International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 772776
Section 3.2.4. Molecular symmetry
U. Müller^{b}

In this section, by `molecule' we mean a separate set of closely bonded atoms; this definition includes molecular ions. The term `molecule' as used here also includes molecular aggregates held together by hydrogen bonds or other important molecular interactions. We consider only the symmetry of isolated molecules, i.e. we act as if the molecule had no surroundings or a completely isotropic surrounding (even if it is a molecular ion that in reality has nearby counter ions), and we do not consider any vibrational distortions. Of course, in crystals the surroundings are not isotropic; their anisotropy is reflected in the site symmetry of the molecule. The site symmetry of the molecule in the crystal may not be higher than the molecular symmetry, and, in most cases, it is lower, i.e. it is a subgroup of the point group of the isolated molecule. From experience we know that the actual deviation of the molecular symmetry in crystals as compared to the symmetry in an isotropic surrounding is usually unimportant; generally, only conformation angles are affected significantly.
According to the statistical data for organic molecules (molecules having C—C or C—H bonds), the point group 1 is the most frequent (70.9%), followed by (8.1%), 2 (7.5%), m (6.5%), 2/m (2.1%) and mm2 (1.7%); all others are 1% (Pidcock et al., 2003). In crystals, the site symmetry of organic molecules is 1 with a frequency of 80.7%, followed by (11%). Centrosymmetric molecules ( and higher centrosymmetric point groups) almost always crystallize in centrosymmetric space groups (99%) and lie on centrosymmetric sites ( and higher) with a frequency of 97% (Pidcock et al., 2003; Müller, 1978).
Since all real molecules are objects of threedimensional space, we consider only threedimensional symmetry groups. This choice also applies to socalled `zero', `one' and `twodimensional' molecules; these misleading terms are sometimes used for, respectively, finite molecules, polymeric chain molecules and polymeric layer molecules. For chemists who use these terms, the `dimensionality' mostly refers to the linkage by covalent bonds; for physicists, `dimensionality' may refer to some other anisotropic property, e.g. conductivity.
The set of all isometries that map a molecule onto itself is its molecular symmetry. Its symmetry operations form a group which is the point group of the molecule.
The group is finite if the molecule consists of a finite number of atoms and is mapped onto itself by a finite number of isometries. However, the group is infinite for linear molecules like HCl and CO_{2}, because of the infinite order of the molecular axis. For the symmetry of polymeric molecules see below (Section 3.2.4.4).
All symmetry operations of a finite molecule leave its centre of mass unchanged. If this point is chosen as the origin, all symmetry operations can be represented by 3 × 3 matrices , referred to some predefined coordinate system.
According to their equivalence, point groups are classified into pointgroup types. Two point groups and belong to the same pointgroup type if, after selection of appropriate coordinate systems, the matrix groups of and coincide. Why point groups and pointgroup types have to be distinguished is shown in the following example of metaldehyde; however, for most molecules the distinction is usually of only minor importance.
In Section 1.1.7 a general definition is given for an orbit under the action of a group. Applied to molecules, we formulate this definition in the following way:
The set of points symmetryequivalent to a point X of a molecule is the orbit of X under .
A point X in a molecule has a definite site symmetry (sitesymmetry group). It consists of all those symmetry operations of the point group which leave the point fixed. The sitesymmetry group is always a subgroup of the point group of the molecule: .
A point X is on a general position if its site symmetry consists of nothing more than the identity, . Otherwise, if , the point is on a special position.
The multiplicity of a point X in a molecule is equal to the number of points which are symmetryequivalent to this point, i.e. the number of points in its orbit. If the point is on a general position, its multiplicity is equal to the group order . If is the order of the site symmetry of a point on a special position, the multiplicity of X is .
Two points X and Y of a molecule belong to the same Wyckoff position if the sitesymmetry groups and are conjugate subgroups of , i.e. if and are symmetryequivalent under a symmetry operation of ; expressed mathematically, this is with . The special case is included.
Example
A molecule of metaldehyde [tetrameric acetaldehyde, 2,4,6,8tetramethyl1,3,5,7tetraoxacyclooctane, (H_{3}CCHO)_{4}] has the point group () of order , consisting of the symmetry operations 1, 4^{+}, 2, 4^{−}, m_{10}, m_{01}, m_{11} and .
The atoms C1^{1}…C1^{4}, the four H atoms bonded to them (not labelled in the figure and largely hidden), C2^{1}…C2^{4} and H2^{1}…H2^{4} make up four sets of symmetryequivalent atoms and thus form four orbits. All these atoms are placed on the mirror planes and ; their sitesymmetry groups are or . and are conjugate subgroups of 4mm because they are equivalent by a fourfold rotation, i.e. . All of these atoms belong to one Wyckoff position, labelled 4c in Table 3.2.3.2. The symmetryequivalent atoms O^{1}…O^{4} make up another orbit which belongs to a different Wyckoff position, labelled 4b, with the conjugate site symmetries and . Although the site symmetries of both 4b and 4c consist of the identity and one reflection, they are not conjugate because there is no symmetry operation of 4mm that relates them. The subgroups and are different point groups of the same pointgroup type.
These four sitesymmetry groups all have order 2. Accordingly, the multiplicity for each of the atoms discussed above is 8/2 = 4. The atoms H3^{1}…H3^{8}, however, are on a general position, their site symmetry is 1 and their multiplicity corresponds to the order 8 of the point group 4mm. Their orbit belongs to the Wyckoff position 8d. The Wyckoff position 1a on the fourfold rotation axis is not occupied.
In Table 3.2.3.2 all crystallographic point groups are listed, i.e. all point groups with axes of orders 1, 2, 3, 4 and 6. As described in Section 3.2.1.2.3, the tables also contain the crystal forms which refer to the faces of crystals. As far as molecular symmetry is concerned, only the first four columns and the names of the polyhedra (point forms) in the fifth column given in italics are of interest.
The polyhedra listed in the fifth column result when the points of one orbit of the corresponding Wyckoff position are connected by lines. As for their names, see the remarks on `point forms' in Sections 3.2.1.2.3 and 3.2.1.2.4.
In Table 3.2.3.2 each point group is headed by its short and, if it differs, full Hermann–Mauguin symbol and by its Schoenflies symbol. Each of the subsequent entries corresponds to one Wyckoff position, in order of increasing site symmetry. The last Wyckoff position of the list is always the one with the highest site symmetry and a multiplicity of 1. The sequence of the coordinate triplets follows the sequence as for the symmorphic space groups.
The columns of Table 3.2.3.2 contain the following data:
Since there is no lattice, point groups of molecules, unlike point groups of crystals, have no `natural' coordinate system. It is common for coordinates of atoms in molecules to refer to a Cartesian xyz coordinate system; with the exception of trigonal and hexagonal point groups, the coordinates given in Table 3.2.3.2 can be taken as Cartesian coordinates with arbitrary unit length. For point groups with a floating origin, the origin 0, 0, 0 has to be assigned to an arbitrarily chosen point of the Wyckoff position 1a. The following conventions are used for the orientation of the coordinate axes:
For the trigonal and hexagonal systems, the following pairs of pointgroup types are distinguished in Table 3.2.3.2: 321 & 312, 3m1 & 31m, & and & . In fact, each of these pairs represents only one molecular pointgroup type, the difference being only the arbitrarily chosen direction of the x axis. The members of each pair can be interconverted by turning the coordinate system by 30° about the z axis (referred to hexagonal axes). The same applies to the tetragonal pair & , the turning angle being then 45°.
The terms triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic designate the crystal systems, i.e. the point groups of crystal lattices. As far as point groups of molecules are concerned, these terms only have the meaning of a classification by analogy with the crystal classes. For those who are accustomed to using Schoenflies symbols, it may seem odd to term the point groups and hexagonal rather than trigonal, because the Schoenflies symbolism relies on rotoreflection operations rather than rotoinversion operations. In fact, some properties of and are rather more `trigonal' than `hexagonal'; for example, they have only one kind of degenerate irreducible representation (E), rather than two ( and ) as do the other hexagonal point groups.
For the sake of treating molecular symmetries in the most uniform way, some recommendations for the preferred directions of the coordinate axes have been proposed (Joint Commission for Spectroscopy, 1955). These recommendations include:
Molecular symmetry is not restricted to the crystallographic point groups. No detailed tables are given for noncrystallographic point groups, but the symbols for them can be found in Table 3.2.1.6. For the two icosahedral point groups see Section 3.2.1.4.2 and Table 3.2.3.3.
Certain physical properties depend on symmetry. In particular, a molecule can have a dipole moment only if it is polar, i.e. if it has no rotoinversion axis ( excepted) and if it has at most one symmetry axis of order and no reflection plane perpendicular to it. The polar point groups are N (), m, Nm (N odd) and Nmm (N even). The corresponding crystallographic point groups are those marked with a + sign under the heading `Firstrank tensor' in Table 3.2.2.1.
Examples
For the example of metaldehyde mentioned in the preceding section, we find in Table 3.2.3.2, point group 4mm, the Wyckoff positions 8d, 4c, 4b and 1a. Looking at the image of the molecule, we choose a coordinate system with z along the fourfold axis and x and y perpendicular to the mirror planes marked and . Some value has to be assigned to the unit of the (Cartesian) coordinate system, say 1 Å. The origin must be chosen at some arbitrary site on the Wyckoff position 1a, say such that z = 0 for the C1 atoms. There is no atom on the Wyckoff position 1a . The orbit consisting of the four oxygen atoms is on position 4b , , , , with x = 0.831, z = 0.586; these atoms form a square and have the site symmetry ..m, which in the image is on the mirror planes and . Position 4c , , , has three occupied orbits: C1 with x =1.710 and (the arbitrary) z = 0, C2 with x = 3.194, z = 0.294 and H2; each of these orbits again forms a square and their site symmetry is .m. on the mirror planes and . The orbit of the eight H3 atoms is on the general position 8d; they form a truncated square. (The numerical values have been adapted from the coordinates of Barnett et al., 2005).
The fullerene C_{70} is an example of a molecule with noncrystallographic symmetry, (). This pointgroup type is not listed in Table 3.2.3.2, but it can be found in Table 3.2.1.6.
Polymeric molecules actually consist of a finite number of atoms, but it is more practical to treat them as parts of infinitely extended molecules in the same way as crystals are treated as parts of ideal infinite crystals. If an infinitely long ideal molecule has translational symmetry in one direction, then its symmetry group is a rod group. If the molecule forms a layer with translational symmetry in two dimensions, its symmetry group is a layer group (not to be confused with a plane group, where space is restricted to two dimensions, i.e. to an infinitely thin plane).
Rod groups can have noncrystallographic symmetries, with axes of any order, such as the 10_{3} screw axes in polymeric sulfur or the 29_{9} axes in polyoxymethylene. Symbols for rod groups correspond to Hermann–Mauguin spacegroup symbols, but begin with a script style . The unique direction c is the direction with the translational symmetry; lattice vectors exist in this direction only and are integral multiples of c. This is also the direction of any rotation or screw axis with order . There is no natural length for the a and b vectors. If there are (twofold) rotation axes perpendicular to c or normals of mirror or glide planes perpendicular to c, these directions serve to determine the directions of a and b; otherwise, their directions are arbitrary.
Molecules of chain polymers tend to be entangled, in which case they have no overall symmetry; symmetry is then restricted to the local symmetry in the immediate surrounding of an atom. In crystalline polymers the chains are forced to align themselves and to adopt a symmetric conformation. This symmetry can be crystallographic, but often the symmetry of the single molecule within the crystal matrix is noncrystallographic.
Crystalline chain polymers often adopt helical molecular structures. In polymer science helices are designated according to the `helical point net system' by N/r (`N/r helix') where N is the number of repeating units within one translation period and r is the corresponding number of helical coil turns along the molecular chain (Alexander, 1969; Compendium of Polymer Terminology and Nomenclature (2008); Spruiell & Clark, 1980).^{25} Provided that all repeating units in the chain are symmetryequivalent, the corresponding Hermann–Mauguin screw axis symbol N_{q} can be calculated fromwhere n () and q () are integers to be chosen such that the equation is satisfied. The chemical handedness of the helix does not follow from the N/r symbol, but is specified by the letters M (minus; or L) and P (or R) for left and right, respectively.
Example
Isotactic poly4methyl1pentene (form I) consists of helical 7/2 chains (7 monomers per two chain windings). The equation above is fulfilled either as 7 × 1 + 1 = 2 × 4 or 7 × 1 − 1 = 2 × 3, i.e. q = 4 or q = 3, and the corresponding Hermann–Mauguin screw operation symbol is either 7_{4} or 7_{3}, depending on chirality. The helix is shown in Fig. 3.2.4.1. The N/r symbol cannot be deduced uniquely from the Hermann–Mauguin symbol, because a 7/9 or any other 7/(2 modulo 7) Phelix and any 7/(5 modulo 7) Mhelix also has symmetry.
Layer groups can have axes of orders of 1, 2, 3, 4 and 6 only. There exist 80 layergroup types. Layergroup symbols refer to a crystallographic coordinate system with c (of arbitrary length) perpendicular to the layer defined by the basis vectors a and b. Layergroup symbols are like spacegroup symbols, but begin with a lowercase p or c specifying the centring. Since planegroup symbols also begin by p or c and can therefore be confused with layergroup symbols, it is advisable to always mention what kind of group is meant.
Rod and layer groups with axis orders of 1, 2, 3, 4 and 6 are the subject of International Tables for Crystallography (2010), Volume E, cf. Section 1.7.3 . Table 3.2.4.1 lists the rodgroup types for any axis order.

Definition: An object is chiral if it cannot be superposed by pure rotation and translation on its image formed by inversion through a point.
The symmetry group of a chiral object contains no symmetry operations of the second kind, i.e. no inversion, rotoinversion, reflection or glide reflection. As a consequence, a chiral object can occur in two different forms, which are related in the same way as are a right and a left hand. Further terms in this context are (Flack, 2003; Moss, 1996):
Possible point groups for chiral molecules correspond to the crystal classes marked in Table 3.2.2.1 with a + sign under the heading `Enantiomorphism'. In addition, the noncrystallographic point groups N, N2 (N odd, ), N22 (N even, ) and 235 are possible. Chiral crystal structures are compatible only with space groups that have no inversion centres, rotoinversion axes, reflection or glidereflection planes. For a chiral molecular compound, these symmetry elements would generate the opposite enantiomer and the compound would be a racemate. Chiral crystals belong to one of the 11 crystal classes allowing enantiomorphism (Table 3.2.2.1); they can adopt one out of 65 spacegroup types. These are called the 65 Sohncke spacegroup types [after L. Sohncke who was the first to derive them; Flack (2003). Julian (2015) calls them `proper space groups']. The Sohncke spacegroup types comprise the eleven pairs of enantiomorphic spacegroup types (cf. Section 1.3.4.1 ) and further 43 nonenantiomorphic spacegroup types. Whereas an enantiomorphic space group (e.g. or ) is chiral in itself, which means that its Euclidean normalizer is noncentrosymmetric, the space groups of the other 43 Sohncke spacegroup types (e.g. ) are not chiral themselves; they have centrosymmetric Euclidean normalizers. Keep in mind that the symmetry of a molecule is specified by its point group; the symmetry of a crystal by its space group; and the symmetry of a space group by its Euclidean normalizer (cf. Section 3.5.1.2 ).^{26} However, the arrangement of the atoms in any Sohncke space group is always chiral. In other words, an enantiomorphic space group is a sufficient but not a necessary condition for a chiral crystal structure. Chirality in a nonchiral space group results if the molecules or other building blocks of the crystal consist of only one kind of enantiomers or because the molecules or building blocks, without having to be chiral themselves, are arranged in a chiral manner in the crystal.
Examples
Trigonal selenium exhibits two enantiomorphic forms that consist of either righthanded or lefthanded helical chain molecules (rod groups and , respectively). The molecules as well as the space groups and are enantiomorphic and the helical molecules of righthanded selenium are incompatible with the space group of lefthanded selenium.
The case of selenium can be contrasted to that of NaP (details for NaP are given at the end of Section 3.5.3.2 in the Example for case 1). Crystalline NaP contains one kind of enantiomeric, helical ions that are either or helices (rod groups and , respectively). In this case the space group is a nonenantiomorphic (achiral) Sohncke space group. It is therefore compatible with either of the two different NaP enantiomorphs.
The building blocks of NaClO_{3}, namely spherical Na^{+} ions and pyramidal ions (point group 3m), are not chiral, but their arrangement in the crystal is chiral. Both enantiomorphs adopt the same nonenantiomorphic Sohncke space group (Abrahams & Bernstein, 1977; BrukeLaing & Trueblood, 1977).
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