1.3 Summary of Symmetry Operations, Symmetry Elements, and Point Groups.

Rotation axis. A rotation by 360˚/n that brings a three-dimensional body into an

equivalent configuration comprises a Cn symmetry operation. If this operation is performed a

second time, the product CnCn equals a rotation by 2(360˚/n), which may be written as Cn2. If n

is even, n/2 is integral and the rotation reduces to Cn/2. In general, a Cnm operation is reduced

by dividing m and n by their least common divisor (e.g., C96 = C32). Continued rotation by

360˚/n generates the set of operations:

Cn, Cn2, Cn3, Cn4, ... Cnn

where Cnn = rotation by a full 360˚ = E the identity. Therefore Cnn+m = Cnm. Operations

resulting from a Cn symmetry axis comprise a group that is isomorphic to the cyclic group of

order n. If a molecule contains no other symmetry elements than Cn, this set constitutes the

symmetry point group for that molecule and the group specified is denoted Cn. When additional

symmetry elements are present, Cn forms a proper subgroup of the complete symmetry point

group. Molecules that possess only a Cn symmetry element are rare, an example being

Co(NH2CH2CH2NH2)2Cl2+, which possesses a sole C2 symmetry element.

Cl

NCl

NN

N

Co

One special type of Cn operation exists only for linear molecules (e.g., HCl). Rotation by

any angle around the internuclear axis defines a symmetry operation. This element is called

C• axis and an infinite number of operations C•f are associated with the element where f

denotes rotation in decimal degrees.

We already saw that molecules may contain more than one rotation axis. In BF3,

depicted below, a three-fold axis emerges from the plane of the paper intersecting the center of

the

21

21

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 end52 66 281 23 40 80 20 20 chemdict begin SP /bs[[1 1 6200 1120 6200 1696][1 1 6100 980 5692 572][1 1 6200 1120 6580 620]]d [0 2 I 6200 1696 DSt [1 I 5692 572 DSt Db gr end

B

F

F F

triangular projection. Three C2 axes containing each B-F bond lie in the plane of the molecule

perpendicular to the three-fold axis. The rotation axis of highest order (i.e., C3) is called the

principal axis of rotation. When the principal Cn axis has n even, then it contains a C2 operation

associated with this axis. Perpendicular C2 axes and their associated operations must be denoted

with prime and double prime superscripts.

Reflection planes. Mirror planes or planes of reflection are symmetry elements whose

associated operation, reflection in the plane, inverts the projection of an object normal to the

mirror plane. That is, reflection in the xy plane carries out the transformation (x,y,z) ∅ (x,y,-z).

Mirror planes are denoted by the symbol s and given the subscripts v, d, and h according to the

following prescription. Planes of reflection that are perpendicular to a principal rotation axis of

even or odd order are named sh (e.g., the plane containing the B and 3F atoms in BF3). Mirror

planes that contain a principal rotation axis are called vertical planes and designated sv. For

example, in BF3 there are three sv planes, each of which contains the boron atom, fluorine atom,

and is perpendicular to the molecular plane. Notice that for an odd n-fold axis there will be n sv

mirror planes which are similar. That is if we have one sv plane, then by operation of Cn a total

of n times, we generate n equivalent mirror planes. The n sv operations comprise a conjugate

class. For even n the sv planes fall into two classes of n/2 sv and n/2 sd planes, as illustrated

earlier in the chapter.

Rotoreflection axes or improper rotation axes. This new operation is best thought of as

the stepwise product of a rotation, and reflection in a plane perpendicular to the rotation axis.

We need to introduce it since a product such as C6 sh is not equal to either a normal rotation or

reflection. To satisfy group closure a new symmetry operation must be introduced. An

22

22

inversion operation is the simplest rotoreflection operation and is given the name S2 or i . That

S2 = i is immediately apparent because i .(x,y,z) = (-x,-y,-z) and C2(z)s(xy).(x,y,z) =

C2(z).(x,y,-z) = (-x,-y,-z). Because Cn and sh commute we have Sn = Cn sh for the general

rotoreflection operation. Since shn = E for even m and for m odd shm = sh, an Sn axis requires

simultaneous independent existence of a Cn/2 rotation axis and an inversion center when n is

even. For example, an S6 axis generates the operations for the point group S6. They are shown

below.

S6, S62 = C32 sh2 = C32, S63 = S2 = C2 sh = î, S64 = C32, S65 = C65 sh, S66 = E.

An odd order rotoreflection axis requires independent existence of a Cn axis and a sh symmetry

element. An S7 axis generates the following operations.

S7, S72 = C72, S73, S74 = C74, S75, S76 = C76, S77 = C77 sh7 = sh, S78 = C78 =

C7, S79 = C72 sh, S710 = C73, S711 = C74 sh, S712 = C75, S713 = C76 sh, S714 = E

The S7 axis generates a cyclic group of order fourteen and requires the independent existence of

C7 and sh. This group is denoted C7h and not S7.

Conjugate Symmetry Elements and Operations. In the symmetry group of BF3 the

sequence of operations C3´-1svC3 means to rotate by 120 ˚ perform sv, then rotate back 120˚.

This sequence of operations is equivalent to reflection in a plane rotated 120˚ from the initial

one, as shown below. The similarity transformation (C32)-1svC32 generates an operation, which

is the third vertical mirror plane. The three equivalent sv symmetry operations belong to the

same conjugate class. Symmetry equivalent symmetry operations (i.e., operations belonging to

symmetry elements that are interchanged by symmetry operations of the group) belong to the

same conjugate class. In BF3 all the operations associated with the equivalent C2 elements

belong to the same class. This may be verified by the similarity transformations C3-1C2C3 and

(C32)-1 C2C32 operating on any of the three C2 axis.

23

23

^^^

sv 1

23

conjugate sv3

1 2

C3-1svC3

112

22

3 33

1

generatedfrom a

conjugate sv

sn sn sn

When the principal axis is of even order two types of vertical reflection planes sv and sd

are possible. Consider the planar ion PtCl42-, which contains one four-fold axis normal to the

molecular plane and four perpendicular C2 axis as shown. Clearly the pair of axis labeled C2´

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 end145 125 53 60 40 1296 20 20 chemdict begin SP 3400 3520 2980 3080 2 Ar 3440 2920 3140 3240 2 Ar /bs[[1 2 1200 2480 2352 2480][1 2 2520 2480 3672 2480][1 2 2352 2480 2352 1328][1 2 2352 2480 2352 3632][1 1 2240 2380 1832 1972][1 1 2352 2480 2880 3000][1 1 2240 2560 1840 2975][1 1 2480 2340 2905 1950][1 2 1832 1972 1340 1480][1 2 1680 3040 1300 3420][1 2 2905 1950 3360 1480]]d gs dL [0 I 2 2352 1328 DSt [1 I 3672 2480 DSt [3 I 2352 3632 DSt [8 I 1340 1480 DSt [9 I 1300 3420 DSt [10 I 3360 1480 DSt gr [4 I 1832 1972 DSt [5 I 2880 3000 DSt [6 I 1840 2975 DSt [7 I 2905 1950 DSt Db gr end

Pt C2"

C2"

Cl

ClCl

Cl

C2' C2'

x

y

are equivalent, as are the pair of axis C2" since the four-fold rotation interchanges elements

among the pairs; however, the C2´ and C2" axis are not equivalent. There is no operation that

will interchange these elements. Primes designate C2 axes perpendicular to an even principal

axis. The C42 operation associated with the four-fold axis is called C2. You might be surprised

to find that a C4 axis and one perpendicular C2´ axis generate the remaining three "primed" two-

fold axes. Action of C4 on one C2´ axis generates the other C2´ axis. With the application of C4

or C2 no new axes are generated; however, consider the product of operators C4C2´ acting on

the point (x,y,z). Take one C2´ axis to lie along the x axis and choose C4 coincident with the z

24

24

axis. The operation C2´ on the general point (x,y,z) produces a new point at (x,-y,-z). The C4

operation does not change z and rotates the molecule in the x,y plane. So C4 (x,y,z) ∅

^• C4 x

•

y

(-y,x,z)

x

(-y,x,z) and the product C4C2´(x,y,z) = C4(x,-y,-z) = (y,x,-z) is equivalent to rotation around the

C2" axis bisecting the xy quadrant of the coordinate system because C2" (x,y,z) ≠ (y,x,-z).

•

y

(x,y,z)x

C2"

o

This proves that the combination of a C2´ operation with C4 generates the C2" operation, and

that the two C2´ belong to the same conjugate class. You may prove that the two C2" comprise a

second class.

There are likewise two classes of vertical reflection planes. The two planes containing

the C4 axis and one C2´ axis are called sv. Those mirror planes containing the C2" axis are

denoted sd. This illustrates an interesting consequence of a vertical reflection plane. Consider

the effect of sv or sd on rotation around the principal axis. The effect of sv is to change the

sense of

currentpoint 192837465

% 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 end136 98 77 294 40 1296 20 20 chemdict begin SP 1940 6120 1760 6140 12 Ar 3720 6080 3900 6080 6 Ar 3220 6660 2380 6660 2 Ar /bs[[1 1 1700 6040 1700 7200][1 1 3900 6020 3900 7180]]d [0 I 1700 7200 DSt [1 I 3900 7180 DSt Db gr end

sv

Cnm Cn

-m

25

25

rotation from counterclockwise to clockwise. This means that rotation by 360˚/n and -360˚/n are

equivalent. In other words a rotation Cnm and its inverse Cn-m belong to the same conjugate

class when a mirror plane contains the rotation axis. For example, the operations of a C7 axis

fall into the classes (C7, C76); (C72, C75); (C73, C74) and for a C6 axis the classes are (C6, C65);

(C3, C32); (C2) when a sv or sd mirror plane also exists.

Molecular symmetry groups. When you search for molecular symmetry elements, look

for rotations, reflections and rotoreflection elements, which interchange equivalent atoms in a

molecule and leave others untouched. The short hand notation for the various point groups uses

abbreviations, which often specify the key symmetry elements present. The complete set of

symmetry operations of a molecule defines a group, since they satisfy all the necessary

mathematical conditions. When confronted with a molecule whose symmetry group must be

determined, the following approach can be used.

Special symmetry groups usually are self-evident. The group D•h is the point group of

homonuclear diatomic molecules and other linear molecules that possess a sh plane (e.g., CO2).

For linear molecules an infinite number of sv planes always exist. The symbol D in a group

name always means that there are n-two-fold axes perpendicular to the n-fold principal axis. The

presence of sh and sv planes in a molecule always requires the presence of n^C´2 axes, since

one can show that sh sv = C´2; however, the converse is not true. Therefore, there are an infinite

number of C2 axes perpendicular to the C• axis of the molecule. The C• axis and sh generate

an infinite number of S•f operations; one is S2 = î.

If a linear molecule does not contain a sh plane, e.g., HCl or OCS, then it belongs to the

C•v point group. This point group contains a C• axis and an infinite number of sv planes.

Clearly C•v is a proper subgroup of D•h.

A free atom belongs to the symmetry group Kh, which is the group of symmetry

operations of a sphere. We shall not be concerned with this point group; however, a

26

26

consequence of spherical symmetry is the requirement that the angular wavefunctions of atoms

behave like spherical harmonics.

There are seven special symmetry groups, which contain multiple high order (Cn, n > 3)

axes. These are the only possibilities for 3-dimensional objects. The groups may be derived

from

Figure 1.16. The Five Platonic Solids and Their Point Groups, with The High Order C n Axes Listed.

Oh3 C 4

Td4 C 3

Oh3 C 4

octahedron tetrahedron cube

dodecahedron icosahedron

Ih12 C 5 axes

Ih12 C 5 axes

27

27

the symmetry operations of the five Platonic solids (octahedron, tetrahedron, cube, icosahedron,

dodecahedron). Platonic solids are solids, shown in figure 1.16, whose edges and verticies are

all equivalent and whose faces are all some regular polygon. The symmetry operations of the

tetrahedron comprise the group Td. The cube and octahedron possess equivalent elements and

their operations define the Oh group.

The pentagonal dodecahedron and icosahedron likewise define Ih. We will restrict our

discussion to the Oh group and its subgroups, O, Td, Th, and T, since Ih and its subgroup I are

encountered less frequently. The collection of groups {O, Td, Th, and T} are often referred to as

cubic groups. The symmetry elements belonging to an octahedron are: three C4 axes that pass

through opposite verticies of the octahedron; four C3 axes that pass through opposite triangular

faces of the octahedron; six C2 axes that bisect opposite edges of the octaedron; four S6 axes

coincident with the C3 axes; three S4 axes coincident with the C4 axes; an inversion center or S2

axis coincident with C4; three sh which are perpendicular to one C4 axis and contain two others.

Therefore, a single plane defined as sh, also can be thought of as sv with respect to the other C4

axis. Similarly, there are six planes called sd, which are also sv planes containing a C3 axis.

Subgroups of Oh are generated by removing the following symmetry elements:

O = lacks i, S4, S6, sh and sd and is called the pure rotation subgroup of Oh.

Td = this group lacks C4, i and sh and is the group of tetrahedral molecules, e.g., CH4.

Th = this uncommon group is derived from Td by removing S4 and sd elements.

T = the pure rotation subgroup of Td contains only C3 and C2 axes.

If a molecule belongs to none of these special groups, then focus on the highest order

rotation axis. Those rare molecules that possess only one n-fold rotation axis and no other

symmetry elements belong to the Cn symmetry group, which is cyclic. If only a rotation axis

exists and no mirror planes or other elements are obvious, a last check should be made for an Sn

axis of order higher than the obvious rotation axis. When a higher order Sn axis can be found,

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then the Sn groups (n = even) are obtained. Naturally n must be even since Snn = sh when n is

odd and both Cn and sh exist independently. These groups (n = odd) are conventionally

designated Cnh. Also, when n = 2, the S2 group (remember S2 = i) is conventionally called Ci.

If we add to a Cn group nsv (n odd) or n/2sv and n/2sd (n even) mirror planes, but no other

rotation axes, then the Cnv groups are generated. If to the elements of Cn we add a horizontal

mirror plane, then we generate the Cnh groups. The product of Cn and sh also generate an Sn

symmetry element and its associated operations.

When additional n-two-fold axes perpendicular to Cn are present, but no mirror planes,

then the Dn groups are generated. Addition of a sh plane to the rotation symmetry elements of a

Dn group generates the Dnh groups. Similar to the Cnh groups an Sn symmetry axis is also

produced (if n is even an inversion center will be present). The product C2 sh (where C2 is

associated with n ^ C2 axes in the Dnh group) generates vertical mirror planes so that the Dnh

group also contains nsv planes (n odd) or n/2sv and n/2sd planes (n even). If in addition to the

elements of a Dn group an S2n axis is present, then the Dnd groups are obtained. The product

SnC2 (C2 belongs to a C2 axis perpendicular to the n-fold axis) yields the operation sd so that an

additional n dihedral mirror planes are required.

If a molecule contains as the only symmetry element a mirror plane, then the group is

called Cs. Finally, if no symmetry elements, other than E, exist, then the molecular symmetry

group is the trivial one called C1. Table I.1 summarizes the groups we just discussed and their

associated elements. It provides a useful reminder of all the necessary elements when assigning

symmetry point groups.

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Table 1.1 Common Point Groups and Their Symmetry Elements

Point Group Symmetry Elements Present

C1 E

Cs E, sh

Ci E, i

Cn E, Cn

Dn n = odd E, Cn, n^C2

Dn n = even E, Cn, n/2^C2´, n/2^C2´´

Cnv n = odd E, Cn, nsv

Cnv n = even E, Cn, n/2sv, n/2sd

Cnh n = odd E, Cn, sh, Sn

Cnh n = even E, Cn, sh, Sn, i

Dnh n = odd E, Cn, sh, n^C2, Sn, nsv

Dnh n = even E, Cn, sh, n/2^C2´, n/2^C2´´, Sn, n/2sv, n/2sd, i

Dnd n = odd E, Cn, n^C2, i, S2n, nsd

Dnd n = even E, Cn, nC2´, S2n, nsd

Sn n = even only E, Sn, Cn/2 and i if n/2 odd

T E, 4C3, 3C2

Th E, 4C3, 3C2, 4S2n, i, 3sh

Td E, 4C3, 3C2, 3S4, 6sd

O E, 3C4, 4C3, 6C2

Oh E, 3C4, 4C3, 6C2, 4S6, 3S4, i, 3sh, 6sd

I E, 6C5, 10C3, 15C2

Ih E, 6C5, 10C3, 15C2, i, 6S10, 10S6, 15s

Kh E, infinite numbers of all symmetry elements

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1.4 Site Symmetry of Molecules and Ions in Solids

Most of the elements, as well as most inorganic compounds, exist as solids. The ideas

about symmetry point groups can be applied to molecules or ions in crystal lattices in the context

of site symmetry and pseudosymmetry. A crystal results from the periodic repetition of a

specific arrangement of atoms through space. For a regular infinite array of objects the act of

translation in space can be a valid symmetry operation, in addition to those described previously.

Consider the infinite one-dimensional lattice

---,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,---

Translation to the right or left by an integral multiple of the lattice spacing yields an

indistinguishable physical configuration. This operation is important in considering atomic

arrays in crystals. Seven three dimensional boxes (or parallelpipeds) used to summarize three-

dimensional packing are given in Table I.2. Each box can be classified by the lengths of its three

edges a, b, and c, which are assigned using the convention for a right-handed coordinate system.

_____________________________________________________________________________

Table I.2: The Seven Crystal Systems

System Conditions on Cell Parameters Lattice Symmetry

triclinic a ≠ b ≠ c; a ≠ b ≠ g Ci or _1

monoclinic a ≠ b ≠ c; a = g = 90˚, b > 90˚ C2h or 2/m

orthorhombic a ≠ b ≠ c; a = b = g =90˚ D2h or mmm

tetragonal a = b ≠ c; a = b = g = 90˚ D4h or 4/mmm

trigonal (rhombohedral setting)* a = b = c; a = b = g ≠90˚ D3d or _3 m

hexagonal a = b = c; a = b = 90˚; g = 120˚ D6h or 6/mmm

cubic a = b = c; a = b = g = 90˚ Oh or m3m_____________________________________________________________________________*A hexagonal setting is also possible.

The angles between the vectors are defined as a = between b and c, b = between a and c and g =

between a and b. These unit cells summarize the fundamental packing patterns possible in

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31

solids. The rightmost column in this table lists the symmetry point group of the crystal system in

conventional (Schoenflies) notation, along with the Hermann-Maugin equivalent used by

crystallographers and solid state chemists. In the latter notation Cn rotation axes are denoted by

n, mirror planes by m and sh mirror planes are denoted n/m. An inversion center is specified by

a bar above the principal rotation axis. In a formal approach to space groups the addition of

centered lattices and the possible subsymmetries within each unit cell would be developed to

yield the 230 space groups. For an excellent discussion of these considerations we recommend

the text by Stout and Jensen listed in the references for this chapter. Extensive tabulations for the

symmetry elements of each space group can be found in the reference work, International Tables

for X-ray Crystallography, available in most libraries. The discussion here will be limited to the

relation between the lattice symmetry for the solid and the point group symmetry of an

individual ion or molecule in the unit cell.

It is important to recognize that for molecules in crystals there is not necessarily a

correlation between molecular symmetry and the symmetry of the unit cell. Highly symmetrical

molecules may crystallize in low symmetry unit cells and molecules of low symmetry may

crystallize in unit cells of high symmetry. Atoms, molecules, or ions in a crystal may occupy

two types of positions in a unit cell. The first is called a general position, which exists for all

unit cells. There is no crystal symmetry associated with a general position. When a high

symmetry molecule, e.g., an octahedral metal complex, occupies a general position then it

rigorously possesses no symmetry because the environment of the crystal about it is asymmetric.

The asymmetry of the local environment distorts the site symmetry to C1 or 1; however, an

octahedral pseudosymmetry remains. The low symmetry crystal environment can often be

detected spectroscopically from a splitting of degenerate E or T energy levels. Perturbation of a

molecule's symmetry in the crystal from its isolated symmetry, as in the gas phase, may be great

or small. The magnitude of the perturbation depends on the strength of interactions between the

molecule and its environment. In a molecular crystal, where the forces will primarily be Van der

Waals in nature, the deviation from the idealized gas phase molecular symmetry will be small.

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In ionic crystal, where strong electrostatic forces are present, the distortion of the idealized

symmetry is expected to be greater.

The other kind of position in a unit cell is called a special position. These special

positions generally have unit cell (x, y, z) coordinates of 0, 1/2, 1/4, or 1/3 and are listed for each

space group in the International Tables for X-ray Crystallography. When a molecule or ion

occupies a special position then it possesses a rigorous site symmetry, which is determined by

the crystal environment. Several different site symmetries are possible in a unit cell, the highest

being the maximum symmetry of the cell. Thus cubic unit cells may have special positions of

Oh symmetry, which an octahedral molecule could occupy. If an octahedral molecule occupies

such a position, then the symmetry is rigorously Oh and splitting of degenerate energy levels by

a low symmetry crystal environment does not occur. Coupling between molecules in the unit

cell may, however, give exciton splittings, as noted later.

A simple example illustrates the essential points. Consider the case of triclinic crystals.

Two space groups are possible. One, denoted _

P1 , has the maximum symmetry indicated in

Table I.2. The other P1, a subgroup of _

P1 , has no symmetry. In P1 every position in the unit

cell has a crystal environment with complete asymmetry. There is generally only one molecule

(more rigorously 1 asymmetric unit since a pair or e.g., a hydrogen bonded cluster of molecules

can sometimes form a packing unit) per unit cell in this space group. In _

P1 , which possesses an

inversion center at the origin of the unit cell, an arbitrary point or general position (x, y, z) is

transformed to (-x, -y, -z) by the inversion operation. Thus, there must be two molecules

(asymmetric units) per unit cell if they occupy a general position. It is possible for molecules

that contain an inversion center to occupy the special position (0. 0, 0). Then there need be only

one asymmetric unit per cell, with the added condition that it possess rigorous inversion

symmetry. Similar conditions hold for higher symmetry space groups, but the basic concept

resembles that illustrated above.

Another concern, beyond that of the site symmetry in a crystal, is the possibility of

coupling between asymmetric units in a crystal. In ionic solids the forces that interconnect all

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33

the asymmetric units in a cell makes it imperative that these couplings be considered if one wants

to understand vibrations in the solid state. Even molecular crystals can exhibit unusual splittings

from such phenomena. One example occurs for coupling between excited states in crystals and

this is often referred to as exciton coupling. For example, the excited states q1 and q2 of two

asymmetric units in _

P1 could interact to yield composite sum and difference net states (q1 + q2

and q1 - q2). In this way a single excited state of the isolated molecule gives rise to two crystal

states of different energy. The student with a serious interest in solid-state spectroscopy will

need to delve into this in more detail. For most readers this section serves as a caveat should

they attempt to interpret spectra of solids.

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Additional Readings:

Atkins, P. W.; Child, M. S.; Phillips, C. S. G. Tables for Group Theory; Oxford University Press:

London, 1970.

Bunker, P. R. Molecular Symmetry and Spectroscopy; Academic Press, New York, 1979.

Burdett, J. K. Molecular Shapes: Theoretical Models of Inorganic Stereochemistry; Wiley:

New York, 1980.

Burns, G. Introduction to Group Theory with Applications; Academic Press: New York, 1977.

Cotton, F. A. Chemical Applications of Group Theory; 2nd Ed., Wiley-Interscience: New York,

1971. A good introduction for chemists, with many worked examples.

Donaldson, J. D.; Ross, S. D. Symmetry and Stereochemistry; Wiley: New York, 1972.

Describes space groups and solid state symmetries.

Dorain, P. B. Symmetry in Inorganic Chemistry; Addison-Wesley: Reading, Mass., 1965.

Fackler, J. P., Jr. Symmetry in Chemical Theory; the Application of Group Theoretical

Techniques to the Solution of Chemical Problems; Dowden, Hutchinson and Rossi:

Stroudsburg, Pennsylvania, 1973.

Fackler, J. P., Jr. Symmetry in Coordination Chemistry; Academic Press, New York, 1971.

Ferraro, J. R.; Ziomek, J. S. Introductory Group Theory and its Application to Molecular

Structure; 2nd Ed., Plenum Press: New York, 1975.

Griffith, J. S. The Irreducible Tenson Method for Molecular Symmetry Groups; Prentice-Hall:

Englewood Cliffs, New Jersey, 1962.

Hall, L. H. Group Theory and Symmetry in Chemistry; McGraw-Hill: New York, 1969.

Hamermesh, M. Group Theory and its Application to Physical Problems; Addision-Wesley:

Reading, Massachusetts, 1962. A rigorous mathematical development with advanced

applications to atomic and nuclear energy levels. Develops symmetric and continuous groups

as well as irreducible tensor methods.

Hochstrasser, R. M. Molecular Aspects of Symmetry; W. A. Benjamin: New York, 1966.

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35

Jaffe, H. H.; Orchin, M. Symmetry in Chemistry; R. E. Krieger Pub. Co.: Huntington, New

York, 1977.

Kettle, S. F. A. Symmetry and Structure; Wiley: New York, 1986.

Lax, M. J. Symmetry Principles in Solid State and Molecular Physics; Wiley: New York, 1974.

McWeeny, R. Symmetry; An Introduction to Group Thoery and its Applications; Pergamon

Press: Oxford, 1963.

Piepho, S. B.; Schatz, P. N. Group Theory in Spectroscopy: With Applications to Magnetic

Circular Dichroism; Wiley: New York, 1983.

Schonland, D. S. Molecular Symmetry: An Introduction to Group Theory and Its Uses in

Chemistry; Van Nostrand: Princeton, New Jersey, 1965.

Slater, J. C. Symmetry and Energy Bands in Crystals; Dover: New York, 1972. Applications of

Group Theory to Wavefunctions of Crystalline Solids.

Stout, G. H.; Jensen, J. H. X-ray Structure Determination: A Practical Guide; 2nd Ed., Wiley:

New York, 1989. Contains a discussion of crystal symmetries with applications to X-ray

crystallography.

Tinkham, M. Group Theory and Quantum Mechanics; McGraw-Hill: New York, 1964. A

concise development of theory with applications to atomic, molecular, and solid state

electronic wavefunctions.

Vincent, A. Molecular Symmetry and Group Theory: A Programmed Introduction to Chemical

Applications; Wiley: New York, 1977.

Wells, A. F. Structural Inorganic Chemistry, 5th ed., Clarendon Press: Oxford, 1984.

Wherrett, B. S. Group Theory for Atoms, Molecules, and Solids; Prentice-Hall International:

Englewood Cliffs, New Jersey, 1986.

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1.6 Problems

1) Determine which of the following are groups. For those, that are not groups, specify the

missing conditions.

a) The set of all positive integers with group multiplication defined as normal addition.

b) The set of all integers (± and 0) with group multiplication defined as normal addition.

c) The set of all integers (± and 0) with group multiplication defined as normal

multiplication.

2) Prove that all groups of prime order are isomorphic to the cyclic group of that order.

3) Give the point group to which the following molecules belong.

a) MnO42- c) WF6

b) Rh6(CO)16 d) Mo6Cl12 (8 face capping, 4 terminal Cl)

e) methane f) cubane

g) benzene h) Mn2(CO)10

i) Re2Cl82- j) OsCl4N-

k) H2S l)

currentpoint 192837465

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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 end123 130 46 421 40 1296 20 20 chemdict begin SP 2220 10040 2320 9860 46 Ar /bs[[1 1 2320 9300 1840 9560][1 1 1840 9560 1840 10106][1 1 1840 10106 2320 10392][1 1 2320 10392 2801 10131][1 1 2801 10131 2801 9584][1 1 2320 9300 2801 9584][1 1 1840 9560 1400 9300][1 1 955 10071 1400 10331][1 1 1840 10106 1400 10331][1 1 2320 10392 2320 10940][1 1 3283 10964 3283 10416][1 1 2801 10131 3283 10416][1 1 2801 9584 3275 9310][1 1 2801 8457 2320 8741][1 1 2320 9300 2320 8741]]d [0 5 4 11 10 I 1 2 9 2320 10940 DSt [3 I 2801 10131 DSt [6 I 1400 9300 DSt [7 I 8 1840 10106 DSt [12 I 3275 9310 DSt [13 I 14 2320 9300 DSt Db gr end

BN

BN

B

NH

H

H

(planar conformation ignore H

on carbons)

m) Co(en)33+ n) cis-MoCl2(CO)4

o) trans-MoCl2(CO)4 p) fac-Mo(CO)3(PPh3)3 (coordination symmetry only,

i.e., ignore Ph groups)

q) mer-Mo(CO)3(PPh3)3 r) Al2Cl6

s) BrCl2F t) CO2

37

37

u) S8 (cyclic crown structure) v) PPh3 (with propeller conformation of phenyl rings)

w) HOCl x) cyclo-(BCl)4(NCH3)4

y) Ce(NO3)62- (the coordination environment of Ce is icosahedral-each nitrate is bound in

a

planar bidentate fashion such that trans NO3- groups are eclipsed).

z) PF5

4) Consider the 2-dimensional packing of regular polygons of three-fold through eight-fold

symmetry. Which ones can form a close-packed planar network with no overlapping or

gaps in the plane?

5) Based on work in 4 what are the allowed rotational symmetry axes in unit cells of 3-

dimensional crystalline solids?

6) Consider the S2 operation around the z axis. Show that its effect on the vector (x, y, z) is

the same as the inversion operation.

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38

Key to Problems

1) a) Not group - identity element missing and inverse missing.

b) Yes

c) Not group - inverse elements missing.

2) Proof: Consider how one generates a group of prime order. Lagrange's theorem tells us

the elements of a group must be an integral divisor of the order of the group. If n = prime

then the elements are of order n. Since the # of elements = n, then one element a, and its

powers generate the group

a, a2, ..., an

This is just the cyclic group of order n. Q.E.D.

3) a) Td; b) Td; c) Oh; d) Oh; e) Td; f) Oh; g) D6h; h) D4d; i) D4h; j) C4v;k)

C2v; l) C3h; m) D3; n) C2v; o) D4h; p) C3v; q) C2v; r) D2h; s) C1; t) D•h; u) D4d;

v) C3; w) Cs; x) S4; y) Th; z) D3h

4) It is possible to pack triangles, squares, and hexagons in a close packed 2-dimensional array

but not pentagons: See figure on following page.

5) C2, C3, C4, and C6; note: the C4 requires the possibility of C42 = C2.

39

39