A real time dynamical calculation of H2− photodissociation

13 June 1997

ELSEVIER

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 271 (1997) 204-208

A real time dynamical calculation of H 2 photodissociation Hong Zhang, Ke-Li Han *, Yi Zhao, Guo-Zhong He, Nan-Quan Lou State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, PR China

Received 11 March 1997; in final form 1 April 1997

Abstract

A one-dimensional wavepacket study of a two-state system (H ~-), coupled by a laser pulse, is presented. The wavepacket time evolution on a repulsive excited potential surface is simulated for a weak laser field as well as for a strong one. The complex lifetime of H 2 photodissociation is calculated to be about 8.5 fs. The deformation of the ground state wavepacket under a strong laser field has been discussed and it is found that the strong field did not only induce the electronic state excitation but also resulted in the vibrational excitation of the electronic ground state in H 2 photodissociation.

1. Introduction

Recent advances in the physics and chemistry of laser interactions with atoms and molecules have brought wavepackets and their dynamics into focus. A whole new realm of phenomena has opened since femtosecond pulse technology emerged. With ultrashort pulses one can now prepare a molecular wavepacket and probe its evolution in the time domain experimentally [1,2] and theoretically [3-7]. This makes it possible to chart the path of a chemical process taking place in terms of intermediate states. Our final goal is the manipulation of the reaction in order to achieve precise control over the output products. In other words, we wish to control the wavepacket processes through adjusting laser pulse parameters such as intensity, frequency, duration, timing, shape, etc. Photodissociation with a fem-

* Corresponding author.

tosecond pulse is an ideal field for investigating such wavepacket processes in real time. Utilizing lasers to manipulate molecules has a long history [8-11]. A wavepacket under an intense laser field has shown some interesting features such as the dynamical 'hole' phenomena reported by Kosloff and co-workers [12-14]. A detailed investigation of the mechanisms causing the phenomena is necessary for controlling intramolecular dynamics and for an adequate understanding of all aspects of light-matter interaction. Under a weak laser field the shape of the ground state wavepacket is unchanged after laser excitation, but in this case, only a minor fraction of the population can be excited. With an increase in laser intensity, it becomes possible to achieve larger excitation. If the pulse area is larger than 7r, the excited state population profile shows oscillations (so-called Rabi oscillations). The population is transferred up and down between the ground state and the excited state surface. Two extreme situations are 7r pulses and 2-tr

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H. Zhang et al. / Chemical Physics Letters 271 (1997) 204-208

pulses on the basis of the area theorem. For Ir pulses, all the populations become excited, while 2"rr pulses leave the ground state population unchanged. In most circumstances, the transferred population is between 1 and 0. We know that a short exciting pulse can be approximated by a 6 function in time. In this case, the ground state wavepacket has no time to deform, and the excited wavepacket appears as a well localized wavepacket. However, a real exciting pulse has width, and thus the discussion of the influence of a finite pulse width is of practical interest. What is the behavior of the ground state and excited state wavepacket under such a finite strong laser field? What is the difference between the weak and strong laser cases? What causes distortion of the ground state wavepacket? To answer these questions, we choose H 2 as an example to investigate the wavepacket processes in the presence of a light field (see Fig. 1). The photodissociation and multiphoton ionization of He and H~ have been widely investigated for their simplicity and fundamental status in quantum mechanics using a time-dependent method [10,15,16]. However, H 7 has not drawn much attention. Because the excited state potential of H~ is steep and the excited state wavepacket slides down the slope rapidly, the whole dissociation process (including laser excitation, the excited state wavepacket evolution and the asymptotic product) can be simulated on the condition that an ultrashort pulse (we choose FWHM = 1 fs) is exploited. This Letter is arranged as follows. A description of the V(a.u.) 1

0.6 0.4

0.2 ~ i

00

method of calculation is given in Section 2 and a discussion of the results follows in Section 3. The conclusion is summarized in Section 4.

2. Method of calculation The two-surface dynamics, coupled by a laser pulse, are calculated through the exact time-dependent quantum mechanical method. We ignore rotational motion in this calculation, which can be achieved for molecular beam experiments with low rotational temperature. The ground state is taken to be Morse potential, Vg = D[1 - exp( - / 3 A r)] 2 + 7.35 × 10 -2,

fi=cl +c2Ar,

,

,

1

f ,

;

2

,

I

3 4 R(a.u.)

I

5

Fig. l. Potential-energy surfaces for H~ states.

6

ground and excited

A r = r - - r o,

(1)

here D = 0 . 0 6 6 1 au, r o = 1.52 au, c I and c 2 are constants: c I = 1.35, c 2 = 0.02. The excited state potential is an exponential, V~ = b e x p ( - c r )

+ 0.14,

(2)

with b = 3.0 au, c = 1.7. The above potential parameters chosen are obtained through a reasonable fit to known spectroscopic data [17]. The two surfaces are coupled by a transition dipole operator /,. E(t), where /x is estimated through ab initio calculation. E(t) is treated by a semiclassical approximation: E(t) = ei'°'A(t) with central frequency oJ and pulse shape A(t). We choose A(t) as a Gaussian pulse. The key question is to solve the coupled time-dependent SchriSdinger equation: i~Tt(~)=

0.8

205

(/*.HgE(t)

tz.E(t))(::),//~.

(3)

where q,~ is the projection of the wavefunction on the upper surface, &g is the projection of the wavefunction on the ground surface, and Hg, He are the ground/lower surface Hamiltonian, respectively, where Hg = p2/2m + Vg, H e = P2/2m + Ve. The above time-dependent Schr~Sdinger equation (3) is solved numerically by a split operator method which uses the grid method and fast Fourier transformation (FFT) for computing the Laplacian operator [18-23]. The bound vibrational eigenfunctions of the ground electronic state are calculated by the discrete variable representation (DVR) method [24,25]. In the

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calculation, we choose the wavefunction of the vibrational ground state of the lower electronic state as the initial wavepacket. To check the convergence, we compare the initial energy of FFT with that of the DVR method to see whether both are consistent. The grid size is chosen as 512, the spacing span is 0.05 au and the time propagation is 0.52 au. Only the on-resonance situation was considered in all the calculations.

surface at the Franck-Condon point R = R e, it instantaneously begins to slide down the repulsive potential surface. Because the excited state wavepacket moves with just a little dispersion, we can obtain the complex lifetime from the wavepacket evolution on the excited state potential. We take R~o = 6.0 au, R~ is the point from where the force -OV/OR approximates to stay zero. It takes 8.5 fs for the excited state wavepacket to move from R 0 to Ro~, which can be considered as the complex lifetime. The above weak field excitation only transfers less than 1% of the population to the excited state. For larger excitation, we use a high-power laser pulse (a = 0.065 au). Fig. 3 shows the population profiles of the excited state and ground state. About 50% of the population is transferred to the excited state. However, under such a strong laser field case, some nonlinear phenomena appear. Fig. 4 shows the excited and ground state evolving wavepackets. Compared with the results under weak field, we found that the excited state wavepackets are almost the same aside from the amplitude difference but the ground state wavepacket in the case of a strong laser

3. Results and discussion The time evolution of H ] on a weak laser field (a = 0.0005 au) was simulated using the method described above. Fig. 2 shows the evolving wavepackets on the excited and ground states. The ground state wavepacket keeps the shape of the vibrational ground state without any dispersion and deformation. The excited state wavepacket has no distortion but with a little dispersion (still localized). From Fig. 2 we can see after the laser promotes the ground state wavepacket onto the excited state

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Fig. 2. The evolving excited state wavepacket [~b~(R, t)l and ground state wavepacket I~bg(R, t)l under a weak laser field. (a), (b) and (c) corresponding to the excited state, while (d), (e) correspond to the ground state. Note the difference in units.

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H. Zhang et al. / Chemical Physics Letters 271 (1997) 204-208

excitation. Fig. 5 implies that the strong laser field produces substantial vibrational excitation. Kosloff et al. have investigated the photodissociation of 13 and CH3I under a strong laser field with an emphasis on ~ pulses and 2at pulses [12-14] and found the ground state dynamical 'hole' phenomenon. They give a rigorous definition of the dynamical 'hole' in density matrix language and a method of measuring them. The ground state wavepacket deformation in H~ is probably a kind of diminished dynamical 'hole' phenomenon with pulse area < "rr in essence. What is the essence of the deformation of the ground state wavepacket? It is apparent that the phenomenon is a kind of vibrational energy randomization [26] under a strong laser field. This phenomenon is much like a macroscopic analogue: a bomb is thrown into a pool, which not only gives rise to water splashing onto the bank (like electronic excitation), but also results in a huge perturbation of the water in the pool (ground state vibrational excitation). This may result from the self-interference of

population

1 0.8 0.6 0.4 0.2 /

0

(4)

(2)"o

2

;

~

8 '1o

time(fs)

Fig. 3. Time evolution of the ground and excited-state populations for strong laser excitation. The dashed line ( - - - ) represents the ground state while the solid line ( - - - - ) state.

represents the excited

field is apparently distorted. To explain this phenomenon, we calculated the projection of the ground state wavepacket onto the different vibrational eigenstates. Fig. 5 illustrates the time-dependent behavior of the population of 11 low-lying vibrational levels of the ground electronic state during and after laser

3E-8

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t=6fs

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4

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8

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(d)

10

12 14

0

2

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(e)

Fig. 4. Same as in Fig. 2 but for a strong laser field. Note that the excited state wavepacket is almost the same aside from the amplitude difference but the ground state wavepacket has evident distortion compared with the weak laser case.

208

14. Zhang et al. / Chemical Physics Letters 271 (1997) 204-208

I,

O.8

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Acknowledgements

We would like to thank the Chinese Natural Science Foundation for financial support. The authors thanks Dr. Y.-J. Yan for useful discussions and advice.

0.6

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i

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and may be the result of ground state wavepacket self-interference under a strong laser field.

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References

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.

6

(c)

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8

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0

1

2

3

4

5

6

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(d)

Fig. 5. Time developmentof populationof the vibrational levels of the ground electronic state under a strong laser field. the ground state wavepacket. The phenomenon is much like the situation of a p u m p - d u m p [27,28] experiment where two laser pulses are exploited. In that case, the first pulse promotes the ground state wavepacket onto the excited surface. After a time delay, the second pulse pulls the excited state wavepacket down again. The interference of these two wavepackets results in ground state vibrational excitation, which gives rise to the ground-state wavepacket distortion.

4. C o n c l u s i o n s

We have utilized a time-dependent approach to show the wavepacket behavior and population in the ground and excited states for two different laser pulses. Detailed direct photodissociation processes (including laser excitation, wavepacket evolution on a repulsive potential surface and the asymptotic products region) of H~- are simulated. The dissociation stage lasts only about 8.5 fs. and the excited state wavepacket stays on well-localized during three stages. The deformation of the ground state wavepacket under strong laser excitation of H 2 is simulated, which is probably a kind of dynamical 'hole' phen o m e n o n found by Kosloff et al. in 13 photodissociation [ 13,14]. This is a nonlinear optical phenomenon

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