Symmetry & Group Theory

Transcription

1 ymmetry & Group Theory MT Chap. Vincent: Molecular ymmetry and group theory ymmetry: The properties of self-similarity 1

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3 Re(CO)10

4 W(CO)6 C C60

5 ymmetry: Construct bonding based on atomic orbitals Predict Raman & IR spectra Access reaction pathway Determine optical activity ymmetry Operation: Movement of an object into an equivalent or indistinguishable orientation ymmetry Elements: A point, line or plane about which a symmetry operation is carried out 5

6 5 types of symmetry operations/elements Identity: this operation does nothing, symbol: E Element is entire object Proper Rotation: Rotation about an axis by an angle of /n 1 m C n Rotation m/n C C n n n1 n E C n C PtCl 6

7 C The highest order rotation axis is called the principle axis. H O NH How about: NO? 7

8 Identity E Proper Rotation Cn Reflection: reflection through a mirror plane NH H O 8

9 a mirror plane containing a principle rotation axis is labeled v B n n E( n even) ( n odd) a mirror plane normal to a principle rotation axis is labeled h B 9

10 Inversion: i inversion center or center of symmetry (x,y,z) (-x,-y,-z) i i n n E( n even) i( n odd) Difference between inversion and -fold rotation 10

11 Inversion? Improper rotation: n rotation about an axis by an angle of /n followed by reflection through a perpendicular plane. (C n, h symmetry are not necessary for n to exist) 11

12 6 1

13 1 5 P P 5 1 Contain C, h In general: n with n even molecule contains C n/, nn =E n with n odd molecule contains C n + h ; nn = h, n n =E 1

14 1 P 1 5 P 1 5 Contain C, h P 1 5 P P 1 5 P 1 5 P 1 5 = h =E 6 E C i C

15 Xe E ', i, h, v, v, C, C,C ',C", C C " ' v C C B ' v E,,, C,C, h 15

16 Group Theory Definition of a Group: A group is a collection of elements which is closed under a single-valued associative binary operation which contains a single element satisfying the identity law which possesses a reciprocal element for each element of the collection. Mathematical Group 1. Closure: A, B G AB G. Associativity: A, B, C G A(BC)= (AB)C. Identity: There exists E G such that AE=EA=A for all A G. Inverse: A G there exists A -1 G such that AA -1 = A -1 A=E Order of a group: the number of elements it contains 16

17 Example: 1. set of all real number, under addition, order = Closure: x + y G Associativity: x + (y +z) =(x+y) +z Identity: x +0 =0+x =x Inverse: x +(-x) =(-x)+x =0. set of all integers, under addition. {set of all real number}-{0}, under multiplication Closure: x * y G Associativity: x * (y *z) =(x*y) *z Identity: x *1 =1*x =x Inverse: x *(1/x) =(1/x)*x =1. {+1, -1} 5. { 1, i} ymmetry of an object point group (symmetry about a point) {E, C, v, v' } = point group C v Binary operation: one operation followed by another Multiplication Table Cv E C v v E E C v v Closure: Associativity: Identity: Inverse: C C E v v v v v E C v v v C E 17

18 Cv E C v v E E C v v C C E v v v v v E C v v v C E Rearrangement Theorem: each row and each column in a group multiplication table lists each of the elements once and only once. A B C Proof: suppose AB=AC, i.e. two column entries are identical then: 1 1 A B C AA AB AC A AB EB=EC B=C A AC Cv E C v v E E C v v C C E v v v v v E C v v v C E A group is Abelian if AB=BA ( the multiplication is completely commutative). Not all groups are abelian. vc C v 18

19 Any object (or molecule) may be classified into a point group uniquely determined by its symmetry. Groups with low symmetry: {E}=C 1, chönflies ymbol/notation {E,} =C s {E, i} =C i ONCl, Cs 19

20 H Cl Cl H Ci 0

21 Groups with a single C n axis {E, C n, C n, C n.c n n-1 } =C n HO 1

22 Groups with a single C n axis plus a perpendicular h plane: C nh N { E, C,, } i O C h h

23 C h { E, C,, i} h H Cl C = C Cl H C h { E, C, C, h,, 5 }

24 Groups with a single C n axis plus n vertical v planes: C nv C v O H { E, C, v, v' } C v NH { E, C, C, v, v', v' '}

25 C v Br { E, C,,,,, } 5 C C v v' d, d ' d : dihedral reflection planes (bisects v or C) quare pyramidal Br n-gonal pyramidal shape: Cnv C v H { E, C,... v...} 5