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5 Field-Induced Surface-Hopping Method (FISH) for Simulation and Control of Ultrafast Photodynamics

5 Field-Induced Surface-Hopping Method (FISH) for Simulation and Control of Ultrafast Photodynamics

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R. Mitri´c et al.

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 Nonadiabatic Dynamics in Complex Systems


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 } −


Im (μ i j · E(t)ρ ji )

h¯ ∑





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¯ k=i, j


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


ρii (q, p,t) =


Ntra j

∑ θik (t) δ (q − qik(t; q0 , p0 ))δ (p − pik (t; q0 , p0 ))



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


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.


R. Mitri´c et al.

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)



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).



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


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


R. Mitri´c et al.

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


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 )


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


R. Mitri´c et al.

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


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


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

17 Nonadiabatic Dynamics in Complex Systems


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.


R. Mitri´c et al.

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.).


1. Car R, Parrinello M (1985) Phys Rev Lett 55:2471

2. Wigner E (1932) Phys Rev 40:749

3. Hillery M, O’Connel RF, Scully MO, Wigner EP (1984) Phys Rep 106:121

4. Hartmann M, Pittner J, Bonaˇci´c-Kouteck´y V (2001) J Chem Phys 114:2123

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5 Field-Induced Surface-Hopping Method (FISH) for Simulation and Control of Ultrafast Photodynamics

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