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proach

The ionization of atoms in strong laser fields has long been a subject of investigation for theorists [100, 101, 102, 103, 104, 105]. Classical approaches to calculate the ionization rate of atoms are the tunneling ionization model of Ammosov-Delone-Krainov (ADK) [102] and the strong-field approximation (SFA) [106]. Their extensions to the molecular case intro- ducing molecular orbitals (MOs) lead to the MO-SFA [107] and MO-ADK [67, 108] models that could describe the ionization rate of small linear molecules. Both approaches provide orientation-dependent ionization rates for ionization from a given molecular orbital, e.g. the HOMO, within the one-center approximation. It is important to have a method capa- ble to calculate the angle-dependent ionization rates of larger molecules. In general, such molecules are not necessarily symmetric, so that one-center approximation is no longer valid. In addition, smaller energetic spacing between outer molecular orbitals might in- troduce a multi-orbital contribution to the angle-dependent ionization probability. Recent strong-field experiments provide evidence of ionization from energetically close-lying va- lence orbitals, so that ionization occurs not only from the HOMO but also from lower-lying orbitals, e.g. the HOMO-1 [24, 109, 110, 111].

This section outlines an approach to calculate angle-dependent ionization probabilities that is applicable to complex molecular systems. It is based on electronic structure calcu- lations and supports ionization from more than a single MO. A detailed description of the method can be found in Refs. [58, 97], where it was in particular shown that for diatomic heteronuclear molecules it is important to go beyond the one-center approximation in or- der to achieve good agreement with experimental data. In this context, the ability of the method was demonstrated comparing the theoretical results with MO-ADK calculations and with experimentally measured data for angle-dependent ionization probabilities of the

diatomic molecules D2, N2, O2, and CO. In the experiments, the angular distribution of ionic fragments from the ionization of molecules in strong (about 1014 W cm−2) and sufficiently short laser pulses (here 4–5 fs) reflects the angle-dependent ionization prob- abilities [58]. In this case RCE and post ionization alignment of the molecules do not significantly change the angular distributions of detected fragments [112].

The theoretical calculations are performed as follows. The orientation-dependent ion- ization rate w(θ, t) of a molecule in an intense laser field can be calculated as the induced electron flux through the barrier of the combined molecular and external electric fields [101]:

ω(θ, t) =−d dt Z V ρ(~r, θ, t)dV (4.1) where ρ(~r, θ, t) = P n6nHOM O

|ψn(~r, θ, t)|2 is the electron density and ψn(~r, θ, t) are the MOs

of the molecule in the presence of the external field. The angle θ is the angle between the intramolecular axis and the external field. Within the axial recoil approximation this angle corresponds to the angle θ in the experimental images, under which the fragments formed by the dissociation of the molecule are observed. The MOs are calculated in the adiabatic approximation using a static electric field with the quantum chemistry package Molpro [92] and are represented on a three dimensional grid.

In order to evaluate the tunneling probability numerically the electronic wave function with and without the external field are necessary. In this sense Eq. 4.1 was integrated from the initial time ti corresponding to a time where the electric field is zero to the final time

t, where the field and therefore the ionization rate is maximal:

T(t, θ;S) = Z V ρ(~r, θ, ti)dV − Z V ρ(~r, θ, t)dV. (4.2)

The integral in Eq. 4.2 is taken over the volumeV which is confined by the surface S. For the surfaceSit is convenient to choose a plane perpendicular to the direction of the electric field. As the ionization takes place only at the edges of the electronic wave function, it was chosen to follow the basic idea of MO-ADK [67] to calculate the induced electron flux, so the surface S is spanned by all the outer turning points ψn(~r, θ, t). Beyond these points

the exponentially decaying term in the electron wave function becomes dominant and the wave function enters the classical forbidden region relevant for the tunneling process. The surface S can be determined on the three dimensional grid by numerical evaluation of the

4.1 Attosecond control of electron dynamics in carbon monoxide 45

Figure 4.4: Angle-dependent ionization probability for (a) the HOMO (3σg) of N2 and (b) linear combination of HOMOs (1π_x∗ and 1π_y∗) of O2. Red crosses connected by the red lines indicate results of calculations by the method described in the text. Obtained values are compared directly to ionization probabilities measured in the experiment (blue lines) and calculated with the MO-ADK model (green lines). All lines were normalized by their maximum values. The laser polarization is vertical. Adapted from [58].

first and second derivative ofψn(~r, θ, ti) starting from maximal ~r values.

In order to make a direct comparison with an experiment where molecules were not aligned or oriented, the calculations were performed for series of orientations from 0◦ to 360◦ with steps of 10◦. Figs. 4.4a and b illustrate a comparison of experimentally measured angle-dependent ionization rates (blue lines) for N2 and O2 molecules and angle-dependent ionization probabilities of these molecules calculated with the method introduced above (red lines) and with MO-ADK theory (green lines). The laser polarization is vertical. Ex- perimental N+ and O+ momentum distributions were obtained by dissociative ionization of N2 and O2 molecules in the presence of 5 fs laser pulses at the central wavelength of 730 nm, 1kHz repetition rate and intensity of 1.6×1014 W cm−2 with a VMI spectrome- ter. Resulting ion distributions were integrated over the momentum ranges of (1.2–2.2) × 10−22 N·s for N₂ and (1.1–2.2)× 10−22 N·s for O₂ thereby excluding the background signal from doubly ionized parent ions. Detected N+ and O+ fragments and their doubly ion- ized parent ions N2+₂ and O2+₂ have the same mass-to-charge ratio and thus the same TOF. These species can not, therefore, be separated from each other with the VMI spectrometer. In the case of N2, the electric field strength was set to 0.067 a.u., equivalent to the experimental peak intensity of 1.6× 1014 W cm−2. Hartree Fock and MCSCF calcula-

tions for N2 carried out with basis sets larger than STO-3G predict the wrong energetic ordering of the valence orbitals. The HOMO of the N2 molecule is the 3σg orbital (not

the 1πu), so that the ground state of N+2 has X2Σ+g symmetry. To overcome the wrong

energetic ordering the HF/STO-3G level of theory was used to calculate correct ioniza- tion probabilities for N2. This calculated angular distribution looks very similar to the experimental one. The parallel vs. perpendicular ionization ratio is about 3.5 from the calculations (see red line in Fig. 4.4a) and about 4.5 for the experimental data (blue line in Fig. 4.4a) in excellent agreement with the published value of 4.5 [113]. The shape of the angle-dependent ionization probabilities obtained from MO-ADK theory (green line) also follows the experimental data. However, the predicted parallel vs. perpendicular ratio of 10.5 doesn’t fit the experimental value well.

The electronic wave functions for the O2molecule are calculated on the CASSCF(12,10)/ 6-311+G* level of theory and an electric field strength of 0.067 a.u. Molecular oxygen is a stable diradical. It has two unpaired electrons occupying two degenerate 1π∗_x and 1π∗_y

molecular orbitals in its electronic ground state. The degeneracy of the two orbitals is not broken in the presence of the electric field and ionization from both orbitals can occur. Taking into consideration the fact that tunneling from both orbitals is allowed with the same probability, the ionization probability from the linear combination of both HOMOs of O2 was calculated. The resulting angle-dependent ionization rate is shown in Fig. 4.4b as red crosses fitted by the red line. This result is in good agrement with the experimental data (Fig. 4.4b blue line) and the literature [114].

For the MO-ADK calculations, consistent with the literature [67, 114], only the πg

orbital lying in the plane perpendicular to the rotation axis was used. In general, the result of MO-ADK is also in good agreement with the experimental data (Fig. 4.4b green and blue lines). However, the minima in the ionization rate along and perpendicular to the laser polarization direction are overestimated. This arises mainly from the assumption of an ionization only from one HOMO [58]. Taking into account both HOMOs provides good agreement with the experimental data, especially with respect to the minima at ±0◦ and ±90◦.