Point Symmetry Elements and Point Groups

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Point Symmetry Elements and Point Groups

Point Symmetry, Point Groups of Molecules** Symmetry elements: A symmetry operation is an action that leaves the molecule seemingly unchanged. Associated to each symmetry operation is the symmetry element (line, plane, point). With this element the symmetry operation is carried out. All operations leave at least one point of the molecule unchanged. • Identity: The identity operation leaves the whole molecule unchanged. Every molecule has at least this symmetry element. • Rotation axis: A n-fold rotation rotates the molecule by 360°/n around the rotation axis. The molecule is apparently unchanged after rotation by 360°/n. Rotation by 360° (n=1) is equal to identity. • Mirror plane: The corresponding symmetry operation is a reflection. A mirror plane exists when two halves of a molecule can be interconverted by reflection across the mirror plane. • Center of inversion: Each atom is projected in a straight line through the center of inversion and then out to an equal distance on the other side. • Alternating axis: This symmetry operation is a composite operation. At first there is a rotation by 360°/n around a rotation axis, followed by a reflection perpendicular to that axis. • Inversion axis: This operation is also a combined symmetry operation. At first there is a rotation by 360°/n around a rotation axis, followed by an inversion through an inversion center. Symmetry element Hermann-Mauguin symbols Schönflies symbols (used in crystallography) (used in spectroscopy) Indentity ---- E C (n = 2,3…) Rotation axis n= 2,3,… n C1 = E σh (h = horizontal) Mirror plane m σv (v = vertical)aa σd (d = dihedral)a Center of inversion -1 i S (n = 1,2,3…) Alternating axis n ----- S = σ (rotoreflection axis) 1 S2 = i Inversion axis -n ---- (rotoinversion axis) Point groups: A point group is a set of symmetry elements. To assign a molecule to a particular point group, you have to list all symmetry elements of this molecule and compare that list with the list that defines each point group. To facilitate the identification of the point group of a molecule there exists a decision tree*. With the help of this decision tree the point group is determined by answering successively questions about the symmetry of a given molecule . * see: Inorganic Chemistry, Shriver & Atkins, 3rd edition p. 122 or in the web: http://www.scs.uiuc.edu/~chem315/point_groups/point_group_flowchart_shriver.jpg Questions: 1) List all symmetry operations you know with the corresponding Hermann- Mauguin and Schönflies symbols and give at least one example for every symmetry element. 2) List all symmetry elements of ammonia and define the point group by using a decision tree. ** Kohlhaas 2007 .

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