MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.80 Lecture #29 Fall, 2008 Page 1 of 8 pages Lecture #29: A Sprint Through Group Theory Bernath 2.3-4, 3.3-8, 4.3-6. I'll touch on highlights Symmetryodd vs. even integrands → 0 integralsselection rules for matrix representation of any operator* transition moment * H ← block diagonalization generation of symmetry coordinates how to deal with totality of exact ⎡⎣O, H⎤⎦ = 0 approx. ⎡⎣O, H°⎤⎦ = 0 convenient C2, H ROT a,b,c ⎡⎣⎤⎦ = 0 symmetries Chapter 2: Molecular Symmetry rotation C n (axis) rotation by 2π n radians about (specified) axis ( C n C n = C n 2 etc.) reflection σ (plane) reflect thru plane σv vertical (includes highest order Cn axis) σh horizontal (⊥ to highest order Cn axis) σd dihedral (also vertical, bisects angle between 2 C2 axes ⊥ to Cn) contrast to I - inversion in lab (parity) inversion in body ˆi = C 2 σ h inversion (C2 axis ⊥ to plane of σ h)improper rotation Sn = σ hC n = C n σ h (Cn axis ⊥ to plane of σ h) [i = S2]identity E do nothing Groups: Closure Associative MultiplicationIdentity ElementInverse of every element R. Rigid isolated molecules — point groups — all symmetry elements intersect at one point[distinct from translational symmetries — periodic latticesCNPI - nonrigid molecules (Complete Nuclear Permutation-Inversion)MS - (Molecular Symmetry Group) subgroup of CNPI, isomorphic with point group, butmore insightful (especially when dealing with transitions between different point-group structures)] Point Group notationCs , Ci , Cn, Dn , Cnv , Cnh , Dnh , Dnd ↓↓ ↓↓↓↓ ↓ 1 plane inversion nC2⊥Cn nσv Cn+ σh Cn + nC2⊥ + σh Cn + nC2⊥ + σdFall, 2008 Page 2 of 8 pages Sn Td Oh Ihtetrahedral octahedral icosohedral [Flow Chart: Figure 2.11, page 52 of Bernath] Khspherical Bernath Chapter 3. Matrix Representations ⎛ ⎞ x ⎜ ⎜⎜⎝ ⎟ ⎟⎟⎠ which means r = xˆi + yˆj + zkˆ = xi eˆi ∑ r = y z i eˆ1 eˆ2 eˆ3 convenient notationx1x2 x3 Apply symmetry operator, R , to coordinates of an atom (“Active”) ⎛ ⎛ ⎞ ⎟ ⎟⎟⎠ = ⎜ ⎜⎜⎝ x′ 1 x′ 2 x′ 3 ⎛ ⎞ ⎟ ⎟⎟⎠ ( ) = D R ⎜ ⎜⎜⎝ x1 x2 x3 ⎞ ⎟ ⎟⎟⎠ x1 x2 x3 ⎜ ⎜⎜⎝ R D R symmetry operator.( ) is a 3 × 3 matrix representation of the R1 0 0 ⎛ ⎞ ⎜ ⎜⎜⎝ ⎟ ⎟⎟⎠−1 cθ sθ 0 D(σ(12))= 0 1 0 0 0 ⎛ ⎞ ⎜ ⎜⎜⎝ ⎟ ⎟⎟⎠ ( D C)= (3) 3 axis −sθ cθ 0 0 0 1 θθ → –θ U–1 = U† What is the inverse of D C) ?( (3)What are the characteristics of a unitarytransformation?( )=D C θ (3)−1 * normalized rows and columns * rows (and columns) are orthogonal ⎞ ⎟⎟⎟⎟⎟⎟⎠⎜⎜⎜⎜⎜⎜⎝⎛? 5.80 Lecture #29Fall, 2008 Page 3 of 8 pages −1 0 0 ⎛ ⎞ ⎜ ⎜⎜⎝ D( ) ˆi = 0 −1 0 0 0 −1 ⎛ cθ sθ 0 ⎞ ⎜ ⎟ ⎟ ⎟⎟⎠D S( θ (3) )= 0 difference between Sθ and C−sθ cθ⎜ −1⎠ ⎟⎟ θ ⎜⎝ 0 0 1 0 0 ⎛ ⎞ ⎜ ⎜⎜⎝ ⎟ ⎟⎟⎠ ( ) =D E0 1 0 0 0 1 We have been considering the effect of symmetry operations on coordinates of a point. We generatedmatrices which represent the symmetry operations by producing the intended effect on coordinates.These matrices have the same multiplication table as the symmetry operations themselves. The matrices form a representation of the group that includes these symmetry operations . We can form a matrix representation of any group by selecting any set of:BASIS VECTORS;coordinates of each atom in molecule;each equivalent bond;each equivalent angle;anything convenient. (over-complete is OK) Before generating lots of matrix representations, we must consider ACTIVE vs. PASSIVE coordinatetransformations. ACTIVE: move the object (r → r′). Change the coordinates of the object. PASSIVE: move the axis system. (eˆ→ eˆ′) Equivalence of the two kinds of transformation: the coordinates of the untransformed object in the newaxis system are identical to the coordinates of the transformed object in the old coordinate system. in matrix notationr = ∑eˆixi ⇒ r = etx  r = et ( ) x⎤⎦ = etr′ = R⎡⎣D Rx′ active (transformationapplied to the object) ( ) passive (transformation= ⎡⎣etD R⎤⎦x applied to the= e′tx coordinate system) 5.80 Lecture #29Fall, 2008 Page 4 of 8 pages e′t ≡ etD R( ) take transpose ( ) e′ =[etD R]t = Dt(R)e = D(R−1 ) e ! same as inverse for unitary matrix R acts on the coordinate system in the inverse sense to the way it acts on the object. We are now ready to construct 3N × 3N dimension matrix representations of effects of symmetryoperations on an N-atom molecule. We are going to simplify things soon to the traces or characters of these matrices, χ(R) : 3N ( ) ≡ ( )ii i=1 χ R∑D R( ) !Keep this in mind when we focus on only what appears along the diagonal of D RIf a symmetry operation causes 2 atoms α, β to be permuted, all information about this is in the α, β off-diagonal 3 × 3 block. ⎛ ⎜⎜⎜⎜⎜ α α,β  ⎝β ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠  no contribution to character, χ 5.80 Lecture #29Fall, 2008 Page 5 of 8 pages NON-LECTURE What about the effect of a symmetry operation on a function?f(x) = a numberactive: move the function f(x′) = a different #passive: move the coordinate system, which changes the function so that f′(x) ≠ f(x) [but it mustbe true that f′(x) = f(x′)]We want to find out what f′(x) is in terms of a complete orthogonal set of basis functions. How do we do this? We require that f(x) = f′(x′). The new function operating in the new coordinate system gives thesame number as the old function operating in the old coordinate system.See pages 75-76 in Chapter 3 of Bernath for how to derive the new functions in terms of oldcoordinates f(x,y,z) = xyz for example OC3(z)f(x,y,z) = f (x,y,z) =(− 31 / 2 / 2 )x12 + 31 / 2 / 2 )x2 ′( ( 2 + x1x2 )x3 / 2 So we know how to derive a matrix representation of any symmetry operation.NOT unique, but it doesn't matter because regardless of what set of orthogonal basis vectors we use togenerate our matrices, the matrices* have the same trace (sum of eigenvalues)* have