On laser Coulomb explosion imaging of proton motion

23 March 2001

Chemical Physics Letters 336 (2001) 518±522

www.elsevier.nl/locate/cplett

On laser Coulomb explosion imaging of proton motion Andre D. Bandrauk *,1, Szczepan Chelkowski Laboratoire de Chimie Th eorique, Facult e des Sciences, Universit e de Sherbrooke, Sherbrooke, Que., Canada J1K 2R1 Received 17 October 2000; in ®nal form 11 December 2000

Abstract From exact non-Born±Oppenheimer simulations of the dissociative-ionization dynamics of H‡ 2 in an intense …I > 1015 W=cm2 †, ultrashort (tp < 5 fs) k ˆ 800 nm laser pulse, one can measure the time evolution of the probability distribution jw…R; t†j2 of the initial proton wave packet. It is shown that best imaging is obtained for ultrashort pulses (tp < 3 fs) at 4  1015 W=cm2 or much shorter wavelengths, k 6 30 nm. The relevance of this imaging criterion is discussed in relation to current laser Coulomb explosion imaging of proton transfer in large molecules. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Temporal imaging of chemical reactions is one of the major challenges of modern science. Recent advances in laser technology allow for the creation, with table-top lasers of ultrashort (tp < 5 fs), 2 intense …I P 1015 W=cm † pulses in the near IRvisible region; k ˆ 800 nm [1]. This allows one to freeze virtually all nuclear motion, especially that of the elusive proton (tp ' 15 fs) in molecules on the time scale for ionization with such pulses [2]. We have recently proposed a new imaging technique, laser Coulomb explosion imaging (LCEI) of molecular dynamics using such pulses [2,3]. This proposal was based on exact non-Born±Oppenheimer solutions of the time-dependent Schroe-

*

Corresponding author. Fax: +819-821-8017. E-mail address: [email protected] (A.D. Bandrauk). 1 Present address: Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128, Montreal, Que., Canada H3C 3J7.

dinger equation (TDSE) for dissociative ionization of the simplest molecule H‡ 2 [4]. Since both the laser [1] and the ion imaging technology [5,6] are now fully developed, then ultrashort, ultrafast LCEI experiments should now be feasible, even to follow the motion of the lightest nuclei, the proton whose vibrational time scale is 10 6 tp 6 15 fs. Multiphoton ionization and dissociation studies of large molecules has opened up new ways of looking at elementary chemical processes on the femtosecond time scale [7±9] since with shorter, more intense pulses radiative excitation rates exceed intra-molecular relaxation. LCEI with ultrashort pulses allows therefore in principle for temporal imaging of nuclear dynamics because of the excessively fast ionization rates achievable. Recently the LCEI technique has been used by Castleman et al. [9,10] to measure the excited state double proton transfer (ESDPT) reaction in the 7-azaindole dimer in order to con®rm the stepwise transfer mechanism proposed previously by Zewail et al. [11]. The LCEI experiments of Castleman have been recently discussed critically by Kasha et al. [12] who emphasize the invasive nature of the

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 1 1 3 - 0

A.D. Bandrauk, S. Chelkowski / Chemical Physics Letters 336 (2001) 518±522

LCEI technique in disputing the evidence for sequential versus concerted double proton transfer. In the present Letter, we present new results based on simulations of the exact TDSE for dissociative-ionization of H‡ 2 [4] as a function of laser wavelength and pulse duration in order to establish the conditions for optimal LCEI. Our recent work on intense femtosecond laser pulse ionization of molecules at the wavelength of 800 nm has led to the discovery of charge resonance enhanced ionization (CREI) in odd [13,14] and even [15] electron systems. Thus large ionization rates are predicted to occur at large critical distances Rc which can be explained using quasistatic ®eld models [14±18]. Most ecient ionization and Coulomb explosion is therefore obtained with long pulses tp > 10 fs in order to allow dissociative  However long fragments to reach Rc ' 3±5 A. pulses at high intensities introduce severe distortion via laser-induced avoided crossings of `dressed' molecular potentials which are coupled strongly by charge resonance e€ects [19]. We use therefore an exact non-Born±Oppenheimer simulation of the intense laser pulse dissociative-ionization to establish the conditions for best LCEI. Our general conclusion is that pulse durations: tp < 3 fs at long wavelengths (k ˆ 800 nm) or shorter wavelengths (k 6 30 nm) should produce the most reliable imaging at intensities I P 1015 W=cm2 .

519

2

1015 W=cm . We assume that at t ˆ 0 the H‡ 2 molecule is prepared in a speci®c vibrational state, the v ˆ 2 state as an example, so that the total initial state is described by the vibronic wavefunction: w…z; R; 0† ˆ vV …R†w1rg …z; R†

…3†

vV …R† is the initial vibrational wavefunction and 1rg is the exact 1rg ground state H‡ 2 electronic molecular orbital. The envelope f …t† is a Gaussian with varying half-widths from tp ˆ 3:2 to 12.8 fs (see Figs. 1 and 2). The TDSE (1) is propagated in time using highly ecient split-operator numerical algorithms which we have developed for current high intensity molecular problems [20]. The resulting non-separable electron±proton wavefunction is projected onto proton Coulomb waves, wc …E; R†, after integrating out the electron coor-

2. Methodology and results We have solved numerically the complete threebody 1-D, TDSE for the H‡ 2 molecule in the ®eld of strong laser pulses following the method described in [4], iow…z; R; t† ˆ H …z; R; t†w…z; R; t†; ot

…1†

where R is the internuclear distance, z is the electron position with respect to the proton center of mass. H …z; R; t† is the 3-body Hamiltonian in 1-D for H‡ 2 parallel to the electric ®eld: E…t† ˆ Eo f …t† sin xt; …2† where f …t† is the pulse envelope and Eo is the maximum amplitude such that Io ˆ cEo2 =8p ˆ 4 

Fig. 1. LCEI-H‡ simulations for a Gaussian pulse at 2 k ˆ 800 nm, I ˆ 4  1015 W=cm2 , pulse duration (half-width) tp : (a) 3.2 fs; (b) 4.5 fs. Solid (±±) line calculated image jwCEI …R†j2 ; dotted (


) line: initial state distribution jwin …R†j2 for v ˆ 2. Right inset: laser pulse E…t†.

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A.D. Bandrauk, S. Chelkowski / Chemical Physics Letters 336 (2001) 518±522

Fig. 2. LCEI-H‡ simulations for a Gaussian pulse at 2 k ˆ 800 nm, I ˆ 4  1015 W=cm2 , pulse duration (half-width) tp (a) 6.4 fs; (b) 12.8 fs. Solid (±±) line: calculated image jwCEI …R†j2 ; dotted (


) line: initial state distribution jwin …R†j2 for v ˆ 2. Right inset: laser pulse E…t†.

dinate, z. The proton kinetic energy spectrum of exploding protons, S…E†, where E is the total proton kinetic energy at the center of mass, is calculated from appropriate momentum Coulomb wavefunctions [2,3]. In general, there is no simple mapping between the calculated (theoretical) or measured (experimental) proton kinetic energy spectrum S…E† and the initial proton wavefunction vV …R†. However assuming a classical explosion occurs for an Rdependent Coulomb explosion image (CEI) density distribution jwCEI …R†j2 , we obtain a simple relation between this distribution and the proton spectrum S…E†: 2

jwCEI …R†j ˆ S…E†q2 =R2 2

…4† 2

where the factor q =R is the Jacobian jdE=dRj for the Coulomb potential E ˆ q2 =R. Quantum

corrections to this classical procedure occur at the turning points of the initial state as shown earlier [21]. Typical LCEI results are shown in Figs. 1 and 2 for an initial v ˆ 2 state at k ˆ 800 nm, I ˆ 4 1015 W=cm2 and pulse half-widths tp ˆ 3:2; 4:5, 6:412:8 fs. The corresponding pulses are shown in the ®gure in-sets. We observe in the above ®gures that the shortest pulse, Fig. 1a with tp ˆ 3:2 fs, reproduces correctly the node positions of the initial v ˆ 2 proton wavefunction. The two inner (R ˆ 2:2 a.a.) and outer (R ˆ 3:4 a.u) maxima correspond to turning points of the wavefunction. The corresponding image slightly di€ers from the exact maxima in the initial state due to quantum corrections [21]. What is clearly evident is the gradual displacement (movement) of the total image with increasing pulse length (width). Thus in  is Fig. 2b a displacement of about 0.5 a.u (0.2 A) observed for a 12.8 fs pulse. Geometrical deformation and modi®cation of simple molecules has now been observed at in2 tensities I P 1015 W=cm for CO2 [22,23] and H2 O [24]. These are generally attributed to bond angle softening on laser-induced dressed potential energy surfaces. In the case of H‡ 2 , bond-softening has been clearly established as due to the strong radiative coupling via charge resonance e€ects between the 1rg and 1ru electronic states. These states become degenerate at large internuclear distances with a divergent transition moment, R=2, leading to very large dynamic Stark shifts of the ground state [19]. One can avoid such large Stark shifts of the ground state by going directly into the electronic (ionization) continuum by single photon absorption. This is illustrated in Fig. 3 for the same pulse length, tp ˆ 6:4 fs and intensity I ˆ 4 2 1016 W=cm as in Fig. 2a, but now with a much shorter wavelength k ˆ 32 nm leading to direct one-photon ionization. Shorter wavelengths have two advantages: (a) circumventing multiphoton transitions to charge resonance states with inherent large transition moments [19]; (b) eliminating large laser induced ponderomotive electron energies which can be back-transferred to the protons [4]. One disadvantage of shorter wavelengths is the distortion of the image by the strongly R-depen-

A.D. Bandrauk, S. Chelkowski / Chemical Physics Letters 336 (2001) 518±522

Fig. 3. LCEI-H‡ simulation for a Gaussian pulse at 2 k ˆ 32 nm, I ˆ 4  1016 W=cm2 , pulse duration (half-width) tp ˆ 6:4 fs. Solid (±±) line: calculated image; dotted (


) line: initial v ˆ 2 state distribution. Right inset: pulse envelope f …t†.

dent ionization rate at the shorter distance, R ' 2 a.u, Fig. 3. The larger wavelength images, Figs. 1 and 2 do not su€er from this distortion since at the high intensities used here, CREI is absent, i.e., the molecule behaves like an atom with an R-independent ionization rate [2,3]. 3. Discussion Current laser technology is pushing laser pulse duration, tp , towards near-femtosecond time resolution and intensities approaching the atomic unit of electric ®eld, Eo ˆ 5  109 V/cm or the 2 corresponding intensity Io ˆ 3  1016 W=cm . We have shown in the present work from exact nonBorn±Oppenheimer solutions of the TDSE for a 1-D H‡ 2 parallel to the laser ®eld that LCEI allows for imaging of proton wavefunctions from the proton kinetic energy spectrum S…E†, Eq. (4). Excellent agreement between the initial vibrational 2 distribution jvV …R†j and the Coulomb explosion 2 image jwCEI …R†j is obtained for pulses of duration 2 tp 6 3 fs at the intensity I ˆ 4  1015 W=cm and wavelength k ˆ 800 nm. Such high intensities are necessary to achieve high ionization rates in order to produce high charged states in many-electron molecules (e.g., see our recent ab-initio calculation on N3‡ 2 [25]). However such high intensities produce bond-softening and severe distortions of the

521

ground state via charge-resonance e€ects in both odd [14±17] and even electron systems [15]. As shown in Figs. 2 and 3, these distortions produce movements of the nuclei in the ground state to larger internuclear distances, which can only be limited using pulses shorter than 5 fs. One solution to avoid this laser induced geometry changing effect is to use shorter pulses, tp < 3 fs or shorter wavelengths k 6 30 nm. In the latter case, since ionization proceeds directly by one photon processes, one avoids multiphoton electronic excitation in the long wavelength region and therefore `dressing' of the ground state, which leads to severe geometry distortions by bond softening [19]. Both shorter pulses and shorter wavelengths at high intensities are now being achieved [26] to render LCEI therefore more reliable as a probe of molecular dynamics. A further comparison of LCEI in both regimes, multiphoton at long wavelength and single photon at short wavelength is being undertaken to resolve more clearly this issue [27]. The simplicity of LCEI due to its experimental feasability using `table-top' lasers makes it an attractive tool for monitoring timeresolved molecular dynamics. However the present analysis based on exact numerical simulations suggests that near-femtosecond pulses (tp 6 3 fs) are necessary to reliably probe proton dynamics using LCEI. Acknowledgements We thank the following colleagues for `illuminating' discussions on the following subjects: M. Kasha (proton transfer), P.B. Corkum and F. Krausz (ultrashort intense pulses). References [1] C. Spielman et al., Science 278 (1997) 661. [2] S. Chelkowski, P.B. Corkum, A.D. Bandrauk, Phys. Rev. Lett. 82 (1999) 3416. [3] A.D. Bandrauk, S. Chelkowski, P.B. Corkum, Int. J. Quantum Chem. 75 (1999) 951. [4] S. Chelkowski, C. Foisy, A.D. Bandrauk, Phys. Rev. A 57 (1998) 1176. [5] A.J.R. Heck, D.W. Chandler, Ann. Rev. Phys. Chem. 46 (1995) 335.

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