Exact factorization of the time-dependent electron-nuclear 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)

)r,R(V)r(W)r(T)R(W)R(TH eneeennn

with

e ne

nen

N

1j

N

1 j

en

N

kjk,j kj

ee

N

,nn

N

1i

2i

e

N

1

2

n

Rr

ZV

rr

1

2

1W

RR

ZZ

2

1W

m2T

M2T

Full Schrödinger equation: R,rER,rH

convention:Greek indices nucleiLatin indices electrons

Hamiltonian for the complete system of Ne electrons with coordinates

and Nn nuclei with coordinates

, masses M1 ··· MNn and charges Z1 ··· ZNn

.

rrreN1 RRR

nN1

Born-Oppenheimer approximation

.Rfor each fixed nuclear configuration

rΦ rΦ)R,r(V)r(V)r(W)r(T BOR

BORen

exteeee R BO

solve

RχrR,rΨ BOBO

R BO

Make adiabatic ansatz for the complete molecular wave function:

and find best χBO by minimizing <ΨBO | H | ΨBO > w.r.t. χBO :

Nuclear equation

R EχRχ rdrΦRT rΦ BOBOBO

Rn*BO

R

)(-i

M

1)R(V)R(W)R(T ext

nnnn RA BOυ R BO

Berry connection

rdrΦ)(-i rΦRA BORυ

* BOR

BOυ

C

BOBO RdRACγ

is a geometric phase

In this context, potential energy surfaces and the Berry potential are APPROXIMATE concepts, i.e. they follow from the BO approximation.

RA BO

R BO

Nuclear equation

R EχRχ rdrΦRT rΦ BOBOBO

Rn*BO

R

)(-i

M

1)R(V)R(W)R(T ext

nnnn RA BOυ R BO

Berry connection

rdrΦ)(-i rΦRA BORυ

* BOR

BOυ

C

BOBO RdRACγ

is a geometric phase

In this context, potential energy surfaces and the Berry potential are APPROXIMATE concepts, i.e. they follow from the BO approximation.

RA BO

R BO

“Berry phases arise when the world is approximately separated into a system and its environment.”

GOING BEYOND BORN-OPPENHEIMER

Standard procedure:

J

JK,BO

J,RK RχrΦR,rΨ

and insert expansion in the full Schrödinger equation → standard

non-adiabatic coupling terms from Tn acting on .BOJ,R rΦ

• 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.

Drawbacks:

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

rΦ BO

R1,

RE BO1

0 00 01Ψ r,R χ R χ R rΦ BO

R0, rΦ BO

R1,

rΦ BO

R0, RE

BO0

Potential energy surfaces are absolutely essentialin 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:

R

The first excited state is a weakly bound covalent state with a shallow minimum and atomic fragments.

Na + I

Na+ + I-

PE

R

PE

The non-crossing rule

“Avoided crossing”

Na++ I-

Na + I

ionic

covalent

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 somewill keep oscillating on the upper adiabatic surface.

Na++ I-

Na + 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)

300

311

321

339

λpump/nm

Different periods indicative of anharmonic potential

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

GOAL: Show that 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

RχrR,rΨ R

Theorem I

The exact solutions of

can be written in the form

R,rER,rH

RχrR,rΨ R

where 1 rΦrd2

R for each fixed .R

First mentioned in: G. Hunter, Int. J.Q.C. 9, 237 (1975)

Immediate consequences of Theorem I:

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

with

R 2

Rχ

in this sense, can be interpreted as a proper nuclear wavefunction. Rχ

proof: 222

R

2 RχRχ rΦrd R,rΨrdRΓ

1

2. and are unique up to within the “gauge transformation” Rχ rΦ R

rΦe:rΦ~

RRiθ

R Rχe:Rχ~

Riθ

proof: Let and be two different representations of an exact eigenfunction i.e.

~~

Ψ r,R Φ r χ R Φ r χ R R R

Rχ~Rχ

rΦ

rΦ~

R

R RG rΦ rΦ~

RR RG

2

rΦ~

rd R

1

2

rΦrd R 2RG

1

R iθeRG1RG

rΦ rΦ~

RR

Riθe Rχ Rχ~ e Riθ

Eq.

nN

ν

2νν

νen

exteeee Ai

2M

1VVWT

BOH

rΦ rΦAiAχ

χi

M

1

nN

νννν

ν

νRR

R

Eq. REχRχ VWAi2M

1 extnnn

N

ν

2νν

ν

n

R

where rdrriRA ν*

ν RR

Theorem II: satisfy the following equations: R r and R

G. Hunter, Int. J. Quant. Chem. 9, 237 (1975).N.I. Gidopoulos, E.K.U.G. arXiv: cond-mat/0502433

• Eq. and are form-invariant under the “gauge” transformation

ΦeΦ~

Φ Riθ

RR~R

χeχ~χ Riθ

RθAA~

A νννν

• is a (gauge-invariant) geometric phase the exact geometric phase

C

RdA:Cγ

Exact potential energy surface is gauge invariant.

OBSERVATIONS:

• Eq. is a nonlinear equation in

• Eq. contains selfconsistent solution of and required

rΦR

Rχ

• Neglecting the terms in , BO is recovered• There is an alternative, equally exact, representation

(electrons move on the nuclear energy surface) rχRΦΨ r

νM1

2

R : e dr r,R iS R

S R

Proof of Theorem I:

Choose:

Ψ r,R

with some real-valued funcion

Φ r : Ψ r,R / χ RR

Given the exact electron-nuclear wavefuncion

1 rΦrd2

RThen, by construction,

Proof of theorem II (basic idea)

first step:

Find the variationally best and by minimizing the total energy under

the subsidiary condition that . This gives two Euler equations: 1 rΦrd2

R

Rχ rΦR

Eq.

Eq.

prove the implication

, satisfy Eqs. , := satisfies H=E

second step:

0rΦrdRΛRdχΦχ

χHΦχ

rδΦ

δ 2

*

R

R

0ΦχΦχ

ΦχHΦχ

Rδχ

δ

How do the exact PES look like?

+ +

-5 Å +5 Å

0 Å

xy

R

+

– –(1) (2)

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

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

*

R RA R dr r i r

i RR : e R

R r r,R / R

Exact Berry connection

Insert:

2*

νA R Im dr r,R r,R / R

2

A R J R / R R

with the exact nuclear current density Jν

* 2R R R

1 A R i dr r r i dr r

2

2R

idr r 0

2 Eqs. , simplify:

2

BO R R

1H r rR

2M 2M

ext

n nn n ˆ ˆ ˆT W V R R E R

Consider special cases where is real-valued (e.g. non-degenerate ground state DFT formulation)

rΦ R

Density functional theory beyond BO

What are the “right” densities?

first attempt e n

2N 1 N

en r N d r d R r,R e n

2N N 1

nN R N d r d R r,R A HK theorem is easily demonstrated (Parr et al). nN,V,V 11ext

eextn

This, however, is NOT useful (though correct) because, for one has:

, V0V exte

extn

easily verified using ψeΨ CMRik

n = constantN = constant

next attempt CMRrn~ CMRRN~

NO GOOD, because spherical for ALL systems

Useful densities are:

2

R,rΨrd:RΓ (diagonal of nuclear DM)

RΓ

R,rΨrdN:rn

2

e -1N

R

e

is a conditional probability density

Note: is the density that has always been used in the DFT within BO rnR

now use decomposition

then

Ψ r,R Φ r χ R R

2 2 2Γ R dr Φ r χ R χ R R

1

e

e

2 2N -12e R N -1

R e R2

N d r r Rn r N d r r

R

(like in B.O.)

HK theorem 1 1gs gs

R e nn r , R v r,R , v R

KS equations

nuclear equation stays the same

ext

n nn n ˆ ˆ ˆT W R V R R R E R

Electronic equation:

r Rηr Rr,v2m

2

j,Rjj,RKS

Rr, n, vrvRr,vRr,v R HxcexteenKS

Electronic and nuclear equation to be solved self-consistently using:

R3

R n, ErdRr,vrn-RηR

HxcHxc

N

1jj

e

Time-dependent case

Time-dependent Schrödinger equation

tcostfERZrt,R,rV

t,R,rt,R,rVR,rHt,R,rt

i

e nN

1 j

N

1 jlaser

laser

)r,R(V)r(W)r(T)R(W)R(TH eneeennn

with

e ne

nen

N

1j

N

1 j

en

N

kjk,j kj

ee

N

,nn

N

1i

2i

e

N

1

2

n

Rr

ZV

rr

1

2

1W

RR

ZZ

2

1W

m2T

M2T

Hamiltonian for the complete system of Ne electrons with coordinates

and Nn nuclei with coordinates

, masses M1 ··· MNn and charges Z1 ··· ZNn

.

rrreN1 RRR

nN1

Theorem T-I

The exact solution of

can be written in the form

where

ti r,R, t H r,R, t r,R, t

Rr,R, t r, t R, t

2

Rdr r, t 1 for any fixed .R, t

A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010)

Eq.

nN

ν

2νν

νen

exteeee t,RAi

2M

1R,rVt,rVWT

tH BO

t,rΦirΦt,RAit,RA

t,Rχ

t,Rχi

M

1

n

t

N

νννν

ν

νRR

Eq.

t,Rχit,Rχt,Rt,RVRWt,RAi2M

1t

extnnn

N

ν

2νν

ν

n

Theorem T-II

t,R and t,r R satisfy the following equations

A. Abedi, N.T. Maitra, E.K.U.G., PRL 105, 123002 (2010)

t,rΦit,RAi

2M

1tH t,rΦ rdt,R R

N

νt

2νν

νBO

*R

n

EXACT time-dependent potential energy surface

EXACT time-dependent Berry connection

*

ν ν A R, t i r,t r, t dr R R

Example: H2+ in 1D in strong laser field

exact solution of :tR,r,Ψ HtR,r,Ψi t

Compare with:

• Hartree approximation:

• Standard Ehrenfest dynamics

• “Exact Ehrenfest dynamics” where the forces on the nuclei are calculated from the exact TD-PES

Ψ(r,R,t) = χ(R,t) ·φ(r,t)

The internuclear separation < R>(t) for the intensities I1 = 1014W/cm2 (left) and I2 = 2.5 x 1013W/cm2 (right)

Dashed: I1 = 1014W/cm2 ; solid: I2 = 2.5 x 1013W/cm2

Exact time-dependent PES

Summary:

• is an exact representation of the complete electron-nuclear wavefunction if χ and Φ satisfy the right equations

• 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

RχrR,rΨ R

(namely Eqs. , )