POINT GROUPS Molecular Symmetry Symmetry element Point GroupsLET’S GO
Molecular Symmetry All molecules can be described in terms of their symmetrySymmetry operation Reflection, rotation, or inversion Symmetry elements such as mirror, axes of rotation, and inversion centers
There are two naming systems commonly used when describing symmetry elements:1. The Schoenflies notation used extensively by spectroscopists2. The Hermann-Mauguin or international notation preferred by crystallographers Symmetry elements Symmetry element Notation Hermann-Manguin Schönflies (crystallography) (spectroscopy) Point Symmetry Identity 1 for 1-fold rotation C Rotation axes n Cn Mirror planes m σh, σv, σd Centres of Ī i inversion(centres of symmetry) Sn Axes of rotary inversion (improper rotation) Space symmetry Glide plane n, d, a, b, c - Screw axis 21, 31, etc -
Symmetry Elements Identitas (C ≡E atau 1) 1 Rotation axes (Cnatau n) Centres of inversion (centre of symmetry (i atau ) 1 inversion axes (axes of rotary inversion) Mirror planes ( atau m)
1. Identity (C1 ≡ E or 1) Rotasi dengan sudut putar 360° melalui sudut z sehingga molekul kembali seperti posisi semula. Putaran seperti ini diberi simbol dengan C1 axis atau 1. Schoenflies: C1 Hermann-Mauguin: 1 for 1- fold rotation Operation: act of rotating molecule through 360° Element: axis of symmetry (i.e. the rotation axis).
2. Rotation (Cn or n) Rotasi melalui sudut selain 360°. Operation: act of rotation Element: rotation axis Symbol untuk symmetry element yang mana rotasinya adalah rotasi dari 360°/n Schoenflies: Cn Hermann–Mauguin: n. Molekul mempunyai n- fold axis dari symmetry.
a. Two-fold rotation A Symmetrical Pattern = 360o/2 rotation to reproduce a motif in a 6 symmetrical pattern 6
Operationa. Two-fold rotation = 360o/2 rotation Motif to reproduce a motif in a 6 symmetrical Element pattern= the symbol for a two-fold rotation 6
a. Two-fold rotation = 360o/2 rotation to reproduce a motif in a 6 first operatio n step symmetrical pattern= the symbol for a two-fold rotation second 6 operatio n step
b. Three-fold rotation = 360o/3 rotation to reproduce a motif in a symmetrical pattern
b. Three-fold rotation = 360o/3 rotation to reproduce a step 1 motif in a symmetrical pattern step 3 step 2
Symmetry Elements Rotation 6 6 6 6 6 6 6 6 1-fold 2-fold 3-fold 4-fold 6-foldObjects with symmetry: a identity Z t 9 d 5-fold and > 6-fold rotations will not work in combination with translations in crystals (as we shall see later). Thus we will exclude them now.
5. Axes of rotary inversion (improper rotation Snor An improper rotation involves a combination of rotation and n) reflection The operation is a combination of rotation by 360°/n (Cn) followed by reflection in a plane normal ( σh) to the Sn axis Molecule does not need to have either a Cn or a σh symmetry element
Combinations of symmetry elements are also possibleTo create a complete analysis of symmetry about a point inspace, we must try all possible combinations of thesesymmetry elementsIn the interest of clarity and ease of illustration, wecontinue to consider only 2-D examples
Try combining a 2-fold rotation axis with a mirror
Try combining a 2-fold rotation axis with a mirrorStep 1: reflect(could do either step first)
Try combining a 2-fold rotation axis with a mirrorStep 1: reflectStep 2: rotate (everything)
Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)No! A second mirror is required
Try combining a 2-fold rotation axis with a mirrorThe result is Point Group 2mm“2mm” indicates 2 mirrors
Now try combining a 4-fold rotation axis with a mirror
Now try combining a 4-fold rotation axis with a mirrorStep 1: reflect
Now try combining a 4-fold rotation axis with a mirrorStep 1: reflectStep 2: rotate 1
Now try combining a 4-fold rotation axis with a mirrorStep 1: reflectStep 2: rotate 2
Now try combining a 4-fold rotation axis with a mirrorStep 1: reflectStep 2: rotate 3
Now try combining a 4-fold rotation axis with a mirrorAny other elements?
• Now try combining a 4-fold rotation axis with a mirrorAny other elements?Yes, two more mirrorsPoint group name??4mm
3-fold rotation axis with a mirror creates point group 3m
6-fold rotation axis with a mirror creates pointgroup 6mm
Point groups Most molecules will possess more than one symmetry element. All molecules characterised by 32 different combinations of symmetry elements: POINT GROUPS There are symbols for each of the possible point groups These symbols are often used to describe the symmetry of a moleculeFor example: rather than saying water is bent, you can say that water has C2v point symmetry
THE GROUPSThe groups C1, Ci and CsC1: no element other than the identityCi: identity and inversion aloneCs:identity and a mirror plane alone The groups Cn, Cnv and Cnh Cn: n-fold rotation axis Cnv: identity, Cn axis plus n vertical mirror planes σv Cnh: identity and an n-fold rotation principal axis plus a horizontal mirror plane σhThe groups Dn, Dnh and DndDn: n-fold principal axis and n two-foldaxes perpendicular to CnDnh: molecule also possesses a horizontalmirror planeDnd: in addition to the elements of Dnpossesses n dihedralmirror planes σd
The groups SnSn: Molecules not already classifiedpossessing one Sn axisMolecules belonging to Sn with n > 4 arerareS2 ≡ Ci The cubic groups Td and Oh: groups of the regular tetrahedron (e.g. CH4) and regular octahedron (e.g. SF6), respectively. T or O: object possesses the rotational symmetry of the tetrahedron or the octahedron, but none of their planes of reflection Th: based on T but also contains a centre of inversionThe full rotation groupR3: consists of an infinite number ofrotation axes with allpossible values of n. A sphere and anatom belong to R3,but no molecule does.