First-principles insights into ultrashort laser spectroscopy of molecular nitrogen
FFirst-principles insights into ultrashort laser spectroscopy of molecular nitrogen
Mohammad Reza Jangrouei and S. Javad Hashemifar ∗ Department of Physics, Isfahan University of Technology, 84156-83111 Isfahan, Iran (Dated: January 1, 2019)In this research, we employ accurate time-dependent density functional calculations for ultrashortlaser spectroscopy of nitrogen molecule. Laser pulses with different frequencies, intensities, anddurations are applied to the molecule and the resulting photoelectron spectra are analyzed. It isargued that relative orientation of the molecule in the laser pulse significantly influence the orbitalcharacter of the emitted photoelectrons. Moreover, the duration of the laser pulse is also found to bevery effective in controlling the orbital resolution and intensity of photoelectrons. Angular resolveddistribution of photoelectrons are computed at different pulse frequencies and recording times. Byexponential growth of the laser pulse intensity, the theoretical threshold of two photons absorptionin nitrogen molecule is determined.
I. INTRODUCTION
The recent progress in the field of ultra-short laserpulses has provided novel opportunities to capture fastdynamics of atoms and electrons in chemical reactionsand photo ionization phenomena [1–4]. The studies ofAhmed Hassan Zewail, on the dynamic of chemical re-actions by using femtosecond spectroscopy, awarded himthe Nobel Prize of chemistry in 1999 [1]. In 2001, Fer-enc Krausz succeeded to generate attosecond laser pulses[2], which provide invaluable abilities to investigate andeven control electron dynamics in photo ionization phe-nomena [5]. Haessler and coworkers used a train of at-tosecond laser pulses in presence of a weak infrared fieldto ionize nitrogen molecule [6]. They identified two ion-ization channels in the system correspond to the groundstate and excited state of the ionized molecule. Kelkens-berg and others applied the same method to ionize hy-drogen molecule and observe changes in charge distribu-tion of the system on attosecond time scales [7]. Siu etal . found that the time delay between attosecond pulsetrain and a corresponding infrared field may be used tocontrol the dissociative ionization of oxygen molecule [8].Penka and others applied time dependent density func-tional theory in the nonlinear nonperturbative regimeto investigate laser induced photo ionization in CO andH CO molecules [9]. They found that the interplay be-tween the ionization potential, the orbital shape, and thelaser polarization axis significantly influence the ioniza-tion process.In the present work, we employ time-dependent ab-initio calculations to study photo ionization of N molecule under irradiation of short laser pulses. The ef-fects of frequency and intensity of the pulse on the po-larization will be investigated. ∗ [email protected] II. COMPUTATIONAL METHOD
Our calculations have been performed in the frame-work of time dependent Kohn-Sham (TDKS) densityfunctional theory [10], which provides a proper single-particle description of many-body systems in the pres-ence of time dependent external potentials (e.g. an elec-tromagnetic pulse) Adiabatic local density approxima-tion (ALDA) is adapted for description of the time de-pendent exchange correlation functional in this approach.It is already argued to be the proper functional for de-scription of atomic clusters under intense electromag-netic fields [11]. We used the Octopus package to solvethe TDKS equations by employing the norm-conservingpseudo potential technique [12]. The KS orbitals are ex-panded on a real space grid defined inside geometricalboxes around atoms or around whole system. Two gen-eral approaches are implemented in this package for solv-ing the TDKS equations: linear response and explicit realtime propagation methods. In the linear response regime,which is used to address the effects of a weak uniformwhite electromagnetic noise, the absorption spectra andthe character of electronic excitations of the system aredetermined. While in the presence of strong laser pulses,explicit propagation of KS orbitals in real time domainis considered.In order to calculate the emitted photoelectron spec-tra of a sample after strong laser irradiation, a detec-tor region is defined around the system and then theWigner quasi-probability distribution function in thephase space: ω ( R , p , t ) = (cid:90) d s π e i p · s ρ ( R + s , R − s , t )is used to integrate the photoelectrons in the detectorregion. In the above equation, ρ ( r , r (cid:48) , t ) is a two bodydensity matrix and R and s are the center of mass andrelative coordinates. The momentum resolved photoelec-tron spectrum is then given by: P ( p ) = lim t →∞ (cid:90) d R ω ( R , p , t ) a r X i v : . [ phy s i c s . a t m - c l u s ] D ec TABLE I. Obtained molecular orbitals of the nitrogenmolecule. The Occupied orbitals are highlighted.state energy (eV) state energy (eV) σ )
11 0.53 σ ∗ )
12 1.05 π )
13 1.12 π )
14 1.12 σ )
15 1.136 -2.32 ( π ∗ ) 16 1.137 -2.32 ( π ∗ ) 17 1.218 0.03 ( σ ∗ ) 18 1.659 0.52 19 1.6510 0.53 20 1.66 In this equation the integral is calculated in the detec-tor region after a sufficiently long time to ensure contribu-tion of all photoelectrons. In the Kohn-Sham approach,the two body density matrix is defined by the followingsum over occupied states: ρ KS ( r , r (cid:48) , t ) = occ. (cid:88) i ψ i ( r , t ) ψ i ( r (cid:48) , t )This procedure needs calculation area of hundredsAngstrom to give reliable photoelectron spectra. In orderto reduce the required calculation area, a mask region isdefined before the detector region [13]. III. RESULTS AND DISCUSSIONS
First, we performed some static DFT calculations toidentify the equilibrium properties of N molecule. Theequilibrium bond length and binding energy of nitro-gen was found to be about 1.09 ˚A and 8.89 eV, respec-tively, which agree with the measured data (1.1 ˚A and9.79 eV) [14]. The energy gap between the highest oc-cupied molecular orbital (HOMO) and the lowest unoc-cupied molecular orbital (LUMO) was determined to beabout 8.2 eV (table I). Comparing this parameter withthe experimental energy gap of N is questionable, be-cause of the frozen character of orbitals in the staticDFT calculations, while in practice; electron excitationhas non-trivial influences on orbital energy levels. Thisproblem is well resolved in time dependent DFT, whereorbitals are allowed to relax during electronic excita-tions. The obtained absorption spectrums of N by us-ing TDDFT within the Casida linear response and realtime propagation approaches are presented in Fig. 1. Theagreement between these two spectra is acceptable, espe-cially in lower energies. In higher energies, the accuracyof the linear response approach decreases and hence thereal time propagation approach is more reliable. Energy (eV) S t r e ng t h f un c ti on ( / e V ) Casida TP FIG. 1. Calculated absorption spectra of N2 by using theCasida linear response and real time propagation (TP) meth-ods.
The obtained energy gap within both Casida andreal time approaches is about 11.0 eV. In order to com-pare the obtained absorption spectra with experiment,we note the complex absorption spectrum of nitrogenmolecule [15], which includes weak dipole-forbidden tran-sitions from 6 to 12.4 eV and strong dipole-allowed tran-sitions from 12.4 to 18.8 eV [16]. The first 20 electronicexcitations, identified within the Casida approach, arelisted in table II. Our results confirm that the peaks be-low 11 eV have negligible dipole moment and very weakstrength while strong dipole allowed peaks occur abovethis threshold. Moreover, we may conclude that the ob-served experimental transitions below 11 eV are likelynon-electronic excitation transition (rotational or vibra-tional transitions). The ionization energy of nitrogenmolecule, the difference of the minimized energy of theneutral and ionized molecule, was found to be 16.02 eV,which compares well with the measured value of 15.80 eV[17].For calculations of photoelectron spectra, we usedspherical boxes around the molecule with an optimuminternal radius of 12 ˚A for region A, an external radius of22 ˚A for detector region and a sine-mask function. Theoptimum grid spacing in the atomic spheres was found tobe 0.18 ˚A while the time step for real time evolution wasset to 1 m¯ h /eV ( ∼ h/ eV (2.63 fs). The fre-quency of the extreme ultraviolet (xuv) laser pulses wasset to some specific odd (9-17 th ) multiples of a fundamen-tal frequency of 1.565 eV. These odd harmonics have beenalready produced by propagating intense laser pulses ina gas jet and then used for photo ionization of nitrogenmolecule [6]. The 12 th multiple was also considered formore accurate inspection.The calculated photoelectron spectra at the desiredpulse frequencies and two different geometries are pre-sented in Fig. 2. In these geometries, the molecule is ei-ther parallel or perpendicular to the direction of the laserpulse propagation. First, we focus on the perpendicular TABLE II. Identified characteristic parameters of the first 20 electronic excitations of N , within the Casida linear responseapproach, energy (eV): excitation energy, dipole (˚A): Cartesian components of the excitation dipole moment, strength (1/eV):excitation strength, character: major involved transitions between molecular orbitals, the corresponding probability domainsare written in the parenthesis and molecular orbital numbers are consistent with table I. The dipole allowed transitions arehighlighted.energy dipole strength character1 9.19 1.74E-09 5.48E-06 4.58E-06 4.10E-11 5 → → → → → → → → → → → → → → → → →
10 (0.12), 5 →
11 (0.99)9 11.00 →
10 (0.99), 5 →
11 (-0.12)
10 11.56 9.73E-07 1.36E-10 5.08E-10 9.57E-13 5 →
12 (0.97), 5 →
17 (0.25)11 11.61 3.08E-10 4.52E-08 8.81E-07 7.91E-13 5 →
14 (1.00)12 11.61 5.60E-11 8.81E-07 4.52E-08 7.91E-13 5 →
13 (1.00)13 11.62 6.49E-10 3.93E-11 4.41E-11 4.31E-19 5 →
15 (1.00)14 11.62 3.20E-10 2.33E-11 1.69E-11 1.05E-19 5 →
16 (1.00)15 11.83 3.38E-07 1.65E-10 1.19E-09 1.19E-13 5 → →
12 (-0.24), 5 →
17 (0.96)
16 12.04 → →
18 (-0.11)17 12.04 → →
19 (-0.11)18 12.13 → →
19 (0.99), 5 →
25 (0.11)19 12.13 → →
18 (0.99), 5 →
24 (-0.11)20 12.14 2.51E-01 →
20 (0.99), 5 →
23 (-0.16) Perpendicular0 4 8 12 16Photoelectron energy (eV)012 P ho t o e m i ss i on i n t e n s it y ( x - ) Parallelp0 4 8 12 16 20Photoelectron energy (eV)
FIG. 2. Calculated photoelectron spectra of N at the perpendicular and parallel geometries at the pulse intensity of10 W/cm and at six different laser pulse frequencies from 14.08 to 26.60 eV. DistanceI P o t e n ti a l LaserTotalNucleus Intensity24681012 E n e r gy ( e V ) | |⊥ Molecular orbital levels σ -10.45 eV π -11.96 eV σ * -13.38 eV tunneling K i ne t i c B i nd i ng Ground state of molecular nitrogen
FIG. 3. left: schematic representation of photo ionization via electron tunneling in a strong laser field. The letter I standfor ionization potential. Middle: calculated photoelectron spectra in the perpendicular (solid line) and parallel (dashed line)geometries at pulse frequency of 18.78 eV and pulse intensity of 10 W/cm . The spectrum of perpendicular geometry is50% enlarged to be clearer. The photoelectrons kinetic energy is identified by deconvolution of the spectra in two Gaussianfunctions. The kinetic energy of the photoelectrons is subtracted from the laser pulse energy to determine the binding energyof photoelectrons. Right: the highest occupied energy levels of molecular nitrogen in the ground state. geometry. It is seen that at the lowest pulse frequency(14.08 eV), two peaks are appeared in the spectra, indi-cating emission of two different kinds of photoelectronsfrom the system. We will argue that these peaks arelikely attributed to the two sigma molecular orbitals ofthe nitrogen molecule. Taking into account the calcu-lated ionization energy of N (16.02 eV), it seems that alaser pulses with frequency of 14.08 eV should not be ableto create any photoelectrons. Therefore, the observedvery weak ionization is either due to the multi-photonabsorption or electron tunneling in strong laser field. Thelow kinetic energy of photoelectrons ( ∼ , one π molecular orbital located betweentwo σ orbitals (table I, Fig. 3). These orbital levels ex-hibit very good consistency with the obtained bindingenergy of the photoelectrons. This consistency enablesus for a brief anatomy of the calculated photoelectronspectra. The σ orbitals are mainly distributed along themolecular axis, while in the case of π orbital, out of axisdistribution may also play a significant role. In the per-pendicular geometry, the pulse electric field is along themolecular axis and hence mainly σ and σ ∗ photoelec-trons are emitted from the system. In the parallel geom-etry, the pulse field is perpendicular to axis and hencemainly ionizes the π orbital of the molecule, with a mi-nor contribution from the σ orbital. These argumentspresumably explain occurrence of one (two) peaks in thephotoelectron spectra of the molecular nitrogen in theparallel (perpendicular) geometries. -4 -3 -2 -1 P ho t o e m i ss i on i n t e n s it y I = 9 x I = 9 x I = 9 x I = 9 x I = 9 x FIG. 4. Calculated photoelectron spectra of N at five dif-ferent laser pulse intensities from 10 to 10 W/cm . Thepulse frequency was set to 18.78 eV. The intensity of the photoelectron spectra exhibits dif-ferent trends in the parallel and perpendicular geometries(Fig. 2). In the parallel geometry, we observe that byincreasing the pulse frequency, the photoelectron spec-tra intensity increases smoothly. But in the perpendic-ular geometry, a more complicated trend is seen. Thephotoelectron intensity decreases in the frequency rangeof 14.08 to 18.78 eV and then increases from 20.34 to26.60 eV. In fact, theoretical description of strong fieldionization has already shown that tunneling rate is acomplicated function of the laser pulse frequency and in-tensity [18].In order to investigate feasibility of multi photon ab-sorption in molecular nitrogen, we considered laser pulseswith frequency of 18.78 eV and five different intensi-ties from 10 to 10 W/cm . The obtained resultsare presented in Fig. 4. For the intensities lower than10 W/cm , the general feature of the spectra does notchange by increasing the pulse intensity. The spectrahave two peaks and the photoelectron intensity is lin-early scaled by the pulse intensity. At the pulse intensityof 10 W/cm , a third peak appears in the spectra witha kinetic energy of about 25 eV. Adding this value tothe binding energy of σ orbital (10.47 eV) gives a min-imum required energy of about 35.5 eV for emission ofthe corresponding photoelectrons, which is almost twicethe energy of a single laser photon (18.78 eV). Hence,we conclude that at this high intensity two photons ab-sorption happens in the system. At the pulse intensityof 10 W/cm , the intensity of the third peak increasesabout two order of magnitudes compared with the pre-vious one. Hence, the two photons absorption intensityscales with the square of the laser pulse intensity, in wellagreement with theoretical description of this nonlinearoptical phenomenon [19]. In the presence of high inten- I n c r ea s i n g p h o t o e l e c t r o n d e n s i t y w = 14.08 eV w = 20.34 eV w = 26.60 eV T = ħ / e V w = . e V T = 3 ħ/eV T = 5 ħ/eV T = 4 ħ/eV FIG. 5. Effects of pulse frequency and recording time on theangular distribution of photoelectrons emitted from nitrogenmolecule. The top row shows the distributions at three dif-ferent pulse frequencies at the recording time of 5 ¯ h /eV. afterpulse irradiation. The bottom row shows the photoelectrondistributions at a single pulse frequency and at three differentrecording times. The laser pulse intensities are the same asFig. 2. sity laser pulses, the pondermotive energy may also in-fluence the results. In the case of long pulses the ponder-motive energy has a well defined constant value (cid:15) / ω where (cid:15) is the electric field amplitude [13]. However, inthe case of intense ultrashort pulses it is argued that thepondermotive energy is not constant and calculation ofthis parameter is not straightforward [20]. Moreover, itis discussed that in this situation the Stark shift maysubstantially compensate the pondermotive shift [21].Hence, we ignore consideration of non-constant ponder-motive energy in the current work.The angular distribution of the photoelectrons resultedfrom the laser pulse frequencies of 14.08, 20.34 and26.60 eV are presented in Fig. 5. Obviously, increasingfrequency of the incident pulse enhances the kinetic en-ergy of the photoelectrons and hence speeds up its propa-gation away from the molecule. The laser pulse is propa-gating perpendicular to the molecule and the electric fieldpolarization is parallel to the molecule. Hence, we ob-serve that photoelectrons are propagating in the moleculedirection. We have also investigated time evolution of thephotoelectron distribution. At the laser pulse frequencyof 23.47 eV, the distribution is recorded after three prop-agation times (3, 4, and 5 ¯ h /eV). We observe that beforearriving to the detector wall, the photoelectron densityhas a smooth distribution, however after incident to thedetector wall, many fluctuations appear in the distribu-tion density.Throughout this project, we have mainly used ultra-short laser pulses with a duration of 4 ¯ h /eV ( ∼ . ∼ . Time (hbar/eV)-2-1012 P u l s e a m p lit ud e
12 16 20 24Pulse frequency (eV)00.10.2 P u l s e a m p lit ud e I n t e n s it y ( x - ) x2 FIG. 6. Effects of pulse duration on the photoelectrons spec-tra of molecular nitrogen. Two pulse lengths of 4 and 16 ¯ h /eV,their Fourier transform, and the resulting photoelectron spec-tra have been compared together. The frequency of the pulsesis 18.78 eV. The photoelectron spectrum of the shorter pulseis enlarged twice to be clearer. energy uncertainty principle, ∆ E ∆ t ≤ ¯ h , applies thatincreasing the pulse duration should decrease the energytolerance. As it is obvious in the figure, the Fourier trans-form of the longer pulse has much narrower dispersionaround the central frequency of 18.78 V. Therefore, the resulting photoelectron spectrum has more resolution interms of orbital character of the emitted photoelectrons.We observe that the two peaks of the corresponding spec-trum are significantly sharper, compared with the pho-toelectron spectrum of the shorter pulse. Moreover, fre-quency tolerance of the pulse exhibits nontrivial influenceon the intensity of the photoelectrons. A laser pulse withsharper frequency distribution is clearly much more effi-cient for electron emission from the sample. IV. CONCLUSIONS
Real time propagation of the single particle Kohn-Sham orbitals within adiabatic local density approxima-tion was applied to study photoelectron spectra of nitro-gen molecule in short laser pulses. It was argued thatwhen direction of the pulse propagation is perpendicu-lar to the molecule, σ photoelectrons are mainly emittedfrom the system, while in the parallel geometry the high-est occupied π orbital is more ionized. It was seen thatlonger laser pulses, with lower frequency dispersion, aremore efficient for creation of well orbital resolved pho-toelectrons. Angular resolved distributions were plottedto observe real space propagation of photoelectrons atdifferent pulse frequencies and propagation time. It wasargued that at 10 W/cm pulse intensities and higher,some new photoelectrons with much higher kinetic en-ergy are emitted from the molecule which indicate oc-currence of two photons absorption phenomenon. ACKNOWLEDGMENTS
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