Elsevier

Chemical Physics

Volume 504, 26 March 2018, Pages 22-30
Chemical Physics

Ionisation of H2O by a strong ultrashort XUV pulse: a model within the single active electron approximation

https://doi.org/10.1016/j.chemphys.2018.02.014Get rights and content

Abstract

We present and discuss a new computationally inexpensive method to study, within the single active electron approximation, the interaction of a complex system with an intense ultrashort laser pulse. As a first application, we consider the one photon single ionisation of the highest occupied molecular orbital of the water molecule by a laser pulse. The ionisation yield is calculated for different orientations of the molecule with respect to the field polarization axis and compared against predictions of another single active electron approach.

Introduction

The recent development of coherent sources of light such as attosecond lasers [1], high-order-harmonic generation (HOHG) sources [2], [3] or free-electron lasers [4], [5] has opened the route to the study of the interaction of matter with intense femtosecond and even sub femtosecond radiation pulses in the XUV regime. Such studies allow one to analyze electron dynamics in atoms, or electron and nuclei dynamics in molecules with an unprecedented degree of temporal and spatial resolution. Within this context, the interaction of the water molecule with such radiation pulses is of particular relevance in medical physics, for example, radiotherapy, since water is one of the main components of most living tissues.

In the present contribution, we develop a new computationally inexpensive method to study, within the single active electron (SAE) model and in the non-relativistic regime, the single ionisation of atoms and molecules by an intense femtosecond or sub femtosecond XUV pulse. Specifically we aim to address the total ionisation yield as function of field frequency (or of the photon energy, as they are equal in atomic units), intensity and orientation dependence of the polarisation vector with respect to the molecule. We apply this approach to the water molecule in its ground state while paying attention to the validity of the assumptions we make in such a treatment. We assume that the pulse duration is small compared to the characteristic time of the vibrational motion of the molecule and that, during the interaction, the geometry of the molecule is not modified as we employ the fixed nuclei approximation.

Recently, by measuring the ratio of H2O and D2O high harmonic yields, Farrell et al. [6] managed to characterize the nuclear motion in the molecular states of H2O+. They showed that by contrast to the ionisation of the highest occupied molecular orbital (HOMO), the single ionisation of the second least bound orbital HOMO-1 triggers a fast nuclear dynamics of the molecular ion through a strong bending motion of the molecule. As a result, we only consider here, the single ionisation of the orbital HOMO 1b1 which leaves the geometry of the molecule practically unchanged during the interaction with the pulse. In fact, the period of the fastest oscillation in the water molecule, namely the asymmetrical stretching of the OH bonds, is 8.9 fs [7] which is much longer than the pulse durations we consider here. In other words, we can assume that the molecule is frozen during its interaction with the pulse. However, it is important to note that experimentally, it is impossible to know, a priori, from which orbital the electron is ejected. Farrell’s results show that ionisation from HOMO-1 3a1, which sends the molecule into the A2A1 state of H2O+, strongly excites the bending mode at photon energies around 0.54 a.u. This puts some limitation on the photon energy used in the present work and requires us to pay attention to the bandwidth of the pulse. Furthermore, for the frequencies we use later on, the inner shell ionisation has to play a significant role [8]. We do not consider this type of ionisation, as our model is a SAE model, but it could be a serious correction when a more complete calculation appears.

Our approach is based on a model that was first developed to treat the interaction of atomic hydrogen with an electromagnetic pulse [9]. We work in the momentum space and use the velocity gauge. The main idea is to replace the kernel of the Coulomb potential by a sum of N symmetric separable potentials, each of them supporting a bound state of the system. This method, which we denote by SPAM for Separable Potentials for Atoms and Molecules, allows one to reduce the 4-dimensional time-dependent Schrödinger equation (TDSE) to a system of N 1-dimensional Volterra integral equations depending only on time. As a result, the integration over the spatial coordinates which, in some cases, requires prohibitively large grids or bases, is completely avoided. Each separable potential may be calculated from the exact wave function of the atomic state it supports. However, its analytical expression is not always unique. We developed a procedure to calculate these separable potentials. It provides results for the electron energy spectra that compare very well with those obtained by solving the TDSE with the exact Coulomb potential in situations where the number of essential atomic states playing a significant role is low. By moving from the momentum space to the configuration space, it is easy to show that the separable potentials have a finite range. Let us note that once the separable potentials are determined, the continuum states are automatically defined and, being solutions of the same equation as the one satisfied by the exact bound states taken into account in the calculations, they are orthogonal to these bound states. To generalize to more complex systems such as the water molecule, we proceed along the same lines. We first generate the HOMO in terms of gaussian type orbitals by means of the well established quantum chemistry software package GAMESS(US). It is then straightforward to move to the momentum space and to define the corresponding separable potential which is unique in this case. As for atomic hydrogen, the final step involves solving a 1-dimensional Volterra integral equation. For the sake of completeness, it is important to mention that our approach is not gauge invariant as it is the case for most of the approximate treatments. The problem of the gauge and the delicate question of the possible existence of a privileged gauge are discussed in detail in the context of the present model by Galstyan et al. [10].

The problem of the interaction of the water molecule with a femtosecond or sub femtosecond XUV pulse has almost never been treated theoretically up to now. As far as we know, most of the theoretical calculations have been performed at a wave length of 800 nm (0.057 a.u. photon energy). However, it is interesting from the methodological point of view to mention three different contributions. In the first one, Borbély et al. [11] study the ionisation of the water molecule by an intense ultrashort half-cycle electric pulse. They performed both quantum mechanical and classical calculations within the single active electron and frozen core approximation. They considered the ejection from the HOMO 1b1. Since this orbital is mainly constructed from the 2pz orbital of the oxygen atom when the molecule is lying in x-y plane, they modelled it by an hydrogenic 2pz orbital with an effective charge chosen to reproduce the ionization energy of the HOMO. They found good agreement between the classical and quantum mechanical calculations at high field intensity where the over-the-barrier ionisation mechanism is dominant. In the second contribution, Della Picca et al. [12] study the orientation-dependent single ionisation of fixed-in-space H2O by a short laser pulse for two wave lengths: 800 nm and 76 nm well in the XUV regime (0.057 a.u. and 0.6 a.u. photon energies respectively). In their calculation, which is based on the strong field approximation (SFA) [13], the initial and final states are described by single-Slater determinants of spin-orbitals, the spatial part of it being calculated by means of the same quantum chemistry software package as in our case. They take into account the five occupied molecular orbitals. The SFA is a first order theory in the sense that the ionisation results from the absorption of a “virtual” photon that is supposed to describe tunneling emission. In the high frequency regime, we have shown [13] that the SFA gives qualitatively good results. In their contribution, Della Picca et al. showed that the HOMO-1 dominates the single-electron emission process when the laser is polarized along the symmetry axis of the molecule and that the electron emission is in general favored in the direction along the laser polarization direction. In the third contribution Petretti et al. [14] apply the single active electron approximation time dependent Schrödinger equation (SAE-TDSE) method to the water molecule. They solve a 4-dimensional TDSE within the SAE approximation to treat the orientation-dependent ionisation of H2O in few-cycle 800 nm linear-polarized laser pulses. The molecular orbitals are Kohn–Sham orbitals obtained by using the LB94 exchange–correlation functional. They showed that the HOMO dominates the overall ionsation behaviour except in the nodal plane of this orbital where the dominant contribution comes from the HOMO-1. The role of the carrier envelope phase is also investigated.

Our contribution is organized as follows. After this introduction, we present our method. First, very briefly in the case of atomic hydrogen and then, in more detail, in the case of the water molecule. In the third section, we first present some tests of our method in the case of atomic hydrogen. Before the conclusions and perspectives we present our results for the water molecule and compare them against the predictions of the SAE-TDSE method described in detail by Petretti et al. [14].

Atomic units (a.u.) combined with the Gaussian system for the electromagnetic field are used throughout unless otherwise specified.

Section snippets

Theoretical model

In this section, we define the pulse and describe our model. For the sake of clarity, we first consider briefly the case of atomic hydrogen. Details of the calculations are found in Galstyan et al. [10]. We then show, in more detail, how it can be generalized to a more complex system such as the water molecule.

Results and discussions

This section is divided in two subsections. In the first one, we present some tests of the adequacy of our model in the case of atomic hydrogen exposed to a XUV pulse. Results for the water molecule are presented and discussed in the second subsection.

Conclusions and perspectives

We have extended a computationally inexpensive model, previously developed for atomic hydrogen to the treatment, within the SAE approximation, of the interaction of a complex quantum system with a high frequency ultrashort laser pulse. As a first application, we have applied this model to the single ionisation of the HOMO of the water molecule by an ultrashort XUV pulse. We studied the dependence of the single ionisation yield on the pulse frequency, the peak intensity and the orientation of

Acknowledgments

The authors thank Prof. Clément Lauzin for many useful discussions. A.G. is “aspirant au Fonds de la Recherche Scientifique (F.R.S.-FNRS)”. Yu.P. thanks the Université catholique de Louvain (UCL) for financially supporting several stays at the Institute of Condensed Matter and Nanosciences of the UCL. F.M.F. and P.F.O’M. gratefully acknowledge the European network COST (Cooperation in Science and Technology) through the Action CM1204 “XUV/X-ray light and fast ions for ultrafast chemistry”

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