ki2141-2015 sik lecture06 molecularsymmetry

7/23/2019 KI2141-2015 SIK Lecture06 MolecularSymmetry

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Molecular SymmetryAchmad Rochliadi, MS., PhD.

and Dr. Veinardi suendo


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Molecular Symmetry2

Contents

What and Why? The symmetry elements and operations

The symmetry classification

Consequences of symmetry Linear algebra in symmetry

Group representation and character tables


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Molecular Symmetry3

What is symmetry ?

According to Webster Dictionary Correspondence in size, shape and relatie

position of parts that are on opposite sides of a

diiding line or median plane or that are

distributed about a center of a!is" #olecular symmetry

$f a molecule has t%o or more orientation that

are indistinguishable then the molecule

possesses symmetry"


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Molecular Symmetry4

Why symmetry important in chemistry

The symmetry of the molecule tells us %hetherthe molecule is chiral, and %hether it has a

dipole moment"

&ymmetry %ill allo% us to interpret

spectroscopic measurements on molecules" $t is

particularly important %hen %e come to

interpreting the infrared 'ibrational( spectra of

molecules"


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Molecular Symmetry5

&ymmetry is important in interpreting the crystalstructures of molecules" #odern )*ray diffraction

methods use symmetry in order to interpret the spectra

obtained and determine the absolute position of atoms

%ithin a crystalline solid, and hence its structure"

&ymmetry is crucial both in understanding the

electronic structure of molecules '#olecular orbital, or

#+ theory(" $t is crucial in simplifying the other%ise

computationally intensie calculations that need to be

carried out in order to find the energies of moleculesand hence predict their structure and the chemical

reactions that can be carried out on them


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Molecular Symmetry6


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Symmetry Elements and Operations

Symmetry Elements A point, line or plane in the molecule about

%hich the symmetry operation tae out" There is

only - symmetry elements related to molecule

symmetry"Symmetry Operations

&ome transformations of the molecule such as a

rotation or reflection %hich leaes the molecule

in a configuration in space that is

indistinguishable from its initial configuration"



Symmetry Elements

Elements Symbol Operation$dentity E Leaes each particles in its

original position

.*fold proper

a!is

Cn /otation about the a!is by

01223n 'or by multiply(

4lane /efle!ion in plane

$nersion

center

i $nersion through center

.*foldimproper a!is

Sn /otation by 01223nfollo%ed

by refle!ion in a planeperpendicular to the a!is



1. The identity operators, E

The simplest symmetry operation is no%n as theidentity operator, gien the symbolE'Efrom the

German %ord 5inheit, meaning unity(" TheEoperator

basically means 6do nothing to the molecule"7

5idently, if you do nothing to the molecule it %ill loothe same as %hen you started" The identity operation

%ill thus %or on all molecules"

#any non*symmetrical molecules,

such as the amino acid alanine

sho%n here, contain only the

Eoperator"



2. The n-fold rotation operators, Cn

/otation a!es hae the nomenclature Cn%hich means6rotate the molecule around the specified a!is throughan angle of 01283n" Thus, a C2a!is means rotate by9:28, C3by 9;28, C4by



&ome of the rotation symmetry elements of a cube" The t%ofold,threefold and fourfold a!es are labeled %ith the conentionalsymbols"



'a( An .=0molecule has a threefold 'C3( a!is and 'b( =;+ molecule has at%ofold 'C2( a!is"

&ymmetry modeling > http>33%%%"ch"ic"ac"u3local3symmetry3



Successive rotations



enzene hae C6, C2, the principal a!is is the si!fold

a!is that perpendicular to the he!agonal ring"



. The reflections operators,

A mirror plane 'symbol ( is a symmetryelement that results in the reflections of the

molecule through a mirror plane"

$f the plane is parallel to the principal a!is, it is

called @ertical and denoted v"

$f the plane is perpendicular to the principal

a!is, it is called @horizontal and denoted h

A ertical mirror plane that bisect 'diide( theangle bet%een t%o C2a!es is called a @dihedral

plane and denoted d"


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Molecular Symmetry17

An =;+ molecule has t%o mirror planes" They are both

ertical so denoted vand v"


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Dihedral mirror planes 'd( bisect the C2a!es

perpendicular to the principal a!is"


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. The inversion operators, i

$magine taing each point in a molecule,moing it to the centre of the molecule, and

then moing it out the same distance on the

other side"

#oe eery atom at position '!,y,z( to position

'*!,*y,*z(" $f the molecule still loos the same,

then it contains a centre of inersion"


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A /egular octahedron has a centre of inersion


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$nersion of benzene, notice that the three of the C*=

groups hae been color coded" When the inersion is

performed, these groups moe to the mirrored side"


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!. The n-fold improper rotation, Sn

An improper is the most comple! symmetryelement to understand" An improper rotation

consist of TW+ steps and neither operation

alone needs to be a symmetry operation>

The rotation lie Cn, %here the molecule is

rotate around the a!is"

/eflection through a plane perpendicular to the

a!is of that rotation,


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'a( A C=Bmolecule has a fourfold improper rotation a!is 'S4( themolecule is indistinguishable after a


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To classify the molecules according to theirsymmetries, the molecule symmetry elements is

listed and collect together the molecules %ith

the same list of elements"

The name of the group is determined by thesymmetry elements it possesses"

T%o system of notation

The &choenflies system, more common" The =ermann*#auguin system3$nternational

system, e!clusiely used in crystal symmetry"

The symmetry classification "#roup Theory$


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What is point %roup

A point group is a collection of symmetryoperations that together are specific to a %ide

number of different molecules" These

molecules are from a symmetry ie%point,

equialent" or e!ample, both %ater and cis*dichloroethene are members of the C2vpoint

group" +nce you hae learned about the arious

symmetry operations, go to the lin on point

groups to find out more about this concept"


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#roup Theory

$n essence, group theory is a set ofmathematical relationships that allo% us to

study symmetry" An in depth and rigorous

study of group theory requires an e!tensie

no%ledge of matri! algebra" As chemists, %ecan usually concern ourseles less %ith the

details of the math, and more on isualizing

ho% symmetry operations transform molecules

in three dimensional space"


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The diagram for

determining the point

group of a molecule


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&ummary of shapes corresponding to

different point groups"


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#roup C1& Ci& Cs

#olecule belong to C1if has no other elementthan the identity '9(" Ciif has identity and

inersion '0(, and Csif it has identity and a

mirror plane alone 'B("


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#roup Cn& Cnv& Cnh

Cn> possess n*fold a!is '-(Cnv> possess n*fold E v'=;+ .=0(

Cnh> possess n*fold E h'1( * 'F(


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#roup 'n& 'nv& 'nh

Dn> possess n*fold a!is nt%ofold a!esperpendicular to Cn

Dnh> possess n*fold a!is n*t%ofold a!es

perpendicular to Cn E h ':, possessDnE ndihedral mirror planes d

'9;" 90(


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#roup S2n

$ts the molecule that has not classified into oneof the group aboe but possess oneS2na!is"


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The cu(ic %roups

#olecule that possess more than one principala!is belong to the cubic groups"

Tetrahedral groups > Td'a(, T 'a(, and Th 'a7(

+ctahedral groups > Oh

'b( and O 'b(

$cosahedral group >Ih 'c( andI'c(


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(a) T (b) O

(c) ) (a) Th


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The full rotation %roup

The molecule rotational group,R3, consists ofinfinite number of rotation a!es %ith all

possible alues of n"

&phere and an atom belong toR3

CC


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Cvand 'hpoint groups

Asymmetrical diatomics 'e"g" =, C+ and C.H*( and linear

polyatomics that do not possess a centre of symmetry 'e"g" +C& and=C.( possess an infinite number of vplanes but no hplane or

inersion centre" These species belong to the Cvpoint group"

&ymmetrical diatomics 'e"g" =;, +;H;*( and linear polyatomics that

contain a centre of symmetry 'e"g" .0H*, C+;, =CIC=( possess a h

plane in addition to a C

a!is and an infinite number ofv

planes"

These species belong to theDhpoint group"

#roups of hi%h symmetry


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#roups of hi%h symmetry


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'5h '5d

E*ercise 1


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Conse+uences of symmetry

4olarity $f the molecule belongs to group Cn%ith n J 9,

it cannot possess a charge distribution %ith a

moment dipole perpedicular to the symmetry

a!is but it may hae one parallel to the a!is" #olecule belong to Cn, Cnv, Csmay be polar"

All other group such C3h,D, ets there are

symmetry operations that tae one end otmolecule into the other"


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'a( A molecule %ith a Cn

a!is cannot hae a dipole

perpendicular to the a!is,

but 'b( it may hae one

parallel to the a!is"


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Conse+uences of symmetry

Chirality A chiral molecule is a molecule that cannot be

superimposed on its mirror image" A chiralmolecule is an optic actie molecule"

A molecule may be chiral only if it does notposses an a!is of improper rotation,Sn"

Tae notice thatSnoperation could be present

under different symmetry element" 5!ample>

molecules belonging to the groups Cnhposses anSna!is implicity because the possess both Cn

and h, %hich are the t%o components of an

improper rotation a!is" &o as the molecule haeielements"


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&ome symmetryelements are implied bythe other symmetry

elements in a group"Any moleculecontaining an inersionalso possesses at leastanS2element because i

andS2are equialent


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S4


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E*ercise 2

inear al%e(ra in symmetry


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inear al%e(ra in symmetry

Th ff t t i f h i di t t


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The effect on a matri* of a chan%e in coordinate system


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Traces and determinants

The trace of a matri! is defined as the sum ofthe diagonal elements"

The trace of a matri! is defined as the sum of

the diagonal elements>


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#roup representation and character ta(les


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E*ample C2vpoint %roup

C2operation

v(xz)operation


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E*ample 1 C2vpoint %roup "2O$

C2 v(xz)operation

4roof the follo%ing statements>

C2 C2=E

v(xz) v(yz) = C2

v(yz) v(yz) =E


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Characters

The !hara!ter, defined only for a square matri!, is the trace of the

matri!, or the sum of the numbers on the diagonal from upper leftto lo%er right" or the C2vpoint group, %e can obtained the

follo%ing characters>

We can say that this set of characters also forms a representation,%hich is an alternate shorthand ersion of the matri!representation"

Whether in matri! or character format, this is called a red"!iblerepresentation, a combination of more fundamental irred"!iblerepresentations"

/educible representations are designated %ith a capital gamma'G("


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/educi(le and irreduci(le representations


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Character ta(les

0roperties of characters of irreduci(le


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representations in point %roups

Also i

i Eh ('


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representations in point %roups


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E*ample 2 Cvpoint %roup "$


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Transformation matrices


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$n the C3v

point group c'C3

2( K c'C3

(, %hich means that

they are in the same class and described as 2C3in

character table" $n addition, the three reflections hae identical characters

and are in the same class, as described as 3v"

The transformation matrices for C3 and C32cannot be

bloc diagonalized into 9 9 matrices because the C3

matri! has off*diagonal entries" =o%eer, they can bebloc diagonalized into ; ; and 9 9 matrices, %ith all

other matri! elements equal to zero"

Transformation matrices


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Character ta(les of Cvpoint %roup

The C3matri! must be bloced this %aybecause the 'x,y( combination is needed for thene%xandy, %hile the other matrices mustfollo% the same pattern for consistency across

the representation" The set ; ; matrices has the characters

corresponding to theErepresentation, %hile theset of 9 9 matrices matches theA9

representation" TheA;representation can be found using the

defining properties of a mathematical group asin preious e!ample"


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Character ta(les of Cvpoint %roup 1


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Character ta(les of Cvpoint %roup E

AntisymmetricSymmetric asymmetric

f f C


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0roperties of the characters for Cvpoint %roup


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ddition feature of character ta(les

The e!pression listed to the right of the charactersindicate the symmetry of mathematical functions of thecoordinatesx,yandzand of rotation about the a!es 'Rx,

Ry,Rz("

This can be used to find the orbitals that match therepresentation" or e!ample>x%ith 'E( and '-( directionmatches thepxorbital %ith 'E( and '-( lobes in thequadrants in thexyplane the product !y %ith alternatingsigns on the quadrants matches lobes of the dxyorbital"


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$n all cases, the totally symmetricsorbitalmatches the first representation of in the group,one of theAset"

The rotational functions are used to describe therotational motion of the molecule"

$n the C3ve!ample, thexandycoordinates

appeared together in theEirreduciblerepresentation %ith notation 'x,y(" This means

thatxandytogether hae the same symmetryproperties as theEirreducible representation"Consequently, thepxandpyorbitals together

hae the same symmetry as the 5 irreduciblerepresentation in this point group"


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#atching the symmetry operations of a molecule%ith those listed in the top ro% of the charactertable %ill confirm any point group assignment"

$rreducible representations are assigned labelsaccording to the follo%ing rules, in %hichsymmetric means a character of 9 andantisymmetric a character of *9" Letter are assigned according to the dimension of

the irreducible representation"

ddi i f f h (l


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This might also gie us information about degeneracies as

follo%s> A and 'or a and b( indicate non*degenerate 5 'or e( refers to doubly degenerate T 'or t( means triply degenerate

&ubscript 9 designates a representation symmetric to a C2

rotation perpendicular to the principal a!is, and subscript ;designates a representation of antisymmetric to the C2" $fthere are no perpendicular C2a!es, 9 designates arepresentation symmetric to ertical plane, and ;designates a representation antisymmetric to a ertical

plane"

&ubscriptg'gerade( designates symmetric to inersion,and subscript u'ungerade( designates antisymmetric toinersion"

&ingle prime '@( are symmetric to shand double prime '6(are antisymmetric to sh%hen a distinction bet%eenrepresentations is needed 'C

3h, C

5h,D

3h,D

5h("

3 l l i( ti O "C $


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3olecular vi(ration 2O "C2v$

Degree of freedom

3 l l i( ti O "C $


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ull C2

operation of =;

+

The =aand =bentries are not on the principal diagonalbecause =aand =be!change each other in a C2rotation,andx'=a( K -x'=b(,y'=a( K -y'=b( andz'=a( Kz'=b("

+nly o!ygen contribute to the character for thisoperation, for total of -9"



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/educin% representations to irreduci(le representations


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/educin% representations to irreduci(le representations

S t l l ti f t


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Symmetry molecular motion of 4ater

)/ Spectra of O


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)/ Spectra of 2O

Experimental values are 3756, 3657 an 1595 cm!1

"alculate #$ spectrum o% &aseous '2(

(ent triatomic O "C $


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(ent triatomic 2O "C2v$



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;satomic orbital of the + 'a9(



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;pxatomic orbital of the + 'b9(



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;pyatomic orbital of the + 'b;(



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;pzatomic orbital of the + 'a9(



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9satomic orbital of the = 'a9and b;(

and

a1 b2



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summation



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Normaliation



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9satomic orbital of the = 'a9and b

;(



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/eferences and further readin%s


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/eferences and further readin%s

4" Atins and M" de 4aula,A!ins Physical "hemisry,:th5dition, +!ford Nniersity 4ress, +!ford, ;221"

&""A" Oettle, Symmery and Srucure# Readable$roup %heory &or "hemiss, 0rd5dition, Mohn Wiley P&ons, Chichester, ;22F"

A"#" Les,'nroducion o Symmery and $roup%heory &or "hemiss, Olu%er Academic 4ublishers,Dordrecht, ;22B"

C"5" =ousecroft and A"G" &harpe, $norganic Chemistry,0rd5dition, 4earson 5ducation Limited, =arlo%, ;22:"

G"L" #iessler and D"A" Tarr, $norganic Chemistry, Bth5dition, 4earson 5ducation Limited, =arlo%, ;292"

ki2141-2015 sik lecture06 molecularsymmetry

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