⎡ 1 ⎤ x z : up Transformation Matrices; xˆ = ⎢ 0 ⎥ y ⎢ ⎥ ⎢ 0 ⎥ Geometric and Otherwise ⎣ ⎦ φ1 φ3 N x • ⎡ 0 ⎤ σv2 σv1 As examples, consider the transformation ⎢ ⎥ 2 yˆ = 1 φ3 C ⎢ ⎥ H 3 matrices of the C group. The form of y φ1 φ2 3v ⎣⎢ 0 ⎦⎥ φ2 σv3 C3 these matrices depends on the we ⎡ 0 ⎤ zˆ = ⎢ 0 ⎥ Cartesian basis: choose. Examples: ⎢ ⎥ 1 • Cartesian vectors: xˆ , yˆ , zˆ ⎣⎢ ⎦⎥ ⎡ 1 3 ⎤ ⎡ 1 3 ⎤ ⎡ ⎤ − 2 − 2 0 − 0 1 0 0 ⎢ ⎥ ⎢ 2 2 ⎥ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎢ ⎥ 3 1 2 ⎢ ⎥ 3 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ E = 0 1 0 C3 = 2 − 2 0 C3 = ⎢ − − 0 ⎥ xˆ = 0 yˆ = 1 zˆ = 0 ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 ⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ 0 0 1 ⎦⎥ ⎢ 0 0 1 ⎥ ⎣⎢ 0 ⎦⎥ ⎣⎢ 0 ⎦⎥ ⎣⎢ 1 ⎦⎥ ⎣ ⎦ ⎡ − 1 − 3 0 ⎤ ⎡ − 1 3 0 ⎤ ⎡ ⎤ • p orbitals on the N atom of NH3 ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ 1 0 0 3 1 3 1 ⎢ ⎥ σ v1 = ⎢ − 0 ⎥ σ v2 = ⎢ 0 ⎥ σ v3 = 0 −1 0 • the three 1s orbitals on the hydrogen atoms of NH3 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 ⎥ ⎣⎢ 0 0 1 ⎦⎥ ⎣⎢ 0 0 1 ⎦⎥ ⎣ ⎦

x z : up y φ Example Example, φ1 3 N σv1 φ1 φ3 σv2 N Answers 2 x σv2 σv1 H φ3 C3 φ 2 φ φ1 2 H φ3 C3 2 φ2 σv3 φ2 y φ1 C3 σv3 C3 ⎡ 1 0 0 ⎤ ⎡ 0 0 1 ⎤ ⎡ 0 1 0 ⎤ Three 1s orbitals on the hydrogen atoms of NH3 E ⎢ 0 1 0 ⎥ C ⎢ 1 0 0 ⎥ C 2 ⎢ 0 0 1 ⎥ = ⎢ ⎥ 3 = ⎢ ⎥ 3 = ⎢ ⎥ ⎣⎢ 0 0 1 ⎦⎥ ⎣⎢ 0 1 0 ⎦⎥ ⎣⎢ 1 0 0 ⎦⎥ ⎡ 1 0 0 ⎤ ⎡ 0 0 1 ⎤ ⎡ 0 1 0 ⎤ ⎢ 0 0 1 ⎥ ⎢ 0 1 0 ⎥ ⎢ 1 0 0 ⎥ σ v1 = ⎢ ⎥ σ v2 = ⎢ ⎥ σ v3 = ⎢ ⎥ ⎣⎢ 0 1 0 ⎦⎥ ⎣⎢ 1 0 0 ⎦⎥ ⎣⎢ 0 0 1 ⎦⎥ Transformation of d orbitals Group Representations

d (l = 2,m = 0) ∝ 1 5 (3cos2 θ −1) ⎫ 0 l 4 π ⎪ •Representation: A set of matrices that 1 15 ±iϕ ⎪ see, e.g., Atkins & de Paula, d±1 (l = 2,ml = ±1) ∝ (∓) 2 sinθ cosθe ⎬ 2π Physical Chemistry “represent” the group. That is, they behave 1 15 2 ±2iϕ ⎪ d±2 (l = 2,m = ±2) ∝ 4 sin θ e ⎪ 2π ⎭ in the same way as group elements when 1 1 15 1 iϕ −iϕ 1 15 1 15 d = [−d + d ] ∝ sinθ cosθ [e + e ] = sinθ cosθ cosϕ ∝ × 2xz xz 2 1 −1 2 π 2 2 π 4 π products are taken. −1 1 15 1 iϕ −iϕ 1 15 1 15 d = [d + d ] ∝ sinθ cosθ [e − e ] = sinθ cosθ sinϕ ∝ × 2yz yz i 2 1 −1 2 π 2i 2 π 4 π •A representation is in correspondence with 1 1 15 2 1 2iϕ −2iϕ 1 15 2 1 15 2 2 d 2 2 = [d + d ] ∝ sin θ [e + e ] = sin θ cos2ϕ ∝ × x − y x − y 2 2 −2 4 π 2 4 π 4 π ( ) the group multiplication table. 1 1 15 2 1 2iϕ −2iϕ 1 15 2 1 15 d = [d − d ] ∝ sin θ [e − e ] = sin θ sin2ϕ ∝ × 2xy xy i 2 2 −2 4 π 2i 2 π 4 π 1 5 2 1 15 1 2 2 •Many representations are in general possible. d 2 = d ∝Y ∝ 3cos θ −1 ∝ × 3z − r z 0 20 4 π ( ) 4 π 3 ( ) 1 1 •The order () of the matrices of a d1 = − 2 [dxz + idyz ] ; d−1 = 2 [dxz − idyz ] 1 1 2 2 2 2 representation can vary. d2 = 2 [dx − y + idxy ] ; d−2 = 2 [dx − y − idxy ]

Example - show that the matrices Reducible and Irreducible Reps. found earlier are a representation • If we have a set of matrices, {A, B, C, ...}, that form a representation of a group and we can ⎡0 0 1⎤ ⎡0 1 0⎤ ⎡1 0 0⎤ find a transformation , say Q, that eg., C C 2 = ⎢1 0 0⎥ ⎢0 0 1⎥ = ⎢0 1 0⎥ = E serves to “block factor” all the matrices of 3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢0 1 0⎦⎥ ⎣⎢1 0 0⎦⎥ ⎣⎢0 0 1⎦⎥ this representation in the same block form by similarity transformations, then {A, B, ⎡1 0 0⎤ ⎡0 0 1⎤ ⎡1 0 0⎤ ⎡0 1 0⎤ C, ...} is a reducible representation. If no (σ )−1C σ = ⎢0 0 1⎥ ⎢1 0 0⎥ ⎢0 0 1⎥ = ⎢0 0 1⎥ = C 2 v1 3 v1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3 such similarity transformation is possible ⎢0 1 0⎥ ⎢0 1 0⎥ ⎢0 1 0⎥ ⎢1 0 0⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ then {A, B, C, ...} is an irreducible representation. Similarity Transformation Reducing a Representation by maintains a Representation Similarity Transformations Suppose the group multiplication rules are ⎡A1 ⎤ such that AB = D, BC = F , etc ... ⎢ A ⎥ ⎢ 2 ⎥ • Q−1AQ = ⎢ ⎥ Now perform similarity transforms using ⎢ ⎥ A ⎡C1 ⎤ ⎢ 3 ⎥ the transformation matrix Q: ⎢ C ⎥ ⎣⎢ ⎦⎥ ⎢ 2 ⎥ -1 –1 –1 Q−1CQ = ⎢ ⎥ A´ = Q AQ, B´ = Q BQ, C´ = Q CQ, etc. ⎢ ⎥ C ⎢ 3 ⎥ • ⎡B1 ⎤ Multiplication rules preserved: ⎣⎢ ⎦⎥ ⎢ B ⎥ –1 –1 –1 ⎢ 2 ⎥ A´B´ = (Q AQ)(Q BQ) = (Q DQ) = D´ Q−1BQ = ⎢ ⎥ ⎢ ⎥ –1 –1 –1 B B´C´ = (Q BQ)(Q CQ) = (Q FQ) = F´, etc. ⎢ 3 ⎥ ⎣⎢ ⎦⎥

“Blocks” of a Reduced Rep. are Recall: also Representations This must be true because any group AB = C multiplication property is obeyed by the subblocks. If, for example, AB = C, then a11 a12 a13 a14 a15 b11 b12 b13 b14 b15 c11 c12 c13 c14 c15 A1B1 = C1, A2B2 = C2 and A3B3 = C3. a21 a22 a23 a24 a25 b21 b22 b23 b24 b25 c21 c22 c23 c24 c25 a31 a32 a33 a34 a35 b31 b32 b33 b34 b35 = c31 c32 c33 c34 c35 Example: Show that the ⎡ 1 −1 1 ⎤ matrix at left, Q , can reduce ⎢ 2 6 3⎥ a41 a42 a43 a44 a45 b41 b42 b43 b44 b45 c41 c42 c43 c44 c45 the matrices we found for the Q = ⎢−1 −1 1 ⎥ a51 a52 a53 a54 a55 b51 b52 b53 b54 b55 c51 c52 c53 c54 c55 ⎢ 2 6 3⎥ representation given earlier. ⎢ 0 2 1 ⎥ ⎣ 6 3⎦ a41 × b12 + a42 × b22 + a43 × b32 + a44 × b42 + a45 × b52 = c42 A Block Factoring Example Significance of Transformations

★ Irreducible Representations are of pivotal ⎡0 0 1⎤ ⎡ 1 −1 0 ⎤ ⎡ 1 −1 1 ⎤ importance ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ 2 6 3⎥ ⎢ ⎥ −1 ⎢ ⎥ ⎢ ⎥ Q C Q −1 −1 2 1 0 0 −1 −1 1 ★ Chosen properly, similarity transformations can 3 = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 6 6 6 ⎢ ⎥ 2 6 3 ⎢ 1 1 1 ⎥ ⎢ 2 1 ⎥ reduce a reducible representation into its ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ 3 3 3⎦ ⎣⎢0 1 0⎦⎥ ⎣ 6 3⎦ irreducible representations ★ With the proper first choice of basis, the ⎡ −1 3 0⎤ ⎡ 1 − 3 0⎤ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ transformation would not be necessary −1 ⎢ ⎥ −1 ⎢ ⎥ Q C Q − 3 −1 Q Q − 3 −1 ★ 3 = ⎢ 2 2 0⎥ σ v1 = ⎢ 2 2 0⎥ Important Future goal: finding the basis functions ⎢ 0 0 1⎥ ⎢ 0 0 1⎥ for irreducible representations ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

Great Theorem Great Orthogonality Theorem - again • ∗ h Vectors formed from matrix elements from the [ (R) ][ (R) ] th th ∑ Γi mn Γ j m′n′ = δijδmm′δnn′ m rows and n columns of different irreducible R l l i j representations are orthogonal: ∗ 2 Proofs: Eyring, H.; Walter, J.; Kimball, G.E. Quantum Chemistry; Wiley, 1944. [ (R) ][ (R) ] 0 if i j http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter4.pdf ∑ Γi mn Γ j mn = ≠ R Γ (R) — matrix that represents the operation R in the ith representation. i • Such vectors formed from different row-column Its form can depend on the basis for the representation. th th sets of the same irreducible representation are [Γi (R)mn ] — matrix element in m row and n column of Γi (R) th orthogonal and have magnitude h/li : li — the dimension of the i representation ∗ h — the order of the group (the number of operations) [ (R) ][ (R) ] (h l ) ∑ Γi mn Γi m′n′ = i δmm′δnn′ δ = 1 if i=j, 0 otherwise ij R The First Sum Rule Second Sum Rule The sum of the squares of the characters in The sum of the squares of the dimensions any irreducible representation equals h, the of the irreducible representations of a [ (R)]2 h group is equal to the order of the group, order of the group ∑ χi = that is, R 2 2 2 2 “Proof”— From the GOT: li = l1 + l2 + l3 + ⋅⋅⋅ = h ∑ ∗ [ (R) ][ (R) ] (h l ) i ∑ Γi mn Γi m′n′ = i δmm′δnn′ this is equivalent to: R 2 ∗ let m=m’=n=n’: [Γ (R) ][Γ (R) ] = (h l ) χi (E) = h ∑ i mm i mm i ∑[ ] R i

Characters of Different Irreducible Proof Representations are Orthogonal Setting m = n in first GOT statement:

∑Γi (R)mm Γ j (R)mm = 0 if i ≠ j The vectors whose components are the R characters of two different irreducible representations are orthogonal, that is, compare this to the statement (i ≠ j):

⎧⎡ ⎤ ⎡ ⎤⎫ ∑ χi (R)χ j (R) = ∑⎨⎢∑Γi (R)mm ⎥ ⎢∑Γ j (R)mm ⎥⎬ R R ⎩⎣ m ⎦ ⎣ m ⎦⎭ ∑ χi (R)χ j (R) = 0 when i ≠ j R ⎡ ⎤ = ∑⎢∑Γi (R)mm Γ j (R)mm ⎥ = 0 m ⎣ R ⎦ Matrices in the Same Class have # of Classes = # of Irred. Reps. Equal Characters The number of irreducible representations of a group is equal to • This statement is true whether the the number of classes in the group. representation is reducible or irreducible • This follows from the fact that all ∑ χi (R)χ j (R) = hδij elements in the same class are conjugate R and conjugate matrices have equal if the number of elements in the mth class characters. is gm and there are k classes, k ∑ χi (Rp )χ j (Rp )gp = hδij p=1