How to make the Born-Oppenheimer approximation exact: A fresh look at potential energy surfaces and Berry phases

Transcription

1 How to make the Born-Oppenheimer approximation exact: A fresh look at potential energy surfaces and Berry phases E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)

2 Hamiltonian for the complete system of N e electrons with coordinates r r and N n nuclei with coordinates ( ) ( ) r 1 N e 1 N n Ĥ = Tˆ n () + Ŵnn () + Tˆ e(r) + Ŵee(r) + Vˆ en (, r) with Tˆ n = N n = 1 2M 2 Tˆ e = N e i= 1 2 i 2m Ŵ nn = 1 2 N n Z µ Z µ µ, µ Ŵ ee = 1 2 N e j,k j k r j 1 r k Vˆ en = N N e n r j= 1 = 1 j Z Stationary Schrödinger equation Ĥ Ψ ( r, ) = EΨ( r,)

3 Hamiltonian for the complete system of N e electrons with coordinates r r and N n nuclei with coordinates ( ) ( ) r 1 N e 1 N n Ĥ = Tˆ n () + Ŵnn () + Tˆ e(r) + Ŵee(r) + Vˆ en (, r) with Tˆ n = N n = 1 2M 2 Tˆ e = N e i= 1 2 i 2m Ŵ nn = 1 2 N n Z µ Z µ µ, µ Ŵ ee = 1 2 N e j,k j k r j 1 r k Vˆ en = N N e n r j= 1 = 1 j Z Time-dependent Schrödinger equation i Ψ( r,, t) = H( r, ) + V ( r,, t) laser ψ r,, t t V laser ( ) ( ) N e N n ( r,, t) = rj Z E f ( t) cosωt j = 1 = 1

4 Born-Oppenheimer approximation solve ( ) ext BO( ) BO Tˆ (r) Ŵ (r) + Vˆ (r) + Vˆ (r, ) Φ r Φ ( r) BO + ( ) e ee e en = for each fixed nuclear configuration. Make adiabatic ansatz for the complete molecular wave function: Ψ BO ( ) BO ( ) BO r, = Φ r χ ( ) and find best χ BO by minimizing <Ψ BO H Ψ BO > w.r.t. χ BO :

5 Born-Oppenheimer approximation solve ( ) ext BO( ) BO Tˆ (r) Ŵ (r) + Vˆ (r) + Vˆ (r, ) Φ r Φ ( r) BO + ( ) e ee e en = for each fixed nuclear configuration. Make adiabatic ansatz for the complete molecular wave function: Ψ BO ( ) BO ( ) BO r, = Φ r χ ( ) and find best χ BO by minimizing <Ψ BO H Ψ BO > w.r.t. χ BO :

6 Tˆ Nuclear equation n ext BO BO () Ŵnn () Vˆ n () A ( ) (-i υ ) + υ ( ) M υ υ + Φ BO BO BO ( r) Tˆ ( ) Φ ( r) dr χ ( ) Eχ ( ) BO * n = Berry connection A BO υ γ BO BO * BO ( ) Φ ( r) (-i υ )Φ ( r) = dr BO ( C) = A C ( ) d is a geometric phase ( ) BO In this context, potential energy surfaces and the vector potential follow from an APPOXIMATION (the BO approximation). BO A ( )

7 Tˆ Nuclear equation n ext BO BO () Ŵnn () Vˆ n () A ( ) (-i υ ) + υ ( ) M υ υ + Φ BO BO BO ( r) Tˆ ( ) Φ ( r) dr χ ( ) Eχ ( ) BO * n = Berry connection A BO υ γ BO BO * BO ( ) Φ ( r) (-i υ )Φ ( r) = dr BO ( C) = A C ( ) d is a geometric phase ( ) BO In this context, potential energy surfaces and the vector potential follow from an APPOXIMATION (the BO approximation). BO A ( )

8 Geometric Phases Concept of geometric phase: Discovered by S. Pancharatnam (1956) Proc. Indian Acad. Sci. A 44: In the context of quantum mechanics: Michael V. Berry (1984) Proc. oyal Society 392 (1802),

9 Whenever the Hamiltonian of a quantum system depends on a vector of parameters,, the Berry phase is defined as: where the line integral is along a closed loop, C, in parameter space. A non-vanishing value of γ only appears when C encircles some non-analyticity.

10 Standard representation of the full TD wave function Expand full molecular wave function in complete set of BO states: ( ) BO ( ) ( ) Ψ r,,t = Φ,J r χj,t J and insert expansion in the full Schrödinger equation standard BO non-adiabatic coupling terms from T n acting on Φ ( r).,j

11 Plug Born-Huang expansion in full TDSE: ( ) ( ) ( ) ( ) i χ,t = T χ,t + χ,t t k n k k k 2 BO BO + φ,k i φ,j i χj,t α α jα Mα NAC-1 ( ( )) 2 + φ φ χ α jα 2Mα BO 2 BO,k,j j (,t) NAC-2 The dynamics is "non-adiabatic" when the NAC terms cannot be neglected

12 BO Φ1, ( r) E BO ( ) 1 BO E ( ) Φ ( r ) 0 BO 0, Ψ ( ) ( ) BO, t,t ( ) + (, t) BO r, χ Φ r χ Φ ( r) , 01 1, When only few BO-PES are important, the BO expansion gives a perfectly clear picture of the dynamics

13 Example: NaI femtochemistry Na + + I - Na + I

14 Example: NaI femtochemistry Na + + I - Na + I emitted neutral Na atoms

15 Effect of tuning pump wavelength (exciting to different points on excited surface) λ pump /nm Different periods indicative of anharmonic potential T.S. ose, M.J. osker, A. Zewail, JCP 91, 7415 (1989)

16 For larger systems one would like to (one has to) treat the nuclei classically.

17 Trajectory-based quantum dynamics

18 For larger systems one would like to (one has to) treat the nuclei classically. But what s the classical force when the nuclear wave packet splits??

19 For larger systems one would like to (one has to) treat the nuclei classically. But what s the classical force when the nuclear wave packet splits??

20 For larger systems one would like to (one has to) treat the nuclei classically. But what s the classical force when the nuclear wave packet splits?? There is only one correct answer!

21 Outline Show that the factorisation Ψ ( r, ) Φ ( r) χ( ) = can be made exact Concept of exact PES and exact Berry phase Concept of exact and unique time-dependent PES Mixed quantum-classical treatment

22 THANKS Axel Schild Ali Abedi Federica Agostini Yasumitsu Suzuki Seung Kyu Min Neepa Maitra (Hunter College, CUNY) yan equist Nikitas Gidopoulo (Durham University, UK)

23 Theorem I The exact solutions of Ĥ Ψ ( r, ) = EΨ( r, ) can be written in the form ( ) =Φ ( ) ( ) Ψ r, r χ 2 = ( ) where dr Φ r 1 for each fixed. N.I. Gidopoulos, E.K.U. Gross, Phil. Trans.. Soc. 372, (2014), arxiv:cond-mat/ (2005)

24 Proof of Theorem I: Given the exact electron-nuclear wavefuncion Ψ( r,) Choose: ( ) ( ) ( ) 2 is χ : = e dr Ψ r, with some real-valued funcion S( ) ( ) = ( ) ( ) Φ r : Ψ r, / χ Then, by construction, dr Φ ( ) 2 r = 1

25 Proof of Theorem I: Given the exact electron-nuclear wavefuncion Ψ( r,) Choose: ( ) ( ) ( ) 2 is χ : = e dr Ψ r, with some real-valued funcion S( ) ( ) = ( ) ( ) Φ r : Ψ r, / χ Then, by construction, dr Φ ( ) 2 r = 1 Note: If we want χ() to be smooth, S() may be discontinuous

26 Immediate consequences of Theorem I: ( ) 1. The diagonal Γ of the nuclear N n -body density matrix is identical with χ( ) proof: Γ ( ) = dr Ψ( r,) = dr Φ ( ) ( ) ( ) 2 r χ = χ 2 in this sense, χ( ) 1 can be interpreted as a proper nuclear wavefunction.

27 Theorem II: ( ) χ( ) Φ r and satisfy the following equations: ext Eq. Tˆ + Ŵ + Vˆ + Vˆ + n ( i A ) e ee e Ĥ BO en N 1 2M N n 1 i χ + + A ( i A ) Φ ( r) ( ) Φ ( r) M χ 2 = Nn Eq. 1 2 ( i A ) Ŵ Vˆ ext + + nn + n + ( ) χ( ) = Eχ( ) 2M * where A ( ) i Φ ( r) Φ ( r) = dr N.I. Gidopoulos, E.K.U. Gross, Phil. Trans.. Soc. 372, (2014), arxiv:cond-mat/ (2005)

28 Theorem II: ( ) χ( ) Φ r and satisfy the following equations: ext Eq. Tˆ + Ŵ + Vˆ + Vˆ + n ( i A ) e ee e Ĥ BO en N 1 2M N n 1 i χ + + A ( i A ) Φ ( r) ( ) Φ ( r) M χ 2 = Nn Eq. 1 2 ( i A ) Ŵ Vˆ ext + + nn + n + ( ) χ( ) = Eχ( ) 2M * where A ( ) i Φ ( r) Φ ( r) = dr Exact PES Exact Berry potential N.I. Gidopoulos, E.K.U. Gross, Phil. Trans.. Soc. 372, (2014), arxiv:cond-mat/ (2005)

29 How do the exact PES look like?

30 MODEL S. Shin, H. Metiu, JCP 102, 9285 (1995), JPC 100, 7867 (1996) (1) (2) Å -5 Å +5 Å x Nuclei (1) and (2) are heavy: Their positions are fixed

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33 Exact Berry connection ( ) * ( ) ( ) ( ) A dr r i r = Φ Φ Insert: Φ ( r) =Ψ( r, ) / χ( ) ( ) ( ) iθ χ = χ : e ( ) * ( ) ( ) ( ) { } ( ) 2 A = Im dr Ψ r, Ψ r, / χ θ ( ) = ( ) χ( ) θ( ) A J / 2 with the exact nuclear current density J

34 Another way of reading this equation: 2 ( ) = χ ( ) ( ) + θ( ) J {A } Conclusion: The nuclear Schrödinger equation Nn 1 2 ( i A ) Ŵ Vˆ ext + + nn + n + ( ) χ( ) = Eχ( ) 2M yields both the exact nuclear N-body density and the exact nucler N-body current density A. Abedi, N.T. Maitra, E.K.U. Gross, JCP 137, 22A530 (2012)

35 Question: Can the true vector potential be gauged away, i.e. is the true Berry phase zero?

36 Question: Can the true vector potential be gauged away, i.e. is the true Berry phase zero? Look at Shin-Metiu model in 2D: (1) (2) + + +

37 BO-PES of 2D Shin-Metiu model

38 BO-PES of 2D Shin-Metiu model conical intersection with Berry phase

39 Non-vanishing Berry phase results from a non-analyticity in the electronic wave function BO r as function of. Φ ( ) Such non-analyticity is found in BO approximation.

40 Non-vanishing Berry phase results from a non-analyticity in the electronic wave function BO r as function of. Φ ( ) Such non-analyticity is found in BO approximation. Does the exact electronic wave function show such non-analyticity as well (in 2D Shin-Metiu model)? Look at D ( ) = rφ ( ) r dr as function of nuclear mass M. S.K. Min, A. Abedi, K.S. Kim, E.K.U. Gross, PL 113, (2014)

41 D() M =

42 D() M =

43 Question: Can one prove in general that the exact molecular Berry phase vanishes?

44 Question: Can one prove in general that the exact molecular Berry phase vanishes? Answer: No! There are cases where a nontrivial Berry phase appears in the exact treatment.. equist, F. Tandetzky, EKU Gross, Phys. ev. A 93, (2016).

45 Time-dependent case

46 Theorem T-I The exact solution of t ( ) ( ) ( ) i Ψ r,,t = H r,,t Ψ r,,t can be written in the form ( r,,t) ( r,t ) (,t) Ψ =Φ χ where ( ) 2 dr Φ r,t = 1,t for any fixed. A. Abedi, N.T. Maitra, E.K.U.G., PL 105, (2010) JCP 137, 22A530 (2012)

47 Φ ( r, t) and χ(, t) Theorem T-II satisfy the following equations Eq. Tˆ e N n + Eq. ( r, t) + Vˆ ( r, ) + ( i A (, t) ) ext + Ŵ + Vˆ n 1 M ee i e ( t) Ĥ BO χ(, t) χ(, t) + en A N 1 2M (, t) ( i A ) (, t) Φ ( r) = i Φ ( r, t) 2 t N n 1 2M 2 ext ( i + A (, t) ) + Ŵ ( ) + Vˆ (, t) + (, t) χ(, t) = i χ(, t) nn n t A. Abedi, N.T. Maitra, E.K.U.G., PL 105, (2010) JCP 137, 22A530 (2012)

48 Φ ( r, t) and χ(, t) Theorem T-II satisfy the following equations Eq. Tˆ e N n + ( r, t) + Vˆ ( r, ) + ( i A (, t) ) ext + Ŵ + Vˆ n 1 M ee i e ( t) Ĥ BO χ(, t) χ(, t) + en A N 1 2M (, t) ( i A ) (, t) Φ ( r) = i Φ ( r, t) 2 t Eq. Exact Berry potential Exact TDPES N n 1 2M 2 ext ( i + A (, t) ) + Ŵ ( ) + Vˆ (, t) + (, t) χ(, t) = i χ(, t) nn n t A. Abedi, N.T. Maitra, E.K.U.G., PL 105, (2010) JCP 137, 22A530 (2012)

49 How does the exact time-dependent PES look like?

50 Example: Nuclear wave packet going through an avoided crossing (Zewail experiment) A. Abedi, F. Agostini, Y. Suzuki, E.K.U.Gross, PL 110, (2013) F. Agostini, A. Abedi, Y. Suzuki, E.K.U. Gross, Mol. Phys. 111, 3625 (2013)

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72 New MD scheme: Perform classical limit of the nuclear equation, but retain the quantum treatment of the electronic degrees of freedom. A. Abedi, F. Agostini, E.K.U.Gross, EPL 106, (2014) S.K. Min, F Agostini, E.K.U. Gross, PL 115, (2015) F. Agostini, S.K. Min, A. Abedi, E.K.U. Gross, JCTC 12, 2127 (2016)

73 Eq. Theorem T-II Tˆ e N n + Eq. ( r, t) + Vˆ ( r, ) + ( i A (, t) ) ext + Ŵ + Vˆ n 1 M ee i e ( t) Ĥ BO χ(, t) χ(, t) + en A N 1 2M (, t) ( i A ) (, t) Φ ( r) = i Φ ( r, t) 2 t N n 1 2M 2 ext ( i + A (, t) ) + Ŵ ( ) + Vˆ (, t) + (, t) χ(, t) = i χ(, t) nn n t

74 Eq. Theorem T-II Tˆ e N n + Eq. ( r, t) + Vˆ ( r, ) + ( i A (, t) ) ext + Ŵ + Vˆ n 1 M ee i e ( t) Ĥ BO χ(, t) χ(, t) + en A N 1 2M (, t) ( i A ) (, t) Φ ( r) = i Φ ( r, t) 2 t N n 1 2M 2 ext ( i + A (, t) ) + Ŵ ( ) + Vˆ (, t) + (, t) χ(, t) = i χ(, t) nn n t

75 Shin-Metiu model

76 Propagation of classical nuclei on exact TDPES

77 Measure of decoherence: Quantum: dd CC 1, tt 2 CC2, tt 2 χ, tt 2 Trajectories NN 11 tttttttt II CC 11 II tt 22 CC 22 II (tt) 22

78 Algorithm implemented in:

79 The "right" electron-phonon interaction

80 electron-phonon interaction ( ) ( ˆ ˆ ) Hˆ = M k,q cˆ cˆ b + b e ph λ k q k qλ qλ k,q, λ k-q q + k-q -q k k

81 electron-phonon interaction ( ) ( ˆ ˆ ) Hˆ = M k,q cˆ cˆ b + b e ph λ k q k qλ qλ k,q, λ k-q q k MM kk, qq + k-q -q k MM kk, qq In a genuine ab-initio description, what is the exact coupling MM kk, qq?

82 LITEATUE on MM kk, qq : What everybody uses: ( p p ) ξ np V n' p' qλ KS qλ = δg, p' p+ q 12 M n,n' ' ( 2MN ω ) c qλ obert van Leeuwen, PB 69, (2004): ( ) ( ) 12 Z 1 i 0, α M λ, ω= 2MN cω / r q r qλ d r1εe ( rr, 1 ; ωξ ) q, λ e Many textbooks neglect ϵ -1 α r1 0, α completely (no screening). Higher-order terms (Marini, Ponce, Gonze, PB 91, (2015) (using DFPT): (2) e ph qλ, q' λ' k kqq - - ' -qλ qλ - q' λ' q' λ' k, qλ, q ' λ' ( ) = θ ˆˆ ( ˆ + ˆ )( ˆ + ˆ ) H ˆ (k)c c b b b b

83 Theorem II: ( ) χ( ) Φ r and satisfy the following equations: ext Eq. Tˆ + Ŵ + Vˆ + Vˆ + n ( i A ) e ee e Ĥ BO en N 1 2M N n 1 i χ + + A ( i A ) Φ ( r) ( ) Φ ( r) M χ 2 = Nn Eq. 1 2 ( i A ) Ŵ Vˆ ext + + nn + n + ( ) χ( ) = Eχ( ) 2M Exact phonons Expand ϵ() around equilibrium positions (to second order): Ĥph = ˆ ˆ ω qλ k bqλbqλ + qλ Eq. ( ) 1 2

84 Theorem II: ( ) χ( ) Φ r and satisfy the following equations: ext Eq. Tˆ + Ŵ + Vˆ + Vˆ + n ( i A ) e ee e Ĥ BO en N 1 2M N n 1 i χ + + A ( i A ) Φ ( r) ( ) Φ ( r) M χ Exact el-ph interaction 2 = Nn Eq. 1 2 ( i A ) Ŵ Vˆ ext + + nn + n + ( ) χ( ) = Eχ( ) 2M Exact phonons Expand ϵ() around equilibrium positions (to second order): Ĥph = ˆ ˆ ω qλ k bqλbqλ + qλ Eq. ( ) 1 2

85 ξ ˆ qλ np u VKS n' p' α M n p,n' p' =δ 1+ 3 n p,n' p' n p,n' p' ( ) ( 2MNnωqλ ) ( f ) qλ g, p' p+ q 12 HXC Traditional term

86 Summary on exact factorisation ( t) =Φ ( t) ( t) Ψ r,, r, χ, is exact A. Abedi, N.T. Maitra, E.K.U. Gross, PL 105, (2010) Exact Berry phase vanishes S.K. Min, A. Abedi, K.S. Kim, E.K.U. Gross, PL 113, (2014) TD-PES shows jumps resembling surface hopping A. Abedi, F. Agostini, Y. Suzuki, E.K.U.Gross, PL 110, (2013) mixed quantum classical algorithms S.K. Min, F Agostini, E.K.U. Gross, PL 115, (2015) correct electron-phonon interaction shows new terms (in addition to standard DFT expression)

87 SFB 450 SFB 685 SFB 762 SPP 1145