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.[1] [2] [3] [4] [5] [6] 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.
The point group symmetry of a molecule is defined by the presence or absence of 5 types of symmetry element.
\tfrac{360\circ}{n}
\tfrac{360\circ}{n}
The five symmetry elements have associated with them five types of symmetry operation, which leave the geometry of the molecule indistinguishable from the starting geometry. 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 in opposite directions (90° and 270°), a Ĉ2 rotation (180°) and Ĉ1 (0° or 360°). Because Ĉ1 is equivalent to Ê, Ŝ1 to σ and Ŝ2 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 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.
| E | C3 | C32 | |
|---|---|---|---|
| E | E | C3 | C32 |
| C3 | C3 | C32 | E |
| C32 | C32 | E | C3 |
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 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). 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 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.
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 of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins.[11] Point groups describe the geometrical symmetry of a molecule whereas permutation-inversion groups describe the energy-invariant symmetry.
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.[12] 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.[9] 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.[13] This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.
The following table lists many of the point groups applicable to molecules, labelled using the Schoenflies notation, which is common in chemistry and molecular spectroscopy. The descriptions include common shapes of molecules, which can be explained by the VSEPR model. In each row, the descriptions and examples have no higher symmetries, meaning that the named point group captures all of the point symmetries.