Point Group Group Theory Assigning the Point Group of a Molecule A. Low Symmetry Groups
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1 Point Groups Point Group = the set of symmetry operations for a molecule Group Theory = mathematical treatment of the properties of the group which can be used to find properties of the molecule Assigning the Point Group of a Molecule 1. Determine if the molecule is of high or low symmetry by inspection A. Low Symmetry Groups
2 B. High Symmetry Groups
3 2. If not, find the principle axis 3. If there are axes perpendicular to C n the molecule is in D If not, the molecule will be in C or S 4. If h perpendicular to C n then D nh or C nh If not, go to the next step 5. If contains C n then C nv or D nd If not, D n or C n or S 2n 6. If S 2n along C n then S 2n 7. If not C n
4 The determination of point groups of molecules only one rotational axis = two but no σ h mirror planes means point group is v The point group of the water molecule is v
5 Naming point groups: The name of the point group has information about the symmetry elements present. The letter is the rotational group and the subscript number after the letter indicates the order of the principal rotational axis (e.g. 3-fold or 4 fold etc.): A C indicates only one rotational axis A D indicates an n-fold principal rotation axis plus n 2-fold axes at right angles to it C 3 C 3v D 4d D 4h 3-fold rotational has but 4-fold d = no h indicates axis no σ h mirror principal σ h mirror a σ h mirror planes in a C group axis plane plane
6 Naming point groups (contd.): A subscript h means that there is a σ h mirror plane at right angles to the n-fold principal axis: C 4 principal axis C 3 principal axis only one of the three planes is shown σ h D 4h D 3d A subscript d (or v for C groups) means there is no σ h mirror plane, but only n mirror planes containing the principal C n axis.
7 Naming platonic solids: Platonic solids: Five special polyhedra: Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron Their faces are all exactly the same. The same number of faces meet at each vertex. T = tetrahedral = 4 three-fold axes O = octahedral = 3 four-fold axes I = icosahedral = 6 five-fold axes T d O h I h C 60 bucky-ball or Fullerene
8 Flow chart for determining point groups.
9 The point group of the carbon dioxide molecule i C We start at the top of the flow-chart, and can see that the CO 2 molecule is linear, and has a center of inversion D h (i) so it is D h. Note the C principal rotation axis.
10 Other linear molecules: The top row of linear molecules all have a center of inversion (i) and so are D h. i D h i N 2 O 2 F 2 H 2 HC N HI C O The bottom row have no i and so are C v C v All have a C axis
11 The Platonic solids: tetrahedron octahedron icosahedron T d O h I h C 60 buckyball
12 The C s point group: I σ C s F F C Cl chloro-difluoro-iodomethane
13 Most land animals have bilateral symmetry, and belong to the C s point group: C s Mirror planes (σ)
14 The C 1 point group: Molecules that have no symmetry elements at all except the trivial one where they are rotated through 360º and remain unchanged, belong to the C 1 point group. In other words, they have an axis of 360º/360º = 1-fold, so have a C 1 axis. Examples are: C 1 Br I C Cl F I Cl N H C 1 Bromo-chloro-fluoro-iodomethane chloro-iodo-amine
15 The division into C n and D n point groups: After we have decided that there is a principal rotation axis, we come to the red box. If there are n axes at right angles to the principal axis, we have a D n point group, If not, it is a C n point group. D n C n
16 The C n point groups: The C n point groups all have only a single rotational axis, which can theoretically be very high e.g. C 5 in the complex [IF 6 O] - below. They are further divided C 5 groups O symmetry iodine the C nv point groups also n mirror planes F containing the C n rotational axis, while the C nh point F into C n, C nv, and C nh point groups. The C n point have no other elements, have groups also have a σ h mirror plane at right angles to the [IF 6 O] - principal rotational axis.
17 The point group of the water molecule We start at the top of the flow-chart, and can see that the water molecule is not linear, and is not tetrahedral (T d ), octahedral (O h ), or icosahedral, (I h ) so we proceed down the chart
18 Yes, there is a principal C n axis, so we proceed down the chart, but in answer to the next question, there are no further axes at right angles to the principal axis, which is the only axis, so we proceed down the chart
19 there is no σ h plane at right angles to the axis, but there are two planes containing the axis. The point group of the water molecule is v
20 Other C nv molecules: water ammonia v C 3v C 4v V Vanadyl tetrafluoride (VOF 4 )
21 Some more v molecules: P S C Phosphorus iodo- sulfur tetra- carbonyl tetrafluoride (PF 4 I) fluoride (SF 4 ) chloride (COCl 2 )
22 The C n point groups: These have a C n axis as their only symmetry element. Important examples are (hydrogens omitted for clarity): triphenyl phosphine viewed down C 3 axis C 3 C 3 Cobalt(III) tris-glycinate viewed down C 3 axis C 3 C 3 C 3 C 3 triphenyl phosphine viewed from the side Cobalt(III) tris-glycinate viewed from the side
23
24 The D nh point groups: C 4 principal axis four axes at rt. angles to C 4 axis mirror plane at rt. angles to C 4 axis σ h D 4h
25 Examples of molecules belonging to D nh point groups: C 3 C 3 C 3 C 4 C 4 D 2h D 3h D 3h D 3h C 5 C 5 D 4h D 4h D 5h D 5h
26 Benzene, an example of the D 6h point group: C 6 principal axis C 6 σ h D 6h C 6 principal axis C 6 principal axis
27 The D n point groups: these have a principal n-fold axis, and n 2-fold axes at right angles to it, but no mirror planes. principal axis N C N Cu [Cu(en) 2 ] 2+ complex with H-atoms omitted for clarity. (en = ethylene diamine) D 2
28 Some further views of the symmetry elements of [Cu(en) 2 ] 2+, point group D 2 : principal axis principal axis D 2 principal axis [Cu(en) 2 ] 2+ complex with H-atoms omitted for clarity. (en = ethylene diamine) principal axis C2
![Some views of the symmetry elements of [Co(en) 3 ] 3+, point group D 3.](/docs-images/93/114313372/images/29-0.jpg "C 3 principal axis C 3 principal axis view down the C 3 axis of [Co(en) 3 ] 3+ showing the three axes.")
29 Some views of the symmetry elements of [Co(en) 3 ] 3+, point group D 3. C 3 principal axis C 3 principal axis view down the C 3 axis of [Co(en) 3 ] 3+ showing the three axes. D 3 axis view down one of the three axes of [Co(en) 3 ] 3+ at right angles to C 3
30 Other examples of the D 3 point group C 3 principal axis D 3 D 3 [Co(oxalate) 3 ] 3- [Co(bipyridyl) 3 ] 3+
31 Molecules belonging to the D nd point groups These have mirror planes parallel to the principal axis, but not at right angles to it. C 3 axis planes C 5 axis contain the principal axis D 3d D 5d Staggered form of ethane Staggered form of ferrocene
![The D 4d point group: C 4 principal axis C 4 principal axis [ZrF 8 ] 4- Square antiprism C 4 principal axis D 4d As predicted by VSEPR, the [ZrF 8 ] 4- anion has a square anti-prismatic structure.](/docs-images/93/114313372/images/32-0.jpg "At left is seen the C 4 principal axis. It has four axes at right angles to it, so it has D 4 symmetry. One axis is shown side-on (center).")
32 The D 4d point group: C 4 principal axis C 4 principal axis [ZrF 8 ] 4- Square antiprism C 4 principal axis D 4d As predicted by VSEPR, the [ZrF 8 ] 4- anion has a square anti-prismatic structure. At left is seen the C 4 principal axis. It has four axes at right angles to it, so it has D 4 symmetry. One axis is shown side-on (center). There are four mirror planes (right), but no mirror plane at right angles to C 4, so the point group does not rate an h, and is D 4d.
![[K(18-crown-6)] +, an example of a D 3d point group: C 3 principal axis K + C 3 principal axis D 3d The complex cation [K(18-crown-6)] + above is an important structure that has D 3d symmetry.](/docs-images/93/114313372/images/33-0.jpg "It has a C 3 principal axis with 3 axes at right angles to it, as well as three mirror planes that contain the C 3 axis, but no σ h mirror plane (because it s not flat, as seen at center), so is D")
33 [K(18-crown-6)] +, an example of a D 3d point group: C 3 principal axis K + C 3 principal axis D 3d The complex cation [K(18-crown-6)] + above is an important structure that has D 3d symmetry. It has a C 3 principal axis with 3 axes at right angles to it, as well as three mirror planes that contain the C 3 axis, but no σ h mirror plane (because it s not flat, as seen at center), so is D 3d.
34 Some Point groups
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