Double ionization of the water molecule: Influence of the target orientation on the secondary-electron angular distributions

Double ionization of the water molecule: Influence of the target orientation on the secondary-electron angular distributions

D. Oubaziz,¹H. Aouchiche,¹and C. Champion^2,*

1Laboratoire de M´ecanique, Structures et Energ´etique, Universit´e Mouloud Mammeri de Tizi-Ouzou, BP 17, Tizi-Ouzou 15000, Alg´erie

2Universit´e Paul Verlaine-Metz, Laboratoire de Physique Mol´eculaire et des Collisions, ICPMB (FR CNRS 2843), 1 bd Arago, F-57078 Metz, Cedex 3, France

(Received 28 September 2010; published 24 January 2011)

Fivefold differential cross sections for electron-induced double ionization of isolated oriented water molecules are reported. The theoretical investigation is performed within the first Born approximation by describing the initial molecular state by means of single-center wave functions. The contributions of each final state to the double-ionization process, i.e., with target electrons ejected from similar and/or different molecular subshells, are studied and compared in terms of shape and magnitude. Furthermore, for the particular target orientations investigated, we identify clearly the signature of the main scenarios involved in (e,3e) reactions, namely, the shake-off and the two-step 1 mechanisms.

DOI:10.1103/PhysRevA.83.012708 PACS number(s): 34.70.+e, 82.30.Fi

I. INTRODUCTION

Water is known as a molecule of fundamental importance owing to its abundance on Earth and in space under various phases (gas, liquid, and ice). Consequently, numerous funda- mental studies on its main properties (electronic structure, chemical reactivity, etc.) have been reported by many au- thors in various fields such as astrophysics, plasma physics, biophysics, and, more particularly, in radiobiology, where water is commonly used for modeling the biological medium, essentially due to its presence in the cellular environment (for

∼80 % in mass). Monte Carlo simulations were then success- fully developed for modeling the biological consequences of cellular irradiations (see, for example, Refs. [1–5] and [6–10]

for electron- and ion-track simulations, respectively). In these numerical codes, the electron history is described step by step, interaction after interaction, by means of differential and total cross sections in order to provide the most detailed energetic cartography (see, for example, theCELLDOSEcode developed by Champion, Zanotti-Fregonara, and Hindi´e [11], which is devoted to the modeling of the microscopic energetic distribution received by normal tissues or cancer cells in order to assess the relative merits of specific radiopharmaceuticals).

Thus, ionization, electronic excitation, and elastic scattering have been investigated independently via theoretical and/or semiempirical approaches in order to providein fine useful input data (cross sections) for simulating all the electronic (primary as well as secondary) trajectories (see, for example, Champion [12] and references therein). However, in the major part of the available existing codes, the ionization process is seen as a succession of single (binary) collisions except in the few codes devoted to heavy-ion track-structure simulations, where multiple processes have to be considered. For this particular topic, we refer the reader to the full-differential code developed by Champion et al. [6]—called TILDA— where multiple ion-induced processes (ionization, capture, and transfer ionization) have been fully described. Furthermore,

*champion@univ-metz.fr

let us note that up to now, to the best of our knowledge, only few experiments have been devoted to multiple ionization of molecules by heavy-charged projectiles (see, for example, Refs. [13] and [14] and references therein).

For the electron-induced double ionization—usually desig- nated by the (e,3e) process—only rare theoretical predictions are available in the literature. In fact, since the first mea- surements of fivefold differential cross sections performed by Lahmam-Bennani et al. [15,16] on neon and krypton, several works were performed, essentially in the first and second Born approximations. In addition, recent experiments on helium [17,18] (this target being the simplest two-electron system and then the best candidate for theoretical benchmarks) have renewed interest in (e,3e) studies (see, for example, the works of Dal Cappello and co-workers [19–22] for a review).

Finally, note that additional interesting candidates such as H⁻ and Li⁺ ions have been recently proposed by Nath and Sinha [23] and Kshamata Muktavat and Srivastava [24] for studying, in particular, the role played by the incoming particle description (see, for example, Ref. [25] and references therein).

For molecular targets, experimental works remain rare and for the major part limited to (e,3–1e) studies, i.e., experiments where only one of the two ejected electrons is detected in coincidence with the scattered electron (see, for example, the experimental study of Lahmam-Bennani, Duguet, and Roussin [26] on molecular hydrogen). Such experiments have pointed out clearly a number of puzzling aspects of the full Coulomb four-body problem and prompted several theoretical attempts to explain the experimental observations. Among them, let us cite the well-known convergent close-coupling (CCC) model successfully used by Lahmam-Bennani et al.

[27] for predicting the angular distributions for high-incident electron reactions on helium and the more recently used J-matrix method reported by Zaytsev, Knyr, and Popov [28].

Furthermore, a more sophisticated model such as the well- known 6C approach (a wave function that treats all six two-body Coulomb interactions between the four particles exactly; see, for example, Jones and Madison [29]) was suc- cessfully used in the case of double ionization. In this model, the pairwise interactions between all the particles existing

in the final state, namely, the scattered and the two ejected electrons and the double-ionized target, have been taken into account. However, a large disagreement (∼1.5−2) concern- ing the order of magnitude has been reported. Finally, let us also mention the first-order plane-wave Born approximation (PWBA) model, which is, up to now, the only theoretical model used for treating heavier targets. In this approach, the two ejected electrons are described in terms of Coulomb waves together with a Gamow factor to take into account the repulsion between them, with the incident and the scattered electrons being described by a plane wave. This model was successfully used for describing the shape (but not the magnitude) of the angular distribution of the first (e,3e) experiments by Lahmam-Bennani, Dupr´e, and Duguet [15], particularly for high-incident energy (e,3e) experiments where noticeable disagreements were observed [30]. The authors explained the discrepancies in this simple model by the lack of second-order mechanism contributions, which ideally should be taken into account but which are only partially introduced in the distorted-wave Born approximation (DWBA) [15] via the use of a distorted wave (which continuously takes into account the interaction between the scattered electron and the residual target ion) to describe the scattered electron. However, even this more sophisticated model fails to reproduce the experimental data at lower momentum transfer [31].

Concerning the water molecule, let us note that, contrary to the photo-double-ionization experiments, which have been extensively reported on [32–35], electron-double-ionization measurements remain up to now, to the best of our knowledge, scarce and limited to the very recent (e,3-1e) data reported by Jones et al.[36]. Similar to the above cited H2 experiments [26], here the existing measurements are also performed in terms of (e,3−1e) momentum spectroscopy experiments for 2055-eV incident electrons. On the theoretical side, the literature remains poor and we only find three works, all based on the first Born approximation. The first one was proposed by Kada et al. [22], who studied the double ionization of the 1b1water molecular state: fully differential cross sections were then reported, exhibiting in particular the signature of the main mechanisms involved in the double-ionization process, namely, the shake-off (SO) and the two-step 1 (TS1) mechanisms (see the following for more details). More recently, Mansouriet al.[37] have treated the case of the four outermost molecular shells (1b₁, 3a₁, 1b₂, and 2a₁) by using the CCC approach. In this work, the authors reported fivefold differential cross sections for 1-keV electron impact versus the ejected polar angles for particular energetic conditions, namely, for equal and unequal energy sharing between the two ejected electrons and for a fixed scattering angle of 1^◦. Finally, we have recently reported in Ref. [38] a detailed study of the water double ionization within the first Born approximation, and then reported a strong dependence of the fivefold differential cross sections with respect to the target molecular orientation.

In the present work, we extend our previous study by evaluating the contribution of each intermolecular shell to the double-ionization process, namely, the final molecular states referred to as (1b1)⁻¹(3a1)⁻¹, (1b1)⁻¹(1b2)⁻¹, (1b1)⁻¹(2a1)⁻¹, (3a1)⁻¹(1b2)⁻¹, (3a1)⁻¹(2a1)⁻¹, and (1b2)⁻¹(2a1)⁻¹. Indeed, in Ref. [38] we have only considered ejected electrons

extracted from the same molecular orbital, namely, the final states, hereafter referred to as (1b₁)⁻², (3a₁)⁻², (1b₂)⁻², and (2a1)⁻², whereas here we consider all the final configurations, i.e., with secondary electrons ejected from any orbitals, be they similar or not. In view of this, to the best of our knowledge, the present work appears as the first complete theoretical study dedicated to the water molecule electron-induced double ionization in terms of multidifferential cross sections.

In the sequel, we deal with the theoretical model developed here for calculating the (e,3e) fivefold differential cross sections. The results obtained are then reported and analyzed in Sec. III, and conclusions regarding the influence of the molecular orientation on the double-ionization process are discussed in Sec.IV. In the following sections, atomic units (a.u.) are used throughout unless indicated otherwise.

II. THEORY

The double ionization of a single (oriented) water molecule may be schematized by

e⁻_i +H2O→e⁻_s +e⁻₁ +e⁻₂ +H2O²⁺, (1) where e_i⁻,e_s⁻,e₁⁻, ande⁻₂ refer to the incident electron, the scattered electron, and the two outgoing electrons, respec- tively, with H₂O²⁺being the residual ion. The corresponding momentaki,ks,k₁, andk₂are linked to the electron kinetic en- ergies via the relationski =√

2Ei,k₁=√

2E1,k₂=√ 2E2, and ks=

2(Ei−E₁−E₂−I²⁺), where I²⁺ denotes the double-ionization threshold and varies according to the two molecular orbitals involved in the collision (see TableI).

In the present work as well as in our previous studies devoted to single and double water ionization by charged particles [38,40–43], the frozen-core approximation was used in order to reduce the N =10 electron problem to a two-electron system and then to consider only two active ejected electrons. In addition, we also assume that the TABLE I. Binding energies of the various final states of the double-ionized water molecule. The data have been taken from Ref. [39].

Molecular final states Binding energies (eV) State multiplicity

1b₁⁻² 39.7 Singlet

3a₁⁻² 44.4 Singlet

1b₂⁻² 52.2 Singlet

2a₁⁻² 83.3 Singlet

(1b1)⁻¹(3a1)⁻¹ 41.3 Singlet

38.6 Triplet

(1b1)⁻¹(1b2)⁻¹ 44.9 Singlet

43.0 Triplet

(1b1)⁻¹(2a1)⁻¹ 63.9 Singlet

57.1 Triplet

(3a1)⁻¹(1b2)⁻¹ 47.1 Singlet

44.9 Triplet

(3a1)⁻¹(2a1)⁻¹ 65.2 Singlet

58.8 Triplet

(1b₂)⁻¹(2a₁)⁻¹ 70.3 Singlet

63.9 Triplet

remaining electrons in the doubly charged ion core are un- affected by the ionization process. Consequently, the potential V ≡V(r0,r1,r2) involved in the transition matrix element may be written as

V(r0,r1,r2)= −2 r0

+ 1

|r0−r1| + 1

|r0−r2|, (2) wherer0denotes the position vector of the incident particle, whereasriis the position vector of theith bound electron with respect to the center of the molecule, i.e., the oxygen nucleus.

The initial state of the water target molecule is described here by means ofan accurate molecular wave functiontaken from [44] constructed with molecular orbitals that are the best possible linear combination available from a basis set consisting of functions all centered upon the heaviest nucleus.

These wave functions are computed by a general program for several arbitrary positions of the nuclei employing different basis sets, and the retained wave functions refer to equilibrium configurations calculated by the self-consistent-field (SCF) method. In particular, a good agreement was found with some of the experimental data such as the electric dipole moment, the ionization potential, the binding length O—H, the equilibrium distance H—H, and the molecular angle H—O—H (see Ref. [44] for more details).

Under these conditions, the ten bound electrons of the water target molecule are distributed among five molecular wave functions corresponding to the five molecular orbitals denoted as 1b1, 3a1, 1b2, 2a1, and 1a1, each of them being expressed as

υj(r)=

N_at(j) k=1

aj k^ξ_n^{j k}

j kl_{j k}m_{j k}(r), (3)

where the radial and angular parts are given by Slater functions and by real solid harmonics, respectively (see Ref. [45] for more details). In Eq. (3),N_at(j) is the number of Slater func- tions used in the development of thejth molecular orbital and aj k is the weight of each real atomic component^ξ_n^{j k}

j kl_{j k}m_{j k}(r) (more details can be found in Refs. [40] and [44], where all the necessary coefficients and quantum numbers are reported).

Finally, note that the electronic (structural) correlation is not included in the present water description. Indeed, we have exhibited in our recent paper [37] that a Hartree-Fock target wave function was generally sufficient to get accurate results.

Thus, we have shown that a highly correlated function for the initial state gave practically the same results (except a minor difference in magnitude) as those obtained with the usual Hartree-Fock wave function in the case of double ionization of neon (which has ten electrons as the water molecule).

However, let us note that the relatively good agreement between experimental data and first Born predictions generally reported by authors for double ionization of simple targets is still currently not clearly understood as to what emphasizes the complexity of the electron-impact double-ionization process as compared to single ionization, and stresses the necessity of producing more experimental data under various kinematical conditions and of pursuing theoretical efforts for a better understanding of this problem.

In this context, the fivefold differential cross sections σ⁽⁵⁾(₁,₂,_s,E₁,E₂) for the water double ionization,

denoted 5DCS in the following, are given by d⁵σ

d₁d₂dsdE₁dE₂ ≡σ⁽⁵⁾(1,₂,s,E₁,E₂)

= 5 j1=1

5 j2j1

σ_j⁽⁵⁾

1j₂(₁,₂,s,E₁,E₂)

=(2π)⁴k₁k₂k_s k_i

5 j1=1

5 j2j1

Tj₁j₂²,

(4) where_sdenotes the solid angle of the scattered electron and ₁and₂refer to the solid angles of each of the two ejected electronse⁻₁ ande⁻₂, respectively.

Here, the transition-matrix elementT is denoted asTj₁j₂, also referring to the simultaneous ejection of two electrons from two molecular orbitals labeled j₁ andj₂, respectively.

Thus, according to the j₁ and j₂ values, we will suc- cessively study the following: (i) the four states (1b₁)⁻², (3a1)⁻², (1b2)⁻², and (2a1)⁻² that correspond to the ejec- tion of electrons coming from the same molecular orbital (j1=j₂), and (ii) six other states, (1b1)⁻¹(3a1)⁻¹, (1b1)⁻¹ (1b₂)⁻¹, (1b₁)⁻¹(2a₁)⁻¹, (3a₁)⁻¹(1b₂)⁻¹, (3a₁)⁻¹(2a₁)⁻¹, (1b₂)⁻¹(2a₁)⁻¹, corresponding to secondary electrons ejected from two different shells (j₁=j₂). Note that the inner shell 1a1was not considered in the present work owing to its minor contribution in the reported 5DCS.

Moreover, as underlined in our previous works, the above- cited wave functions describing the initial bound states of the water molecule refer to a particular molecular orientation given by the Euler angles (α,β,γ). In these conditions, the matrix elementTj₁j₂ also depends on the target orientation and may be rewritten as

T_j1j2≡T_j1j2(α,β,γ)

=

_f^j1j2(ks,k1,k2;r0,r1,r2;α1,α2)V(r0,r1,r2)

×_i^j1j2(ki;r0,r1,r2;α1,α2;α,β,γ)

, (5)

where _i^j1j2(ki;r0,r1,r2;α1,α2;α,β,γ) represents the initial state of the system {e⁻_i +H2O} and _f^j1j2 (ks,k₁,k₂;r₀,r₁,r₂;α1,α2) stands for the final state of the system, comprising the scattered electron, the two ejected electrons, and the residual molecular ions {e_s⁻+e⁻₁ +e⁻₂ +H2O²⁺}. The corresponding wave functions may then be written as

_i^j1j2(ki;r₀,r₁,r₂;α1,α2;α,β,γ)

=φ(ki;r0)ϕ_i^j1j2(r1,r2;α,β,γ)

|α1,α2, (6)

_f^j1j2(ks,k₁,k₂;r₀,r₁,r₂;α1,α2)

=

φ(ks;r0)ϕ_f^j1j2(k1,k2;r1,r2)α1,α2|,

where the vectors|α1,α2indicate the spin of the two active electrons. Four possibilities can then be identified, namely, (u,u), (u,d), (d,u), and (d,d), whereu(d) refers to a spin up

(down). Thus, we have

|α1,α2 =

⎧⎪

⎪⎨

⎪⎪

⎩

√1

2(|ud − |du) for a singlet state

|uu

√1

2(|ud + |du)

|dd

⎫⎬

⎭for a triplet state (7)

The functions φ(ki;r₀) and φ(ks;r₀) refer to the plane- wave functions associated with the incident electron and the scattered electron, respectively, whereas the func- tion ϕ^j_i^1j2(r1,r2;α,β,γ) refers to the initial wave function, namely,

ϕ_i^j1j2(r₁,r₂;α,β,γ)≡

ϕ_i^j1j2(r₁,r₂;α,β,γ)_±

= υj₁(r1;α,β,γ)υj₂(r2;α,β,γ)±υj₁(r2;α,β,γ)υj₂(r1;α,β,γ)

√2 . (8)

Thus, ϕ_i^j1j2(r₁,r₂;α,β,γ) is symmetric ([ϕ_i^j1j2(r1,r2;α,β,γ)]⁺) when the considered vector

|α1,α2 refers to a singlet state and it is antisymmetric ([ϕ_i^j1j2(r₁,r₂;α,β,γ)]⁻)otherwise.

Finally, the functionsυ_j1,j2(r;α,β,γ) are given by υj₁,j₂(r;α,β,γ)=

Nat(j) k=1

fj k(r)

lj k

µ=−l_{j k}

Dµ,m^{lj k} _{j k}(α,β,γ)S^µ_{lj k}(ˆr), (9)

where S^µ_l

j k(ˆr) and D^l_µ,m^ik

j k(α,β,γ) refer to the spherical har- monics in their real form [45] and to the rotation matrix, respectively, the latter being defined by

Dµ,m^{lj k} _{j k}(α,β,γ)=e^{−imj kα}dµ,m^{lj k} _{j k}(β)e^−iµγ, (10) where the quantity dµ,m^{lj k} _{j k}(β) is given by the Wigner formula,

dµ,m^{lj k} _{j k} = τ

t=0

(−1)^t

(lj k+µ)!(lj k−µ)!(lj k+m_{j k})!(lj k−m_{j k})!

(lj k+µ−t)!(lj k−m_{j k}−t)!t!(t−µ+m_{j k})!·ξ^{2lj k+µ−mj k−2t}·η^{2t−µ+mj k}, (11)

withξ =cos(β/2) andη=sin(β/2).

Furthermore, in the final state—described by the wave function ϕ^j_f^1j2(k1,r1;k2,r2)—the two ejected electrons are both described by a Coulomb wave function with a Som- merfeld parameter η₁=Z₁/k₁ (η2=Z₂/k₂) whereZ₁ (Z2) corresponds to the effective charge [46]. Note that these may be taken as either equal to the value of the residual nuclear charge or as dependent on the different momentaks,k1, and k2, as reported by Berakdar [47] or by Zhang [48] for (e,2e) collisions or more recently by El Azzouziet al.[49] for (e,3e) collisions. In the present study, as in our previous work [38], we consider the first case, i.e.,Z₁=Z₂=2.

Under these conditions and by using the well-known partial-wave expansion of the plane wave, as well as that of the interaction potential_|r ¹

0−r_i|, we get the following expression for the 5DCS:

d⁵σ^±(α,β,γ) d₁d₂dsdE₁dE₂

=(2π)⁴k₁k₂k_s k_i gG

5 j1=1

5 j2j1

nˆ

j1

(α,β,γ;k1)

×

j₂(α,β,γ;k₂)−ˆ

j₂(α,β,γ;k₂)

±ˆ

j1

(α,β,γ;k2)

j2

(α,β,γ;k1)−ˆ

j2

(α,β,γ;k1) +

j₁(α,β,γ;k₁)

j₂(α,β,γ;k₂)

±

j1

(α,β,γ;k2)

j2

(α,β,γ;k1)², (12)

withσ⁻andn=3 for the triplet state andσ⁺andn=1 for the singlet state, where the various elements are given by

j₁,j₂(α,β,γ;k₁)

= 2 qk₁

2 π

Nat(j) k=1

l,m

l₁,m₁

X^l,l_{j k}¹(k1,q)i^l−l1e^iσl1(η1)

×Y_l^m1

1 ( ˆk₁)Y_l^m∗( ˆq)lj k,m1−m,mj k(α,β,γ)(−1)^m1

×

lˆ₁lˆlˆj k

4π

l₁ l lj k

0 0 0

l₁ l lj k

−m₁ m m₁−m

, (13) where the momentum transfer is defined byq=ki−ks, and

ˆ

j1,j2

(α,β,γ;k1)= 1 π qk₁

2 π

N_at(j) k=1

l_{j k}

m1=−lj k

×Xˆ_{j k}^{lj k}(k1)l_{j k},m₁,m_{j k}(α,β,γ)Y_l^m_{j k}¹( ˆk₁)i^{−lj k}e^iσljk(η1), (14) where

l_{j k},m_{j k},µ(α,β,γ)= D^l_µ,^{j k}_{−mj k}(α,β,γ)−D^lµ,m^{j k} _{j k}(α,β,γ)

√2 ifj =1,i.e.,for the 1b1orbital,

l_{j k},m_{j k},µ(α,β,γ)=iDµ,m^{lj k} _{j k}(α,β,γ)+D_µ,^{lj k}_{−mj k}(α,β,γ)

√2 ifj =3,i.e.,for the 1b2orbital,

l_{j k},m_{j k},µ(α,β,γ)= D^lµ,m^{j k} _{j k}(α,β,γ)+D_µ,^{lj k}_{−mj k}(α,β,γ)

√2

×δ_{mj k,2}+D^lµ,m^{j k} _{j k}(α,β,γ)δmj k,0otherwise.

(15) The radial parts X_{j k}^l,l1(k,q) and Xˆ^l_{j k}^{j k}(k) introduced in Eqs. (13) and (14) are expressed as

X^l,l_{j k}¹(k,q)= _∞

0

dr rFl₁(k,r)jl(qr)fj k(r),

(16) Xˆ^l_{j k}^{j k}(k)=

_∞

0

dr rFl_{j k}(k,r)fj k(r),

where Fl(k,r) and jl(qr) denote the radial hypergeometric function and the Bessel function, respectively, while fj k(r) refers to the jkth component of the radial part of the target wave function (for more details we refer the reader to our previous works in Refs. [40–43] and to Ref. [44]).

Finally, note that at low ejected electron velocities, the Coulomb interaction between the two ejected electrons in the final state becomes significant and therefore cannot be ignored.

Thus, taking into account the electrostatic repulsion between the two outgoing electrons and investigating all the angular distributions, we introduced into Eq. (12) the well-known Gamov factor gG, whose expression depends on the polar (θ1,θ₂) and azimuthal (ϕ1,ϕ₂) angles of the wave vectorsk1

andk₂, namely,

gG(k₁,k₂)= ν

e^ν−1 withν= 2π

|k1−k2|. (17) III. RESULTS AND DISCUSSION

In this section, we report the fivefold differential cross sections for double ionization of an isolated water molecule oriented in three particular directions, namely, those reported in Ref. [38], i.e., those referred to by the Euler angle triplets (0,0,0), (0,π/2,0), and (0,π/2,π/2) (see Fig.1). In the first configuration, the water molecule is positioned in theyzplane with thezaxis parallel to the bisecting line of the molecule, whereas the second (third) configuration refers to a water molecule placed in the xy (xz) plane with a bisecting line along the x axis (for more details we refer the reader to Ref. [38]). Furthermore, let us note that in all the geometries investigated here, the incident momentumkiremains collinear to thez axis while the molecular plane is either parallel or perpendicular to the collision plane. The calculated 5DCSs are then reported versus the ejected anglesθ₁andθ₂for a scattering angle θ_s =0^◦ in a coplanar geometry (ϕ₁=ϕ₂=ϕ_s=0^◦).

The incident energy was chosen to be equal to 1 keV and an equal energy sharing between the two ejected electrons, namely,E₁=E₂=10 eV, was considered. Let us note that this kinematics has been studied extensively both theoretically and experimentally [27,37,38]. Finally, let us note that in all the cases reported in this study, the obtained 5DCSs exhibit an expected symmetry axis given by θ₁=θ₂. Under these conditions, the following analysis will be focused essentially on the description of one of the two parts of the figures, the second part being deduced simply from the first one by means of an axial symmetry with regard to theθ₁=θ₂direction.

x

z

x

z

x

z

(α,β,γ) = (0,π/2,π/2) (α,β,γ) = (0,π/2,0)

(α,β,γ) = (0,0,0) x

y y y

FIG. 1. (Color online) Schematic representation of the three particular water molecule orientations investigated in the present work.

In Fig.2, the 5DCSs corresponding to the (α,β,γ)=(0,0,0) configuration are shown for the final states referred to as (1b1)⁻¹(3a1)⁻¹,(3a1)⁻¹(2a1)⁻¹, and (1b1)⁻¹(2a1)⁻¹ [corre- sponding to Figs. 2(a), 2(b), and 2(c), respectively]. We clearly observe four hills: two major ones delimited by a θ₁ angle varying from 0^◦ to∼50^◦ (with aθ₂ angle ranging from 160^◦ to 310^◦) and from 50^◦ to ∼160^◦ (with a θ₂ angle ranging from 310^◦ to 360^◦). Two minor hills are also reported, respectively characterized by 80^◦θ₁140^◦(with 160^◦θ₂220^◦) and by 120^◦θ₁200^◦ (with 220^◦ θ₂280^◦). The corresponding maxima are then observed for (θ₁=20^◦, θ₂ =270^◦) and (θ₁=90^◦, θ₂=340^◦) for the ma- jor domains, whereas they are located at (θ₁=95^◦, θ₂=205^◦) and (θ1=150^◦, θ₂=260^◦) for the two minor hills. Then, this orientation clearly reveals the signature of the TS1 mechanism, |θ₁−θ₂| ∼=110^◦−120^◦, during which a first electron is ejected in a privileged direction and a second electron is ejected during a quasielastic collision, leading to a |θ₁−θ₂| angle of greater than 90^◦, generally of the order of 110^◦−120^◦ (see Schr¨oter et al. [50]). However, note that contrary to our previous work [38], where the θq(=0) direction played an important role in the TS1 scenario, here we observe that the privileged direction is not that of the transfer momentum q but that corresponding to an ejected direction of θ =270^◦+110^◦/2=20^◦−110^◦/2= 325^◦ andθ=340^◦+110^◦/2=90^◦−110^◦/2=35^◦ for the major hill and that corresponding to the angles θ=95^◦+ 110^◦/2=205^◦−110^◦/2=150^◦ and θ=150^◦+110^◦/2= 260^◦−110^◦/2=205^◦ for the minor ones [see Fig. 2(a)].

Similar observations can be made from Fig. 2(b), where the 5DCSs corresponding to the final state (3a1)⁻¹(2a1)⁻¹ are reported. Here, the signature of the TS1 mechanism is still evident with particular maxima located at (θ1=15^◦, θ₂= 253^◦) and (θ₁ =107^◦, θ₂ =345^◦) for the major domains and at (θ₁=0^◦,θ₂=112^◦) and (θ₁=248^◦, θ₂=360^◦) for the minor domains, leading to a|θ₁−θ₂|angle of the order of 112^◦. On the contrary, Fig.2(c)shows two major hills whose maxima reveal a|θ₁−θ₂|angle of 180^◦, which is commonly attributed to the SO process, during which the two ejected electrons are emitted in opposite directions. Finally, let us note that the other states((1b1)⁻¹(1b2)⁻¹,(3a1)⁻¹(1b2)⁻¹,(1b2)⁻¹(2a1)⁻¹)are not reported because they exhibit a negligible contribution to this particular target orientation.

As already reported in our previous work [38], when molecule rotations are applied, the role played by the different

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320 360

(90°,340°)

(20°,270°)

(a)

θ₁(degrees) θ2(degrees)

0 0.00625 0.0125 0.0187 0.0250 0.0312 0.0375 0.0437 0.0500

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320 360

(130°,310°)

(50°,230°)

(c)

θ₁(degrees) θ2(degrees)

0 0.00375 0.00750 0.0113 0.0150 0.0188 0.0225 0.0263 0.0300

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320 360

(248°,360°) (107°,345°)

(0°,112°) (15°,253°)

(b)

θ₁(degrees) θ2(degrees)

0 0.00175 0.00350 0.00525 0.00700 0.00875 0.0105 0.0123 0.0140

FIG. 2. (Color online) Fivefold differential cross sections (5DCS) corresponding to the double ionization of a water molecule oriented in the (α,β,γ)=(0,0,0) direction and impacted by a 1-keV electron beam. The 5DCSs are reported as a function of the ejected anglesθ1andθ2

with respect to the incident direction. The kinematical conditions are given byθs =0^◦,ϕs =ϕ1=ϕ2=0^◦, andE1=E2=10 eV. Three final states are reported here, namely, (a) (1b₁)⁻¹(3a₁)⁻¹,(b) (3a₁)⁻¹(2a₁)⁻¹, and (c) (1b₁)⁻¹(2a₁)⁻¹.

subshells is modified, leading to a shifting, decrease, increase, and/or disappearance of the above observed maxima. Thus, by keeping in mind that the 1b₁,3a₁, and 1b₂water molecule orbitals are mainly governed by 2p₊₁, 2p0, and 2p₋₁atomic orbitals, respectively, which corresponds—in the present water molecule description based on real solid harmonics—to orbitals respectively collinear to the x-,z-, and y-molecular axes (denoted asP_x,P_z, andP_yin the following; see Ref. [38]

for more details), the theoretical predictions obtained for the different target orientations then may be explained by considering the successive rotations studied here. Thus, for the configuration (α,β,γ)=(0,π/2,0), we have

(α,β,γ)=(0,π/2,0):

⎧⎨

⎩

Px →Pz

P_y →P_y P_z→P_x

. (18)

Under these conditions (see Fig. 3), the contributions of the molecular states (1b₁)⁻¹(1b₂)⁻¹,(3a₁)⁻¹(1b₂)⁻¹, and

(1b2)⁻¹(2a1)⁻¹ still remain negligible in the present case, essentially due to the fact that the 1b2 molecular orbital—

which is mainly governed by a Py atomic orbital in the initial orientation—remains perpendicular to the collision plane. Thus, in Fig.3(a), the (1b₁)⁻¹(3a₁)⁻¹molecular state exhibits four major structures whose maxima are characterized by|θ₁−θ₂| ∼=262^◦ and|θ₁−θ₂| ∼=153^◦. Whereas the first group may be undoubtedly attributed to the TS1 mechanism in comparison to Fig.2(a), and according to the transformations reported in Eq. (18), the second group is less evident even if it shows characteristics close to those of the SO process. Indeed, in the present case, the 1b1 orbital behaves as aPz orbital, i.e., as the 3a1orbital in the (α,β,γ)=(0,0,0) case, whereas the 3a1 orbital behaves as aPx orbital, i.e., as the 1b1 orbital in the (α,β,γ)=(0,0,0) case. Thus, here the (1b₁)⁻¹(3a₁)⁻¹ molecular state exhibits a 5DCS that is globally unchanged in comparison to the initial configuration. Regarding the contribution of the final state (3a1)⁻¹(2a1)⁻¹ [Fig.3(b)], we

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320

360 (a)

(180°,333°) (83°,345°)

(15°,277°)

(27°,180°)

θ₁(degrees) θ2(degrees)

0 0.00300 0.00600 0.00900 0.0120 0.0150 0.0180 0.0210 0.0240

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320

360 (c)

(180°,360°)

(0°,180°)

θ₁(degrees) θ2(degrees)

0 0.00275 0.00550 0.00825 0.0110 0.0137 0.0165 0.0192 0.0220

0 40 80 120 160 200 240 280 320 360 0

40 80 120 160 200 240 280 320 360

(b)

(35°,221°) (138°,324°)

(205°,332°)

(28°,155°)

θ1(degrees) θ2(degrees)

0 0.00437 0.00875 0.0131 0.0175 0.0219 0.0262 0.0306 0.0350

FIG. 3. (Color online) Same as in Fig.2but with (α,β,γ)=(0,π/2,0).

observe two major maxima localized at (θ1,θ₂)∼=(35^◦,221^◦) and (θ1,θ₂)∼=(138^◦,324^◦), which leads to a |θ₁−θ₂| value of ∼180^◦, i.e., the signature of the SO mechanism. Small contributions of the TS1 process are nevertheless noticeable [see the minor maxima localized at (θ₁,θ₂)∼=(28^◦,155^◦) and (θ₁,θ₂)∼=(205^◦,332^◦) leading to|θ₁−θ₂| ∼=127^◦]. These observations are coherent with the fact that in the present configuration, the 3a1orbital (initially governed by aPzatomic orbital) exhibits an overallPxbehavior [see Eq. (18)] whereas the 2a₁orbital—which is mainly governed by a spherical 2s component—does not affect the modifications. Under these conditions, the final state (3a1)⁻¹(2a1)⁻¹looks like the initial state (1b1)⁻¹(2a1)⁻¹ and then shows a SO signature that is similar to that reported in Fig. 3(b). In Fig. 3(c), the (1b₁)⁻¹(2a₁)⁻¹ clearly evidences a SO process with maxima characterized by|θ₁−θ₂| ∼=180^◦, while the overall behavior should be governed mainly by a (Pz)⁻¹(S)⁻¹, i.e., similar to that reported in Fig.2(b): This particularity may be linked to the fact that the “spherical atomic orbital” 2a1 exhibits a low p₀ contribution, i.e., a P_z contribution for the (0,0,0)

orientation that leads to a Px contribution in the present case. Under these conditions, the overall behavior of the final state (1b1)⁻¹(2a1)⁻¹ is close to that reported in Fig. 2(b) for the final state (1b₁)⁻¹(2a₁)⁻¹ in the (α,β,γ)=(0,0,0) orientation. Furthermore, note that the privileged direction here corresponds to that of the momentum transfer, i.e., θq=0^◦.

Let us now consider Fig.4, corresponding to the configura- tion (α,β,γ)=(0,π/2,π/2), according to which we have the following transformations:

Px →Py, Py →Pz, P_z→P_x.

(19)

At first, note that, in the present case, the negligible contributions are those corresponding to the final states (1b1)⁻¹(3a1)⁻¹,(1b1)⁻¹(1b2)⁻¹, and (1b1)⁻¹(2a1)⁻¹, the 1b1

molecular orbital here being perpendicular to the collision plane (ϕs=ϕ₁=ϕ₂=0^◦).