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 ^ ^ ^ 2 second time, the product CnCn equals a rotation by 2(360˚/n), which may be written as Cn . If n ^ ^ m is even, n/2 is integral and the rotation reduces to Cn/2. In general, a Cn operation is reduced ^ 6 ^ 2 by dividing m and n by their least common divisor (e.g., C9 = C3 ). Continued rotation by 360˚/n generates the set of operations: ^ ^ 2 ^ 3 ^ 4 ... ^ n Cn, Cn , Cn , Cn , Cn ^ n ^ ^ n+m ^ m where Cn = rotation by a full 360˚ = E the identity. Therefore Cn = Cn . 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. N Cl N Co Cl N N ^ 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 ^ f C• axis and an infinite number of operations C• 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 52/bs[[1[1]dgr [01end66 61006200 2 2811I 62006200 980112023 169640 11205692 6580 80 DSt 620020572] 620] 20[1 1696] Ichemdict 5692 572 begin DSt Db SP %userdict/chemdictL/gr/grestoreL/tr/transformxl0pplSARA}{6-1st}b/OrA{py-8dppycpnpgsfilleo5awyInx w 02 dp/cYcwpy1-1gbWChemDrawCopyRight dpp cmfillpA sc8DA}{cwgr-1pp}{sqrt0m2p2oix dv1p l 2 -9.6r lt{1-2mv m p2px scsmv180 cmHA}{dL0gr}{pp}{gs a}b/PT{8-1dp exec}{aldvsms{dp dvm gmvdv dprO dv-.6 x WIpy -1 exec0sc smnegb2 lp 12stCAnp wF5 0pn mvpy0 at bdp LB 1.2scpxneg grm a}{ex1986, pmvest 2.251-9.6Laser plOA}{1emarcn 2L/gs/gsavep clip}b/Ct{bspp naLneg}if/pypexlcpgssl0L/xl/translatenpOBatmvgr 5 pmst}b/HA{lW lwy12 0op mv 1 3.375px-1 0 fillmvn/ex81ro gsnSA16.8 90lppy npl1451st}{0 accounttomark{bs 1987,Prep OB/bLwxaR bWixsg -1dpbd rpyscpx ac gr0.3 DA}{dL0.6pxro rad8 dv1 aL fillatsc dictxm10cv0pn 0SAend}b/Db{bs{dpdp eq{DD}{DS}ie DLBx py 1negey bd1.2ne{bWlpm mtCambridge80p py120L/ie/ifelsegr -9.6 xdp px rotDA}{cw scm l}forputm/w-1mvrad rO n/ey 180 bWp 1.5cm2-1 ro p eL/S{sfpy 0b10 CAg0 dv1ac cptl 021.6chemdict180 tr/dy0lt{-1 p -10st2bs sm0-.6 SAaAdpacx a}ie}b/WW{gsa}ie}b/BW{wD pyx px120lOA}{1ne{bW dv}{bd}ie p 5npgr}{gs xl cpscdx rote 8 pyDA}{dL smm}b/dA{[3st}{Asc6 p2.23m-8 py10L/ix/indexg Scientificx/dx sc50180 0DLB-1 cml-1n/dx alp scg360 sl2 px-11624.6 0px p-1negsqrttype[]type gs2 4 12rbegin/versionac2.25r neg}if/px smp arcmv gs rpy s0dv/bds xgi0 1 scLBarcwFmv o x20 OB180e31 -4.80dp rO SAst}{00.5 al pCB0.60dy eq{DB}{DS}iepy90 gs neg1SAac wDlWComputing,mS]}b/dL{dA begings w l cw1 pprev{neg}ifac L/l/linetoDA}{2.25 r pxn/dyCB 6lp 0.5sgCA x}ifcW1622.2DA}{cwsc2 25.8e1 rObegin eeq{dp -1px-2 o m xpydvs sg0 1fill xl}{xlnp 0OA}{1 acXsgbsnp enp[{pyp prdpsetgray0dvx lp161wy fill cv0 gr lmvsbW23 l ne0filllxe 0o cmbs rlinetoaccp0arc5p SA-1.61 g/wb0wxdpgrsc}b/Ov{OrAcmpp L/mt/matrix 0n lypp}{2def/b{bind-1 0gopInc.mlgr 0 e1 rOpxcm sm16 pw gsOA}{1.5cmppycY2mv 0ne{bW pp}ifelseatg acsmslclippathgs sc mvl 2 l27 eq{gsaclp 3 2 div1smcmx stne pxmt1.5WIcp sm cmm14 270cm lW st}{Asc4 es bs0bL 1.6gr}b/OB{/bS 0gi or{4p duppywfill}b/SA{aFgipst}{px0rosm 0 SAst2smm psme bdldp SAall pAA}{1 0al 27dv/bd trrOcpdef}bind g/bblcp1grfill put st1 DA}{270mvlL/mv/movetopp 2py01st}]e st4ppOA}{11 wy1 radgs 0.4 fill4.8acZLBgr cmSlW0r mg S grrO 0DT}]opx 4lt{pp o-1 pa/pysetgray xpx1dv cptend w2setdash}d/cR x}ifdv39sc sm}b/CB{np[{[{CS}{CS}{cB}{CW}{pp}{cB}{CW}{pp}{cB}]o 125.8 onpgsapac wb0eq{dL}if SA dvsc-1pm xr}ifdpdef/L{loadxl py0round CAs pxe 0.5cp12-0.4mvl gr}b/In{px0x 3 lW18040ag OA}{14atsc rad pxcvsc 1S}if/lppp wxgix pyDA}{180lWg/cX1execst}{1.0 Bd-1 1sgro}ie}b/AA{npm/aL nHZLBg 8cm 30 L/m/mul-0.4 ppwAA}{1 0lp6a/px2-2 exec}b/CS{p gr}b/wDr24.6fillix360 rO 1-1n p4o3 accurrentmatrixxgsgr}ie}b/Cr{0 o/cXx dv xls{nH papgiodef}b/d/def-1x12 gracAc}{0.5wbsc sclp pxdxx 1 larclt{e}if-1aLCA9.63 pal lw cmpxrLB gsd/wFsg0 2.25 5-1x emv pyne{pys/wx ppdv1scL/n/neg2 cmpDA}{12018 21.6g/cYn dp/cYgscX0fill sm aR mr mv2 pp120dprOeDS Ac}{1.0210rad sdictSA sm 1.5mv1cYdv grnws 0xst}{0 3 dy cvi/nSqdp360bd09.6sgx gr}{dp m0 n OA}{1 s/wyrOAA}{10xl 1L/a/addcmpx gsxppp 0d/aFd/WI{wx bb 0 spL/np/newpath npfill rlinetoradmrCA o lnp p16.8 ac 0pxpnpsglp elOv}{0.5cwsmlW 4iX wFl w pgr0 apsx DA}{90 0px 180a10-1 gfill dxarc xeqslmv2oplW bs st gopy0cmmcv cX mv bLL/al/aload ommvpy px4.0scd/aR gr s3lpdx0 gro cW st}b/DS{npe 180opx 9.6pat eq dv/m1Ov}{Ascsmdpeq{2 nSqpyiY11.5 p90g aclWst}b/Asc{OrAo radgs lne{wysetmiterlimitCA ro2 py0mv2lpbb eqp 12 0.25rOn 6st}{Ascpx AA}]e2 m 0.3pSA4 2dp mv0pyDA}{3 L/pp/pop4pp -1 dv 5and{ppIn apac8gi ma py pgixl l OA}{1 gdym 1p LB sl8ncpd/aAL/at/atan cmpr aall pydv cpl alcYg2 sclapgs 4.8p erOlppsp sLB gspxppRA}{3 st}b/Ar1{gs execdvlxarc ppm1mvdp cmsml}fordp wx0eq cm ppsl}b/cB{12a -1pp}b/DT{gs gs45 14.8 ex1lp4 0px 1np}b DT}{pp}ie}ie}{pp}ie}forall}b/I{counttomark scpyL/r/rollmgs psmst}{gsand{bb pxsgdx 27-1 12cp}{psmg1gr}b/ac{arctosc scmd/nH HA}{1sl pynSq{.135a cv lWcmL/cp/closepath 01sgst}b/DD{gs filllym sst}{rO dvr1 sqrt0 end st smps cp}b/ZLB{LB}b/Ar{dpdv/m1lp 3 ey2 SAs/b1 fill360 bWgrsm l}ie}repeatg dpmv gr}b/CA{aF6-1dv25L/ro/rotateac-1 sste OBdp OA}{270cmgr d/o{1 g1 0py stsc}b/LB{9pnp 2rpxarca scgr}{pp}{DB}{gs x}if bbr ne{bW}{5 s/lxbSgs0l 4rO gr}b/RA{lWx.667 smp 3ex PT 2700py0 gigs wy dL3 mv4{pp}repeat}b/pAcmdvixlx aix}b/rot{3p p st}{Ascn/excmx g}{bbalpx 0.5ppp 3 l.865m10DS mvCAdp CAs/lyrO smL/cv/curvetolWpx ppl ixL/sc/scalene{ly lp-6sm rOst}b/SP{gs/sf gr}b/DB{gssg lW OA}{180rOwx0m/aLDA}{1 x xlpyst 5 s4 x mv p.667lW LBstey m/wfill 2m}iepy-1 3ggrxl 0pmlx 12l -1 39mgr}b/Bd{[{pp}{[{DS}{DD}{gs -1bb -2n/eygs cmbW21 l0lp grnprxdv/m2 s/b1-1 px OB1lt{pp exec}{alr}b/x{epy slxlt{Ar1}{gsaL r0.5 dv5ac -10cmsmCA 2L/sg/setgray-0.40 sscnp lp0g}ie x L/cw/currentlinewidthnppx 1 dv12rcurvetocma3 32x}if 0 sgsm dx180OA}{120cwslW x/lWaRradpy27 1x 0py pwF py-0.4OB cY d/rO{4 smfill d}b/cmrly lxn/dx2 bWlWppstsmvlW -8 ew rOCA pm m}if/bds/ly mey bW5exx/bW gr st}{pys 2 2p l dp270 dppx 1oac.135 2o DA}{1cms dv dvcmmvne{lyx ZLBCAne{bW}{bd 0roxlW dvom2 scL/sl/setlinewidth0py1 pxdy mto ne{bW}{2wyocX aatsmx/cW dpsm wpcvgt{2OA}{90 m}b/Ac{0rado-.667gs LBo F F B F currentpoint 192837465 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.