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Point Groups C1 : no symmetry Cs : only a plane of symmetry Ck : only a k rotational axis Ci : only an inversion center Ckh : a k rotational axis and σh Ckv : a k rotational axis and k σv Dk : only Ck and k C2 rotational axes Dkh : operations of Dk and σh which implies k σv Dkd : operations of Dk and k σd which bisect the angles of C2 Sk : only the improper rotation Sk Td : tetrahedral Oh : octahedral H O B N H H O O H H H C3h C3v Point group representations A point group representation is a basis set in which the irreducible representations are the basis vectors. Shown below is the C2v point group. It has four dimensions. The dimensions are the symmetry operations. There are four basis vectors, which are also known as irreducible representations. Dkh point groups N N Fe M N N D2h D3h D4h Dkd point groups D3d D5h D6h Dkh character table The D4h point group is a second example. It has 16 dimensions. Note that the symmetry operations are listed by class. A class refers to a given type of operation, e.g. reflection, inversion, rotation, etc. Dkh character table The D6h point group is a third example. The molecule benzene belongs to this point group. High symmetry point groups: Tetrahedral and octahedral Tetrahedron (Th) Octahedron (Oh) The tetrahedral point group The Td point group is a high symmetry group. Many transition metal complexes belong to this group, but also molecules such as methane, CH4. Examples of point group assignment Determining the point group to which a molecule belongs will be the first step in a treatment of the molecular orbitals or spectra of a compound. It is important that this be done systematically. The flow chart in the figure is offered as an aid, and a few examples should clarify the process. Systematic assignment of a molecule to a point group Symmetry properties are used to determine the molecular orbitals and spectral features of a molecule. It is important to have a systematic approach to assignment of the point group. The scheme gives a systematic series of questions that lead to the point group assignment. What are the point groups for the following Pt(II) ions? Cl Pt Cl Cl 2- Cl Cl Pt Cl A Pt Cl Br Cl 2- Cl Cl Pt Cl B Pt Br Br Br 2- Cl Pt Cl C Cl Examples of point group assignment Cl Cl Pt Pt Cl Cl Cl Cl 2- Examples of point group assignment Cl Cl Pt Pt Cl Cl Cl Cl 2- Examples of point group assignment Cl Cl Pt Pt Cl Cl Cl Cl 2- Examples of point group assignment Cl Cl Pt Pt Cl Cl 2- Cl Cl D2h Examples of point group assignment Cl Cl Pt Pt Br Cl Cl Br 2- Examples of point group assignment Cl Cl Pt Pt Br Cl Cl Br 2- Examples of point group assignment Cl Cl Pt Pt Br Cl 2- Cl Br C2v Examples of point group assignment Cl Cl Pt Pt Br Br Cl Cl 2- Examples of point group assignment Cl Cl Pt Pt Br Br Cl Cl 2- Examples of point group assignment Cl Cl Pt Pt Br Br 2- Cl Cl C2h What are the point groups for the following Pt(II) ions? Cl Cl Cl 2Cl Cl Cl 2Cl Cl Br 2Pt Cl Pt Cl Pt Cl Br Pt Cl Pt Br Br Pt Cl Cl C B A A contains three C2 axes, i.e., [Ck?] is yes with k=2. It contains a plane of symmetry so [σ?] is yes. The three C2 axes are perpendicular, i.e, there is a C2 axis and two perpendicular C2's which means that [⊥C2?] is yes. There is a plane of symmetry perpendicular to the C2 so [σh?] is yes and we arrive at the D2h point group. B contains only one C2 axis, no ⊥C2's, no σh, but it does have two σv's and is therefore a C2v ion. C contains a single C2 axis and a horizontal plane (the plane of the ion) and therefore has C2h symmetry.
