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Competition between Light-Induced and Intrinsic Nonadiabatic Phenomena in Diatomics András Csehi,†,‡ Gábor J. Halász,§ Lorenz S. Cederbaum,∥ and Á gnes Vibók†*,†,‡ †

Department of Theoretical Physics, University of Debrecen, PO Box 400, H-4002 Debrecen, Hungary ELI-ALPS, ELI-HU Non-Profit Ltd., Dugonics tér 13, H-6720 Szeged, Hungary § Department of Information Technology, University of Debrecen, PO Box 400, H-4002 Debrecen, Hungary ∥ Theoretische Chemie, Physikalish-Chemisches Institut, Universität Heidelberg, H-69120 Heidelberg, Germany ‡

ABSTRACT: Nonadiabatic effects arise due to avoided crossings or conical intersections that are either present naturally in field-free space or induced by a classical laser field in a molecule. Recently, it was demonstrated that nonadiabatic effects in diatomics can also be created in an optical cavity. Here, the quantized radiation field mixes the nuclear and electronic degrees of freedom. We show the equivalence of using the cavity’s quantized field and the classical laser field as usually done for molecules. This is demonstrated for NaI, which exhibits a pronounced natural (intrinsic) avoided crossing that competes with the avoided crossing induced by the field. Furthermore, rotating molecules exhibit lightinduced conical intersections (LICIs) in classical laser light, and we also investigate the impact of these intersections. For NaI, we undoubtedly demonstrate a significant difference between the impact of the laser-induced avoided crossing and that of the LICI on the dynamics of the molecule.

C

the photofragments of the D+2 molecule has been found that serves as a direct signature of the LICI,20 and also a recent experiment has provided clear evidence for the existence of the LICI in the H+2 molecule.25 Due to the presence of several vibrational degrees of freedom, LICIs are ubiquitous in polyatomic molecules, and they can exist even without rotations.28 This provides new opportunity for manipulating and controlling nonadiabatic effects by light.23,24,26−28 In this Letter, we theoretically investigate how a diatomic molecule that already possesses an intrinsic (natural) avoided crossing is affected by another object causing nonadiabatic effects that is induced by the laser light. This can either be a lightinduced avoided crossing (LIAC) or a LICI, depending on the level of theoretical description. In a restricted one-dimensional (1D) description, the molecular rotational angle is only a parameter and thus can only mimic an avoided crossing situation. However, in the full two-dimensional (2D) description, the rotational angle is a dynamic variable, the inclusion of which leads to a LICI explicitly included in the description. Our showcase example is the sodium iodine molecule. This molecule has already served as a prototype system in several basic photochemical studies.29,30 Recently, two remarkable contributions have discussed the nonadiabatic photochemical dynamics of this system. Bandrauk and co-workers34 have successfully studied the direct and indirect photofragment distributions during the dissociation reaction. In

onical intersections (CIs) are not present in diatomics, but they can already be found in triatomic molecules and are abundant in large polyatomic systems. At a CI, the nonadiabatic couplings become infinite, resulting in dramatic nonadiabatic effects intensively studied in the literature.1−12 In several important chemical dynamical phenomena such as vision, photosynthesis in plants, photochemistry of DNA, fluorescence of proteins, molecular electronics, molecular light-harvesting processes, and so forth, CIs serve as an efficient and ultrafast (typically on the femtosecond time scale) channel for the process at hand. In polyatomics, naturally occurring CIs (“natural CIs”) are not isolated points in the nuclear configuration space but rather form a seam. The position of the CIs and strength of the nonadiabatic effects are inherent properties of the involved molecular electronic states and are difficult to manipulate.31−33 New features emerge when molecules are exposed to a strong resonant laser field. So-called “light-induced conical intersections” (LICIs) are formed by the light, which are present even in the case of diatomic systems.13,14 The angle θ between the laser polarization and the molecular axis provides the missing dynamical coordinate that together with the vibrational mode spans the “branching” space in which the LICI can exist in a diatomic. The positions of these LICIs are determined by the laser frequency, while the laser intensity controls the strength of the nonadiabatic coupling. In other words, the LICIs are rather easy to manipulate. Several theoretical14−22,34 and experimental25 works have demonstrated that the LICIs have strong impact on the spectroscopic and dynamical properties of diatomic molecules. Among others, a very robust effect in the angular distribution of © XXXX American Chemical Society

Received: February 19, 2017 Accepted: March 23, 2017 Published: March 23, 2017 1624

DOI: 10.1021/acs.jpclett.7b00413 J. Phys. Chem. Lett. 2017, 8, 1624−1630

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The Journal of Physical Chemistry Letters the other work,36 the natural avoided crossing has been manipulated by putting the molecule into a nanoscale cavity discussing the dynamics by means of the quantized radiation field. We note here that photochemical reactions of molecules in quantized electromagnetic fields have also been the subject of several other considerable works.35,37−41 The goals of this Letter are two-fold. First, efforts are made to demonstrate the close similarity between the classical and the cavity radiation field description of the light. We show to what extent these different physical scenarios, a molecule in a laser field and a molecule in a cavity, provide the same results. The discussion of this issue is in 1D as the 2D Hamiltonian for a molecule in the cavity and its evaluation are not yet available. Second, the NaI is an interesting system by itself, and we would like to study its nonadiabatic dynamics in a laser field. Such a study must be done in 2D where LICIs appear. This will allow us to reveal the difference between the impact of the laser-induced avoided crossings found in 1D and of CIs that appear in the full

description of the problem. NaI exhibits a natural avoided crossing, and its competition with the LICI is of interest. We study the action of the molecule in long laser pulses. This relates to the situation in the cavity and also complements the investigation of Bandrauk and co-workers34 who studied NaI in short laser pulses. This molecule is rather heavy, and applying short and long pulses leads to quite different dynamics. The two relevant electronic states of the NaI, which are considered in the simulations, are the ground ψe1 and the first excited ψe2 states of the molecule. The respective potential curves V1(R) and V2(R) are shown in Figure 1. These curves exhibit an avoided (covalent−ionic) crossing at Rc ≈ 8 Å at which the nonadiabatic coupling3−7 τ(R), also shown in Figure 1A, peaks. The NaI molecule has a permanent dipole moment that averages out in high-frequency laser fields.28 As done in former works on this molecule,34,36 it will not be included here as well. The resulting matrix Hamiltonian describing NaI in a laser field reads

⎛ 1 ∂2 ⎞ L2 ⎜− ⎟ + 0 2 −ε0f (t )d(R ) cos θ cos ωLt + K (R )⎞ V1(R ) 2μR2 ⎜ 2μ ∂R ⎟ ⎛ ⎟ + ⎜⎜ H=⎜ ⎟ ⎟ 2 2 − − ( ) ( ) cos cos ( ) ( ) ε θ ω f t d R t K R V R ∂ 1 L ⎠ L 2 ⎜ ⎟ ⎝ 0 − + 0 ⎜ ⎟ 2μ ∂R2 2μR2 ⎠ ⎝ (1)

Here, R is the molecular vibrational coordinate, μ is the reduced mass, and L denotes the angular momentum operator of the nuclei. θ is the angle between the polarization direction and the direction of the transition dipole and thus the angle of rotation of the molecule. The two electronic states are coupled by a laser pulse of frequency ωL, amplitude ε0, and envelope f(t), and d(R) (=−⟨ψe1|∑j rj|ψe2⟩) is the transition dipole matrix element (e = me = ℏ = 1 atomic units are used throughout the article). The states are coupled nonadiabatically by the intrinsic operator3−7

Nq

ϕj(q)(q , t ) = q

(2τ(R) ∂∂R + τ(2)(R)). It can be well represented 1 ∂ ∂ K (R ) ≈ 2μ (2τ(R ) ∂R + ∂R τ(R )) as done in other

K (R ) =

by

internuclear separation R. The rotational degree of freedom was described by Legendre polynomials { PJ(cos θ)}j=0,1,2,...,Nθ. These so-called primitive basis sets (χ) were used to represent the single-particle functions (ϕ), which in turn were used to represent the wave function

1 2μ

∑ c (j ql )(t )χl(q) (q) nR

ψ (R , θ , t ) =

42

nθ

∑ ∑ A j ,j (t )ϕj(R)(R , t )ϕj(θ)(θ , t ) R θ

jR = 1 jθ = 1

35−37

on NaI. For the sake of comparison, we also keep studies this form. The Hamiltonian (eq 1) governs the dynamics of the system, and we will use it in our numerical calculations as it stands. Before turning to the numerical calculations, we briefly elucidate the appearance of a LICI. It has been shown that for long pulses a transformation to the Floquet picture is appropriate.17 By transforming to a Floquet picture (see, e.g., ref 17), the Hamiltonian takes on a form as in (eq 1) but with f(t) cos ωLt = constant and V2(R) replaced by V2(R) − ℏωL. Around the crossing of V2(R) and V2(R) − ℏωL, the intrinsic nonadiabatic coupling is negligible (see Figure 1B), and diagonalization of the potential energy part of the Floquet Hamiltonian gives rise15 to the light-induced adiabatic potential energy surfaces Vlower and Vupper (see Figure 1C). These two surfaces cross each other at a single point (R, θ) where cos θ = 0 (θ = π/2) and V1(R) = V2(R) − ℏωL are simultaneously fulfilled, giving rise to a LICI.14 Before discussing results on NaI, we present the method of computations. The MCTDH (multiconfiguration time-dependent Hartree) method is applied to solve the time-dependent Schrödinger equation (TDSE).44−48 To characterize the vibrational degree of freedom, we have used a sin-DVR with NR basis elements distributed within the range of 2.1−60 au of the

q = R, θ

q

l=1

R

θ

(2)

The actual numbers of primitive basis functions in the numerical simulations were chosen to be NR = 2048 for the vibrational and Nθ = 61, 201, and 301 for the rotational degrees of freedom, respectively. On both adiabatic surfaces and for both degrees of freedom, a set of nR = nθ = 5, 8, 20, and 30 single-particle functions was applied to form the nuclear wave packet of the system. The various values of Nθ and nR = nθ were chosen depending on the peak field intensity I0 of the laser pulse used. Attention has been paid to proper choice of the basis to ensure that convergence has been reached in each propagation. The solution of the TDSE is used to calculate the time-dependent population P Vi(t ) = =

∫0

ψ Vi(R , θ , t )|ψ Vi(R , θ , t )

π

dθ·sin θ

∫0

∞

dR ·ψ Vi(R , θ , t )*·ψ Vi(R , θ , t )

(i = 1, 2)

(3)

on the ground state and the first excited state. We have also computed the adiabatic potential energy curves, the nonadiabatic coupling matrix element, and the transition dipole moment. For that purpose, we used the program package MOLPRO43 on the MRCI/CAS(6/7)/aug-cc-VQZ level of 1625


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The Journal of Physical Chemistry Letters

same quantum chemical input quantities for the dynamical calculation as those in ref 36. The considered external electric field is linearly polarized and has a constant frequency, ωL E(t ) = ε0f (t ) cos(ωLt )

(4)

where ε0 is the amplitude of the electric field and ωL has a value of 0.815 eV, which corresponds to the energy difference between the V1(R) and V2(R) potential curves at R ≈ 6 Å. The laser field is switched on during the whole propagation process (tfinal = 3000 fs). f(t) is the envelope function of the pulse and has been set constant to a value of 1 to simulate a continuous laser wave. In the simulations, several values of the intensity (from I = 109 to 1012 W/cm2) were considered. We consider the following scenario. At time t = 0, the vibrational and rotational ground state of the electronic ground potential is promoted to the first excited electronic state, and as a result, the wave packet starts to oscillate on the excited state’s potential curve, reaching the avoided crossing region (see Figure 1A). Due to the nonadiabatic coupling, population of the wave packet will be transferred back to the ground state whenever it comes close to the avoided crossing region. Once the laser light is on, the light-induced nonadiabaticity will compete with the intrinsic one. Let us begin with the field-free dynamics with the laser off. To compare with the results of Kowalewski et al.36 who studied the dynamics in the cavity, the initial wave packet at time t = 0 is located at R = 3.4 Å on the upper adiabatic potential. The results for the population of the nuclear wave packet on the ground electronic state are shown as a function of time in Figure 2.

Figure 1. Potential energies of the NaI molecule. (A) Adiabatic energies V1(R) and V2(R) of the ground and first excited states of the NaI molecule are displayed by solid black and red lines, respectively. The inset shows the avoided crossing, and the nonadibatic coupling term τ(R) (green line) is shown below the crossing with its scale is on the right side. (B) Energies of the ground V1(R) and the field-dressed excited V2(R) − ℏωL states of the NaI are displayed by solid black and solid red lines, respectively. Note the different scale of the R axis. The curves now cross at a much smaller distance than that of the avoided crossing in (A). This crossing becomes the LICI once the transition dipole moment is taken into account in the (R, θ) plane. Shown is also a cut at θ = 0 (parallel to the field) through the light-induced adiabatic surfaces in this plane (see C) depicted by dotted black (lower adiabatic state) and dotted red (upper adiabatic state) lines. (C) Light-induced potential energy surfaces as a function of the interatomic distance R and the angle θ between the molecular axis and the laser polarization direction. The applied energies and intensity of the laser pulse are ℏωL = 0.815 eV and I = 1013 W/cm2, respectively.

Figure 2. Population of the ground adiabatic electronic state as a function of time in the field-free case (laser off). At time t = 0, the nuclear ground state of the electronic ground-state potential is promoted to the first excited electronic state, and as a result, the wave packet starts to oscillate on the excited state’s potential curve, reaching the avoided crossing region (see Figure 1A) and transferring population back to the ground electronic state. The results of the 1D and 2D calculations are the same. They are also identical with the cavity switched-off situation in Figure 3a of ref 36.

Appreciable nonadiabatic transfer of population to the ground electronic state occurs when the wave packet is in the vicinity of the avoided crossing. As the rotational quantum number of the initial wave packet is J = 0, the 1D and 2D numerical calculations provide identical results, as expected. These results also perfectly reproduce the respective results in the case of the cavity; see Figure 3a in ref 36. In the cavity, the quantized radiation field couples to the electronic degrees of freedom by a cavity coupling constant g. The results shown in Figure 2 for the field-free

theory with an effective core potential for iodine (ECP46MWB) as done in ref 36. As a result, we could successfully reproduce the 1626


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Figure 3. Population of the ground adiabatic electronic state as a function of time computed in 1D, setting the parameter θ in the Hamiltonian (1) to a constant in order to be able to reproduce the results found for the cavity.36 Results are shown for different laser intensities I (for θ = 0) chosen to reproduce the results of ref 36. (A) Results for the weak intensities, where the evolution of the population is still similar to that found in the field-free case I = 0 (black). (B) Results for the other intensities, where the findings become already qualitatively different from those in the first panel. (C) Relationship between the laser amplitude and the coupling strength g of the cavity.

carry out this comparison, we have to fix the value of θ in the Hamiltonian (eq 1) as θ does not appear in the 1D cavity study. Then, there is a linear relation between the laser amplitude and g. The results obtained are shown in Figure 3. It can be seen that by increasing the light intensity one can reproduce the time evolution of the ground-state population found in the cavity for increasing values of g as depicted in Figure 3b,c of ref 36. The relationship between the laser amplitude and g is also shown in Figure 3. We may conclude that the θ = constant 1D semiclassical description of the light−molecule interaction is fully consistent with the cavity results of refs 35−37, 39, and 40). We now return to the Hamiltonian (eq 1) for NaI in the laser field and stay in 1D but now where θ appears as a parameter. As can be seen in eq 3, the correct 1D calculation to describe a molecule in a laser field requires averaging over numerous computations done for different values of the parameter θ. This approach is also necessary for comparison with the full (i.e., 2D) solution of the problem discussed below. The results are shown in Figure 4 for several values of the laser intensity. Compared to the populations shown in Figure 3, the variation with the laser intensity are

dynamics correspond to g = 0. We now demonstrate the close similarity between the classical and the cavity radiation field description of the light. We show that these different physical scenarios, a molecule in a laser field and a molecule in a cavity, provide the same physics. The discussion is in 1D as the 2D Hamiltonian for a molecule in the cavity has not yet been derived and evaluated. When the quantized radiation field couples to the electronic degrees of freedom, g ≠ 0. The quantity g in the quantized radiation field formalism plays a similar role as the field amplitude times the electronic transition dipole moment in the semiclassical light−molecule treatment. We note here that cavity effects are usually perturbative, whereas classical fields allow one to describe nonperturbative effects. We performed 1D calculations for several values of the laser intensity. In these calculations, the angular momentum operator L in eq 1 is set to 0 and the molecular orientation θ becomes a parameter. We first investigated to what extent the results are similar to those obtained from the calculations for the quantized radiation field performed for different values of g. Both the semiclassical and cavity approaches describe a LIAC situation. To 1627


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Figure 4. Comparison of the population of the ground adiabatic electronic state as a function of time computed in 2D and 1D. Each panel on the lefthand side (A−I) shows the 2D and 1D results for a given laser intensity; from top to bottom I = 109 (A), 1010 (C), 1011 (E), 5 × 1011 (G), and 1012 W/ cm2 (I). Panels (B), (D), (F), (H), and (J) on the right-hand side depict the difference of the respective 2D and 1D curves. For the weak intensities, I = 109 and 1010 W/cm2, the evolution of the population in 2D is similar to that found in 1D, which still resembles that found in the field-free case (see Figure 2). Clear deviations are already found for I = 1011 W/cm2, and a completely different behavior are found for I = 5 × 1011 and 1012 W/cm2.

the laser light compared to that in the field-free molecule (see Figure 1B). Increasing the laser intensity, the splitting between the light-induced states can strongly vary, resulting in an increased nonadiabatic population transfer and trapping of population, which prevent the molecules from dissociating. In turn, this is equivalent to the statement that the number of

somewhat less pronounced. This is due to averaging over θ, which implies that effectively weaker light-induced couplings are involved for each value of the laser intensity. The underlying dynamics is mainly governed by the interplay of the LIAC created by the resonant coupling of the participating electronic states and of the natural avoided crossing, which is, however, modified by 1628


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degrees of freedom to form such intersections, which can be used to selectively manipulate the dynamics.28

dissociating particles gradually decreases and hence the lightinduced nonadiabatic phenomena lead to suppression of dissociation. In reality, molecules in laser fields are subject to rotation, and this implies that θ is a dynamic variable like R and not just a parameter. Inclusion of rotations is the next major goal of our work and will allow us to study the influence of the LICI. The ground-state populations computed in 2D are shown in Figure 4 for the same laser intensities as those depicted there for 1D. Obviously, the impact of the LICI can be strong. To better distinguish between the 1D and 2D results, we compare in Figure 4 these populations for each intensity separately. For low intensities (I = 109−1010 W/cm2), one sees no remarkable differences between the two approaches (panels A−D). Here, as we have seen above, the influence of the light-induced phenomena is small. Upon further increasing the intensity (I = 1011 W/cm2), visible differences start to appear at longer times t (panels E and F). This trend changes dramatically at the next set of intensities (I = 5 × 1011 and 1 × 1012 W/cm2). The differences between the 1D and 2D populations are now eye-catching (panels G−J). The pronounced oscillations found in 1D are significantly squeezed in 2D. The higher the intensity, the more prominent the effect. The explanation is obvious. LICIs give rise to much stronger nonadiabatic effects than do LIACs, as is the case for the corresponding natural objects. In 2D, a greater portion of the molecules can dissociate than in the 1D simulation as the transfer of population to the ground state is better realized via LICIs than via LIACs. Although the present example is more complex than that studied in our previous works on diatomics due to the existence of a natural avoided crossing in the field-free molecule, the resulting effect is similar to that obtained in one of our previous studies20 in that LICIs provide more efficient pathways for extremely fast population transfer between the electronic states than avoided crossings. Once the laser intensity is sufficiently large (about I = 5 × 1011 W/cm2 or larger), the LICI dominates the dynamics of the system, in particular, the nonadiabatic dynamics. We showed that one can manipulate by laser light the impact of the naturally occurring avoided crossing present in NaI. By means of 1D and 2D quantum dynamical calculations, one can simulate the impact of a LIAC and a LICI, respectively. Applying these two light-induced phenomena, the time-dependent state populations and the photodissociation process of NaI were studied for several different field intensities. The 1D results were found to coincide with those obtained before for NaI in a cavity with quantized light. The impact of the LICI was demonstrated to be dramatic at sufficiently large laser intensities. The results clearly demonstrated that by increasing the intensities the dissociation yield decreased, but significantly differently for the LICI and the LIAC. The difference is related to the finding that the LICI provides more efficient pathways for ultrafast population transfer between the involved electronic states than does the LIAC. As found for natural avoided crossings and CIs,3−7 there is also a qualitative difference between LICIs and LIACs. The latter can, however, be controlled and used to manipulate the dynamics, for instance, to diminish or to enhance the impact of the former. We hope that our findings will stimulate laser as well as photochemical cavity experiments and also extension of the theory in a cavity to include rotation. We mention that there is much potential in studying LICIs in polyatomic molecules by lasers or in a cavity without rotations as there are many nuclear

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Á gnes Vibók†: 0000-0001-6821-9525 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support by the Deutsche Forschungsgemeinschaft (Project ID CE10/50-3). The ELIALPS project (GOP-1.1.1-12/B-2012-000, GINOP-2.3.6-152015-00001) is supported by the European Union and cofinanced by the European Regional Development Fund. The supercomputing service of NIIF has been used for this work. This material is based upon work supported in part by the U.S. ARL and the U.S. ARO under Grant No. W911NF-14-1-0383. The authors thank Markus Kowalewski and Péter Domokos for the fruitful discussions.

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REFERENCES

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