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III-1 III. Symmetry and Group Theory Levine – Chap 12 McHale – Appendix C More focused texts: Cotton – Chap 3 Harris & Bertolucci – Chap 1 A. Categorize Symmetry – operations that put molecule into equivalent state/orientation [exchange equivalent atoms by rotation/reflection/inversion] O Water example H1 H2 rot o 180 O H2 H1 H1, H2 indistinguish cos sin x Mathematical representation of rotation: sin cos y 1 0 x - x for rotation about z of = 180o: 0 -1 y - y x y Another operation: reflection through (xz) plane: (x,y,z) → (x,-y,z) Operations (symbols) Cn – rotation about on axis by 2 n – reflection through plane – vertical h – horizontal - Cn d – dihedral – bisect 2 C2 axes i – inversion: (x,y,z) → (-x,-y,-z) Sn – improper rotation: combine: ( 2 rotation + h reflection Sn) n z N NH3 example: C3 – z-axis 2 = 120º rotation - 2 of them, i.e. 2 3 3 – contain each N-H bond, exchange other 2 (always get nC for Cn → C3v , if there is a v) CH4 example: C2 – bisect each CH2 (3 of them x,y,z) S4 – Same - 90º + flip H’s (6 of them x,y,z) C3 – along each C-H bond (8 - exchange 3 H’s 2 ) 3 – contain each CH2 (6 - 2 on each x,y,z) SF6 example: C4, C2, S4 – each F-S-F bond pair (x,y,z) ‘s on axes (contain xy, yz, xz) and d’s between axes plus i – all invert through center H1 y H3 x H2 H H C H H III-2 Categorize these symmetry operations – Point Group – mathematical entity with defined properties, relations between operations Molecules that have same group (set of operations) will have common properties Comes from – [H, sym] = 0 i.e. energy is uncharged if molecules same Basic operations: Cn, Sn, , n, d, i, + E (do nothing) “identity” 1 – Find all symmmetry elements by inspection (models help) 2 – Use flow chart (handout, e.g. Levine) to determine point group a. Special → Spherical → full rotation group (atoms) Cubic → T d, On, In … (2 or more n ≥ 3 Cn) Linear → C∞, D∞h ( has i) b. Cn determine group → highest order rotation → it n ≥ 3 This will be the principle axis for symmetrical top (if none then C1, Ci, Cs → depend on other elements) c. If n C2 Cn → then a D group (otherwise C) if n → Dnh or Cnh n → Dnd or Cn none → Dn or Cn III-3 N N C C Pt Practice (Handout) , C , N C , FeCl5Br , FeCl5H2O , N O FCH3 , B2H6 , BH3 , H2C CH2 , Fe(Cp)2 –ferrocene III-4 Value – symmetry characterization can tell if QM integral = 0 or ≠0 How to handle → collection of operations forms a Group Group Theory can be used to evaluate properties ( =0 or ≠0 ) without knowing function [note: analogy to angular momentum → YJM are symmetry representations of full rotation group] Read PJS handout (mathematical start—on line, Web Site) Cotton, Chap 2; Harris & Bertolucci, Chap 1; Douglas & Hollingsworth, Chap 2 B. Group – Definitions: set of elements {} = G under an operation Properties: 1) Closure – A B G, A, B G (don’t leave the group) 2) Identity – E A = A, All A < G, E G (one does nothing) 3) Reciprocal (Inversion) – A-1 A = E and A-1 G if A G (can undo anything, A-1 is in G) 4) Associative – (A B) C = A (B C) Note: order maintained but can combine pairs either way Mathematical examples: a) {1, -1} = multiplication E = 1, self inverse b) {negative, positive integers} = addition, E = 0, -N = N-1 (inversion) c) {E, A, A2} successive application A, E = A3 (cyclic), A-1 = A2 Symmetry example (H20) {E, C2, , } each is self inverse Multiplication Table E C2 E E C2 When working out keep track of sign of x,y,z x -x C2 y y → C2 C2 E E C2 C2 E Note: a) not require A B = B A (commutation) but if true Abelian Group b) each element occurs only once in multiplication Table Ai, AK G, Ar = AiAK-1 G, ArAK = Ai Ai in column but h elements Ai and h entries in a column → occur only once Form of multiplication table will be limited E A = {A,E}; h = 2 A E E A B h = 3 A B E B A A cyclic group A A A 2; A 2 A A 2 E only one 2 if A E then B A A B E III-5 Construct Multiplication Table NH3 → note self-inversion define class x-1 AX class but C3-1 = C32 NH3 – C3 – (C3, C32) – class {A,B} class conjugate (1,2,3) – class complete set See Multiplication Table blocks out by classes Sub Group {E, C3, C32} cyclic – step in itself Mirrors: self inverse 1-1 = 1 but ij = C3m (rotation) {E i} also sub group – 3 of them III-6 C. Representation Since multiplication tables limited characteristic of interconversion of elements – choose simplest one to represent the rest (easier manipulation) 2 1 2 3 4 E C2 1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1-4 are representation of the group all have same multiplication table as the 2 Class X-1AX = B A, B conjugate If {A,B} – complete the class i) every element conjugate to itself X-1XX = X ii) if A conj B then B conj A X-1AX = B XB = AX XBX-1 = A iii) If A conj B and C, then B conj C X-1BX = A = Y-1CY YX-1BXY-1 = C XY-1 = (XX-1)-1 B conj C iv) E in class by itself (X-1 EX = E always) order of the class is integral factor of order of group, h Representation behaves like vectors C2: {1 1 1 1}, {1 -1 1 -1}, {1 1 -1 -1}, {1 -1 -1 1} Note – these are orthogonal (scale pred = 0) Now any four vector is linear ???? of these (a b c d) = c i i → i spare space i 1 0 , 1 0 , 1 0 , 1 0 , etc Also matrices: 5 0 1 0 - 1 0 - 1 0 1 so this is possible to extend to larger and larger reps 5 1 0 not independent, combine i i = 1-4 but note: 4 General similarity transform {E,A,B…} set of matrices {A-1E Z, Z-1A Z, Z-1B Z, …} = {E´, A´, B´, …} These also representation of the group (same multiplication table) A ·B = D A´B´ = (Z-1A Z)(Z´B Z) = Z-1A B Z -1 = Z D Z = D´ For any matrix Rep there exist unitary matrix Z or unitary def: UU+ = E, U-1 = U+, Uij-1 = Uji+ + – adj out which does similarly transform of rep to block diagonal For the smallest blocks → irreducible representation a 0 1 (blocks do not mix on multiplication: 0 a 2 b 0 a b 0 1 1 1 0 b 0 a b 2 2 2 III-7 maintain shape on multiplication (Cotton, Pg 66) each block has multiplication table of group → rep of group Grand Orthogonality Theory i (R)nm j (R)nm * R h ij nm mm i j R → element in the group (symmetry op) i → irreducible representation (j another one) n,m → indices of matrix elements in representation ℓi → order (dimension) of matrix in i h → order of the group i j Ex: C2 4 1 ij 1 1 = 1+1+1+1 = 4 23 = 1-1-1+1 = 0 R Impact a) Products of elements of irreducible rep. restricted b) Irreducible rep behave as orthogonal and vector ( δ ij ) c) Magnitude normalized to h i for i d) Elements of matrices orthogonal for different rep.: nn , mm e) All of this limits number and size irreducible rep. sum: i 2 h – clear dimensional construct – also affects “shape” of rep. ex: C2 - h = 4 must have E – 1111 ℓi = 1 solution ℓi = 2 (only one), ℓi = 1 (4 of them) C2 = h = 6 solution: ℓi = 1 (6 of them), ℓi = 1 (-- 2), ℓj = 2 ( – one), only alteration now need rule to choose → classes But if ℓi = 1 – 6 vectors must be orthogonal –– can you do it? Characters → simplify matrix rep by Trace Trace of matrix invariant to similarity transform [eg: reducible rep → sum irreducible rep → character same] i i (R) Tr i (R) i (R) ← diagonal sum of matrix element 1 Now all elements in class related by similarity transform so have same character [condense Tables] GOT recall the mm´ nn´ → only diagonals contribute * i (R)mm j (R)nn R m n h i j ij nm mn III-8 since ij nn i i j i nm left ij → orthogonal really only i = j contribute * i (R) j (R) h ij R characters for h-dimensional orthogonal vector sort by class * i (CK ) j (CK ) NK ij h K CK – class, Xi (CK) character of ith rep in CK class NK = number of elements in class Reduction of the space → reduced dimension = number classes since all elements of class same character → vectors dependent or there are not h independent vectors number classes = number of independent components (character element) = number of independent vectors can form number classes = number of irreducible rep → character Tables Grand Orthogonality Examples C2 1 2 3 4 E C2 1 1 1 1 1 -1 1 -1 1 1 -1 -1 e.g. i i R 1 -1 -1 1 41 2 2 [ (1 1 (-1)(-1) 1 1 (-1)(-1) ] 4 if i j i j 0 R try 2 3 [ (1 1 (1) 1 1 (-1) (-1)(-1)] Note: the 5 , 6 rep’s are not irreducible n i2 h C2: h = 4 could have ℓi = 1, n=4 or ℓi = 2, n = 1 i -1 but must have one ℓi = 1 for simple rep: 1,1,1,1 ℓi = 1, only 4 irreducible rep C3: h = 6 can do: ℓi = 1, n=6 are there 6 independent 6-vectors n = 5, 4 will not work ex: n = 5 i2 5 i 8 for ℓi = 1 for ℓi = 1, ℓ5 = 1 III-9 n = 3: ℓ1 = ℓ2 = 1, ℓ3 = 2 i2 1 1 4 6 n = 8 – ℓi = 1 C4: h = 8 trivial 2 N = 5 – ℓi = 1, ℓ5 = 2 C2 Representations 1 2 3 4 E 1 1 1 1 C2 1 -1 1 -1 8 Multiplication Table (yz) ´(xz) 1 1 1 -1 -1 -1 -1 1 E C2 E E C2 C2 C2 E ´ ´ ´ (yz) ´(xz) ´ ´ E C2 C2 E 1 0 1 0 1 0 1 0 1 0 5 , , , 0 1 0 - 1 0 - 1 0 1 0 4 1 0 eg. 0 - 1 like: C2 · 1 0 1 0 0 - 1 0 1 ´ = z O H1 y H H2 Interchange matrix; operate on 1 vector 1 0 0 1 E 0 1 1 0 C2 0 1 1 0 xz H2 1 0 0 1 x Rotation matrix; operate on y vector z 1 0 0 0 1 0 0 0 1 E - 1 0 0 0 - 1 0 0 0 1 C2 -1 0 0 0 1 0 0 0 1 1 0 0 0 -1 0 0 0 1 yz III-10 cos sin 0 - sin cos 0 0 0 1 = 180º Chem 406 We have seen that a set of matrices of dimension n can form a representation of the group if it has the same multiplication table Note: if n = 1 have set of integers, these will only work if each i 1 In general can convert this set of matrices to another with similarity transform. i.e. (E, A, B …) → (E', A', B' …) where A' = Z-1 A Z and Z is a unitary matrix Z Z+ = E (identity matrix) or Z-1 = Z+ , + → conjugate transpose or adjunct aside Hermitian: H = H+ , When Z is chosen so that (E, A, B …) become “most block diagonal” then the blocks are irreducible representation and original matrices A i A , etc. j i.e. A A These matrices can represent symmetry effects on a molecule: O H1 H1 as vector H2 H2 interchange of H1, H2 rep 1 0 0 1 E: Test 0 1 0 1 1 0 : ´: 1 0 1 0 0 1 C2: 1 0 0 1 1 0 C2 0 1 1 0 0 1 interchange of x y z (rotation of axes) x y vector z III-11 cos sin 0 C2: - sin cos 0 0 0 1 1 0 0 E: 0 1 0 0 0 1 = - 1 0 0 0 - 1 0 0 0 1 1 = 1 1 1 ´ = - 1 1 Note – these are all block diagonal if multiplication matrices product block diagonal and elements on diagonal of product correspond to products of diagonal → each block is a representation - same multiplication Examples z N H1 y H3 x H2 NH3 → C3 Elements: C3 about z , 3 (N – H) Operations: E; C3, C32 ; 1, 2, 3 Note: C3-1 = C32 i = i-1 self inverse Group Theory should reflect similarity of ’s, C3 Multiplication Table h=6 E C3 C32 1 2 3 E C3 E C3 C3 C32 1 2 3 C32 3 1 2 C32 C32 E E C3 2 3 1 1 1 2 3 E C3 2 2 3 1 C32 C3 3 3 1 2 X-1 A X: E C3 E = C3 C32 E C3 C32 C3 C3 C32 = C3 C3-1 = C32 1 C3 1 = 1 • 3 = C32 2) {E, 1} also SUBGROUP for each i, order = 2 E (self-conjugate) C32 C3 C3 = C3 Note: 1) {E, C3, C32} form a cyclic group SUBGROUP, order = 3 is a dvisor of h = 6 Class formed (C3, C32) same follow for III-12 2 C3 2 = 2 • 1 3 C3 3 = 3 • 2 = C32 = C32 i = i-1 (1, 2, 3) Mullikan notation Labeling irreducible rep as i is okay but gives no clue as to composureMullikan notation designed to denote dimension and symmetry (sym/anti) with r / t center open Dimension: 1D 2D 3D 4D A or B E T (old/phys – F) U (old/G) Principal axis 1D – A symmetrical B anti symmetrical C2 Principal subscript 1D 1 - symmetrical (or vertical plane) 2 – anti symmetrical note ambiguity if 2 classes h - horizontal plane superscript all dimensional ′ – symmetrical ″ – anti symmetrical inversion center subscript all g – symmetrical u – anti symmetrical 1 1 1 1 1 → A1 2 1-1 1-1 → B1 3 1-1-1 1 → B2 4 1 1 -1 -1 → A2 Note all groups need A total symmetry Ex. C2 - all 1D C3 C4 1 - 1 1 1 1 → A1 2 - 1 1 1 1 → A2 3 - 2 - 1 0 → E A1, A2, B1, B2, E Special Case – Abelian Groups Abelian → operations (el of G) commute Now if AB = BA for all el < G X-1AX = X-1XA = A therefore every element is in a class by itself Thus given our rules: number of class = number of irreducible rep h dimension abelian group → h rep from i h i 1 for all irreducible rep III-13 So Abelia has h 1-D irreducible rep and momentum Example – Cyclic Group: E, A, A2… Ah-1 So for 1D (E) 1 E = Ah 1 ( A ) (1 ) h E A A1 1 1 A2 Ah-1 1 1 11 Easy solution: p ( A ) e2i p h , p (E) [e2i p h ] h e In this sense A1 = h p = h all rep e2i = 1 HANDOUT Example: C3 C3 C32 C32 = E 1 2 3 e2i/3 e4i/3 e2i = 1 e4i/3 e8i/3 (e2i)2 = 1 e6i/3 = 1 e12i/3 = 1 (e2i)2 = 1 1 1 1 E 1 1 1 C32 C32 1 1 e2i/3 e4i/3 e4i/3 e2i/3 (e2i=1) so then 3 = A1 = rearrange A1 → clockwise → counter Note: e4i/3 = e-2i/3 = (e2i/3=1) * A1 E E 1 1 1 C32 C32 1 * 1 * The second two representations are basically the same thing but correspond to rotations clockwise and counter-clockwise – eventually they will behave alike → in quantum mechanics → give the symbol E – degenerate but separable if have ???? of ???? HANDOUT / Example What do all of these things have to do with molecules? III-14 Consider: to describe bonding we will need to discuss O H H O: 2s, 2pz, H: 1s1, 1s2 2s : 2pz : 2px : 2py E 1 C2 1 ´ 1 1 1 1 1 1 1 -1 -1 1 2px, 2py i.e. no effect A1 -B1 (try) H H1 + H2 : 1 1 1 1 A H1 - H2 : 1 -1 -1 1 -B1 –- doesn’t work H1 → H2 or 2p = 2px + 2py + 2pz = 3 = A1 + B1 + B2 aA1 = 1 4 aA2 = 1 4 aB1 = 1 4 aB = 1 2 4 -1 1 1 (3 • 1 + -1 • 1 + 1 • 1 = 4 9 (3 • 1 + -1 • 1 + 1 – -1 + 1 = 1 (3 – 3) = 0 4 (3 • 1 + (-1)(1) + 1 • -1 + 1 = 4 = 1 4 (3 • 1 + (-1)(1) same or consider vibration of H2O bond stretches, angle bonds = OH = OH + OH = OH - OH = or 2OH = E 1 C2 1 ´ 1 1 A1 1 1 don’t work A1 -B1 1 1 1 -1 1 -1 2 0 0 2 = 1 (2 • 1 + 2 • 1) = 1 4 aB1 = 1 (2 • 1 + 2 • 1) = 1 4 aA 1 HANDOUT Cotton, Chap 5 2nd first Product of two symmetry species transforms as the direct product of their representation III-15 1-D – this is easy because rep = character = number hence 1 2 = X1(R1) X2(R1) , X1(R2) X2(R2) , … , X1(Rh) X2(Rh) Examples – use C2 A1 A2 B1 B2 yz E C2 xz 1 1 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 A2 B1 = 1•1, 1• -1, -1•1, -1• -1 = 1, -1, -1, 1 = B2 Rules: Since all 1-D can use shortcut tricks: A2 B2 = B1 B1 B2 = A2 etc. A•A=B•B=A A•B=B•A=B 1•1=2•2=1 2•1=1•2=2 similarly 1 • 1 = 11 • 11 = 1 1 • 11 = 11 • 1 = 11 g•g=u•u=g u•g=g•u=u HANDOUT Degenerate representations not so trivial Direct product matrix a11b11 a11b12 a12b11 a1nb1m a11 a12 a1n b11 b12 b1m a11b21 a b 21 21 a21b11 a21b12 a ann bm1 bmm n1 a a a b n1 nn nn mm n n m m mn mn new rep is much bigger and consists of series of m x n matrices – can see this will block out In our rotation A(R) B(R) = AB(R) in general AB will be a reducible rep n x n times III-16 ex. 1 3 1 0 2 2 , 1 0 , E ~ , 1 0 1 3 0 1 2 2 E C3 x2 In C3 1 0 E•E = 0 0 XE•E = 0 1 0 0 4 1 0 0 4 3 0 0 4 , 1 0 3 4 0 1 3 4 3 4 1 4 3 4 3 4 3 4 3 4 1 4 3 4 3 4 1 0 0 3 4 , 0 1 0 0 0 1 3 4 0 0 0 1 4 1 0 0 0 1 0 Ex – Cotton, Chap 5 1) Two most commonly used integrals for Quantum Mechanics calculations: * * i Ĥ j d dij and i j d Sij Rules above say 2nd one Sij = 0 unless i = j true also 1st one, Hij because Ĥ transforms as A1 so Ĥ j since XH(R) Xj(R) = Xj(R) j i.e. character product of rep = product character and character A1 = 1 for all R or Hij = Hii ij 2) Dipole selection rules – from time dependent perturbation th/semi ??? you learned i* ˆ j d 0 for electric dipole allowed transition n qi rˆi sum over charges in system, q’s constant i 1 transforms in same way as ̂ = (x, y, z) For non-zero integral need: i xyz j A 1 or if i A , common for ground state then i x, y,z (reducible in general) 1