A fresh look at potential energy surfaces and Berry phases

How to make the Born-Oppenheimer approximation exact:

A fresh look at potential energy surfaces and Berry phases

Max-Planck Institute of Microstructure Physics

Halle (Saale) E.K.U. Gross

Process of vision

Light-induced isomeriztion

"Triad molecule": Candidate for photovoltaic applications

C.A. Rozzi et al, Nature Communications 4, 1602 (2013) S.M. Falke et al, Science 344, 1001 (2014)

TDDFT propagation with clamped nuclei

"Triad molecule": Candidate for photovoltaic applications

C.A. Rozzi et al, Nature Communications 4, 1602 (2013) S.M. Falke et al, Science 344, 1001 (2014)

Moving nuclei

Stationary Schrödinger equation

) r , R ( V ˆ )

r ( W ˆ )

r ( T ˆ )

R ( W ˆ )

R ( T ˆ H ˆ

en ee

e nn

n

   



with











  









 

  









 



 

 



 

 

 





e n

e

n e

n

N

1 j

N

1 j

en N

k j

k ,

j j k

ee

N

, nn

N

1 i

2 i e

N

1

2 n

R r

Vˆ Z r

r 1 2

Wˆ 1

R R

Z Z 2

Wˆ 1 m

Tˆ 2 M

Tˆ 2

Hamiltonian for the complete system of N_e electrons with coordinates and N_n nuclei with coordinates









  ^{r , R E}   ^{r , R}

H ˆ   

Time-dependent Schrödinger equation

         





^{ }

V

t , R , r t

, R , r V

R , r H t

, R , t r

i

e n

N

1 j

N

1 j

laser

laser











 



 









 



 

) r , R ( V ˆ )

r ( W ˆ )

r ( T ˆ )

R ( W ˆ )

R ( T ˆ H ˆ

en ee

e nn

n

   



with











  









 

  









 



 

 



 

 

 





e n

e

n e

n

N

1 j

N

1 j

en N

k j

k ,

j j k

ee

N

, nn

N

1 i

2 i e

N

1

2 n

R r

Vˆ Z r

r 1 2

Wˆ 1

R R

Z Z 2

Wˆ 1 m

Tˆ 2 M

Tˆ 2

Hamiltonian for the complete system of N_e electrons with coordinates and N_n nuclei with coordinates









Born-Oppenheimer approximation

.

for each fixed nuclear configuration

R

 ^{Tˆ ( r ) W ˆ ( r ) V ˆ ( r ) V ˆ ( r , R )}  ^{Φ   r Φ}

^{  r}

R en

ext e ee

e

   

 

solve

  ^{r , R}   ^{r χ}   ^R

Ψ

 



Make adiabatic ansatz for the complete molecular wave function:

and find best χ^BO by minimizing <Ψ^BO | H | Ψ^BO > w.r.t. χ^BO :

Born-Oppenheimer approximation

.

for each fixed nuclear configuration

R

 ^{Tˆ ( r ) W ˆ ( r ) V ˆ ( r ) V ˆ ( r , R )}  ^{Φ   r Φ}

^{  r}

R en

ext e ee

e

   

 

solve

  ^{r , R}   ^{r χ}   ^R

Ψ

 



Make adiabatic ansatz for the complete molecular wave function:

and find best χ^BO by minimizing <Ψ^BO | H | Ψ^BO > w.r.t. χ^BO :

Nuclear equation

     

 

 

Φ^BO_R ^* _n ^BO_R ^BO  ^BO









    _ 

 



) R ( Vˆ )

R ( Wˆ )

R (

Tˆ ^ext

n nn

n ^{AB O}_υ

 

 

Berry connection

 



 

 

A^BO_υ ^BO_R ^* _υ ^BO_R

 



 

BO

BO C A R dR

γ  

is a geometric phase

In this context, potential energy surfaces and the vector potential follow from an APPROXIMATION (the BO approximation).

 

A ^BO

 

B O

Nuclear equation

     

 

 

Φ^BO_R ^* _n ^BO_R ^BO  ^BO









    _ 

 



) R ( Vˆ )

R ( Wˆ )

R (

Tˆ ^ext

n nn

n ^{AB O}_υ

 

 

Berry connection

 



 

 

A^BO_υ ^BO_R ^* _υ ^BO_R

 



 

BO

BO C A R dR

γ  

is a geometric phase

In this context, potential energy surfaces and the vector potential follow from an APPROXIMATION (the BO approximation).

 

A ^BO

 

B O

Geometric Phases

Concept of geometric phase:

Discovered by S. Pancharatnam (1956) Proc. Indian Acad. Sci. A 44: 247–262.

In the context of quantum mechanics:

Michael V. Berry (1984) Proc. Royal Society 392 (1802), 45–57.

Whenever the Hamiltonian of a quantum system depends on a vector of parameters, R, the Berry phase is defined as:

where the line integral is along a closed loop, C, in parameter space.

A non-vanishing value of  only appears when C encircles some non-analyticity.

Standard representation of the full TD wave function

  ^ _

    ^

J

Ψ r, R,t Φ r χ R,t

and insert expansion in the full Schrödinger equation → standard non-adiabatic coupling terms from T_n acting on ^ΦBO_R,J

 

Expand full molecular wave function in complete set of BO states:

Plug Born-Huang expansion in full TDSE:

       

t k n k k k

i   R, t   T R, t   R  R, t

   

2

BO BO

R ,k R R , j R j

j

i i R, t

M

 

 

         

 



2

 

BO 2 BO

R ,k R R , j j

j

2M

R, t

 

 

       

 



NAC-2 NAC-1

The dynamics is "non-adiabatic" when the NAC terms cannot be neglected

 

BO

Φ1,R r

 

BO

E1 R

 ^{, t}  ^

  ^{, t}

  ^

 

^{, t}

 

0

r ,R χ

R Φ

r χ R Φ r

Ψ

 

BO

Φ0,R r ^EBO0

 

When only few BO-PES are important, the BO expansion gives a perfectly clear picture of the dynamics

Na⁺+ I^- Na + I

Example: NaI femtochemistry

Na⁺+ I^- Na + I

Example: NaI femtochemistry

emitted neutral Na atoms

Effect of tuning pump wavelength (exciting to different points on excited surface)

300 311

321 339 λ_pump/nm

Different periods

indicative of anharmonic potential

T.S. Rose, M.J. Rosker, A. Zewail, JCP 91, 7415 (1989)

For larger systems one would like to (one has

to) treat the nuclei classically.

Trajectory-based quantum dynamics

For larger systems one would like to (one has to) treat the nuclei classically.

But what’s the classical force when the nuclear

wave packet splits??

For larger systems one would like to (one has to) treat the nuclei classically.

But what’s the classical force when the nuclear

wave packet splits??

For larger systems one would like to (one has to) treat the nuclei classically.

But what’s the classical force when the nuclear wave packet splits??

There is only one correct

answer!

Outline

• Show that the factorisation

can be made exact

• Concept of exact PES and exact Berry phase

• Concept of exact and unique time-dependent PES

• Mixed quantum-classical treatment

  ^{r , R}     ^{r χ R}

Ψ  



Ali Abedi Axel Schild

Federica Agostini

Yasumitsu Suzuki

Seung Kyu Min

Neepa Maitra

(Hunter College, CUNY)

Ryan Requist Nikitas Gidopoulo

(Durham University, UK)

THANKS

Theorem I

The exact solutions of

can be written in the form

  ^{r , R E}   ^{r , R}

H ˆ   

     

Ψ r, R  

r χ R 

where

_ ^{d r Φ}

  ^r

^{ 1}

Phil. Trans. R. Soc. 372, 20130059 (2014)

  ^{R : e}

^dr  ^{r, R} 

 

 

 

S R

Proof of Theorem I:

Choose:

 

Ψ r, R

with some real-valued funcion

     

Φ

r : Ψ r, R / χ R 

Given the exact electron-nuclear wavefuncion

  ^{r 1}

r Φ

d





Then, by construction,

  ^{R : e}

^dr  ^{r, R} 

 

 

 

S R

Proof of Theorem I:

Choose:

 

Ψ r, R

with some real-valued funcion

     

Φ

r : Ψ r, R / χ R 

Given the exact electron-nuclear wavefuncion

  ^{r 1}

r Φ

d





Then, by construction,

Note: If we want (R) to be smooth, S(R) may be discontinuous

Immediate consequences of Theorem I:

1. The diagonal of the nuclear N_n-body density matrix is identical with

 

 

 in this sense, can be interpreted as a proper nuclear wavefunction.^χ

 

proof: ^Γ

 

_

 

_

 

 

 

1

Eq.  _

 



    



ν

2 ν ν

ν en

ext e ee

e i A

2M Vˆ 1

Vˆ Wˆ

Tˆ

Hˆ BO





 

 

χ A χ i M

1

Nn

ν

ν ν

ν ν

ν

R

R 



 



 



 



   





 

Eq. 





 

 

2M

1 _ext

n nn

N

ν

2 ν ν

ν

n  



 







 

where ^A_ν

 

_

 

 

Theorem II:

^

  ^{r and }   ^R

N.I. Gidopoulos, E.K.U. Gross,

Phil. Trans. R. Soc. 372, 20130059 (2014)

Eq.  _

 



    



ν

2 ν ν

ν en

ext e ee

e i A

2M Vˆ 1

Vˆ Wˆ

Tˆ

Hˆ BO





 

 

χ A χ i M

1

Nn

ν

ν ν

ν ν

ν

R

R 



 



 



 



   





 

Eq. 





 

 

2M

1 _ext

n nn

N

ν

2 ν ν

ν

n  



 







 

where ^A_ν

 

_

 

 

Theorem II:

^

  ^{r and }   ^R

Exact PES

Exact Berry potential

N.I. Gidopoulos, E.K.U. Gross,

Phil. Trans. R. Soc. 372, 20130059 (2014)

How do the exact PES look like?

+ +

-5 Å +5 Å

0 Å

x R

+

–

(1) (2)

MODEL

Nuclei (1) and (2) are heavy: Their positions are fixed

S. Shin, H. Metiu, JCP 102, 9285 (1995), JPC 100, 7867 (1996)

 

  ^{ }

 

A

R   dr  r    i

r

  ^{R : e}

  ^R

  

     

R

r r, R / R

   

Exact Berry connection

Insert:

  

     ^{ }

Aν R  Im



     

 

A

R  J

R /  R   

R

with the exact nuclear current density J_ν

Another way of reading this equation:

   

   

J

R   R {A

R   

R }





 

 

2M

1 _ext

n nn

N

ν

2 ν ν

ν

n  



 







 

Conclusion: The nuclear Schrödinger equation

yields both the exact nuclear N-body density and the exact nucler N-body current density

A. Abedi, N.T. Maitra, E.K.U. Gross, JCP 137, 22A530 (2012)

Question: Can the true vector potential be gauged away,

i.e. is the true Berry phase zero?

Question: Can the true vector potential be gauged away, i.e. is the true Berry phase zero?

Look at Shin-Metiu model in 2D:

+ + +

–

(1) (2)

BO-PES of 2D Shin-Metiu model

BO-PES of 2D Shin-Metiu model

conical intersection with Berry phase

• Non-vanishing Berry phase results from a non-analyticity in the electronic wave function as function of R.

• Such non-analyticity is found in BO approximation.

 

BO r

_R

• Non-vanishing Berry phase results from a non-analyticity in the electronic wave function as function of R.

• Such non-analyticity is found in BO approximation.

Does the exact electronic wave function show such non-analyticity as well (in 2D Shin-Metiu model)?

Look at

as function of nuclear mass M.

 

BO r

_R

   

D ^R   ^{r Φ}

^{r dr}

S.K. Min, A. Abedi, K.S. Kim, E.K.U. Gross, PRL 113, 263004 (2014)

M = ∞ D(R)

M = ∞ D(R)

Question: Can one prove in general that the exact molecular Berry phase vanishes?

Question: Can one prove in general that the exact molecular Berry phase vanishes?

Answer: No! There are cases where a nontrivial Berry phase appears in the exact treatment.

R. Requist, F. Tandetzky, EKU Gross, Phys. Rev. A 93, 042108 (2016).

Time-dependent case

Theorem T-I

The exact solution of

can be written in the form

where

     

i  

r, R, t  H r, R, t  r, R, t

 ^{r, R, t} 

    ^{r, t R, t}

   

 

dr 

r, t  1



A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010) JCP 137, 22A530 (2012)

Eq. 

       



    



ν

2 ν

ν ν

en ext

e ee

e i A R,t

2M R 1

, r Vˆ t

, r Vˆ Wˆ

Tˆ

 t Hˆ

BO

   

 



    

 

t , χ R

t , χ R i

M 1

n

t N

ν

ν ν

ν ν

ν

R

R  





 







 







   





Eq. 

 





 

     

 

2M 1

t ext

n nn

N

ν

2 ν

ν ν

n   

 







Theorem T-II

 

 

R 

 satisfy the following equations

A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010) JCP 137, 22A530 (2012)

Eq. 

       



    



ν

2 ν

ν ν

en ext

e ee

e i A R,t

2M R 1

, r Vˆ t

, r Vˆ Wˆ

Tˆ

 t Hˆ

BO

   

 



    

 

t , χ R

t , χ R i

M 1

n

t N

ν

ν ν

ν ν

ν

R

R  





 







 







   





Eq. 

 





 

     

 

2M 1

t ext

n nn

N

ν

2 ν

ν ν

n   

 







Theorem T-II

 

 

R 

 satisfy the following equations

A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010) JCP 137, 22A530 (2012)

Exact TDPES Exact Berry potential

How does the exact

time-dependent PES look like?

Example: Nuclear wave packet going through an avoided crossing (Zewail experiment)

A. Abedi, F. Agostini, Y. Suzuki, E.K.U.Gross, PRL 110, 263001 (2013)

F. Agostini, A. Abedi, Y. Suzuki, E.K.U. Gross, Mol. Phys. 111, 3625 (2013)

New MD scheme:

Perform classical limit of the nuclear equation, but retain the quantum treatment of the electronic degrees of freedom.

A. Abedi, F. Agostini, E.K.U.Gross, EPL 106, 33001 (2014)

Eq. 

       



    



ν

2 ν

ν ν

en ext

e ee

e i A R,t

2M R 1

, r Vˆ t

, r Vˆ Wˆ

Tˆ

 t Hˆ

BO

   

 



    

 

t , χ R

t , χ R i

M 1

n

t N

ν

ν ν

ν ν

ν

R

R  





 







 







   





Eq. 

 





 

     

 

2M 1

t ext

n nn

N

ν

2 ν

ν ν

n   



 







Theorem T-II

Eq. 

       



    



ν

2 ν

ν ν

en ext

e ee

e i A R,t

2M R 1

, r Vˆ t

, r Vˆ Wˆ

Tˆ

 t Hˆ

BO

   

 



    

 

t , χ R

t , χ R i

M 1

n

t N

ν

ν ν

ν ν

ν

R

R  





 







 







   





Eq. 

 





 

     

 

2M 1

t ext

n nn

N

ν

2 ν

ν ν

n   



 







Theorem T-II

Shin-Metiu model

Propagation of classical nuclei on exact TDPES

𝑑𝐑 𝐶₁ 𝐑, 𝑡 ² 𝐶₂ 𝐑, 𝑡 ² χ 𝐑, 𝑡 ²

𝑵_{𝒕𝒓𝒂𝒋}^−𝟏

𝑰

𝑪_𝟏^𝑰 𝒕 ^𝟐 𝑪_𝟐^𝑰 (𝒕) ^𝟐

Measure of decoherence:

Quantum:

Trajectories

Algorithm implemented in:

The "right" electron-phonon interaction

electron-phonon interaction

 

 

e ph k q k q q

k,q,

ˆ ? ?

H

M

k, q c

c b

b



  

k-q

k

q

k-q

k

+

electron-phonon interaction

 

 

e ph k q k q q

k,q,

ˆ ? ?

H

M

k, q c

c b

b



  

In a genuine ab-initio description, what is the exact coupling 𝐌 𝐤, 𝐪 ?

k-q

k

q

k-q

k

+

𝐌 𝐤, 𝐪 𝐌 𝐤, 𝐪

    

(2) ? ?

e ph , ' ' k - - ' - - ' ' ' '

, ' '

? ?

ˆ ?

H _{  } (k)c c b _ b _ b _ b _

 

 





k, q q

R

LITERATURE on 𝐌 𝐤, 𝐪 :

• What everybody uses:

• Robert van Leeuwen, PRB 69, 115110 (2004):

Many textbooks neglect ϵ^-1 completely (no screening).

• Higher-order terms (Marini, Ponce, Gonze, PRB 91, 224310 (2015) (using DFPT):

 





g, ' 1 2

c

n V n ' ' M n , n ' '

2MN



  



  

  

q

q p p q

q

p p

p p

  



^

^

1 0,

M _ , 2MN _ / ^ d ^ , ; _ Z e ^{ }

 

      



q r q r r r q

r R



Eq.  _

 



    



ν

2 ν ν

ν en

ext e ee

e i A

2M Vˆ 1

Vˆ Wˆ

Tˆ

Hˆ BO





 

 

χ A χ i M

1

Nn

ν

ν ν

ν ν

ν

R

R 



 



 



 



   





 

Eq. 





 

 

2M

1 _ext

n nn

N

ν

2 ν ν

ν

n  



 







 

Theorem II:

^

  ^{r and }   ^R

Exact phonons Expand ϵ(R) around equilibrium positions (to second order):

Eq.  ph q

 

q

? 1

ˆH k b b

   2



 

    

 

