BOHMIAN MECHANICS: a complementary computational tool to...

Guillem Albareda

Fritz Haber Institute [email protected]

BOHMIAN MECHANICS:

a complementary computational tool to describe molecular dynamics

Ringberg July 2012

D. Bohm L. De Broglie J.S. Bell

Ringberg July 2012 2

1. Motivation: A wave equation for a classical single particle

4. The use of Bohmian Mechanics in Molecular Dynamics

5. Conclusions and Future Work

Outline:

2. A short overview of Bohmian Mechanics: Analytic Bohmian Mechanics Synthetic Bohmian Mechanics

3. The Conditional wavefunction

3

1. Motivation: A wave equation for a classical single particle

( ) ( ) ( ) ( ) ( )( ) ( )

2 222 2

2

, /, ,, , ,

2 2 ,clcl cl

cl clcl

x t xx t x ti V x t x t x t

t m x m x tψψ ψα αα ψ ψψ

∂ ∂∂ ∂= − + +

∂ ∂

Classical wave equation for a single particle: [N. Rosen, Found. Phys. 16, 687 (1968)]

( ) ( ) ( ), , exp ( , ) /cl x t R x t iS x tψ α=

( ) ( )2

2, 1 ( , ) , 0x

R x t S x t R x tt m x

∂ ∂ +∇ = ∂ ∂

( ) ( )2, 1 ( , ) , 0

2S x t S x t V x t

t m x∂ ∂ + + = ∂ ∂

Local Conservation law:

Hamilton-Jacobi Equation:

( ) ( ),1,S x t

v x tm x∂

=∂

Since S(x,t) does not depend on R(x,t), v(x,t) is independent of the shape of clψ

Polar form of the wavefunction

Ringberg July 2012

Velocity field:

[ ] [ ] [ ]0

0 ' 'tj j j

tx t x t v t dt= + ∫( ) [ ]( )2

0 01

1, limM

jcl M j

x t x x tM

ψ δ→∞

=

= −∑Initial distribution of trajectories:

Can we do something similar with quantum mechanics?

“Extrinsic “ uncertainty

4

1. Motivation: A wave equation for classical single particle


Ringberg July 2012




Outline:

( ) ( ) ( )2

2, ,1 , 0R x t S x t

R x tt x m x

∂ ∂ ∂+ = ∂ ∂ ∂

( ) ( ) ( ) ( )22

2

, ,, ,

2Q Q

Q

x t x ti V x t x t

t m xψ ψ

ψ∂ ∂

= − +∂ ∂

( ) ( ) ( ), , exp ( , ) /x t R x t iS x tψ =

Quantum wave equation for a single particle: [D. Bohm, Phys. Rev. 85, 166 & 180 (1952)]

Local Continuity Equation:

Q. Hamilton-Jacobi Equation:

( ) ( ),1,S x t

v x tm x∂

=∂

Since S(x,t) does depend on R(x,t), v(x,t) dependent of the shape of Qψ

Polar form of the wavefunction

( ) ( ) ( ) ( )( )

2 2 22, , , /1 , 02 2 ,

S x t S x t R x t xV x t

t m x m R x t∂ ∂ ∂ ∂

+ + − = ∂ ∂

( ),Q x t≡

Velocity field: “Analytic” Bohmian Mechanics “Synthetic”

Bohmian Mechanics

[ ] [ ] [ ]0

0 ' 'tj j j

tx t x t v t dt= + ∫

2. A short overview of Bohmian Mechanics: Single-particle Bohmian Mechanics

5 Ringberg July 2012

( ) [ ]( )20 0

1

1, limM

j

M jx t x x t

Mψ δ

→∞=

= −∑Initial distribution of trajectories

“Intrinsic “ uncertainty

Pilot-wave mechanics

( ) ( ) ( )2

2

1

, ,1 , 0N

k k k

R x t S x tR x t

t x m x=

∂ ∂ ∂+ = ∂ ∂ ∂ ∑

( ) ( ) ( ) ( )22

21

, ,, ,

2

NQ Q

Qk k

x t x ti V x t x t

t m xψ ψ

ψ=

∂ ∂= − +

∂ ∂∑

( ) ( ) ( ), , exp ( , ) /x t R x t iS x tψ =

Quantum wave equation for N many particles: [D. Bohm, Phys. Rev. 85, 166 & 180 (1952)]

Local Continuity Equation:

Hamilton-Jacobi Equation:

( ) ( ),1,kk

S x tv x t

m x∂

=∂

Polar form of the wavefunction:

( ) ( ) ( ) ( )( )

2 2 22

1 1

, , , /1 , 02 2 ,

N Nk

k kk

S x t S x t R x t xV x t

t m x m R x t= =

∂ ∂ ∂ ∂ + + − = ∂ ∂ ∑ ∑

( )1

,N

kk

Q x t=

≡∑

Velocity field:

Since does depend on , each dependent of the shape of the whole wavefunction Qψ

( ),S x t ( ),R x t

( ),kv x t

[ ] [ ] [ ]0

0 ' 'tj j j

k k ktx t x t v t dt= + ∫

2. A short overview of Bohmian Mechanics: Many-particle Bohmian Mechanics


( ) [ ]( )20 0

1 1

1, limNM

jkM j k

x t x x tM

ψ δ→∞

= =

= −∑∏Initial distribution of trajectories

Evaluation of Observables

“…In any case, the basic reason for not paying attention to the Bohm approach is not some sort of ideological rigidity, but much simpler…It is just that we are all too busy with our own work to spend time on something that doesn’t seem likely to help us make progress with our real problems”. Steven Weinberg (private comunication with Shelly Goldstein)

Main criticism against Bohmian Mechanics formalism:

( ) ( ) ( ) *

ˆ ˆA A

ˆRe , , ,xx t A x i x t dx

ϕ ϕ

ϕ ϕ+∞

−∞

=

= − ∇ ∫

where ( ) ( ) ( )ˆ ˆA , , ' ', 'x t A x x x t dxϕ ϕ+∞

−∞= ∫

( )( ) ( ) ( )

( ) ( )

*

*

ˆ, , ,A , Re

, ,x

B

x t A x i x tx t

x t x t

ϕ ϕ

ϕ ϕ

− ∇ =

“Local expectation value”: Expectation values:

( ) ( )

[ ]( )

2

1

A R , A ,

1lim A ,

B

Mj

BM j

x t x t dx

x t tM

+∞

−∞

→∞=

=

=

∫

∑

Examples for common observables:

( )Bx x x= ( ) ( ), , /Bp x t S x t x= ∂ ∂ ( ) ( ) ( )2

,1, ,2B

S x tK x t Q x t

m x∂

= + ∂ ( ) ( ), ,BJ x t v x t=

2. A short overview of Bohmian Mechanics: Evaluating Observables


8

1. Motivation: A wave equation for classical single particles



2. A short overview of Bohmian Mechanics: “Analytic” Bohmian Mechanics “Synthetic” Bohmian Mechanics

3. The Conditional Wavefunction

Ringberg July 2012

Outline:

The Conditional wave equation:

( ) [ ]( )20 0

1 1

1, limNM

jkM j k

x t x x tM

ψ δ→∞

= =

= −∑∏ [ ] ( )[ ]

( )[ ]0 0

0 0

1 1

, ,1 1,...,j j

jo

N Nx x t x x t

S x t S x tv t

m x m x= =

∂ ∂ = ∂ ∂

Initial Trajectory’s Positions obey: Initial Trajectory’s Velocity obey:

( )0,x tψ [X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]

( ) ( ) ( )2

2

1

,, ,

2

N

kk

x ti V x t x t

t mψ

ψ=

∂ = − ∇ + ∂ ∑

[ ] ( )[ ]

( )[ ]0

1 1

, ,1 1,...,j j

j

N Nx x t x x t

S x t S x tv t

m x m x= =

∂ ∂ = ∂ ∂

[ ] [ ] [ ]

00 '

tj j j


M grid points

configuration points NM

3. The Conditional wave function


The Conditional wave equation:

( ) [ ]( )20 0

1 1

1, limNM

jkM j k

x t x x tM

ψ δ→∞

= =

= −∑∏ [ ] ( )[ ]

( )[ ]0 0

0 0

1 1

, ,1 1,...,j j

jo

N Nx x t x x t

S x t S x tv t

m x m x= =

∂ ∂ = ∂ ∂

Initial Trajectory’s Positions obey: Initial Trajectory’s Velocity obey:

( )0,x tψ

( ) ( ) ( )2

2,, ,

2k k

k eff k k k

x ti V x t x t

t mϕ

ϕ∂

= − ∇ + ∂

[ ] ( )[ ]

( )[ ]1 01

1 1

1 1

, ,1 1,...,j j

N N

N Nj

N Nx x t x x t

S x t S x tv t

m x m x= =

∂ ∂ = ∂ ∂

[ ] [ ] [ ]

00 '

tj j j


M grid points

configuration points M N⋅

[X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]



M grid points

configuration points

Bad points :

2 2

2

( , ) ( , [ ], ) ( , [ ], ) · ( , [ ], ) ( , )2

a aa a b a b a b a a

a a

x ti V x x t t G x x t t i J x x t t x tt m x

ϕ ϕ ∂ ∂

= − + + + ∂ ∂

The interacting potential from (classical-like) Bohmian trajectories

Good points :

The terms G and J depend on the many-particle wave-function

This difficulty reminds the one it is found in the DFT (or TD-DFT)

An exact procedure for computing many-particle Bohmian trajectories where the correlations are introduced into time-dependent potentials:

There is a real potential to account for “non-classical” correlations

There is a imaginary potential to account for non-conserving norms

( ) ( ) ( ) ( ) ( ) [ ]( )1

,, , , , , ,

N

a a b b b k k kk kk a

S x tG x x t V x t K x t Q x t v x t t

x=≠

∂ = + + − ∂

∑

( ) ( ) [ ]( ) ( ) ( )2 2

1

, , ,, , ,

N

a a b kk k k k kk a

R x t R x t S x tJ x x t v x t t

x x m x=≠

∂ ∂∂ = − ∂ ∂ ∂ ∑



[X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]



1 1

2 221 1 1 11 1 1 1 1 2

0 1 2 1 1 [ ]

( , ) ( , )( , ) ; [ ] [ ]2 * 4 [ ] | ( , ) |

o

t

ot r r t

r t q J r ti r t r t r t dtt m r r t r tπ ε ε

=

∂Ψ = − ∇ + Ψ = + ∂ − Ψ ∫

2 2

2 222 2 2 22 2 2 2 2 2

0 1 2 2 2 [ ]

( , ) ( , )( , ) ; [ ] [ ]2 * 4 [ ] | ( , ) |

o

t

ot r r t

r t q J r ti r t r t r t dtt m r t r r tπ ε ε

=

∂Ψ = − ∇ + Ψ = + ∂ − Ψ ∫

A simple case example: two Coulomb interacting electrons

0.0 0.1 0.2 0.3 0.40.8

1.0

1.2

1.4

(b)

ER FROZEN

ER FROZEN

SELF-CONSISTENT POTENTIALS

SELF-CONSISTENT POTENTIALS

Our approachVisual guided line

Fano

Fac

tor

Applied bias (Volts)

0

1

2

3

4

5

6(a)

Visual guided line# Particles collector# Particles emitter

Our approach

Cur

rent

(µA)

0.0 0.1 0.2 0.3 0.40.8

1.0

1.2

(b)

FROZEN POTENTIALS

FROZEN POTENTIALS

Our approachButtiker results [6]

Fano

Fac

tor

Applied bias (Volts)

0

1

2

3

4(a)

EFE

EFC

x=Lx=0L

ColectorEmitterER

Esaki results [16]# Particles collector# Particles emitterOur approach

Cur

rent

(µA)



Application to nanoelectronic device simulation:

2 2

2

( , ) ( , [ ], ) ( , [ ], ) · ( , [ ], ) ( , )2

a aa a b a b a b a a

a a

x ti V x x t t G x x t t i J x x t t x tt m x

ϕ ϕ ∂ ∂

= − + + + ∂ ∂

G. Albareda et al., Phys. Rev. B 79, 075315 (2009).

G. Albareda et al., Phys. Rev. B 82, 085301 (2010).

BITLLES: Bohmian Interacting Transport in non-equiLibrium eLEctronic Structures

Bowling pins

F.L.Traversa et al., IEEE Trans. Elect. Dev. 58, 2104 (2011).

14






Ringberg July 2012

Outline:

ˆ ˆ ˆmol el nuc el el el nuc nuc nucH K K V V V− − −= + + + +( ) ( )

, , ˆ , ,mol

x X ti H x X t

t

ψψ

∂=

∂

Conditional wavefunction of the electrons: ( ) [ ]( ) [ ]( ) [ ]( ) ( ), ˆ , , , , , , ,el

el el el el

x ti H x X t t G x X t t iJ x X t t x t

tϕ

ϕ∂

= + +∂

( ) ( ) ( )( )

( ) [ ]2

2 22

1

, , , , / , ,1, ,2 2 , ,

nucNn

el nn n n

S x X t R x X t X S x X tG x X t v t

M X M XR x X t=

∂ ∂ ∂ ∂ = − − ∂ ∂ ∑

( ) ( )( ) [ ] ( ) ( )2 2

21

, , , , , ,, ,

2 , ,

nucN

el nn n n n

R x X t R x X t S x X tJ x X t v t

X X M XR x X t=

∂ ∂∂ = − ∂ ∂ ∂ ∑

Transmission of probability density between electrons and nuclei:

Non-classical correlations between electrons and nuclei:

MQCB & Ehrenfest dynamics for the electronic part:

( ) [ ]( ) ( ), ˆ , , ,elel el

x ti H x X t t x t

tϕ

ϕ∂

=∂

where

( ) [ ]( ) ( ), ˆ , , ,elel el

d x ti H x X t t x t

dtϕ

ϕ=



[E. J. Heller J. Chem. Phys. 62, 1544 (1975)] [E. Gindensperger J. Chem. Phys. 113, 1 (2000)]

Conditional wavefunction of the nuclei a:

( ) [ ] [ ]( ) [ ] [ ]( ) [ ] [ ]( ) ( ), ˆ , , , , , , , , , ,a aa a b a a b a a b a

X ti H x t X X t t G x t X X t t iJ x t X X t t x t

t∂Ω

= + + Ω∂

( ) ( ) ( )( )

( ) [ ]

( ) ( )( )

( ) [ ]

22 22

1

22 22

1

, , , , / , ,1, ,2 2 , ,

, , , , / , ,12 2 , ,

el

nuc

Nj

a jj j j

Nn

nn n nn a

S x X t R x X t x S x X tG x X t v t

m x m xR x X t

S x X t R x X t X S x X tv t

M X m XR x X t

=

=≠

∂ ∂ ∂ ∂ = − − ∂ ∂ ∂ ∂ ∂ ∂ + − − ∂ ∂

∑

∑

( ) ( )( ) [ ] ( ) ( )

( )( ) [ ] ( ) ( )

2

21

2

21

, , , , , ,, ,

2 , ,

, , , , , ,

2 , ,

el

nuc

N

a jj j j j

N

nn n n nn a

S x X t R x X t S x X tJ x X t v t

x x m xR x X t

S x X t R x X t S x X tv t

X X M XR x X t

=

=≠

∂ ∂∂ = − ∂ ∂ ∂ ∂ ∂∂ + − ∂ ∂ ∂

∑

∑

Transmission of probability density from the nuclei a to all the other particles:

Non-classical correlations between the nuclei a and all the other particles:

ˆ ˆa a el el el a nuc aH K V V V− − −= + + +where



( ) [ ] [ ]( ) [ ] [ ]( ) [ ] [ ]( ) ( ), ˆ , , , , , , , , , ,a aa a b a a b a a b a

X ti H x t X X t t G x t X X t t iJ x t X X t t x t

t∂Ω

= + + Ω∂

Conditional wavefunction of the nuclei a:

( ) ( ) ( )( ), , exp , /a a a a a aX t r X t is X tΩ = a×∇

( ) ( ) ( ) ( ) ( ) , , , , ,a a a el a a nuc a a a a nuc aM r X t V X t V X t Q X t G X t− −⋅ = −∇ + + +

Ehrenfest dynamics for the nuclei:

( ) ( ) ( ) ( ) ( ), , , , ,a a a el el a a el nuc a aM r X t x t V X t x t V X tϕ ϕ− − ⋅ = −∇ +

MQCB dynamics for the nuclei:

( ) ( ) ( ), , ,a a a el a a nuc a aM r X t V X t V X t− −⋅ = −∇ +



[E. J. Heller J. Chem. Phys. 62, 1544 (1975)]

[E. Gindensperger J. Chem. Phys. 113, 1 (2000)]

18






Ringberg July 2012

Outline:

Coherence in the SSH model Hamiltonian:

In particular, Bohmian Mechanics together with the concept of Conditional wavefunction allows a rigorous derivation of the MQCB and Ehrenfest approaches to molecular dynamics.

The exact splitting of electronic and nuclear degrees of freedom in terms of the Conditional wavefunction leads to a rigorous starting point for making approximations, specially for mixed quantum-classical approaches.

SSH el el ph phH H H H−= + +

( )1

† †0 1, , , 1,

1 1

N

el n s n s n s n sn s

H t c c c c−

+ += =±

= − × +∑∑

( ) ( )1

† †1 1, , , 1,

1 1

N

el ph n n n s n s n s n sn s

H u u c c c cα−

− + + += =±

= − × +∑∑

( )2 1

21

1 12 2

N Nn

ph n nn n

p kH u uM

−

+= =

= + −∑ ∑

4. Conclusions and Future work


Ringberg July 2012

ACKNOWLEDGMENTS:

Heiko Appel Angel Rubio

Ignacio Franco Xavier Oriols

Tanja Dimitrov Johannes Flick René Jestädt

Jessica Walkenhorst Kurt Baarman Wael Chibani

Alexander Kegeles

THANK YOU!

BOHMIAN MECHANICS: a complementary computational tool to...

Documents

Transcript of BOHMIAN MECHANICS: a complementary computational tool to...