Exact factorization of the time-dependent electronnuclear wave function: A fresh look on potential
energy surfaces and Berry phases
Co-workers:
Ali Abedi (MPI-Halle)
Neepa Maitra (CUNY)
Nikitas Gidopoulos (Rutherford Lab)
Hamiltonian for the complete system of Ne electrons with coordinates
r1 rN   r and Nn nuclei with coordinates R1  R N   R , masses
M1 ··· MNn and charges Z1 ··· ZNn.
e
n
ˆ  Tˆ ( R )  W
ˆ ( R )  Tˆ ( r )  W
ˆ (r)  V
ˆ (R , r)
H
n
nn
e
ee
en

with
ˆ 
T

n
 1
ˆ 
W
ee
1
2
Ne
2
Nn
2M 
Ne
r
j, k
j k
ˆ 
T

e
1
j
 rk
convention:
Full Schrödinger equation:
i 1
i
2
ˆ 
W
nn
2m
1
Nn

Z Z 
2  , R   R 

Ne
Nn
ˆ 
V
 
en
j1  1
Z
rj  R 
Greek indices  nuclei
Latin indices  electrons
ˆ   r, R   E  r, R 
H
Born-Oppenheimer approximation
solve
Tˆ (r)  Wˆ

ˆ ext ( r )  V
ˆ ( r, R ) ΦBO  r  BO  R  ΦBO  r 
(
r
)

V
ee
e
en
R
R
e
for each fixed nuclear configuration R .
Make adiabatic ansatz for the complete molecular wave function:
Ψ
BO
 r, R     r   χ  R 
BO
R
BO
and find best χBO by minimizing <ΨBO | H | ΨBO > w.r.t. χBO :
Nuclear equation

1
BO
ˆ (R)  W
ˆ (R)  V
ˆ ext ( R ) 
T
(-i

)



R
R

 M A BO
nn
n

 n
υ



  ΦR
BO *
r  Tˆ  R Φ r d r  χ  R   Eχ  R 
n
BO
R
BO
BO

Berry connection
BO
Aυ
γ
R    Φ
BO
 C  
C
BO *
R
r  (-i
BO
) ΦR
υ
r dr
 BO

A  R   dR is a geometric phase
 BO
 R  and the Berry potential A  R 
In this context, potential energy surfaces 
are APPROXIMATE concepts, i.e. they follow from the BO approximation.
BO
Nuclear equation

1
BO
ˆ (R)  W
ˆ (R)  V
ˆ ext ( R ) 
T
(-i

)



R
R

 M A BO
nn
n

 n
υ



  ΦR
BO *
r  Tˆ  R Φ r d r  χ  R   Eχ  R 
n
BO
R
BO
BO

Berry connection
BO
Aυ
γ
R    Φ
BO
 C  
C
BO *
R
r  (-i
BO
) ΦR
υ
r dr
 BO

A  R   dR is a geometric phase
 BO
 R  and the Berry potential A  R 
In this context, potential energy surfaces 
are APPROXIMATE concepts, i.e. they follow from the BO approximation.
BO
“Berry phases arise when the world is approximately separated into a system and
its environment.”
GOING BEYOND BORN-OPPENHEIMER
Standard procedure:
Expand full molecular wave function in complete set of BO states:
Ψ K r, R    ΦR , J r   χ K, J  R 
BO
J
and insert expansion in the full Schrödinger equation → standard
non-adiabatic coupling terms from Tn acting on Φ BO
 r .
R ,J
Drawbacks:
• χJ,K depends on 2 indices: → looses nice interpretation as
“nuclear wave function”
• In systems driven by a strong laser, hundreds of BO-PES can be
coupled.
Φ1,R r 
BO
BO
E1
Φ
BO
0,R
r 
BO
E0
R 
R 
BO
BO
Ψ 0  r,R   χ 00  R Φ0,

χ
R
r  01  Φ1,R r 
R
Potential energy surfaces are absolutely essential
in our understanding of a molecule
.... and can be measured by femto-second pump-probe spectroscopy:
Zewail, J. Phys. Chem. 104, 5660, (2000)
fs laser pulse (the pump
pulse) creates a wavepacket,
i.e., a rather localised object
in space which then spreads
out while moving.
The NaI system (fruit fly of femtosecond spectroscopy)
The potential energy curves:
The ground, X-state is ionic in character
with a deep minimum and 1/R potential
leading to ionic fragments:
Na+ + IPE
R
PE
Na + I
R
The first excited state is a weakly
bound covalent state with a shallow
minimum and atomic fragments.
The non-crossing rule
ionic
covalent
Na++ INa + I
“Avoided crossing”
ionic
NaI femtochemistry
The wavepacket is launched on the
repulsive wall of the excited surface.
As it keeps moving on this surface it
encounters the avoided crossing at
6.93 Ǻ. At this point some molecules
will dissociate into Na + I, and some
will keep oscillating on the upper
adiabatic surface.
Na++ INa + I
The wavepacket continues sloshing about on the
excited surface with a small fraction leaking out each
time the avoided crossing is encountered.
I.
Probing Na atom products:
Steps in the production of Na
as more of the wavepacket
leaks out each vibration into
the Na + I channel. Each step
smaller than last (because
fewer molecules left)
Effect of tuning pump wavelength (exciting to
different points on excited surface)
λpump/nm
300
311
321
339
Different periods
indicative of anharmonic
potential
T.S. Rose, M.J. Rosker, A. Zewail, JCP 91, 7415 (1989)
GOAL: Show that Ψ  r, R    R  r   χ  R  can be made EXACT
• Concept of EXACT potential energy surfaces (beyond BO)
• Concept of EXACT Berry connection (beyond BO)
• Concept of EXACT time-dependent potential energy surfaces
for systems exposed to electro-magnetic fields
• Concept of ECACT time-dependent Berry connection
for systems exposed to electro-magnetic fields
Theorem I
The exact solutions of
ˆ   r, R   E  r, R 
H
can be written in the form
Ψ  r, R    R  r   χ  R 
where
 dr Φ r 
R
2
 1 for each fixed R .
First mentioned in: G. Hunter, Int. J.Q.C. 9, 237 (1975)
Immediate consequences of Theorem I:
1. The diagonal R  of the nuclear Nn-body density matrix is identical
2
with χ R 
proof: ΓR    dr Ψr, R 
2
  dr ΦR r 
2
χ R   χ R 
2
2
1
 in this sense, χ R  can be interpreted as a proper nuclear wavefunction.
2. Φ R  r and χ R are unique up to within the “gauge transformation”
 
~
iθ  R 
Φ R  r  : e
ΦR r 
 iθ  R 
~
χ  R  : e
χ R 
~
proof: Let    and   ~
 be two different representations of an exact eigenfunction
 i.e.
Ψ  r, R   Φ R  r  χ  R   Φ R  r  χ  R 

~
Φ R r 
Φ R r 
χ R 
G R

~
χ R 
~
  dr ΦR r 
 
2
 GR 
1

GR   1
2

 
~
Φ R r  G R Φ R r
 dr Φ r 
2
R
1
 GR   e
~
 Φ R r   eiθ R  Φ R r 
iθ  R 
~
χ R   e  iθ R  χ R 
Theorem II: R  r  and   R  satisfy the following equations:
Eq. 
Nn

1
2
ˆ W
ˆ V
ˆ ext  V
ˆ 
T



i


A
 2M
ee
e
en
ν
ν
 e
ν
ν

ˆ
H
BO
Nn

ν
Eq. 
where


1   i ν χ
 A ν  i ν  A ν Φ R  r    R  Φ R  r 


Mν 
χ


 Nn 1

2
ext
ˆ
ˆ

 i ν  A ν   Wnn  Vn   R χ R   Eχ R 

 ν 2M ν

A ν  R   i   R  r  ν  R  r d r
*
G. Hunter, Int. J. Quant. Chem. 9, 237 (1975).
N.I. Gidopoulos, E.K.U.G. arXiv: cond-mat/0502433
OBSERVATIONS:
• Eq.  is a nonlinear equation in Φ R r 
• Eq.  contains χ R   selfconsistent solution of  and  required
• Neglecting the 1 M ν terms in , BO is recovered
• There is an alternative, equally exact, representation Ψ  Φ r  R  χ  r 
(electrons move on the nuclear energy surface)
• Eq.  and  are form-invariant under the “gauge” transformation
~
iθ R 
ΦΦ e
Φ
χ ~
χ e
 iθ R 
χ
~
A ν  A ν  A ν   ν θR 
~  R    R 
  R  
Exact potential energy surface is gauge invariant.


• γC : A  dR is a (gauge-invariant) geometric phase

C
the exact geometric phase
Proof of Theorem I:
Given the exact electron-nuclear wavefuncion Ψ  r , R 
Choose:
  R  : e
iS R 
 dr   r, R 
2
with some real-valued funcion S  R 
Φ R  r  :  Ψ  r, R  / χ  R 
Then, by construction,
 dr Φ r 
R
2
1
Proof of theorem II (basic idea)
first step:
Find the variationally best Φ R r  and χ R  by minimizing the total energy under
the subsidiary condition that
Eq. 
δ
δΦR r 
*
Eq. 
δ
δχ R 
 dr Φ r 
R
2
 1. This gives two Euler equations:
 Φχ H

ˆ χ
2







d
R
Λ
R
d
r
Φ
r
R

0


 Φχ χ



 Φχ H
ˆ Φχ


 Φχ Φχ



0


second step:
prove the implication
,  satisfy Eqs. , 
 := satisfies H=E
How do the exact PES look like?
MODEL
(S. Shin, H. Metiu, JCP 102, 9285 (1995), JPC 100, 7867 (1996))
R
(1)
(2)
–
+
–
+
+
0Å
-5 Å
+5 Å
x
y
Nuclei (1) and (2) are heavy: Their positions are fixed
Exact Berry connection
A  R    dr  R  r 
*
Insert:
 i 

R  r 
 R  r     r, R  /   R 
  R  : e
A ν  R   Im  dr 
*
i  R 
 r, R 
 R
    r, R  /   R     
2
A  R   J   R  /   R      R 
2
with the exact nuclear current density Jν
Consider special cases where Φ R  r  is real-valued (e.g. non-degenerate ground
state  DFT formulation)

A

R   i d r  R
*

r    R

r    i dr
 
i
2
1
2
  R
2
  dr  R
2
 r
 r
Eqs. ,  simplify:


2



1    

ˆ

 H BO  

      R  r    R   R  r 
 
 2M 
 2M  


 Tˆ
n

ˆ V
ˆ ext    R    R   E  R 
W
nn
n

0
Density functional theory beyond BO
What are the “right” densities?
first attempt
n r  Ne  d
N e 1
N R   Nn  d
A HK theorem
V
ext
n
ext
, Ve
Ne
r d
r d
Nn
R 
N n 1
R 
 r, R 
 r, R 
2
2

   N, n  is easily demonstrated (Parr et al).
11
This, however, is NOT useful (though correct) because, for
one has:
n = constant
N = constant
easily verified using Ψ  e  ikR CM ψ
ext
Vn
 0  Ve
ext
,
next attempt
~
n r  R CM

~
N  R  R CM

NO GOOD, because spherical for ALL systems
Useful densities are:
Γ R  :  d r Ψ r, R 
n R r  :
Ne   d
N e -1
2
(diagonal of nuclear DM)
r Ψ r, R 
Γ R 
2
is a conditional probability density
Note: n R r  is the density that has always been used in the DFT within BO

now use decomposition Ψ r, R
then
Γ  R    dr ΦR  r 
2
  Φ  r χ  R 
R
χ R 
2
 χ R 
2
1
nR r  
Ne   d
HK theorem
Ne -1
r R  r 
R 

2
 Ne   d
2
nR r,
gs
 R 
2
gs
R

Ne -1

r R  r 
2
(like in B.O.)
 v e  r, R  , v n  R 
1 1

KS equations
nuclear equation stays the same


ext
ˆ
ˆ
ˆ
Tn  Wnn  R   Vn  R     R    R   E  R 
Electronic equation:
 2

 
 v KS r, R   R , j r   η j R   R , j r 
 2m

v KS r, R   v en r, R   v e
ext
r   v Hxc
, n R
r, R 
Electronic and nuclear equation to be solved self-consistently using:
 R  
Ne
 η j R  -  n R r v Hxc r, R d r  E Hxc , n R
3
j1
Time-dependent case
Hamiltonian for the complete system of Ne electrons with coordinates
r1 rN   r and Nn nuclei with coordinates R1  R N   R , masses
M1 ··· MNn and charges Z1 ··· ZNn.
e
n
ˆ  Tˆ ( R )  W
ˆ ( R )  Tˆ ( r )  W
ˆ (r)  V
ˆ (R , r)
H
n
nn
e
ee
en

with Tˆ n   
 1
ˆ 
W
ee
1
2
Ne
2
Nn
2M 
Ne
r
j, k
j k
1
j
 rk
ˆ 
T

e
i 1
i
2
ˆ 
W
nn
2m
1
Nn

Z Z 
2  , R   R 

Ne
Nn
ˆ 
V
 
en
j1  1
Z
rj  R 
Time-dependent Schrödinger equation
i

t
V laser
 r , R , t   H r , R   V laser r , R , t   r , R , t 
 Ne
r , R , t     r j 
 j1

 Z  R    E  f  t   cos  t
 1

Nn
Theorem T-I
The exact solution of
i t 
 r, R , t   H  r, R , t 

 r, R , t 
can be written in the form

 r, R , t     r, t    R , t 
where
R
 d r   r, t 
R
2
 1 for any fixed R , t
.
A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010)
Theorem T-II
 R  r, t  and  R , t 
satisfy the following equations
Eq. 
Nn

1
2
ext
ˆ W
ˆ V
ˆ r, t   V
ˆ r, R  
T





i


A
R
,
t
 2M
ee
e
en
ν
ν
 e
ν
ν

ˆ t 
H
BO
Nn

ν


1   i ν χ  R , t 

 A ν  R , t  i ν  A ν    R , t Φ R  r   i t Φ R  r , t 



Mν 
χ  R, t 


Eq. 
 Nn 1

2
ˆ R  V
ˆ ext  R , t    R , t χ R , t   i χ R , t 






i


A
R
,
t

W
ν
ν
nn
n
t


2M
ν
ν


A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010)
Nn


1
2
*

 i ν  A ν R, t   i t ΦR  r, t 
  R , t    d r Φ R  r, t   H BO t   
ν 2M ν


EXACT time-dependent potential energy surface
 
A ν  R, t   i   R r,t  ν  R  r, t  dr
*
EXACT time-dependent
Berry connection
Example: H2+ in 1D in strong laser field
exact solution of
i  t Ψ  r, R, t   H Ψ  r, R, t  :
Compare with:
• Hartree approximation:
Ψ(r,R,t) = χ(R,t) ·φ(r,t)
• Standard Ehrenfest dynamics
• “Exact Ehrenfest dynamics” where the forces on the nuclei are
calculated from the exact TD-PES
The internuclear separation < R>(t) for the intensities
I1 = 1014W/cm2 (left) and I2 = 2.5 x 1013W/cm2 (right)
Exact time-dependent PES
Dashed: I1 = 1014W/cm2 ; solid: I2 = 2.5 x 1013W/cm2
Summary:
• Ψ  r, R    R  r   χ  R  is an exact representation
of the complete electron-nuclear wavefunction if χ and Φ
satisfy the right equations (namely Eqs. ,  )
• Eqs. ,  provide the proper definition of the
--- exact potential energy surface
--- exact Berry connection
both in the static and the time-dependent case
• Multi-component (TD)DFT framework
• TD-PES useful to interpret different dissociation meachanisms