Symmetry elements, operations and point groups ( in the molecular world )

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1 ymmetry elements, operations and point groups ( in the molecular world ) ymmetry concept is extremely useful in chemistry in that it can help to predict infra-red spectra (vibrational spectroscopy) and optical activity. It can also help in describing orbitals involved in bonding, and in interpreting electronic spectra of molecules. The five fundamental symmetry elements (operators) or operations: The following symbols (choenflies) are used to describe both symmetry elements as well as the symmetry operations (application of the symmetry element) E (Identity) all molecules have E (or C 1 ); causes no change in the molecule C n (Rotation axis, clockwise by definition) 360 (or pi)/n rotation about a rotation axis ighest order or principal rotation axis: the axis with the highest n value e.g. C 3 (in C 3 ), C 6 (snowflake, benzene); C (linear molecules) σ (Reflection plane) mirror plane (exchange of left and right ) four different types of reflections: σ v, σ h, σ d, σ σ v : vertical mirror plane (contains principal rotation axis) example: B 3 (planar) σ v B B B σ v σ v σ h : horizontal mirror plane (plane perpendicular to principal axis) example: B 3 (trivial because its planar) σ d : dihedral mirror plane (planes that cut the angle between C axes in half) example: allene C ' C ' C C C C σ d σ d

2 σ: Mirror plane, but no rotational axis ex: aniline (because the hydrogen atoms on N are out of plane) N σ i (Inversion centre) position of all atoms remains unchanged after their reflection through the inversion centre ex: [Pt 4 ] - Pt n (Mirror-rotation or rotation-reflection axis or improper axis) 360 /n rotation followed by a reflection through a plane perpendicular to the rotation axis Ex: staggered ethane 6 1. rotation by60deg. reflection through plane Note: 1 σ; i

3 Multiplication of symmetry operations One of the important features of symmetry operations is that they can be performed multiple times and they can also be combined: 1. Repetition of symmetry operations: Example 1: C E (the superscript indicates how many times C is performed) Obviously C does not lead to a new symmetry operation Another example of this: σ h E Example : C 3 is a new symmetry operation (whereas C 3 3 E) the symmetry element C 3 creates the symmetry operations C 3 and C 3 C 3 C C 3 Example 3: C 6 (benzene): only C 6 and C 6 5 are symmetry operations of C 6 C 6 = C 3, C 6 3 = C, C 6 4 = C 3, C 6 5, C 6 6 = E. Combination of different sym. operations: B x A (A first, B second) Example 1: C σ v ' σ v Example : n = σ h x C n 3

4 3. Inverse operation: or each operation there is one that brings the molecule back into its original position. Example 1: C -1 (counterclockwise rotation by 180 deg), C -1 = C and C x C -1 = E Generally: C n -1 = C n n-1 Example : C 3-1 = C 3 Example 3: σ -1 = σ; i -1 = i 4

5 Point Groups Each molecule has at least one point which is unique and which remains unchanged, no matter how many or what type of symmetry operations are performed. uch point is termed singular point. (e.g. centre of the benzene molecule) Point group = The complete set of symmetry operations that characterize a molecule s overall symmetry. pace groups = ymmetry classes characterizing entities with translational symmetry. [pace groups don t have singular points!] Because there is a limited number of symmetry elements (at least in the molecular world, because n is usually <10) and their combinations, there must be a limited number of point groups! Therefore, so many different molecules belong to the same point group: Example: O and C both C v (E, C, σ v, σ v ) Three point groups of low symmetry: Point group Description Example C 1 asymmetry CBr C s only 1 symmetry (mirror) plane C Br Br C i only 1 inversion centre (very rare) 1,-dibromo-1,- dichloroethane (staggered) x Br Br

6 ive point groups of high symmetry: Point group Description Example C v linear - infinite number of rotations infinite number of reflection planes containing the principal axis (easy criterion: linear but no i) D h linear (like C v but with additional σ h, C, and i) CO (easy criterion: linear + i) T d tetrahedral symmetry C 4 Contains 4 symmetry operations (E, 4C 3, 4C 3, 3 4, 3 3 4, 3C, 6σ d ) O h octahedral symmetry (48 symmetry operations) 6 I h icosahedral symmetry (10 symmetry operations) B dodecahydridododecaborane

7 Groups of high symmetry ymmetry operations for high symmetry point groups and their rotational subgroups 3

8 Diagram of the Point Group Assignment Method 4

9 C n and n point groups Point groups without any mirror planes: C n the only symmetry element is C n ; symmetry operations: C n, C n, C n 3, C n n-1 C 1 complete asymmetry C, C 3, are (extremely) rare and difficult to visualize Example of C : O (C cuts angle between the two OO planes, 94 deg, in half) 94 deg O O C n (only possible with n =, 4, 6, ) symmetry operations: n, n, n 3, n n-1 C i example: staggered 1,-dibromo-1,-dichloroethane 4 : Rare! Contains symmetry operations 4, 3 4 ( C ), 4 Example: 1,3,5,7-tetrafluorocyclooctatetraene 4 Point groups with mirror planes: C nv symmetry operations: C n,..., C n n-1, n x σ v C 1v C s

10 C v (very common): O, 1,-dichlorobenzene C σ v ' O σ v C 3v (very common): trigonal-pyramidal molecules (N 3, C 3 ) N C 4v : square-pyramidal molecules (Br 5 ) Br C 5,6,..v : very rare C v : all linear molecules without inversion centre (, ) C nh symmetry elements: C n, σ h, i if n even, n C h : has C, σ h and i example: difluorodiazene (planar) N N

11 C 3h : B(O) 3 boric acid (planar) O B O O C 4,5...h : extremely rare D point groups Criterion: in contrast to C n groups, D groups have n C axes perpendicular to C n! D n : symmetry operations: C n,..., C n-1 n, n x C (rare point group!) D : one of the excited states of ethylene (slight deviation from planarity!) C v fragment C v fragment fragmentation method: D can be regarded as consisting of identical C v fragments joined back to back, so that one half is rotated (with respect to other) by any degree (except m x π/n with m =,4,6,..; n = order of C) D 3 : [Co(en) 3 ] 3+ en = ethylenediamine 3

12 D nd : symmetry elements: C n, n x C, n, n x σ d D d : allene C ' C ' σ d C 4 C C C σ d D 3d : symmetry operations: C 3, C 3, 3 C, 6, 6 3 = i, 6 5, 3 σ d example: staggered ethane ( C 3v fragments: 3 C, rotation by 60 deg) C σ d C σ d σ d C D 4d : 8 D 5d : errocene e 4

13 D nh : symmetry elements: C n, n x C, σ h, n, n x σ, i if n is even Note: the horizontal mirror plane is usually easily identifiable! D h : e.g. C 4 (has 3 equivalent C axes; therefore: assigning a v or h subscript to σ is redundant) D 3h : P 5 (trigonal-bipyramidal), B 3 (planar) P B D 4h : molecules with the geometry of a square (Xe 4 ) Xe D 5h : planar with 5-fold rotational symmetry (e.g. cyclopentadienyl anion) D 6h : planar with 6-fold rotational symmetry (benzene) D h : linear symmetrical molecules (N, C ) Three tips for assigning point groups 1. Most of the low and high symmetry point groups are relatively easy to assign.. When it comes to D vs. C or : look for n C axes perpendicular to C n. 3. Then look for σ h. Chirality Group-theoretical criterion for chirality: Absence of n! T d, O h, C nh, D nd, D nh possess n. Molecules belonging to these point groups can therefore not be chiral! Note: Molecules are usually chiral when neither i ( ) nor σ h is present (σ h 1 ) 5