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Interference and non-Franck-Condon effects in ionization of H2 molecules by photon impact

We analyze theoretically interference effects in the spectra of electrons emitted in the H2 photoionization by high energy linearly polarized photons. Molecular bound and continuum states are accurately described by means of B-spline basis allowing the inclusion of the nuclear degrees of freedom. One interesting feature is observed: the usual Franck-Condon behavior is not followed when the H2 internuclear axis is parallel to the polarization direction. Moreover, this is related to the fact that for this molecular orientation and under certain conditions, the electron cannot be emitted in the direction of the radiation field. On the contrary, for H2 molecules perpendicular to the polarization direction, the angular distribution of electrons is analogous to the one observed in the two slits Young's experiment.


Introduction
Nowadays, the study of interference effects in the electronic emission from diatomic molecules is a very active subject of research (see for instance [1]). In the sixties, Cohen and Fano explained anomalies in the photoelectron spectra of diatomic molecules [2]. They argued that the coherent emission from both molecular centers produced interferences when the de Broglie wavelength of the ejected electrons is of the order of the internuclear distance of the molecule in analogy with the Young's two slit experiment. They elaborated a simple model based on the following assumptions: i) ionization is basically a one-electron process, ii) the lowest molecular orbital ψ g is well described in the Linear combination of atomic orbitals (LCAO) approximation (ψ g ∼ 1s A +1s B , where 1s A and 1s B are atomic orbitals centred in nucleus A and B, respectively) and iii) the ionized electron is well described by a single-centre plane wave. In this way, Cohen and Fano showed that the total photoionization cross section is given by the analytical form σ = σ H [1 + sin(k e R)/(k e R)] /(1 + S) (1) where σ H is the photoionization cross section of a hydrogen atom with effective charge Z ef f , R is the equilibrium internuclear distance, S is the overlap between 1s A and 1s B , and k e is the electron wave vector. This formula shows the typical oscillatory behaviour associated with interferences. However, it was not until recently that this kind of interference was experimentally detected in the differential cross sections of ionization of diatomic molecules by heavy ion impact [3]. The detection of these effects is not an easy task as the cross section as a function of the energy decreases quickly. In order to circumvent this inconvenient in those experiments, the ratio between molecular cross sections and adequate atomic cross sections (two isolated atoms for instance, for which no interferences are produced) was considered. The coherent emission was then traced by analyzing oscillations in the mentioned ratios. In the same way, interferences effects were observed subsequently in the case of electronic impact [4,5]. Moreover, the case of photoionization of hydrogen molecules was revisited in order to provide a realistic description of the photoionization spectra including the influence of the nuclear motion. In this work we review some of the remarkable results obtained for the case of impact of linearly polarized photons on hydrogen molecules. The Cohen and Fano predictions were studied concluding that they are valid at high emission energies. At low energies, deviations are observed mainly due to the electronic correlation and the nature of the molecular potential [6]. Interestingly, it was also observed that the usual Franck-Condon behavior is not followed when the H 2 internuclear axis is parallel to the polarization direction for linear polarized photons [7]. The origin of this anomaly may be related to interference and confinement effects [8]. Such effects are more easily evidenced in the electron angular distributions associated with specific vibrational states of the residual H + 2 ion. For H 2 molecules oriented perpendicular to the polarization direction, the distributions are close to the ones obtained in the Young's double-slit experiment. In the case of molecules parallel to the polarization direction and under certain conditions, the electron cannot be emitted in the classical direction given by the radiation field.
Atomic untis are used unless otherwise explicitly stated.

Theoretical methods
In this section, the theoretical methods employed in the previous works are described [7]. Photoionization cross sections have been evaluated in the framework of the dipole and the adiabatic Born-Oppenheimer (BO) approximations. The fully differential photoionization cross section, i.e., differential in both the energy and direction of the ejected electron and the energy and orientation of the residual H + 2 molecular ion, is given by Dill's formula [9]: with In this equation, µ 0 = 0 for linearly polarized light and µ 0 = ±1 for circularly polarized light, α denotes the electronic state of the residual molecular ion,hω is the photon energy, ε is the photoelectron energy, Ω e = (θ e , φ e ) is the photoelectron emission direction in the molecular frame (θ e and φ e are the polar angles), Ω n = (θ n , φ n ) is the polarization direction with respect to the molecular axis z, c is the speed of light, C(j 1 , j 2 , j; m 1 , m 2 , m) denotes a Clebsch-Gordan coefficient, Y M L is a spherical harmonic,σ ℓ (ε) is the Coulomb phase shift and T αℓmµ (ε) is the transition dipole matrix element given by: where Ψ gν is the ground molecular state of energy W gν , Ψ + αυαℓmε is the final molecular state of energy W υα + ε representing a molecular ion in the υ α vibronic state (either dissociative or non dissociative) and an ionized electron of energy ε and angular momentum ℓm, r represents the electronic coordinates, R is the internuclear distance, e µ is the photon polarization vector, and D is either i r i (length gauge) or (hω) −1 i ∇ i (velocity gauge). Energy conservation implies that W gν +hω = W υα + ε. Neglecting rotational effects, the wave functions Ψ gν and Ψ + αυαℓmε are evaluated in the adiabatic (Born-Oppenheimer, BO) approximation where ψ n and χ vn are the usual electronic and nuclear BO wave functions [10,11]. For each value of R, the electronic continuum states must satisfy the usual outgoing boundary conditions of electron-molecule scattering.
Integrating equation (2) over one or several differential magnitudes leads to partially differential or total cross sections, each one representing a specific experimental situation [12]. In particular, integrating equation 2 over the solid angle Ω e , leads to the cross section differential in the nuclear solid angle and in the energy of the ejected electron irrespective of the electron emission direction [12]: where P 2 is the Legendre polynomial of order 2, dσ µ 0 α (ω)/dε is the cross section differential in the energy of the ejected electron, and β µ 0 α,n (ε) is the nuclear asymmetry parameter where M i is the projection of the initial-state angular momentum. As we pointed out before, in this paper, we restrict our study to linearly polarized light, i.e., µ 0 = 0, and to H 2 molecules initially in the ground state X 1 Σ + g , i.e., M i = 0. The above equations can also be used to obtain the cross sections differential in the energy of the residual molecular ion (non dissociative case) or the ejected proton (dissociative case) by using the energy conservation relation that relates ε to W υα . Since the ground state of H 2 has 1 Σ + g symmetry, the dipole selection rule implies that only continuum states of 1 Σ + u and 1 Π u symmetries can be populated. Hence, the cross section given in Eq. (6) can be written as the sum of 1 Σ + u and 1 Π u cross sections In this paper, we will analyze in detail two particular molecular orientations: parallel (θ n = 0) and perpendicular (θ n = π/2) to the polarization vector. Hence, from Eq. (5), it can be easily and Summation (integration) in Eqs. (9), (10) and (11) over the energy of the ejected electron when the ionized molecule if left in a non dissociative (dissociative) state leads to the integrated cross section and the corresponding ones for molecules oriented parallel and perpendicular to the polarization vector, respectively, and The total photoionization cross section is obtained by summing over all open channels α the integrated cross sections given in Eq. (12): Starting again from equation (2), one can integrate over the solid angle Ω n to obtain the cross section differential in the energy and solid angle of the ejected electron irrespective of the molecular orientation [13,14]: where dσ µ 0 α (ω)/dε is the cross section differential in the energy of the ejected electron given in Eq. (6) and β µ 0 α,e (ε) is the electron asymmetry parameter The evaluation of the wave functions involved in the T αℓmµ matrix elements requires the use of computational techniques. It is worthy to be mentioned that such techniques have been employed with succes to analyze many ionization reactions of H 2 targets. Among them, we cite resonant dissociative photoionization [11,15,16] and ion impact ionization [17]. Moreover, they allowed the obtention of the first numerical solution of the complete photo-induced break-up of H 2 [18]. More specifically, the vibrational (bound and dissociative) wave functions have been expanded in a basis of 280 B-splines of order k = 8 contained in a box of 12 a.u.. The electronic wave functions have been evaluated as described in detail in Refs. [10,11]. Briefly, the ground state Ψ g results from a configuration interaction (CI) calculation in which the H 2 hamiltonian has been diagonalized in a basis of 321 configurations built from products of one-electron H + 2 orbitals and pseudo-orbitals. The calculated energy at the equilibrium internuclear distance is -1.8865023 a.u., to be compared with the exact non relativistic value -1.88876138 a.u. [19]. All these orbitals have been represented through a one-center expansion that includes spherical harmonics up to ℓ = 16. The corresponding radial parts have been expanded in a basis of 310 B-splines of order k = 8 in a box of radial length of 60 a.u.. The final electronic continuum state Ψ + αυαℓmε results from a close coupling calculation that includes all partial waves with ℓ ≤ 7 associated with the four lowest ionization thresholds of H 2 : . For every value of R, these continuum states satisfy the usual boundary conditions corresponding to (i) one electron in a bound electronic state of H + 2 and (ii) the other electron in a single outgoing spherical wave with a well defined value of the angular momentum ℓ plus a combination of incoming spherical waves for all accessible electronic states of H + 2 and all possible values of the angular momentum of the ejected electron compatible with the molecular symmetry [10] (At the photon energy considered in this work, for a given energy, there is always a continuum state for each electronic state α of the residual H + 2 ion and angular momentum ℓ of the ionized electron).
In this way, we represent in a realistic way the molecular nature including electron correlation and the two-center character of the molecular potential in all bound and continuum wavefunctions. Moreover, interferences among the various ionization thresholds and angular momenta of the ejected electron was taken into account through Ψ + αυαℓmε . The convergence of the close coupling expansion was verified by performing calculations (for a reduced number of photon energies) in which six additional ionization thresholds were considered [6]. Those tests do not exhibit relevant variations in the dominant channel, i.e., the 2 Σ + g (1sσ g ) one.

Validity of the Cohen and Fano model
To analyze the validity of equation (1), we calculated cross sections ratios dividing H 2 results by twice the ones of atomic H [6]. The result is shown in Fig. 1 as a function of photon energy. Strong oscillations are clearly visible in both 4-and 10-channel calculations. We include also in Fig. 1 a fitting to the Cohen-Fano formula (1), leaving a 0 = σ H /(1 + S) and a 1 = R as free parameters. The fit produces a 0 = 1.46 and a 1 = 1.57. At high photon energies, the fitting backs the Cohen-Fano model. Extensions of this model have been proposed to interpret similar oscillatory behaviours observed in ionization of D 2 by fast heavy ions [3] and electrons [4]. As in the Cohen-Fano model [2], these theories are based on a one-electron description of ionization and an LCAO representation of the initial molecular state. This has led to results in reasonable agreement with experiments for large k e . However, none of these theories have been able to reproduce the fall of the experimental σ/2σ H ratio when k e decreases (see Fig. 1). The present results do show such a fall, in excellent agreement with the measurements of Samson and Haddad [23] (see Fig. 1). This is the first theoretical evidence of these observations. The failure to reproduce the observed behaviour at relatively low photon energies using the fitting function given in equation (1) indicates that at least one of the basic assumptions of the  Cohen-Fano model does not hold. To clarify this fact, we show in Fig. 1 B-spline results for a fictitious one-electron diatomic molecule with nuclear charges Z = 0.5 and equilibrium distance R = 1.4 au. This system differs from the real H 2 molecule in the absence of correlation and screening whereas all the other aspects are treated as in H 2 . At high electron energies, this model calculation reproduces qualitatively those of H 2 but at lower energies it fails to reproduce the fall of the σ/2σ H ratio. Therefore, the origin of this fall must be electron correlation and/or screening. Indeed, at low k e , the ejected electron is more sensitive to details of the potential near the nuclei. In particular, in a real H 2 molecule, a slow electron feels a charge larger than 0.5 due to incomplete screening of the inner electron, which must lead to a decrease of the ionization cross section. Electron correlation is expected to play a significant role only in the vicinity of the doubly excited states of H 2 , where the nuclear motion cannot be neglected. Fig. 1 shows some structure near the ionization threshold on the curve of the fictitious one-electron molecule. This is probably provoked by the two-centre nature of the electronic continuum that is absent in the Cohen-Fano model but must be visible for electrons with small kinetic energy.

Confinement effects
In Fig. 2, we show results for the different partial waves associated with the lowest ionization channel 2 Σ + u (1sσ g ) for the 1 Σ + u . It can be seen that results including the nuclear motion do differ significantly from the FNA ones [6]. Discrepancies are more evident at photon energies of ∼ 3.5 a.u. and > 20 a.u. where the ℓ = 1 and ℓ = 3 partial waves contributions exhibit a minimum, respectively. Analogous minima were observed for FNA results for the case of H + 2 [21]. For photon energies around 3.5 a.u., it can be seen from 2 that the minimum observed for the ℓ = 1 wave in the H 2 FNA results is more pronounced than in calculations including the nuclear motion. On the contrary, its position is only slighlty changed by the nuclear motion. As shown in our previous work [8], such minima are expected to appear when the momentum of the ejected electron satisfies the condition: k e R e ∼ ℓπ (ℓ = 2n + 1) (18) where k e and R e are the ejected electron momentum and the equilibrium internuclear distance,respectively. The condition requires that the product k e R e be equal to an odd number times π. Incidentally, the l angular momenta associated to the partial waves used in the description of the final continuum states must be also odd due to the symmetry requirements imposed by the selection rules. So, we use these odd l values to express the mentioned condition.  According to this simple formula, the minimum in the ℓ = 1 partial wave should appear at a photon energy of ∼ 3.1 a.u. and that of the ℓ = 3 one at ∼ 23 a.u.. These values are in reasonable agreement with the actual ones observed in Fig. 2.
The electron energies, ε = k 2 e /2, obtained by using Eq. 18 are the same as the ones of an electron confined in a one-dimensional infinite-square well potential of width R e . This coincidence inspired us the word 'confinement' to refer to this interference effect. In fact, for the particular k e values given by Eq. 18, a system of standing waves is created in the inner region |r| < R e leading to destructive interferences outside this region and producing consequently no propagating waves. At high energies, the minima location is also the one found in the transmission function for a one-dimensional potential made of two delta functions separated by a distance R e . The k e R e = ℓπ condition obtained from these two models where l = 2n + 1 indicates an odd number (of course, in all these analogies, l is not any angular momentum, it is just an odd number) can be easily understood considering that in three dimensions (a real situation), the electrons tend to move in the directions given by the electric field of the incident radiation. As shown by Eq. (13), the 1 Σ + u cross section represents photoionization of H 2 molecules oriented parallel to the polarization axis, then electrons are mostly forced to move along the internuclear axis. Owing to the fact that the ejected electron has a large kinetic energy, it will only be affected by the potential in the vicinity of the two nuclei, which is similar to what happens in a one-dimensional infinite-square well or in the scattering by two delta functions. Hence the larger k e the more accurate the formula k e R e ∼ ℓπ. This is indeed what happens in H 2 photoionization even for the lowest value of k e that arise from this formula (ℓ = 1) and also, as shown in [6] and [24], in H + 2 photoionization. In Fig. 3, it can be seen that the reduced electron continuum wave function at R e = 1.4 a.u (the equilibrium internuclear distance of H 2 ) associated with the ℓ = 1 partial wave is very similar in fashion to the 2pσ u orbital of H + 2 (with R e = 2 a.u.) in the region between both molecular centers. In turn, the reduced continuum wave function associated with the ℓ = 3 partial wave has the same nodal structure as the 4pσ u orbital in between the nuclei. This similarity between reduced electron continuum waves and the H + 2 molecular orbitals gives additional evidence for the image of partial-wave confinement. In Ref. [24], the minima in the 1 Σ + u cross section have been interpreted as Cooper-like minima similar to those found in atomic photoionization. The continuum wave functions of H + 2 were unfortunately compared with the 1sσ g orbital, which does not have the correct nodal symmetry imposed by the dipole selection rule. Moreover, this interpretation does not bring a simple prediction for the minima as the one given by our Eq. 18.

Non Franck-Condon effects
In what follows, we consider only non dissociative photoionization that is the most important contribution to the total cross section.
At low photon energies (e.g., 20 eV), the vibrational distribution of residual H + 2 follows approximately a Franck-Condon distribution, which predicts that the υ = 2, 3 vibrational levels should be the most populated ones.
In Fig. 4 we show cross sections differential in the vibrational energy of the residual H + 2 ion (viz. the reverse of the energy of the ejected electron) for the non dissociative case and for both parallel ( 1 Σ + u ) molecular orientations. It can be seen that the population of the final vibrational states does not follow a Franck-Condon behavior, even for a photon energy of 2 a.u.. At this energy, the ℓ = 1 partial wave is dominant for almost all υ, but, surprisingly, the ℓ = 3 partial wave dominates for the lowest υ's. As the photon energy increases reaching the region in which the ℓ = 1 partial wave has a ion in different vibrational states are shown by three dimensional plots in blue. For a better visualization, these distributions have been renormalized so that their maximum value is always 1. The polarization direction is indicated by the double arrow (green). The two nuclei are indicated by two small spheres (red). The angle-integrated cross sections, including the partial wave decomposition, as a function of the final quantum vibrational number υ are shown in the two-dimensional plots. minimum (see Fig. 2), the ℓ = 3 partial wave becomes dominant at higher and higher υ until, at energies above 6 a.u., the ℓ = 1 partial wave becomes again the dominant one. A more careful inspection of Fig. 4 shows that the ℓ = 1 partial wave has a minimum around υ = 5 at a photon energy of 3.5 a.u.. This is a photon energy that lies in the minimum of the cross section (see Fig. 2). Another minimum is clearly visible for the ℓ = 5 partial wave at photon energies of 16.5 and 20 a.u but now located around ν = 4. From this analysis one may conclude that this deviation from the FC predictions is related to the confinement effects discussed above.

Interference effects in the angular distribution of electrons
For the perpendicular orientation, one can see that the electronic emission is mainly produced, as expected, in the direction of the polarization vector. As the photon energy increases, additional lobes appear producing interference patterns analogous to the ones of the Young's double-slit experiment given by R sin θ e = nλ e [8].
In Fig. 5, we show the electron angular distributions for non-dissociative photoionization of H 2 for molecules parallel (Σ + u symmetry) to the polarization direction at a photon energy of 2.5 a.u.. The corresponding angular distributions are only shown for the case of H + 2 ions left in the v = 0, 2, 4, and 12 vibrational levels. The first three values of v correspond to vibrational levels that are close to or within the FC region, while the v = 12 vibrational level is clearly outside this region. The distribution has an almost perfect f shape (ℓ = 3) when H + 2 remains in a low vibrational level. As the vibrational quantum number of the remaining H + 2 increases, a complicated angular pattern emerges as the result of the interference between the ℓ = 1 and ℓ = 3 partial waves. For υ = 4, interferences lead to almost no emission in the direction of the polarization vector, whereas, for υ = 12, the distribution is almost p-like. In Ref. [8], this sudden change of the angular distribution with υ has been attributed to confinement because, around this photon energy, the ℓ = 1 partial component of the 1 Σ + u integrated cross section exhibits a pronounced minimum (see Fig. 2).
In the case of H + 2 photoionization, the angular distributions approximately follow the formula [25] (e µ · k e ) 2 cos 2 (k e · R/2). If e µ and k e are parallel to the molecular axis, this formula leads to zero when k e R = π, 3π, ..., i.e., no electron emission along the molecular axis in agreement with the image of confinement.

Conclusions
Photoionization of H 2 ionization by photons of a few hundred eV has been studied. By using Bspline functions to evaluate bound and continuum states and within the framework of the Born-Oppenheimer approximation, we obtain a realistic description of the molecular sates including its nuclear degrees of freedom.
The Cohen and Fano predictions were analyzed showing that they are valid at sufficiently high photon energies. Failures are related to electronic correlation and/or screening [6].
In the case of molecules oriented parallel to the polarization direction, the partial-wave cross sections exhibit pronounced minima at electron energies determined closely by k e R e = ℓπ, where ℓ is an odd number and R e is the equilibrium internuclear distance [7,8].
The fully differential angular distribution presents in a clear way the effects of confinement for molecules aligned parallel to the polarization direction. Young's double slit interferences are clearly visible in the perpendicular case when the electron wavelength is comparable to the internuclear distance [7,8] .
From the analysis of the vibrational distribution of the residual H + 2 ion for the dominant nondissociative channel, it is found that molecules parallel to the polarization direction transitions do not follow the typical Franck Condon (FC) distribution. Deviations from the FC distribution are more pronounced in the regions where confinement occurs. On the contrary, no deviations are observed for molecules perpendicular to the polarization direction [7].