-2013 - 3D Structures of Biological Macromolecules Point Group Determination for Molecules

1. -2013 - 3D Structuresof Biological Macromolecules Point Group Determination for Molecules Jürgen Sühnel jsuehnel@fli-leibniz.de Leibniz Institute for Age Research, Fritz Lipmann Institute, Jena Centre for Bioinformatics Jena / Germany Supplementary Material: www.fli-leibniz.de/www_bioc/3D/

2. Symmetry Elements and Operations • The symmetry of a molecule can be described by 5 types of symmetry elements. • Symmetry axis: an axis around which a rotation by results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system. • Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). A third type of symmetry plane exists: if a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σd). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz). • Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF4) where the inversion center is at the Xe atom, and benzene (C6H6) where the inversion center is at the center of the ring. • Rotation-reflection axis: an axis around which a rotation by  , followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn, with n necessarily even. Examples are present in tetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis. • Identity, abbreviated to E, from the German 'Einheit' meaning Unity.This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, its consideration is necessary for the group-theoretical machinery to work properly. It is so called because it is analogous to multiplying by one (unity).

3. Point Group Determination of the Water Molecule

4. Naming Point Groups D4h D3d

5. Naming Point Groups D4d D4h mirror

6. Naming Point Groups – Platonic Solids

7. Point Group Determination of Molecules

8. Point Group Determination of Molecules

9. Point Group Determination of Molecules

10. Point Group Determination: CO2

11. Point Group Determination: Other Linear Molecules

12. Point Group Determination: The CsPoint Group

13. Point Group Determination: The C1Point Group C1

14. Point Group Determination: The DnPoint Groups

15. Point Group Determination: Comparison of C3 and D3tris(chelates) D3 C3

16. Point Group Determination: Examples of Molecules Belonging to DnhPoint Groups D2h D3h D4h D4h D4h D4h D5h D5h

17. Point Group Determination: Benzene an Example of the D6h Point Group

18. Point Group Determination: [K(18-crown-6)]+, an Example of a D3d Point Group The complex cation [K(18-crown-6)] is an important structure that has D3d symmetry. It has a C3principal axis with 3 C2axes at right angles to it, as well as three svmirror planes that contain the C3 axis, but no shmirror plane (because it is not flat, as seen at the center), so it is D3d.

19. Point Group Determination: The D4d Point Group The [ZrF8]4- anion has a square anti-prismatic structure. At left is seen the C4 principal axis. It has four C2axes at right angles to it, so it has D4symmetry. One C2axisis shown side-on (center). There are four sv mirror planes (right), but no mirror plane at right angles to C4, so the point group does not rate an h, and is D4d.

20. Point Group Determination: Molecules Belonging to Dnd Point Groups D3d D5

21. Point Group Determination: Some Point Groups

22. What is the determination of the symmetry of molecules good for? • Establishment of selection rules for spectral transitions of all kinds • examples: occurrence of lines in IR or Raman spectra • Chirality of molecules; Optical activity (rotation) • Better understanding of molecular wavefunctions • Reduction of computer power required for geometry optimization of molecules (if one stays within a certain symmetry) • ….