Molecular symmetry

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can 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 Woodward-Hoffmann 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 X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.

Formaldehyde symmetry elements
Symmetry elements of formaldehyde. C2 is a two-fold rotation axis. σv and σv' are two non-equivalent reflection planes.

Symmetry concepts

The study of symmetry in molecules makes use of group theory.

Examples of the relationship between chirality and symmetry
Rotational
axis (Cn)
Improper rotational elements (Sn)
  Chiral
no Sn
Achiral
mirror plane
S1 = σ
Achiral
inversion centre
S2 = i
C1 Chiral sym CHXYZ Chiral sym CHXYRYS Chiral sym CCXRYRXSYS
C2 Chiral sym CCCXYXY Chiral sym CHHXX Chiral sym CCXYXY

Elements

The point group symmetry of a molecule can be described by 5 types of symmetry element.

  • Symmetry axis: an axis around which 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 axis in water and the C3 axis in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is aligned with the z-axis in a Cartesian coordinate system.
  • 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 and abbreviated σ (sigma = Greek "s", from the German 'Spiegel' meaning mirror).[6] Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed verticalv) and one perpendicular to it horizontalh). 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 dihedrald). 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) correspond to identical objects. 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 where the inversion center is at the Xe atom, and benzene (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 tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis. An S1 axis corresponds to a mirror plane σ and an S2 axis is an inversion center i. A molecule which has no Sn axis for any value of n is a chiral molecule.
  • Identity, abbreviated to E, from the German 'Einheit' meaning unity.[7] This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, it must be included in the list of symmetry elements so that they form a mathematical group, whose definition requires inclusion of the identity element. It is so called because it is analogous to multiplying by one (unity). In other words, E is a property that any object needs to have regardless of its symmetry properties.[8]

Operations

Xenon-tetrafluoride-3D-vdW
XeF4, with square planar geometry, has 1 C4 axis and 4 C2 axes orthogonal to C4. These five axes plus the mirror plane perpendicular to the C4 axis define the D4h symmetry group of the molecule.

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, Ĉn 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 C4 axis of the square xenon tetrafluoride (XeF4) molecule is associated with two Ĉ4 rotations (90°) in opposite directions and a Ĉ2 rotation (180°). Since Ĉ1 is equivalent to Ê, Ŝ1 to σ and Ŝ2 to î, all symmetry operations can be classified as either proper or improper rotations.

Symmetry groups

Groups

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 C4 rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C4. 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, yG, 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 zG)
(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 xG )
(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 xG there is a yG such that x*y=y*x= e for every xG )

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

Pt Group chart 2
Chart for determining Point Group

The successive application (or composition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, a C2 rotation followed by a σv reflection is seen to be a σv' symmetry operation: σv*C2 = σv'. ("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 (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. The symmetry classification of the rotational levels, the eigenstates of the full (rovibronic nuclear spin) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins[9]. The relation between point groups and permutation-inversion groups is explained in this pdf file Link .

Examples of point groups

Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl3, POF3, XeO3, and NH3 all share identical symmetry operations.[10] They all can undergo the identity operation E, two different C3 rotation operations, and three different σv plane reflections without altering their identities, so they are placed in one point group, C3v, with order 6.[11] Similarly, water (H2O) and hydrogen sulfide (H2S) also share identical symmetry operations. They both undergo the identity operation E, one C2 rotation, and two σv reflections without altering their identities, so they are both placed in one point group, C2v, 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]

Common point groups

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
C1 E no symmetry, chiral Chiral
bromochlorofluoromethane (both enantiomers shown)
Lysergic acid
lysergic acid
Leucine-ball-and-stick
L-leucine and most other α-amino acids except glycine
Cs E σh mirror plane, no other symmetry Thionyl-chloride-from-xtal-3D-balls-B
thionyl chloride
Hypochlorous-acid-3D-vdW
hypochlorous acid
Chloroiodomethane-3D-vdW
chloroiodomethane
Ci E i inversion center Tartaric-acid-3D-balls
meso-tartaric acid
Mucic acid molecule ball
mucic acid (meso-galactaric acid)
(S,R) 1,2-dibromo-1,2-dichloroethane (anti conformer)
C∞v E 2C ∞σv linear Hydrogen-fluoride-3D-vdW
hydrogen fluoride (and all other heteronuclear diatomic molecules)
Nitrous-oxide-3D-vdW
nitrous oxide
(dinitrogen monoxide)
Hydrogen-cyanide-3D-vdW
hydrocyanic acid
(hydrogen cyanide)
D∞h E 2C ∞σi i 2S ∞C2 linear with inversion center Oxygen molecule
oxygen (and all other homonuclear diatomic molecules)
Carbon dioxide 3D spacefill
carbon dioxide
Acetylene-3D-vdW
acetylene (ethyne)
C2 E C2 "open book geometry," chiral Hydrogen-peroxide-3D-balls
hydrogen peroxide
Hydrazine-3D-balls
hydrazine
Tetrahydrofuran-3D-balls
tetrahydrofuran (twist conformation)
C3 E C3 propeller, chiral Triphenylphosphine-3D-vdW
triphenylphosphine
Triethylamine-3D-balls
triethylamine
Phosphoric-acid-3D-balls
phosphoric acid
C2h E C2 i σh planar with inversion center, no vertical plane Trans-dichloroethylene-3D-balls
trans-1,2-dichloroethylene
(E)-Dinitrogen-difluoride-3D-balls
trans-dinitrogen difluoride
Azobenzene-trans-3D-balls
trans-azobenzene
C3h E C3 C32 σh S3 S35 propeller Boric-acid-3D-vdW
boric acid
Phloroglucinol-3D
phloroglucinol (1,3,5-trihydroxybenzene)
C2v E C2 σv(xz) σv'(yz) angular (H2O) or see-saw (SF4) or T-shape (ClF3) Water molecule 3D
water
Sulfur-tetrafluoride-3D-balls
sulfur tetrafluoride
Chlorine-trifluoride-3D-vdW
chlorine trifluoride
C3v E 2C3v trigonal pyramidal Ammonia-3D-balls-A
ammonia
Phosphoryl-chloride-3D-vdW
phosphorus oxychloride
HCo(CO)4-3D-balls
cobalt tetracarbonyl hydride, HCo(CO)4
C4v E 2C4 C2vd square pyramidal Xenon-oxytetrafluoride-3D-vdW
xenon oxytetrafluoride
Pentaborane-3D-balls
pentaborane(9), B5H9
Nitroprusside-anion-from-xtal-3D-balls
nitroprusside anion [Fe(CN)5(NO)]2−
C5v E 2C5 2C52v 'milking stool' complex CpNi(NO)
Ni(C5H5)(NO)
Corannulene3D
corannulene
D2 E C2(x) C2(y) C2(z) twist, chiral Biphenyl 3D
biphenyl (skew conformation)
Twistane-3D-balls
twistane (C10H16)
cyclohexane twist conformation
D3 E C3(z) 3C2 triple helix, chiral Lambda-Tris(ethylenediamine)cobalt(III)-chloride-3D-balls-by-AHRLS-2012
Tris(ethylenediamine)cobalt(III) cation
Delta-tris(oxalato)ferrate(III)-3D-balls
tris(oxalato)iron(III) anion
D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz) planar with inversion center, vertical plane Ethylene-3D-vdW
ethylene
Pyrazine-3D-spacefill
pyrazine
Diborane-3D-balls-A
diborane
D3h E C3 3C2 σh 2S3v trigonal planar or trigonal bipyramidal Boron-trifluoride-3D-vdW
boron trifluoride
Phosphorus-pentachloride-3D-balls
phosphorus pentachloride
Cyclopropane-3D-vdW
cyclopropane
D4h E 2C4 C2 2C2' 2C2 i 2S4 σhvd square planar Xenon-tetrafluoride-3D-vdW
xenon tetrafluoride
Octachlorodirhenate(III)-3D-balls
octachlorodimolybdate(II) anion
Trans-dichlorotetraamminecobalt(III)
Trans-[CoIII(NH3)4Cl2]+ (excluding H atoms)
D5h E 2C5 2C52 5C2 σh 2S5 2S53v pentagonal Cyclopentadienide-3D-balls
cyclopentadienyl anion
Ruthenocene-from-xtal-3D-SF
ruthenocene
Fullerene-C70-3D-balls
C70
D6h E 2C6 2C3 C2 3C2' 3C2‘’ i 2S3 2S6 σhdv hexagonal Benzene-3D-vdW
benzene
Bis(benzene)chromium-from-xtal-2006-3D-balls-A
bis(benzene)chromium
Coronene3D
coronene (C24H12)
D7h E C7 S7 7C2 σhv heptagonal Tropylium-ion-3D-vdW
tropylium (C7H7+) cation
D8h E C8 C4 C2 S8 i 8C2 σhvd octagonal Cyclooctatetraenide-3D-ball
cyclooctatetraenide (C8H82−) anion
Uranocene-3D-vdW
uranocene
D2d E 2S4 C2 2C2' 2σd 90° twist Allene3D
allene
Tetrasulfur-tetranitride-from-xtal-2000-3D-balls
tetrasulfur tetranitride
Diborane(4) excited state
diborane(4) (excited state)
D3d E 2C3 3C2 i 2S6d 60° twist Ethane-3D-vdW
ethane (staggered rotamer)
Dicobalt-octacarbonyl-D3d-non-bridged-from-C60-xtal-2009-3D-balls
dicobalt octacarbonyl (non-bridged isomer)
Cyclohexane-chair-3D-sticks
cyclohexane chair conformation
D4d E 2S8 2C4 2S83 C2 4C2' 4σd 45° twist Cyclooctasulfur-above-3D-balls
sulfur (crown conformation of S8)
Dimanganese-decacarbonyl-3D-balls
dimanganese decacarbonyl (staggered rotamer)
Square-antiprismatic-3D-balls
octafluoroxenate ion (idealised geometry)
D5d E 2C5 2C52 5C2 i 3S103 2S10d 36° twist Ferrocene 3d model 2
ferrocene (staggered rotamer)
S4 E 2S4 C2 Tetraphenylborate
tetraphenylborate anion
Td E 8C3 3C2 6S4d tetrahedral Methane-CRC-MW-3D-balls
methane
Phosphorus-pentoxide-3D-balls
phosphorus pentoxide
Adamantane-3D-balls
adamantane
Th E 4C3 4C32 i 3C2 4S6 4S65h pyritohedron
Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6hd octahedral or cubic Sulfur-hexafluoride-3D-balls
sulfur hexafluoride
Molybdenum-hexacarbonyl-from-xtal-3D-balls
molybdenum hexacarbonyl
Cubane-3D-balls
cubane
Ih E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ icosahedral or dodecahedral Buckminsterfullerene-perspective-3D-balls
Buckminsterfullerene
Dodecaborane-3D-balls
dodecaborate anion
Dodecahedrane-3D-sticks
dodecahedrane

Representations

The symmetry operations can be represented in many ways. A convenient representation is by matrices. For any vector representing a point in Cartesian coordinates, left-multiplying 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 C2v example this is:

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations.

Character tables

For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square.

The table itself consists of characters that represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But, for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric).

The representations are labeled according to a set of conventions:

  • A, when rotation around the principal axis is symmetrical
  • B, when rotation around the principal axis is asymmetrical
  • E and T are doubly and triply degenerate representations, respectively
  • when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion.
  • with point groups C∞v and D∞h the symbols are borrowed from angular momentum description: Σ, Π, Δ.

The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the righthand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.

The character table for the C2v symmetry point group is given below:

C2v E C2 σv(xz) σv'(yz)
A1 1 1 1 1 z x2, y2, z2
A2 1 1 −1 −1 Rz xy
B1 1 −1 1 −1 x, Ry xz
B2 1 −1 −1 1 y, Rx yz

Consider the example of water (H2O), which has the C2v symmetry described above. The 2px orbital of oxygen has B1 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 C2 and a σv'(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 B1 irreducible representation. Likewise, the 2pz orbital is seen to have the symmetry of the A1 irreducible representation (i.e.: none of the symmetry operations change it), 2py B2, and the 3dxy orbital A2. These assignments and others are noted in the rightmost two columns of the table.

Historical background

Hans Bethe used characters of point group operations in his study of ligand field theory in 1929, and Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy.[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]

Molecular nonrigidity

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 Longuet-Higgins has introduced a more general type of symmetry group suitable not only for rigid molecules but also for non-rigid 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 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, or a combination of the two.

For example, ethane (C2H6) 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 C3 axis. Although each conformation has D3d 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 G36.

Similarly, ammonia (NH3) has two equivalent pyramidal (C3v) 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 NH3 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 permutation-inversion group to be used in this situation is D3h(M) which is isomorphic with the point group D3h.

Additionally, as examples, the methane (CH4) and H3+ molecules have highly symmetric equilibrium structures with Td and D3h 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 permutation-inversion groups required for the complete study of CH4 and H3+ are Td(M) and D3h(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]

See also

References

  1. ^ Quantum Chemistry, Third Edition John P. Lowe, Kirk Peterson ISBN 0-12-457551-X
  2. ^ Physical Chemistry: A Molecular Approach by Donald A. McQuarrie, John D. Simon ISBN 0-935702-99-7
  3. ^ The chemical bond 2nd Ed. J.N. Murrell, S.F.A. Kettle, J.M. Tedder ISBN 0-471-90760-X
  4. ^ Physical Chemistry P.W. Atkins and J. de Paula (8th ed., W.H. Freeman 2006) ISBN 0-7167-8759-8, chap.12
  5. ^ G. L. Miessler and D. A. Tarr Inorganic Chemistry (2nd ed., Pearson/Prentice Hall 1998) ISBN 0-13-841891-8, chap.4.
  6. ^ "Symmetry Operations and Character Tables". University of Exeter. 2001. Retrieved 29 May 2018.
  7. ^ LEO Ergebnisse für "einheit"
  8. ^ a b c d e Pfenning, Brian (2015). Principles of Inorganic Chemistry. John Wiley & Sons. ISBN 9781118859025.
  9. ^ a b Longuet-Higgins, H.C. (1963). "The symmetry groups of non-rigid molecules". Molecular Physics. 6 (5): 445–460. Bibcode:1963MolPh...6..445L. doi:10.1080/00268976300100501.
  10. ^ Pfennig, Brian. Principles of Inorganic Chemistry. Wiley. p. 191. ISBN 978-1-118-85910-0.
  11. ^ pfennig, Brian. Principles of Inorganic Chemistry. Wiley. ISBN 978-1-118-85910-0.
  12. ^ Miessler, Gary. Inorganic Chemistry. Pearson. ISBN 9780321811059.
  13. ^ Group Theory and its application to the quantum mechanics of atomic spectra, E. P. Wigner, Academic Press Inc. (1959)
  14. ^ Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables Randall B. Shirts J. Chem. Educ. 2007, 84, 1882. Abstract
  15. ^ Rosenthal, Jenny E.; Murphy, G. M. (1936). "Group Theory and the Vibrations of Polyatomic Molecules". Rev. Mod. Phys. 8: 317–346. Bibcode:1936RvMP....8..317R. doi:10.1103/RevModPhys.8.317.
  16. ^ Philip R. Bunker and Per Jensen (2005), Fundamentals of Molecular Symmetry (Institute of Physics Publishing) ISBN 0-7503-0941-5
  17. ^ Watson, J.K.G (1971). "Forbidden rotational spectra of polyatomic molecules". Journal of Molecular Spectroscopy. 40 (3): 546–544. Bibcode:1971JMoSp..40..536W. doi:10.1016/0022-2852(71)90255-4.
  18. ^ Oldani, M.; et al. (1985). "Pure rotational spectra of methane and methane-d4 in the vibrational ground state observed by microwave Fourier transform spectroscopy". Journal of Molecular Spectroscopy. 110 (1): 93–105. Bibcode:1985JMoSp.110...93O. doi:10.1016/0022-2852(85)90215-2.
  19. ^ Altmann S.L. (1977) Induced Representations in Crystals and Molecules, Academic Press
  20. ^ a b Flurry, R.L. (1980) Symmetry Groups, Prentice-Hall, ISBN 0-13-880013-8, pp.115-127

External links

Carbon trioxide

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 electron-irradiated ices of carbon dioxide.

Chirality (chemistry)

Chirality is a geometric property of some molecules and ions. A chiral molecule/ion is non-superposable 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 right-handed or left-handed. 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 plane-polarized light, a common technique in studying chirality.

Cyclopropane

Cyclopropane 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, cyclopropane-oxygen mixtures may explode.

Dibromine monoxide

Dibromine 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#15656-19-6 ).

Durene

Durene, or 1,2,4,5-tetramethylbenzene, 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 (1,2,3,4-tetramethylbenzene, m.p. −6.2 °C) and isodurene (1,2,3,5-tetramethylbenzene, m.p. −23.7 °C). Durene has an unusually high melting point (79.2 °C), reflecting its high molecular symmetry.

Hafnium acetylacetonate

Hafnium 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 Hf-O 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 chirality

In 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 3-D space. The term was first coined by Volker Boehmer in a 1994 review, to describe the chirality of calixarenes arising from their non-planar structure in 3-D 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 chemistry

Inorganic chemistry deals with the synthesis and behavior of inorganic and organometallic compounds. This field covers all chemical compounds except the myriad organic compounds (carbon-based compounds, usually containing C-H 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.

LIESST

In chemistry and physics, LIESST (Light-Induced Excited Spin-State 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 d4-d7 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 point

The 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.

Neopentane

Neopentane, also called 2,2-dimethylpropane, is a double-branched-chain 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 n-pentane 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 24-cell or "octacube". Because a real 24-cell is four-dimensional, the artwork is actually a projection into the three-dimensional 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 metal-work. Octacube was funded by an alumna in memory of her husband, Kermit Anderson, who died in the September 11 attacks.

Oliver Indris

Oliver 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 chemistry

Organouranium 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 air-sensitive compound. In it the U-Cl 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 d-orbitals 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 half-sandwich 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 exclusion

The 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−).

Sigma-pi and equivalent-orbital models

The σ-π model and equivalent-orbital model refer to two possible representations of molecules in valence bond theory. The σ-π model differentiates bonds and lone pairs of σ symmetry from those of π symmetry, while the equivalent-orbital model hybridizes them. The σ-π 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 equivalent-orbital 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 bent-bond 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 ester

A 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 operation

In 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 acetylacetonate

Zirconium 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 Zr-O 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 1,1,1-trifluoroacetylacetonate.

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