Molecular symmetry in [1]^{[2]}^{[3]}^{[4]}^{[5]}
While various frameworks for the study of molecular symmetry exist, group theory is the predominant one. This framework is also useful in studying the symmetry of molecular orbitals, with applications such as the Hückel method, ligand field theory, and the WoodwardHoffmann rules. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.
Many techniques for the practical assessment of molecular symmetry exist, including Xray crystallography and various forms of spectroscopy, for example infrared spectroscopy of metal carbonyls. Spectroscopic notation is based on symmetry considerations.
Symmetry concepts
The study of symmetry in molecules is an adaptation of mathematical group theory.
Elements
The symmetry of a molecule can be described by 5 types of symmetry elements.

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 nfold rotational axis and abbreviated C_{n}. Examples are the C_{2} in water and the C_{3} 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 assigned the zaxis in a Cartesian coordinate system.

Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. 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. There may or may not be an atom at the center. 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 \tfrac{360^\circ} {n} , 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.

Identity, abbreviated to E, from the German 'Einheit' meaning unity.^{[6]} This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, 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
XeF_{4}, with square planar geometry, has two C
_{4} rotations in opposite directions and a C
_{2} rotation.
The 5 symmetry elements have associated with them 5 types of symmetry operations, which leave the molecule in a state indistinguishable from the starting state. 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 (90°) in opposite directions and a Ĉ_{2} rotation (180°). Since 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.
Molecular symmetry groups
Groups
The symmetry operations of a molecule (or other object) form a group, which is a mathematical structure usually denoted in the form (G,*)) consisting of a set G and a binary combination operation say '*' satisfying certain properties listed below.
In a molecular 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 molecular 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.
(2) 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)
(3) 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 )
(4) existence of inverse property:
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 group
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. 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. For some symmetries, an entire axis or an entire plane are fixed.
The symmetry of a crystal, however, is described by a space group of symmetry operations, which includes translations in space.
Examples
(1) The point group for the water molecule is C_{2v}, consisting of the symmetry operations E, C_{2}, σ_{v} and σ_{v}'. Its order is thus 4. Each operation is its own inverse. As an example of closure, 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}'. (Note that "Operation A followed by B to form C" is written BA = C).
(2) Another example is the ammonia molecule, which is pyramidal and contains a threefold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an NH bond and bisects the HNH bond angle opposite to that bond. Thus ammonia molecule belongs to the C_{3v} point group that has order 6: an identity element E, two rotation operations C_{3} and C_{3}^{2}, and three mirror reflections σ_{v}, σ_{v}' and σ_{v}".
Common point groups
The following table contains a list of point groups with representative molecules. The description of structure includes common shapes of molecules based on VSEPR theory.
Point group

Symmetry operations

Simple description of typical geometry

Example 1

Example 2

Example 3

C_{1}

E

no symmetry, chiral

bromochlorofluoromethane

lysergic acid


C_{s}

E σ_{h}

mirror plane, no other symmetry

thionyl chloride

hypochlorous acid

chloroiodomethane

C_{i}

E i

inversion center

(R,R) 1,2dichloro1,2dibromoethane (anti conformer)



C_{∞v}

E 2C_{∞} ∞σ_{v}

linear

Hydrogen fluoride

nitrous oxide
(dinitrogen monoxide)


D_{∞h}

E 2C_{∞} ∞σ_{i} i 2S_{∞} ∞C_{2}

linear with inversion center

oxygen

carbon dioxide


C_{2}

E C_{2}

"open book geometry," chiral

hydrogen peroxide



C_{3}

E C_{3}

propeller, chiral

triphenylphosphine



C_{2h}

E C_{2} i σ_{h}

planar with inversion center

trans1,2dichloroethylene



C_{3h}

E C_{3} C_{3}^{2} σ_{h} S_{3} S_{3}^{5}

propeller

boric acid



C_{2v}

E C_{2} σ_{v}(xz) σ_{v}'(yz)

angular (H_{2}O) or seesaw (SF_{4})

water

sulfur tetrafluoride

sulfuryl fluoride

C_{3v}

E 2C_{3} 3σ_{v}

trigonal pyramidal

ammonia

phosphorus oxychloride


C_{4v}

E 2C_{4} C_{2} 2σ_{v} 2σ_{d}

square pyramidal

xenon oxytetrafluoride



D_{2}

E C_{2}(x) C_{2}(y) C_{2}(z)

twist, chiral

cyclohexane twist conformation



D_{3}

E C_{3}(z) 3C_{2}

triple helix, chiral

Tris(ethylenediamine)cobalt(III) cation



D_{2h}

E C_{2}(z) C_{2}(y) C_{2}(x) i σ(xy) σ(xz) σ(yz)

planar with inversion center

ethylene

dinitrogen tetroxide

diborane

D_{3h}

E 2C_{3} 3C_{2} σ_{h} 2S_{3} 3σ_{v}

trigonal planar or trigonal bipyramidal

boron trifluoride

phosphorus pentachloride


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(III) anion


D_{5h}

E 2C_{5} 2C_{5}^{2} 5C_{2} σ_{h} 2S_{5} 2S_{5}^{3} 5σ_{v}

pentagonal

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


D_{2d}

E 2S_{4} C_{2} 2C_{2}' 2σ_{d}

90° twist

allene

tetrasulfur tetranitride


D_{3d}

E C_{3} 3C_{2} i 2S_{6} 3σ_{d}

60° twist

ethane (staggered rotamer)

cyclohexane chair conformation


D_{4d}

E 2S_{8} 2C_{4} 2S_{8}^{3} C_{2} 4C_{2}' 4σ_{d}

45° twist

dimanganese decacarbonyl (staggered rotamer)



D_{5d}

E 2C_{5} 2C_{5}^{2} 5C_{2} i 3S_{10}^{3} 2S_{10} 5σ_{d}

36° twist

ferrocene (staggered rotamer)



T_{d}

E 8C_{3} 3C_{2} 6S_{4} 6σ_{d}

tetrahedral

methane

phosphorus pentoxide

adamantane

O_{h}

E 8C_{3} 6C_{2} 6C_{4} 3C_{2} i 6S_{4} 8S_{6} 3σ_{h} 6σ_{d}

octahedral or cubic

cubane

sulfur hexafluoride


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
The symmetry operations can be represented in many ways. A convenient representation is by matrices. For any vector representing a point in Cartesian coordinates, leftmultiplying it gives the new location of the point transformed by the symmetry operation. Composition of operations corresponds to matrix multiplication. In the C_{2v} example this is:

\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_{v}} = \underbrace{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }_{\sigma'_{v}}
Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations.
Character tables
For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square.
The table itself consists of characters that represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But, for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric).
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 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, 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.^{[7]} 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.^{[8]} The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.^{[9]}
See also
References

^ Quantum Chemistry, Third Edition John P. Lowe, Kirk Peterson ISBN 012457551X

^ Physical Chemistry: A Molecular Approach by Donald A. McQuarrie, John D. Simon ISBN 0935702997

^ The chemical bond 2nd Ed. J.N. Murrell, S.F.A. Kettle, J.M. Tedder ISBN 047190760X

^ Physical Chemistry P.W. Atkins and J. de Paula (8th ed., W.H. Freeman 2006) ISBN 0716787598, chap.12

^ G. L. Miessler and D. A. Tarr Inorganic Chemistry (2nd ed., Pearson/Prentice Hall 1998) ISBN 0138418918, chap.4.

^ LEO Ergebnisse für "einheit"

^ Group Theory and its application to the quantum mechanics of atomic spectra, E. P. Wigner, Academic Press Inc. (1959)

^ Correcting Two LongStanding Errors in Point Group Symmetry Character Tables Randall B. Shirts J. Chem. Educ. 2007, 84, 1882. Abstract

^ Group Theory and the Vibrations of Polyatomic Molecules Jenny E. Rosenthal and G. M. Murphy Rev. Mod. Phys. 8, 317  346 (1936) doi:10.1103/RevModPhys.8.317
External links

Point group symmetry @ Newcastle University Link

Molecular symmetry @ Imperial College London Link

Molecular Point Group Symmetry Tables

Symmetry @ Otterbein
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