C h a p t e r4

Some importanttools of theory

4.1 Perturbationtheory and the variationalmethod

In most practicalapplicationsof quantummechanicsto n.rolecular problerns.
one is faced with the harshreality tirat the Schrcidinger
equationpertinentto
the prclblemat handcannot be solvedexactly.To illustratehou,desperate this
situationis, I notethatneitherof the followingtwo Schrcidinger
equationshave
everbeensolvedexactly(n.reaning analytically):
(i) TheSchrodinger
equation
for thet*,oelcctrons
moringabouttheHenucleus:
I tt _- /?: _. ]c- l.,r .,: I
_ _
l__ \: V:
'
_ _ _ _ + _ | 1 ! ,_ F j,, (4.t)
L :lllr ltllt t'1 t'. /.r I

(ii) The Schrcidingcr equationfor the trvoelectronsrn,,r

iniin an H1 rnoleeule evenif
the locationsof thetwo nuclej(labejedA and B) areheldclamoccl:
'
[
v :- ; h, :" j -
fi--,
-: -; (,:
- ; cr]
l t,: E,,, ,.r,
L-;. ; ;
Thesetwo problernsare examplesof u'hat is called the ..three-bodyproblern",
meaningsolving for the behaviorof threebodiesmoving relativeto one another.
Motionsof thesun,earth,andmoon (evenne_electing all theotherplanetsandtheir
moons)constituteanotherthree-bodyproblem.None of theseproblems,eventhe
classicalNewton equationfor the sun. earth,and moon, haveever been solved
exactly.So, what doesone do when faced u'ith trying to study real molecules
uslng quantummechanics?
There are two very powerful tools that one can use to "sneak up" on the
solutionsto thedesiredequations by first solvingan easier.,rnodel,'problemand
then using the solutionsto this problem to approximatethe solutionsto the real
Schrodingerproblemof interest.For example,to solvefor the energiesand r,,,ave
functionsof a boronatom,one could usehydrogenicl s orbitals(but u,ith Z = 5)
and hydrogenic2s and2p orbitals with Z : 3 ro accountfor the screeningofthe
full nuclearchargeby the two is electronsas a startingpoint. To solve for the
vibrationalenergies of a diatomicmoleculewhoseenergy.,,s. bond lengthE(R)
is known, one could usethe Morse oscillatorwave functionsas startingpoints.

r t o
 P e r t u r b a t i o tnh e o r y a n d t h e v a r i a t i o n am
l ethod 117

"starting point"
But. once one has decidedon a reasonable model to use,horv
doesone connectthis model to the real systemof interest?Perfurbationtheory
and the variationalmethod are the two tools that are most commonly used for
thisPurPose.

4 . 1 . 1P e r t u r b a t i otnh e o r y
In this method one has availablea set of equationsfor -uenerating a sequence
of approximationsto the true energy E and true wave function /. I will now
briefly outline the derivationof theseworking equationsfor you. First. one de-
composes the true HamiltonianH into a so-calledzerorhorder part H0 (this is
theHamiltonianof the modelproblemone haschosento useto representthe real
s.v-stem)and the difference( H - H 0l ivhich is calledthe perturbationand often
denotedtr/:

H:H"+l/. (4.3)

The fundamentalassumptionof perturbationtheory is that the wavefunctions

andenergiescan be expandedin a Taylor seriesinvolvingvariouspowersofthe
perturbation.That is. one expandsthe energy E and the wave function ry' into
zeroth.first.secondetc..orderpieceswhich form the unknownsin this method:

E : E " - E l - E : , E ' . . . . (.+.+)

' l t : t l t ' -' , l t ' - , l t :+ , ! t ' + . . . (,1.5
)

Next. one substitutes theseexpansionsfor E of H and of t! into H$ : EtL.

Thisproducesone equationrvhoseright-and lett-handsidesboth containterms
of various"powers"in theperturbation. Forexantple. termsof theform Et {2 and
Vt!2 andE0lt) areall of third power(alsocalledthird order).Next.oneequates
the terms on the lefi and right sidesthat are of the sameorder.This producesa
setof equations.eachcontainingall the termsof a givenorder.The zeroth,first,
andsecondorclersuchequationsaregivenbelow:

H t ) t f r r- F o t y \ . (4.6)
H"rlrt + Vtlrt): 5o',1tt+ E' ,lrn. (4.1)
H",ltt + v,lt' : E"t: + E ' , l t t+ E t r l t " (4.8)

Thezerothorderequationsimplirinstructsusto solvethezerothorderSchrodinger
equationto obtainthe zerothorderwave function ry'0and its zerothorder energy
80. In the first order equation.the unknownsare ry'r and El (recall that Z is
assumedto be known becauseit is the differencebetweenthe Hamiltonianone
wantsto solveandthe modelHamiltonianF1')).
To solve the first order and higher order equations.one expandseachofthe
correctionsto thewavefunction ry'K in termsof thecompletesetof wavefunctions
118 S o m ei m p o r t a n t o o l so f t h e o r y

of the zerothorder problem {V/l,}.This meansthat one must be able to solve

Aur!t!) : Elrltj notjust for the zeroth order stateone is interestedin (clenoted
ry''above)but for all of the other(e.-e..
excitedstatesif r/r(ris the groundstate)
zerothorderstates{ry'l}.For example.expandingry'l in this mannergives

r' :\"r, ('1e)

Nou',the unknownsin the first orderequationbecomeEl andthe C) expansron

this expansiontnto Ht)rftt -, l'ryo: E0lrl + Etr1r0
coefficients.Substitutin_e
and solving for theseunknownsproducesthe folloil.ing final first ordcr lvorking
eouations:

E': i,\t"ll,l1t/). ( , +I .0 )
) ,1' l) l( -8"El) l
'l ''= I,l ' i lft,t' 1r (4.1
I)

wherethe index J is restrictedsuchthat ry'l not equalthe stater/r0vou are inter-

estedin. Thesearethe fundamentalworking equationsolfirst orderperturbation
theory.They instructus to computethe averagevalue ofthe perturbationtaken
over a probabilitydistributionequalto tlrrt^
,yo to obtainthe first ordercorrection
to the energyf l. They alsotell us hou'to computethe first ordercorrectronto
the u,avefunctionin termsof coefficients multiplyingvariouscltherzerothorder
wave functionsry'j.
An analogous approachis usedto solvethesecondandhigherorderequations.
Although modernquantummechanicsdoesindeeduse high order perturbation
theoryin somecases.muchof whatthe studentneedsto knou,iscontainedin the
first and secondorder resultsto which I will thereforerestrictour attention.The
expressionfor the secondorder energycorrectionis found to be

E,:tl,t/r,/,,)l'/(Eu-E:). (1.12\

where again,the index J is restrictedas noted above.Let's now consideran

exampleproblemthat illustrateshow perturbationtheory is used.

Example problem for perturbation theory

As rve discussedearlier,an electronmoving in a conjugatedbond framework
can be modeledas a particle-in-a-box.An externally applied electric field of
strengthe interactswith the electronin a fashionthat can be describedby adding
the perturbation V : e€(x - ! t to ttre zerothorder Hamiltonian.Here,x is the
positionof the electronin the box, e is the electron'scharge,and Z is the length
ofthe box.
First, we will computethe first order correctionto the energyof the r = I
stateand the first order wave function for the r : I state.In the wave function
calculation,we will only compute the contribution to ry' made by riil (ttris is
just an approximationto keepthings simple in this example).Let me now do all
 P e r t u r b a t i otnh e o r y a n d t h e v a r i a t i o n am
l ethod 119

thestepsneededto solve this part of the problem.Try to make sureyou can do

the algebrabut also make sureyou understandhow we are using the first order
perturbationequations.

r:ee('-:) . /nft.f 1
*;':(;)' srn(
/ /'

E:a,:tlI.
:(*;:,1
E,,,'=, nlwl'_:,) -
=(*;i1",(,
/ 1 r
f),-;:;
f l . / 7 - r r
:(;J/ s i n r ( z )/ * ( ' L- l; ) o '
= |/ 2,e.r 1 f t s i. n
rrr.\.r
- ( i ) ' -o('t;2 le ir ft ,I rL -.ay
''n'1,)ar
//'
The first integralcan be evaluatedusingthe following identity with a : {.

r/,r:; - {j#t! - ryP =+

,in'to,\,r
1,.,' f,
The secondintegralcan be evaluatedusing the following identity with 0 :
f
anddo : f d.r:

: sinz
e tte
/',i",(f )a, : 1,"
f - - | ! r 7
"
/ s i n . e , l H= _ _ s i n t : r t + l l :
Jrt 4 2lt) 2
Making all of these appropriatesubstitutionswe obtain:

L- r r r= (/ 2rt r 1 1 1 , : L L t \
/(T-1;t)=o
/ w l _. "i, '..t . - |_ t lI w_:-t\v1 .,
r r , , r _r \ ' "
_ _ F .
La=1 - t)r=.

( i ) / l s i n ( * ) . . r (- . r
vl: : ! ) s i n ( r ) , 1( ri ) ! r , " ( r f : \
\L/ \ r t
#i.r- ,
The two integralsin the numeratorneedto be evaluated:

/" 'in(+),,"(?)r' an<r

1,,"*(Tl),r"(f)a.,
Using the integral
[,cos(ax)clx: lcos(a;r) f ] sin(ax), and the integral
; cos(a.v)ri.r- -Lsin(a.r),we obtainthe following:

l,',^(#),,"(?)
d,::f/,.". _lo,
(T)u. "o,(+)
a.rf
l
: ltl,"(f),',,-
*''"(+):] -n
120 S o m e i m p o r t a n t o o l so f t h e o r y

: :11,"..,
/'".',rn(?).'"(?)r,
-lu"'"'(T)"]
(| ),r.
l l t L t L . ,
: - - c o sr (7 \ \ , 7 . \ , \ 1
; | |
l l t . t ' \ ;L- l t- r - : 7r n { - ' L ' 1I I, ,

- ( # . . . ( + ) - + .J" ( +
tL: /-l.zt\ Lr 1 . 1 . - ,1. 1. ' f

1,2 1t
: - ( c- o s : z - c o s 0 ) * ^ sinr -0
llT lir
_2L2 _2L2 L) L: 8l:
: - :
2". 18", W-7:-r)".

Making all of theseappropriate

substitutions
we obtain

( i ) r " ' r ( - f f - a t o i/)2 \ r /2rr\

\z/ ''nl,-/
Jltlt L cF
slnl - l
21 h1;ra

Now. let's computethe induceddipole momentcausedby the polarizationof

the eiectrondensitydue to the electricfield effectusingthe equationl/inducetl:
-e V.1x - r/r with V now being the sum of our zerothand first order
I ilV
wave functions.In computingthis integral.we neglectthe term proportionalto
e2 becausewe are interestedin oniy the term linear in e becausethis is what
gives the dipole moment. Again. allow me to do the algebraand see if you
can follow.

t / Z\
Fnduced:
-"/ *-
(t
- - l vrtr. where* : (*lo,+ qrl,,)
2)
7L ,
- ' v l ') a r
: -n,/n
r.,,nou."o (*10'* r y l ' ) _ l
).(" l) (*''+
: -n *1n'.
rL
- " *1"(.,- I) *\"0'
(' l) v','a,
/

J" I,'
rL /
-'/ vl')'(.' -,I,'*1"-("
l)*1''r' -t)*\"0,
The first integralis zero (seethe evaluationof this integralfor.Ell) above).The
fourth integral is neglectedsince it is proportionalto e2. The secondand third
integralsare the sameand are combinedto give

rL
L1,.. I

Vi"' ["-
/
r \ wi"dx.
lt,nau*a=-2? |
JO \ t)
 Perturbation
theoryandthevariational
method 121

S u b s r i t u tw n ': ( i ) , s i n ( f t a n dt t l l " :
i nl g s i n r f ) .w eo b t a i n
Y;titi 2

. r . 3 2 t r t L ' le2r \ f , rt.ytl Ll llz.r\

/ / n , r : r c -, rz =t ' ; i t ' i l;l I stn(-)(,--,
/sin(;),/(

Thesc integre,r.**r,,,rr rr"rln., *.;,"'."r;;. *.',0",", rhemwe

finallyobtain

/ , r J u=e_e.u" tt /ln!- 1r' ! ' "f, ,+

L./
) f _ 'gt t -)
\ \ /

mLae2e 2to
_
'
h 2t 6 3-i

Nor.vlet's computethe polarizability,a, of the electronin the il : I stateof

the box, and try to understandphysicallywhy a should dependas it doesupon
the length of the box z. To computethe polarizabiiity,we need to know that
o : #1.:6. Using our inducedmomentresultabove,we thenfind

":ft) -nrL'e)2t"
\,JFl_,) h:t,.Ji'

Noticethat this finding sLr-sgests

that the largerthe box (molecule).the more
polarizable
rhe electrondensity.This resLrltalso suggeststhatthe polarizability
ofconjugatedpolyenes shouldvarynon-linearll,with thelengthofthe conjugated
chain.

4 . 1 . 2f h e v a r i a t i o n aml e t h o d
Let us now turn to the othermethodthat is usedto solveSchrddingerequarions
approximatel,v.
thevariational method.In thisapproach.onemustagainhavesome
reasonable waveflnction iy'trthatis usedto approximatethetrue,,vave
function.
within this approximatewavefunction,one imbedsone or morevariables
{a.7}
thatonesubsequently variesto achievea minimumin theenergyof ry'0computed
asan expectation 'nalueof the trueHamiltonianF1:

E ( { d J l )= l | t t l H j l / a ) 1 H . , , , 1 . , 1 r , , ) . ( 4 .1 3 )
The optimal valuesof the cyl parametersare determineciby makine

JE1da.,:Q ( + l.+ )
to achievethedesiredenergyminimum(n.b.,we alsoshouldverifzthatthesecond
derivativematrix 1i)16/da.t dar) hasall positiveeigenvalues).
The theoreticalbasis underlyingthe-variationalmethod can be understood
throughthe fbllor.vingderivation.Suppose- that someoneknew the eract eigen-
states(i.e..true w1' andtrue 6r) of the true HamiltonianH. Thesestatesobev

HVu: Er'Po (-l.ls)

122 S o m e i m p o r t a n t o o l so f t h e o r y

Becausethesetrue statesform a cornpleteset (it can be shoi.vnthat the eigen-

functionsof all tlie Hamiltonianoperatorswe everencountcrhavethispropertv).
ourso-called"trial r.l'ave
function"tfrt)can.in principle.be expandecl
in termsof
t h e s eV 6 :

/":fcrvr ( : 1 .61)

Beforeploceeding further.allor.r'nre to overcomeoneIikelvrnisconception. what

I arn going throu-shnou is onlv a derir,ationof the u,orkingformula of tlie
variationalmethod.The final formulawill not requireus ro e\/crknoil'theexact
v6 or the exact81 . but rl'e are allor.r,ed to usether.r.r as tools in our derivation
becauserveknou'they exist evenif u,ener,,er knou' ther"u.
with the aboveexpansionof our trial funcrionin lernrsof the exacteigen-
f u n c t i o n sI.e r u s n o r vs u b s t i t u tteh i s i n t o t h e q u a n t i t y\ r l r r l H l { r , t ) l ( 7 r r ) 1 r 2
t h{ a
) ;t
thevariationalmethodinstructsus to colnDute:

E : ( r l r t ' H l l r oi,V) l,l'l=,r)(, )F . ^ * ^ i r l f , *r)l

(F . ^a ^f , ,* , ) ( . +l 7 )

Usingthefactthatthe W6 obel HW6 : f x Vr, andthatthe W1.areorthonorr.nal

(l hopevou remember thispropertl'of solutionsto all Schrodinser
cquations
thal
we discussed earlier)

( V , iI W r ) = 6 r z . (4.18)

the above expression reduces to

r : f ( c { vr t H t C r * , ,( f { c " v " c ^w ( ) )
I
= f tc,t'r,
/ +,r,t . ( 4 .l 9 )

one of the basicpropertiesof the kind of Hamiltonianwe encounter is thatthey

have a lowest-energystate.Sometilneswe say they are boundedfrom belou,.
which n.reans their energystatesdo not continueall the u'ay,to minus infinity.
There are systemsfor which this is not the case,but \\,e lvill now assuinethat
we arenot dealingwith suchsystems. This allowsus to introducethe inequality
E x > Eo u'hich saysthat all of the energiesarehigherthanor equalto the e'erg1.,
of the lou,eststatewhich we denore86. Introducingthis inequalityinto the above
expressionsives

r = p tcrr,E,, : Eo
/ F,.^ t2
(4.20)
 P e r t u r b a t i o tnh e o r y a n d t h e v a r i a t i o n am
l ethod lz5

Thismeansthat the variationalenerg)'. computedas qr1t01H1ty0)l(,10I /0) will

lie abovethe true ground-stateenergyno matterwhat trial function ry'0we use.
The significanceof the aboveresultthat 6 > Es is as follows.we areallowed
to imbedinto our trial wavefunctionry'0parametersthat we can varv to make E,
cornputed as 1rp01n1y0) lhlt't 1,y'o),ar lor.vas possiblebecause
rveknowthatwe
cannever make (ry'01
Hlrlrq)lilt\ | ry'o)lower than the true ground-stateenergy.
Thephilosophythenis to vary theparametersin ry'0to renderE aslow aspossible,
because the closerE is to ,00the "better" is our variationalwavefunction.Let me
now demonstratehow the variationalmethodis usedin sucha mannerby solving
an erampleproblem.

Exam pl e va ri atio n al problem

SupposeyoLlare given a trial wavefunction of the form

-r"', .1
o= =
nd,,
,*p(-4.,,,
\ ..r., ) .*o(
' \
/ crr, /

to representa two-electronion ofnuclearchargeZ and supposethatyou arelucky

enoughthat I havealreadyevaluatedthe Q/0lH)lto)l0lr,, i ry'O)integral"which
I'll call l/. tbr you and found

n = ( t . , - 2 z1 . .
;r.) ;
Now.let'sfind the optimum valueof the variationalparameterZ, for anarbitrary
nuclearchargcZ by settingdIl'ldZ":0. After hnding the optimal value of
2., we'll then find the optimal ener_qy by pluggingthis Z. into the aboveW
expression. I'll do the algebraand seeif you can fbllow

/ 5 \ - :
il - l/.. _222,-,.2.]l'_.
\ 11 "1,, /
t lI I '
/-_ 5rel
_ = f\ 2 2 , - 2 2 - - l _ : { ) ,
tL, U,/ rr,r
5
22"-22*;:0.
- -
5
-.
z.L( _L

Z":Z-;: z-0.312s

(n.b..0.3I 25 represents
the shieldingfactorof one I s electronro the other).
Noq using this optimal Z. in our energyexpressiongives

r y: z e ( t . - t , - l ) ' :
\ 8./ a,,
= ( ' - ,116 ,lfL)f\z - : \ - r r * : l 1
\ t6/ 8Jro
a a A
S o m e i m p o r t a n t o o l so f t h e o r y

\ r r
/
- - at":
\ t n ) \ - t * ' J;
- ( z- 1 \ ( r - i ) l : - ( z - I ) ' l
\ 16l\ 16/u,, \ 16/ cre
- ( z - 0 . 3 1 2 5 y r 1 217I c. 2v

( n . b . "s i n c ea 6 i s t h eB o h rr a d i u s0 . 5 2 9A , e 1l u o - 2 7 . 2 1e y ) .
Is this energv"anygood"?The total energies of sometwo-electron atomsand
ions havebeenexperimentally determinedto be:

Atom E n e r g y( e V )

i H- - | 4.5C
2 He -78.98
3 Li* - 198.02
4 tse-' -J / t.5
5 B-3 - 599.3
/-r4 -aJat t.o

7 N*5 -1218.3
I n+6

Using our optimizedexpressionfor W, let"snow calculatethe estirnatedtotal

energiesof eachof theseatoms and lons as well as the percentageerror in our
estimatefor eachion.

Z Atom (eV) o/oError

E x p e r i m e n t a(le V ) C a l c u l a t e d

1 H - - 14.35 -12.86 10.38

2 H e -78.98 -77.46 t.Jz
? ti+ - 198.02 - 196.46 0.79
4 Be+2 -Jl t.5 -JOV.UO 0.44
5 B+3 _6qq ? -597.66 0.27
A a+4 -881.6 -879.86 0.19
7 N+5 -1218.3 -1216.48
g 0*6 - 1609.5 - 1607.46 u. t5

The energyerrors are essentiailyconstantover the range of Z, but producea

largerpercentageerror at small Z.
In 1928,whenquantummechanicswasquiteyoung,it wasnot knownwhether
the isolated gas-phase
hydride ion, H-, was stablewith respectto dissociation
into a hydrogenatom and an electron.Let's compareour estimatedtotal enersv
 P o i n tg r o u p s y m m e t r v
125

for H- to the ground-stateenergyof a hydrogenatom and an isolatedelectron

-
lrvhichis knownto be 13.60ev). when we useour expression for Ii andtake
Z - l. we obtain W : - 12.86eV which is greaterthan_
13.6eV (H * e- ), so
thissimpler''ariational calculationerroneously predictsH- to be unstable.More
cornplicated variational treatments givea groundstateenergyof H- of - 14.35eV
i n a g r e e m e nr vr i t he x p e r i m e n t .

4.2 Point group symmetry

It is assumedthat the readerhaspreviouslylearned.in undergra<iuate inorganic
or physicalchemistryclasses,how symmetryarisesin molecularshapesand
structuresand what symmetryelementsare (e.g.,planes,axesof rotation,cen-
tersof inversion,etc.). For the readerwho feels,after readingthis section.that
additionalbackgroundis needed.the texts by Eyring, walter. and Kimball or
by Atkins and Friedmancan be consulted.we review and teachhere only that
materialthat is of directapplicationto symmerryanalysisof molecularorbitals
and vibrationsand rotationsof molecules.we use a specificexample.the arn_
moniamolecule,to introduceand illustratethe importantaspects of point group
symmetry.

4 . 2 . 1T h e C 3 us l m m e t r y g r o u p o f a m m o n i a- a n e x a m p l e
The ammoniamoleculeNH.rbelongs.in its ground-state equilibriumgeometry,
to-thec;,. point group.lts symmetryoperationsconsistof two C-rrotations,c-i,
cr2(rotationsby 120 and 240'. respectivel.v, aboutan axispassingthroughthe
nltrogenatomandlyingperpendicular to theplaneformedby thethreehydrogen
atoms),threeverticalreffections, o,, o,l, oj', and the identityoperation.corre-
spondingto thesesix operations are symmetryelements:the three-fbldrotarron
arrs' C3 a rd the threesymmetryplanesou. ol and o,' rharcontainthe three
NH
bondsandthe :-axis (seeFig.4. I ).

Ammonia
m o l e c u l ea n d i t s
symmetryelements.
tzo Some important tools of theory

Thesesix symmetry operatioirsform a mathernaticalgroup. A group is defined

a s a s e l o f o b i e c t s s a t i s f v i n gf o u r p r o p e r t i e s .

(i) A combinationrule is definedthroughwhich trvogroupelementsarecombinedto

givea resultq'hichu'e call the product.The productofnvo elementsin the group
mustalsobe a memberof the group(i.e.,the grouprs closedunderthe
combinationrule).
(ii) One specialmemberof the group.when combinedu,ithan!'othermemberof the
group.mustleavethe groupmemberunchanged (i.e..the groupconlalnsan
identityelement).
(iii) Everv group membermust havea reciprocalin the group. When any'group
memberis combinedwith its reciprocal.the productis the rdentitlelement.
(iv) The associative
law musthold u'hencombiningthreegroupmenrbers (i.e.,(AB)C
mustequalA(BC)).

The members of symmetry groups are symmetry operations; the cornbination

rule is a successiveoperation. The identity element is the operation of doing noth-
ing at all. The group propenies can be demonstrated by forming a multiplication
table. Let us label the rows of the table by the first operation and the columns by
the second operation.Note that this order is important becausemost croups are
not commutative.The C3, group multiplication table is as follou's:

Secondoperation

E E Cr C: o\ oi oi'
L.l C s C 4 E o i o i o \
Cr' E C3 oi' o' oy
o\ o\ oi o, E C'3 C3
t' oi 6\ oi' C3 E C21
6, ol' oi o\ Ci C3 E
First
operation

Note the reflectionplanelabelsdo not move.That is, althoughwe startwith Hr

in the o., plane,H2 in oj', and H3 in oj', if H1 movesdue to the first symmetry
operation,o,. remainsfixed and a differentH atom lies in the o, plane.

4.2.2 Matricesas group representations

In usingsymmetryto help simplify molecularorbital (m.o.)or vibrationiroration
energylevel identifications,the following strategyis followed:

(i) A setof M objects

belonging to theconstituent
atoms(or molecular fragments,in a
moregeneralcase)is introduced.Theseobjectsaretheorbitalsofthe individual
 Point group symmetry
127

atoms(or of the fragments)in the m.o. case:they are unit vectorsalongthe -r, -v,
and: directionslocatedon eachofthe atoms,and represent displacementsalong
eachofthese directions.in the t,ibrationirotationcase.
(ii) Symmetrytools are usedto combinetheseM objectsinto M nervobjectseachof
u'hichbelongsto a specificsymmetryof the point group. Becausethe Hamiltonian
(electronicin the m.o. caseand vibrationi'rotationin the lattercase)commuteswith
the symmetryoperationsof the point group, the matrix representation of H within
the symmetryadaptedbasiswill be "block diagonal".That is. objectsof differenr
symmetrywill not interact;only interactionsamongthoseof the samesymmetry
needbe considered.

To illustrate such symmetry adaptation, consider symmetry adapting the 2s

orbital of N and the three I s orbitals ofthe three H atoms. we begin by determining
hou' these orbitals transform under the symmetrv operations of the c3u point
group. The act of each of the six symmetry operations on the four atomic orbitals
can be denoted as follows:

E
(.tN.
sr,.t,,sr) (sN.
sr.s.,s.)
C-r
+ (Sr. Sr..tr. S:)

C.r
e (Sr. S:, S,. Sr)

o\
+ (5\, Sr, Sr, S.)

o\
- ( . t \ . S j . S r . S r)

o\
- ( S N , . t 2 ,S r. S r ) (4.21)

Herewe are usingthe activeview that a c3 rorationrotatesthe moleculeby 120..

The equivalent passiveviervis thatthe ls basisfunctionsare rotated-120.. In
the C3 rotation,S3endsup where 51 began,51 endsup where 52 began,and 52
endsup where53 began.
Thesetransfbrmationscan be thought of in terms of a matrix multiplyrng a
vectorwith elements(Srr,Sr. S:. S:) For example,'tpt+t (C:) is the represen_
tation matrix giving the c3 rransformation,then the aboveaction of cr on the
fbur basisorbitalscan be expressedas

[r o o nl [r,l
D " 'I 'lc=,lli; : ; l i l i [s"l
[r*l
=l;i (4.22)

Ls,J Lo o I oJLs,j Ls,l

128 S o m e i m p o r t a n t o o l so f t h e o r v

we canlikewiservriternatrixrepresentations
for eachof the symmetryoperatlons
of the C.r, point group:

t-l : : ll [r o (' ol
, , ( c=:l):: ; ? l a , , , r r; :?l l: l
L o r o o J f o o o r j

,,.,,",):l:
; ; il
f r o o o l
,,,,n,,:ll
;:;l
f r o o o l

L o or o l L nr o o _ ]
^:1,,,f;:?:l
D,1)(o,,):lo
I o ol e.n)
L;o;;l
It is easyto verify that a Cr rotation foilowed by a o, reffection
is equi'alent to
a o] reffectionalone.In otherr.vords

sl Sr
o.Cr : o'. or. c3 d\
t4 )J\

S: .l Sr .t: S: Sl

Note that this samerelationshipis carriedby the matrices:

[r o o ol[r o o ol [r o o ol
D , o , 1 o , ;1 ;p?al,l6:I ,:: ;l l: : l ;
: i ;l
L o o I oJLo o r oJ [o r o o]
- Dt4)@).
(.251

Likewisewe canveri$, thatC3o,. : oj, directlyandwe cannotice

thatthematrices
also show the sameidentity:

[r o o ol[, o o ol [r o o ol
p , r ' { c r r o , o , r0o ,0l :'1l l9o I 0 ol-lo o | 0l
l o I o o l l oo o 'l-lo I o ol
L o o I o J L oo I oJ [o o o r]
Dt4)G:'). (4.26)

In fact, one finds that the six matrices,D6)(R),when

multipiied togetherin all
36 possibleways obey the same multiplication table
as did the six symmetry
 P o i n tg r o u p s y m m e t r y 129

operations.we say the matricesform a representationof the group becausethe

matriceshaveall the propertiesof the group.

Cha racte rs of rep resentati on s

one importantpropertyof a matrix is the sum of its diagonalelementswhich is
calledthe traceof the matrix D and is denotedTr(D):

Tr(D): ^. (4.27)
)o,,:
So,x is calledthe traceor characterof the matrix. In the aboveexamDle

x ( E ): a . (4.28)
x,c): x (c,'):1. (4.2e)
X@):x@)=X1o,,1:2. (1.30)
The importanceof the charactersof the symmetryoperationslies in the fact that
they do not dependon the specificbasisused to form them. That is, they are
invariantto a unitary or orthogonaltransformationofthe objectsusedto define
thematrices.As a result,they containinformationaboutthe symmetryoperation
itselfandaboutthesp./cespannedby the setofobjects.The significance ofthis
observationtbr our symmetryadaptationprocesswill becomeclearlater.
Note that the charactersof both rotationsare the same as are those of all
threereflections.Collectionsofoperationshavingidenticalcharacters arecalled
classes,Eachoperationin a c'lassof operationshas the samecharacteras other
membersofthe class.The characterofa classdependson the spacespanned by
thebasisof functionson which the symmetryoperations act.

Another basis and another representation

Above we used (S5, "91 , S:,.t3) as a basis. If, alternatively.we use the one_
dirnensionalbasisconsistingof the ls orbitalon the N atom,we obtaindiilbrent
characters.as we now demonstrate.
The act of the six symmetry operationson this Sy can be representedas
follo."vs:

E C ; Li
Sx --' SN. Sr'; + Sy. Sr --+ '!ru,
( 4 . r3)
o\ o, ov
Snr --+ Su. Sr - Sx, Sr.r- Sr.r.

we can representthis group of operationsin this basisby the one-dimensional

setof matrice-s:

D\' (E): l. D ( r ) 1 C , , 1l .: D ( ' ) ( C r r: ) I ,

G.32)
D l r ) ( o , . )- l, p r , 1 1 6 , , , )l:, D (r ) ( . r J-) l .
130 S o m ei m p o r t a n t o o l s o f t h e o r y

Againwe have

D ( t t 1 o , 7 D t t ) 1 C: - . )I . t - r t r t 1 o , " )
a n d D \ t ' 1 C . . 1 D ' t ' 1 o: , ) I . I : I ) ' r ) ( o , ' ) . r-1.-ljr
Thesesix matricesform anotherrepresentation of the -eroup.In this basis.each
characteris equal to unity. The representationforrned by,allo*'ing the six sym-
metry operatrons to act on the 1sN-atom orbitalis clearlynot thc sameas that
formedwhen the samesir operationsactedon the ( S^-. St. S: . 5i ) basis.We nou,
needto learnhou'to further analyzethe information contentof a specificrepre-
sentationof the group formed when the symmetryoperationsact on any specific
setof objects.

4 . 2 . 3R e d u c i b l ea n d i r r e d u c i b l er e p r e s e n t a t i o n s
A reducible representation
Note that everymatrix in the four-dimensionalgroup representation
labcledD(a)
hasthe so-calledblock diaconalforrn

3 x 3 matrix

This meansthat these D(a) matricesare really a combinationof two separate

group representations
(mathematically,it is called a direct sum representatlon.).
We saythat D(a)is reducibleinto a one-dimensional representationD(l)and a
three-dimensional
representationformed by the 3 x 3 submatricesthat we will
call DQ).

[t o' ol [o o rl [o I o-l
D ( 3 ) ( E )=
l 0 0 orl l . D'r)1C,;: o l , D . ( c: il )o o t l
I r oI 0-J
L0 L0 0 0l Ll
[ r oo 'o1l . lo o ll [o i ol
D,,,(o,.)-
l 0 r 0_.1 D13)1o];:
lot ol, D ( 3 ) ( c ' Jo ' o
) :l l l
L0 Lr o oJ [o o
,,0]ro,
T h e c h a r a c t e rosf D ( 3 ) a r eX ( E ) : 3 , X ( 2 C ) : 0 , X ( 3 o , ) : 1 . N o t e t h a tw e
would haveobtainedthis D(3)representation directly if we had originallychosen
 P o i n tg r o u p s y m m e t r y '131

roexaminethebasis(Sr. S:, 5l): alsonotethatthesecharacters

areequalto those
o f D l 1 ) m i n u st h o s eo f D ( l ) .

A change in basis
Now let us convertto a neu, basisthat is a linear combinationof the original
( S r .S : . S 3 )b a s i s :

Ir:Sr*S:+S:. (4.35)
l=2Sr-S:-S:. (4.36)
l:S:-Sr. (4.37)
(Don't worry abouthow.we constructTy. 72.and7l yet. As will be demonstrated
Iater.we form them by using symmetryprojectionoperatorsdefinedbelow:)we
determinehow the "T" basis functionsbehaveunder the group operationsby
allorvingthe operationsto act on the s, and interpretingthe resultsin terms of
the 4. In particular

( r r .l . n ) 3 t r , .n . _ n l . ( T tT. z . r ,77 g , . T . . r ; .
( T t .T ) .T l a f s , + . S :* ^ ! r , 2 . t r- S : - S r , S : - S r )
: (. - j ,. -3.r,,-
jn * jn),
( T t .T : . 7 ) a t S ,+ . ! r * 5 ' . .2 5 :- . S r - S : . . t r- S r )
: ( . - j r . + 3 r r , l nj ^. )
(Tt.T2.Tt)!tS,+Sr*S:. l!,-Sr -S:,.tr --t:)
= (n - j r. _ 3rr, -
Ir. )n)
(n, l. n)j ,t, * sr+ s,. 2s:- s, - s'r..tr- "tr)
: (n - j r.+ 3rr,.
-)r. - (1.i8)
)r.,)
So thematrixrepresentations
in the new.I basisare

. [r o ol
D'.'(E):10
[ t _o+ o . l
I 01. ao)1c,;:10 _ll
lo o ,J [o *j _;i
=fj i j*l D,,,(*):
,,,,,.,, [; o?-'J
:.l
-] -jl [o lo
ol
[ '
D ( ' ) ( o j ) : 1-0j - i I [r o ol
D , r , 1 o , , ) :_1+0 + + l (4.3e)
[o -l *j] Lo+i .J.l
132 S o m e i m p o r t a n t o o l so f t h e o r v

Reduction of the reducible representation

Thesesix matricescan be verifiedto multiplyjust as the symmetry
operations
do: thustheyform anotherthree-dimensional representation of thegroup.we see
that in the 7l basisthe matricesare block diagonal.This
meansthat rhe space
spannedby the f functions,which is the samespaceas the
s, span.forms a
reduciblerepresentation that can be decomposed into a one-climensional space
and a two-dimensional space(r,ia formationof the 7] functions).Note that the
c h a r a c t e rnsr a c e so) f t h e m a t n c e sa r en o t c h a n g c db y r h e
c h a n g ci n b a s e s .
The one-dimensionar part of the abovereduciblethree-dirnensionar represen_
tation is seento be the sameas the totally symmetricrepresentation
we ar.ved
at before,D( r). The two-dimensional representation that is reft can be shownto
be in'educ'ible;it hasthe follou,ing matrix represenrations:

D , , , ( E ) : ol tl . D ( 2 ' 1 c .:1
[ _ 1 _ :] tl
I ,t o( :) ( cl)
: tl
r
|
I
1
, j-l
r l

L0 ]j L r l '
- ;
- - t .
l r ; I l - l; t
l - i I
. [r ol T I
:
.t-t
l - - :1 _ r t
D ' - ' 1 o , ) -| | to,) ,:
2t(2t tI : I
I D ( 2 ) 1 o :i , 1| r l
LO -IJ ll - 1) r. l 1I | - t . r - 1l '
l . r l

(4.40)
The characterscan be obtainedby sun-rmingdiagonalelements:

x ( E :\ 2 x(2Ct\: -1.
x 1 3 oI, : g ( 4 . 4t )

Rotations as a basis
Another one-dimensional representationof the group can be obtainedby taking
rotation about the :-axis (the C3 axis) as the obiect on
ivhich the symmetry
operationsact:

c?
n--3 R,, n.-S R,, R_i
R,3 -R., R. 5 -R,, R . a- R " (4.42)
In writing theserelations,we use the fact that reflection
reversesthe senseof
a rotation. The matrix representarionscorrespondingto this one-dimensional
basisare

DtttE) - l, D ( r ) ( C 3 -) l , D o r ( C r 2 :) l .
D ( r ) ( o ":) - 1 , D(r)1o]= '; -1, D(r)1o;;: _1. (4.43)
Theseone-dimensional matricescan be shownto multipry togetherjust like the
symmetry operationsof the c3, group. They form an
irreduc,iblerepresentation
of the group (becauseit is one-dimensional,it can not be further reduced).Note
that this one-dimensionalrepresentationis not identical
to that found abovefor
the 1sN-atom orbital,or the | function.
 P o i n tg r o u p s y m m e t r v
133

Overview
\l'e ha'e found three distinct irreducibrerepresentations for the C;u slmmetr!
sroup;t$'o differentone-dimensional and one two-dimensional.ap..r.nrur,onr.
Arc'rhereanymore?An importanttheoremof grouptheoryshowsthatthenumber
of irreducible representations of a groupis equalto thenumberof classes.
Since
rhereare three classesof operation(i.e.. E. C3 and ou), we
have found alr the
irreduciblerepresentations of the C:u point group. Thereare no more.
Theirreduciblerepresentations havestandardnames;thefirst D( r)
lthat arrsing
from the 11and ls1' orbitals)is calledA1, the D( r) arisingfrom R,-is cailed
42
anilD1:)is ca'ed E (not to be confusedwith the rdentityoperation
t). we will
seeshortlywhereto find and identify thesenames.
Thus.our original D(a)representation was a combinationof two A1 represen-
tationsand one E representation. we say that D(a) is a direct ,u. ..pr...n,u_
tion:D(a):2Ar e E. A consequence is that the charactersof the cornbinatron
representation D(a) can be obtainedby adding the characters
of its constltuent
irreduciblerepresentations.

E 3o,

I I I
A. I I I
E -l 0

?A,a tr

How to decompose reducible representations

in general
supposevou were-eivenonly the characters (4.1.2). How canyou find out how
manytimesA1, E. and 42 appearwlienyou
reduceD(a)to its irrecrucible
parts,/
Yourvantto find a linearcombinationof the
charactersof A1. 42 andE thatadd
up (4.1,2). You can treatthe characters
ofmatricesas vectorsand takethe dot
productof ,,1r with Dt+t

1 E
I c]
[ r I r r r '] I
I : 4 + l - l * ' + r - ) - r ?
lEC. o. 2 6\
(4.44)
L '

2
)
T h e v e c t o r( l . l . l . r . l . r ) i s n o t n o r m a r i z e d ;
h e n c et o o b t a i nt h e c o m p o n e not f
{ 4 , 1 . 1 , 22, , 2 ) a l o n ga u n i tv e c t o irn t h e( 1 , 1 . 1 . 1 , 1 . 1 )
d i r e c t i o no,n em u s td i v i d e
by the norm of(l.l.l,l.r,l): this
n o r m i s 6 . T h e r e s u r ti s t h a t t h e r e d u c i b r e
representation containsr2/6 :2 A1 components. Anaiogousprojectionsin the
134 S o m e i m p o r t a n tt o o l so f t h e o r y

E ar.rdA.2directionsgive componentsof I and 0" respectively.

In seneral,to
determinethe number,?rof times irreduciblerepresentatior.r
I' appearsin the
reduciblerepresentationwith characters one calculates
Xred.
l

rt7: ! f Zrtn)x,..,r(R'). t 1 . . 1r5

q -

r"'hereg is the orderof the group and 1p(R) are the characters
of the fth irre-
duciblerepresentation.

Commonly used bases

Wecouldtakeanl setof functionsasa basisfor a grouprepresentation. Cornmonlv
used setsinclude:coordinates (r..t'.:) locatedon the atomsof a polvatomic
molecule(their symmetrytreatmentis equivalentto that involvedin treating
a set of p orbitalson the sarneatoms).quadraticfunctionssuch as d orbitals
-,r,r,, ,1,:.,r:, .r2 -.r'2. :2, as well as rotationsaboutthe ,r. and : axes.The
-i'
transformationpropertiesof thesevery comlnonly used basesare listed in the
charactertablesshownin the Aopendix.

Summary
The basicideaof symmetryanalysisis that any basisof orbitals.displacements.
rotations.etc. transformseither as one of the irreduciblerepresentations or as
a direct sum (reducible)representation.Symmetrytools are usedto first deter-
mine hou' the basistransformsunder action of the symmetry operations.They
are then usedto decomposethe resultantrepresentations into their irreducible
components.

4.2.4 Anotherexample
The2p orbitalsof nitrogen
For a function to transform accordingto a specificirreducible representation
meansthatthefunction,whenoperateduponby a point-groupsymmetryoperator,
yields a linear combinationof the functions that transform accordingto that
irreduciblerepresentation. For example,a 2p.-orbital(z is the C3 axis of NH3) on
the nitrogenatom belongsto the .A1representation becauseit yields unity times
itself when Ct, Cl, ou, o!, o,'.'orthe identity operationact on it. The factor of I
meansthat 2p, hasA1 symmetrysincethe characters (thenumberslistedopposite
.A1and below E, 2C3, and3ou in the C3ucharactertableshownin the Appendix)
of all six symmetryoperationsare I for the A1 irreduciblerepresentation.
The 2p_*and2p, orbitalson thenitrogenatomtransformasthe E representation
since C3, C?, o,, oi, oi' andthe identity operationmap2p, and 2p, amongone
 P o i n tg r o u p s y m m e t r y
135

another.SPecificallY.
F A

c ,l ' p .I = fcos120' - sin120'

L : P ' J Isin120 cos120"
t r l
l[il:] (.+.46)

E l ' p , l [r
= 2a0' - sin240
c' i l - p ' l [cos
L:P'J lsin 240' cos2210''
]

= o.l[zp.l
l[;;:] (4.47)

(4.18)
L2p.l lo'jLro_j
lzo I [- r o.l[,0,j
(4.4e)
L:P'J I o r]L:p,l
i *l +;l
T f

" ': l t P ' | : [:p,l (1.50)

L z p ' J L*: -i JLzp J
T f
*i --"1
l |
o' " l t P ' = lroI ( 4 . 5)1
L : P ' l L-+ -i JL:p J
The2 x 2 matrices.which indicatehow eachsymmetryoperationmaps2p. and
2p. intosomecombinations of 2p.,and2p,.,aretherepresentation matrices1Drrnr';
for thatparticularoperationandfor this particularirreduciblerepresentation
(lR).
Forexample.
T , I - 1 - 'Ti - l
I T. I
l - - l = D,t,{o, } (45)r
1 , , 5 - ; r I
l rr I

This set of matriceshave the samecharactersas the D(l) matricesobtainecl

earlierr'vhenthe 7] displacementvectorswereanalyzerlbut the individualmatrix
elementsare different becausewe useda ciifferentbasisset (here 2p.. and 2p,,,;
aboveit rvasI and 11).This illustratesthe invarianceofthe traceto the specific
representation:the traceonly dependson the spacespanneclnot on the specific
mannerin which it is spanned.

A short-cut
A short-cutdeviceexistsfor evaluatingthe traceofsuch representation
matrices
(thatis. fbr computingthe charaoers).The diagonalelements
of the representa-
tlon matncesare the proiectionsalong eachorbital of the effect
of the symme-
try operationacting on that orbital. For example,a diagonalelemenr
of the C3
matrix is the componentof Cs2p, along the 2p,, direction.More rigorously,
it is
I Z p :. C , 2 p , c. l t . T h u s .t h ec h a r a c i t e rtohfeC 3m a r r i xi sr h es u mo l J ' 2 p i Cj 2 p ,d r
and / 2pl C3 2p, dt . In general,the character of any ,y**"i.y
X operationS
canbe computedby allowing S to operateon eachorbital
@;,thenprojecting.!@i
along@;(i.e.,forming O: SO,rtt), andsummingrheseterms,
I

o:to Ltt: x(Sl ( 4 . 5 )3

\ |
136 S o m e i m p o r t a n tt o o l s o f t h e o r y

If theserulesareappliedto the2p. and2p,.orbitalsof nitrogen'"vithin

the Cr,.
p o i n tg r o u p .o r r eo b r a i n s

x(El:2. x ( C : ) : x ( C ; ): - t xto,t: x@il: rto,l:0. (4.54)

This setof characters is the sameas rrl) aboveand agreesu,ith thoseof the E
representation lor theCj..pointgroup.Hence.2p.,and2p,.belongto or transform
as the E representation. This is why (.r..1')is to the riglrtof the row of characters
for the E representation in the Cq, charactertable shou,nin the Appendix.In
similarfashion,the Cr, character table(pleasereferto this tablenow) statesthat
d,,-,,. and d., orbitalson nitrogentransformas E, as do d.,. and d,.--.but d-,
transformsas A1.
Earlier.u,econsidered in sornedetailhor.",the threeI sp1 orbitalson thehydrogen
atomstranslbrm.Repeating this analysisusingthe short-cutrulejust described
the traces(characters) of the 3 x 3 representatiorr rnltricesare cornputedby
a l l o w i n gE , 2 C t , a n d3 o , t o o p e r a t e
o n l s s , . l s p . . a n d 1 s s .a n dt h e nc o m p u t i n g
thecomponentof theresultingfunctionalongthe originalfunction.The resulting
c h a r a c t ear rsey . l E) = 3 . 2 { 6 . y : X t C i ) : 0 . a n d X ( 6 \ l : y t o , y : / l o , ' l :
l, in agreement u'ith whatwe calculated before.
Using the orthogonalityofcharacterstakenasvectors\\rccanreducethe above
setof characters to A1* E. Hence,we saythatour orbitalsetof threelss orbitals
forms a redLrcible representation consistingof the sum of A1 and E IRs. This
meansthat the tl.rreelss orbitals can be con.rbinedto yield one orbital of ,A1
symmetryand a pair that transfornraccordingto the E representation.

4 . 2 . 5P r o j e c t i o no p e r a t o r ss: y m m e t r y - a d a p t eldi n e a r
c o m b i n a t i o n so f a t o m i co r b i t a l s
To -eeneratethe aboveA'1and E symmetry-adapted orbitals,we make useof so-
called symmetryprojectionoperatorsP6 and Pa,. Theseoperatorsare given in
termsof linearcombinationsof productsof characterstimeselementarysymme-
trv oDerationsas follows:

^ s -
r', : )_ Xe(J15. r4 551
5'

P' . : tL " - y r ( , S ) S . (4.56)

.l

where,Srangesover C3,C! . o,,. o, ando| and the identity operation.The result

o f a p p l y i n gP a , t o s a yl s s , i s

P a ,l s s , : l s H , * l s 6 , * l s s , * l s p . 1 l s s , * l s s ,
- 2 ( l s y 1*, l s H .+ 1 s " , )- ( 45 7 )
dn'.
F
F:
P o i n tg r o u p s y m m e r r y
t5t

r'
i
s.hichis an (unnormalized)orbital havingAl symmetry.Clearly,this same@a,
by Ps, actingon lsg, or 1s11..
rYouldbe -generated Hence,only one Al orbital
Likewise,
exists.
P p l s 1-12, x l s g , - l S r r : - l s u , : d e . r . (4.5g)
*hich is one of the symmetry-adapted
orbitals havin_eE symmetry.The other E
orbitalcanbe obtainedby allowingpE to act on lsH, or lss.:
P s l s p , - 2 . l s g ,- l s s , - l s s . :
@E.;. (4.59)
P E 1 s 1 1=. 2 . l s l r , - l s H r l s u , = (4.60)
de.:.
It might seemas though three orbitalshaving E symrnetrywere generated.but
only two of theseare really independentfunctions.For example,
@E3 is related
to @r-r and dr : as follows:

Qt.::-ltr.t*Qr.z). ( 4 . 6r )
Thus,onry /s 1 and Q2.2afeneededto spanthe two-tlimensionalspace
of the E
representation. If we include@p. 1 in our setof orbitalsandrequireour orbitalsto
beorthogonal, thenwe mustfindnumbersa andb suchthat : aQp.:f
@i b@g.3 is
orthogonal to Qv.1:.f 0o0o.ttlt :0. A straightforwarcl
givesa : _b
calculation
or Qu: a( lss. - lsH,) which agreeswith what we usedearlier
to constructthe
?l functionsin terms of the Si functions.

4 . 2 . 6S u m m a r y
Letusnow summarizewhatwe havelearned.Any given
setofatomic orbitars{@;}.
atom-centered displacements or rotationscanbe usedasa basisfbr thesymmetry
operations of the point group of the molecule.The characters
X(S) bllung,ng
to the operationsS of this point group within any
suchspacecan be founclby
summingthe inte-urals o; so, rlt overall the atomic
I orbitals(or corresponding
unit vectoratomicdisplacements). The resultantcharactersrvill. in ,9eneral,
be
reducibleto a combinationof the characters of the irreduciblereprJsentations
xi(5'). To decomposethe characters x(s) of the reduciblerepresentation to a
sumof characters X;(S) of the irreciucible
representatron

x(S):Ir,x,(S). (4.62)

tt is necessary
to determinehow manytimes,n;. thei th irreducible
representatron
occursln the reduciblerepresentation.
The expression for n; is

,,=:I.x(s)xi(s), (.+.63
)
tn rvhich g is the order of thc point group -
the total number of symmetry
operatrons in the group(e.g.,g: 6 fbr C:").
138 S o m e i m p o r t a n t o o l so f t h e o r y

Forexample.thereduciblerepresentationy,(E ) : 3. / ( (': ) : 0"andXlo, ) :

I formedby thethreelss orbitalsdiscussed
abovecanbedecornposed asfollou's:

I
r ^ : - ( i . l + 2 . 0 . 1 + l . 'l l ) : l . (4.64)
l
/ r A :r .l '(-l)):0 (.1.6-51
6(3.1+2.0.1+-l
I
i r ,= - 1 1 . 1 - 1 . ( l . r l-r + 1 . l . { ) ) : I (.:1.66)

Theseequations orbitalscanbe combinedto give oneA1

statethatthe three 1s11
orbitaland sinceE is de-generate.
one pair abovc.With
of E orbitals.asestablished
knolvledseof the n;, the symmetrl'-adapted orbitalscanbe formedby allou'ing
the projectors

/,,: I x;(s)s ( 1 . 6 7|

to operateon eachof the primitiveatomicorbitals.Hou'this is carriedout was

illustratedfor the lsu orbitalsin our earlierdiscussion.Thesetools allow a
symmetrydecomposition of anv setof aton.ricorbitalsinto appropriatesylnmetry-
adaptedorbitals.
Beforeconsidering otherconceptsand group-theoretical machinery. it should
once againbe stressed that thesesar.ne tools can be usedin symtnetryanall'-
sis of the translational. vibrationaland rotationalmotionsof a molecule.The
tweivemotior.rs of NH: (threetranslatior.rs, threerotations,six vibratior.rs)can be
describedin termsof combinations of displacements of eachof the four atoms
in eachof three(.r,_r,. --1directions.Hence.unit vectorsplacedon eachatom
directedin the x.-r'. and: directionsform a basisfor actionby the operations
{S} of the point group.In the caseof NHr, the characters of the resultantl2 x
I2 representation matricesform a reduciblerepresentation in the C2..point group:
X@):12. X€): X ( C : r :) 0 . X ( o ,) : X @ , \ : X @ l ' ) : 2 . F o r e x a m p l e u n -
der o.,.the H2 and H3 atomsare interchangeclso unit vectorson eitherone will
not contributeto the trace.Unit:-vectors on N and H1 remainunchangedas well
asthe correspondingt,-r,'ectors. However,the x-vectorson N and H1 arereversed
in sign. The total characterfor o] of the H2 and H3 atomsare interchangedso
unit vectorson either one will not contributeto the trace.Unit ;-vectors on N
and H1 remainunchangedas well as the correspondingl'-vectors.However,the
r-vectors on N and H1 0r€ reversedin sign. The total characterfor o, is thus
4 - 2 :2. This representation canbe decomposed as follows:

I
n n ,= .1.12+2.1.0+3'1.2]=3. (4.68
)
6[1
I
no=
, '1.122
+.1.0+3.(-1).2]:1. (4.6e)
A[1
I
n E= 1 2+ 2 . ( . - l ) . 0 + 3 . 0 . 2 1 : 4 . (4.70)
6t1.2.
 P o i n tg r o u p s y m m e t r v
139

Fromtheinformationon therightsideof thec-i,.character table.translations of all

fburatomsin the:. -vand-r,directions transformasA 1(:) andE(.v.-r.,), respectively.
* h c r e a sr o t a t l o n a
s b o u tt h e : ( R - - ) . . r ( R , )a. n dr - ( R , ) a . r e st r a n s f b r m
asA: and
E. l{ence.of thetrvelvemotions.threetranslations haveA1 and E symmetryand
threerotationshaveA2 and E symmetry.This leal'essix vibrations,of r.vhichtu,o
ha'e 41 svmmetrv.none haveAl symmetry.ancltwo (pairs) haveE symmery.
\!'e couldobtainsymmeffy-adapted'ibrationar and rotationalbasesbyallowing
svlnnlctrvprojectionoperatorsof the irredrrciblerepresentation svlrmetriesto
operateon variouselementarycartesian(r.,r.:) atomicdisplacement vectors.

4 . 2 . 7D i r e c tp r o d u c tr e p r e s e n t a t i o n s
Directproductsin N-electronwavefunctions
we now turn to the syrnmetryanalysisof orbital products.Such
knowledgeis
importantbecauseone is routinelyfacedwith constructingsl,mmetry_adapted
N-electronconfigurations that consistof productsof N individualspinorbitars,
onefor eachelectron. A point-groupsvrnmetryoperators. whenactingon strch
a
productof orbitals.gir,estheproductof s actin,eon eachof theindividualorbrtals
S ( t b t r b z e.)..d r ) = ( S d r) ( " t d :X . t @): . ( . ! @ r ( 4 . 7| )
)
For example'reflectionof an ,v-orbitalprotructthrou_sh the ou pranein NH-q
appliesthe reflqctionoperationto all ,V electr.ons.
Justas the individLrar orbitaisformed a basist.r actionof the poinr--rroup
operators, the configr-rrations (.V_orbital prociucts) fbrm a basisfor the actronof
thesesamepoint--eroup operotors.Hence.the variouselectr.nicconfiguratrons
canbe treatedastirnctions on rvhichs operates. andthemachineryillustrated ear_
lier for decornposing orbitarsymmetrycanthenbe usedto carrvout a svmnletry
analysisof confi-gLrrations.
Anothershorr-cutmakesthis taskeasier.Sincethe synrmetry-adapted
indi-
t i d u a l o r b i t asl { Q i .i - l . . . . . , t 1 }t r a n s f o r ranc c o r d i n g
t o i r r e d u c i b l ree p r e s e n -
tations,the representation r.natrices for the .v-termproductsshor.vn aboveconsist
of productsof the rnatricesberon-uing to cach@;.This rnatrixproductrs not a
simpleproductbut what is calleda directproduct.To
computethe characters of
thedirectprodtrctmatrices.onernultipliesthe
characters of the i6dividLral matri-
cesof the irreduciblerepresentations of'the,\ orbitalsthatappearin theelectron
configuration. The direct-product representation fbrmedb1,theorbitalproducts
cantherefbrebe s.u-rnmetry analyzed(reduced)usingthe sametoolsas we used
earlier.
Forerample.if oneis interested i. knowingthesvmmetryof an orbitalproduct
of the lorm alaier lnote: lower caseletters
are useclto denotethe symmetry
of electronicorbitals.whereascapitallerters
are reservedto label the o'crall
confi-quration's symmetry)in Clv symmetrl,.11.,. fbllowing procedureis used.For
140 S o m e i m p o r t a n t o o l so f t h e o r v

eachof the six symmetryoperationsin the C2, point group.

tr"te
ptrxruct of the
characters associated
with eachof the.rrrspinorbitals(orbitarmultipriedby a
or
p spin)is formed:

x ( s ) = f l x i ( S y= ( x . r , r S ) () xr n . ( . ! ) () x: r ( s ) ) : . (.72)

I n t h e s p e c i f i c a s ec o n s i d e r ehde r e ,
X ( E ) : 4 . X e C t ) : l . a n dy ( 3 o , ) : 6 .
Notice that the contributionsof any doubiv occupiednon_degenerate
orbitals
(e.g..ai a'<1aj) to thesedirectproducrchara*ers
x(s) areur.ritybecauseror a//
operators(xr(s))2 : I fbr any one-di'ensio'al irreducible
represenratron. As
a result,only the singly occupiedor degenerate orbiralsneedto be consrdered
whenfbrnrinqthecharacters ofthe reducibledirect-product representarion x (J ).
Forthis examplethisnreansthatthe direct-procluct characters canbe dererrrrrned
from the characters x6(s) of the two active(i.e..non-crosed-shell) orbitars_ the
e 2o r b i t a l sT. h a ti s . X ( ^ g ): X r . ( S ).
Xn(S).
Fror' the direct-product characters x(s) belongingto a particularelectronic
configuration te.g..a]aie:;. onemusrstill deconposethis list of characters into a
sumof irreducible characters. Fortheexampleat hand.thedirect-pr.ductcharac_
ters1(s)decomposeint. oneA1, ore 42. anclore E representation.
This means
that the e2 configuration containsAr. A:. and E symnrerryeiements.projection
operatorsanalo-qous to thoseintroducedearlier.fororbitalscan be usedto
lbrnr
slimmetry-adapted orbital productsfrom the individualbasisorbital products
of the fornr afaiell'ei'. *'here ,r and,nr' denotethe occupation(
l or 0) of'the
trvo degenerate orbitalse,, and e, . when dealingu,ith indistinguishabre
parrr-
clessuchas electrons. it is alsonecessary to furtherprojectthe resultingorbital
productsto makethem a'tisymmetric (for fernrrons)
or sy'metric (for bosons)
with respectto interchangeof any pair of particles.This
stepreducesthe set of
'v-electronstatesthat can arise.For example,in the above
e2configurationcase.
o n l y 3 A 2 ,l A r , a n d l E s t a t e sa r i s e t; h e t E , t A , . a n d r 4 2 p o s s i b i i i t - i e s
disappear
when the antisvmmetryprojector is applied.In contrast,
for an ele'l configura-
tion, all statesariseevenafter the wave functio' has
beenrnadeantisymmetric.
The stepsinvolvedin combiningthe point-groupsymmetry
rvith permutationar
antisynrmetryare illustratedin chapter l0 of my
eMIC text.In Appendix III
of Electronic Spectra and Electr.onic Structure of pol),atornic
Molecules,
G. Herzberg,Van NostrandReinholdCo., Ner.r,york,
N.y. (1966),the resolu_
tion ofdirect productsarnong'arious represenrations u,ithin 'rany point grcups
are tabulated.

Direct products in selection rules

Two statestlroand1Lbthatareeigenfunctionsof a
HamiltonianH{rin the absence
of someexternalperturbation(e.g.,erectromagnetic
field or staticerectricfierdor
potentialdueto surroundingligands)can "coupred"
be by theperturbationv only
 P o i n tg r o u p s y m m e t r y 141

if the symmetriesof v and of the two wave functionsobey a so-calledselectron

rule.In particular,only ifthe couplingintegral

,/':v,L,a,: r,.n (4.73)

[
is non-vanishing wiil the two statesbe coupledby V.
The role of symrnetryin determiningwhethersuchintegrarsarenon-zerocan
bedemonstrated by notingthatthe integrandconsideredasa whole,mustcontain
a componentthat is invariantunder all of the group operations(i.e.,belongsro
thetotally symmetricrepresentation of the group) if the integralis to not vanish.
In termsof the projectorsintroducedabovewe must have

I sl.r.Yg1,
x.,,rsl (4.74)

not vanish.Here the subscriptI denotesthe totally symmetricrepresentation of

whateverpoint groupapplies.The symmetryof theproduct,lrivrltt is. according
to what was coveredearlier.given by the direct product of the symmetriesof
'ltj of Y and of /a. So. the conclusionis that the integralwill vanishunlessthis
tripledirectproductcontains,whenit is reducedto its irreduciblecomponents, a
componentof the totally symmetricrepresentation.
To seehorvthisresultis usedconsidertheintegralthatarisesin formulatingthe
interactionof electromagnetic radiationwith a moleculewithintheelectric-dinole
approximation:

1,t,,:,*,,,, (4.7sl

Here,r is the vector-riving.togetherwith e, the unit charge.the quantumme-

chanicaldipolemomentoperator

.:of ZnR,-nI.,. (4.76)

where2,, and R,, are the chargeand position of the nth nucleusand r, is the
positionof the7th electron.Now,considerevaluating this inte-eral
fbr the singlet
n "->n* transitionin fbrmaldehyde. Here.the closed-shell groundstateis of I A 1
symmetryand the singletexcitedstate,rvhichinvolvespromotingan electron
from thenon-bonding b2 lonepair orbitalon the oxygeninto thez*b1 orbitalon
theCO moiety,is of rAz symmetry(b1x b: : a:). The directproductof tnet,rvo
rvavefunctionsvmmetriesthuscontainsonly a2symmetry.The threecomponents
(,r.1,.and;) of the dipoleoperatorhave.respectively. b1,b2,anda1 stmmetr1l.
Thus'the tripledirectproductsgive riseto the fbllorvingpossibilities:

a 2 X b l : f . (4.77)
a2xbl:!1 (4.78)
a : X a t = a l (4.7e)
1At S o m e i m p o r t a n t o o l so f t h e o r Y

Thereis no componentof a1 svmmetryin the triple directproduct,so the integral

vanishes. This allou'sus to concludethatthen -'- 7t*excitationin formaldehyde
is electricdipoleforbidden.

4 . 2 . 8O v e r v i e w
We have shown hou,'to make a symmetrv decompositionof a basis of atornic
orbitals(or Cartesiandisplacements or orbital products)into irreduciblerep-
resentationcompouents.This tool is ver,vhelpful when studying spectroscopy
and when constructingthe orbital correlatioudiagrar.ns that form the basis of
the Woodward-Hoffnrannrules.We also learnedhor"'to form the direct-product
symmetries thatarisewhenconsideringconfiguratiotls consistingof productsof
symmetry-adapted spinorbitals.Finally.u,elearned directproductanaly-
hou'tl.re
sisallowsoneto determineu'hetheror not integralsof productsof u'avefunctrons
with operatorsbetweenthem vanish.This tool is of utmost importancein deter-
mining selectionrulesin spectroscopy and for determiningthe effectsof external
perturbations on the statesofthe speciesunderinvestigation.