Controlled Dynamics at an Avoided Crossing Interpreted in Terms of

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Controlled Dynamics at an Avoided Crossing Interpreted in Terms of Dynamically Fluctuating Potential Energy Curves Simona Scheit, Yasuki Arasaki, and Kazuo Takatsuka* Department of Basic Science, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902 Tokyo, Japan ABSTRACT: The nonadiabatic nuclear wavepacket dynamics on the coupled two lowest 1Σ+ states of the LiF molecule under the action of a control pulse is investigated. The control is achieved by a modulation of the characteristics of the potential energy curves using an infrared field with a cycle duration comparable to the time scale of nuclear dynamics. The transition of population between the states is interpreted on the basis of the coupled nuclear wavepacket dynamics on the effective potential curves, which are transformed from the adiabatic potential curves with use of a diabatic representation that diagonalizes the dipole-moment matrix of the relevant electronic states. The basic feature of the transition dynamics is characterized in terms of the notion of the collision between the dynamical crossing point and nuclear wavepackets running on such modulated potential curves, and the transition amplitude is mainly dominated by the off-diagonal matrix element of the time-independent electronic Hamiltonian in the present diabatic representation. The importance of the geometry dependence of the intrinsic dipole moments as well as of the diabatic coupling potential is illustrated both theoretically and numerically.

1. INTRODUCTION In the past years the continuous advance in laser technology generating shorter widths and higher intensities makes it possible to control molecular processes such as chemical reactions.16 When a molecule is placed in a laser field, its electronic states, particularly those of ionic character, can be significantly affected by the time-dependent electric field, which is analogous to the well-known Stark effect in a static electric field. Among the many possible control scheme making use of the oscillating electric fields, the so-called dynamic Stark control (DSC), which makes direct use of the energy-level shift, seems particularly promising. Depending on the intensity, phase, shape, and frequency of the laser field applied, various types of DSC regimes can be identified: resonant and nonresonant, and perturbative and nonperturbative. Among others, a very promising control is of the nonresonant and nonperturbative type, in which the DSC is used suppressing the excitation of other electronic states, which would otherwise result in the presence of competing processes. Within this regime, one can further distinguish between Raman and dipole subregimes.7 In the former case, the system response follows the field envelope, while in the latter it follows the instantaneous electric field. The DSC based on the Raman-coupled DSE is believed to be one of the most efficient schemes even though control is achievable also in the dipole-coupling case.8 The nonresonant nonperturbative DSC has been successfully applied, for example, to the control of nonadiabatic dynamics in IBr between two states whose diabatic potential energy curves are not dipole coupled.9 Both in refs 8 and 9, the results have been explained by referring to the simple argument based on the LandauZener formula in terms of the energy gap between potential energy curves modified by the laser and of the wavepacket velocity at the r 2011 American Chemical Society

crossing point. In particular, Stolow et al. noted that, in the diabatic picture, the control laser pulse induces a Stark shifting of the potential energy curves, which in turn results in the shifting of the crossing point between states, thereby affecting the velocity of the wavepackets at the crossing.9 With this modulation of the speed of the wavepacket, one should be able to control the nonadiabatic dynamics. In this paper we will further elaborate the notion of dynamical crossing and of its collision with a nuclear wavepacket by theory and numerical calculations in a diabatic representation of the nuclear wavepacket dynamics on the two lowest Σ states of LiF, which have an avoided crossing. We survey how such nonadiabatic dynamics can be controlled with a control pulse having a cycle length comparable to the time-scale of the nuclear dynamics. The concepts of light induced potential energy surface (due to the so-called dressed state) and of field-modified potentials have been widely recognized and applied to the description of nuclear wavepacket dynamics under strong laser fields.1014 Besides, the advance in laser technology now makes it possible to generate very short few-cycle pulses, which is particularly convenient to explore the very basic and elementary processes involved in the dynamical Stark effect. In such a fewcycle pulse environment, the modulation of the potential energy curves follows not the envelope function of the pulse but directly Special Issue: Femto10: The Madrid Conference on Femtochemistry Received: July 27, 2011 Revised: November 13, 2011 Published: November 16, 2011 2644

dx.doi.org/10.1021/jp2071919 | J. Phys. Chem. A 2012, 116, 2644–2653

The Journal of Physical Chemistry A

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the cycles of the applied field, and it is on these curves that nuclear dynamics take place. To carry out such nuclear wavepacket dynamics, we represent the potential energy curves in a diabatic representation. A diabatic representation treating the external field as a time-dependent modification of the potential energy surfaces proved helpful in describing and gaining insight into the controlled dynamics at a conical intersection in a previous communication. 15 Here in this article we choose a representation so that the transition dipole moment between the two electronic states becomes exactly zero at any nuclear distance. Through such a diagonalization of the dipole-moment matrix, the effect of the time-dependent electric field is well incorporated into the potential curves, particularly into those of ionic character. We survey the crucial characteristics of this diabatic representation rather precisely to highlight the physical reason why this particular representation gives a particularly convenient view of nonadiabatic dynamics. Thus, the results and detailed analyses obtained here are complementary to those by Han et al.8 and Stolow et al.9,16,17 on the utility of DSC. The paper is organized as follows: Section 2 introduces the theoretical framework and the main concepts that are needed for the interpretation of the control-driven nuclear dynamics at the dynamically fluctuating crossing point between the two states. The characteristics arising from the present diabatic representation will be described in section 3. Section 4 discusses the main effects induced by the control on state population and wavepacket dynamics, which is followed by further discussion on the laser parameters in section 5. This paper concludes with some remarks in section 6.

2. DYNAMICALLY FLUCTUATING POTENTIAL ENERGY CURVES AND QUANTUM TRANSITION AMONG THEM We begin the present report on the nonadiabatic nuclear wavepacket dynamics with the description of our adopted diabatization of the system and its characteristics in the potential energy curves and the interaction potential among them. 2.1. Dynamically Fluctuating Potentials and Shift of Crossing Point. The coupled time-dependent Schr€ odinger equation

for the vibrational wavepackets Ψ1(R,t) and Ψ2(R,t) on the electronic states 1 and 2, respectively, in a diabatic representation is given as ! ! ∂ Ψ1 ðR, tÞ ^ Ψ1 ðR, tÞ ip ¼H ð1Þ Ψ2 ðR, tÞ ∂t Ψ2 ðR, tÞ ^ is given by where the Hamiltonian H ^ ^ H ¼ T N 1 þ VðRÞ þ V F ðR, tÞ

ð2Þ

^ N the nuclear kinetic energy operator. The potential with T energy matrix V(R) is ! V11 ðRÞ V12 ðRÞ VðRÞ ¼ ð3Þ V12 ðRÞ V22 ðRÞ where V11(R) and V22(R) are the diabatic PECs and V12(R) is the diabatic coupling potential between them. An avoided crossing in the adiabatic representation becomes literally a crossing of the curves V11(R) and V22(R) in the diabatic

Figure 1. Dynamically fluctuating potential curves of the two low lying states of LiF under control laser. Upper panel: the extreme positions eff reached by the effective PECs Veff 1 (R,t) and V2 (R,t) under the control pulse of section 4.1. Lower panel: time evolution of the dynamical crossing point RX(t) during the first half of the control pulse for t0 = 51 fs in eq 23.

representation. The magnitude of V12(R) near the crossing point Rcross indicates the strength of state mixing at the avoided crossing. A larger V12(R) there induces more population transfer between the diabatic states, thereby making the dynamics more adiabatic. A smaller V12(R) there induces less population transfer between the states. For a linearly polarized external field in the dipole approximation, the time-dependent interaction VF(R,t) is given by ! μ11 ðRÞ μ12 ðRÞ FðtÞ ð4Þ V F ðR, tÞ ¼ μ12 ðRÞ μ22 ðRÞ where μij (R) is a dipole moment matrix element between the states i and j in the direction of field polarization, and F(t) represents the time-dependent amplitude of the external field. For the case of significant dipole moment values we neglect the less contributing higher order terms for simplicity. In the present work, we consistently choose a diabatic representation such that the off-diagonal dipole matrix elements vanish at any R, that is, μ12 (R) = 0 by diagonalization of the dipole matrix. (We discuss the practical diabatization of the LiF molecule more in section 3.) The electric field generated by the classical vector potential of the electromagnetic field of a pulse laser to be used in eq 8 is of the standard form FðtÞ ¼ Ef ðt t0 Þ sinðωðt t0 Þ þ ϕÞ þ

E df cosðωðt t0 Þ þ ϕÞ ω dt

ð5Þ

where we2 use a Gaussian envelope function f(t t 0 ) = 2 e a (tt 0 ) with a full width at half-maximum (fwhm) τ = (2(log 2)1/2 )/a throughout the paper. The field energy pω, center of the pulse t 0 , field amplitude E (in terms of intensity I = cε0 E 2 /2, where c is the speed of light, and ε0 is the electric constant), and phase ϕ are the parameters of the control field. 2645

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The time integral of the electric field associated with the pulse is equal to zero. 18 It will prove convenient to rewrite the Hamiltonian, eq 2, as ^ ^ H ¼ T N 1 þ V eff ðR, tÞ

ð6Þ

the following phase factors

i K2 ðr, s; Ri , 0Þ = A2 ðr, s; Ri , 0Þ exp S2 ðr, s; Ri , 0Þ p and

eff

where the effective potential matrix V is given by V eff ðR, tÞ ¼ VðRÞ þ V F ðR, tÞ

Vieff ðR, tÞ ¼ Vii ðRÞ þ μ ii ðRÞFðtÞ

i K1 ðRf , t; r, sÞ = A1 ðRf , t; r, sÞ exp S1 ðRf , t; r, sÞ p

ð7Þ

This representation is useful because we are considering an external field oscillating with a time-scale comparable to that of the nuclear motion. The diagonal matrix elements of eq 7 are defined as ð8Þ

ð9Þ

that is, they are obtained by a time- and coordinate-dependent energy shift of the original PECs by the control field through the intrinsic dipole moments. We may therefore regard the nuclear wavepacket dynamics as that taking place on dynamically fluctuating (on the same time scale as the wavepacket eff dynamics) effective potentials V eff i (R,t) and V 12 (R,t) rather than a dynamics on the static PECs Vij(R). This implies that not just the PECs, but also the crossings between PECs, change in position as a function of time (or the crossing may even disappear for certain times), as seen in Figure 1. In other words the static crossing, located at Rcross, is replaced by a dynamical crossing, and at the dynamically crossing position RX(t) (driven by the control field), a propagating wavepacket on one state may split and be transferred to the other state, mediated by V eff 12(R,t) there. 2.2. Factors to Determine the State Transition: Relative Location and Velocity between the Dynamical Crossing Point and Running Wavepackets. Since μ12(R) is chosen to be zero in the effective interaction V eff 12(R,t), the static interaction V12(R) is solely responsible for causing the state transition. Besides, such a transition takes place most effectively only in the close vicinity of the dynamical crossing point RX(t). This is due to the principle of smooth continuation of quantum phase (this is not a new concept, see ref 19 as an example) as described below. Let a wavepacket Ψ2(Ri,0) begin to run on V eff 2 (R,t) starting from Ri at time zero, assuming there is no wavepacket on V eff 1 (R,0). As long as this packet runs without undergoing state transition, it evolves in time as Z dRi K2 ðR, t; Ri , 0ÞΨ2 ðRi , 0Þ ð10Þ Ψ2 ðR, tÞ ¼ where K2(R,t;Ri,0) is the Feynman kernel for the dynamics on eff V eff 2 (R,t). Suppose that V12(R,t) induces a transition of Ψ2(R,t) to eff V 1 (R,t) creating a new wavepacket component Ψnew 1 (Rf,t), Ψnew 1 (Rf,t) should be written as Ψ1new ðRf , tÞ ¼ þ

1 ip

Z t 0

ds

ZZ

Z

dRi K1 ðRf , t; Ri , 0ÞΨ1 ðRi , 0Þ

eff ðrÞK ðr, s; R , 0ÞΨ ðR , 0Þ dRi drK1 ðRf , t; r, sÞV12 2 i 2 i

ð11Þ

which is an integral over pieces of transition at a position r (variable) and at a time s (variable). It is well-known in the semiclassical theory that the individual kernels are accompanied by

ð12Þ

ð13Þ

A2, for instance, is the amplitude factor, while S2 is the action integral arising from a classical trajectory, if any, which runs on V eff 2 at R and connects the two end points in space-time. These action integrals are to be taken along all the possible paths and should satisfy the condition ∂ S2 ðr, s; Ri , 0Þ ¼ p2 ðr, sÞ ∂r

and the off-diagonal elements as eff ðR, tÞ ¼ V ðRÞ þ μ ðRÞFðtÞ V12 12 12

ð14Þ

where p2(r,s) is a classical momentum of the trajectory at spacetime (r,s). Rewriting eq 11, featuring only these phases in Ψnew 1 (Rf,t), we have Ψ1new ðRf , tÞ ¼

1 Z t ZZ i i ds dRi dr exp S1 ðRf , t; r, sÞ f12 exp S2 ðr, s; Ri , 0Þ ip 0 p p

ð15Þ where f12 denotes all the other factors collectively (recall Ψ1 (Ri,0) = 0). Note that the phase terms in eq 15 are generally rapidly oscillatory due to the presence of the Planck constant p. Thus, only the stationary phase point(s) in the integrand of eq 15 can make a significant contribution to the integral. In other words, Ψnew 1 (Rf,t) survives only through such specific transition that satisfies the stationary phase condition ∂S1 ðRf , t; r, sÞ ∂S2 ðr, s; Ri , 0Þ þ ¼ p2 ðr, sÞ p1 ðr, sÞ ¼ 0 ∂r ∂r ð16Þ which requires that the momenta before and after jumping (transition) are exactly the same. Obviously, this condition is eff satisfied only by a point at which V eff 1 (r,s) = V 2 (r,s) is fulfilled. This is just the dynamical crossing point RX(t). Therefore, the state transition can be realized only at crossing points in the semiclassical limit. In the full quantum mechanical calculations as ours, however, this condition is softened, and we should say that the transition takes place in the close vicinity of the dynamical crossing point RX(t). Now that the quantum transition takes place predominantly in the close vicinity of the dynamical crossing point RX(t), the entire transition probability should be dominated by the number of encounters of RX(t) and the wavepackets. Since each transition causes a wavepacket bifurcation, this phenomenon can give rise to a series of interference among thus generated wavepackets, and therefore, the interference among the individual transition amplitudes may result in an interesting quantum effect.20 Note also that the interference among the individual pieces of the wavepackets can be an important factor in determining the entire quantum dynamics. For instance, two or more wavepackets may encounter each other on a single potential energy curve due to reflection at the turning points, which are also time-dependent. The magnitude of each transition should be affected by the electronic coupling matrix element at RX, that is, V12(RX) should be the leading factor for transition probability. The relative speed 2646



Figure 2. PECs and selected properties of the two lowest 1Σ+ states of LiF. (a) Diabatic PECs V11(R) and V22(R) and the coupling potential V12(R) between them. Nonadiabatic vibrational wavepacket motion initiated with a pump pulse is illustrated schematically. (b) Diabatic intrinsic dipole moments. (c) Adiabatic to diabatic transformation angle α(R) (defined by eq 18). (d) Time evolution of the wavepacket positions of the excited state wavepacket initiated with the pump pulse but without the control pulse. Thick curves show the center of wavepacket; thin curves indicate the width of the wavepacket.

between a wavepacket and RX(t) is also a key factor. The instantaneous velocity of RX(t) is determined by the time derivative of the control field amplitude. As already noted by other authors,8,9 the shift of the PECs modulates the partition between kinetic and potential energies, thereby changing the wavepacket velocity. As a particularly interesting example, it is possible to choose the pulse such that dynamical crossing and wavepacket move together keeping their contact for a long time. In such a case, the duration time for the wavepacket mixing is far longer than in the control-free case, and the wavepacket thus generated can be quite characteristic in that it loses its memory of the initial condition very easily. Note also that the semiclassical analysis totally breaks down there. Notice that all these factors can be controlled by the laser parameters: timing of switching the field, spatial direction of the electric field induced, frequency of the change of the field, intensity of the amplitude, and so on. The rest of this paper is devoted to the presentation of very basic controlled dynamics under laser fields having as simple parameters as possible.

3. DIABATIC REPRESENTATION OF THE TWO LOWEST 1 + Σ STATES OF THE LiF MOLECULE In the rest of this paper, we will investigate with full quantum mechanical calculations how a control pulse, represented in the potential function VF(t) in eq 4, can modify the nonadiabatic nuclear dynamics due to the avoided crossing between the two lowest 1Σ+ states of LiF.

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3.1. Diabatic Potential Energy Curves. We consider the controlled dynamics of the LiF molecule between the two lowest 1 + Σ states. The diabatic PECs V11(R) and V22(R) of the two electronic states considered are plotted in Figure 2a together with the diabatic coupling potential V12(R) between them in this representation. These quantities have been taken from very accurate calculations given in ref 21. Diabatization of the adiabatic potential energy curves obtained by means of quantum chemistry is done by the method of diagonalizing the dipole moment matrix, often used with alkali-halides.22,23 It is based on the observation that the values of the adiabatic dipole moment matrix vary rapidly at the avoided crossing, but a rotation of the matrix elements so as to eliminate the transition matrix elements connects the resultant curves across the nonadiabatic spatial region smoothly to make the curves slowly varying over the nuclear coordinate as should be for a diabatic representation.24 State 1, which supports a bound state, is of ionic character, while state 2 is of neutral and globally repulsive character, leading to dissociation into the ground state of Li and F atoms. The crossing between the diabatic PECs is located at Rcross = 7.18 Å . Figure 2b shows the intrinsic dipole moments (defined as pointing in the direction from F to Li) in the present diabatic representation. μ22(R) is very small and approaches zero asymptotically as R f ∞. μ11(R) is much larger, always positive, and moreover increases almost linearly with R. μ12(R) is zero by definition. Therefore, when considering the effective timedependent potential curves defined by eq 8, the control pulse eff affects Veff 1 (R,t) very strongly while it affects V2 (R,t) only eff minimally. Because the shift of V1 (R,t) increases with R and Veff 2 (R,t) remains almost constant, the crossing position RX(t) moves in a wide range according to its amplitude at each instant. Note again that Veff 12(R,t) = V12(R) for all times. The numerical results in this paper are very much dependent on the shape of the diabatic coupling potential curve V12(R) on which there is some disagreement in the literature. The curve we use from ref 21 has a maximum at about 2.5 Å, which is much shorter than Rcross. For R > 2.5 Å, V12(R) is monotonically decreasing and its value becomes relatively small in the vicinity of Rcross. This shape agrees with that given in ref 22, and with the diabatic coupling potential curve given for NaI,23 another alkali halide. In earlier studies a monotonically decreasing exponential form was used for the V12(R) behavior near the avoided crossing.25 However, in other studies, often for coupled nonadiabatic wavepacket dynamics, a V12(R) localized near the avoided crossing has often been assumed.2628 A local V12(R) description is convenient in studies where the mixing of states happens only near the avoided crossing and the dynamics is adiabatic elsewhere. However, the present paper treats the case where the dynamical crossing can move around, so we do not assume a priori that V12(R) is at its maximum at the avoided crossing and keep the obtained wide spatial distribution of V12(R). We note in passing that the system and diabatization model chosen for this study has particularly convenient features that allow clear interpretation through dynamically fluctuating potential energy curves, but such a view is not confined to this system; it may provide additional insight even when applied to more complex systems.15 3.2. Relation to the Adiabatic Potential Energy Curves and Nuclear Derivative Coupling. Before proceeding to the wavepacket calculation, we further discuss the validity of the functional shape of the diabatic interaction potential V12(R) for the 2647



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interaction of the form V Fp ðR, tÞ ¼

! μ 11 ðRÞ μ 12 ðRÞ F ðtÞ μ 12 ðRÞ μ 22 ðRÞ p

ð21Þ

to the Hamiltonian of eq 6, where Figure 3. Pump pulse (eq 21) and control pulses (eq 23) with t0 = 80 fs.

present system, which may seem rather unusual in that it is mostly localized in a range of rather short internuclear distance (peaked at 2.5 Å), as seen in Figure 2. The transformation matrix U(R) connects the adiabatic and diabatic representations (indicated with superscript a and d, respectively) such that ! ! a Ψd1 ðR, tÞ † Ψ1 ðR, tÞ ¼ UðRÞ ð17Þ Ψd2 ðR, tÞ Ψa2 ðR, tÞ This orthogonal transformation matrix U is defined as29 0 1 cos αðRÞ sin αðRÞ A UðRÞ ¼ @ sin αðRÞ cos αðRÞ

ð18Þ

in terms of the adiabatic-to-diabatic mixing angle α(R). Figure 2c shows the computed α(R) for the curves from ref 21 for the present diabatic representation, which consequently gives birth to this functional form of V12(R). To examine the validity of this mixing angle, we carried out a multiconfiguration self-consistent field calculation30,31 in the spirit of ref 22 with the cc-pVTZ32 and aug-cc-pVTZ33 basis sets for the Li and F atoms, respectively, using the Molpro quantum chemistry package34 and numerically evaluated the nonadiabatic coupling element X12(R) as * + a a dΨ2 X12 ðRÞ ¼ Ψ1 ð19Þ dR and obtained the mixing angle α(R),29 αðRÞ ¼ αðR∞ Þ

Z R ∞

X12 ðR 0 ÞdR 0

ð20Þ

with α(R∞) = π/2 for our case. It is confirmed that α(R) obtained in this way is practically the same as the one in Figure 2, obtained by a diagonalization of the adiabatic dipole moment matrix. As is expected, the global behavior of the mixing angle of Figure 2c indicates that the mixing of the adiabatic states is actually very small in the range of R < 7.0 Å and R > 8.0 Å . In particular, there is virtually no statemixing in the area of R > 8.0 Å . Therefore, in this region, the diabatic potential curves are mostly close to the adiabatic counterparts. In the range 7.0 < R < 8.0 Å, the steep rise of α(R) realizes a reconnection of the states so as to keep continuity of the state character in the region of the avoided-crossing as in Figure 2a. The V12(R) curve with its maximum at a bond angle much less than the avoided crossing distance is fully consistent with this view. 3.3. Creating the Excited State Wavepacket Dynamics by Pumping. To have the wavepacket dynamics begin on thus coupled diabatic PECs, we pump the ground vibrational eigenstate on electronic state 1 to state 2. We add a pump pulse

Fp ðtÞ ¼ Ep fp ðtÞ sinðωp tÞ

ð22Þ

and ωp is the pulse frequency (pωp = 6.94 eV), Ep is the amplitude of the electric field (intensity Ip = 4.8 1012 W cm2), and fp(t) is a Gaussian envelope function with full width at half-maximum (fwhm) τp = 20 fs. The center of the pump pulse defines time t = 0. We neglect the less significant (in comparison with the electric dipole moment) polarizability and other higher order terms. We also confine our study to two state interaction, neglecting the presence of other excited states and ionization that may become significant at higher intensities. The field shape of the pump pulse is shown in Figure 3. Because there is a large frequency difference between the pump and the control pulses, the pump pulse is not included in Veff(R,t). A naive question arises why and how the optical excitation by laser is possible under the circumstance of μ12(R) = 0. Here again we should consider two states having the energy V eff 1 (R,t) = V11(R) + μ11(R)Fp(t) and V eff 2 (R,t) = V22(R) + μ22(R)Fp(t), where we remind that Fp(t) is for the pumping, as specified in eq 22. An appropriate choice of the frequency ωp can lift V eff 1 (R,t) to make it cross V eff 2 (R,t), through which the state mixing can undergo mediated only by the static interaction V12(R) of eq 3. Therefore, it is crucial for V12(R) to have a large magnitude in the area of the initial vibrational eigenfunction for the initial excitation to be sufficiently large. In other words, the spatial distribution of V12(R) of the present system, which looks peculiar at first sight as in Figure 2, is indeed natural and consistent with the discussion made in section 3.2.

4. BASIC QUANTUM DYNAMICS UNDER SINGLE-CYCLE CONTROL PULSE; A FEW ENCOUNTERS OF THE DYNAMICAL CROSSING AND WAVEPACKETS The control is exercised by means of a few-cycle control pulse field with a cycle duration comparable to the time-scale of the nuclear dynamics and a short pulse width such that the control can be switched on/off at the right moment with respect to the position of the evolving wavepacket. First we survey the effects the moving crossing has on the nuclear wavepacket dynamics by looking at the results of a simplified single-cycle control pulse of the form Fðt, t0 Þ ¼ Ef ðt t0 Þsinðωðt t0 Þ þ ϕÞ

ð23Þ

with ϕ = 0 and π only. The time integral of the electric field associated with the pulse is approximately zero for these particular phases. The pulse intensity chosen is I = 1.69 1013 W cm2, pulse energy is pω = 0.1 eV, and the pulse width is fwhm τ = 20 fs. The Schr€odinger equation for the wavepackets is solved with the split-operator method.3537 The wavepacket is represented on a grid (2048 points between 0.9 and 18.9 Å) and the kinetic energy operator is applied using the Fourier transform, while the nondiagonal (2 2) potential energy operator is diagonalized at each R for each short time step (Δt = 0.001 fs). The initial vibrational eigenfunction is readily generated by the energy screening technique.38 2648



Figure 4. (a) Time-evolution of the state 1 population P1(t) under the action of the pump and control pulses. The population in absence of control is shown in black for reference. (b) RX(t) for several t0 seen in (a) and position of Ψ2(R,t) in the absence of control (the same as Ψ2(R,t) in Figure 2d). Thick dashed curve indicates the center of wavepacket; thin dotted curves indicate the width of the wavepacket. RX(t) is shown only for the first half-cycle of the control pulse.

4.1. Early Encounter in the Dynamics under Single-Cycle Pulse O = 0. We first consider the control pulse of eq 23 with ϕ = 0.

The field shape (for t0 = 80 fs as an example) is shown as the thick solid curve in Figure 3, consisting of two symmetric half-cycles with zero field amplitude at the pulse center. The upper panel of Figure 1 shows the effective potential curves V eff i (R,t) obtained under the control field (t0 = 51 fs). The plotted curves at t = 44 fs represent the maximum shifts of the PECs in the first half-cycle of the control pulse, and those at t = 58 fs represent the maximum shifts in the later half-cycle of the control pulse. In the first halfcycle of the pulse, the control field, through the positive transition amplitude function μ11(R), acts to shift the V11(R) potential curve upward, and thus, the dynamical crossing position RX(t) is first shifted to the left. It reaches the left-most position Rleft at the height of the control field, and then is shifted right and restored to Rcross by t = t0. The lower panel of Figure 1 shows the resulting time-dependence of RX(t). The fact that the dynamical crossing is slower when going from Rcross to Rleft (taking 15 fs) than when returning from Rleft to Rcross (taking 5 fs) has important consequences on the controlled wavepacket dynamics. In the later half-cycle of the control pulse, the field acts to lower V11(R), and thus, RX(t) is shifted to the right. For a certain time range, eff V eff 1 (R,t) is shifted so much that it is below V 2 (R,t) for all R and the crossing disappears. After disappearing for 15 fs, the crossing

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is reestablished and is restored at Rcross when the control field is turned off. Figure 4a shows the time dependence of the diabatic state 1 population P1(t) for various delay times t0 of the control pulse between 30 and 120 fs. In each panel, the black curve shows P1(t) computed without application of the control pulse. The curve shows a drop in population between 20 to 20 fs; this is the pump pulse excitation. The diabatic representation and the significant V12(R) around the initial wave function position results in the oscillatory features on top of the population drop seen in this time range (these oscillations are not seen in the adiabatic representation). Then the pump-excited state population decreases by 13% (from P2(t) = 0.31 down to 0.27) around t = 110 fs; this is due to the static crossing at Rcross. Without the control pulse, this is the only population transfer seen. The colored curves in Figure 4a show P1(t) under control pulses with various t0. The pump pulse excitation (t < 20 fs) is common to all the cases, and the control pulse affects the dynamics after the pump. We see the final population (P1(t) at t = 250 fs) increase from t0 = 30 up to 60 fs, reach the maximum around 70 fs, then decrease as t0 is made larger, and become even less than the final population found for the case without control by t0 > 100 fs. For t0 > 120 fs, the effect of the control pulse becomes smaller, and it completely vanishes for t0 > 140 fs (not shown in Figure). The initial rise in P1(t) increases as t0 is increased from 30 to 65 fs, and then decreases as t0 is increased from 65 to 120 fs. For t0 < 80 fs, we see an immediate drop in P1(t) following the initial rise. Then after some time (t g 100 fs), we see the population increase at Rcross that is also present without the control pulse. For 80 e t0 < 100 fs, we instead see a slow but lasting decrease some time after the initial rise. To see the geometrical relation between the dynamical crossing point and the wavepacket propagation, we show in Figure 4b the time-dependence of RX(t) during the first half-cycle of the control pulse, for t0 = 30, 51, 65, and 90 fs, together with an indication of the position of Ψ2(R,t) computed for the case without application of a control pulse. The thick dashed curve is the expectation value ÆΨ2(R,t)|R|Ψ2(R,t)æ. The thin dotted curves indicate the range of |Ψ2(R,t)| > 0.006, which is 1/10 of the maximum height of Ψnew 1 (R,t) created at Rcross (without the control pulse). The wavepacket is roughly Gaussian-shaped, and the indicated range is meant to suggest where the dynamical crossing may start to interact with the wavepacket. When RX(t) first being shifted toward the left by the control pulse meets the rightmost tail of Ψ2(R,t), the transfer of the wavepacket from electronic state 2 to state 1 starts, resulting in a new component Ψnew 1 (R,t). This is the initial rise in P1(t). For t0 = 30 fs, Ψ2(R,t) is located mostly toward the left of RX(t) when RX(t) reaches Rleft, and there is little population transfer. As t0 is increased from 30 fs, the amount of the wavepacket contacting RX(t) increases until t0 = 65 where the whole wavepacket is involved in the initial transfer, and the initial rise in P1(t) increases through this range of t0. For t0 g 65 fs, Ψ2(R,t) and the dynamical crossing interact where RX(t) is moving faster and faster and V12(R) smaller and smaller, causing the initial rise to diminish. The created Ψnew 1 (R,t) initially inherits momentum from Ψ2(R,t) but slows down because of the steeper slope of V eff 1 (R,t) as compared to Veff 2 (R,t). The difference in position of new Ψ1 (R,t) and Ψ2(R,t) widens as time passes (see Figure 2d). The slowing down of Ψnew 1 (R,t) is larger when the control field 2649



Figure 5. Same as Figure 4 but for the control pulses in section 4.2. RX(t) is shown only for the second half-cycle of the control pulse.

amplitude is larger (and positive) because of the larger deformation of the PEC. After the initial transfer, RX(t) keeps going to the left and reaches Rleft and then starts to come back toward the right. When RX(t) now moving toward the right overtakes Ψnew 1 (R,t), a transfer from state 1 to state 2 is caused. This is the drop in P1(t) following the initial rise. We call this process the backtransfer. Because Ψ2(R,t) on the dissociating curve moves faster than Ψnew 1 (R,t) on the bound curve, the two wavepacket components do not mix during the back-transfer. Nevertheless, when the interaction between the states continues for a sufficiently long time by RX(t) taking some time to overtake Ψnew 1 (R, t), interference between the wavepacket components on the two states is observed as small oscillations in P1(t) following the backtransfer. The ratio of population transferred in the back-transfer to that in the initial transfer becomes less as t0 is increased. As t0 is made larger, RX(t) meets Ψnew 1 (R,t) at larger speed and smaller V12(R). For t0 = 30 fs, the back-transfer cancels out the initial transfer, but by t0 = 80 fs, the back-transfer is hardly seen. During the later half-cycle of the control pulse, either RX(t) > Rcross or the crossing has vanished and the wavepackets propagate undisturbed toward larger R. For t0 < 70 fs, by the time Ψ2(R,t) reaches Rcross, the crossing position has already been restored there and Ψ2(R,t) is partially transferred to state 1 with the same efficiency as in the case without the control pulse. This transfer starts at a time in between 100 and 130 fs depending on the value of t0, gradually shifting toward later times as t0 is increased. The delayed onset is due to Ψ2(R,t) slowing down

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during the population transfer between states. This deceleration is accentuated the longer the transfer lasts. The transfer at Rcross after the original crossing is restored is no longer very effective at t0 = 70 fs and for t0 > 70 fs is substituted by a slow but continuous population transfer in the opposite direction from state 1 to state 2. For t0 = 71 fs, there is less time between the end of the initial-transferback-transfer process and the wavepackets reaching Rcross, and both Ψ2(R,t) and Ψnew 1 (R,t) go through the restored crossing at the same time. Transfer takes place in both directions and we do not see a net population change. For 71 < t0 e 90 fs, Ψnew 1 (R,t) remains partly localized around Rcross for a long time, while Ψ2(R,t) goes through quickly to the right, leading to the slow population transfer from state 1 to state 2. For 90 < t0 e 120 fs, Ψ2(R,t) reaches Rcross just before the control field is switched on. The transfer at Rcross thus lasts for a much shorter time than in the control-free case, and the generated Ψnew 1 (R,t) is very small. Furthermore, when RX(t) is returning from Rleft to Rcross and overtakes the wavepackets, RX(t) is moving too quickly for any relevant population transfer to take place. For t0 > 120 fs, Ψ2(R,t) will have already reached the original crossing before the control pulse is switched on so that the effect of the control pulse is small and completely without effect for t0 > 140 fs. 4.2. Late Encounter of Wavepacket and Dynamical Crossing Point: Dynamics under the Single-Cycle Pulse O = π Next we study the dynamics under a control pulse defined by eq 23 but with ϕ = π. The pulses in this section and in the previous section differ only in their sign, and therefore the effect of the ϕ = π pulse on eff V eff 1 (R,t) will be the opposite of that of the ϕ = 0 pulse, i.e., V1 (R,t) will be first shifted downward, then upward and finally again downward until it coincides with V11(R) when the pulse is switched off. Figure 5a and b show P1(t) and RX(t), respectively, for the control pulse with various t0. RX(t) is shifted first from Rcross toward the right until it disappears for about 15 fs, and then it reappears and moves toward the left until reaching Rleft. Finally it moves again to the right until it resumes its original position. In other words, the dynamical crossing first moves away from the wavepacket and meets the wavepacket during the second half-cycle of the control pulse as opposed to during the first as in the previous section. Of course the extreme positions eff reached by V eff 1 (R,t) and V 2 (R,t), plotted in Figure 1, are the same for the two pulses. Most of the effects appearing are analogous to those observed for the ϕ = 0 pulse. For the same t0 the interaction is delayed because now RX(t) meets the wavepacket in the second instead of the first half-cycle of the control pulse. The interferences between the wavepackets on states 1 and 2 are more pronounced because RX(t) is moving more slowly when heading toward the right, compared to the case in the previous section. For 65 e t0 e 80 fs, the initial transfer is less efficient than in the case of Section 4.1. RX(t) now meets the wavepacket at larger R and at a faster speed. Later, when returning toward the right, RX(t) meets Ψ2(R,t) for a second time and because of its slower speed, causes a second transfer from state 2 to state 1, resulting in the step-like structure seen for these t0. If the RX(t) arrives at Rcross earlier than Ψ2(R,t) does, one additional transfer takes place, as is the case for t0 = 71 fs, where three transfers are observed. For t0 = 95 fs, RX(t) is moved away from Ψ2(R,t) just as the wavepacket is reaching Rcross, so that Ψ2(R,t) will see no crossing when it arrives at Rcross, and no transfer will take place. RX(t) and Ψ2(R,t) meet later when the relative speed between them is large and V12(R) is small and again no transfer takes place, resulting in a complete suppression of transfer between the diabatic states. 2650



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Figure 6. Upper panel: field shapes for three different intensities I1 = 3.5 1010, I2 = 1.7 1013, and I3 = 6.2 1013 W cm2. pω = 0.2 eV, t0 = 80 fs, and ϕ = π. Lower panel: corresponding P1(t). For reference, also the population obtained in the absence of control is plotted.

5. DEPENDENCE ON INTENSITY AND FREQUENCY OF THE CONTROL PULSE The examples so far presented have shown that, for a given (single-cycle) pulse, very different control scenarios are obtained by varying the delay time t0 used, which is essentially equivalent to controlling the phase. In the following, we will briefly discuss what roles the other pulse parameters, that is, the pulse intensity and frequency, have on the controlled dynamics. 5.1. Intensity Dependence. The intensity determines how much the effective PECs of the system will be shifted with respect to the static PECs. In the following we compare the results obtained for three different pulses, I1 = 3.5 1010, I2 = 1.7 1013, and I3 = 6.2 1013 W cm2, all with pω = 0.2 eV, t0 = 80 fs, τ = 20 fs, and ϕ = 0. The upper panel of Figure 6 shows the field shapes for the three cases. The lower panel of Figure 6 shows P1(t) for the three cases. The more intense the field is, the more the dynamical crossing is shifted in position from Rcross. Because the field of intensity I1 shifts the dynamical crossing only slightly, the final population resulting is not much different from the one obtained in the absence of control, even though we already see a transfer back-transfer process at this intensity. For higher intensities, the displacement of the crossing position is considerably larger. As a consequence, the final population deviates more from the control-free one. The state 1 population at the end is not a monotonic function of the intensity. A larger modification in P1(t) with respect to the control-free case is obtained with the I2 pulse, even though the I3 pulse shifts the dynamical crossing more than the I2 pulse. Because during the initial transfer process the crossing goes through the region occupied by Ψ2(R,t) more quickly in the I3 pulse case than in the I2 pulse case, the I3 pulse results in a less intense transfer. Thus a large shift in the crossing position alone does not automatically result in a strong modification of dynamics or population. As we have seen in the previous sections, it is the relative position and speed of the wavepackets and RX(t)

Figure 7. |Ψ2(R,t)|2 at different times, obtained with the control pulse of Figure 6 for intensity I1 (red), I2 (blue), and that obtained without the control pulse (black). The curve for I1 obscures the curve obtained without control for t = 65 fs.

that gives the final picture. Therefore, to magnify the effect of the control pulse on the dynamics, it is not sufficient to simply increase the pulse intensity, but attention must be paid to how the field shape affects the geometrical location and relative velocity of the dynamical crossing point with respect to the wavepackets. The distribution and phase of the wave function must also be a key factor to determine the transition dynamics. It is therefore interesting to see how the nuclear wavepackets themselves, and not just the populations, are affected by the control field. Figure 7 compares the time evolution of the wavepacket Ψ2(R,t) for the control pulses with intensities I1 and I2 with the one obtained in the absence of control. Shown is the |Ψ2(R,t)|2 as a function of R for the times t = 65, 90, and 225 fs. Ψ2(R,t) is very little altered by the low intensity pulse, but with the higher intensity pulse, the shape of Ψ2(R,t) changes dramatically. For this intensity the initial transfer takes place already at about 4 Å and at later times Ψ2(R,t) is split into different components, reflecting the presence of further interactions at shifted crossing positions between Ψ2(R,t) and Ψ1(R,t). These are also responsible for the oscillatory structure visible on Ψ2(R,t) (a thorough discussion of the effect of the control on the nuclear wavepackets structure is given in ref 20). Thus, the best control is to tune the fields that place the dynamical crossing in suitable places with respect to the wavepacket of fluctuating shape. Incidentally, the authors have been investigating the use of femtosecond time-resolved photoelectron spectroscopy to the study of nonadiabatic dynamics,27,3941 and such a shaped wavepacket should be a suitable candidate for real-time observation. Hence, integrating all these techniques, we may be able to find the optimal way to control in the future. 2651



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when a multicycle pulse is active, RX(t) is shifted in position a number of times and the wavepackets toward the left of Rcross have more opportunity to interact with the crossing. Nevertheless, the final population also depends on how fast the transfer is: several transfers of short durations are not necessarily more effective than a single, slow transfer. A rough arithmetic suggests that a single wavepacket may bifurcate into pieces as many as 2M in the end, with M being the number of encounters of the packet and the dynamical crossing point. It will be only good calculations that can project the final results accurately of such a complicated situation.

Figure 8. Upper panel: representative multicycle control pulses. (a) ω = 0.3 eV, t 0 = 30 fs, ϕ = 0; (b) ω = 0.3 eV, t 0 = 30 fs, ϕ = π; and (c) ω = 0.4 eV, t 0 = 40 fs, ϕ = 0. For all cases, I = 1.7 10 13 W cm 2 and τ = 20 fs. Lower panel: time evolutions of state 1 populations obtained under the respective pulses.

5.2. Multicycle Pulses. The discussion so far made has been restricted to the case of single-cycle pulses. This was to identify in a clear way the effects generated by the control pulse. In the following, we want to briefly address the case of multicycle pulses, where the number of cycles contained in a pulse is determined by the field frequency. The principles illustrated for single-cycle pulses can be applied also to interpret the results obtained with multicycle pulses, since the effects induced are analogous in the two cases. When multicycle pulses are used, though, the effective PECs are moving upward and downward more frequently, depending on the number of cycles. Also, the leftmost and rightmost points reached by the crossing at each cycle may vary, depending on the maximum and minimum amplitude of the field at the associated field cycle, and this influences how the field affects the dynamics. Therefore, the interpretation of the results becomes more involved: some of the effects might be washed out, others amplified, but the effect in the end may be thought of as a coherent sum of the basic interactions. Figure 8 shows some representative results obtained for fewcycle pulses defined by eq 23 with different ω, t0, and ϕ. The upper panel shows the shape of three control fields: (a) ω = 0.3 eV, t0 = 30 fs, ϕ = 0; (b) ω = 0.3 eV, t0 = 30 fs, ϕ = π; and (c) ω = 0.4 eV, t0 = 40 fs, ϕ = 0. For all cases, I = 1.7 1013 W cm2 and τ = 20 fs. The lower panel shows P1(t) under the respective control fields. In P1(t) for pulses (a) and (c), we recognize the presence of more than one transferback-transfer processes, each of them being associated with one-half-cycle with positive sign of the pulse. The rapid oscillations in the curves for (b) and (c) reflect the fact that the wavepackets Ψ1(R,t) and Ψ2(R,t) have more opportunities to interfere than in the single-cycle case. The interference is also reflected in the structure of the nuclear wavepackets, which split into several components, some with positive and some with negative momentum. We notice that multicycle pulses more likely increase the final state 1 population, enhancing the overall population transfer from state 2 to state 1. A plausible explanation for this is that,

6. CONCLUDING REMARKS As the first survey of a series of our studies on the dynamical Stark control of chemical reactions, we have reported the basic nuclear wavepacket dynamics on the coupled two lowest 1Σ+ states of LiF under control pulse laser field. The field energy used in the calculations varies from 0.1 to 0.4 eV and has been chosen such that the field oscillations occur in a time scale comparable to that of nuclear dynamics. In the present diabatic representation, in which the dipole transition matrix is diagonalized at each nuclear configuration (μ12 = 0), the state transition dynamics is dominated by the collision between the dynamical crossing point and nuclear wavepackets together with the static electronic Hamiltonian element V12(R). Therefore, the population dynamics is determined in principle in terms of (i) the number of encounters of RX(t) and the wavepackets, (ii) the amplitude of the interaction V12(R) at the crossing and the relative velocity between the wavepacket and the crossing, and (iii) interference among the individual transition amplitude and that of wavepackets. These dynamical events may be controlled by calibrating the laser parameters. As the first step of our study, we have studied some very basic population dynamics arising from only a few or less encounters between the dynamical crossing point and the wavepackets by varying the switching-on time of the laser t0. The results indicate that, at least for single- or few-cycle pulses, the sign of the pulse, and therefore its phase, plays an important role. Furthermore, they suggest that the control is more easily achieved if a semicycle pulse of positive or negative sign, depending on the system characteristics, could be used. In this sense, the statement that DSC in the Raman coupling scheme is a better controlling tool than the DSC in dipole coupling scheme is justified and supported by the results presented here. Nevertheless, as shown in section 5.2 and in agreement with the results of ref 8, the control is possible also when few-cycle pulses are used. Clearly, even better results can be obtained by using appropriately designed pulses. Alternatively, for fixed t0, pulse intensity, and frequency, the position and time at which one or more transfers take place can be influenced by varying the phase, which determines the relative weight of field components of positive and negative sign. Our results clearly show the importance and role played by the pulse intensity, which determines how much the dynamical crossing is shifted with respect to the original crossing position due to the pulse. While analogous studies have already been performed by other groups, specifically in relation to the dynamic Stark control, the main emphasis of our work has been placed in identification of the relation between nuclear wavepacket dynamics and outcome of the control, aiming at a deeper understanding of the mechanisms at the basis of the control itself. We here make a 2652


The Journal of Physical Chemistry A brief comparison of our results with those of previous works, in particular, those of refs 8 and 9. While our conclusions are along the same lines as those obtained there, we believe that our interpretation gives important insights on the control mechanisms and justifies some of the statements given there. For example, in refs 8 and 9, it is stated that the two times at which the control laser is most effective are during initial excitation (i.e., using simultaneous pump and control pulses, case not treated in this paper) and during traversal of the (original) crossing. While this statement is correct for the systems studied in those works, it is by no means of general validity. This is immediately clear if one thinks of the controlled dynamics in terms of the present dynamical picture of diabatic PECs and crossing presented in our paper. Also, the present study suggests that, by using a control pulse, it may be possible to probe the functional form of V12(R) experimentally.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Professor J€orn Manz for valuable discussions. One of the authors (S.S.) thanks the Japanese Society for the Promotion of Science (JSPS) for postdoctoral scholarship for foreign scientists. This work has been supported by Grant-in-Aid for Scientific Research from JSPS. ’ REFERENCES (1) Assion, A.; Baumert, T.; Bergt, M.; Brixner, T.; Kiefer, B.; Seyfried, V.; Strehle, M.; Gerber, G. Science 1998, 282, 919. (2) Rice, S. A.; Zhao, M. Optical Control Of Molecular Dynamics; Wiley: New York, 2000. (3) Shapiro, M.; Brumer, P. Principles Of The Quantum Control Of Molecular Processes; Wiley-Interscience: Hoboken, NJ, 2003. (4) Wollenhaupt, M.; Engel, V.; Baumert, T. Annu. Rev. Phys. Chem. 2005, 56, 25. (5) Lozovoy, V. V.; Dantus, M. Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 2006, 102, 227. (6) Calvert, C. R.; Bryan, W. A.; Newell, W. R.; Williams, I. D. Phys. Rep. 2010, 491, 1. (7) Underwood, J. G.; Spanner, M.; Ivanov, M. Y.; Mottershead, J.; Sussman, B. J.; Stolow, A. Phys. Rev. Lett. 2003, 90, 223001. (8) Han, Y.-C.; Yuan, K.-J.; Hu, W.-H.; Cong, S.-L. J. Chem. Phys. 2009, 130, 44308. (9) Sussman, B. J.; Townsend, D.; Ivanov, M. Y.; Stolow, A. Science 2006, 314, 278. (10) Molecules in Laser Fields; Bandrauk, A. D., Ed.; Marcel Dekker: New York, 1994. (11) Bandrauk, A. D.; Sink, M. L. J. Chem. Phys. 1981, 74, 1110. (12) Zavriyev, A.; Bucksbaum, P. H.; Squier, J.; Saline, F. Phys. Rev. Lett. 1993, 70, 1077. (13) Giusti-Suzor, A.; Mies, F. H.; DiMauro, L. F.; Charron, E.; Yang, B. J. Phys. B: At., Mol. Opt. Phys. 1995, 28, 309. (14) Niikura, H.; Corkum, P. B.; Villeneuve, D. M. Phys. Rev. Lett. 2003, 90, 203601. (15) Arasaki, Y.; Takatsuka, K. Phys. Chem. Chem. Phys. 2010, 12, 1239. (16) Sussman, B. J.; Underwood, J. G.; Lausten, R.; Ivanov, M. Y.; Stolow, A. Phys. Rev. A 2006, 73, 53403. (17) Sussman, B. J.; Ivanov, M. Y.; Stolow, A. Phys. Rev. A 2005, 71, 51401(R). (18) Doslic, N. J. Phys. Chem. A 2006, 110, 12400.

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(19) Takatsuka, K.; Nakamura, H. J. Chem. Phys. 1986, 85, 5779. (20) Scheit, S.; Arasaki, Y.; Takatsuka, K., to be published. (21) Giese, T. J.; York, D. M. J. Chem. Phys. 2004, 120, 7939. (22) Werner, H.-J.; Meyer, W. J. Chem. Phys. 1981, 74, 5802. (23) Alekseyev, A. B.; Liebermann, H.-P.; Buenker, R. J.; Balakrishnan, N.; Sadeghpour, H. R.; Cornett, S. T.; Cavagnero, M. J. J. Chem. Phys. 2000, 113, 1514. (24) Macas, A.; Riera, A. J. Phys. B: At., Mol. Opt. Phys. 1978, 11, 1077. (25) Faist, M. B.; Levine, R. D. J. Chem. Phys. 1976, 64, 2953. (26) Engel, V.; Metiu, H.; Almeida, R.; Marcus, R. A.; Zewail, A. H. Chem. Phys. Lett. 1988, 152, 1. (27) (a) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. Phys. Rev. Lett. 2003, 90, 248303. (b) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. J. Chem. Phys. 2003, 119, 7913. (28) Varandas, A. J. C. J. Chem. Phys. 2009, 131, 124128. (29) Heil, T. G.; Dalgarno, A. J. Phys. B: At., Mol. Opt. Phys. 1979, 12, L557. (30) Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1985, 82, 82. (31) Knowles, P. J.; Werner, H.-J. Chem. Phys. Lett. 1985, 115, 259. (32) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (33) Kendall, R. A.; Dunning, T. H. J. Chem. Phys. 1992, 96, 6796. (34) Werner, H.-J.; Knowles, P. J.; Lindh, R.; Manby, F. R.; Sch€utz, M.; Celani, P.; Korona, T.; Rauhut, G.; Amos, R. D.; Bernhardsson, A.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Lloyd, A. W.; McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nickla, A.; Palmieri, P.; Pitzer, R.; Schumann, U.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.MOLPRO, a package of ab initio programs. (35) Feit, M. D.; Fleck, J. A.; Steiger, A. J. Comput. Phys. 1982, 47, 412. (36) Kosloff, R. J. Phys. Chem. 1988, 92, 2087. (37) Alvarellos, J.; Metiu, H. J. Chem. Phys. 1988, 88, 4957. (38) Takatsuka, K.; Hashimoto, N. J. Chem. Phys. 1995, 103, 6057. (39) (a) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. Chem. Phys. Lett. 1999, 302, 363. (b) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. J. Chem. Phys. 2000, 112, 8871. (c) Takatsuka, K.; Arasaki, Y.; Wang, K.; McKoy, V. Faraday Discuss. 2000, 115, 1. (d) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. J. Electron Spectrosc. Relat. Phenom. 2000, 108, 89. (e) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. J. Chem. Phys. 2001, 114, 7941. (40) (a) Arasaki, Y.; Yamazaki, K.; Varella, M. T. do N.; Takatsuka, K. Chem. Phys. 2005, 311, 255. (b) Varella, M. T. do N.; Arasaki, Y.; Ushiyama, H.; McKoy, V.; Takatsuka, K. J. Chem. Phys. 2006, 124, 154302. (c) Varella, M. T. do N.; Arasaki, Y.; Ushiyama, H.; Takatsuka, K.; Wang, K.; McKoy, V. J. Chem. Phys. 2007, 126, 054303. (41) (a) Arasaki, Y.; Takatsuka, K. Chem. Phys. 2007, 338, 175. (b) Arasaki, Y.; Takatsuka, K.; Wang, K.; McKoy, V. J. Chem. Phys. 2010, 132, 124307. (c) Arasaki, Y.; Wang, K.; McKoy, V.; Takatsuka, K. Phys. Chem. Chem. Phys. 2011, 13, 8681.

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Controlled Dynamics at an Avoided Crossing Interpreted in Terms of

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