Direct quantum dynamics using
variational Gaussian wavepackets.
Application to laser-driven control of
benzene photochemistry.
Benjamin Lasorne,a Fabrizio Sicilia,a Graham Worth,c
Michael J. Bearpark,a Lluís Blancafort,b
and Michael A. Robb.a
Department of Chemistry, Imperial College London,
South Kensington, London SW7 2AZ, UK
Institut de Química Computacional and Departament de Química,
Universitat de Girona, E-17071 Girona, Spain
c School of Chemistry, University of Birmingham, Edgbaston,
Birmingham B15 2TT, UK
a
b
Standard MCTDH for non-adiabatic organic photochemistry?
Vibronic coupling Hamiltonian model
 Size? Benchmark: pyrazine: 10 atoms (24D) / 3 coupled electronic states
[Raab et al. JCP (1999) 110:936]
 Photochemistry? No, spectroscopy
 Limitation? Local expansion for small amplitude motions
 Why? Global PESs and NACs on a large grid: difficult and expensive
Possible strategy: direct dynamics with Gaussian WPs
‘On-the-fly’ calculation of coupled PESs
 How? Centre of localised functions: trajectories  moving local expansions
 Applications:
Butatriene+: red. dim. (4D) & full dim. (18D)  comparison vibronic model vs. direct
[Worth et al. Farad. Discuss. 127 (2003) 307]
NOCl: full dim. (9D Cart. or 3D vib.)  comparison analytic vs. direct
[Lasorne et al. PCCP 9 (2007) 3210]
Benzene: red. dim. (5D, 9D, 11D)  S1/S0 photochemistry
Formaldehyde: full dim. (6D)  S1/S0 photochemistry
Thymine: full dim. (39D)  S2/S1 photochemistry
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
[Lasorne et al. JCP (2008) in press]
[PhD thesis of Marta L. Araujo]
[PhD thesis of David Asturiol Bofill]
2
Methodology
 Direct quantum dynamics
 Automatic generation of active coordinates
for photochemistry
Application
 S1/S0 benzene photochemistry
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
3
Methodology
The vMCG ansatz
(variational Multi-Configuration Gaussian wavepacket)
 G-MCTDH with only parameterised Gaussian functions as SPFs
(Cf. Graham Worth’s talk)
( s)
( s)
! ( x,t ) = " Aj (t ) g j
j
(x,t) s
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
4
Methodology
The vMCG equations of motion
 Variational, time-dependent basis set (~ SPFs):
frozen Gaussian functions (non-orthogonal!)
2
(
,
*
*
s
s)
(
)
(
"
%
*
*
x
!
x
t
p
t
x
(
)
(
)
$
'
*
*
s
()
j
j
*
*
g j ( x,t ) = N exp )! $
' +i
*
2!
! *
$
'
*
*
$#
'&
*
*
*
*
+
.
 Evolution of all coefficients and parameters are coupled
⇒ ‘quantum trajectories’
( s)
Aj (t )
{x
( s)
j
(t),p (t)}
( s)
j
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
5
Methodology
The vMCG Hamiltonian
Easy thanks to
!2
T̂ = "!r 2
 Rectilinear coordinates
!x r
r
!
 Local harmonic approximation
V !# x j (t )$& ,V ' !# x j (t )$& ,V '' !# x j (t )$&
"
%
"
%
"
%
⇒ Analytical integrals: Gaussian moments
gk (t ) x n g j (t )
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
n=0,1,2,3
6
Methodology
Direct dynamics implementation
S1
S1
 Diabatic picture for dynamics
S0
S0
 On-the-fly adiabatic PESs (quantum chemistry)
!Ĥ
# 00
#
#" Ĥ10
!V ( x )
$
0
Ĥ01$& !T̂ 0 $
&U
† # 0
#
&
=
+
U
#
&
& #
&
Ĥ11 &% " 0 T̂ %
V1 ( x )&
#" 0
%
 Diabatic transformation (U matrix):
regularised model (Köppel et al.)
V00
V1
V11
2V01
V0
[On-the-fly U? to be done]
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
V11
V00
7
Time t :
! ( x,t ) = " Aj (t ) g j ( x,t )
j
g j ( x,t )
Local
Harmonic
Approximation
s
n
u
r
l
e
ll
a
r
Pa
}
j
{
x1
x2
! x (t ),x (t )$
2j
#" 1j
&%
GAUSSIAN
Database
Methodology
{A (t + !t)}
vMCG
{g (x,t + !t)}
j
j
Potential energy,
gradient, Hessian
(for each el. state)
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
8
Methodology
Automatic generation of
active coordinates for photochemistry

FC point (pseudo-branching plane):
gradient difference (x1) // breathing mode
(driving force)
derivative coupling (x2) // Kékulé mode
(2nd-order JT effect)

CoIn point (branching plane):
gradient difference (x1) // prefulvenic mode
derivative coupling (x2) // ~ Kékulé mode

Need quadratic expansion at the FC point
W2
FC
!E = !H + 4W " !H + 2
!E
>0
FC
!E
%
'
1 28
FC
'
!H " !E + !!Q x + $ !"i Q i2
'
1
2 i=1
#'
&
'
SS
'
W " ! 1 0Qx
'
2
'
(
2
2
(
)
prefulvene
E
benzene
x1
x2
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
9
!E " !E FC + !!Q x +
1

1
!"i Q i2
#
2 i=1
28
! )
(
+2
S1S0
!E
Methodology
2
FC
Q x2
Photoactive modes :
 3 out-of-plane modes
(chair, boat, twist)
on C6 rather than C6-H
2
2nd-order terms (3 cases):
4
E
ΔE ~ constant
E
FC
ΔE decreases
FC
IC
Bath mode
E
ΔE increases
16
16
 4 in-plane modes
(rectangle, parallelogram,
trapezes)
Photoactive mode
6
6
18
18
FC
Photo-inactive mode
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
10
Application
Application: S1 benzene photochemistry (channel 3)
254 nm
S0 (1A1g) → S1 (1B2u)
benzene
Vibrational excess
~3000 cm-1
⇒ fluorescence
extinction
prefulvenoid
geometries
TS
benzvalene
S1
Internal conversion
via S1/S0 conical intersection
S0
prefulvenoid
plateau
reaction mode (Cs-preserving)
distortion modes
(Cs-breaking)
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
The quantum
yield
of benzvalene
is wavelength
dependent
but remains
weak
0.016 @ 253 nm
0.037 @ 237 nm
11
Application
Objective: internal conversion yield / selectivity

How? Targeting specific regions of the S1/S0 intersection seam
S1
S1
S1
S1
S1
S0
prefulvene
S0
benzene
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
S0
benzvalene
12
Application
Reactive directions: photoactive modes
reactive path
nonreactive
path
FC
S1/S0 intersection seam
E
Qphotoactive
Q1
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
13
Application
5D model: role of the out-of-plane modes
Mode 1: breathing (1a1g) D6h → D6h
 // Franck-Condon gradient (S1)
(driving force: initial motion)
Mode 15: Kékulé (1b2u) D6h → D3h
 // derivative coupling
(2nd-order Jahn-Teller effect)
Out-of-plane modes 4 and 16: chair (1b2g) + boat/twist (1e2u 2e2u)
combination  half-chair (prefulvenoid)
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
14
Application
Searching for the optimal direction
S1
S0
Optimal direction:
 out-of-plane modes
 target prefulvenoid geometries
(TS and CoIn)
 breathing mode
 target TS
(to avoid bobsleigh effect)
S0 ← S1 population transfer:
50% 32% 33% 32%
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
15
Application
Effect of the number of GWPs in the basis set
Previous case with the 33 %
population transfer: 1 GWP on
each electronic state
Comparison: 5 GWPs per state
 Very little influence before the
non-adiabatic transition
 After: oscillations in the
population (structure in the
global WP), and spreading and
slowing down for some
coordinates
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
16
Application
Effect of the in-plane modes
Previous case with the 33 % population
transfer in 5D
Comparison: 9D, same momentum along
modes 1, 4 et 16x, additional components
along 6x et 18x (prefulvenoid target)
6x
6y
18x
18y
 Direction: more refined
 Potential energy: less steep
(better description after the crossing)
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
17
Application
Simulation: shape of the initial wavepacket
S1 ← S0 transition is forbidden:
Herzberg-Teller selection rules
 Various S1 vibrational states can be populated
 possible excitation of non-totally symmetric
vibrations (photoactive modes)
 Modulation of the initial wavepacket:
 effect of a laser pulse (control)
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
18
Application
Herzberg-Teller initial wavepacket
Stationary state
v !!;S0
Laser pulse Free evolution of the wavepacket
! (t = 0) = # cv " v ";S1 $ ! (t )
Energy
Laser pulse spectrum
Energy
t = –T
v"
t=0
Example with two S1 vibrational states:
v ! = 2;S1
phase
amplitude
t
v ! = 1;S1
! (t = 0) = T2 e 2 c2ref v " = 2;S1
v ! = 0;S1
+T0 e 0 c0ref v " = 0;S1
i!
i!
!
#transmission factors T0 ,T2
control parameters: #
"
#
#
$phase delays !0 ,!2
v !! = 0;S0
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
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Application
[…]
4
16x
16y
[…]
[…]
[T. A. Stephenson, P. L. Radloff and S. A. Rice, J. Chem. Phys. 81 (1984) 1060]
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
20
Application
Influence on population transfer
4
1
2
2
Initial wavepacket: a 6 x,y 4 + b 16 xy ,x 2 !y 2
16x
16y
 36 combinations tested
 Different responses: possible control  being tested
experimentally
1
2
1
3
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
61x 4 2 ! i
61y 4 2 !
1
2
162x 2 !y 2
2
162xy
3
21
Conclusions
 Limitation: accuracy
quantum chemistry level
local harmonic approximation
constrained basis set
 Applicability to organic photochemistry
large number of degrees of freedom
large amplitude motions
grid free
 Chemically relevant sampling
database (~ 25000 geometries)  interpolation
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
22
Acknowledgements
Computational photochemistry group 
of Prof. Michael A. Robb
and Dr. Michael J. Bearpark
(Imperial College, London)
Energy difference analysis: Dr. Fabrizio Sicilia (PhD thesis)
+ coll. Dr. Lluís Blancafort (Universitat de Girona)
Direct dynamics vMCG: group of Dr. Graham A. Worth
(University of Birmingham)
+ coll. Dr. Irène Burghardt (ENS, Paris)
Experiments: group of Prof. Helen H. Fielding
(University College, London)
Funding:
The Engineering and Physical Sciences Research Council
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
23
References
Direct dynamics vMCG:
G. A. Worth, M. A. Robb and I. Burghardt, Faraday Discuss. 127 (2004) 307
B. Lasorne, M. J. Bearpark, M. A. Robb and G. A. Worth, Chem. Phys. Lett. 432 (2006) 604
B. Lasorne, M. A. Robb and G. A. Worth, Phys. Chem. Chem. Phys. 9 (2007) 3210
G. A. Worth, M. A. Robb and B. Lasorne, Mol. Phys. (2008) submitted
Energy difference analysis:
F. Sicilia, L. Blancafort, M. J. Bearpark and M. A. Robb, J. Phys. Chem. A 111 (2007) 2182
F. Sicilia, M. J. Bearpark, L. Blancafort and M. A. Robb, Theor. Chem. Acc. 118 (2007) 241
Benzene:
B. Lasorne, M. J. Bearpark, M. A. Robb and G. A. Worth, in B. Lasorne and G. A. Worth (Ed.),
Coherent Control of Molecules (2006, CCP6, Daresbury)
B. Lasorne, F. Sicilia, M. J. Bearpark, M. A. Robb, G. A. Worth & L. Blancafort, J. Chem. Phys. (2008) in press
B. Lasorne, M. J. Bearpark, M. A. Robb and G. A. Worth, J. Phys. Chem. A (2008) to be submitted
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
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Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
25
Additional information…
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
26
Methodology
At the Franck-Condon point

1st-order description:
gradient difference (x1) // breathing mode
= S1 gradient  driving force
~ mode 1 (a1g)
FC
CoIn
ΔE
!E " !E FC + !!Q x
1
Q x1
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
27
Methodology
At the conical intersection point

Degeneracy lifted at 1st-order along two directions (branching plane)
! E IC 0 $
# S0
&
#
&
# 0 E SIC &
"!###"###1$%
!H
# SS
' ' ' ' ' ' ''
( # 00
# HS S
" 10
geometrical displacement
frozen electronic states
HS S $&
0 1
&
HS S &
1 1 %
CoIn geometry
%
) diagonalisation
!E
$
# S0 0 &
#
&
# 0 ES &
"!###"###1$%
further away
!E = E S " E S = !H 2 + 4W 2
1
0
(!E
IC
=0
)
%!H = H " H $ !!Q
'
S1S1
S0 S0
x1
'
#'
&
SS
'
W = HS S = HS S $ ! 1 0 Q x
'
1 0
0 1
2
'
(
x1: !H gradient direction
x2 : W gradient direction
Example: S1/S0 benzene
prefulvene
x1:
S1− S0
gradient
difference
x2:
S1/S0
coupling
gradient
E
benzene
x1
x2
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
28
Methodology
From the FC point to the CoIn point

Generalisation of the 1st-order branching plane (x1, x2) at the FC point
+ 2nd-order representation of ΔE (quadratic expansion)
! E FC 0 $
# S0
&
#
&
# 0 E SFC &
"!####"###1#$%
!H
# S0S0
geometrical displacement
''
'
'
'
'
''
(
#
frozen electronic states
# HS S
" 10
HS S $&
0 1
&
HS S &
1 1 %
FC geometry
%
) diagonalisation
!E
$
# S0 0 &
#
&
# 0 ES &
!" ###"###1$%
further away

W2
!E = !H + 4W " !H + 2
!E FC > 0
FC
!E
'%'
1 28
FC
2
''!H " !E + !!Q x1 + $ !"i Q i
2 i=1
#&
''
SS
''W " ! 1 0 Q x2
(
2
2
(
)
x1: !H gradient direction
x2 : W gradient direction
i: normal directions of
the projected !E-Hessian
2nd-order Jahn-Teller effect (squared 1st-order term):
derivative coupling (x2) // Kékulé mode
Direct quantum dynamics using variational Gaussian wavepackets – La Grande Motte – 26/02/08
~ mode 15 (b2u)
29