Mixing Quantum and Classical Mechanics: A Partially Miscible Solution

Transcription

1 Mixing Quantum and Classical Mechanics: A Partially Miscible Solution R. Kapral S. Nielsen A. Sergi D. Mac Kernan G. Ciccotti

2 quantum dynamics in a classical condensed phase environment how to simulate quantum-classical dynamics nonadiabatic chemical rate processes calculation of reaction rate constants

3 dynamics of open quantum systems in classical environments quantum system quantum subsystem + classical bath ^ q,m ^ q,m ^ Q,M (R,P),M equations of motion for mixed quantum-classical systems non-adiabatic dynamics in terms of surface hopping trajectories R. Kapral and G. Ciccotti, J. Chem. Phys., 110, 8919 (1999)

4 quantum-classical dynamics quantum mechanics in the partial Wigner representation quantum Liouville equation: ˆρ t = ī h [Ĥ, ˆρ] Hamiltonian: Ĥ = ˆP 2 2M + ˆp2 2m + ˆV (ˆq, ˆQ) single out degrees of freedom to be treated classically: partial Wigner transform  W (R, P ) = dze ip z/ h R z 2  R + z 2 Imre, et al., J. Math. Phys., 5, 1097 (1967)

5 quantum Liouville equation in partial Wigner form ˆρ W (R, P, t) t = ī h (ĤW e hλ/2iˆρ W (t) ˆρ W (t)e hλ/2i Ĥ W ) Poisson bracket operator Λ = P R R P scale velocities of light and heavy particles to have comparable magnitudes; lengths measured in terms of de Broglie wavelengths of light particles scaled equation involves mass ratio µ = ( ) m 1/2 M ˆρ W (R, P, t) t = i (ĤW e µλ/2iˆρ ) W (t) ˆρ (t)e hλ/2i W Ĥ W

6 passage to the quantum-classical limit light quantum particles and heavy bath particles: M m λ m λ M thermal wave length: λ M = ) 1/2 ( ( h 2 β M = m ) 1/2λm M expand evolution operator to order µ = ( ) m 1/2 M

7 mixed quantum-classical Liouville equation ˆρ W (R, P, t) t = ī h [Ĥ W, ˆρ W (t)] i ˆLˆρ W (t) ( Ĥ W, ˆρ W (t) ) ({ĤW, ˆρ W (t) } {ˆρ W (t), Ĥ W }) partially Wigner transformed hamiltonian Ĥ W (R, P ) = P 2 2M + ˆp2 2m + ˆV W (ˆq, R) similar evolution equation for operators

8 nice features of quantum-classical dynamics density matrix dynamics of quantum subsystem and classical bath not equivalent to Schrödinger description energy conservation of ensemble guaranteed some unusual features of quantum-classical dynamics the Jacobi identity valid to O( h): (Â W, ( ˆB W, Ĉ W )) + (Ĉ W, (Â W, ˆB W )) + ( ˆB W, (Ĉ W, Â W )) = O( h) in quantum and classical mechanics if Ĉ = Â ˆB then Ĉ(t) = Â(t) ˆB(t) quantum-classical analog true only to O( h) time translation invariance in usual form valid to O( h) R. Kapral and G. Ciccotti, A Statistical Mechanical Theory of Quantum Dynamics in Classical Environments, (Springer-Verlag, 2002) rkapral

9 trajectory picture of quantum-classical dynamics Ĥ W (R, P ) = P 2 2M + ĥ W (R) adiabatic eigenvalue problem : ĥ W (R) α; R = E α (R) α; R quantum-classical Liouville operator in adiabatic basis il αα,ββ = ( iω αα il αα )δ αβ δ α β + J αα,ββ evolution on αα quantum transitions plus bath momentum changes

10 evolution of density matrix ρ αα W (R, P, t) = ββ (e i ˆLt ) αα,ββ ρ ββ W (R, P ) iterate to obtain solution in terms of surface-hopping trajectories ρ αα W (R, P, t) = e (iω αα +il αα )t ρ αα 0 (R, P ) + ββ t + 0 dt e (iω αα +il αα )(t t ) J αα ββ e (iω ββ +il ββ )t ρ ββ 0 (R, P )

11 hybrid molecular dynamics-monte Carlo method for exact surface hopping trajectories sample times at which transitions occur and quantum state from suitable distributions quantum transitions and bath momentum changes determined by J αα,ββ J αα,ββ P ( M d αβ ) 2 S αβ δ α β P which depends on the non-adiabatic coupling matrix element S αβ = (E α E β )d αβ ( P M d αβ) 1 d αβ = α; R / R β; R evolve system either on single adiabatic surfaces or coherently under mean Hellmann-Feynman forces between transition events

12 schematic representation of a surface-hopping trajectory P αα αβ ββ (R,P ) (R,P) t t t 0 R evolution on coherent evolution evolution on α surface on mean of β surface α and β surfaces

13 an ensemble of points at t = 0 contributes to the density matrix at time t ρ αα (R,P,t) (R,P ) (R,P) t (R",P") t=0

14 example: two-level system coupled to a one-dimensional bath points at t = 0 contributing to density matrix at time t 1 11 P R

15 points at t = 0 contributing to density matrix at a much longer time t

16 density matrix evolution and decomposition into jump contributions ρ(-1,1,t) t ρ(-1,1,t) t

17 simulating quantum-classical dynamics: sequential short time propagation divide the time interval t into N segments of lengths t j jth segment t j t j 1 = t j so that the e -il t 0 e - ilt t ( e ilt ) s 0 s N = N s 1 s 2...s N 1 j=1 ( ) e il(t j t j 1 ) s j 1 s j index s is associated with a pair of quantum states (αα )

18 ( e ilt ) s 0 s N = N s 1 s 2...s N 1 j=1 ( ) e il(t j t j 1 ) s j 1 s j suppose that the time intervals t j t j 1 = t are sufficiently small ( ) N e ilt e il0 s j 1 (t j t j 1 ) (δsj s 0 s s j 1 tj sj 1 s j N ) s 1 s 2...s N 1 j=1 in the limit N, t 0 with N t = t we recover the iterated form of the integral propagator

19 algorithm for short time sequential propagation sums over quantum indices and integrals over phase space variables evaluated by Monte Carlo sampling propagate dynamics through one time interval: update the initial positions and momenta at time t by applying e il s 0 t define the probability, Π, of a non-adiabatic transition Π P s I 1, t M and sample from it d s I 1 s I (R si 1, t) t

20 calculation of expectation value of an observable Ô W (R, P, t) O(t) = s 0 drdp O s 0 W (R, P, t)ρs 0 W (R, P ) short time sequential segments can be concatenated to obtain nonadiabatic evolution for finite times

21 example: spin-boson system with a 10-oscillator bath Ĥ = hωˆσ x + N j=1 ( ˆP 2 j 2M j M jω j2 ˆR 2 j c j ˆR jˆσ z ) two-level system, with states { >, >}, separated by an energy gap 2 hω, bilinearly coupled to a harmonic bath with coupling characterised by an Ohmic spectral density quantum-classical dynamics is equivalent to full quantum mechanics for this model

22 evolution of average population difference σ z (t) = Tr drdp ˆσ z (R, P, t)ˆρ W (R, P, 0) O(t) t ξ = ξ = 0.1

23 momentum-jump approximation to J: since S αβ = E αβ ˆd αβ ( P M ˆd αβ ) 1 with E αβ = E α E β, we may write ( S αβ P ) = 1 + E αβ M (P ˆd αβ ) 2 action on any function f(p ) of the momentum if E αβ M is small yields the momentum jump approximation: ( S αβ ) f(p ) e E αβm / (P ˆd αβ ) 2 f(p ) f(p + 1 P 2 S αβ)

24 evolution of trajectories in the momentum jump approximation schematic representation of a surface-hopping trajectory P αα αβ ββ (R,P ) (R,P) t t t 0 R evolution on coherent evolution evolution on α surface on mean of β surface α and β surfaces

25 comparison with other surface hopping schemes Tully s method assume a classical trajectory (R(t), P (t)) and expand the wave function for the quantum subsystem in adiabatic states depending on the instantaneous values of coordinates Ψ[t; R(t)] = α c α (t) α; R(t) diagonal density matrix elements determine the probabilities of quantum transitions ρ αα (t) = c α (t)c α (t)

26 ansatz for classical evolution ββ (R,P ) P αα 0 (R,P) t t R classical trajectories in current formulation P αα αβ ββ (R,P ) (R,P) t t t 0 R

27 comparison of quantum dynamics, quantum-classical dynamics with Tully s surface hopping scheme for three-oscillator spin boson model Santer, Manthe and Stock, J. Chem. Phys. 114, 2001 (2001)

28 nonadiabatic reaction dynamics two-level system coupled to a nonlinear oscillator which is embedded in a bath of 50 harmonic oscillators system hamiltonian in diabatic basis H = ( ) Vn (R 0 ) + γ 0 R 0 Ω + Ω V n (R 0 ) γ 0 R 0 N j=0 P 2 j 2 + V b(r) γ b R 0 N c j R j I j=1 nonlinear oscillator potential and bath hamiltonian V n (R 0 ) = a 4 R4 0 b 2 R2 0, V b(r, P ) = N j=1 1 2 ω2 j R2 j Ohmic spectral density

29 adiabatic free energy potential surfaces N E 1,2 (R 0, R) = V n (R 0 ) Ω 2 γ 20 R20 + V b(r) γ b R 0 c j R j j= E W E e βw 1,2(R 0 ) R 0 dr e βe 1,2(R 0,R)

30 diabatic free energy potential surfaces N E d 1,2 (R 0, R) = V n (R 0 ) ± γ 0 R 0 + V b (R) γ b R 0 c j R j j= W 0 2 Ω R 0

31 phenomenology of reaction dynamics C W A B R 0 A k AB B k BA A k AC C k CA B k BC k CB C

32 linear theory of reaction rates entropy production σ = C I=A dn I dt ( µi k B T ) = B I=A dn I dt flux ( ) µc µ I k B T force linear phenomenological laws in terms of affinities, A JC = µ C µ J dn I dt = B J=A L IJ ( µj µ C k B T ) B = L IJ βa JC J=A rate law L AJ = (k JA δ JA )N eq J dn A dt similar equation for dn B /dt = L AA βa AC L AB βa BC

33 microscopic description species operators A : ˆχ A (R) = 1; R > θ(r 0 ) < 1; R B : ˆχ B (R) = 1; R > θ( R 0 ) < 1; R C : ˆχ C (R) = 2; R >< 2; R note: ˆχ A + ˆχ B + ˆχ C = 1

34 linear response derivation of rate law Ĥ(t) = Ĥ W B I=A ˆχ I A IC (t) = Ĥ W ˆχ A(t) microscopic expression for rate constant k A k AB + k AC = (βn eq A ) 1 t 0 dt Tr drdp ˆχ A (R, P, t)(ˆχ A (R),ˆρ W e (R, P )) involves quantum-classical bracket and equilibrium density

35 equilibrium density is stationary under quantum-classical dynamics: i ˆLˆρ W e = 0 solution to O( h) in the adiabatic basis ρ αα W e = Z0 1 W [δαα i h P M d αα ( β 2 (1 + α α 1 e βe e βe α α ) + )(1 δ αα )] E αα +O( h 2 ) non-adiabatic coupling matrix element: d αα = α; R R α ; R

36 adiabatic reaction dynamics on ground state surface W A -1 E R 0 B κ(t) t rate constant ka 1 eq (t) = (NA ) 1 drdp θ(r 0 (t)) P M δ(r 0)ρ 11 ST W e (R, P ) = kt A κ(t)

37 nonadiabatic reaction dynamics k A (t) = k d A (t) + ko A (t) ka d eq (t) = (NA ) 1 drdp χ 11 A (R, P, t) P M δ(r 0)ρ 11 W e (R, P ) contributions from even numbers of nonadiabatic transi- (R, P, t): tions dominant contribution χ 11 A arises from off-diagonal part of equilibrium density matrix ka o eq (t) = (NA ) 1 drdp χ 12 A (R, P, t)f(r, P )ρ11 W e (R, P ) (R, P, t): contributions from odd numbers of nonadiabatic transitions small contribution χ 12 A

38 rate kernel for nonadiabatic dynamics κ(t) t

39 decomposition of rate kernel contributions for nonadiabatic dynamics 0.2 κ i (t) j 2j t

40 two-jump nonadiabatic trajectory contributing to the rate kernel C W B A R 0

41 dynamics on ground state and mean of two adiabatic surfaces

42 Remarks quantum-classical Liouville equation provides a route to study nonadiabatic dynamics in condensed phase systems numerically exact solutions of the quantum-classical evolution equations can be obtained that account for energy transfer between the quantum and classical subsystems

43 iterated Dyson form of the solution of the quantum-classical Liouville equation can be simulated by hybrid MD-MC methods when used with the momentum jump approximation each realization of the dynamics is a single trajectory composed of incoherent and coherent evolution segments coherent segments in evolution differ from standard surface hopping schemes method can be used to simulate nonadiabatic rate processes applicable to complex many-body solvents