Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional...

Stochastic Quantum Molecular Dynamics:a functional theory for electrons and nuclei

dynamically coupled to an environment

Heiko Appel

University of California, San Diego

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 1 / 38

Outline

Open Quantum Systems

I Different approaches to deal with decoherence and dissipation

I Stochastic current density-functional theory

I Application: Stochastic simulation of (1,4)-phenylene-linkedzincbacteriochlorin-bacteriochlorin complex

Stochastic Quantum Molecular Dynamics: Theory and Applications

I Including nuclear motion: Stochastic Quantum Molecular Dynamics

I Application: Stochastic quantum MD of 4-(N,N-Dimethylamino)benzonitrile

Outlook: future prospects of SQMD

Outline

Why Open Quantum Systems?

General aspects:

I Cannot have perfectly isolated quantum systems

I Dissipation and Decoherence

I Every measurement implies contact with an environmentOne actually needs to bring a system into contact with an environment (i.e.measurement apparatus), in order to perform a measurement→ environment as continuos measurement of the system.

Research fields:

I Quantum computing/Quantum information theory

I (time-resolved) transport and optics

I Driven quantum phase transitions

I Quantum measurement

General aspects:

Research fields:

General aspects:

Research fields:

General aspects:

Research fields:

General aspects:

Research fields:

General aspects:

Research fields:

General aspects:

Research fields:

Open Quantum System

S +B : HS ⊗HB ,Ψ, ρ

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Hamiltonian of combined system

H = HS ⊗ IB + IS ⊗ HB + HSB

Unitary time evolution

i∂tΨ(t) = H(t)Ψ(t)d

dtρ(t) = −i

hH(t), ρ(t)

Open Quantum System

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Hamiltonian of combined system

H = HS ⊗ IB + IS ⊗ HB + HSB

Unitary time evolution

i∂tΨ(t) = H(t)Ψ(t)d

dtρ(t) = −i

hH(t), ρ(t)

iHeiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 4 / 38

Reduced system dynamics

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Tracing over bath degrees of freedom

ρS = TrB ρ

dtρS(t) = −iTrB

hH(t), ρ(t)

I ρS(t) represents in general no pure state.

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Tracing over bath degrees of freedom

ρS = TrB ρ

dtρS(t) = −iTrB

hH(t), ρ(t)

iI ρS(t) represents in general no pure state.

Nakajima-Zwanzig projection operator technique

Projection Operators

P ρ = trB{ρ} ⊗ ρB ≡ ρS ⊗ ρBQρ = ρ− P ρ

Properties

P + Q = I

P 2 = P

Q2 = Q

P Q = QP = 0

S. Nakajima, Progr. Theor. Phys., 20, 948-959 (1958). R. Zwanzig, J. Chem. Phys., 33, 1338-1341 (1960).

Total Hamiltonian

H = HS + HB + αHSB

Liouville von Neumann equation in interaction picture

∂tρ(t) = −iα

hHSB , ρ(t)

i≡ αLSB(t)ρ(t)

Apply Projection Operators

∂tP ρ(t) = αP LSB(t)P ρ(t) + αP LSB(t)Qρ(t)

∂tQρ(t) = αQLSB(t)P ρ(t) + αQLSB(t)Qρ(t)

Formally solve for Qρ(t)

Qρ(t) = G(t, t0)Qρ(t0) + α

dsG(t, s)QLSB(s)Qρ(s),

G(t, s) = T exp

QLSB(s′)ds′–, G(s, s) = I

Nakajima-Zwanzig equation

∂tP ρ(t) =αPLSB(t)P ρ(t) +

Source Termz }| {αPLSB(t)G(t, t0)Qρ(t0)

PLSB(t)G(t, s)QLSB(s)P ρ(s)ds| {z }Memory Term

Typical approximations for the Nakajima-Zwanzig equationI Perturbation theory in α, e.g. Born approximation (up to α2).

I Linked cluster/cumulant expansions.

I Markov approximation. Z t

... ρ(s)ds =⇒Z t

... ρ(t)ds

Problem:

Approximations to the Nakajima-Zwanzig equation can lead to unphysical states(loss of positivity)

Similar problems:

Redfield equations, Caldeira-Legget equation, ...

... ρ(s)ds =⇒Z t

... ρ(t)ds

Problem:

Similar problems:

... ρ(s)ds =⇒Z t

... ρ(t)ds

Problem:

Similar problems:

... ρ(s)ds =⇒Z t

... ρ(t)ds

Problem:

Similar problems:

Lindblad theorem

Lindblad equation

dtρS(t) = LρS(t)

Most general form for the generator of a quantum dynamical semigroup

LρS(t) = −ihH, ρS(t)

„VkρS(t)V †k −

2V †k VkρS(t)− 1

2ρS(t)V †k Vk

«On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976).

Semigroup properties:I Q(t) is completely positiveI Q(t) Q(s) = Q(t+ s)I Q(0) = I

Semigroup preserves:I Hermiticity (probabilities are real numbers)I Trace (conservation of norm)I Positivity (probabilities are positive)

Dynamical semigroup only for time-independent Hamiltonians⇒ Problem for TDDFT formulation

Lindblad theorem

Lindblad equation

dtρS(t) = LρS(t)

2ρS(t)V †k Vk

Lindblad theorem

Lindblad equation

dtρS(t) = LρS(t)

2ρS(t)V †k Vk

Lindblad theorem

Lindblad equation

dtρS(t) = LρS(t)

2ρS(t)V †k Vk

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

=⇒ Use density operator ρS

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

=⇒ Use state vector ΨS

I No need to work with composite objects like density matrices

I Can trace out bath directly on the level of state vectors

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

=⇒ Use state vector ΨS

I No need to work with composite objects like density matrices

I Can trace out bath directly on the level of state vectors

Feshbach Projection-Operator Method

H = HS + HB + αHSB , HBχn(xB) = εnχn(xB)

Expand total wavefunction in arbitrary complete and orthonormal basis of the bath

Ψ(xS , xB ; t) =Xn

φn(xS ; t)χn(xB)

P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n

|χk 〉〈χk |

Apply to TDSE

i∂tPΨ(t) = P HPΨ(t) + P HQΨ(t)

i∂tQΨ(t) = QHQΨ(t) + QHPΨ(t)

P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).

Ψ(xS , xB ; t) =Xn

φn(xS ; t)χn(xB)

P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n

|χk 〉〈χk |

Apply to TDSE

Ψ(xS , xB ; t) =Xn

φn(xS ; t)χn(xB)

P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n

|χk 〉〈χk |

Apply to TDSE

Effective equation for P Ψ (still fully coherent)

i∂tPΨ(t) =P HP PΨ(t) +

Source Termz }| {P HQe−iQHQtQΨ(0)

−iZ t

dτP HQeiQHQ(t−τ)QHP PΨ(τ)| {z }Memory Term

=⇒ Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al.

Non-Markovian Stochastic Schrodinger equation

I perturbative expansion to second order in αHSBI random phase approximation, dense bath spectrum, bath in statistical equilibrium

i∂tψ(t) =HSψ(t) + αXα

ηα(t)Vαψ(t)

− iα2

dτXαβ

Cαβ(t− τ)| {z }Bath correlation functions

V †αe−iHS(t−τ)Vβψ(τ) +O(α3)

−iZ t

ηα(t)Vαψ(t)

− iα2

dτXαβ

−iZ t

ηα(t)Vαψ(t)

− iα2

dτXαβ

Markovian Stochastic Schrodinger equation

δ-correlated bath

Cαβ(t− τ) = Dαβδ(t− τ)

Stochastic Schrodinger equation in Born-Markov approximation

ηα(t)Vαψ(t)

− iα2Xαβ

DαβV†α Vβψ(t) +O(α3)

Statistical average:

ρS(t) =|ψ(t) 〉〈ψ(t) |〈ψ(t) |ψ(t) 〉

I Unravelling of Lindblad equation for static (linear) HamiltoniansI Does not rely on semigroup propertyI Valid for time-dependent HamiltoniansI Gives always physical statesI Sound starting point to formulate stochastic TDDFT

TDDFT for Open Quantum Systems

I Approach in terms of density matricesK. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, 146803 (2005).

I Density matrix evolution may not obey positivity during time-evolution

I Scales as O(N2) due to usage of density matrix

I Approach in terms of stochastic Schrodinger equationsM. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).

I Positivity is guaranteed by construction

I Scales as O(N) since only state vectors are used

I Comparison to classical stochastic systems:

Fokker-Planck equation ⇐⇒ Langevin equation

Stochastic Time-Dependent Current-Density-Functional Theory

Can prove: For fixed bath operators V and initial state

j(r, t)1:1←→ A(r, t)

M. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).

Mapping of fully interacting stochastic TDSE to stochastic TDKS equations

i∂tψj(r, t) =

26664HKS(t)− 1

2iV †V| {z }

damping

+ l(t)V| {z }fluctuations

37775ψj(r, t)l(t) : stochastic process

l(t) = 0, l(t)l(t′) = δ(t− t′)

Assumes

I Factorization at initial time: ψ(t0) = ψS(t0)× ψB(t0)

I Markovian approximation: no bath memory

I Weak coupling to the bath (second order in HSB)

j(r, t)1:1←→ A(r, t)

i∂tψj(r, t) =

26664HKS(t)− 1

2iV †V| {z }

damping

37775ψj(r, t)

l(t) : stochastic process

l(t) = 0, l(t)l(t′) = δ(t− t′)

Assumes

j(r, t)1:1←→ A(r, t)

i∂tψj(r, t) =

26664HKS(t)− 1

2iV †V| {z }

damping

37775ψj(r, t)l(t) : stochastic process

l(t) = 0, l(t)l(t′) = δ(t− t′)

Assumes

Choice of bath operators: a simple model

Order-N scheme and bath operators which obey Fermi statistics

V jkk′(r) = δkj(1− δkk′)pγ(r)fD(εk) |ψj(r) 〉〈ψk′(r) |

Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, 054302 (2008).

Fermi-Dirac distribution

fD(εk) =

»1 + exp

„εk − µkBT

«–−1

Local relaxation ratesγkk′(r) = |ψk(r) 〉γ0〈ψk′(r) |

I Operators ensure that Fermi statistics is obeyed.

I If a steady state is reached, it will be a thermal state.

Choice of bath operators: a simple model

Order-N scheme and bath operators which obey Fermi statistics

V jkk′(r) = δkj(1− δkk′)pγ(r)fD(εk) |ψj(r) 〉〈ψk′(r) |

Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, 054302 (2008).

Fermi-Dirac distribution

fD(εk) =

»1 + exp

„εk − µkBT

«–−1

Local relaxation ratesγkk′(r) = |ψk(r) 〉γ0〈ψk′(r) |

I Operators ensure that Fermi statistics is obeyed.

I If a steady state is reached, it will be a thermal state.

Quantum jump algorithm

1) Draw uniform random number ηj ∈ [0, 1]

2) Propagate auxilary wave function under non-Hermitian Hamiltonian

i∂tΦ = H0Φ− iV †V Φ

3) Propagate system wave function under norm-conserving Hamiltonian

i∂tΨ = H0Ψ− iV †VΨ + i||VΨ||2Ψ

4) If norm of auxilary wave function drops below ηj , act with bath operator

||Φ(tj)|| ≤ ηj , Ψ(tj) = VΨ(tj), Φ(tj) = Ψ(tj)

5) Go to step 1)

cf. H.P. Breuer and F. Petruccione, Theory of Open Quantum Systems, Oxford University Press

=⇒ Leads to piecewise deterministic evolution

Average over stochastic realizations:

ρ = |Ψj 〉〈Ψj | guarantees always physical state!

Practical Implementation

Full-potential, all-electron code with local orbitals

http://www.fhi-berlin.mpg.de/aims/

Practical Implementation

Expansion of TD orbitals in non-orthorgonal atom centered basis functions φk

ψj(t) =Xk

cjk(t)φk

Overlap matrixSjk = 〈φj |φk 〉

TDKS equations

∂tcj(t) = HKS(t) cj(t)

Exponential midpoint approximation for short-time propagator

U(t+ ∆t, t) = exph−iS−1HKS(t+ ∆t/2)∆t+O(∆t3)

i= S−1/2 exp

h−iS−1/2HKS(t+ ∆t/2)∆tS−1/2 +O(∆t3)

Application

(1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex

Application: zincbacteriochlorin-bacteriochlorin complex

Free Propagation of HOMO

Free propagation of HOMO:

ψHOMO(t) = ψGSHOMO × e−iεHOMO t

Re{ψHOMO(t)} = ψGSHOMO × cos(εHOMOt)

(for real-valued orbitals)

Linear combination of HOMO and LUMO

Initial state

ψTDKSHOMO(t = 0) =

GSHOMO + e−i

π2 ψ

GSLUMO

iClosed quantum system

Linear combination of HOMO and LUMO

Initial state

ψTDKSHOMO(t = 0) =

GSHOMO + e−i

π2 ψ

GSLUMO

iIncluding coupling to environment

Outline

I Different approaches to deal with Decoherence and Dissipation

Molecular Dynamics for Open Systems

Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:

I Electronic degrees of freedom are treated with closed system approach

I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)

However:

Electrons are the first to experience energy transfer to a bath

Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei

Need open quantum theory for both electrons and nuclei

However:

Stochastic Quantum Molecular Dynamics

Extension of stochastic TDCDFT to include nuclear degrees of freedom

i∂tΨ = H(t)Ψ− 1

2iV †VΨ + l(t)VΨ

H(t) =Te(r, t) + Wee(r) + Uext,e(r, t)+

Tn(R, t) + Wnn(R) + Uext,n(R, t)+

Wen(r,R)

Total current〈 J(x, t) 〉 = 〈 j(r, t) 〉+ 〈 J(R, t) 〉, x = (r,R)

For given initial state Ψ(x, t = 0) and bath operators Vα(x, t)

〈 J(x, t) 〉 ←→ A(x, t)

Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, 212303 (2009).

Stochastic Quantum Molecular Dynamics: practical scheme

Extension of stochastic TDCDFT to include nuclear degrees of freedom

i∂tΨ(x, t) = H(t)Ψ(x, t)− 1

2iV †VΨ(x, t) + l(t)VΨ(x, t)

In practice: resort as approximation to classical nuclei

Bath operators

V jkk′(r,R(t); t) = δkj(1− δkk′)pγ(r,R(t); t)fD(εk) |ψj(r,R(t); t) 〉〈ψk′(r,R(t); t) |

Ehrenfest forces as approximation for classical nuclei

MαRα(t) = −Z

Ψ∗∇RαHeΨdr

Wavefunctions are stochastic =⇒ stochastic force on the nuclei

Vibronic excitation of Beryllium dimer, moving nuclei

closed quantum systemstretched initial condition

open quantum systemrelaxation rate τ = 300 fs

Maxwell-Boltzmann velocity distribution atjumps

f(~v) =“ m

”3/2exp

„−m~v2

Average over 15 stochastic realizations

Vibronic excitation of Beryllium dimer, moving nuclei

closed quantum systemstretched initial condition

open quantum systemrelaxation rate τ = 300 fs

Maxwell-Boltzmann velocity distribution atjumps

f(~v) =“ m

”3/2exp

„−m~v2

Average over 15 stochastic realizations

Dual-Fluoresence in 4-(N,N-Dimethylamino)benzonitrile

polar solventacetonitrile

non-polar solventn-hexane

ε(−45K) = 50.2ε(+75K) = 30.3

Experiment: J. Phys. Chem. A 2006, 110, 2955-2969

Stochastic Quantum MD simulation for4-(N,N-Dimethylamino)benzonitrile

Electron Localization Function

Rotated dimethyl group asinitial condition

Stochastic Quantum MD simulation for4-(N,N-Dimethylamino)benzonitrile

t=0 fsT=0 K

t=137 fsT=0K

t=273 fsT=0K

0102030

t=550 fsT=0K

-25 0 25angle [deg]

t=46 fsT=300K

-25 0 25angle [deg]

t=137 fsT=300K

-25 0 25angle [deg]

t=273 fsT=300K

-25 0 25angle [deg]

20t=550 fsT=300K

0 100 200 300 400 500 600 time [fs]

-30-20-10

0102030

closed system @ 0Kopen system @ 0Kopen system @ 300K

Conclusions

I SQMD allows to describe ab-initio relaxation and dephasing of electronic andnuclear degrees of freedom

I In constrast to standard QMD, stochastic QMD allows also coupling of electrons toa heat bath

I Nuclei can feel the heat bath via electron-nuclear coupling

Within reach of Stochastic Quantum Molecular Dynamics

I Themopower/Thermoelectric effects

I Excited state relaxation in solution

I Decoherence and dephasing in pump-probe experiments

I ....

Conclusions

I ....

Conclusions

I ....

Conclusions

I ....

Future work

I Relaxation times from system-bath interaction Hamiltonian

I Non-markovian dynamics: bath-correlation functions which are not delta correlated

I Functionals: Dependence of XC-functional on bath operators

Future work

Thanks!

Massimiliano Di VentraYonatan DubiMatt KremsJim Wilson

Roberto D’Agosta

Thank you for your attention!

Positivity

A density matrix which obeys positivity is a non-negative operator in Hilbert space

〈Ψ | ρ |Ψ 〉 ≥ 0, ∀ |Ψ 〉

This guarantees that the expectation value of a positive operator A

〈 A 〉(t) =Xi

pi〈Ψ(t) | A |Ψ(t) 〉 = Tr{ρ(t)A} ≥ 0

and the variance of any operator B

〈 B2 〉(t)− 〈 B 〉2(t) ≥ 0

Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional...

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