Laser Tunnel Ionization from Multiple Orbitals in HCl

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Science  11 Sep 2009:
Vol. 325, Issue 5946, pp. 1364-1367
DOI: 10.1126/science.1175253


Tunneling, one of the most striking manifestations of quantum mechanics, influences the electronic structure of many molecules and solids and is responsible for radioactive decay. Much of the interaction of intense light pulses with matter commences with electrons tunneling from atoms or molecules to the continuum. Until recently, this starting point was assumed to be the highest occupied orbital of a given system. We have now observed tunneling from a lower-lying state in hydrogen chloride (HCl). Analyzing two independent experimental observables allowed us to isolate (via fragment ions), identify (via molecular frame photoelectron angular distributions), and, with the help of ab initio simulations, quantify the contribution of lower-lying orbitals to the total and angle-dependent tunneling current of the molecule. Our results bolster the emerging tenet that the coherent interaction between different orbitals—which can amplify the impact of lower orbitals—must be considered in tunneling processes.

When a particle penetrates a barrier without having the necessary energy, it is said to have tunneled. As a fundamental quantum phenomenon that challenges classical intuition, tunneling has been a testing ground for our understanding of quantum mechanics (1). Simultaneously, tunneling is having a tremendous impact on technology through applications such as the scanning tunneling microscope (STM) (2). The exponential dependence of the tunneling current on the distance between a surface and an STM tip allows for imaging of atomic-scale structure of the surface. By analogy, tunneling of an electron from a molecule by an intense infrared laser pulse can image the electronic structure of single molecules (3). In such a “molecular STM,” the exponential decay of the wave function in the barrier means that small variations in the ionization potential have a large influence on the ionization probability. Therefore, in atoms and molecules that are subjected to intense laser pulses, the highest electronic state dominates the tunneling current exponentially.

Recently, there has been mounting evidence to suggest that lower-lying levels contribute to the total tunneling current (411). Most evidence arises from structure in the high harmonic spectrum (611). However, because high harmonic radiation is linked to tunneling only through the intermediate steps of continuum propagation and recombination, the interpretation of high harmonic spectra requires sophisticated theories, making quantitative conclusions difficult (6, 12, 13). Here, by applying our molecular STM technique to the HCl molecule, we observed a 0.2% contribution to the total tunneling current from an orbital with a 30% greater ionization potential than the highest occupied orbital (I0 ~ 1.4 × 1014 W/cm2). According to our calculations, such a lower-lying orbital can contribute in excess of 10% to the tunneling probability along the molecular axis under otherwise identical conditions. Thus, our measurements support the finding of McFarland et al. (10) that tunneling from a lower-lying orbital can dominate the high harmonic spectrum for certain molecular alignments. Our results further imply that attosecond hole dynamics resulting from tunneling will be ubiquitous in atoms and molecules (11) as well as in solids (14).

We chose HCl as an exemplary system because it offers an opportunity to isolate the tunneling current contribution of the highest occupied molecular orbital (HOMO) from that of the next-highest-energy orbital (HOMO-1) (inset, Fig. 1A). The HOMO of HCl is a lone-pair 3p orbital of the Cl atom aligned perpendicular to the molecular axis, whereas the HOMO-1 accounts for the molecular bond and is formed by the overlap of the 1s orbital of the hydrogen atom and a chlorine 3p orbital along the molecular axis. Tunneling from the HOMO-1 weakens the bond and, in the laser field, leads to fragmentation of the molecular ion. Hence, the breakup of an HCl molecule after ionization is a direct signature for tunneling from a lower-lying orbital.

Fig. 1

(A) Schematic diagram for the tunnel ionization and subsequent bond-softening processes of HCl in an intense near-infrared laser field. Data for the potential energy curves were obtained from (24) and (32) for ion and neutral states, respectively. The ground state of singly charged HCl+ ion (X2Π), produced by ionization from the HOMO of the HCl molecule, does not dissociate in the laser field. In contrast, the first electronic excited state of the HCl+ ion (A2Σ+), produced by ionization from the HOMO-1, dissociates through the bond-softening process by coupling with the (2)2Σ+ state. (B) Sketch of electron and fragment ion momenta in the plane of the circularly polarized laser pulse in the present experiment.

Figure 1A illustrates this point using the potential energy surfaces of HCl and HCl+. Here, we consider only ionization to the ground X2Π and first excited A2Σ+ states of the HCl+ ion, as the second excited state of HCl+ requires an additional five photons (i.e., ~7.75 eV). Whereas the ground (X) state is produced by tunnel ionization from the HOMO, the A state has its origin in the HOMO-1. The A state couples strongly with the dissociative (2)2Σ+ state, allowing HCl+ fragmentation through bond softening (15). The X state is produced in a transition perpendicular to the molecular axis and does not possess such a direct pathway to dissociation. However, as indicated in Fig. 1A, there is a weak secondary pathway to bond softening (see below).

By measuring the fragment ion and the tunneled electron in coincidence, we determine the molecular frame photoelectron angular distribution (MFPAD) that identifies the HOMO-1 of HCl. The concept, introduced in (16), is illustrated in Fig. 1B (17). In a cold target recoil ion momentum spectrometer (COLTRIMS) (18) an intense, circularly polarized laser pulse singly ionizes and subsequently dissociates an unaligned molecule. The ejected electron drifts perpendicular to the electric field direction at the moment of ionization (19); the fragment ion trajectory reflects the orientation of the molecule.

To ensure that only a single electron had tunneled from the molecule and that the fragmentation occurred rapidly enough to preserve the molecular orientation, we needed to establish HCl+ bond softening as the origin of the detected H+ and Cl+ ions. We ruled out a dissociative channel of HCl2+ as the origin of the measured fragments because we did not observe a correlation between the H+ and Cl+ momenta. Moreover, the measured kinetic energy spectrum of H+ and Cl+ exhibited discrete peaks. Bond softening often produces multiple peaks in the kinetic energy spectrum of the ion fragments according to the absorbed number of photons (15). Finally, we found that H+ and Cl+ ions are emitted within ±30° to ±15° of the polarization plane in the intensity range of 1 × 1014 to 2 × 1014 W/cm2. A small angular spread is further characteristic of bond softening (15).

In Fig. 2, A and B, the kinetic energy spectra of Cl+ and H+ are shown for two intensities: 1 × 1014 W/cm2 (dashed line) and 2 × 1014 W/cm2 (solid line). The energies of vibrational levels corresponding to the different dissociation pathways are superimposed as vertical lines. Chlorine ions are produced only with a single characteristic kinetic energy in the studied intensity range. As suggested by the vertical lines, the Cl+ energy is consistent with the net-two-photon bond-softening channel, as the dissociation limit of the H + Cl+(1D) channel is 2.58 eV above the vibrational ground state of HCl+ (A2Σ+, v = 0). Hence, the energy of a laser photon with 1.55 eV is insufficient for net-one-photon dissociation in the H + Cl+ channel. The proton distribution also has a single peak at an intensity of 1 × 1014 W/cm2. However, at higher intensity, the hydrogen background in our chamber contributes to the signal. The peak at 100 meV is well known from H2+ bond softening. Background hydrogen also contributes to the peak at 500 meV. Therefore, in our analysis of protons produced via the H+ + Cl channel, we only include the net-two-photon peak located near 1.2 eV.

Fig. 2

(A) Cl+ kinetic energy at 1 × 1014 W/cm2 (dashed line) and 2 × 1014 W/cm2 (solid line). (B) Same as (A) for protons. (C) Intensity dependence of the ratio of total yields of the molecular ion HCl+ and the dissociation fragments H+ and Cl+. The error bars in the intensity direction correspond to ±1σ determined by a least-squares fit of electron momentum, whereas the bars in the yield ratio direction indicate ±20% error (17). Solid line: wHOMO-1/wHOMO relative atomic tunneling probabilities for orbitals with the corresponding ionization potential. Star: wHOMO-1/wHOMO predicted by our molecular DFT calculation. Dashed line: wX-A/wHOMO relative probability of bond softening from the X state.

Figure 2C shows the laser intensity dependence of the experimentally determined dissociation probability wdiss=NH+,C35l+/NHC35l+(1) whereNH+,C35l+=[NH+NHC35l+/ (NHC35l++NHC37l+) ]+NC35l+(2)takes the Cl isotope ratio into account for the number of detected protons. The probability to detect dissociated HCl molecules is 9 (±2) × 10−4 at the lowest intensity of 1 × 1014 W/cm2, and increases to 5 (±1) × 10−3 at the highest intensity experienced by the neutral molecule, 2 × 1014 W/cm2.

Using atomic tunneling theory (20), we estimate the ionization probabilities wHOMO and wHOMO-1 for the respective equilibrium ionization energies of HOMO (12.747 eV) and HOMO-1 (16.265 eV) (21). The calculated ratio wHOMO-1/wHOMO is shown as the solid curve in Fig. 2C. Within experimental error, the observed ratio coincides with the theory. However, in making this comparison we have assumed that bond softening of the A state is 100% efficient and that the laser field does not alter the excited-state population ratios created by tunneling. For comparison, bond softening in H2+ dissociates up to 50% of all molecules at similar intensities. Hence, we might expect that the experimentally presented ratio systematically underestimates the total tunneling probability from the lower-lying orbital HOMO-1 by at least a factor of 2. In addition, tunneling theory does not account for other shake-up processes that will contribute (4). The ratio determined by a density functional theory (DFT) calculation, described below, is shown as the red star.

We now examine the tunneling probability in the molecular frame. Because tunneling projects a filtered version of the ionizing momentum wave function of the orbital into the continuum (3, 20), the angle-dependent tunneling probability can uniquely differentiate the very different structures of the HOMO and HOMO-1. Figure 3 shows the electron momentum space and the corresponding radially integrated angular distributions when we select electrons that are correlated with either Cl+ (Fig. 3, A and C) or H+ (Fig. 3, B and D) at an intensity of 2 × 1014 W/cm2. Both fragmentation channels yield almost identical molecular frame photoelectron angular distributions. Thus, the electron distributions unambiguously identify the HOMO-1 [Fig. 3F, calculated by the full valence CASSCF procedure using the TZV basis sets with polarization functions (22)] as the origin of the fragmentation channel.

Fig. 3

(A and B) Electron momentum distribution in the molecular frame for the Cl+ and the H+ channels, respectively. (C and D) MFPADs derived from (A) and (B). Molecules are subjected to a circularly polarized 800-nm laser pulse for 50 fs at a peak intensity of 2 × 1014 W/cm2. (E and F) Two-dimensional representation of the HOMO and HOMO-1 of HCl.

To quantify the contribution of tunneling from HOMO-1, we compared our experiment to a DFT (23) analysis of angular tunneling ionization probability for both HOMO and HOMO-1 in a static field corresponding to an intensity of 1.4 × 1014 W/cm2. Our calculations generally followed the method of (23). We found (Fig. 4A) that the theoretical angle-dependent ionization probability follows the basic structure of the orbital as expected. A second important observation from both theory and experiment is that the ionization probability along the molecular axis (θ = 0°, 180°) is asymmetric, although this is not obvious in Fig. 4A because of the dominating asymmetry in the ionization rates parallel to and perpendicular to the molecular axis. For tunneling from the HOMO, our calculation predicts that the probability to ionize from the proton end of the molecule [wHOMO(θ = 0°)] is 1.14 times the probability from the chlorine end of the molecule [wHOMO(θ = 180°)]. For the HOMO-1 this ratio rises to a factor of 1.56, in excellent agreement with the experimental ratio of 1.6 obtained from Fig. 3, C and D, as a ratio of electron yields ejected from the proton end (θ = 0° ± 18°) and from the chlorine end (θ = 180° ± 18°).

Fig. 4

(A) DFT calculated angular ionization probability of HOMO [wHOMO(θ), dashed line] and HOMO-1 [wHOMO-1(θ), solid line] at an intensity of 1.4 × 1014 W/cm2. (B) Ratio wHOMO-1/wHOMO from (A). (C) Comparison of experiment and theory for the HCl angular ionization probability. The experimental curves are normalized to θ = 0°. Solid line, calculated MFPAD of HOMO-1 convoluted with the experimental angular uncertainty (±30°); dotted line, same as solid line but including 0.1% of the HOMO angular distribution.

The calculated angle-resolved ratio of wHOMO-1/wHOMO (Fig. 4B) suggests that the HOMO-1 contribution can exceed 10% for aligned molecules. Angular integration yields a ratio of wHOMO-1/wHOMO = 0.6%. However, for comparison with our measurement, an alignment factor must be taken into account (17), decreasing the ratio to 0.3% (red star in Fig. 2C), in good agreement with the experimental data.

In Fig. 4C the angular tunneling probabilities of the HOMO-1 obtained from experiment and simulation are compared, the theory (solid line) being convoluted with the experimental angular uncertainty (±30°). The measured distributions for all intensities are included. Notably, the experimental data are not intensity dependent. However, there is one major discrepancy with theory: The minimum at 90° is less pronounced in the experiment, and it cannot be accounted for by the experimental resolution alone. We attribute this effect to a contribution from the perpendicular coupling between the ion’s ground state and its first excited state through a perpendicular transition (as indicated in Fig. 1A). Using the X-A transition moment matrix theoretically obtained by Pradhan et al. (24), we estimate the X-A transition fraction of the ion’s ground state wX-A/wHOMO to be 0.1% at an intensity of 1.4 × 1014 W/cm2, including the geometrical weight factor of ½ (17). The corrected angular distribution (dotted line in Fig. 4C) is in nearly perfect agreement with the experiment.

This experimental asymmetry may be amenable to a qualitative explanation, as follows: The DFT calculation suggests that the main mechanism responsible for the difference in the ionization rate when the field points toward the hydrogen (θ = 0°) versus the chlorine (θ = 180°) atom is the angle-dependent Stark shift of the neutral molecule (16). The ground state of the molecule undergoes a Stark shift because of both the permanent and induced dipole moments. In the tunneling limit, the Stark shift adds directly to the ionization potential for the molecule. The ground-state shift is given byΔVg=(μpg+12μig)E(3)where the induced dipole moment is μig = αE, determined by the polarizability tensor α and the laser field E, and μpg is the permanent dipole moment of the ground state of the neutral molecule.

A Stark shift of the ion also changes the ionization potential. However, the change in the ionization potential created by the induced dipole in the ion is almost exactly compensated by the modification to the tunneling barrier (16, 25), whereas the ion Stark shift created by the permanent dipole is canceled to ~50% by the modification to the tunneling barrier that it creates. Therefore, the influence of the ion’s Stark shift is muted; the neutral molecule plays the dominant role.

On the basis of these considerations, the effective ionization potential IPeff is simply given by IPeff(E)=IP0(μpg12μpi+12μig)E(4)where μpi is the permanent dipole moment of the A state of the ion. Inserting IPeff into the common quasi-static atomic tunneling equations for linearly polarized light [e.g., equation 7 in (20)], we find that the electron is 1.4 times as likely to be ejected when the field points toward the H atom as when the field points toward the Cl atom. This is in qualitative agreement with the experimental asymmetry of 1:1.6.

Our results have broader implications. First, the difference in ionization potential between the HOMO and lower-lying states in larger molecules is generally smaller than the 4 eV of HCl. Therefore, tunneling from two or more orbitals should occur frequently. Depending on how the experiment is performed, these channels can interfere. In our experiment, by measuring the ion, we can distinguish the orbital from which the electron emerged. Like a two-slit experiment where we measure the slit through which the electron passes, there can be no interference [except for a post-tunneling shake-up process (4) or the weak contribution through the X-A coupling]. However, in many experiments the tunneling orbital is not distinguishable. Electron wave packets must escape through the tunnel within a very restricted time window near each field maximum. Their coherent superposition will influence the angle-dependent rates. This kind of interference may be responsible for the very sharply peaked tunneling distribution found for CO2 (26).

Second, ionization creates an electron and a correlated ion (27). Simultaneous tunneling from two or more orbitals locks the phase of the ion’s states (11). Hence, a correlated bound electron wave packet is created in the ion that can be probed by the tunneled electron through recollision. This process is the analog to the creation of vibrational wave packets in molecules by tunneling (28, 29). Whereas vibrational wave packets typically evolve on the femtosecond scale, bound electron wave packets exhibit true attosecond dynamics and are expected to be a ubiquitous feature of laser-driven tunneling in both molecular gases and solids (14).

Third, we have shown that tunneling selectively occurs with a specific orientation within a homogeneous sample. The only requirement for probing orbitals (and orbital dynamics) in two-color experiments (30, 31) is that ionization labels the orbital. Our experiment shows that, in principle, molecules could be selected on the basis of their dipole moment.

Supporting Online Material

Materials and Methods

Figs. S1 to S3


References and Notes

  1. See supporting material on Science Online.
  2. Supported by the U.S. Air Force Office of Scientific Research, the Alexander von Humboldt Foundation, the Deutsche Forschungsgemeinschaft, the Canadian Institute for Photonics Innovation, and the Natural Sciences and Engineering Research Council of Canada. We thank B. Avery for technical assistance and M. Zimmer for help in preparing some of the graphics.
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