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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

Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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Page 1: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Page 2: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

Page 3: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

Page 4: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

Page 5: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 6: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 7: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 8: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 9: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 10: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 11: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

Page 12: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

i

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

Page 13: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

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

Page 14: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Reduced system dynamics

S +B : HS ⊗HB ,Ψ, ρ

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Tracing over bath degrees of freedom

ρS = TrB ρ

d

dtρS(t) = −iTrB

hH(t), ρ(t)

i

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 5 / 38

Page 15: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Reduced system dynamics

S +B : HS ⊗HB ,Ψ, ρ

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

Tracing over bath degrees of freedom

ρS = TrB ρ

d

dtρS(t) = −iTrB

hH(t), ρ(t)

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 5 / 38

Page 16: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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).

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 6 / 38

Page 17: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

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)

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 7 / 38

Page 18: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

Formally solve for Qρ(t)

Qρ(t) = G(t, t0)Qρ(t0) + α

Z t

t0

dsG(t, s)QLSB(s)Qρ(s),

where

G(t, s) = T exp

»α

Z t

s

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)

+ α2

Z t

t0

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

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 8 / 38

Page 19: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

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

t0

... ρ(s)ds =⇒Z t

t0

... 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, ...

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38

Page 20: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

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

t0

... ρ(s)ds =⇒Z t

t0

... 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, ...

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38

Page 21: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

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

t0

... ρ(s)ds =⇒Z t

t0

... 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, ...

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38

Page 22: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Nakajima-Zwanzig projection operator technique

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

t0

... ρ(s)ds =⇒Z t

t0

... 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, ...

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38

Page 23: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Lindblad theorem

Lindblad equation

d

dtρS(t) = LρS(t)

Most general form for the generator of a quantum dynamical semigroup

LρS(t) = −ihH, ρS(t)

i+Xk

γk

„VkρS(t)V †k −

1

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38

Page 24: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Lindblad theorem

Lindblad equation

d

dtρS(t) = LρS(t)

Most general form for the generator of a quantum dynamical semigroup

LρS(t) = −ihH, ρS(t)

i+Xk

γk

„VkρS(t)V †k −

1

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38

Page 25: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Lindblad theorem

Lindblad equation

d

dtρS(t) = LρS(t)

Most general form for the generator of a quantum dynamical semigroup

LρS(t) = −ihH, ρS(t)

i+Xk

γk

„VkρS(t)V †k −

1

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38

Page 26: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Lindblad theorem

Lindblad equation

d

dtρS(t) = LρS(t)

Most general form for the generator of a quantum dynamical semigroup

LρS(t) = −ihH, ρS(t)

i+Xk

γk

„VkρS(t)V †k −

1

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38

Page 27: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Reduced system dynamics

S +B : HS ⊗HB ,Ψ, ρ

S : HS ,ΨS , ρS

System

B : HB ,ΨB , ρB

Environment

=⇒ Use density operator ρS

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 11 / 38

Page 28: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Reduced system dynamics

S +B : HS ⊗HB ,Ψ, ρ

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 12 / 38

Page 29: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Reduced system dynamics

S +B : HS ⊗HB ,Ψ, ρ

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 12 / 38

Page 30: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Projection Operators

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).

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

Page 31: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Projection Operators

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).

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

Page 32: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Projection Operators

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).

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

Page 33: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Feshbach Projection-Operator Method

Effective equation for P Ψ (still fully coherent)

i∂tPΨ(t) =P HP PΨ(t) +

Source Termz }| {P HQe−iQHQtQΨ(0)

−iZ t

0

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

Z t

0

dτXαβ

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

V †αe−iHS(t−τ)Vβψ(τ) +O(α3)

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38

Page 34: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Feshbach Projection-Operator Method

Effective equation for P Ψ (still fully coherent)

i∂tPΨ(t) =P HP PΨ(t) +

Source Termz }| {P HQe−iQHQtQΨ(0)

−iZ t

0

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

Z t

0

dτXαβ

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

V †αe−iHS(t−τ)Vβψ(τ) +O(α3)

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38

Page 35: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Feshbach Projection-Operator Method

Effective equation for P Ψ (still fully coherent)

i∂tPΨ(t) =P HP PΨ(t) +

Source Termz }| {P HQe−iQHQtQΨ(0)

−iZ t

0

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

Z t

0

dτXαβ

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

V †αe−iHS(t−τ)Vβψ(τ) +O(α3)

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38

Page 36: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Markovian Stochastic Schrodinger equation

δ-correlated bath

Cαβ(t− τ) = Dαβδ(t− τ)

Stochastic Schrodinger equation in Born-Markov approximation

i∂tψ(t) =HSψ(t) + αXα

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 15 / 38

Page 37: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38

Page 38: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38

Page 39: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38

Page 40: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38

Page 41: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38

Page 42: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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)

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38

Page 43: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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.

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 18 / 38

Page 44: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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.

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 18 / 38

Page 45: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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!

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 19 / 38

Page 46: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Practical Implementation

Full-potential, all-electron code with local orbitals

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 20 / 38

Page 47: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

iS∂

∂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)

iS1/2

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 21 / 38

Page 48: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Application

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 22 / 38

Page 49: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Application: zincbacteriochlorin-bacteriochlorin complex

HOMO

LUMO

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 23 / 38

Page 50: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Free Propagation of HOMO

Free propagation of HOMO:

ψHOMO(t) = ψGSHOMO × e−iεHOMO t

Re{ψHOMO(t)} = ψGSHOMO × cos(εHOMOt)

(for real-valued orbitals)

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 24 / 38

Page 51: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Linear combination of HOMO and LUMO

Initial state

ψTDKSHOMO(t = 0) =

1√2

GSHOMO + e−i

π2 ψ

GSLUMO

iClosed quantum system

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 25 / 38

Page 52: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Linear combination of HOMO and LUMO

Initial state

ψTDKSHOMO(t = 0) =

1√2

GSHOMO + e−i

π2 ψ

GSLUMO

iIncluding coupling to environment

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 26 / 38

Page 53: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 27 / 38

Page 54: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38

Page 55: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38

Page 56: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38

Page 57: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38

Page 58: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38

Page 59: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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).

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Page 60: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Note:

Wavefunctions are stochastic =⇒ stochastic force on the nuclei

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 30 / 38

Page 61: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

2πkT

”3/2exp

„−m~v2

2kT

«

Average over 15 stochastic realizations

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 31 / 38

Page 62: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

2πkT

”3/2exp

„−m~v2

2kT

«

Average over 15 stochastic realizations

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 31 / 38

Page 63: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

polar solventacetonitrile

non-polar solventn-hexane

ε(−45K) = 50.2ε(+75K) = 30.3

DMABN

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 32 / 38

Page 64: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Electron Localization Function

Rotated dimethyl group asinitial condition

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 33 / 38

Page 65: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

CH3CH

CH3CH

C NN

δ

0

20

40

coun

ts

t=0 fsT=0 K

t=137 fsT=0K

t=273 fsT=0K

0102030

t=550 fsT=0K

-25 0 25angle [deg]

0

20

40

coun

ts

t=46 fsT=300K

-25 0 25angle [deg]

t=137 fsT=300K

-25 0 25angle [deg]

t=273 fsT=300K

-25 0 25angle [deg]

0

10

20t=550 fsT=300K

0 100 200 300 400 500 600 time [fs]

-30-20-10

0102030

angl

e [d

eg]

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

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 34 / 38

Page 66: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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 Driven quantum phase transitions

I ....

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38

Page 67: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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 Driven quantum phase transitions

I ....

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38

Page 68: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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 Driven quantum phase transitions

I ....

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38

Page 69: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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 Driven quantum phase transitions

I ....

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38

Page 70: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38

Page 71: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38

Page 72: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38

Page 73: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

Thanks!

Massimiliano Di VentraYonatan DubiMatt KremsJim Wilson

Roberto D’Agosta

Thank you for your attention!

Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 37 / 38

Page 74: Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment Heiko Appel University

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

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