Nonadiabatic Dynamics current methods and challenges Benjamin LASORNE [email protected]Department of Theoretical Chemistry Institut Charles Gerhardt CNRS – Université de Montpellier Summer School EMIE-UP Multiscale Dynamics in Molecular Systems August 2019
102
Embed
Lasorne Emie-Up 2019 - IRAMISiramis.cea.fr/meetings/Ecole_EMIE_UP/presentations/Lasorne.pdf · the promotion to the excited electronic state , if possible to the formation of products
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Nonadiabatic Dynamicscurrent methods and challenges
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
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
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
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
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
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
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
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
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
Summer School – Emie-Up – Aug. 2019
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
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
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
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
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
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)
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
49
Diabatisation by ansatz: smooth Hamiltonian matrix
� 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
∇∇ =
−
�
�
� � ��� ��
� �
Summer School – Emie-Up – Aug. 2019
61
Photostability vs. photoreactivity
� Internal conversion can regenerate the electronic ground state in the reactant or product regions
S0
Reaction coordinate
E
S1
Summer School – Emie-Up – Aug. 2019
Non-adiabatic processes
62
Role of the crossing topography
Non-adiabatic processes
Summer School – Emie-Up – Aug. 2019
63
Retinal photoreactivity
Isomerisation coordinate
Pote
nti
al energ
y
trans
cis
Energy transfer
NH+
2nd step
Introduction
Summer School – Emie-Up – Aug. 2019
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
Summer School – Emie-Up – Aug. 2019
4. Methods
65
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
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
Summer School – Emie-Up – Aug. 2019
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
� 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= =
=∑∑��������� �������������������
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
Summer School – Emie-Up – Aug. 2019
� 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
Summer School – Emie-Up – Aug. 2019
� 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
Summer School – Emie-Up – Aug. 2019
� 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
Summer School – Emie-Up – Aug. 2019
� 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
Summer School – Emie-Up – Aug. 2019
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
Summer School – Emie-Up – Aug. 2019
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
� 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=∑∑� � �
Summer School – Emie-Up – Aug. 2019
� 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)
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
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
Summer School – Emie-Up – Aug. 2019
� 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)
Examples of Application
( ) ( )0
σ ω cI ω I e
−= ℓ
Summer School – Emie-Up – Aug. 2019
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
Summer School – Emie-Up – Aug. 2019
Using all 24 degrees of freedom� spectrum converged vs. experiment
90
S2 S0 pyrazine absorption spectrum (MCTDH)
Examples of Application
Summer School – Emie-Up – Aug. 2019
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
� 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
Summer School – Emie-Up – Aug. 2019
93
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
The S1 TS is directly connected to the S1/S0 conical intersection
Summer School – Emie-Up – Aug. 2019
94
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Influence of initial conditions
Summer School – Emie-Up – Aug. 2019
95
S1/S0 formaldehyde photodissociation (DD-vMCG)
Examples of Application
Distribution of products (final geometry and electronic state)
Summer School – Emie-Up – Aug. 2019
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
� 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...)
Summer School EMIE-UPMultiscale Dynamics in Molecular Systems
Quantum Chemistry and Dynamics of Excited States: Methods and Applications
Edited by Leticia González and Roland
Lindh
(to be released, 2019)
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.