Luca Argenti
email: argenti@physto.se

Roberto Moccia
email: moccia@ipcf.cnr.it

Nondipole effects in helium photoionization

Keywords: numbers: 32, 80, Fb

come

Introduction

For photoionization processes in the sub-keV X-ray energy range, the wavelength of the impinging radiation is significantly larger than the characteristic size of an atomic target in its ground state. This generally justifies the use of the time honored dipole approximation. In 1969, though, a net forward drift of the electrons ejected in xenon photoionization with lowenergy X-rays was detected [1]. For linearly polarized radiation, the dipole approximation (DA) predicts a symmetric angular distribution of the photoelectrons about the polarization vector. Beyond the DA, this symmetry no longer holds, possibly resulting in a net excess of photoelectrons along the direction of propagation of the impinging light. Therefore, the measurement reported in [1] was readily recognized as the first clear evidence of a deviation from the DA in atomic photoionization at low energies [2]. This and other similar results [3] were the basis for a systematic investigation of nondipole effects at energies so low that the wavelength of the radiation is orders of magnitude larger than the characteristic size of the target atom. The limits of the DA are emphasized whenever the dipole transition amplitude is vanishingly small, as it may occur near the zeroes of Fano profiles, or near Cooper minima, Confidential: not for distribution. Submitted to IOP Publishing for peer review 8 October 2010 a common feature in the spectra of the heavier rare gases. Large nondipolar transition amplitudes, found for example near quadrupolar resonances, also cause the breakdown of the DA. Indeed, resonant nondipole effects at low photon energies can be comparable to nonresonant ones in the keV energy region [4,5]. Nondipole effects were eventually reported also in the photoionization of helium [6], the lightest of all rare gases and therefore arguably the least prone to give rise to observable nondipole effects. In that work, the authors were able to reveal the detailed resonant profile of the backward/forward asymmetry parameter γ [7] in the vicinity of a dipole forbidden quadrupole-allowed resonance lying close to one which is dipole allowed.

In this paper we present the results of an accurate calculation of the asymmetry of the photoelectron angular distribution in the photoionization of helium in a large energy range below the N=2 threshold comprising several closely spaced 1 P o and 1 D e doubly excited states. The results for the lowest couple of these resonances compare fairly well to the available experimental data [6]. Discrepancies between theory and experiment, though, are found in correspondence to the minimum of the total cross section σtot, where the spectrum of γ shows a strong, spiked feature. It is argued that the observed discrepancy is of instrumental origin, due to the particularly strong dependence of γ on the experimental uncertainties on σtot whenever the latter is vanishingly small. Indeed, with a realistic uncertainty for σtot near its minimum, the agreement between theory and experiment becomes very satisfactory.

Theoretical method

The differential photoionization cross-section of helium, with the parent ion He + being left in the state α and the photoelectron having linear momentum ke, is given by

dσα dΩ = 4π 2 cω X mασασ | ψ (-) α, ke σ |ˆ • 2 X i=1 e i k• r i pi|φ0 | 2 , (1)
where ω = ck is the photon energy (atomic units, = 1, me = 1, e = 1, will be used throughout unless otherwise stated), ˆ is the polarization of the impinging light, φ0 is the initial atomic state (average over the spin and spatial orientation of the initial state is omitted as we focus on the photoionization of the 1 S ground state); mα and σα are the magnetic quantum number and the spin projection of the residual electron in the parent ion α; σ is the spin projection of the photoelectron, and ψ (-) α, ke σ is the final single-ionization scattering state, fulfilling incoming boundary conditions [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF]. If the exponential in the transition operator in Eq. ( 1) is expanded in a Taylor series

exp(i k • r) 1 + i k • r + 1 2 (i k • r) 2 + . . . (2)
the transition amplitude results in a sum of terms of increasing order in the photon wavenumber k:

∞ X n=0 1 n! ψ (-) α, ke σ |ˆ • 2 X i=1 " i k • ri " n pi|φ0 . (3)
The ground state of helium is strongly localized, with a characteristic size of 1 a.u. Thus, for photon energies of the order of one atomic unit, the wavenumber k is of the order of 0.01 a.u. and the series (3) converges rapidly. The dipole approximation is obtained by retaining the first term only (E1), which does not depend on k, while the largest correction to the dipole approximation comes from the second term, linear in k [START_REF] Bechler | [END_REF]10] which yield two terms identified as the magnetic dipole moment (M1) and the electric quadrupole moment (E2).

The largest corrections to the photoionization cross section, beyond the pure electric dipole E1-E1, arise from the interference terms: electric dipole/magnetic dipole E1-M1 and electric dipole/electric quadrupole E1-E2. The correction to the photoelectron angular distribution due to these terms is proportional to k. If only the terms beyond the pure dipole which are linear in k are retained, the partial differential cross section (PDCS) is fully described by just two additional nondipole parameters beyond the well known dipole total cross section σ and asymmetry parameter β. Several equivalent parametrization of the PDCS have been proposed in the past [2,[10][11][12][13][14][15][16][17]; the most common parametrization factors out the total cross section [15] and reads, for a linearly polarized radiation:

dσ dΩ = σtot 4π ˘1 + β P2(cos θ) + ˆδ + γ cos 2 θ ˜sin θ cos φ ¯, (4)
where θ is the angle between the direction of observation Ω and the polarization vector ˆ , while φ is the angle between the planes spanned by the vectors (ˆ , k) and (ˆ , Ω). The case which is relevant here is the photoionization of the ground state of helium below the N=2 threshold. Since the ground state of He is an S state, β = 2 identically and the transition matrix element due to the magnetic dipole is zero. The only contribution linear in k comes from the E1-E2 interference term. According to the notation of the Appendix, the parameter γ has the following expression

γ = k -12 √ 2 π 2 cω Re " φ0 V0,0 1 ψ (-)P o 1s Ep φ0 V1,0 2 ψ (-)1D e 1s E d * « σtot (5)
which is divergent at the possible zeros of σtot. In the present case, if only the E1-E1 contribution to σtot is retained, such divergence occurs close to each and every 1 P o resonance below the N=2 threshold. It may be noted, however, that the measurement of the asymmetry parameter γ involves two separate experimental determinations: that of the total cross section σtot, on the one side, and that of the difference between the cross sections for the photoelectron being ejected in the forward and in the backward direction with respect to k, on the other side. Therefore, to compare with experiments, the finite resolution of the experimental equipment should be taken into account. This was accomplished by the usual method of convoluting separately the calculated values of the numerator and of the denominator of eq. 5 with a normalized Gaussian function with an appropriate width. In fact, as detailed in the next section, a wide range of reasonable values for the experimental width were used. Yet, for energies very close to the minimum of σtot, the agreement between the resulting γ(E) and the experiment did not show any sizeable improvement. Since, in the neighborhood of very small values of σtot, γ(E) is very sensitive to the accurate evaluation of the total cross section, the attention was focused on the possible shortcomings of the expression used to calculate σtot.

As discussed in the next section, in order to improve the agreement with the experiment, the value of σtot close to its minimum should be larger than its dipolar value. Thus, we considered those terms in σtot which are neglected in the DA. The largest contributions arise from: i) the terms proportional to k 2 ; ii) the singlet component of the dominant 3 P o J=1 triplet state due to spin-orbit coupling. With the notation introduced in the appendix, the expression for the total cross section truncated after the electric quadrupole contribution and retaining only the terms up to

k 2 is σtot = 4π 2 3cω » ˛ φ0 V0,0 1 ψ (-)P o 1s Ep ˛2 + 9 10 k 2 ˛ φ0 V1,0 2 ψ (-)D e 1s E d ˛2 + 2k 2 φ0 V0,0 1 ψ (-)P o 1s Ep ψ (-)P o 1s Ep V0,1 1 + 1 √ 2 V2,0 1 φ0 - . (6)
In the energy region where the pure electric dipole amplitude E1 φ0 V0,0

1 ψ (-)P o 1s
Ep is vanishingly small, the second line in Eq. ( 6) can be neglected, so that the most important contribution is due to the E2-E2 term ˛ φ0 V1,0

2 ψ (-)D e
1s E d ˛2. The contribution of the higher multipoles to the total cross section can of course be safely ignored. As mentioned above, another possible contribution to the total cross section may come from the mixing between closely spaced doubly excited states of different multiplicity, due to the spin-orbit interaction [18][19][20][21][22][23][24]. It is well known, for example, that high values of for the external electron may induce a sizable mixing of the triplet and singlet manifolds [25,26]. In the present case, though, the orbital angular momentum involved is essentially = 1, so the spin-orbit mixing is rather small. If c1 and c2 are the coefficients of the 1 P o and of the 3 P o in the dominant 3 P o J=1 state around an isolated resonance, c1 is proportional to the ratio between the spin-orbit matrix element between resonances with different multiplicity and the energy difference between the closest singlet and triplet doubly excited states. The order of magnitude of the former is arguably the same as the fine structure of the lowest 1 P o J multiplet, that is ∼ 10 -5 a.u. [27], while the energy difference between doubly excited states for principal quantum numbers of the outer electron n ≤ 8 is of the order of 10 -2 a.u. [20]. This means that the triplet contribution to the background cross section can be estimated to be six orders of magnitude smaller than the singlet-dipole background, and is therefore completely negligible.

Results and discussion

The continuum wave functions necessary to evaluate the electric dipole and electric quadrupole amplitudes in Eqs. ( 5) were computed with the B-spline K-matrix method. In the following we will summarize it briefly. Full details can be found in the literature [28,29].

A stationary wave function ψ P αE in the continuum is written as linear combination of partial-wave channel (PWC) functions φ βE and of localized channel (LC) functions φj.

In each PWC, a well defined state of the parent ion is coupled to a single-electron wave function with assigned angular momentum, to give the required total angular momentum and spin. The radial part of the single-electron function is otherwise free. The LC is formed by antisymmetrized products of localized orbitals, and it serves the purpose of completing the description of the many-particle wave function at short radii. The hamiltonian is separately diagonalized in all the PWC's and in the LC. In this way, a complete set of independent stationary solutions to the secular problem at any given total energy E can be sought in the form

ψ P αE = φαE + X γ X Z d φγ P E - Kγ ,αE (7)
where K is the off-shell reaction matrix [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF]. The index α runs over all the channels that are open at the energy E, while the index γ runs over all the available channels (open and closed) including the LC. The requirement φ β |E -H|ψ P αE = 0 ∀β, , translates into a system of integral equations for the off-shell K matrix, which can be discretized and solved numerically.

The scattering solutions ψ ± E = (ψ ± 1E , . . . , ψ ± nE ) fulfilling outgoing/incoming boundary conditions are obtained through the transformation

ψ ± E = ψ P E [1 ± iπK(E)] -1 , ψ ± E |ψ ± E = δ(E -E ) 1.
For the phase conventions on the ψ ± states, see the appendix. Resonance parameters are determined from the positions of the poles of the on-shell scattering matrix S = (1 -iπK)(1 + iπK) -1 in the lower half of the complex energy plane. The radial component of all the single-particle wave functions are expanded on a B-spline set. B-splines of order 7 and a maximum orbital angular momentum max = 8 ensured a good accuracy. The radial space is partitioned in a localized sector and in a continuum/Rydberg sector. The former is defined on a subset of the spline space where the knots are optimized so as to reliably represent short-range correlations and bound states of the parent ion. The latter is capable of representing both the oscillating tail of the continuum states and the long range behavior of the Rydberg satellites up to a principal quantum number n ∼ 20. The localized sector comprises 33s, 32p, 31d, 30f , 29g, 28h, 27i, 27j, 27k orbitals, confined within a radius R1 ∼ 70 a.u. The diffuse states are such that the continuum part is reliably represented up to a radius R2 ∼ 170 a.u. while the Rydberg satellites stretch outwards up to few hundreds a.u.

Since photoelectron anisotropy parameters are known to be rather delicate quantities, it is important to ensure that all the calculated quantities that influence them are satisfactorily converged. To this end, an extensive set of pertinent results was checked by comparison with the most accurate theoretical and experimental data available in the literature.

Bound states

The energies of the hydrogen-like parent ions involved are virtually exact (within the electrostatic approximation), the error with respect to the hydrogen-like series for all the states involved being less than 2 • 10 -10 a.u. The energy of the ground state is E = -2.903679895 a.u. (accurate electrostatic limit -2.903 724 377 a.u. [30]). This value is intermediate between the accurate extrapolated limit for max = 7, E = -2.903 674 59 and for max = 8, E = -2.903 689 47 a.u. [31].

To ascertain the capability of the basis set to represent the far reaching tails of the more excited Rydberg states and, arguably, of the doubly excited states below the N=2 threshold, the energies of the first ten Rydberg states in the 1 S, 1 P o , 1 D e manifolds were compared with accurate electrostatic values reported in the literature [START_REF] Drake | Atomic, Molecular, & Optical Physics Handbook[END_REF] (par. 11.3). Since the values agree up to the ninth digit, the long range tail of the Rydberg satellites is certainly well represented.

Another stringent test is the comparison of the oscillator strengths for the transitions between the ground state and the first few 1 P o Rydberg states with the corresponding most accurate values available in the literature [START_REF] Drake | Atomic, Molecular, & Optical Physics Handbook[END_REF] (par. 11.5). Also for these quantities, a very satisfactory agreement, to the fifth or sixth digit, is found. Furthermore, the matrix elements obtained in the length and acceleration gauges differ from those in the velocity gauge, on average, by only ∼ 0.003% and ∼ 0.01% respectively. Therefore, it can be concluded that, at least for the bound states, even the correlated short range part of the wave function is accurately represented.

Doubly excited states

The resonance series are classified according to the Stark-Lin notation [N1N2m] A . See the original paper by Lin [START_REF] Lin | [END_REF] for tables of approximate quantum numbers for the autoionizing series in helium, and the review by Rost et al [34] for a comparison between different classification schemes. The resonances of the 1 P o and of the 1 D e manifolds below the N = 2 threshold were calculated up to energies very close to the threshold (n ∼ 15). The energies and widths of the autoionizing states are obtained with a fit of the total phaseshift as a function of the energy, as detailed in [35]. As an example, in Tables 1 and2 our results for the first few [001] + and [010] -resonances of the 1 P o manifold and the first few [010] + and [001] 0 resonances of the 1 D e manifold are compared with the best data from the literature. A similar accuracy is found also for the higher terms (n ≤ 15) in all the 1 P o and 1 D e resonance series. The much narrower [100] 0 1 P o and 1 D e resonances are not listed since, at the experimental resolutions considered in this work, they are not visible. On the other hand, for photon energies above 64eV, the resonant features in the nondipole anisotropy parameter are essentially a repetition of those between 62eV and 64eV. Therefore, the listed resonances are sufficient to interpret all the qualitative features of the spectra examined in the present work. The agreement of energies, widths, and the q parameters of the Fano profiles is very good -→[...](the energies, both from our calculations and from the literature, which were available in atomic units, have been converted to electronvolts by means of the transformation E(eV)= (E(a.u.)+2.90369376)*27.20765438, where 27.20765438 is the Rydberg constant for the He + -e -system).

Photoionization spectra As anticipated in the the previous section, the γ parameter is particularly sensitive to the precise value of σtot in the neighborhood of the total cross section minima. For this reason, we also report the parameter σ drag , defined as [38], Lindroth (1994) [39], Ho and Bhatia (1991) [40], and DeHarak et al [41] (experimental results).

σ drag = k • Z dσ dΩ Ω dΩ, (8)
[ which quantifies the net photoelectron flux in the forward direction. In the simple case studied here, β = 2, δ = 0, so σ drag = (4π/15)σtotγ is the only anisotropy parameter required for an alternative parametrization of the differential cross section where the total cross section is not factored out:

dσ dΩ = 3 4π σtot cos 2 θ + 15 4π σ drag cos 2 θ sin θ cos φ, (9)
and therefore it does not have the same strong dependence on the experimental uncertainties of σtot as γ.

Fig. 1 shows all the quantities involved in the evaluation of the nondipole anisotropy parameter measured by Krässig et al [6]: the dipole total cross section (Fig. 1.a), the quadrupole total cross section (Fig. 1.b), the drag cross section (Fig. 1.c), and the γ parameter itself (Fig. 1.d). In Fig. 1 each quantity has been convoluted with several Gaussian weight functions in order to simulate the effect of an experimental slit function on the measured quantities. The following fwhm have been used: 20, 30, 40, 50, 60 meV, the smaller widths corresponding to the sharpest profiles. In Fig. 1.d, the ratio in the expression ( 5) is taken after the numerator and the denominator are independently convoluted with the slit function, with a given fwhm, assuming a zero background. The present spectra for γ are compared with the experimental values by Krässig et al [6], obtained by digitalizing the plot in the original paper (the error introduced by the digitalization is significantly smaller than the experimental error bars). In the total cross section shown in Fig. 1 For a closer comparison with the experimental γ spectrum (Fig. 1.d), our theoretical prediction was shifted at lower energies by 10 meV, well within the energy uncertainty of 20 meV declared by the authors. In order to separately assess the accuracy of the energy calibration of the present theoretical spectra, we compared the total cross section with the experimental spectra by Schultz et al [36], with an absolute uncertainty in the energy of just 1.7 meV, in the region of the n = 20, 21, 22 multiplets, where the experimental points could be accurately extracted from the original published plots. With an energy shift of just 0.5 meV, well below the experimental uncertainty, our spectrum is in excellent agreement with the experimental one, see Fig. 2. Experimental data of comparable accuracy at lower energies are not readily available, but it is not unreasonable to assume a similar precision therein as well.

The comparison of the calculated γ with the experimental values of Krässig et al [6] shows a fairly good agreement except at the energies close to the minimum of σtot. In the small region between 60.15eV and 60.3 eV, and particularly above 60.2 eV, none of the present convoluted theoretical profiles is really in good agreement with the experimental data. In particular, the theoretical values corresponding to resolution of 20 meV, as declared in [6], display an exaggeratedly spiked behavior. This is most likely due to a considerable difference between the theoretical and the experimental values of the total cross section in that critical region.

A comparison between Figs. 1.a and 1.b shows that, on average, the dipole cross section dominates over the quadrupole total cross section by as much as four orders of magnitude, as expected. As a consequence, it turns out that, with the present experimental resolution, the contribution of the E2-E2 term to σtot close to the minima of the latter is completely negligible with respect to the contribution due to the finite energy resolution itself. In fact, we estimated that in order to be able to resolve the contribution of the E2-E2 term, a fwhm of 1 meV or less would be required. In conclusion, the contribution of the quadrupole cross section to the total cross section near its minima is not really of any relevance whenever the energy resolution, as in the present case, is worse than few meV. The cause of the observed discrepancy must be sought elsewhere. [36]. The agreement is very good as the uncertainty in the experimental energy calibration in this energy region is declared to be 1.7meV.

A possible origin of the discrepancy may be a background signal of entirely instrumental origin. As a matter of fact, a background signal in the measurement of the total cross section due to experimental offsets, like stray light, is generally to be expected. For example, in a measurement of the helium total photoionization cross section below the N=2 threshold performed in 2003 [42], a background cross section as large as 0.16 Mb (i.e. just one order of magnitude smaller than the non-resonant cross section), is reported. In order to test the effect of a possible uncertainty in the experimental values of σtot, in Fig. 3 the experimental γ parameter is compared with two theoretical curves, both with an energy resolution of 40 meV: one which assumes no background (dashed curve, green on-line), the other which assumes a constant background equal to 0.2 Mb (continuous curve, blue on-line), which is comparable to that reported in [42]. With the inclusion of such background, the large disagreement with the experimental results in the critical region disappears. Moreover, the inclusion of the background improves the agreement also in the non-resonant regions.

In [43] the measurement of γ reported in [6] are compared to a semiempirical, convoluted profile for the non-dipole parameter. The resonant feature in the non-dipole amplitude used in such profile is obtained by including in the theoretical RPAE background a Fano profile whose parameters are taken from the literature. The shape of the bandpass is unspecified, while the resonant profile in the xenon spectrum reported in Fig. 4 seems to confirm the existence of a non-zero background signal. In [43] the agreement found at photon energies around 60.2 eV is good, while close to the the 1 D e resonance large deviations are observed. At variance with this latter finding, our fully theoretical calculation is in very good agreement with the experiment in that region.

In the present case, near the energies where the total cross section almost vanishes, it may be presumed that the details of the peaks displayed by γ might depend upon the experimental uncertainties of the energy resolution and of the background. The drag cross section reported in Fig. 1.c appears to be much less sensitive to the experimental resolution. In particular, it is not affected by the uncertainty on the offset of σtot.

Sizable resonant effects in the nondipole asymmetry parameter due to the other terms belonging to the [001] + 1 P o and [010] + 1 D e series, as well as to the terms belonging to the [010] -1 P o and [001] 0 1 D e , should be visible at higher energies. In Fig. 4, the dipole (a), the quadrupole (b), and the drag cross section (c) are reported below the N=2 threshold, convoluted with a Gaussian weight function with fwhm = 5 meV. Already at this resolution, which is reasonable by today's standards, several other multiplets should be clearly visible. Figure 5 shows a detail of the same quantities close to the threshold.

Summary

We computed the nondipole anisotropy parameter γ in the photoionization of helium below the N=2 threshold; we compared our results with the experimental data by Krässig et al [6] between 59.5 eV and 60.6 eV. In general, fairly good agreement between theoretical and experimental results was found, except near the minimum of the total photoionization cross section at ∼ 60.2 eV. An analysis of the approximations involved excluded any significant role of either electric quadrupole E2-E2 effects, or of the spin-orbit mixing between states of different multiplicity. The observed discrepancies might be instead attribuited to the experimental uncertainty in the total photoionization cross section around its minimum.

In fact, because of its very definition, the nondipole parameter γ is very sensitive to the uncertainties in the total cross section when the latter attains vanishingly small values. The agreement between theory and experiment is indeed completely restored when a realistic value for the background signal in the measurement of σtot is assumed. A similar behaviour of the nondipole anisotropy parameter is to be expected also at energies closer to the N=2 threshold. In the present work, several resonant structures arising from the interplay between 1 P o and 1 D e doubly excited states have been calculated and should be experimentally detectable. For those cases which are similar to the one considered here, that is whenever the total cross section takes on values smaller than the instrumental sensitivity, it is advisable to use a parameter other than γ (e.g. the drag photoionization cross section) to measure the forward/backward anisotropy of the photoelectron distribution.

In the LS coupling when V is used in the expression for the partial differential cross section, Eq. ( 1), the partial differential single photoionization cross section, averaged over the orientations of the initial target state φ0 and summed over the orientations of the final parent ion state α and the spin projection of the photoelectron, may be written as Here k = [2(E -Eα)] 1/2 and Φ Γ α, is obtained by coupling φα to the angular and spin component of the N -th electron so as to give a total symmetry Γ.

In the total cross section, only the square moduli of the individual multipoles appear:

σα,E = 2π 2 c ω(2L0 + 1)

X T L α  ˛q T 2T +1 φ0 V T +1 T ψ (-)Γ αE α + q T +1 2T +1 φ0 V T -1 T ψ (-)Γ αE α ˛2 + + ˛ φ0 V T T ψ (-)Γ αE α ˛2ff , (A.8)
where the first square modulus refers to the electric multipoles, while the second refers to the magnetic multipoles. For photoionization processes in which only photons with large wavelengths are involved, the transition matrix elements can be conveniently expanded in powers of the argument kr of the spherical Bessel function [45]

V J T τ = ∞ X ν=0
V J,ν T τ k J+2ν where (A.9) .10)

V J,ν T τ = N X i=1 √ 4π i J r J+2ν i Y J T,τ (i) (-2) ν ν!(2J + 2ν + 1)!! . ( A

2 1 2 1

 22 .a, only the well isolated Fano profile of the [001] + P o resonance is evident. The closely lying [010] + D e resonance is visible in the much smaller electric-quadrupole contribution to the total cross section, shown in Fig. 1.b. The effect of both resonances is, on the contrary, apparent on the drag cross section shown in Fig. 1.c as well as in the γ reported in fig. 1.d.

Figure 1 .

 1 Figure 1. Parameters involved in the calculation of the nondipole asymmetry parameters in helium photoionization below the N=2 threshold. a) dipole total cross section, b) quadrupole total cross section, c) drag cross section, d) nondipole asymmetry parameter γ. Each parameter has been convoluted with Gaussian weight functions with five different fwhm: 20, 30, 40, 50 and 60 meV. In d) the theoretical prediction is compared with the experimental data by Krassig et al [6], with a declared uncertainty of 20 meV.

Figure 2 .

 2 Figure 2. Energy calibration. Continuous (red) line: present results convoluted with a Gaussian function with fwhm=1 meV and shifted 0.5 meV at higher energies. Dashed (blue) line/points: experimental data by Schulz et al digitalized from[36]. The agreement is very good as the uncertainty in the experimental energy calibration in this energy region is declared to be 1.7meV.

Figure 3 . 2 1 2 1

 322 Photon Energy (au)

Figure 4 .Figure 5 .

 45 Figure 4. Dipole (a), quadrupole (b), and drag (c) cross sections below helium N=2 threshold.

(

  normalized continuum wave function with incoming boundary conditions, energy E, total angular momentum L, parity Π and spin S specified by the collective index Γ. To be explicit, if the parent ion α, in state φα, has charge Z and energy Eα, ψ (-)Γ αE is defined in such a way that, for large values of one of the N electron radial variables, say rN , its only outgoing component is h ψ

Table 1 .

 1 Parameters for the first few [001] + and [010] -1 P o resonances in helium, below N=2 threshold (in brackets we report the dominant configuration of the lowest lying resonance). The present values (first rows) are compared with those by Rost et al[34] and to the experimental values bySchulz et al (1996) [36] and byDomke et al (1996) [37].

	[N 1 N 2 m] A n	Ref. E (eV)	Γ(meV) q
	[001] + 2	60.144 650	37.359	-2.770
	(∼ 2s2p)	[34] 60.144 121	37.364	-2.77
		[36] 60.1503(40) 37.6(2)	-2.73(4)
		[37] 60.147	37	-2.75
	[001] + 3	63.655 354	8.194	-2.580
		[34] 63.655 261	8.195	-2.58
		[36] 63.6575(30) 8.3(5)	-2.53(4)
		[37] 63.658	10	-2.5
	[010] -3	62.757 719	0.10460 -4.252
		[34] 62.757 719	0.1046	-4.25
		[36] 62.7610(20) 0.11(2)	-4.1(4)
		[37] 62.758	0.5(3)	-3.5

Table 2 .

 2 Parameters of the first few [010]

+ and [001] 0 1 D e resonances below the N=2 threshold (in brackets we report the dominant configuration of the lowest lying resonance). The present values (first rows) are compared with experimental and theoretical results by other authors:

Chen (1999)

  N 1 N 2 m] A

	n	Ref.	E(eV)	Γ(meV)	q
	[010] + 2 (∼ 2p 2 )	59.9048 [38] 59.9050	64.319 64.24	-0.030
		[39] 59.9044	64.26
		[40] 59.9044	64.267
		[41] 59.9038(8) 52(21)
	[010] + 3		63.5156	15.129	0.249
		[38] 63.5157	15.17
		[39] 63.5156	15.1
		[40] 63.51553	15.1
		[41] 63.515(9)	12(8)
	[001] 0 3		63.8636	0.54530 -2.102
		[38] 63.8636	0.5491
		[39] 63.864	0.544
		[40] 63.86353	0.547
		[41] 63.855(8)	8(8)

Acknowledgments

L.A. acknowledges the financial support by the Göran Gustafsson Foundation and the Swedish science research council (VR).

Appendix A. Differential cross section

In this appendix a general expression is derived for the atomic partial differential cross section for the absorption of one photon. The transition operator for the absorption of a photon by an atom, within the field of a monochromatic plane electromagnetic wave, is

where ˆ indicates the polarization. Using the well known expansion of a plane wave in spherical components [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF], Eq. (A.1) is readily expanded in electric (J = T ±1) and magnetic (J = T ) multipoles:

where the irreducible transition operator V J T τ is defined as

Here jJ (kr) is a spherical Bessel function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], Y M J = Y * JM , and the vector spherical harmonics and tensor product conform to the following definitions [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF][START_REF] Manakov | [END_REF]]


K Kooser

D T Ha

S Granroth

E Itälä

L Partanen

E Nõmmiste

H Aksela

E Kukk

Resonant Auger electronphotoion

Keywords: numbers: 33.80, 33.20, 33.15, 31.15

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Acrylonitrile (also known as cyanoethylene, propenenitrile, and vinyl cyanide) may be regarded as a derivate of ethylene (C 2 H 4 ) by substituting one of the hydrogen atoms with the cyano group (C ≡ N ) or in other words it is olefinic nitrile containing one double bonded C = C moiety together with a triple bonded CN group. This molecule is an important industrial chemical: it is widely used as a monomer or comonomer for synthetic fibers, resins, plastics, elastomers and rubber. Acrylonitrile contributes heat, chemical, solvent, and weathering resistance to polymers. It is used in fumigants as well. On the other hand acrylonitrile plays an important role in astrochemistry, since this molecule and its fragments caused by cosmic radiation are present in interstellar medium. This has been a stimulus for investigations of the ion-molecule chemistry of acrylonitrile and its fragments [1,2,3,4]. There is a number of experimental and theoretical studies concerning the inner-and valence-shell excitations [5,6,7,8,9,10] and the core-level photoemission of gaseous and adsorbed acrylonitrile [11,12,13,14,15,16] as well. In the case of solid acrylonitrile the non-radiative (electronic) decay channels at the nitrogen K-edge have been studied by resonant Auger electron spectroscopy (RAES) [17] and for the gas phase acrylonitrile have been published the mass spectrometric investigation results of ionization and appearance energies for all detected cations [18]. Recently the kinetic energy release distribution for the metastable decay reactions of ions formed by electron impact on acrylonitrile has been reported [19]. The possible molecular and radical elimination channels in acrylonitrile photodissociation in the vacuum-ultraviolet region have been investigated by many different experimental techniques -flash photolysis-kinetic absorption spectroscopy [20], laser-induced fluorescence spectroscopy (LIF) [21], transient frequency-modulation spectroscopy [22], photofragment translational spectroscopy (PTS) [23], time-resolved Fourier transform infrared emission spectroscopy [24]. Also the theoretical calculations of reaction pathways for the unimolecular fragmentation of acrylonitrile in the ground state were thoroughly examined and it was found that the H-atom migration reactions are rapid and influence significantly the following dissociation processes [25,26].

In the present study, the photodissociation and possible geometrical rearrangements of acrylonitrile molecule induced by the core level electron excitations are explored and discussed depending on the character of the preceding electronic relaxation channels and/or the kinetic energy of ejected Auger (or photo-)electrons. The experimental data of recently published article about the fragmentation patterns of doubly charged acrylonitrile following C 1s core ionization has been collected with the same experimental setup as the spectra of present work [27].

Next, we present the background to the processes involved. Firstly, consider the evolution of events following the interaction with soft x-rays. The subsequent description is confined to core-level photoexcitation only, as is the case of this study. The photoexcitation of a molecule promotes an electron from the core level to an initially unoccupied orbital creating a high-energy intermediate core-excited state, the lifetime of which is a few femtoseconds. The subsequent relaxation process leads to the autoionization of the excited neutral molecule and may be followed by photodissociation of the molecular cation (see Figure 1):

M + hν → M * → m + + n + e - Auger ,
where M * is the excited molecule and e - Auger is the resonant Auger electron with the characteristic kinetic energy. The dissociation products in the present case are a fragment cation and a neutral fraction (m + and n, respectively). In terms of energy redistribution, the above process is represented as

E int + EX = E hν -E kin (e - Auger )
, where E int is the internal energy of the system (molecular ion or ionized and neutral fraction) and EX is the adiabatic ionization energy of the neutral precursor (or appearance energy of an ion fragment). This formula implies that it is possible to extract very versatile information from the electron-energy-resolved (E kin ) electron-ion coincidence experiments about the formation and fragmentation dynamics of molecular cations formed by single ionization process (see, e.g., refs. [28,29] and articles in ref. [START_REF]Frontiers of Coincidence Experiments[END_REF]).

We have investigated the photodissociation of acrylonitrile (C H H 2 = C C H -C N ≡ N ) molecules following resonant core excitations of nitrogen and carbon 1s electrons to the molecular orbital of the localized π * C≡N character using the electron-energy-resolved photoelectron-photoion coincidence (PEPICO) technique (the subscript letters used in the acrylonitrile formula indicate the non-equivalent carbon atoms). Also the total ion yields at the N and C 1s absorption edges were recorded in order to determine the exact energy values of the corresponding excitation. In order to unravel the ambiguous pairs in the mass spectrum, also the experimental data from triply deuterated (CD 2 = CD -C ≡ N ) and singly deuterated (CH 2 = CD -C ≡ N ) acrylonitrile samples was recorded.

In addition, ab initio molecular orbital calculations for appearance energies of dissociated fragments were carried out and the local equilibrium and the transition conformations of the acrylonitrile molecule have been calculated for some dissociation pathways.

Experimental setup

The detailed description of the joint PEPICO setup of Oulu and Turku Universities used for the coincidence measurements is given in Reference [31]. In brief, the experimental PEPICO setup consists of a modified Scienta SES-100 electron energy analyser [32] and a homemade Wiley-McLaren-type ion time-of-flight detector with a 400 mm long drift tube. The original CCDcamera-based detection system of the electron spectrometer was replaced by a resistive anode detector (Quantar) in order to achieve fast single electron position-sensitive response signal in nanosecond scale. The ion time-of-flight(TOF) spectrometer is equipped with a 77 mm active area Hamamatsu MCP detector followed by 10 concentric anode rings to enable a registration of the longitudinal and radial time-of-flight components of ions relative to the symmetry axis of the ion spectrometer tube. The ion detection electronics is based on a 1 GHz waveform digitizer card (Signatec PDA 1000). The PEPICO measurements were performed in pulsed field mode, where the detection of electron by the electron analyzer triggers the clock for the TOF of the ions (true coincidences), within a certain time window. However, the ion TOF spectrometer detects also fragments (false coincidences or noncoincidences) which are not created from the initially ionized molecule. For extracting the coincidence and noncoincidence events the experimental setup contains also an additional pulse generator which provides artificial extraction pulses at a constant frequency. During the PEPICO experiment both -the electron and artificially triggered events, and the coincidence and noncoincidence data maps, respectively were recorded. Finally, a custom made Igor Pro software based data analysis macros [START_REF] Kukk | Spectral Analysis by Curve Fitting Macro Package /SPANCF[END_REF] were used to subtract the false coincidence background from the initial coincidence map and retrieve the real coincidence data pattern. The experiment was executed at the undulator based soft X-ray beamline I411 at MAX-II synchrotron radiation storage ring (Lund University, Sweden) [START_REF] Bässler | [END_REF]. The synchrotron radiation coming from undulator was monochromatized by a Zeiss SX-700 plane grating monochromator. The PEPICO system is operated in the pulsed extraction field mode and in the present experiment the extraction pulse voltage was ±100 V across the sample region, with the drift tube held at -525 V. The entrance slit of the electron spectrometer was 1.6 mm and the pass energy was 200 eV, which gives a resolution 1.2 eV. In order to minimize the signal of the false coincidences the low counting rates of about 10 electrons/s have been used. The average number of ions per electron trigger was 0.8 and number of ions per artificial trigger 0.2 (including detector noise counts). The base pressure in the end-station chamber was less than 5 × 10 -7 mbar. During the experiment the chamber pressure was maintained below 5 × 10 -6 mbar. The measurements were made at room temperature. The samples of acrylonitrile and its isotopologues were purchased from Sigma-Aldrich with purity greater than 98 %. The further purification process was performed only through the degassing the liquid sample by multiple freeze-pump-thaw cycles before leading the sample vapor into the measurement chamber.

Results

Photoexcitation and resonant non-radiative decay

The acrylonitrile molecule is planar in its electronic ground state and belongs to the C S point group. In the following notations the z axis is chosen perpendicular to the plane (x, y) of the molecule. The occupied valence configuration of the ground electronic state of acrylonitrile is given by 5a 2 6a 2 7a 2 8a 2 9a 2 10a 2 11a 2 1a 2 12a 2 2a 2 and the energy ordering of the unoccupied molecular orbitals is:

(3a ) 0 (13a ) 0 (4a ) 0 .
According to the earlier studies [35,[START_REF] Kimura | Handbook of He I Photoelectron Spectra of Fundamental Organic Molecules[END_REF]6,9,8] the orientation, localization and orbital type of these states are as follows:

the bonding MOs :

1) 5a , 6a , 7a -inner valence σ character MOs, 2) 8a -s-type orbital with C 2s character localized in cyano group, 3) 9a , 10a , 11a -correspond to σ C-C , σ CH 2 and σ C≡N , respectively, 4) 1a -mixed character of π z (C ≡ N ) and π z (C = C), 5) 12a -localized π xy (C ≡ N ) character, 6) 2a -delocalized conjugation of π z (C ≡ N ) and π z (C = C) orbitals;
the antibonding MOs :

7) 3a -π * z orbital, delocalized along the C H = C C -C N ≡ N chain, 8) 13a -π *
xy orbital, localized on cyano (C ≡ N ) group and 9) 4a -the highest π * z orbital, delocalized along the C H = C C -C N ≡ N chain. On the basis of the character of abovementioned MOs and theoretical calculations [8], it is clear, that core electron excitations to the initially unoccupied 13a orbital can take place mostly from the C N and N 1s orbitals, since there is considerable overlap only between the 1s wavefunctions of these atoms and the final wave function of the excited electron. It is also noteworthy to mention that only these core level transitions (C N (and

N ) 1s → π * xy (C ≡ N )) have localized nature.
In the total ion yield spectra of acrylonitrile at the nitrogen and carbon K-edges (Figure 2), we observed strong resonances at 399.84 eV and at 286.80 eV which correspond to the transitions

N 1s → π * xy (C ≡ N ) and C N 1s → π * xy (C ≡ N )
, respectively. The beamline monochromator resolution was 0.3 eV and 0.4 eV for incident radiation at corresponding C and N K-edges.

Resonant Auger decay spectra of these two excitations are shown in Figure 3 in comparison with a direct photoionization spectrum taken at ionization energy of 60 eV and with 25 meV photon energy resolution. Both resonant decay spectra possess a similar structure with two clearly separated maxima. The structure in the binding energy range between 11 and 15 eV corresponds to the participator transitions, where the photoexcited electron is directly involved in the deexcitation process leading to the single-hole final states as in the direct ionization from 11a and 12a orbitals. The high binding energy side of the deexcitation spectra has a broad structure from 16 up to 30 eV with the maximum at around 21 eV. In this region the nonresonant electron energy spectrum has only weak inner valence and electron-correlation satellite ionization states. This spectral region can be attributed to the different spectator decay channels, which are in more detail described in Ref. [17]. This decay channel leads to a two hole-one electron final state where the excited electron acts as a localized spectator and screens the final two hole states. pattern has been expounded in the experimental setup section. The ion TOF spectrum shows a large variety of different dissociation fragments including the precursor molecular ion C 3 H 3 N + . Unfortunately the equal mass fragment pairs like

Photofragmentation pathways

(CH + 2 , N + ), (CH + 3 , NH + ), (C 2 H + 2 , CN + ), (C 2 H + 3 , CNH + ), (CNH + 2 , N + 2 ), (C 3 H + 2 , C 2 N + ) and (C 3 H + 3 , C 2 NH + )
are not discernible in time-of-flight ion spectrum. In order to identify the ambiguous fragments from the coincidence ion TOF, we performed also the PEPICO measurements of the deuterated acrylonitrile which has different fragment pairing. Using the curve fitting analysis [START_REF] Kukk | Spectral Analysis by Curve Fitting Macro Package /SPANCF[END_REF], we determined the intensities of all TOF peaks for both samples and composed the systems of linear equations, where the variables were the abundancies of the possible fragments with the same mass-to-charge ratio. The last step was to use least-squares minimization to achieve the best agreement between the calculated abundancies and the curve-fitted TOF intensities. The list of dissociation products according to this analysis is shown in Table 1. The light H + and D + fragments are not added to this table due to the unreliability of quantitative intensity information, which is caused by the low detection efficiency of these fast fragments. excitations can be divided into the heavy C 3 NH + n and the light C 2 H + n groups, where n = 1 -3. The relative weights of these fragmentation groups constitute approximately 77 % and 66 % of the recorded fragmentation pattern of acrylonitrile following the corresponding resonant excitations. The comparison of relative intensity weights of fragment pairs (C 2 H + , CNH + 2 ), (C 2 H + 2 , HCN + ) and (C 2 H + 3 , CN + ) also implies that the singly ionized fragment cations are formed more probably from the moieties containing vinyl (CH 2 = CH-) than from the cyano (-C ≡ N ) group.

Electronic relaxation channel dependent fragmentation

The main panels of Figures 4 and5 display the fragment and molecular ion intensities depending on the energy of electrons. These coincidence fragmentation patterns also allow to study, how the fragmentation pathways depend on the atomic site of the core electron excitation, as the initially unoccupied π * xy (C ≡ N ) is strongly localized in the neighborhood of the cyano group. Examination of both PEPICO patterns indicates no major variances between their dissociation products (see also Table 1). This leads to the conclusion that, even in the present case where the initial perturbation has a well localized nature, the atomic site of initial core excitation has no major influence to the following fragmentation process. In other words, the C N 1s and N 1s core level excitations lead to the final electronic states with very similar populations (see the two upper decay spectra in Figure 3) as a starting point of the dissociation.

In order to explore in detail the energy dependence of the creation of fragment cations, the data of the PEPICO map in Figure 4 have been processed to the form of the coincidence ion yields (CIYs), which represent the ion intensities as a function of the coincident electron s binding energy (or as a function of the total available excess internal energy of the parent molecular cation). Also the experimental appearance energies for the main fragments have been estimated by linear interpolation of the coincidence ion yield curves. In Figure 6 are shown the CIYs of the acrylonitrile precursor ion and five main fragment cations, following the N 1s -1 π * xy (C ≡ N ) excitation and in coincidence with the corresponding resonant KV V Auger electrons. The ionization of the parent molecule starts at 10 eV binding energy and has the peak intensity at 12.4 eV, which coincides with the maximum of the participator structure of the electron spectrum and is in resonance with the direct ionization final state (12a ) -1 . The participator transitions lead also to the fragmentation of the acrylonitrile molecule. The pattern of the C 3 H 2 N + ion appears at around 12 eV and concurrent loss of a neutral H atom starts to take place. The intensity distribution of the C 3 H 2 N + ion extends up to 23 eV in the binding energy scale and has two maxima at 13.3 and 17.4 eV. The first one coincides with the state (11a ) -1 and the second occurs in the low binding energy side of spectator Auger region. The C 3 HN + fragment appears at 11.5 eV with very slow intensity increase up to 18.5 eV binding energy and then the intensity of this fragment starts to follow the shape of the spectator structure of the electron spectrum. The co-product of this dissociation channel is molecular H 2 which is energetically more favored compared to the release of two atomic H.

In addition to the production of heavy fragments at the energy region of the participator transitions, the rise of lighter C 2 H + 2 ion pattern and the loss of neutral hydrogen cyanide (HCN) molecule starts. The appearance threshold of this fragment is at 11.8 eV and the first intensity maximum is located at 13.3 eV. It can be seen from Figure 6 that the elimination channels of the H atom and the HCN molecule are competing at the high binding energy shoulder of the participator peak in the electron spectrum. The creation of H can be explained by simple rupture of the bond between hydrogen and carbon (C H or C C ) atoms if the molecular ion has got enough excess energy for this process. The HCN elimination can be explained as follows: when the core electron is promoted to the π C≡N orbital, the hybridization of the C N atom changes from sp to sp 2 and thus gives rise to the change of geometry of the acrylonitrile molecule [START_REF] Ilakovac | [END_REF]. Subsequently, the bond between H and C C (or C H ) atoms weakens and the hydrogen atom starts to form partial bond(s) with the C N (and N ) atom(s). Finally, the bond between the migrating H atom and vinyl group (C 2 H 2 ) breaks and the HCN molecule separates from the parent acrylonitrile molecular ion.

In the binding energy range corresponding to the spectator transition, three strong fragments are observable:

C 3 HN + , C 2 H + 2 and C 2 H + 3 .
The co-product of the latter is the neutral CN radical caused by the simple bond cleavage between vinyl and cyano group.

Nuclear rearrangements caused by the core level excitations and metastability of fragments

This section will discuss the acrylonitrile fragmentation preceded by the cyano group core level excitations, concerning the possible geometrical changes during the evolution of these processes. In addition, ab initio molecular orbital calculations for appearance energies of dissociated fragments, carried out by the unrestricted Hartree-Fock (UHF) theory with the split-valence 6-311(dp) basis set [38] containing polarization functions for all atoms will be reported. Computational studies presented in this work were executed with GAMESS (The General Atomic and Molecular Electronic Structure System) program [39]. The theoretical appearance energies were estimated by substracting the total energy of the neutral parent compound from the sum of total energies of its isolated fragments. These fragments are the isomers corresponding to the conformations which have the (globally) lowest energy. The careful exploration and analysis of PEPICO maps together with calculations shows the relevance of geometrical readjustment to the coincidence dissociation pathways.

Molecular dissociation pathway: (C

2 H + 2 , HCN)
Let us turn back to Figure 6, where every CIY is presented together with the corresponding ionization or appearence energy (AE) bar. The overall agreement between experiment and calculation is excellent, except in the case of the strongest fragment (C 2 H + 2 ), for which we calculated AEs for two possible configuration -acetylene (H -C ≡ C -H) and vinylidene (: C = CH 2 ). It has been observed earlier that the acetylene molecular ion induced by the ionization of carbon 1s electron often rearranges from the acetylene configuration to the vinylidene form prior to the dissociation in a time scale estimated to be less than 60 fs [START_REF] Osipov | [END_REF]. According to the calculations [START_REF] Ilakovac | [END_REF] the periods of the C -H vibrational stretching modes in acrylonitrile are comparable to the intrinsic lifetime of the core excited molecule (5-10 fs). The non-zero probability for a hydrogen atom migration may cause the complete geometrical rearrangement of excited molecule during the processes following the initial excitation.

In order to ascertain the possible geometrical configurations of C 2 H + 2 , we performed the coincidence measurements also with singly deuterated acrylonitrile (CH 2 = CD -CN or acrylonitrile-2-d). The curves of CIYs of main fragments and parent ion are depicted in Figure 7. The analysis of experimental data (which method is already given in subsection 3.2) shows that the abundancy ratios of fragment pairs

{(C 3 HDN + ) M/Q=53 /(C 3 H 2 N + ) M/Q=52 } and {(C 2 HD + ) M/Q=27 /(C 2 H + 2 ) M/Q=26
} approach the ratio of 2 : 1, which can be deduced by calculating the number of the 2-combinations of the 3-element set (H, H, D). Secondly, the experimental appearence energies of fragments M/Q = 26 and M/Q = 27 are equal. The last circumstance indicates that the geometrical structure of the precursor molecular ion has undergone significant rearrangement process before the dissociation to these two fragments.

Our calculations confirm clearly this interpretation. Two possible dissociation pathways with minima and corresponding characteristic transition states are traced in Figure 8. The upper pathway conforms to the fragmentation channel, which leads to the creation of C 2 H + 2 and lower corresponds to C 2 HD + fragment product.

It is quite obvious that the hydrogen bonded with C C carbon would most probably attach itself to CN part in the C 2 H + 2 and HCN dissociation process (Figure 8a). However, the strong CIY corresponding to M/Q = 27 in Figure 7 together with the theoretical appearance energy indicate that the above mentioned dissociation channel is not the only possibility. This implies that the parent compound should go through an isomerization process before the fragmentation (Figure 8b). The calculated appearance energy matching to the experimental appearance threshold of M/Q=27 was acquired by substracting the total energy of the neutral parent molecule from the sum of total energies of geometry-optimized singly charged acetylene (HCCD + ), which is the global minimum of all the possible C 2 H + 2 conformations) and neutral HCN. Accordingly, using modelling result of HCCH + compound to attempt to describe the singly deuterated one (HCCD + ) is reasonable, since the effect of deuterating the sample is assumingly minimal [4] and of course negligible in sense of the Hartree-Fock theory. The isomerization process can be qualitatively modelled by examining the potential hypersurface of the molecule. The hypersurface is the function of (3*7-6=) 15 degrees of freedom representing nuclear coordinates of the atoms in the molecule. Here, our main interest lies in the local equilibrium and the transition conformations of the parent ion and also the potential barriers between the stationary points. The equilibrium conformations were located by quasi-Newton algorithm [40] and the transition structures were found by following the minimum energy path on the potential surface of the parent ion. The intermediate structures were also verified by a vibrational frequencies analysis -the vibrational motion associated with the negative frequency is the one oscillating back and forth between the initial and the final equilibrium state. Figure 8 illustrates two of the possible internal rearrangements of the dissociating acrylonitrile ion, which lead to the same final state, but the hydrogen atom initially located at central carbon atom (C C ) ending up in different fragments.

Kinetic energy release of fragments depending on the electronic state

The kinetic energy release (KER) during the dissociation results in an additional velocity spread of the fragment ions and this velocity change is reflected in the broadening of the TOF peak of fragment. The coincidence measurements permit to examine the KERs of different cations depending on the excess internal energy of parent ion. In the present case we have determined the FWHMs of TOF peaks for the C 3 H 2 N + and C 2 H + 2 fragments; both of them were detected in participator and also in spectator range of the electron spectrum. We used the pseudo-Voigt function for curve fitting [START_REF] Kukk | Spectral Analysis by Curve Fitting Macro Package /SPANCF[END_REF] of the experimental data, since the shapes of these TOF peaks resemble more a Gaussian than flat-topped (or dish-shaped) distributions that arise at high KER values. This curve fitting approximation gives one representative value of the KERs, possibly combining several channels each with a different KER [41]. The KERs were obtained by comparing the experimental widths of peaks with Monte-Carlo simulations of the ion TOF spectra using the values of experimental extraction and acceleration voltages and the corresponding dimensions of the ion-TOF spectrometer s regions. The results of KER values of these two fragments are presented in Table 2. The simulation also shows that under the present experimental conditions the C 2 H + 2 fragments stay in the spectrometer s extraction region only during the first 0.37 µs, which is a small fraction of the whole time-of-flight. The similar observation is also valid for the C 3 H 2 N + fragment. According to general classification, fragment ions can be divided to three groupsstable, metastable and unstable, where the decomposition rate for metastable ions lies between 10 5 and 10 6 s -1 [START_REF] Gross | Mass Spectrometry. A Textbook[END_REF]. So, stable ions will reach the detector without fragmentation, but instead of unstable ions their fragments will be detected. In terms of TOF measurements, the metastable ion is a fragment dissociating in flight from the extraction region to the field-free region of the mass spectrometer. In the case of acrylonitrile we have detected an abrupt increase of FWHM of time-of-flight peaks at around 15 eV in binding energy scale (Table 2). We suggest to attribute the increase in KERs of C 3 H 2 N + and C 2 H + 2 fragments in spectator energy range of electron spectrum to the existence of metastable states, which cause the decomposition of corresponding fragments in flight from the extraction region to the field-free region of TOF analyzer. In both cases the KER values are below 100 meV, which indicates the absence of reverse activation barrier and simple bond cleavage dissociation reaction. The metastable state of protonated cyanoacetylene (HC ≡ CH -CN) (or cyanovinylidene (CH 2 = Ċ -CN)) is probably the precursor for C 3 HN + fragment, which appears to PEPICO map in this excess energy region. In the case of metastable fragmentation of C 2 H + 2 , we expect the following reaction

C 2 H + 2 → C 2 H + + H.
This channel is most likely , if we consider the experimental and theoretical appearance energy of the C 2 H + fragment on PEPICO map. In other words, the metastable decay of the C 2 H + 2 fragment appears if the internal excess energy exceeds the appearance energy of the C 2 H + . This assumption is also confirmed by earlier theoretical calculations and PEPICO experiment for C 2 H 2 molecule [44,45], where it was concluded that C 2 H + 2 ions with internal energies ≥ 17.40 eV decompose overwhelmingly into C 2 H + in a temporal scale < 12µs.

Conclusions

The investigation of acrylonitrile dissociation following the core level excitations of cyano group atoms to the π * xy (C ≡ N ) orbital have been performed. The comparison of the electron-ion coincidence maps of the N 1s -1 π * and C 1s -1 π * Auger decay shows that in both cases the fragmentation channels are the same and the energy dependency on the initial site of core hole creation is very weak. On the other hand, the main features of these PEPICO maps strongly depend on the final cationic state reached by Auger decay of the core excited states. The fragmentation pattern clearly depends on the nature of resonant decay channel and/or the presence of excited electron. Also the migration of light H atoms during the excitation and after that plays an important role for the geometrical rearrangement of parent ion. Additionally, we have reported the existence of metastable states of acrylonitrile fragments, which depend on character of electronic decay channels and appear with the opening of the spectator Auger channels.

Figure 1 .

 1 Figure 1. Single photoexcitation of the neutral molecule (M ) to excited state (M ), the following relaxation to the ionized state (M + ) or/and to the fragments (m + + n) depicted in the potential energy space. The symbols IE(M + ) and AE(m + ) denote the ionization energy of the precursor molecule and appearance energy of an ion fragment, respectively. Horizontal axis corresponds to the internuclear coordinate.

Figure 2 .

 2 Figure 2. Upper panel: total ion yield of acrylonitrile at nitrogen K-edge. Lower panel:total ion yield of acrylonitrile at carbon K-edge. The labels indicate the corresponding excited states.

Figures 4

 4 Figures 4 and 5 depict the PEPICO maps with corresponding integrated ion TOF and electron energy spectrum recorded at the N and C N 1s → π * xy (C ≡ N ) resonant excitation energy, respectively. The spread of electron kinetic energy window, which is determined by the pass energy of electron spectrometer, was chosen to cover the range of corresponding resonant KV V Auger structures. The coincidence measurement setup and extraction of real coincidence dissociation

Figure 3 .

 3 Figure 3. Electron spectra of gaseous acrylonitrile measured at marked excitation energies. The labels above the lower spectrum indicate the ionization energies of acrylonitrile valence molecular orbitals. Two upper spectra correspond to the resonant Auger decay after N 1s → π * C≡ N and C N 1s → π * C≡ N excitations, respectively.

Figure 4 .

 4 Figure 4. Electron-ion coincidence (PEPICO) map of acrylonitrile from the Auger decay of the N 1s -1 π * xy resonance at hν=399.84 eV. The electron energy spectrum (top panel) and the ion TOF spectrum (right-hand panel) are both integrated from the data of the PEPICO map (main panel). The labels with arrows denote the main dissociation fragments.

Figure 5 .

 5 Figure 5. Electron-ion coincidence (PEPICO) map of acrylonitrile from the Auger decay of the C 1s -1 π * xy resonance at hν=286.8 eV. The electron energy spectrum (top panel) and the ion TOF spectrum (right-hand panel) are both integrated from the data of the PEPICO map (main panel).

Figure 6 .

 6 Figure 6. Coincidence ion yields (CIY) of acrylonitrile main fragments following the Auger decay of the N 1s -1 π * resonance at hν=399.84 eV and the electron energy spectrum on top panel. The calculated appearance energy values for each CIY curve are given by vertical (black) bars. Labeled appearance energy bars (A), (V) correspond to the acetylene (H -C ≡ C -H) and vinylidene (: C = CH 2 ) isomeric configurations of C 2 H + 2 cation, respectively. The horizontal bars denote the zero level of signal for each CIY curve.

Figure 7 .

 7 Figure 7. Coincidence ion yields (CIY) of acrylonitrile-2-d (CH 2 = CD -CN ) main fragments following the Auger decay of the N 1s -1 π * resonance at hν=399.84 eV and the electron energy spectrum on top panel. The calculated appearance energy values for each CIY curve are given by vertical (black) bars. Labeled appearance energy bars (A), (V) correspond to the acetylene (H -C ≡ C -H) and vinylidene (: C = CH 2 ) isomeric configurations of C 2 H + 2 cation, respectively. The horizontal bars denote the zero level of signal for each CIY curve.

Figure 8 .

 8 Figure 8. Two possible photofragmentation and isomerization pathways of the acrylonitrile parent ion leading to the creation of the C 2 H + 2 ion and the loss of HCN molecule. The minima (Min) and corresponding characteristic transition states (TS) are given together with their total energy values in electronvolts and hartrees. The hydrogen atom (replaced with D in singly deuterated sample) initially located at central carbon atom (C C ) is denoted with asterisk ( ) .

Table 1 .

 1 Photofragmentation products of the C 3 H 3 N and C 3 D 3 N molecules following the C1s → π * and N 1s → π * core excitations. H or D are commonly denoted by X. Masses of the deuterated fragments are given in brackets. Abundancies are given in percent together with error bars in brackets.

			Abundancy
	Fragment Mass (a.m.u.) C1s → π * N 1s → π *
	CX +	13(14)	1.2 (0.1)	0.9 (0.1)
	N +	14	0.1 (0.2)	0.2 (0.3)
	CX + 2	14(16)	1.9 (0.4)	1.3 (0.4)
	NX +	15(16)	1.3 (0.1)	0.8 (0.1)
	CX + 3	15(18)	0.0 (0.1)	0.0 (0.1)
	NX + 2	16(18)	1.1 (0.1)	0.9 (0.1)
	C 2 X +	25(26)	7.4 (0.1)	5.2 (0.1)
	C 2 X + 2	26(28)	23.7 (0.7) 28.5 (0.8)
	CN +	26	2.2 (0.5)	0.0 (0.5)
	C 2 X + 3	27(30)	7.1 (0.3)	10.9 (0.4)
	CNX +	27(28)	2.0 (0.5)	2.2 (0.6)
	CNX + 2	28(30)	2.6 (0.1)	3.3 (0.1)
	C 3 X +	37(38)	4.0 (0.1)	2.4 (0.1)
	C 3 X + 2	38(40)	1.8 (0.5)	1.1 (0.5)
	C 3 X + 3	39(42)	0.0 (0.1)	0.0 (0.1)
	C 2 N +	38	3.9 (0.4)	2.3 (0.3)
	C 2 NX +	39(40)	2.0 (0.2)	2.1 (0.2)
	C 2 NX + 2	40(42)	0.3 (0.1)	0.7 (0.1)
	C 2 NX + 3	41(44)	0.3 (0.1)	0.1 (0.1)
	C 3 NX +	51(52)	12.6 (0.3) 18.5 (0.3)
	C 3 NX + 2	52(54)	5.7 (0.2)	5.8 (0.2)
	C 3 NX + 3	53(56)	9.8 (0.2)	7.9 (0.2)
	C +	12	1.8 (0.1)	1.0 (0.1)
	C + 2	24	1.2 (0.1)	0.8 (0.1)
	C + 3	36	0.5 (0.1)	0.0 (0.1)
	C 3 N +	50	3.9 (0.1)	3.0 (0.1)

Table 2 .

 2 KERs values of C 2 H + 2 and C 3 H 2 N + fragments determined from the data depicted in Figure4. The error bars are given in brackets.

	Fragment Binding energy (eV) FWHM (ns) KER (meV)
	C 2 H + 2	27 .. 19	46(3)	60(12)
	C 2 H + 2	15 .. 12	20(1)	10(2)
	C 3 H 2 N +	20 .. 15	29(2)	9(2)
	C 3 H 2 N +	15 .. 12	18(1)	0(1)

[45] K. Norwood, C. Y. Ng, J. Chem. Phys. 91, 2898 (1989).

Acknowledgments

The experimental and theoretical work of this project was supported by the Academy of Finland, Nordforsk Infrastructure Network "Advanced spectroscopy using MAX-laboratory in Lund" and the EU "Transnational Access to Research Infrastructures" programm. Members of the electron spectroscopy group of the University of Oulu are acknowledged for their role in developing the PEPICO setup and sharing the equipment. We thank A. Caló and S. Urpelainen for their help during the experiment. Also the help of the staff of MAX-lab is greatly appreciated.


Lilianna Bryjko

Tanja Van Mourik

Amar Dora

Jonathan Tennyson

R-matrix calculation of low-energy electron collisions with phosphoric acid

Electron collision calculations are performed on two conformers of H 3 PO 4 , a weakly dipolar form with all OH groups pointing up, and a strongly dipolar form where one OH group points down. Strong evidence is found for a broad shape resonance at about 7 eV for both conformers, although the precise parameters of this resonance are sensitive to the details of the target wavefunction used. 10-state close-coupling calculations suggest the presence of very narrow, Feshbach resonances in a similar energy region. Again both conformers behave similarly. Elastic and electronically inelastic cross sections are calculated for both conformers.

Introduction

Low-energy electrons have been shown to be a major cause of radiation damage in biological systems (Boudaiffa et al. 2000). A number of theoretical studies have considered electron collisions with DNA (Caron et al. 2008, Caron et al. 2009) or DNA constituents (Tonzani & Greene 2006a, Tonzani & Greene 2006b, Winstead et al. 2007, Gianturco et al. 2008, Winstead & McKoy 2008, Dora et al. 2009). However the backbone of DNA also contains phosphate, and as it has been suggested that electron attachment to the phosphate group (Berg et al. 2002) also contributes to DNA strands breaks, the study of electron collisions with phosphoric acid, H 3 PO 4 , is also of potential importance. Li et al. (2003) performed calculations on a sugar-phosphate-sugar model system using the ONIOM layer method (using B3LYP/6-31+G* as the high level and AM1 as the low level) and found that the activation barrier for bond rupture of the anion's phosphate-sugar C-O bond is only 0.5 eV, indicating that very-low energy electrons can induce DNA strand breaks. Berdys et al. (2004a) argued that electrons with kinetic energies near 0 eV cannot directly (i.e. vertically) attach to the phosphate group, but electrons with energies in the 2-3 eV range can. These attach to the phosphate's P = O π * orbital, forming a metastable anion.

Like many large and flexible molecules, phosphoric acid can occur as several conformers which are fairly close in energy. Here we consider two possible conformers, a weakly dipolar form with C 3 point group symmetry, described below as conformer uuu, and an asymmetric, strongly dipolar form described below as conformer uud, see Fig. 1.

There is only limited data on electron collisions with phosphoric acid and previous studies appear each to have focussed on a single conformer. Calculation on low-energy collisions with H 3 PO 4 were performed by Winstead & McKoy (2008), who studied electron collisions with conformer uuu, and Tonzani & Greene (2006b), who appear to have studied an asymmetric geometry somewhere between the uuu and uud minima. Related studies have considered electron collisions with trimethyl phosphate (TMP), (CH 3 ) 3 PO 4 , (Aflatooni et al. 2006, Burrow et al. 2008), higher substituted phosphates (König, Kopyra, Bald & Illnberger 2006) or indeed the salt NaH 2 PO 4 (Pan & Sanche 2005). The focus of most of this work has been on detecting behaviour associated with low-energy resonances. The results remain somewhat inconclusive; while Winstead & McKoy (2008) and Tonzani & Greene (2006b) both reported resonances close to the ionisation limit of H 3 PO 4 , Burrow et al. (2008) found no evidence for resonances in TMP and, by extension, did not expect them to appear in the phosphate group of DNA.

In this work we present R-matrix calculations of low-energy collisions for both the main conformers of phosphoric acid performed using a variety of models. We report cross sections for both elastic collisions and electron impact excitation, as well as results for low-lying resonances.

Method

The basic idea of the R-matrix method is the division of space into two regions: a spherical inner region of radius a which is designed to enclose all the charge density of the target, and an outer region where interactions between the electron and the target can be represented by long-range moments. The major advantage of the method is that within the inner region the problem is solved independent of scattering energy. Within this region the wavefunction can be written:

ψ N +1 k = A ij a ijk Φ N i (x 1 ...x N )u ij (x N +1 ) + i b ik χ N +1 i (x 1 ...x N +1 ) , (1)
where the target contains N electrons and functions are labelled as N or N +1 according to whether they refer to the target or the compound scattering system respectively. In this work all orbitals underlying the wave functions of eq. ( 1) were constructed from Gaussian Type Orbitals (GTOs). In eq. ( 1), Φ N i is the wave function of the i th target state. u ij are the extra orbitals introduced to represent the scattering electron (Faure et al. 2002). The operator A ensures that the product of these two terms obeys the Pauli principle. The second summation in eq. ( 1) involves L 2 configurations which have no amplitude on the Rmatrix boundary and where all electrons are placed in orbitals associated with the target. Such configurations, the choice of which is quite subtle (Tennyson 1996), are essential in even the simplest scattering model applied here, Static Exchange (SE), as they allow for relaxation of the orthogonalization between the continuum orbitals and those belonging to the target given that the continuum orbitals are forced to be orthogonal to the target ones. In the more sophisticated Static Exchange plus Polarization (SEP) model extra L 2 configurations are obtained by considering single excitation from the target wavefunction which allows for the effects of target polarization.

Eq. ( 1) contains most of the physics of the problem; the three models considered below are determined by the choices made for this equation. While the SE and SEP models only consider a single target state in the first sum, usually represented by a Hartree-Fock wavefunction, the close-coupling (CC) model runs over a sum of such states which here are represented using configuration interaction (CI). One of us has recently written a comprehensive review of the R-matrix method for electron-molecule collisions (Tennyson 2010); the reader is referred to this for more details of the method.

All calculations were performed using the fixed nuclei approximation except for the Born correction discussed below which included treatment of rotational motion within the rigid rotor approximation.

Calculations

Target models

As a first step we performed a series of calculations to determine which geometries of H 3 PO 4 we should consider. The three hydroxyl groups can be in up or down position with respect to the P=O group. Starting structures were created by taking into account all possible combinations of the positions of the hydroxyl groups. These were optimised with the B3LYP method (Becke 1993, Lee et al. 1988, Vosko et al. 1980, Stephens et al. 1994) and the 6-31+G* basis set (Hehre et al. 1972) using the Gaussian 03 computational package (Frisch et al. 2003). Three stationary points were found: the first one with two hydrogens up and one hydrogen down (uud) with C 1 symmetry, the second one with three hydrogens up (uuu) with C 3 symmetry, see figure 1, and one transition state with one imaginary frequency and C 3v symmetry and three hydrogens down (ddd). The uuu conformer was found to have a lower electronic ground state energy (-644.15488 Hartree), and a smaller dipole moment (0.40 D) than the uud conformer (-644.15388 Hartree, 3.34 D, respectively). These results agree with the recent CCSD(T)/aug-cc-pV(T+d)Z calculations by Yekutiel et al. (2010), which showed that the uuu conformer is about 0.06 eV lower in energy than the uud conformer. Calculations are presented for both of these conformers below. The geometries of these minima are given as supplementary data.

Subsequent target calculations on these two conformers were performed with several different basis sets and with a variety of configuration interaction (CI) models using a mixture of MOLPRO [START_REF] Werner | MOLPRO, version[END_REF]) and the less flexible target calculation facility provided in the UK molecular R-matrix code (Morgan et al. 1998). Only a brief summary of target models considered are discussed here. Unless otherwise stated, calculations were performed using the cc-pVTZ target basis set; however some differences in the results obtained with calculations performed with different target basis sets were observed and these will be noted below.

Restricted Hartree-Fock (RHF) wavefunctions were used for the SE and SEP calculations presented below. The CC calculations used a complete active space CI (CAS-CI) model. In this model 36 "core" electrons were frozen in their RHF orbitals whereas the remaining 14 "active" electrons were distributed over 10 orbitals. A variety of different orbital sets were tested; all CAS-CI results presented here are based on the use of state-averaged CASSCF orbitals obtained for the lowest 5 singlet and triplet electronic states.

Our CAS-CI model gives 4950 singlet configurations and 6930 triplet configurations for both conformers. We note that none of the codes we used in this study are able to explicitly treat the C 3 symmetry of the uuu conformer, so no symmetry was used in any of our calculations. The results of the CAS-CI models are summarised in table 1. The CAS-CI results show lower energies for the uud than for the uuu conformer, in apparent disagreement with the B3LYP results. However, in our state- averaged CASSCF calculations the orbitals were optimised for an average of 10 states, and these calculations therefore do not yield accurate ground-state energies. We note that the lowest two triplet states and the lowest two excited singlet states occur as quasi-degenerate pairs for the uuu conformer which almost certainly correspond to 3 E and 1 E states respectively. For the lower symmetry uud conformer, the corresponding states lie close to each other but with an increased splitting. All higher states appear to be non-degenerate and therefore correspond to A states in C 3 symmetry.

Scattering

A variety of calculations were performed to test both the numerical stability of our models and their convergence with respect to various parameters. These tests showed that an R-matrix box of radius at least a = 15 a 0 was required to get realistic results: calculations with a = 10 a 0 gave eigenphase sums which were clearly unphysical while calculations with a = 13 a 0 appeared physically correct but gave resonance structures that disappeared when a was increased. In all cases GTO continuum basis sets were used which followed the prescription of Faure et al. (2002).

A second key parameter was found to be the number of virtual orbitals retained in the calculation. These unoccupied target orbitals can be thought of as providing low-lying unoccupied orbitals which temporarily trap the electron in resonance states as well as compensating the truncation of the angular part of the continuum basis by providing high angular momentum functions in the region of the nuclear singularities. these orbitals [START_REF] Winstead | [END_REF], Dora et al. 2009). Tests showed that to get converged results it was necessary to retain at least 15 virtual orbitals in all calculations and the results presented here were performed on this basis. Calculations which retained a larger set of virtual orbitals were essentially indistinguishable from the ones presented here and, in particular, showed no new resonance structures. For the CC calculations the 10 lowest electronic states, as listed in table 1, were retained in the CC expansion.

It is well known that the representation of low-lying resonances depends critically on

Molecules with dipole moments have cross sections that become very large at low scattering energies and low scattering angles. These cross sections are difficult to determine experimentally and require special treatment theoretically (Zhang et al. 2009).

Here allowance was made for the contribution due to high (l > 4) partial waves using the Born approximation and code BORNCROSS (Baluja et al. 2000); this treatment also includes the effects of target rotational motion which acts to reduce the effects of the asymptotic dipole potential. Rotational constants were calculated from the conformer geometries and only motions about the A-axis was considered. In practice the uuu conformer is very close to being a spherical top and the uud conformer is only slightly further from this high symmetry limit. Given the significant difference between the polarity of the two conformers considered here, inclusion of this effect will be important for evaluating any relative properties.

Results

Resonances

An important part of the present study is the identification of any possible resonance features. For this we used the automatic resonance detection program RESON (Tennyson & Noble 1984) and visual inspection of the eigenphase sums. Fig. 2 gives the eigenphase sums as a function of model for a target calculation with the cc-pVTZ basis. The eigenphases plotted show a feature in the region above 6 eV. For the SE and SEP models, a broad resonance feature is shown; for the CC model this broad feature is disrupted by the opening of channels associated with excited target electronic states which also show associated narrow resonance features (see insets). For the other target basis sets a broad resonance is found for both conformers in the 6 to 9 eV region. The predicted position of this resonance appears to vary little between the two conformers but to be sensitive to the target basis and model chosen, see table 2.

Previous studies (Winstead & McKoy 2008, Tonzani & Greene 2006b) also identified broad resonances in this region. Tonzani & Greene (2006b) found a resonance at 7.70 eV with a width of about 2.0 eV; Winstead & McKoy (2008) identify broad resonances directly from peaks in the integral cross sections. They find a peak at about 8.5 eV for the SE model and 7 eV for SEP. These results are very similar to ours obtained with the cc-pVDZ target basis. Winstead & McKoy (2008) used their cross sections to identify two further, higherenergy resonances. Our cross sections (Fig. 3) show some structure, although the strong dipole-scattering background cross section tends to obscure these. We are however cautious about assigning resonances on the basis of features in the cross section alone. Our conclusion is that H 3 PO 4 does indeed show a broad resonance(-like) feature at about 7 eV. This feature appears to be approximately the same for both conformers which can be understood from the fact that the two lowest unoccupied molecular orbitals (LUMOs) in the system are associated with P-O bonds and are therefore, presumably, not very sensitive to the isomerisation. Some of our SEP calculations also showed rather narrow, higher-energy resonances not shown in the corresponding SE studies. Such resonances, which are likely to be Feshbach in nature, are better characterised by CC calculations in which the "parent" state is included in the close-coupling expansion. Table 2 summarises the Feshbach resonances found below 10 eV for each conformer in the CC calculations. These resonances are all extremely narrow. For the uuu conformer the lowest-two (quasi-)degerenate resonances are associated with the two (quasi-)degerenate triplet excited states at 7.5 eV while the next two, also (quasi)-degenerate, resonances lie almost on top of the (quasi-)degerenate singlet excited states at 7.85 eV. The two uud conformer resonances can also be associated with the two lowest triplet states. The increased separation between these target states is reflected in an increased splitting between the resonances. For uud, RESON did not find a pair of resonances associated with the two lowest singlet excited states. However, given the likely proximity of these resonances to the target state it is entirely possible that the fit missed them or, indeed, that the resonances have become unbound.

Cross sections

Figure 3 presents our total cross sections, which are essentially elastic cross sections, for the two conformers considered as a function of the different models used. Both conformers show cross sections which increase rapidly at low energy which is characteristic of electron collisions from targets with large permanent dipole moments. As could be anticipated the Born correction significantly increases the resulting cross sections; for simplicity this correction is only shown augmenting the CC calculations, but a very similar correction could be applied to the other models. While both conformers show an increased low-energy cross section when the Born correction is added, this is very much larger for the strongly polar uud conformer (note the 10 % scaling of this curve in the figure). Our uuu cross sections are about 10 % larger than those of Tonzani & Greene (2006b) and show similar structures at lower energies. Below 5 eV, both cross sections are significantly larger than those calculated by (Winstead & McKoy 2008); it is unclear why Winstead and McKoy's results do not show the low-energy peak characteristic of polar molecules.

Figure 4 presents electron impact electronic excitation to the two lowest singlet and triplet excited states of each conformer. For the singlet states a Born correction →2 1 A+Born Corr. has been added to the cross section based on our calculated transition dipole; however this correction is small in all cases. As we found for uracil (Dora et al. 2009), the electronic excitation cross sections are fairly small compared to the elastic cross section. The cross sections show near-threshold structure, albeit weakly except for excitation of the uuu triplet states whose structures can be attributed to the broad, near-threshold shape-resonance-like feature discussed above.

→3 1 A →2 1 A →2 3 A X 1 A→1 3 A uuu σ (

Conclusions

We have performed electron collision calculations on the two main conformers of phosphoric acid. Both conformers display a broad feature at about 7 eV which may be associated with a shape resonance, as well as some very much narrower Feshbach resonances. The broad feature had been found in previous static exchange plus polarization studies (Winstead & McKoy 2008, Tonzani & Greene 2006b); the Feshbach resonances are a feature of a close-coupling treatment and appear not to have been identified previously. In common with the other electron scattering calculations on isolated H 3 PO 4 molecules, we found no evidence for very low resonances or zero-energy states that could be the key to DNA strand breaks due to collisions with very low energy electrons. We note that our calculations do not give a definitive, converged value for the resonance positions since these are very sensitive to the treatment of polarisation in the calculation. The molecular R-matrix with pseudo-states method (Gorfinkiel & Tennyson 2004, Gorfinkiel & Tennyson 2005) gives a systematic procedure for addressing this issue but, due to computational considerations, has yet to be applied to a molecule as large as phosphoric acid.

In general the behaviour of the two conformers is notably similar with resonance positions close to each other and cross sections that match quite closely. This suggests that calculations on either conformer would be appropriate input for modelling the phosphate group in studies of low-energy collisions with DNA and related biological molecules. However one area where the electron scattering between the two conformers may differ is that the uuu conformer is actually chiral. Scattering of spin-polarized electrons from chiral systems can lead to enantiomer-dependent scattering (Blum & Thompson 1998), an effect known as electron circular dichroism.

Figure 1 .

 1 Figure 1. Conformers of phosphoric acid studied in this work: (A) uud, (B) uuu.

Figure 2 .

 2 Figure 2. Eigenphase sums for electron scattering from phosphoric acid as a function of model (see text for details): Upper panel uuu conformer, lower panel uud conformer. The insets show the narrow Feshbach resonances in the CC calculation.

Figure 3 .

 3 Figure 3. Total electron impact cross sections for phosphoric acid as a function of model (see text for details): Upper panel uuu conformer, lower panel uud conformer. The insets illustrate the influence of the narrow Feshbach resonances in the CC calculations without Born correction.

→3 1

 1 A+Born Corr.

Figure 4 .

 4 Figure 4. Electron impact excitation cross sections for excitation of the lowest two singlet and two triplet excited states of phosphoric acid; calculations were performed using the close-coupling (CC) model; the upper curves have a Born correction: Upper panel uuu conformer, lower panel uud conformer.

Table 1 .

 1 The X 1 A ground state energy (in Hartree), vertical excitation energies (in eV) for singlet and triplet electronic states for the two conformers of phosphoric acid considered. Ground state dipole moments are also given.

	State		uuu			uud
		6-31G	cc-pVDZ	cc-pVTZ	6-31G	cc-pVDZ	cc-pVTZ
	X 1 A -641.75014 -642.07314 -642.21334 -641.75239 -642.07420 -642.22751
	1 3 A	8.01	8.51	7.46	7.86	8.43	7.84
	2 3 A	8.05	8.54	7.51	7.98	8.55	7.96
	2 1 A	8.75	9.13	7.82	8.63	9.10	8.35
	3 1 A	8.81	9.17	7.87	8.69	9.15	8.42
	3 3 A	9.30	9.52	8.76	9.25	9.55	9.17
	4 3 A	9.76	10.02	9.05	9.58	9.97	9.50
	4 1 A	10.04	10.31	9.22	9.77	10.19	9.70
	5 3 A	9.85	10.11	9.63	10.09	10.34	10.01
	5 1 A	10.24	10.50	9.53	10.29	10.66	10.20
	µ / D	1.35	0.39	0.35	3.31	2.96	3.08

Table 2 .

 2 Positions (and widths), both in eV, of low-lying shape resonances found for phosphoric acid using the SE and SEP models for several target basis sets.

	6-31G		cc-pVDZ		cc-pVTZ
	SE	SEP	SE	SEP	SE	SEP	CC
			Conformer uuu
	6.90 (1.12) 5.71 (0.81)	8.59 (1.63)	7.31 (1.11) 8.55 (1.64) 7.54 (1.26) 7.21 (0.021)
							7.26 (0.017)
							7.85 (0.040)
							7.87 (0.0007)
			Conformer uud
	7.07 (1.52) 5.75 (1.07)	8.71 (1.77)	7.39 (1.26) 8.64 (1.83) 7.51 (1.42) 7.74 (0.019)
							7.88 (0.014)

Acknowledgements

We thank Stefano Tonzani for providing information on his calculations. This project was funded by the UK Engineering and Physical Sciences Research Council. LB and TvM thank EaStCHEM for computational support via the EaStCHEM Research Computing Facility.