5 Field-Induced Surface-Hopping Method (FISH) for Simulation and Control of Ultrafast Photodynamics
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Fig. 17.5 Schematic representation of the photodynamics of furan obtained from nonadiabatic
dynamics. The S2 /S1 and S1 /S0 internal conversions and the corresponding time scales are shown
in red while the products in the ground state are indicated by black arrows (main product bold
arrow, other two products thin arrows). The minor channels have energies of 1.2 (II) and 2.1 eV
(III) above the main channel (I)
field has been described using perturbation theory these methods are limited only
to processes in relatively weak fields. For this reason, new theoretical approaches
for the simulation of dynamics driven by moderately strong laser fields (below the
multielectron ionization limit) are particularly desirable. Such fields open a rich
variety of pathways for the control of ultrafast dynamics in complex systems.
Therefore, we present here our semiclassical “Field-Induced Surface Hopping”
(FISH) method [59] for the simulation and control of the laser-driven coupled
electron-nuclear dynamics in complex molecular systems including all degrees of
freedom. It is based on the combination of quantum electronic state population
dynamics with classical nuclear dynamics carried out “on the fly”. The idea of the
method is to propagate independent trajectories in the manifold of adiabatic electronic states and allow them to switch between the states under the influence of the
laser field. The switching probabilities are calculated fully quantum mechanically.
The application of our FISH method will be illustrated in Sect. 17.6 on the example
of optimal dynamic discrimination (ODD) of two almost identical flavin molecules.
The starting point for the description of laser-driven multistate dynamics is the
semiclassical limit of the Liouville-von Neumann (LvN) equation for the quantum
mechanical density operator ρˆ ,
i¯hρ˙ˆ = [Hˆ 0 − μ · E(t), ρˆ ].
(17.13)
17 Nonadiabatic Dynamics in Complex Systems
313
Hˆ 0 represents the field-free nuclear Hamiltonian for a molecular system with several
electronic states in the Born-Oppenheimer approximation, and the interaction with
the laser field E(t) is described using the dipole approximation. The semiclassical
limit can be straightforwardly derived in the framework of the Wigner phase space
representation [2, 3] of quantum mechanics. The equations of motion for the phase
space representation of the density matrix elements involving an arbitrary number
of electronic states then read:
ρ˙ ii = {Hi , ρii } −
2
Im (μ i j · E(t)ρ ji )
h¯ ∑
j
(17.14)
i
i
ρ˙ i j = −iωi j ρi j + μ i j · E(t) (ρ j j − ρii ) + ∑ μ ik · E(t)ρk j − μ k j · E(t)ρik
h¯
h¯ k=i, j
(17.15)
where the diagonal elements ρii determine the quantum mechanical state populations and the off-diagonal elements ρi j describe the coherence. The curly braces
denote the Poisson brackets, Hi are the Hamiltonian functions for the respective
electronic state i. The quantity ωi j is the energy gap and μ i j the transition dipole
moment between the electronic states i and j.
In order to connect Eqs. 17.14–17.15 with classical molecular dynamics “on the
fly” the diagonal density matrix elements ρii (q, p,t) which are functions of the coordinates q and momenta p can be represented by independent trajectories propagated
in the ground and excited electronic states, respectively. Thus, employing a number
of Ntra j trajectories, ρii (q, p,t) can be represented by a swarm of time-dependent δ
functions
ρii (q, p,t) =
1
Ntra j
∑ θik (t) δ (q − qik(t; q0 , p0 ))δ (p − pik (t; q0 , p0 ))
(17.16)
k
where (qik , pik ) signifies a trajectory propagated in the electronic state i and the
parameter θik (t) is one if the trajectory k resides in the state i and zero otherwise
[56]. The population transfer between the electronic states is achieved by a process
in which the trajectories are allowed to switch between the states. This procedure is
related to Tully’s surface hopping method [13] for field-free nonadiabatic transitions
in molecular systems. However, in our case the coupling between the states is
induced by the applied laser field. The probabilities for switching the electronic
state can be calculated according to Eq. 17.5 given in Sect. 17.2.
The simulation of the laser-induced dynamics in the framework of our FISH
method using the above derived approach is performed in the following three
steps:
1. We generate initial conditions for an ensemble of trajectories by sampling e.g.
the canonical Wigner distribution function (cf. Eq. 17.12) or a long classical
trajectory in the electronic ground state.
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2. Along each trajectory R(t) which is propagated in the framework of MD “on
the fly”, we calculate the density matrix elements ρi j by numerical integration.
If the initial electronic state is a pure state as it is in our case, the set of
equations 17.14–17.15 is equivalent to the time-dependent Schrăodinger equation
in the representation of adiabatic electronic states:
ihci (t) = Ei (R(t))ci (t) − ∑ μ i j (R(t)) · E(t)c j (t)
(17.17)
j
where ci (t) are the expansion coefficients of the electronic wavefunction from
which the density matrix elements can be calculated as ρi j = c∗i c j .
If the intrinsic nonadiabatic coupling Di j of the electronic states (cf. Eq. 17.3)
also has to be taken into account, the Eq. 17.17 can be generalized to
i¯hc˙i (t) = Ei (R(t))ci (t)− ∑ [μ i j (R(t)) · E(t) + i¯hDi j (R(t))] c j (t).
(17.18)
j
In this way, after the duration of the applied field is over, field free multistate
nonadiabatic dynamics can be further carried out. The Eqs. 17.17 or 17.18 are
solved numerically using e.g. the fourth order Runge-Kutta procedure.
The nuclear trajectories R(t) are obtained by solution of Newton’s equations
of motion where the necessary forces are obtained from the energy gradients in
the actual electronic state in which the trajectory is propagated.
In contrast to field-free nonadiabatic dynamics, in the presence of electric
fields the energy of a molecular system is not conserved due to the interaction
with the field. Therefore, when exposed to a long intense laser pulse, molecules
can accumulate energy and eventually get heated, which for isolated molecules
can finally lead to fragmentation. However, if the molecule is interacting with an
environment such as solution, the excess thermal energy can be dissipated. For
approximate inclusion of these effects dissipative Langevin dynamics instead of
Newtonian dynamics can be employed. The solution of the Newton or Langevin
equations of motion provides continuous nuclear trajectories which reside in
different electronic states according to the quantum mechanical occupation
probabilities given by ρii .
3. In order to determine in which electronic state the trajectory is propagated we
calculate the hopping probabilities under the influence of the field and decide
if the trajectory is allowed to change the electronic state by using a random
number generator. For a general number of states the hopping probability can
be calculated according to Eq. 17.5.
Notice that while the trajectories jump between the electronic states at a given
time, all density matrix elements are propagated continuously over the entire time
according to Eqs. 17.14–17.15, or alternatively either Eq. 17.17 or 17.18. Although
the individual trajectory is allowed to jump, the fraction of trajectories in a given
state, which represents ρii as an ensemble average, is also a continuous function of
time. The phase of the electronic wavefunction is preserved and our procedure gives
17 Nonadiabatic Dynamics in Complex Systems
315
rise to the full quantum mechanical coherent state population, therefore being able
to mimic laser-induced processes such as coherent Rabi oscillations between the
electronic states [59].
Our semiclassical FISH method is a valuable tool for the simulation of ultrafast
laser-driven coupled electron-nuclear dynamics involving several electronically
excited states in complex molecular systems. It can be applied to simulate spectroscopic observables [77] as well as to control the dynamics employing shaped
laser fields and thus to steer molecular processes. Since the laser field enters the
equations for population dynamics directly, the combination with the optimal control theory is straightforward. The electric field can be iteratively optimized using
e.g. evolutionary algorithms [58, 89] as it has been illustrated in Ref. [59]. For the
propagation of classical trajectories the whole spectrum of methods ranging from
empirical force fields, semiempirical to ab initio quantum chemical methods can
be employed. Moreover, in addition to isolated systems in the gas phase, molecular
systems interacting with different environments such as solvent, bioenvironment,
surfaces or metallic nanostructures can be also treated. The FISH method allows
not only to obtain optimized pulses but also to analyze their shapes on the basis of
molecular dynamics “on the fly”. In this way the comparison between theoretically
optimized laser fields with those obtained from experiments, e.g. using the CLL
procedure, allows to assign the underlying processes to the specific forms of the
pulses. By this means the inversion problem can be addressed and important parts
of the PES could be constructed. Altogether, the FISH method opens new avenues
to perform the optimization of laser pulses for different exciting applications as it
will be illustrated in Sect. 17.6.
17.6 Application of the FISH Method for the Optimal
Dynamic Discrimination
We wish to reveal the mechanism for the optimal dynamic discrimination between
the very similar biochromophores riboflavin (RBF) and flavin mononucleotide
(FMN) using optimally shaped laser fields. Our FISH simulations utilize experimentally optimized laser fields and show that the fluorescence depletion ratio
between two molecules can be manipulated with such fields, eventually achieving
discrimination between them. Moreover, these results validate for the first time the
experimental optimal control technique applied on complex systems [63].
The general concept of the optimal dynamic discrimination (ODD) has been
recently proposed by Rabitz and Wolf et al. [60, 61]. The idea of the ODD relies
on a theoretical analysis which has shown that quantum systems differing even
infinitesimally may be distinguished by means of their dynamics when a suitably
shaped ultrafast control field is applied. In the case of the two similar flavins
(differing only by replacement of H by PO(OH)2 in the side chain) the controlled
depletion of the fluorescence signal has been used as a discriminating observable
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Fig. 17.6 Schematic illustration of the discrimination of FMN and RBF by fluorescence depletion.
Excitation with a shaped UV laser pulse leads to transition from S0 to S1 state, as indicated by the
light grey arrow. After a time-delay Δ t during which dynamical processes take place, an unshaped
IR pulse is applied. In the case of FMN (left part of the figure), this leads to transitions to higher
excited states where irreversible processes such as ionization can occur (dark arrow), consequently
the fluorescence gets depleted (crossed dark arrow). For RBF (right part of the figure), excitation
to higher states is less favorable (crossed dark arrow), and fluorescence will remain stronger than
in FMN (dark arrow). With differently shaped UV pulses, also the reverse situation is possible
[62]. The schematic representation of the discrimination process is presented in
Fig. 17.6. In general, a shaped ultraviolet (UV) pulse excites both molecules to the
S1 state and induces ultrafast dynamics which can follow slightly different pathways
in both molecules. After a specified time delay Δ t a second unshaped infrared
(IR) pulse excites the molecule further to higher excited states and can induce
dissipative processes such as ionization which lead to irreversible depopulation of
the S1 state, and thus to depletion of the fluorescence signal in one of the species
(cf. left part of Fig. 17.6) and not in the other one (cf. right part of Fig. 17.6). Since
for both molecules depletion can be minimized and maximized independently, the
total fluorescence yield can be used to quantitatively determine the amounts of
both species [62]. Although in this study only flavins have been considered, the
results should be broadly applicable to control systems whose static spectra show
essentially indistinguishable features. In particular, this should allow in the future
for the selective identification of target molecules in the presence of structurally and
spectroscopically similar background. This is an important issue in multiple areas of
science and engineering. Our FISH method offers a unique opportunity not only to
perform multistate dynamics “on the fly” and to optimize the laser pulses but also to
apply directly the experimentally optimized pulses and thus to reveal the processes
which enable discrimination of similar chromophores.
Our simulation of ODD between RBF and FMN is based on FISH dynamics
“on the fly” in the ground and the nine lowest excited singlet states (S0 –S9 )
under the influence of the experimentally optimized laser fields. We describe the
electronic structure using the semiempirical PM3 CI method [90] and calculate
17 Nonadiabatic Dynamics in Complex Systems
317
the nonadiabatic couplings and transition dipole moments between all electronic
states using the method of Thiel et al. [91, 92] The nuclear dynamics is performed
employing the Langevin equation in order to approximately account for dissipative
effects of the water environment present in the experiment [93]. Along the nuclear
trajectories, the time-dependent Schrăodinger equation (17.18) is integrated and the
hopping probabilities are obtained from the electronic state populations according to
Eq. 17.5. The shaped UV laser fields with a central wavelength of 400 nm employed
in the simulation are reconstructed from the experimental spectral amplitudes An ,
phases φn and frequencies ωn [62] according to E(t) = ∑n An exp [i (ωnt + φn )]. The
pulses obtained in this way have a duration of ∼5 ps and a maximum amplitude of ∼
6 · 1011 W cm−2 . The unshaped IR probe pulse with a wavelength of 800 nm has a
maximum amplitude of ∼3 · 1012 W cm−2 and a Gaussian envelope with a width of
100 fs (cf. Fig. 17.8).
The irreversible processes such as ionization, which lead to fluorescence depletion, are approximately introduced in Eq. 17.18 by adding an imaginary component
iΓ to the energy of the highest excited state S9 which lies close to the experimentally
determined ionization limit in water [94]. In this way irreversible population decay
from the S9 state is introduced. Subsequently, the time-dependent coefficients along
the trajectories are recalculated and the hopping from the S9 state to the ionized
state is accounted for. Thus, an ionized population Pion is obtained by averaging
over all trajectories, which can be used as a measure for the decrease of the excited
state population and thus for fluorescence depletion. In the experiment, the latter is
quantified by the fluorescence intensities after application of the UV pulse alone,
F(UV ), and application of both the UV and IR pulses, F(UV + IR), according
to Dexp = [F(UV ) − F(UV + IR)]/F(UV ). In order to calculate the equivalent
depletion signal from our ionized populations Pion , we determine the fluorescence
depletion D as
Pion (UV + IR) − Pion(UV )
D=
1 − Pion(UV )
For further computational details see Ref. [59, 74].
The RBF and FMN molecules represent particularly challenging systems for the
optical discrimination since they have nearly identical stationary absorption and
fluorescence spectra. The electronic spectroscopy of flavins is primarily associated
with their common chromophore π − π ∗ type transitions at 400 nm localized on
the isoalloxazine ring and is influenced only very slightly by the terminal chemical
moieties (H versus PO(OH)2 ) on the side chains. The optimal UV pulses allowing
for discrimination have been obtained experimentally by closed-loop optimization
of the fluorescence depletion ratio of FMN over RBF and vice versa. Specifically,
the maximization of the FMN over RBF ratio yielded a shaped UV/IR pulse
pair (termed pulse 1 in the following) that leads to distinguishable fluorescence
depletions values of 12.6% for RBF and 16.4% for FMN (cf. Fig. 17.7a). Oppositely,
minimization of the FMN over RBF ratio has yielded a second pulse pair (pulse 2)
that achieves approximately the same level of discrimination but reverses the
ordering of the depletion signals (cf. Fig. 17.7b). In contrast, the excitation with
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Fig. 17.7 Absolute experimental RBF and FMN depletion signals for optimized UV pulse shapes
for maximizing (a) and minimizing (b) the ratio D(FMN)/D(RBF). Reprinted with permission
from Roth et al. [62]. Copyright 2009 by the American Physical Society. The time delay for the IR
pulse is 500 fs. Absolute depletions induced by the transform-limited pulse for both RBF (black)
and FMN (grey) are statistically equivalent at 26%. Optimal pulses pull apart the RBF and FMN
distributions to achieve discrimination between the two molecules
Fig. 17.8 (a) Upper panel: Temporal structure of the shaped pulse 1 for maximization of
D(FMN)/D(RBF) (blue) and of the unshaped IR probe pulse (red). Middle panel: Time-dependent
populations of the electronic states S0 (black), S1 (red), and S2 –S9 (orange) in RBF driven by the
pulses shown in the upper panel. Lower panel: Time-dependent populations of the electronic states
S0 (black), S1 (red), and S2 –S9 (orange) in FMN driven by the pulses shown in the upper panel.
(b) The same as (a), but for pulse 2
an unshaped UV component leads to indistinguishable fluorescence depletion
signals of 26% for RBF and FMN as also shown in Fig. 17.7. In order to
discover the processes responsible for ODD of RBF and FMN, these experimentally optimized pulses for maximization and minimization of the depletion ratios
[62] (pulses 1 and 2, respectively) have been used in our FISH simulations. The
population dynamics induced by both pulses in RBF and FMN is shown in Fig. 17.8
17 Nonadiabatic Dynamics in Complex Systems
319
Fig. 17.9 Upper panel: Temporal structure of pulse 1 for maximization (left) and of pulse 2 for
minimization (right) of the depletion ratio D(FMN)/D(RBF). Middle panel: Ionized populations
Pion of RBF (black) and FMN (red) due to pulse 1 (left) and pulse 2 (right). Lower panel:
Fluorescence depletion D of RBF (black) and FMN (red) due to pulse 1 (left) and pulse 2 (right)
(Reprinted from Ref. [63]. Copyright 2010 by the American Physical Society)
together with the temporal pulse structure. Both pulses 1 and 2 lead to a smaller
population of the higher excited states (S2−9 ) in RBF compared to FMN before the
IR component has been applied. After the IR pulse, the S2−9 population raises for
both pulses and both molecules. Although transient differences in the excited state
populations are present during the pulses, the population returns in all cases from
the higher excited states to the S1 state after the pulses have ceased, if no irreversible
processes such as ionization from these states are taken into account. Therefore, in
order to describe fluorescence depletion the irreversible population decay from the
higher excited states is modeled by adding an imaginary component to the energy of
highest excited state S9 as described above. The value of the imaginary component
has been calibrated such that with an unshaped UV pulse both molecules exhibit
identical depletion ratios. The ionized state populations Pion obtained in this way
are shown in Fig. 17.9. It can be seen that the IR subpulse is mainly responsible for
the ionization, which in the case of RBF sets in at about +0.5 ps. Although for FMN,
there is some ionization at earlier times, the main part of the ionized population is
also generated at about +0.5 ps. The ionization yield of RBF is lower for pulse
1 than for pulse 2, whereas the reversed effect is found for FMN, proving that
the shaped laser fields can selectively and independently modulate the ionization
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R. Mitri´c et al.
Fig. 17.10 Left: Average transition dipole moments for S1 → S2 –S9 transitions for the dynamics
driven by pulse 1 (a) and 2 (b) for FMN (red) and RBF (black). The average is performed over the
states S2 –S9 and over the ensemble of trajectories. Right: Selected averaged ground state normal
mode displacements for RBF (c) and FMN (d) induced by pulse 1 (black) and pulse 2 (red)
efficiency. The fluorescence depletion D (cf. lower part of Fig. 17.9) relies upon the
relative decrease of the excited state population (S1 –S9 ) due to both the UV and
IR pulses compared to the UV pulse alone. It is initiated for both pulses and both
molecules by the IR subpulse at +0.5 ps. For pulse 1, D is systematically larger
for FMN than for RBF, whereas for pulse 2 after 1 ps it becomes larger for RBF
than for FMN. The final depletion ratios D(FMN)/D(RBF) after the pulses have
ceased are calculated to be 1.4 for pulse 1 and 0.4 for pulse 2. These values are in
good agreement with the experimental ODD values of 1.3 for pulse 1 and 0.7 for
pulse 2 [62], thus confirming the experimental optical discrimination between FMN
and RBF.
Our simulations offer a unique opportunity not only to reproduce the experimental findings but to gain an insight into the mechanism of dynamical processes
responsible for discrimination. The fluorescence depletion is directly related to
the ionization yield, which depends on the efficiency of populating excited states
above S1 . Therefore the averaged transition dipole moments between S1 and the
higher excited states along the trajectories driven by the optimal laser fields have
been calculated. It can be seen from Fig. 17.10a that pulse 1 induces dynamical
pathways which exhibit systematically larger transition dipole moments for FMN
than for RBF, indicating the stronger ionization and accordingly stronger depletion
of fluorescence in FMN. In contrast, for pulse 2 (cf. Fig. 17.10b) at times after
+0.5 ps this behavior is reversed leading to higher transition dipole moments for
RBF, and thus in this case the fluorescence depletion should become stronger in
RBF. These findings are consistent with the ionization yields presented in Fig. 17.9.
In order to establish the connection between higher transition dipole moments
and the structural changes during the dynamics the averaged time-dependent normal
mode displacements along the trajectories have been analyzed. In general, the
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321
Fig. 17.11 Schematic illustration of the optimal dynamic discrimination by shaped laser fields on
the example of shaped pulse 1 maximizing the FMN/RBF fluorescence depletion ratio (Reprinted
from Ref. [63]. Copyright 2010 by the American Physical Society)
conformational differences induced by the discriminating pulses are localized
mainly in the polar side chains of both molecules. In Fig. 17.10c and d one prototype
low-frequency normal mode of each molecule exhibiting large displacements
induced by pulses 1 and 2 is shown. The pulse 1 invokes smaller deviations
for the normal coordinate Q2 in RBF and larger deviations after 0 ps for Q2 in
FMN compared to pulse 2. Thus, the excitation of low-frequency normal modes
leads to conformations which have systematically higher or lower transition dipole
moments to higher excited states leading to ionization, depending on which of two
discriminating pulses is acting. Since RBF and FMN only differ in the side chain,
differences in the dynamical behavior are expected to occur here due to the interplay
between the effect of the heavy phosphorus atom in FMN and the differences of the
vibrational density of states in both molecules.
In summary, the discrimination mechanism can be depicted as shown in
Fig. 17.11: UV excitation of the molecule induces dynamical processes in excited
states which mainly affect low-frequency tail vibrational modes. The discriminating
pulse efficiently drives one of the molecules to regions of the PES where the
transition dipole moments to higher excited states are large, such that the ionization
and thus the fluorescence depletion are enhanced (pulse 1 for FMN). The same
pulse acting on the other molecule (RBF, cf. right part of Fig. 17.11) suppresses
the ionization (depletion of fluorescence) by keeping it in regions of the PES
with lower transition dipole moments. Thus, in general, the shaped pulses can
take advantage of minute differences in vibrational dynamics and exploit them
to manipulate observables such as transition dipoles allowing for the selective
molecular discrimination. This mechanism represents a general feature that can be
exploited for the discrimination between similar molecules and offers a promising
tool for using optimally shaped laser pulses in bioanalytical applications, thus
increasing the selectivity beyond the current capability.
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17.7 Conclusions and Outlook
We have presented a general theoretical approach for the simulation and control
of ultrafast processes in complex molecular systems. Our methodological developments are based on the combination of quantum chemical nonadiabatic dynamics
“on the fly” with the Wigner distribution approach for simulation and control of
laser-induced ultrafast processes. Specifically, we have developed an approach for
the nonadiabatic dynamics in the framework of TDDFT using localized basis sets,
which is applicable to a large class of molecules and clusters.
Furthermore, the FISH method is introduced, allowing to include laser fields
directly into the nonadiabatic molecular dynamics simulations and thus to realistically model their influence on ultrafast processes. In particular, this approach can
be combined with genetic algorithms allowing to design shaped laser pulses which
can drive a variety of processes.
The applications of our approaches have been illustrated on selected examples
which serve to demonstrate their scope as well as the ability to accurately simulate
experimental ultrafast observables and to assign them to underlying dynamical
processes. In particular, a general approach for the simulation of TRPES has
been developed, representing a powerful tool to identify nonadiabatic processes.
Moreover, we have demonstrated for the first time that in the framework of the
FISH method experimentally optimized laser fields can be directly used to reveal
dynamical processes behind the optimal control. In addition, the FISH method
combined with the optimal control theory allows to predict forms of laser fields
capable to steer molecular dynamics in complex systems such as large molecules
and nanosystems in different environments. Altogether, our approaches based on
the classical molecular dynamics accounting for electronic transitions induced
by both nonadiabatic effects as well as by light open new avenues for studying
femtochemistry of attractive molecular and nano-systems which were not accessible
earlier due to their complexity.
Acknowledgements We wish to acknowledge the contribution of our experimental partner Prof.
T. Suzuki. We extend our thanks to Prof. J.-P. Wolf and Prof. H. Rabitz for stimulating cooperation
and for providing us with experimental results on the discrimination of flavin molecules. Prof. W.
Thiel we thank for providing us with the nonadiabatic MNDO code. Finally, we would also like
to acknowledge the financial support from the Deutsche Forschungsgemeinschaft in the frame of
SPP 1391, FOR 1282, the Emmy Noether Programme, MI-1236 (R.M.), as well as the Fonds der
Chemischen Industrie (J.P.).
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