In chemistry, molecular symmetry describes the symmetry present in molecules 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, 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. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the Woodward–Hoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.
There are many techniques for determining the symmetry of a given molecule, including X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.
Molecular symmetry concepts + Examples of the relationship between chirality and symmetry Rotationalaxis (Cn) Improper rotational elements (Sn) Chiralno Sn Achiralmirror planeS1 = σ Achiral inversion centreS2 = i C1 C2
Elements There are three types of symmetry elements, these being the point, the lines or the planes about which symmetry operations take place. The important feature of symmetry elements is that they are invariant in symmetry operations: they are the coordinates that do not move as similar atoms are permuted during symmetry transformations. Three basic symmetry operations can be defined to manipulate objects about the three elements. Rotational motions about axes and reflections in planes leave lines and planes invariant while spatial inversion (parity inversion) leaves just one centre point invariant. Molecular motions about a point in a three dimensional space can be completely described by rotational symmetry and either one of the reflection or spatial inversion operations. Two notations are widely used to describe molecular symmetry. The first of these was published by Schoenflies in 1891 and generally uses rotational symmetry combined with mirror reflections but departs from this pattern in a few cases. The second was developed by Hermann and Mauguin in 1931 and uses a structure based on rotational operations and spatial inversion. (the mirror symbols in Hermann-Mauguin notation are required for spatial symmetry)
Operations
Schoenflies defined the following five symmetry operations which leave the appearance of the molecule indistinguishable from the starting state 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). A symmetry plane parallel with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σd). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz). Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. In other words, a molecule has a center of symmetry when the points (x,y,z) and (−x,−y,−z) of the molecule always look identical. For example, whenever there is an oxygen atom in some point (x,y,z), then there also has to be an oxygen atom in the point (−x,−y,−z). There may or may not be an atom at the inversion center itself. Rotation-reflection axis: an axis around which a rotation by , followed by a reflection in the 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. 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 C1 proper rotation. 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).
Rotational symmetry operations are often described as "proper" operations because they could be physically performed on a model in three dimensional space. All other operations involving or including reflections and inversions are said to be improper. Rotations about axes are always assumed by chemists to move clockwise when viewed from above, i.e. from along the +z axis, and this is standard in molecular chemistry texts. A counter clockwise convention is common in other subject areas. For example, a square xenon tetrafluoride (XeF4) molecule has 4 symmetry operations associated with an obvious 4-fold axis: clockwise rotations of 90°, 180°, 270° and 360°. The last of these operations gets the molecule back to where it started and is equivalent to the identity operation. If the 4-fold axis is aligned with the z-axis (running vertically in the image) then further 2-fold rotational axes at right angles to the main rotational axis are obvious. One of the molecular 2-fold axes can be positioned along the y axis then the other three occur at 45° intervals around the vertical axis. This means that all of the fluorine atoms sit on two of the horizontal rotational axes while none is positioned on the other two axes. The 4-fold rotation about the vertical axis, generated by a clockwise rotation of 90°, is an example of cyclic symmetry and the additional four 2-fold horizontal rotations provide an example of dihedral symmetry. This molecule also has horizontal, vertical and diagonal mirror planes of symmetry together with a centre of symmetry. Only one of these operations is required to expand the 8 proper dihedral transformations to the maximum 16 operations and Schoenflies notation uses the horizontal plane reflection. Rotational axial orders run from 1 to infinity but in practice are either small numbers or the infinite orders associated with linear molecules.
Symmetry groups Molecular symmetry groups are sets of the operations listed above together with the binary operation of consecutive application. If A, B and C are group operations and the application of operation A to the molecule "followed by" operation B results in an equivalent result to operation C the process may be shown as or . For example, a C2 rotation followed by a σv reflection is seen to be a σv′ symmetry operation: . The binary combination consists of applying first one symmetry operation and then the other. Operations are applied from right to left. Another example is the sequence of a C4 rotation about the z-axis and a reflection in the xy-plane, denoted . The composition (repeated application) of symmetry operations is sometimes shown as to emphasise the fact that binary operation is being performed but the most common practice in modern literature is just right-to-left placement. Molecular symmetry groups are also called point groups 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 summarises all symmetry operations that all molecules in that category have. The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.
The symmetry operations of a molecule provide a specific example of a mathematical group, a concept from linear algebra combining a set of operations with a binary operation that obeys the following three axioms there exists an identity element that in a binary operation with another element simply reproduces that element: for all there exists such that there exists an inverse element for every element in the group: for all there exists such that combinations of elements are associative; for all , . ie the order in which binary operations are applied does not matter so long as the overall order is maintained.
Assigning each molecule to a point group classifies molecules into categories with similar symmetry properties. Schoenflies symbols for point groups evolved during the 19th century from the detailed examination of crystals and minerals but are used now only in atomic and molecular work. Hydrogen peroxide (H2O2) has the structure shown with hydrogen atoms attached to two oxygen atoms in a 'V' shape relative to the oxygen–oxygen bond. It provides an example of the simplest non-trivial point group C2, containing just the identity element E and a 2-fold rotational symmetry about the vertical axis, the set . Clearly, , , and . This molecule has two enantiomers that cannot be superimposed on each other by rotational motions in three dimensional space. Each enantiomer has hydrogen atoms placed on the opposite side of the V shape. Any molecule described completely by a rotational group has enantiomers but it is possible that the two might be interconverted by rotations about bonds. A molecule which has no improper axis for any value of n is a chiral molecule. So, molecules belonging to rotational groups Cn, Dn, T and O are potentially enantiomeric
Formaldehyde is a planar molecule with two hydrogen atoms and an oxygen atom attached to a carbon atom as shown in the diagram. It has a very clear 2-fold rotational axis through the carbon and oxygen atoms and so exhibits C2 symmetry. There are also vertical mirror symmetry planes labelled σv and σv' at right angles to each other so the molecule exhibits the higher order C2v group symmetry. Binary combinations of these four operations are shown in the C2v operation table where a slightly different notation is used for the reflections because in low symmetry molecules of this kind the mirror planes can be identified with the x, y and z axes. Operation σxz in the top row "followed by" operation C2 in the left hand column is equivalent to σyz read from the body of the table. In this point group the order in which operations are applied makes no difference to the outcome but for the vast majority of groups it does make a difference. Water (H2O) and hydrogen sulfide (H2S) molecules share identical symmetry operations with formaldehyde and therefore belong to the same symmetry group and are subject to the same four symmetry operations with order 4.