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Molecular symmetry in
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
describes the
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
present in
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s 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
A chemical property is any of a material's properties that becomes evident during, or after, a chemical reaction; that is, any quality that can be established only by changing a substance's chemical identity.William L. Masterton, Cecile N. Hurley ...
, 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
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. This involves classifying the states of the molecule using the
irreducible representations
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
from the
character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters ...
of the symmetry group of the molecule. Symmetry is useful in the study of
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and 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 ...
s, with applications to the
Hückel method
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electro ...
, to
ligand field theory
Ligand field theory (LFT) 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 valen ...
, and to the
Woodward-Hoffmann rules. Many university level textbooks on
physical chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...
,
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
,
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
and
inorganic chemistry
Inorganic chemistry deals with synthesis and behavior of inorganic and organometallic compounds. This field covers chemical compounds that are not carbon-based, which are the subjects of organic chemistry. The distinction between the two disci ...
discuss symmetry. Another framework on a larger scale is the use of
crystal system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their po ...
s to describe
crystallographic
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The word ...
symmetry in bulk materials.
There are many techniques for determining the symmetry of a given molecule, including
X-ray crystallography
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
and various forms of
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
.
Spectroscopic notation
Spectroscopic notation provides a way to specify atomic ionization states, atomic orbitals, and molecular orbitals.
Ionization states
Spectroscopists customarily refer to the spectrum arising from a given ionization state of a given element by ...
is based on symmetry considerations.
Point group symmetry concepts
Elements
The point group symmetry of a molecule is defined by the presence or absence of 5 types of
symmetry element In chemistry and crystallography, a symmetry element is a point, line, or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane, an axis of rotation (either proper and improper), or a center of ...
.
* Symmetry axis: an axis around which a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
by
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
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
and the ''C''
3 axis in
ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous was ...
. 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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
* 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
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of refle ...
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
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to it. A symmetry plane
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
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) are identical. For example, if there is an oxygen atom in some point (x,y,z), then there is an oxygen atom in the point (−x,−y,−z). There may or may not be an atom at the inversion center itself. Examples are
xenon tetrafluoride
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:
: Xe + 2 →
This reaction is exothermic, rele ...
where the inversion center is at the Xe atom, and
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, ...
() 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 ''S''
''n''. Examples are present in tetrahedral
silicon tetrafluoride
Silicon tetrafluoride or tetrafluorosilane is a chemical compound with the formula Si F4. This colorless gas is notable for having a narrow liquid range: its boiling point is only 4 °C above its melting point. It was first prepared in 1771 ...
, with three ''S''
4 axes, and the
staggered conformation
In organic chemistry, a staggered conformation is a chemical conformation of an ethane-like Moiety (chemistry), moiety abcX–Ydef in which the substituents a, b, and c are at the maximum distance from d, e, and f; this requires the torsion ang ...
of
ethane
Ethane ( , ) is an organic chemical compound with chemical formula . At standard temperature and pressure, ethane is a colorless, odorless gas. Like many hydrocarbons, ethane is isolated on an industrial scale from natural gas and as a petr ...
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
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
, 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 In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...
, which leave the geometry of the molecule indistinguishable from the starting geometry. They are sometimes distinguished from symmetry elements by a
caret
Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofreade ...
or
circumflex
The circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from la, circumflexus "bent around"a ...
. 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
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
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:
: Xe + 2 →
This reaction is exothermic, rele ...
(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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
. In mathematics, a group is a set with a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
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:
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 C3, there are three symmetry operations: rotation by 120°, ''C''3, rotation by 240°, ''C''32 and rotation by 360°, which is equivalent to identity, ''E''.
:
This table also illustrates the following properties
- ''
Associative property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
'':
- ''existence of
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
'' property:
- ''existence of
inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
'':
The ''order'' of a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.
Point groups and permutation-inversion groups
![Determination_du_groupe_de_symetrie](https://upload.wikimedia.org/wikipedia/commons/1/1b/Determination_du_groupe_de_symetrie.svg)
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
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
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 summarizes all symmetry operations that all molecules in that category have.
The symmetry of a crystal, by contrast, is described by a
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
of symmetry operations, which includes
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
in space.
One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (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, requires the use of the appropriate permutation-inversion group as introduced by
Longuet-Higgins.
Point groups describe the geometrical symmetry of a molecule whereas permutation-inversion groups describe the energy-invariant symmetry.
Examples of point groups
Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl
3, POF
3, XeO
3, and NH
3 all share identical symmetry operations. They all can undergo the identity operation ''E'', two different ''C''
3 rotation operations, and three different σ
v plane reflections without altering their identities, so they are placed in one point group, ''C''
3v, with order 6.
Similarly, water (H
2O) and hydrogen sulfide (H
2S) 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
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
applicable to molecules, labelled using the
Schoenflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
, which is common in chemistry and molecular spectroscopy. The descriptions include common shapes of molecules, which can be explained by the
VSEPR model
Valence shell electron pair repulsion (VSEPR) theory ( , ), is a model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. It is also named the Gillespie-Nyholm th ...
. In each row, the descriptions and examples have no higher symmetries, meaning that the named point group captures ''all'' of the point symmetries.
Representations
A set of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
that multiply together in a way that mimics the multiplication table of the elements of a group is called a
representation of the group. For example, for the ''C''
2v point group, the following three matrices are part of a representation of the group:
:
Although an infinite number of such representations exist, the
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s (or "irreps") of the group are all that are needed as all other representations of the group can be described as a linear combination of the irreducible representations.
Character tables
For any group, its character table gives a tabulation (for the classes of the group) of the characters (the sum of the diagonal elements) of the matrices of all the irreducible representations of the group. As the number of irreducible representations equals the number of classes, the character table is square.
The representations are labeled according to a set of conventions:
* A, when rotation around the principal axis is symmetrical
* B, when rotation around the principal axis is asymmetrical
* E and T are doubly and triply degenerate representations, respectively
* when the point group has an inversion center, the subscript g (german: gerade or even) signals no change in sign, and the subscript u (''ungerade'' or uneven) a change in sign, with respect to inversion.
* with point groups ''C''
∞v and ''D''
∞h the symbols are borrowed from
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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:
Consider the example of water (H
2O), which has the ''C''
2v symmetry described above. The 2''p''
x orbital of oxygen has B
1 symmetry as in the fourth row of the character table above, with x in the sixth column). It is oriented perpendicular to the plane of the molecule and switches sign with a ''C''
2 and a σ
v'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus , corresponding to the B
1 irreducible representation. Likewise, the 2''p''
z orbital is seen to have the symmetry of the A
1 irreducible representation (''i.e''.: none of the symmetry operations change it), 2''p''
y B
2, and the 3''d''
xy orbital A
2. These assignments and others are noted in the rightmost two columns of the table.
Historical background
Hans Bethe
Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel Prize ...
used characters of point group operations in his study of
ligand field theory
Ligand field theory (LFT) 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 valen ...
in 1929, and
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
used group theory to explain the selection rules of
atomic spectroscopy
Atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have characteristic (signature) spectra, atomic spectroscopy, specifically the electromagnetic spectrum or mass spectrum, is appl ...
. The first character tables were compiled by
László Tisza
László Tisza (July 7, 1907 – April 15, 2009) was a Hungarian-born American physicist who 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-flui ...
(1933), in connection to vibrational spectra.
Robert Mulliken
Robert Sanderson Mulliken Note Longuet-Higgins' amusing title for reference B238 1965 on page 354 of this Biographical Memoir. The title should be "Selected papers of Robert S Mulliken." (June 7, 1896 – October 31, 1986) was an American ph ...
was the first to publish character tables in English (1933), and
E. Bright Wilson
Edgar Bright Wilson Jr. (December 18, 1908 – July 12, 1992) was an American chemist.
Wilson was a prominent and accomplished chemist and teacher, recipient of the National Medal of Science in 1975, Guggenheim Fellowships in 1949 and 1970, the ...
used them in 1934 to predict the symmetry of vibrational
normal mode
A normal mode of a dynamical 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 free motion described by the normal modes takes place at fixed frequencies. ...
s. The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.
Molecular rotation and molecular nonrigidity
As discussed above in the section Point groups and permutation-inversion groups, 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 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 (called ''versions'') and to allow for the distorting effects of molecular rotation. These groups are known as ''permutation-inversion'' groups, because the symmetry operations in them are energetically feasible permutations of identical nuclei, or inversion with respect to the center of mass (the ]parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
operation), or a combination of the two.
For example, ethane
Ethane ( , ) is an organic chemical compound with chemical formula . At standard temperature and pressure, ethane is a colorless, odorless gas. Like many hydrocarbons, ethane is isolated on an industrial scale from natural gas and as a petr ...
(C2H6) has three equivalent staggered conformation
In organic chemistry, a staggered conformation is a chemical conformation of an ethane-like Moiety (chemistry), moiety abcX–Ydef in which the substituents a, b, and c are at the maximum distance from d, e, and f; this requires the torsion ang ...
s. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the ''C''3 axis. Although each conformation has ''D''3d symmetry, as in the table above, description of the internal rotation and associated quantum states and energy levels requires the more complete permutation-inversion group ''G''36.
Similarly, ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous was ...
(NH3) has two equivalent pyramidal (''C''3v) conformations which are interconverted by the process known as nitrogen inversion In chemistry, pyramidal inversion (also umbrella inversion) is a fluxional process in compounds with a pyramidal molecule, such as ammonia (NH3) "turns inside out". It is a rapid oscillation of the atom and substituents, the molecule or ion passin ...
. This is not the point group inversion operation ''i'' used for centrosymmetric rigid molecules (i.e., the inversion of vibrational displacements and electronic coordinates in the nuclear center of mass) since NH3 has no inversion center and is not centrosymmetric. Rather, it is the inversion of the nuclear and electronic coordinates in the molecular center of mass (sometimes called the parity operation), which happens to be energetically feasible for this molecule. The appropriate permutation-inversion group to be used in this situation is ''D''3h(M) which is isomorphic with the point group ''D''3h.
Additionally, as examples, the methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...
(CH4) and H3+ molecules have highly symmetric equilibrium structures with ''T''d and ''D''3h point group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational
centrifugal distortion. The permutation-inversion groups required for the complete study of CH4 and H3+ are ''T''d(M) and ''D''3h(M), respectively.
In its ground (N) electronic state the ethylene molecule C2H4 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 permutation-inversion group ''G''16.
A second and less general approach to the symmetry of nonrigid molecules is due to Altmann.[Altmann S.L. (1977) ''Induced Representations in Crystals and Molecules'', Academic Press][Flurry, R.L. (1980) ''Symmetry Groups'', Prentice-Hall, , pp.115-127] In this approach the symmetry groups are known as ''Schrödinger supergroups'' and consist of two types of operations (and their combinations): (1) the geometric symmetry operations (rotations, reflections, inversions) of rigid molecules, and (2) ''isodynamic operations'', which take a nonrigid molecule into an energetically equivalent form by a physically reasonable process such as rotation about a single bond (as in ethane) or a molecular inversion (as in ammonia).[
]
See also
*
*
*
*
*Character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters ...
*Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...
*Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries tha ...
*Symmetry of diatomic molecules Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physic ...
*Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the ...
References
{{reflist
External links
Point group symmetry
@ Newcastle University
Molecular symmetry
@ Imperial College London
Molecular Point Group Symmetry Tables
Molecular Symmetry Online
@ The Open University of Israel
An internet lecture course on molecular symmetry @ Bergische Universitaet
DECOR – Symmetry
@ The Cambridge Crystallographic Data Centre
Details of the relation between point groups and permutation-inversion groups
by Philip Bunker
Philip R. Bunker is a British-Canadian scientist and author, known for his work
in theoretical chemistry and molecular spectroscopy.
Education and early work
Philip Bunker was educated at Battersea Grammar School in Streatham. He received a b ...
Symmetry
Theoretical chemistry