 Molecular symmetry

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions (based on selection rules such as the Laporte rule). Virtually every university level textbook on physical chemistry, quantum chemistry, and inorganic chemistry devotes a chapter to symmetry.^{[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. Spectroscopic notation is based on symmetry considerations.
Contents
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 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 (XeF_{4}) 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.
 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, its consideration is necessary for the grouptheoretical machinery to work properly. It is so called because it is analogous to multiplying by one (unity).
Operations
The 5 symmetry elements have associated with them 5 symmetry operations. They are often, although not always, distinguished from the respective elements by a caret. 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. Since C_{1} is equivalent to E, S_{1} to σ and S_{2} to i, all symmetry operations can be classified as either proper or improper rotations.
Point groups
A point group is a set of symmetry operations forming a mathematical group, for which at least one point remains fixed under all operations of the group. A crystallographic point group is a point group that is compatible with translational symmetry in three dimensions. There are a total of 32 crystallographic point groups, 30 of which relevant to chemistry. Their classification is based on the Schoenflies notation.
Group theory
A set of symmetry operations form a group, with operator the application of the operations itself, when:
 the result of consecutive application (composition) of any two operations is also a member of the group (closure).
 the application of the operations is associative: A(BC) = (AB)C
 the group contains the identity operation, denoted E, such that AE = EA = A for any operation A in the group.
 For every operation A in the group, there is an inverse element A^{1} in the group, for which AA^{1} = A^{1}A = E
The order of a group is the number of symmetry operations for that group.
For example, the point group for the water molecule is C_{2v}, with 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).
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 elements Simple description, chiral if applicable Illustrative species C_{1} E no symmetry, chiral CFClBrH, lysergic acid C_{s} E σ_{h} mirror plane, no other symmetry thionyl chloride, hypochlorous acid C_{i} E i Inversion center anti1,2dichloro1,2dibromoethane C_{∞v} E 2C_{∞} σ_{v} linear hydrogen chloride, dicarbon monoxide D_{∞h} E 2C_{∞} ∞σ_{i} i 2S_{∞} ∞C_{2} linear with inversion center dihydrogen, azide anion, 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 D_{5h} E 2C_{5} 2C_{5}^{2} 5C_{2} σ_{h} 2S_{5} 2S_{5}^{3} 5σ_{v} pentagonal ruthenocene, eclipsed ferrocene, C_{70} fullerene D_{6h} E 2C_{6} 2C_{3} C_{2} 3C_{2}' 3C_{2} i 3S_{3} 2S_{6}^{3} 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 C_{60}, B_{12}H_{12}^{2} 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:
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 0124575510
 ^ 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 0471907600
 ^ Physical Chemistry P. W. Atkins ISBN 0716728710
 ^ G. L. Miessler and D. A. Tarr “Inorganic Chemistry” 3rd Ed, Pearson/Prentice Hall publisher, ISBN 0130354716.
 ^ 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
 Molecular symmetry @ University of Exeter Link
 Molecular symmetry @ Imperial College London Link
 Molecular Point Group Symmetry Tables
 Symmetry @ Otterbein
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