Download - Co-workers : Ali Abedi (MPI-Halle) Neepa Maitra (CUNY) Nikitas Gidopoulos (Rutherford Lab)
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. , )