1. Walton Section 2

8/3/2019 1. Walton Section 2

1/32

What use is group theory?

There s no doubt hatgroup heory s of pivotal mportancen chemistry. n mostundergraduatehemistrycourses,hegroup heory ourse omes bout alfway hrough hewholeof the chemistry ourse.By this imethebasicconcepts sed n chemistrywill havebeen overed-especiallyhe deasof atomicorbitals, ondingtheories,molecular ibrations,molecular hape nd electronic onfigurations. o take hese deas urther tbecomesecessaryo understandomething bout roup heory. ndeed, ourses hich ollow hegroup heorycourse re ikely o buildon heconceptsand erminology)fgroup heory uiteextensively.Unfortunately,t is true hat somestudentsan ind their first encounter ith group heoryan uncomfortableone. In manycaseshis canbe traced ack o a ackof familiaritywith themathematicalonceptshatgrouptheoryuses, nd n othercasest is related o thedifficulty in 'seeing'the symmetry perationsnvolvednexamining molecule'sverall ymmetry.Even or thestudents hoare amiliarwith themathematicsndcanperformhree-dimensionalanipulationsn theirheadwith ittledifficulty, hewhole easonor groupheory anbe unclear.Whatever ourstarting ointwhen t comeso understandingroup heory, t is importanthatyoucanatleast segroup heory o solvesome hemical roblems, nd hatyouare amiliarwith the erminology, hichpops p again ndagarnn otherareas f chemistry.Hopefullyhisbookcanhelpanychemistry tudent tudyinggroup heory or the irst time. If youareonewho findsgroup heory npalatableecause f the mathematics,then oushould now hatusinggroupheoryn chemistry nly equ ires imple rithmetic: omore! (Of course,a fullerunderstandingf group heory equires athematics.)n fact, heexercisesn thisbookcanbe completedwithouthaving o read he'mathematical'bit Section ) at all. If youareoneof thosewho finds t difflcult o'see'thesymmetry roperties f a molecule,hen ealisehat practicemakes erfect'.Theworkedexercisesnthisbookaredesigned xactly or thispufpose,o makeyoumoreconfidentn handling he hree-dimensionalshape f molecules.And, inally, f youareonewho ustcannot eewhere roup heoryhts n andwhere t al lleads, hen ome f thesimpleexplanationsn thisbook-particularly hosen Section -will hopefully elp omake hingsclearer. Certainly,doing heworkedexercises ill help you to see he useof group heory nchemisUy.Thisbook s meant o be written n. Doing heexercisess an mportant artof completinghebook,and1'ou hould avea pencilat hand. Thereare spacesor you to write yoru answerso theproblems.The ul lanswers re hengiven mmediately elow. Try to resist he emptationo lookat heanswer efore ouattempttheproblem;t mighthelp o cover'upheanswer ith apieceof paper. n sayinghis,do notexpecto geteveryarswer ompletelyight irst ime, t wouldbe remarkablef youdid. It will alsobe veryuseful o havea smallmolecularmodelling it nearby some Blu-Tack'and strawsarea usefulsubstitutef you do not possessmolecularmodelling it). Of course, ome ections ill take onger o completehanothers, ut t is probablyrealistico try ard complete ach ectionn onesitting.Try tounderstandhe deas overedn each ection eforeproceedilg o the nextone,and watchout for the erminology,whichwill keeppoppingup. By theendyouwillhavecovered earlyall of the usesof group heoryn chemistry,o a f,irst evelof understalding.Thisshould eenoughat least o usegroup heory o solveproblemshat you are ikely to encountern chemistry, ut shouldalsoprovidea firm foundationfyou are nterestedn takinggroup treory urther.


2/32

SECTION2

Symmetry-a start

2.1 An introduction to symmetry2.2 Rotat ion axes2.3 Reflection planes2 .4 Centres of inversionZ 5 Improper rotation axes2.6 Successiveoperations, identi ty, inverse and2.7 Point groups2 .8 Summ4ry

class

447

1318)

,/o'\Br -Br

Brr

Br

.B rb .Br

Br.oA

,,/o'- -r.


11/32

Beginning roup heoryor chemistry

SketchheAuB{ molecule nd heo, ando7 planes hicharenotshownn thediagram bove'

Answer

Br/

oy plane

Apart from the o, ando7 reflectionplanes-shown bove, here s onemorereflectionplane n $e AuBromolecule. Figure ou, *lir" the new plane ies, and sketch he plane on a diagram of the molecule'


12/32

Svmmetry- a start 13

lne remaining reflection plane is shown above. This plane is perpendicular to the principal axis, and theh. ' t t . i c r p f l pc t i nn::#ffi:';;;ilt;;;i"J*i,r,i" ,rre lane.nracr,he vmmerrvperarionefinedv his eflection- n ^ ^ - : + ^ + L : . i r i c c f i l l c c \ / m m e f f v.;;,;;F.;y*rj :;J;T1i::j1"^1"*::::::ll:l.t",Yl1l',':,3.?"n#,*l?,il|,fJ',#,TTIil:J:l#k.;;; ^rt sgiven he ymbol h,whereftistandsfor horizontal'.It turns ut hathiss an^ f l ^ ^ r l n n n l a n o-:;;il";:;;";;;"", ,Ju" breodentirv'.tr oun1::1"1:il:':.0^1:T:-':::?;3::ti1:Ttlii;-:'"'l'Ji: ,l;il ,ffi;;';",*r,";;i;; mooerfAuB{,.:o hu: .:l_.-T^:::,he.morecure

n three-:nensions. The key thiig to noiice s ttrat he oh plane s perpe*ndicularo the principal axis'

2.4 Centres f inversion,i'e now come to anothersymmetryelementwhich can be used o describe he symmetryof a molecule' This...mmetryelemenr s cailed a centreof inversion(or inversioncentre)and t is given the symbol i' The good-;ng about his symmetryelement s that,unlike rotationaxesandreflectionplanes'a singlemoleculecan only':\.0 onesuchcentre,andwe only need o decidewhethera moleculehasgot one or not' Thg bad thing about t. lhat the actualsymmetry element tself is sometimesdifficult to .See'. Certainly, this is one of the cases.. .e e practicemakesPerfect.Therearetwowaystodescr ibeaninversioncentre.Somepeoplef indi teasier tounderstandonewayrather--.n the other. The first way of ,seeing' an inversion centre s by inspection; his gets easierwith practice':.:rera while, this is the quickestway of spottingan inversioncentre' Take AuBr ' as the exampleagain' To,:.e the inversioncentre n AuB{ by inspectio"n, ne has to imagine moving e*ach tom in a straight line.*ards he Au atom, putting ,nfougtt tit" Au atom, and then moving away from the Ax atom in the same-.:ection o a point wfreie he atom s at the samedistanceaway rom the Au atomas t wasbefore he symmetry:erationbegan. The inversioncentre ies on the Au atom'

SketchtheAuBr,molecule nd abeleachBr atom. Mark thepositionof the nversion entrewith an;;;.' fu..n ouf,n. inversion perationsdescribedbove, ndsketchhe esult.


13/32

Beginning roup heoryor chemistry

,/ \ \Br ' \ -Brd \ c \ iof course,carrylng out this operationon the Au atom doesnot change ts position at all' The bromine

atoms'however, are al l interchangedwith their 'opposite' bromine atoms' SinCe he final result

after inversion isindistinguishable fromtir. &iginuf , then AuB{ has a centreof inversion as a symmetry element'The following two exarnples houldalsohelf in identifyingan nversioncentfe'using he methoddescribedabove.

For NH3, Z)-r,2-dichroroerhene,E)-1,2-dichloroethenear"::: l j : (octahedral),sing he methoddescribed bove, easonwhetheror not eachmolecule as an n'Ersioncentreof symmetry' If themoleculeoes avean nversion entIe,abel heatoms ndsketchhe esu-ltf the nversion peration'


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Symmetry- a start 1 5

AnswerOnly (E)-1,2-dicNoroethenendCoCf]- have nversion entres f symmetry.The nversion entres rein themiddleof thedouble ondanddn theCo atom espectively. arrying ut he operationsn hesemoleculess shown elow.

inversion+ob

O - inversionentre

Inversron---+

Thesecond ay to identifyan nversion entres to usea slightlymoremathematicalescription.If the:athematicsworries ou here, henyou canuse he qualitative efinitiongivenabove,t works ust aswell.l"rwever, f possible, ou should ead hispartanyway o see f you canunderstandt.) The mathematical-:scriptiongoesas follows: f the nversion entre s placedon the origin of a three-dimensionalartesian- tordinate etof axes, ndatomsareat general ositions ivenby the coordinates"r, , z), thenan nversion:entres a symmetry lement f thatmoleculef a molecule aving he same tomswith coordinates-x, -y,-:) is ndistinguishablerom heoriginalmolecule.An example hould elp o make hisclearer.Thediagram.: the op of page16 shows he AuBro molecule lacedon a setof cartesian xes.The Au atomsits on the:rgin 0,0, 0). Eachof theBr atompbsitionss givenby a unique etof coordinates;o, or Bru etus define:e coordinatess -ru,!a, z). The nversion entre peration, ith thesymmetry lement laced t theorigin,;arsforms ach f thecoordinatess ollows:-

(' ' Y'') -:-'(-'' - Y' -' ):rrr example,he Br" atom coordinates fter he originaloperation ecome -x",-y., -zc). Since he new:.rordinatesf eachof theBr atoms ave dentrcal alues o coordinatesf oneof the otherBr atoms-so. for:rample(-xc, -yc, -zc) = (x^,ya,za)-then the molecule fter he symmetry perations indistinguishable:om theoriginal.Therefore,he nversion entres a synrmetry lementor AuB{.

CI

CI


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l o Beginning roup heoryor chemistry

1' Brto y aYr .a

as ndividualsymmetry lements'

Br-inversion c+Br-..4"fr lvo V^ ,/ '

(x., y", z")

./i"f. I Xb v" , , a

A t h i r d , b u t s o m e t i m e s u n r e l i a b l e w a y i s t o r e a l i s e t h a t a n i n v e r s i o n c e n t r e i S e x a c t l y t h e scombinafionof specificotation xisan du-rp!"in.

reflection lane.Thiscombinations rotation y 180'fottowedbyreflectionn a reflection laneperpendicularo therotationaxis. In otherwords, f a moleculecontainsac2axisand a reflection lanepe'peodlcutaro-thisaxis' hen t must'by definition'alsohavealinversion entre f symmetry. Thecz axrsneednotbe theprincipalaxis.) The nversion enffeelement

splaced t themeetlng oint of theaxisand tr.9eflection lane,But,BEWARE, hepresencef an nversioncentre f symmetry oesnotnecessarilyru" ,r,"i,rr" .or.iur. hasa cyaxisandaperpendiculareflection

lane

(-x., -Y^, r^) = (x", f", z")

Usingthemethodsonpreviouspages,reason_whetherornottheH2omoleculehasaninversioncenffeaasymmetrylement? o helm.e f9r lufro nor Oltlq*-ltLthe moleiuleand hekeysvmmetrvelementswhich help you decidewhether t ha3an inversion centreor not'


16/32

.H."o\,o ' \o"'

Svmmetry- a start

l ' s rve rl : HlO molecule has a C2 axis and two mi rror planes,but neither of these mirror planes is-:::i: ' idicular to the C2 axis. Despite the fact that the molecule does not have a mirror plane-,:- i:dicular to a C2 axis,we cannotsay for definite that the moleculedoesnot havean inversioncentre:. xrfletry. We must use the other dehnitions to decide. Using the first definition: if , the origin isI ::3J irn the oxygen atom, and the hydrogenatomsare'moved'towards the oxygenatom and then-- ',ed' away from the oxygen atom n the samedirectionto the samedistanceaway from the O a tom,--::. e different orientationof the water molecule s obtained,see igure. In fact, it is impossible o pick.:'. rri-einwhereperforming an inversioncentreoperationgives a molecule ndistinguishnble romthe^._l;nal.Therefore,he water moleculedoesnot havean nversioncentreof symmetry.

^ HoO* *o

ica\,

inversionboutO

,lHr -rHoAuBro doeshave a C2 axis (coaxial with the principal C4 axis) and it doeshave a reflectionplane.Jrne?s


17/32

18 Beginning,roupheoryor chemistry

AnswerH2O2 only has an inversion centre when the molecule s in a planar E (or trans) configuration. Theinversioncentre s then n the middle of the O-O bond.Benzene-in the centreof the molecule.Chair-fgrmcyclohexane-in the centreof themolecule.Ni(CNI (square lanar)----onheNi atom.

It is important hat you can dentify an inversioncentrebefore you progresswith the restof this book, as t willbe referred o again and again. If you have had difficulty in 'seeing' the inversion centre, hen it is well worthbuilding someof the molecules escribedn thequestionabove,andseeing f you can ind the nversioncentre nthreedimensions.

2.5 lmproperotation xesThis now brings us to the hnal symmetry element and operation) hat we need o know. I am often askedwhythis particularsymmetryelement s needed n group theory. The best way of answering his is to consideranexample rom a qualitativepoint of view. Consider he simple, etrahedralmolecule,CHa. It is easy o see hatal l of thehydrogenatoms n this moleculeare n identicalchemicalenvironments. n other words, hehydrogenatomscan be relatedby some single symmetryoperation. That is, theremustbe a single symmetry operationthat we can carry out which will transform each of the hydrogen atoms to all of the other hydrogen atompositions. This is exactly analogouso the C2 axis n water relating the equivalenthydrogenatoms,and he C3axis n NH3 relatingal l of thehydrogenatoms.However, f one attempts o relateall of the hydrogenatoms n methaneusing our existingknowledgeof rotation axesan d reflectionplanes th e moleculedoesnot containan inversioncentreof sy mmetry) t isimpossible o find a single symmetry element which will allow us to transform each of the hydrogen atompositions nto all of the other hydrogen atom positions. So, for example, he C3 axis which is presentas asymmetryelement(seediagram below), describesa symmetry operationwhich transforms only three of thehydrogenatompositionsonto eachother; the fourth hydrogenatom on the rotationaxis, cannotbe transformedontg the other hreepositionsby this operation. Clearly, we needanothersymmetryelementwhich can describea symmetryoperation o relateal l hydrogenatomspositions o eachother.

Enter the improperrotation axis. This is a single symmetryelement, he operationof which can transformeachof the hydrogenatompositions nto all of the otherhydrogen atom positions n CH4. Unfortunately, hissymmetryelement s themost difficult to identify, and requiresquitea bit of practicebeforeone can be confidentusing t. As before, he bestway of approaching his is to startwith an example.Considermethaneagain. The moleculehas two typesof rotation axes,C3 and C2. One of the C3 axes sidentifiedabove.


18/32

e I

I_lwil l

s

s

Symmetm- a starl 1 9

Using the C2 axis, only Hu and Hc (o r H6 and H6 ) atoms can be rnterchanged. t is impossible o- .:rchangehe H" and H5 atoms. For instance, clockwise otationof 90' (not a symmetryoperationn itself: CHa) doesnot give an indistinguishableesult,as he H atomsare on opposite idesof theplaneof thepaper- :r where hey would give an ndistinguishableesult see igurebelow).

u;-Q-H90'rotat ion-+

i loIIH' ' . . . . . ' 'C ' ' . . . . - ' 'Hc laIIHb

H"

HcBut, a rotationof 90oalmostgives the right result. If we now combine he rotationby 90" with reflection- .l reflectionplaneperpendicularo the axis of rotation(so, n the figure the reflection plane s in the planeof-- ; paper), henwe obtain a moleculewhich is indistinguishable rom the original (see igure below). In fact,-'..s ombination of operationsgives us a single operationwhich is a symmetry operation for the molecule.fr :e.however, hat the moleculedoesnot necessarily avea eitherC4 axis of rotation,or a reflectionplaneas

>ietch he methanemolecule, nd dentify hepositionof one of theCz axesof symmetry. A small:.olecularmodelmay be useful here.) Label the hydrogenatomsandsketch he effect of the Cr.\ mmetry peratlon.

Answer()neof the C2 txes can be seen f the molecule s viewed with the carbonatom n theplaneof the paper,:nd tw o hydrogenatomsgoing nto the planeof the paper,an d wo hydrogenatomscoming out of th e:lane of the paper. Th e view is looking right down the Cz axis. Notice that rh e C2 axis bisects heH-C-H ansle.l"i: Hi ':2

H;-Q-H -'4d : " bui"b:diHc H^


19/32

20 Beginninggroup th'eoryor chemistryindividual symmetry elements. (This is analogous o the combination of C2 followed by reflection in areflectionplane for an inversioncentreof symmetry. The presence f an inversioncentredoesnot necessarilyimply that hereare ndividual C2and eflectionplaneelements f symmetry.)

90'rotat ionand eflection

!o"H-C-Hd : b:

:,H c

ci '" b

The combinationof a rotationandreflection n a perpendiculareflectionplanegives a new symmetryelement,which is called an improper rotation axis. In the example given above, his axis is given the symbol 54,becausehe rotation s 90", and 360190 4.Beforewe move on to try someexamples,here s a coupleof importantpoints about mproper otationaxes. First, we need o study the effectsof successive pplicationsof an improperrotation on a molecule. Forinstance, ake the methaneexampleabove;what is ttreeffect of applying the 54 operationonce, twice, threetimes or four times in succession?As with the rotation axes,we can use the following notation to represenlsuccessiveoperatlons:one operation = S41, two operationsS42, three operationsS43 and so on. The 541operations shown n the figure above.

Using the methaneexampleshown above,sketch he moleculewith labelledhydrogenatomsand showtheeflect otS+2 and S43operations.


20/32

Answer

H-C-Ho : i:

Symmetry- a start 21

H,: o:i

H"

Hc

so'

so,H;-Q-H

-- . 54 2 operation s exactly the sameas the C21 operation,an d the S4a operation s the sameas th e E::rrion. Therefore, he only uniquesymmetry operations rom an 54 axis are S41and S43. Notice, therefore,

: .. ,n 54 axis must have a coaxial C2 axisas a symmetryelement n the molecule. In fact, without taking the: ,: much further, t is true that for any even-orderS, axis, heremust be a coaxial Cn12axis.For an 56 axis, he successive pplication f the improper otationalso eads o operations hich can be-. - - hc dby other ymmery operations.

?ertormsuccessive 6 operations n ageneralmoleculeand igure out which operations an be described:r othersymmetry operations, .g. 562 s the sameas C3r

Answer 562 is ttre sameas C3563 s the sameas564 is ttre sameas C32566 is the sameas E

Only 56l and56 5ar eunique

The Sz operation s unusual nsofar as t is exactly he sameas carrying out an inversionoperation,since t- ::espondso a rotationby 180'followed by reflection n a perpendiculareflectionplane. (See he sectionon.: r-jgSof inversion.) Accordingly, he52 operationdoesnot concernus as chemists.

Improper otationaxeswith odd orders e.g.S:, SS,S2...) equirea little morecare. Taking53 asan: , ::rple. Theproblemhere s that he S33 operations not thesameas heE operation. n fact, heS33_ ;ation onespondso a o operation. ou can see'this f you ealisehatperforminghe mproperotation n.,:: number f times,meanshatanoddnumber f reflectionsasbeen erformed,hichnecessarilyeanshat--: resultof S33 s a reflectionn a planeperpendicularo the mproper otationaxis. However, he Slb


21/32

22 Beginninggroup theory or chemistryoperation i.e. a double rotation) is the same as the E operation,sincenow an even number of reflectionshasbeen applied. Without going into the details of which other symmetry operations are the same as S3noperations,he corresponding perations re istedbelow.

S3 2 s the sameas C32S33 s ttresameas oS3a s the sameas C3536 s the sameas

Only 531 and S35 are uniquesymmetry operations.A similar argument applies o other odd-order mproper rotation axes. Theseare not describedhere,but the'take-homemessage' s that many of the improper rotation operationscan be describedby other symmetryoperations, nd hatwith odd-ordermproper otation axes,S12'is the sameas E.Now try the following examples:-

PC15has a trigonal bipyramidalstructure. Sketch he molecule and dentify the improper rotation axis.By labelling he fluorineatomsshowthe effectof theS operation.(Hint, first identify any rotationaxes.)


22/32

Ttr.'

AnswerThe 53 axis s coaxial

a*Ua

with the C3 axis.C, and S.

Symmetry- a starl 23

F"F" In

d' ' , \t \ FFasrt

Fb

e lF

Fasrt

Tt,"Fa

srt

F' itr'Fb

s,tIo

dF F

F f 'F

Fbseo

Identify the 56 axis n CoCl]- (an octahedral omplex). This can be a difficult axis to 'see'. This is aneven-ordermproper otation" xis and t must havea coaxialCn12axis. Therefore,t is worth looking for'-hecorrespondingC3 axis first. A molecular model is useful for this example. Sketch the molecule','reweddown the C3 axis, and by labelling he chlorine atoms,showthe effectof an 56l operation.


23/32

AnswerI f t h e m o d e l i s v i e w e d i n t h e c o r r e c t o r i e n t a t i o n , t h e n t h e s 6 a x i s c a n b e s e e n q u i t e c l e a r l y . S e e t h ediagrambelow, whictr ts viewed directly d.*;;; s; (and i:) axis'

The effect of a 56 operatton sshown'

This bringsus to the end of the symmetryelementsandoperationswhich areused o describe he symmetrl

ofmorecuresouanowescribe,T.r;di]:Tl5i*:rjuJ:;TJfiT:il:T#J"J1';il;f molecules. ou cannowdescrtbenesyrlllrlsu ur @rJrv^v-..'- - :, and t is worthsymmerry lemenrs. i;ili; of idenirfylng y*t.W elementsetseasierwith practtctperseveringith,unrrr ;;;" confidentn 'seeiig'al l of thesytmet'y etementsndoperations' heexampiesbelowaredesignedo giveyoupracrice t d";;i?;; tfiti':li]ements' Beforeattemptinghese xamplesmakesure hatyou*oJrro"o whateachor trresymmJtry perationsoes.Thesymmetry lementshatwehavestudied re:Rotation xes sYmbol r)ReflectionlanessYmbol)InversionentressYmbol)ImproPerotationssYmbol ,)

AnswerPCl5has he followrngsymmetryl:TT':-E' C3'3C2'o7x' 3 and3o'' (Morecompletely'

herearetwoC3operations,t"ct' 'areC3 f an dc*'aniini'"ut"two 53operationsr1andSl)')

(E)-1,2-dibromoethaneas he ollowing ymmetry lements:' C2' ando;t'

CH4hasthefo l lowrngSymmetrye lements : ! , , , ,C2,54ando4( theseared ihedra lp laness incetbisecttheH-c-Hangles).ThereareseveraloreactrofthesymmetryelementsexceptE'

Identrfyand is t as many symmetryelements syou can n the following molecules:

PCl5 (trigonalbiPYramidal)(E) 1,2-dibromoethaneCH+


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Symmetry- a start 252.6 Successiveperations,dentity,nverse ndclass3eforewe go further t is important hat we look atrolecule. These roperties il l be requiredn section-nderstandingf group heory.In thispartwe will studysomeof theproperties f symmetry perations.Most of theresultswe come:Jross eresimplyhave o beremembered,heydo not have o beproved.

2.6.1 Successiveperations-ie first question o ask s: what happens f we carry out successive ymmetry operations? We have already seen-^te uccessive ombination of rotations with reflections o describe mproper rotations(rotation followed by-:tlection), but what about any combination of symmetry operations; or example, s thereany difference between. foration ollowedby a reflection and a reflection ollowedby a rotation? In otherwords, s the order n which.ieperform operations mportant? Again, this is probably best illustrated by an example.

Phosphine, H3,hasa pyramidal tructureike NH3. Sketchhe structure ndelements f the structure.

the symmetry

AnswerPH3hasE, C3 and3o, symmetryelements.

someof thepropertiesf symmetry perationsn a3, wherewe will build on them o develop urtherour

the hydrogen atoms show the effect of performing(Use the same orientation of reflection plane inAgain,for PH3, sketch the structu-re, nd by labelling

the following combinationsof symmetry operations.eachcase.)a) oy afterC3l.U) C:l after ou.


25/32

AnswerThePH3molecules sketchedelowwith abels n hehydrogen toms'The eflection lane s theonethat ies n theplaneof thepaper.The opschemehows , performed fterC31and hebottom chemeshowsC3l performed fteror .

P-. . . .\HbHc

. 'Hb

6 v+ -P. . . .,/ \ HcH aH

P-. . . .\HbHc

6 v p . . .+. / ' \ HH\ca to

cu' D . _+./=\ nH.\o H.a

1 0 Beginningroup heoryor chemistry

This can be exPressedn asign, henwe get.

Theexercise hows hat he resultof performing ymmetry perationsn a differentorderdoesnol giveidenticalesults. ndeed,his s ageneralule or combining ymmetry perations:heorderof combinationsimportant. aking hispointa ittle further,withtheabove xample ehave hown hat:C3l performedaftero, doesnot give the same esultas o, performedafterC3l

more mathematicalway. If we replace he 'performedafter' with a multiplication

C31performedafterou can be written as:C31x ouand

o, performedafter C31 canbe written as: o, x C31

If we say that'sameas' is replacedby an equalssign '=', and not the-same s' is replacedby a not equalssign.*, -also, let us dispensewith the multiplications-ign as n normal algebra)-then we can write the following

for this example for otherexamples his may not necessarily e true):C 3 l o , * o u C 3 l

Thereare wo things o rememberabout his particular esult' First,we haveexpressedhe symmetryoperationsin a moremathematicalway. (Notice here hat, n theequation,Lre perationwhich is performedsecondswritten first. This may sound ike an odd way of doing things,but it is easier o understandf the symmetryoperation ymbol s readas operateson whatever s ti tt e right'' So, for example'C3l c' canbe readas C3operates n the resultof ou. This way of writing symmetryoperationsakessomegettingused o')

second, for the mathematicians, his g.nrrut property of the order of performing an operationbeingimportant s called a non-commutativeproperty. we will comeback t0 this in the next section'and t simplyneeds o be remembered or the moment'So, we have seen hat the order of performing symmetry operations s important' The combinationofsymmetryelementsalso leads o the following importantresult' The result of any combinationof symmetryoperations an be described y a single y*rn"tty operationof the molecule' So' for the exampleabovewe cansay he following:


26/32

Symmetry^- a starl6 y C 3 l = ou '

aldC31 cu = ou "

Whereov' ando, " are heresultsof thesymmetry perationsf the other eflectionplanesn themolecule.Theeffectof o, C3l is illustrated elow rememberhat hehrstoperationo perform s the oneon theright)

crt P. . . .\HaHb

P . . . .\HbHc

+

\' l6v\

"Ha

Thediagram howshat heoperation f C31 ollowed y reflectionn ougivesan denticalesult o reflectionin or' (or' is the eflectionplanewhich containsheoriginalP-H6bondandbisects heH.-P-H" angle). niact, t is ageneralesult f symmetry perationsf amoleculehatanycombinationf symmetry perationswithina singlemolecules thesame sperforming single ymmetry peration. hiswill be ef t asageneralresult o beremembered-wewill need hisresult n thenext section. f you are n anydoubtabout he ruth offte statement,ry a few examplesor yourself; Cl5 s agoodmoleculeo try in this espect.So ar wehavecombinedwo symmefyoperations, hataboutperforminghreesuccessiveymmetryoperations?

For the PH3 examplegiven above,what is the effect of combining he following operations?Rememberthat ov is the reflectionplane which lies in the planeof the paper,and ou' is the reflectionplane whichconta:ns he original P-H6 bond before any synrmetry operationsare carried out (i.e. this reflection planedoesnot 'move' with the P-H6 bond during the operations).The parenthesesn the last two examplesmean that the combination of the operations n the parenthesesmust be carried out before any othercombinations, ollowing the samerule when combining two operations hat the one on the right isperformedhrst. (Hint: with the parentheses,irst hgure out what single symmetry operation s the sameas the combination of the two operationswithin the parentheses). t will probably help to sketch hemoleculewith labelledhydrogen atoms n each nstance.C 3 1o u ' o uo r ' o , c 3 l(C3l or) or 'Cgl (ou ou')

21

t\H


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28 Beginning roup heoryor chemistry

Althoughtheanswersaboveareforaspecif iccase,theyi l lustratetwoimportant 'and'asi t turnproperties f symmetryoferatlonsn moiecules.First, he order n which operations recombineds stillimportant-i.e. hey u." ion-.ornmutative.Second,he ast wo examplesho w hat he sameanswersobtainedCzL nthiscase) espite orkingoutdifferent artsn a different rder. n general,hismeanshat hefollowings true where , B andC denote nypaflicuiar ynrmetryperation):A(BC) = (AB)C

To mathematicians,his meanshat he combination f symmetry perationss associative'This s simplyanameforthistypeofmathematicalbehaviour,andjustneedstoberememberedforthemoment.2.6.2 ldentitYwe havealreadyestablishedhat the operation f doingnothing'E' is an important ar t of describinghesymmetryof a molecule. For instanceE canbe seenas the resultof combinations f othersymmetn'operat,ions.his act,again, eedsoberememberedor now,aSwewill buildon t in thenextsection,

2.6.3 InverseAnotherpropertyof symmetryelementss that the following is always rue (whereA is any particulalsymmeul;;;;;;i;"";; Al l is atsoa svmmetrvoperationof themolecule):

Answerc3 1o r ' o ,Splittinghisup. We hrstneedo figureoutwhatsingle perationdone y sketchingut hemolecule s ollows:

o,'o, corresPondso. This can be

o\H

6v, /r\ i l .bH' ' a Ho

Ioy-+ ,P - . . . ., / \ HcbH aH

cH

From the ftgure, t can seen hat ov ' 6v = C32'Therefore,we can

"ytrtu'^c]f"'\t" J czrt*'which can similarly be worked out as = E' so' the

answers: -Ca1or ' 6v = EThe otheranswersare:-ou' ou C31 = C32C31= g(C31or ) ou 'C3 (o , ou') = C3 l C3 l = Cz z

ooerationsasan nverse.

A-rA = E


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SymmetrY-a start 29

Write he nverse f the ollowing ymmetry peratlons

2.6.4Class3l.now'youarehopeful lyfamil iarwiththenotionthatt l resymmetryofamoleculecanbeexpressedintermso. :ollecdon r .yn}rn"ty operationsl1:1,.r.;;;;;il* :itt'e

motecuie'These vmm:ffvoperationsan

:-.en eused o erp edne r,esymmetry t tt e motecule. or ."u,nprJ,itr"-tvr1i1vôPeratronsn amolecule

.ie NH3 ar e, C3,,*,o",o, and ; *1.' " qualitativeointof view (not'ap,9ol, several f these

.\'mmetlyoperationsrJsimltar n thesymme;;;J;;", tnut r'"y;;'i;;;.C31alrdC32arcsimilar,and

: '. "r; .itn br" arealsosimilar' E is by itsel{' nathematicalrocedure'.alled similarity',:;:H::it#nt*uili";"#T$,ffi:,T"',T[i,',{*ffiliur*,voudoneed'[ok--.at imilarsymmeryoperarionsanbe og"n.u t"il t^"'' s..''

Nn'"f-d ;oeed' anymoleculewhichhas he.'me Symmet,yp".u,l6o,) anbe ,'o]:.1'"]"-* iorio*ing classes i,v'In"ov

opjT.l:i''- E,2C3and3ou..rherethe,2 ' andthe3' denote owmany nOiviOuatp"'ution'ut" 'n

'ftu'clasi-C31 andC32ue \n the2C3

- rss Norehathis 'au.J""'*t:ft^-,t::Tl,J:";::i:?ifr:Jrog'uurng*11"-:I:r or he vmmetrvPuttingv*"ttj operationsntoclassest"l^::ii:J;tfiit#;*ti*v operationsithinaclass lso::rl,Hi,,i"fr:#$?ffi*'3,''l'';1"J:irx'n";T'Jl"'#ff13il:Tl*;u'" r'in1he3\t sectlon'

AnswerE has nverse (i t is ts own nverse)'Ca1has nverseC3zczznas"]:T"-t7i-în,ir s rsownnverse).o, has nverseov (agal

ff immetryoperationn?lul inverseopêrationwhichisalsoasymmetryoperationofthe:...rleculesageneral"'uf' undwewill

usehisn thenext ectlon'

2.7 Point roups: ] i s i snowthe las tpar to f th issec t ion ,Here ,saqu ick recaponwhatwehavecoveredsofar :.Rota t ionaxes , re f lec t ionp lanes , , " l : iYcent resand improper ro ta t ionaxesarea l l symmetrye lemen. n;"*XX;3ffi:11'J'1"-r'S":tT1":iio.o,n"rmsritsvmmetrvperations'r. Thesymmetry"nJ;l#;il" .o*uro"o?"ra*,", *rrain rules,o grurumarhemaricallike

ehavrour'

]:ese factsnow provide a foundation lor loôkingat the Symmetryof molecules n more detail' and we will make

:Jpearedseof rhem. f" ;i;;;"*r .f ".t;*sil.. 6;;;',Y.::,*;;t'canbecumbersomeohaveowrite

.ir 'orr,eymmetrvperationsl: *::'j; eu"'vim": f ry.'t11*lti},1#'ffi ;:':tT"jl-lil:. a shorthandu, oi ,lpr-.r"otiog he symmetry f molecules' nrr. rfihafld method works'


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Beg nninggroup t teo y o r chemis ry

List all of theunique ymmetry perationshatyoucan ind n a PCl5moleculetrigonalbipyramidal).It will help o sketchhemolecule, nda smallmolecular odelmayalsohelp.

AnswerThesymmetry perationsre:

E, Cl , Czz,Cz Cz'Cz",oh, 531 35,o, oy' oy"Notice hat he edundantymmetry perations,uchasS32which s thesame sC31, avebeenmissed ut.Thesymmetry lements reshown elow.

.:) C, and S,axesa

CI CI ohCr' axis /;;-r ;'i\CI CICraxis

6v

lclCI

To describehe symmetry f a molecule y listingal l of its symmetry perationss clearly oo time-consuming.Therefore,he followingctassir,calioovtttl is used' It relieson beingable o identifykey


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i i ) ::ES

Symmelry- a start 31...:.metry elementswithin a molecule. The key symmetry elements hen define a particulargroup, which has.;'.ir&l differentsymmetryelements. So, fo r example, o classify he PCl5 molecule, t is only necessaryo-;:riiy the principal axis of rotation, the perpendiculu C2 axes of rotationand the oh reflectionplane. This:\3s away he drudgeryof having to identify everysinglesymmetryoperation.Al l of theseclassifications ave,::r workedout beforehand, nd all we need o do is to follow the rules or getting o theparticularclassification:-'ebelow).

Eachclassifications given a symbol, which uniquely dentifies hat classification. Eachsymbol stands or. : ilection of symmetry operations. The symbol alsorepresentswhat is called a point group. The two words- ,: , andgroup eachmean something. Point means hat ali of the symmetry elementsassociatedwith the.:.metry operations as s hrougha singlepoint n space.This point is not changedn posit ionby any of th e-..:,metryoperations.Fo r example, he 'point ' in the PCl5 molecule ies directly on the phosphorus tom.3:n are,however, t is not necessaryor the 'point' of the group o lie on an atom, e.g. H2O2) The word group.:ins that we havea groupof symmetry operations.We shall see n the next section hat we can definea group:',rematically,but for the moment t is sufficient to recognise hat we are dealing with a group of symmetry- J:.1tionsn eachpoint group.We classify a molecule nto a point groupby answering omesimplequestions bout he molecule.

. - ;st ion1.:itennlecule is one of the ollowing'recognisable'groupsGo to question2octahedral, iven the point group symbol07,tetrahedral, iven thepoint group symbolT7linear,given the point group symbolC-v (i f it doesnot havean I symmetryelement)linear,given the point group symbolD-2 (i f it alsohasan I symmetry element)

- jcst ion .- ,. es the moleculepossessa rotation axis of order 2 2?':ES: Go to question\il: If it has no other symmetryelements, hen t is given the point group symbol C1

If it has onereflectionplaneof symmetry, hen t is given thepoint groupsymbol C,If it hasa centreof inversion, hen t is given thepoint group symbolC;

{ iest ion3.:,.ts he molecule more han one rotation axis?':-ES: Go to part4.\t-): If it has no other symmetry elements, hen t is given thepoint group symbol C,(wheren = Ihe orderof theprincipalaxis, e.g. C3).

If it alsohas one o,6, hen t is given the point group symbol Cn7,(wheren = the order of theprincipal axis,e.g.C2p).If it hasn oy, then t is given the point group symbol Cru(wheren = the order of theprincipalaxis, e.g. C3r).If it hasan 52, axis coaxial with the principal axis, hen t is given the point group symbol 52,

l, ,nolecutecan be assigneda point group as ollows:If it hasno othersymmeffy lements,hen t is given hepointgroup ymbolDn(where = theorderof theprincipal xis,e.g.D3).If it hasSot 64 reflection lanes isectingheC2 axes,hen t is given hepointgroup ymbolDn7(where = theorder f theprincipal xis,e.g.D4l.If it alsohasoneo7r,hen t is given hepointgroupsymbolDn7,(where = theorderof theprincipal xis,e.g.D31).

Thisseries f questions llowsus to identifyall of the common oint groups ncounteredn chemistry.llere areotherpossible ointgroups, ut fortunatelyhese re ue. As a starting oint s it importanthatyou-.n recogniseome peciltc ointgroups;hese re he07, foroctahedral),Tafor etrahedral),-v for a inear--rleculewithouta centreof symmetrye.g.HCN) andD-7, for a linearmoleculewith a centre f symmetry


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32 Beginning roup heoryor chemistry(e.g.COZ). Hopefullyall of these ases reeasily ecognisable,nd t should e a fairly simpleexercisenassigninghepointgroups.Note,however,hat or a moleculeo be n theO7, r theT4 pointgroupt mustbeperfectty ctahedralr tefahedralespectively,fit is not then t should e assignedo a different ointgroup.Identifyinghese roups uicklygets etterwithpractice,nd treexerciseselowshould elp.

Thepointgroups reshorthandor thesymmetry lements ithina molecule. orexample,f amoleculesassignedo a C3, pointgroup, hen t hasall of the followingsymmetry lements: , 2C3,3ou. It is a very'important artof group heory n chemistryhatyoucanassignhepointgroupof a molecule ccurately.t is

Using he questions bove, ssignelementsn PCl5 hatareshownn

the point group of PCl5.the previousexercise.

It mav helo to refer back to the symmetry

Using the questionsabove,assign he point group of NH3.

AnswerPCl5Question .PCl5hasatrigonal ipyramidalhape nddoes ot all nto anyof thespecial ointgroups.

Question .YES: heres a C: axiswhichpasseshroughhe inealF-P-F paltof themolecule.Question .YES: here e C2axes hichareperpendicularo theprincipai xis.Question .Themoleclrle asseveral thersymmetry lements.t has trree , planes.However,t alsohasa o7,plane.Thereicre,t is assignedo pointgroupD37t.

N HsQuestion .NH3hasapyramidal hape nddoes ot all ntoanyof thespecial ointgroups.Question .YES: heres a C3axis.Question .NO: there reno other otation xes ther han heprincipalC3axis.It doeshaveothersymmetry lements. t doesnothavea o7,plane, ut t doeshave hreeoy planes.Therefore,t is assignedo pointgroupC3r.


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Symmetry- a starl- - .:leiy worth spending ime practising this aspectof the subject until you are confident of assigning. .::.: point groupeach ime. Try to assign hepoint groups or all of the molecules n the exercisebelow.

rrsi-Qn oint groups o the following molecules:

1 O , P H 3 , S O 2 , H C l , A u B r , , C o C l i ,4 0 (E)- I,2-dibromoethane), benzene, methylbenzene,? rCo(en ', * (en = l,24iaminoethane), -O 2CCH 2NH ]:ehloromerhane.O-, SO1-, HCCH,BzHo,3 4

AnswerHlO (C2u),PH3 C3u),SO2 C2r),HCI (D-ft), AuBro D+il, CoCrl- (On),@)-1,2{ibromoerhaneC27,), enzeneDoa),methylbenzeneC2r), trichloromethaneC3u),NO" (Dra), SOI- f a) , HCCHD-1), B2H6 D22),Co(enl]*Oll, glycine dependsn theconformation,ut f th e O2CCNpart s

'i l planar, ndoneof thehydrogen toms f theammonium art s also n thisplane,hen hemolecule:as a single eflection laneand t is assigned r, otherwiset is Cl.)

2 3 Summary-- :ls sectionwe haveseen hat our qualitativeunderstanding f the symmetry of moleculescal be mademore--:rtitative by determininghow the molecule s symmetricwith respect o severalsymmetry operations. Each- .eculecan be said to contain one or more synrmetry elements,which completely describe he symmetry of the- .ecule. Moleculescan also be classified n point groups. A point group is simply shorthan d or th e- ..:ction of symmetryelements hat describe he symmetry of a molecule.We candetermine hepoint groupof- :. ,,lecule y answering everalquestions bout he symmetryelements f the molecule.We arenow in a positionwhere we can classifymoleculesaccording o their symmetryand also describe he":.netry elementsof a molecule. The symmetryoperations ssociated ith tiese symmetryelements ppear ow mathemitical rules like multiplication. The questionnow is can we make the transition into fully- j::isenting the symmetry operationsof a moleculemathematically? f we can,can we then make quantitative-: : jlcdons abouthow symmetryaffectsvariousenergy evels n the molecule? The next sectionshowshow this

- . le done.What are he main noints of this section?The energy evels of a molecule are inked, in someway, to the symmetry of the molecule.The symmetry of a molecule can be describedwith symmetryelements.S1'rnmetryperations, escribed y symmetryelements, an be carriedout on a molecule.Rotationaxes, eflectionplanes,nversioncentres nd mproper otation xes aresymmetryelements.Thesymmetryoperations f a moleculehave mathematicallikeproperties.Eachsetof symmetryoperations as an nverseand dentity.Thecombinationof symmery operations n a molecule s non-commutative, ut it is associative.Every moleculecan be classified nto a point groupwhich completelydescribest symmetry.

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1. Walton Section 2