simetri lengkap 2 sks

8/18/2019 Simetri Lengkap 2 Sks

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Page 1

Symmetry and Group Theory

Feature: Application for Spectroscopy

and Orbital Molecules

Dr. Indriana Kartini


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Page 2

P. H. Walton “Beginning Group Theory for Chemistry”

Oxford Uniersity Press !n".# $e% &or'# ())*

!+B$ ,()*--)/

0.1.Cotton “ Chemi"al 0ppli"ations of Group Theory”

!+B$ ,/2(-(,)/2

3. 4. Carter “5ole"ular +ymmetry and Group Theory”

6ohn Wiley 7 +ons# !n".# $e% &or'# ())*

8ettle# +.1.0.”+ymmetry and +tru"ture”

6ohn Wiley and +ons# Chi"hester# ()*-

Text books


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Page 3

Marks

• exam: – 50% mid

– 50% ina!

• "#!!ab$s re&mid: Prinsi dasar

– 'erasi dan $ns$r simetri – "iat gr$ titik dan k!asiikasi mo!ek$! da!am s$at$ gr$ titik

– Matriks dan reresentasi simetri

– Tabe! karakter

• "#!!ab$s Pas(a&mid: )!ikasi – rediksi sektra *ibrasi mo!ek$!: I+ dan +aman – rediksi siat otik mo!ek$!

– rediksi orbita! mo!ek$! ikatan mo!ek$!


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Page ,

-ns$r simetri dan oerasi simetri mo!ek$!

• Operasi simetri – "$at$ oerasi #ang dikenakan ada s$at$ mo!ek$!

sedemikian r$a seingga mem$n#ai orientasi

bar$ #ang seo!a&o!a tak terbedakan denganorientasi a/a!n#a

• Unsur simetri – "$at$ titik garis ata$ bidang sebagai basis

oerasi simetri


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Page 5

"imbo! -ns$r 'erasi

-ns$r identitas Membiarkan ob#ek tidak

ber$ba

n"$mb$ rotasi +otasi se$tar s$mb$ dengan

deraat rotasi 4n63708s$d$trotasi 4n ada!a bi!anganb$!at

σ 9idang simetri +e!eksi me!a!$i bidang simetri

i P$sat8titik in*ersi Pro#eksi me!e/ati $satin*ersi ke sisi seberangn#adengan arak #ang sama dari$sat

"n"$mb$ rotasi tidakseati 4Improperrotational axis

+otasi mengitari s$mb$ rotasidiik$ti dengan re!eksi adabidang tegak !$r$s s$mb$

rotasi


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9 !mperial College 4ondon

Operasi +imetri


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Page

B B

3otate (:,O

1( 1(

1:1;

1;1:

Operation rotation by 360/3around C3 axis (element)

BF3

Rotations 360/n where n is an integer


8/146

Page ;

H 1

H 2

H 1

H 2

H 1 H 2

σ

σ

=

y

x

x is out of the plane

3efle"tion is the operation

σ element is plane of symmetry

H2O

+e!e(tions


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Page <

+e!e(tions or =2'


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Page 10

+e!e(tions

• Prin(i!e 4igest order axis is deined as > axis – )ter M$!!iken

σ4x? in !ane erendi($!ar to mo!e($!ar !aneσ4#? in !ane ara!!e! to mo!e($!ar !ane

bot exam!es o σ*

σv : re!e(tion in !ane (ontaining igest order axis

σh : re!e(tion in !ane erendi($!ar to igest

order axis

σd : diedra! !ane genera!!# bise(ting 2


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@enera! ass$mtion orσv dan σd

-nt$k σv1. Mengand$ng s$mb$ $tama

2. Mem$at atom terban#ak

-nt$kσd

1.Diantara d$a rotasi 2

2.Diantara d$a rotasiσv

3.Membagi d$a s$d$t da!am mo!ek$! sama besar

,.Aika sete!a diana!isis dg oint 414243 tern#ata σd 6 σv maka $tamakan σv

Page 11


12/146

Page 12

X e

F F

F F

X e

F F

F F

X e

F F

F F

3efle"tions σvσh

σ

d

σ

d

XeF4


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Page 13

XeF4


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Page 1,

Z

Y

X

Z

Y

X

0tom at


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Page 15

C

H H

HH

+/ !mproper 3otation

3otate aout C/ axis and then refle"t

perpendi"ular to this axis

+/

C/σ


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Page 17

+/ !mproper 3otation


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Page 1

s$((essi*e oeration


18/146

Page 1;

K-BI)= MIC@@- II

T'+I @+-P


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Page 1<

athemati!al "e#inition$ %roup &heory

A group is a collection of elements having certain properties

that enables a wide variety of algebraic manipulations to be

carried out on the collection

Be"ause of the symmetry of mole"ules they "an

e assigned to a point group


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Page 20

"tes to (!assi# a mo!e($!e into a oint gro$

$estion 1:

• Is te mo!e($!e one o te o!!o/ing re(ognisab!egro$s E

C': @o to te $estion 2F":'(taedra!oint gro$ s#mbo! '

Tetraedra! oint gro$ s#mbo! Td

Binear a*ing no i∞υBinear a*ing i D∞


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Page 21


$estion 2:

• Does te mo!e($!e ossess a rotation axis o order ≥ 2 E

F": @o to te $estion 3

C':I no oter s#mmetr# e!ementsoint gro$ s#mbo! 1

I a*ing one re!e(tion !ane oint gro$ s#mbo! s

I a*ing i

i


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Page 22


$estion 3:

• =as te mo!e($!e more tan one rotation axis E

F": @o to te $estion ,

C':

I no oter s#mmetr# e!ements oint gro$ s#mbo! n 4n is te order ote rin(i!e axis

I a*ing n σ oint gro$ s#mbo! nI a*ing n σ* n*I a*ing an "2n axis (oaxia! /it rin(ia! axis "2n


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Page 23


$estion ,:

• Te mo!e($!e (an be assigned a oint gro$ as

o!!o/s:

Co oter s#mmetr# e!ements resent Dn

=a*ing n σd bise(ting te 2 axes Dnd=a*ing one

σ. Dn.


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Page 2,

5ole"ule

4inear D

iDE∞h C∞ : or moreCn# nF:D

iD

Td

C-

D!h

Oh

CnD

'ele!t Cn with highest n

nC perpendi!ular to Cn*

σhDEnh

nσdDEnd En σDCs

iDCi C(σhDCnh

nσDCn

+:nD+:n Cn

F

C


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Page 25

Ben=ene

4inear D

iDE∞h C∞ : or moreCn# nF:D

iD

Td

C-

D!h

Oh

CnD

'ele!t Cn with highest n

nC perpendi!ular to Cn*

σhDEnh

nσdDEnd En σDCs

iDCi C(σhDCnh

nσDCn

+:nD+:n Cn

F

C

n Ben=ene

is Eh


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Page 27

T$gas I: "#mmetr# and Point @ro$s

Tent$kan $ns$r simetri dan gr$ titik adamo!ek$!

a. C2G2

b. P'!3

(. "2'36

@ambarkan geometri masing&masing

mo!ek$! terseb$t


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Page 2;

Basic Properties of Groups

• )n# ombination o 2 or more e!ements o te (o!!e(tion m$st be

eH$i*a!ent to one e!ement /i( is a!so a member o te (o!!e(tion )9 6 /ere ) 9 and are a!! members o te (o!!e(tion

• Tere m$st be an IDCTITF BMCT 4

) 6 ) or a!! members o te (o!!e(tion

(omm$tes /it a!! oter members o te gro$

)6 ) 6)

• Te (ombination o e!ements in te gro$ m$st be )""'I)TI

)49 6 )94 6 )9

M$!ti!i(ation need not be (omm$tati*e 4ie: )≠)

• *er# member o te gro$ m$st a*e an IC+" /i( is a!so amember o te gro$.

))&1 6


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Page 2<

B

O

OO

H

H

H

+xample o# %roup ,roperties

B; elongs to C; point group

!t has @# C; and C;:

symmetry operations


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Page 30

I0ny Comination of : or more elements of the "olle"tion must e eJuialent to

one element %hi"h is also a memer of the "olle"tion

0B C %here 0# B and C are all memers of the "olle"tion

B

O 2

O 3

O 1

H 2

H 1

H 3

B

O 1

O 2

O 3

H 1

H 3

H 2

B

O 3

O 1

O 2

H 3

H 2

H 1

C;C;

OerallA C; follo%ed C; gies C;:


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Page 31

IThere must e an !E@$T!T& @4@5@$T

0@ 0 for all memers of the "olle"tion

@ "ommutes %ith all other memers of the group 0@ @0 0

B

O 2

O 3

O 1

H 2

H 1

H 3

B

O 3

O 1

O 2

H 3

H 2

H 1

B

O 1

O 2

O 3

H 1

H 3

H 2

C;: C;

:

@. C;:

C;

: and C;:. C; @ and C;

:. C;

: C;


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Page 32

IThe "omination of elements in the group must e 0++OC!0T!K@

0 0B 0BC

5ultipli"ation need not e "ommutatie

C; .

C;:

C;.C;

: @ L C; .@ C;C; .C; C;

: L C;: .C;

: C;

Operations are asso"iatie and @# C; and C;: form a group


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Page 33

@ro$ M$!ti!i(ation Tab!e

! " ! !#

" 3 32

! 3 32

!# 3

2 3

Order of the group ;

I@ery memer of the group must

hae an !$K@3+@ %hi"h is also

a memer of the group.

00?( @

The inerse of C;: is C;

The inerse of C; is C;:


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Page 3,

K-BI)= MIC@@- I&


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J Imeria! o!!ege Bondon 35

Mat 9ased

Matrix mat is an integra!

art o @ro$ Teor#

o/e*er /e /i!! o($s on

a!i(ation o te res$!ts.

Gor m$!ti!i(ation:

C$mber o *erti(a! (o!$mns in te

irst matrix 6 n$mber o orisonta!

ro/s o te se(ond matrix

Prod$(t:

+o/ is determined b# te ro/ o

te irst matrix and (o!$mns b# te

(o!$mn o te se(ond matrix


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J Imeria! o!!ege Bondon 36

Mat based

L1 2 3

1 0 0

0 &1 0

0 0 1

6


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Page 3

$epresentations of Groups

• Diagrams are ($mbersome

• +eH$ire n$meri(a! metod – )!!o/s matemati(a! ana!#sis

– +eresent b# T'+" or Matemati(a! G$n(tions

– )tta( artesian *e(tors to mo!e($!e

– 'bser*e te ee(t o s#mmetr# oerations on tese *e(tors

• e(tors are said to orm te basis o te representation each

symmetry operation is expressed as a transformation matrix

[New coordinates] = [matrix transformation] x [old coordinates]


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Page 3;

+

O O

=

y

x

Constru!ting the Representation

Put unit e"tors on ea"h atom

C:A M@# C:# σx=# σy=N

These are useful to des"rie mole"ular irations

and ele"troni" transitions.


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Page 3<

+

O O

+

O O

C:

0 unit e"tor on ea"h atom represents translation in the y dire"tion

C:.


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Page ,0

+

O O


0 unit e"tor on ea"h atom represents rotation around the = axis

C:. 3 = @ . 3 =

σy= .


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Page ,1

onstructin% the $epresentation

#v 2 σ4x? σ4#?

N1 N1 N1 N1 T&

N1 N1 &1 &1 $&

N1 &1 N1 &1 T'($y

N1 &1 &1 N1 Ty($'


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Page ,2

+

O O


Use a mathemati"al fun"tion

@gA py orital on +

#v 2 σ4x? σ4#?

N1 &1 &1 N1 Ty($'

py has the same symmetry properties as Ty and 3 x e"tors


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Page ,3


0u0u

0u

σh

σh.Md x2-y2N .Md x2-y2N

C4.Md x

2

-y

2

N


44/146

Page ,,


)*h 2, 2 22O 22 I 2", σ 2σ* 2σd

N1 &1 N1 N1 &1 N1 &1 N1 N1 &1

@ffe"ts of symmetry operations generate the

T30$+1O35 50T3!

1or all the symmetry operations of E/h on Md x2-y2N

We haeA

+imple examples so far.


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Page ,5

Constru!ting the Representation$

The T30$+1O350T!O$ 50T3!

@xamples "an e more "omplexA

e.g. the px and py oritals in a system %ith a C/ axes.

&

C/ px pxQ ≡ py

py pyQ ≡ px

−=

y

x

y

x

p

p

p

p

,(

(,

R

R!n matrix formA 0 :x: transformation

matrix


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Page ,7


• e(tors and matemati(a! $n(tions (an be $sed to b$i!d a

reresentation o oint gro$s.

• Tere is no !imit to te (oi(e o tese.

•'n!# a e/ a*e $ndamenta! signii(an(e. Tese (annot bered$(ed.

• Te I++D-I9B +P+"CT)TI'C"

• )n# +D-I9B reresentation is te "-M o te set oI++D-I9B reresentations.


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Page ,

;;;:;(

:;:::(

(;(:((

aaa

aaa

aaa

;;

:::(

:(((

,,

,

,

b

bb

bb


!f a matrix elongs to a redu"ile representation it "an e transformed so that =ero elements are distriuted aout the diagonal

+imilarity Transformation

0 goes to B

The similarity transformation is su"h that

C?( 0C B %here C?(C@


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Page ,;

A

n B

B

B

..

..

:

(


Generally a redu"ile representation 0 "an e redu"ed su"h

That ea"h element Bi is a matrix elonging to an irredu"ile representation.

0ll elements outside the Bi lo"'s are =ero

This "an generate ery large matri"es.

Ho%eer# all information is held in the "hara"ter of these matri"es


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Page ,<

Chara!ter &ables

;;;:;(

:;:::(

(;(:((

aaa

aaa

aaa

Chara"ter # χ a(( a:: a;;.

∑==n

inma( χ !n general

0nd only the "hara"ter χ# %hi"h is a numer is reJuired and $OT the %hole matrix.


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Page 50

haracter Tables an "'ample !v : +,F!-

!v " !.

!#

σ

vσ

vσ

v

1 1 1 1 1 1 T?

1 1 1 &1 &1 &1 +?2 &1 &1 0 0 0 4TxT# or 4+x+#

This simplifies further. +ome operations are of the same "lass and al%ays hae the

same "hara"ter in a gien irredu"ile representation

C3- C3

- are in the same !lass

σv σv σv are in the same !lass


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Page 51

haracter Tables an "'ample !v : +,F!-

!v " #! σv

)1 1 1 1 T? x2 N #2

)2 1 1 &1 +?

2 &1 0 4TxT# or 4+x+# 4x2 #2 x# 4#? ?x

There is a nomen"lature for irredu"ile representationsA ulli.en +ymols

is single and + is douly degenerate


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Page 52

ote$

1ou will not be as.ed to generate !hara!ter tables2

&hese !an be brought/supplied in the examination


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Page 53

K-BI)= MIC@@- I&II&III


54/146

Page 5,

%eneral #orm o# Chara!ter &ables$

4ists the "hara"ters# for all irredu"ile representations for ea"h "lass

of operation.

+ho%s the irredu"ile representation for %hi"h the six e"tors

Tx# Ty# T=# and 3 x# 3 y# 3 =# proide the asis. +ho%s ho% fun"tions that are inary "ominations of x#y#=

proide ases for "ertain irredu"ile representation.

4ist "onentional symols for irredu"ile representationsA

ulli.en symols


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Page 55

ulli.en symolsA 4aelling

0ll one dimensional irredu"ile representations are laelled or 2

0ll t%o dimensional irredu"ile representations are laelled +2


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Page 57


0 one dimensional irredu"ile representation is laelled if it is symmetri"%ith respe"t to rotation aout the highest order axis Cn.

(>

:>

lli. l 4 lli


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Page 5


+us"ripts g and u are gien to irredu"ile representationsThat are symmetri" and anti?symmetri" respe"tiely# %ith respe"t to inersion

at a "entre of symmetry.

+upers"ripts and 7 are gien to irredu"ile representations that are symmetri"and anti?symmetri" respe"tiely %ith respe"t o refle"tion in a σh plane.

;>

/>

oteA Points (> and :> apply to one?dimensional representations only.Points ;> and /> apply eJually to one?# t%o?# and three? dimensional representations.


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Page 5;

+

O O

=:

y:

x:

%enerating Redu!ible Representations

x(

xsy(

ys

=s

=(

σx=

1or the symmetry operation σx=

x(→ x: x:→ x( xs→ xs

y(→ ?y: y:→ ?y( ys→ ?ys

=(→ =: =:→ =( =s→ =s


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Page 5<


−

−

−

=

−

−

−

=

s

s

s

s

s

s

s

y

s

xz

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

(

(

(

:

:

:

:

:

:

(

(

(

:

:

:

(

(

(

>< .

(,,,,,,,,

,(,,,,,,,

,,(,,,,,,

,,,,,,(,,

,,,,,,,(,

,,,,,,,,(

,,,(,,,,,

,,,,(,,,,

,,,,,(,,,

σ

!n matrix form


60/146

Page 70

−

−

−

=

−

−

−

=

s

s

s

s

s

s

s

y

s

xz

z

y x

z

y

x

z

y

x

z

y x

z

y

x

z

y

x

z

y x

z

y

x

z

y

x

(

(

(

:

:

:

:

:

:

(

(

(

:

:

:

(

(

(

>< .

(,,,,,,,,

,(,,,,,,,,,(,,,,,,

,,,,,,(,,

,,,,,,,(,

,,,,,,,,(

,,,(,,,,,

,,,,(,,,,

,,,,,(,,,

σ

Only reJuire the "hara"tersA The sum of diagonal elements

1or σ χ (


61/146

Page 71

−

−

−

=

−

−

−

=

s

s

s

s

s

s

s

y

s

yz

z

y

x z

y

x

z

y

x

z

y

x z

y

x

z

y

x

z

y

x z

y

x

z

y

x

:

:

:

(

(

(

:

:

:

(

(

(

:

:

:

(

(

(

>< .

(,,,,,,,,

,(,,,,,,,

,,(,,,,,,,,,(,,,,,

,,,,(,,,,

,,,,,(,,,

,,,,,,(,,

,,,,,,,(,

,,,,,,,,(

σ

1or σ χ ;


62/146

Page 72

=

=

s

s

s

s

s

s

s

y

s

z

y

x

z

y

x

z y

x

z

y

x

z

y

x

z y

x

z

y

x

z

y

x

z y

x

E

:

:

:

(

(

(

:

:

:

(

(

(

:

:

:

(

(

(

.

(,,,,,,,,

,(,,,,,,,

,,(,,,,,,

,,,(,,,,,

,,,,(,,,,

,,,,,(,,,

,,,,,,(,,,,,,,,,(,

,,,,,,,,(

1or @ χ )


63/146

Page 73

−

−

−−

−−

=

−

−

−−

−−

=

s

s

s

s

s

s

s

y

s

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

C

(

(

(

:

:

:

:

:

:

(

(

(

:

:

:

(

(

(

: .

(,,,,,,,,

,(,,,,,,,

,,(,,,,,,

,,,,,,(,,

,,,,,,,(,

,,,,,,,,(

,,,(,,,,,

,,,,(,,,,

,,,,,(,,,

.

1or C: χ ?(


64/146

Page 7,


Cv

Γ ;n

@ C: σ σ

) ?( ( ;

+ummarising %e get that Γ ;n for this mole"ule isA

#v 2 σ4x? σ4#?

)1 N1 N1 N1 N1 T? x2 #2 ?2

)2 N1 N1 &1 &1 +? x#

91 N1 &1 N1 &1 Tx +x x?

92 N1 &1 &1 N1 T# +# #?

To redu"e this %e need the "hara"ter tale for the point groups

Redu!ing Redu!ible Representations


65/146

Page 75

Redu!ing Redu!ible Representations

We need to use the redu"tion formulaA

( ) R Rn g a p R R p χ χ >.


66/146

Page 77

#v 2 4x? 4#?

)1 N1 N1 N1 N1 T? x2 #2 ?2

)2 N1 N1 &1 &1 +? x#

91 N1 &1 N1 &1 Tx +x x?

92 N1 &1 &1 N1 T# +# #?

Cv

Γ ;n

@ C:σ σ

) ?( ( ;

1or C: L g / and n3 ( for all operations

( C @ C σ σ


67/146

Page 7

a0( M < (x)x(> N ;

( ) R Rn g

a p R

R p χ χ >. N (

aB( M < (x)x(> N :

aB:

M < (x)x(> N ;

Γ ;n ;0( 0: :B( ;B:

: hi#t d t # C t ti R d ibl R t ti


68/146

Page 7;

a0(

M < (x8x(> N ;

:nshi#ted atoms #or Constru!ting Redu!ible Representations

The terms in lue represent "ontriutions from the un9shi#ted atoms

Only these a"tually "ontriute to the tra"e.

!f %e "on"entrate only on these un?shifted atoms %e "an

simplify the prolem greatly.

1or +O: < 9- ( x –1> and < 3 ; x 1>

umber o# un9shi#ted atoms Contribution from these atoms

" # σ'& σy&

RR $ 8 9- - 3

!dentity @


69/146

Page 7<

!dentity @

@

=

z

y

x

z

y

x

.

(,,

,(,

,,(

(

(

(

1or ea"h un?shifted atom

χ ;

=

y

x

=(

y(

x(

!nersion i


70/146

Page 0

!nersion i

=

y

x=(

y(

x(

i


χ ?;

−−

−=

z

y

x

z

y

x

.

(,,

,(,

,,(

(

(

(

3efle"tion σ


71/146

Page 1


=(

y(

x(x

=

y

σ

< p g >

χ (

−=

z

y

x

z

y

x

.

(,,

,(,

,,(

(

(

(

3 t ti C


72/146

Page 2

θθ

;,Sn

x(y(

=(=

y

x

Cn

3otation Cn

−

=

z

y

x

nn

nn

z

y

x

.

(,,

,;,

"os;,

sin

,;,sin;,"os

(

(

(

χ ( :."os

!mproper rotation axis# +nQ


73/146

Page 3

−

−

=

z

y

x

nn

nn

z

y

x

.

(,,

,;,

"os;,

sin

,;,

sin;,

"os

(

(

(

χ ?( :."os

Cnσ=

y

x

=Q

yQxQ

y(x(

=(

+ummary of "ontriutions from un?shifted atoms to Γ;


74/146

Page ,

+ummary of "ontriutions from un shifted atoms to Γ ;n

$χ+$-

N3i &3

σ N11N 2.(os43708n 2 &1

1N 2.(os43708n 3 32 0

1N 2.(os43708n , ,3 N1

&1 N 2.(os43708n "31"3

2 &2

&1 N 2.(os43708n ",1

",2

&1&1 N 2.(os43708n "7

1"75 0

O

Wor'ed exampleA POCl


75/146

Page 5

P

C lC l

C l

Wor'ed exampleA POCl;

3 χ

@

σ

:C;

;

(

,

C3v@ 3σ

0(

0:

@

( ( (

( ( ?(

: ?( ,

C;

Un?shifted

atoms

Contriution

Γ3n

- : ;

; , (

-; 0 3

$umer of "lasses#

Order of the group#

g

3edu"ing the irredu"ile representation for POCl;


76/146

Page 7

3edu"ing the irredu"ile representation for POCl;

( ) R Rn

g

a p R

R p χ χ >. N (S M(- , ?)N (

a (SM N (SM;, , , N -

Γ3n < =- 4 4 ;+

1or POCl; n - therefore the numer of degrees of freedom is ;n (-.

@ is douly degenerate so Γ3n has (- degrees of freedom.


77/146

Page

K-BI)= MIC@@- IQ&Q&QI&QII )PBIK)"I T'+I @+-P


78/146

Page ;

#v 1 12 1σ4x? 1σ4#?

)1 N1 N1 N1 N1 T? x2 #2 ?2

)2 N1 N1 &1 &1 +? x#

91 N1 &1 N1 &1 Tx +x x?

92 N1 &1 &1 N1 T# +# #?

Cv

Γ ;n

@ C:σ σ

) ?( ( ;

Γ3n < 3- 4 4 - 4 3

Group Theory and Kirational +pe"tros"opyA +O:

Group Theory and Kirational +pe"tros"opyA +O


79/146

Page <

Group Theory and Kirational +pe"tros"opyA +O:

Γ3n < 3- 4 4 - 4 3 < 3 4 - 4 4 3 < 8 < 3n

#v 1 12 1σ4x? 1σ4#?

)1 N1 N1 N1 N1 T? x2 #2 ?2

)2 N1 N1 &1 &1 +? x#

91 N1 &1 N1 &1 Tx +x x?

92 N1 &1 &1 N1 T# +# #?

1or non linear mole"ule there are ;n? irational degrees of freedom

Γ rot 0: B( B:Γ trans 0( B( B:

Γ i Γ ;n Γ rot Γ trans

Γ

vib < - 4

Γ ;n ;0( 0: :B( ;B:

O

Group Theory and Kirational +pe"tros"opyA POCl;


80/146

Page ;0

P

C lC l

C l

p y p py ;

Γ3n < =- 4 4 ;+

Γtrans < - 4 +

Γrot < 4 +

Γvibe < 3- 4 3+

There are nine irational modes .

The @ modes are douly degenerate and

"onstitute TWO modes

There are ) modes that transform as 3- 4 3+.

These modes are linear "ominations of the three e"tors

atta"hed to ea"h atom.

@a"h mode forms a B0+!+ for an !33@EUC!B4@ representation

of the point group of the mole"ule


81/146

Page ;1

>rom Γ3n to Γvibe and 'pe!tros!opy

$o% that %e hae Γ ie %hat does it meanD

We hae the symmetries of the normal modes of irations.

!n terms of linear "ominations of Cartesian "o?ordinates.

We hae the numer and degenera"ies of the normal modes.

Can %e predi"t the infrared and 3aman spe"traD

&es

Applications in spectroscopy: /nfrared Spectroscopy


82/146

Page ;2

Applications in spectroscopy: /nfrared Spectroscopy

• ibrationa! transition is inrared a(ti*e be(a$se o intera(tion

o radiation /it te:

molecular dipole moment( µ0

• Tere m$st be a (ange in tis dio!e moment

• Tis is te transition dio!e moment

• Probabi!it# is re!ated to transition moment integra! .

In#rared 'pe!tros!op


83/146

Page ;3

ψ f

ψ i

τ ψ µ ψ τ ψ µ ψ d d TM f i f i ∫ ∫ =∝ G

In#rared 'pe!tros!opy

µ !s the transition dipole moment operator and

has "omponentsA µx# µy# µ=.

Waefun"tion final state

Waefun"tion initial state

$oteA !nitial %aefun"tion

is al%ays real

/nfrared Spectroscopy


84/146

Page ;,

/nfrared Spectroscopy

• Transition is orbidden i TM 6 0

• 'n!# non ?ero i dire(t rod$(t: ψ µ ψ i (ontains te tota!!# s#mmetri( reresentation.

• i.e a!! n$mbers or χ in reresentation are N1

• Te gro$nd state ψ i is a!/a#s tota!!# s#mmetri(

• Dio!e moment transorms as Tx T# and T?.

• Te ex(ited state transorms te same as te *e(tors tat des(ribete *ibrationa! mode.


85/146

C)? @ C: σ σ


86/146

Page ;7

/- ( ( ( ( @

/) ( ( −( −( R @

3- ( −( ( −( x( R y

3) ( −( −( ( y( R x

/- × 3- × /- ( −( ( −( ≡3-

/- × 3) × /- ( −( −( ( ≡3)/- × /- × /- ( ( ( ( ≡/-

/- × 3- × 3) ( ( −( −( ≡/)

/- × 3) × 3) ( ( ( ( ≡/-

/- × /- × 3) ( −( −( ( ≡3)

&he "IR+C& ,RO":C& representation


87/146

Page ;

=•

•

(

:

(

(

(

:

(

(

A

B

B

A

A

B

B

A

=•

•

:

(

:

:

(

:

(

(

B

A

A

B

A

B

B

A

Group theory predi"ts only 0( and B: modes

Both of these dire"t produ"t representations "ontain

the totally symmetri" spe"ies so they are symmetry allo%ed.

This does not tell us the intensity only %hether they are allo%edor not.

Γvib < - 4 We predi"t three ands in

the infrared spe"trum of +O:


88/146

Page ;;

In#rared 'pe!tros!opy $ %eneral Rule

!f a irational mode has the same symmetry properties

as one or more translational e"tors for that

point group# then the totally symmetri" representation is

present and that transitions %ill e symmetry allo%ed.

ote$

+ele"tion rule tells us that the dipole "hanges during a iration

and "an therefore intera"t %ith ele"tromagneti" radiation.

Inrared se(tra


89/146

Page ;<

gaseo$s =2'

4"artan O0,

©Ra*e$n(tion In(. 2003

!iH$id =2'

xerimenta! absortion *a!$es:

3756,3657and 15


90/146

Page


91/146

Page


92/146

Page


93/146

Page


94/146

Page


95/146

Page


96/146

Page


97/146

Page


98/146

Page

nalysis o# ?ibrational odes$


99/146

Page


100/146

Page 100

O O

Cv

r (

@ C:σ σ

r (

r :

r :

r (

Pi"' a generating e"tor egA r (

Ho% does this transform under symmetry operationsD

5ultiply this y the "hara"ters of 0( and B:

1or 0( this giesA r ( r : r : r ( :r ( :r :

$ormalise "oeffi"ients and diide y sum of sJuaresA

><:

(:( r r +=

'ymmetry dapted Ainear Combinations


101/146

Page 101

1or B: this giesA r (


102/146

Page 102

+

O O3emaining mode “li'ely” to e a end

Cv

Γ end

@ C:σ σ

( ( ( (

By inspe"tion this end is 0( symmetry

+O: has three normal modesA

0( stret"hA 3aman polarised and infrared a"tie

0( endA 3aman polarised and infrared a"tie

B: stret"hA 3aman and infrared a"tie

nalysis o# ?ibrational odes$ 'O experimental data2


103/146

Page 103

I+4ao$r8(m&1 +aman4!iH$id8(m&1 "#m Came

51; 52, )1 bend ν1

1151 11,5 )1 stret( ν2

1372 1337 92 stret( ν3

l i # ?ib ti l d 'O i t l d t


104/146

Page 10,

nalysis o# ?ibrational odes$ 'O experimental data2

otes$

+tret"hing modes usually higher in freJuen"y than ending modes

Eifferen"es in freJuen"y et%een !3 and 3aman are due to

differing phases of measurements

“$ormal” to numer the modes 0""ording to ho% the 5ulli'en term

symols appear in the "hara"ter tale# ie. 0( first and then B:

nalysis o# ?ibrational odes$ ,OCl3


105/146

Page 105

P

O

C l

C l

C l

P

O

C lC l

C lP

O

C lC l

C l

P

; @ irations !3 a"tie 3aman a"tie < x: ? y: # xy>

'ix bands 'ix !o9in!iden!es



106/146

Page 107

Γvibe < 3- 4 3+C; @ :C; ;σ

Γ PO str Γ P?Cl str Γ

end

( ( (

; , (

, :

Using redu"tion formulae or y inspe"tionA

Γ PO str 0( and Γ P?Cl str 0( @

Γ end Γ ie 9 Γ PO str ? Γ P?Cl str ;0( ;@ :0( @ 0( :@

3edu"tion of the representation for ends giesA Γ end :0( :@



107/146

Page 10

Γ end Γ ie 9 Γ PO str ? Γ P?Cl str ;0( ;@ :0( @ 0( :@

3edu"tion of the representation for ends giesA Γ end :0( :@

One of the 0( terms is 3@EU$E0$T as not

all the angles "an symmetri"ally in"rease

Γ end 0( :@

$oteA

!t is adisale to loo' out for redundant "o?ordinates and thin'

aout the physi"al signifi"an"e of %hat you are representing.

3edundant "o?ordinates "an e Juite "ommon and "an lead to a

doule “"ounting” for irations.



108/146

Page 10;

I+ 4!iH8 (m&1 +aman 8(m&1 Des(rition "#m Babe!

12


109/146

Page 10<

1 )!! o!arised bands are +aman )1 modes.

2 =igest reH$en(ies robab!# stret(es.

3 P&! stret(es robab!# o simi!ar reH$en(#.,Do$b!e bonds a*e iger reH$en(# tan simi!ar sing!e bonds.

)1 modes irst. P6' – igest reH$en(#

Ten P&! stret( ten deormation.

5;1 simi!ar to P&! stret( so ass#m. stret(.

+emaining modes m$st tereore be deormations

o$!d no/ $se ")Bs to !ook more (!ose!# at te norma! modes

Symmetry( Bondin% and "lectronic Spectroscopy


110/146

Page 110

• -se atomi( orbita!s as basis set.

• Determine irred$(ib!e reresentations.

• onstr$(t -)BIT)TI mo!e($!ar orbita! diagram.

• a!($!ate s#mmetr# o e!e(troni( states.

• Determine Sa!!o/edness o e!e(troni( transitions.

'ymmetry onding and +le!troni! 'pe!tros!opy


111/146

Page 111

'

= =

N '

= =

N0 20 σx?0 σ#x

' 2s orbita!

C)? + C) x@ y@

O)s +1 +1 +1 +1 a(

σ onding in 0n mole"ules e.g. A %ater

Ho% do :s and :p oritals transformD



112/146

Page 112

s?oritals are spheri"ally symmetri" and %hen at the

most symmetri" point al%ays transform as the totallysymmetri" spe"ies

1or ele"troni" oritals# either atomi" or mole"ular#

use lo%er "ase "hara"ters for 5ulli'en symols

Oxygen :s orital has a- symmetry in the C: point group



113/146

Page 113

'= =

0 20 σx?0 σ#x

' 2%? orbita!

−

+

'= =−

+

C)? + C) σx@ σy@

O)p@ +1 +1 +1 +1 a(

Ho% do the :p oritals transformD


114/146



115/146

Page 115

Ho% do the :s and :p oritals transformD

Oxygen :s and :p= transforms as a(:px transforms as ( and :py as :

$eed a set of σ?ligand oritals of "orre"t symmetry to intera"t%ith Oxygen oritals.

Constru"t a asis# determine the redu"ile representation#

redu"e y inspe"tion or using the redu"tion formula# estimate oerlap#

dra% 5O diagram


h i l h h d


116/146

Page 117

'

= =

= 1s orbita!s

N N

φ1 φ2

?

#

x

C)? + C) σx@ σy@

σ

2 0 0 2 a( :

Use the (s oritals on the hydrogen atoms



117/146

Page 11

0ssume oxygen :s oritals are non onding

Oxygen :p= is a(# px is ( and py is :

4igand oritals are a( and :

Whi"h is lo%er in energy a( or :D

Guess that it is a( similar symmetry etter intera"tionD

Oritals of li'e symmetry "an intera"t

Oxygen :px is “%rong” symmetry therefore li'ely to e non?onding

Yualitatie 5O diagram for H:O

a(G


118/146

Page 11;

:s

O )B

a(

a( ( :a( :

non?onding

non?onding

a(

a(

a(G

:

:G

(

B)O


!s symmetry suffi"ient to determine ordering of a and oritalsD


119/146

Page 11<

!s symmetry suffi"ient to determine ordering of a( and : oritalsD

Constru"t +04C and asses degree of oerlap.

Ta'e one asis that maps onto ea"h other

Use

φ1 or

φ2 as a generating fun"tion.

Osere the effe"t of ea"h symmetry operation on the fun"tion

5ultiply this ro% y ea"h irredu"ile representation of the point

Group and then normalise.


120/146

Page 120

φ( φ( φ: φ: φ(

a( +1 +1 +1 +1

+um and normalise φ( φ: φ: φ( a( H (S√:

: +1 −1 −1 +1

+um and normalise φ( −φ: −φ: φ( : H (S√:

'

= =N N

φ1 φ2

N

−

Ψ

'

= =− N

φ1 φ2

Ψ

− +

p=

py


121/146

Yualitatie 5O diagram for H:O

a(G


122/146

Page 122

:s

O )B

a(

a( ( :a( :

non?onding

non?onding

a(

a(

a(

:

:G

(

B)O

'ymmetry o# +le!troni! 'tates #rom O9"+%++R&+ Os2


123/146

Page 123

The ground ele"troni" "onfiguration for %ater isA

:

::

,,

The symmetry of the ele"troni" state arising from this "onfiguration

is gien y the dire"t produ"t of the symmetries of the 5OQs of all

the ele"trons

: a(.a( 0(

: :.: 0(

: (.( 0(

1or 1U44 singly degenerate

5OQs# the symmetry is 04W0&+0(



124/146

Page 12,

1or 1U44 singly degenerate 5OQs# the symmetry is 04W0&+ 0(

1or oritals %ith only one ele"tronA

( 0(

# ( B:

# ( B(

General ruleA

1or full 5OQs the ground state is al%ays totally symmetri"


h h if l D


125/146

Page 125

What happens if %e promote an ele"tronD

a(

(

:

:

a(

Bonding

$on onding

0nti Bonding

1irst t%o ex"itations moe an ele"tron form ( non onding

!nto either the : or a( anti?onding oritals .

Both of these transitions are

non onding to anti onding

transitions. n?π

Dhat ele!troni! states do these new !on#igurations generate*


126/146

Page 127

:

:(

(,

:

:(

,(

0(.0(.B(.B: 0:

0(.0(.B(.0( B(

!n these states the spins "an e paired or not.

!@A + the TOT04 ele"tron spin "an eJual to , or (.

The multipli"ity of these states is gien y :+(

These "onfigurations generateA;0: #

(0: and;B( #

(B( ele"troni" states.

ote$ if + Z then %e hae a doulet state



127/146

Page 12

a(

(

:

:

a(

5ole"ular Oritals

(0(

(B(

;B((

0:

;0:

@le"troni" +tates



128/146

Page 12;

Triplet states are al%ays lo%er than the related singlet statesEue to a minimisation of ele"tron?ele"tron intera"tions and

thus less repulsion

Bet%een %hi"h of these states are ele"troni" transitions

symmetry allo%edD

$eed to ealuate the transition moment integral li'e %e did for

infrared transitions.

τ ψ µ ψ τ ψ µ ψ d d TM f i f i ∫ ∫ =∝ G

Dhi!h ele!troni! transitions are allowed*


129/146

Page 12<

@le"troni"

τ ψ µ ψ τ ψ ψ τ ψ ψ d d d TMI f e

ie f S

iS f V

iV #

#G

##

G

##

G

∫ ∫ ∫ ••≈ Kirational +pinTo first approximation

µ

"an only operate on the ele"troni" part

of the %aefun"tion.

Kirational part is oerlap et%een ground and ex"ited state nu"lear

%aefun"tions. 1ran"'?Condon fa"tors.

+pin sele"tion rules are stri"t. There must e $O "hange in spin

Eire"t produ"t for ele"troni" integral must "ontain the totally

symmetri" spe"ies.



130/146

Page 130

0 transition is allo%ed if there is no "hange in spin and the

ele"troni" "omponent transforms as totally symmetri".The intensity is modulated y 1ran"'?Condon fa"tors.

The ele"troni" transition dipole momentµ

transforms as the

translational spe"ies as for infrared transitions.



131/146

Page 131

1or the example of H:, the dire"t produ"ts for the

ele"troni" transition are

=•

•

(

:

:

:

(

:

(

(

B

B

A

A

A

B

B

A

=•

•

:

(

(

(

(

:

(

(

A

A

B

B

A

B

B

A

The totally symmetri" spe"ies is only present for the transition

to the B( state. Therefore the transition to the 0: state is“ symmetry forbidden”

Transitions et%een singlet states are spin a!!o"ed”#

transitions et%een singlet and triplet state are spin forbidden”#



132/146

Page 132

(0(

(0:

(B(

;B(

;0:

+ymmetry

foridden

+pin foridden+ymmetry

allo%ed


133/146

ore bonding #or E6 mole!ules / !omplexes


134/146

Page 13,

!n the "ase of Oh point groupA

d x:?y: and d=: transform as eg

dxy# dy= and d=x transform as t:g

px# py and p= transform as t(u

Γσ a(g eg t(u

Γπ t(g t:g t(u t:u

t(u

0 for Oh


135/146

Page 135

t(u

a(g

eg t:g

t(u

a(g

a(g

eg

eg

t:g

a(g eg t(u

/p

/s

;d

+le!troni! 'pe!tros!opy o# d8 !omplex$


136/146

Page 137

MCuN: is a d) "omplex. That is approximately Oh.

Ground ele"troni" "onfiguration isA ;

@x"ited ele"troni" "onfiguration is A -/

The ground ele"troni" state is :@g

@x"ited ele"troni" state is :T:gUnder Oh the transition dipole moment transforms as t(u

0re ele"troni" transitions allo%ed et%een these statesD



137/146

Page 13

$eed to "al"ulate dire"t produ"t representationA

:@g . .:T:g

' ;3

727,

32 i7",

;"7 3σ 7σd

T2g 3 0 1 &1 &1 3 &1 0 &1 1

t1$ 3 0 &1 1 &1 &3 &1 0 1 1

g 2 &1 0 0 2 2 0 &1 2 0

DP 1; 0 0 0 2 &1; 0 0 &2 0

@le"troni" +pe"tros"opy of d) "omplexA

DP 1; 0 0 0 2 &1; 0 0 &2 0


138/146

Page 13;

Use redu"tion formulaA ( ) R Rn g

a p R

R p χ χ >. N ,

The totally symmetri" spe"ies is not present in this dire"t produ"t.

The transition is symmetry foridden.

We 'ne% this any%ay as g?g transitions are foridden.

Transition is ho%eer spin allo%ed.


Groups theory predi"ts no allo%ed ele"troni" transition


139/146

Page 13<

Groups theory predi"ts no allo%ed ele"troni" transition.

Ho%eer# a %ea' asorption at 2),nm is osered.

There is a phenomena 'no%n as ironi" "oupling %here the

irational and ele"troni" %aefun"tons are "oupled.

This effe"tiely "hanges the symmetry of the states inoled.

This %ea' transition is $ibroni%a!!y ind&%ed and therefore is partially

allo%ed.

• )re #o$ ami!iar /it s#mmetr# e!ements oerationsE

• an #o$ assign a oint gro$E

• an #o$ $se a basis o 3 *e(tors to generate Γ 3n E


140/146

Page 1,0

# g 3

• Do #o$ kno/ te red$(tion orm$!aE

• Rat is te dieren(e bet/een a red$(ib!e and irred$(ib!ereresentationE

• an #o$ red$(e Γ 3n E• an #o$ generate Γ *ib rom Γ 3n E• an #o$ redi(t I+ and +aman a(ti*it# or a gi*en mo!e($!e $sing

dire(t rod$(t reresentationE

• an #o$ dis($ss te assignment o se(traE

• an #o$ $se ")Bs to des(ribe te norma! modes o "'2E

• an #o$ dis($ss M' diagram in terms o ")B"E

• an #o$ assign s#mmetr# to e!e(troni( states and dis($ss /etere!e(troni( transitions are a!!o/ed $sing te dire(t rod$(t

reresentationE• @i*en and inrared and +aman se(tr$m (o$!d #o$ determine te

s#mmetr# o te mo!e($!eE


141/146

Page 1,1

• tt:88///.(emso(.org8exem!ar(em8entries8200,8$!!boot8ino8/ebro.tm

• tt:88///.$!!.a(.$k88(sab8s#mmetr#

Use(tros(o#8o1.tm!

• tt:88///.eo!e.o$(.b(.(a8smsnei!8s#m

m8s#mmg.tm


142/146

Page 1,2

Konse b!o(k

diagona!i?ed


143/146

Page 1,3

transormasi 2 da!am notasi matriks:

koordinat bar$ koordinat asa!matriks

transormasikoordinat bar$ d!m term

koordinat asa!

σ* 4x? σ* 4#? s$ms$m

karakter ++


144/146

Page 1,,

b!o(k diagona!i?edsetia matriks transormasi die(a menadi

matriks !ebi ke(i! seanang diagona!

$taman#a


145/146

Page 1,5

)2 ar$s mem$n#ai χ 4 6 χ 42 6 1 dan χ 4σx? 6 χ 4σ#? 6 &1

9agaimana b!o(k diagona!i?ed $nt$k 3* E


146/146

&10

&10

simetri lengkap 2 sks

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