Chapter 16. Molecular Symmetry

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1 I. Smmetr Chpter 6. Moleculr Smmetr Elements xis mirror plne inversion center... Opertions rottion bout n xis reflection thru plne inversion thru center Five smmetr elements nd corresponding opertions: i. Doing nothing, identit E ii. Rottion bout n n-fold xis A rottion of 36 o /n. C n C 3 C C 6 Axis with the lrgest n is the principl xis. n smmetr opertions: n = 3 C 3 C 3 3 C3 E 3C3 C3 3 3 C, C C E

2 iii. Inversion through center of smmetr i iv. Reflection through mirror plne Verticl plne v : prllel to principl xis. Horizontl plne h : perpendiculr to principl xis Dihedrl plne d : bisects two C xes perpendiculr to the principl xis. v. Improper rottion bout n xis of improper rottion S n Rottion + reflection

3 Smmetr groups Group: A collection of elements (smmetr opertions) tht stisf the following conditions:. There is lws n identit element. b. Ever element hs n inverse. c. An products of two elements re lso elements of the group. d. Multipliction of elements is ssocitive A(BC)=(AB)C. Point group: t lest point unchnged. A molecule belongs to point group. Spce group: point group + trnsltionl smmetries. C : E C i : E, i C s : E, 3

4 C n : E, C n C nv : E, C n, n v C nh : E, C n, h D n : E, C n, nc (-fold xes perpendiculr to C n )

5 D nh : E, C n, nc, h D nd : E, C n, nc, n d S n : E, S n. C i = S Tetrhedrl groups T, T h nd T d Octhedrl groups O, O h nd O d Rottionl group R 3. Group cn be determined b flow digrm. 5

6 Consequences of smmetr i. Polrit A polr molecule (with permnent electric dipole) belongs to one of the groups C n, C nv nd C s. CO belongs to C v nd is polr. N belongs to D h nd is non-polr. ii. Chirlit Chirl molecule: cnnot be superimposed b its mirror imge. A chirl molecule must not hve i or. Chirl molecules cn chnge the polriztion of light. Usefulness: i. Clssif molecules. ii. Sve computtionl efforts. iii. Determine selection rules. 6

7 II. Chrcter tble Representtion nd chrcter tble Representtion: mthemticl elements representing smmetr opertions. Exmple: H O (C v group, four elements) E: (x z) (x z) C : (x z) (-x - z) v : (x z) (x - z) v ': (x z) (-x z) In prticulr, x z s bses v C (x z) = v (-x - z) = (-x z) = v '(x z) Multipliction tble st opertion nd opertion E, C, v, v ' E E, C C, v, v ' C, E, v ', v v v, v ', E, v ' v ', v, C, E C 7

8 8 Mtrix rep: E: z x z x C : z x z x v : z x z x v ': z x z x Verifiction: v v C 3-D rep (3) spnned b (x z)

9 Trce of mtrix (sum of digonl elements) is clled chrcter E C v v ' 3 - ( (3) ) (3) cn be reduced to direct sum of two mtrix reps: (3) = () + () () is spnned b z (irreducible representtion, irrep). E C v v ' ( () ) () cn be further reduced to the direct sum of two D reps spnned b x nd. Chrcter tble: list of chrcters of ll its irreps. E C v v ' bsis A z A - - x B - - x B - - The lst column is the bsis for different irreps. E C x x, x x, v x x( ) v, x ( x) 9

10 Smmetr species (lbel of irrep) A: -D irreps with + under principl rottion. B: -D irreps with E: -D irreps. T: 3-D irreps. Subscript determined b for v or perpendiculr C xis. Dimension of n irrep (d j ): size of mtrix of the irrep. Order (h): totl number of opertions. Clss: opertions of the sme kind. Properties of chrcters for irreps i. Chrcters re unique, independent of bsis. ii. Chrcters of elements in the sme clss re identicl. iii. Order of group is relted to dimensions of irreps b j d j h iv. Chrcters form set of mutull orthogonl vectors (Grnd Orthogonlit Theorem). ( O) ( O) O i j h ij where O denotes the smmetr opertions.

11 A A : ( ) = A A : [ + + (-)+ (-)] = v. Number of irreps = number of clsses. vi. An rep cn be decomposed into irreps: ( O ) i ( O) i i ( O) i( O) h O i vii. To construct the bsis for prticulr irrep (i), define projection opertor: d Pˆ i i i ( O) Oˆ, h O where d i is the dimensionlit of the irrep. Exmple: A mtrix rep for C v group hs the following chrcters: =,,,. Determine how mn times ech irrep is contined in it. A A ( ) ( )

12 B B ( ) ( ) ( ) ( ) In other words: A A B III. Applictions Clssifiction of MOs (H O): : =c (sh shb) + c so + c 3 po z becuse this MO is invrint under ll smmetr opertions in C v. Thus, it is bse for the A irrep. Similrl, b : =c (sh shb) + c po Orbitl degenerc is determined b under E. Vnishing integrls nd SALC: Onl AOs with the sme smmetr species form MOs becuse otherwise the overlp integrl is zero.

13 Consider n overlp integrl I = f f d f f must contin the totl smmetric irrep A if I is non-zero. Exmple: Judge whether I is zero if f = p x nd f = p for H O. i. Find the irrep ech function belongs to nd write its chrcters. f : B - - f : B - - ii. Multipl them together b column f f : - - iii. Find out if it contins A. If not, I =. The chrcters belong to the A irrep. So I is zero. A spectrl trnsition is forbidden if the trnsition dipole is zero. Exmple. Judge whether I = f f f 3 d is zero if f = p x, f = p nd f 3 = x for H O. 3

14 The chrcters for the bses re f : B - - f : B - - f 3 : A - - The product: f f f 3 : So, the chrcters belong to the A irrep. So I m be non-zero (but could be ver smll). Smmetr-dpted liner combintion LCAO-MO with moleculr smmetr is clled smmetrdpted liner combintions (SALC). Projection opertor method: The projection opertor for prticulr irrep (i) is defined s follows: ˆ di P ( ) ˆ i i O O h O For the A irrep in C v for H O, we hve

15 Pˆ A E C v v nd Pˆ A sh E C v v sh Pˆ sh sh sh sh sh sh sh sh sh A b b ˆ PA so E C v P ˆ p O= A x P ˆ p O= A b b b so so so so so v so So P ˆ p O=p O A z z : =c (sh shb) + c so + c 3 po z 5

16 For the B irrep, we hve nd Pˆ B E C v v Pˆ B sh E C v v sh sh sh sh sh sh sh b b b Pˆ so B P ˆ p O= B x P ˆ p O=p O B So P ˆ p O= B z b : =c (sh shb) + c po Tbultion method: i. Tbulte results of ll opertion on AOs, ii. Multipl the chrcters to ech column, iii. Add together ll the results. sh sh b so p x O p O p z O E sh sh b so p x O p O p z O 6

17 C sh b sh so -p x O -p O p z O v sh b sh so p x O -p O p z O v ' sh sh b so -p x O p O p z O For A species, multipl ( ) to the first column: = (sh + sh b + sh b + sh )/ = (sh + sh b )/ second column: = (sh b + sh + sh + sh b )/ = (sh + sh b )/ third column: = so fourth nd fifth columns: = nd the sixth column: = p z O So, the SALC-MO with A smmetr (the orbitl) is the sum of ll the bove: : =c (sh shb) + c so + c 3 po z 7

18 For the B irrep, we hve for st (nd nd ) column = (sh - sh b - sh b + sh )/ = (sh - sh b )/ 3 rd nd th columns = 5 th column = p O 6 th column = So b : =c (sh shb) + c po 8