Molecular Geometry and Point Groups

1 Symmetry Operations and Elements The goal for this section of the course is to understand how symmetry arguments can beapplied to solve physical problems of chemical interest. To achieve this goal we must identify and catalogue the complete symmetry of a system andsubsequently employ the mathematics of Groups to simplify and solve the physical problemin question. Asymmetry element is an imaginary geometrical constructabout which a symmetryoperationisperformedoperationisp erformed. Asymmetry operation is a movement of an object about a symmetry elementsuch that theobject's orientation and position before and after the operation are titiltitth Asymmetry operation carries every pointintheobjectintoanequivalentpointort heidentical group Symmetry All symmetry elements of a molecule pass through a central Point within the molecule. The symmetry of a molecule or ion can be described in terms of the complete collection ofsymmetry operations it possesses. The total number of operations may be as few as one or as many as operations a molecule has, the higher its symmetry is.

2 Regardless of the number of operations, all will be examples of only five Identity Operation (E) The simplest of all symmetry operations isidentity, given the symbolE. Every object possesses identity. If it possesses no other symmetry, the object is said to beasymmetric. As an operation, identity does nothing to the molecule. It exists for every object, because theobject itself exists. The need for such an operation arises from the mathematical requirements of group theory. Inadditionidentityisoftentheresultofcarr yingoutaparticularoperationsuccessivelya Inaddition,identityisoftentheresultofcar ryingoutaparticularoperationsuccessively acertain number of times, , if you keep doing the same operation repeatedly, eventually you may bring the objectback to the identical (not simply equivalent) orientation from which was started. When identifying the result of multiple or compound symmetry operations they aredesignated by their most direct single equivalent. Thus, if a series of repeated operations carries the object back to its starting Point , the Rotation Operation (C) The operation ofrotationis designated by the symbolCn.

3 If a molecule has rotational symmetryCn,rotationby2 /n= 360 /nbrings the object into an equivalentposition. The value ofnis theorder of an n fold rotation. If the molecule has one or more rotational axes, theone with the highest value ofnis theprincipal axis SuccessiveC4clockwise rotations ofaplanarMX4molecule about an axis perpendicular to the plane ofthe molecule (XA=XB=XC=XD). Multipleiterationsaredesignatedbyasupers cript,Multipleiterationsaredesignatedbya superscript, three successiveC4rotations are identified asC43 TheC42andC44operations are preferably identified asthesimplerCandEoperationsrespectivelyt hesimplerC2andEoperations,respectively. There are four otherC2axes in the place of the molecule. TheC2'andC2"axesofaplanarMX4moleculeTheC 2andC2axesofaplanarMX4molecule. As these twofold axes are not collinear with the principalC4rotational axis they aredistinguished by adding prime ( ) and double prime ( ) to their symbols. Onlytwonotationsareneededforthefouraxesb ecausebothC axesaresaidtobelongto Onlytwonotationsareneededforthefouraxes, becausebothC2axesaresaidtobelongtothe sameclass, while the twoC2 axes belong to a separate , bothC2 axes are geometrically equivalent to each other and distinct fromC2.

4 In listing the complete set of symmetryoperations for a molecule, operations of thesame class are designated by a single notationpreceded by a coefficient indicating the numberof equivalent operations comprising the for the square planar structure herediscussedofDsymmetrytherotationaldis cussed,ofD4hsymmetry,therotationaloperat ions grouped by class are2C4(C4andC43),C2(collinear withC4) d The C2'and C2" axes of a planar ,and2C2 .General Relationships for CnCnn =EC2nn= C2(n = 2, 4, 6, )Cnm= Cn/m(n/m = 2, 3, 4, )Cn 1=CCn=Cn 1 Cnn+m=Cnm(m < n) Every n fold rotational axis has n 1 associated operations (excluding Cnn=E). Remember, the rotational operationCnmispreferably identified as the simplerCn/moperation wherem/nis an integer Reflection Operation ( ) The operation ofreflectiondefines bilateral symmetry about a plane, called amirror planeorreflection plane. Foreverypointadistanceralonganormaltoami rrorplanethereexistsanequivalentpoint Foreverypointadistanceralonganormaltoami rrorplanethereexistsanequivalentpointat points, equidistant from a mirror plane , related by reflection.

5 For a Point (x,y,z), reflection across a mirror plane xytakes the Point into (x,y, z).xy Each mirror plane has only one operation associated with it, since 2= , Vertical, and Dihedral Mirror Planes A hplane is defined as perpendicularto the principal axis of rotation. If no principal axis of rotation exists, his defined as the plane of the molecule. vand dplanes are defined so as tocontain a principal axis of rotation andtobeperpendiculartoa planetobeperpendiculartoa hplane. When both vand dplanes occur inthe same system, the distinctionbetween the types is made by defining vto contain the greater number ofatoms or to contain a principal axis of areference Cartesian coordinate system(x or y axis). Any dplanes typically will containbond angle bisectors. ThefivemirrorplanesofasquareThefivemirro rplanesofasquareplanar molecule MX4are grouped intothree classes ( h,2 v,2 d).The Inversion Operation (i ) The operation ofinversionis defined relative to the central Point within the molecule,through which all symmetry elements must pass,egtypicallytheoriginoftheCartesianc oordinatesystem(xyz=000) ,typicallytheoriginoftheCartesiancoordin atesystem(x,y,z=0,0,0).

6 If inversion symmetry exists, for every Point (x,y,z) there is an equivalent Point ( x, y, z). Molecules or ions that have inversion symmetry are said to becentrosymmetric. Each inversion center has only one operation associated with it, sincei2= of inversion (i) on an octahedral MX6molecule (XA= XB= XC= XD= XE= XF).Inversion Center of Staggered Ethane Ethane in the staggered configuration. The inversion center is at the midpoint along the C Cbond. Hydrogen atoms related byinversion are connected bydotted lines,which intersect atygyy,the inversion center. The two carbon atoms are also related by Improper Rotation Operation (Sn) The improper rotation operationSnis also known as therotation reflectionoperation and, asits name suggests, is a compound operation. Rotation reflection consists of a proper rotation followed by reflection in a planeperpendicular to the axis of rotation. nrefers to the improper rotation by 2 /n= 360 db(i)bithbj ttilt Snexistsifthe movementsCnfollowedby h(or vice versa)bringtheobjecttoanequivalentpositi on.

7 If bothCnand hexist, thenSnmust ,S4collinear withC4in planar MX4. NeitherCnnor hneed exist forSnto ,S4collinear withC2in tetrahedral tetrahedral MX4molecule inscribed in a , collinear with anS4axis, passes throughthecentersofeachpairofoppositecub efacesthecentersofeachpairofoppositecube facesand through the center of the , each axis bisects one of the M X rotation of a tetrahedral MX4molecule(X=X=X=X)Theimproperaxisisper pendicular(XA=XB=XC=XD).Theimproperaxisi sperpendicularto the page. Rotation is arbitrarily taken in aclockwise direction. Note that neitherC4nor haregenuine symmetry operations of tetrahedral MX4. SuccessiveS4operations on a tetrahedralMX4molecule(XA=XB=XC=XD)MX4mo lecule(XA=XB=XC=XD). Rotations are clockwise, exceptS4 1,whichis equivalent to the clockwise operationS43. SuccessivelycarryingouttwoSoperations SuccessivelycarryingouttwoS4operationsis identical to the result of a singleC2operation about the same ,S42=C2,42 Similarly,S44=E. Thus, there are only two operationsbelongingtothisclassforthetetr ahedralbelongingtothisclassforthetetrahe dralMX4molecule (S4andS43) about this axis.

8 In the highly symmetric tetrahedral systemthere are three equivalent andindistinguishableS4axes. Consequently, each axes gives rises to twoS4operations resulting in a class designated6S(3S3S3)as6S4(3S4+3S43).Non Genuine SnOperations: S1= h S2=iGeneral Relations of Sn Equivalences of successiveSnoperations: Ifnis even,Snn=En Ifnis odd,Snn= andSn2n=E IfiSmCmChdSmCmnh Ifmiseven,Snm=Cnm=Cn/mwhenm<nandSnm=Cnm nwhenm>n IfSnwith evennexists, thenCn/2exists IfSnwith oddnexists, then bothCnand perpendicular Find all symmetry elements and operations in the following:Defining the Coordinate System Molecules are conventionally oriented relative to a right hand Cartesian coordinate system: The following conventions of axis orientation are usually observed:1. Theoriginof the coordinate system is located at the central atom or the center of Thezaxis is collinear with the highest order rotational axis (the principal axis). If thereareseveralhighestorderrotationalaxe s,zisusuallytakenastheaxispassingthrough areseveralhighestorderrotationalaxes,zis usuallytakenastheaxispassingthroughthe greatest number of , for a tetrahedral molecule, thex,y,andzaxes aredefined as collinear with the threeC2axes (collinear with the threeS4axes).

9 Defining the Coordinate System (contd.)3. For planar molecules, if thezaxis as defined above is perpendicular to the molecularplane, thexaxis lies in the plane of the molecule and passes through the greatestnumber of the z axis lies in the plane of the molecule, then thexaxis stands perpendicular to the Coordinate System (contd.)4. For non planar molecules, once thezaxis has been defined, thexaxis is usually chosenso that thexzplane contains as many atoms as possible. If there are two or more suchplanes containingidentical sets of atoms,anyone maybe taken as ,yypWhere a decision about the orientation of thexaxis cannot be made on this basis, thedistinction betweenxandyis usually not important or is not generally fixed Symmetry Operations (Multiplication) Multiplication of symmetry operations is the successive performance of two or moreoperations to achieve an orientation that could be reached by a single ,i2=E;S4S4=S42=C2;C4 h=S4etc. The order in which successive different symmetry operations are performed can affect theresult.

10 Multiplication of symmetry operations is not commutativein general, although certaincombinations may be. In writing multiplications of symmetry operation we use a "right to left" notation: BA=X"Doing A then B has the same result as the operationX." We cannot assume that reversing the order will have the same result. ItmaybethateitherBA ABorBA=AB ItmaybethateitherBA ABorBA=AB. Multiplication of symmetry operations is associative:C(BA) =(CB)ATheorderofperformingSand shownhereforatetrahedralMXmoleculeaffect stheresultThe order of performing S4and v , shown here for a tetrahedral MX4molecule, affects the final positions in each case are not the same, but they are related to each other by v v S4butC2 v S4= S4 v We will now consider the complete set of symmetry operations for a particular moleculeand determine all the binary combinations of the symmetry operations it possesses. The symmetry elements of the CBr2Cl2molecule are shown below. This molecule has atetrahedral geometryNote:tetrahedralgeometrydoesnota utomaticallyimplytetrahedralsymmetry!