Exact factorization of the electron-nuclear wave function and the concept of exact forces in MD

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1 Exact factorization of the electron-nuclear wave function and the concept of exact forces in MD E.K.U. Gross Max-Planck Institute for Microstructure Physics Halle (Saale)

2 OUTLINE Thanks Exact factorisation of electronic and nuclear degrees of freedom Exact time-dependent PES An alternative to Tully surface hopping Ali Abedi Federica Agostini Yasumitsu Suzuki Seung Kyu Min Neepa Maitra (CUNY) Nikitas Gidopoulos (utherford Lab) Ivano Tavernelli (EPFL Lausanne)

3 Hamiltonian for the complete system of N e electrons with coordinates ( r and N n nuclei with coordinates, masses 1 rn ) r ( 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 = 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 convention: Greek indices nuclei Latin indices electrons ( ) ( ) 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 are APPOXIMATE concepts, i.e. they follow from 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 are APPOXIMATE concepts, i.e. they follow from 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: Zewail, J. Phys. Chem. 104, 5660, (2000)

9 Example: NaI femtochemistry Na + + I - Na + I Nuclear wavepacket is created by a femto-second laser pulse on the repulsive wall of the excited surface. As it keeps moving on this surface it encounters the avoided crossing at 6.93 Ǻ. At this point some molecules will dissociate into Na + I, and some will keep oscillating on the upper adiabatic surface. The wavepacket continues sloshing about on the excited surface with a small fraction leaking out each time the avoided crossing is encountered.

10 I. Probing Na atom products: Steps in the production of Na as more of the wavepacket leaks out each vibration into the Na - I channel.

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 GOAL: Show that Ψ ( r, ) Φ ( r) χ( ) = can be made EXACT Concept of EXACT potential energy surfaces (both static and TD) Concept of EXACT Berry phase (both static and TD)

13 Theorem I The exact solutions of Ĥ Ψ ( r, ) = EΨ( r, ) can be written in the form Ψ ( r, ) Φ ( r) χ( ) = 2 = ( ) where dr Φ r 1 for each fixed. First mentioned in: G. Hunter, Int. J.Q.C. 9, 237 (1975)

14 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~

15 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~

16 Eq. ( ) n N 2 en ext e ee e A i 2M 1 Vˆ Vˆ Ŵ Tˆ Ĥ BO ( ) ( ) ( ) r Φ r Φ A i A χ χ i M 1 N n = + + ( ) Eq. ( ) ( ) ( ) Eχ χ Vˆ Ŵ A i 2M 1 ext n nn N 2 n = ( ) where ( ) ( ) ( ) Φ Φ = r d r r i A * Theorem II: satisfy the following equations: ( ) ( ) r and Φ χ

17 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 = M χ ( ) 2 ( i A ) Φ ( r) Φ ( r) N 1 ext nn n ( ) 2M n 2 Eq. ( i + A ) + Ŵ + Vˆ + χ( ) = Eχ( ) * where A ( ) i Φ ( r) Φ ( r) = dr Exact PES Exact Berry potential

18 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

19 How do the exact PES look like?

20 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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23 Exact Berry connection ( ) * ( ) ( ) ( ) = Φ Φ A dr r i r Insert: Φ ( r) =Ψ( r, ) / χ( ) ( ) : e ( ) iθ χ = χ ( ) * ( ) ( ) ( ) { } ( ) 2 A = Im dr Ψ r, Ψ r, / χ θ ( ) = ( ) χ( ) θ( ) A J / 2 with the exact nuclear current density J

24 Time-dependent case

25 Hamiltonian for the complete system of N e electrons with coordinates ( r and N n nuclei with coordinates, masses 1 rn ) r ( 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 j, k 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

26 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)

27 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)

28 ( ) ( ) 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

29 Example: H 2 + in 1D in strong laser field exact solution of i Ψ( r,,t) H Ψ( r,,t): Compare with: t = Hartree approximation: Ψ(r,,t) = χ(,t) φ(r,t) Standard Ehrenfest dynamics Exact Ehrenfest dynamics where the forces on the nuclei are calculated from the exact TD-PES

30 The internuclear separation < >(t) for the intensities I 1 = W/cm 2 (left) and I 2 = 2.5 x W/cm 2 (right)

31 Exact time-dependent PES Dashed: I 1 = W/cm 2 ; solid: I 2 = 2.5 x W/cm 2

32 Second TD example: Molecular motion without laser, but initial state is a wavepacket (i.e. not an eigenstate)

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58 New MD scheme: Perform classical limit of the nuclear equation, but retain the exact forces from the exact electronic equation

59 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

60 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.

61 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

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64 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 TD-PES useful to interpret different dissociation meachanisms when few PES are involved: Jumps resembling surface hopping mixed quantum classical algorithm with non-stochastic surface hopping

65 SFB 450