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 be used to predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions. Many university level textbooks on physical chemistry, quantum chemistry, and inorganic chemistry devote a chapter to symmetry.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}
The predominant framework for the study of molecular symmetry is group theory. Symmetry is useful in the study 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.
The study of symmetry in molecules makes use of group theory.
Rotational axis (C_{n}) 
Improper rotational elements (S_{n})  

Chiral no S_{n} 
Achiral mirror plane S_{1} = σ 
Achiral inversion centre S_{2} = i  
C_{1}  
C_{2} 
The point group symmetry of a molecule can be described by 5 types of symmetry element.
The five symmetry elements have associated with them five types of symmetry operation, 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.
The symmetry operations of a molecule (or other object) form a group. In mathematics, a group is a set with a binary operation that satisfies the four properties listed below.
In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C_{4} rotation about the zaxis and a reflection in the xyplane, 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.
(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.
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).^{[8]} Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, where S is the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.
This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarizes all symmetry operations that all molecules in that category have.^{[8]} The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations 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, 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 (vibrationelectronic) 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. The symmetry classification of the rotational levels, the eigenstates of the full (rovibronic nuclear spin) Hamiltonian, requires the use of the appropriate permutationinversion group as introduced by LonguetHiggins^{[9]}. The relation between point groups and permutationinversion groups is explained in this pdf file Link .
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.^{[10]} 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.^{[11]} Similarly, water (H_{2}O) and hydrogen sulfide (H_{2}S) also share identical symmetry operations. They both undergo the identity operation E, one C_{2} rotation, and two σ_{v} reflections without altering their identities, so they are both placed in one point group, C_{2v}, with order 4.^{[12]} 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.^{[8]}
The following table contains a list of point groups labelled using the Schoenflies notation, which is common in chemistry and molecular spectroscopy. The description of structure includes common shapes of molecules, which can be explained by the VSEPR model.
Point group  Symmetry operations  Simple description of typical geometry  Example 1  Example 2  Example 3  
C_{1}  E  no symmetry, chiral  bromochlorofluoromethane (both enantiomers shown) 
lysergic acid 
Lleucine and most other αamino acids except glycine  
C_{s}  E σ_{h}  mirror plane, no other symmetry  thionyl chloride 
hypochlorous acid 
chloroiodomethane  
C_{i}  E i  inversion center  mesotartaric acid 
mucic acid (mesogalactaric acid) 
(S,R) 1,2dibromo1,2dichloroethane (anti conformer)  
C_{∞v}  E 2C_{∞} ∞σ_{v}  linear  hydrogen fluoride (and all other heteronuclear diatomic molecules) 
nitrous oxide (dinitrogen monoxide) 
hydrocyanic acid (hydrogen cyanide)  
D_{∞h}  E 2C_{∞} ∞σ_{i} i 2S_{∞} ∞C_{2}  linear with inversion center  oxygen (and all other homonuclear diatomic molecules) 
carbon dioxide 
acetylene (ethyne)  
C_{2}  E C_{2}  "open book geometry," chiral  hydrogen peroxide 
hydrazine 
tetrahydrofuran (twist conformation)  
C_{3}  E C_{3}  propeller, chiral  triphenylphosphine 
triethylamine 
phosphoric acid  
C_{2h}  E C_{2} i σ_{h}  planar with inversion center, no vertical plane  trans1,2dichloroethylene 
transdinitrogen difluoride 
transazobenzene  
C_{3h}  E C_{3} C_{3}^{2} σ_{h} S_{3} S_{3}^{5}  propeller  boric acid 
phloroglucinol (1,3,5trihydroxybenzene) 

C_{2v}  E C_{2} σ_{v}(xz) σ_{v}'(yz)  angular (H_{2}O) or seesaw (SF_{4}) or Tshape (ClF_{3})  water 
sulfur tetrafluoride 
chlorine trifluoride  
C_{3v}  E 2C_{3} 3σ_{v}  trigonal pyramidal  ammonia 
phosphorus oxychloride 
cobalt tetracarbonyl hydride, HCo(CO)_{4}  
C_{4v}  E 2C_{4} C_{2} 2σ_{v} 2σ_{d}  square pyramidal  xenon oxytetrafluoride 
pentaborane(9), B_{5}H_{9} 
nitroprusside anion [Fe(CN)_{5}(NO)]^{2−}  
C_{5v}  E 2C_{5} 2C_{5}^{2} 5σ_{v}  'milking stool' complex  Ni(C_{5}H_{5})(NO) 
corannulene 

D_{2}  E C_{2}(x) C_{2}(y) C_{2}(z)  twist, chiral  biphenyl (skew conformation) 
twistane (C_{10}H_{16}) 
cyclohexane twist conformation  
D_{3}  E C_{3}(z) 3C_{2}  triple helix, chiral  Tris(ethylenediamine)cobalt(III) cation 
tris(oxalato)iron(III) anion 

D_{2h}  E C_{2}(z) C_{2}(y) C_{2}(x) i σ(xy) σ(xz) σ(yz)  planar with inversion center, vertical plane  ethylene 
pyrazine 
diborane  
D_{3h}  E C_{3} 3C_{2} σ_{h} 2S_{3} 3σ_{v}  trigonal planar or trigonal bipyramidal  boron trifluoride 
phosphorus pentachloride 
cyclopropane  
D_{4h}  E 2C_{4} C_{2} 2C_{2}' 2C_{2} i 2S_{4} σ_{h} 2σ_{v} 2σ_{d}  square planar  xenon tetrafluoride 
octachlorodimolybdate(II) anion 
Trans[Co^{III}(NH_{3})_{4}Cl_{2}]^{+} (excluding H atoms)  
D_{5h}  E 2C_{5} 2C_{5}^{2} 5C_{2} σ_{h} 2S_{5} 2S_{5}^{3} 5σ_{v}  pentagonal  cyclopentadienyl anion 
ruthenocene 
C_{70}  
D_{6h}  E 2C_{6} 2C_{3} C_{2} 3C_{2}' 3C_{2}‘’ i 2S_{3} 2S_{6} σ_{h} 3σ_{d} 3σ_{v}  hexagonal  benzene 
bis(benzene)chromium 
coronene (C_{24}H_{12})  
D_{7h}  E C_{7} S_{7} 7C_{2} σ_{h} 7σ_{v}  heptagonal  tropylium (C_{7}H_{7}^{+}) cation 

D_{8h}  E C_{8} C_{4} C_{2} S_{8} i 8C_{2} σ_{h} 4σ_{v} 4σ_{d}  octagonal  cyclooctatetraenide (C_{8}H_{8}^{2−}) anion 
uranocene 

D_{2d}  E 2S_{4} C_{2} 2C_{2}' 2σ_{d}  90° twist  allene 
tetrasulfur tetranitride 
diborane(4) (excited state)  
D_{3d}  E 2C_{3} 3C_{2} i 2S_{6} 3σ_{d}  60° twist  ethane (staggered rotamer) 
dicobalt octacarbonyl (nonbridged isomer) 
cyclohexane chair conformation  
D_{4d}  E 2S_{8} 2C_{4} 2S_{8}^{3} C_{2} 4C_{2}' 4σ_{d}  45° twist  sulfur (crown conformation of S_{8}) 
dimanganese decacarbonyl (staggered rotamer) 
octafluoroxenate ion (idealised geometry)  
D_{5d}  E 2C_{5} 2C_{5}^{2} 5C_{2} i 3S_{10}^{3} 2S_{10} 5σ_{d}  36° twist  ferrocene (staggered rotamer) 

S_{4}  E 2S_{4} C_{2}  tetraphenylborate anion 

T_{d}  E 8C_{3} 3C_{2} 6S_{4} 6σ_{d}  tetrahedral  methane 
phosphorus pentoxide 
adamantane  
T_{h}  E 4C_{3} 4C_{3}^{2} i 3C_{2} 4S_{6} 4S_{6}^{5} 3σ_{h}  pyritohedron  
O_{h}  E 8C_{3} 6C_{2} 6C_{4} 3C_{2} i 6S_{4} 8S_{6} 3σ_{h} 6σ_{d}  octahedral or cubic  sulfur hexafluoride 
molybdenum hexacarbonyl 
cubane 

I_{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 
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. Within a point group, a multiplication of the matrices of two symmetry operations leads to a matrix of another symmetry operation in the same point group.^{[8]} For example, 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.
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:
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 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 {1, −1, 1, −1}, corresponding to the B_{1} irreducible representation. Likewise, the 2p_{z} orbital is seen to have the symmetry of the A_{1} irreducible representation (i.e.: none of the symmetry operations change it), 2p_{y} B_{2}, and the 3d_{xy} orbital A_{2}. These assignments and others are noted in the rightmost two columns of the table.
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.^{[13]} 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.^{[14]} The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.^{[15]}
Point groups are useful for describing rigid molecules which undergo only small oscillations about a single equilibrium geometry, and for which the distorting effects of molecular rotation can be ignored, so that the symmetry operations all correspond to simple geometrical operations. However LonguetHiggins has introduced a more general type of symmetry group suitable not only for rigid molecules but also for nonrigid molecules that tunnel between equivalent geometries (called versions) and which can also allow for the distorting effects of molecular rotation.^{[9]}^{[16]} These groups are known as permutationinversion groups, because the symmetry operations in them are energetically feasible permutations of identical nuclei, or inversion with respect to the center of mass, or a combination of the two.
For example, ethane (C_{2}H_{6}) has three equivalent staggered conformations. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C_{3} axis. Although each conformation has D_{3d} symmetry, as in the table above, description of the internal rotation and associated quantum states and energy levels requires the more complete permutationinversion group G_{36}.
Similarly, ammonia (NH_{3}) has two equivalent pyramidal (C_{3v}) conformations which are interconverted by the process known as nitrogen inversion. This is not an inversion in the sense used for point group symmetry operations of rigid molecules (i.e., the inversion of vibrational displacements and electronic coordinates in the center of mass) since NH_{3} has no inversion center. Rather it the inversion of all nuclei and electrons in the center of mass (close to the nitrogen atom), which happens to be energetically feasible for this molecule. The appropriate permutationinversion 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 (CH_{4}) and H_{3}^{+} 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.^{[17]}^{[18]} The permutationinversion groups required for the complete study of CH_{4} and H_{3}^{+} are T_{d}(M) and D_{3h}(M), respectively.
A second and less general approach to the symmetry of nonrigid molecules is due to Altmann.^{[19]}^{[20]} 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).^{[20]}
Carbon trioxide (CO3) is an unstable oxide of carbon (an oxocarbon). Three possible isomers of carbon trioxide, with molecular symmetry point groups Cs, D3h, and C2v, have been most studied by theoretical methods, and the C2v state has been shown to be the ground state of the molecule. Carbon trioxide should not be confused with the stable carbonate ion (CO32−).
Carbon trioxide can be produced, for example, in the drift zone of a negative corona discharge by reactions between carbon dioxide (CO2) and the atomic oxygen (O) created from molecular oxygen by free electrons in the plasma. Another reported method is photolysis of ozone O3 dissolved in liquid CO2, or in CO2/SF6 mixtures at 45 °C, irradiated with light of 253.7 nm. The formation of CO3 is inferred but it appears to decay spontaneously by the route 2CO3 → 2CO2 + O2 with a lifetime much shorter than 1 minute. Carbon trioxide can be made by blowing ozone at dry ice (solid CO2), and it has also been detected in reactions between carbon monoxide (CO) and molecular oxygen (O2). Along with the ground state C2v isomer, the first spectroscopic detection of the D3h isomer was in electronirradiated ices of carbon dioxide.
Chirality (chemistry)Chirality is a geometric property of some molecules and ions. A chiral molecule/ion is nonsuperposable on its mirror image. The presence of an asymmetric carbon center is one of several structural features that induce chirality in organic and inorganic molecules. The term chirality is derived from the Ancient Greek word for hand, χείρ (cheir).
The mirror images of a chiral molecule or ion are called enantiomers or optical isomers. Individual enantiomers are often designated as either righthanded or lefthanded. Chirality is an essential consideration when discussing the stereochemistry in organic and inorganic chemistry. The concept is of great practical importance because most biomolecules and pharmaceuticals are chiral.
Chiral molecules and ions are described by various ways of designating their absolute configuration, which codify either the entity's geometry or its ability to rotate planepolarized light, a common technique in studying chirality.
CyclopropaneCyclopropane is the cycloalkane molecule with the molecular formula C3H6, consisting of three carbon atoms linked to each other to form a ring, with each carbon atom bearing two hydrogen atoms resulting in D3h molecular symmetry. The small size of the ring creates substantial ring strain in the structure.
Cyclopropane is an anaesthetic when inhaled. In modern anaesthetic practice, it has been superseded by other agents. Due to its extreme reactivity, cyclopropaneoxygen mixtures may explode.
Dibromine monoxideDibromine monoxide is the chemical compound composed of bromine and oxygen with the formula Br2O. It is a dark brown solid which is stable below −40°C and is used in bromination reactions. It is similar to dichlorine monoxide, the dioxide of its halogen neighbor one period higher on the periodic table. The molecule is bent, with C2v molecular symmetry. The Br−O bond length is 1.85Å and the Br−O−Br bond angle is 112°. A related diatomic molecule bromine monoxide (CAS#15656196 ).
DureneDurene, or 1,2,4,5tetramethylbenzene, is an organic compound with the formula C6H2(CH3)4. It is a colourless solid with a sweet odor. The compound is classified as an alkylbenzene. It is one of three isomers of tetramethylbenzene, the other two being prehnitene (1,2,3,4tetramethylbenzene, m.p. −6.2 °C) and isodurene (1,2,3,5tetramethylbenzene, m.p. −23.7 °C). Durene has an unusually high melting point (79.2 °C), reflecting its high molecular symmetry.
Hafnium acetylacetonateHafnium acetylacetonate, also known as Hf(acac)4, is a coordination compound with formula Hf(C5H7O2)4. This white solid is the main hafnium complex of acetylacetonate. The complex has a square antiprismatic geometry with eight nearly equivalent HfO bonds. The molecular symmetry is D2, i.e., the complex is chiral. It is prepared from hafnium tetrachloride and acetylacetone, and base. Zr(acac)4 is very similar in structure and properties.
Inherent chiralityIn chemistry, inherent chirality is a property of asymmetry in molecules arising, not from a stereogenic or chiral center, but from a twisting of the molecule in 3D space. The term was first coined by Volker Boehmer in a 1994 review, to describe the chirality of calixarenes arising from their nonplanar structure in 3D space.
This phenomenon was described as resulting from "the absence of a place of symmetry or an inversion center in the molecule as a whole". Boehmer further explains this phenomenon by suggesting that if an inherently chiral calixarene macrocycle were opened up it would produce an "achiral linear molecule". There are two commonly used notations to describe a molecules inherent chirality: cR/cS (arising from the notation used for classically chiral compounds, with c denoting curvature) and P/M. Inherently chiral molecules, like their classically chiral counterparts, can be used in chiral host–guest chemistry, enantioselective synthesis, and other applications. There are naturally occurring inherently chiral molecules as well. Retinal, a chromophore in rhodopsin. exists in solution as a racemic pair of enantiomers due to the curvature of an achiral polyene chain.
Inorganic chemistryInorganic chemistry deals with the synthesis and behavior of inorganic and organometallic compounds. This field covers all chemical compounds except the myriad organic compounds (carbonbased compounds, usually containing CH bonds), which are the subjects of organic chemistry. The distinction between the two disciplines is far from absolute, as there is much overlap in the subdiscipline of organometallic chemistry. It has applications in every aspect of the chemical industry, including catalysis, materials science, pigments, surfactants, coatings, medications, fuels, and agriculture.
LIESSTIn chemistry and physics, LIESST (LightInduced Excited SpinState Trapping) is a method of changing the electronic spin state of a compound by means of irradiation with light.
Many transition metal complexes with electronic configuration d4d7 are capable of spin crossover (and d8 when molecular symmetry is lower than Oh). Spin crossover refers to where a transition from the high spin (HS) state to the low spin (LS) state or vice versa occurs. Alternatives to LIESST include using thermal changes and pressure to induce spin crossover. The metal most commonly exhibiting spin crossover is iron, with the first known example, an iron(III) tris(dithiocarbamato) complex, reported by Cambi et al. in 1931.
For iron complexes, LIESST involves excitation of the low spin complex with green light to a triplet state. Two successive steps of intersystem crossing result in the high spin complex. Movement from the high spin complex to the low spin complex requires excitation with red light.
Melting pointThe melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at a standard pressure such as 1 atmosphere or 100 kPa.
When considered as the temperature of the reverse change from liquid to solid, it is referred to as the freezing point or crystallization point. Because of the ability of some substances to supercool, the freezing point is not considered as a characteristic property of a substance. When the "characteristic freezing point" of a substance is determined, in fact the actual methodology is almost always "the principle of observing the disappearance rather than the formation of ice", that is, the melting point.
NeopentaneNeopentane, also called 2,2dimethylpropane, is a doublebranchedchain alkane with five carbon atoms. Neopentane is a flammable gas at room temperature and pressure which can condense into a highly volatile liquid on a cold day, in an ice bath, or when compressed to a higher pressure.
Neopentane is the simplest alkane with a quaternary carbon, and has achiral tetrahedral symmetry. It is one of the three structural isomers with the molecular formula C5H12 (pentanes), the other two being npentane and isopentane. Out of these three, it is the only one to be a gas at standard conditions; the other being liquids.
Octacube (sculpture)The Octacube is a large, stainless steel sculpture displayed in the mathematics department of Pennsylvania State University in State College, PA. The sculpture represents a mathematical object called the 24cell or "octacube". Because a real 24cell is fourdimensional, the artwork is actually a projection into the threedimensional world.
Octacube has very high intrinsic symmetry, which matches features in chemistry (molecular symmetry) and physics (quantum field theory).
The sculpture was designed by Adrian Ocneanu, a mathematics professor at Pennsylvania State University. The university's machine shop spent over a year completing the intricate metalwork. Octacube was funded by an alumna in memory of her husband, Kermit Anderson, who died in the September 11 attacks.
Oliver IndrisOliver Indris is a chemist. He discovered the molecular symmetry group G24.He studied chemistry at the University of Kiel, Germany. Then he worked in the group of Wolfgang Stahl on the microwave spectra of organometallic carbonyls.
Organouranium chemistryOrganouranium chemistry is the science exploring the properties, structure and reactivity of organouranium compounds, which are organometallic compounds containing a carbon to uranium chemical bond. The field is of some importance to the nuclear industry and of theoretical interest in organometallic chemistry.
The development of organouranium compounds started in World War II when the Manhattan Project required volatile uranium compounds for 235U/238U isotope separation. For example, Henry Gilman attempted to synthesize compounds like tetramethyluranium and others worked on uranium metal carbonyls but none of the efforts met success due to organouranium instability. After the discovery of ferrocene in 1951, Todd Reynolds and Geoffrey Wilkinson in 1956 synthesized the uranium metallocene Cp3UCl from sodium cyclopentadienide and uranium tetrachloride as a stable but extremely airsensitive compound. In it the UCl bond is an ionic bond and the uranium bonds with the three cyclopentadienyl ligands are covalent of the type found in sandwich compounds with involvement of the uranium 5f atomic orbitals.
Ernst Otto Fischer in 1962 discovered tetracyclopentadienyluranium Cp4U by reaction of NaCp with UCl4 (6% yield) as a compound stable in air as a solid but not in solution. A zero molecular dipole moment and IR spectroscopy revealed that it was also a sandwich compound with uranium in a tetrahedral molecular geometry. In 1970, Fischer added Cp3U to the list of known organouranium compounds by reduction of Cp4U with elemental uranium.
In 1968, the group of Andrew Streitwieser prepared the stable but pyrophoric compound uranocene (COT)2U which has an atom of uranium sandwiched between two cyclooctatetraenide anions (D8h molecular symmetry). The uranium f orbitals interact substantially with the aromatic rings just as the dorbitals in ferrocene interact with the Cp ligands. Uranocene differs from ferrocene because its HOMO and LUMO are centered on the metal and not on the rings and all reactions thus involve the metal often resulting in ligand  metal cleavage.
Uranocenes show ease of reduction of U(IV) compounds to U(III) compounds; otherwise they are fairly unreactive. A close relative that does have sufficient reactivity, obtained by reaction of uranocene with uranium borohydride is the halfsandwich compound (COT)U(BH4)2 discovered in 1983 by the group of M.J. Ephritikhine. Compounds of this type react in many different ways, for instance alkylation at uranium with organolithium reagents or conversion to hybrid sandwich compounds.
Other organouranium compounds are inverted uranocenes with a COT ligand in between two uranium atoms or uranium sandwich compounds with pentalenide ligands instead of COT ligands.
Rule of mutual exclusionThe rule of mutual exclusion in molecular spectroscopy relates the observation of molecular vibrations to molecular symmetry. It states that no normal modes can be both Infrared and Raman active in a molecule that possesses a centre of symmetry. This is a powerful application of group theory to vibrational spectroscopy, and allows one to easily detect the presence of this symmetry element by comparison of the IR and Raman spectra generated by the same molecule.The rule arises because in a centrosymmetric point group, IR active modes, which must transform according to the same irreducible representation generated by one of the components of the dipole moment vector (x, y or z), must be of ungerade (u) symmetry, i.e. their character under inversion is 1, while Raman active modes, which transform according to the symmetry of the polarizability tensor (product of two coordinates), must be of gerade (g) symmetry since their character under inversion is +1. Thus, in the character table there is no irreducible representation that spans both IR and Raman active modes, and so there is no overlap between the two spectra.This does not mean that a vibrational mode which is not Raman active must be IR active: in fact, it is still possible that a mode of a particular symmetry is neither Raman nor IR active. Such spectroscopically "silent" or "inactive" modes exist in molecules such as ethylene (C2H4), benzene (C6H6) and the tetrachloroplatinate ion (PtCl42−).
Sigmapi and equivalentorbital modelsThe σπ model and equivalentorbital model refer to two possible representations of molecules in valence bond theory. The σπ model differentiates bonds and lone pairs of σ symmetry from those of π symmetry, while the equivalentorbital model hybridizes them. The σπ treatment takes into account molecular symmetry and is better suited to interpretation of aromatic molecules (Hückel's rule), although computational calculations of certain molecules tend to optimize better under the equivalentorbital treatment. The two representations produce the same total electron density and are related by a unitary transformation of the occupied molecular orbitals; different localization procedures yield either of the two. In a 1996 review, Kenneth B. Wiberg concluded that "although a conclusive statement cannot be made on the basis of the currently available information, it seems likely that we can continue to consider the σ/π and bentbond descriptions of ethylene to be equivalent. Ian Fleming goes further in a 2010 textbook, noting that "the overall distribution of electrons [...] is exactly the same" in the two models. Nevertheless, as pointed out in Carroll's textbook, at lower levels of theory, the two models make different quantitative and qualitative predictions, and there has been considerable debate as to which model is most useful conceptually and pedagogically.
Sulfite esterA sulfite ester is a functional group with the structure (RO)(R'O)SO. They adopt a trigonal pyramidal molecular geometry due to the presence of lone pairs on the sulphur atom. When substituents R and R' differ, the compound is chiral owing to the stereogenic sulphur centre; when the R groups are the same the compound will have idealised Cs molecular symmetry. They are commonly prepared by the reaction of thionyl chloride with alcohols. The reaction is typically performed at room temperature to prevent the alcohol being converted into a chloroalkane. Bases such as pyridine can also be used to promote the reaction:
2 ROH + SOCl2 → (RO)2SO + 2 HClThe pesticide endosulfan is a sulfite ester. Some simple members include ethylene sulfite, dimethyl sulfite, and diphenylsulfite. Many examples have been prepared from diols, such as sugars. Sulfite esters can be powerful alkylation and hydroxyalkylation reagents.
Symmetry operationIn the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state.
Two basic facts follow from this definition, which emphasize its usefulness.
Physical properties must be invariant with respect to symmetry operations.
Symmetry operations can be collected together in groups which are isomorphic to permutation groups.Wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.
Zirconium acetylacetonateZirconium acetylacetonate is the coordination complex with the formula Zr(C5H7O2)4. It is a common acetylacetonate of zirconium. It is a white solid that exhibits high solubility in nonpolar organic solvents, but not simple hydrocarbons.The complex is prepared by treating zirconium oxychloride with acetylacetone:
ZrOCl2 + 4 Hacac → Zr(acac)4 + 2 HCl + H2OThe complex has a square antiprismatic geometry with eight nearly equivalent ZrO bonds of length 2.19 Å. The molecular symmetry is D2, i.e. the complex is chiral. Compounds of high coordination number tend to be stereochemically nonrigid as indicated by the observation of one methyl signal by proton NMR spectroscopy.More volatile than Zr(acac)4 is the related complex of 1,1,1trifluoroacetylacetonate.
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