What s the correct classical force on the nuclei

Transcription

1 What s the correct classical force on the nuclei E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)

2 What s the correct classical force on the nuclei or: How to make the Born-Oppenheimer approximation exact E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)

3 Hamiltonian for the complete system of N e electrons and N n nuclei Ĥ = Tˆ n () + Ŵnn () + Tˆ e(r) + Ŵee(r) + Vˆ en (, r) with ( r r ) r ( ) 1 N e 1 N n 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 j, k k r j 1 r k Vˆ en = N N e n r j= 1 = 1 j Z ( ) ( ) Full Schrödinger equation: Ĥ Ψ r, = EΨ r,

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 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 Berry potential follow from an APPOXIMATION (the BO approximation). BO A ( )

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 Berry potential follow from an APPOXIMATION (the BO approximation). BO A Berry phases arise when the world is approximately separated into a system and its environment. ( )

7 GOING BEYOND BON-OPPENHEIME Standard procedure: Expand full molecular wave function in complete set of BO states: Ψ K ( ) = BO r, Φ ( r) χ ( ) J and insert expansion in the full Schrödinger equation standard,j K,J BO non-adiabatic coupling terms from T n acting on Φ ( r).,j Drawbacks: χ J,K depends on 2 indices: looses nice interpretation as nuclear wave function In systems driven by a strong laser, many BO-PES can be coupled.

8 BO Φ 1, ( r) ( ) BO E 1 BO Φ 0, ( r) ( ) BO E 0 ( ) ( ) BO Φ r + ( ) Ψ r, χ χ ( ) BO( ) 0, Φ 1, r Potential energy surfaces are absolutely essential in our understanding of a molecule... and can be measured by femto-second pump-probe spectroscopy: A. Zewail, J. Phys. Chem. 104, 5660, (2000)

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

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

11 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)

12 But what s the classical force when the nuclear wave packet splits??

13 OUTLINE Show that the factorisation Ψ ( r, ) Φ ( r) χ( ) = can be made exact Concept of exact PES and exact Berry phase Concept of time-dependent PES for nuclear motion Concept of time-dependent PES for electronic motion Mixed quantum-classical treatment

14 OUTLINE Thanks Show that the factorisation Ψ ( r, ) Φ ( r) χ( ) = can be made exact Concept of exact PES and exact Berry phase Concept of time-dependent PES for nuclear motion Concept of time-dependent PES for electronic motion Ali Abedi Federica Agostini Yasumitsu Suzuki Seung Kyu Min Neepa Maitra (CUNY) Nikitas Gidopoulos (utherford Lab) Mixed quantum-classical treatment

15 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. First mentioned in: G. Hunter, Int. J.Q.C. 9, 237 (1975)

16 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. ( ) Φ r ( ) χ 2. and are unique up to within the gauge transformation Φ ~ ( ) iθ( ) r : e Φ ( r) = ( ) iθ ( ) ( ) : = e χ χ~

17 φ χ ~ φ χ ~ proof: Let and be two different representations of an exact eigenfunction Ψ i.e. Ψ r, = Φ r χ = Φ r χ Φ ( ) ( ) ( ) ( ) ( ) ( r) χ( ) G ( ) ( r) χ~ ( ) Φ ~ = ( r) G ( ) Φ ( r) Φ ~ = d r Φ ~ ( ) 2 r = G ( ) 2 d r Φ ( r) ( ) ( ) iθ( ) = 1 G e G = Φ ~ ( r) Φ ( r) ( iθ e ) = χ( ) iθ( ) e ( ) = χ~

18 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 Proof in: N.I. Gidopoulos, EKUG, arxiv:cond-mat/

19 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 connection Proof in: N.I. Gidopoulos, EKUG, arxiv:cond-mat/

20 OBSEVATIONS: Eq. is a nonlinear equation in Eq. contains χ ( ) Φ ( r) selfconsistent solution of and required Neglecting the 1 M terms in, BO is recovered There is an alternative, equally exact, representation (electrons move on the nuclear energy surface) Ψ = Φ r ( ) χ( r) Eq. and are form-invariant under the gauge transformation Φ χ A γ Φ ~ χ ~ = A ~ = e e iθ iθ = ( ) Φ ( ) χ A ( ) ~ ( ) = ( ) ( C) : A = d C + θ ( ) Exact potential energy surface is gauge invariant. is a (gauge-invariant) geometric phase the exact geometric phase

21 How do the exact PES look like?

22 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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25 Exact Berry connection ( ) * ( ) ( ) ( ) A dr r i r = Φ Φ Insert: Φ ( r) =Ψ( r, ) / χ( ) ( ) ( ) iθ χ : = e χ ( ) * ( ) ( ) ( ) { } ( ) 2 A = Im dr Ψ r, Ψ r, / χ θ ( ) = ( ) χ( ) θ( ) A J / 2 with the exact nuclear current density J

26 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

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

28 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) + + +

29 BO-PES of 2D Shin-Metiu model

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

31 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 (for the 2D Shin-Metiu model) in the BO approximation.

32 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 (for the 2D Shin-Metiu model) in the 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.

33 D() M =

34 M =

35 QUESTION: Can one prove in general that the exact molecular Berry phase vanishes? Are there systems where the exact Berry phase does not vanish?

36 Time-dependent case

37 Hamiltonian for the complete system of N e electrons with coordinates ( r1 rn ) r and N n nuclei with coordinates (, masses e 1 N ) M 1 M N n and charges Z 1 Z N n. n Ĥ = Tˆ n () + Ŵnn () + Tˆ e(r) + Ŵee(r) + Vˆ en (, r) with Tˆ n Ŵ ee = n = 1 = N M N e j,k j k Time-dependent Schrödinger equation i Ψ t V r laser j 1 r k Tˆ e = ( r,, t) = ( H( r, ) + V ( r,, t ) ψ( r,, t) N e i= 1 Vˆ en 2 i 2m = N e n N e N n ( r,, t) = rj Z E f ( t) cosωt j = 1 = 1 N Ŵ j= 1 = 1 j laser nn r = Z 1 2 N n Z µ Z µ, µ µ

38 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)

39 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)

40 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 TD Berry connection 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)

41 ( ) ( ) n * 2, t = dr Φ r, t H ( t) + ( i A (, t) ) i BO t Φ ( r, t) 2M N 1 EXACT time-dependent potential energy surface ( ) ( ) * = Φ Φ ( ) A,t i r,t r,t dr EXACT time-dependent Berry connection N-body version of unge-gross theorem guarantees that ϵ(,t) and A(,t) are UNIQUE (up to within a gauge transformation)

42 How does the exact time-dependent PES look like?? Example: Nuclear WP 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. (2013)

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64 New MD scheme: Perform classical limit of the nuclear equation, but retain the quantum treatment of the electronic degrees of freedom.

65 Nuclear wavefunction ( ) i S,t ( ) ( ) χ,t = e χ,t Classical limit ( ) 2 (,t) c ( t) S(,t) P (t) χ δ c Hence i χ χ 0 P (t) c

66 Expand the exact electronic wave function in the adiabatic basis: ( r,t) c (,t) BO ( r) Φ = ϕ j,j j Insert this in the (exact) electronic equation of motion: ( ) ( ) { } ( ) ( { } { 2 ( )}) c,t = f c,t, c,t, c,t j j k k k in the classical limit: 2 ( ) ( ) c,t, c,t 0 k k i.e. in this limit the c k (,t) become independent of.

67 In practice we solve the following equations: i ( j) ( ) ( ) ( ) ( I c t = ε V + iv ) c ( t) c ( t) D j BO eff eff j k jk k 2 2 P A V c cc d ( I) BO * ( 2) = ε + + eff j, j j k jk j M M j < k 2 V = cc d M I < ( ) * ( 1) eff j k jk j k P i D = d d d M 2M ( ) ( 1) ( 1) ( 2) jk jk jk jk ( ) ( ) d = ϕ ϕ 1 BO BO jk,j,k ( ) ( ) d = ϕ ϕ 2 BO BO jk,j,k 2 P and classical EoM for the nuclear Hamiltonian: ( ) HN = + Veff 2M

68 Shin-Metiu model

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70 Summary: Ψ ( r, ) Φ ( r) χ( ) = is exact Eqs. of motion for Φ ( r ) and χ( ) lead to --- exact potential energy surface --- exact Berry connection both in the static and the time-dependent case Exact Berry phase may vanish when BO Berry phase 0 TD-PES shows jumps resembling surface hopping mixed quantum classical algorithm

71 SFB 450 SFB 685 SFB 762 SPP 1145