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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
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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 $tamakan σv
Page 11
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Page 12
X e
F F
F F
X e
F F
F F
X e
F F
F F
3efle"tions σvσh
σ
d
σ
d
XeF4
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Page 13
XeF4
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Page 1,
Z
Y
X
Z
Y
X
0tom at
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Page 15
C
H H
HH
+/ !mproper 3otation
3otate aout C/ axis and then refle"t
perpendi"ular to this axis
+/
C/σ
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Page 17
+/ !mproper 3otation
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Page 1
s$((essi*e oeration
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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
"tes to (!assi# a mo!e($!e into a oint gro$
$estion 2:
• Does te mo!e($!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
"tes to (!assi# a mo!e($!e into a oint gro$
$estion 3:
• =as te mo!e($!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
"tes to (!assi# a mo!e($!e into a oint gro$
$estion ,:
• Te mo!e($!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
Constru!ting the Representation
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($'
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Page ,2
+
O O
Constru!ting the Representation
Use a mathemati"al fun"tion
@gA py orital on +
#v 2 σ4x? σ4#?
N1 &1 &1 N1 Ty($'
py has the same symmetry properties as Ty and 3 x e"tors
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Page ,3
Constru!ting the Representation
0u0u
0u
σh
σh.Md x2-y2N .Md x2-y2N
C4.Md x
2
-y
2
N
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Page ,,
onstructin% the $epresentation
)*h 2, 2 22O 22 I 2", σ 2σ* 2σd
N1 &1 N1 N1 &1 N1 &1 N1 N1 &1
@ffe"ts of symmetry operations generate the
T30$+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
onstructin% the $epresentation
• 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
Constru!ting the Representation
!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
..
..
:
(
Constru!ting the Representation
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
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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
ulli.en symolsA 4aelling
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
ulli.en symolsA 4aelling
+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<
%enerating Redu!ible Representations
−
−
−
=
−
−
−
=
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
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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 (
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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 ;
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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 )
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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 ?(
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Page 7,
%enerating Redu!ible Representations
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
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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 >.
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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
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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
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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 @
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Page 7<
!dentity @
@
=
z
y
x
z
y
x
.
(,,
,(,
,,(
(
(
(
1or ea"h un?shifted atom
χ ;
=
y
x
=(
y(
x(
!nersion i
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Page 0
!nersion i
=
y
x=(
y(
x(
i
1or ea"h un?shifted atom
χ ?;
−−
−=
z
y
x
z
y
x
.
(,,
,(,
,,(
(
(
(
3efle"tion σ
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Page 1
1or ea"h un?shifted atom
=(
y(
x(x
=
y
σ
< p g >
χ (
−=
z
y
x
z
y
x
.
(,,
,(,
,,(
(
(
(
3 t ti C
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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
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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;
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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
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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;
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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.
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Page
K-BI)= MIC@@- IQ&Q&QI&QII )PBIK)"I T'+I @+-P
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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
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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;
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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
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Page ;1
>rom Γ3n to Γvibe and 'pe!tros!opy
$o% that %e hae Γ ie %hat does it meanD
We hae the symmetries of the normal modes of irations.
!n terms of linear "ominations of Cartesian "o?ordinates.
We hae the numer and degenera"ies of the normal modes.
Can %e predi"t the infrared and 3aman spe"traD
&es
Applications in spectroscopy: /nfrared Spectroscopy
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Page ;2
Applications in spectroscopy: /nfrared Spectroscopy
• ibrationa! transition is inrared a(ti*e be(a$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
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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
$oteA !nitial %aefun"tion
is al%ays real
/nfrared Spectroscopy
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Page ;,
/nfrared Spectroscopy
• Transition is orbidden i TM 6 0
• 'n!# non ?ero i dire(t rod$(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.
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C)? @ C: σ σ
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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
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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:
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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
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Page ;<
gaseo$s =2'
4"artan O0,
©Ra*e$n(tion In(. 2003
!iH$id =2'
xerimenta! absortion *a!$es:
3756,3657and 15
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Page
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Page
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Page
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Page
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Page
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Page
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Page
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Page
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Page
nalysis o# ?ibrational odes$
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Page
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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 :
$ormalise "oeffi"ients and diide y sum of sJuaresA
><:
(:( r r +=
'ymmetry dapted Ainear Combinations
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Page 101
1or B: this giesA r (
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Page 102
+
O O3emaining mode “li'ely” to e a end
Cv
Γ end
@ C:σ σ
( ( ( (
By inspe"tion this end is 0( symmetry
+O: has three normal modesA
0( stret"hA 3aman polarised and infrared a"tie
0( endA 3aman polarised and infrared a"tie
B: stret"hA 3aman and infrared a"tie
nalysis o# ?ibrational odes$ 'O experimental data2
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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
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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
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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
nalysis o# ?ibrational odes$ ,OCl3
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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( :@
nalysis o# ?ibrational odes$ ,OCl3
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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( :@
$oteA
!t is adisale to loo' out for redundant "o?ordinates and thin'
aout the physi"al signifi"an"e of %hat you are representing.
3edundant "o?ordinates "an e Juite "ommon and "an lead to a
doule “"ounting” for irations.
nalysis o# ?ibrational odes$ ,OCl3
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Page 10;
I+ 4!iH8 (m&1 +aman 8(m&1 Des(rition "#m Babe!
12
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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/ $se ")Bs to !ook more (!ose!# at te norma! modes
Symmetry( Bondin% and "lectronic Spectroscopy
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Page 110
• -se atomi( orbita!s as basis set.
• Determine irred$(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
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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
'ymmetry onding and +le!troni! 'pe!tros!opy
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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
'ymmetry onding and +le!troni! 'pe!tros!opy
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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
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'ymmetry onding and +le!troni! 'pe!tros!opy
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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
'ymmetry onding and +le!troni! 'pe!tros!opy
h i l h h d
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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
'ymmetry onding and +le!troni! 'pe!tros!opy
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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
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Page 11;
:s
O )B
a(
a( ( :a( :
non?onding
non?onding
a(
a(
a(G
:
:G
(
B)O
'ymmetry onding and +le!troni! 'pe!tros!opy
!s symmetry suffi"ient to determine ordering of a and oritalsD
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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.
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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
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Yualitatie 5O diagram for H:O
a(G
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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
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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(
'ymmetry o# +le!troni! 'tates #rom O9"+%++R&+ Os2
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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"
'ymmetry o# +le!troni! 'tates #rom O9"+%++R&+ Os2
h h if l D
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Page 125
What happens if %e promote an ele"tronD
a(
(
:
:
a(
Bonding
$on onding
0nti Bonding
1irst t%o ex"itations moe an ele"tron form ( non onding
!nto either the : or a( anti?onding oritals .
Both of these transitions are
non onding to anti onding
transitions. n?π
Dhat ele!troni! states do these new !on#igurations generate*
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Page 127
:
:(
(,
:
:(
,(
0(.0(.B(.B: 0:
0(.0(.B(.0( B(
!n these states the spins "an e paired or not.
!@A + the TOT04 ele"tron spin "an eJual to , or (.
The multipli"ity of these states is gien y :+(
These "onfigurations generateA;0: #
(0: and;B( #
(B( ele"troni" states.
ote$ if + Z then %e hae a doulet state
Dhat ele!troni! states do these new !on#igurations generate*
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Page 12
a(
(
:
:
a(
5ole"ular Oritals
(0(
(B(
;B((
0:
;0:
@le"troni" +tates
Dhat ele!troni! states do these new !on#igurations generate*
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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*
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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 $O "hange in spin
Eire"t produ"t for ele"troni" integral must "ontain the totally
symmetri" spe"ies.
Dhi!h ele!troni! transitions are allowed*
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Page 130
0 transition is allo%ed if there is no "hange in spin and the
ele"troni" "omponent transforms as totally symmetri".The intensity is modulated y 1ran"'?Condon fa"tors.
The ele"troni" transition dipole momentµ
transforms as the
translational spe"ies as for infrared transitions.
Dhi!h ele!troni! transitions are allowed*
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Page 131
1or the example of H:, the dire"t produ"ts for the
ele"troni" transition are
=•
•
(
:
:
:
(
:
(
(
B
B
A
A
A
B
B
A
=•
•
:
(
(
(
(
:
(
(
A
A
B
B
A
B
B
A
The totally symmetri" spe"ies is only present for the transition
to the B( state. Therefore the transition to the 0: state is“ symmetry forbidden”
Transitions et%een singlet states are spin a!!o"ed”#
transitions et%een singlet and triplet state are spin forbidden”#
Dhi!h ele!troni! transitions are allowed*
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Page 132
(0(
(0:
(B(
;B(
;0:
+ymmetry
foridden
+pin foridden+ymmetry
allo%ed
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ore bonding #or E6 mole!ules / !omplexes
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Page 13,
!n the "ase of Oh point groupA
d x:?y: and d=: transform as eg
dxy# dy= and d=x transform as t:g
px# py and p= transform as t(u
Γσ a(g eg t(u
Γπ t(g t:g t(u t:u
t(u
0 for Oh
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Page 135
t(u
a(g
eg t:g
t(u
a(g
a(g
eg
eg
t:g
a(g eg t(u
/p
/s
;d
+le!troni! 'pe!tros!opy o# d8 !omplex$
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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
+le!troni! 'pe!tros!opy o# d8 !omplex$
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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
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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.
+le!troni! 'pe!tros!opy o# d8 !omplex$
Groups theory predi"ts no allo%ed ele"troni" transition
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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$E
• an #o$ $se a basis o 3 *e(tors to generate Γ 3n E
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Page 1,0
# g 3
• Do #o$ 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($!eE
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Page 1,1
• tt:88///.(emso(.org8exem!ar(em8entries8200,8$!!boot8ino8/ebro.tm
• tt:88///.$!!.a(.$k88(sab8s#mmetr#
Use(tros(o#8o1.tm!
• tt:88///.eo!e.o$(.b(.(a8smsnei!8s#m
m8s#mmg.tm
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Page 1,2
Konse b!o(k
diagona!i?ed
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Page 1,3
transormasi 2 da!am notasi matriks:
koordinat bar$ koordinat asa!matriks
transormasikoordinat bar$ d!m term
koordinat asa!
σ* 4x? σ* 4#? s$ms$m
karakter ++
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Page 1,,
b!o(k diagona!i?edsetia matriks transormasi die(a menadi
matriks !ebi ke(i! seanang diagona!
$taman#a
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Page 1,5
)2 ar$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
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&10
&10
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