Non-Resonant Dynamic Stark Control at a Conical Intersection: The

Article pubs.acs.org/JPCA

Non-Resonant Dynamic Stark Control at a Conical Intersection: The Photodissociation of Ammonia Gareth W. Richings and Graham A. Worth* School of Chemistry, University of Birmingham, Edgbaston, B15 2TT, U.K. ABSTRACT: Control of the photodissociation of ammonia, by the nonresonant dynamic Stark effect, has been studied theoretically by the numerically exact propagation of wavepackets on ab initio potential energy surfaces. An assessment of the feasibility of controlling the proportion of the wavepacket which dissociates to produce ground or electronically excited state NH2 fragments, mediated by a conical intersection, has been made by use of a simple two-dimensional, two-state model. It was found that modest control was possible for nonresonant pulses applied during and after excitation, and that the control was caused not by alteration of the position or nature of the conical intersection but by modification of the energy surfaces around the Franck−Condon region. This is made possible by the predissociative nature of the mechanism for hydrogen ejection. The control effect is frequency independent but dependent on pulse, i.e., electric field, strength, indicating that it is indeed due to the Stark effect. Analysis of the control is, however, complicated by the presence of vibrational effects which can come into play if the control pulse frequency is not carefully chosen. By systematically varying the excitation energy, it was also found that the capacity for control is only significant at low energies.

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INTRODUCTION It has been a long held goal in chemistry to be able to control chemical reactions as they proceed, generating products not normally available under standard reaction conditions. With the advent of ultrafast laser pulses, it has become possible to apply electromagnetic fields on a short enough time scale to influence the propagation of the system in question at a molecular level.1−3 The shaping and timing of these pulses can be precisely tuned to maximize control, but the majority of work in this area has been done in the weak field limit.4 As such, the control can be modeled by adding a perturbation to the system’s Hamiltonian. In the work presented here, we are interested in control in the intermediate field strength regime, where the field is too strong for the perturbative assumption to hold but too weak to effect multiphoton ionization. At the intermediate pulse intensities under scrutiny here, the electric field of the photons can cause shifting of the molecular potential energy surfaces through the Stark effect. As the field is oscillating, we are here dealing with the dynamic Stark or Autler−Townes effect,5 an effect that was initially observed in molecules by Quesada et al.,6 where the spectral lines of hydrogen were split. Since that paper, it has been studied in a selection of small molecules such as Na2,7−10 Li2,11−13 N2,14 Cs2,15 NaK,16 NO,17,18 CH2,19 and IBr.20,21 When the photons generating the dynamic Stark effect are not in resonance with an electronic transition in the molecule, we are concerned with the nonresonant dynamic Stark effect (NRDSE). This has been used to control the branching of dissociating IBr into ground and excited state Br atoms.20,21 This particular experiment has also been the subject of theoretical studies in an attempt to understand the control seen.22,23 In those works, it was found that control was achieved © 2012 American Chemical Society

via the different polarizabilities possessed by the states on which the dissociating wavepacket traveled. The differences meant that the NRDSE could shift the states by varying amounts, hence moving the position of the avoided crossing between the 3 Σ0+ and 3Π0+ states and thus allowing control. Similar ideas have also been applied in theoretical studies on the control of the torsional motion in 4-methylcyclohexylidene24 and 1,1,difluoroethylene25 (both using a one-dimensional model), in controlling the dissociations of LiF26 and ICl−,27,28 and in the modification of the spin−orbit coupling of Rb2.29 In the work presented here, we extend our one-dimensional studies on IBr22 to a two-state, two-dimensional model of the dissociation of ammonia, augmented by the use of full dipole and polarizability surfaces (the IBr work used constant, representative values for the whole range of bond lengths). While the use of this very simple model may not provide the most realistic of simulations of the dissociation of ammonia, it will allow the assessment of the feasibility of NRDSE control on surfaces with a conical intersection rather than on curves with an avoided crossing, and provide a stepping stone to studies using more accurate models and on other systems. The validity of studying the dissociation of a single hydrogen atom, rather than all three together, has been justified by Seideman.30 Ammonia has been chosen as a test system because its dissociation has been extensively studied and characterized, both experimentally31−54 and theoretically.30,55−70 It is wellSpecial Issue: Jörn Manz Festschrift Received: May 29, 2012 Revised: July 29, 2012 Published: August 3, 2012 11228

dx.doi.org/10.1021/jp305216v | J. Phys. Chem. A 2012, 116, 11228−11240

The Journal of Physical Chemistry A

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known that ammonia adopts a C3v, pyramidal geometry in its electronic ground state and that the lowest energy structure in its first electronically excited state has a planar, D3h geometry.31 This change leads to a well-defined progression in the absorption spectrum, assigned to the umbrella-bending mode31−34 (denoted v2 in the literature). Excitation to this state is taken as a classic example of vibrational predissociation, as there is a barrier in the dissociation coordinate which causes the state to be quasi-bound.31,32 It is found that the rate of dissociation is strongly affected by the level of umbrella mode excitation in the electronically excited state. In the lowest two levels (usually denoted v2′ = 0, 1), dissociation occurs by tunnelling through the barrier,51,52,67 the rate for the higher vibrational state being less than that for the lower state due to the increased barrier height at geometries further from planar57 and the node in the wave function at planar geometries.51−53 With greater vibrational excitation, the molecule has enough energy to surmount the barrier and hence to directly enter the dissociation channel.46,67 Moving along the dissociation channel, there is a conical intersection between the ground and first excited electronic states at planar geometries (at planar geometries, the states have the same symmetry, but away from planar, they are of different irreducible representations and hence form avoided crossings). In this way, it is possible for the molecule to dissociate in two different channels: the wavepacket can avoid the intersection and move adiabatically to form slow H and excited NH2, or it can pass through the intersection, nonadiabatically, producing fast H and ground state NH2.46 It has been found, experimentally, that most of the dissociation occurs nonadiabatically48 but that higher excitation of the umbrella mode promotes a greater proportion of the wavepacket going along the adiabatic channel.45,46 In theoretical studies, these conclusions are supported,59 but it has also been found that using model systems of lower dimension, to study the dynamics, also results in a greater proportion of dissociation through the adiabatic channel.70 The ratio of adiabatic to diabatic dissociation is also affected by excitation of the N−H stretching modes, the asymmetric stretch promoting adiabatic dissociation more significantly than the symmetric stretching vibration does.45,48−50 In the first section of this paper, we describe the computational procedure we used to study the NRDSE control of dissociating ammonia in a two-dimensional model including the calculation of the necessary energy and property surfaces, and the method for calculating the dynamics of the system. In the Results and Discussion section, we first describe the calculated surfaces, then move on to discuss the dynamics of dissociation in our model, before analyzing control of the ratio of adiabatic to nonadiabatic dissociation by variation of the properties of the laser pulses applied to the system. In the final section, we will draw our conclusions.

center-of-mass of the NH2 fragment with length r, making an angle of θ with the x-axis. The ground and first singlet excited state diabatic energies were calculated at various points along each mode using the state-averaged CASSCF(8,7) method, implemented in Molpro71 and a 6-31++G(3df,3pd) basis. The quasi-diabatization was carried out using the MRCI-based method described by Simah et al.72 but with the necessary keyword set in Molpro to ensure only a CASSCF diabatization. This uses orbitals rotated to have maximum overlap with the orbitals at a reference point (where adiabatic and diabatic orbitals are assumed to be the same), here taken to be the geometry with r = 0.6 Å and θ = 0 rad. The bond coordinate, r, was varied from 0.6 to 5.0 Å in 0.1 Å steps, and the angle, θ, was altered between 0 and 5π/6 rad in π/18 rad steps. By symmetry, we thus get grids for angles between −5π/6 and 5π/6 rad. In addition to the energy grids, dipole moments and numerical polarizabilities for both ground and excited states were calculated at the same level, and at the same geometries as the energies, in Gaussian 03.73 Transition dipoles and numerical transition polarizabilities were also calculated, in Gaussian 03, using the CIS approach and the same basis set. The lower level method was used for the transition components in order to get a reasonable set of surfaces relatively easily. It is understood that this approach will not produce the most accurate values for the transition properties, but the approximation is justified on the grounds that, for a first investigation on the feasibility of control through a conical intersection, the general features will be qualitatively, if not quantitatively, correct. These molecular property grids displayed some numerical irregularity and as such were smoothed using Gorry’s variant74 of the Savitzky−Golay75−77 smoothing procedure. A five-point smoothing procedure with a fifth degree polynomial smoothing function was found to be satisfactory in ameliorating the spurious points. The properties were then individually rotated to the diabatic representation using the transformation matrices produced by Molpro when calculating the energy surfaces, before a second smoothing procedure. In this way, without claim to high accuracy, fair representations of the dipole and polarizability grids were produced. It should be noted that potential energy surfaces exist for ammonia with more accurate quantum chemical methods.58,65,66,78 It was, however, necessary to calculate new diabatic potential energy surfaces rather than use pre-existing ones in order to have the rotation angles necessary to represent the dipoles and polarizabilities diabatically, and it was felt that the lower accuracy method would provide adequate initial insight. In particular, the neglect of dynamic correlation was accepted as a reasonable sacrifice in order to obtain an acceptable topographic representation of the energy and property surfaces. Having produced the necessary range of energy and dipole/ polarizability grids, they were put to use in dynamics calculations to simulate the effects of performing an NRDSE experiment on ammonia. This was done using the numerically exact method of calculation within the MCTDH package,79,80 which sets up the wavepacket on the full product primitive basis. The details of the basis function grids on which the wavepackets were propagated are presented in Table 1 (it should be noted that the grids were extended, using constant energy and property values along the θ mode to allow absorption of the dissociating wavepackets by complex

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COMPUTATIONAL DETAILS In this study, we use a simple two-state, two-dimensional model30,70 to investigate the dissociation of ammonia. The model is based around a static NH2 fragment (bond lengths of 1.008 Å and bond angle of 120°) in the xy-plane and bisected by the x-axis. The modes of the dissociating hydrogen are taken relative to the center-of-mass of the fragment, with the departing atom moving in the xz-plane. The hydrogen’s position is taken as a vector in this plane with origin at the 11229



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⎛ p00 p01⎞ ⎟(εe(t )2 + εc(t )2 ) Pmn = −⎜ ⎝ p10 p11⎠

Table 1. Details of the Grids Used to Represent Each Electronic Surface in the Dynamics Calculations on Dissociating Ammonia mode

basis used

Nc

xid

xfd

θ r

exponentiala FFTb

121 100

−3.14 1.43

3.14 9.45

The m and n superscripts for the property matrices refer to the x-, y-, and z-components for the dipole−field interaction and the xx-, xy-, yy-, xz-, yz-, and zz-components for the polarizability−field interaction. The effect of the excitation and control laser pulses on the molecule is mediated by the dipole and polarizability surfaces, as shown in eqs 4 and 5. Respectively, these pulses have the form of the Gaussian-shaped cosine functions as defined in our earlier work on IBr.22 It was assumed that the molecule had a fixed orientation with respect to the laser pulses, with the NH2 fragment lying on the xy-plane and the pulses applied in the zdirection, perpendicular to this plane. As such, only the zcomponent of the dipole and zz-component of the polarizability were required.39,50,54 Before propagation, it was necessary to ensure that the initial wavepacket was in its lowest energy state on the ground state surface. This was ensured by performing a relaxation in imaginary time79,81 so as to damp out any higher energy components. Doing so, the wavepacket successfully relaxed to be evenly distributed between the two degenerate wells at r ≈ 2 bohr and θ ≈ ±1 rad. It was desirable to see the effect of the nonresonant control pulse on the dissociation when applied at various times in relation to the excitation to the first electronically excited state. In order to do this, it was necessary to assess where the propagating wavepacket was moving as it dissociated, so to measure the proportion of the wavepacket dissociating on the upper and lower diabatic surfaces, flux operators at dividing surfaces were defined along each mode for each surface.22,79 The fluxes through the surfaces were then calculated perpendicular to the angular mode at rc = 5.5 bohr and also at θc = ±2.5 rad for motions perpendicular to the dissociation coordinate. The latter were measured so as to account for the very small amount of dissociative motion with high angular component. The fluxes through the dissociation channels were recorded every 0.5 fs. At the end of each propagation, the fluxes through each channel were integrated to give total proportions of the wavepacket dissociating via the high and low energy channels. In addition to the flux operators, complex absorbing potentials were placed at appropriate points on each surface to

a

Exponential = an exponential discrete variable representation (DVR) which uses plane waves as basis functions.79,80 bFFT = fast Fourier transform type basis.79,80 cN = number of functions used along each mode. dxi and xf refer to the coordinates of the first and last DVR functions, respectively, on the grid for each mode. Both are given in atomic units (bohr for r and radians for θ).

absorbing potential functions). The equations of motion were integrated using the eighth-order Runge−Kutta method. In these calculations, it was assumed that all rotational quantum numbers affecting the nuclear framework were zero, so the full kinetic energy operator of the system is30,67,68,70 T ̂ (r , θ ) = −

1 ∂2 1⎛ 1 ∂ ∂ ⎞ − ⎜ sin θ ⎟ 2 ⎝ 2μr ∂r 2I sin θ ∂θ ∂θ ⎠

(1)

where μr = mHmNH2/mNH3 and I−1 = (μrr2)−1 + (mR2)−1 with m = 2mHmN/mNH2 and R = 0.9599808 bohr, the distance from the nitrogen atom to the center of mass of the static H2 fragment, (all units au). We also note that mi denotes the mass of fragment i. The potential terms of the Hamiltonian (both on- and offdiagonal) were provided by natural spline fits to the diabatic energy and property grids calculated as described above, and the full Hamiltonian reads ̂ + V + Dm + Pmn Ĥ = T1

(2)

where the diabatic potential energy matrix is ⎛ V00 V01⎞ V = ⎜⎜ ⎟⎟ ⎝ V10 V11 ⎠

(3)

and the dipole−field interaction is given by ⎛ d00 d01⎞ ⎟⎟(εe(t ) + εc(t )) Dm = −⎜⎜ ⎝ d10 d11 ⎠

(5)

(4)

and the polarizability−field interaction is

Figure 1. Diabatic potential energy surfaces of NH3 calculated using the CASSCF(8,7) method and 6-31++G(3df,3pd) basis in Molpro:71 (a) X̃ state; (b) Ã -state. Contour lines have been added to highlight important features and higher energy parts of the surfaces at r < 1.5 bohr omitted for clarity. The energy scale is taken relative to the energy of the lowest energy geometry of the X̃ -state (−1529.83 eV). See main text for details of the coordinate system. 11230



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Figure 2. Diabatic polarizability surfaces for the X̃ - and Ã -state of NH3 (parts a and b, respectively) calculated using the CASSCF(8,7) method and 6-31++G(3df,3pd) basis in Gaussian 03.73 See main text for details of the coordinate system. NB 1 au = 0.2798 Å2.

which play a role in mediating the population transfer between the two states under consideration. Dynamics of Dissociation. We begin with a brief discussion of the dynamics of the photodissociation of ammonia, within the two-state, two-mode model and on the energy surfaces described above. A propagation was carried out with an initial wavepacket fully relaxed in its electronic and vibrational ground state. The propagation was carried out from −300 to 1500 fs, and a 50 fs wide, Gaussian excitation pulse centered at 0 fs, of energy 4.7 eV, was applied. This particular excitation energy was chosen after a scan of possible excitation energies and was found to give the greatest population transfer from the X̃ -state to the Ã -state consistent with significant dissociation along the upper channel (the energy gap between the upper dissociation limit and the wavepacket in its relaxed state spread across the double well in the X̃ -state was found to be between 4.65 and 4.83 eV). The pulse had an intensity of 0.0025 au (1 au = 1.4 × 1015 W cm−2), firmly in the intermediate field strength regime4 where multiphoton ionization is not significant. The flux of the wavepacket through the upper and lower dissociation channels was calculated every 0.5 fs, as described above, and a plot of those fluxes against propagation time is presented in Figure 6, one curve for each channel. At t = 0 fs, the population of the upper state is 4%. The first thing to note is that the dissociation begins along both channels almost immediately on excitation, which is unsurprising, as the 4.7 eV pulse provides enough energy to surmount the small barrier in the dissociation coordinate of the upper state. However, the

prevent reflection of the wavepacket from the ends of the grid (both in the radial and angular modes).

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RESULTS AND DISCUSSION Surfaces. The potential energy surfaces, in the diabatic representation, used in the propagations, are shown in Figure 1. In our notation, we denote the adiabatic surfaces, in order of ascending energy as S0 and S1, while the diabatic surfaces are labeled X̃ and Ã . Around the Franck−Condon point, S0 and X̃ correspond, as do S1 and Ã , while beyond the diabatic crossing locus and out toward the dissociation limit, the correspondences are swapped to S0/Ã and S1/X̃ . The diabatic surfaces exhibit the salient features noted in previous calculations; the intersection between them at r ≈ 4 bohr is present, as is the double well in the electronic ground state forming degenerate global minima, and also the small barrier in the dissociation coordinate on the higher energy surface yielding a small well centered around r ≈ 2 bohr at planar geometry. In more detail, we note that the lowest energy geometry was located at r = 2.08 bohr and θ = ±1.05 rad with energy −1529.83 eV. At this Franck−Condon point, the excitation energy was 4.817 eV. The minimum energy structure in the small Ã -state well was at planar geometry with r = 2.08 bohr (E = −1525.59 eV). Vertical excitation at the same geometry was 4.052 eV. The barrier to dissociation on the Ã -state was found to be 0.37 eV. As such, while we cannot claim to have quantitative agreement with experiment or more accurate calculations, we have a set of surfaces which exhibit the crucial, qualitative features seen in previous work allowing a reasonable assesment of the feasibility of Stark control in a multidimensional model system. As noted above, we only apply laser pulses perpendicular to the NH2 fragment, in this case defining the z-direction. As such, only the z-components of the dipole and polarizability arrays were needed in the calculations. The zz-components of the polarizability matrices of each state, rotated into the diabatic representation, are presented in Figure 2. It should be noted that, at large r values, the magnitudes of the polarizabilities are similar for both states. Both have the greatest magnitude around the Franck−Condon points, the polarizability of the X̃ state reaching just beyond −30 au and that of the Ã -state falling just shy of −300 au (1 au = 0.2798 Å2), nearly an order of magnitude greater. In both cases, the negative peaks are separated by a region of far lesser polarizability at planar geometries. These features will be crucial in understanding that which follows. In Figures 3 and 4 we present the diabatic coupling and transition dipole surfaces respectively, both of

Figure 3. Diabatic coupling of the X̃ - and Ã -states calculated using the CASSCF(8,7) method and 6-31++G(3df,3pd) basis in Molpro.71 See main text for details of the coordinate system. 11231



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Figure 4. Transition dipole moment in the z-direction, μz, between the X̃ - and Ã -states calculated using the CIS method and 6-31+ +G(3df,3pd) basis in Gaussian 03.73 See main text for details of the coordinate system.

Figure 6. Total dissociation through upper and lower channels at each time during propagation relative to an excitation of 4.7 eV at 0 fs. The solid line represents flux through the upper, adiabatic channel, while the dashed line shows flux through the lower, diabatic channel.

dissociation does not occur with all of the wavepacket density moving outward at once. At the end of the propagation, there is still some dissociation occurring, a full 1.5 ps after excitation. Between excitation and this end point, the flux plots exhibit a decaying profile. The reason for this profile is the nature of the motion of the wavepacket on excitation into the weakly bound well of the upper state. Excess energy given by the pulse mainly goes into the angular vibrational mode, the minimum in the upper state is at a planar geometry, while those of the ground state are at angles of ∼ ±1 rad. As the wavepacket oscillates in this mode, a proportion escapes over or through the barrier into the dissociation channel as a planar geometry is passed. This can most easily be seen in Figure 5 where the dynamics following excitation from only the positive θ-region of the X̃ state (achieved by projecting out half of the relaxed wavepacket before running a propagation as descibed above) are exhibited as plots of wavepacket density, at representative times, on a portion of the Ã -state potential surface. Excitation begins at negative times, so in the first plot (t = −15 fs), we see the wavepacket appearing at the Franck−Condon point on the Ã state. As the density appears on the upper state, it starts to move down the well in the θ-direction. In plot b (t = +13 fs), a significant proiportion of the density has reached the negative θ side of the well and density is moving into the dissociation channel at this side. After 17 fs, the density has moved back to the original side and density is escaping over the barrier to dissociation in the positive θ region. This oscillation continues for the duration of the dissociation. Immediately after excitation, the amount of the wavepacket in the excited state well is at a maximum, so more can dissociate, leading to the peaks in the flux in both channels. As more and more leaves, there is less and less remaining to dissociate, so the fluxes decay. The oscillatory nature of the dissociation is particularly apparent in the plot of the flux through the upper channel. Integrating the fluxes over the propagation time for each channel, we can calculate the branching ratio of the proportion of the wavepacket which dissociates adiabatically and that which dissociates diabatically. The value obtained for the 4.7 eV excitation is 1.57; i.e., 61% of the wavepacket leaves adiabatically along the upper channel. This compares well to

Figure 5. Plots showing wavepacket density on the Ã -state potential surface at representative times during and after excitation from a relaxed wavepacket projected onto the positive θ half of the X̃ -state potential surface: (a) t = −15 fs; (b) t = 13 fs; (c) t = 30 fs. 11232



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peak, the relative branching ratio is 0.462, meaning 67% adiabatic dissociation, a small but significant effect. In both regimes, application of the second pulse thus has a controlling effect on the branching ratios. It is clear that the dissociation branching ratio can be controlled by application of the nonresonant pulse, but why? Taking the control at positive times, we note that the effect decays much like the flux plots do in Figure 6. This is reasonable, the amount of the wavepacket dissociating or waiting to dissociate in the excited state well reduces over time so the level of control that is possible reduces. The shape of this plot can be compared to that seen in our earlier work on IBr,22 where the postexcitation control was characterized by a welldefined peak in the relative branching ratio due to the direct dissociation nature of the dynamics in that system. The predissociation seen in ammonia yields the broader feature seen here. Observing the dynamics of the system, both with and without application of the nonresonant pulse, and the shape of the potential surfaces during application of the control pulse, the reason for the effect becomes clear. When applied at the given intensity, the effect of the control pulse on the energy surfaces is dominated by the polarizabilities. Of main interest here is the excited state around the Franck−Condon point. The two large, negative peaks in the 1.5−2.0 bohr, ±1.5−1.6 rad region of the polarizability surface (Figure 2b) raise the Ã -surface up in energy significantly. At the Franck−Condon point, the shift is about 0.9 eV when the control pulse is at maximum intensity, and only about 0.3 eV around planar geometries. This has the effect of making the sides of the well, behind the dissociation barrier, steeper in the θ-coordinate. In this way, the energy of the umbrella bending mode is increased in the excited state, an effect which is seen in the dynamics of the dissociating wavepacket. Figure 8 shows wavepacket density plots 200 fs after excitation in the region of the Ã diabatic state between the barrier in the dissociation coordinate and the conical intersection, for three limiting cases of control. We observe that, for 4.7 eV excitation and no control pulse (Figure 8a), there are two ridges of density perpendicular to the angular coordinate at about 0.5 rad on either side of the planar geometry, i.e., dissociating in a region which avoids the conical intersection, and through to the upper channel. However, there is also a similar amount dissociating in the planar geometry, i.e., capable of passing through the intersection to the lower channel. The ∼60% adiabatic dissociation is thus qualitatively rationalized. With application of the control pulse at 97 fs postexcitation (Figure 8b), distinct ridges at ±0.5−1.0 rad are seen with small outliers of the wavepacket further from planar. There is relatively less density at the planar geometries. The wavepacket can thus avoid the conical intersection more effectively when the control pulse comes after excitation; more adiabatic dissociation results. When the control pulse is centered at t = 0 (Figure 8c), the two ridges of density on either side of planar are closer in, at around ±0.3 rad, and there is a more significant ridge along the zero radian line. The density is thus moving in regions closer to the conical intersection, making it far more likely that the diabatic pathway will be taken. We postulate that, because the Ã -state’s energy surface is distorted when excitation is occurring, the levels energetically accessible from the ground state are more closely concentrated around the planar geometries. Therefore, dissociation takes place toward the conical intersection and hence via the diabatic route. The

the value of 66% obtained for a vertical excitation of the gerade wavepacket in the 2D model of Giri et al.70 Dynamic Stark Control. Initial Calculations. Having provided a brief analysis of the photodissociation dynamics of ammonia, we turn to the control of that dissociation, in particular control of the branching ratio between the two possible dissociation channels. To do this, propagations, as described in the previous section on the Dynamics of Dissociation, were carried out with the additional feature that a single 150 fs wide, 0.7 eV laser pulse was applied at a particular time relative to the excitation. Propagations were carried out with this pulse centered at times ranging in steps of 9.7 fs between 242 fs before and 968 fs after excitation (1 fs ≈ 41 atomic units of time). Fluxes through the upper and lower channels were integrated over the entire propagation time for each calculation, and the branching ratios were evaluated. The ratio for the calculation with no control pulse was then subtracted to give relative ratios. The results of these calculations against the timing of the control pulse are presented as the solid line in Figure 7.

Figure 7. Effect of 150 fs wide, 0.7 eV control pulse applied at times before, during, and after application of a 50 fs wide, 4.7 eV excitation pulse, on the NH2*/NH2 branching ratio in the dissociation of ammonia. The ratio is relative to the natural branching ratio, i.e., that obtained with no control pulse. The solid line represents calculations done with all available dipole and polarizability surfaces, while the dashed line is the plot for calculations carried out using constant polarizability surfaces (11.8 and 13.5 au for the X̃ - and Ã -states, respectively).

Beginning with the effect of applying a nonresonant pulse prior to the resonant excitation, we note that there is no effect; the relative branching ratio is zero. This is to be expected: the pulse excites no vibrational or electronic modes, so once it is turned off, the system returns to its initial state. As such, the dynamics, on excitation, are the same as that as if no control pulse had been applied. At t = 0, i.e., with the control pulse coincident with excitation, there is a large negative peak, indicating an increase in the proportion of the wavepacket dissociating along the lower, diabatic channel. In fact, the relative branching ratio is −1.269. The diabatic dissociation is thus increased to 77% of the total from 39% with no control. When the control pulse is applied after excitation, we get a broad, positive feature which decays at later times, meaning that a greater proportion of the wavepacket dissociates along the upper, adiabatic channel. At its 11233



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control and with control applied after excitation, the wavepacket has four nodes, whereas, with coincident control and excitation, there are only two. The wavepacket is thus in a lower energy umbrella-vibrational state than when the control is applied later or not at all, and as such is channelled along the planar regions of the potential energy surface, toward the conical intersection. At this point, it is perhaps appropriate to mention the previous work on the Stark control of dissociating molecules as a comparison to the discussion above. In those works, onedimensional models were used and control was achieved by the shifting of the position of the intersection of two states, either to affect the branching ratio22,23 as here or to prevent unwanted dissociation.24 As such, it was decided to investigate whether this effect was causing any alteration in the branching ratios of dissociating ammonia. To isolate the effect of the shifting of the conical intersection induced by the polarizabilities of both surfaces, control calculations were performed as described above but with the full polarizability surfaces replaced by constant values for all geometries. The constants chosen were the values of the polarizabilities at the calculated gridpoint closest to the intersection (r = 3.97 bohr and θ = 0 rad), and for the X̃ - and Ã -states respectively were 11.8 and 13.5 au. The coupling polarizability between the surfaces was set to 0.07 au. The full dipole surfaces were maintained in these calculations, and the results are given by the dashed line in Figure 7. It is immediately clear that the control is miniscule compared to that given when using the full α-surfaces; the line is almost flat on the scale used. From this null result, we can conclude that, in this case, there is essentially no control of the branching ratio induced either by the dipole moments of the surfaces or by the shifting of the position of the conical intersection through the polarizability differential in its neighborhood. Effect of Control Pulse Strength. Having established a likely control mechanism for the dissociation of ammonia, we can begin to look at other aspects around the effect. To that end, we performed a series of calculations using the same excitation and control frequencies as in the previous section but with a systematic variation of the intensity of the control pulse. This was achieved by the modification of the normalization constant, N2, in eq 6 of our earlier work.22 For a pulse of width σ2, N2 = 2s2(πσ2)−1/2, where s2 is a strength parameter which we can vary. For the control pulse of 0.02 au intensity (equal to 2.8 × 1013 W cm−2) used above, s2 = 1.4. As such, we varied s2 from 1.0 to 2.0 in steps of 0.2 to see the effect on the plot of relative branching ratios. The results of this are shown in Figure 9. It is immediately apparent that the decaying peak at t > 0 monotonically increases in maximum height (from 0.3582 to 0.6681) as s2 is increased. This is easily rationalized: as the pulse strength rises, the Stark effect on the surfaces is magnified by the greater electric fields present. Because of this, the distortion of the well sides in the Ã state, as discussed previously, increases and so does the amount of energy added to the umbrella mode. Therefore, the proportion of dissociation along channels at bond angles away from the planar, and thus along the adiabatic pathway, is increased. The increase in the magnitude of the negative peak at t = 0 with a rise in the value of s2 from 1.0 to 1.6 can also be explained in a similar way. The enhanced Stark shaping of the Ã -surface which raises the energy around the Franck−Condon point means that portions of the surface, with angles ever closer to planar, become energetically inaccessible from the ground state. Hence, the excited wavepacket becomes more localized

Figure 8. Representative contour plots of wavepacket density, at t = 200 fs, on a portion of the Ã -state, between the barrier in the dissociation coordinate (at r ≈ 2.5 bohr) and the conical intersection. The location of the conical intersection is marked by the X in each plot. See Figure 1 for a more detailed picture of the energy surfaces. (a) Density plot with no control pulse applied. (b) Density with a control pulse applied 97 fs after excitation. (c) Density with a control pulse applied at t = 0 fs, i.e., coincident with excitation.

negative peaks in the polarizability surface of the X̃ -state, despite being in the same region as those of the upper state, are an order of magnitude smaller, and therefore there is no corresponding rise in the energy of the ground state to maintain resonance with the higher angle parts of the Ã -state. This is borne out by the nodal structure of the wavepacket in the region of the energy surface at short bond lengths: with no 11234



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Figure 9. Effect of increasing the intensity of the control pulse on the branching ratio of dissociating ammonia (see Figure 7 for further details).

Table 2. Natural Branching Ratios and Percentage Adiabatic Dissociation for Ammonia as a Function of Excitation Energy excitation (eV)

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.4

5.5

5.6

branching ratio % adiabatic

1.57 61.06

2.02 66.91

4.35 81.31

4.25 80.97

8.13 89.05

9.86 90.79

9.52 90.49

10.25 91.11

9.89 90.82

7.61 88.38

Figure 10. Effect of the energy of the excitation pulse on the capacity for nonresonant Stark control of the branching ratio of the dissociation of ammonia (see Figure 7 for further details).

around the planar geometries, encouraging a greater level of dissociation through the conical intersection onto the S0 surface. At s2 = 1.8−2.0, we also get a splitting of the peak at t = 0 into two, with the branching ratio at t = 0 itself becoming more positive until at s2 = 2.0 it rises above 0. A possible reason for this would be that we are entering a regime where the pulse is intense enough for a multiphoton effect to become significant. It was noted previously that we were interested in the intermediate field-strength regime, where such effects are negligible, but increasing the intensity of the field takes us further away from this ideal; in effect, the model breaks down. By calculating the Stark shifting of the states, when the control pulse is applied, it is apparent that in the region of the potential surfaces, where the delocalized wavepacket is excited, the excitation energies range from 5 to 7 eV (dependent on the exact position). The sum of the excitation and control pulse energies is 5.4 eV (6.1 eV if two control photons contribute), adequate to excite high up the sides of the well in the Ã -state, giving a portion of the wavepacket with high excitation in the

umbrella mode. This would encourage dissociation at higher bond angles, leading to a greater proportion of dissociation along the upper channel. This mechanism would be in competition with the single photon mechanism seen at lower pulse intensities. At t < 0, we see no control effects, as would be expected from the explanation in the previous section. Any change in the shape of the surfaces, within reason, is of no consequence before excitation has occurred if the change has subsided in time. Effect of Excitation Energy. Moving on from the effect of the intensity of the control pulse, we engaged in a systematic study of the consequences of changing the frequency of the excitation pulse on the branching ratios and the capacity for controlling those ratios. To do this, calculations were carried out with the pulse properties described in the section discussing the initial Stark control calculation but with the excitation energy varied from 4.7 to 5.6 eV in 0.1 eV steps. The same intensity for the excitation pulse was used as before in order to keep constant any Stark shaping that could be caused by that 11235



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Figure 11. Effect of neglecting components of the Hamiltonian on the relative branching ratios for 5.5 eV excitation energies, specifically the polarizability via the excitation with or without the dipole moments via the control pulse (see Figure 7 for further details).

adiabatically, so it appears that any effect of the extra vibrational energy, due to the control pulse, is negligible. The maximum branching ratio is achieved because no more density can effectively be pushed away from the conical intersection. Between 5.4 and 5.6 eV, there is some negative control at both positive and negative times, meaning that the second pulse is causing an increased preference for the diabatic channel. The shape of the t = 0 peak also becomes more complex, exhibiting splitting into doublets and triplets. The latter effect can be rationalized by looking at the potential surfaces under the influence of Stark shifting by the control pulse. At the Franck− Condon point, the excitation energy is around 5.5 eV at maximum distortion (coincident with the peak of the Gaussian pulse); hence, we get maximum population transfer in this region. The dynamics on the Ã -state thus proceeds with the bulk of the wavepacket starting, with high energy, at geometries far from planar. As the Stark distortion weakens with the decaying pulse, the wavepacket will have high excitation in the θ-mode and as such will be able to dissociate in the channels further from the conical intersection. The triplet shape seen at 5.4 eV is explained similarly, except that here the distortion which brings the transition into resonance is less than the maximum possible. As such, application of the control pulse slightly before or after excitation means that the lesser distortion necessary for resonance occurs coincident with excitation. At 5.6 eV, the control pulse is never strong enough for resonance to be induced, so we see a lesser effect (there is still some splitting due to the uncertainty in the frequencies produced by the pulses, i.e., down to 5.5 eV). The first effect (negative control at negative and positive times) is surprising, as we have seen that the effect of the control pulse is usually to direct wavepacket density away from the conical intersection. As such, further sets of calculations were carried out in an attempt to rationalize the behavior. Using the standard intensity excitation and control pulses, of energies 5.5 and 0.7 eV, respectively, two sets of calculations were carried out, both of which neglected the effect of the polarizability through the excitation pulse in the Hamiltonian

pulse. Even though the maximum populations of the upper state are reduced due to movement away from optimum resonance, they were still adequate to get reliable branching ratios (at t = 0, population of the upper state was 4.0% for an excitation of 4.7 eV, monotonically reducing to 0.15% at 5.6 eV). Before describing the potential for Stark control of the branching ratios at different excitations, it will prove useful to discuss the ratios without application of the control pulse. The numbers are presented in Table 2, both in raw form and in terms of the percentage dissociation along the upper, adiabatic channel. The general trend to note is the overall increase in the branching ratio with excitation energy up to a maximum of around 90% adiabatic dissociation. By examining the dynamics of the wavepackets, it is clear that the rise in the excitation energy magnifies the proportion of the density seen dissociating in channels corresponding to bond angles up to 1 rad from planar. The excess energy goes into the umbrella vibrational mode, encouraging dissociation away from the conical intersection, minimizing the possibility of passing down to the lower surface. Moving on to the possibility of control with higher excitation energies, we plot the relative branching ratios against timing of the control pulse for each excitation energy under scrutiny in Figure 10. Concentrating on the t > 0 regime, the decaying, positive control seen for the 4.7 eV excitation is also seen for excitation energies up to and including 5.0 eV. Referring back to Table 2, these are the excitations which give the relatively low branching ratios when no control pulse is applied, in other words a comparatively high proportion of diabatic dissociation through the conical intersection. The control pulse thus has the same effect as with the initial calculation at 4.7 eV, squeezing of the Ã -state well causing increased umbrella mode motion which directs the wavepacket away from the conical intersection and toward the adiabatic channel. In contrast, between 5.1 and 5.3 eV inclusive, there is little or no control after excitation. These excitations give natural branching ratios where ∼90% of the wavepacket dissociates 11236



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Figure 12. Effect of the frequency of the control pulse on the capacity for nonresonant Stark control of the branching ratio of the dissociation of ammonia (see Figure 7 for further details).

eq 2, αexc, and one of those also omitted the effect of the dipoles (static and transition), μcont, through the control pulse. The results of these calculations (longer dashes for just polarizability removed, shorter dashes for both polarizability and dipole), in relation to the standard plot under these conditions from Figure 10 (solid line), are presented in Figure 11. The first point to note is that the change due to removing only αexc is very small, producing just minimal changes in the intensity of features present in the plot from the full calculation. This is to be expected from the relatively weak excitation pulse which can only induce minor changes in the topography of the energy surfaces through the Stark effect. In contrast, a major change is achieved by neglecting the dipole effect through the control pulse. Both at t < 0 and t > 50 fs, the negative control becomes washed out and returns to essentially 0, as seen with the medium-valued excitation energies. However, the most noticeable change is around t = 0 where the complex double, negative peak becomes a small, positive peak. Both of these changes indicate that the control at this high excitation energy is due to dipolar effects rather than pulse-envelope following, polarizability-induced energy surface shaping. The decaying negative control feature seen at t > 50 fs in the full picture is thus possibly due to stimulated emission causing de-excitation of the wavepacket when resonance conditions are fulfilled around the conical intersection22 (as they must be at some points). There is still some negative control around t = 50 fs, indicating a small Stark control effect which pushes a small amount of extra density toward the conical intersection through surface distortion. The positive control around t = 0 must also be due to the polarizability-induced Stark effect, perhaps an extra jolt to the θ-mode causing extra avoidance of the conical intersection. It should also be noted that a calculation with excitation of 4.7 eV and all other pulse features being as standard was carried out with both the αexc and μcont components removed from the Hamiltonian. In this case, the change in the plot from that in Figure 7 was negligible and so is not reproduced here. It follows that, at low excitation energies, the polarizability-mediated Stark effect dominates the control of the branching ratio but that at higher energies the picture is complicated by dipolar interactions, especially when the excitation and control pulses overlap. It is not immediately clear why there are two different control regimes for the lower and higher excitation energies, but a possible reason can be seen by considering the

magnitudes of the static dipole moments at different geometries. At low excitation energies, the wavepacket can only be excited to the Ã -state at geometries close to planar (especially when the upper surface has been Stark shifted by the polarizability interaction), but at these geometries, the magnitudes of the dipoles (for both states) are close to 0 in the z-direction; hence, any subtle effect caused by dipoleinduced shifting of the potential surfaces is minimal. With higher excitation energies, the parts of the wavepacket which are excited are those where the surfaces are further apart, i.e., those with greater θ-values. At these geometries, the dipole moments have greater magnitude; in fact, at r = 2.08 bohr, the maximum absolute dipole for the X̃ -state is 1.14 D at θ = ±1.2 rad and for the Ã -state is 1.98 D at ±1.9 rad. As such, any effects mediated by the dipole moments at time close to excitation will be of greater significance for higher excitation energies. Controlling the dissociation of ammonia is thus not a trivial matter where one effect always makes the major contribution: a detailed study of each system is needed to predict which effect would dominate in which excitation/control regime. Control Pulse Frequency. As a final insight into the control of the branching ratio of dissociating ammonia, we look at the effect of varying the frequency of the control pulse. As such, calculations using the 4.7 eV excitation pulse were carried out using 0.02 au intensity control pulses of frequencies corresponding to energies between 0.1 and 0.8 eV. The branching ratio plots for each set of calculations are shown in Figure 12. The first point to note from the figure is the similarity of the plots with control pulses of ≥0.4 eV. All have the same basic structure as the plot in Figure 7: there is little or no control for pulses applied prior to excitation, moderate negative control for the pulses overlapping excitation, and decaying, positive control at times after excitation. The maxima of the decaying tails, at around 70 fs after excitation, have relative branching ratios between 0.46 and 0.62, which correspond to 67−69% adiabatic dissociation, a minor difference. The frequency independence of these plots is evidence for the assertion that the control is mediated by the Stark effect, shaping the potential surfaces through the intensity of the applied electric field rather than through any resonant medium. Moving to lower frequencies, we see some differences in the plots. Starting with the 0.3 eV control pulse, there is again no control at t < 0, nor is there any at t = 0. The major difference, 11237



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though, is the very large, decaying peak at t > 0, indicating a major favoring of the adiabatic channels during dissociation. Having said that the Stark control effect is frequency independent, what is the cause of this extra effect? The assumption used in applying lasers to cause a Stark shift was that the pulse would be nonresonant. This is apparently not the case when a 0.3 eV pulse is applied. Calculating the vibrational frequencies of the whole molecule (with Molpro at the same level of theory as used for the energy surfaces) when in the quasi-bound region of the excited state shows that there is a degenerate antisymmetric stretching mode at 3201 cm−1 (0.397 eV) with an intensity relative to the maximum intensity mode of 31.15%. It is thus possible that, within our two-dimensional model, a stretching mode is being excited at about 0.3 eV. It has been noted previously45 that excitation of the antisymmetric stretching mode in the ground state strongly promotes adiabatic dissociation, so excitation of a similar mode in the excited state would appear to be a plausible reason for the large peak giving up to 83% product in the adiabatic channel. In effect, a vibrational control overlays the Stark control induced by the presence of the electric field. This vibrational effect is much more significant than the Stark control alone (which increased the adiabatic portion from 61% to at most 67%). The disappearance of the negative peak at t = 0 can be ascribed to the width of the control pulse. The control pulse has a width of 150 fs, so even if centered at t = 0, there is significant overlap with the t > 0 regime. The large positive control, at times postexcitation, counteracts the negative control around excitation. The small sharp peaks just before and after excitation, with the 0.4 eV control pulse, suggest some of this vibrational control is present in this case as well (due to slight overlapping of the pulse energy with the vibrational resonance). We turn next to the 0.1 eV control pulse, where we see a different deviation from the plots seen with high control pulse energies. The positive tail at t > 0 is in place, indicating just the normal Stark control effect in that region. The major difference in the plot is at times prior to excitation, where there is a positive, oscillating control effect centered around a relative branching ratio of 0.4. Clearly, this is not due to the Stark effect. As we have already noted, the distortion of the energy surfaces prior to excitation has little or no effect on the branching ratio. The effect must be vibrational. This is seen by observing the dynamics of the system in the ground state after application of a 0.1 eV pulse to the relaxed wavepacket, the result of which is vibrational motion in the angular coordinate. Molpro calculations of the frequencies of the ground state also indicate that the most intense mode is the umbrella vibrational mode (with energy of 0.142 eV), corresponding to our angular coordinate. Thus, we have a pre-excitation of this vibrational mode prior to excitation to the electronically excited state. This angular motion is thus transferred to the upper state where it enhances dissociation along the adiabatic pathway, much as is achieved by the Stark shaping of the upper surface after excitation. Within the resolution provided by our spacing of the control pulse on the plots, the peaks in this oscillating control feature are approximately 40 fs apart, the same separation in time of consecutive peaks of a light pulse with 0.1 eV energy. The exact amount of control is thus determined by where in the vibrational motion the wavepacket is when electronic excitation occurs, which is, in turn, decided by the exact timing of the control pulse. This effect carries through to the t = 0 regime where vibrational and electronic excitation occur simultaneously.

The only plot remaining to mention is that with a control pulse of 0.2 eV. The general shape of the purely Stark controlled plots with ≥0.4 eV control pulses is seen here, but there is a small, sharp peak just after excitation. This would appear to be due to some small overlap of the pulse energy with the vibrational mode close to 0.3 eV: we thus get a residue of the vibrational control. It is apparent from the preceding that there are two types of control mechanisms which can be exploited by judicious choice of control pulse frequency; the nonresonant Stark effect which is present for all frequencies and also vibrational control pathways in both ground and electronically excited states when a resonant pulse is chosen. In the latter case, both resonant and nonresonant pathways compete or complement one another as appropriate (vide t = 0 and t > 0 for the 0.3 eV control pulse).

■

CONCLUSIONS In this paper, we have presented a theoretical study of the control by nonresonant laser pulses of the branching of a molecule dissociating through a conical intersection based on ammonia which dissociates into ground and electronically excited state NH2 fragments. It was found that, for all excitation energies under investigation, the majority of the wavepacket passes along the adiabatic channel to produce excited NH2 and a slow hydrogen atom. This preponderance is generally enhanced by increasing the excitation energy, the excess energy going into exciting the angular vibrational mode in the Ã -state which gives the wavepacket more opportunity to avoid the conical intersection, leading to the diabatic route. We have demonstrated that it is possible to use the nonresonant Stark effect to control the branching ratio through the shaping of the potential energy surfaces. In contrast to earlier work, where the control had been obtained by the shifting of the position of the conical intersection as a function of the molecular coordinates, the control observed in our case was through the shaping of the Ã -state around the Franck− Condon point. This shaping affected the angular motion of the wavepacket, enabling it better to avoid the intersection and hence enhancing the proportion of the wavepacket dissociating adiabatically. The similarity of the polarizabilities of the two states at and around the intersection means that its movement is minimal, and it was shown that control by this mechanism was barely noticeable. In addition to the Stark control, which was shown to be independent of the frequency of the control pulse, other effects can come into play depending on the conditions chosen. If a control pulse resonant with a vibrational mode in the ground state is chosen, we can see control effects if the pulse is applied before excitation. In a similar manner, where the hydrogen removal follows a vibrationally predissociative path and there are bound vibrational modes in the excited state, we can get major enhancement of the control effect (and possibly cancellation with the Stark effect in other systems). In the case of this model system, the effect is greater than the Stark control alone. It was also found that with high excitation energies, giving lower populations in the excited state, the observed control was not due to the polarizability interaction as expected from an NRDSE control model but due to dipole interactions with the control field. These would include the laser-type de-excitation effects seen previously.22 The predissociative nature of the mechanism also means that control can be exerted over a long time period, as the majority of the wavepacket remains in the upper well as only small 11238



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(10) Ahmed, E. H.; Qi, P.; Beser, B.; Bai, J.; Field, R. W.; Huennekens, J. P.; Lyyra, A. M. Phys. Rev. A 2008, 77, 053414/1− 053414/7. (11) Schlöder, U.; Deuschle, T.; Silber, C.; Zimmermann, C. Phys. Rev. A 2003, 68, 051403/1−051403/4. (12) Qi, J.; Lazarov, G.; Wang, X.; Li, L.; Narducci, L. M.; Lyyra, A. M.; Spano, F. C. Phys. Rev. Lett. 1999, 83, 288−291. (13) Salihoglu, O.; Qi, P.; Ahmed, E. H.; Kotochigova, S.; Magnier, S.; Lyyra, A. M. J. Chem. Phys. 2008, 129, 174301/1−174301/8. (14) Girard, B.; Sitz, G. O.; Zare, R. N.; Billy, N.; Vigué, J. J. Chem. Phys. 1992, 97, 26−41. (15) Tolra, B. L.; Drag, C.; Pillet, P. Phys. Rev. A 2001, 64, 061401/ 1−061401/4. (16) Sweeney, S. J.; Ahmed, E. H.; Qi, P.; Kirova, T.; Lyyra, A. M.; Huennekens, J. J. Chem. Phys. 2008, 129, 154303/1−154303/10. (17) Duxbury, G.; Kelly, J. F.; Blake, T. A.; Langford, N. J. Chem. Phys. 2012, 136, 174318/1−174318/6. (18) Duxbury, G.; Kelly, J. F.; Blake, T. A.; Langford, N. J. Chem. Phys. 2012, 136, 174319/1−174319/13. (19) Kim, Y.; Hall, G. E.; Sears, T. J. Phys. Chem. Chem. Phys. 2006, 8, 2823−2825. (20) Sussman, B. J.; Ivanov, M. Y.; Stolow, A. Phys. Rev. A 2005, 71, 051401/1−051401/4. (21) Sussman, B. J.; Townsend, D.; Ivanov, M. Y.; Stolow, A. Science 2006, 314, 278−281. (22) Sanz-Sanz, C.; Richings, G. W.; Worth, G. A. Faraday Discuss. 2011, 153, 275−291. (23) Marquetand, P.; Richter, M.; González-Vázquez, J.; Sola, I.; González, L. Faraday Discuss. 2011, 153, 261−273. (24) Kinzel, D.; Marquetand, P.; González, L. J. Phys. Chem. A 2012, 116, 2743−2749. (25) González-Vázquez, J.; González, L.; Sola, I. R.; Santamaria, J. J. Chem. Phys. 2009, 131, 104302/1−104302/5. (26) Scheit, S.; Arasaki, Y.; Takatsuka, K. J. Phys. Chem. A 2012, 116, 2644−2653. (27) Chang, B. Y.; Shin, S.; Sola, I. R. J. Chem. Phys. 2009, 131, 204314/1−204314/6. (28) Chang, B. Y.; Shin, S.; Santamaria, J.; Sola, I. R. J. Chem. Phys. 2009, 130, 124320/1−124320/9. (29) González-Vázquez, J.; Sola, I. R.; Santamaria, J.; Malinovsky, V. S. Chem. Phys. Lett. 2006, 431, 231−235. (30) Seideman, T. J. Chem. Phys. 1995, 103, 10556−10565. (31) Douglas, A. E. Discuss. Faraday Soc. 1963, 35, 158−174. (32) Ashfold, M. N. R.; Bennett, C. L.; Dixon, R. N. Faraday Discuss. Chem. Soc. 1986, 82, 163−175. (33) Vaida, V.; McCarthy, M. I.; Engelking, P. C.; Rosmus, P.; Werner, H.-J.; Botschwina, P. J. Chem. Phys. 1987, 86, 6669−6676. (34) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Chem. Phys. 2002, 116, 9315−9325. (35) Reid, J. P.; Loomis, R. A.; Leone, S. R. J. Chem. Phys. 2000, 112, 3181−3191. (36) Reid, J. P.; Loomis, R. A.; Leone, S. R. J. Phys. Chem. A 2000, 104, 10139−10149. (37) Reid, J. P.; Loomis, R. A.; Leone, S. R. Chem. Phys. Lett. 2000, 324, 240−248. (38) Loomis, R. A.; Reid, J. P.; Leone, S. R. J. Chem. Phys. 2000, 112, 658−669. (39) Mordaunt, D. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1998, 109, 7659−7662. (40) Mordaunt, D. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1996, 104, 6460−6471. (41) Mordaunt, D. H.; Dixon, R. N.; Ashfold, M. N. R. J. Chem. Phys. 1996, 104, 6472−6481. (42) Woodbridge, E. L.; Ashfold, M. N. R.; Leone, S. R. J. Chem. Phys. 1991, 94, 4195−4204. (43) Henck, S. A.; Mason, M. A.; Yan, W.-B.; Lehmann, K. K.; Coy, S. L. J. Chem. Phys. 1995, 102, 4772−4782. (44) Henck, S. A.; Mason, M. A.; Yan, W.-B.; Lehmann, K. K.; Coy, S. L. J. Chem. Phys. 1995, 102, 4783−4792.

proportions dissociate. As such, wavepacket density is present for a long time in a region where it can be affected. The importance of using full polarizability surfaces has been demonstrated in this work. Our earlier study22 relied on taking a single, representative values of α for the entire surface, but this means one can miss the important effect of the gradient of the energy surface being altered, which can then alter the vibrational motion of the molecule. It appears that the Stark effect can wield its influence in a variety of ways. The complexity of analyzing the control of systems such as this has become apparent, even for what should be a simple model of dissociation. There are several interactions which have an effect, and isolating the most important in a particular case is not straightforward. This was made especially difficult by the strongly delocalized nature of the dissociating wavepacket due to the predissociation. It was noted in the Introduction that this study was intended as an exploratory work to assess the feasibility of Stark control on multidimensional surfaces using a qualitative, rather than high-accuracy, model of ammonia. As such, it is not the definitive work on the control of the photodissociation of ammonia, but the results are adequately promising such that it would be of great interest to extend the study to the full sixdimensional model (using higher level methods to generate the surfaces), not least because of the large effect extension to the higher dimensions has on the branching ratio.70 In these situations, where the diabatic pathway becomes ever more dominant, the ability to control dissociation by increasing the adiabatic proportion, as seems possible from the work here, would be of great interest. Extension to models including a larger number of states, particularly the diabatic state which causes the barrier in our Ã -state, would also be of significant value. It is also a matter of significant curiosity as to what the results would be if rotation of the molecule and its fragments could be included in the calculations, as it has been found that the conical intersection provides a torque to the dissociating wavepacket,37,40,39 thus promoting rotational excitation in the NH2 fragment.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Rice, S. A.; Zhao, M. Optical Control of Molecular Dynamics; John Wiley: New York, 2000. (2) Brumer, P. W.; Shapiro, M. Principles of Quantum Control of Molecular Processes; John Wiley: New York, 2003. (3) Brixner, T.; Gerber, G. ChemPhysChem 2003, 4, 418−438. (4) Worth, G. A.; Sanz, C. S. Phys. Chem. Chem. Phys. 2010, 12, 15570−15579. (5) Autler, S. H.; Townes, C. H. Phys. Rev. 1955, 100, 703−722. (6) Quesada, M. A.; Lau, A. M. F.; Parker, D. H.; Chandler, D. W. Phys. Rev. A 1987, 36, 4107−4110. (7) Garcia-Fernandez, R.; Ekers, A.; Klavins, J.; Yatsenko, L. P.; Bezuglov, N. N.; Shore, B. W.; Bergmann, K. Phys. Rev. A 2005, 71, 023401/1−023401/10. (8) Yi, P.; Song, M.; Liu, Y.; Field, R. W.; Li, L.; Lyyra, A. Opt. Commun. 2004, 233, 131−138. (9) Ahmed, E.; Hansson, A.; Qi, P.; Kirova, T.; Lazoudis, A.; Kotochigova, S.; Lyyra, A. M.; Li, L.; Qi, J.; Magnier, S. J. Chem. Phys. 2006, 124, 084308/1−084308/13. 11239



Article

(45) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Chem. Phys. 2002, 116, 4955−4961. (46) Biesner, J.; Schnieder, L.; Ahlers, G.; Xie, X.; Welge, K. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1989, 91, 2901−2911. (47) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Chem. Phys. 2003, 118, 7144−7145. (48) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Phys. Chem. A 2003, 107, 10490−10496. (49) Bonhommeau, D.; Valero, R.; Truhlar, D. G.; Jasper, A. W. J. Chem. Phys. 2009, 130, 234303/1−234303/17. (50) Hause, M. L.; Yoon, Y. H.; Crim, F. F. J. Chem. Phys. 2006, 125, 174309/1−174309/7. (51) Ziegler, L. D. J. Chem. Phys. 1985, 82, 664−669. (52) Biesner, J.; Schnieder, L.; Schmeer, J.; Ahlers, G.; Xie, X.; Welge, K. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1988, 88, 3607− 3616. (53) Ashfold, M. N. R.; Bennett, C. L.; Dixon, R. N. Chem. Phys. 1985, 93, 293−306. (54) Wells, K. L.; Perriam, G.; Stavros, V. G. J. Chem. Phys. 2009, 130, 074308/1−074308/5. (55) Harshbarger, W. R. J. Chem. Phys. 1970, 53, 903−911. (56) Avouris, P.; Rossi, A. R.; Albrecht, A. C. J. Chem. Phys. 1981, 74, 5516−5520. (57) McCarthy, M. I.; Rosmus, P.; Werner, H.-J.; Botschwina, P.; Vaida, V. J. Chem. Phys. 1987, 86, 6693−6700. (58) Rosmus, P.; Botschwina, P.; Werner, H.-J.; Vaida, V.; Engelking, P. C.; McCarthy, M. I. J. Chem. Phys. 1987, 86, 6677−6692. (59) Bonhommeau, D.; Truhlar, D. G. J. Chem. Phys. 2008, 129, 014302/1−014302/15. (60) Dixon, R. N. Chem. Phys. Lett. 1988, 147, 377−383. (61) Dixon, R. N.; Hancock, T. W. R. J. Phys. Chem. A 1997, 101, 7567−7575. (62) Tang, S. L.; Imre, D. G. Chem. Phys. Lett. 1988, 144, 6−10. (63) Tang, S. L.; Abramson, E. H.; Imre, D. G. J. Phys. Chem. 1991, 95, 4969−4976. (64) Lin, H.; Thiel, W.; Yurchenko, S. N.; Carvajal, M.; Jensen, P. J. Chem. Phys. 2002, 117, 11265−11276. (65) Nangia, S.; Truhlar, D. G. J. Chem. Phys. 2006, 124, 124309/1− 124309/13. (66) Li, Z.; Valero, R.; Truhlar, D. Theor. Chem. Acc. 2007, 118, 9− 24. (67) Lai, W.; Lin, S. Y.; Xie, D.; Guo, H. J. Chem. Phys. 2008, 129, 154311/1−154311/7. (68) Lai, W.; Lin, S. Y.; Xie, D.; Guo, H. J. Phys. Chem. A 2010, 114, 3121−3126. (69) Yarkony, D. R. J. Chem. Phys. 2004, 121, 628−631. (70) Giri, K.; Chapman, E.; Sanz, C. S.; Worth, G. J. Chem. Phys. 2011, 135, 044311/1−044311/13. (71) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M.; et al. Molpro, version 2009.1; http://www.molpro.net: Cardiff, U.K., 2009. (72) Simah, D.; Hartke, B.; Werner, H.-J. J. Chem. Phys. 1999, 111, 4523−4534. (73) Frisch, M. J.; et al. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. (74) Gorry, P. A. Anal. Chem. 1990, 62, 570−573. (75) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1964, 36, 1627−1639. (76) Steinier, J.; Termonia, Y.; Deltour, J. Anal. Chem. 1972, 44, 1906−1909. (77) Madden, H. H. Anal. Chem. 1978, 50, 1383−1386. (78) Zhu, X.; Ma, J.; Yarkony, D. R.; Guo, H. J. Chem. Phys. 2012, 136, 234301/1−234301/12. (79) Beck, M. H.; Jäckle, A.; Worth, G. A.; Meyer, H. D. Phys. Rep. 2000, 324, 1−105. (80) Worth, G. A.; Beck, M. H.; Jackle, A.; Meyer, H. D. The MCTDH Package, version 10.2; http://www.pci.uni-heidelberg.de/ cms/mctdh.html, University of Heidelberg: Heidelberg, Germany, 2012. (81) Kosloff, R.; Tal-Ezer, H. Chem. Phys. Lett. 1986, 127, 223−230. 11240


Non-Resonant Dynamic Stark Control at a Conical Intersection: The

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