Conical Intersections: Electronic Structure,Dynamics and Spectroscopy, Yarkony eds. World Scientific Publishing, Singapore, (2004) 1/20/2006 1

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1 Conical Intersections: Electronic Structure,Dynamics and Spectroscopy, Wolfgang Domcke,, Horst Köppel, K and David R. Yarkony eds. World Scientific Publishing, Singapore, (2004) 1/20/2006 1

2 Characterizing Conical Intersections David. R. Yarkony Johns Hopkins University Seungsuk Han with Help from: Spiridoula Matsika Hermann Weyl,, Alden Mead 1/20/2006 2

3 Odd number of Electrons- An example OH(A 2 Σ + )+H 2 OH(X 2 Π)+H 2 H 2 O(X 1 A 1 ) +H Without SOC:C 2v (A 1 -B 2 ), C v (Σ-Π), C s (A -A ) seams H 2 +OH(A 2 Σ + )--> > H 2 + OH(X 2 Π) 1/20/2006 3

4 2 Σ - 2 Π system 1 2 A, 2 2 A, 1 2 A in C s symmetry 2 Σ + 2 Π +H SO 2 Π 1/2 +HSO +geom changes (Σ, Π) 1/2 Π 1/2 2 Σ 1/2 2 Π 3/2 2 Π 3/2 Σ 1/2, Π3/2 1/20/2006 4

5 Degeneracy can be established but more parameters must be sacrificed Seam dimension=n int - dim of branching space In C s symmetry N int =5 Seam dimension: 5-2=3 without spin-orbit interaction 5-3=2 with spin-orbit interaction In no symmetry N int =6 Seam dimension: 6-2=4 without spin-orbit interaction 6-5=1 with spin-orbit interaction 1/20/2006 5

6 Odd-electron systems Time reversal symmetry changes, in an essential manner, the nature of conical intersections 1/20/2006 6

7 Time Reversal Symmetry Kramers' ' Degeneracy and NonCrossing Rule C. A. Mead, J. Chem. Phys. 70, 2276 (1979). 1/20/2006 7

8 Time Reversal (T)Symmetry and Kramers' ' Degeneracy Odd electron systems: T 2 =-1 ψ α and Tψ α linearly independent, degenerate (Kramers degeneracy) Complex-valued Hamiltonian matrix Even electron systems: T 2 =1 ψ α and Tψ α the same Real-valued Hamiltonian matrix 1/20/2006 8

9 Conical intersection is an intersection of 4 states, not 2 I J TI TJ To understand the nature of the intersection, need the eigenvalues and eigenvectors of a 4x4 Hamiltonian 1/20/2006 9

10 Require: same basic tools as in nonrelativistic case near a conical intersection Need analytic expressions for eigenvalues and eigenvectors of H, to Characterize the singularity in the derivative couplings and hence deduce a rigorous diabatic basis, Also need a continuous representation of the branching space along the seam 1/20/

11 Consequences of Time Reversal Symmetry φ ψ = Tφ Tψ T is antiunitary Ψ HTΨ 2 = TΨ HT Ψ = TΨ HΨ = 0 Ψ HTΦ = TΨ HT 2 Φ = TΨ HΦ = ΦHTΨ Ψ HΦ = TΨ THΦ = TΨ HTΦ 1/20/

12 h II h IJ h ITI h ITJ h IJ h ITI h ITJ h JJ h JTI h JTJ h JTI h JTJ h TITI h TITJ h TITJ h TJTJ Ψ HΦ = TΨ THΦ = TΨ HTΦ 1/20/

13 h II h IJ h ITI h ITJ h IJ h ITI h ITJ h h h JJ JTI JTJ h h II IJ h JTI h JTJ h IJ h JJ Ψ HTΨ = TΨ HT 2 Ψ = TΨ HΨ = 0 1/20/

14 h h 0 h II IJ ITJ h h 0 JJ JTI 0 h h h JTI II IJ 0 h h IJ JJ h IJ h ITJ Ψ HTΦ = TΨ HT 2 Φ = TΨ HΦ = ΦHTΨ 1/20/

15 Hamiltonian withtime Reversal Symmetry H is a T-type T matrix h II h IJ 0 h ITJ h h 0 JJ ITJ 0 h h h ITJ II IJ 0 h h IJ JJ h IJ h ITJ 1/20/

16 Two non relativistic states Geometric phase = π(1 sin 2θ ) 0,a E 1 2 A' y +ih 1 2 A'2 2 A' y ih 1 2 A'2 2 A' 0,a E 2 2 A' nonrelativistic seam closed loop z 2θ ρ point of conical intersection including spinorbit coupling Ω solid angle subtended by loop 'Odd Electron' Molecules 1/20/

17 Diagonalization 1/20/

18 U s s that preserve form of H Transformations that mix direct and time-reversed states equivalently u 11 u u u u u 21 u 12 u 22 1/20/ = [u,0]

19 U s s that preserve form of H Transformations that permute direct and time-reversed states = [ 1 0, 0 0 ] /20/

20 Diagonalization using T-T preserving U's h II h IJ h ITI h ITJ h II h IJ 0 h ITJ h IJ h ITI h ITJ h h h JJ h JJ JTIITJ JTJ 0 h JTI 0 h ITJ h JTJ h TITI h TITJ h TITJ h II h IJ h TJTJ 0 h IJ h JJ U 11 U 12 U 13 U 14 U 21 U 22 U 23 U 24 U 31 U 23 U 33 U 34 U 41 U 42 U 43 U 44 1/20/ UHU -1 =ε

21 h h 0 h h h h h h h h h 12 h 14 Step 1 u 11 u h(1) u u u u 21 u 12 u 22 1/20/

22 ε 0 0 h ε h h ε ε 2 h 14 Step /20/

23 ε h 14 0 ε 2 h 14 ` 0 0 h 14 ε 1 0 h ε = ε 1 h h 14 ` ε ε 1 h 14 h 14 ε = ε 1 h h 14 ε ε 1 h h 14 ε 2 1/20/

24 ε h h ε ε h h ε 14 2 Step 3 u 11 u h(2) u u u u 21 1/20/ u 12 u 22

25 ( ε 2 + a 2 ) 1/ ( ε 2 + a 2 ) 1/ ( ε 2 + a 2 ) 1/ ( ε 2 + a 2 ) 1/ ε = ε 2 ε 1 a=h 14 E ± =±[(h 11 h 22 ) 2 + h a 2 ] 1/2 Step 4! 2 = ' 2 = det H = det H' 1/20/

26 Conditions for a conical intersection including the spin-orbit interaction Ψ 1 Ψ 2 TΨ 1 TΨ 2 H 11 H 12 0 H 1T2 H 12 H 22 H 1T 2 0 H 11 H 12 H 12 H 22 In general 5 conditions need to be satisfied. H 11 =H 22 Re(H 12 )=0 Im(H 12 )=0 Re(H 1T 2 )=0, satisfied in C s symmetry Im(H 1T 2 )=0, satisfied in C s symmetry The dimension of the seam is N int -5 or N int -3 C.A.Mead J.Chem.Phys., 70, 2276, (1979) 1/20/

27 Eigenvectors of H 1/20/

28 (1) u 11 (1) u 21 (1) u 12 (1) u (1) 0 0 u 11 u 12 (1) (1) 0 0 u 21 u 22 (1) (2) u 11 (2) u 21 (2) u 12 (2) u 22 (2) 0 0 u 11 (2) 0 0 u (2) u 12 (2) u 22 u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) u (1) u (2) /20/

29 U (D) cosθ (1 ) cosθ (2) cosθ (1) e iγ (2) sinθ (2) e iγ (1) sin Θ (1) cosθ (2) e iγ (1 ) sin Θ (1) e iγ (2) sin Θ (2) e iγ (1) sinθ (1) cosθ (2 ) e iγ (1) sinθ (1) e iγ (2) sinθ (2) cosθ (1) cosθ (2) cosθ (1) e iγ (2) sinθ (2) e iγ (1) sinθ (1 ) e iγ (2) sinθ (2) e iγ (1) sinθ (1) cosθ (2 ) cosθ (1) e iγ (2) sinθ (2 ) cosθ (1) cosθ (2) cosθ (1) e iγ (2) sinθ (2 ) cosθ (1) cosθ (2) e iγ (1) sinθ (1) e iγ (2) sinθ (2 ) e iγ (1 ) sinθ (1) cosθ (2) 1/20/

30 Near the conical Intersection Taylor Series 1/20/

31 h II h IJ 0 h ITJ h h 0 JJ ITJ 0 h h h ITJ II IJ 0 h h IJ JJ h IJ h ITJ g ij = h ii h jj = v (1) : gz h ij = h r,ij + ih i,ij = v (2) + iv (3) : h r x + ih i y h itj = h r,itj + ih i,itj = v (4) + iv (5) : t r v + it i w 1/20/

32 H near a conical intersection z x iy 0 v + iw x + iy z v iw 0 H = 0 v + iw z x+ iy v iw 0 x iy z a b b a [a,b] gz h r x ih i y 0 t r v + it i w h r x + ih i y gz t r v it i w 0 0 t r v + it i w gz h r x + ih i y t r v it i w 0 h r x ih i y gz If H hermitian a is hermitian b antisymmetric 1/20/

33 Algorithms have been developed for the location of these conical intersections E ij + g ji δr = 0 Re h ji δr = 0 Im h ji δr = 0 S. Matsika and D. R. Yarkony, J. Chem. Phys., (2001), 115, 2038 S. Matsika and D. R. Yarkony, J. Chem. Phys., (2001), 115, 5066 S. Matsika and D. R. Yarkony, J. Chem. Phys., (2002), 116, /20/

34 Location Energy (cm -1 ) OH(A 2 Σ + ) + H Relativistic crossing non relativistic minimum energy crossing OH(X 2 Π) + Η 2 H 2 O + H /20/2006 Reaction Coordinate 34

35 Coordinate Systems Representation of U (D) z' = gz = ρ (1) cosθ (1) x' = h r x = ρ (1) sinθ (1) cosφ (1) y' = h i y = ρ (1) sinθ (1) sinφ (1) v' = t r v = ρ' cosφ (2) w' = t i w = ρ' sinφ (2) h (1) = ρ (1) cosθ (1) e iφ (1 ) sinθ (1) e iφ(1) sinθ (1) cosθ (1) h (2) = ρ (1) (2) iφ ρ' e ρ' e iφ (2) ρ (1) 1/20/

36 Hyperspherical Coordinates ρ (1) = ρ (2) cosθ (2) and ρ' = ρ (2) sinθ (2) h (2) = ρ (2) cosθ (2) sinθ (2) e iφ (2) sinθ (2) e iφ( 2) cosθ (2) 1/20/

37 U (1) and U (2) cosθ (i ) e iγ (i) sinθ (i ) e iγ ( i) sinθ (i ) cosθ (i ) 1/20/

38 U (D) cosθ (1 ) cosθ (2) cosθ (1) e iγ (2) sinθ (2) e iγ (1) sin Θ (1) cosθ (2) e iγ (1 ) sin Θ (1) e iγ (2) sin Θ (2) e iγ (1) sinθ (1) cosθ (2 ) e iγ (1) sinθ (1) e iγ (2) sinθ (2) cosθ (1) cosθ (2) cosθ (1) e iγ (2) sinθ (2) e iγ (1) sinθ (1 ) e iγ (2) sinθ (2) e iγ (1) sinθ (1) cosθ (2 ) cosθ (1) e iγ (2) sinθ (2 ) cosθ (1) cosθ (2) cosθ (1) e iγ (2) sinθ (2 ) cosθ (1) cosθ (2) e iγ (1) sinθ (1) e iγ (2) sinθ (2 ) e iγ (1 ) sinθ (1) cosθ (2) Inverse transformation produces a locally diabatic basis 1/20/

39 fkl f Tj,i = e iγ (2 ) (- Θ (2) + i γ (2) /2 sin2θ (2) ) f ji = e iγ (1) ( cos2θ (2) Θ (1) +i[sin 2Θ (1) cos2θ (2) γ (1) /2]) f Tii = e i (γ (1) +γ (2)) ( sin 2Θ (2) Θ (1) + i[sin 2Θ (1) sin 2Θ (2) γ (1) /2]) f ii = i( γ (1) sin 2 Θ (1) + γ (2) sin 2 Θ (2) ) f jj = i( γ (1) sin 2 Θ (1) γ (2) sin 2 Θ (2) ) f Tj,j = e i(γ (1) γ (2) ) (sin2θ (2) Θ (1) + i[sin 2Θ (1) sin 2Θ (2) γ (1) /2]) 1 ρ' kl f γ (2), ρ' kl ρ (2)2 f Θ (2) or ρ(1) kl f ρ (2 )2 Θ (2), 1 kl ρ (1) f Θ (1 ) 1/20/ , 1 kl ρ (1) f γ (1)

40 Branching space and orthogonality 1/20/

41 Hamiltonian near conical intersection, R x H = (E i (R x ) + s ij δr) I + g ji δr h ji δr 0 h jti δr h ji δr g ji δr h jti δr 0 g ji δr h ji δr h ji δr g ji δr Magnitude x direction gz= g ij h r x+ih i y=h ij t r v+it i w=h itj δr ij δr itj δr ij is energy difference gradient Readily kx first order coupling of states k and x evaluated g ij h kx 1/20/

42 Branching Space g ij = h ii h jj = v (1) h ij = h r,ij + ih i,ij = v (2) + iv (3) h itj = h r,itj + ih i,itj = v (4) + iv (5) Branching space or cone is 3 dimensional (h( itj =0 C s symmetry) 1/20/ or 5 dimensional ( in general ) Alden Mead

43 Rotated Nascent g g H 1 O H 2 Orthogonal Vectors so with C s symmetry H so H 3 h i h i Characteristic vectors for branching space continuous along seam reflect symmetry h r h r Rotation easy for H so with C s - but 1/20/

44 Orthogonal Rep of Branching Space H can be expressed as a linear combination of 5 basic hermitian matrices Γ H=Γ i The U s that preserve the form of H satisfy UHU -1 = H = Γ This defines R so that R(U) = Can show that R(U) is a n element of O(5), the rotation group in 5-space 1/20/

45 Basic Ideas continued These U are isomorphic to the ten parameter symplectic group of order 4 and R is a homomorphism between Sp(4) and O(5) This identification enables the continuous representation of the conical intersection to be determined for free 1/20/

46 A counting argument U k,l = Ψ Ψ k l K,l=1-4 U ii U ij U iti U itj U ii U ij U iti U itj U ji U jj U jti U jtj = U ji U jj U jti U jtj U Tii U Tij U TiTi U TiTj U iti U itj U ii U ij U Tji U Tjj U TjTi U TjTj U jti U jtj U ji U jj 16 parameters; 2 normalization 2 orthogonality constraints 10 parameters 1/20/

47 H can be expressed as a linear combination of 5 basic hermitian matrices 1/20/

48 H=Γ Γ k = σ k 0 = [σ k,0] 0 σ k k =x,y,z 0 iσ y Γ v = = [0,iσ iσ y 0 y ] 0 σ y Γ w = = [0,σ σ y 0 y ] H = ( Γ x,γ y,γ z,γ v,γ w ) = x y z v w 1/20/

49 Weyl Notions The most general T-preserving U ; a homomorphism R; and R(U) 1/20/

50 A homomorphism H = ( Γ x,γ y,γ z,γ v,γ w ) U 1 HU = ( Γ z,γ x,γ y,γ v,γ w ) ' = x y z v w x' y' '= z' v' w' R(U) = ' R(U 1 )R(U 2 ) = R(U 1 U 2 ) 1/20/

51 Determine U s U s that preserve the form of H ie UHU - 1 = H =Γ H 1/20/

52 Basic U,R(U) U (xy) (φ) = [u (xy) (φ /2),0];u (xy) (φ /2) = eiφ /2 0 0 e a b b a [a,b] iφ /2 cosφ sinφ sinφ cosφ ;R(U) = cosφ 0 sinφ 0 0 cos(φ /2) sin(φ /2) U (xz) (φ) = [u (xz) (φ /2),0];u (xz) (φ /2) = ;R(U) = sinφ 0 cosφ 0 0 sin(φ /2) cos(φ /2) /20/

53 Basic U, R(U)- continued U ( p ) = 1 0, 0 0 ;R(U ( p ) ) = /20/

54 Most General U U (2) (φ,θ) = U (zx) (θ /2)U (xy) (φ /2) U ( p,2) (φ,θ) = U ( p) U (2) (φ,θ) 4 i=1 U (10) (φ,θ) = U (2) (φ 5,θ 5 ) U ( p,2) (φ i,θ i ) 1/20/

55 Relation to Symplectic Group of Order 4 The symplectic group of order 4 is a subgroup of SU(4) which leaves a skew-symmetric symmetric bilinear form invariant U t GU = G G = [0,1] = /20/

56 Equivalently GG t = UU = I U t GU = G; U t GUG t = GG t U U t GUG t = U GG t GUG t = U 1/20/

57 if U is T-type T then GUG t = U <--if GUG t = U then U is T-typeT > 0 1 a b 0 1 = 0 1 b a a b = 1 0 b a a b b a < a b 0 1 = 0 1 c d b a d c = d c a b = b a c d 1/20/

58 Symplectic Group of Order 4 The name 'complex group' formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric linear forms, has become more and more embarrassing through collision with the word 'complex' in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective symplectic. -Hermann Weyl in The Classical Groups 1/20/

59 Orthogonal Representation H = Γ k (v (k) δq) = Γ k k k=1,5 k=1,5 now rotate the electronic states by U which is equivalent to replacing by ' R(U) = ' which becomes v' (k) δq = v' (k) = i=1,5 R(U) k,i v (i) δq Choose R(U) to diagonalize the 5x5 overlap matrix R(U) k,i v (i) S =v (k) 1/20/2006 k=1,5 k,l v (l) 59

60 An Invariant g x h = g x h y g y h x = g x h x g y h y = gxh gxh r h i = g x g y h x r h x r h x i h y i g z h y r h z i 1/20/

61 An Invariant g x h x r h x i t x r t x i g,h r,h i,t r,t i = g y g z h z i t z i g v g w h w i t w i 1/20/

62 Summary Spin-orbit interaction can be treated within the same adiabatic states method that has been used so successfully in the nonadiabatic case Conical intersection characterized Local diabatic representation exists 1/20/

63 Perturbation Theory Converts big electronic structure problem to 4x4 problem, exactly- through 1st order in displacement and provides a Quasi diabatic basis 1/20/

64 Limits η=3 Θ(2),γ(2)=0 η=2 γ(1)=0 f Tj,i = 0 f ji = e iγ (1) ( Θ (1) + i[sin2θ (1 ) γ (1) /2])) f Tii =0 f ii = i( γ (1) sin 2 Θ (1) ) f jj = i( γ (1) sin 2 Θ (1 ) ) f Tj,j =0 f Tj,i = 0 f ji = ( Θ (1) )) f Tii =0 f ii =0 f jj = =0 f Tj,j =0 1/20/

65 Relation to COlumbus CI gradients and matrix elements needed to locate and characterize conical intersections and perform energy minimzation 1/20/

66 Relativistic CI Time reversal adapted bases Odd electron systems φ=tψ <ψ Tψ>=0 Tχ A, S, M = i( 1) S+ M χ A, S, M χ A, S, M, = i( 1) S+ M (χ A, S, M iχ A, S, M )/ 2 χ A, S, M,+ = (χ A, S, M + iχ A, S, M )/ 2 1/20/

67 Relativistic CI Time reversal adapted bases Even electron systems ψ=tψ T + =1/ 2(αα + ββ) T = i / 2(αα ββ) 1/20/

68 Relativistic CI Odd electron systems r, TRA CSF H H i, TRA CSF t H H i, TRA CSF r, TRA CSF cr, I c i, I = E I c r, I c i, I v j A, S, M, w = j ' TRA CSF A', S ', M ', w' H j, S, M, w, A;j ', M ', w', A' v j ' v β = [ so,β (s,α a, b h so 1 a, b + s 0,β,α h 0 ) 0,β a, b + (S,α I a, b a, b, c, d g a, b, c, d )]c α a;b a;b, c, d 1/20/

69 Evaluation of CI gradients ( H TRA CSF E I )c I = (H TRA CSF E I ) c I c I ( H TRA CSF )c I = E I c J ( H TRA CSF )c I = (E J E I )c J c I 1/20/

70 Gradients c k ( R α H TRA CSF )c l = dh k, l,α + L k, l U α dh k, l,α = (γ a, b + a, b 0, k, l (Γ a, b, c, d a;b a;b, c, d α g a, b, c, d ) 1/20/