Le,..........................cture 10.pdf

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1. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Lecture no: 10
Chapter: Symmetry elements and operations
Contents:
➢Crystal systems
➢Symmetry elements
➢Symmetry Operations
➢Point groups
1. High order point groups
2. Low order point groups
➢Assigning point groups

2. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
CrystalSystems
Crystalsystemisamethodofclassifyingcrystallinesubstancesonthebasisoftheir
unitcell.Therearesevenuniquecrystalsystems.ACrystalSystemreferstooneof
themanyclassesofcrystals,pointgroups,spacegroups,andlattices.

3. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Symmetry elements: Discussed before in previous lecture.
Symmetry Operations: Asymmetry operationis an action that leaves an object looking the
same after it has been carried out.
1.Reflectionisasymmetryoperationwhichcausesasetofpointstobemirroredacrossa
plane.Wecallthisplanethe“mirrorplane.”Anypoint(X,Y,Z)becomes(-X,Y,Z)ifthereis
amirroraxisperpendiculartotheXdirection.

4. 2.Rotationisasymmetryoperationwhichcausesasetofpointstoberotatedaround
apoint.Wecallthispointan“axisofrotation.”Inpolarcoordinates,anypoint(R,θ,
φ)becomes(R,θ+360º/n,φ)forann-foldrotationaxisperpendiculartoθ.
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal

5. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
➢This is an axis passing through the crystal such that if the crystal is rotated around it through
some angle, the crystal remains unvaried.
➢The axis is called `n-fold, axis’ if the angle of rotation is 360
o
/n
➢If equivalent configuration occurs after rotation of 180º, 120º , 90º and 60º the axes of
rotation are known as two-fold, three-fold, four-fold and six-fold axes of symmetry,
respectively.
N-Fold Axis of Rotation Symmetry

6. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
➢If n=1, the crystal has to be rotated through an angle = 360º, about an axis to achieve
self coincidence. Such an axis is called an `identity axis’. Each crystal possesses an
infinite number of such axes.
➢If n=2, the crystal has to be rotated through an angle = 180º about an axis to achieve
self coincidence. Such an axis is called a `diadaxis’. Since there are 12 such edges
in a cube, the number of diadaxes is six.
➢If n=3, the crystal has to be rotated through an angle = 120º about an axis to achieve
self coincidence. Such an axis is called is `triad axis’. In a cube, the axis passing
through a solid diagonal acts as a triad axis. Since there are 4 solid diagonals in a
cube, the number of triad axis is four.
➢If n=4, for every 90º rotation, coincidence is achieved and the axis is termed `tetrad
axis.
➢If n=6, the corresponding angle of rotation is 60º and the axis of rotation is called a
hexad axis. A cubic crystal does not possess any hexad axis.
➢Crystalline solids do not show 5-fold axis of symmetry orany other symmetry axis
higher than `six’, Identical repetition of a unit can take place only when we consider
1,2-,3-,4-and 6-fold axes.

7. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Let us consider a lattice P Q R S as shown in figure
➢Letthislatticehasn-foldaxisofsymmetryandthelatticeparameterbeequalto‘t’.
➢LetusrotatethevectorsQPandRSthroughanangleɷ=360°/nintheclockwiseandanti
clockwisedirections,respectively.
➢Afterrotation,theendsofthevectorsbeatAandD.
➢SincethelatticePQRShasn-foldaxisofsymmetry,thepointsAandDshouldbethelattice
points.
➢FurtherthelineADshouldbeparalleltothelinePQRS.Therefore,thedistanceADmust
equaltosomeintegralmultipleofthelatticeparameter‘t’say,mt
➢Where,m=0,±1,±2,±3,….
Why not 5-fold
symmetry?

8. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Again,AB = CD, Thus, AD = t+ 2AB
mt = t+2t cos ɷ ………… (1)
From equation (1), 2t cos ɷ = m t –t
2cos ɷ = (m-1)
N= (m-1), where N is again 0, ±1, ±2, {since (m-1) is also an integer say, N}
cos ɷ= N/2 ………………..(2)

9. 3.Inversionisasymmetryoperationwhichpullseverypointthroughan“inversioncenter”to
theotherside.Anypoint(X,Y,Z)becomes(-X,-Y,-Z)ifthereisaninversioncenteratthe
origin.Youcancombinerotationwithinversiontoproducetheroto-inversionsymmetry
operation
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal

10. 4.Translationisasymmetryoperationthatmovesasetofpointsthroughspace.Pointgroups
DONOTinvolvetranslation.Anypoint(X,Y,Z)becomes(X+a,Y,Z)fortranslationoflength
aintheX-direction.
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal

11. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
MolecularPointGroups
Itisonlypossibleforcertaincombinationsofsymmetryelementstobepresentinamolecule
(oranyotherobject).Asaresult,wemaygrouptogethermoleculesthatpossessthesame
symmetryelementsandclassifymoleculesaccordingtotheirsymmetry.Thesegroupsof
symmetryelementsarecalledpointgroups(duetothefactthatthereisatleastonepointin
spacethatremainsunchangednomatterwhichsymmetryoperationfromthegroupisapplied).
Therearetwosystemsofnotationforlabelingsymmetrygroups,calledtheSchoenfliesand
Hermann-Mauguin(orInternational)systems.Thesymmetryofindividualmoleculesis
usuallydescribedusingtheSchoenfliesnotation,whichisusedbelow.
Low Symmetry Point Groups
GroupDescription Example
C1 only the identity operation (E) CHFClBr
Cs only the identity operation (E) and one mirror planeC2H2ClBr
Ci only the identity operation (E) and a center of inversion (i) C2H2Cl2Br

12. HighSymmetryPointGroups
HighsymmetrypointgroupsincludetheT
d,O
h,I
h,C
∞v,andD
∞hgroups.Thetablebelow
describestheircharacteristicsymmetryoperations.Thefullsetofsymmetryoperations
includedinthepointgroupisdescribedinthecorrespondingcharactertable.
GroupDescription Example
C∞v
linear molecule with an infinite number of rotation axes and vertical
mirror planes (σv)
HBr
D∞h
linear molecule with an infinite number of rotation axes, vertical mirror
planes (σv), perpendicular C2axes, a horizontal mirror plane (σh), and an
inversion center (i)
CO2
Td
typically have tetrahedral geometry, with 4 C4axes, 3 C2axes, 3
S4axes, and 6 dihedral mirror planes (σd)
CH4
Oh
typically have octahedral geometry, with 3 C4axes, 4 C3axes, and an
inversion center (i) as characteristic symmetry operations
SF6
Ih
typically have an icosahedral structure, with 6 C5axes as characteristic
symmetry operations
B12H12
2-
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal

13. Table : Common Point Groups for Molecules
Nonaxial
groups
C1 Cs Ci - - - - - -
Cngroups C2 C3 C4 C5 C6 C7 C8 - -
Dngroups D2 D3 D4 D5 D6 D7 D8 - -
Cnvgroups C2v C3v C4v C5v C6v C7v C8v - -
Cnhgroups C2h C3h C4h C5h C6h - - - -
Dnhgroups D2h D3h D4h D5h D6h D7h D8h - -
Dndgroups D2d D3d D4d D5d D6d D7d D8d - -
Sngroups S2 - S4 - S6 S8 S10 S12
Cubic
groups
T Th Td O Oh I Ih - -
Linear
groups
C∞v D∞h - - - - - - -
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
*** Some of the point groups share their names with symmetry operations and or
elements, so be careful you do not mix up the two. It is usually clear from the context
which one is being referred to.

14. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
1. C
1-contains only the identity (a C
1rotation is a rotation by 360°and
is the same as the identity operation) e.g. CHDFCl.
2. C
i -contains the identity E and a center of inversion i.
3. C
s-contains the identity E and a plane of reflection σ .
4. C
n-contains the identity and an n -fold axis of rotation.
5. C
nv-contains the identity, an n -fold axis of rotation, and n vertical
mirror planes σ
v.
Point groups –Symmetry operations

15. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
6. Cnh-contains the identity, an n-fold axis of rotation, and a horizontal
reflection plane σh(note that in C2h this combination of symmetry
elements automatically implies a center of inversion).
7. Dn-contains the identity, an n-fold axis of rotation, and n 2-fold
rotations about axes perpendicular to the principal axis.
8. Dnh-contains the same symmetry elements as Dnwith the addition of a
horizontal mirror plane.
9. Dnd-contains the same symmetry elements as Dnwith the addition of n
dihedral mirror planes.
10.Sn-containstheidentityandoneSnaxis.NotethatmoleculesonlybelongtoSnifthey
havenotalreadybeenclassifiedintermsofoneoftheprecedingpointgroups(e.g.S2isthe
sameasCi,andamoleculewiththissymmetrywouldalreadyhavebeenclassified).

16. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
11.Td-containsallthesymmetryelementsofaregulartetrahedron,
includingtheidentity,4C3axes,3C2axes,6dihedralmirrorplanes,and
3S4axese.g.CH4.
12.T-asforTdbutnoplanesofreflection.
13.Th-asforTbutcontainsacenterofinversion.
14.Oh-thegroupoftheregularoctahedrone.g.SF6.
15.O-asforOh,butwithnoplanesofreflection
Onceyoubecomemorefamiliarwiththesymmetryelementsandpointgroupsdescribed
above,youwillfinditquitestraightforwardtoclassifyamoleculeintermsofitspoint
group.Inthemeantime,theflowchartshownbelowprovidesastep-by-stepapproachto
theproblem.

17. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Flowchart to identify molecular point groups using Schoenfliesnotation system.

18. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
Identifying Symmetry Elements and Point group:
Practice session
(a) HCN
Symmetry elements: E, C∞, σv
Point group = C∞v
(b) H
2O
Symmetry elements:
E, C
2, 2σ
v
Point group: C
2v
Linear ?
Yes
Centre of inversion, i?
No
C∞v
Linear ?
No
2 or more Cn where n>2 ?
No
Cn ?
Yes, C
2
n C
2┴ to principle Cn ?
No
σh?
nσv?
Yes, 2σ
v
No
CnvC
2v

19. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
(c)
Linear ?
No
2 or more Cn where n>2 ?
N
o
Cn ?
No
σh?
No
C
1
Centre of inversion, i?
No
Symmetry elements: E
Point group: C
1
(d) Ethane (Eclipsed)
Symmetry
Elements:
E, Principle C
3axis
of rotation, 3
perpendicular C
2
axes. 3σ
v, σ
h
σ
v
σ
v
σ
v
σ
h
3D representation of horizontal and vertical mirror plane
Problem: Prepare the flow chart to find the point
groups of Ethane (eclipsed and staggered).

20. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
(e) Benzene
Symmetry Elements:
E, principle axis C
6 , 6C
2, 3σ
v, σ
h , 3σ
d
Prepare the flow chart to
determine the point group of
benzene.

21. (f) SF
6
Symmetry Elements:
E, 3 C
4, 4 C
3, 9 C
2, 4 S
6, 3 S
4, 3 σ
h, 6 σ
d
and a centreof inversion i.
Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal

22. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal
•HCl
•CO
2
•H
3C-CH
3
•NH
3
•CH
4
•CHFClBr
•H
2C=CClBr
•SF
6
•H
2O
2
•C
4H
4
•CH
3Cl
•CO
Determine symmetry elements and assign point
groups of the following compounds

23. Dept. of Chemistry , CH-402, Solid State Chemistry
Nahida Akter, Assistant Professor, Dept. of Chemistry, University of Barisal