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# Time-dependent view of an isotope effect in electron-nuclear nonequilibrium dynamics with applications to N_{2}

Contributed by R. D. Levine, May 1, 2018 (sent for review March 14, 2018; reviewed by Joern Manz and Thomas E. Markland)

## Significance

Wide-range isotopic anomalies found in extraterrestrial sources suggest that we seek better mechanistic insights on photochemical processes induced by far UV radiation. To be able to follow the process, we simulate the progress in time of an N_{2} molecule excited by an ultrafast pulse in the vacuum UV (VUV). Such a short pulse necessarily initiates a nonstationary state of the molecule that we follow in silico. In the VUV, N_{2} is pumped to a valence excited and Rydberg states. The ultrashort pulse creates a coherent combination of these electronic states, localized in the Franck–Condon region, leading to a dynamical isotope effect.

## Abstract

Isotopic fractionation in the photodissociation of N_{2} could explain the considerable variation in the ^{14}N/^{15}N ratio in different regions of our galaxy. We previously proposed that such an isotope effect is due to coupling of photoexcited bound valence and Rydberg electronic states in the frequency range where there is strong state mixing. We here identify features of the role of the mass in the dynamics through a time-dependent quantum-mechanical simulation. The photoexcitation of N_{2} is by an ultrashort pulse so that the process has a sharply defined origin in time and so that we can monitor the isolated molecule dynamics in time. An ultrafast pulse is necessarily broad in frequency and spans several excited electronic states. Each excited molecule is therefore not in a given electronic state but in a superposition state. A short time after excitation, there is a fairly sharp onset of a mass-dependent large population transfer when wave packets on two different electronic states in the same molecule overlap. This coherent overlap of the wave packets on different electronic states in the region of strong coupling allows an effective transfer of population that is very mass dependent. The extent of the transfer depends on the product of the populations on the two different electronic states and on their relative phase. It is as if two molecules collide but the process occurs within one molecule, a molecule that is simultaneously in both states. An analytical toy model recovers the (strong) mass and energy dependence.

There is considerable interest in the ultrafast photochemistry of highly excited electronic states (see ref. 1 and references therein). This is driven by the recent experimental developments of ultrashort pulses (2, 3) supplemented and complemented by advances in the theory (4⇓⇓⇓⇓–9). A particular need for advanced theory arises because an ultrashort light pulse is necessarily broad in frequency and can excite several electronic states. The molecule is thus initiated in a superposition of excited electronic states that are spatially localized in the Franck–Condon region because there is not enough time during the excitation for the atoms to move substantially. The molecule is thus born in a state of coherent disequilibrium. It is not a mixture of states. It is coherent because each molecule by itself is in a linear combination of several electronic states. In the Born–Oppenheimer regime, the electronic states are stationary for a given electronic state, but not so after an ultrafast excitation. The electronic states are in a state of disequilibrium with one another and with the nuclei (10).

Several well-designed experimental methods have been developed to probe the state of a molecule shortly after excitation. These include transient absorption (3, 11, 12), photoelectron spectroscopy technique (4, 13⇓⇓⇓–17), X-ray absorption (5, 18), or electron-diffraction imaging (19, 20). An efficient tool to unravel the effect of the nuclear motion on the nonequilibrium electron dynamics is isotope labeling (21⇓⇓⇓⇓–26), a methodology widely used in the studies of the reaction mechanisms in chemical kinetics (27⇓–29) and photochemistry (see refs. 30⇓–32 for the special case of N_{2}). In the steady state, there is clear evidence (33) for an isotope effect in the high-resolution spectrum of N_{2}. Direct photochemical steady-state studies on the photodissociation of N_{2} with rather spectacular wavelength-selective isotope effect were recently reported (32). Such results were analyzed in ref. 34 as due to isotope shifts in vibrational levels affecting the coupling of the Rydberg and valence states. Here, we seek a time-dependent view relevant for early time dynamics. To observe the dynamical, nonequilibrium, effect, one needs a fast probe. This is possible in a pump–probe configuration where the probe is also ultrafast. The measurement is the spectral response to the probe as a function of the delay between the probe and the pump. For the specific case of N_{2}, this is actually possible as has been demonstrated in the pump–probe University of California, Berkeley experiment (3, 11) and computationally in our ref. 34.

In a short time range after the excitation, photochemical dynamics is the simultaneous correlated motion of the wave packets on different coupled electronic states. We computationally explore the effect of mass on the nonequilibrium electron dynamics in isotopomers of nitrogen molecule. Extensive study of the predissociation pathways for different isotopomers of N_{2} shows that dissociation occurs following spin-orbit coupling to a triplet state (see review of the subject in refs. 34⇓–36). Here, we discuss a shorter time range after the excitation where there is considerable exchange between the bound, directly excited, singlet electronic states.

We focus on the earliest stage toward predissociation to examine a possible interplay between the coupling of the singlet electronic states and the mass of the N atoms. Excitation of the aligned molecules with short extreme-UV pulses gives rise to the population of several dipole-allowed electronic states of *R*, but it is not mass or velocity independent.

Two diabatic states of each symmetry are of Rydberg character and one is valence excited (41). The two kinds of states differ markedly in the strength of the N–N binding (Fig. 1). The Rydberg states are almost as tightly bound as the ground state and are therefore localized in the Franck–Condon region. The valence excited potential is significantly more shallow. The diabatic states do not diagonalize the electronic Hamiltonian and are coupled by local potential terms as also shown in Fig. 1. An important point to note is that the coupling of the diabatic states rapidly declines as the distance increases beyond the Franck–Condon region, which is shaded in Fig. 1.

The difference in the bonding character of the potentials of the excited states is another key to the strong isotope effect. With an ultrashort pulse, all excited states are localized in the Franck–Condon region when the N–N distance is as in the ground state. Following the excitation, the localized wave functions describing the vibrational motion on the different states can move out. The wave functions on the two more tightly bound Rydberg states rather soon move back to the Franck–Condon region. The shallower potential of the valence state allows its vibrational wave function to move much further out. This is a key point. Shortly after the nuclei start to move, the wave packet on the valence excited state is localized in a different range of N–N distances than the wave packets on the Rydberg states. During that early time, the wave packet on the valence excited state has hardly any overlap with the vibrational wave functions on the Rydberg states. It takes almost three vibrational periods of the Rydberg states before the vibrational wave packet on the valence state comes back to the Franck–Condon region. At that point, the vibrational wave functions on the different electronic states overlap. This is when the significant population transfer between the valence and Rydberg states takes place. We shall show directly from the full quantum-mechanical equations of motion that this transfer is proportional to the product of the populations on each electronic state. It is a strictly intramolecular transfer but its kinetic behavior is as if it is two molecules that are taking part. The extent of transfer and particularly its direction, meaning from valence to Rydberg or the other direction, depends critically on the relative phase of the two vibrational wave functions. This result of the numerical computations is discussed below as a key aspect of the dynamical isotope effect. Using the toy model described in detail in *SI Appendix*, we analytically compute the change of the relative phase and show that the quadrant of the phase determines the initial direction of the transfer. Shortly after the transfer, the wave function on the valence excited state will move out of the Franck–Condon region and will not return for a while. The key point is the localization of the coupling of the diabatic states essentially in the Franck–Condon region. The wave packet on the valence excited state moves back into and then out of this region, effectively switching the isotope effect on and off. Acting against this orderly picture is a dephasing of the initially localized vibrational states. The potentials are anharmonic, particularly that of the valence excited state. Such potentials tend, in a few vibrations, to delocalize an initially localized wave packet (42). Then the wave functions on different states overlap in a stationary manner.

## Results and Discussion

### Dynamics.

The wave function *a* apart. The Hamiltonian that is used to solve the time-dependent Schrödinger equation is defined on this grid with the explicit form of the laser pulse included (see *SI Appendix*, Fig. S2, for the time-dependent profiles of the field). We use a localized form for the Hamiltonian (discussed in detail in *SI Appendix*, section 1) that propagates from a given point on the grid only to its near neighbors. The form of the Hamiltonian allows us to use a near-classical imagery while doing accurate quantal dynamics. The electronic basis is a set of diabatic (40) electronic states *SI Appendix*, Eqs. **S2**–**S5**) are given in *SI Appendix*. Our use of a diabatic basis for the electronic states means that the terms that couple different electronic states are local on the grid (Fig. 1). Only the mass-dependent nondiagonal terms of the kinetic energy operator,

### Isotope Effect in N 2 14 and N 2 15 .

In the early time (10–30 fs) range following the excitation as the wave packet on the valence state is out of the coupling region, the isotope effect on the population dynamics is found to be small (Fig. 2 and *SI Appendix*, Figs. S3–S5). It is a transfer from the Rydberg states that have a population in the region of strong diabatic coupling to the valence excited state whose wave packet is away at larger separations. It is a limited isotope effect of the order of a few percents (Fig. 2*A*). A significant isotope-sensitive step is observed after about 50 fs of the dynamics (Fig. 2*B*), and it results in meaningfully different populations of electronic states in the two isotopomers. Results for the population in the *SI Appendix*, Fig. S6. Note that at certain times the isotopic fractionation can exceed 100%.

The use of an ultrashort pulse is so that the nuclei hardly move during the excitation. The dynamics following a longer pulse, 8.3 fs (width in energy of 0.5 eV), also exhibit a strong isotope effect with an onset in about the same time as shown in *SI Appendix*, Figs. S7 and S8. This narrower in energy pulse does build a coherent superposition of vibrational states because it is still a shade shorter than a vibrational period of the Rydberg states. It is ipso facto shorter than the period on the valence excited states. Several vibrational levels are therefore coherently excited even for this longer pulse (*SI Appendix*, Fig. S9). For such a longer pulse with a narrower bandwidth, the carrier frequency is more relevant. It needs to be in an energy range where the pulse width suffices to overlap both a Rydberg and a valence excited state (Fig. 1). That this can be achieved is shown in Movie S1. The short 1-fs pulse has a width of 4 eV in energy and coherently excites a wider range of vibrational levels. The intrastate beating between different vibrational levels seen in the population dynamics for the 1-fs pulse is not evident for the longer pulse (*SI Appendix*, Fig. S5*B*). Such oscillations could be observed using an ultrashort probe pulse at different delay times (3, 11). The ratio between the two vibrational frequencies due to the different mass, 1.035, is too small to be seen in the graph. The intrastate oscillations are superposed on the somewhat slower interstate beating caused by the coupling of the diabatic states.

Mechanistically, the time-dependent isotope effect arises from an interaction of wave packets localized on the same N–N distance range in two different electronic states. So even a longer pulse can give rise to this isotope effect provided that the pulse length is shorter than a vibrational period and that it builds a coherent superposition of two electronic states. This brings the process to the realm of femtosecond lasers as generated by modern parametric amplifiers. The molecule needs to have more than one excited electronic state within the bandwidth of the laser: a conical intersection near the Franck–Condon region is a suitable example.

### Quantum Dynamics of Populations.

We analyze the key factors responsible for the effect of mass using the coupled equations of motion for the population of different electronic states for adjacent points on the grid. In the Hamiltonian localized on the grid, we distinguish between the kinetic energy at a given point on the grid and the terms, coupling constant *b* at the grid point *i* is shown in *SI Appendix*, section 3, to be given by the following:*SI Appendix*, Eqs. **S11** and **S12**. Equations of motion for the two distinct types of coherences that are in the right-hand side of Eq. **1** are given in *SI Appendix*, Eqs. **S13**–**S15**.

The change of the overall electronic state population on the valence state **1**) will be exactly zero (see *SI Appendix*, Eq. **S16** for the details). Only interelectronic couplings contribute to the overall population dynamics, which is rather obvious: in the diabatic basis, the kinetic operator couples only neighboring grid points of the same electronic state, so it cannot cause population transfer between the electronic states. As one can see from Eqs. **1** and **2**, the coupling terms between different electronic states have a similar form. To illustrate the essential aspects of the time-dependent isotope effect we use a model system of only two coupled electronic states (*b* and *c*) of different character, where**2** and also its two coupled-state version, Eq. **3**, show very clearly that the transfer rate between two electronic states depends directly on the product of their amplitudes

For very strong diabatic coupling, there arises another aspect due to a nonlinear dynamic effect. The equation of motion for the local coherence between two electronic states has a term proportional to the diabatic coupling and to the local population difference (*SI Appendix*, Eq. **S13**). Therefore, even if the wave packet on the valence state is mostly away from the coupling region, there can be an effective transfer into the valence state from the Rydberg states. Unlike the earlier-discussed case of a bistate transfer, here the rate of transfer is unistate, analogous to a unimolecular process. We will separately discuss the effect of the mass on the local population and on the relative phase. The latter plays a crucial role for the case of bistate transfer and leads to the time-dependent isotope effect.

### Newtonian Effect of Mass for Multielectronic States.

The Newtonian role of mass as the measure of resistance to motion is seen through the role of the off-diagonal kinetic energy operator where mass enters explicitly as the coupling constant

For the case of nonnegligible diabatic coupling, the motion along the grid has an indirect effect on the rate of overall population transfer (Eq. **2**). It changes the local population in the coupling region, and accordingly the local coherence in the coupling terms of Eq. **3**. This effect can be examined through the dynamics of the mean value of *R* for the coupled electronic states. The dynamics of the population, *R* value, *b* wave packet in the coupling region around 50–60 fs. At this time, we are in the regime of the bistate transfer and the isotope effect on the population dynamics becomes significant. When the diabatic coupling is rather strong (200% compared with ref. 40) (rightmost panel in *SI Appendix*, Fig. S10 and also *SI Appendix*, Fig. S11), there arises a unistate transfer at early times as discussed in connection with Eq. **3**. This transfer from Rydberg to the valence electronic state, when initial valence wave packet is already further away, reduces the mean value of *R* by adding a component wave packet in the coupling region. The isotope effect both on the population and the mean *R* value is small and monotonic for the early times unistate transfer. At later times, after the initial valence state wave packet is coming back to the coupling region, the nonlinear bistate isotope effect occurs.

### Quantum Origin of the Isotope Effect.

The strong time-dependent isotope effect appears in the case of bistate regime of transfer, when the wave packets on different electronic states revisit and overlap in the coupling region. Before the recurrence in the coupling region, the two initial wave packets on the Rydberg and valence potentials have different paths, which give rise to different phases and thereby to the interference terms in the Eqs. **1** and **2**:*SI Appendix*, section 5).

To highlight the effect of mass, we present a plot of the population dynamics for the model of two coupled excited electronic states with realistic (Fig. 1) strength of diabatic coupling (the reported value of ref. 40) for the states of *A* the direction of initial transfer changes, going from mass 12 to 13 to 15 coming back to the same direction by mass 17. We quantify this effect in the toy model.

### The Toy Model: Interference as the Origin of the Isotope Effect.

We discuss the mass effect in a simple toy model that shows it to originate from the interference between nuclear wave packets on two coupled electronic states. A simple model is needed because of the strong diabatic coupling between electronic states. With time, this coupling spawns wave packets by transfer from the Rydberg *c* and *o* states to the valence state. The wave packets born initially localized in the Franck–Condon region are gradually delocalized by this spawning. So we develop a toy model where the wave functions on each electronic state remain bell-shaped Gaussians. The model neglects the early time, unistate, population transfer where the isotope effect is not large and monotonic. It focuses on the first recurrence of the overlap of two wave packets where the isotope effect is dominant (Fig. 4).

In the toy model, we use Gaussian wave functions as proposed by Heller (43), and modify the valence state potential to be harmonic (*SI Appendix*, Fig. S12) so that the Gaussian form is retained in time. The value of the electronic coupling between the two electronic states is one-half the value of Fig. 1. A single Gaussian function on each electronic state (*SI Appendix*, Eq. **S18**) has an overall time-dependent phase. Heller (43) uses the notation *k* is the index of the electronic state. It is this angle that will make a key contribution to the interference term in the rate of transfer as shown in *SI Appendix*, Eq. **S33**. All of the computational details for this model are given in *SI Appendix*, section 5.

We propagate the dynamics followed by excitation with a 0.5-fs pulse (*SI Appendix*, Fig. S13) for a set of isotopomers with reduced masses spanning the range of 10–20 a.m.u. We simulate the dynamics either via a single Gaussian on each excited electronic state or using propagation on a grid within the Fourier method. When we use a single Gaussian for each state, the equation for the rate of population transfer, *SI Appendix*, Eq. **S5** for the propagation on the grid, is modified. As discussed in *SI Appendix*, section 5, the major role of the diabatic coupling is when the overlap of the two Gaussians is large, suggesting a Condon-like approximation:*SI Appendix*, Eq. **S17**. *SI Appendix*, section 5. Initially, the two wave packets have the same phase. It is the rate of change of the phase on each state:*k*. Effective transfer requires a stationary phase as is shown in Fig. 5 *A* and *B* by comparing the two panels vertically. The computed phase difference is varying quite rapidly with time at all but the stationary phase region (see *Bottom*). There is also a large change in the rate even for fractionally small changes in the mass. A detailed discussion of computing the phase difference is in *SI Appendix*, section 5.

A summary of the mass dependence and its origin in wave functions overlap is suggested by the heatmaps in Fig. 6. These are contour plots of the rate of population transfer between two states, the valence excited *b* and the Rydberg *c* state, plotted vs. mass in the range of 10–20 a.m.u. and time. The plots are for the specific time interval that the dynamics suggest as the time range of maximal overlap. On the *Left* are numerically exact grid calculations for the case of realistic Rydberg and valence potentials and a pulse duration of 8.3 fs. The localization of the maximal rate of transfer in time and the mass-dependent switch in the direction of transfer (Fig. 5) are evident. Very similar trends and patterns are seen for shorter pulse durations. The *Right* panel is generated by propagating in time the Gaussian wave functions of the toy model with a pulse duration of 0.5 fs. The model clearly captures the essence. It emphasizes that such differences that can be seen between the two panels in Fig. 6 are primarily due to the approximate valence state potential used in the toy model. In *SI Appendix*, Fig. S16*B*, we show also a third panel. It is a numerically exact grid propagation for the Hamiltonian of the toy model. This plot is very similar to the results shown here for the Gaussian propagation (Fig. 6, *Right*).

## Conclusions

A significant effect of the mass of the atoms on the electron-nuclear dynamics following an excitation of N_{2} with an ultrashort vacuum-UV pulse was examined. A time-dependent isotope effect was demonstrated after a short time interval following excitation, when the quantum electron dynamics is still not adjusted to the instantaneous positions of the nuclei and the nuclear wave functions are localized. We used a diabatic basis for the electronic states, states of definite electronic character such as valence or Rydberg states where the kinetic energy is diagonal. In this basis, interelectronic coupling is independent of mass and varies only as a function of the internuclear distance. Equations for the rate of population transfer between the electronic states allowed us to identify the terms that are responsible for the mass effect on the dynamics. We reported analytical derivations complemented and supplemented with numerical simulations of quantum dynamics for different isotopomers of N_{2} and strengths of the diabatic interelectronic coupling. Within the early time range, coherent nuclear motion of the wave packet dynamics on several potentials is accompanied by passing several times through the range of distances with strong electronic diabatic coupling. Coupled valence and Rydberg states have qualitatively different shape of the potential. Therefore, the initially created wave packets revisit the coupling region at different times and acquire different phase shifts during their paths. The isotope-sensitive step is observed after the first vibrational period of the wave packet on the valence state, when there is a large overlap with the wave packets of the other excited states in the coupling region. The interstate electronic coupling is influenced by the phase matching between the wave packets of the two electronic states. It is also proportional to the product of the local vibrational populations on the two different electronic states. The coupling results in significant dependence of the rate of transfer of population upon the mass—exhibiting a time-dependent isotope effect on the electron-nuclear dynamics.

## Methods

The time-dependent Schrödinger equation for several coupled electronic states is solved on a grid. The electronic Hamiltonian is defined for each grid point with diabatic potentials, transition dipoles and diabatic coupling terms taken from ref. 40. The molecules are taken to be aligned in the direction of the laser pulse or perpendicular to it so states of either *SI Appendix*, sections 1 and 5, respectively.

## Acknowledgments

F.R. thanks Fonds National de la Recherche Scientifique for its support. This work was supported by US Department of Energy, Office of Science, Basic Energy Sciences Award DE-SC0012628 and by Fonds de la Recherche Fondamentale Collective (T.0132.16 and J.0012.18). We benefited from our participation in COST Action CM 1405 “Molecules in Motion.”

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: rafi{at}fh.huji.ac.il.

Author contributions: K.G.K., F.R., and R.D.L. designed research; J.S.A. and K.G.K. performed research; J.S.A., K.G.K., F.R., and R.D.L. analyzed data; and J.S.A., K.G.K., F.R., and R.D.L. wrote the paper.

Reviewers: J.M., Freie Universität Berlin; and T.E.M., Stanford University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1804455115/-/DCSupplemental.

Published under the PNAS license.

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