This lists the character tables for the more common molecular point groups used in the study of molecular symmetry. These tables are based on the group-theoretical treatment of the symmetry operations present in common molecules, and are useful in molecular spectroscopy and quantum chemistry. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references. [1] [2] [3] [4] [5]
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Zn: cyclic group of order n, Dn: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, Sn: symmetric group on n letters, and An: alternating group on n letters.
The character tables then follow for all groups. The rows of the character tables correspond to the irreducible representations of the group, with their conventional names, known as Mulliken symbols, [6] in the left margin. The naming conventions are as follows:
All but the two rightmost columns correspond to the symmetry operations which are invariant in the group. In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.
The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations. The symbol i used in the body of the table denotes the imaginary unit: i 2 = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes complex conjugation.
The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (Rx, Ry and Rz), and functions of the quadratic terms of the coordinates(x2, y2, z2, xy, xz, and yz).
A further column is included in some tables, such as those of Salthouse and Ware [7] For example,
, , | , , , | , , , , , | |||
, , | , | , , , |
The last column relates to cubic functions which may be used in applications regarding f orbitals in atoms.
These groups are characterized by a lack of a proper rotation axis, noting that a rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse.
In the group , all functions of the Cartesian coordinates and rotations about them transform as the irreducible representation.
Point Group | Canonical Group | Order | Character Table | |||||||||||||||
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The families of groups with these symmetries have only one rotation axis.
The cyclic groups are denoted by Cn. These groups are characterized by an n-fold proper rotation axis Cn. The C1 group is covered in the nonaxial groups section.
Point Group | Canonical Group | Order | Character Table | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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C2 | Z2 | 2 |
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C3 | Z3 | 3 |
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C4 | Z4 | 4 |
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C5 | Z5 | 5 |
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C6 | Z6 | 6 |
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C8 | Z8 | 8 |
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The reflection groups are denoted by Cnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) a mirror plane σh normal to Cn. The C1h group is the same as the Cs group in the nonaxial groups section.
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C2h | Z2× Z2 | 4 |
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C3h | Z6 | 6 |
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C4h | Z2× Z4 | 8 |
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C5h | Z10 | 10 |
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C6h | Z2× Z6 | 12 |
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The pyramidal groups are denoted by Cnv. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n mirror planes σv which contain Cn. The C1v group is the same as the Cs group in the nonaxial groups section.
Point Group | Canonical group | Order | Character Table | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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C2v | Z2× Z2 (=D2) | 4 |
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C3v | D3 | 6 |
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C4v | D4 | 8 |
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C5v | D5 | 10 |
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C6v | D6 | 12 |
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The improper rotation groups are denoted by Sn. These groups are characterized by an n-fold improper rotation axis Sn, where n is necessarily even. The S2 group is the same as the Ci group in the nonaxial groups section. Sn groups with an odd value of n are identical to Cnh groups of same n and are therefore not considered here (in particular, S1 is identical to Cs).
The S8 table reflects the 2007 discovery of errors in older references. [4] Specifically, (Rx, Ry) transform not as E1 but rather as E3.
Point Group | Canonical group | Order | Character Table | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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S4 | Z4 | 4 |
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S6 | Z6 | 6 |
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S8 | Z8 | 8 |
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The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.
The dihedral groups are denoted by Dn. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn. The D1 group is the same as the C2 group in the cyclic groups section.
Point Group | Canonical group | Order | Character Table | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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D2 | Z2× Z2 (=D2) | 4 |
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D3 | D3 | 6 |
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D4 | D4 | 8 |
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D5 | D5 | 10 |
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D6 | D6 | 12 |
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The prismatic groups are denoted by Dnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) a mirror plane σh normal to Cn and containing the C2s. The D1h group is the same as the C2v group in the pyramidal groups section.
The D8h table reflects the 2007 discovery of errors in older references. [4] Specifically, symmetry operation column headers 2S8 and 2S83 were reversed in the older references.
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D2h | Z2×Z2×Z2 (=Z2×D2) | 8 |
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D3h | D6 | 12 |
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D4h | Z2×D4 | 16 |
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D5h | D10 | 20 |
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D6h | Z2×D6 | 24 |
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D8h | Z2×D8 | 32 |
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The antiprismatic groups are denoted by Dnd. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) n mirror planes σd which contain Cn. The D1d group is the same as the C2h group in the reflection groups section.
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D2d | D4 | 8 |
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D3d | D6 | 12 |
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D4d | D8 | 16 |
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D5d | D10 | 20 |
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D6d | D12 | 24 |
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These symmetries are characterized by having more than one proper rotation axis of order greater than 2.
These polyhedral groups are characterized by not having a C5 proper rotation axis.
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T | A4 | 12 |
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Td | S4 | 24 |
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Th | Z2×A4 | 24 |
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O | S4 | 24 |
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Oh | Z2×S4 | 48 |
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These polyhedral groups are characterized by having a C5 proper rotation axis.
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I | A5 | 60 |
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Ih | Z2×A5 | 120 |
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These groups are characterized by having a proper rotation axis C∞ around which the symmetry is invariant to any rotation.
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C∞v |
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D∞h |
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In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
In geometry, an improper rotation is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group is a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations, tiles and physical wallpaper.
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected.
In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.
The Schoenfliesnotation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation.
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.
A regular octahedron has 24 rotational symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dihn.
In chemistry and crystallography, a symmetry element is a point, line, or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane, an axis of rotation, or a center of inversion. For an object such as a molecule or a crystal, a symmetry element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the object unchanged. The set containing these operations form one of the symmetry groups of the object. The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 µm.
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1⁄3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element.
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.
Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. 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. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.
To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum.