Group Theory - Part 2 Symmetry Operations and Point Groups

1 1C734b Symmetry Operations and Point groups1 Part II: Symmetry Operations Part II: Symmetry Operations and Point Groupsand Point GroupsC734bC734b Symmetry Operations and Point Operations : leave a set of objects in indistinguishableindistinguishableconfig urations said to be equivalent-The identity operator, E is the do nothing operator. Therefore, its final configuration is not distinguishable from the initial one, but identicalidenticalwith element: a geometrical entity (line, plane or Point ) with respect to which one or more Symmetry Operations may be carried kinds of Symmetry elements for molecular symmetryFour kinds of Symmetry elements for molecular symmetry1.)1.)Plane operation = reflection in the plane2.)2.)Centre of Symmetry or inversion centre:operation = inversion of all atoms through the centre3.)3.)Proper axisoperation = one or more rotations about the axis4.

2 4.)Improper axisoperation = one or more of the sequence rotation about the axis followed by reflection in a plane perpendicular ( ) to the rotation Symmetry Operations and Point groups31. Symmetry Plane and Reflection1. Symmetry Plane and ReflectionA plane must pass through a body, not be = . The same symbol is used for the operation of reflecting through a plane mas an operation means carry out the reflection in a plane normal to m . Take a Point {e1, e2, e3} along ()z ,y ,x x{e1, e2, e3} = {-e1, e2, e3} 321e,e,eOften the plane itself is specified rather than the normal. x= yzmeans reflect in a plane containing the y- and z-, usually called the yz olaneC734b Symmetry Operations and Point groups4-atoms lying in a plane is a special case since reflection through a plane doesn t move the atoms. Consequently all planar molecules have at least one plane of Symmetry molecular planeNote:Note: produces an equivalent configuration.

3 2= produces an identical configuration with the original. 2= E n= E for n even; n = 2, 4, 6,.. n= for n odd; n = 3, 5, 7,..3C734b Symmetry Operations and Point groups5 Some molecules have no -planes:SFClOLinear molecules have an infinite number of planes containing the bond molecules have a number of planes which lie somewhere between these two extremes:Example:Example:H2O2 planes; 1 molecular plane + the other bisecting the H-O-H groupOHHC734b Symmetry Operations and Point groups6 Example:Example:NH3 NHHH3 planes containing on N-H bond and bisecting opposite HNH groupExample:Example:BCl3 BClClClExample:Example:[AuCl4]-square planarAuClClClCl-4 planes; I molecular plane + 3 containing a B-Cl bond and bisecting the opposite Cl-B-Cl group5 planes; 1 molecular plane + 4 planes; 2 containing Cl-Au-Cl+ 2 bisecting the Cl-Au-Cl Symmetry Operations and Point groups7 Tetrahedral molecules like CH4have 6 molecules like SF6have 9 planes in totalSFFFFFFC734b Symmetry Operations and Point groups82.

4 Inversion Centre2.) Inversion Centre-Symbol = i- operation on a Point {e1, e2, e3}i{e1, e2, e3} = {-e1, -e2, -e3}-e2-e1-e3e3e2e1iyxzzyx5C734b Symmetry Operations and Point groups9-If an atom exists at the inversion centre it is the only atom which will not move upon other atoms occur in pairs which are twins . This means no inversion centre for molecules containing an odd number of more than one species of ii = E in= En evenin= in oddC734b Symmetry Operations and Point groups10 Example:Example:AuClClClCl-1122344iiAuCl ClClCl-4411322butHCHHHHHHHH-orhave no inversion centre even though in the methane case the number of Hs is even6C734b Symmetry Operations and Point groups113. Proper Axes and Proper Rotations3. Proper Axes and Proper Rotations- A proper rotation or simply rotation is effected by an operator R( ,n) which means carry out a rotation with respect to a fixed axis through an angle described by some unit vector n.

5 For example:For example:R( /4, x){e1, e2, e3} = {e1 , e2 , e3 }R( /4, x)45o45oe2'e1'e3'e3e2e1yxzzyxC734b Symmetry Operations and Point groups12 Take the following as the convention: a rotation is positive if looking down axis of rotation the rotation appears to be common symbol for rotation operator is Cnwhere n is the order of the axis. Cnmeans carry out a rotation through an angle of = 2 /n R( /4) C8R( /2) = C4R( ) = also called a binary Symmetry Operations and Point groups13 Product of Symmetry operators means: carry out the operation successively beginning with the one on the right . C4C4= C42= C2= R( , n)(Cn)k= Cnk= R( , n); = 2 k/n Cn-k= R(- , n); = -2 k/nCnkCn-k= Cnk+(-k)= Cn0 E Cn-kis the inverse of CnkC734b Symmetry Operations and Point groups14 Example:Example:112233C31 C3= rotation by 2 /3= 120o113322C3axis is perpendicular to the plane of the equilateral C32= rotation by 4 /3= 240o112233 IIIIII8C734b Symmetry Operations and Point groups15 But112233332211C3-1= rotation by -2 /3= -120oIIIIIII C32= C3-1112233332211C33= rotation by 2 = 360oIII C33= EC734b Symmetry Operations and Point groups16 What about C34?

6 112233332211C34= C3C3C3C3 IIIII C34= C3 only C3, C32, E are separate and distinct operationsSimilar arguments can be applied to any proper axis of order n9C734b Symmetry Operations and Point groups17 Example:Example:CC66::C6; C62 C3C63 C2C64 C6-2 C3-1C65 C6-1C66 ENote:Note:for Cnn odd the existence of one C2axis perpendicular to or containing Cnimplies n-1 more separate (that is, distinct) C2axes or planesHHHHH-C5axis coming out of the page12345C734b Symmetry Operations and Point groups18* When > one Symmetry axis exist, the one with the largest value of n PRINCIPLE AXISPRINCIPLE AXIST hings are a bit more subtle for Cn, n evenTake for example C4axis:C2(1)C2(2)1122334410C734b Symmetry Operations and Point groups19C2(2)C2(1) ; C2(1) C2(2); C2(2) C2(1)C2(1)C2(2)33441122C4 Total = C42= ; C2(1) C2(1); C2(2) C2(2)C734b Symmetry Operations and Point groups20C2(1)C2(2)11223344C4 Total C43= ; C2(1) C2(2); C2(2) C2(1)Conclusion:Conclusion:C2(1) and C2(2) are not distinctConclusion:Conclusion:a Cnaxis, n even, may be accompanied by n/2 sets of 2 C2axes11C734b Symmetry Operations and Point groups21a1a2b1b24 C2axes: (a1, a2) and (b1, b2).

7 Cnrotational Groups are AbelianC734b Symmetry Operations and Point groups224. Improper Axes and Improper Rotations4. Improper Axes and Improper RotationsAccurate definitions:Improper rotation is a proper rotation R( , n) followed by inversion iR( , n)Rotoreflection is a proper reflection R( , n) followed by reflection in a plane normal to the axis of rotation, hCalled Sn= hR( , n) = 2 /nCotton and many other books for chemists call Snan improper rotation, and we will Symmetry Operations and Point groups23 Example:Example:staggered ethane (C3axis but no C6axis)C611223664455221133665544 h665544112233C734b Symmetry Operations and Point groups24 h11223664455446655332211C6665544112233 Note:Note: Sn= hCnor Cn hThe order is Symmetry Operations and Point groups25 Clear if Cnexists and hexists Snmust :HOWEVER:Sncan exist if Cnand hdo example above for staggered ethane is such a element Sngenerates Operations Sn, Sn2, Sn3.

8 However the set of Operations generated are different depending if n is even or evenn even{Sn, Sn2, Sn3, .., Snn} { hCn, h2Cn2, h3Cn3, .., hnCnn}But hn= E and Cnn= E Snn= Eand therefore: Snn+1= Sn, Snn+2= Sn2, etc, and Snm= Cnmif m is even. C734b Symmetry Operations and Point groups26 Therefore for S6, Operations are:S6S62 C62 C3S63 S2 iS64 C64 C32S65 S6-1S66 EConclusion:Conclusion:the existence of an Snaxis requires the existence of a , n even, are Symmetry Operations and Point groups27n oddn oddConsider Snn= hnCnn= hE= h hmust exist as an element in its own right, as must as an example an S5axis. This generates the following Operations :S51= hC5S52= h2C52 C52S53= h3C53 hC53S54= h4C54 C54S55 = h5C55 hS56= h6C56 C5S57= h7C57 hC52S58= h8C58 C53S59 = h9C59 hC54S510= h10C510 EIt s easy to show that S511= S5 The element Sn, n odd, generates 2ndistinct operationsSngroups, n odd, are not AbelianC734b Symmetry Operations and Point groups28a) A body or molecule for which the only Symmetry operator is E has no Symmetry at all.

9 However, E is equivalent to a rotation through an angle = 0 about an arbitrary axis. It is not customary to include C1in a list of Symmetry elements except when the only Symmetry operator is the identity ) = 2 /n; n is a unit vector along the axis of Symmetry Operations and Point groups29 Products of Symmetry OperatorsProducts of Symmetry Operators- Symmetry operators are conveniently represented by means of a stereogramstereogramor stereographic with a circle which is a projection of the unit sphere in configuration space (usually the xy plane). Take x to be parallel with the top of the Point above the plane (+z-direction) is represented by a small filledsmall Point below the plane (-z-direction) is represented by a larger openlarger general Point transformed by a Point Symmetry operation is marked by an Symmetry Operations and Point groups30 For improper axes the same geometrical symbols are used but are not filled Symmetry Operations and Point groups31C734b Symmetry Operations and Point groups3217C734b Symmetry Operations and Point groups33 Example:Example:Show that S+( ,z) = iR-( - , z)Example:Example:Prove iC2z= zC734b Symmetry Operations and Point groups34 The complete set of Point - Symmetry operators including E that are generated from the operators {R1, R2.}

10 } that are associated with the Symmetry elements {C1, i, Cn, Sn, } by forming all possible products like R2R1satisfy the necessary Group properties:1)Closure2)Contains E3)Satisfies associativity4)Each element has an inverseSuch Groups of Point Symmetry operators are called Point GroupsPoint Groups18C734b Symmetry Operations and Point groups35 Example:Example:construct a multiplication table for the S4point Group having the set of elements: S4= {E, S4+, S42= C2, S4-} ++ + +4422444244244 SSCCSSSCSEESCSESC omplete row 2 using stereograms: S4+S4+, C2S4+, S4-S4+ (column x row)C734b Symmetry Operations and Point groups3619C734b Symmetry Operations and Point groups37 Complete TableComplete Table2444442242444244244 CSESSSESCCESCSSSCSEESCSES+ + ++ + +C734b Symmetry Operations and Point groups38 Another example:an equilateral triangleChoose C3axis along zThe set of distinct operators are G = (E, C3+, C3-, A, B, C}aabbccxyBlue lines denote Symmetry Symmetry Operations and Point groups39 Let 1=aabbcc E 1=aabbccC3+ 1= ccaabbC3- 1=bbccaa A 1= aaccbb B 1= ccbbaa C 1=bbaacc= 1= 2= 3= 4= 5= 6C734b Symmetry Operations and Point groups40 Typical binary products:C3+C3+ 1= C3+ 2bbccaa C3- 1 C3+C3+= C3-C3+C3- 1 = C3+ 3aabbcc E 1 C3+C3-= EC3+ A 1 = C3+ 4bbaacc C 1 C3+ A= C AC3+ 1= C3+ 2ccbbaa B 1 AC3+= BNote:Note.