plane, 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$$^2$ and rotation by 360°, which is equivalent to identity, E.C2v point group multiplication tab
    Point group C$_3$ Multiplication table EC$_3$C$_3$$^2$ E EC$_3$C$_3$$^2$ C$_3$ C$_3$C$_3$$^2$E C$_3$$^2$ C$_3$$^2$EC$_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

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.

### 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 | L-leucine and most other α-amino acids except glycine |
| C$_s$ | E σ | mirror plane | thionyl chloride | hypochlorous acid | chloroiodomethane |
| C$_i$ | E i | inversion center | meso-tartaric acid | mucic acid (meso-galactaric acid) | 1,2 dibromo 1,2 dichloroethane |
| 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 | trans-1,2-dichloroethylene | trans-dinitrogen difluoride | trans-azobenzene |
| C$_{2v}$ | E C$_2$ σ$_v$(xz) σ$_v$'(yz) | angular (H$_2$O) or see-saw (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 (1,3,5-trihydroxybenzene) | benzotrifuroxan |
| 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$$^2$ | five-fold rotational symmetry | C-reactive protein | [Fe(Me$_5$-Cp)(P$_5$)] | Corannulene derivative |
| C$_{5v}$ | E 2C$_5$ 2C$_5$$^2$ 5σ$_v$ | 'milking stool' complex | Cyclopentadienyl nickel nitrosyl (CpNiNO) | corannulene | Pentamethylcyclopentadienyl nickel nitrosyl (Cp*NiNO) |
| D$_2$ | E C$_2$(x) C$_2$(y) C$_2$(z) | twist, chiral | biphenyl (skew conformation) | twistane (C$_{10}$H$_{16}$) | (δ,δ)-trans-[Co(en)$_2$Cl$_2$]$^+$ |
| D$_3$ | E C$_3$(z) 3C$_2$ | triple helix, chiral | Tris(ethylenediamine)cobalt(III) cation | tris(oxalato)iron(III) anion | tris(en)cobalt(III) |
| 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−}$) anion | uranocene | bis(cot)thorium(IV) |
| D$_{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 (non-bridged 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}$$^3$ 2S$_{10}$ 5σ$_d$ | 36° twist | ferrocene (staggered rotamer) | Ruthenocene (stag) | fulleride ion |
| S$_4$ | E 2S$_4$ C$_2$ | | 1,2,3,4-tetrafluorospiropentane (meso isomer) | tetramethyl-cot | bis(dth)copper(I) |
| T$_d$ | E 8C$_3$ 3C$_2$ 6S$_4$ 6σ$_d$ | tetrahedral | methane | phosphorus pentoxide | adamantane |
| T$_h$ | E 4C$_3$ 4C$_3$$^2$ i 3C$_2$ 4S$_6$ 4S$_6$$^5$ 3σ$_h$ | pyritohedron | [Fe(C$_5$H$_5$N)$_6$]$^{2+}$ | [Th(NO$_3$)$_6$]$^{2-}$ | [Fe(H$_2$O)$_6$]$^{2+}$$^/$$^{3+}$ |
| O$_h$ | E 8C$_3$ 6C$_2$ 6C$_4$ 3C$_2$ i 6S$_4$ 8S$_6$ 3σ$_h$ 6σ$_d$ | octahedral or cubic | sulfur hexafluoride | molybdenum hexacarbonyl | cubane |
| I | E 12C$_5$ 12C$_5$$^2$ 20C$_3$ 15C$_2$ | chiral icosahedral or dodecahedral | Rhinovirus | snub dodecahedron | human polio virus |
| 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 and their characters

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. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to the C$_{2v}$ point group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiar C2 operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 x 3 representation for this operation.
    ${\begin{bmatrix}1\\3\\2\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\\\end{bmatrix}}\times {\begin{bmatrix}1\\2\\3\\\end{bmatrix}}_$
This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1's on the leading diagonal (top left to bottom right) and 0's elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves.
    $\underbrace {\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\\\end{bmatrix}} _{C_{2}}\times \underbrace {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}} _{\sigma _{\text{v}}}=\underbrace {\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\\\end{bmatrix}} _{\sigma '_{\text{v}}}$
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 direct sum of the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the traces or characters of the four matrices. Asymmetric point groups such as C$_{2v}$ only have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters.
| 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 |
Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C$_2$ representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A$_1$+ B$_1$
Robert Mulliken was the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C$_{2h}$ example below. (Subscript 3 also appears in D$_2$ ) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector x transforms as irrep B$_1$ under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors.
Point group C$_{2h}$ has the operations {E, C$_2$, i, σ$_h$ } and the 1,5-dibromonapthalene (C$_{10}$H$_6$Br$_2$) shown in the figure belongs to this symmetry group. It is possible to construct four 18 x 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 1D irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two toms alone during the molecular C$_2$ rotation is
    ${\begin{bmatrix}5\\1\\\end{bmatrix}}={\begin{bmatrix}0&1\\1&0\\\end{bmatrix}}\times {\begin{bmatrix}1\\5\\\end{bmatrix}}_$
with character 0. When this computation is carried out for each of the operations above the characters obtained are (2,0,0.2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two1D irreps, the second column (C$_2$) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are A$_g$ + B$_u$. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(A$_g$ + B$_u$).
| C$_{2h}$ | E | C$_2$ | i | σ$_h$ | | |
| A$_g$ | 1 | 1 | 1 | 1 | R$_z$ | xy, x$^2$, y$^2$, z$^2$ |
| B$_g$ | 1 | -1 | 1 | −1 | Rx, R$_y$ | xz,yz |
| A$_u$ | 1 | 1 | −1 | −1 | z | |
| B$_u$ | 1 | −1 | −1 | 1 | x, y | |
Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps.
Dihedral point groups contain a cyclic group of the same rotational order: so group D$_n$ always contains group C$_n$ as an index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E$_{−x}$ and E$_{+x}$ cyclic irreps combine to form single E$_x$ 2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60 degree rotation about the main axis becomes equivalent to a (360 - 60) degree rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions, Σ, Π, Δ that follow from early linear molecule calculations.
In summary, 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.

## Historical background

All of the group operations described above and the symbols for crystallographic point groups themselves were first published by Arthur Schoenflies in 1891 but the groups had been applied by other researchers to the external morphology of crystals much earlier in the 19th century.
In 1914 Max von Laue published the results of experiments using x-ray diffraction to elucidate the internal structures of crystals producing a limited version of a table of "Laue classes" shown to the right. When adapted for molecular work this table first divides point groups into three kinds: asymmetric, symmetric and spherical tops. These are categories related to the angular momentum of molecules, having respectively 3, 2 and 1 distinct values of angular momentum. A further sub-division into systems is defined by the rotational group G in the leftmost column then into rows of Laue classes. Every point group in a Laue class has exactly the same abstract group structure except the centred group in the rightmost column which is the direct product of the rotational group with inversion. It follows that all groups in a Laue class have the same order except the centred group which is twice that of the others. Laue found that x-ray diffraction was unable to distinguish between point groups of a Laue class.
E. Bright Wilson used character tables in 1934 to predict the symmetry of vibrational normal modes.
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. It is important to note that, since all the point groups of a Laue class have the same abstract structure, they also have exactly the same irreducible representations and character tables. As in x-ray crystallography many properties in molecular work are decided by the Laue class.
The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy

## Symmetry of molecular orbitals

When Schrodinger's 3D wave equation is applied to a one-electron atom it provides a number of solutions called wave functions that are then used to label the allowed energy levels in that atom. Exact solutions of this kind are usually described by three quantum numbers, n, l and m from which the probable radial and angular distribution of the electron around the atom can be computed. This type of deduction leads to the familiar s, p, d, f,.... description of atomic orbitals based on the l and m quantum numbers. Each solution is a base vector from which more complex structures may be constructed. Descriptions of many-electron atoms use the one-electron model to build models that are sometimes pictured as multiple electrons in the simple structure. Molecular orbitals then take linear combinations of atomic orbitals (LCAOs) to explain the distribution of electrons over multiple atoms within a molecule. Atomic orbital symmetry follows from the angular part of the wave function which increases in complexity in the series s,p,d,f,... so that s orbitals only have radial symmetry while p orbital base vectors have a symmetry identical to that of the Cartesian polar base vectors.
Consider the example of water (H$_2$O), which has the C$_{2v}$ symmetry described above. The 2p$_x$ orbital of oxygen has, like the x base vector, B$_1$ symmetry. 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 are noted in the rightmost columns of the table.
Each molecular orbital also has the symmetry of one irreducible representation. For example, ethylene (C$_2$H$_4$) has symmetry group D$_{2h}$, and its highest occupied molecular orbital (HOMO) is the bonding pi orbital which forms a basis for its irreducible representation B$_{1u}$.

## The molecular symmetry group

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 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, can be achieved through the use of the appropriate permutation-inversion group (called the molecular symmetry group), as introduced by Longuet-Higgins.

## Symmetry of vibrational modes

Each normal mode of molecular vibration has a symmetry which forms a basis for one irreducible representation of the molecular symmetry group. For example, the water molecule has three normal modes of vibration: symmetric stretch in which the two O-H bond lengths vary in phase with each other, asymmetric stretch in which they vary out of phase, and bending in which the bond angle varies. The molecular symmetry of water is C$_{2v}$ with four irreducible representations A$_1$, A$_2$, B$_1$ and B$_2$. The symmetric stretching and the bending modes have symmetry A$_1$, while the asymmetric mode has symmetry B$_2$. The overall symmetry of the three vibrational modes is therefore Γ$_{vib}$ = 2A$_1$ + B$_2$.

### Vibrational modes of ammonia

The molecular symmetry of ammonia (NH$_3$) is C$_{3v}$, with symmetry operations E, C$_3$ and σ$_v$. For N = 4 atoms, the number of vibrational modes for a non-linear molecule is 3N-6 = 6, due to the relative motion of the nitrogen atom and the three hydrogen atoms. All three hydrogen atoms travel symmetrically along the N-H bonds, either in the direction of the nitrogen atom or away from it. This mode is known as symmetric stretch (v₁) and reflects the symmetry in the N-H bond stretching. Of the three vibrational modes, this one has the highest frequency.
In the Bending (ν₂) vibration, the nitrogen atom stays on the axis of symmetry, while the three hydrogen atoms move in different directions from one another, leading to changes in the bond angles. The hydrogen atoms move like an umbrella, so this mode is often referred to as the "umbrella mode".
There is also an Asymmetric Stretch mode (ν₃) in which one hydrogen atom approaches the nitrogen atom while the other two hydrogens move away.
The total number of degrees of freedom for each symmetry species (or irreducible representation) can be determined. Ammonia has four atoms, and each atom is associated with three vector components. The symmetry group C$_{3v}$ for NH$_3$ has the three symmetry species A$_1$, A$_2$ and E. The modes of vibration include the vibrational, rotational and translational modes.
Total modes = 3A$_1$ + A$_2$ + 4E. This is a total of 12 modes because each E corresponds to 2 degenerate modes (at the same energy).
Rotational modes = A$_2$ + E (3 modes)
Translational modes = A$_1$ + E
Vibrational modes = Total modes - Rotational modes - Translational modes = 3A$_1$ + A$_2$ + 4E - A$_2$ - E - A$_1$ - E = 2A$_1$ + 2E (6 modes).

### More examples of vibrational symmetry

* W(CO)$_6$ has octahedral geometry. The irreducible representation for the C-O stretching vibration is A$_{1g}$ + E$_g$ + T$_{1u}$ . Of these, only T$_{1u}$ is IR active.
* B$_2$H$_6$ (diborane) has D$_{2h}$ molecular symmetry. The terminal B-H stretching vibrations which are active in IR are B$_{2u}$ and B$_{3u}$.
* Fac-Mo(CO)$_3$(CH$_3$CH$_2$CN)$_3$, has C$_{3v}$ geometry. The irreducible representation for the C-O stretching vibration is A$_1$ + E. Both of which are IR active.

## Molecular rotation and molecular nonrigidity

As discussed above in § The molecular symmetry group, point groups are useful for classifying the vibrational and electronic states of rigid molecules (sometimes called semi-rigid molecules) which undergo only small oscillations about a single equilibrium geometry. Longuet-Higgins introduced the molecular symmetry group (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 non-rigid (or fluxional) molecules that tunnel between equivalent geometries and to allow for the distorting effects of molecular rotation. The symmetry operations in the molecular symmetry group are so-called 'feasible' permutations of identical nuclei, or inversion with respect to the center of mass (the parity operation), or a combination of the two, so that the group is sometimes called a "permutation-inversion group".
Examples of molecular nonrigidity abound. 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. The molecule 2-butyne (dimethylacetylene) has the same molecular symmetry group (G$_{36}$} as ethane but a very much lower torsional barrier. Similarly, ammonia (NH$_3$) has two equivalent pyramidal (C$_{3v}$) conformations which are interconverted by the process known as nitrogen inversion.
Additionally, the methane molecule (CH$_4$) and Trihydrogen cation {H$_3$$^+$) have highly symmetric equilibrium structures with T$_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.
Sometimes it is necessary to consider together electronic states having different point group symmetries at equilibrium. For example, 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 molecular symmetry group G$_{16}$.