The determination of point groups of moleculesonly one rotationalaxis = C2two v but no h mirror planes means point group is C2vThe point group of the water molecule is C2v

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 orderof the principal rotational axis (e.g. 3-fold or 4 fold etc.):C3 C3v D4d D4hA C indicates only one rotational axisA D indicates an n-foldprincipal rotation axisplus n 2-fold axes at right angles to it3-fold rotational has v but 4-fold d = no h indicates axis no h mirror principal h mirror a h mirror planes in a C group axis plane plane

Naming point groups (contd.):A subscript h means that there is a h mirror plane at right angles to the n-fold principal axis:D4hC4 principal axishC3 principal axisvA subscript d (or v for C groups) means there is no h mirrorplane, but only n v mirror planes containing the principal Cn axis.only oneof the threev planes is shownD3d

Naming platonic solids:Platonic solids: T = tetrahedral = 4 three-fold axes O = octahedral = 3 four-fold axes I = icosahedral = 6 five-fold axes Td Oh IhC60bucky-ballor Fullerene

Flow chart for determining point groups.

The point group of the carbon dioxide moleculeWe start at the top of theflow-chart, and can see thatthe CO2 molecule is linear, and has a center of inversion(i) so it is Dh. Note the C principal rotation axis.iC

Dh

Other linear molecules: HCN HI CON2 O2 F2 H2DhCviiThe top row of linear molecules all have a center ofinversion (i) and so are Dh.The bottom row have noi and so are CvAll have a C axis

The Platonic solids:Td Oh Ih C60 buckyballtetrahedron octahedron icosahedron

The Cs point group:Cschloro-difluoro-iodo-methaneIFClCF

Most land animals have bilateral symmetry, and belong to the Cs point group:Mirror planes ()CsCs

The C1 point group:Molecules that have no symmetry elements at all except the trivial one where they are rotated through 360 and remain unchanged, belong to the C1 point group. In other words, they have an axis of 360/360 = 1-fold, so have a C1 axis. Examples are:Bromo-chloro-fluoro-iodo- chloro-iodo-amine methaneIBrFClCIClHNC1C1

The division into Cn and Dn point groups: After we have decided thatthere is aprincipal rotat-ional axis, wecome to thered box. If thereare n C2 axesat right anglesto the principalaxis, we have aDn point group,If not, it is a Cnpoint group.DnCn

The Cn point groups:The Cn point groups all have only a single rotational axis, which can theoretically be very high e.g. C5 in the complex [IF6O]- below. They are further dividedinto Cn, Cnv, and Cnh point groups. The Cn point groups have no other symmetry elements, the Cnv point groups have also n mirror planes containing the Cn rotational axis, while the Cnh point groups also have a h mirror plane at right angles to the principal rotational axis.C5iodine[IF6O]-OFF

The point group of the water moleculeWe start at the top of theflow-chart, and can see thatthe water molecule is notlinear, and is not tetrahedral (Td),octahedral (Oh), or icosahedral,(Ih) so we proceed down the chart

C2Yes, there is a principal Cn axis,so we proceed down the chart, butin answer to the next question, thereare no further C2 axes at right anglesto the principal axis, which is the onlyaxis, so we proceed down the chart

The point group of the water molecule is C2vthere is no h planeat right angles tothe C2 axis, butthere are two vplanes containingthe C2 axis. C2C2C2vv

Other Cnv molecules:C2vC3vC4vammoniawatervvvVanadyl tetrafluoride (VOF4)V

Some more C2v molecules:vvvvvvC2C2C2Phosphorus iodo- sulfur tetra- carbonyltetrafluoride (PF4I) fluoride (SF4) chloride (COCl2)CSP

The Cn point groups:These have a Cn axis as their only symmetry element. They generally resemble propellers which have the front and back different. Important examples are (hydrogens omitted for clarity):C3C3C3C3C3C3 triphenyl phosphineviewed down C3 axis Cobalt(III)tris-glycinateviewed down C3 axis triphenyl phosphineviewed from the side Cobalt(III)tris-glycinateviewed from the side

The Dnh point groups:C2hfour C2axes atrt. anglesto C4 axisC2C2C2 C4 principal axismirror planeat rt. anglesto C4 axisD4h

Examples of molecules belonging to Dnh point groups:D2hD3hD3hD3hD4hD4hD5hD5hC2C3C3C3C4C4C5C5

C6principal axisC2C2C2C6C2vvBenzene, an example of the D6h point group:h C6principal axis C6principal axisD6h

The Dn point groups:C2C2C2 principal axisD2these have a principaln-fold axis, and n2-fold axes at rightangles to it, but nomirror planes. [Cu(en)2]2+ complex with H-atomsomitted for clarity. (en = ethylene diamine)CuNNC

Some further views of the symmetry elements of [Cu(en)2]2+, point group D2 :C2[Cu(en)2]2+ complex with H-atomsomitted for clarity. (en = ethylene diamine)C2C2C2C2C2C2 principalaxisC2 principal axisC2 principalaxisC2 principalaxisC2C2D2

Some views of the symmetry elements of [Co(en)3]3+, point group D3.C2C2C2C3 principal axisC3 principal axisC2 axisview down the C3 axisof [Co(en)3]3+ showingthe three C2 axes.D3view down one of the three C2 axes of [Co(en)3]3+ at right angles to C3

Other examples of the D3 point group[Co(oxalate)3]3- [Co(bipyridyl)3]3+C2C2C2C2C2C2C3 principal axisD3D3

Some cobalt(III) complexes belonging to the D3 point group:tris(ethylenediamine) tris(2,2-bipyridyl)tris(acetylacetonato)cobalt(III) cation cobalt(III) cationcobalt(III)D3C2C2C2

Comparison of C3 and D3 tris(chelates)D3C3C2no C2 axis atthis pointtris(ethylenediamine)cobalt(III) tris(glycinato)cobalt(III)

Molecules belonging to the Dnd point groupsThese have mirror planes parallel to the principal axis, but not at right angles to it.C3 axisC5 axisStaggered form of ethaneStaggered form of ferrocenev planescontain theprincipalaxisD3dD5d

The D4d point group:C2C2C2C2C4 principal axisC4 principal axisC2vvvvC4 principal axis [ZrF8]4-Square antiprismAs predicted by VSEPR, the [ZrF8]4- anion has a square anti-prismaticstructure. At left is seen the C4 principal axis. It has four C2 axes atright angles to it, so it has D4 symmetry. One C2 axis is shown side-on(center). There are four v mirror planes (right), but no mirror plane atright angles to C4, so the point group does not rate an h, and is D4d.D4d

[K(18-crown-6)]+, an example of a D3d point group:The complex cation [K(18-crown-6)]+ above is an important structure thathas D3d symmetry. It has a C3 principal axis with 3 C2 axes at rightangles to it, as well as three v mirror planes that contain the C3 axis,but no h mirror plane (because its not flat, as seen at center), so is D3d.D3dvvK+C3 principal axisC3 principal axisvC2C2C2C2C2C2

Some Point groups