Theoretical Investigation of the Competitive Mechanism Between

Article pubs.acs.org/JPCA

Theoretical Investigation of the Competitive Mechanism Between Dissociation and Ionization of H2+ in Intense Field Hongbin Yao†,‡ and Guangjiu Zhao*,† †

State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China ‡ Department of Basic Courses, Xinjiang Institute of Engineering, Urumqi 830091, China ABSTRACT: The competitive mechanism between dissociation and ionization of hydrogen molecular ion in intense field has been theoretically investigated by using an accurate non-Born−Oppenheimer method. The relative yield of fragments indicates that the dissociation and ionization channels are competitive with the increasing laser intensity from 5.0 × 1013 to 2.0 × 1014 W/cm2. In the case of intensity lower than 1.0 × 1014 W/ cm2, the dissociation channel is dominant, with a minor contribution from ionization. The mechanism of dissociation includes the contributions from the bond softening, bond hardening, below-threshold dissociation, and above-threshold dissociation, which are strongly dependent on the laser intensity and initial vibrational state. Furthermore, the ionization dominates over the dissociation channel at the highest intensity of 2.0 × 1014 W/cm2. The reasonable origin of ionization is ascribed as the above-threshold Coulomb explosion, which has been demonstrated by the space-time dependent ionization rate. Moreover, the competition mechanism between dissociation and ionization channels are displayed on the total kinetic energy resolved (KER) spectra, which could be tested at current experimental conditions. W/cm2; the fragment of dissociation was separated from the Coulomb explosion on the reflection time-of-flight mass spectrometry.13 Pavičić et al. discussed the Coulomb explosion channel of H2+ and D2+ with femtosecond laser pulses (100 fs, I = 5.0 × 1013 − 1.0 × 1015 W/cm2), and suggested that ionization was enhanced at critical internuclear distances.14 Among these molecular species, hydrogen molecular ion is an ideal system for a detailed understanding of the mechanism of dissociation and ionization, since the simplest molecular configuration makes it the best choice for theoretical and experimental studies. Interacting with an intense laser field, H2+ exhibits two fragment channels, dissociation

I. INTRODUCTION The dynamics of atoms and molecules exposed to intense laser fields continues to attract attention due to breeding a multitude of interesting strong-field processes, such as high-order harmonic generation (HHG),1−4 above threshold ionization (ATI),5 Autler-Towner (AT) splitting.6−9 HHG is first detected in 1987 by employing the rare gases interaction with subpicosecond 248 nm laser field at the intensity of 1016 W/ cm2.3 The mechanism of HHG can be well-understood by the means of semiclassical “three-step” model,4 consisting of ionization, acceleration, and recombination. ATI is discovered by Agostini et al. in 1979 via multiphoton ionization of xenon atoms, and the model based on inverse bremsstrahlung gives a reasonable interpretation of the ATI process.5 AT splitting is first observed in an atomic beam of sodium6 and then widely studied in quantum dots7 and the molecular system.8,9 Up to now, the dissociation and ionization dynamics have been widely investigated in many molecular species. For instance, Lavancier et al. reported the multiphoton dissociation and ionization of N2 molecule with the intensity up to 1014 W/ cm2, and the essential characteristics were discussed at 694 and 347 nm.10 Gibson et al. presented strong-field ionization of the I2 molecule using a 33 fs laser pulse; the channels of chargeasymmetric and charge-symmetric dissociation were demonstrated on the ion time-of-flight spectra.11 Seideman et al. found that the rate of nonlinear ionization was strongly enhanced when an A2n+ molecular ion was stretched beyond its equilibrium position.12 Liu et al. studied the dissociation and ionization of the CH3I molecule at the intensity of 6.6 × 1014 © 2014 American Chemical Society

H2+ + nℏω → H + + H

(1)

and ionization followed by Coulomb explosion (CE) H2+ + n′ℏω → H + + H + + e−

(2)

These two fragment channels incorporate some important molecular processes, such as bond softening (BS),15 bond hardening (BH),16,17 above threshold dissociation (ATD),18,19 below threshold dissociation (BTD),20,21 and charge resonance Special Issue: International Conference on Theoretical and High Performance Computational Chemistry Symposium Received: March 26, 2014 Revised: May 7, 2014 Published: May 7, 2014 9173

dx.doi.org/10.1021/jp5030153 | J. Phys. Chem. A 2014, 118, 9173−9181

The Journal of Physical Chemistry A

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enhanced ionization (CREI).12,22 BS, BH, ATD and BTD describe the dissociation channel (eq 1), while CREI describes the ionization channel (eq 2). Experimentally, the details of these processes could be detected by kinetic energy resolved (KER) spectra of fragments. The KER spectra have been found to be sensitive to the peak intensity, pulse duration, carrierenvelope phase and wavelength of the laser field.23−27 Especially, the progress in experimental techniques has made it possible to directly separate the fragments of dissociation from ionization via the coincidence three-dimensional momentum imaging.28−30 In recent years, the strong-field fragments have been controlled by gating the dissociation pathways with few-cycle laser pulses.31−35 Meanwhile, these molecular processes have been considered in numerous theoretical investigations. Strictly speaking, an accurate description of these dynamics needs to solve the timedependent Schrödinger equation (TDSE). Plenty of theoretical investigations have introduced Born−Oppenheimer (BO) approximation,36−39 which separates the nuclear motion from electronic motion. These studies with BO approximation have a disadvantage when describing both processes of the dissociation and ionization simultaneously. In order to consider both processes of H2+ in intense fields, Feuerstein et al. employed a reduced-dimensionality model by introduction of a modified “soft-core” Coulomb potential.40 Both the nuclear and electronic motions were restricted to one degree of freedom with non-Born−Oppenheimer coupling. This one-dimensional (1D) theoretical simulation demonstrated that relative probabilities for dissociation and ionization depend on the initial vibrational state of H2+. Subsequently, He et al. used an accurate three-dimensional (3D) time-dependent wave packet approach to discuss the dissociation and ionization of H2+ in an intense field (800 nm, 25 fs, 2 × 1014 W/cm2).41 They found that the ratio of dissociation to ionization was different to the 1D simulation, and the disagreement between the 3D and 1D calculations was due to the nonexact potential curves used in the 1D method. However, the mechanism of CREI is the most common explanation for the origin of ionization of H2+. That is, the ionization is strongly enhanced when the dissociating H2+ moves to the internuclear distances between 5 and 10 au.22,40 This ionization mechanism is an extension of the atomic tunneling picture, whereas the description of dissociation in terms of the dressed states is a multiphoton one.42 Therefore, a new model is needed to describe both the dissociation and ionization processes on equal footing. Esry and Ben-Itzhak first put forward a multiphoton model of above-threshold Coulomb explosion (ATCE) to demonstrate the origin of ionization.43 This novel model well-explained the experimental results of ionization peaks. In addition, the most powerful feature of ATCE is to predict the critical internuclear distances where the ionization occurs, which is a good complement to the mechanism of CREI. The primary aim of this paper is to consider the competitive mechanism between the dissociation and ionization of hydrogen molecular ion at three different intense fields. By analyzing the KER structure of the dissociation channel and the nuclear probability density, we can gain a deeper understanding of the intriguing dissociation mechanisms and their contributions on the total dissociation fragments at the given intensities. Using the model of ATCE, the ionization mechanism is directly connected with the dissociation processes. By analyzing the KER structure of the ionization channel and the space-time

dependent ionization rate, we can reveal which dissociation processes are most important to ionization and where the ionization channel opens at the given intensities. Many of these critical internuclear distances can be interpreted as a manifestation of CREI, leading to enhancements in the KER spectra of the ionization channel.

II. THEORETICAL METHOD We employ the accurate non-Born−Oppenheimer approach to calculate the quantum dynamics of hydrogen molecular ion in an intense laser field. The details of this method are given in previous publications.41,44,45 Here, we summarize it briefly for the current calculations. The time-dependent Schrö dinger equation (TDSE) is (atomic units are used throughout) i

∂ ψ (R, r, t ) = [H0 + W (t )]ψ (R, r, t ) ∂t

(3)

where H0 is the field-free ions Hamiltonian and W(t) describes the interaction of the electrons with the laser field. R is the internuclear vector, and the electron coordinate r is measured from the center of mass of the nuclei. A full solution to eq 3 requires the propagation of a six-dimensional wave function. Unfortunately, this task is beyond our current capability. However, if we consider the alignment that the molecular ion parallels with the electronic field and ignore the rotation of the molecular on the femtosecond timescale, the sixdimensional space is then reduced to three: (R, ρ, z),46−48 where R is the internuclear distance and (ρ, z) are the cylindrical coordinates of the electron. The field-free Hamiltonian in the three-dimensional model can be written as H2+

H0 = TR + Tz + Tρ + Vc(R , ρ , z)

(4)

TR = −

1 ∂2 mp ∂R2

(5)

TR = −

2mp + me ∂ 2 4mpme ∂z 2

(6)

Tρ = −

2mp + me ⎛ ∂ 2 1 ∂ ⎞ ⎜ 2 + ⎟ 4mpme ⎝ ∂ρ ρ ∂ρ ⎠

(7)

Vc(R , ρ , z) =

1 − R −

1 2

ρ + (z − R /2)2 1

ρ2 + (z + R /2)2

(8)

where, mp and me are the masses of proton and electron, respectively. The interaction of electron with the laser field is treated in the dipole approximation (length gauge), ⎛ ⎞ me ⎟z W (t ) = −E(t )⎜⎜1 + 2mp + me ⎟⎠ ⎝

(9)

with E(t) = E0 f(t) cos(ωt), where E0 is the laser peak amplitude, f(t) is the envelope of the pulse duration, and ω is the angular frequency. The laser field is chosen to be linearly polarized along the molecular axis. 9174



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To solve the above TDSE, we combine the sine discrete variable representation49 in the R and z directions and the Crank−Nicholson method50,51 in the ρ direction. In order to economize on computation time, we arrange the time evolution operator to take advantage of the disparity in the timescales of the nuclear and electronic motion, and the resulting operator can be expressed as e−iHδt ≈ e−iTRδt /2{UeSPO(δt /N )}N e−iTRδt /2

(10)

where UeSPO(δt ) = e−iTzδt /2e−iVδt e−iTzδt /2

(11)

represents the electronic part, TR and TZ denote the nuclear and electronic kinetic energy operators, V is the interaction potential including all the potential energy of the system, and N is the ratio between the time steps of nuclear and electronic. Initially, H2+ resides on the 1sσ electronic state of H2+, and the equilibrium distance Re of the nuclei is about 2.0 au. We employ the imaginary time propagation method to obtain the initial wave packet of H2+. The electron may be ionized in an intense field, and the ionization potential is the Coulomb repulsion of two H+ with the function of 1/R. Flux operator and virtual director method49,52 are used for detecting the fragment emission probability and kinetic energy release distributions, respectively. A “binning” procedure53 is necessary to derive the KER spectra across the whole interaction time of the laser pulse. Our numerical grids extend from 0 to 30 au in R, from −45 to 45 au in z and from 0 to 15 au in ρ. The grid points for these three coordinates are 300, 450, and 30, respectively. The time steps in the R direction is 1 au and in z and ρ directions are 0.05 au. Absorption potentials in R, z, and ρ are implemented in order to prevent the reflection of the outgoing wave packets at the border of the grids.

Figure 1. Laser intensity dependence of the dissociation and ionization probabilities for the lowest 10 vibrational states (ν = 0−9). (a) 5.0 × 1013 W/cm2, (b) 1.0 × 1014 W/cm2, and (c) 2.0 × 1014 W/cm2.

does not happen at intensity lower than 1.0 × 1014 W/cm2. For the highest intensity of 2.0 × 1014 W/cm2, shown in Figure 1c, the ionization from the ν = 3−9 states becomes more favorable at the expense of dissociation, and the maximum dissociation probability is down to the ν = 2 state. The dissociation processes can be described by the photondressed states.42 To maintain consistency, the origin of ionization is explained by applying the ATCE model. In the next sections, we will reveal the physical mechanisms of dissociation and ionization at the given intensities and discuss the competition between them on the total KER spectra. B. Mechanism of Dissociation at the Given Intensities. The KER spectra of dissociation channel for vibrational states ν = 3 to 9 are displayed in Figure 2. The left (a−g), middle (h− n) and right (o−u) columns correspond to the intensities of 5.0 × 1013, 1.0 × 1014, and 2.0 × 1014 W/cm2, respectively. For simplicity, the KER spectra are divided into two parts. One is low-energy below 0.6 eV, another is high-energy above 0.6 eV. In accordance with the theory of photon-dressed states, these peaks with high-energy derive from the ATD process, which is caused by the three-photon (net two-photon) absorption at an intense field. However, the situation is more complex for these low-energy peaks, which could be dissociated via the BS, BH, and BTD processes. In consideration of the KER spectra of 5.0 × 1013 W/cm2 (shown in Figure 2, panels a−g), the main feature is that dissociation peaks are mainly from the ν = 5 to 9 states with low energy. As shown in Figure 2 (panels a and b), we cannot observe any peaks on the KER spectra of ν = 3 and 4 states because these states are far away from one-photon crossing and also cannot exit through the three-photon crossing at this low intensity. The schematic diagram of one-photon and threephoton crossings is shown in Figure 3a. In order to further describe the feature of these dissociation processes, we calculate the time evolution of the nuclear probability density. The formula of the nuclear probability density is

III. RESULTS AND DISCUSSIONS We perform calculations for dissociation and ionization on the lowest 10 vibrational states (ν = 0−9) of H2+. The laser pulse carrier wavelength is taken to be 790 nm, and the pulse envelope has a Guassian profile with 40 fs duration. The peak intensities I0 are chosen to be 5.0 × 1013, 1.0 × 1014, and 2.0 × 1014 W/cm2, respectively. A. Competition between Dissociation and Ionization: Intensity and Vibrational State Dependence. The dissociation and ionization probabilities for the lowest 10 vibrational states are shown in Figure 1 (panels a, b, and c), corresponding to the peak intensities of 5.0 × 1013, 1.0 × 1014, and 2.0 × 1014 W/cm2, respectively. We can see that both the dissociation and ionization probabilities are strongly dependent on the laser intensity and initial vibrational state of H2+. For instance, at the lowest intensity of 5.0 × 1013 W/cm2, shown in Figure 1a, there is a considerable dissociation from the ν = 5 to 9 states, and the ionization channel does not open for the lowest 10 vibrational states. When the laser intensity increases, the competition between the dissociation and ionization channels appears on the high states. As shown in Figure 1b, the dissociation probabilities for the ν = 5 to 9 states are reduced at 1.0 × 1014 W/cm2. These reduced populations are converted into the fragments of ionization, and the maximum ionization probability has enhanced to 45% on the ν = 8 state. Whereas for the low states (ν = 3−4), the dissociation probabilities are enhanced enormously, and the competition

P(R , t ) =

+z υd

∫−z ∫0 υd

ρ

|ψ (R , ρ , z , t )|2 ρdρ dz

(12)

where zvd is the border of z grid, on which the absorbing potential begins to act. In the density plot of Figure 4, the dissociation appears as a jetlike structure. This structure 9175



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Figure 2. KER distributions of dissociation channel for each vibrational state. The left, middle, and right column corresponds to the laser intensities of 5.0 × 1013, 1.0 × 1014, and 2.0 × 1014 W/cm2, respectively.

For the ν = 6 to 8 states, it may have more than one dissociation pathway. As shown in Figure 2 (panels d−f), the dissociated KER spectra include the contributions from the BS and BH processes. The BS process shifts the peak to even higher energy with the increasing level, while the BH process produces a constant peak centered around 0.1 eV. The interpretation of BH peak must be elaborated from the adiabatic potentials.17 It is worth noting that, according to ref 29, BH only emerges on the vibrational states higher than ν = 10. For these states ν = 6−8, the BH process is subject to neglect because it is hardly discriminated from BS on the KER spectra. Here, the different features between the BS and BH processes are exhibited clearly in the density plot. As shown in Figure 4c, the 2-fold jetlike structure appears during the dissociation process, corresponding to the different mechanism. As the intensity increases on the leading edge of laser pulse, the gap at the one-photon crossing widens, leading to the BS dissociation. Meanwhile, a fraction of nuclei are trapped in a laser-induced potential well, leading to a counterintuitive stabilization of the molecular bond. These nuclei dissociate on the falling edge due to its inertia, corresponding to the BH process. For the ν = 9 state, it has the maximum energy centered about 0.45 eV in Figure 2g. The dissociation mechanism for ν = 9 is the BS process. As shown in Figure 3a, ν = 9 state is located very close to the one-photon crossing, so it is subject to absorb one photon and directly dissociate to the |2pσ − 1ω⟩ state. For the BS process, the dissociation happens on the leading edge of laser pulse, corresponding to the jetlike structure shown in Figure 4d. At the higher intensity of 1.0 × 1014 W/cm2 (shown in Figure 2, h−n), KER spectra have significant changes. First, high-energy peaks are present on the ν = 3 to 5 states, which

Figure 3. (a) The dressed potentials for electronic 1sσ and 2pσ states of H2+ ions. Each state is dressed in energy by an integer number of photons. (b) The same as (a) but including the ionization potentials.

corresponds to the distinct peaks on the KER spectra. For the ν = 3 state, we cannot observe any jetlike structure in Figure 4a. For the ν = 5 state, there is a distinct peak centered around 0.28 eV in Figure 2c. In accordance with the energy of the onephoton dissociation limit, this state is very hard to dissociate via the BS. Also, it is unlikely to dissociate via the ATD at 5.0 × 1013 W/cm2. The only way left is the BTD process. It can be observed from the density plot of Figure 4b, the jetlike structure occurs on the falling edge of the laser pulse, corresponding to the BTD process. Here, we emphasize that the released energy via the BTD is strongly dependent on the duration of laser pulse. The BTD peak will shift toward higher energy when the pulse length becomes shorter. This pulselength effect has been confirmed by experiment.17 9176



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Figure 4. Nuclear probability density for the vibrational states ν = 3, 5, 7, and 9. The left, middle, and right column corresponds to the laser intensities of 5.0 × 1013, 1.0 × 1014, and 2.0 × 1014 W/cm2, respectively. The dash line corresponds to the peak intensity of the laser pulse.

can be seen in Figure 2 (panels h, i, and j). These peaks correspond to the ATD process. The highest peak of them appears at ν = 3 with the energy of 1.1 eV. Because this level is close to the three-photon crossing shown in Figure 3a, it will preferentially dissociate via ATD. From the density plot of Figure 4e, we can see that the jetlike structure appears around the peak intensity. It means that dissociation via ATD becomes favorable at the peak intensity. Second, the BTD peak of ν = 5 disappears in Figure 2j. Comparing the density plot in Figure 4 (panel f with panel b), we can find that almost all the dissociated nuclei flow back to initial bound state around the peak intensity of 1.0 × 1014 W/cm2. Adding to the pulse-length effect, we can conclude that the released energy via the BTD is strongly dependent on the pulse duration, and its population is dependent on laser intensity. In addition, according to Figure 2 (panels k−n), the dissociation peaks from the ν = 6−9 states decay obviously on the KER spectra. From the density plots of Figure 4 (panels g and h), we can see that these jetlike structures decay around the peak intensity. It is due to the subsequent ionization following the BS and BH processes. For the highest intensity of 2.0 × 1014 W/cm2 (shown in Figure 2, panels o−u), the dramatic change on the KER spectra is that the ATD peaks from the ν = 3−5 states drop remarkably in Figure 2 (panels o, p, and q). Additionally, the BS and BH peaks from the ν = 6−9 states almost disappear in Figure 2 (panels r−u). It coincides with the jetlike structure shown in

the density plot of Figure 4 (panels i−l), which almost disappears at the peak intensity. It indicates that the dissociation processes of BS, BH, and BTD can be neglected as the laser intensity higher than 2.0 × 1014 W/cm2. In the next section, we will elaborate how these dissociating H2+ converts into the subsequent ionization. C. Mechanism of Ionization at the Given Intensities. The KER spectra of ionization channel for vibrational states ν = 3 to 9 are displayed in Figure 5. The left (Figure 5, panels a−g) and right (Figure 5, panels h−n) columns correspond to the intensities of 1.0 × 1014 and 2.0 × 1014 W/cm2, respectively. The situation of 5.0 × 1013 W/cm2 is not shown since it is hard to ionize at this low intensity. As shown in Figure 5 (panels a− g), the ionization peaks mainly come from the ν = 5 to 9 states at 1.0 × 1014 W/cm2, and these peaks shift from 5.2 to 3.1 eV with the increasing level. As the intensity increases to 2.0 × 1014 W/cm2, shown in Figure 5 (panels h−n), the ionization peaks enhance enormously on each state. In the following, we will apply the model of ATCE43 to explain the origin of these ionization peaks. In H2+ ions, shown in Figure 3a, the electronic 1sσ and 2pσ states are “dressed” in energy by an integer number of photons absorbed or emitted (i.e., |1sσ − nω⟩ or |2pσ − nω⟩). As we discussed above, high levels can dissociate from the initial |1sσ − 0ω⟩ state via the one-photon crossing to the |2pσ − 1ω⟩ state, leading to the BS and BH processes. Low levels near 9177



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Figure 6. Space-time dependent ionization rate for the vibrational states ν = 3, 5, 7, and 9. The left and right column corresponds to the laser intensities of 1.0 × 1014 and 2.0 × 1014 W/cm2, respectively.

Figure 5. KER distributions of ionization channel for each vibrational state. The left and right column corresponds to the laser intensities of 1.0 × 1014 and 2.0 × 1014 W/cm2, respectively.

dissociation and ionization from the KER spectra. For the ν = 3 state, they center at 0.8 and 4.5 eV, respectively. In accordance with eq 13, the calculated Rc equals 7.4 au, corresponding to the 13ω crossing of ATD completely. As shown in Figure 6e, the calculated Rc is consistent with the internuclear distance where the maximum of ionization rate appears. For the ν = 5 state, the dissociating H2+ are trapped in the laser-induced potential well at 1.0 × 1014 W/cm2, so it preferentially ionizes via the BH pathway. In accordance with the ionization rate shown in Figure 6b, the ionization peak centered at 5.3 eV is associated with the 13ω crossing of BH. As the intensity increases to 2.0 × 1014 W/cm2, a fraction of nuclei transfers into the ATD pathway. Hence, another ionization peak arises around 4.5 eV, which is initiated by the 13ω crossing of ATD. It can be clearly observed in Figure 6f, the ionization channel opens around two important regions of 4.8 and 7.4 au, corresponding to the 13ω crossings of BH and ATD, respectively. It is also noted that the contribution from the pathway of BH is stronger than ATD, since most of the nuclei are trapped in the bound state of ν = 5. For the higher states (ν = 6, 7, and 8), the dissociating H2+ moves along the BS and BH pathways. At 1.0 × 1014 W/cm2, the ionization peak is associated with the 12ω crossing of BS, and the ionization pathway via the BH does not open. As shown in Figure 6c, the maximum of ionization rate locates at R = 7.3 au, which is in accordance with the 12ω crossing of BS. At 2.0 × 1014 W/cm2, an additional ionization pathway opens around R = 4.1 au in Figure 6g, corresponding to the 14ω crossing of BH. But the contribution from the 14ω crossing is very small, since it requires the larger number of photons compared with the 12ω crossing of BS. For the ν = 9 state, the dissociating H2+ has a single ionization pathway: BS. At 1.0 × 1014 W/cm2, the ionization peak centered at 3.2 eV is initiated by the 12ω crossing of BS,

three-photon crossing will preferentially dissociate to |1sσ − 2ω⟩ state via ATD process. In Figure 3b, states representing ionization are intruded, which are exactly analogous to the dissociation states. These ionization states, 1/R − nω, cut across the dissociation states. It is clear from Figure 3b that critical values of R occur where the ionization curves cross the dissociation curves. These crossings, where the ionization channels open, will correlate to enhancements in the KER spectrum of the ionization channel. Due to the complex network of crossing in Figure 3b, there are a number of different ionization pathways. For simplicity, we categorize these pathways into three forms: (i) BS (ionization from |2pσ − 1ω⟩ state), (ii) BH (ionization from |1sσ − 0ω⟩ state), and (iii) ATD (ionization from |1sσ − 2ω⟩ state). The critical values of Rc can be deduced from the simple formula: 1 Rc = E ion(ν) − Ediss(ν) (13) where the terms of Ediss(ν) and Eion(ν) describe the release energies from the dissociation and ionization channels, shown in Figures 2 and 5, respectively. Next, we will reveal which ionization pathways and curve crossings are most important to each vibrational state at the given intensities. To confirm the inference, we plot the space-time dependent ionization rate of v = 3, 5, 7, and 9 states at 1.0 × 1014 W/cm2 (Figure 6, panels a− d) and 2.0 × 1014 W/cm2 (Figure 6, panels e−h). For the ν = 3 and 4 states, the dissociating H2+ moves along the ATD pathway. At 1.0 × 1014 W/cm2, we cannot observe any ionization peak on these two states since the ionization channel does not open in Figure 6a. At 2.0 × 1014 W/cm2, the ionization peak appears around 4.5 eV. To determine which crossing is most important, we extract the released energies of 9178



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dissociation process, and the ionization channel does not open at this low intensity. At 1.0 × 1014 W/cm2 (shown in Figure 7b), the low-energy peak descends and the high-energy peak ascends obviously. It means that the mechanism of ATD become important to the dissociation channel with the increasing intensity. These reduced low-energy fragments are converted into the subsequent ionization following the BS and BH pathways. Consequently, the structure of ionization KER spectrum consists of two peaks. According to the discussion in Mechanism of Ionization at the Given Intensities, the left peak centered at 3.6 eV comes from the ionization of the BS pathway crossed by 12ω and the right peak located at 5.2 eV mainly derives from the BH pathway crossed by 13ω. For the highest intensity of 2.0 × 1014 W/cm2 (shown in Figure 7c), the branching ratio between the dissociation and ionization channels has the dramatic change. The proportion of dissociation is reduced remarkably on the KER spectrum, especially for the low-energy peak. On the contrary, the proportion of ionization is enhanced enormously. The left ionization peak located at 4.1 eV is initiated by the BS and ATD pathways, and the right ionization peak centered about 5.6 eV is mainly from the BH pathway.

corresponding to the maximum ionization rate shown in Figure 6d. The situation has changed with the increasing intensity. At 2.0 × 1014 W/cm2, the contribution from the 11ω crossing of BS is dominant during the ionization process. As shown in Figure 6h, the maximum ionization rate reaches to R = 11.9 au, which is very close to the 11ω crossing of BS. It is due to the BS dissociation barrier being lower and the nuclei moving faster at a higher intensity. Moreover, the curve crossing is widened with the increasing intensity, allowing ionization to occur over a larger range of R. In a comparison of Figure 6 (panel h with panel g), it is surprising that the maximum ionization rate on the ν = 9 is less than ν = 7. It may be that the transition dipole moment becomes smaller at the asymptotic region. D. Competition between the Dissociation and Ionization on the Total KER Spectra. Dissociation and ionization from a single vibrational state, however, is not a situation currently available, since such molecular target is difficult to prepare experimentally. If H2+ is produced with some fast process, such as impact ionization of the neutral molecule, the initial distribution can be well-approximated with a Franck−Condon (FC) distribution.28,29 For this reason, we assume the initial states are satisfied with FC distribution and sum the corresponding KER spectra of the lowest 10 vibrational states (ν = 0−9) with FC weights. The total KER spectra of the dissociation and ionization channels are shown in Figure 7. The total dissociation KER spectra can be divided into

IV. CONCLUSION AND OUTLOOK Using the accurate non-Born−Oppenheimer approach, we have carried out the theoretical study of the competition between the dissociation and ionization of H2+ in the intense field (790 nm, 40 fs, 0.5−2.0 × 1014 W/cm2). At 5.0 × 1013 W/cm2, dissociation channel is predominant for the states ν = 5−9, and the ionization channel does not open for the lowest 10 states. The mechanism of dissociation includes the contributions from the BS, BH, and BTD processes. As the intensity increases to 1.0 × 1014 W/cm2, the contribution from ATD process enhances, while the BS, BH, and BTD processes decay on the dissociation channel. These reduced populations of dissociation convert into the ionization, which is mainly associated with the 12ω crossing of BS and 13ω crossing of BH. For the highest intensity of 2.0 × 1014 W/cm2, the proportion of dissociation continues to decline, especially for the BS and BH processes. In contrast, the proportion of ionization enhances enormously. The origin of the enhanced ionization is ascribed to the ionization of the BS, BH, and ATD pathways. In view of the possible applications for quantum control, this work has primary importance, since it will improve the efficient population transfer between the dissociation and ionization. For instance, the branching ratio between the dissociation and ionization could be controlled by regulating the intensity of the laser pulse. Furthermore, another intense field could be added to control the transition between dissociation and ionization when the dissociating H2+ moves to a particular crossing. Of course, we hope that the present results will encourage future experimental works.

Figure 7. Total KER spectra of the dissociation and ionization channels after FC weights of each vibrational state. (a) 5.0 × 1013; (b) 1.0 × 1014; and (c) 2.0 × 1014 W/cm2.

two parts: low energy (0−0.6 eV) and high energy (0.6−2 eV). These fragments with low-energy include the contributions from the BS, BH and BTD processes and the released highenergy origins from the ATD process. For the ionization channel, the KER distributions are larger than 2 eV. At 5.0 × 1013 W/cm2 (shown in Figure 7a), the released energy of fragments mainly distributes on the region of low energy. In accordancde with the discussion in Mechanism of Dissociation at the Given Intensities, we can deduce that the mechanisms of BS, BH, and BTD are predominant during the

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The authors declare no competing financial interest. 9179



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ACKNOWLEDGMENTS This work was supported by Scientific Research Program of the Higher Education Institution of Xinjiang (Grant XJEDU2012S41) and Youth Science and Technology Innovation Talents Project of Xinjiang (Grant 2013731008). G.J.Z. thanks the NSFC (Grants 20903094 and 20833008) and NKBRSF (Grants 2009CB220010 and 2013CB834604) for financial support. G.J.Z. also thanks the financial support from CAS Youth Innovation Promotion Association, Hanse-Wissenschaftskolleg (Institute for Advanced Study) Fellowship, University of Bremen Fellowship, and the Frontier Science Project of the Knowledge Innovation Program of CAS.

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