Molecular and ensemble symmetry

  sum e = EJL R S β General objective: Reduce the complexity of the analysis by exploiting symmetry. Specific Objectives: 1. The molecular symmetry matrix S. How to populate it. 2. Relationships between the character tables for transitions of different symmetry and the molecular tensor. 3. The surface response for a uniaxial ensemble as a function of molecular symmetry. 4. Extension to other linear and nonlinear optical processes. 1 Let’s consider a simple example of a uniaxial surface assembly of “rod-like” molecules with a molecular tensor dominated by just the βzzz element. For C∞ surface symmetry, we have 4 unique elements for vibrational SFG. RZz = − sin θ cos φ ; RYz = cos θ , RXz = sin θ sin φ χ ZZZ : β zzz = N s cos3 θ β zzz 2 2 = χ ZXX : β zzz N= s cos θ sin θ cos φ β zzz 2 2 = χ XXZ : β zzz N= s cos θ sin θ cos φ β zzz Ns cos θ sin 2 θ β zzz 2 Ns cos θ sin 2 θ β zzz 2 χYZX : β zzz = N s − cos θ sin 2 θ sin φ cos φ β zzz = 0 2 Bottom up: Identification of the total set based on symmetry. C2v E Worked example: C2v A1 1 -The A1, B1, and B2 A2 1 symmetries all have B1 1 nonzero elements for B2 1 both µ and α. ( ) ∑  S n (ω2 ) α 0(1n) ⊗ µn 0 β 00ijk −ω3 ; ω1 , ω= 2 of unique molecular tensor elements C2 (z) σv(xz) σv(yz) Lin. Quad. 1 1 1 z 1 -1 -1 -1 1 -1 x xz -1 -1 1 y yz ( xy ) ∑ ,ω β 00ijk −2ω ; ω=  S m ( 2ω ) µ0 m ⊗ α m(10) m n SFG x2,y2,z2 αijµk SHG µiαjk A1 βxxz, βyyz, βzzz A1 βzxx, βzyy, βzzz A2 - A2 - B1 βxzx = βzxx B1 βxxz = βxzx B2 βyzy = βzyy B2 βyyz = βyzy 3 SFG αijµk A1 βxxz, βyyz, βzzz A2 - B1 βxzx = βzxx B2 βyzy = βzyy For a B1 transition in vibrational SFG, one unique and two nonzero tensor elements survive.  xxx   xxy     xxz     xyx   xyy     xyz   xzx     xzy   xzz     yxx     yxy   yxz     yyx   yyy     yyz   yzx     yzy     yzz   zxx     zxy   zxz     zyx   zyy     zyz   zzx     zzy     zzz  0  0    0    0  0  Symmetry matrix S,   0  populating the full set of 1    27 elements from the 0  0  subset of independent   0  tensor elements (27×1 in   0  this case). 0    0   0  ⋅ [ β ] = S ⋅β xzx   0  0    0  Independent, nonzero   0   elements within the 1    molecular tensor. 0  0    0  0    0  0    0    4 0  Top down: A more general approach exemplified for D4 Set of matrices corresponding to the C2(z) and C4(z) symmetry operations cos ( 2π / n ) − sin ( 2π / n ) 0    C ( z ) =  sin ( 2π / n ) cos ( 2π / n ) 0   0 0 1  1 n D4 E 2C4(z) Character C2(z) 2C' 2C"2 table 2 Lin. Quad. A1 1 1 1 1 1 A2 1 1 1 -1 -1 B1 1 -1 1 1 -1 x2-y2 B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 x2+y2,z2 z (x,y) - (xz,yz) Linear transformations (µ) 0 −1 0   µ x      1 = µ ' C4 ( z ) ⋅= µ 1 0 0  ⋅  µ= y 0 0 1   µ z  −µ y  µ   x   µ z   −1 0 0   µ x   − µ x     1 µ ' =C2 ( z ) ⋅ µ = 0 −1 0  ⋅  µ y  = − µ y   0 0 1   µ z   µ z  -For A2 symmetry, the transition moment is unchanged by this operation, consistent with a transition moment with µz as the only nonzero element. -For E symmetry, each invert in sign for the C2 operation, consistent with the negative sign for that operation. Both (µx,µy) transform together. 5 Top down: A more general approach exemplified for D4 Set of matrices corresponding to the C2(z) and C4(z) symmetry operations cos ( 2π / n ) − sin ( 2π / n ) 0    C ( z ) =  sin ( 2π / n ) cos ( 2π / n ) 0   0 0 1  1 n   α ' = C41 ( z ) ⊗ C41 ( z ) ⋅ α  α yy −α xy  =   −α yx α xx  α yz −α xz   α zy   −α zx  α zz   α ' = C21 ( z ) ⊗ C21 ( z ) ⋅ α  α xx  =   α xy  −α xz  α yx α yy −α yz −α zx   −α zy  α zz  D4 E 2C4(z) Character C2(z) 2C' 2C"2 table 2 Lin. Quad. A1 1 1 1 1 1 A2 1 1 1 -1 -1 B1 1 -1 1 1 -1 x2-y2 B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 x2+y2,z2 z (x,y) - (xz,yz) Quadratic transformations (α) -For A1symmetry, the α matrix is unchanged by this operation. Only the diagonal elements survive both C41 and C43 operations with no sign change. Since the operation swaps αxx and αyy, the two must be equal. -For B1 symmetry, swapping αxx and αyy through the C4 operation results in a sign change, such that the two must be equal and opposite. Analogous operation of the other C2 rotations separate out the (xy) and (xz,yz) off-diagonal elements. 6 Top down: A more general approach exemplified for D4 This process can be extended all the way to the complete molecular tensor.  D4 E A1 1 1 1 1 1 A2 1 1 1 -1 -1 B1 1 -1 1 1 -1 x2-y2 B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 β ' = C41 ( z ) ⊗ C41 ( z ) ⊗ C41 ( z ) ⋅ β  γ ' = C41 ( z ) ⊗ C41 ( z ) ⊗ C41 ( z ) ⊗ C41 ( z ) ⋅ γ etc. Lin. Quad. x2+y2,z2 z (x,y) - (xz,yz) Qubic transformations (β) and higher   2C4(z) Character C2(z) 2C' 2C"2 table 2 All parametric processes (starting and ending in the same state with no net energy transfer), including SFG and SHG, transform with the totally symmetric representation. The nonzero tensor elements and equalities between them within the D4 point group can be found by performing each of the individual symmetry operations and finding the set of equalities that is consistent with all. 7 Top down: A more general approach exemplified for D4 SHG SFG β(-ωsum, ω1, ω2) χiso µ α β(-2ω, ω, ω) z (x,y) (xx+yy), zz (xx-yy) xy (xz,yz) xyz=xzy=-yxz=-yzx (χXYZ=χYZX=χZXY=0) - - xzy=zxy=-yzx=-zyx (χXYZ=χYZX=χZXY=0) - C2, 2 A µ z β(-2ω, ω, ω) zxx, zyy, zzz, zxy=zyx β(-ωsum, ω1, ω2) xxz, yyz, zzz, xyz=yxz B x,y α xx, yy, zz, xy yz, xz xxz=zxz, xyz=xzy, yxz=yzx, yyz=yzy (χZZZ=-2χZXX) xzx=zxx, xzy=zxy, yzx=zyx, yzy=zyy (χZZZ=-2χXXZ) √ χiso √ √ New equalities emerging between surface tensor elements D4 (422) – D∞ A1 A2 B1 B2 E (E1) 8 How are these equalities between surface tensor elements generated? The Cartesian tensor in the surface frame.   χ C = N s Rθψφ S β   χ= N s Rθψφ ⊗ Rθψφ ⊗ Rθψφ β C   χ C = N s Q ⋅ ( Rθ ⋅ Rψ ) ⊗ ( Rθ ⋅ Rψ ) ⊗ ( Rθ ⋅ Rψ ) β Q= Rφ ⊗ Rφ ⊗ Rφ Since the distribution in φ is known in advance to be uniformly distributed in a uniaxial assembly, the averages can be separately evaluated in this expanded matrix notation. Given the constraints on β, what constraints emerge on the elements of χC? In crystals or assemblies of other symmetry, Q will have a different form, but can nevertheless serve a similar purpose. 9   sum e = EJL R S β 1. Character tables indicate the nonzero elements in µ and α, from which the nonzero elements of β can be populated. 2. Additional equalities between the nonzero elements can be deduced by performing the symmetry operations of the point group on µ, α, and β. 3. From these equalities, the symmetry matrix S populates the full set of 27 tensor elements from the smaller subset of unique, nonzero values. 4. For ensembles with C∞ symmetry, equalities between the macroscopic surface tensors are indicative of molecular symmetry. References: -Begue, N. J.; Moad, A. J.; Simpson, G. J. “Nonlinear optical Stokes ellipsometry (NOSE) Part I: Theoretical Framework.” J. Phys. Chem. C. 2009, 113, 10158-65. -Moad, A. J., Moad, C. W.; Perry, J. M.; Wampler, R. D.; Begue, N, J.; Shen, T.; Goeken, G. S.; Heiland, R.; Simpson, G. J. “NLOPredict: Visualization and Data Analysis Software for Nonlinear Optics” J. Computational Chem. 2007, 28, 1996-2002. 10