Molecular symmetry in
chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
describes the
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
present in
moleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s 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 propertiesA chemical property is any of a material's properties that becomes evident during a chemical reaction; that is, any quality that can be established only by changing a substance's chemical identity...
, such as its dipole moment and its allowed
spectroscopic transitionsSpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
(based on selection rules such as the
Laporte ruleThe Laporte rule is a spectroscopic selection rule. It states that electronic transitions that conserve either symmetry or asymmetry with respect to an inversion center — i.e., g → g, or u → u respectively—are forbidden...
). Virtually every university level textbook on
physical chemistryPhysical chemistry is the study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems in terms of physical laws and concepts...
,
quantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
, and
inorganic chemistryInorganic chemistry is the branch of chemistry concerned with the properties and behavior of inorganic compounds. This field covers all chemical compounds except the myriad organic compounds , which are the subjects of organic chemistry...
devotes a chapter to symmetry.
While various frameworks for the study of molecular symmetry exist,
group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
is the predominant one. This framework is also useful in studying the symmetry of
molecular orbitalIn chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s, with applications such as the
Hückel methodThe Hückel method or Hückel molecular orbital method proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene,...
,
ligand field theoryLigand field theory describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes. A transition metal ion has nine valence atomic orbitals, five d, one s, and three p orbitals...
, and the Woodward-Hoffmann rules. Another framework on a larger scale is the use of
crystal systemIn crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...
s to describe
crystallographicCrystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
symmetry in bulk materials.
Many techniques for the practical assessment of molecular symmetry exist, including
X-ray crystallographyX-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...
and various forms of
spectroscopySpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
.
Spectroscopic notationSpectroscopic notation provides various ways to specify atomic ionization states, as well as atomic and molecular orbitals.-Ionization states:Spectroscopists customarily refer to the spectrum arising from a given ionization state of a given element by the element's symbol followed by a Roman numeral...
is based on symmetry considerations.
Symmetry concepts
The study of symmetry in molecules is an adaptation of mathematical
group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
.
Elements
The symmetry of a molecule can be described by 5 types of
symmetry elementA symmetry element is a point of reference about which symmetry operations can take place. In particular, symmetry elements can be centers of inversion, axes of rotation and mirror planes.-See also:* Symmetry* Group theory* Crystallography...
s.
- Symmetry axis: an axis around which a rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
by 
results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
and the C3 in ammoniaAmmonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...
. 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 z-axis in a Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
.
- 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
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...
. Water has two of them: one in the plane of the molecule itself and one perpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to it. A symmetry plane parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). 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 dihedralIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
(σd). 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
Xenon tetrafluoride is a chemical compound with chemical formula . It was the first discovered binary compound of a noble gas. It is produced by the chemical reaction of xenon with fluorine, , according to the chemical equation:...
(XeF4) where the inversion center is at the Xe atom, and benzeneBenzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
(C6H6) where the inversion center is at the center of the ring.
- Rotation-reflection 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 n-fold improper rotation axis, it is abbreviated Sn. Examples are present in tetrahedral silicon tetrafluorideSilicon tetrafluoride or Tetrafluorosilane is the chemical compound with the formula SiF4. This tetrahedral molecule is notable for having a remarkably narrow liquid range...
, with three S4 axes, and the staggered conformation of ethaneEthane is a chemical compound with chemical formula C2H6. It is the only two-carbon alkane that is an aliphatic hydrocarbon. At standard temperature and pressure, ethane is a colorless, odorless gas....
with one S6 axis.
- Identity, abbreviated to E, from the German 'Einheit' meaning Unity. 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 group-theoretical 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 operationIn the context of molecular symmetry, a symmetry operation may be defined as a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state....
s. They are often, although not always, distinguished from the respective elements by a
caretCaret usually refers to the spacing symbol ^ in ASCII and other character sets. In Unicode, however, the corresponding character is , whereas the Unicode character named caret is actually a similar but lowered symbol: ....
. 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 groupIn geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
is a set of symmetry operations forming a mathematical
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, for which at least one point remains fixed under all operations of the group. A
crystallographic point groupIn crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
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 notationThe Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...
.
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
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
: A(BC) = (AB)C
- the group contains the identity operation
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
, 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
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
A-1 in the group, for which AA-1 = A-1A = E
The
order of a groupIn group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
is the number of symmetry operations for that group.
For example, the point group for the
waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
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
ammoniaAmmonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...
molecule, which is pyramidal and contains a three-fold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an N-H bond and bisects the H-N-H 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
32, 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 theoryValence shell electron pair repulsion theory is a model in chemistry used to predict the shape of individual molecules based upon the extent of electron-pair electrostatic repulsion. It is also named Gillespie–Nyholm theory after its two main developers...
.
|
Point group
|
Symmetry elements >
Simple description, chiral A chiral molecule is a type of molecule that lacks an internal plane of symmetry and thus has a non-superimposable mirror image. The feature that is most often the cause of chirality in molecules is the presence of an asymmetric carbon atom....
if applicable |
Illustrative species
>-
|C1 |
E |
no symmetry, chiral |
lysergic acid Lysergic acid, also known as D-lysergic acid and -lysergic acid, is a precursor for a wide range of ergoline alkaloids that are produced by the ergot fungus and some plants. Amides of lysergic acid, lysergamides, are widely used as pharmaceuticals and as psychedelic drugs...
>-
|Cs |
E σh |
mirror plane, no other symmetry |
thionyl chlorideThionyl chloride is an inorganic compound with the formula SOCl2. It is a reactive chemical reagent used in chlorination reactions. It is a colorless, distillable liquid at room temperature and pressure that decomposes above 140 °C. Thionyl chloride is sometimes confused with sulfuryl...
, hypochlorous acidHypochlorous acid is a weak acid with the chemical formula HClO. It forms when chlorine dissolves in water. It cannot be isolated in pure form due to rapid equilibration with its precursor...
>-
|Ci |
E i |
Inversion center |
|
>-
|C
∞v
E 2C∞ σv |
linear |
hydrogen chlorideThe compound hydrogen chloride has the formula HCl. At room temperature, it is a colorless gas, which forms white fumes of hydrochloric acid upon contact with atmospheric humidity. Hydrogen chloride gas and hydrochloric acid are important in technology and industry...
, dicarbon monoxideDicarbon monoxide is an extremely reactive molecule that contains two carbon atoms and one oxygen atom. Dicarbon monoxide, covalently bonded, is a product of the photolysis of carbon suboxide. It is closely related to CO, CO2 and C3O2, and other oxocarbons.It is stable enough to observe reactions...
>-
|D∞h |
E 2C∞ ∞σi i 2S∞ ∞C2 |
linear with inversion center |
azideAzide is the anion with the formula N3−. It is the conjugate base of hydrazoic acid. N3− is a linear anion that is isoelectronic with CO2 and N2O. Per valence bond theory, azide can be described by several resonance structures, an important one being N−=N+=N−...
anion, carbon dioxideCarbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
>-
|C2 |
E C2 |
"open book geometry," chiral |
hydrogen peroxideHydrogen peroxide is the simplest peroxide and an oxidizer. Hydrogen peroxide is a clear liquid, slightly more viscous than water. In dilute solution, it appears colorless. With its oxidizing properties, hydrogen peroxide is often used as a bleach or cleaning agent...
>-
|C3 |
E C3 |
propeller, chiral |
triphenylphosphine Triphenylphosphine is a common organophosphorus compound with the formula P3 - often abbreviated to PPh3 or Ph3P. It is widely used in the synthesis of organic and organometallic compounds. PPh3 exists as relatively air stable, colorless crystals at room temperature...
>-
|C2h |
E C2 i σh |
planar with inversion center |
trans-1,2-dichloroethylene |
|
|C
3h >
E C3 C32 σh S3 S35 |
propeller |
Boric acid Boric acid, also called hydrogen borate or boracic acid or orthoboric acid or acidum boricum, is a weak acid of boron often used as an antiseptic, insecticide, flame retardant, as a neutron absorber, and as a precursor of other chemical compounds. It exists in the form of colorless crystals or a...
>-
|C2v |
E C2 σv(xz) σv'(yz) |
angular (H2O) or see-saw (SF4) |
sulfur tetrafluorideSulfur tetrafluoride is the chemical compound with the formula SF4. This species exists as a gas at standard conditions. It is a corrosive species that releases dangerous HF upon exposure to water or moisture...
, sulfuryl fluorideSulfuryl fluoride is the inorganic compound with the formula SO2F2. This easily condensed gas has properties more similar to sulfur hexafluoride than sulfuryl chloride, being resistant to hydrolysis even up to 150 °C...
>-
|C3v |
E 2C3 3σv |
trigonal pyramidal |
ammoniaAmmonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...
, phosphorus oxychloride
>-
|C4v |
E 2C4 C2 2σv 2σd |
square pyramidal |
xenon oxytetrafluoride Xenon oxytetrafluoride is an inorganic chemical compound. As are all xenon compounds, it is extremely reactive and unstable, and hydrolyses in water to give dangerously hazardous and corrosive products:...
>-
|D2 |
E C2(x) C2(y) C2(z) |
twist, chiral |
cyclohexane twist conformation A cyclohexane conformation is any of several three-dimensional shapes that a cyclohexane molecule can assume while maintaining the integrity of its chemical bonds....
>-
|D3 |
E C3(z) 3C2 |
triple helix, chiral |
Tris(ethylenediamine)cobalt(III) cationTriscobalt chloride is a coordination complex with the formula [Co3]Cl3 . This complex was important in the history of coordination chemistry because of its stability and its stereochemistry. Many different salts have been described...
>-
|D2h |
E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz) |
planar with inversion center |
ethyleneEthylene is a gaseous organic compound with the formula . It is the simplest alkene . Because it contains a carbon-carbon double bond, ethylene is classified as an unsaturated hydrocarbon. Ethylene is widely used in industry and is also a plant hormone...
, dinitrogen tetroxideDinitrogen tetroxide is the chemical compound N2O4. It is a useful reagent in chemical synthesis. It forms an equilibrium mixture with nitrogen dioxide; some call this mixture dinitrogen tetroxide, while some call it nitrogen dioxide.Dinitrogen tetroxide is a powerful oxidizer, making it highly...
, diboraneDiborane is the chemical compound consisting of boron and hydrogen with the formula B2H6. It is a colorless gas at room temperature with a repulsively sweet odor. Diborane mixes well with air, easily forming explosive mixtures. Diborane will ignite spontaneously in moist air at room temperature...
>-
|D3h |
E 2C3 3C2 σh 2S3 3σv |
trigonal planar or trigonal bipyramidal |
boron trifluorideBoron trifluoride is the chemical compound with the formula BF3. This pungent colourless toxic gas forms white fumes in moist air. It is a useful Lewis acid and a versatile building block for other boron compounds.-Structure and bonding:...
, phosphorus pentachloride
>-
|D4h |
E 2C4 C2 2C2' 2C2 i 2S4 σh 2σv 2σd |
square planar |
xenon tetrafluoride Xenon tetrafluoride is a chemical compound with chemical formula . It was the first discovered binary compound of a noble gas. It is produced by the chemical reaction of xenon with fluorine, , according to the chemical equation:...
>-
|D5h |
E 2C5 2C52 5C2 σh 2S5 2S53 5σv |
pentagonal |
ruthenoceneRuthenocene is an organoruthenium compound with the formula 2Ru. This pale yellow, volatile solid is classified as a sandwich compound and more specifically, as a metallocene.-Structure and bonding:...
, eclipsed ferroceneFerrocene is an organometallic compound with the formula Fe2. It is the prototypical metallocene, a type of organometallic chemical compound consisting of two cyclopentadienyl rings bound on opposite sides of a central metal atom. Such organometallic compounds are also known as sandwich compounds...
, C70 fullereneA fullerene is any molecule composed entirely of carbon, in the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are also called buckyballs, and they resemble the balls used in association football. Cylindrical ones are called carbon nanotubes or buckytubes...
>-
|D6h |
E 2C6 2C3 C2 3C2' 3C2 i 3S3 2S63 σh 3σd 3σv |
hexagonal |
benzene Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
, bis(benzene)chromiumBischromium is the organometallic compound with the formula Cr2. It is sometimes called dibenzenechromium. The compound played an important role in the development of sandwich compounds in organometallic chemistry and is the prototypical complex containg two arene ligands.-Preparation:The...
>-
|D2d |
E 2S4 C2 2C2' 2σd |
90° twist |
allene An allene is a compound in which one carbon atom has double bonds with each of its two adjacent carbon centres. Allenes are classified as polyenes with cumulated dienes. The parent compound of allene is propadiene. Compounds with an allene-type structure but with more than three carbon atoms are...
, tetrasulfur tetranitrideTetrasulfur tetranitride is an inorganic compound with the formula S4N4. This gold-poppy coloured solid is the most important binary sulfur nitride, which are compounds that contain only the elements sulfur and nitrogen. It is a precursor to many S-N compounds and has attracted wide interest for...
>-
|D3d |
E C3 3C2 i 2S6 3σd |
60° twist |
ethaneEthane is a chemical compound with chemical formula C2H6. It is the only two-carbon alkane that is an aliphatic hydrocarbon. At standard temperature and pressure, ethane is a colorless, odorless gas....
(staggered rotamer), cyclohexane chair conformationA cyclohexane conformation is any of several three-dimensional shapes that a cyclohexane molecule can assume while maintaining the integrity of its chemical bonds....
>-
|D4d |
E 2S8 2C4 2S83 C2 4C2' 4σd |
45° twist |
dimanganese decacarbonyl Dimanganese decacarbonyl is the chemical compound with the formula Mn210. This metal carbonyl is an important reagent in the organometallic chemistry of manganese.-Synthesis:...
(staggered rotamer)
>-
|D5d |
E 2C5 2C52 5C2 i 3S103 2S10 5σd |
36° twist |
ferroceneFerrocene is an organometallic compound with the formula Fe2. It is the prototypical metallocene, a type of organometallic chemical compound consisting of two cyclopentadienyl rings bound on opposite sides of a central metal atom. Such organometallic compounds are also known as sandwich compounds...
(staggered rotamer)
>-
|Td |
E 8C3 3C2 6S4 6σd |
tetrahedralIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
methane Methane is a chemical compound with the chemical formula . It is the simplest alkane, the principal component of natural gas, and probably the most abundant organic compound on earth. The relative abundance of methane makes it an attractive fuel...
, phosphorus pentoxidePhosphorus pentoxide is a chemical compound with molecular formula P4O10 . This white crystalline solid is the anhydride of phosphoric acid. It is a powerful desiccant.-Structure:...
, adamantaneAdamantane is a colorless, crystalline chemical compound with a camphor-like odor. With a formula C10H16, it is a cycloalkane and also the simplest diamondoid. Adamantane molecules consist of three cyclohexane rings arranged in the "armchair" configuration. It is unique in that it is both rigid...
>-
|Oh |
E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd |
octahedralIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
or cubic |
cubaneCubane is a synthetic hydrocarbon molecule that consists of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom. A solid crystalline substance, cubane is one of the Platonic hydrocarbons. It was first synthesized in 1964 by Philip Eaton, a...
, sulfur hexafluorideSulfur hexafluoride is an inorganic, colorless, odorless, and non-flammable greenhouse gas. has an octahedral geometry, consisting of six fluorine atoms attached to a central sulfur atom. It is a hypervalent molecule. Typical for a nonpolar gas, it is poorly soluble in water but soluble in...
>-
|Ih |
E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ |
icosahedralIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
C60A fullerene is any molecule composed entirely of carbon, in the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are also called buckyballs, and they resemble the balls used in association football. Cylindrical ones are called carbon nanotubes or buckytubes...
, B12H122-Caesium dodecaborate is an inorganic compound with the formula Cs2B12H12. It is a salt, with caesium cations and [B12H12]2− anions.-Structure:...
>- |
Representations
The symmetry operations can be
represented in many waysIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
. A convenient representation is by
matricesIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. For any vector representing a point in Cartesian coordinates, left-multiplying 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
orbitalAn atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
, 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 ( 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
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
description: ΣSigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...
, ΠPi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...
, ΔDelta is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet...
.
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 |
C2 |
σv(xz) |
σv'(yz) |
|
|
>-
| A
1
1 |
1 |
1 |
1 |
|
> x
2, y
2, z
2
|-
| A
2
1 |
1 |
−1 |
−1 |
Rz |
|
>-
| B
1
1 |
−1 |
1 |
−1 |
x, Ry |
|
>-
| B
2
1 |
−1 |
−1 |
1 |
y, Rx |
yz |
Consider the example of water (H
2O), which has the C
2v symmetry described above. The 2p
x orbitalAn atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
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 BetheHans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...
used characters of point group operations in his study of
ligand field theoryLigand field theory describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes. A transition metal ion has nine valence atomic orbitals, five d, one s, and three p orbitals...
in 1929, and Eugene Wigner used group theory to explain the selection rules of
atomic spectroscopyAtomic spectroscopy is the determination of elemental composition by its electromagnetic or mass spectrum. Atomic spectroscopy is closely related to other forms of spectroscopy. It can be divided by atomization source or by the type of spectroscopy used. In the latter case, the main division is...
. The first character tables were compiled by
László TiszaLászló Tisza was Professor of Physics Emeritus at MIT. He was a colleague of famed physicists Edward Teller, Lev Landau and Fritz London, and initiated the two-fluid theory of liquid helium.-United States:...
(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 modeA normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
s. The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.
External links