Lasorne Emie-Up 2019 - IRAMISiramis.cea.fr/meetings/Ecole_EMIE_UP/presentations/Lasorne.pdf · the...

Nonadiabatic Dynamicscurrent methods and challenges

Benjamin LASORNE

[email protected]

Department of Theoretical ChemistryInstitut Charles Gerhardt

CNRS – Université de Montpellier

Summer School EMIE-UPMultiscale Dynamics in Molecular Systems

August 2019

1. Introduction

2. Beyond the Born-Oppenheimer Approximation

3. Non-Adiabatic Processes

4. Methods

5. Examples of Application

6. Conclusions and Outlooks

2Summer School – Emie-Up – Aug. 2019

1. Introduction

3

R

Reaction coordinate

Po

ten

tia

l en

erg

y

CoIn TS

PP’

S0

S1

P*

R*

TS*

FC

−hν +hν

Adiabatic

photochemical

reaction

Non-adiabatic

photochemical

reaction


August 2019

4

“Nonadiabatic dynamics”: a tentative definition

Summer School – Emie-Up – Aug. 2019

FieldComputational and theoretical description of molecular processes induced upon UV-visible light absorption and starting in electronic excited states

MethodsQuantum/semiclassical molecular dynamics (time evolution of the molecular geometry governed by potential energy surfaces and non-adiabatic couplings)and electronic structure (set of coupled excited states)� both challenging compared to ground-state simulations

ObjectiveSimulation of time/energy-resolved processes at the molecular level from the promotion to the excited electronic state, if possible to the formation of products or regeneration of reactants back in the electronic ground state

ApplicationsElectronic spectroscopy (photoabsorption, photoionisation)Photochemistry, photophysics, chemiluminescence [atto/femto]

Introduction

� Franck-Condon approximation� Electric dipole approximation� Wavepacket evolution (time)� Absorption spectrum (energy)

5

Electronic spectroscopy

S0

S1

Q1

S0

S1

Q1

S0

S1

Q1

Example: ππ* benzene (vibrational progression in mode 1: totally symmetric breathing)

E

Introduction


� Intramolecular vibrational redistribution� Internal conversion� Photostability vs. photoreactivity

6

Photochemistry

benzene prefulvenoidgeometries

benzvalene

254 nm

S0 (1A1g) → S1 (1B2u)

S0

Reaction coordinate

E

S1

Introduction


7

Spectroscopic picture (photophysics): Jablonski diagram

0

1

2

3

4

5

6

7

8

9

10

S0

0123456789

10

S1

10

0123456789

T1

AF PIVR

IVR

IVR

IC ISC

Radiative: Absorption, Fluorescence, Phosphorescence

Radiationless: Intramolecular Vibrational Redistribution, Internal Conversion, InterSystem Crossing

Introduction


8

Molecular theoretical chemistry: role of the geometry

Quantum chemistry Molecular dynamics

Electronic energy(at various positions of the

nuclei)

� Static approach

Nuclear motion(in the mean field of the

electrons)

� Dynamical approach

Potential

energy

surface

Reciprocal influence betweenthe electronic structure and the molecular geometry

Structure (thermodynamics, spectroscopy) and reactivity (mechanism, kinetics)

RP

TSV

Q

Introduction


9

Interplay between topography and motion

Electronic structure Molecular geometry

Energy landscape

( ) ( )0 0V Q V Q δQ→ +

0 0Q Q δQ→ +0 0Φ; Φ;Q Q δQ→ +

Introduction


10

Mechanistic picture (photochemistry): reaction path

Adiabatic reaction: radiative deactivation after excited-state product is formed

Non-adiabatic reaction: radiationless decay to ground-state product through conical intersection (CoIn)

R

Reaction coordinate

Po

ten

tia

l en

erg

y

CoIn TS

PP’

S0

S1

P*

R*

TS*

FC

−hν +hν

Adiabatic

photochemical

reaction

Non-adiabatic

photochemical

reaction

Introduction


11

Photochemistry vs. thermal chemistry

� The system does not start from around an equilibrium geometry

� The slope on S1 creates an initial driving force (skiing rather than hiking)

� The system will develop momentum as it escapes the Franck-Condon region

� The trajectory can easily deviate from the minimum energy path (bottom of the valley connecting R to P through TS)

� molecular dynamics simulations

� State crossings are likely to happen

� non-adiabatic (vibronic) effects

� quantum dynamics

S0

S1

Introduction


2. Beyond the Born-Oppenheimer

Approximation

12


August 2019

a. Adiabatic Electronic States

b. Adiabatic Partition of the Molecular Schrödinger Equation

c. Non-Adiabatic Couplings

d. Conical Intersections

e. Diabatic Electronic States

13

Beyond BOA


14

Formalism: prerequisites

( ) ( ) ( ) ( ) ( )

2

eigenvalues 2

2

2

2

2 2

00 0

02

uv u v u v u v

a c a b a be c

c b

a ba be e c

c

±

+ −

′′ ′′ ′ ′ ′′= + +

+ − → = ± +

− =− − = ⇔ + = ⇔ =

Beyond BOA


15

Common ground: the molecular Hamiltonian

� Non-relativistic

� Electrostatic

� Electronic (r) and nuclear (R) coordinates

� Direct solution extremely expensive and rarely useful

Beyond BOA

2

mol1

2

1 e

21

1 0

21

1 0

2

1 1 0

ˆ2

2

1

4

1

4

1

4

J

j

N

RJ J

n

rj

N NJ K

J K JJ K

n n

j k j j k

N nJ

J jJ j

HM

m

Z Z e

πε R R

e

πε r r

Z e

πε R r

=

=

−

= >

−

= >

= =

=− ∆

− ∆

+−

+−

−−

∑

∑

∑∑

∑∑

∑∑

�

�

ℏ

ℏ

� �

� �

� �


16

The adiabatic partition: electronic/nuclear separation

� Nuclei at least 1800 heavier than electrons

� Time scale separation

Electronic ~ 0.1 fs

Vibrational ~ 10 fs

� Energy scale separation

Electronic ~ 50 000 cm-1

Vibrational ~ 500 cm-1

�Problem solved in two sequential steps

1) Electronic, relaxed at each geometry (quantum chemistry)

2) Nuclear, in the electronic mean field (molecular dynamics)

Beyond BOA


17

( ) ( ) ( )

( ) ( ) ( ) ( )

el

el

mo nul

el

cˆ, , ;

ˆ ,

ˆˆ

Φ, ;

,

Φ ;

,

;

q q

q

Q Qq H q

H

Q T Q

Q q q

H

EQ qQ Q Q

= +

∀ =

∂ ∂∂

∂

∂

Beyond BOA

Starting point: Born-Oppenheimer (BO) approximation


18


( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

nucm el

el el

el

olˆ , ˆ, , ;

ˆ , ; Φ ;,

,

,

Φ ;

ˆQ Qq q

q

Q T Q

Q Q Q Q

q H q

H q q E q Q

Q E Q

H

Q V

= +

∀

∂

=

∂

∀

∂

≡

∂

∂

Beyond BOA


19


( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

mol

m

el

el el

e

ol,

BO

nuc

22

l

ˆ, , ;

ˆ , ; Φ ; Φ ;

ˆˆ

,

,

2

,

,

q qQ Q

q

Q v v v

Q T Q

Q Q Q Q Q

Q

q H q

Q V Q

V Q φ Q φ

H q q E q

E

H

mE Q

= +

∀ =

∀ ≡

+ =

∂ ∂ ∂∂

∂

∂

−��

ℏ

Beyond BOA


20

Quantum or classical dynamics: equations of motion

( )�

( ) ( )

( ) ( )( )

potential energy surface

2

local for

2

ce

2

nuclear wavepacket

, ,

nuclear trajector

2

y

Q

Q Q

t

t t

t iV Q ψ Q ψ Qm

tm Q

t

V Q

+− ∂ = ∂

∂ ∂ =−

ℏℏ

��

H transfer with tunnelling: non-classical behaviour

X-H…YX…H-Y

X-H…YX…H-Y

X-H…YX…H-Y

Beyond BOA


21

Potential energy surface: explicit function of Q

( )el, samplingQ QE∀ →

Q

Beyond BOA


22


( ) ( )el, sampling and fittingQ QE V Q∀ ≡ →

Q

Beyond BOA


23


( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

0

2

0 0 0

2

0 0 0 0

22

el

mol,

BO

, sampling and fitting

Example: linear harmonic oscillator

Parameters: , ,

1

2

2

Q Q

Q v v vE

Q Q V Q

V Q V k Q Q

Q V V Q k V Q

V Q φ Q φ Qm

E∀ ≡ →

+ =

= + −

=

= ∂

− ∂��

ℏ

��

Q

Beyond BOA


24

Potential energy surface: many-(many!)-body expansion

( ) ( )

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

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

0

1 3 6 0

3 60 0

1

3 6 3 60 0 0

1 1

3 6 3 6 3 60 0 0 0

1 1 1

, ,

1

2

1

6

N

N

j j jj

N N

jk j j k kj k

N N N

jkl j j k k l lj k l

V Q Q V

V Q Q

V Q Q Q Q

V Q Q Q Q Q Q

−

−

=

− −

= =

− − −

= = =

=

+ −

+ − − +

+ − − − +

∑

∑∑

∑∑∑

⋯

⋯

Beyond BOA

� Unknown form (not 1/r12) � sampling and fitting

� More than two-body terms: multidimensional integrals <ϕ|V|ϕ>, not limited to (µν||λη)

� The very bottleneck in quantum dynamics!


25

Photochemistry often involves radiationless decay

Isomerisation coordinate

Pote

nti

al energ

y

trans

cis

Energy transfer

NH+

2nd step

� Example: first step of the vision process:

retinal photoisomerisation � no fluorescence!

Internal conversion

Beyond BOA


26

Why and where does it happen?

� When the Born-Oppenheimer approximation is no longer valid

� Vibronic coupling between the electronic states and the nuclear motion: non-adiabatic coupling

� Strong effect when the electronic energy separation is small (~ vibrational, so similar time scales)

� Explanation of internal conversion (if same spin)

� NB: intersystem crossing (if different spins) due to spin-orbit coupling (relativistic origin)

Beyond BOA


27

Beyond Born-Oppenheimer

Beyond BOA


( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )n

mol

luc e

elnuc

nuc

m

ol

el

ol

m

or Ψ , , , , ( . Federica Agostini's Φ ; t k al )

Φ ;

Φ ;

Ψ , , , BO approximation

Beyond BO Ψ , , ,

s

s

s

ss

q t t t cQ ψ Q Qq f

t t

t t

Q ψ Q Q

Q ψ

q

Q

q

q Qq

=

= ⇒

⇒ =

∑

ɶɶ

��

��

��

��

��

��

��

��

( ) ( )

( ) ( ) ( )

e

22

nuc

l : no longer replaced by a single of its eigenvalues,

matrix ( ) of "potential energies" wrt.

: acts on both , and

non-adiabatic-coup

ˆ , ;

indices

ˆ

lin

Φ

g mat

, ;2

r

q s

Q Q s s

Q Q

Q

T Q ψ Q Qm

H q V

s

t

s

q

∂

∂ −

∂

→

→

′

ℏ∼

indicesix ( ) wrt. s Qs′

“Exact factorisation”

28

Refresher: Riemannian vector derivative

( ){ }( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

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

orthonormal basis set (~rotating frame):parametri

2

2 ,

2

c

,

s

s s s ss

s s s s s ss

s s p s p p sp

u

v v u v

x

x x x x x x

x x x x x x x

x x x

u v

uv u v u v

uv u v u v u v

v v u v u v u

u v v v u u vx x u ux x x

′ ′ ′

′′ ′

= = ⋅

= +

= + +′ ′ ′ ′′

′′ ′′ ′ ′ ′= + +

⋅ = + ⋅ + ⋅

′

′′ ′′ ′ ′ ′′

∑

∑

∑

�

� � � �

� � � �

� � � � � �( )( )p

p

x∑

Beyond BOA


29

Non-adiabatic coupling (general principle)

{ }( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

Φ ; orthonormal basis set (~rotating frame):

Ψ; Φ ; , Φ ; Ψ;

Ψ

par

; Φ ; 2 Φ ; Φ ; ,

Φ ; Ψ ; 2 Φ ; Φ ; Φ ; Φ ;

ametrics

s s s ss

s s s s s ss

s s s s p s s pp p

x

x x x x xψ ψ

ψ ψ ψ

ψ ψ ψ

x

x x x x x x x

x x x x x x x x x

′′ ′′ ′ ′ ′′

′′ ′′ ′ ′

= =

= + +

= ′′+ +

∑

∑

∑ ∑

Beyond BOA


30

Representation of the wavefunctions

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( )( ) ( )

nuc

22

nu

mo

el

el

el e

m

c

l

lΨ , , ,

,

Representation explicit with respect to

but implicit w

Φ ;

ˆ

ith respect to (co

; Φ ; Φ ;

Φ ; Φ ;

uld be

ˆ

,

2

)

ˆ;

Ψ

ˆ

s ss

s s s

Q

s

Q

s

q

Q ψ Q Q

Q Q Q Q Q

Tm

Q

q q

H q q V q

q p k

q

H q

Q Q

t

Q

t

H Q

=

∀ =

→

→

∂ =− ∂

=

∂

∑��

ℏ

��

ℏ

( ) ( )nuc

ol

el

, , Φ ;s s

s

Q ψ Q Qt t=∑��

Beyond BOA


31

Vibronic non-adiabatic couplings

( ) ( )

( ) ( ) ( )( ) ( )

( ){ }

( )

( )

( ) ( ) ( ) ( )

( )

BO

22

mol

mol mol m

22

ol

mol

l

*

ˆ

2 2

e

2

ˆ

Φ ;

Φ ; Φ ;

Φ ;

ˆ

ˆ Ψ , Ψ ,

Ψ , ,

, ,

Φ ; Φ ; Φ ;Λ̂2

,

2

Λ̂2

s

Q

Q

s ss

s s

q

s s sp p sp

sp s p s

Q

Q Q Q p

t

t

H

H t i t

t

Q Qm

Q Q Q

Q ψ Q Q

Q Q

V Q ψ Q Q ψ Q ψ Qm

Q Q Q Q Qm

t

t

m

dq

t

H

t i

q

= +

= ∂

=

=

+ + = ∂

− ∂

=

− ∂

− ∂ ∂ − ∂

∫

∑

∑

ℏ

ℏ

��

⋯ ⋯

��

ℏ

ℏ

ℏ ℏ

��H

Beyond BOA


32

Non-adiabatic photochemistry: a non-classical field

Radiationless decay mechanisms � the nuclei behave as quantum-mechanically as the electrons (~WE, ~Wt)

� Nuclear wavepacket (Q,t) with two (or more) components

� Vibronic non-adiabatic coupling (non-Born-Oppenheimer)

� electronic population transfer

( )( )

( )( )

( )( )

000 01

11

BO0 00

BO1 11 0 11

ˆ, ,0

ˆ, ,0

ˆ ˆ ,Λ Λ

ˆ ˆ ,Λ Λt

ψ Q t ψ Q ti

ψ Q t ψ Q t

ψ Q t

ψ Q t

∂ = +

ℏ

H

H

S0

S1

S0

S1

( ) ( ) ( ){ }

2

,s

s

Q

P t ψ Q t dQ= ∫

Beyond BOA


33

Matrix Hellmann-Feynman theorem (general principle)

†

ˆ Φ Φ

ˆ ˆ

ˆΦ Φ

Φ Φ

with respect to anderivative external paramete :r

Φ Φ Φ Φ Φ Φ 0

b b b

a a a

a b ab

a b a b a b

H E

H H

H E

δ

=

′ ′ ′

=

=

=

= + =

Beyond BOA


34


( )

ˆΦ Φ

ˆΦ Φ

ˆ ˆ ˆΦ Φ Φ Φ Φ Φ

ˆΦ Φ Φ Φ Φ Φ

ˆΦ Φ Φ Φ

a b ab a

a b

a b a b a b

b a b a a b a b

a b a b a b ab a

H δ E

H

H H H

E E H

E E H δ E

′

′ ′ ′

′

=

=

+ + =

+ ′ ′

′ =′

+ =

′− +

Beyond BOA


35


ˆ: Φ Φ

ˆΦ Φ: Φ Φ

a a a

a b

a b

b a

a b E H

Ha b

E E

=

′ =

′

′

=

−

′

≠

Beyond BOA


36

Link with 1st-order perturbation and response theories

�energy correction

wav

0

0 0

efunction response

0 0

0

0 0 0

0 0

0 0

0

0 0

ˆ ˆ ˆ

ˆ Φ Φ

ˆ Φ Φ

Φ Φ Φ Φ Φ

ˆΦ Φ

ˆΦ ΦΦ Φ

s s s

s s s

s s s

s s p p sp

s s s

p s

p s

s p

δx

δx

δ

H H H

H V

H V

V V V

V H

H

x

V V

′

′

′

′

= +

=

=

= + +

= + +

= ′

′′ =

−

∑

…

��

…

�

Beyond BOA


37

Matrix Hellmann-Feynman theorem here

( ) ( )( ) ( ) ( )

( ) ( )

( )( ) ( )

adiabatic gradient

1st-order no

el

e

n-a

l

e

diabatic coupling

l

ˆΦ ; Φ ; Φ ; Φ ;

ˆΦ ; Φ ;

ˆ

:

:Φ ; Φ ;

Φ ; Φ ;

Q Q Q

Q

s p s p s p sp s

s s s

s p

s

Q

Q

p

Q p

s

HV Q V Q

s p

Q Q Q Q Q V Q

V Q Q Q Q

Q Q QQ Q

V V

H

Q

δ

Hs

Qp

− + =

= =

≠ =

∂ ∂ ∂

∂

−

∂

∂∂

��

��

Beyond BOA


38

Analytic derivatives

( ) ( ) ( ) ( )( ){ }

( )ext

*

el el-nuc

transition density ;

nuc-nuc

ˆΦ ; Φ ; Φ ; Φ ; ;Q Q

Q

s p s p

q v q

sp Q

H q q V q dq

δ

Q Q Q Q Q Q

V Q

→ →

∂

∂

=

+

∂∫ ��

Beyond BOA


39

Two-state model

( )

0100

00 0

11

11

11 11

11 11

0

00

0

22

0,10 00

01

0101

01

1 0 20 12

2

eigenvalues

2 2

H

H

H H

H

HH H

H

H

H

H

HV

H

H

H

H

− − + = + −

↓

+ − = ± +

Beyond BOA


40

Conical intersection

Adiabaticsurfaces

V1

V0

x(1)

Degeneracy lifted at first order along 2D branching space

- x(1) || gradient of (H11 − H00)/2- x(2) || gradient of H01

x(2)

( )2

2

01

0

01 0 01

1 0

1

1

1

001

2 2

00

02

V V

V H

HH

V H

H

H − − = +

− =− = ⇔ =

Beyond BOA


41

Conical intersections: divergence and cusp

V

�1�2

( ) ( ) ( )( ) ( )( )2 2

1 21 0 0 0

gradient: ill-defined (cusp)

02

V δ V δδ δ

+ − += + ⋅ + ⋅ +

Q Q Q Qx Q x Q ⋯

��

( )( ) ( )

0

0

1 0

el 1

1

ˆΦ ; Φ ;Φ ; Φ ;

V

H

V

∇∇

−= →∞Q

QQ

QQ Q

Q

Q Q

Beyond BOA


42

Non-adiabatic photochemistry around conical intersections

� Adiabatic representation (BO states)

� Divergent non-adiabatic coupling and cusp of the potential energy surfaces at the conical intersection

� Problem for QD: integrals require regular functions of Q

Beyond BOA


43

Molecular symmetry

� Diatom (one coordinate): crossing only if different symmetry (Wigner non-crossing rule)

� Jahn-Teller:

two degenerate electronic

states (E)

� two equivalent vibrations (E)


Beyond BOA

44

Molecular symmetry

� Symmetry-induced conical intersection: allowed crossing between states of different symmetries (ΓA and ΓB), coupling of ΓA ⊗ ΓB symmetry � non-zero when symmetry gets broken (states mix)

� Accidental conical intersection: probable because large number of degrees of freedom (often related to high-symmetry Jahn-Teller prototype)

el el

el el

ˆ ˆ

ˆ ˆ

A H A A H B

B H A B H B


Beyond BOA

45

Getting rid of singularities: diabatisation

� Unitary transformation minimising the non-adiabatic coupling

� Electronic Hamiltonian matrix no longer diagonal

†

cos sin

sin cos

φ φ

φ φ

− =

=

U

H UVU

00 01 0 1 1 0

10 11

cos2 sin2

sin2 cos22 2

H H φ φV V V V

H H φ φ

− −+ − = + − 1

0 1Φ ; Φ ;Q

Q Q∂ →∞ 0 1Φ ; Φ ; 0Q

Q Q′ ′∂ ≈

Beyond BOA


46

Two states: explicit relationships

� Rotation angle

� Condition to make the diabatic derivative coupling zero

� Never fully achieved in practice: various types of quasi-diabatic states, all based on a smoothness condition of the wavefunction or the energy, the dipole moment, etc.

( )

( )

0 1 0 1

0

0 1

Φ ; Φ ; Φ ; Φ ;

Φ ; Φ ;

Q Q Q

Q Q

Q Q Q Q φ Q

φ Q Q Q

≈

′ ′∂ = ∂ −∂

⇒ ∂ ≈− ∂

��

( )( )

( ) ( )01

11 00

2tan2

H Qφ Q

H Q H Q=−

−

Beyond BOA


47

Why quasi-diabatic states?

0 1

2

0 1 0 1

2

0 1 0 10

2

0 1 0 10

Φ ; Φ ;

Φ ; Φ ; Φ ; Φ ;

Φ ; Φ ; Φ ; Φ ; Φ ; Φ ;

Φ ; Φ ; Φ ; Φ ; Φ ; Φ ;

Q Q

Q Q Q

Q Q s s Qs

Q Q s s Qs

Q Q

Q Q Q Q

Q Q Q Q Q Q

Q Q Q Q Q Q

=

=

∞

∞

∂ ∂

= ∂ + ∂ ∂

= ∂ + ∂ ∂

= ∂ − ∂ ∂

∑

∑

Beyond BOA


� Infinite basis set required

(unless isolated Hilbert subspace)

48

Diabatisation a priori or a posteriori

Working space(N configurations)

Target subspace(n adiabatic states)

Model subspace(n diabatic states)

diag

diab

� Configuration contraction wrt. static correlation yielding well-behaved model states equivalent to the target states (n out of N)

� effective Hamiltonian

diab

Beyond BOA


49

Diabatisation by ansatz: smooth Hamiltonian matrix

� Electronic structure preserved wrt. geometry variation

diabatic

Q

adiabatic

V

Beyond BOA


50

Consequence: smoother PES with no cusp

Adiabaticsurfaces

Diabaticsurfaces

H00

H11

H11

H00

V1

V02|H01|

x(1)

Diabaticpotential coupling

Degeneracy lifted at first order along 2D branching space

- x(1) || gradient of H11 − H00

- x(2) || gradient of H01

( )2

21 0 11 00012 2

V V H HH

− − = +

x(2)

Beyond BOA


51

Intuitive chemical interpretation

� Walsh

correlation

diagrams

q

V

Beyond BOA


52

Photochemistry: ES and GS multiple wells and crossings

Beyond BOA


53

Thermal chemistry: GS local coupled states

� Marcus theory of electron transfer reactions

� Valence bond: transition barriers as avoided crossings

Beyond BOA


3. Non-adiabatic Processes

54

S0

S1

S0

S1


August 2019

a. Non-Adiabatic Quantum Dynamics

b. Radiationless Decay

c. Electronic State Crossings

55

Non-adiabatic processes


56

Born-Oppenheimer quantum dynamics

� Time-dependent molecular Schrödinger equation within the Born-Oppenheimer approximation

� Nuclear kinetic energy operator

and potential energy surface

� Wavepacket for the nuclear motion (R,t)

� amplitude of probability to be at geometry R at time tExample: H transfer with tunnelling

X-H…YX…H-Y

X-H…YX…H-Y

X-H…YX…H-Y

( ) ( ) ( )( ) ( )

BO

BO adia

ˆ, ,

ˆ ˆ

s

R

s

sR R

i ψ R t H ψ R tt

H T V R

∂=

∂

= +

�

� �

� �ℏ

�



57

Non-adiabatic quantum dynamics

� Time-dependent molecular Schrödinger equation for two vibronically coupled singlet electronic states, S0 and S1,

� Two-component wavepacket for the nuclear motion (R,t)

� Non-adiabatic coupling terms (non-Born-Oppenheimer)

� transfer of electronic population

S0

S1

S0

S1

Example: photodissociation with internal conversion

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

0 000 01

10 111 1

, ,ˆ ˆ

ˆ ˆ, ,

R R

R R

ψ R t ψ R ti

t ψ R t ψ R t

∂ = ∂

� �

� �

� �

ℏ � �H H

H H



58

Representations of the molecular Hamiltonian

Diabatic electronic states

General form

Zero kineticcoupling (Λ(01))� Easier for quantum

dynamics(no singularity at conical intersection)

Adiabatic electronic states Zeropotentialcoupling (H01)

� Output of quantum chemistry

( ) ( )

( ) ( )

( ) ( )

( ) ( )

00 00

00

01 01

01

10 1010

11 1111

ˆˆˆ Λ

ˆ ˆˆ Λ

0

0

ˆˆ Λ

ˆˆ Λ HT

T

H

H H = + +

H H

H H

( ) ( )

( ) ( )

( ) ( )

( ) ( )

01 01

10 11 1101

0 0

1

0 0

0

ˆ

ˆˆ Λ

ˆˆ Λ

00

00

ˆˆ Λ

ˆ

ˆ

ˆ Λ

VT

VT

= + + H

H

H

H

( ) ( )

( ) ( )

00

0

01

01

11

0

10

1011 ˆˆ

ˆ 0

ˆ

ˆ ˆ

0 HT H

HT H = +

H

H

H

H

NB: diagonal scalar corrections Λ(00), Λ(11) (Born-Huang approximation)

( ) ( ) ( )mol elˆˆ ˆ ˆˆ ; ; Λ ; ;

ss ss

ssR R Rs R H s R T δ s R H R s R

′ ′′′ ′= = + +� � �

� � � � �H



59

Electronic population

� Time-resolved population of each electronic state = integral of the density within the electronic state

� Normalisation condition

� Transfer of population due to non-adiabatic coupling

( ) ( ) ( ) ( ){ }

2

,s s

R

P t ψ R t dτ= ∫�

�


( ) ( ) 1s

s

P t =∑

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )conservation term

(Born-Oppenheim creation/dissipation terms(non-adiabatic couplin

er)

)g

ˆ ˆ, , ,sss ss s s

sR R

s

i ψ R t ψ R t ψ R tt

′ ′

′≠

∂= +

∂ ∑� �

� � �ℏ

��

H H


60

Radiationless decay

� Internal conversion = population transfer between singlet states

� No light emission: vibronic (non-adiabatic) coupling

� Very efficient at or near conical intersections


S0

S1

S0

S1

( )( ) ( )

adia adia

eladia adia

adia adia

1 0

ˆ0 ; 1 ;0 ; 1 ;

R

R

R H R RR R

V R V R

∇∇ =

−

�

�

� � ��

� �


61

Photostability vs. photoreactivity

� Internal conversion can regenerate the electronic ground state in the reactant or product regions

S0

Reaction coordinate

E

S1



62

Role of the crossing topography



63

Retinal photoreactivity

Isomerisation coordinate

Pote

nti

al energ

y

trans

cis

Energy transfer

NH+

2nd step

Introduction


64

DNA photostability

Proton transfer coordinate

Pote

nti

al energ

y

Locally

excited

stateCharge-transfer

state

Ground state

N

N

H

H

N

N

H

H

N

N-

H

N+

N

H

H

H

Introduction


4. Methods

65


August 2019

a. Grid-Based Methods

“Exact” methods

MCTDH

b. Direct (on-the-Fly) Methods

Gaussian-Based Quantum Dynamics

Trajectory-Based Dynamics � cf. F. Agostini’s talk

66

Methods


Grid-based quantum dynamics � electronic spectroscopyVibronic coupling Hamiltonian model (local expansion of the diabatic potential

surfaces and couplings)

� Benchmark: pyrazine: 10 atoms (24D) / 3 coupled electronic states� Valid only for small amplitude motions� Global potential energy surfaces on a large grid: difficult and expensive

Trajectory-based mixed dynamics � photochemistrySwarm of classical trajectories + probability of electronic transfer

� Most applications up to date (not accurate but useful for mechanistic purposes)� Approximate treatment of non-adiabatic events� ‘On-the-fly’ calculation of the potential energy or force field

+ A more quantum strategy: direct quantum dynamics� Moving Gaussian functions (centre follows a ‘quantum trajectory’)� Approximate quantum dynamics with ‘on-the-fly’ potential energy surfaces

67

State of the art

Methods


68

Hierarchy of methods

Methods


NA-MQC : nonadiabatic mixed quantum-classical approximations � trajectory-based methods

(courtesy: Mario Barbatti)

� Initial condition = initial wavepacket

� Electric dipole + Franck-Condon approximation

� Heller picture: sudden projection on S1 of the vibrational ground state in S0

69

First step: initial condition

Methods

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

1 0

0

0

, 0 , 0 ;

, 0

, 0 0

s

s

ψ R t ψ R t s R

ψ R t φ R

ψ R t

= = =

= =

= =

∑� � �

� �

�

S0

S1

� Possible to include t-dependent laser pulse

and R-dependent electric dipole


� Solution of the time-dependent Schrödinger equation

� The wavepacket evolves in time driven by the molecular Hamiltonian (often in a diabatic representation)

� “Exact” propagator (numerical integration)

� MCTDH (variational convergence)

70

Second step: propagation

Methods

( ) ( )ˆ, ,R

i ψ R t ψ R tt

∂=

∂�

� �ℏ H

( ) ( )ˆ W

, W ,R

it

ψ R t t e ψ R t−

+ =�

ℏ� �H

( ) ( )ˆ, ,R

δψ R t i ψ R tt

∂−

∂�

� �ℏH

( )

,

ˆ ˆ; ;ss

R Rs s

s R s R′

′

′=∑� �

� �HH


71

Approximate propagator

Methods

� Example: split-operator (Fourier transform R-space/K-space)

� Other typical propagator: Chebyshev polynomial expansion

( ) ( )( ) ( )

( )

( )

( )

( )

( )

( )

adia0

adia1

adia0

adia1

†

adia dia dia a

1

1

2

1

2

1

adia 0 W

adia 1 W

W

W

W

W

1

2

1

2

dia

, W 0

, W 0

0

0

0

0

R K

iV R t

iV R t

iT K t

iT K t

K R

R K K R

iV R t

iV R t

ψ R t t e

ψ R t t e

e

e

e

e

−

−

−

−

−

← ←

−

← ←

−

−

← ←

+ ≈ +

U U� ��

�

ℏ

�

� ��

ℏ

�

ℏ

�

ℏ

�

ℏ

�

ℏ

�

�

F F

F F

( ) ( )( ) ( )

adia 0

adia 1

,

,

ψ R t

ψ R t

�

�

( )2

2 J JJ J

T K K KM

= ⋅∑� � �ℏ


72

MultiConfiguration Time-Dependent Hartree (MCTDH)

Methods

� Nuclear coordinates � f internal degrees of freedom

� Similar to one-state case

Hartree product of SPF ~ configuration

SPF ~ molecular orbital

expansion of the SPF in a primitive basis set ~ LCAO

� Extra electronic degree of freedom (s) in the equations of motion

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1

1

1

,

, ,11 1

SPFcoefficient

Hartree product

, , ;

, ,f

f κ

f

nn fκ

j j j κκj

s

s

s s s

j

ψ t ψ t

ψ t A t φ Q t

s

== =

=

=

∑

∑ ∑ ∏

Q Q Q

Q…

⋯��

��

{ }1, ,f

R Q Q→ =Q�

…


73

Comparison: 3D example

Methods

� Standard methods (equivalent to full CI expansion)

� Large basis set � N1N2N3 configurations (3D functions) and expansion coefficients

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )31 2

1 2 3 1 2 3

1 2 3

1 2 3

1 2 3 1 2 31 1 1

coefficient configuration

, , ,NN N

j j j j j jj j j

ψ Q Q Q t A t f Q f Q f Q= = =

=∑∑∑��


( ) ( ){ } ( ) ( ){ } ( ) ( ){ }1 2 3

1 2 31 2 3

1 2 3

1 2 31 1 1

primitive basis set (mathematical functions, equivalent to AOs):N N N

j j jj j j

f Q f Q f Q= = =⊗ ⊗

74


Methods

� MCTDH method (equivalent to MCSCF)

� Contraction scheme � n1n2n3 << N1N2N3 configurations (3D functions) and expansion A-coefficients, but extra “LCAO” C-coefficients

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )31 2

1 2 3 1 2 3

1 2 3

1 2 3

1 2 3 1 2 31 1 1

coefficient configuration

, , , , , ,nn n

k k k k k kk k k

ψ Q Q Q t A t φ Q t φ Q t φ Q t= = =

=∑∑∑��


( ) ( ){ } ( ) ( ){ } ( ) ( ){ }1 2 3

1 2 31 2 3

1 2 3

1 2 31 1 1

single-particle functions (variational, equivalent to MOs):

, , ,n n n

k k kk k k

φ Q t φ Q t φ Q t= = =⊗ ⊗

( ) ( ) ( ) ( ) ( ) ( ),

1

, ; 1,2,3κ

κ

κ κ κ

κ

Nκ κ k κ

k κ j j κj

φ Q t C t f Q κ=

= =∑

75


Methods

� ML(multilayer)-MCTDH method (hierarchical contraction)

� Smaller vectors but more complicated equations of motion

( ) ( ) ( ) ( ) ( ) ( ){ }{ }

312

12 3 12 3

12 3

12 3

1 2 3 1 2 31 1

coefficient first layer 1,2 3

, , , , , ,nn

k k k kk k

ψ Q Q Q t A t φ Q Q t φ Q t= =

=∑∑��


( ) ( ){ } ( ) ( ){ }12 3

12 312 3

12 3

1 2 31 1

intermediate correlated functions (groups of coordinates):

, , ,n n

k kk k

φ Q Q t φ Q t= =⊗

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }{ }

1 2

12

12 1 2 1 2

1 2

12 12, 1 2

1 2 1 21 1

coefficient second layer 1 2

, , , ,m m

k

k l l l ll l

φ Q Q t B t ξ Q t ξ Q t= =

=∑∑��

QD: rigorous and accurate but...

PES required as an analytical expression fitted to a grid of data points

� grid-based methods

Representing a multiD function is expensive� Exponential scaling

Fitting procedures are complicated and system-dependent� Constrained optimisation techniques

Much grid space is wasted� High or far regions: seldom explored

� trajectory-based methods

PES calculated on-the-fly as in classical MD

76

Quantum dynamics: the curse of dimensionality

Methods


� The variational multiconfiguration Gaussian (vMCG) method

� Time-dependent Gaussian basis set (local around centres)

� Coupled ‘quantum trajectories’: position, momentum, and phase at centre of Gaussian functions

77

Gaussian-based quantum dynamics

Equations of motion implemented in a development version (QUANTICS, London) of the Heidelberg MCTDH

package

( ) ( ) ( ), ,j j

j

ψ Q t A t g Q t=∑

Methods


� Diabatic picture for dynamics

� Adiabatic picture for on-the-fly quantum chemistry

� Diabatic transformation (U):

V1

V0

H00

H11

H11

H00

2|H01|

78

Direct dynamics implementation (DD-vMCG)

( ) ( )

( ) ( )

11

01

10

00

0 †

dia adia adia dia11

00

1

01

10

0ˆ

ˆ

ˆ

ˆ

0

00ˆ

ˆ

H

H H

H

T

T

V

V← ←

= + U U

��

H H

HH

S0

S1

S0

S1

Methods


� x(1) and x(2) (branching space): lift degeneracy at a selected conical intersection (Qref)

� Simplest diabatic Hamiltonian

79

Diabatisation: start with linear vibronic-coupling model

( ) ( )( ) ( )

( ) ( )( )

1 ref

1

2 ref

2

1

11 00

2

01

1

2

kQ

λQ

H H

H

= ⋅ −

= ⋅ −

= ∇ −

=∇

Q

Q

x Q Q

x Q Q

x

x

V1

V0

x(1)

x(2)

( ) 1 2

2 1

kQ λQ

λQ kQ

− = W Q

Methods


� First-order expansion of the adiabatic energy difference and rescaling

80

And generate regularised Hamiltonian

0 †

dia adia ad

111

ia d

1

0

a

0

i

0 01 0

0

V

V

H H

H H ← ←

=

U U

( )( )( ) ( )

( ) ( )

( )( ) ( )

( )( ) ( )

( ) ( )( ) ( )

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

1

1

0 1 1 0

1 22 2

1 1 1 2

2 1

ˆˆ Σ

Σ , 2 2

,

T

V V V V

δ δδ δ

δ δ

∆= + +

∆

+ −= ∆ =

− ⋅ ⋅ = ∆ = ⋅ + ⋅ ⋅ ⋅

Q Q

Q1 Q 1 W Q

Q

Q Q Q QQ Q

x Q x QW Q Q x Q x Q

x Q x Q

H

V1

V0

H00

H11

H11

H00

2|H01|

Methods


81

Advantage: effective Hamiltonian

� Exact eigenvalues (adiabatic data) at any point

� Approximate rotation angle φ(1) and non-adiabatic coupling (first-order contribution inducing the singularity around Qref)

( )( )

( )( )( ) ( )

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

( )( )

1

1

1 † 1 10

adia dia dia adia11

0

0

0Σ

0

V

V← ←

−∆ ∆

∆ = + ∆ Q

Q

Q QQ 1 U Q W Q U Q

Q Q ��

( ) ( )1

0 1Φ ; Φ ;φ∇ ≈− ∇

Q QQ Q Q

Methods


82

Potential energyand derivatives

Database

Quantu

mChem

istry

R1R2

LocalHarmonicApproximation

Interface quantum chemistry – quantum dynamics

Quantu

m D

ynam

ics

Methods

( ) ( )1 2,j j

R t R t

( ){ }( ){ }

,j

j

g R t

A t

�

( ){ }( ){ }

, W

W

j

j

g R t t

A t t

+

+

�


� Non-orthogonal basis set: ambiguous attribution of how much of the density is given by each Gaussian function

� Solution: Mulliken-like population (Cf. atomic orbitals, not orthogonal)

83

Gross Gaussian populations

Methods

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

* *

,

, , ,s s s s s

k j k jk j

ψ R t A t A t g R t g R t=∑� � �

( ) ( ) ( ) ( ) ( ){ }

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

*

*

, ,s s

kj k j

R

s s s

j k j kjk

S t g R t g R t dτ

GGP t A t A t S t

=

= ℜ

∫

∑

�

� �

( ) ( ) ( ) ( )s s

jj

P t GGP t=∑ ( ) ( ) 1s

js j

GGP t =∑∑Summer School – Emie-Up – Aug. 2019

� Similar expansion as vMCG

� Difference: quantum evolution for the coefficients but classical equations of motion for the centres of the Gaussian functions

� Drawback: convergence is slower (classical motion)

� Advantage: the basis set can increase (spawning) when necessary � when the wavepacket reaches a conical intersection

84

Ab initio Multiple Spawning (AIMS)

Methods

( ) ( ) ( ) ( ) ( ), , ;s s

j js j

ψ R t A t g R t s R=∑∑� � �


� Statistical sampling (positions and velocities) of the initial wavepacket

� Swarm of trajectories driven by Newton’s classical equations of motion far from conical intersection

� Diabatic state-transfer probability when the energy gap becomes small (surface hopping) calculated from approximate quantum equations (e.g., Landau-Zenerformula)

85

Mixed quantum-classical (semiclassical) trajectories

Methods

( ) ( ) ( ) ( ) ( ){ }, ,k k

kψ R t R t V t→� � �

( ) ( )( ) ( )( )

( ) ( ) ( ) ( )( )

2

01

1 0

11 00

2exp

k

k

R tk

k

R R t

Hπ

P tV t H H

→

= −

⋅∇ −

�

� �

� �ℏ


5. Examples of Application

86


August 2019

� Franck-Condon approximation� Electric dipole approximation� Wavepacket evolution� Absorption spectrum: Fourier transform of the autocorrelation function

87

Electronic spectroscopy: absorption spectrum

Examples of Application

S0

S1

Q

E

ωℏ S0

S1

Q

E

S0

S1

Q

E

( ) ( ) ( ) ( ) ( )0

, where Ψ 0 Ψ

Ei ω t

C tω e tω t tσ dt C

+∞ +

−∞

= =∝ ∫ ℏ

E0


� Vibronic (non-adiabatic) coupling between S2 and S1

� Intensity borrowed by dark state from bright state� Benchmark for MCTDH with vibronic coupling Hamiltonian model

10 atoms (24D) / 3 electronic states

88

S2 S0 pyrazine absorption spectrum (MCTDH)


( ) ( )0

σ ω cI ω I e

−= ℓ


89



� Quadratic vibronic coupling Hamiltonian model

� Parameters fitted to ab initio calculations after diagonalisation

� Four groups of coordinates (G1 to G4) depending on irreducible representations in D2h

1 2

3 4

2dia 2

2

,

,

1 0 W 0ˆ

0 1 0 W2

00

0 0

00

0 0

ii

i i

iji

i i ji G i j Gi ij

iji

i i ji G i j Gi ij

ωQ

Q

aaQ Q Q

b b

ccQ Q Q

c c

∈ ∈

∈ ∈

−∂ = − + + ∂ + + + +

∑

∑ ∑

∑ ∑

Q

ℏH


Using all 24 degrees of freedom� spectrum converged vs. experiment

90




91

Heme photodissociation with SO coupling (ML-MCTDH)


Summer School – Emie-Up – Aug. 2019 179 el. states (S/T/Q), 15 vib. modes, 1 ps simulations

� After S1 photoexcitation (nO,π*CO), formaldehyde dissociates toH + HCO or H2 + CO

� At low energy, Fermi-Golden-Rule decay to S0 (slow decay at large energy gap) followed by ground-state reactivity

� At medium energy, involvement of the triplet T1

� At higher energy, possible to overcome the S1 TS followed by direct decay to S0 products through conical intersection

92

S1/S0 formaldehyde photodissociation (DD-vMCG)



93



The S1 TS is directly connected to the S1/S0 conical intersection


94



Influence of initial conditions


95



Distribution of products (final geometry and electronic state)


Some recent test cases with new diabatisation (DD-vMCG)

96



� Butatriene cation2/3/4-state 2-dimensional diabatic matrix (global): smooth� surfaces and populations depend on number of states

� Formamidic acid6-state 12-dimensional diabatic matrix (global): smooth� 24 Gaussian functions: 80% dissociation (various fragments)

� Ozone8 Gaussian functions with CASPT2: spectrum in correct place� took a student only one week compared to one year for the original work with grid-based quantum dynamics!

6. Conclusions and Outlooks

97

Excited-state dynamics is still a growing field of theoretical and physical chemistry (applications to laser-driven control, influence of

the environment in large systems or condensed matter...)

Developments are still required to treat photochemical reactivity with as much accuracy as electronic spectroscopy (cheaper quantum

chemistry methods for excited states, general procedures to produce accurate potential energy surfaces and couplings for large-amplitude

deformations of the geometry...)


August 2019

Outlooks: attophysics and attochemistry

98

Neutral

State 1

State 2

Attosecondpulse

Cation

Conclusions and Outlooks


99

Beyond WPs: dissipative quantum dynamics (system-bath)


Conclusions and Outlooks

100

Some recent reviews…


101

… and books

ISBN: 978-1119417750


Quantum Chemistry and Dynamics of Excited States: Methods and Applications

Edited by Leticia González and Roland

Lindh

(to be released, 2019)

102

Further reading


• Michl, J. and Bonačić-Koutecký, V., Electronic Aspects of Organic Photochemistry, New York, Wiley, 1990.• Klessinger, M. and Michl, J., Excited States and Photochemistry of Organic Molecules, New York ; Cambridge, VCH, 1995.• Turro, N. J., Ramamurthy, V. and Scaiano, J. C., Principles of Molecular Photochemistry: an Introduction, Sausalito, CA., University Science Books, 2009.• Olivucci, M., Computational Photochemistry, Amsterdam, Elsevier, 2005.• Szabo, A. and Ostlund, N. S., Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Mineola, N.Y, Dover ; London : Constable, 1996, 1989.• Shavitt, I. and Bartlett, R. J., Many-Body Methods in Chemistry and Physics : MBPT and Coupled-Cluster Theory, Cambridge, Cambridge University Press, 2009.• Shaik, S. S. and Hiberty, P. C., A Chemist's Guide to Valence Bond Theory, Hoboken, N.J., Wiley-Interscience ; Chichester : John Wiley, 2008.• Douhal, A. and Santamaria, J., Femtochemistry and Femtobiology: Ultrafast Dynamics in Molecular Science, Singapore, World Scientific, 2002.• Bersuker, I. B., The Jahn-Teller Effect, Cambridge, Cambridge University Press, 2006.• Köppel, H., Yarkony, D. R. and Barentzen, H., The Jahn-Teller Effect: Fundamentals and Implications for Physics and Chemistry, Heidelberg, Springer, 2009.• Baer, M., Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections, Hoboken, N.J., Wiley, 2006.• Domcke, W., Yarkony, D. R. and Köppel, H., Conical Intersections: Electronic Structure, Dynamics & Spectroscopy, Singapore, World Scientific, 2004.• Domcke, W., Yarkony, D. and Köppel, H., Conical Intersections : Theory, Computation and Experiment, Hackensack, N. J., World Scientific, 2011.• Heidrich, D., The Reaction Path in Chemistry: Current Approaches and Perspectives, Dordrecht ; London, Kluwer Academic Publishers, 1995.• Mezey, P. G., Potential Energy Hypersurfaces, Amsterdam ; Oxford, Elsevier, 1987.• Marx, D. and Hutter, J., Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge, Cambridge University Press, 2009.• Wilson, E. B., Cross, P. C. and Decius, J. C., Molecular Vibrations. The theory of Infrared and Raman Vibrational Spectra, New York, McGraw-Hill Book Co., 1955.• Bunker, P. R. and Jensen, P., Molecular Symmetry and Spectroscopy, Ottawa, NRC Research Press, 1998.• Schinke, R., Photodissociation Dynamics: Spectroscopy and Fragmentation of Small Polyatomic Molecules, Cambridge, Cambridge University Press, 1993.• Tannor, D. J., Introduction to Quantum Mechanics: a Time-Dependent Perspective, Sausalito, CA., University Science Books, 2007.• Robb, M. A., Garavelli, M., Olivucci, M. and Bernardi, F., "A computational strategy for organic photochemistry", in Reviews in Computational Chemistry, Vol 15, K. B.

Lipkowitz and D. B. Boyd, New York, Wiley-VCH, Vol. 15, 2000, p. 87-146.• Daniel, C., "Electronic spectroscopy and photoreactivity in transition metal complexes", Coordination Chemistry Reviews, Vol.238, 2003, p. 143-166.• Braams, B. J. and Bowman, J. M., "Permutationally invariant potential energy surfaces in high dimensionality", International Reviews in Physical Chemistry, Vol.28 n°4,

2009, p. 577-606.• Lengsfield III, B. H. and Yarkony, D. R., "Nonadiabatic interactions between potential-energy surfaces", Advances in Chemical Physics, Vol.82, 1992, p. 1-71.• Köppel, H., Domcke, W. and Cederbaum, L. S., "Multi-mode molecular dynamics beyond the Born-Oppenheimer approximation", Advances in Chemical Physics, Vol.57,

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