Introduction A Point Group describes all the symmetry operations that can be performed on a molecule that result in a conformation indistinguishable from the original. Point groups are used in Group Theory, the mathematical analysis of groups, to determine properties such as a molecule's molecular orbitals. Assigning Point Groups While a point group contains all of the symmetry operations that can be performed on a given molecule, it is not necessary to identify all of these operations to determine the molecule's overall point group. Instead, a molecule's point group can be determined by following a set of steps which analyze the presence (or absence) of particular symmetry elements. Determine if the molecule is of high or low symmetry. If not, find the highest order rotation axis, C n . Determine if the molecule has any C 2 axes perpendicular to the principal C n axis. If so, then there are n such C 2 axes, and the molecule is in the D set of point groups. If not, it is in either the C or S set of point groups. Determine if the molecule has a horizontal mirror plane (σ h ) perpendicular to the principal C n axis. If so, the molecule is either in the C nh or D nh set of point groups. Determine if the molecule has a vertical mirror plane (σ v ) containing the principal C n axis. If so, the molecule is either in the C nv or D nd set of point groups. If not, and if the molecule has n perpendicular C 2 axes, then it is part of the D n set of point groups. Determine if there is an improper rotation axis, S 2n , collinear with the principal C n axis. If so, the molecule is in the S 2n point group. If not, the molecule is in the C n point group. Example \(\PageIndex{1}\) Find the point group of benzene (C 6 H 6 ). Solution Highest order rotation axis: C 6 Benzene is neither high or low symmetry There are 6 C 2 axes perpendicular to the principal axis There is a horizontal mirror plane (σ h ) Benzene is in the D 6h point group. Low Symmetry Point Groups Low symmetry point groups include the C 1 , C s , and C i groups Group Description Example C1 only the identity operation (E) CHFClBr Cs only the identity operation (E) and one mirror plane C2H2ClBr Ci only the identity operation (E) and a center of inversion (i) C2H2Cl2Br High Symmetry Point Groups High symmetry point groups include the T d , O h , I h , C ∞v , and D ∞h groups. The table below describes their characteristic symmetry operations. The full set of symmetry operations included in the point group is described in the corresponding character table. Group Description Example C∞v linear molecule with an infinite number of rotation axes and vertical mirror planes (σv) HBr D∞h linear molecule with an infinite number of rotation axes, vertical mirror planes (σv), perpendicular C2 axes, a horizontal mirror plane (σh), and an inversion center (i) CO2 Td typically have tetrahedral geometry, with 4 C4 axes, 3 C2 axes, 3 S4 axes, and 6 dihedral mirror planes (σd) CH4 Oh typically have octahedral geometry, with 3 C4 axes, 4 C3 axes, and an inversion center (i) as characteristic symmetry operations SF6 Ih typically have an icosahedral structure, with 6 C5 axes as characteristic symmetry operations B12H122- D Groups The D set of point groups is classified as D nh , D nd , or D n , where n refers to the principal axis of rotation. Overall, the D groups are characterized by the presence of n C 2 axes perpendicular to the principal C n axis. Further classification of a molecule in the D groups depends on the presence of horizontal or vertical/dihedral mirror planes. Group Description Example Dnh n perpendicular C2 axes, and a horizontal mirror plane (σh) benzene, C6H6 is D6h Dnd n perpendicular C2 axes, and a vertical mirror plane (σv) propadiene, C3H4 is D2d Dn n perpendicular C2 axes, no mirror planes [Co(en)3]3+ is D3 C Groups The C set of point groups is classified as C nh , C nv , or C n , where n refers to the principal axis of rotation. The C set of groups is characterized by the absence of n C 2 axes perpendicular to the principal C n axis. Further classification of a molecule in the C groups depends on the presence of horizontal or vertical/dihedral mirror planes. Group Description Example Cnh horizontal mirror plane (σh) perpendicular to the principal Cn axis boric acid, H3BO3, is C3h Cnv vertical mirror plane (σv) containing the principal Cn axis ammonia, NH3, is C3v Cn no mirror planes P(C6H5)3 is C3 S Groups The S set of point groups is classified as S 2n , where n refers to the principal axis of rotation. The S set of groups is characterized by the absence of n C 2 axes perpendicular to the principal C n axis, as well as the absence of horizontal and vertical/dihedral mirror planes. However, there is an improper rotation (or a rotation-reflection) axis collinear with the principal C n axis. Group Description Example S2n improper rotation (or a rotation-reflection) axis collinear with the principal Cn axis 12-crown-4 is S4 See Also Symmetry Gallery