Time-Resolved Photoelectron Imaging of Molecular Coherent

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Time-Resolved Photoelectron Imaging of Molecular Coherent Excitation and Charge Migration by Ultrashort Laser Pulses Kai-Jun Yuan* and André D. Bandrauk* Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1 ABSTRACT: Charge migration is a fundamental and important process in the photochemistry of molecules and has been explored by time-resolved photoelectron angular distributions. A scheme based on UV pump and polarized soft X-ray probe techniques shows that photoelectron diffraction effects enable us to reconstruct electronic coherences encoding the information of the charge migration with extreme time resolutions. We discuss how probe pulse helicity influences the probing photoelectron spectra in the presence of molecular nonspherical Coulomb potentials. This phenomenon is analyzed theoretically and simulated via ab initio calculations for the molecular hydrogen ion, offering a reliable approach for measurements of charge migration and for the exploration of molecular structure in attosecond science.

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INTRODUCTION Imaging and control of molecular reaction dynamics associated with nuclear motion using ultrafast laser pulses in the femtosecond (1 fs = 1015 s) time regime have been widely studied in the chemical sciences.1−3 Upon projection of an initial electronic state onto accessible excited states with an ultrashort pump laser pulse in a molecule, its time evolution can be monitored after a variable time delay with ultrafast probe pulses, leading to time-resolved photoelectron spectroscopy which has been shown to be a powerful method for examining electron dynamics in chemical reactions.4−6 Ultrashort laser pulses have led to imaging pump−probe techniques such as laser Coulomb explosion imaging7−9 for nuclear motion and laser-induced electron diffraction (LIED)10−12 for coupled electron−nuclear motion. Electrons have a great potential for probing the time-resolved transient structure of molecules, materials, and even biological systems via ultrashort electron diffraction. Insights into reaction dynamics through electron imaging can now be obtained by time-resolved photoelectron spectroscopy on the attosecond (1 as = 1018 s) time scale and subnanometer size.13−16 Currently, 43 as linearly polarized pulses are the shortest available pulses for such new optical imaging techniques.17 Charge migration is a fundamental quantum process in photoinduced chemical reactions involving electron dynamics of reactivity on the attosecond time scale.18−31 It arises from a coherent population in multiple electronic states and is likely responsible for selective laser excitation in molecules, leading to the concept of charge-directed chemistry.32,33 Ultrafast charge migration offers an approach to controlling chemical reactions and probing molecular orbitals. It has been shown that quantum control of electron flux during intramolecular charge migration is possible by designing ultrafast laser pulses that prepare the system in selective electronic states.34 It has been © XXXX American Chemical Society

found that the electron currents are functions of the polarization and helicity of the pump laser pulse, i.e., the symmetry property of the excited state.28 Laser control of electron localization in molecular charge migration by ultrafast laser pulses has also been presented recently.35,36 Exploring ultrafast charge migration dynamical processes requires demanding spatiotemporal resolutions. Recent experiments have, for the first time, measured attosecond charge migration with linearly polarized high-order harmonic generation (HHG) spectroscopy in ionized iodoacetylene via attosecond electron recollision processes.37 Angular electronic fluxes have also been used to reconstruct attosecond charge migration in excited benzene by preparing a coherent electron superposition state.24 Electronic ring currents prepared by ultrashort circularly polarized pulses give rise to ultrafast molecular magnetism38,39 and new quantum devices,40 which can be tracked with probing time-dependent photoelectron spectra.41 We have also proposed that one can monitor coherent excitation and attosecond charge migration by electron diffraction in molecular photoelectron spectra, which encode information on the symmetry of electron orbitals and molecular bonding.42 In the present work we propose a new method to image the coherent excitation and charge migration of linear molecules by photoelectron diffraction patterns in photoelectron spectra excited by linearly/circularly polarized soft X-ray probe pulses. Molecular frame photoelectron angular distributions (MFPADs) exhibit a signature asymmetric ionization with respect to the molecular symmetry axis, which is shown to be sensitive to the helicity of probe pulses. We attribute this to the resulting effects of the asymmetric electron Received: November 28, 2017 Revised: February 6, 2018 Published: February 6, 2018 A

DOI: 10.1021/acs.jpca.7b11669 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

the ionized wave packet from the boundary, a mask function with the form cos1/8[π(ρ − ρa)/2ρabs] is used at ρabs = ρmax − ρa = 24 au with ρmax = 128 au. Photoelectron spectra, i.e., MFPADs, are calculated by a Fourier transform of the 3D time-dependent electronic wave function ψ(ρ, θ, z, t), which exactly describes the electron dynamics in the continuum:46

density distributions due to effects of coherent excitation and nonspherical molecular Coulomb potentials.

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COMPUTATIONAL METHODS We use the molecular ion H2+ aligned with the internuclear R axis along the x axis to study molecular charge migration in ultrafast laser fields. The excitation and ionization dynamics of H2+ can be exactly described by numerically solving the threedimensional (3D) time-dependent Schrö dinger equation (TDSE), thus allowing us to describe ultrafast electron dynamics in molecular reaction processes.43,44 For an x-aligned H2+, the corresponding TDSE with a static nucleus frame described in cylindrical coordinates (ρ, θ, z) with (x = ρ cos θ, y = ρ sin θ) as the molecular plane can be written as ⎡ ℏ2 2 ⎤ ∂ i ℏ ψ (r , t ) = ⎢ − ∇r + Vc(r) + VL(t )⎥ψ (r, t ) ∂t ⎣ 2me ⎦

1l(θ , z , E)| ρf = 1r(θ , z , E)| ρf = ⎡1 1(θ , E) ∼ Re⎢ ⎣ 2i

1 ∂2 ℏ2 2 ℏ2 ⎡ 1 ∂ ⎛ ∂ ⎞ ∂2 ⎤ ⎢ ∇r = − ⎜ρ ⎟ + 2 2 + 2 ⎥ 2me 2me ⎣ ρ ∂ρ ⎝ ∂ρ ⎠ ∂z ⎦ ρ ∂θ

(2)

(3)

and for the linearly/circularly polarized probe pulse ⎧ ⎪ ex̂ E0 f (t − t pr0) cos[ωpr (t − t pr0)] Epr(t ) = ⎨ ⎪ ⎩ eŷ ξE0 f (t − t pr0) sin[ωpr(t − t pr0)]

p

∞ p

ψ (θ , z , t )| ρf eiEt dt

∂ψ (θ , z , t ) | ρf eiEt dt ∂ρ ⎤

∫ 1 *l (θ , z , E)|ρ 1r(θ , z , E)|ρ dz⎦⎥ f

f

where tp is the time after the pulse is turned off and ρf = 100 au, which is an asymptotic point before the wave packet is absorbed. E = pe2/2me is the kinetic energy of an ionized electron with wave vector k = pe/ℏ = 2π/λe, and pe = (px2 + py2)1/2 is the momentum of a photoelectron of wavelength λe. Since the ionization occurs in the laser polarization (x, y) molecular plane, we define θ as the angle between the electron momentum pe and the x polarization axis. With the transformation of px = pe cos θ and py = pe sin θ, we then obtain the two-dimensional (2D) momentum distributions of photoelectrons from eq 5. MFPADs at photoelectron kinetic energy Ee are obtained by integrating over the one-photon energy, where the spectral width of the probe pulse is Δω ≈ ωpr/3

The field-molecule interaction VL(t) = er·E(t) is treated in the length gauge and dipole approximation, since by exact solutions of the TDSE, numerical results are gauge invariant.45 The laser fields E(t) = Epu(t) + Epr(t) always propagate along the z axis, perpendicular to R, with the forms, for the linearly polarized pump pulse Epu(t ) = ex̂ / yE0 f (t ) cos[ωpu(t − t pu0)]

∫t

∞

(5)

(1)

The kinetic energy operator reads as −

∫t

1 Ee(θ ) =

Ee + ωpr /3

∫E −ω /3 e

pr

dE 1(θ , E) (6)

corresponding to the main one-photon ℏωpr frequency of absorption. In this work, the wavelength of photoelectron λe > R leads to photoelectron diffraction effects10 in MFPADs obtained from eq 6. Therefore, photoelectron diffraction refers to photoelectron imaging spectra in molecules.

(4)

where êx/y are the laser field polarization vectors. ωpu and ωpr are frequencies of the pump and probe pulses, respectively. ξ is the helicity of the probe pulse. ξ = ±1 and 0 denote right/lefthanded circularly and linearly polarized pulses in the (x, y) plane of the molecule. A temporal slowly varying envelope f(t) = sin2(πt/Tpu/pr) is used with duration Tpu/pr = nτpu/pr, where τpu/pr = 2π/ωpu/pr, in one optical cycle of pump/probe pulses. Multiple optical cycle duration ensures that the field area ∫ E(t) dt = 0 to eliminate effects of static fields and carrier envelope phases of pulses. tpu0 and tpr0 are the initial times of the pump and probe pulses, respectively. The time delay is the time interval between the peak times of the pump Epu(t) and probe Epr(t) pulses, which is defined as Δτ = tpr0 − tpu0 − (Tpu − Tpr)/2. In this work we always set the initial time as tpu0 = 0. Vc(r) is the two-center molecular electron Coulomb potential, where a soft parameter is used to remove the Coulomb singularity and accurately produce the electronic state potential energies of H2+.46 We numerically solve the 3D H2+ TDSE by using a five-point finite difference method and a fast Fourier transform technique combined with high-order split-operator methods.47,48 The time step is fixed at Δt = 0.01 au (atomic units, 1 au = 24 as), and the spatial discretization is Δρ = Δz = 0.25 au (1 au = 1 a0, bohr radius) for radial grid sizes 0 ≤ ρ ≤ 128 au and |z| ≤ 64 au and angle grid size Δθ = 0.01 rad. The grid sizes largely exceed the maximum field-induced electron displacements αd = E0/ ωpu/pr2. To prevent unphysical effects due to the reflection of

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THEORETICAL MODELS AND NUMERICAL SIMULATIONS We study the electron dynamics for a hydrogen molecular ion at equilibrium as a benchmark model that is excited by a pump pulse with photon energy ℏωpu = ΔE = Eu − Eg to create coherent excitation and charge migration of the 1sσg−2pσu and 1sσg−2pπu electronic state combinations. This leads to nonsymmetric charge distributions, as illustrated in Figure 1, where the electron density distributions are obtained from numerical solutions of TDSEs in eq 1 excited by linearly polarized UV pump pulses. The ground state of the molecular ion can be obtained from ionization of the neutral31,37,49 molecule or by X-ray resonant Raman processes.50 An ultrafast UV pump pulse is used to build the coherent superposition state of the two electronic states without unexpected higher excited states for a multiphoton process. We then develop electron diffraction effects to temporally resolve the charge migration of the molecule by a soft X-ray attosecond pulse. It is found that the resulting effect of nonspherical cylindrical molecular Coulomb potentials and asymmetric photoelectron distributions give rise to a distortion of electron diffraction patterns in the probe spectra, as illustrated in Figure 1. We B


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The Journal of Physical Chemistry A ((r, t ) = ⟨ψ0(r, t )|ψ0(r, t )⟩ = cσg 2⟨ψσ (r)|ψσ (r)⟩ + cσu 2⟨ψσ (r)|ψσ (r)⟩ g

g

u

u

+ 2cσgcσuℜ[⟨ψσ (r)|ψσ (r)⟩ e−iΔEt / ℏ] g

(8)

u

with 9 being the real part. The coherent electron dynamics are composed of two electronic state densities, ((g)(r) = cσg 2⟨ψσ (r)|ψσ (r)⟩ and ((u)(r) = cσu 2⟨ψσ (r)|ψσ (r)⟩, g

g

u

u

and their time-dependent interfering superposition ((g,u)(r, t ) = 2cσgcσuℜ[⟨ψσ (r)|ψσ (r)⟩ e−iΔEt / ℏ]. Of note is g

Figure 1. Illustrations of molecular Coulomb potentials Vc(x, y, z = 0) (white curves) and coherent electronic densities ((x , y , t ) (density plots) at different times (a and c) t = 11τpu and (b and d) 11.5τpu for (a and b) σg−σu and (c and d) σg−πu coherent resonant electron excitation in the molecule H2+ aligned along the x−R axis excited by linearly polarized pump pulses.

-(pe , t ) = (n ·e)

measure the dependence of photoelectron spectra on the helicity of the soft X-ray ionizing probe pulse in an ultrafast pump−probe scheme. With high-frequency pump−probe pulses, multiphoton ionization processes dominate since the Keldysh parameter γ ≫ 1 and the ponderomotive energy Up is very weak.51 Therefore, the modification of the ionization potential by laser-induced Stark shifts can be ignored. Moreover, the dipole approximation, in which the spatial dependence and magnetic component of the external field are neglected, remains valid. We first consider σg−σu parallel resonant pump excitation from the electronic ground 1sσg state, |ψσg(r)⟩ with the eigenenergy Eσg, to the electronic excited 2pσu state, |ψσu(r)⟩ with the eigenenergy Eσu. A coherent superposition of the two electronic states created by a frequency ωpu = (Eσu − Eσg)/ℏ linearly polarized pump laser pulse is obtained due to a strong charge-resonance excitation52

1 (2π )3/2

2

∫

dr e−i pe·r e−i F·rψ0(r, t )

(9)

The wave function ψσg/σu is a linear combination of a hydrogenic 1s orbital localized at ±R/2, i.e., ψσg(r) = [ψ1s(−R/2) + ψ1s(+R/2)]/√2 and ψσu(r) = [ψ1s(−R/2) − ψ1s(+R/2)]/√2. e is the laser polarization vector, and the unit vector is n = pe/pe, with |(n·e)|2 = 1 for circular polarization and cos2 θ/sin2 θ for linear parallel/perpendicular polarization. The coherent state transition amplitude, i.e., the momentum distributions of the photoelectron in eq 9, reads as -(pe , t ) ∼ |n ·e|2 {|cσg cos[(pe + F) ·R/2]|2 + |cσu sin[(pe + F) ·R/2]|2 + cσgcσu sin[(pe + F)·R /2] cos(ΔEt /ℏ)}ψ1s 2(pe )

(10)

From eq 10 one sees that, in the ultrafast photoionization model, the effect of the delta-function pulse is to shift the total photoelectron distributions by the amount of the field vector F. For homonuclear diatomic systems, the initial orbitals are peaked equally at both nuclei, separated by R. These terms (pe + F)·R/2 present the interference effects of the two outgoing electron wave packets with identical amplitudes emanating from the two nuclear centers. This is reminiscent of photoelectron diffraction through a double slit. Therefore, the photoelectron distributions encode information on the timedependent electron motion in molecules, which is much faster than nuclear motion. In this work, the pulse field strength F can be ignored to describe the effect on the electron movement in high-frequency photoionization since the chosen intensities (2.75 × 1013 W/

|ψ0(r, t )⟩ = cσg(t )|ψσ (r)⟩ e−iEσgt / ℏ + cσu(t )|ψσ (r)⟩ e−iEσut / ℏ g

u

that ((g)(r) and ((e)(r) are symmetric and do not depend on time. The time-dependent superposition term ((g,u)(r, t ) describes the attosecond electronic coherence with the oscillation period Δτ(0) = 2πℏ/ΔE = 331 as = τpu, where ΔE = Eσu − Eσg = 0.43 au. The electron density probability ((r, t ) oscillates between the two molecular centers with time t, Figure 1(a,b). The asymmetry of ((r, t ) illustrates the electron motion in the molecules. At time t = (2n + 1)τ/4, n is an integer and the superposition term cos(ΔEt/ℏ) = 0, leading to symmetric electron density distributions ((r, t ) in eq 8. We next derive MFPADs from the coherent electronic superposition state |ψ0⟩. According to the ultrafast two-center photoionization delta-function pulse model,10,53 the instantaneous transition amplitude to the continuum state of momentum pe can be written as

u

(7)

where cσg(t) and cσu(t) are the occupation coefficients. We assume that at the initial time t = 0, the molecular system is in the stationary state |ψ0(r, t = 0)⟩ = |ψσg(r)⟩. The phase of the superposition state, i.e., the asymmetry of the electronic density distributions in Figure 1, is determined by the pump pulse. We focus on the probe process after the pump pulse where cσg(t) and cσu(t) are constant (see also Figure 4). Then the corresponding time-dependent coherent electron density distributions are described by C


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The Journal of Physical Chemistry A cm2) in the simulations produce negligible ponderomotive energies Up = e2E02/4meωpr2 ≪ Ip and F ≪ pe. The transition amplitude of the coherent state can be simply expressed as

in Figure 2(b). For both cases, MFPADs are symmetric with respect to the molecular (parallel) axis since -(pe , θ , t ) ∼ cos θ . Of note is that in eq 11 and Figure 2 the effects of the electron and nucleus attraction are not taken into account in the two-center ultrafast photoionization model.10 We next compare the results in the presence of molecular Coulomb potentials. We show in Figure 3 simulated results of the molecular excitation and photoionization with a time delayed by Δτ, eq 4, and a linear polarization pump and helicity-dependent probe scheme from numerical solutions of TDSEs of H2+ in eq 1 where the molecular Coulomb potential is included. The time

-(pe , θ , t ) = - (g)(pe , θ ) + - (u)(pe , θ ) + - (g,u)(pe , θ , t ) 2 ⎡ ⎛1 ⎞ ∼ |n ·e|2 ⎢ cσg cos⎜ pe R cos θ ⎟ ⎝2 ⎠ ⎢⎣

⎛1 ⎞ + cσu sin⎜ pe R cos θ ⎟ ⎝2 ⎠

2

⎤ + cσgcσu sin(pe R cos θ ) cos(ΔEt /ℏ)⎥ψ1s 2(pe ) ⎥ ⎦ (11)

where θ is the ejected angle of the photoelectron between the molecular x axis and the electron momentum pe direction. The total photoelectron distributions are also composed of two direct components from the ground 1sσg and excited 2pσu electronic states and their coherent superposition term. From eq 11 one sees that the superposition term cos(ΔEt/ℏ) determines the evolution of the photoelectron distributions -(pe , θ , t ). At time t = (2n + 1)τpu/4, cos(ΔEt/ℏ) = 0 gives rise to symmetric MFPADs, - (g)(pe , θ ) + - (u)(pe , θ ). We next focus on the asymmetric photoelectron distributions in molecular charge migration. Figure 2 displays predicted MFPADs obtained from eq 11 with pe = 4.0 au and |n·e| = 1 originating from the coherent

Figure 2. MFPADs from a σg−σu coherent state (Figure 1(a,b) and eq 8 at |cσg|2 = |cσu|2 = 0.5) versus angle θ predicted from eq 11 at moments (a) t = 11τpu and (b) t = 11.5τpu for a photoelectron with momentum pe = 4.0 au and |n·e|2 = 1. The molecular R or x axis is θ = 0 or 180°, and arbitrary units of photoelectron distributions are used.

superposition state |ψ0⟩ in eq 7 of H2+ at R = 2 au. We set |cσg|2 = |cσu|2 = 0.5 and the ionization times at t = 11τpu (3.64 fs) and 11.5τpu (3.81 fs). It is found that multiple angular nodes are produced in MFPADs due to diffraction effects of a photoelectron with λe = 1.57 au < R. In Figure 2, MFPADs exhibit a signature of asymmetry with respect to the molecular center, which is a function of the time t. The interference term ∼ cos(ΔEt/ℏ) leads to oscillations of the MFPADs with time with a period of Δτ(0) = 2π/ΔE = τpu = 331 as. In Figure 2(a) at t = 11τpu, the angle nodes appear at θ = ±(0 + 18°), ±(90° − 11°), and ±(180° − 55°) . The distributions along the negative momentum (θ = 180°) direction are suppressed. A reverse process occurs at t = 11.5τpu, a 1/2 cycle later (165 as), as seen

Figure 3. Probing MFPADs of the σg−σu state versus angle θ in the molecule H2+ aligned along the x axis by pump−probe pulses at different time delays (a, b, and c) Δτ = 6τpr and (d, e, and f) 6.5τpr, corresponding to the probe times t = 11.0τpu and 11.5τpu, respectively. A λpu = 100 nm linearly polarized pump pulse with its field polarization vector along the molecular axis leads to a σg−σu resonant excitation. The helicity of the soft X-ray λpr = 5 nm probe pulse is (a and d) ξ = 0, (b and e) −1, and (c and f) +1 for linear, left-handed, and righthanded circular polarizations, respectively. The pulse intensities and durations are I0 = 2.75 × 1013 W/cm2 (E0 = 2.8 × 10−2 au) and Tpu/pr = 10τpu/pr, i.e., Tpu = 3.31 fs (1.655 fs fwhm, full width at halfmaximum) for the pump pulse and Tpr = 165.4 as (83 as fwhm) for the probe pulse. D


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helicity ξ, as predicted in eq 11. Varying the time delay from Δτ = 6.0τpu to 6.5τpu, i.e., the probe times t = 11τpu and 11.5τpu, the positive and negative momentum photoelectron distributions are mirror images of each other for all pulse helicities ξ. The dominant angular nodes along the molecular axis evolve from the position at θ = 0 in opposite angular directions to the other position at 180°. This illustrates the electron coherence in molecules induced by the pump pulses. It is found, furthermore, that the photoelectron spectra are sensitive to the helicity ξ of the soft X-ray probe pulse. When comparing to the σg−σu state in Figure 2 prepared with coefficients |cσg|2 = | cσu|2 = 0.5, one sees that the pump−probe time-delayed photoelectron diffraction patterns in MFPADs are suppressed in the simulations in Figure 3. At Δτ = 6.0τpu, i.e., the probe time t = 11τ for linearly polarized probe pulses with ξ = 0, angular nodes at θ = ±18° and ±79° due to photoelectron diffraction predicted in eq 11 are not produced in MFPADs in Figure 3(a). For the linear parallel probe process, the angular scale factor is n·e = cos θ. As a result, MFPADs are mainly localized along the laser polarization direction. For the processes induced by circularly polarized ξ = ±1 soft X-ray probe pulses, we see that MFPADs are also asymmetric with respect to the molecular axis, as a function of the helicity ξ. As shown in Figure 3(b,c), the angular nodes at θ = 18° and −18° vanish. Similar phenomena occur at Δτ = 6.5τpu. For both the right and left circular polarization cases with helicity ξ = ±1, the angular scale factor is |n·e|2 = |exp(±iθ)|2 = 1.54 The helicity dependence therefore illustrates the effects of the nonspherical electron distribution. For the superposition state |ψ0(r,t)⟩ of the σg−σu resonant transition (eq 7) created by the x-polarized pump pulse Epu(t) (eq 3) the density probabilities ((x , y , t ) (eq 8) are asymmetric due to the time-dependent electron migration between the protons. As illustrated in Figure 1(a) at t = 11τpu, the coherent electronic wave packets mainly lie on the left nuclear center at (x, y) = (−1, 0) . According to the photoelectron diffraction model in eq 11, the dominant ionization occurs in the right half planes. The molecular Coulomb potential is not spherical but cylindrical with strong asymmetry in the orthogonal directions, parallel (x) and perpendicular (y) to the molecular axis. The orientation photoelectron distributions from the right molecular center are therefore modulated by the helicity ξ of the probe pulse. As a consequence, the distributions rotate, leading to asymmetric structure with respect to the molecular axis (see Figure 3). Along the molecular R/x axis, the photoelectron diffraction angular nodes at θ = 18° for ξ = −1 and −18° for ξ = +1 predicted in eq 11 are absent, reflecting the strong effects of the molecular potential. At t = 11.5τpu, the coherent electron wave packets are mainly localized on the right molecular center, thus leading to reverse processes of MFPADs. Comparing the simulations in Figure 3 of a parallel σg−σu coherent excitation to the predictions in eq 11 and Figure 2, one sees that, for both the right-handed (ξ = +1) and lefthanded (ξ = −1) circularly polarized probe processes, the photoelectron spectra are influenced by the molecular Coulomb potential, leading to a distortion of MFPADs. The annihilated diffraction angular nodes depend on the helicity of the probe soft X-ray pulse. The distortion of MFPADs results from the asymmetry of the electronic density distributions of the coherent superposition state. The electron motion in molecules gives rise to the evolution of the asymmetric

delay defines Δτ = tpr0 − (Tpu − Tpr)/2, i.e., the time interval between the peak times of the pump and probe pulse, as illustrated in Figure 4. The molecular ion H2+ aligned along the

Figure 4. Evolutions of the orbital occupation |cσg(t)|2 and |cσu(t)|2 in the ground 1sσg state and the excited 2pσu state of the molecular ion H2+ aligned along the x axis by the linearly polarized λpu = 100 nm pump pulse with its field vector polarized along the molecular axis, cf., Figure 3. Dashed lines present the (magenta) pump and (purple) probe laser envelopes. Δτ is the time delay between the pump and probe pulses. The cyan line denotes the probe time.

x axis is first excited by a λpu = 100 nm (ωpu = 0.455 au) linearly polarized pump pulse with its field vector along the molecular internuclear axis. The initial ground 1sσg state is calculated by propagating the zero-field Hamiltonian in imaginary time.46 In our numerical models the molecular ionization potentials for the ground 1sσg state and the excited 2pσu state are Ip(1sσg) = 1.08 au and Ip(2pσu) = 0.65 au, respectively. The linearly polarized UV pump pulse at λpu = 100 nm [ℏωpu = 0.455 au ≃ Ip(2pσu) − Ip(1sσg) = 0.43 au] gives rise to the σg−σu resonant excitation. Figure 4 illustrates the time evolution of the orbital occupations cσg = ⟨ψσg(r)|ψ(r)⟩ and cσu = ⟨ψσu(r)|ψ(r)⟩ in the ground 1sσg state and the excited 2pσu state of H2+ by the λpu = 100 nm pump pulse. After the pump excitation, constant populations lie on the two electronic states, |cσg|2 = |cσu|2 = 0.5. The coherent superposition state of the two electronic states, the 1sσg state and the 2pσu state, as indicated in eq 7 is therefore created by the pump pulse. Subsequently, a timedelayed Δτ attosecond soft X-ray probe pulse of different helicities ξ, eq 4, ionizes the excited molecules to produce highfrequency MFPADs. The pulse intensity is 2.75 × 1013 W/cm2 (E0 = 2.8 × 10−2 au) and the pulse durations are Tpr = 10τpr = 6.89 au = 165.4 as (83 as fwhm) for the probe pulse Epr(t) and Tpu = 10tpu = 138 au = 3.31 fs (1.655 fs fwhm) for the pump pulse Epu(t) . For the purpose of the present work to explore the electron coherence by polarized soft X-ray spectra, we consider only the distribution around the photoelectron kinetic energy Ee (momentum pe = 2meEe ) in one ωpr photon ionization, eq 4. Figure 3 shows that asymmetric photoelectron diffraction patterns are produced in MFPADs, which depend on the time delay Δτ between the pump−probe pulses and the probe E


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The Journal of Physical Chemistry A MFPADs with time. The angular node is determined by the molecular bond length R. Therefore, the time-resolved MFPADs describe the coherent electron dynamics in molecular charge migration well, providing information on the molecular orbital symmetry and chemical bonding. The duration of the soft X-ray probe pulse also influences the imaging processes of the coherent charge migration. Figure 5(a−c) displays results of photoelectron spectra with a long

molecules. Moreover, for the long duration soft X-ray probe processes, the distortion of MFPADs disappears due to the symmetry of the averaged density distributions of the coherent state. We finally present results in a perpendicular σg−πu resonant transition process by a perpendicular linearly polarized pump pulse. The coherent electronic state is now a configuration of σg and (πu+,πu−) orbitals.55 Figure 6 displays MFPADs in the

Figure 5. Probing MFPADs versus angle θ in the σg−σu coherent resonant excitation in the molecule H2+ aligned along the x axis by 5 nm soft X-ray pulses Epr(t) with long durations Tpr = 25τpr = 413.6 as and a time delay Δτ = 6.0τpu. The helicity of the probe pulse is (a) 0, linear polarization, (b) −1, left-handed, and (c) +1, right-handed circular polarization. (d) Predicted MFPADs of the sum of - (σg) and - (σu) in eq 11 at t = 11τpu and pe = 4 au, without interfering effects.

Figure 6. Probing MFPADs versus angle θ in the σg−πu resonant excitation of the molecule H2+ aligned along the x axis by a λpu = 70 nm linearly polarized pump pulse Epu(t) with its field polarization vector perpendicular to the molecular axis and time-delayed Δτ = 6.0τpu and λpr = 5 nm soft X-ray probe pulses Ept(t) with helicities (a) ξ = 0, (b) −1, and (c) +1. The pulse intensities and durations are I0 = 2.75 × 1013 W/cm2 (E0 = 2.8 × 10−2 au) and Tpu/pr = 10τpu/pr, i.e., Tpu = 2.32 fs (1.16 fs fwhm) for the pump pulse and Tpr = 165.4 as (83 as fwhm) for the probe pulse. (d) MFPADs from a σg−πu coherent state, Figure 1(c) and eq 12 at |cσg|2 = 0.7 and |cπu|2 = 0.3, versus angle θ predicted from eq 14 at moment t = 11τpu for a photoelectron with momentum pe = 4.0 au.

duration of probe pulses at Tpr = 25τpr = 413.6 as. The time delay between the pump−probe pulses is also taken as Δτ = 6.0τpu, i.e., the initial probe time t = 11τ. The other laser parameters are the same as those used in Figure 3. A coherent superposition state |ψ0⟩ of the 1sσg and 2pσu electronic states, eq 7, is created again in the molecule H2+ aligned along the x axis by the λpu = 100 nm pump pulse. From Figure 5 we see that symmetric MFPADs with respect to the molecular center are produced for both linear and circular polarization probe processes. The coherent excitation and charge migration are exclusively due to the motions of electrons. By the long duration probe pulse, since Tpr > Δτ(0) = 313 as, the averaged density probabilities of the coherent superposition state are symmetric during the probe processes. Therefore, the time evolution of the electron between the protons cannot be resolved in MFPADs. The photoelectron distributions are simply the sum of the distributions originated from the ground state and the excited electronic state, i.e., - = - (g) + - (u), as shown in Figure 5(d). This confirms the importance of the short duration and high frequency of the probe pulse in the dynamical reconstruction of electron charge migration in

photoelectron spectra obtained by a linearly polarized λpu = 70 nm (ωpu = 0.65 au) pump pulse Epu(t) with its field vector perpendicular to the molecular R axis in combination with a time-delayed λpr = 5 nm linearly/circularly polarized soft X-ray probe pulse Epr(t) . The pulse intensities and durations are I0 = 2.75 × 1013 W/cm2 (E0 = 2.8 × 10−2 au) and Tpu/pr = 10τpu/pr, i.e., Tpu = 2.32 fs (1.16 fs fwhm) for the pump pulse and Tpr = 165.4 as (83 as fwhm) for the probe pulse. The time delay between the pump−probe pulses is fixed at Δτ = 6.0τpu, and the helicities of the probe pulse are (a) ξ = 0, (b) −1, and (c) +1, corresponding to linear, left-handed, and right-handed circular polarizations, respectively. The occupation coefficients are |cσg|2 = 0.7 and |cπu|2 = 0.3. With the 70 nm laser pulse, a perpendicular resonant excitation between the ground 1sσg state and the excited 2pπu electronic state is triggered. As a result, a coherent superposition state |ψ0(r,t)⟩ is created F


Article

The Journal of Physical Chemistry A ψ0(r, t )⟩ = cσg(t )|ψσ (r)⟩ e−iEσgt / ℏ + cπu(t )|ψπ (r)⟩ e−iEπut / ℏ g

distributions in the upper half plan are dominant. The resulting effect of the nonspherical molecular Coulomb potential and the circularly polarized laser fields thus leads to a rotation of MFPADs with respect to the molecular symmetric axis, which is sensitive to the helicity ξ of the probe pulse. Altering the time delay Δτ varies the asymmetry of the coherent electron wave packet distributions, thus giving rise to different asymmetries of photoelectron diffraction patterns in MFPADs, as predicted in eq 14. We simulate molecular excitation and ionization dynamics within the fixed nuclear frame. The vibrational motion of nuclei may reduce the electronic coherence.56 Of note is that the time delay Δτ between the pump and probe pulses is very short, Δτ ∼ 6τpu (∼2 fs), which is far less than the molecular vibrational period of ∼15 fs.57 Therefore, the simulation of MFPADs for tracking coherent electron wave packets remains valid. Even considering the inhomogeneous dephasing of the electronic coherence caused by vibrational wave packet width, electronic oscillations should be be experimentally observable at these short time delays. As previously discussed in H2+, the effects of the nuclear movement are to lower the amplitude of oscillations as a function of the time delay.58 This also allows us to measure the decay of the electronic coherence by probing MFPADs.

u

(12)

After the pump pulse is over, the occupation coefficients cσg(t) and cπu(t) of the ground 1sσg and the excited 2pπu electronic states are constant. The corresponding electron density is given by ((r, t ) = cσg 2⟨ψσ (r)|ψσ (r)⟩ + c πu 2⟨ψπ (r)|ψπ (r)⟩ g

g

+ cσgc πu⟨ψσ (r)|ψπ (r)⟩ e g

u

u

−iΔEt / ℏ

u

+ cσgc πu⟨ψπ (r)|ψσ (r)⟩ e+iΔEt / ℏ u

g

(13)

where ψπu is the wave function of the excited 2pπu electronic state with eigenenergy Eπu. The energy difference ΔE = Eπu − Eσg = 0.65 au, which is the photon energy of the 70 nm pump pulse. The phase η of the excited electronic state is determined by the helicity of the pump pulse. Figure 1(c,d) illustrates evolutions of the coherent electron density distributions with time by the 70 nm pump pulse. As illustrated in Figure 1(c,d), the wave packets oscillate perpendicularly with respect to the molecular axis with a period of Δτ(0) = 2πℏ/ΔE = 232 as, in agreement with the predictions in eq 13. According to the ultrafast photoionization model,10 photoelectron distributions of the coherent electronic state in eq 12 can be expressed as

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CONCLUSIONS We present a scheme to monitor electron coherent excitation and charge migration in molecular reaction processes by MFPADs with attosecond time resolutions. The aligned molecular ion H2+ is excited by a linearly polarized UV pump pulse with its field vector parallel/perpendicular to the molecular axis, leading to coherent resonant excitation between the ground 1sσg and the excited 2pσu/2pπu electronic states. Subsequently, a time-delayed attosecond soft X-ray polarized probe pulse of helicity ξ = ±1 or 0 ionizes the excited molecules. Our theoretical analysis from ultrafast photoionization models 10 has shown that the photoelectron diffraction patterns in MFPADs can reconstruct the timedependent charge migration processes and extract the information on the electron coherence including the molecular orbital symmetry and chemical bondings. The present demonstration exploring coherent electronic dynamics paves the way to image and control charge migration in molecular reaction processes by ultrafast laser pulses. We have validated the approach via reconstruction by numerical simulations from solutions of TDSEs and found that the dynamics can be described by MFPAD spectra. We show how the nonspherical molecular Coulomb potential leads to a distortion of MFPADs in the presence of the soft X-ray probe pulse. In general, for high-frequency circular polarization probe processes being initiated from a single electronic state of homonuclear molecules, no rotation or distortion of MFPADs can be generated.59,60 The dependence of MFPADs on the probe pulse helicity and the time delay illustrates the instantaneous charge migration information. The new experiments relating to time delays in attosecond photoionization of molecules61 and our simulations should serve to create movies of attosecond electron motion in molecules by time-dependent electron diffraction.62,63 These new results, in principle, provide a guide for exploring coherent electron dynamics which can be extended to other complex molecular systems for future research in ultrafast physics and chemistry.64,65

2 ⎧ ⎛1 ⎞ -(pe , θ , t ) ∼ (n · e)2 ⎨ cσg cos⎜ pe R cos θ ⎟ψ1s(pe ) ⎝2 ⎠ ⎩

⎛1 ⎞ + c πu cos⎜ pe R cos θ ⎟ψ2p (pe ) ⎝2 ⎠ ±

2

+ cσgc πu[1 + cos(pe R cos θ)]

⎫ × cos(ΔEt /ℏ + η)ψ1s(pe )ψ2p (pe )⎬ ± ⎭

(14)

where ψ2p± = ψ2pz ± iψ2py. The photoelectron distributions in eq 14 are asymmetric with respect to the molecular axis, as illustrated in Figure 6(d), which also evolves with the ionization time t, i.e., the time delay Δτ with a period Δτ(0) = 232 as. From Figure 6(a−c) we see that similar phenomena are obtained for the σg−πu excitation process, as in Figure 3 for the σg−σu parallel transition. MFPADs are asymmetric with respect to the molecule axis, and photoelectron diffraction patterns with multiple angular nodes are produced in the photoelectron spectra. As compared to the ultrafast model in Figure 6(d), the simulated MFPADs by soft X-ray probe pulses are distorted again. For the linear ξ = 0 process in panel (a) where the probe laser field Epr(t) has its polarization direction perpendicular to the molecular axis, the distributions are mainly localized along the perpendicular (θ = ±90°) direction, whereas the amplitudes of the nodes at angles (n ± 2/9)π are strongly suppressed due to n·e = sin θ. For the circular polarization processes ξ = ±1, MFPADs also show asymmetry with respect to the perpendicular polarization direction. At ξ = −1 in Figure 6(b), the amplitude of the angular node at θ = 320° (−2π/9) is larger than that of the left plane at θ = 220° (11π/9). A reverse process occurs at ξ = +1 in Figure 6(c), where the node at θ = 220° is stronger than that at θ = 320°. Again, the distortion of angular distributions illustrates the asymmetry of the coherent electron wave packet. During the perpendicular resonant excitation and charge migration processes, the electron moves around the molecule. As shown in Figure 1(c) at time t = Tpu/2 + Δτ = 11τpu, the electron G


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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Kai-Jun Yuan: 0000-0002-0338-6995 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Compute Canada for access to massive parallel computer clusters and NSERC and FQRNT for support of this research.

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REFERENCES

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Time-Resolved Photoelectron Imaging of Molecular Coherent

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