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Dynamique Quantique
et Non-Adiabatique“Photochimie Théorique”
Benjamin LASORNE
Equipe CTMM – Institut Charles GerhardtCNRS – Université de Montpellier
Ecole ThématiqueModPhysChemOctobre 2017
2
Some selected references…
Ecole Thématique – ModPhysChem – Oct. 2017
• 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.• Meyer, H.-D., Gatti, F. and Worth, G., Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Weinheim, Wiley-VCH, 2009.• Gatti, F., Molecular Quantum Dynamics: From Theory to Applications, Berlin ; Heidelberg, Springer-Verlag, 2014.• 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.
3
… and a new book
Ecole Thématique – ModPhysChem – Oct. 2017
springer.com/fr/book/9783319539218
E-book ISBN: 978-3-319-53923-2Hardcover ISBN: 978-3-319-53921-8
DOI: 10.1007/978-3-319-53923-2
1. Introduction
2. Beyond the Born-Oppenheimer Approximation
3. Non-Adiabatic Processes
4. Quantum and Semiclassical Dynamics Methods
5. Examples of Application
6. Conclusions and Outlooks
4Ecole Thématique – ModPhysChem – Oct. 2017
1. Introduction
5
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
Ecole ThématiqueModPhysChemOctobre 2017
6
“Theoretical photochemistry”: a tentative definition
Ecole Thématique – ModPhysChem – Oct. 2017
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
Introduction
� Franck-Condon approximation� Electric dipole approximation� Wavepacket evolution (time)� Absorption spectrum (energy)
7
Electronic spectroscopy
S0
S1
Q1
S0
S1
Q1
S0
S1
Q1
Example: ππ* benzene (vibrational progression in mode 1: totally symmetric breathing)
E
Introduction
Ecole Thématique – ModPhysChem – Oct. 2017
� Intramolecular vibrational redistribution� Internal conversion� Photostability vs. photoreactivity
8
Photochemistry
benzene prefulvenoidgeometries
benzvalene
254 nm
S0 (1A1g) → S1 (1B2u)
S0
Reaction coordinate
E
S1
Introduction
Ecole Thématique – ModPhysChem – Oct. 2017
9
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
Ecole Thématique – ModPhysChem – Oct. 2017
10
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)
� Dynamic approach
Potential
energy
surface
Reciprocal influence betweenthe electronic structure and the molecular geometry
Structure (thermodynamics, spectroscopy) and reactivity (mechanism, kinetics)
RP
TSV
Q
Introduction
Ecole Thématique – ModPhysChem – Oct. 2017
11
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
Ecole Thématique – ModPhysChem – Oct. 2017
12
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
Ecole Thématique – ModPhysChem – Oct. 2017
13
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
Ecole Thématique – ModPhysChem – Oct. 2017
a. Adiabatic Electronic States
b. Adiabatic Partition of the Molecular Schrödinger Equation
c. Non-Adiabatic Couplings
d. Conical Intersections
e. Diabatic Electronic States
15
Beyond BOA
Ecole Thématique – ModPhysChem – Oct. 2017
16
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
±
+ −
′′ ′′ ′ ′ ′′= + +
+ − → = ± +
− =− − = ⇔ + = ⇔ =
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
17
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
� �
� �
� �
Ecole Thématique – ModPhysChem – Oct. 2017
18
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
Ecole Thématique – ModPhysChem – Oct. 2017
19
( ) ( ) ( )
( ) ( ) ( ) ( )
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
= +
∀ =
∂ ∂∂
∂
∂
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
Starting point: Born-Oppenheimer (BO) approximation
20
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
= +
∀
∂
=
∂
∀
∂
≡
∂
∂
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
21
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
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
22
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
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
23
Potential energy surface: explicit function of Q
( )el, samplingQ QE∀ →
Ecole Thématique – ModPhysChem – Oct. 2017
Q
Beyond BOA
24
Potential energy surface: explicit function of Q
( ) ( )el, sampling and fittingQ QE V Q∀ ≡ →
Ecole Thématique – ModPhysChem – Oct. 2017
Q
Beyond BOA
25
Potential energy surface: explicit function of Q
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
0
2
0 0 0
20 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
�����
Ecole Thématique – ModPhysChem – Oct. 2017
Q
Beyond BOA
26
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
−
−
=
− −
= =
− − −
= = =
=
+ −
+ − − +
+ − − − +
∑
∑∑
∑∑∑
⋯
⋯
Ecole Thématique – ModPhysChem – Oct. 2017
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!
27
Photochemistry often involves radiationless decay
Isomerisation coordinate
Pote
nti
al e
nerg
y
trans
cis
Energy transfer
NH+
2nd step
� Example: first step of the vision process:
retinal photoisomerisation � no fluorescence!
Internal conversion
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
28
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
Ecole Thématique – ModPhysChem – Oct. 2017
29
Beyond Born-Oppenheimer
( ) ( ) ( )
( ) ( ) ( )
( )
mol
mol
el
el
e
nuc
nuc
n
l
uc
Ψ , , , BO approximation
Beyond BO Ψ , , ,
: no longer a single eigenvalue
matrix ( ) of potential energi
Φ ;
Φ ;
ˆ , ;
inde e
ˆ
r x s fo
s ss
q
q q
q q
tQ ψ Q Q
Q
t
t ψ Q Q
Q
t
H q
T
s
Q
= ⇒
⇒ =
∂
→
∑
�������
��
�������
���� ����� � ��� ���
( ) ( ) ( )2
2 : acts on both , and
non-adiabatic-coupling matr2
ix ( )
Φ ;
index for
s sQ Qt qψ Q Q
m
Qs→
∂ =− ∂ℏ
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
30
Refresher: rotating frame and vector derivative
( )( )
( ) ( )
( ) ( )( ) ( )
( )( ) ( ) ( ) ( )
Example: polar coordinates
cos sin
sin cos
chain rule: r r
r θ r
θ r θ
r x y
θ x y
x y r
rx y r θr
u θ θu θu
u θ θu θu
r xu y u ru θ
r xu y u ru
u θ u θθ
t t θ
u θ θ u θ
u θ u θ
ru θ rθu θθ ru θ
∂ ∂∂=
∂ ∂ ∂=
= +
=− +
=
∂
∂ =
+ =
= + = + = +
� �
� �ɺ ɺ
�
� � �
� � �
� � � �
� � � �
�
� �ɺ ɺ�ɺ ɺ ɺɺ ɺ
Ecole Thématique – ModPhysChem – Oct. 2017
θ
r
xu�
yu�
ru�
θu�
Beyond BOA
31
Non-adiabatic coupling (general principle)
{ }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
fixed orthonormal basis set: s s s
s
s s
s s s ss s
s s s s
u v v u
v u v
v v u v v u
v
x x x x
x xu v v u vx x
=
= ⋅
= =
=
′′ ′′
′′ ⋅ ′⋅ ′=
∑
∑ ∑
� � �
� �
� � � �
� � � �
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
32
Non-adiabatic coupling (general principle)
( ){ }( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
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∑
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
33
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
′′ ′′ ′ ′ ′′
′′ ′′ ′ ′
= =
= + +
= ′′+ +
∑
∑
∑ ∑
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
34
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=∑��������� �������Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
35
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
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
36
Non-adiabatic photochemistry: a non-classical field
Radiationless decay mechanisms � the nuclei behave as quantum-mechanically as the electrons (~\E, ~\t)
� 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
Ecole Thématique – ModPhysChem – Oct. 2017
S0
S1
S0
S1
( ) ( ) ( ){ }
2,s
s
Q
P t ψ Q t dQ= ∫
Beyond BOA
37
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
=
′ ′ ′
=
=
=
= + =
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
38
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
′
′ ′ ′
′
=
=
+ + =
+ ′ ′
′ =′
+ =
′− +
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
39
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
=
′ =
′
′
=
−
′
≠
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
40
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
′
′
′
′
= +
=
=
= + +
= + +
= ′
′′ =
−
∑
…
������������������
…
�
HDR viva — Montpellier — 15/01/2016
Beyond BOA
41
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
− + =
= =
≠ =
∂ ∂ ∂
∂
−
∂
∂∂
���������
�����������������
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
42
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
→ →
∂
∂
=
+
∂∫ ����������������������������
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
43
Two-state model
Ecole Thématique – ModPhysChem – Oct. 2017
( )
0100
00 0
11
11
11 11
11 11
0
00
0
22
0,10 00
01
0101
01
1 0 20 12
2eigenvalues
2 2
H
H
H H
H
HH H
H
H
H
H
HV
H
H
H
H
− − + = + −
↓
+ − = ± +
Beyond BOA
44
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
Ecole Thématique – ModPhysChem – Oct. 2017
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
45
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 ⋯
�����������������������
( )( ) ( )
00
1 0
el 11
ˆΦ ; Φ ;Φ ; Φ ;
V
H
V
∇∇
−= →∞Q
QQ Q
Q
Q Q
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
46
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
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
47
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)
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
48
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
Ecole Thématique – ModPhysChem – Oct. 2017
Beyond BOA
49
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
Ecole Thématique – ModPhysChem – Oct. 2017
50
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
Ecole Thématique – ModPhysChem – Oct. 2017
51
Why quasi-diabatic states?
0 1
20 1 0 1
20 1 0 1
0
20 1 0 1
0
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
Ecole Thématique – ModPhysChem – Oct. 2017
� Infinite basis set required
(unless isolated Hilbert subspace)
52
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
Ecole Thématique – ModPhysChem – Oct. 2017
53
Diabatisation by ansatz: smooth Hamiltonian matrix
� Electronic structure preserved wrt. geometry variation
diabatic
Q
adiabatic
V
Beyond BOA
Ecole Thématique – ModPhysChem – Oct. 2017
54
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
Ecole Thématique – ModPhysChem – Oct. 2017
55
Intuitive chemical interpretation
� Walsh
correlation
diagrams
q
V
Beyond BOA
Ecole Thématique – ModPhysChem – Oct. 2017
56
Photochemistry: ES and GS multiple wells and crossings
Beyond BOA
Ecole Thématique – ModPhysChem – Oct. 2017
57
Thermal chemistry: GS local coupled states
� Marcus theory of electron transfer reactions
� Valence bond: transition barriers as avoided crossings
Beyond BOA
Ecole Thématique – ModPhysChem – Oct. 2017
a. Non-Adiabatic Quantum Dynamics
b. Radiationless Decay
c. Electronic State Crossings
59
Non-adiabatic processes
Ecole Thématique – ModPhysChem – Oct. 2017
60
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
Ecole Thématique – ModPhysChem – Oct. 2017
61
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
Ecole Thématique – ModPhysChem – Oct. 2017
62
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
Ecole Thématique – ModPhysChem – Oct. 2017
63
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
Ecole Thématique – ModPhysChem – Oct. 2017
64
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 adiaeladia adia
adia adia1 0
ˆ0 ; 1 ;0 ; 1 ;
R
R
R H R RR R
V R V R
∇∇ =
−
�
�
� � ��� ��
� �
Ecole Thématique – ModPhysChem – Oct. 2017
65
Photostability vs. photoreactivity
� Internal conversion can regenerate the electronic ground state in the reactant or product regions
S0
Reaction coordinate
E
S1
Ecole Thématique – ModPhysChem – Oct. 2017
Non-adiabatic processes
66
Role of the crossing topography
Non-adiabatic processes
Ecole Thématique – ModPhysChem – Oct. 2017
67
Retinal photoreactivity
Isomerisation coordinate
Pote
nti
al e
nerg
y
trans
cis
Energy transfer
NH+
2nd step
Introduction
Ecole Thématique – ModPhysChem – Oct. 2017
68
DNA photostability
Proton transfer coordinate
Pote
nti
al e
nerg
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
Ecole Thématique – ModPhysChem – Oct. 2017
a. Grid-Based Methods
“Exact” methods
MCTDH
b. Trajectory-Based Methods
Direct Quantum Dynamics
Semiclassical Dynamics
70
Methods
Ecole Thématique – ModPhysChem – Oct. 2017
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 semiclassical 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
71
State of the art
Methods
Ecole Thématique – ModPhysChem – Oct. 2017
� Initial condition = initial wavepacket� Electric dipole + Franck-Condon approximation
� Heller picture: sudden projection on S1 of the vibrational ground state in S0
72
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 pulseand R-dependent electric dipole
Ecole Thématique – ModPhysChem – Oct. 2017
� 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)
73
Second step: propagation
Methods
( ) ( )ˆ, ,R
i ψ R t ψ R tt
∂=
∂�
� �ℏ H
( ) ( )ˆ \, \ ,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
Ecole Thématique – ModPhysChem – Oct. 2017
74
“Exact” 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 \
adia 1 \
\
\
\
\
1
2
1
2
dia
, \ 0
, \ 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
Ecole Thématique – ModPhysChem – Oct. 2017
75
MultiConfiguration Time-Dependent Hartree (MCTDH)
Methods
� Nuclear coordinates � f internal degrees of freedom
� Similar to one-state caseHartree product of SPF ~ configurationSPF ~ molecular orbitalexpansion 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�
…
Ecole Thématique – ModPhysChem – Oct. 2017
76
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= = =
=∑∑∑������� �����������������������
Ecole Thématique – ModPhysChem – Oct. 2017
( ) ( ){ } ( ) ( ){ } ( ) ( ){ }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= = =⊗ ⊗
77
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= = =
=∑∑∑������� �����������������������������
Ecole Thématique – ModPhysChem – Oct. 2017
( ) ( ){ } ( ) ( ){ } ( ) ( ){ }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=
= =∑
78
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= =
=∑∑������� �����������������������
Ecole Thématique – ModPhysChem – Oct. 2017
( ) ( ){ } ( ) ( ){ }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
79
Quantum dynamics: the curse of dimensionality
Methods
Ecole Thématique – ModPhysChem – Oct. 2017
� 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
80
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
Ecole Thématique – ModPhysChem – Oct. 2017
� Diabatic picture for dynamics
� Adiabatic picture for on-the-fly quantum chemistry
� Diabatic transformation (U):
V1
V0
H00
H11
H11
H00
2|H01|
81
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
Ecole Thématique – ModPhysChem – Oct. 2017
� Approximate rotation angle, φ(1):
first-order expansion of the adiabatic energy difference
82
Diabatic transformation: regularised states
0 †dia adia ad
111ia d
1
0a
0i
0 01 0
0
V
V
H H
H H ← ←
=
U U
( )( )( ) ( )
( ) ( )
( )( ) ( )
( )( ) ( )
( ) ( )( ) ( )( ) ( )
( ) ( ) ( ) ( )
1
1
0 1 1 0
2 21 11 2
1 2
2 1
ˆˆ Σ
Σ , 2 2
,
T
V V V V
k X λ Xk X λ X
λ X k X
∆= + +
∆
+ −= ∆ =
− = ∆ = +
Q Q
Q1 Q 1 W Q
Q
Q Q Q QQ Q
Q QW Q Q Q Q
Q Q
H
V1
V0
H00
H11
H11
H00
2|H01|
Methods
Ecole Thématique – ModPhysChem – Oct. 2017
83
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 the conical intersection)
( )( )
( )( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )( )
( )( )
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 Q
Q Q Q
Methods
Ecole Thématique – ModPhysChem – Oct. 2017
84
Potential energyand derivatives
Database
Quantum
Chem
istry
R1R2
LocalHarmonicApproximation
Interface quantum chemistry – quantum dynamics
Quantum
Dynam
ics
Methods
( ) ( )1 2,j j
R t R t
( ){ }( ){ }
,j
j
g R t
A t
�
( ){ }( ){ }
, \
\
j
j
g R t t
A t t
+
+
�
Ecole Thématique – ModPhysChem – Oct. 2017
� 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)
85
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 =∑∑Ecole Thématique – ModPhysChem – Oct. 2017
� 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
86
Ab initio Multiple Spawning (AIMS)
Methods
( ) ( ) ( ) ( ) ( ), , ;s s
j js j
ψ R t A t g R t s R=∑∑� � �
Ecole Thématique – ModPhysChem – Oct. 2017
� 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)
87
Semiclassical trajectories
Methods
( ) ( ) ( ) ( ) ( ){ }, ,k k
kψ R t R t V t→� � �
( ) ( )( ) ( )( )
( ) ( ) ( ) ( )( )
201
1 0
11 00
2exp
k
k
R tk
k
R R t
Hπ
P tV t H H
→
= −
⋅∇ −
�
� �
� �ℏ
Ecole Thématique – ModPhysChem – Oct. 2017
a. Pyrazine Absorption Spectrum(Grid-Based Quantum Dynamics)
b. Formaldehyde Photodissociation(Direct Quantum Dynamics)
c. Thymine Photorelaxation(Semiclassical Dynamics)
89
Examples of Application
Ecole Thématique – ModPhysChem – Oct. 2017
� Franck-Condon approximation� Electric dipole approximation� Wavepacket evolution� Absorption spectrum: Fourier transform of the autocorrelation function
90
Electronic spectroscopy: absorption spectrum
Examples of Application
S0
S1
Q
E
ωℏ S0
S1
Q
E
S0
S1
Q
E
( ) ( ) ( ) ( ) ( )0
, where Ψ 0 ΨE
i ω t
C tω e tω t tσ dt C
+∞ +
−∞
= =∝ ∫ ℏ
E0
Ecole Thématique – ModPhysChem – Oct. 2017
� 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
91
Example 1: S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
( ) ( )0
σ ω cI ω I e
−= ℓ
Ecole Thématique – ModPhysChem – Oct. 2017
92
Example 1: 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 \ 0ˆ0 1 0 \2
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
Ecole Thématique – ModPhysChem – Oct. 2017
Using all 24 degrees of freedom� spectrum converged vs. experiment
93
Example 1: S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
Ecole Thématique – ModPhysChem – Oct. 2017
� 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
94
Example 2: S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Ecole Thématique – ModPhysChem – Oct. 2017
95
Example 2: S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
The S1 TS is directly connected to the S1/S0 conical intersection
Ecole Thématique – ModPhysChem – Oct. 2017
96
Example 2: S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
The probability to access S1 TS from the Franck-Condon region is not zero: let us start from there and check if we can control the H + HCO : H2 + CO branching ratio with the initial momentum of the wavepacket
� parallel to transition vector with varying sign and size
4 Gaussian functions in the wavepacketexpansion
Look at the most representative trajectory
Extrapolated direction of the transition vector
Negative momentum:go back to reactant
Zero momentum:form H2 + CO
Positive momentum:form H + HCO
Ecole Thématique – ModPhysChem – Oct. 2017
97
Example 2: S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Better-converged result with 8 Gaussian functions
Ecole Thématique – ModPhysChem – Oct. 2017
98
Example 2: S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Distribution of products (final geometry and electronic state)
Ecole Thématique – ModPhysChem – Oct. 2017
� Excitation to the lowest π,π* state � Competition between various pathways� Direct decay from π,π* to ground state through (Eth)X conical intersection� Indirect decay through π,π* minimum, π,π* TS, and π,π*/ n,π* (dark state)
conical intersection
99
Example 3: S2/S1/S0 thymine photorelaxation (semiclassical)
Examples of Application
Ecole Thématique – ModPhysChem – Oct. 2017
100
Example 3: S2/S1/S0 thymine photorelaxation (semiclassical)
Examples of Application
Semiclassical dynamics study� From Franck-Condon: trajectories get trapped in S1 minimum for more
than 500 fs (too long for simulations to be stable + ab initio level possibly too low)
� From S2(π,π*) TS (towards S2(π,π*)/S1(n,π*) conical intersection)
Missed hops at 5 and 16 fs(oscillation around the seam)
Successful hop at 28 fs
Oscillation of C4−O8 bond length (coordinate connecting TS and conical intersection)
Critical value at the conical intersection
Trajectory ending up on S1(n,π*)
Ecole Thématique – ModPhysChem – Oct. 2017
101
Example 3: S2/S1/S0 thymine photorelaxation (semiclassical)
Examples of Application
Trajectories restarted on S1 after S2/S1 hops
C4−O8 now remains ~ 1.2 Å
Coordinates inducing the S1/S0 crossing:C5−C6 stretched up to 2 Å then compressed backMethyl bent out of the plane (100°-120°)
Example from S1(π,π*)
Ecole Thématique – ModPhysChem – Oct. 2017
102
Example 3: S2/S1/S0 thymine photorelaxation (semiclassical)
Examples of Application
� 14 trajectories started from S2(π,π*) TS and restarted on S1
after S2/S1 hops
� 9 end up on S1(π,π*), then decay to S0 within 15-200 fs
� 5 populate the S1(n,π*) dark state, 3 of them decay to S0
within 10-100 fs (the other 2 stay too long for the calculation to be stable)
� The indirect decay pathway seems about as fast as the direct pathway
� The slow decay component may be due to population trapped in the S1(n,π*) dark state
Ecole Thématique – ModPhysChem – Oct. 2017
6. Conclusions and Outlooks
103
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...)
Ecole ThématiqueModPhysChemOctobre 2017
Outlooks: attophysics and attochemistry
104
Neutral
State 1
State 2
Attosecondpulse
Cation
Conclusions and Outlooks
Ecole Thématique – ModPhysChem – Oct. 2017
105
Announcement: 16-ICQC (2018) and satellites
Event
Menton, France, 18-23 June, 2018, Palais de l’EuropeWebsite: icqc16.sciencesconf.orgContact: [email protected]
Chair: Odile EisensteinPre-registration is open! (no payment to be made yet)
Ecole Thématique – ModPhysChem – Oct. 2017