Effective-mode approach to multidimensional
vibronic-coupling problems
Etienne Gindensperger
Laboratoire de Chimie Quantique, Institut de Chimie, Université Louis Pasteur, Strasbourg
E. Gindensperger – La Grande Motte – 02/2008
- p. 1
Content
● Content
■
Effective-mode approach
Fluorobenzene cation
Conclusion
■
■
Effective-mode approach
◆ conical intersections and system-environment complexes
◆ Model Hamiltonian
◆ Construction of a hierarchy of effective Hamiltonians
◆ Sequential dynamics
◆ Illustrations
Some applications on the fluorobenzene radical cation.
Conclusion
E. Gindensperger – La Grande Motte – 02/2008
- p. 2
Conical intersections
● Content
■
CIs are particular topologies of potential energy surfaces:
Effective-mode approach
● Conical intersections
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
adiabatic states are degenerate along the
intersection seam → kinetic couplings
diverge.
◆
diabatic representation is appropriate:
transforms kinetic couplings into -smoothpotential couplings.
!
W11 (q) W12 (q)
H = TN 1 +
W21 (q) W22 (q)
2
1.5
1
0.5
0
● Illustration: model complex
● Illustration: results
◆
-0.5
● "Important" modes
● HRE
Fluorobenzene cation
4
vg
3
2
Conclusion
1
0
-1
-2 -10
-5
adiabatic surfaces
E. Gindensperger – La Grande Motte – 02/2008
0
vu
5
10
- p. 3
System-environment complexes
● Content
Effective-mode approach
macrosystems ⇔ SYSTEM - ENVIRONMENT complexes
SYSTEM = a few modes supposed to dominate the dynamics.
what about the collective effects of the ENVIRONMENT on the
SYSTEM
?
● Conical intersections
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
● "Important" modes
● HRE
SYSTEM
(few modes)
l
ENVIRONMENT
(many modes)
Fluorobenzene cation
Conclusion
E. Gindensperger – La Grande Motte – 02/2008
- p. 4
Model Hamiltonian
■
Hamiltonian of the complex (diabatic) : H = HS + HB
● Content
Effective-mode approach
◆
● Conical intersections
SYSTEM
(NS modes) :
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
HS =
● Hierarchy
● Hierarchy
E1 + TS + W11
W21
W12
E2 + TS + W22
!
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
● "Important" modes
◆
● HRE
Fluorobenzene cation
ENVIRONMENT
(NB modes):
NB
X ωi 2
(pi + qi2 )1 +
HB =
2
i=1
Conclusion
PNB (1)
κi qi
Pi=1
NB
i=1 λi qi
!
PNB
λi qi
.
PNi=1
(2)
B
i=1 κi qi
HB is described by the linear vibronic coupling model (LVC).
(H. Köppel et al., Adv. Chem. Phys. 57, 59 (1984))
■
Idea: constructing effective modes for the environment.
transform HB = H1 + Hr1
E. Gindensperger – La Grande Motte – 02/2008
- p. 5
Effective Hamiltonian
Effective Hamiltonian H1 (3 modes) :
● Content
3
X
Ωi 2
H1 =
(Pi + Q2i )1
2
i=1
!
P3
(1)
(1) P3
λ̄ i=1 Λi Qi
κ̄
i=1 Ki Qi
+
P
P
(2)
λ̄ 3i=1 Λi Qi
κ̄(2) 3i=1 Ki Qi
Effective-mode approach
● Conical intersections
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
Residual Hamiltonian Hr1 (NB − 3 modes):
● "Important" modes
● HRE
Hr1
Fluorobenzene cation
Conclusion
■
■
■
NB
NB
3 X
X
X
Ωj 2
dij (Pi Pj + Qi Qj )1
(Pj + Q2j )1 +
=
2
i=1 j=4
j=4
Hr1 diagonal: does not couple the electronic states.
H1 contains all the "topology" of the environment.
Unitary transformation: physics unchanged.
L.S. Cederbaum, E. Gindensperger, and I. Burghardt, Phys. Rev. Lett. 94, 113003, (2005)
E. Gindensperger – La Grande Motte – 02/2008
- p. 6
moments, dynamics and spectra
● Content
Effective-mode approach
Approximation: remplace H = HS + HB par H ′ = HS + H1 .
NS + NB modes → NS + 3 modes !
Quantum dynamics manageable.
● Conical intersections
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
This approximation is related to the moments of H.
■ autocorrelation function: P (t) = h0|e−iHt |0i.
P∞ (−it)k
■ moments : P (t) =
Mk ; Mk = h0|H k |0i
k=0
k!
■
spectrum: Fourier transform of P (t)
● "Important" modes
● HRE
Fluorobenzene cation
Conclusion
H ′ reproduce exactly the moments M0 , . . . , M3 of H.
→ short-time dynamics and band shapes of macrosystems.
E. Gindensperger, I. Burghardt, and L.S. Cederbaum, J. Chem. Phys. 124, 144103 and 144104 (2006).
E. Gindensperger – La Grande Motte – 02/2008
- p. 7
Hierarchy of effective Hamiltonians
■
● Content
■
Effective-mode approach
■
● Conical intersections
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
● Hierarchy
short-time dynamics → H1 is enough
dynamics at longer times → include Hr1 (?)
idea: constructing additional effective modes for Hr1 :
"
!
NB
NB
3
X Ωj 2
X
X
2
Hr1 =
Pi
dij Pj + Qi
(Pj + Qj )1 +
2
j=4
i=1
j=4
NB
X
j=4
dij Qj
!#
1
● Sequential dynamics
● molecular complex
● Illustration: model complex
■
thus Hr1 (NB − 3) = H2 (3) + Hr2 (NB − 6)
Fluorobenzene cation
■
Conclusion
■
property: Hr2 have the same mathematical form as Hr1 !
successive transformations: Hr2 (NB − 6) = H3 (3) + Hr2 (NB − 9), etc.
● Illustration: results
● "Important" modes
● HRE
E. Gindensperger, H. Köppel, et L.S. Cederbaum, J. Chem. Phys. 126, 034106 (2007).
E. Gindensperger – La Grande Motte – 02/2008
- p. 8
Hierarchy of effective Hamiltonians
■
full transformation of H:
● Content
H = HS +
Effective-mode approach
N
X
Hm
m=1
● Conical intersections
● molecular complex
with N the total number of effective Hamiltoniens, with
● Model Hamiltonian
● Effective Hamiltonian
● Properties
● Hierarchy
H1 =
● Hierarchy
● Sequential dynamics
3
X
i=1
● molecular complex
● Illustration: model complex
Ωi 2
(Pi + Q2i )1 +
2
and Hm , m > 1,
● Illustration: results
● "Important" modes
P3
(1)
κ̄
i=1 Ki Qi
P3
λ̄ i=1 Λi Qi
(1)
!
P3
λ̄ i=1 Λi Qi
(2)
(2) P3
κ̄
K
i Qi
i=1
● HRE
Fluorobenzene cation
Hm =
Conclusion
3m
X
j=3(m−1)+1
Ωj 2
(Pj + Q2j )1 +
2
3(m−1)
X
3m
X
dij (Pi Pj + Qi Qj )1
i=3(m−2)+1 j=3(m−1)+1
each Hm is comprised of 3 effective modes only and couples to Hm−1 .
■ only H1 couples directly to HS , via the electronic subsystem.
Approach valid for nel states → each Hm is comprised of nel (nel + 1)/2 modes
■
E. Gindensperger – La Grande Motte – 02/2008
- p. 9
Sequential dynamics
■
in practice: truncate the hierarchy at the order n.
● Content
H = HS +
Effective-mode approach
● Conical intersections
● Model Hamiltonian
● Properties
■
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
● "Important" modes
● HRE
moment analysis:
Pn
theorem: using the Hamiltonian truncated at the order n, HS + m=1 Hm ,
suffices to reproduce exactly all the moments of the full macrosystem up to
the order 2n + 1 included.
Analytical proof, valid for nel electronic states, whatever HS is, and for an
arbitrary large value of NB .
E. Gindensperger and L.S. Cederbaum, J. Chem. Phys. 127, 124107 (2007)
Fluorobenzene cation
Conclusion
Hm
m=1
● molecular complex
● Effective Hamiltonian
n<N
X
■
sequential dynamics: each Hm comes into play at a later time than Hm−1 .
E. Gindensperger – La Grande Motte – 02/2008
- p. 10
System-environment complexes
The
H AMILTONIANS subsumes all the environment
on a given time-scale.
HIERARCHY OF EFFECTIVE
● Content
Effective-mode approach
● Conical intersections
SYSTEME
● molecular complex
● Model Hamiltonian
(few modes)
● Effective Hamiltonian
● Properties
l
● Hierarchy
● Hierarchy
● Sequential dynamics
PRIMARY
● molecular complex
● Illustration: model complex
● Illustration: results
EFFECTIVE MODES
● "Important" modes
l
● HRE
Fluorobenzene cation
Conclusion
SECONDARY
EFFECTIVE MODES
..
.
3
Q, P
3
Q, P
E. Gindensperger – La Grande Motte – 02/2008
...
- p. 11
Illustration: model complex
2
■
1.5
model complex:
2 states – 22 modes
◆ system, HS : prototype of CI
(2 modes νg , νu )
◆ environment, HB : 20 modes
vg
(weakly coupled)
calculations up to 22 dimensions (MCTDH).
1
0.5
0
● Content
-0.5
Effective-mode approach
● Conical intersections
4
● molecular complex
● Model Hamiltonian
● Effective Hamiltonian
● Properties
■
● Hierarchy
3
2
1
0
-1
-2 -10
0
-5
vu
10
5
system (adiabatic surfaces)
● Hierarchy
● Sequential dynamics
● Illustration: results
● "Important" modes
● HRE
P(t)
Fluorobenzene cation
sys + 20
sys
0.6
P(E)
● Illustration: model complex
Conclusion
PE spectrum (40 meV)
autocorrelation function (state 1)
1
sys + 20
sys
0.8
● molecular complex
0.4
0.2
0
9.5
0
E. Gindensperger – La Grande Motte – 02/2008
50
100
Time [fs]
150
200
10
10.5
11
11.5
Energy [eV]
- p. 12
Illustration: results
1
(40 meV)
● Content
0.5
Effective-mode approach
● Conical intersections
● molecular complex
0
● Model Hamiltonian
● Effective Hamiltonian
● Properties
0.1
● Hierarchy
● Hierarchy
(20 meV)
0.05
● molecular complex
[arb. unit]
● Sequential dynamics
0
● Illustration: model complex
1
● Illustration: results
● "Important" modes
● HRE
Fluorobenzene cation
0.5
Conclusion
(10 meV)
0
0.1
9.9
10
0.05
0
0
20
40
60
80
100
Time [fs]
E. Gindensperger – La Grande Motte – 02/2008
120
140
160
180
9.5
10
10.5
11
11.5
Energy [eV]
- p. 13
Effective vs. "important" modes
● Content
Comparison of the effective-mode and "conventional" approaches.
Effective-mode approach
1
● Conical intersections
sys + 20
sys + 3 eff
sys + 10
sys + 5
sys
● molecular complex
● Model Hamiltonian
0.8
● Effective Hamiltonian
● Properties
● Hierarchy
P(t)
● Hierarchy
● Sequential dynamics
● molecular complex
0.6
0.4
● Illustration: model complex
● Illustration: results
● "Important" modes
0.2
● HRE
Fluorobenzene cation
0
Conclusion
0
10
20
30
40
50
Time [fs]
To truncate the original environment does not reproduce even the low-order
moments!
E. Gindensperger – La Grande Motte – 02/2008
- p. 14
Hartree residual environment approximation
The new form of H
● Content
H = HS +
Effective-mode approach
● molecular complex
● Effective Hamiltonian
autocorrelation function
Hm + Hrm .
m=1
● Conical intersections
● Model Hamiltonian
n
X
0.3
allows new approximations.
0.2
Example: use a single configuration for the
modes entering Hrm (TDH approximation):
!
(1)
(1)
ΨS+ef f ΨRE
Ψ(t) =
(2)
(2)
ΨS+ef f ΨRE
0.1
● Properties
● Hierarchy
● Hierarchy
● Sequential dynamics
● molecular complex
● Illustration: model complex
● Illustration: results
● "Important" modes
● HRE
Fluorobenzene cation
Conclusion
0
0.03
0.02
- very small numerical requirement
- straightforward implementation in MCTDH
0.01
0
0
E. Gindensperger – La Grande Motte – 02/2008
20
40
60
Time [fs]
80
100
- p. 15
Fluorobenzene radical cation
Experimental finding: NO fluorescence in the first excited states of FBz+ .
However, optical transitions are allowed.
● Content
Effective-mode approach
Fluorobenzene cation
● Fluorobenzene radical cation
● Hamiltonian
● Linear and quadratic models
● XA subsystem
● BCD subsystem
● XABCD - population
dynamics
● XABCD - PE spectrum
Conclusion
→ Ultrafast electronic relaxation process.
The 5 lowest electronic states of the cation,
X̃ 2 B1 , Ã2 A2 , B̃ 2 B2 , C̃ 2 B1 , D̃2 A1 ,
are divided in two bands: X̃ − Ã and B̃ − C̃ − D̃.
Photoelectron spectrum of FBz (G. Bieri et al., J. Electron Spectr. Relat. Phenom. 13, 281 (1981))
12
E. Gindensperger – La Grande Motte – 02/2008
10
Energie [eV]
8
- p. 16
Hamiltonian
5-state Hamiltonian, 19 modes (out of 30).
H = (TN + V0 ) 1 +
EX + κX Q + gX Q2
B
B
B
X
B
(X,A)
B
Qj
λ
B
j
B
j∈B
B
2
B
B
0
B
B
B
B
B
B
0
B
B
B
B
@
0
0
X
j∈B2
(X,A)
Qj
λ
j
0
EA + κA Q + gA Q2
X
j∈B1
X
j∈B1
(A,B)
λ
Qj
j
0
(A,B)
Qj
λ
j
EB + κB Q + gB Q2
X
0
(B,C)
Qj
λ
j
j∈A2
X
(B,D)
Qj
λ
j
j∈B2
0
0
X
j∈A2
(B,C)
λ
Qj
j
EC + κC Q + gC Q2
X
j∈B1
(C,D)
Qj
λ
j
1
0
0
X
(B,D)
λ
Qj
j
j∈B2
X
(C,D)
Qj
λ
j
j∈B1
ED + κD Q + gD Q2
with
κα Q ≡
X
i∈A1
(α)
Qi ,
κ
i
gα Q
2
≡
X
i∈A1
(α) 2
Qi .
g
i
ab initio data : J. Franz.
3 examples: XA, BCD, and XABCD.
E. Gindensperger – La Grande Motte – 02/2008
- p. 17
C
C
C
C
C
C
C
C
C
C
C,
C
C
C
C
C
C
C
C
C
A
Linear and quadratic models
Linear vibronic model
Köppel, Chem. Phys. 329, 65 (2006)).
(I. Bâldea, J. Franz, P. Szalay, and H.
16
~2
D A1
● Content
Effective-mode approach
Fluorobenzene cation
→ CI(Ã − B̃) inactive.
14
o
o o
~2 o
C B1
12
● Fluorobenzene radical cation
● Hamiltonian
~2
A A2
● Linear and quadratic models
● XA subsystem
———–
● BCD subsystem
10
● XABCD - population
dynamics
● XABCD - PE spectrum
Conclusion
-6
Linear model augmented by the quadratic
contributions for the totally symmetric modes
~2
B B2
~2
X B1
o
-4
-2
0
2
4
8 Qeff
6
16
~2
D A1
(E. Gindensperger, I. Bâldea, J. Franz, et H. Köppel, Chem. Phys. 338, 207 (2007))
14
o
→ CI(Ã − B̃) energetically accessible.
o o
~2
A A2
10
-6
E. Gindensperger – La Grande Motte – 02/2008
o
~2
C B1
12
-4
o
~2
B B2
~2
X B1
o
-2
0
2
4
6
- p. 18
8 Qeff
XA subsystem
● Content
Effective-mode approach
3-mode system (2 tuning, 1 coupling)
+ 12 modes "environment" (all LVC).
A
Fluorobenzene cation
● Fluorobenzene radical cation
XA - population of state A
● Hamiltonian
● XA subsystem
● BCD subsystem
exact
Hs + H 1
Hs + H 1 + H 2
Hs + H 1 + H 2 + H 3
0.9
● XABCD - population
dynamics
● XABCD - PE spectrum
0.8
Conclusion
0.6
intensity [arb. unit]
● Linear and quadratic models
1
0.7
B
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time [fs]
150
200
11.5
11
10.5
energy [eV]
E. Gindensperger – La Grande Motte – 02/2008
- p. 19
BCD subsystem
● Content
no system’s modes ; 19-mode environment (LVC)
Effective-mode approach
Fluorobenzene cation
BCD - population of state D
1
exact
H1
H1 + H 2
H1 + H 2 + H 3
● Fluorobenzene radical cation
● Hamiltonian
● Linear and quadratic models
● XA subsystem
● BCD subsystem
● XABCD - population
dynamics
● XABCD - PE spectrum
Conclusion
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
Time [fs]
E. Gindensperger – La Grande Motte – 02/2008
- p. 20
XABCD - population dynamics
XABCD - system(2) + effective modes(10)
1
X
A
B
C
D
0.8
● Content
Effective-mode approach
Fluorobenzene cation
● Fluorobenzene radical cation
● Hamiltonian
2-mode system (2 tuning, QVC)
17-mode environment (LVC).
———————————————12-mode system (QVC)
0.6
0.4
● Linear and quadratic models
● XA subsystem
0.2
● BCD subsystem
● XABCD - population
dynamics
● XABCD - PE spectrum
0
0
50
16
100
Time [fs]
150
200
XABCD - system(12)
Conclusion
~2
D A1
1
X
A
B
C
D
14
o
o o
~2 o
C B1
12
~2
A A2
10
0.8
o
~2
B B2
0.6
~2
X B1
o
0.4
-6
-4
-2
0
2
4
6
8 Qeff
0.2
0
0
E. Gindensperger – La Grande Motte – 02/2008
50
100
Time [fs]
150
- p. 21
200
XABCD - PE spectrum
● Content
Effective-mode approach
Fluorobenzene cation
● Fluorobenzene radical cation
● Hamiltonian
● Linear and quadratic models
● XA subsystem
● BCD subsystem
● XABCD - population
dynamics
● XABCD - PE spectrum
Conclusion
14
E. Gindensperger – La Grande Motte – 02/2008
13
12
11
Energy [eV]
10
9
8
- p. 22
Some remarks
● Content
Effective-mode approach
Fluorobenzene cation
- product form of H → adapted for MCTDH (original and transformed)
- best convergence criterium: compare results for original and fully transformed
H.
Conclusion
● Some remarks
● Thanks to
The effective-mode approach allows to compute the quantum dynamics of truly
large molecular complexes involving CIs on a given time-scale.
E. Gindensperger – La Grande Motte – 02/2008
- p. 23
Thanks to
● Content
■
■
co-workers
◆ Pr. Lorenz S. Cederbaum (Heidelberg)
◆ Dr. Irene Burghardt (CNRS/ENS, Paris)
◆ Pr. Horst Köppel (Heidelberg)
◆ Pr. H.-Dieter Meyer (Heidelberg)
and
◆ M. Basler (HRE)
◆ Dr. J. Franz (FBz+ )
◆ Dr. I. Bâldea (FBz+ )
Alexander von Humboldt Stiftung (www.avh.de)
DFG – Deutschen Forschungsgemeinschaft (www.dfg.de)
CNRS (www.cnrs.fr)
■
... and Fabien Gatti for inviting me.
Effective-mode approach
Fluorobenzene cation
Conclusion
● Some remarks
● Thanks to
■
■
E. Gindensperger – La Grande Motte – 02/2008
- p. 24