# 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. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to symmetry.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}

The framework for the study of molecular symmetry is provided by group theory, and in particular irreducible representation 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.

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.

For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any angle Φ is a symmetry operation.

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 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.

(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 element*:

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).^{[9]} 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.^{[9]} 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 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^{[10]} 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 (rotation-vibration-electronic) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins.^{[11]}

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.^{[12]} 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.^{[13]} 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.^{[14]} 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.^{[9]}

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.

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.^{[9]} For instance, 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 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.

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.^{[17]} 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.^{[18]} The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.^{[19]}

As discussed above in the section **Point groups and permutation-inversion groups,** point groups are useful for classifying the vibronic states of *rigid* molecules (sometimes called *semi-rigid* molecules) which undergo only small oscillations about a single equilibrium geometry. Longuet-Higgins has introduced a more general type of symmetry group suitable not only for classifying the rovibronic states of rigid molecules but also for classifying the states of *non-rigid* (or *fluxional*) molecules that tunnel between equivalent geometries (called *versions*^{[20]}) and which can also allow for the distorting effects of molecular rotation.^{[11]} 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 operation), 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 permutation-inversion 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 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 NH

_{3}has no inversion center and is not centrosymmetric. Rather it 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 (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.^{[21]}^{[22]} The permutation-inversion 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.^{[23]}^{[24]} 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).^{[24]}