Probing in Space and Time the Nuclear Motion Driven by

Article pubs.acs.org/JPCA

Probing in Space and Time the Nuclear Motion Driven by Nonequilibrium Electronic Dynamics in Ultrafast Pumped N2 J. Ajay,† J. Šmydke,*,†,‡ F. Remacle,†,§ and R. D. Levine†,∥ †

The Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ‡ Department of Radiation and Chemical Physics, Institute of Physics, Academy of Sciences of the Czech Republic, 18221 Praha 8, Czech Republic § Département de Chimie, B6c, Université de Liège, B4000 Liège, Belgium ∥ Crump Institute for Molecular Imaging and Department of Molecular and Medical Pharmacology, David Geffen School of Medicine and Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, United States S Supporting Information *

ABSTRACT: An ultrafast electronic excitation of N2 in the vacuum ultraviolet creates a nonstationary coherent linear superposition of interacting valence and Rydberg states resulting in a net oscillating dipole moment. There is therefore a linear response to an electrical field that can be queried by varying the time delay between the pump and a second optical probe pulse. Both the pump and probe pulses are included in our computation as part of the Hamiltonian, and the timedependent wave function for both electronic and nuclear dynamics is computed using a grid representation for the internuclear coordinate. Even on an ultrafast time scale there are several processes that can be discerned beyond the expected coherence oscillations. In particular, the coupling between the excited valence and Rydberg states of the same symmetry is very evident and can be directly probed by varying the delay between pulse and probe. For quite a number of vibrations the nuclear motion does not dephase the electronic disequilibrium. However, the nuclear motion does modulate the dipolar response by taking the wave packet in and out of the Franck−Condon region and by its strong influence on the coupling of the Rydberg and valence states. A distinct isotope effect arises from the dependence of the interstate coupling on the nuclear mass.

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INTRODUCTION In the Born−Oppenheimer description, the electrons are at equilibrium at any given position of the nuclei. As the nuclei move to a new position, the light electrons can track the change adiabatically. This is the Born−Oppenheimer approximation. A gradual change in the physicochemical state of the electrons is possible, e.g., from ionic to covalent, but the change requires that the nuclei move and that the electrons adiabatically track this motion. For a faster motion of the nuclei, the electrons can change their adiabatic state, a so-called nonadiabatic transition. If this occurs, one speaks of a breakdown of the Born− Oppenheimer approximation because more than one electronic state is possible at the same nuclear configuration. Here we discuss the pumping and probing electron−nuclear dynamics where even at a given nuclear position the electrons are not at equilibrium. To be able to treat the nuclear dynamics at a high level, we deal with a diatomic molecule, specifically N2. We will show that the nonequilibrium electronic distribution is reflected in an oscillating dipole moment of this homonuclear molecule. © XXXX American Chemical Society

That there can be a nonequilibrium distribution of electronic charge was suggested, for example, by the pioneering experiments of Weinkauf, Schlag, and co-workers.1−3 They systematically studied the fragmentation of peptides in a mass spectrometer using a two-photon resonant ionization of an aromatic amino acid. Two observations in particular required attention. One is that even large peptides dissociated at a rate fast enough to be detected in the limited time window of a mass spectrometer.4,5 The other is that unlike as expected from a statistical theory, it was not necessarily the weakest bond of the cation that broke.4 This leads to the notion of a charge-directed reactivity,6,7 meaning that initially a hole, the positive charge, is located on the chromophore. Subsequently the charge hops along the backbone of the peptide, and dissociation occurs where the charge is at the given instant. Special Issue: Ronnie Kosloff Festschrift Received: January 6, 2016 Revised: February 16, 2016

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DOI: 10.1021/acs.jpca.6b00165 J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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SIMULATIONS The Hamiltonian we use is in a mixed representation. The electronic part is a matrix in the indices of the electronic states. The matrix elements are obtained by integration over the coordinates of the electrons and are functions of the N−N bond distance, R. The diagonal elements Ei(R) are the potentials of the different states, a ground-state Morse potential48 and excited states computed from quantum chemistry.18 We here use diabatic states in the sense that the states have a definite electronic provenance, most importantly Rydberg and valence for the states that are optically coupled to the ground electronic states. The off-diagonal matrix elements of the Hamiltonian are of two kinds. One kind is Vi,j(R), the electronic coupling of different diabatic states.18 The other kind is the coupling of states induced by the oscillating pump and the probe optical fields, E(t). The field is linearly coupled to the transition dipole μij(R) connecting ground and excited states as computed by Spelsberg and Meyer.18 There are also bilinear and higher-order couplings of the field to the different multipoles of the nonequilibrium charge distribution. The nuclear part of the Hamiltonian is the kinetic energy operator for a nonrotating molecule, a term that is diagonal in the electronic indices but not diagonal between different points of the grid. In summary, the matrix elements of the Hamiltonian in the electronic basis are

Examination of a sudden ionization of an electron from the highest occupied molecular orbital (HOMO) of a tetrapeptide shows that the (positive) charge is localized on the aromatic chromophore. Further analysis shows that the state produced is a linear combination of the HOMO and the HOMO−1 of the cation.8 For the cation, this is a nonstationary electronic density and it leads to ultrafast coherent charge migration along the backbone of the peptide. Other modular systems exhibited the same effect.9−11 A similar charge migration is seen in localized core excitation in modular systems; see ref 12 for a general review. The development of attosecond lasers (see refs 13−15) provided a technology that could experimentally pump electronically nonstationary states. A short pulse is necessarily broad in energy and can coherently access more than one excited state. Accessing the lowest excited states of N2 is a particularly interesting case because for either 1∏u or 1∑+u symmetry there are three states close in energy: two Rydberg and one valence excited.16,17 The usual notation18 for the diabatic states is b, c, and o for the 1∏u states and b′, c′, and e′ for the 1∑+u states, where b and b′ are the valence excited states. The three diabatic states of given symmetry interact and give rise to localized perturbations in the observed spectrum.19 High-resolution spectroscopy20−24 and high-level quantum chemical computations18,25−27 provide a detailed picture. Early analysis28 also shows that an attosecond pulse can excite a linear combination of the states. In particular, the oscillation of the electronic state between the diffuse Rydberg states and the more compactly bound valence state gives rise to a strongly time-varying polarizability. The excited molecule can therefore undergo a strong spontaneous or induced Raman scattering. The interaction of the polarizability of the molecule with an electric field is a process that is second-order in the electric field. In this paper we examine in detail the first-order, linear interaction of the molecule with a probe field.15,29−35 This is possible for a homonuclear molecule such as N2 because the nonstationary electronic state created by an ultrafast pump pulse will exhibit an oscillatory dipole moment. One has several other options for probing the excited states rather than by an optical probe. Computational studies of the angular distribution of photoelectron emission have been reported.36−38 Also, the photochemical implications of an ultrafast excitation of N2 are of interest39,40 and have recently been studied experimentally.41,42 See ref 43 for an experimental and theoretical study of H2. Probing the dissociation and ionization of N2 with ultrafast lasers has previously been demonstrated44−46 but at a higher energy of excitation where the optically accessible electronically excited states are quite dense and include states of the N2+ cation.27 Examination of the ultrafast dynamics of pumped N2 shows a rather early in time onset of the coupling of the valence and Rydberg states, a coupling that is quite sensitive to the mass.47 This coupling gives rise to perturbations in the spectrum that are very localized in energy (see ref 19, for example). For the different isotopomers of N2, the energies of the vibrational states shift with the mass in the expected way. In addition, the mass-dependent mixing of the electronic states leads one to expect a strong isotope effect at such energies where there is a strong perturbation in the spectrum. The isotope effect can therefore offer yet another probe in time of the nonequilibrium electronic state distribution.

⎛ ℏ2∂ 2 ⎞ H i , j (R , t ) = ⎜ − ⎟ + Ei(R )δi , j + Vi , j(R ) − E(t ) ·μij (R ) ⎝ 2μ∂R2 ⎠ (1)

The electronic Hamiltonian can be diagonalized at a given value of the N−N distance, R. This will yield the familiar Born−Oppenheimer stationary states. The localized coupling of the diabatic states will result in Born−Oppenheimer states that have a rapid R dependence around the crossing point of the diabatic states. As is well understood, the disadvantage of a diagonal electronic Hamiltonian is that the nuclear kinetic energy will couple the adiabatic states. It is possible to compute such, so-called nonadiabatic, coupling terms. They depend on the momentum of the nuclei. In our work on N2 we prefer to keep the nuclear kinetic energy diagonal and to use a nondiagonal diabatic electronic Hamiltonian. We propagate the solution of the time-dependent Schrödinger equation in space and time using a grid basis49,50 for the N−N coordinate R. Our Hamiltonian is explicitly time-dependent, so we cannot easily take advantage of more sophisticated integrators for the time part. For the relatively short propagation times that we need, a straightforward Gear predictor corrector method with a short time step for the predictor proves to be fully adequate to deal with the fast oscillations in the coherence of the states and with the oscillations of the electric field. The grid expansion of the wave function allows for an efficient parallelization of the code. The integration of the nuclear motion on a grid was validated against results based on computing a discrete basis for the nuclear motion in every state.39 At the level of the time-dependent dipole moment, the results are the same to within numerical accuracy. The electronic Hamiltonian includes the pump and probe pulses. We allow, as usual, only a coupling to the R-dependent molecular dipole μ to the field E, μ·E. The light pulses have the standard form of a Gaussian envelope defining the width, σ, in time and an oscillating carrier field whose frequency is the mean energy. A prefactor specifies the field strength, E0, and the polarization, e (eq 2). The ground state is of X1∑+g symmetry. B


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Figure 1. Ground-state X1∑+g potential of N2 plotted dressed by the 13.61 eV mean energy of the pump photon with a shading showing a span of ±0.5 eV in energy of the ultrafast pump pulse. The two panels are the optically accessible diabatic potentials of 1∏u or 1∑+u symmetry, adapted from Spelsberg and Meyer18 and using their notation where b, c, and o are the diabatic 1∏u states and b′, c′, and e′ are the diabatic 1∑+u states. b and b′ are the more shallow bound valence excited states. As shown, the diabatic potentials cross one another. The potentials of the 1∏u states as computed by quantum chemistry18 are vertically shifted to match the experimental value of their ground vibrational state.

grid method. We used only the vibrational basis functions to validate the results from the grid basis. In practice we compute all the expectation values over the electronic states, where the integration is over the electronic coordinates, using the results of Spelsberg and Meyer.18 In particular, for the dipole moment of this state we use the transition dipole matrix elements μjl(R) of Spelsberg and Meyer, connecting the different electronic states j and l. The time-independent Franck−Condon factor for this transition is ⟨χjk(R)|μjl(R)|χlm(R)⟩. In the grid method, the matrix elements are localized at grid positions k and m respectively and the Franck−Condon factor is diagonal in these indices. The expression for the mean electronic dipole of a homonuclear molecule in any nuclear basis set is

The polarization e of the pump is here defined in the molecular frame because the molecule is not rotating on the ultrafast time scale. e is taken to be either along the N−N bond, so as to access excited states of ∑u symmetry, or perpendicular to the bond to access excited states of ∏u symmetry. E(t ) = E0 e exp[−(t − t0)2 /2σ 2]cos(ωt + φ)

(2)

The frequency of the pump pulse we report here is 13.61 eV (0.5 au) centered at t0 = 12 fs with a full width at half-maximum (fwhm) in time of 3 fs and a relatively weak intensity of 1011 W/cm2. The carrier−envelope phase is φ. The probe pulse has a variable delay with respect to the pump increasing from 1 fs on. Its carrier wavelength is 750 nm, and it is short in time with a fwhm of 4.7 fs. As suggested to us,51 the intensity of the probe pulse, 1012W/cm2, is higher than that of the pump. The polarization of the probe is the same as the polarization of the pump; because of the ultrafast time scale, the molecule has no time to rotate so that the only allowed emissions to the ground state are those that are allowed in absorption. When both the pump and the probe are applied, the field is the sum of the two terms with the appropriate delay. We integrate the time-dependent Schrödinger equation along a dense grid of 1024 points spanning the R-dependent range of 1.5−8 au. The integration time step of the predictor is short, 0.05 au, to handle the intense pump pulse whose period is about 13 au. The wave function we generate is a linear combination Ψ(r , R , t ) =

C*jk(t )Clm(t )⟨χjk (R )|μjl (R )|χlm (R )⟩

j≠l ,k ,m

(4)

The coefficients Cjk(t) are computed by integrating the timedependent Schrödinger equation. In N2, the transition dipole connects the ground state j = 0 to the allowed excited states so that the actual expression is ⟨Ψ(t )|r |Ψ(t )⟩(t ) =

∑ (C0*k(t )Clm(t ) + C0k(t )Clm* (t )) l ,k ,m

⟨χ0k (R )|μ0l (R )|χlm (R )⟩

(5)

where we assumed that the transition strength matrix element is real. In the Condon approximation, the result is directly determined by the Franck−Condon overlap integrals

∑ Cjk(t )ψj(r ; R)χjk (R) jk

∑

⟨Ψ(t )|r|Ψ(t )⟩(t ) =

(3)

⟨Ψ(t )|r|Ψ(t )⟩(t ) =

where the summation is over all those electronic states j that are included and over all the normalized basis functions χjk(R) for the nuclear motion. r are the electronic coordinates. In general, the basis functions χjk(R) can be chosen for convenience. Most commonly these are either the different vibrational basis functions k computed for the electronic potential j or, in the grid method,49,50 the basis are functions localized at the different grid points k. In the computations reported below, we used the

∑ (C0*k(t )Clm(t ) + C0k(t )Clm* (t ))μ0l (R̅) l ,k ,m

⟨χ0k (R )|χlm (R )⟩

(6)

Here R̅ is the mean distance for which the Franck−Condon factor is large. The implication is that the dipole is the direct measure of the correlation between the ground and electronic states, weighted by the Franck−Condon factor. It is this weighting that makes the matrix element sensitive to the excited-state C


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Figure 2. Nuclear wave packets on the different diabatic potentials plotted versus the interatomic distance, R; scale is the same as that in Figure 1. The pair of panels are for the two symmetries, and the transition strength is higher to the states of 1∑+u symmetry. Panels are shown for different times: 11 fs, during the pump pulse; 20 fs, as the valence states motion exits the Franck−Condon region for the first time; and 34 fs, at about the outer turning point on the valence excited states. The motion on the Rydberg state potentials is confined to the Franck−Condon region.

Figure 3. Dipole function of ultrafast pumped N2 versus time. Shown for states of the 1∏u symmetry. Plotted in the foreground in green is the dipole when there is no coupling of the diabatic states. This shows the regular coherent vibrational motion on each diabatic potential. Plotted in the background in red is the dipole when the coupling of the diabatic states is included in the Hamiltonian. The different rows show the dipole function for particular excited states, showing the fast vibrational motion in the Rydberg states and the long absence of dipolar coupling to the ground state for the valence states. The latter is due to the wave packet being out of the Franck−Condon region, see Figure 2. The total dipole as shown in the bottom row is the algebraic sum of the three contributions of the same symmetry that are shown in the rows above, see eq 7. The plot in the background shows how state mixing allows for intensity sharing. The Supporting Information provides longer time plots for 1∏u states (Figure S1a) and for 1∑u states (Figure S1b).

wave functions only in the Franck−Condon regime. This regime is quite narrow in N219 because the ground-state potential is so very steep. In the grid method, the final result for the dipole is even more transparent ⟨Ψ(t )|r |Ψ(t )⟩(t ) =

∑ (C0*m(t )Clm(t ) + C0m(t )Clm* (t ))μ0l (R m) l ,m

(7) D


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Figure 4. Three-dimensional plots of the coherence between excited electronic states as a function of time and position on the grid. Shown is the correlation between two different set of ∏u states. First shown is ρo,c(Rm;t) + ρc,o(Rm;t) where ρo,c(Rm;t) is an element of the density matrix, * (t)Ccm(t). In an absolute magnitude, the coefficients C are not large because only a rather small fraction of the wave function is ρo,c(Rm;t) = Com excited by our weak pump. The excited states o and c are largely confined to the Franck−Condon region and are moving on very similar potentials, see Figure 1. Therefore, the corresponding wave functions have significant values at the same points of the grids. Also shown is the electronic coherence between the b and c excited states, ρb,c(Rm;t) + ρc,b(Rm;t). Shortly after the pump pulse, the b state wave function moves out of the Franck−Condon region while the c state wave function remains confined. Therefore, their expansion coefficients C, eq 3, are not simultaneously significant. Later, state mixing enhances the correlation in the same region in space.

where μ0l(Rm) = ⟨χ0m(R)|μ0l(R)|χlm(R)⟩ is the value of the transition moment from the ground to the excited state l at the mth point of the grid. The transition moment restricts the summation over the distance index m to such distances where there is overlap with the very localized vibrational wave function of the ground state. Each term is a measure of the population of the electronic state l in the Franck−Condon region where the restriction to that region is the restriction on the range of the distance index m as imposed by the value of the transition dipole.

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fairly long excursion out of the Franck−Condon region in the direction of dissociation. There is not enough energy to dissociate, so the wave function is reflected at the rightmost, outer turning point and returns to the Franck−Condon region after about a 50 fs delay after the pulse. Toward probing the nuclear motion we first show, Figure 3, the expectation value of the dipole moment as a function of time (eq 7). The spectrum in the forefront is a computation when the coupling of the diabatic states is not included. Once the optical pumping is over, each nuclear wave function evolves independently and the dipole due to each diabatic state separately is also shown. Each such term is one term in the sum in eq 7. The very fast oscillations of the dipole are due to the coherent superposition of the excited and the ground state. The long excursion of the motion on the valence excited states b and b′ out and back to the Franck−Condon region is very evident. Also quite clear is the shorter period of the coherent vibrational motion in the Rydberg wells, shown in Figure 3 for state c. A movie of the time-evolving wave packets is available in the Supporting Information. Also shown in Figure 3 in the background is the dipole computed for the physically realistic case when the coupling between the excited Rydberg and valence excited states is included. As is seen, this coupling leads quite early on to intensity borrowing. For example, for both the b and b′ states there are regions that are essentially dark but light up because of the mixing of the Rydberg states that essentially do not exit from the Franck−Condon region. Both the c and the o Rydberg states mix with the b state for states of the 1∏u symmetry. For the 1∑+u states, it is primarily the mixing of c′ with the b′ valence state. On the time scale shown, the anharmonicity of the potentials does not rapidly dephase the vibrational wave packets, as shown by the uncoupled b and b′ state dipoles remaining localized in time with a width about that of the pump pulse. One can expand the nuclear wave function in vibrational components, and we do so when we compare the grid and basis set numerical methods. The localization of the nuclear wave function on a given electronic potential shows that its different vibrational components remain coherent with one another.

RESULTS AND DISCUSSION

The optically accessible diabatic potentials of 1∏u and 1∑+u symmetry are shown in Figure 1; the excited-state potentials are adapted from Spelsberg and Meyer.18 The ground-state Morse potential48 is plotted dressed by the 13.61 eV mean energy of the pump photon with the additional feature that the dressing shows a span in energy of the ultrafast pump pulse. From this figure alone we can expect that at a pump of 13.61 eV mean energy the two symmetries will respond differently. In particular, the o state of 1∏u symmetry is expected to be significantly populated by the pump as opposed to its counterpart, the e′ state of 1∑+u symmetry. Of course, the figure can look qualitatively different for different pump energies so that the mean pump energy is a useful control variable. As seen in Figure 1, the Rydberg states are almost as tightly bound as the ground state while the valence excited b and b′ states are significantly more shallow. As expected from the bond order−bond length correlation, the equilibrium distances of the valence excited states are significantly longer. Upon excitation, the wavepacket propagates along the different potentials. Snapshots of the nuclear wave function as a function of the internuclear distance R, same scale as Figure 1, are shown at different times in Figure 2 plotted separately for the two symmetries. At the early time of 11 fs while the pump is on, the higher transition strength to the states of 1∑+u symmetry is evident. For the Rydberg states, the nuclear wave function remains essentially confined to the Franck−Condon region for absorption from the ground state. That is not the case for the valence excited b and b′ states. The nuclear motion takes a E


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Figure 6. Fractional change of the dipole, eq 9 versus time for the two isotopomers, where the dipole is as defined in eq 7. The change is localized in time because it is large upon switching between the o and other states. The switch can be large in magnitude because the transition moment from the ground to the o state has a different sign than that to the other states.18 The plot is shown at the wavelength of 91 nm (13.6 eV), where we expect that considerable mixing with the o state can occur. In a stationary experiment,52 the isotopic fractionation is largest at this wavelength.

an operator connecting the different electronic states. A dipole is such an operator but subject to selection rules. The simplest choice is the density matrix that, on the grid, has the off diagonal elements ρo,c (R m ; t ) = Co*m(t )Ccm(t )

(8)

The correlation term, (Com * (t)Ccm(t) + Com(t)Ccm * (t)) is the direct measure of the coherence between the excited states o and c on point m of the grid. Figure 4 shows this coherence between the excited states as a time varying function on the grid. The grid points m, m = 1, ..., 1024 are quite dense so the plot appears continuous in space. The plot shows the correlation at time intervals of 10 au. The coherence between states o and c at different points m on the grid is different than that between states b and c that is also shown in Figure 4. Why the difference? Because at early times, once the pulse is over, the nuclear wave function for valence state b moves out of the Franck−Condon region while the nuclear wave function for Rydberg state c is localized in the Franck−Condon region, see Figure 2. Therefore, at early times, the product (Cbm * (t)Ccm(t) + Cbm(t)C*cm(t)) is numerically small on all points of the grid. After about 20 fs, the c nuclear wave function returns back to the Franck−Condon region, in which the diabatic coupling is the strongest,18 and transforms partially into the b state. The b and c states thus better overlap in space. Thereby the coherence of valence and Rydberg states probes the state mixing. The c and o states are strongly correlated because the potentials are very similar; therefore, the two wave packets move in concert. To probe the motions in the excited states we apply a short probe pulse with a variable delay time after the pump pulse. Figure 5 shows the perturbation imposed by this probe on the oscillating dipole at a series of three increasing delay times. Shown in Figure 5 are the total populations in the three excited states. These are the total populations in the electronic state irrespective of the internuclear separation. In the notation of the grid computation, this is the (squared) component of the state j in the total wave function, ∑m|Cjm(t)|2. This is not the same as the probing of the dipole of the state j in the Franck− Condon region where the contribution from the grid point m is, see eq 7, (C0m*Clm + C0mClm * )μ0l(Rm). Because of this

Figure 5. Dipole, ordinate on the left and the populations of the excited states, ordinate on the right, versus time when both the pump and the probe pulses are applied. Shown for the states of the 1∏u symmetry for three different values of the delay time between the pump and probe. As shown, the effect of the probe is confined to the narrow time interval, less than 10 fs, during which it is applied. The excited states respond but do so rather adiabatically. See Figure S2 for similar results for states of the 1∑u symmetry.

In the Supporting Information we show that this vibrational coherence is maintained also for longer times. Another important aspect is the coherence among different excited electronic states. This can be probed by an expectation value of F


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The Journal of Physical Chemistry A difference, the plot of the transition dipole to the state j versus time and the plot of the total population in the electronic state j versus time, both shown in Figure 5, are not closely matched. This is particularly true for the b and b′ valence excited states that before the coupling to the other states sets in are localized mostly outside the Franck−Condon region, as shown in Figure 2. The localization of the probing in space and time means that the strong isotope effect in N2 that we47 attribute in part to strong mixing of the diabatic states can be seen in the dynamics. The mean pump wavelength that we use, 91.1 nm, is in the region where the observed52 effect is large. Figure 6 plots the fractional change in the dipole moment between the light and heavy isotopomers computed using the same eq 7 but for a dynamics with different reduced mass δ⟨μ⟩ = (⟨μ⟩15,15 − ⟨μ⟩14,14 )/(⟨μ⟩15,15 + ⟨μ⟩14,14 )

actions MOLIM CM1405 and CM1204 XLIC. We thank Majdi Hochlaf, Shimshon Kallush, Steve Leone, Wen Li, Gilad Marcus, Harel Muskatel, Dan Neumark, and Daniel Strasser for useful discussions. F.R. is a director of research with FNRS (Fonds National de la Recherche Scientifique), Belgium.

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(9)

A dipole can be both positive and negative, see Figure 5. When the difference in mass makes the motion in one isotopomer lag with respect to the other, the two dipoles can differ in sign. This is the familiar origin of an isotope effect when the difference in the reduced mass changes somewhat the frequency of the vibrational motion. The other origin of the difference is the different sign of the transition dipole for to the Rydberg states o and to all the others.18

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CONCLUDING REMARKS The oscillating dipole moment of N2 with its strong modulation has been shown to offer a clear signature of the rich electronic and vibrational dynamics that can be pumped by an ultrafast ultraviolet pulse. The dipole moment is sensitive to the dynamics in the narrowly confined Franck−Condon region. The electronic coherence between the excited states and the coupling of the diabatic states persists for quite a few vibrational periods. As shown in refs 15 and 30−35, the dipole could be read by its coupling to a probe pulse. When the response to the probe is measured at variable delay times with respect to the pulse, it samples different time scales associated with the nonequilibrium electronic motion.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b00165. Π and Σ dipole plotted up to 180 fs (Figure S1a,b; cf. Figure 3) and action of the probe on the Σ state (Figure S2; cf. Figure 5) (PDF) Movie of the Σ and Π state dynamics during and after the pump pulse (cf. Figure 2a−c) (AVI)

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REFERENCES

(1) Weinkauf, R.; Schlag, E. W.; Martinez, T. J.; Levine, R. D. Nonstationary electronic states and site-selective reactivity. J. Phys. Chem. A 1997, 101 (42), 7702−7710. (2) Weinkauf, R.; Schanen, P.; Yang, D.; Soukara, S.; Schlag, E. W. Elementary processes in peptides - Electron-mobility and Dissociations in peptide Cations in the gas-phase. J. Phys. Chem. 1995, 99 (28), 11255−11265. (3) Weinkauf, R.; Schanen, P.; Metsala, A.; Schlag, E. W.; Bürgle, M.; Kessler, H. J. Phys. Chem. 1996, 100, 18567. (4) Schlag, E. W.; Selzle, H. L.; Schanen, P.; Weinkauf, R.; Levine, R. D. Dissociation kinetics of peptide ions. J. Phys. Chem. A 2006, 110 (27), 8497−8500. (5) Schlag, E. W.; Levine, R. D. On the Unimolecular dissociation of large molecules. Chem. Phys. Lett. 1989, 163 (6), 523−530. (6) Remacle, F.; Levine, R. D.; Schlag, E. W.; Weinkauf, R. Electronic control of site selective reactivity: A model combining charge migration and dissociation. J. Phys. Chem. A 1999, 103, 10149. (7) Remacle, F.; Levine, R. D.; Ratner, M. A. Charge directed reactivity: a simple electronic model, exhibiting site selectivity, for the dissociation of ions. Chem. Phys. Lett. 1998, 285, 25−33. (8) Remacle, F.; Levine, R. D. An electronic time scale in chemistry. Proc. Natl. Acad. Sci. U. S. A. 2006, 103 (18), 6793−6798. (9) Remacle, F.; Levine, R. D. The time scale for electronic reorganization upon sudden ionization of the water and watermethanol hydrogen bonded dimers and of the weakly bound NO dimer. J. Chem. Phys. 2006, 125 (13), 133321−7. (10) Stoychev, S. D.; Kuleff, A. I.; Cederbaum, L. S. On the intermolecular Coulombic decay of singly and doubly ionized states of water dimer. J. Chem. Phys. 2010, 133 (15), 154307. (11) Calegari, F.; Ayuso, D.; Trabattoni, A.; Belshaw, L.; De Camillis, S.; Anumula, S.; Frassetto, F.; Poletto, L.; Palacios, A.; Decleva, P.; Greenwood, J. B.; Martín, F.; Nisoli, M. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. Science 2014, 346 (6207), 336−339. (12) Kuleff, A. I.; Cederbaum, L. S. Ultrafast correlation-driven electron dynamics. J. Phys. B: At., Mol. Opt. Phys. 2014, 47 (12), 124002. (13) Goulielmakis, E.; Loh, Z. H.; Wirth, A.; Santra, R.; Rohringer, N.; Yakovlev, V. S.; Zherebtsov, S.; Pfeifer, T.; Azzeer, A. M.; Kling, M. F.; Leone, S. R.; Krausz, F. Real-time observation of valence electron motion. Nature 2010, 466 (7307), 739−U7. (14) Baltuska, A.; Udem, T.; Uiberacker, M.; Hentschel, M.; Goulielmakis, E.; Gohle, C.; Holzwarth, R.; Yakovlev, V. S.; Scrinzi, A.; Hansch, T. W.; Krausz, F. Attosecond control of electronic processes by intense light fields. Nature 2003, 421 (6923), 611−615. (15) Beck, A. R.; Bernhardt, B.; Warrick, E. R.; Wu, M.; Chen, S.; Gaarde, M. B.; Schafer, K. J.; Neumark, D. M.; Leone, S. R. Attosecond transient absorption probing of electronic superpositions of bound states in neon: detection of quantum beats. New J. Phys. 2014, 16, 113016. (16) Mulliken, R. S. Rydberg and valence-shell character as functions of internuclear distance in some excited-states of CH, NH, H2, and N2. Chem. Phys. Lett. 1972, 14 (2), 141. (17) Ermler, W. C.; McLean, A. D.; Mulliken, R. S. Ab Initio study of valence state potential-energy curves of N2. J. Phys. Chem. 1982, 86 (8), 1305−1314. (18) Spelsberg, D.; Meyer, W. Dipole-allowed excited states of N2: Potential energy curves, vibrational analysis, and absorption intensities. J. Chem. Phys. 2001, 115 (14), 6438−6449. (19) Ajello, J. M.; James, G. K.; Franklin, B. O.; Shemansky, D. E. Medium-resolution studies of extreme ultraviolet emission from N-2

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Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award DE-SC0012628 and in part by the Einstein Foundation of Berlin. We benefited from our participation in the COST G


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The Journal of Physical Chemistry A by Eeectron-impact - Vibrational perturbations and cross-sections of the C′4 1-Sigma-U+- and B′1-sigma-U+- state. Phys. Rev. A: At., Mol., Opt. Phys. 1989, 40 (7), 3524−3556. (20) Stark, G.; Lewis, B. R.; Heays, A. N.; Yoshino, K.; Smith, P. L.; Ito, K. Oscillator strengths and line widths of dipole-allowed transitions in 14N2 between 89.7 and 93.5 nm. J. Chem. Phys. 2008, 128 (11), 114302−10. (21) Sprengers, J. P.; Ubachs, W.; Baldwin, K. G. H.; Lewis, B. R.; Tchang-Brillet, W. U. L. Extreme ultraviolet laser excitation of isotopic molecular nitrogen: The dipole-allowed spectrum of 15N2 and 14N15N. J. Chem. Phys. 2003, 119 (6), 3160−3173. (22) Lewis, B. R.; Gibson, S. T.; Sprengers, J. P.; Ubachs, W.; Johansson, A.; Wahlstrom, C. G. Lifetime and predissociation yield of 14N2 b 1Piu (v = 1) revisited: Effects of rotation. J. Chem. Phys. 2005, 123 (23), 3. (23) Lefebvre-Brion, H.; Lewis, B. R. Comparison between predissociation mechanisms in two isoelectronic molecules: CO and N-2. Mol. Phys. 2007, 105 (11−12), 1625−1630. (24) Wu, C. Y. R.; Fung, H.-S.; Chang, K.-Y.; Singh, T. S.; Mu, X.-L.; Nee, J. B.; Chiang, S.-Y.; Judge, D. L. Fluorescence excitation spectra of the bΠu1, b′Σu+1, cnΠu1, and cn′Σu+1 states of N2 in the 80− 100nm region. J. Chem. Phys. 2007, 127 (8), 084314. (25) Guberman, S. L. Spectroscopy above the ionization threshold: Dissociative recombination of the ground vibrational level of N2+. J. Chem. Phys. 2012, 137 (7), 074309. (26) Little, D. A.; Tennyson, J. An ab initio study of singlet and triplet Rydberg states of N2. J. Phys. B: At., Mol. Opt. Phys. 2013, 46, 145102. (27) Hochlaf, M.; Ndome, H.; Hammoutene, D.; Vervloet, M. Valence-Rydberg electronic states of N-2: spectroscopy and spin-orbit couplings. J. Phys. B: At., Mol. Opt. Phys. 2010, 43 (24), 245101. (28) Muskatel, B. H.; Remacle, F.; Levine, R. D. The post-BornOppenheimer regime: dynamics of electronic motion in molecules by attosecond few-cycle spectroscopy. Phys. Scr. 2009, 80 (4), 048101. (29) Hammerich, A. D.; Kosloff, R.; Ratner, M. A. Chemistry in strong laser fields: An example from methyl-iodide photodissociation. J. Chem. Phys. 1992, 97 (9), 6410−6431. (30) Beck, A. R.; Neumark, D. M.; Leone, S. R. Probing ultrafast dynamics with attosecond transient absorption. Chem. Phys. Lett. 2015, 624, 119−130. (31) Bernhardt, B.; Beck, A. R.; Li, X.; Warrick, E. R.; Bell, M. J.; Haxton, D. J.; McCurdy, C. W.; Neumark, D. M.; Leone, S. R. Highspectral-resolution attosecond absorption spectroscopy of autoionization in xenon. Phys. Rev. A: At., Mol., Opt. Phys. 2014, 89 (2), 023408. (32) Gaarde, M. B.; Buth, C.; Tate, J. L.; Schafer, K. J. Transient absorption and reshaping of ultrafast XUV light by laser-dressed helium. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 83 (1), 013419. (33) Dutoi, A. D.; Cederbaum, L. S. Time-resolved pump-probe spectroscopy to follow valence electronic motion in molecules: Application. Phys. Rev. A: At., Mol., Opt. Phys. 2014, 90 (2), 023414. (34) Bækhøj, J. E.; Yue, L.; Madsen, L. B. Nuclear-motion effects in attosecond transient-absorption spectroscopy of molecules. Phys. Rev. A: At., Mol., Opt. Phys. 2015, 91 (4), 043408. (35) Seidner, L.; Stock, G.; Domcke, W. Nonperturbative approach to femtosecond spectroscopy - General-theory and application to multidimensional nonadiabatic photoisomerization processes. J. Chem. Phys. 1995, 103 (10), 3998−4011. (36) Kus, T.; Mignolet, B.; Levine, R. D.; Remacle, F. Pump and probe of ultrafast charge reorganization in small peptides: A Ccomputational sStudy through sudden ionizations. J. Phys. Chem. A 2013, 117 (40), 10513−10525. (37) Mignolet, B.; Levine, R. D.; Remacle, F. Localized electron dynamics in attosecond-pulse-excited molecular systems: Probing the time-dependent electron density by sudden photoionization. Phys. Rev. A: At., Mol., Opt. Phys. 2012, 86 (5), 053429. (38) Mignolet, B.; Levine, R. D.; Remacle, F. Charge migration in the bifunctional PENNA cation induced and probed by ultrafast ionization: a dynamical study. J. Phys. B: At., Mol. Opt. Phys. 2014, 47 (12), 124011.

(39) Muskatel, B. H.; Remacle, F.; Levine, R. D. Ultrafast predissociation mechanism of the (1)Pi(u) states of N-14(2) and its isotopomers upon attosecond excitation from the ground state. J. Phys. Chem. A 2012, 116 (46), 11311−11318. (40) Muskatel, B. H.; Remacle, F.; Levine, R. D. AttoPhotoChemistry. Probing ultrafast electron dynamics by the induced nuclear motion: The prompt and delayed predissociation of N-2. Chem. Phys. Lett. 2014, 601, 45−48. (41) Okino, T.; Furukawa, Y.; Nabekawa, Y.; Miyabe, S.; Amani Eilanlou, A.; Takahashi, E. J.; Yamanouchi, K.; Midorikawa, K. Direct observation of an attosecond electron wave packet in a nitrogen molecule. Sci. Adv. 2015, 1 (8), e1500356. (42) Medišauskas, L.; Patchkovskii, S.; Harvey, A.; Brambila, D. S.; Neidel, C.; Klei, J.; Rouzée, A.; Vrakking, M. J. J.; Ivanov, M. Y. Initial electronic coherence in molecular dissociation induced by an attosecond pulse. Phys. Rev. A: At., Mol., Opt. Phys. 2015, 92 (5), 053403. (43) Ranitovic, P.; Hogle, C. W.; Riviere, P.; Palacios, A.; Tong, X. M.; Toshima, N.; Gonzalez-Castrillo, A.; Martin, L.; Martin, F.; Murnane, M. M.; Kapteyn, H. Attosecond vacuum UV coherent control of molecular dynamics. Proc. Natl. Acad. Sci. U. S. A. 2014, 111 (3), 912−917. (44) Gagnon, E.; Ranitovic, P.; Tong, X.-M.; Cocke, C. L.; Murnane, M. M.; Kapteyn, H. C.; Sandhu, A. S. Soft X-ray-driven femtosecond molecular dynamics. Science 2007, 317 (5843), 1374−1378. (45) Zhou, X.; Ranitovic, P.; Hogle, C. W.; Eland, J. H. D.; Kapteyn, H. C.; Murnane, M. M. Probing and controlling non-BornOppenheimer dynamics in highly excited molecular ions. Nat. Phys. 2012, 8 (3), 232−237. (46) Strasser, D.; Haber, L. H.; Doughty, B.; Leone, S. R. Ultrafast predissociation of superexcited nitrogen molecules. Mol. Phys. 2008, 106 (2−4), 275−280. (47) Muskatel, B. H.; Remacle, F.; Thiemens, M. H.; Levine, R. D. On the strong and selective isotope effect in the UV excitation of N-2 with implications toward the nebula and Martian atmosphere. Proc. Natl. Acad. Sci. U. S. A. 2011, 108 (15), 6020−6025. (48) Konowalow, D. D.; Hirschfelder, J. O. Morse potential parameters for OO, NN, and NO interactions. Phys. Fluids 1961, 4, 637. (49) Kosloff, R. Propagation methods for quantum molecular dynamics. Annu. Rev. Phys. Chem. 1994, 45, 145−178. (50) Tannor, D. J. Introduction to quantum mechanics. A timedependent perspective; University Science Books: Sausalito, CA, 2007. (51) Warrick, E. R.; Neumark, D. M.; Leone, S. R. Personal communication, 2015. (52) Chakraborty, S.; Muskatel, B. H.; Jackson, T. L.; Ahmed, M.; Levine, R. D.; Thiemens, M. H. Massive isotopic effect in vacuum UV photodissociation of N2 and implications for meteorite data. Proc. Natl. Acad. Sci. U. S. A. 2014, 111 (41), 14704−14709.

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