Quantum Dynamics of H2+ in Intense Laser Fields on Time

Article pubs.acs.org/JPCA

Quantum Dynamics of H2+ in Intense Laser Fields on Time-Dependent Potential Energy Surfaces Manish Garg,† Ashwani K. Tiwari,*,† and Deepak Mathur*,‡ †

Indian Institute of Science Education and Research Kolkata, Mohanpur 741 252, India Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India

‡

ABSTRACT: We have exploited the fully time-dependent Born−Oppenheimer approximation to develop time-dependent potential energy surfaces for the lowest two states of H2+ in the presence of intense, time-varying, few-cycle laser fields of 2−8 fs duration. Quantum dynamics are explored on these field-dressed, time-dependent potentials. Our results show that the potential well in the lowest-energy state of H2+ (i) collapses as the laser pulse reaches its peak amplitude and (ii) regains its form on the trailing edge of the pulse, and (iii) the trapped nuclear wavepacket has a higher probability of leaking out from the well in the case of longer laser pulses. The carrier envelope phase is found to have negligible effect on the nuclear dynamics. the tetramethylsilane cation (TMS+),6 has shown the importance of developing TDPES for bigger molecular systems. The alternative strategy of utilizing approximate methods to compute TDPES was first developed by Kono,7 who used instantaneous eigenvalues of the field-dressed electronic Hamiltonian. This method has been very helpful in rationalizing many strongfield molecular dynamics phenomena. However, the method lacks the exact dynamical features of the laser−molecule interactions. Recently, Cederbaum has generalized the time-independent Born−Oppenheimer approximation to a time-dependent one which leads to an intuitive picture of TDPES.8 Cederbaum’s method appears to be computationally manageable and can be used to compute the TDPES for bigger systems. Quantum nuclear dynamics on TDPES opens the possibility of computing observables such as dissociation probabilities and recently discovered bond formation processes.9,10 We believe that this approximate method should give the same result as the exact method; this is because the time-dependent electron−field interaction in the former is incorporated in the Hamiltonian. However this method, needs to be first benchmarked against exact theoretical calculations as well as against experimental results for the smallest possible molecule, H2+, before using it for larger systems. Therefore, we present here the TDPES of the two lowest electronic states of H2+ using Cederbaum’s approach followed by quantum nuclear dynamics on the computed surfaces in the presence of ultrashort, intense laser pulses of peak intensities in the range 1012−1014 W/cm2 using laser pulse durations of 2, 5, and 8 fs. We obtain results that are counterintuitive in that the potential well in the lowest-energy

I. INTRODUCTION Controlling the dynamics of both electrons and nuclei within molecules by means of attosecond and femtosecond laser pulses is a subject that has attracted increasing attention in recent years.1−3 The traditional theoretical approach to understand these experiments has been to compute potential energy surfaces (PES) for the system using the Born−Oppenheimer (BO) approximation and to carry out dynamical calculations on such surfaces. However, in recent years, there have been experiments that take the “approximation” out of the “BO approximation”. These are the few-cycle, strong-field experiments in which molecules are exposed to very intense laser fields of duration shorter than typical vibrational time periods. Under the influence of intense laser fields, the electronic and nuclear motions become coupled, and therefore, one needs to solve the coupled electron−nuclear Schrödinger equation in order to gain insights into the dynamics under such situations. There have been a few attempts in this direction.4,5 Chelkowski et al. numerically solved the exact coupled electron−nuclear Schrödinger equation to understand the dissociative ionization of H2+ in intense laser fields.4 More recently, Abedi et al. reported a method to compute the time-dependent potential energy surface (TDPES) under the influence of external fields that is based on exact factorization of the time-dependent electron−nuclear wave function.5 However, we note that the methods developed by Chelkowski et al. and Abedi et al. are both computationally very expensive and can, therefore, be applied only for small molecular systems, such as H2+. There is a clear need for methods that are computationally inexpensive and are in consonance with the new generation of molecular dynamics experiments that are becoming possible with the advent of intense, few-cycle laser pulses. Our recent joint experimental and theoretical work on a polyatomic molecular ion, © 2012 American Chemical Society

Received: June 12, 2012 Revised: August 1, 2012 Published: August 3, 2012 8762

dx.doi.org/10.1021/jp305712d | J. Phys. Chem. A 2012, 116, 8762−8767

The Journal of Physical Chemistry A

Article

state of H2+ is seen to collapse as the laser pulse reaches its peak amplitude and starts regaining its original form on the trailing edge of the pulse, albeit with a lower well depth. We also discover that the trapped nuclear wavepacket possesses an increasing probability of leaking out of the well as the laser pulse duration increases. This has implications for ongoing and currently planned experiments using intense, few-cycle and subcycle laser pulses.

Table 1. Grid Parameters Used in the Present Study

ϕe(r , t ; R ) χn (R , t )

∂ ϕ (r , t ; R ) = Heϕe(r , t ; R ) ∂t e

V ̃ (R , t ) =

⎡ ∂ω ⎤ = ⎢Tñ + ⎥χ ⎣ ∂t ∂t ⎦ n where T̃ n is given by

iℏ

(2)

∂χn ∂t

(3)

He(t ) = − −

(4)

Tn = −

(6)

1 ∂2 + Vl(r , t ) − 2me ∂r 2

1 + (r − R /2)2

1 + (r + R /2)2

(12)

ℏ2 ∂ 2 M p ∂R2

(13)

∑ cne−E τ ϕn(0) n

n

∫ ϕe*(r , t ; R) Δϕe(r , t ; R) dr

1

1

ϕ(τ ) = e−Heτ ϕ(0) =

and b(R , t ) =

(11)

where me = (2Mp)/(2Mp + 1) is the electronic reduced mass and Mp is the proton mass. The interaction of the electron with the intense laser field is given by Vl(r,t) = qerE(t), where E(t) is the electric field amplitude and qe is the electronic charge. We have used the time relaxation method to generate the exact electronic wave function ϕe at time t = 0 for each nuclear configuration.11 An arbitrarily chosen Gaussian wavepacket is propagated in the imaginary time by using the following equation:

Tn = −ℏ2/2MΔ, with Δ denoting the Laplacian operator and M being the average nuclear mass, where a and b are timedependent electron−nuclei coupling terms defined as

(14)

where τ = it. In eq 14 the weight of each eigenfunction relaxes to zero at a rate proportional to its eigenvalue. The ground state wave function, which relaxes slowly, persists. Higher states are obtained by numerically subtracting the ground state wave function from the initial wavepacket. Another round of propagation gives the first excited state. Propagation for a long time can mix the eigenfunctions, forcing the initial wavepacket to relax to the ground state. The above propagation scheme is applied at each nuclear

(7)

The gradient and Laplacian operator are derivatives in the nuclear space. We rearrange eq 5 in simpler form as Tñ = −

= [Tn + Ṽ (R , t )]χn

B. Time-Dependent Potential Energy Surfaces. The TDPES for the two lowest electronic states,1sσg and 2pσu , of H2+ were calculated in the presence of linearly polarized, ultrashort, intense laser pulses. Restricting the motion of the single electron and the two nuclei to the direction of the polarization axis of the laser field, we solved the problem using a one-dimensional Hamiltonian featuring soft-core Coulombic interactions:5

i 2i 1 ℏ2 ⎡ 2 Tñ = Tn − ⎢b + Δω + (∇ω) ·a − 2 (∇ω) 2M ⎣ ℏ ℏ ℏ ⎡i ⎤ ⎤ + 2⎢ (∇ω) + a⎥ ·∇⎥ ⎣ℏ ⎦ ⎦ (5)

∫ ϕe*(r , t ; R) ∇ϕe(r , t ; R) dr

(10)

Nuclear dynamics with the above time-dependent PES is determined by the following equation.

∂χn

a(R , t ) =

∂ω ℏ2 [∇·a + a ·a − b] + 2M ∂t 1 + ϕe*(r , t ; R ) ϕe(r , t ; R ) dr R

∫

ϕe(r,t;R) and χn(R,t) are always normalized to unity. Inserting Ψ(r,R,t) into the Schrödinger equation of the full Hamiltonian H and projecting on the nuclear space gives us iℏ

(9)

In the stationary state situation ∂ω/∂t becomes a timeindependent PES. After determining the time-dependent topological phase from the above condition, the time-dependent potential energy surface (TDPES) can be written as

where r denotes the electronic coordinates. ϕe is obtained for each value of R by solving the following equation. iℏ

number of grid points extension of grid along R (dR = 0.02) extension of grid along r (dr = 0.4) time step used in electronic propagation time step used in nuclear propagation

∇ω = iℏa

where He is the electronic Hamiltonian and Tn and Vnn constitute the nuclear Hamiltonian. As demonstrated by Cederbaum,8 inserting a simple product of the electronic and nuclear wave functions, ϕe(r,t;R) χn(R,t), into the Schrödinger equation of the full Hamiltonian H and multiplying the left by ϕe*(r,t;R) gives rise to many complex terms. However, to remedy the situation, the complex electronic wave function can be multiplied by a time-dependent topological phase ω(R,t), which is a function of the nuclear coordinate R. This ansatz for the timedependent Born−Oppenheimer approximation can be written as Ψ(r , R , t ) = e

value (400, 1000) (1.0, 9.0) (−200, 200) 0.244 2.44

T̃ n can be further simplified by eliminating the nuclear momentum coupling operator by using the condition

II. METHODOLOGY A. Time-Dependent Born−Oppenheimer Approximation. The Born−Oppenheimer approximation allows one to separate the fast electronic motion from the slower nuclear motion. The full Hamiltonian of the system can be written as H = He + Tn + Vnn (1)

iω(R , t )/ ℏ

parameter (NR, Nr) (Rmin, Rmax)/a0 (rmin, rmax)/a0 Δte/as Δtn/as

⎤2 i ℏ2 ⎡ ℏ2 a ( ) [∇·a + a ·a − b] ∇ + + ∇ ω + ⎢ ⎦⎥ 2M ⎣ 2M ℏ (8) 8763



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Figure 1. TDPES at specified times (t) for the 1sσg state in the presence of a 2 fs laser pulse (800 nm wavelength). The reference for TDPES in all cases is the energy value in the asymptotic region. The pulse peaks around t = 1.83 fs.

Figure 3. TDPES at specified times (t) for the 1sσg state in the presence of an 8 fs laser pulse. The pulse peaks around t = 7.32 fs.

configuration to generate the exact electronic wave function. We have computed the field-free molecular potential in this soft core potential by using V (R ) = ϕe(r ; R )|Te|ϕe(r ; R ) + ϕe(r ; R )|Vsoft|ϕe(r ; R ) +

ϕe(r ; R )

1 ϕ (r ; R ) R e

where Vsoft is the last two terms of eq 12. We consider a λ = 800 nm laser field, represented by E(t) = 2 2 E0e−2 ln 2(t−t0) /τ cos(ω(t − t0)), with peak intensity I = |E0|2 = 12 10 and 1014 W/cm2. An envelope of the form E(t) = 2 2 E0e−2 ln 2t /τ cos(ω(t)) will have both positive and negative time axes. To overcome this problem, we shift the envelope by t0, where t0 is the time in the negative time axis where the field becomes zero. This way we shift the electric field envelope only in the positive time axis. Hence, t = 0 is the time of start of the laser−molecule interaction. In this form of the pulse envelope, the field will reach its maximum at t = t0. In the present work we have calculated the TDPES for laser pulse durations of 2, 5, and 8 fs, followed by quantum nuclear dynamics on these surfaces. Starting from the initial electronic wave function at each nuclear coordinate, we probe the time evolution of the electronic wave function in the presence of laser fields. The electronic wave functions at discrete time steps are stored so that a and b in eq 10 can be calculated. The topological phase, ω, is calculated from the following equation. Figure 2. TDPES at specified times (t) for the 1sσg state in the presence of a 5 fs laser pulse. The pulse peaks around t = 4.41 fs.

ω(R , t ) = iℏ 8764

∫R

R

a(R ′ , t ) d R ′ min

(15)



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Figure 4. Snapshots, at specified time intervals (t), of the TDPES for the 1sσg state in the presence of a 5 fs pulse along with snapshots of the nuclear wavepacket. The dashed curve is the field-free PES.

III. RESULTS AND DISCUSSION Figures 1−3 show TDPES at different times during laser− molecule interaction of the 1sσg state on exposure to pulses of duration 2, 5, and 8 fs, respectively. The energy reference for the TDPES in every case is the energy value in the asymptotic region. The well depth in the TDPES in each case is seen to start decreasing during the rising edge of the laser pulse, and it collapses around the peak of the pulse, trying to return to its initial form during the trailing edge of the pulse. The PES acquires a new minimum at larger values of R (5−6 au) during the trailing edge of the pulse. The TDPES for the 2pσu state always remains repulsive, with only slight changes in the energy values. Field dressed potentials of H2+ in the presence of

Here, the electron−nuclear momentum coupling term, a, is an imaginary quantity and the topological phase term ω(R,t) is a real quantity. Derivatives of the electronic wave function in nuclear space were calculated by considering higher order terms in the Taylor series expansion to reduce errors in calculations of a and b to ∼ΔR5. We took care in calculating these derivatives as this accumulates error in the calculation of ω. The numerical time evolution of the time-dependent Schrödinger equation (TDSE) was studied using the split operator technique.12 Numerical parameters for solving the TDSE of the electronic Hamiltonian, eq 12, and nuclear dynamics on TDPES, eq 11, are given in Table 1. 8765



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Figure 5. Norm on log10 scale of the nuclear wavepacket in the asymptotic channel (R > 6.0 au) as a function of time for laser− molecule interaction in cases of 2, 5, and 8 fs pulses.

Figure 6. Plot comparing the PES of the 1sσg state at the end of laser− molecule interaction with the corresponding PES in the field-free situation for 2, 5, and 8 fs pulses.

intense laser fields (∼1014 W/cm2) have been earlier used to describe several phenomena including bond hardening and softening.13,14 It has been shown earlier that the potential well of H2+ collapses in the presence of a static field.4 Our calculation shows that the well collapses at the peak intensity of the laser pulse and then regains its form during the trailing edge of the laser pulse. The collapse of the well at the peak of the pulse and then its regaining of the original form during the trailing edge of the pulse seems to be counterintuitive, and it describes the exact dynamical behavior of the PES in time. We note that the TDPES for the 1sσg state for somewhat weaker laser pulses (of intensity 1012 W/cm2) does not feature any collapse of the potential well as the field peaks; only slight changes in energy values are obtained, which is expected as the laser intensity is too weak to have any significant effect on the H2+ potential well. Abedi et al.5 have obtained TDPES by exactly treating the electronic and nuclear degrees of freedom for a one-dimensional system. The features of TDPES obtained with the exact treatment of electronic−nuclear degrees of freedom are the same as those obtained in our work. Though different pulse parameters were considered by Abedi et al., they also reported the collapse of the potential well at the peak of the laser pulse as well as trapping of the classical nuclei. We have carried out quantum nuclear dynamics on the 1sσg state using eq 11. The initial vibrational wave function was calculated by propagating an arbitrarily Gaussian function in imaginary time evaluated using eq 14 in the field-free PES. Snapshots of the time evolving nuclear wavepacket along with the TDPES during laser−molecule interaction for a 5 fs pulse are plotted in Figure 4. It is clear that, as time progresses, the nuclear wavepacket starts spreading and part of the wavepacket reaches the asymptotic region after the pulse is over. A similar

result has been obtained for 8 fs pulses. The spreading of the nuclear wavepacket in the case of 2 fs pulses was found to be very small. The repulsive nature of the TDPES during collapse of the potential well gives the nuclear wavepacket momentum to leak out from the potential well. We have found that momentum gained by the nuclear wavepacket for longer pulses is more than that the shorter pulses. We followed nuclear dynamics for a long time after the laser−molecule interaction. A part of the nuclear wavepacket is seen to leak out toward the asymptotic region in the case of the 5 and 8 fs pulses, whereas for the shorter 2 fs pulse it remains trapped within the well of the PES. The new minimum in PES for the 5 fs pulse manifests itself by trapping the nuclear wavepacket at larger R values. Trapping of the nuclear wavepacket at larger R values supports the elongated bond between the two H-atoms in H2+. The norm of the nuclear wavepacket in the asymptotic region (R > 6.0 au) is plotted on a logarithmic scale in Figure 5 as a function of time in the case of 2, 5, and 8 fs pulses. A larger norm in the asymptotic channel for 8 fs compared to 5 and 2 fs pulses and for 5 fs compared to 2 fs suggests increased leaking of the nuclear wavepacket to the asymptotic region with an increase in pulse duration. Longer rising and trailing edges of adiabatic laser pulses will lead to an increase in the probability of leaking of the initially trapped nuclear wavepacket, resulting in bond rupture of the type that has recently been reported by us in experiments on a polyatomic molecule.6 Very recently, we have used static potentials to demonstrate the leaking of a trapped vibrational wavepacket upon exposure to 5 fs pulses in the polyatomic tetramethylsilane monocation [Si(CH3)4]+.6 Quantum dynamics on its TDPES is expected to reveal the exact dynamical features of the problem. 8766



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(5) Abedi, A.; Maitra, N. T.; Gross, E. K. U. Phys. Rev. Lett. 2010, 105, 123002. (6) Dota, K.; Garg, M.; Tiwari, A. K.; Dharmadhikari, J. A.; Dharmadhikari, A. K.; Mathur, D. Phys. Rev. Lett. 2012, 108, 073602. (7) Kono, H.; Satao, Y.; Tanaka, N.; Kato, T; Nakai, K.; Koseki, S.; Fujimura, Y. Chem. Phys. 2004, 304, 203. (8) Cederbaum, L. S. J. Chem. Phys. 2008, 128, 124101. (9) Garg, M.; Tiwari, A. K.; Mathur, D. J. Chem. Phys. 2012, 136, 024320. (10) Rajgara, F. A.; Mathur, D.; Dharmadhikari, A. K.; Safvan, C. P. J. Chem. Phys. 2009, 130, 231104. (11) Kosloff, R.; Tal-Ezer, H. Chem. Phys. Lett. 1986, 127, 223−230. (12) Feit, M. D.; Fleck, F. A., Jr.; Steiger, A. J. Comput. Phys. 1982, 47, 412. (13) Bandrauk, A. D.; Sink, M. L. J. Chem. Phys. 1981, 74, 1110. (14) Giusti-Suzor, A.; Mies, F. H. Phys. Rev. Lett. 1992, 68, 3869. (15) Znakovskaya, I.; von den Hoff, P.; Schirmel, N.; Urbasch, G.; Zherebtsov, S.; Bergues, B.; de Vivie-Riedle, R.; Weitzel, K. M.; Kling, M. F. Phys. Chem. Chem. Phys. 2011, 13, 8653. (16) Geppert, D.; von den Hoff, P.; de Vivie-Riedle, R. J. Phys. B 2008, 41, 074006.

The utilization of 5 fs long 800 nm wavelength laser pulses (corresponding to barely two optical cycles) in recent experiments6 brings to the fore the importance of considering possible effects that the phase between the intensity “carrier” wave and the pulse envelope might play in determining the overall strong-field molecular dynamics. To explore this, we have also calculated TDPES for different carrier envelope phases (CEP) of a 5 fs pulse. Control of CEP leads to selective electron localization in a dissociating ion.15,16 The effect of CEP on nuclear dynamics has, until now, not been studied. Laser pulses with different values of CEP are expected to attain peak intensities at different times. We find that, in all cases, the TDPES collapses around the peak of the laser pulses. The well depth of the TDPES after the laser field interaction in the Franck−Condon region is found to be approximately the same for different CEP values, and the nuclear dynamics also does not show any significant difference after long-time field-free propagation. Figure 6 compares the PES after the molecule− field interaction is over for 2, 5, and 8 fs pulses with the fieldfree PES of the 1sσg state. We have also plotted the PES for a 5 fs with a defined value of CEP = π/2 in Figure 6. It is also important to note that the response of TDPES to the temporal changes in the field value is quick, so it is difficult for the heavier nuclei to respond to quick variations of the electric field. Our point in studying the effect of CEP was to know if CEP has any effect on nuclear dynamics which would have resulted from varying well depths in PES for different CEP, which is not the case as mentioned above. In summary, we have demonstrated the use of Cederbaum’s time-dependent Born−Oppenheimer (TDBO) approximation to compute TDPES for H2+ in the presence of an external laser field in order to carry out simultaneous electron−nuclear dynamics on attosecond time scales. Our calculations shows that the potential well in the lowest-energy state of H2+ collapses as the laser pulse reaches its peak amplitude; it then regains its form on the trailing edge of the pulse. We find that the trapped nuclear wavepacket has a higher probability of leaking out in the case of longer laser pulses. The carrier envelope phase is found to have negligible effect on the nuclear dynamics. Computing TDPES for multielectron molecules in the presence of a time-varying external field is a challenging task. The TDBO approximation provides a route to computing the TDPES for bigger molecular systems. Currently, we are trying to implement Cederbaum’s method of calculating TDPES for polyatomic molecules.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.K.T.); [email protected] (D.M.). Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Zewail, A. H. Angew. Chem., Int. Ed. 2000, 39, 2586. (2) Niikura, H.; Legare, F.; Hasbani, R.; Ivanov, M. Y.; Villeneuve, D. M.; Corkum, P. B. Nature (London) 2002, 421, 826. (3) Drescher, M.; Hentschel, M.; Keinberger, R.; Ulberacker, M.; Yakoviev, V.; Scrinizi, A.; Westerwalbesioh, T.; Kleinberg, U.; Heinzmann, U.; Krausz, F. Nature (London) 2003, 419, 826. (4) Chelkowski, S.; Foisy, C.; Bandrauk, A. D. Phys. Rev. A 1998, 57, 1176. 8767


Quantum Dynamics of H2+ in Intense Laser Fields on Time

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