36 - university of california, san diegotroglerlab.ucsd.edu/grouptheory224/chap2a.pdfh bond...

36

CHAPTER 2 - APPLICATIONS OF GROUP THEORY

2.1 How Group Theory Applies to a Variety of Chemical ProblemsThe classification of molecules according to their symmetry point groups, provides a

rigorous method for predicting optical activities. For a molecule to exhibit optical activity, it mustbelong to a point group that does not possess an inversion center, mirror plane, or improperrotation axis. The possibility of racemic mixtures must, of course, be considered. Molecules thatbelong to point groups, such as C4 and other pure rotation groups, can exhibit optical activity ifresolved into one optical isomer.

Another helpful symmetry rule in the analysis of diastereomeric protons in NMRspectroscopy is that chemically equivalent atoms (and hence protons with equivalent chemicalshifts) must be interchanged by a symmetry operation of the point group. For example, in Figure2.1 the two protons Ha and Hb in structure A are equivalent because they are interchanged by the

BA

Figure 2.1 Illustration of a structure, A, where Ha andHb are interchanged by a symmetry operation of the pointgroup. This would give rise to a single peak in the NMRspectrum. In structure B, where no such operation exists,Ha and Hb would give rise to separate peaks in the protonNMR spectrum.

C

C

CH3

Cl

H

C

HbHa

CH3

Ha

C

H5C6

H

Cl

CH3

Hb

mirror plane operation, which contains the C-C bond in the plane of the paper and liesperpendicular to the plane of the page. It is important to recognize that rotation around the C-Csingle bond does not interchange Ha and Hb. The conformation produced by this bond rotation isnot equivalent to the original one. The Cl substituent now lies above the plane of the paper and the

37

molecule resides in a different rotomeric configuration (eclipsed vs. the original staggered). Onlythe mirror plane operation interchanges Ha and Hb to produce a configuration indistinguishablefrom the original one. In structure B of Figure 2.1 the substitution of a phenyl group for a methylgroup destroys the mirror plane operation that interchanges Ha and Hb. Now Ha and Hb are nolonger equivalent and they give rise to separate peaks in the proton NMR spectrum. Furtherexamples will appear in Chapter 7 that show the power of this approach for the analysis of chemicalshift nonequivalence in complex structures.

The preceding examples concern straightforward conclusions derived from a considerationof symmetry operations and equivalent configurations; however, the real power of group theoreticalmethods results from their application to equivalent functions in a molecule. Just as group theorycan categorize equivalent atoms in a structure, it can also categorize equivalent functions. Themain applications concern electronic and vibrational wavefunctions in molecules. For example, thetwo 1s atomic functions on hydrogen atoms Ha and Hb are equivalent in Figure 2.1 A. The two C-H bond stretches (vibrational wavefunctions) to these atoms are also equivalent functions. Since allof spectroscopy involves the transition between two states characterized by wavefunctions of onekind or another, it is possible to apply group theory widely. This chapter describes how a set of nequivalent functions can be rearranged into a set of n linear combinations that take advantage ofmolecular point group symmetry. This requires knowledge about group representations. The payoffwill be that this permits the prediction of spectra, selection rules, and molecular orbital diagramswithout the need for detailed quantum mechanical calculations.

2.2 Matrix Representations of Symmetry Groups

Symmetry operations R acting on the point (x,y,z) are defined generally in eqn. 2.2.1.R(x,y,z) (x1, y 1, z 1) (2.2.1)^

Because R preserves the size and shape of objects, it satisfies the requirements for a linearoperator shown in eqn 2.2.2 and 2.2.3. Therefore, it is natural to apply the matrix methods of linearalgebra in the description of symmetry operations.

R(x1, bx2, y 1 + by2, y3 + by3) =

R(ax, ay, az) = aR(x, y, z)^ ^

^

(2.2.2)

(2.2.3)^R(x1, y 1, z 1) + bR(x2, y 2, z 2)

R (x1 + bx2, y1 + by2, z1 + bz2) = R (x1, y1, z1) + bR (x2, y2, z2) (2.2.3)

The action of R on a vector a = axi- + ay-j + az

-k (i-, -j , -k are the usual unit vectors for a

right-handed orthogonal coordinate system) can be represented as in eqn 2.2.4.

38

(2.2.4)-^-b = RaBecause R is a linear operator, eqn 2.2.4 can be written as 2.2.5.

^^^ ------ (2.2.5)bxi + byj + bzk = axRi + ayRj + azRk

The length of b must also equal the length of a (˙-b˙ = ˙ -a˙ ) for a linear operator. The symmetry

transformed vectors Ri-, R-j and R-k are new unit vectors in three dimensional space. They can be

expressed as some linear combination of i-, -j , -k , which are basis vectors for this space.

Rj-

-Ri^

=

=

=

r11i-

-r12i

-r13i

-r21j

-r22j

-r23j

-r31k

-r32k

-r33k

+

+

+

+

+

+

(2.2.6)

Rk-

Substituting in eqn 2.2.5 above and collecting terms yields eqn 2.2.7.

(2.2.7)

+ (axr31 + ayr32 + azr33)k

bxi + byj + bzk = (axr11 + ayr12 + azr13)i + (axr21 + ayr22 + azr23)j- - - --

-

Equating coefficients of i-, -j, and -k yields the set of equations of 2.2.8.

bx = axr11 + ayr12 + azr13 by = axr21 + ayr22 + azr23 (2.2.8)

bz = axr31 + ayr32 + azr33In matrix notation, the set of equations

-b = R -a can be written as shown in 2.2.9. Remember, to

multiply a matrix times a column vector one multiplies each matrix row times the column.

bx

by

bz

=

r11 r12 r13

r21 r22 r23

r31 r32 r33

ax

ay

az

(2.2.9)

The 3 x 3 matrix is called the matrix representation of the linear transformation R.Consider the specific example of a four-fold rotation around the z axis. The effect of this

operation on x,y,z is given by eqn 2.2.10.

C4(z) (x,y,z) (-y,x,z) (2.2.10)^

39

A point also defines a vector from the origin of the coordinate system. The action of thecounterclockwise C4(z) rotation on the vector -a = (x,y,z) therefore yields

-b = (-y,x,z), which can be

pictured as follows:

bx = -y = 0x + (-1)y

by = x = 1x + 0y

bz = z = 0x + 0y

+ 0z

+ 0z

+ 1z

(2.2.11)

y

x(-y,x,z) (x,yz)•

•

Thus eqn 2.2.10 can be abbreviated in matrix form as 2.2.12.

(2.2.12)

z

y

x 0 -1 0

=

z

x

-y

1 0 0

0 0 1

The matrix form of the operator C4(z) is the matrix shown in eqn 2.2.13.

0 0 1

1 0 0

0 -1 0

(2.2.13)C4(z) =

It can be shown that matrices representing symmetry operations are real and orthogonal. Therefore,the transpose of matrix R gives the inverse matrix (R-1 = transpose of R). Recall that the transposeof a matrix is constructed by interchanging corresponding elements across the diagonal of thematrix (rij Æ rji). It is important to remember that the matrix for a coordinate axis transformation isthe inverse of the corresponding transformation for the point (x,y,z). For example, counterclockwiserotation of the coordinate system by C4(z) leads is depicted below.

j

i•

i´

j´•

The reason for the inverse relationship between the matrices of coordinate axes and points is easy tovisualize. Rotation of a coordinate system counterclockwise produces the same effect on a

40

stationary point, from the reference frame of the coordinate system, as if the "stationary point" wererotated clockwise (the inverse transformation).

Consider the matrix representations for common symmetry operations. Counterclockwiserotation (Cn) about the z axis by angle a requires a computation of the rotated unit vectors i´ and j´in terms of their projections on i and j. Simple trigonometry yields:

-

i cos a + j sin a -

-

--

-

---

- -i sin a + j cos a =

=

k´ = k

j´

i´=

==Cn(z)k

Cn(z)j =

Cn(z)i

i

i´j

j-^

^

^

- -

- -

Therefore we can express Cn(z), where a = 360/n, as in eqn 2.2.14

C360/a(z)^

-

-

-i

j

k

-

-1

0

0

00

cos a

sin a

-sin a

cos a

-

=

k

j

i

(2.2.14)

The transformation of the point or vector (x,y,z) uses the inverse of the transformation matrix forthe coordinate system, which is just the transpose (interchange elements off the diagonal) of thepreceding matrix.

z

y

xx

y

z

=

cos a

sin a

-sin a

cos a

0 0

0

0

1

(2.2.15)C360/a(z)

For the corresponding operations Sn, where the Sn axis lies along z, reflection in the x,y planeinverts z (i.e. z' = -z). The matrix for corresponding S360/a operations are the same, except r33 =-1. For sv in a plane containing z and making an angle b with the x axis, sv´ is related to sv(xz) bythe similarity transformation that involves Cb.

41

sv´

y

x

sv´ = Cb-1 s(xz) Cb

^ ^ ^ ^

(2.2.16)

sv´(b) =

cos b

sin b

-sin b

cos b

0 0

0

0

1 1

0

0

0

-1

0

0

1

0 1

0

0

0

cos b

sin b

-sin b

cos b

0

(2.2.17)

b

(2.2.18)

0

cos b

sin b

sin b

-cos b

0

0

0

11

0

0

00

cos b

-sin b

sin b

cos b

=

(cos2 b - sin2 b) 2 cos b (sin b) 0

0sin2 b - cos2 b2 sin b (cos2 b)

0 0 1

= (2.2.19)

(cos2 b - sin2 b ) 2 cosb sinb( ) 02sinb cosb( ) (sin2 b - cos2 b ) 0

0 0 1

Ê

Ë

Á Á

ˆ

¯

˜ ˜

2.2.19( )

But sin 2a = 2 sin a cos a and cos 2a = cos2 a - sin2 a. This leads to eqn 2.2.20, when themirror plane contains z and makes an angle b with the x axis.

sv

1

0

0

00

-cos 2b

sin 2b

sin 2b

cos 2b

= (2.2.20)

2.3 Character Tables and Symmetry Group Representations

42

We have defined a symmetry operation R acting on the point x, y, z and transforming thepoint to some new equivalent location x',y',z' as in eqn 2.3.1.

currentpoint 192837465

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0 aR aL m 180 aA s 180 aA a arc cp fill}b/OA{np0 cw -2 dv mv aL 0 aR aL m 180 aA s 180 arc 0 cw -2 dv rlineto cp fill}b/SA{aF m lW m/aL x aL 1 aR s m np 0 p mv rad 0 p l gs cm sm st gr}b/CA{aF lW m/aL x aL 1 aR s m 2 dv rad dp m o dp m s dp 0 le{pppp pp}{sqrt at 2 m np rad 0 rad 180 6 -1 r s 180 6 -1 r s arc gs cm sm st gr cpt e at ro}ie}b/AA{np rad 0 rad 180 180 6 -1 r a arc gs cm sm st gr}b/RA{lW m/w x np rad w p mv w w p l rad w n p mv w w n p l w 2 m dp p mv 0 0 pl w 2 m dp n p l st}b/HA{lW m/w x np 0 0 p mv w 2 m dp p l w 2 m w p l rad w p l rad w n p l w 2 m w n p l w 2 m dp n p l cp st}b/Ar1{gs 5 1 r 3 ix 3 ix xl 3 -1 r s 3 1 r e s o o at ro dp m e dp m a sqrt/rad x[{2.25 SA DA}{1.5 SA DA}{1SA DA}{cw 5 m sl 3.375 SA DA}{cw 5 m sl 2.25 SA DA}{cw 5 m sl 1.5 SA DA}{270 CA DA}{180 CA DA}{120 CA DA}{90 CA DA}{3 RA}{3 HA}{1 -1 sc 270 CA DA}{1 -1 sc 180 CA DA}{1 -1 sc 120 CA DA}{1 -1 sc 90 CA DA}{6RA}{6 HA}{dL 2.25 SA DA}{dL 1.5 SA DA}{dL 1 SA DA}{2.25 SA OA}{1.5 SA OA}{1 SA OA}{1 -1 sc 2.25 SA OA}{1 -1 sc 1.5 SA OA}{1 -1 sc 1 SA OA}{270 CA OA}{180 CA OA}{120 CA OA}{90 CA OA}{1 -1 sc 270 CA OA}{1-1 sc 180 CA OA}{1 -1 sc 120 CA OA}{1 -1 sc 90 CA OA}{1 -1 sc 270 AA}{1 -1 sc 180 AA}{1 -1 sc 120 AA}{1 -1 sc 90 AA}]e g exec gr}b/ac{arcto 4{pp}repeat}b/pA 32 d/rO{4 lW m}b/Ac{0 0 px dp m py dp m a sqrt 0 360 arc cm sm gs sg fill grst}b/OrA{py px at ro px dp m py dp m a sqrt dp rev{neg}if sc}b/Ov{OrA 1 0.4 sc 0 0 1 0 360 arc cm sm gs sg fill gr st}b/Asc{OrA 1 27 dv dp sc}b/LB{9 -6 mv 21 -10 27 -8 27 0 cv 27 8 21 10 9 6 cv -3 2 -3 -2 9 -6 cv cp}b/DLB{0 0 mv -4.8 4.8 l-8 8 -9.6 12 -9.6 16.8 cv -9.6 21.6 -8 24.6 -4.8 25.8 cv -1.6 27 1.6 27 4.8 25.8 cv 8 24.6 9.6 21.6 9.6 16.8 cv 9.6 12 8 8 4.8 4.8 cv cp}b/ZLB{LB}b/Ar{dp 39 lt{Ar1}{gs 5 1 r o o xl 3 -1 r e s 3 1 r s e o 0 lt o 0 lt ne/rev xdp 0 lt{1 -1 sc neg}if/py x dp 0 lt{-1 1 sc neg}if/px x np[{py 16 div dup 2 S lt{pp 2 S}if/lp x lp 0 p mv 0 0 p l 0 py p l lp py p l px lp s 0 p mv px 0 p l px py p l px lp s py p l cm sm st}{py 16 div dup 2 S lt{pp 2 S}if/lp x lp 0 p mv 0 0 0 py lp ac0 py 2 dv lp neg o lp ac 0 py 2 dv 0 py lp ac 0 py lp py lp ac px lp s 0 p mv px 0 px py lp ac px py 2 dv px lp a o lp ac px py 2 dv px py lp ac px py px lp s py lp ac cm sm st}{py dp 2 dv py 180 pA s 180 pA a arc st np px py s py 2 dvpy pA dp neg arcn st}{0 0 p mv 0 py p l px py p l px 0 p l cp cm sm st}{px lW 2 dv a lW -2 dv p mv rO dp rlineto px lW 2 dv a rO a py lW 2 dv a rO a p l rO lW -2 dv a py lW 2 dv a rO a p l lW -2 dv py lW 2 dv a p l 0 py p l px py p l px 0 p l cp fill0 0 p mv 0 py p l px py p l px 0 p l cp cm sm st}{0 rO p mv 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac 0 0 0 py rO ac cp cm sm st}{rO py p mv rO rO xl 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac rO neg dp xl px py 0 py rO accp fill 0 rO p mv 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac 0 0 0 py rO ac cp st}{1.0 Ac}{0.5 Ac}{1.0 Ov}{0.5 Ov}{Asc LB gs 1 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr gs cm sm st grnp -1 -1 sc LB gs 1 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr gs cm sm st gr np -0.4 -0.4 sc LB gs 1 sg fill gr cm sm st}{Asc LB gs 1 sg fill gr gs cm sm st gr np -0.4 -0.4 dp sc LB gs 0.5 sg fill gr cm sm st}{Asc DLB -1 -1 sc DLB gs 1 sg fill grgs cm sm st gr np 90 ro DLB -1 -1 sc DLB gs 0.5 sg fill gr cm sm st}{Asc gs -1 -1 sc ZLB gs 1 sg fill gr cm sm st gr gs 0.3 1 sc 0 0 12 0 360 arc gs 0.5 sg fill gr cm sm st gr ZLB gs 1 sg fill gr cm sm st}{Asc gs -1 -1 sc ZLB gs 0.5 sgfill gr cm sm st gr gs 0.3 1 sc 0 0 12 0 360 arc gs 1 sg fill gr cm sm st gr ZLB gs 0.5 sg fill gr cm sm st}{0 0 p mv px py p l cm sm st}{gs bW 0 ne{bW}{5 lW m}ie sl 0 0 p mv px py p l cm sm st gr}{gs dL 0 0 p mv px py p l cm sm st gr}{OrA 1 16 dv dp sc0 1 p mv 0 0 1 0 1 ac 8 0 8 -1 1 ac 8 0 16 0 1 ac 16 0 16 1 1 ac cm sm st}]e 39 s g exec gr}ie}b/Cr{0 360 np arc st}b/DS{np p mv p l st}b/DD{gs dL DS gr}b/DB{gs 12 OB bW 0 ne{bW}{2 bd m}ie sl np 0 0 p mv 0 p l st gr}b/ap{e 3 ix ae 2 ix a}b/PT{8 OB 1 sc 0 bd p 0 0 p 3 -1 r s 3 1 r e s e 0 0 p mv 1 0 p l 0 0 p ap mv 1 0 p ap l e n e n 0 0 p ap mv 1 0 p ap l pp pp}b/DT{gs np PT cm sm st gr}b/Bd{[{pp}{[{DS}{DD}{gs 12 OB np bW 0 ne{bW 2 dv/bd x}if dp nH dv dp 3 -1 ro 2 dv s{dp bd p mv bd n p l}for st gr}{gs 12 OB 1 sc np bW 0 ne{bW 2 dv/bd x}if 1 1 nH 1 s{nH dv dp bd m wF m o o p mv n p l}for cm sm st gr}{pp}{DB}{gs 12 OB np 0 lW 2 dv o o n p mv p l bW 0 ne{bW 2 dv}{bd}ie wF m o o p l n p lcp fill gr}{pp}{gs 12 OB/bL x bW 0 ne{bW 2 dv/bd x}if np 0 0 p mv bL bd 4 m dv round 2 o o lt{e}if pp cvi/nSq x bL nSq 2 m dv dp sc nSq{.135 .667 .865 .667 1 0 rcurveto .135 -.667 .865 -.667 1 0 rcurveto}repeat cm sm st gr}]o 1 g 1 s g e 2 4 gi al pp5 -1 r exec}{al pp 8 ix 1 eq{DD}{DS}ie 5 -1 r 2 eq{DB}{DS}ie pp}{2 4 gi al pp DT}]o 0 g g exec}b/CS{p mv p l cw lW cW 2 m a sl sp sl}b/cB{12 OB 0 0 p mv 0 p l cm sm cw bW 0 ne{bW}{bd 2 m}ie cW 2 m a sl sp sl}b/CW{12 OB 1 sc cW lW 2 dva 0 o p mv 0 e n p l bW 0 ne{bW 2 dv}{bd}ie wF m cW a 1 o n p l 1 e p l cp cm sm}b/CB{np[{[{CS}{CS}{cB}{CW}{pp}{cB}{CW}{pp}{cB}]o 1 g 1 s g e 2 4 gi al pp 5 -1 r exec}{al pp p mv p l CS pp pp}{2 4 gi al pp PT cm sm cw cW 2 m sl sp sl}]o0 g 1 s g exec clip}b/Ct{bs rot g bs rot g gs o CB CB 1 setgray clippath fill 0 setgray Bd gr}b/wD 18 dict d/WI{wx dx ne{wy dy s wx dx s dv/m1 x wy m1 wx m s/b1 x}if lx ex ne{ly ey s lx ex s dv/m2 x ly m2 lx m s/b2 x wx dx ne{b2 b1 s m1 m2 s dv}{wx}iedp m2 m b2 a}{ex n dp m1 m b1 a}ie}b/WW{gs wD begin bs e g 2 4 gi al pp o o xl 4 -1 r 3 -1 r s/wx x s/wy x bs e g 2 4 gi al pp 4 -1 r 3 -1 r s/lx x s/ly x 0 bW 2 dv wF m o o wy wx at mt ro tr/dy x/dx x ly lx at mt ro tr n/ey x n/ex x np wxwy p mv WI p l ex n/ex x ey n/ey x dx n/dx x dy n/dy x lx ly p l WI p l cp fill end gr}b/In{px dx ne{py dy s px dx s dv/m1 x py m1 px m s/b1 x}if lx 0 ne{ly lx dv/m2 x ly ey s m2 lx ex s m s/b2 x px dx ne{b2 b1 s m1 m2 s dv}{px}iedp m2 m b2 a}{ex n dp m1 m b1 a}ie}b/BW{wD begin bs e g/wb x bs e g/bb x wb 4 g/cX x wb 5 g/cY x bb 4 g cX eq bb 5 g cY eq and{bb 2 g bb 3 g}{bb 4 g bb 5 g}ie cY s/ly x cX s/lx x/wx wb 2 g cX s d/wy wb 3 g cY s d 0 bW 2 dv ly lx at mt ro tr/ey x/ex x0 bW 2 dv wF m wy wx at mt ro tr/dy x/dx x 0 lW 2 dv wy wx at mt ro tr wy a/py x wx a/px x gs cX cY xl np px py p mv In p l lx ex s ly ey s p l ex n/ex x ey n/ey x dx n/dx x dy n/dy x wx 2 m px s/px x wy 2 m py s/py x lx ex s ly ey s p lIn p l px py p l cp fill gr end}b/Db{bs{dp type[]type eq{dp 0 g 2 eq{gs dp 1 g 1 eq{dL}if 6 4 gi al pp DS gr}{dp 0 g 3 eq{2 4 gi al pp DT}{pp}ie}ie}{pp}ie}forall}b/I{counttomark dp 1 gt{2 1 rot{-1 r}for}{pp}ie}b/DSt{o/iX x dp/iY x o/cXx dp/cY x np p mv counttomark{bs e g 2 4 gi al pp o cX ne o cY ne or{4 1 r 4 1 r}if pp pp o/cX x dp/cY x o iX eq o iY eq and{pp pp cp}{p l}ie}repeat pp st}b/SP{gs/sf x/lW x/bW x/cW x count 7 ge 5 ix 192837465 eq and{ 5 -1 r pp cpt 7 -1 r s e 7 -1 r s e5 -1 r dv neg e 5 -1 r dv neg e cpt xl sc neg e neg e xl}{xl pp pp}ifelse 1 1 S dv dp sc cm currentmatrix pp lW sl 4.0 setmiterlimit np}b end154 20 280 176 40 1296 20 20 chemdict begin SP 7500 3780 6680 3780 2 Ar /bs[]d Db gr end

R(x,y,z) (x´,y´,z´) (2.3.1)Applications to quantum mechanics require knowledge of the symmetry operation OR,

which acts on a wavefunction, y(x,y,z). The operator OR acts on a function so that the new functionORy evaluated at x',y',z' has the same value as (x,y,z).


% ChemDraw Laser Prep% CopyRight 1986, 1987, Cambridge Scientific Computing, Inc.userdict/chemdict 145 dict put chemdict begin/version 23 def/b{bind def}bind def/L{load def}b/d/def L/a/add L/al/aload L/at/atan L/cp/closepath L/cv/curveto L/cw/currentlinewidth L/cpt/currentpoint L/dv/div L/dp/dup L/e/exch L/g/get L/gi/getintervalL/gr/grestore L/gs/gsave L/ie/ifelse L/ix/index L/l/lineto L/mt/matrix L/mv/moveto L/m/mul L/n/neg L/np/newpath L/pp/pop L/r/roll L/ro/rotate L/sc/scale L/sg/setgray L/sl/setlinewidth L/sm/setmatrix L/st/stroke L/sp/strokepath L/s/subL/tr/transform L/xl/translate L/S{sf m}b/dA{[3 S]}b/dL{dA dp 0 3 lW m put 0 setdash}d/cR 12 d/wF 1.5 d/aF 10 d/aR 0.25 d/aA 45 d/nH 6 d/o{1 ix}b/rot{3 -1 r}b/x{e d}b/cm mt currentmatrix d/p{tr round e round e itransform}b/Ha{gs np 3 1 rxl dp sc -.6 1.2 p mv 0.6 1.2 p l -.6 2.2 p mv 0.6 2.2 p l cm sm st gr}b/OB{/bS x 3 ix 3 ix xl 3 -1 r s 3 1 r e s o o at ro dp m e dp m a sqrt dp bS dv dp lW 2 m lt{pp lW 2 m}if/bd x}b/DA{np 0 0 mv aL 0 aR aL m 180 aA s 180 aA a arc cp fill}b/OA{np0 cw -2 dv mv aL 0 aR aL m 180 aA s 180 arc 0 cw -2 dv rlineto cp fill}b/SA{aF m lW m/aL x aL 1 aR s m np 0 p mv rad 0 p l gs cm sm st gr}b/CA{aF lW m/aL x aL 1 aR s m 2 dv rad dp m o dp m s dp 0 le{pppp pp}{sqrt at 2 m np rad 0 rad 180 6 -1 r s 180 6 -1 r s arc gs cm sm st gr cpt e at ro}ie}b/AA{np rad 0 rad 180 180 6 -1 r a arc gs cm sm st gr}b/RA{lW m/w x np rad w p mv w w p l rad w n p mv w w n p l w 2 m dp p mv 0 0 pl w 2 m dp n p l st}b/HA{lW m/w x np 0 0 p mv w 2 m dp p l w 2 m w p l rad w p l rad w n p l w 2 m w n p l w 2 m dp n p l cp st}b/Ar1{gs 5 1 r 3 ix 3 ix xl 3 -1 r s 3 1 r e s o o at ro dp m e dp m a sqrt/rad x[{2.25 SA DA}{1.5 SA DA}{1SA DA}{cw 5 m sl 3.375 SA DA}{cw 5 m sl 2.25 SA DA}{cw 5 m sl 1.5 SA DA}{270 CA DA}{180 CA DA}{120 CA DA}{90 CA DA}{3 RA}{3 HA}{1 -1 sc 270 CA DA}{1 -1 sc 180 CA DA}{1 -1 sc 120 CA DA}{1 -1 sc 90 CA DA}{6RA}{6 HA}{dL 2.25 SA DA}{dL 1.5 SA DA}{dL 1 SA DA}{2.25 SA OA}{1.5 SA OA}{1 SA OA}{1 -1 sc 2.25 SA OA}{1 -1 sc 1.5 SA OA}{1 -1 sc 1 SA OA}{270 CA OA}{180 CA OA}{120 CA OA}{90 CA OA}{1 -1 sc 270 CA OA}{1-1 sc 180 CA OA}{1 -1 sc 120 CA OA}{1 -1 sc 90 CA OA}{1 -1 sc 270 AA}{1 -1 sc 180 AA}{1 -1 sc 120 AA}{1 -1 sc 90 AA}]e g exec gr}b/ac{arcto 4{pp}repeat}b/pA 32 d/rO{4 lW m}b/Ac{0 0 px dp m py dp m a sqrt 0 360 arc cm sm gs sg fill grst}b/OrA{py px at ro px dp m py dp m a sqrt dp rev{neg}if sc}b/Ov{OrA 1 0.4 sc 0 0 1 0 360 arc cm sm gs sg fill gr st}b/Asc{OrA 1 27 dv dp sc}b/LB{9 -6 mv 21 -10 27 -8 27 0 cv 27 8 21 10 9 6 cv -3 2 -3 -2 9 -6 cv cp}b/DLB{0 0 mv -4.8 4.8 l-8 8 -9.6 12 -9.6 16.8 cv -9.6 21.6 -8 24.6 -4.8 25.8 cv -1.6 27 1.6 27 4.8 25.8 cv 8 24.6 9.6 21.6 9.6 16.8 cv 9.6 12 8 8 4.8 4.8 cv cp}b/ZLB{LB}b/Ar{dp 39 lt{Ar1}{gs 5 1 r o o xl 3 -1 r e s 3 1 r s e o 0 lt o 0 lt ne/rev xdp 0 lt{1 -1 sc neg}if/py x dp 0 lt{-1 1 sc neg}if/px x np[{py 16 div dup 2 S lt{pp 2 S}if/lp x lp 0 p mv 0 0 p l 0 py p l lp py p l px lp s 0 p mv px 0 p l px py p l px lp s py p l cm sm st}{py 16 div dup 2 S lt{pp 2 S}if/lp x lp 0 p mv 0 0 0 py lp ac0 py 2 dv lp neg o lp ac 0 py 2 dv 0 py lp ac 0 py lp py lp ac px lp s 0 p mv px 0 px py lp ac px py 2 dv px lp a o lp ac px py 2 dv px py lp ac px py px lp s py lp ac cm sm st}{py dp 2 dv py 180 pA s 180 pA a arc st np px py s py 2 dvpy pA dp neg arcn st}{0 0 p mv 0 py p l px py p l px 0 p l cp cm sm st}{px lW 2 dv a lW -2 dv p mv rO dp rlineto px lW 2 dv a rO a py lW 2 dv a rO a p l rO lW -2 dv a py lW 2 dv a rO a p l lW -2 dv py lW 2 dv a p l 0 py p l px py p l px 0 p l cp fill0 0 p mv 0 py p l px py p l px 0 p l cp cm sm st}{0 rO p mv 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac 0 0 0 py rO ac cp cm sm st}{rO py p mv rO rO xl 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac rO neg dp xl px py 0 py rO accp fill 0 rO p mv 0 py px py rO ac px py px 0 rO ac px 0 0 0 rO ac 0 0 0 py rO ac cp st}{1.0 Ac}{0.5 Ac}{1.0 Ov}{0.5 Ov}{Asc LB gs 1 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr gs cm sm st grnp -1 -1 sc LB gs 1 sg fill gr cm sm st}{Asc LB gs 0.5 sg fill gr gs cm sm st gr np -0.4 -0.4 sc LB gs 1 sg fill gr cm sm st}{Asc LB gs 1 sg fill gr gs cm sm st gr np -0.4 -0.4 dp sc LB gs 0.5 sg fill gr cm sm st}{Asc DLB -1 -1 sc DLB gs 1 sg fill grgs cm sm st gr np 90 ro DLB -1 -1 sc DLB gs 0.5 sg fill gr cm sm st}{Asc gs -1 -1 sc ZLB gs 1 sg fill gr cm sm st gr gs 0.3 1 sc 0 0 12 0 360 arc gs 0.5 sg fill gr cm sm st gr ZLB gs 1 sg fill gr cm sm st}{Asc gs -1 -1 sc ZLB gs 0.5 sgfill gr cm sm st gr gs 0.3 1 sc 0 0 12 0 360 arc gs 1 sg fill gr cm sm st gr ZLB gs 0.5 sg fill gr cm sm st}{0 0 p mv px py p l cm sm st}{gs bW 0 ne{bW}{5 lW m}ie sl 0 0 p mv px py p l cm sm st gr}{gs dL 0 0 p mv px py p l cm sm st gr}{OrA 1 16 dv dp sc0 1 p mv 0 0 1 0 1 ac 8 0 8 -1 1 ac 8 0 16 0 1 ac 16 0 16 1 1 ac cm sm st}]e 39 s g exec gr}ie}b/Cr{0 360 np arc st}b/DS{np p mv p l st}b/DD{gs dL DS gr}b/DB{gs 12 OB bW 0 ne{bW}{2 bd m}ie sl np 0 0 p mv 0 p l st gr}b/ap{e 3 ix ae 2 ix a}b/PT{8 OB 1 sc 0 bd p 0 0 p 3 -1 r s 3 1 r e s e 0 0 p mv 1 0 p l 0 0 p ap mv 1 0 p ap l e n e n 0 0 p ap mv 1 0 p ap l pp pp}b/DT{gs np PT cm sm st gr}b/Bd{[{pp}{[{DS}{DD}{gs 12 OB np bW 0 ne{bW 2 dv/bd x}if dp nH dv dp 3 -1 ro 2 dv s{dp bd p mv bd n p l}for st gr}{gs 12 OB 1 sc np bW 0 ne{bW 2 dv/bd x}if 1 1 nH 1 s{nH dv dp bd m wF m o o p mv n p l}for cm sm st gr}{pp}{DB}{gs 12 OB np 0 lW 2 dv o o n p mv p l bW 0 ne{bW 2 dv}{bd}ie wF m o o p l n p lcp fill gr}{pp}{gs 12 OB/bL x bW 0 ne{bW 2 dv/bd x}if np 0 0 p mv bL bd 4 m dv round 2 o o lt{e}if pp cvi/nSq x bL nSq 2 m dv dp sc nSq{.135 .667 .865 .667 1 0 rcurveto .135 -.667 .865 -.667 1 0 rcurveto}repeat cm sm st gr}]o 1 g 1 s g e 2 4 gi al pp5 -1 r exec}{al pp 8 ix 1 eq{DD}{DS}ie 5 -1 r 2 eq{DB}{DS}ie pp}{2 4 gi al pp DT}]o 0 g g exec}b/CS{p mv p l cw lW cW 2 m a sl sp sl}b/cB{12 OB 0 0 p mv 0 p l cm sm cw bW 0 ne{bW}{bd 2 m}ie cW 2 m a sl sp sl}b/CW{12 OB 1 sc cW lW 2 dva 0 o p mv 0 e n p l bW 0 ne{bW 2 dv}{bd}ie wF m cW a 1 o n p l 1 e p l cp cm sm}b/CB{np[{[{CS}{CS}{cB}{CW}{pp}{cB}{CW}{pp}{cB}]o 1 g 1 s g e 2 4 gi al pp 5 -1 r exec}{al pp p mv p l CS pp pp}{2 4 gi al pp PT cm sm cw cW 2 m sl sp sl}]o0 g 1 s g exec clip}b/Ct{bs rot g bs rot g gs o CB CB 1 setgray clippath fill 0 setgray Bd gr}b/wD 18 dict d/WI{wx dx ne{wy dy s wx dx s dv/m1 x wy m1 wx m s/b1 x}if lx ex ne{ly ey s lx ex s dv/m2 x ly m2 lx m s/b2 x wx dx ne{b2 b1 s m1 m2 s dv}{wx}iedp m2 m b2 a}{ex n dp m1 m b1 a}ie}b/WW{gs wD begin bs e g 2 4 gi al pp o o xl 4 -1 r 3 -1 r s/wx x s/wy x bs e g 2 4 gi al pp 4 -1 r 3 -1 r s/lx x s/ly x 0 bW 2 dv wF m o o wy wx at mt ro tr/dy x/dx x ly lx at mt ro tr n/ey x n/ex x np wxwy p mv WI p l ex n/ex x ey n/ey x dx n/dx x dy n/dy x lx ly p l WI p l cp fill end gr}b/In{px dx ne{py dy s px dx s dv/m1 x py m1 px m s/b1 x}if lx 0 ne{ly lx dv/m2 x ly ey s m2 lx ex s m s/b2 x px dx ne{b2 b1 s m1 m2 s dv}{px}iedp m2 m b2 a}{ex n dp m1 m b1 a}ie}b/BW{wD begin bs e g/wb x bs e g/bb x wb 4 g/cX x wb 5 g/cY x bb 4 g cX eq bb 5 g cY eq and{bb 2 g bb 3 g}{bb 4 g bb 5 g}ie cY s/ly x cX s/lx x/wx wb 2 g cX s d/wy wb 3 g cY s d 0 bW 2 dv ly lx at mt ro tr/ey x/ex x0 bW 2 dv wF m wy wx at mt ro tr/dy x/dx x 0 lW 2 dv wy wx at mt ro tr wy a/py x wx a/px x gs cX cY xl np px py p mv In p l lx ex s ly ey s p l ex n/ex x ey n/ey x dx n/dx x dy n/dy x wx 2 m px s/px x wy 2 m py s/py x lx ex s ly ey s p lIn p l px py p l cp fill gr end}b/Db{bs{dp type[]type eq{dp 0 g 2 eq{gs dp 1 g 1 eq{dL}if 6 4 gi al pp DS gr}{dp 0 g 3 eq{2 4 gi al pp DT}{pp}ie}ie}{pp}ie}forall}b/I{counttomark dp 1 gt{2 1 rot{-1 r}for}{pp}ie}b/DSt{o/iX x dp/iY x o/cXx dp/cY x np p mv counttomark{bs e g 2 4 gi al pp o cX ne o cY ne or{4 1 r 4 1 r}if pp pp o/cX x dp/cY x o iX eq o iY eq and{pp pp cp}{p l}ie}repeat pp st}b/SP{gs/sf x/lW x/bW x/cW x count 7 ge 5 ix 192837465 eq and{ 5 -1 r pp cpt 7 -1 r s e 7 -1 r s e5 -1 r dv neg e 5 -1 r dv neg e cpt xl sc neg e neg e xl}{xl pp pp}ifelse 1 1 S dv dp sc cm currentmatrix pp lW sl 4.0 setmiterlimit np}b end216 19 46 263 40 1296 20 20 chemdict begin SP /bs[]d Db gr end

^ORyR(x,y,z) = O Ry(x´,y´,z´) = y(x,y,z) (2.3.2)Left multiplying by OR-1 and transposing yields eqn 2.3.3.



OR-1yR(x,y,z) = y(x´,y´,z´) (2.3.3)

This definition might seem backwards when compared with the definition of R; however, and theopposite convention (i.e., ORyR(x,y,z) = y(x',y',z')) can be adopted if one is consistent. Because theset of inverses of all the elements of a group give the group back, the convention makes no realphysical difference. We will use the convention of eqn 2.3.4-6, which can also be



ORy(x´,y´,z´) = y(x,y,z) (2.3.4)written as 2.3.5.



^ORy(x´,y´,z´) = y(R-1x´,y´,z´) (2.3.5)Since the primes are arbitrary, let q represent (x,y,z) and one can use the expression 2.3.6.



-^ -ORy(q) = y(R-1q) (2.3.6)This latter form is most useful in applications.

A symmetry group can be regarded as an n-dimensional function space. The behavior of ageneral function y, under the operations of a group OR can be expressed as a combination of thebehavior of "basis functions" that span the possible behaviors of a function belonging to group G.The group Cs (s in the yz plane) and the behavior of various functions f(x) with respect toreflection illustrates this point.

sx = -x antisymmetric fn or odd fnsx2 = (-x)2 = x2 symmetric fn or even fnscos(x) = cos(-x) = cos(x) symmetric fns(x2 + x) = x2 - x linear combination of a symmetric and antisymmetric fns(x4 + x3) = x4 - x3 linear combination of a symmetric and antisymmetric fn

A function is symmetric if it is unchanged when acted on by s and antisymmetric if the functiongoes into minus itself. The properties of symmetry and antisymmetry with respect to s define thefundamental behavior of functions in the Cs group. All linear functions can be decomposed into a

43

linear sum of symmetric and antisymmetric parts. A character table for Cs , which summarizesthese conclusions is shown below.

Cs E s Cs E s

____________________ _________________

symm 1 1 A´ 1 1antisymm 1 -1 A´´ 1 -1

The symmetric and antisymmetric types of functions are more conventionally denoted A´ and A´´.These A´ and A´´ function types are called irreducible representations. They are the twofundamental types of functions one needs to describe symmetry behavior in the two-dimensionalgroup Cs . The characters ±1 show how operations of the group change functions that transformlike these irreducible representations. For example, the functions x and x3 are A´´ and cos x or x2

are A´ in the Cs point group. A linear function, such as x + x2 + x3, can be decomposed into A´ +2A´´ irreducible parts. This intuitive development can be formalized.

Earlier we showed that matrices could be used to represent symmetry operators of a group.It follows that matrix representations multiply just like the symmetry group operations, and form agroup isomorphic to the point group of symmetry operations. In fact, any set of square matricesthat multiply like the elements of a group form a representation for that group. The order of thematrices defines the dimension of the representation. The correspondence between matrices andsymmetry operations need not be one-to-one. A matrix can correspond to more than one groupelement. For example, let the 1 x 1 matrix (1) = all the elements of a group. Thus for Cs .

E s and E x s

(1) (1) (1) x (1) = (1)This trivial one-dimensional representation satisfies all the requirements for a group and is a validgroup representation. The many to less correspondence between the group elements and matrices iscalled a homomorphism, and the matrix representation is termed unfaithful. When a one-onecorrespondence (isomorphism) exists, the representation is called faithful.

When dealing with matrices, it is convenient to define the trace or character of a matrix Rthat is the sum of the diagonal elements in eqn 2.3.7.



iTr(R) = Sa i i (2.3.7)

For symmetry operator matrices that represent a group isomorphically R2-1R1R2 defines a similaritytransformation of coordinates. From matrix theory it can be shown that R2-1R1R2 defines R1 withrespect to a new set of basis vectors. In other words, the operation is the same but just defined inanother basis. This makes sense, since the equivalent symmetry operations in a conjugate class

44

consist of similar kinds of operations that differ only in spatial orientation. The trace or characterof a matrix does not change after a similarity transformation. With this aside, return to theproblem of representations.

1 0 0 0

0 -1 0 0 0 0 1 -1

0 0 -1 -1 1 0 0 0

0 +1 0 0 0 0 1 1

0 0 1 1

For a group of order n, it is possible to find a set of n matrices that faithfully represent thegroup. By similarity transformations, the set of matrices can be arranged in the most reduced form(block diagonal form), as shown above. The number of elements in the submatricies for eachoperation will equal the order of the group. The submatrices are themselves unfaithfulrepresentations of the group. For most purposes, enough information is contained in the trace orcharacter of these matrix representations. Furthermore, group elements belonging to a class can begrouped together because they have identical characters. The character table for the C3 v grouphas the form shown below.

^

^^

(x,y), (R x,R y), (x 2-y2, xy), (xz, yz)0-12E

Rz-111A2

z, z 2, x 2 + y2111A1

3sv2C3(z)EC3 v

The irreducible representations named A1, A2, and E are the fundamental ways that a function canbe classified in the C3 v point group. This example provides some new complexities as compared toCs . The simultaneous effect of C3 and s operations influences the classification of symmetry type.Both the A1 and A2 representations are unchanged (i.e., symmetric) under C3 or C32 rotations.While A1 also does not change under a s operation, the A2 representation is antisymmetric. For thetwo dimensional E representation, the characters alone do not tell us about symmetric orantisymmetric behavior, because they represent the composite trace of 2 x 2 matrices. The rightmostcolumn provides transformation properties for several functions relevant to physical problems. Thex, y, and z coordinates behave just like atomic px, py, and pz orbitals or the x, y, and z components

45

of the electric dipole moment operator, so far as their symmetry properties are concerned. Thesebehaviors can be derived using methods described in section 2.5 and 2.6.

The following theorems hold for the group representations:1) The number of classes in the group = Number of unique irreducible representations2) Let Gi = ith irreducible representation and let XR(Gi) = the character for the R th operator of the

representation Gi. Then

E = order of the group = g2

iX (Gi)S

This means that the number of elements contained in the matrix representations equals the order ofthe group.

2SR

X (Gi)^ = order of the group for any Gi (note S includes sum over R so all elements in each class must be included)

3) ^R

above Gi should be Gi!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

^^SXR(Gi)XR(Gj)R= gdi j (i.e., rows are orthogonal)4)

5) If = g, then Gi irreducible representation. X (Gi)RRS

2

then G i is an irreducible

representation.6) The columns of the character table are orthogonal and normalized to g/Nk (Nk = number of

elements in the class for the k'th column of the table).Consider the application of these theorems to cyclic groups. For C5 , the five elements can bewritten as b, b2, b3, b4, and b5 = e. By application of similarity transforms, one can show that eachelement belongs to its own class. Therefore by theorems 1 and 2 there will be 5 one-dimensionalirreducible representations. From Lagrange's Theorem (section 1.2) the order of the elements of thegroup must be 1 or 5. Since the element of order 1 is e, the others must obey the relation.

Xb5(Gj) = 1c = e2pi(n/g) n = 1,2,3 - - -, g

This equation defines the g th roots of unity, since cg[G(b)] = e2pi = 1. These roots are abbreviatedas e, e2, - - -, eg (e = e2πi/g). For the specific case of C5 , where e = e2pi/5:

46

C5 C5 C52 C53 C54 C55

______________________________________________________________G1 e e2 e3 e4 e5

G2 e2 e4 e6 e8 e10

G3 e3 e6 e9 e12 e15

G4 e4 e8 e12 e16 e20

G5 e5 e10 e15 e20 e25

Next use the fact that:e3 = e2pi3/5 = e2pi(-2/5) = (e2pi(2/5))* = e2* and

en5+j = ej

Here * denotes the complex conjugate, and we recognize that rotation by (+3/5)2p and (-2/5)2p inthe complex plane are equivalent. Continuing the process for the group C5 yields the charactertable:

C5 E C5 C52 C53 C54

G5 A 1 1 1 1 1G1 1 e e2 e2* e*

E1G4 1 e* e2* e2 eG2 1 e2 e* e e2*

E2G3 1 e2* e e* e2

The reason for pairing G1, G4 as E1 and G2, G3 as E2 is that they are complex conjugatepairs. For most physical problems involving real functions, the complex conjugate pairs areeffectively degenerate (not so for magnetic problems like the Zeeman effect where y ~eimf). Thecomplex conjugate pairs can be added [recall eia = cos a + i sin a and e-ia = cosa -i(sina) =(eia)*] and then only the real (cosq) part is retained. This yields a C5 character table that issufficient for solving problems with non-imaginary functions.

47

+

+ 52 p

C5

A

G1 G4 = E1

G3 = E2G2

E1

2

2

C5

1

2cos

2cos 4 p5 5

2 p2cos

2cos

1C5

2

4 p5 5

4 p

C53

1

2cos

2cos 2 p5 5

4 p2cos

2cos

1C5

4

2 p5

2.4 Symbols for Irreducible RepresentationsThe Bethe nomenclature uses G1, G2, G3 - - - to specify irreducible group representations.

This notation is still used for double groups, as discussed in a later chapter. Otherwise the Mullikennotation is usually employed by chemists. For one-dimensional irreducible representations, c(E) =1. One dimensional representations are called A if symmetric with rotation around the principalaxis of symmetry, or B if antisymmetric with rotation around the principal rotation axis of thegroup . Subscripts 1 and 2 denote symmetry or antisymmetry with respect to a ^ C2, or a sv if no^ C2 exists. Primes and double primes specify symmetry or antisymmetry with respect to sh. Ingroups that contain an inversion center, g or u subscripts denote even (g) or odd (u) behaviorwith respect to inversion. Two-dimensional irreducible representations have c(E) = 2 and arecalled E. Three-dimensional irreducible representations, for which c(E) = 3, are named T. Forone-dimensional irreducible representations, the character is the matrix and +1 denotes symmetrywith respect to a given operation, while a character of -1 means antisymmetry.

The totally symmetric representation is conventionally the first A representation in everycharacter table and all of its characters are 1. A function with this symmetry possesses the fullsymmetry of the point group. All the other representations are called non-totally symmetricrepresentations. They represent functions of lower symmetry, since a character ≠ 1 means thefunction does not go into itself on application of that symmetry operation. Character tables for theimportant point groups are collected in Appendix A of this chapter.

2.5 Decomposing Reducible Representations - The Great Orthogonality Theorem.Given a set of characters XR(Gi) for some representation, Gi, of a group, it is desirable to

find out how to reduce the representation to a sum of irreducible representations. The answer isprovided by the great orthogonality theorem. That is, given XR(G) we want to find out G = aGi +bGj + . . . . The number of times the irreducible representation Gi of the group is contained inG may be calculated by the formula:

aGi = 1/g S XR* (Gi) XR (G) (aGi = the number of times the Gi representations occurs in G, g = order of group)R

^ ^

48

For example the representation

0063sv2C3E

in C3 v reduces to A1 + A2 + 2E. The calculation proceeds as follows:aA1 = 1/6 {1(6) + 1(0)2 + 1(0)3} = 1aA2 = 1/6 {1(6) + 1(0)2 - 1(0)3} = 1aE = 1/6 {2(6) + -1 (0)2 + 0(0)3} = 2

Notice that because the sum occurs over all symmetry operations, the 2C3 operations in the sameclass and the 3 equivalent sv must be included explicitly in the calculation.

2.6 The O-H Stretches of Water: Understanding Irreducible RepresentationsSignificance of Irreducible Representations as Basis Sets for Chemical Problems. Suppose

there exists a set of linearly independent functions f1, f2, f3, - - fn. Consider a symmetry operation,OR, that permutes the set of functions, or mixes them with one another to form a new linearlyindependent set of functions with dimension n. The set of matrices for OR form a representation ofthe point group.

OR above!!!!!!!!!

qR

f1.

.

.

.

.fn

. . .a fi +b fj + . . .

reorg . . .

=

Any set of n independent equivalent functions that are transformed into linear combinations of oneanother by the symmetry operations of the group form a basis for an n-dimensionalrepresentation of the group. The trace of the representation matrix yields characters that allow thedecomposition of the n-dimensional representation into a sum of irreducible representations. Thisprocedure permits a classification of electronic, vibrational, rotational or spin wave-functions of amolecule transform in terms of the irreducible representations of the molecular point group. Thedegree of degeneracy of a molecular energy level equals the dimension of the irreduciblerepresentation to which its wavefunction belongs.

Consider what group theory has to say about a simple problem, the O-H stretchingvibrations in the water molecule. The vibrational wavefunctions are classified according to the

49

various irreducible representations of the molecular symmetry group for water. A naive approachmight suggest that both O-H bonds are equivalent, and therefore both would occur at the samestretching frequency in the infrared spectrum. This simple view ignores the coupling of the twovibrations. Stretching of one O-H bond changes the degree of difficulty of stretching the otherbond. Group theory requires that two equivalent functions form the basis for a two-dimensionalgroup representation. Refer back to Figures 1.4 and 1.5, which show that s(xz) and C2 interchangethe two O-H bonds, while s(yz) leaves them unchanged. Each bond stretch can be depicted by adouble headed arrow centered in each bond. Therefore the symmetry operations affect the two O-Hbonds as follows:

^

C2O-HaO-Hb

= O-HbO-Ha

s(xz) O-HbO-Ha

=O-HaO-Hb

O-HaO-Hb

= O-HaO-Hb

s(yz)

^

The matrices for this two-dimensional representation are:

^^

1 00 1

0 11 0

C2CCCCCE s(xz)0 11 0

1 00 1

s(yz)

The characters for this representation, denoted GO-H, are obtained by summing the diagonalelements and can be compared with the C2v character table shown below.

50

C2 v E C2(z) sv(xz) sv(yz)__________________________________________A1 1 1 1 1A2 1 1 -1 -1B1 1 -1 1 -1B2 1 -1 -1 1GO-H 2 0 0 2

Thus GO-H is a reducible representation, which by inspection (or the formula of section 2.5)consists of A1 + B2. Because there are no degenerate representations in C2 v symmetry, the two O-H vibrational wavefunctions cannot be degenerate. The water molecule should show two differentO-H stretching vibrations by infrared spectroscopy (more about selection rules later). Thisprediction of group theory is borne out experimentally, since two such vibrations occur at 3652 cm-1 and 3756 cm-1 in the gas-phase IR spectrum of water. The C2 v character table requires that allwavefunctions of the water molecule be nondegenerate. Any degeneracy that might occur must beby sheer coincidence and is termed accidental degeneracy.

Before proceeding, there are two significant observations. First, in computing the characterfor the two O-H stretches a value of 1 occurs along the diagonal for each bond that goes into itself.Off-diagonal elements of the matrices need not be considered, when calculating the characters.Second, if the coordinate system were selected so the water molecule was in the xz plane, then thecharacters for the O-H stretches would have been (2 0 2 0) = A1 + B1. Either description of the O-H stretches as A1 + B1 or A1 + B2 is correct, they merely correspond to a different choice ofcoordinates. Two scientists describing the IR spectrum of water could appear to be in disagreement,when the difference is merely one of coordinate axes. This illustrates the importance of clarifyingthe coordinate system choice (and being consistent) to avoid confusion. Neither OHa nor OHb

alone behave as an irreducible representation in the C2 v point group.Now consider a more complex case where mixing of functions occurs. For the functions x

and y in C3 v symmetry, the C3 operation mixes the two coordinates according to eqn 2.2.15 (aspecific case of the general rotation matrix derived previously). Because of this mixing, both x andy must be considered together as a two-dimensional representation.

(2.2.15)

svC3(z)Ecos 120 -sin 120sin 120 cos 120

1 00 -1

1 00 1

Computation of the sv matrix takes advantage of the option that the orientation of the x and y axesare arbitrary, as long as they lie perpendicular to z. The calculation is simplified with the assumptionthat sv lies in the xz plane. By reference to eqn 2.2.20, it is easy to see that the trace of the matrixwould indeed be zero regardless of the value of b (i.e., the orientation of sv with respect to the x

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axis). Furthermore, the character is the same for equivalent elements in a class, and so only one ofthe 2C3 and 3sv operations need be considered. The trace of these three matrices (2 -1 0) verifiesthat x and y together form a basis for the E representation of C3 v.

36 - university of california, san diegotroglerlab.ucsd.edu/grouptheory224/chap2a.pdfh bond...

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