Efficient photo-dissociation of CH4 and H2CO molecules with optimized ultra-short laser pulses

S. Rasti, E. Irani, and R. Sadighi-Bonabi

Citation: AIP Advances 5, 117105 (2015); doi: 10.1063/1.4935340 View online: http://dx.doi.Org/10.1063/1.4935340 View Table of Contents: http://aip.scitation.org/toc/adv/5Z11 Published by the American Institute of Physics

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Efficient photo-dissociation of CH4 and H2CO molecules with optimized ultra-short laser pulses

S. Rasti, E. Irani, and R. Sadighi-Bonabia

Department of Physics, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran (Received 3 September 2015; accepted 26 October 2015; published online 3 November 2015)

The fragmentation dynamics of CH4 and H2CO molecules have been studied with ultra-short pulses at laser intensityof up to 1015Wcm-2. Three dimensional molecular dynamics calculations for finding the optimized laser pulses are presented based on time-dependent density functional theory and quantum optimal control theory. A comparison of the results for orientation dependence in the ionization process shows that the electron distribution for CH4 is more isotropic than H2CO molecule. Total conversion yields of up to 70% at an orientation angle of 30o for CH4 and 65% at 900 for H2CO are achieved which lead to enhancement of dissociation probability.© 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4935340]

I. INTRODUCTION

Developments of ultra-short intense laser pulses have attracted increasing attention in the main subject of laser-matter interaction.1,2 Control of quantum dynamics phenomena by ultra-short intense laser pulses is one of the most active topics in physics and chemistry. Studies of the dynamics of intermolecular electrons and nuclear motion in a molecule are useful for understanding of chemical reactions.3 Manipulating the outcome of quantum dynamics by a tailored external laser field is an attractive method. Laser fields can drive the population in ground states to excited states via known mechanisms such as multi-photon ionization,4 electron re-collision,5 tunnel ionization6 and field driven ionization.7

Recent progress on the control of electron wave packets, probing of nuclear and electronic dynamics, imaging of molecular orbitals with broadband ultrafast laser pulses have major roles for coherent control of photochemical reactions.8,9 The electron-nuclear dynamics in dissociative ionization of polyatomic molecules enabled by long-lived charge localization becomes effective for understanding the selective pathways of molecular or ionic fragmentations in strong fields.

The capability of controlling the dynamic of molecules by a laser pulse in order to transfer an initial state to a desired inaccessible state was suggested in 1985.10 The possibility of visualizing the dynamics of chemical bonds soon became reality by real time observations of the transition-state region between reagents and products in 1994.11 The pulse shaping techniques had improved impressively over the last decades, and thus the area of experimental optimal control had translated this main goal from theory to a unique experiment by Judson and Rabitz.12 However, for more complex systems at the high laser intensities these models fail to control properly and a feedback loop, together with an optimization are used to achieve the desired goal.13 Recently, Quantum optimal control theory (QOCT) provides the useful tool to theoretical design of the laser pulses capable of controlling a quantum system towards prescribed reaction paths.14 Since these propagations are unfeasible for many-particle systems, few simplifications and models are postulated when handling these equations. In the earlier works, capability of QOCT for an enhanced ionization yield and suppression of the ionization for only one-electron H2+ molecule with the numerically exact scheme is investigated.15,16 Indeed, this subject is an intriguing research area in the last decade.

Corresponding author: Tel:+982166164526; Fax: +982166022711. Email address: sadighi@sharif.ir

2158-3226/2015/5(11 )/117105/10 5,117105-1 ©Author(s) 2015 I^B

Several experimental results for sub-femtosecond steering of hydrocarbons dissociation through superposition of vibrational modes are reported in 2012 and 2014.1718

Molecular orientation and multielectron responses play a main role in the explanation of the fragmentation dynamics.19-21Recently, the orientation of diatomic and polyatomic molecules in strong laser fields has been determined as an important factor in experimental studies.22

The combination of time-dependent density functional theory (TDDFT) and QOCT is used to optimize the laser field which drives the system from the ground state to the different excited states along different dissociation channels.23-25

Time dependent electron localization function (TDELF) is a perfect tool for studying dynamics of intermolecular electrons and time-resolved visualization of the breaking of chemical bonds.

This paper is devoted to the investigation of the dynamics in the formaldehyde molecule (H2CO) and methane molecule (CH4) generated by an intense, few cycle shaped laser pulses considering different molecular orientations. Although, methane is a clean fuel among other conventional fuels; however, the conversion of methane into higher hydrocarbons and hydrogen has many advantages. Formaldehyde is a toxic molecule and one of the most abundant molecules observed in the interstellar medium.26 Oxygen separation in formaldehyde molecules is a troublesome task due to its strong bonding energy in comparison to the bonding energy of H in this molecule. Due to the extra internuclear degrees of freedom and multi-orbital effects, analysis and controlling of molecular dynamics is considerably complicated. We have previously explained the dissociation of the methane molecule and the controlling of electron and nuclear dynamics of the nitrogen molecule via a shaped laser field with approximate optimization methods.27-29

In the present work, our aim is to present a comprehensive model enable to account for i) time-resolved visualization of the molecular dynamics and non-adiabatic controlling the dissociation pathways, ii) besides the complexity of the molecules, the effect of nuclear motion is considered which makes the analysis of the molecular dynamics more difficult, iii) orientation dependence of strong laser field dissociation, iv) considering the excited electronic states which can significantly effect on the angular dependence of the ionization efficiency. These issues have been only partially tackled in the literature. Indeed, it is expected that by studying the electron-nuclear dynamics of methane and formaldehyde, the acquisition of an optimized laser field via a convolution strategy of TDDFT, non-adiabatic QOCT and Ehrenfest molecular dynamics is feasible. Based on the present convolved approach, one can implement in direct closed experimental control loop to reduce the costs. Therefore, with theoretical design of the laser pulses, a new simple artificial experimental procedure is achieved in a short time. Based on the present approach, one can also reduce the costs in comparison to the expensive and complex experimental equipments. Indeed, our results could be useful to the vast numbers of researchers working in this subject, experimentally.17,18 Comparison of computational results for these two molecules with different molecular structures and chemical bonds are interesting and still intriguing. These results are compared to the available experimental

measurements.30-32

This paper is organized as follows. In Sec. II, time -dependent density functional approach and quantum optimal control theory are explained. In Sec. III the results of electron and nuclear dynamics of methane and formaldehyde molecules exposed to an intense, few cycle pulse in different orientation are demonstrated and compared. The electron transitions are also calculated for both methane and formaldehyde molecules in different molecular orientation with interaction of optimum laser pulse shape that is designed for dissociation of all molecular bonds and the calculated results are explained in detail. The paper is concluded in Sec. IV.

II. THEORY AND METHODOLOGY

The molecular dynamics of H2CO and CH4 are investigated by using the time-dependent density-functional approach in real space and real time. Non-adiabatic QOCT are used for finding the optimal laser pulse that guides the quantum system via predefined pathways towards maximum efficiency. Complete controllability is defined by the variation of intensity, time evolution of the laser field, carrier envelope phase and relative phase between different frequencies.

The time dependent Kohn-Sham (TDKS) equation is defined by

i ^ = -1(r't) + Uks (r, t) tyi (r, t) (1)

ty,(r, t) is the Kohn-Sham orbital and vks(r, t) is the KS potential. The electron density n(r, t) at the electron coordinate rand time t is determined by

n (r, t) = £ 2|tyf (r, t)|2 (2)

i ,2N and tyi (r, t) are orbital index, the number of electrons and single particle wave functions, respectively. The KS potential splits into three terms:

/n (r't)

dV--'— + vxc [n (r, t)] (r, t) (3)

|r - r'|

uext (r, t) is the external potential consisting of a potential generated by the laser field and a potential resultant of nucleus interaction. The term uxc [n (r, t)] (r, t) describes the exchange-correlation potential. Using the dipole approximation and the length gauge, the interaction of the external laser field is expressed as

u (r, t) = E (t) r.p (4)

E (t) is the electric field of laser pulse, and unitary vector p determines the light polarization. The electric field waveform of few-cycle light pulses can be described as:

E(t) = e(t) cos(wt + 0) (5)

Where e(t) is the pulse temporal shape, ro the carrier angular frequency and ^ is the carrier envelope phase (CEP).

The goal of optimization in QOCT is expressed as finding the detailed time dependent shape of optimal laser pulse e (t) which drives the initial state ty of the system in a finite time to the given target state. Since, in the dissociation of a molecular bonding, the time averaged forces acting on nuclei are opposite to each other, target functional of QOCT is defined in a way that maximizes the relative nuclei velocities of the desired bonds in the dissociation process. QOCT maximizes the target functional G[u, ty(u)] with a set of control parameters u which determine the precise shape of the laser pulse. The target functional G[u, ty(u)] splits into two terms14

G [u,ty(u)] = Ji [ty(u)] + J2 [ty(u)] + J [ty(u)] (6)

J1 [ty(u)] is defined as:

Ji [ty (u)] = n ■ (vi (n) - V2 (n)) - 10 2 |V2 (n) - Vi (n)| (7)

Where n is the polarization of laser pulse which is directed along the bond that we want to dissociate, n is the electron density of system, and vi (n) is the respective nuclear velocity. The first and second terms of Eq. (6) are responsible for the maximization of the momenta difference between those atoms that must not remain bound and the minimization of the momenta difference between those atoms that must remain bound, respectively. The factor "10" regulates the weight that is placed on the minimization.

Considering the movement of the atoms and the description of electrons and nuclei in terms of a many-body wave-function is numerically impractical. By assuming the atoms as classical particles described by their positions and velocities the computational cost can be reduced, considerably. The motion of the electrons due to their fast response to the laser field is calculated quantum mechanically. J2[ty(u)] is a penalty functional which makes some constrains on the energy of the pulse so that optimal pulses can be produced in real with pulse shaping systems.

J2[f (u)] = -a

e (t)dt - E0

a is a factor that regulates the weight that is placed on the energy constraint and E0 is responsible for the preferred pulse energy. J3[f(u)] is a functional which fulfills time dependent Schrodinger equation,

Js[^(u)] = -2Im

< K(t)\id - H(t)\f(t) > dt

K(t) acts as Lagrange multiplier and H(t) is Hamiltonian of the system.

The target functional G[u, ^(u)] is solved with NEWUOA algorithm by considering the degrees of freedom of the nuclei and electrons.33

In order to ensure physically meaningful laser pulses in the optimization process, some constraints to the laser pulses are applied which are formulated below:

The first constraint is fulfilling the condition e(0)=e(T)=0, the algorithm expands the pulse temporal shape by Sine bases:

(t) = I e(nnt)

T is the duration of laser pulse. Another constrain is:

dte (t) = 0

This condition makes the coefficient of sine expansion of Eq. (10) to obey following relation:

e(2m+1) (2m + 1)

The cutoff frequency is determined bynN/T. To adjust the significance of the laser intensity, the total fluence of the laser pulse is expressed by

Fo = e2 (t) dt

It is noticed that the optimization algorithm modifies e(t) in a way that both maximizes the target functional G[u, f (u)] and ensures the defined constrains.

To observe the response of the molecule to external potential and trace the fragments after the dissociation process, Ehrenfest molecular dynamics based on the TDDFT formalism is employed. In addition accurate descriptions of HOMO orbitals, excited state energies, and orbital symmetry are considered and are in good agreement with real observations.34,35Combining of QOCT with TDDFT in order to obtain a more accurate energy diagram and indentifying the dissociation pathways for valuable ionic fragmentations are done using octopus code.36,37 The occupation of electronic states P, are computed by the following formula

£\ <ti(t) \ f j(t)>\2

i,j are indices of states and N is the total number of states.

The TDELF is used for time-resolved visualization of the breaking of chemical bonds. Accordingly, the main purpose of using TDELF, is helping our intuition of the time dependent chemistry

concepts of pairs and bonds in molecules. The expression for TDELF takes the form

/7D£LF(r, t) = -1-rr^-, (15)

1 + [D^r, t)/9.11 «55/3]2

n0g, is the homogeneous electron gas density and D0 (r, t) is inverse measurement of localization which is expressed by39

N 1 rn ,r A!2 {27 2 [^;(r; t)V^i(r; t) - ^(r, t W(r; t)]}2

< V in, < i2 1 [vn(r"t)] i=1 ,,,, r(r,t) = ^ |v^i(r,t)| - ^ ^---«(-"t)- (16)

III. RESULTS AND DISCUSSIONS

In the calculations, an equi-distant grid in a three dimensional (3D) real space with 425920 points in a sphere of radius 22a.u. are determined. To avoid an artificial reflection and check the warranting full convergence, an appropriate absorber at the boundary of the sphere is used. A number of 12 distinguished excited electronic states is accounted for, that contribute to the ionization process.

Orientation dependence of the strong laser field ionization

For strong laser fields, the role of inner orbitals and the coupling between them is an interesting and partially unanswered issue. Hence the shapes of the four energetically highest lying occupied molecular orbitals of CH4 and H2CO are visualized in Fig. 1. Methane is a symmetric molecule and the electronic state configuration of the four doubly occupied molecular orbitals is (2a1)2(1t2)6. The valence molecular orbitals of H2CO are composed of (4a1)2(1b1)2(2b2)2. The HOMO of CH4 and H2CO is a type and n type, respectively.

The angular dependence of the ionization probability presents a major effect from the symmetry of the molecular orbitals which can hold the main role on the ionization process Although CH4 is a symmetric molecule; however, orientation dependence of ionization yield might be different. In order to find out the orientation dependence of ionization, the snapshots of TDELF for several orientation angles of jS = 0, 15, 30 and 55 degrees between a molecular bond of the methane molecule and the laser polarization are implemented and the dynamics for each orientation is shown in a separate row at the left panels of Fig. 2. Formaldehyde molecule is planer molecule and the molecular axis is chosen along the C=O bond. The other laser polarizations are in the plane of molecule with different degrees of jS = 0, 45, 90 and 135 with respect to the molecular axis and laser directions which is shown in the right panesl of Fig. 2. The contribution to the ionization yield from HOMO-1 of H2CO is very small in comparison with the HOMO-2 for the laser polarization along the C=O bond. Calculations are extended to a laser pulse of 800 nm wavelength, hyperbolic secant envelope shape with time duration of about 6fs. and laser intensities of 1.96 x 1015Wcm-2 and 3 x 1015Wcm-2 for methane and formaldehyde molecules. The snapshots of TDELF with their nuclear dynamics are obtained from TDDFT calculations for different times of 400 a.u.(9.4fs), 600 a.u.(14.4fs) and 800 a.u.(19.2fs) for an iso-surface of 0.7.

As shown in Fig. 2, at the beginning the system is in the ground state. As the intensity of the laser increases, the system starts to oscillate and then ionizes. Note that the ionized charge leaves the system in fairly localised packets (the blob on the left, and on the right) and then start to widen until it breaks. This can be interpreted as a transition from the bonding states to the non-bonding states. The system then remains in this excited state, and eventually dissociates. The time evolution pattern of the electron localization function is different for various arrangements. By considering the contribution of excited states, the results show that the angular distribution for CH4 is slightly isotropic in comparison to the anisotropic distribution for H2CO molecule. The results denote a strong dependence of the ionization probability on the molecular orientation which leads to fast breaking of C-H bonds of methane and formaldehyde molecules indicated in the snapshots of g,h, i.

FIG. 1. The four highest occupied molecular orbitals HOMO, HOMO-1, HOMO-2 and HOMO-3 of CH4 and H2CO are shown.

However, snapshots a,b,c present the least fragmentations of C-H bonds. The destruction of C=O bond of formaldehyde is abrupt in snapshots j, k, l and d, e, f in comparison to the snapshots given for the methane molecule. The ionization yield of the double bond of formaldehyde is enhanced when the laser polarization is inclined with respect the bond axis. The experimental observations also confirm that the ionization yield for the C=O bond has a peak between 0 and 90 degrees relative to the bond axis and C-H bonds are dissociated much faster.40 Regarding the molecular orbital theory predictions for n orbitals, it is expected to achieve these results. The results show that the angular distribution for CH4 is isotropic in comparison to the H2CO molecule. For formaldehyde, it is noticed that at some specific orientations, the ionization and fragmentation is higher. However, due to the isotropic structure of CH4 there is no considerable change in the ionization and fragmentation pattern for different orientations. The results, reported in Fig. 3, show the time dependent representation of the occupation of molecular states (corresponding to the orientations that are used in Fig. 2). The total number of lower unoccupied molecular orbitals (LUMOs) is 12.

FIG. 2. Snapshots of the TDELF by laser pulse at 800 nm wavelength, 1000 a.u. time duration with different polarization directions and 1.96x 1015Wcm-2 laser intensity for CH4 molecule (left panels) and 3x 1015Wcm-2 laser intensity for H2CO molecule (right panels) which lead to ionization and dissociation. The laser polarization vector is presented in the first snapshot of each orientation in each row. Note that, iso-surfaces of the ELF set at a value of n = 0.7.

From Fig.3, the orientation effects in the control of the time-dependent occupation numbers which transfer the ground state to different excited states are significant. Fig.3 denotes the contribution of each state to the total dissociation amount and also the time dependent depletion of each state. However, the occupations of some states do not depend on the laser orientation. Therefore, comparison of Fig. 2 with Fig. 3 provides a strong relationship between the molecular dynamics, and the HOMO transitions for each orientation. For formaldehyde and methane molecules, the total ionization yields are 66 % and 67%, respectively. However, these values are 70% and 75% for methane and formaldehyde molecules at ¡3 = 300 and ¡3 = 900 which lead to 7% and 12% faster dissociation rates, respectively. These results are in excellent agreement with the results given in Fig. 2.

Several mechanisms are used to explain that the initial ionization follows the electron density of the initial electronic states in intense laser field. In these molecules the electron re-collision and rescattering contribution to the fragmental yields is more remarkable. Figure 3 displays the occupation of a few lowest states during the pulse interaction. The contribution of each state to the total dissociation amount and also the time dependent depletion of each state are shown. The occupation of 6th state is depleted at the earlier time. It means that one electron is smoothly removed from the outer state 6. In the following, the laser field ejects electrons from the inner states. This is the remarkable feature of the presented work in explaining many effective features of molecular orbitals including their proper calculation and realizing their importance in applying of rescattering and re-collision mechanisms together.

Optimal control calculations with different orientations

Molecules are aligned relative to the laser field and it is significance to find the tailored laser pulses for realizing the dependence of the ionization step on the molecular orientation. The orientation-dependent ionization of the inner orbitals similar to the one discussed in the first part of section III, via specially tailored laser pulses are presented. Calculations are carried out for a Ti: sapphire laser pulse at 800 nm with initial values of the laser intensity and frequency maximum range of 0.2a.u., with angles ¡3=0, 15, 30 and ¡3=30, 45, 90 for methane and formaldehyde molecules, respectively. Ehrenfest molecular dynamics model is also used to test the dissociation

FIG. 3. The time variation of wave packet population on the excited states of CH4 molecule (left panels)and H2CO molecule(right panels). The colored lines denote the occupations of 12 distinguished excited electronic states that contribute to the ionization process.

processes considering moving nuclei. By applying temporally shaped femtosecond laser pulses, the vibrational dynamics of a molecule can be accessed and steered directly. This makes it possible to control the ultrafast electronic dynamics in molecules via the different parameters of few-cycle laser pulses as shown in Fig.4 and Fig.5. The calculations continue with an optimized pulse to induce photon absorption in the valence bond and dissociate the molecule for different orientations. Following the excitation of molecules based on the Ehrenfest molecular dynamics approach, the evolution of the nuclei positions is determined also the breaking of molecular bonds and changes in state occupations of states for CH4 molecule are made visible in the Fig.4.

In all mentioned cases, the effect of optimization is based on the tailored laser pulse which is described by a convolution of various frequencies enables to populate a combination of excited states. Due to molecular alignment relative to the laser field, the dependence of the ionization on the molecular orientation is significant. Therefore, the optimization scheme has to be considered for each orientation as presented in Fig.4 and Fig.5. This leads to the optimized production of CH3 as shown in (a) and (c). For CH2 this is presented in (b). Furthermore, strong dependence of dissociation process on the laser characteristics at different orientations is shown and the optimal effective parameters to produce the maximum yields are evaluated. It is noticed that minor changes in the laser parameters at different orientations, show a significantly different dissociation pattern. However, population transfers between the excited states are not considerable. A similar study is presented for H2CO molecule in Fig.5.

The obtained results verify the important of electron localization-assisted enhanced ionization and rescattering processes. Rescattering processes can be illustrated by a three-step model including tunneling ionization, acceleration of electrons and re-collision.41

FIG. 4. Optimal laser pulses (a, b, c) for dissociation CH4 as computed with several orientation angles of 3 = 0, 15, 30 and 55degrees between molecular bonds of methane molecule and the laser directions, respectively. The time variation of corresponding bond length dynamics (d, e, f) and the time evolution of efficient occupation numbers (g, h, i) are shown, respectively.

FIG. 5. Optimal laser pulses (a, b, c) for dissociation of H2CO are computed with laser polarizations for 3=0, 45, 90, respectively. The time variation of corresponding bond length dynamics (d, e, f) and the time evolution of efficient occupation numbers (g, h, i) are shown, respectively.

When an elliptical laser field is used the returning electrons can be adjusted to control multi-electron nonsequential ionization. The fragmentation pattern of formaldehyde is almost similar for the three optimized tailored laser pulses of (a), (b) and (c) resulting in H2CO ^ CO + 2H.

IV. CONCLUSIONS

In this work, a novel approach for time-resolved visualization of the molecular dynamics and non-adiabatic controlling the dissociation pathways with considering the effect of nuclear motion is presented. This procedure relies on convolution strategy of TDDFT, QOCT and Ehrenfest molecular dynamics. Based on this approach, a new artificial experimental procedure is proposed which is different with the experimental pulse shaping equipments.

The dependence of the fragmentation dynamics on the orientation with respect to laser direction for methane and formaldehyde molecules is described by employing in the rescattering model. Dissociation process and control of the reactions are properly explained from the evolution of the occupation numbers of states. Indeed, Chemical reactions pathways are controlled by the variation of intensity, time evolution of the laser field, carrier envelope phase and relative phase between different frequencies. The TDELF method is used to observe the time-resolved modulation and

breaking of C-H bonds of methane and formaldehyde molecules. The main feature in this frame is considering the excited electronic states which can significantly effect on the angular dependence of the ionization efficiency. Total conversion yields of 75% and 70% at orientation angles of 300 and 900 for the semi-isotropic distribution of CH4 and the anisotropic H2CO molecule are achieved, respectively. It is found that the dissociation is by 7% and 12% faster than for other orientations for CH4 and H2CO, respectively. The achieved theoretical results can help to have a better insight into the experimental studies.

ACKNOWLEDGMENTS

The authors would like to acknowledge the Pars Oil and Gas Company of the ministry of Oil for its continuous supporting of this work under Contract number No.1211, and also the research deputy office at Sharif University of Technology.

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