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Nonadiabatic Dynamicscurrent methods and challenges
Benjamin LASORNE
Department of Theoretical ChemistryInstitut Charles Gerhardt
CNRS – Université de Montpellier
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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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
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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
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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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
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� 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
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� 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
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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
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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
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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
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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
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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
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2. Beyond the Born-Oppenheimer
Approximation
12
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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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
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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
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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
� �
� �
� �
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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
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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
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18
Starting point: Born-Oppenheimer (BO) approximation
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
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
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19
Starting point: Born-Oppenheimer (BO) approximation
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
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
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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
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21
Potential energy surface: explicit function of Q
( )el, samplingQ QE∀ →
Q
Beyond BOA
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22
Potential energy surface: explicit function of Q
( ) ( )el, sampling and fittingQ QE V Q∀ ≡ →
Q
Beyond BOA
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23
Potential energy surface: explicit function of Q
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
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
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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!
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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
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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
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27
Beyond Born-Oppenheimer
Beyond BOA
Summer School – Emie-Up – Aug. 2019
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )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”
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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
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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
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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34
Matrix Hellmann-Feynman theorem (general principle)
( )
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
Summer School – Emie-Up – Aug. 2019
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35
Matrix Hellmann-Feynman theorem (general principle)
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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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 Q
Q
Q Q
Beyond BOA
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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)
Summer School – Emie-Up – Aug. 2019
Beyond BOA
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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
Summer School – Emie-Up – Aug. 2019
Beyond BOA
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
� Infinite basis set required
(unless isolated Hilbert subspace)
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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
Summer School – Emie-Up – Aug. 2019
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49
Diabatisation by ansatz: smooth Hamiltonian matrix
� Electronic structure preserved wrt. geometry variation
diabatic
Q
adiabatic
V
Beyond BOA
Summer School – Emie-Up – Aug. 2019
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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
Summer School – Emie-Up – Aug. 2019
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51
Intuitive chemical interpretation
� Walsh
correlation
diagrams
q
V
Beyond BOA
Summer School – Emie-Up – Aug. 2019
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52
Photochemistry: ES and GS multiple wells and crossings
Beyond BOA
Summer School – Emie-Up – Aug. 2019
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53
Thermal chemistry: GS local coupled states
� Marcus theory of electron transfer reactions
� Valence bond: transition barriers as avoided crossings
Beyond BOA
Summer School – Emie-Up – Aug. 2019
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3. Non-adiabatic Processes
54
S0
S1
S0
S1
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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a. Non-Adiabatic Quantum Dynamics
b. Radiationless Decay
c. Electronic State Crossings
55
Non-adiabatic processes
Summer School – Emie-Up – Aug. 2019
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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
�
Non-adiabatic processes
Summer School – Emie-Up – Aug. 2019
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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
Non-adiabatic processes
Summer School – Emie-Up – Aug. 2019
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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
Non-adiabatic processes
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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= ∫�
�
Non-adiabatic processes
( ) ( ) 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
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60
Radiationless decay
� Internal conversion = population transfer between singlet states
� No light emission: vibronic (non-adiabatic) coupling
� Very efficient at or near conical intersections
Non-adiabatic processes
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
∇∇ =
−
�
�
� � ��� ��
� �
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61
Photostability vs. photoreactivity
� Internal conversion can regenerate the electronic ground state in the reactant or product regions
S0
Reaction coordinate
E
S1
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Non-adiabatic processes
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62
Role of the crossing topography
Non-adiabatic processes
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63
Retinal photoreactivity
Isomerisation coordinate
Pote
nti
al energ
y
trans
cis
Energy transfer
NH+
2nd step
Introduction
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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
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4. Methods
65
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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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
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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
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68
Hierarchy of methods
Methods
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NA-MQC : nonadiabatic mixed quantum-classical approximations � trajectory-based methods
(courtesy: Mario Barbatti)
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� 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
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� 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
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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
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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�
…
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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= = =
=∑∑∑������� �����������������������
Summer School – Emie-Up – Aug. 2019
( ) ( ){ } ( ) ( ){ } ( ) ( ){ }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= = =⊗ ⊗
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74
Comparison: 3D example
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= = =
=∑∑∑������� �����������������������������
Summer School – Emie-Up – Aug. 2019
( ) ( ){ } ( ) ( ){ } ( ) ( ){ }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=
= =∑
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75
Comparison: 3D example
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= =
=∑∑������� �����������������������
Summer School – Emie-Up – Aug. 2019
( ) ( ){ } ( ) ( ){ }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= =
=∑∑��������� �������������������
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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
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� 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
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� 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
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� 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
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� 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
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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
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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
+
+
�
Summer School – Emie-Up – Aug. 2019
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� 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
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� 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=∑∑� � �
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� 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)
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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
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5. Examples of Application
86
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
August 2019
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� Franck-Condon approximation� Electric dipole approximation� Wavepacket evolution� Absorption spectrum: Fourier transform of the autocorrelation function
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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
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� 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
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S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
( ) ( )0
σ ω cI ω I e
−= ℓ
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89
S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
� 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
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Using all 24 degrees of freedom� spectrum converged vs. experiment
90
S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
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91
Heme photodissociation with SO coupling (ML-MCTDH)
Examples of Application
Summer School – Emie-Up – Aug. 2019 179 el. states (S/T/Q), 15 vib. modes, 1 ps simulations
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� 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)
Examples of Application
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93
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
The S1 TS is directly connected to the S1/S0 conical intersection
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94
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Influence of initial conditions
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95
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Distribution of products (final geometry and electronic state)
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Some recent test cases with new diabatisation (DD-vMCG)
96
Examples of Application
Summer School – Emie-Up – Aug. 2019
� 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!
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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...)
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Outlooks: attophysics and attochemistry
98
Neutral
State 1
State 2
Attosecondpulse
Cation
Conclusions and Outlooks
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99
Beyond WPs: dissipative quantum dynamics (system-bath)
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Conclusions and Outlooks
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100
Some recent reviews…
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101
… and books
ISBN: 978-1119417750
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Quantum Chemistry and Dynamics of Excited States: Methods and Applications
Edited by Leticia González and Roland
Lindh
(to be released, 2019)
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102
Further reading
Summer School – Emie-Up – Aug. 2019
• 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,
1984, p. 59-246.• Worth, G. A. and Cederbaum, L. S., "Beyond Born-Oppenheimer: Molecular dynamics through a conical intersection", Annual Review of Physical Chemistry, Vol.55, 2004,
p. 127-158.• Worth, G. A., Meyer, H. D., Köppel, H., Cederbaum, L. S. and Burghardt, I., "Using the MCTDH wavepacket propagation method to describe multimode non-adiabatic
dynamics", International Reviews in Physical Chemistry, Vol.27 n°3, 2008, p. 569-606.• Leforestier, C., Bisseling, R. H., Cerjan, C., Feit, M. D., Friesner, R., Guldberg, A., Hammerich, A., Jolicard, G., Karrlein, W., Meyer, H. D., Lipkin, N., Roncero, O. and
Kosloff, R., "A comparison of different propagation schemes for the time-dependent Schrödinger equation", Journal of Computational Physics, Vol.94 n°1, 1991, p. 59-80.• Gatti, F. and Iung, C., "Exact and constrained kinetic energy operators for polyatomic molecules: The polyspherical approach", Physics Reports-Review Section of Physics
Letters, Vol.484 n°1-2, 2009, p. 1-69.• Lasorne, B., Worth, G. A. and Robb, M. A., " Excited-state dynamics ", WIREs Computational Molecular Science, Vol.1, 2011, p. 460-475.