Stephen Harrison

Jonathan Tennyson

Bound and continuum states of molecular anions C 2 H -and C 3 N

Recently a number of molecular anions, closed-shell linear carbon chains of the form C n H -and C n N -, have been detected in space. The molecules C 2 H -and C 3 N - are investigated by using the R-Matrix method to consider electron scattering from the corresponding neutral targets. Initial target calculations are conducted and refined in order to produce target state characteristics similar to the experimental data. A number of different scattering models are tested including static exchange and closecoupling models, and the use of Hartree-Fock or natural orbitals in the close-coupling calculations. The calculations concentrate on bound and resonances states for the anions as well as eigenphase sums, elastic cross-sections and electronic excitation crosssections for electron collisions with the neutral. It is found that electronic resonances are all too high in energy to be important for anion formation in the interstellar medium. However, C 3 N -, unlike C 2 H -, supports a number of very weakly bound excited states, which may well provide the route to electron attachment for this system.

Introduction

The first anion detected in space was C 6 H -by McCarthy et al. (2006). Since then C 4 H - (Gupta et al. 2007), C 8 H - (Brunken et al. 2007), C 3 N - (Thaddeus et al. 2008) and C 5 N - [START_REF] Cernicharo | [END_REF] have also all been detected. CN -, C 3 N -or C 4 H -, and C 5 N - have also recently been identified in the atmosphere of Titan (Vuitton et al. 2009). Since the present work was completed, CN -has also been detected in the interstellar medium (ISM) (Agundez et al. 2010). These anions are all linear multiply-bonded carbon chains with closed shell ground state configurations. The prevailing opinion is that these anions are formed by radiative attachment of an electron to the neutral molecule with the same chemical formula (Millar et al. 2007). These neutral precursors are all linear multiplybonded carbon chains with significant dipole moments which are known to exist in the interstellar medium. The corresponding anions are closed shell molecules whose extra electron is bound by several eV.

Herbst and co-workers (Petrie & Herbst 1997, Terzieva & Herbst 2000, Herbst & Osamura 2008) developed and applied a model for calculating radiative association rates for these C n H -and C n+1 N -(n even) species. This model is based on the formation of low-lying resonances in electron collisions and fairly simple phase-space arguments.

Studies suggest that although this model gives anion formation rates of approximately the right magnitude, the rates it provides cannot actually reproduce the observations (Harada & Herbst 2008).

Although there have been a number of electronic structure studies of these anions (see for example Woon (1995), Natterer & Koch (1995), Botschwina & Oswald (2008), Fortenberry et al. (2010)), there appears to have been no previous study of their continuum states. We have therefore performed a series of studies of electron collisions with both C n H and C n+1 N (n even) targets. Initially our studies focussed on electron collisions with C 4 H and C 3 N as the first member of each series for which the associated anion had been observed in the interstellar medium. However we found it difficult to establish definitive properties for the ground state of C 4 H, a problem that is welldocumented (Fortenberry et al. 2010). Although we have performed a series of electron collision studies with C 4 H, here we report electron collision calculations which focus on C 2 H and C 3 N as we consider the results of these calculations to be more reliable.

Method

Scattering calculations were performed using the R-matrix method [START_REF] Burke | Atomic and Molecular Processes, an R-matrix Approach Institute of[END_REF], whose use for collision calculations involving molecules has recently been extensively reviewed by one of us (Tennyson 2010). 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 longrange 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 is 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); as discussed below a variety of different orbital sets (Hartree-Fock, CASSCF and Natural orbitals) were tested. In this regard one issue is the requirement to represent all target states included in the calculation with a single orbital set. In eq. ( 1), Φ N i is the wave function of 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. As we are dealing with electron collision with strongly dipolar systems, it is necessary to allow for the contribution of electron interactions with the long-range target dipole moment to any cross sections calculated. This was carried out using the program BORNCROS (Baluja et al. 2000), which directly calculates the dipole Born correction to the cross-sections.

The scattering calculations reported here were performed using the UK molecular R-matrix codes (Morgan et al. 1998), mostly starting from the Quantemol-N interface (Tennyson et al. 2007). Target electronic structure calculations were largely performed with MOLPRO [START_REF] Werner | Molpro, version 2008.3, a package of ab initio programs[END_REF], which was also used to provide many of the sets of target molecular orbitals. For both molecules scattering calculations were performed at two levels: static exchange (SE), which is useful for identifying shape resonances, and close coupling (CC). The presence of very low-lying excited target states meant that the widely used one-state static exchange plus polarisation model was deemed inappropriate for this study, as this method is prone to giving pseudoresonances at energies above the first electronically excited target state.

All calculations were performed at the equilibrium geometry of the neutral target and ignored effects due to nuclear motion. We note that although all molecules considered here are linear, neither the polyatomic R-matrix codes nor MOLPRO use C ∞v symmetry. All calculations were therefore performed using C 2v symmetry. In most cases identifying the results in C ∞v is straightforward and these symmetry labels are largely used below.

C 2 H -

In choosing appropriate target wavefunctions to include in the scattering calculation a number of criteria are used. Firstly, the target model must be one for which an appropriate, balanced scattering model can be designed (Tennyson 1996). For this a complete active space configuration interaction (CAS-CI) model is often adopted. Secondly the excitation energy of important target electronic states should be approximately correct. Thirdly other target properties, in particular for strongly polar molecules, the permanant dipole moment of the ground state, should be well represented. Satisfying all these criteria for the systems studied here proved to be difficult and led to some compromises being made. Indeed we note that for C 4 H - electronic structure calculations performed without requirement to use the resulting wavefunction in a scattering calculation struggled to satisfy the other requirements (Fortenberry et al. 2010).

Target calculations

A variety of models were tested when calculating results for the neutral C 2 H target, a selection of these are shown in Table 1. Models 1 -6 were carried out using the R-Matrix software Quantemol-N. These comprised Hartree-Fock (HF) and Complete Active Space Configuration-Interaction (CAS-CI) calculations using each of the 3 basis sets, DZP, cc-pVDZ and cc-pVTZ. The CAS-CI calculations used an orbital space of 7,2,2 orbitals of 7 6 . 2 0 0 0 . 8 3 7 1 . 2 0 4 respectively which maps to 7 σ and 2 π orbitals. Eight electrons were frozen in the 1 -4σ orbitals with five active electrons distributed amongst the 5 -7σ and 1 -2π. Model 7 used this CAS-CI model but the molecular orbitals used in the calculation were produced by MOLPRO, in this case using a multireference CI (MRCI) calculation, and the R-Matrix calculation was carried out using the R-Matrix codes manually, to allow for use of MOLPRO orbitals. These natural orbitals (NOs) were state averaged from 3 states produced one each from 2 A 1 , 2 B 1 and 2 B 2 , ( 2 Σ + and 2 Π) MRCI calculations, using the MOLPRO 'MATROP' facility, with a 13:1:1 weighting. This weighting was chosen to create a target model with good characteristics for the scattering calcuation i.e. a good reproduction of the excitation energies and dipole moment in comparison with the orginal MOLPRO calculations. Use of the pure 2 A 1 MRCI orbitals was found to make the first vertical excitation energy too high in energy by a factor of 2, and the use of more equally weighted state-averaged orbitals was found to reduce the dipole to a value too small to be acceptable in the scattering calculation by up to a factor of 3. Hence model 7 was deemed acceptable in describing both these characteristics without sacrificing one in favour of the other. Whilst we were able to find some experimental data for the bound state energy of the anion, we could not find any data on the neutral, such as the dipole moment.

3 c c -p V D Z S C F H F -7 6 . 1 3 8 0 . 7 8 9 4 c c -p V D Z S C F C A S -C I -7 6 . 2 0 7 0 . 8 0 1 1 . 3 0 4 5 c c -p V T Z S C F H F -7 6 . 1 5 6 0 . 7 9 7 6 c c -p V T Z S C F C A S -C I -7 6 . 2 0 8 0 . 8 0 1 1 . 2 4 8 7 c c -p V T Z N O s C A S -C I -7
There have been a number of previous ab initio studies on C 2 H; Table 1 compares our models with the best of these. As we only use a very limited correlation space, the other studies give lower absolute energies even for calculations which use the same target basis. Our dipole moments are in reasonable agreement with those predicted by Woon (1995). However our predictions for the vertical energy of excitation for the very low-lying A 2 Π state appear to systematically to high. Bound states were calculated using the outer region program BOUND (Sarpal et al. 1991). For these calculations only the low-lying X 2 Σ + and A 2 Π were retained in the outer region of the CC calculations, since experience has shown that strongly closed states can cause such outer region calculations to be numerically unstable. These were propagated to a distance of 30.1 a 0 . Table 2 summarizes the results of these studies. In contrast to C 3 N -discussed below, all models found only a single bound state. The SE calculations give a vertical binding energy of about 1.2 eV whereas for the CC calculations this binding energy is increased to about 2.6 eV. These figures are in broad agreement with the electronic structure calculations of Baker et al. (1986). The CC binding energies are some 0.6 eV less than the measured ones; however the measured binding energies are adiabatic and will therefore be higher than our calculated vertical ones.

S C F C C 2 . 7 1 3 3 c c -p V D Z S C F S E 1 . 1 1 4 4 c c -p V D Z S C F C C 2 . 6 3 7 5 c c -p V T Z S C F S E 1 . 1 5 6 6 c c -p V T Z S C F C C 2 . 6 1 7 7 c c -p V T Z N O s C C 2 .

Resonances

Outer region calculations were performed retaining those target states which lie below 10 eV for the CC calculations, the Quantemol-N default setting. This means 25 states for models 2, 4 and 6 but only 16 for model 7. The results reported below are insensitive to increasing these numbers. Resonances were characterised using a Breit-Wigner fit to the eigenphases, which are shown in Figure 1, as implemented in the automated detection and fitting program RESON (Tennyson & Noble 1984). As noted below, resonances which overlap excitation thresholds were not fitted by this procedure. Only low-lying resonances were considered since higher resonances are unlikely to be important in any radiative association process.

Table 3 summarizes the resonances detected for C 2 H -. The SE calculations all detect a single shape resonance of 3 Π symmetry at about 3.5 eV. At the SE level this is a very broad resonance with a width of about 2 eV. In the CC calculations this resonance postition is lowered to about 3.4 eV and becomes considerably narrower with SCF orbitals and broader with MRCI NO's. The CC calculations based on the use of SCF orbitals introduce Feshbach resonances of 1 Σ -, 3 Σ + and 3 Σ -symmetry. Use of NOs introduce a further narrow 3 Σ + resonance. We note that none of these resonances lie below the A 2 Π first excited state of C 2 H.

Cross-sections

Figure 2 presents the total elastic cross-section of C 2 H for electron scattering energies up to 10 eV, for all 3 models, SCF-SE, SCF-CC and MRCI-CC. For the final MRCI-CC model, a Born correction was applied due to the dipolar nature of the molecule, which acts to increase the cross-section at all energies.

Electron impact electronic excitation cross sections are given by the many-state CC calculations. Figure 3 shows the cross-sections representing the excitation from the 2 Σ + ground state of the neutral to the first three excited states, A 2 Π, a 4 Σ + and b 4 ∆.

Here the Born correction is only applied to the 2 Π excitation, as this is the only dipoleallowed transition. All three excitation cross sections show resonance features. These are particularly pronounced in the CC-MRCI calculations; we note that cross sections approximately average the SCF-CC ones which show fewer resonances. Although it is likely that these cross sections display resonance features, they can only be properly characterised by much larger calculations than the one presented here.

C 3 N -

Calculations on C 3 N and C 3 N -follow closely those on C 2 H and C 2 H -reported above.

Target calculations

Quantemol-N was used to carry out HF and CAS-CI calculations using DZP and cc-pVDZ basis sets. Tests were performed with several other basis sets, including the ANO basis recommended by [START_REF] Widmark | [END_REF] for the C 3 N but for which we were unable to get satisfactory results.

Models 2 and 4, used a 10σ, 3π orbital space, with 8 electrons frozen in the 1 -4σ orbitals and the remaining 17 electron distributed among the 5 -10σ and all the π orbitals. This CAS-CI model was retained for use in model 5, which made use of MOLPRO MRCI orbitals; these NOs were obtained by the state-averaging the lowest 2 A 1 , 2 B 1 , 2 B 2 , and 2 A 2 MRCI states in a 5:2:2:1 weighting, manually chosen in order to produce good target characteristics for the scattering run.

Table 4 compares our target models with the results of previous electronic structure calculations. C 3 N is a more complicated system than C 2 H and there is less agreement 4. Selected calculated target properties of C 3 N. All results are for the equilibrium geometry of C 3 N. The calculated absolute energies of the X 2 Σ + ground state is given in Hartree while the vertical excitation energy to the low-lying A 2 Π is given in eV. µ is the ground state dipole moment, given in Debye. See text for details of models.

Model

Scattering calculations and bound anion states

Again a 48-state calculation was carried out (doublet and quartet states with 6 states per symmetry being included). The 40 target states below 10eV retained in the outer region. 46 in Quantemol-N jobs. As for C 2 H -the bound state calculations retained only the X 2 Σ + and A 2 Π target states, and were propagated to 30.1 a 0 .

Table 5 presents the calculated electronic bound states for all models, as well as previous results from literature. All models find the deeply bound X 1 Σ + state, with CC calculations finding the state between 1 and 1.5 eV deeper, furthermore use of the MRCI-CC calculation bring the bound state even deeper to about 4 eV, closer in line with data from previous results in the literature. This model also finds numerous other weakly bound states of C 3 N -, presented in Table 6. It should be noted that model 3 (SCF-SE calculation using the cc-pVDZ basis set) also finds the 1,3 Π bound state at 0.0002 eV, although this was not found in any other models that use SCF orbitals. Singlet Triplet Σ + 0.0012 0.5490 0.5490 0.6241 0.6220 4.0094 4.0094 Π 0.0003 0.0003 There appears to be no prediction of bound states other than the X 1 Σ + states in the literature.

Resonances

Figure 4 presents the eigenphase sums for C 3 N -calculations. Table 7 summarizes the resonance data for C 3 N -calculations, with the same method being employed as previously described for the C 2 H -resonances. The SE models all find shape resonances of 1 Π and 3 Π symmetries. These appear in all CC calculations at about 0.1 eV lower in electron scattering energy, with the width remaining quite stable for the 1 Π resonance, and at about 0.2 -0.4 eV lower scattering energy with a narrower width for the 3 Π resonance. In all CC calculations, Feshbach resonances appear in 1 Σ -, 1 ∆, 3 Σ + , 3 Σ - and 3 ∆ symmetries. These resonances appear systematicaly at higher energies for the larger cc-pVTZ basis compared to cc-pVDZ when using SCF orbitals. However the use of MRCI NO's leads to lower resonance positions than either SCF-CC models for all symmetries. The Feshbach resonance widths remain stable across all CC models.

Cross-sections

Figure 5 presents the total elastic cross-section of C 3 N for electron scattering energies up to 10 eV, for three models, SCF-SE, SCF-CC and MRCI-CC. Born corrected results for the MRCI-CC model are also shown.

Figures 6 shows inelastic cross-sections representing the excitation from the 2 Σ + ground state of the neutral to the first two excited states, both of which are 2 Π; the Born correction is applied to both excitations. The comparison between the SCF-CC and the MRCI-CC models shows us that the use of MRCI orbitals increases the cross-sections at all energies, although both models share similar features (such as the threshold spike in B 2 Π excitation). Again the extra resonances shown by the MRCI-CC model should be treated with caution.

Conclusions

The presence of electronic resonance features above 1 eV for both molecules suggests that formation of temporary negative ions by electrons trapped in these resonance states is an unlikely mechanism for radiative electron attachment in the ISM. This is because 1 eV converts to a much higher temperature than electrons are found at in the ISM, making it highly unlikely electrons will reach the energies required to become temporarily trapped in the resonance states.

The more likely possibility is that the molecular anions observed may be arise from the very weakly bound anionic states, as found for C 3 N -. These states will support nuclear excited states which lie in the continuum and therefore are resonances. We note that C 2 H -appears not to support such states. If this mechanism does explain the formation of such anions, C 3 N -would be observed in large quantities but C 2 H -would not. This prediction appears to match current observations. We will investigate this proposal further in future work, both for the molecules in this paper, and by investigating other molecules in the C n N -and C n H -series.

Figure 1 .

 1 Figure 1. Electron -C 2 H eigenphase sums for different symmetries for the SCF-SE(dash), SCF-CC(dash-dot) and MRCI-CC(solid) models.

Figure 2 .

 2 Figure 2. Elastic Cross-Section for electron C 2 H collisions

Figure 3 .

 3 Figure 3. Electron impact electronic excitation cross sections for C 2 H final state: A 2 Π, a 4 Σ + and b 4 ∆.

Figure 4 .

 4 Figure 4. Electron -C 3 N eigenphase sums: for SCF-SE(dash), SCF-CC(dash-dot) and MRCI-CC(solid) models.

Figure 5 .

 5 Figure 5. C 3 N Elastic Cross-Section

Figure 6 .

 6 Figure 6. C 3 N electron impact electronic excitation cross sections to A 2 Π and B 2 Π.

Table 1 .

 1 Selected calculated target properties of C 2 H. All results are for the equilibrium geometry of C 2 H. The calculated absolute energies of the X 2 Σ + ground state is given in Hartree while the vertical excitation energy to the low-lying A 2 Π is given in eV. µ is the ground state dipole moment, given in Debye. See text for details of models.

	Model	Basis	Orbitals	Method	X 2 Σ +	µ	A 2 Π
	This work/1	DZP	SCF	HF	-76.142 0.804 1.186
	2	D Z P	S C F	C A S -C I	-

Table 2 .

 2 C 2 H -bound states with energies given in eV. All calculations are for the equilibrium geometry of C 2 H

	Model	Basis	Orbitals	Method	1 Σ + Binding energy (eV)
	This work/1	DZP	SCF	SE	1.186
	2	D Z P

  Our scattering calculations concentrated on finding bound states and resonances for C 2 H -but also provide low-energy electron -C 2 H collision cross sections. Two classes of models were tested: SE based on a single target state and CC models where 48 state calculations (doublet and quartet states, 6 states per symmetry) were performed in the inner region. As discussed below, different numbers of these states were retained in the outer region of the calculation.

					2 8 2
	Baker et al. (1986)	6-311 ++ G	UHF	UHF	1.56
		6-311 ++ G	UHF	UCISD	2.62
	Janousek et al. (1979)			experiment	2.94±0.10
	Ervin & Lineberger (1991)			experiment	2.969±0.006
	Taylor & Xu (1998)			experiment	2.956±0.020
	3.2. Scattering calculations and bound anion states

Table 3 .

 3 C 2 H -low lying resonance positions (and widths) in eV

	Model	1	2	3	4	5	6	7
	1 Σ +							2.957 (0.122)
	1 Σ -		3.876 (1.115)		4.231 (1.056)		3.908 (1.080)	5.130 (1.596)
	3 Σ +		2.589 (0.430)		2.768 (0.377)		2.679 (0.444)	2.923 (0.127)
								3.942 (0.125)
	3 Π	3.522(1.949)	3.49a	3.634(2.038)	3.380(1.360)	3.562(2.021)	3.53a	3.682(2.742)
	3 Σ -		3.374 (0.867)		3.625 (0.814)		3.416 (0.854)	4.495 (1.298)
	a Resonance present in eigenphases but not fitted by RESON.

Table 5 .

 5 C 3 N -bound states with energies given in eV. All results are for the equilibrium geometry of C 3 N

	Model	Basis	Orbitals	Method	X 1 Σ +
	This work/1	DZP	SCF	SE	2.793
	2	D Z P	S C F	C C	3 . 8 9 6
	3	c c -p V D Z	S C F	S E	2 . 2 4 8
	4	c c -p V D Z	S C F	C C	3 . 7 8 8
	5	c c -p V T Z	M R C I	C C	4 . 0 0 9
	Botschwina & Oswald (2008)	VQZ+	RHF	RHF	3.651
		VQZ+	RHF	RCCSD	4.343
		VQZ+	RHF	RCCSD(T) 4.395
	Kolos et al. (2008)	aug-cc-pCV5Z	RHF	RHF	3.650
		aug-cc-pCV5Z	RHF	RCCSD	4.363
		aug-cc-pCV5Z	RHF	RCCSD(T) 4.417
	Zhan & Iwata (1996)	6-31G+(d)	HF	HF	2.519

Table 6 .

 6 Binding energies, in eV, of the states of C 3 N -calculated for the MRCI-CC scheme (model 5).

Table 7 .

 7 C 3 N -low-lying resonance positions (and widths) in eV Resonance present in eigenphases but not fitted by RESON.

	Model	1	2	3	4	5
	1 Π	2.905 (0.471) 2.716 (0.452) 2.989 (0.444) 2.822 (0.475) 2.749 (0.437)
	1 Σ -		2.29 a		2.191 (0.158) 2.094 (0.153)
	1 ∆		2 . 3 0 7 ( 0 . 1 6 1 )		2 . 2 4 8 ( 0 . 1 7 6 ) 2 . 1 6 7 ( 0 . 1 8 4 )
	3 Σ +		0.987 (0.016)		1.327 (0.023) 0.741 (0.034)
	3 Π	1.342 (0.204) 1.167 (0.121) 1.632 (0.220)	1.25 a	1.426 (0.141)
	3 Σ -		1.795 (0.053)		2.163 (0.063) 1.666 (0.075)
	3 ∆		1 . 4 6 0 ( 0 . 0 3 8 )		1 . 8 1 9 ( 0 . 0 4 7 ) 1 . 3 2 9 ( 0 . 0 6 0 )

a

Acknowledgements

The first author would like to thank R. Harrison and E. Brunetti for their support and STFC for a studentship.


R Ward

D Cubric

N Bowring

G C King

F H Read

D V Fursa

I Bray

O Zatsarinny

K Bartschat

Differential cross sections for electron impact excitation of the n = 2 states of helium

Keywords: numbers: 34, 80, Dp Atomic excitation and ionisation 34, 10, +x

    Differential cross sections for electron impact excitation of the n = 2 states of helium at intermediate energies (80, 100 and 120 eV) measured across the complete angular scattering range (0180°) R Ward, D Cubric, N Bowring, G C King, F H Read, D V Fursa, I Bray,

INTRODUCTION

There continues to be a great deal of interest in the study of electron impact excitation of helium, from the standpoints of both theory and experiment. On the theoretical side, helium provides a fundamental test case for the validity of the many theoretical models, especially their description of electron-electron correlations. On the experimental side, there have been numerous studies of electron impact excitation of helium. These include a number of measurements of the differential cross sections (DCS) of the n = 2 states of helium. A summary of these measurements up to 1992 has been given by Trajmar [1]. Further work includes that of Allan [2], Röder et al [3], Asmis and Allan [4], Cubric [5], Allan [6] and Lange et al [7]. Most recently, Hoshino et al [8] have presented DCS for the n = 2 states at near-ionisation threshold energies (23.5 -35 eV). Most of the recent studies have concentrated on the low-energy regime. It is also the case that almost all of the previous experimental studies have presented data up to a maximum angular scattering range of approximately 135°. This is because higher scattering angles are inaccessible to conventional electron spectrometers as the angular range of the electron energy analyser is limited by the presence of the electron monochromator.

A number of theoretical models for DCS of specific interactions exist. Perturbative methods have existed for many years and include Coulomb Projected Born Approximation (CPB) [9]; First-Order Many-Body Theory (FOMBT) [10]; Distorted Wave Born Approximation (DWBA) [11]; Distorted-Wave Polarisation Orbital (DWPO) [12]; Multi-Channel Eikonal Approximation (MCE) [13] and Second-Order Potential Distorted-Wave (SOPDW) [14]. More recently a range of more accurate non-perturbative methods, including R-Matrix with Pseudo-States (RMPS) [15]; B-spline R-Matrix (BSR) [16] and Convergent Close-Coupling (CCC) [17], have emerged as methods that agree well with both themselves and experiment [7]. In the current work all three of these non-perturbative methods have been compared to experimental results, together with a 5-state model based on BSR and available

FOMBT perturbative data

In the present work, DCS have been measured for the 2 3 S, 2 1 S, 2 3 P and 2 1 P states of helium, at the intermediate energies of 80 eV, 100 eV and 120 eV, over the complete angular scattering range 0°to 180°. The full angular scattering range has been achieved by the use of a magnetic angle changer (MAC). The intermediate energy range has been reached by the use of a MAC equipped with a soft-iron core, which allows for the device to work at high electron energies. MACs have been used previously to measure DCS in helium over the complete scattering angular range. Cubric et al [5] measured the DCS of the 2 3 S, 2 1 P, 2 3 P and 2 1 P states of helium at 30, 40 and 50 eV, while Allan [6] measured the DCS of the 2 3 S state from threshold to 24 eV. Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

In addition to the present experimental measurements we have also calculated the corresponding DCS using the CCC, RMPS, and BSR methods. It is the goal of the present work to compare the experimental measurements with the results of these theories. In the following sections we briefly describe the experimental and theoretical techniques used.

Measurements and calculations of the DCS, at the intermediate energies employed, are then presented and discussed in Section 4 of the paper.

APPARATUS AND EXPERIMENTAL PROCEDURES

Measurements of electron angular distributions in atomic and molecular physics have used various experimental configurations. A recent approach has been the adoption of the 'magnetic angle-changing technique' [18,19]. This technique produces a localized magnetic field, within the interaction region, thereby enabling controlled deflection of the electron trajectories. The first practical MAC devices, constructed at Manchester, were limited to low electron energies, below about 20 eV. This was due to the heat produced by the solenoid currents in the MAC and the need to dissipate this heat in an evacuated experimental chamber.

One solution to this problem, developed by Cubric et al [5], is to incorporate a softiron core within the solenoids of the MAC. This significantly increases the magnetic field in the device, and hence greatly extends the range of electron energies that can be used. The present measurements involve higher electron energies, and hence the MAC described by Cubric et al [5] has been used. It is shown schematically in figure 1. It uses two pairs of solenoids and appropriate currents flowing in opposite directions, so that the magnetic dipole moment of the MAC is zero. This, together with zero or minimal contributions from higher magnetic moments, ensures that the magnetic field has negligible effect on the operation of the electron spectrometer used in the measurements. The ferromagnetic material used for the MAC is soft iron with a magnetic permeability of approximately 1000. The magnetic hysteresis curve for soft iron is very narrow. Therefore it can be considered as a linear magnetic material, enabling currents to be applied without residual fields being generated within the angle changer. The magnetic field produced by the MAC has been measured and the results are plotted in figure 2 for a current of 1 A through the inner solenoids and a current of -0.772 A through the outer solenoids. As can be seen, the magnetic field is localized and changes its direction at a distance of about 12 mm from the central axis. Also shown in figure 2 is the magnetic field variation from a comparable MAC without an iron core (which has been multiplied by a factor of 5). The magnetic field produced by the solenoids can also be calculated, using the following equation:

2 0 3 2 2 2 2 0 cos( ) ( , , , ) 4 ( 2 cos( )) I z B z R I d z R R π ρ µ ρ φ ρ φ π ρ ρ φ = + + - , (1)
Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

where R is the solenoid radius and z and ρ are cylindrical polar coordinates representing respectively the axial and radial distances to the coil centre. The resulting magnetic field is due to the contributions from both solenoids. When a charged particle, with kinetic energy E, moves through a magnetic field B, it experiences a deflection proportional to B 2 /E. Hence, by increasing the magnetic field by a factor of 5, using a soft-iron core, the device becomes capable of handling electron energies 25 times larger than the device without the iron core. In the present work, this has enabled the measurements at intermediate energies in the range 80

eV-120 eV. The action of the MAC for inelastic electron scattering is illustrated in figure 3.

The deflection from the MAC is proportional to the beam energy. Therefore the incident beam (originating from the right side, dashed line) is deflected slightly less than the scattered beam (solid line) whose residual energy is 20 eV less. These beams are separated as they exit the MAC as can be seen at the bottom of figure 3. The full angular range of the scattered beam can be detected either by fixing the analyser and varying the MAC currents, or by fixing the currents and moving the analyser. In figure 3, using fixed MAC currents, the scattered beam can be detected across the range of 0°-180°by rotating the electron analyser from -90°t o +90°relative to the incident beam, with the electron monochromator always a significant distance from the electron energy analyser.

The MAC was used in conjunction with an electrostatic electron spectrometer, which has been described in detail previously by Bradford [20]. Briefly, it consists of an electron monochromator and an electron energy analyser, which could be rotated over the angular range from -10 o to 120 o with respect to the direction of the incident electron beam. The construction of the monochromator was based on a hemispherical selector having a mean radius of 50 mm. Two triple-aperture lenses were employed to focus the incident electron beam onto the target gas beam. A second set of two triple-aperture lenses was used to decelerate and focus scattered electrons from the target region onto the entrance aperture of a hemispherical electron analyser, again of 50 mm mean radius. Transmitted electrons were detected by a channel electron multiplier. The incident electron beam current was monitored by a rotatable Faraday cup and typically had a value of 5 nA [21]. The incident electron energy was calibrated against the position of the 2 2 S resonance in helium (19.365 eV [22])

and had an uncertainty of ± 115 meV, while the energy resolution of the present measurements was typically 130 meV (full width at half maximum). The angular resolution of the measurements (full width at half maximum) was estimated to be 6° [23]. This value arose from the angular resolution of the spectrometer and the angular spread induced by the MAC [5]. The target gas beam was produced by a single capillary of internal diameter of 0.5 mm and length 10 mm, and target pressures of typically 4.5 x 10 -3 Pa were employed.

For the present measurements, the MAC was operated in the following way. The electron energy analyser was held at the fixed angle of 90°, with respect to the incident electron beam direction. Appropriate solenoid currents were then used to provide deflection Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

angles for the scattered beam over the range ± 90°, so that a total scattering angular range of 0°to 180°was obtained. To measure a particular DCS, the yield of inelastically scattered electrons, arising from excitation of the state, was measured as a function of the scattering angle. In this procedure, care was taken to ensure that any background contributions were taken into account. These background contributions could arise from electrons scattering off residual gas in the experimental chamber or surfaces close to the interaction region. Total data accumulation times for the DCS measurements were approximately 65 hours.

Once scattering data have been obtained, there are several techniques that can be used to obtain the absolute DCS from them [START_REF] Trajmar | Electron-Molecule Collisions[END_REF]. These include direct normalisation, which requires knowledge of the electron current, gas density etc. [START_REF] Chamberlain | [END_REF], the relative flow technique [26], and normalisation against a known cross section [START_REF] Joyez | Plenum) New measurements of differential and integral cross sections for electron excitation of the n=2 states of helium[END_REF]. For the present study the most appropriate method was the use of a known DCS. The DCS of the 2 1 P state was chosen as the reference, given the agreement in previous values for this state. Values for this DCS were obtained using the CCC method. Such a procedure was previously used with much success by Mercer [23] in low-energy DCS measurements for the n = 2 states in helium. By comparing the theoretical curves for the 2 1 P DCS with the measured electron intensities, the transmission of the electron energy analyser, as a function of scattering angle, could be determined for each value of incident electron energy. These transmission functions were then applied to the scattering data for the remaining states (2 3 P, 2 1 S, 2 3 S), to obtain the DCS. All measurements were obtained under identical conditions of gas flow in the interaction region.

CALCULATIONS OF THE DCS

We have used the CCC, RMPS, and BSR methods to calculate the DCS for the transitions and energies of interest for the present paper. Below these predictions are compared to the current experimental data, previous experimental data, and results from a few selected older theoretical approaches. The principal difference between CCC, RMPS, and BSR compared to discrete-state-only close-coupling or R-matrix approaches, such as the 5state model including only the ground state and the four n = 2 states, is the coupling to both higher-lying discrete states and the ionization continuum. In principle, this coupling can be driven to convergence, and one might expect a computationally "exact" solution of the Schrödinger equation. In practice, of course, the number of states that can be included is finite, and their description is typically subject to some approximations. For example, the CCC calculations have been performed in the frozen-core model, restricting one of the target electrons always to be in the 1s orbital of He+. On the other hand, the RMPS and BSR models use a multi-configuration description of the target states, but they include less pseudostates and hence utilize a coarser discretization of the Rydberg spectrum and the ionization continuum. However, since the remaining shortcomings of these models are Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

essentially independent of each other, the good agreement between the CCC, RMPS, and BSR predictions shown below provides significant confidence in the reliability of these results. Finally, the FOMBT approach effectively corresponds to a 2-state model similar to the DWBA and the many other perturbative approaches used over the years. Channel coupling is neglected in these models, but an attempt is made to account for some of these effects (polarization, absorption, and sometimes exchange) indirectly by choosing what is believed to be an appropriate distortion potential for the calculation of the distorted waves

The application of the CCC method to electron-helium scattering was reviewed by Fursa and Bray [12]. Briefly, the helium target states are obtained by utilising a complete Laguerre basis. In the frozen-core approximation all configurations have the "inner" electrons described by the 1s orbital of He + . This leads to an ionisation energy error of 0.84 eV, but has the advantage of computational simplicity. In the multi-configuration expansion we may obtain a very accurate ground state, but this comes at the expense of subsequent computational complexity in scattering calculations. It has been our experience that for processes, which are dominated by one-electron excitation, the frozen-core approximation suffices, and hence we use it here.

Using the Laguerre basis ensures that all of the target states are square-integrable.

With increasing basis size the negative-energy states converge to the true discrete eigenstates, while the positive-energy states provide an increasingly dense discretization of the continuum. The states are used to expand the total wavefunction, which leads to the closecoupling equations. In the CCC method these take the form of coupled Lippmann-Schwinger equations in momentum space. They are solved utilising a partial-wave expansion, and the required excitation amplitudes are obtained. Convergence in the results of interest is checked by taking a sufficiently large basis. Presently, this is achieved by taking N l = 25-l for target space orbital angular momenta l < 5.

The R-Matrix with Pseudo States (RMPS) approach, developed by Bartschat et al [START_REF] Bartschat | [END_REF]29], uses a similar idea as the CCC method described above, except that the closecoupling equations are solved in coordinate space, with a basis-function expansion for the projectile in the inner region of the R-matrix box. The advantage of using such an expansion is the computational efficiency when results for a large number of collision energies are required. This is typically the case for the low-energy near-threshold regime that is often dominated by resonances. The price to pay, on the other hand, is the fact that the number of pseudostates that can be included is generally less than what can be handled in the CCC approach. Also, the highest energy that can be treated is limited by the highest energy of the one-electron basis functions. Hence, calculations for just a few intermediate energies, as required in the present work, represent a challenge for the R-matrix approach in general.

However, the rapid increase of computational resources has made such calculations possible on a more or less routine basis.

Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

The present RMPS results were generated with the 5+36 model (5 physical states plus 36 pseudostates) described by Bartschat et al [30].The R-matrix radius was set to 27 a 0 , where a 0 = 0.529 x 10 -10 m is the Bohr radius, and 30 continuum orbitals were used per angular momentum to expand the wavefunction of the projectile in the box. We included 10 target states with 1S symmetry, 9 with 3 S, 8 with 1 P and 3 P each, and 3 with 1 D and 3 D each, respectively. This is a relatively small model, but we will see that it is almost sufficient for the situation of the present work.

The B-Spline R-Matrix (BSR) method, developed by Zatsarinny and Fischer [31,32],

with a computer code published by Zatsarinny [16], is an alternative to the standard R-matrix approach developed by the Belfast group [START_REF] Burke | Atomic and Molecular Processes: an R-Matrix Approach[END_REF][START_REF] Berrington | [END_REF]. The BSR method uses a B-spline basis to expand the target states as well as the wavefunction of the projectile inside the R-matrix box.

As piecewise polynomials, the B-splines enable the Schrödinger equation to be solved in a Specifically, the current BSR model included 126 target states, of which the lowest few (including the ground state and the n = 2 states) were described very well by multiconfiguration expansions. A major advantage of the BSR method is the ability to used termdependent one-electron orbitals, i.e., in the present problem the only role of the pseudostates is to approximate the coupling to the higher Rydberg states and the target continuum. In the RMPS approach, on the other hand, the pseudostates also improve -at least to some extentthe target description itself. Specifically, the 126-state BSR model used an R-matrix radius of 40 a 0 , included 59 B-splines to cover this radial grid, and contained 23 bound states, 96 pseudo-states, and the 7 doubly excited autoionizing 2l2l' states. All target states had total orbital angular momenta L = 0, 1, or 2.

The 5-state model is used in the current work to investigate the effect of additional channel coupling compared to other models. Since it employs the same target description for the ground state and the n = 2 states, comparing the results from the 5-state and the 126-state BSR models provides an indication about the importance of the additional coupling to the higher discrete states and the ionisation continuum. Based on previous experience, one would generally expect the largest differences between the 5-state results and those from CCC, RMPS, and BSR with many pseudo-states to appear in the results for the triplet states, with some difference in 2 1 S, and little in 2 1 P.

Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

Trajmar et al [1] have previously provided data using First-Order Many Body Theory (FOMBT) for the 2 1 P state at 80 eV and 100 eV. These have been included in the current work to illustrate the comparison with this earlier modelling, which was originally developed in the 1970s [35]. One would expect the perturbative FOMBT model to do reasonably well for the optically allowed 1 1 S -> 2 1 P transition, since at the energies under consideration the result is mostly determined by the generalized oscillator strength, i.e., an element of the structure rather than the collision model.

RESULTS AND DISCUSSION

DCS results at 80 eV

Measurements of the DCS at 80 eV for the 2 1 P, 2 1 S, 2 3 P and 2 3 S states of helium are shown in figures 4 -7. All the error bars shown on all the spectra represent a one standard deviation estimate of the uncertainty of the measured intensity. The experimental results are presented together with the results of the CCC, RMPS, and BSR calculations, the latter performed with 126 states (i.e., also in the RMPS mode) and only the lowest five physical target states. Also shown for the 2 1 P state are the theoretical and experimental results of Cartwright et al [36] who compared FOMBT calculations to their experimental results. Note that each vertical scale, corresponding to the value of the cross section, is logarithmic.

The 2 1 P data shows excellent agreement between previous experimental data and the current data, and between both experimental data and the BSR and CCC models across the angular range. RMPS deviates slightly from these results at very small angles below 10°and very large angles above 170°. The 5-state and FOMBT models overestimate the cross section beyond 90°and FOMBT also deviates significantly from the other results between 40°and

75°. There is a minor difference in the two experimental data sets below 10°.

The 2 1 S data shows good agreement with the CCC results across the majority of angles (20°-160°) and is consistent in the angular dependence with both BSR and RMPS over this range, where the experimental cross section is consistently slightly lower than predicted by these models. The 5-state model shows very good agreement with experiment below 20°but beyond this differs significantly from the remainder of the data. Beyond 160°t he experimental cross section is lower than the theoretical results, also exhibiting a slight downward slope. The minimum in the DCS at approximately 45º is confirmed together with the shape around this minimum. The observed minimum near 45º and the large-angle dependence of the DCS also agree with previous experimental data [37,1] (not shown in the near 25º and a decreasing cross section below 25º. By contrast, the experimental data increases rapidly below 25º. This discrepancy could be due to larger uncertainties in the experimental data for the 2 3 P state in this angular range. The increase in uncertainty arises because the 2 3 P lies close in energy to the 2 1 P state (250 meV energy separation), which has a much larger signal for forward scattering. Thus, due to the finite resolution of the electron spectrometer (typically 130 meV), the 2 3 P signal may have a contribution from the 2 1 P state.

Several calculations were performed to check the convergence utilising the frozen-core and multi-configuration models, at this and the other considered energies. No substantial variation at the forward angles was found. To simulate the possible effect, adding a 1% contribution of the 2 1 P signal with the RMPS data has also been modelled. The results suggest that the experimental discrepancy at small angles is indeed likely due to a contamination from the 2 1 P signal and explains the discrepancy from theory for pure exchange transitions. There is no clear evidence of the shallow maximum suggested by the RMPS model near 150º. This may be due to insufficient statistical accuracy in the data at large angles, where the signal counting rate falls rapidly. Indeed, the results between 160°and 180°agree well with RMPS and deviate somewhat from both BSR and CCC where the experimental cross section is higher, unlike the 2 1 S DCS where the opposite is true.

The measured 2 3 S DCS shows least agreement with the theoretical results. However, there is general agreement with respect to the overall shape of the DCS and its magnitude.

The largest discrepancy between experiment and theory occurs above about 90º, where the theoretical results lie appreciably above the experimental data. Moreover the theoretical values do not reflect the drop in the DCS beyond 150º that is observed in the experiment. At very small angles (below 10°), the experimental data again are higher than the most sophisticated theoretical predictions, which show very good agreement with each other. In this case, however, it is currently unclear why such a strong contamination should have occurred. This anomaly should be investigated further and is discussed in the sub-section below for the 2 3 S DCS at 100 eV. The significant dip observed in the DCS at approximately 48º is sharper in the experimental data than in the theories, but advances in theoretical modelling are progressing towards the experimental angular dependence in this angular range.

DCS results at 100 eV

The measurements of the DCS at 100 eV for the 2 3 S, 2 1 S, 2 3 P and 2 1 P states of helium are shown in figures 8 -11. The experimental results are again presented together with the theoretical results of the CCC, RMPS, and the two BSR calculations. Also shown for the 2 1 P state are the theoretical and experimental results of Cartwright et al [36] who compared FOMBT predictions to their experimental results. For the 2 3 S, 2 3 P, and 2 1 S states the experimental data of Trajmar et al [1] are also presented.

The 2 1 P data show excellent agreement between the previous and the present experimental data, and also between both experimental data sets and the BSR and CCC Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

predictions across the angular range. RMPS deviates slightly from these results at large angles above 150°, similarly to the behaviour at the lower energy of 80 eV discussed above. The FOMBT model overestimates the cross section beyond 100°, and both the 5-state BSR and the FOMBT results deviate from the other data sets in the range 15°-60°. There is also a minor difference in the experimental and theoretical data sets below 5°.

The 2 1 S DCS shows very good agreement with the theoretical data, apart from the 5state predictions that suggest a higher cross section across the entire angular range. This is a typical problem in models with insufficient channel coupling accounted for. The shape and magnitude of the DCS are in agreement across almost all of the angular range even within the small uncertainties of the experimental values. The only regions with slight but noticeable discrepancies are between 110º and 150º, where the experimental results suggest a slight dip, and below 5°. Generally even at very large and very small angles the agreement is excellent.

The agreement is slightly better with the CCC results, but both BSR and RMPS agree well up to 120°before there is slight divergence both between the models and with the experimental and CCC data, both of which suggest a slightly smaller cross section.

The measured 2 3 P DCS error bars are relatively larger beyond 90°. This means there are limited conclusions that can be drawn from this comparison. However, the current experimental results agree broadly with the theoretical data above 25º. Beyond 120º the experimental data have larger uncertainties because of the low counting rates for the 2 3 P state in this angular range. Looking at the overall trend, however, the theoretical values tend to lie slightly above the experimental data and are in general agreement with each other, again with the exception of the 5-state numbers. Below about 25º the experimental data again deviate strongly from the theoretical predictions, with the latter showing a local maximum near 25º

and decreasing from 25º down to 0º. The present experimental data, on the other hand, increase from 25º towards 0º. Again the discrepancy could be due to a contribution to the measured signal from the neighbouring 2 1 P state, which has been modelled by adding a 1% 2 1 P contribution to the RMPS model. These results also support the previous experimental data of Trajmar et al [1].

The measured 2 3 S DCS again shows the least agreement with the theoretical predictions. The experimental data of both the current and previous work suggest that the cross sections from the theoretical models may be slightly high. For all data, with the exception of the 5-state results, the theoretical angular dependence is similar to the experimentally determined shape, but the absolute data lie above the experimental data points over most of the angular range. The dip near 30º and a suggested dip between 80°and 90°a ppear in both the theoretical and experimental data. The differences at small angles (less than 30°) at all three energies require further investigation elsewhere, but possible causes are discussed briefly here. As the energy increases from 80 eV to 120 eV, the shape of the theoretical and experimental data tends to become flatter over the angular range 30°to 180°

Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

and the steepness shallows for the region 0°to 30°. Whilst the current experimental data follows this general trend, the DCS near 0°dips at 100 eV before increasing again at 120 eV,

The increase at 120 eV is a cause for concern as the expectation would be for this to decrease.

This may be as a result of a small geometrical distortion in the normalisation technique giving a slight difference in line shape at very small angles (below 10°), though there is no clear reason why this should be the case. It may also be due to changes in the overlap of gas and electron beams as the energy increases. The experimental data of Trajmar et al suggests that the current experimental data overestimates the DCS from 0°to 10°, and underestimates the DCS from 10°to 30°. Work on the out-of-plane characteristics of the MAC will be published later, and further modelling of the overlap in the current data will be investigated as part of this work.

DCS results at 120 eV

Measurements of the DCS at 120 eV for the 2 3 S, 2 1 S, 2 3 P and 2 The 2 1 P data show excellent agreement between the current experimental data and the theoretical models across the angular range. The 5-state numbers deviate slightly from these results between 15°and 30°and at large angles above 140°, again similar to the lower energy behaviour discussed above. There is also a minor difference between the experimental data and the BSR and RMPS theoretical results below 5°.

The 2 1 S DCS again shows good agreement with the theoretical results, especially those from the BSR and CCC calculations, above 40°, with the exception of the 5-state results that are higher above 5°. The overall shape and magnitude of the experimental and theoretical DCS are in excellent agreement across almost the entire angular range. Even at very large and very small angles the agreement is good, although there are slight differences in the range between 10º and 30º, where the theoretical values lie above the experimental data. The RMPS cross section tends to be slightly higher up to 150°.

Again the measured 2 3 P DCS error bars are relatively larger beyond 80°. However the current experimental results agree broadly with the predictions above 35º. Given the larger uncertainties in the experimental data the overall trend of the theoretical values tend to lie slightly above the experimental data, but at this energy there are fluctuations between both the models themselves, and between the models and the experimental data. The 5-state model again yields generally higher cross section than the other models at a given scattering angle.

Below about 25º the experimental data again deviate strongly from the theoretical predictions, with the theoretical curves exhibiting a local maximum near 25º and decreasing from 25º down to 0º. The current experimental data, on the other hand, again increase from 25º towards 0º. As before, the discrepancy could be due to a contribution to the measured signal from the Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

neighbouring 2 1 P state. Modelling this effect by adding a 1% 2 1 P contribution to the RMPS results for the 2 1 P state suggests that this may indeed be the case.

Not surprisingly, the 2 3 S DCS shows the least agreement with the theoretical results at this energy as well. The experimental data of the current work again suggest that the cross sections predicted by the theoretical models may be slightly too high or that the experimental data is under-detecting the DCS based on the discussion on 2 3 S at 100 eV. For all data, with the exception of the 5-state results, the theoretical shape of the curve is similar to the experimental shape, but the theoretical numbers lie above the experimental data points over most of the angular range. There appear to be dips near 30º and between 80°and 100°in both the theoretical and experimental results. Below 10º the theoretical values again tend to underestimate the DCS, the peak near zero is suggested to be experimentally erroneous as commented on for 2 3 S at 100 eV. Above 130°there is some divergence between the experimental and theoretical results and amongst the theoretical predictions themselves. The BSR results are surprisingly different from those of the other models and the experimental data. The angular dependence is arguably most similar to the experimental data albeit with higher cross section values.

CONCLUSIONS

We have presented experimental data and calculated results of DCS for the 2 3 S, 2 1 S, 2 3 P, and 2 1 P states of helium for electrons with incident energies of 80, 100, and 120 eV. In general, there is quite good agreement between the experimental DCS and the theoretical predictions with respect to both the magnitude and the overall angular dependence of the cross sections, and good agreement with previous experimental data. Moreover, the level of agreement is similar for all the incident energies investigated. The agreement for the 2 1 S state is particularly good. Where differences do occur, they are found mainly at large angles (above 150°) and small angles (below 30°) in the triplet states 2 3 S and 2 3 P. At small angles these differences in the 2 3 P state are likely the result of limited experimental resolution. The remaining differences, however, particularly for the 2 3 S near the forward direction, require further investigation. These conclusions support similar findings by Lange et al [7] at lower energies. The comparison of multiple theories with experiment at extreme angles provides a rigorous method of investigating these remaining differences, and the successful use of the magnetic angle changer at intermediate energies significantly broadens opportunities for comparing theory with experiment across the entire angular range.

  box, which makes them very effective in forming the R-matrix basis. By diagonalizing the target hamiltonian in the B-spline basis, it is straightforward to generate a set of pseudostates, similar to what is being done in CCC and RMPS. The energy distribution of the pseudostates, however, is very different from those obtained in the Laguerre-based CCC and the Sturmianbased RMPS methods. Once again, we emphasize that agreement between the results obtained in these entirely independent computational implementations for solving the Schrödinger equation for this collision problem provides confidence in the theoretical predictions.

The 2 3 P

 3 DCS agrees broadly with the theoretical models for the angular range between 25º and 160º. The agreement is especially good with the BSR predictions while the CCC results lie somewhat below the experimental data points, and the RMPS data vary significantly between 120°and 160°. Below about 25º the theoretical predictions deviate strongly from the experimental data, with the theoretical results showing a local maximum Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

Figure 1 -

 1 Figure 1 -A quarter cross section of the present iron-cored coil system. The total radius of the coil is 15 mm and the length of one coil is approximately 12 mm. The positions of the iron cores are shown at the top of the diagram by grey rectangles.

Figure 2 -

 2 Figure 2 -The circles indicate magnetic field measurements for the complete coil system. The dotted line represents the calculated fields for the coil using Equation 1.The smooth curve without circles represents the magnetic field variation for a MAC without an iron core, scaled up by a factor of 5.

Figure 3 -

 3 Figure 3 -Motion of electrons under the influence of the MAC for nominal incident and residual energies of 40 eV and 20 eV respectively. The incident beam enters the MAC from the right side of the figure. The effect of the applied currents within the MAC results in the straight-through beam (indicated by the dashed line) exiting the MAC at the bottom of the figure. This is separated from electrons scattered at 0°with lower residual energy (shown by the solid line) which also exit the MAC at the bottom of the figure, but experience a slightly stronger deflection. Electrons scattered at 90°and 180°(solid line to left of the figure and vertically upwards respectively) can also be detected by either moving the analyser, or by fixing the analyser and applying different currents to the MAC.

Figure 4 -

 4 Figure 4 -DCS in Helium for the 2 1 P state at 80 eV.

Figure 5 -

 5 Figure 5 -DCS in Helium for the 2 1 S state at 80 eV.

Figure 6 -

 6 Figure 6 -DCS in Helium for the 2 3 P state at 80 eV.

Figure 7 -

 7 Figure 7 -DCS in Helium for the 2 3 S state at 80 eV.

Figure 8 -

 8 Figure 8 -DCS in Helium for the 2 1 P state at 100 eV.

Figure 9 -

 9 Figure 9 -DCS in Helium for the 2 1 S state at 100 eV.

Figure 10 -

 10 Figure 10 -DCS in Helium for the 2 3 P state at 100 eV.

Figure 11 -

 11 Figure 11 -DCS in Helium for the 2 3 S state at 100 eV.

Figure 12 -

 12 Figure 12 -DCS in Helium for the 2 1 P state at 120 eV.

Figure 13 -

 13 Figure 13 -DCS in Helium for the 2 1 S state at 120 eV.

Figure 14 -

 14 Figure 14 -DCS in Helium for the 2 3 P state at 120 eV.

Figure 15 -

 15 Figure 15 -DCS in Helium for the 2 3 S state at 120 eV.

  1 P states of helium are shown in figures 12 -15. As before, the experimental results are presented together with theoretical predictions from the CCC, RMPS, and two BSR calculations. No further theoretical or experimental results are presented at this energy.

Acknowledgments

The work of RW, DC, NB, GCK and FHR was supported by EPSRC grant GR/K90845/01, the work of DVF and IB was supported by the Australian Research Council, the Australian National Computing Infrastructure Facility and its Western Australian node Ivec, and the Differential cross sections for n = 2 states of helium at intermediate energies (80 eV, 100 eV and 120 eV) measured across the complete angular scattering range (0°-180°).

work of OZ and KB was supported by the United States National Science Foundation under grants PHY-0757755 and PHY-0903818.


Nielka P Van Erp

Sharyn D Baker

Anthe S Zandvliet

Bart A Ploeger

Margaret Den Hollander

Zhaoyuan Chen

Jan Den Hartigh

Jacqueline M C König-Quartel

Henk-Jan Guchelaar

Hans Gelderblom

P H Van Erp
email: p.h.van_erp@lumc.nl

Marginal increase of sunitinib exposure by grapefruit juice

Keywords: Sunitinib, grapefruit juice, pharmacokinetics, CYP3A4 inhibition

Purpose

The drug label of sunitinib includes a warning for concomitant use of grapefruit juice (GJ) but clinical evidence for this drug interaction is lacking. The aim of this study is to determine the effect of GJ, a potent intestinal cytochrome P450 (CYP) 3A4 inhibitor, on steady-state sunitinib pharmacokinetics (PK).

Methods

Sunitinib PK were evaluated in 8 cancer patients receiving sunitinib monotherapy in a "four weeks on -2 weeks off" dose regimen. Serial blood samples for PK analysis of sunitinib were collected on two separate days. On both PK days patients received a single oral dose of 7.5 mg midazolam as a phenotypic probe for assessment of intestinal CYP3A4 activity. The first PK day was at steadystate sunitinib PK (between days 14-20), the second PK day was on day 28. On day 25, 26 and 27, 200 mL GJ was consumed three times a day. The effect of GJ on sunitinib exposure was assessed by comparing sunitinib PK with and without GJ.

Results

Concomitant use of GJ and sunitinib resulted in an 11% increase of the relative bioavailability of sunitinib (P < .05). The effect of GJ on CYP3A4 activity was confirmed by an increase of ~50% of mean midazolam exposure (AUC 0-24hr ) from 122.1 ng*hr/mL to 182.0 ng*hr/mL (P = .034).

Conclusion

GJ consumption results in a marginally increase in sunitinib exposure which is not considered clinically relevant. There is no clinical evidence underscoring the warning in the sunitinib drug label regarding concomitant use of GJ.

INTRODUCTION

Sunitinib malate (Sutent ® ; SU11248) is a multitarget tyrosine kinase inhibitor approved for the first line treatment of metastatic renal cell carcinoma (mRCC) and imatinib-resistant metastatic gastrointestinal stromal tumors (GIST). [START_REF] Motzer | Sunitinib versus interferon alfa in metastatic renal-cell carcinoma[END_REF][START_REF] Demetri | Efficacy and safety of sunitinib in patients with advanced gastrointestinal stromal tumour after failure of imatinib: a randomised controlled trial[END_REF][START_REF] Goodman | Approval summary: sunitinib for the treatment of imatinib refractory or intolerant gastrointestinal stromal tumors and advanced renal cell carcinoma[END_REF] The licensed dosing regimen for sunitinib is a "four weeks ontwo weeks off" schedule. [START_REF] Faivre | Safety, pharmacokinetic, and antitumor activity of SU11248, a novel oral multitarget tyrosine kinase inhibitor, in patients with cancer[END_REF] Sunitinib is absorbed from the gastrointestinal tract to an unknown extent. The intake of food does not affect the pharmacokinetics of sunitinib. [START_REF] Bello | Effect of food on the pharmacokinetics of sunitinib malate (SU11248), a multi-targeted receptor tyrosine kinase inhibitor: results from a phase I study in healthy subjects[END_REF] Sunitinib is in vitro extensively protein bound, has a long half-life of 50 hours and a high apparent volume of distribution of 2000 liters. [3;6] CYP3A4 metabolizes sunitinib into an active metabolite, SU12662, which is further metabolized by CYP3A4 into inactive moieties. [3;7;8] Sunitinib has not been described to be a substrate of any other metabolizing enzymes besides CYP3A4. It was identified in vitro as a moderate substrate of the ATP-binding cassette (ABC) drug transporters ABCG2 and ABCB1 and showed no affinity for organic anion transporting polypeptides (OATPs). However, the clinical relevance of these transporters needs to be investigated in vivo. [9;10] Co-administration of ketoconazol, a potent CYP3A4 inhibitor, results in a 51% increase of the combined area under the concentration time curve (AUC) of sunitinib and SU12662 after a single dose of sunitinib in healthy volunteers. [START_REF] Goodman | Approval summary: sunitinib for the treatment of imatinib refractory or intolerant gastrointestinal stromal tumors and advanced renal cell carcinoma[END_REF] This observation was extrapolated to warnings for the potential effect of strong CYP3A4 inhibitors including grapefruit juice (GJ) in the drug label of sunitinib [8].

GJ contains a rich mixture of several hundred ingredients which may be responsible for the GJdrug interaction effect. [START_REF] Bailey | Grapefruit juice-drug interactions[END_REF][START_REF] Schmiedlin-Ren | Mechanisms of enhanced oral availability of CYP3A4 substrates by grapefruit constituents. Decreased enterocyte CYP3A4 concentration and mechanism-based inactivation by furanocoumarins[END_REF][START_REF] Wang | Inhibition of P-glycoprotein transport function by grapefruit juice psoralen[END_REF][START_REF] Paine | Two major grapefruit juice components differ in intestinal CYP3A4 inhibition kinetic and binding properties[END_REF] However, by administrating the purified forms of the different compounds to human volunteers, the furanocoumarins (most abundant bergamottin (BG) and 6',7'dihydroxybergamottin (DHB)) were confirmed to result in a significant CYP3A4 inhibiting effect and therefore these constituents are held responsible for GJdrug interactions. [START_REF] Goosen | Bergamottin contribution to the grapefruit juice-felodipine interaction and disposition in humans[END_REF][START_REF] Rashid | Quercetin, an in vitro inhibitor of CYP3A, does not contribute to the interaction between nifedipine and grapefruit juice[END_REF][START_REF] Paine | A furanocoumarin-free grapefruit juice establishes furanocoumarins as the mediators of the grapefruit juice-felodipine interaction[END_REF] GJ is an inhibitor of intestinal CYP3A4, with little effect on hepatic CYP3A4 activity. [START_REF] Saito | Undesirable effects of citrus juice on the pharmacokinetics of drugs: focus on recent studies[END_REF] Grapefruit juice also appears to be an inhibitor of ABCB1 and possibly of OATP located in the intestines. [START_REF] Paine | A furanocoumarin-free grapefruit juice establishes furanocoumarins as the mediators of the grapefruit juice-felodipine interaction[END_REF][START_REF] Saito | Undesirable effects of citrus juice on the pharmacokinetics of drugs: focus on recent studies[END_REF][START_REF] Paine | Further characterization of a furanocoumarin-free grapefruit juice on drug disposition: studies with cyclosporine[END_REF][START_REF] Mertens-Talcott | Grapefruit-drug interactions: can interactions with drugs be avoided?[END_REF] Recently, multiple oral anticancer therapies, mainly tyrosine kinase inhibitors, are introduced and since most of them are substrates of CYP3A4, their drug label mentions that grapefruit juice could increase their exposure. Notably, only two studies has specifically determined the clinical effect of GJ on an oral anticancer drugs (etoposide and nilotinib) and inconsequent effects were observed, namely a decrease in etoposide exposure but an increase in nilotinib exposure. [21;22] Therefore the evidence for GJdrug interaction of most anticancer drugs is lacking. Since patients are increasingly being treated with oral anticancer therapy in the recent years, it is highly relevant to better understand and to determine potential GJdrug interactions of oral anticancer drugs. Therefore, in this study the effect of concomitant use of GJ on sunitinib exposure in cancer patients was determined.

MATERIALS AND METHODS

Patients

Patients eligible for study entry were treated with sunitinib at a dose level of 25 -50 mg once daily in a "four weeks on -two weeks off" regimen. The patient should have a tumor for which sunitinib is registered as the first line (metastatic renal cell carcinoma), second line (gastrointestinal stromal tumor) treatment or for which no treatment options were available anymore. All patients were ≥ 18 years old, had a WHO performance status ≤ 2 and a life expectancy of at least 12 weeks. Cytotoxic chemotherapy or radiation therapy within four weeks prior to entering the study protocol was not allowed. Concurrent use of substances known or likely to interfere with the pharmacokinetics (PK) of sunitinib and with CYP3A4 activity, such as ketoconazol, fluconazol, rifampicine and St. John's wort, were not allowed within 14 days before study entry and during the study. All patients had adequate bone marrow, renal and hepatic functions as defined by hemoglobin 6.0 mmol/L, WBC 3.0 × 10 9 /L, platelets 100 × 10 9 /L, creatinine clearance 60 mL/min, bilirubin 1.75 × the upper limit of institutional normal range. Prior to commencing the study, a sample size of 8 patients was determined as sufficient for a paired, two sided analysis to detect a difference of 25% in sunitinib exposure with a power (1-) of 0.8, and a two-sided significance level ( ) of 0.05. The study was approved by the institutional ethics committee (Leiden University Medical Center, The Netherlands), and all patients gave written informed consent before entering the study.

Study design

The study was designed to evaluate the effect of GJ on steady-state sunitinib PK.

All patients were treated with commercially available hard capsule sunitinib malate (Pfizer, Kent, United Kingdom) at an oral dose of 25/ 37.5/ 50 mg once daily in a "four weeks on followed by two weeks off" dose regimen. The study was performed during one sunitinib treatment cycle of six weeks. Patients were admitted to the hospital on two separate PK days. The first PK day was at steady-state sunitinib PK (between day 14 -20) and the second PK day was on day 28. On days 25, 26, and 27, the patients used 200 ml grapefruit juice of a preselected lot of commercially available GJ three times daily. On these three days, sunitinib was simultaneously used with the morning consumption of the grapefruit juice. On both PK days patients were given one midazolam 7.5 mg tablet (Roche, Woerden, The Netherlands) as a phenotypic probe to confirm the inhibitory effect of GJ on intestinal CYP3A4 activity (Fig. 1).

Selection of a grapefruit juice batch

Different batches of GJ show a considerable variability in BG ( 35 fold) and DHB ( 200 fold) concentration. [START_REF] De Castro | Variation of flavonoids and furanocoumarins in grapefruit juices: a potential source of variability in grapefruit juicedrug interaction studies[END_REF] Therefore, selecting a batch with a sufficient amount of BG and DHB to induce a clinically relevant effect on CYP3A4 substrates was necessary before the interaction study was conducted. [START_REF] Goosen | Bergamottin contribution to the grapefruit juice-felodipine interaction and disposition in humans[END_REF] Concentrations of BG and DHB were quantified in various batches of GJ from different brands using a validated high performance liquid chromatographyultraviolet detection (HPLC-UV) method. This assay was based on a previously published method with minor modifications. [START_REF] De Castro | Variation of flavonoids and furanocoumarins in grapefruit juices: a potential source of variability in grapefruit juicedrug interaction studies[END_REF] Briefly, the juice was homogenized by shaking. GJ (0.5mL) was mixed with 10 L internal standard (100 g/mL, in methanol) and 2 mL ethyl acetate. Calibration standards contained 0.2 -4 g/mL BG and 0.1 -2 g/mL DHB were prepared at the start of each analytical run. The standard stock solution contained BG and DHB (100 and 50 g/mL in DMSO:methanol(1:3)). The residue from the organic phase was reconstituted with 100 L of DMSO/acetonitril solution (1:3 v/v) and applied to the HPLC separation system (Unexas 2104, Knauer, Berlin, Germany). The compounds of interest were separated on a Hypersil ODS RP analytical column minutes, respectively. Linearity was confirmed over the range of 0.2 -24 g/mL for BG and 0.1 -12 g/mL for DHB. The within day and between day precision and accuracy were < 15%.

Pharmacokinetic sampling

Blood samples were collected on the first and second PK day of the study for assessing sunitinib and midazolam plasma concentrations. Blood was collected in heparin-containing tubes at the following time points: pre-dose, 10, 20, 40 minutes; 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, and 24 hours after simultaneous intake of sunitinib and midazolam. Blood samples were centrifuged at 3,000 rpm for 10 minutes, the plasma was stored in two separate tubes for the midazolam and sunitinib analysis and both tubes were stored at -80˚C until the day of analysis.

Bioanalysis of sunitinib and midazolam

Sunitinib was measured using a validated liquid chromatographic-tandem mass spectrometric (LC-MS/MS) assay, which has been described earlier. [START_REF] Minkin | Quantification of sunitinib in human plasma by highperformance liquid chromatography-tandem mass spectrometry[END_REF] The calibration curve of sunitinib was linear over the range of 0.2 -500 ng/mL. The within day and between day precision and accuracy were < 8%. The LLQ of the sunitinib assay was 0.2 ng/mL. Midazolam was measured using a validated liquid chromatographic-tandem mass spectrometric (LC-MS/MS) assay. Briefly, 200 l plasma was extracted by adding 500 l of acetonitril containing midazolam D4 (4 g/L) as the internal standard, followed by vortex mixing and centrifugation at 13,000 rpm for minutes at ambient temperature. The supernatant was collected and 10 L was separated on an Atlantis T3 C18 analytical column (2.1 x 50 mm, i.

Pharmacokinetic analysis of sunitinib

Sunitinib plasma concentrations were evaluated by a population pharmacokinetic method using NONMEM (version VI, level 1.0) (Globomax, Hanover, MD, USA).

The First-Order Conditional Estimation (FOCE) method of NONMEM with interaction (INTER) between the interindividual and residual random effects was used. [START_REF] Beal | NONMEM user's guides[END_REF] Discrimination between hierarchical models was based on comparison of the objective function values (OFV) of NONMEM using the likelihood ratio test.

A decrease in ∆OFV of 6.63 was considered statistical significant (P < .01). [START_REF] Wahlby | Assessment of actual significance levels for covariate effects in NONMEM[END_REF] A base model was developed to describe sunitinib PK, using sunitinib concentrations obtained on the first and second PK day. Next, a final model was developed by the introduction of a GJ effect on the relative bioavailability of sunitinib, resulting in an effect on the apparent clearance and apparent volume of distribution and thereby exposure to sunitinib, since it was assumed that GJ exerts its effect only by irreversible inhibition of intestinal CYP3A4 and possibly by inhibition of ABCB1 (Fig. 2). The recovery half life of CYP3A4 activity after GJ consumption was set to 23 hours. [START_REF] Greenblatt | Time course of recovery of cytochrome p450 3A function after single doses of grapefruit juice[END_REF] The model was evaluated by goodness of fit plots, case deletion diagnostics and a numerical predictive check. Moreover, a log-likelihood profile was generated for the effect size of GJ to determine the confidence interval.

The effect of GJ on sunitinib bioavailability was studied for simultaneous intake of sunitinib and grapefruit juice but was also evaluated for sunitinib intake 7 hours, 24 hours, 72 hours and one week after the last GJ consumption.

RESULTS

Patients

Eight patients were enrolled into the study and all were evaluable for PK analysis. Patient characteristics are summarized in Table 1. No severe or unexpected side effects were observed during the three days of GJ or midazolam co-administration on both PK days.

Selection of grapefruit juice

The concentration of BG and DHB was measured in 6 different lots of GJ. BG and DHB concentrations among the lots tested ranged from 2.6 -11.9 mg/L and 0.4 -8.1 mg/L, respectively. The concentration of BG and DHB in the selected lot of GJ was 33.1 mol/L and 2.7 mol/L, respectively, corresponding with 2.2 mg/200mL BG and 0.2 mg/200mL DHB. Due to the expiry date a second lot of the same brand was selected for the last two patients of the study. The concentrations in the second lot selected were 23.5 mol/L BG and 5.7 mol/L DHB, corresponding with 1.6 mg/200mL BG and 0.4 mg/200mL DHB. The concentration of BG in both lots was sufficient to induce a significant drug interaction. [START_REF] Goosen | Bergamottin contribution to the grapefruit juice-felodipine interaction and disposition in humans[END_REF]

Pharmacokinetic analysis of midazolam

Midazolam exposure (AUC 0-24hr ) increased after prior intake of GJ. The midazolam exposure expressed as AUC 0-24hr (± standard error of the mean (SEM)) with and without GJ was 122.1 (± 32.9) ng*hr/mL and 182.0 (± 52.2) ng*hr/mL, respectively (P = .034). Therefore midazolam exposure increased with 50% by GJ and this observation confirms the inhibitory effect of GJ on CYP3A4 activity (Fig. 3)

Pharmacokinetic analysis of sunitinib

A one-compartment model with linear elimination and first-order absorption adequately described the concentration-time profile of sunitinib. The data did not contain sufficient information to support a two-compartment model this was demonstrated by a non significant improvement ( OFV = -0.048) of the model on introduction of a second compartment [START_REF] Houk | A population pharmacokinetic meta-analysis of sunitinib malate (SU11248) and its primary metabolite (SU12662) in healthy volunteers and oncology patients[END_REF]. Inclusion of an absorption lag time significantly improved the base model of sunitinib. Between-subject variabilities of the absorption rate and clearance were large (60-70%). The base model of sunitinib is graphically presented in Figure 2 (left side).

In the final model, CYP3A4 activity was depleted by each GJ consumption (9 in total) and the activity was restored with a half life of 23 hours (Fig. 4A). [START_REF] Greenblatt | Time course of recovery of cytochrome p450 3A function after single doses of grapefruit juice[END_REF] Inhibition of CYP3A4 activity resulted in an increase in the relative bioavailability of sunitinib (Fig. 4B). The individual predicted and measured sunitinib concentrations are depicted for all patients (Fig. 4C). Additionally the predicted value and the individual measured concentration are depicted specifically for the both sampling days (Fig. 5). Introduction of the GJ effect on the relative bioavailability of sunitinib significantly improved the model (∆OFV = -10.01, P < .01) and resulted in the final model (Fig. 2).

The estimated PK parameters in the final model are listed in Table 2. The derived parameters are calculated with the estimated PK parameters and represent the data when GJ and sunitinib are used simultaneously. Goodness-of-fit plots demonstrated that the final model adequately described the time profile of sunitinib concentrations. Case deletion diagnostics demonstrated that the estimated GJ effect was not highly dependent on the data from a single patient (range in relative F = 1.05 -1.14). Moreover, suitability of the final model was confirmed by the results from a numerical predictive check. [START_REF] Karlsson | Diagnosing model diagnostics[END_REF] Out of 268 observed sunitinib plasma concentrations, 21.6% was below the P25-P75 (interquartile) prediction interval, 57.1% was within the interval and 21.3% was above the P25-P75 prediction interval.

Based on the final model it is determined that simultaneous intake of sunitinib and grapefruit juice results in a decrease of intestinal CYP3A4 activity and consequently an increase of sunitinib exposure of 11% (as a result of the increased relative bioavailability 1.11, 95%CI: 1.042-1.182).

Since the intestinal CYP3A4 activity is restored with a half-life of 23 hours, the relative bioavailability of sunitinib is also restored with a half-life of 23 hours. The different time interval evaluations resulted in the following estimates: when GJ is consumed 7 hours before the sunitinib dose, the sunitinib exposure is increased by 8.9%, after 24 hours the effect is 5.3% and after 72 hours it is 1.3%. If sunitinib therapy starts one week after the last GJ consumption the effect of GJ on the exposure to sunitinib is negligible ( 0.07%).

DISCUSSION

This study shows that inhibition of the intestinal CYP3A4 activity by GJ results in a significant but not clinically relevant increase of sunitinib exposure. The drug label of sunitinib mentions that GJ may increase the exposure to sunitinib This remark however, is based on extrapolation of the effect of ketoconazol on sunitinib exposure after single dose administration and clinical evidence for the presumably drug interaction between sunitinib and GJ is lacking. [3;8] Our study is the first to directly investigate the effect of GJ on sunitinib exposure in cancer patients and shows that there is the increase in sunitinib exposure by GJ consumption is clinically not relevant. Moreover, this is the third study investigating an interaction of GJ with oral anticancer therapy the studies show a reverse, relevant and irrelevant effect of GJ. [21;22] Indeed, all eight approved tyrosine kinase inhibitors are substrates of CYP3A4 and therefore their exposure could be increased by the consumption of GJ. Since the tyrosine kinase inhibitors have a very different bioavailability the effect of GJ observed in our study cannot be extrapolated to the other tyrosine kinase inhibitors additional studies determining the effect of GJ on the individual TKI exposure should therefore be conducted.

GJ is a potent inhibitor of intestinal CYP3A4 with little, if any, effect on the activity of hepatic CYP3A4. Indeed, a significant effect of GJ on the exposure to CYP3A4 substrates (e.g. simvastatine, felodipine, triazolam) is seen after oral administration, while the effect is only limited after intravenous administration of these drugs. [15;29-31] GJ is also an inhibitor of the drug transporters ABCB1, OATP1A2 and OATP2B1, which could contribute to the effect of GJ on the exposure of co-administered drugs. [13;32-37] Midazolam is extensively metabolized by CYP3A4 with less affinity for CYP3A5, ABCB1, ABCG2 and OATPs. [START_REF] Franke | Influence of solute carriers on the pharmacokinetics of CYP3A4 probes[END_REF][START_REF] Fuhr | Appropriate phenotyping procedures for drug metabolizing enzymes and transporters in humans and their simultaneous use in the "cocktail" approach[END_REF][START_REF] Kim | Interrelationship between substrates and inhibitors of human CYP3A and P-glycoprotein[END_REF][START_REF] Kurnik | The erythromycin breath test reflects P-glycoprotein function independently of cytochrome P450 3A activity[END_REF] In previous studies, GJ showed a pronounced effect on the exposure of orally administered midazolam. [27;31;42] In our study, midazolam was co-administrated on both PK days as a phenotypic probe to confirm the decreased activity of intestinal CYP3A4 by the selected batch of GJ.

The patients in our study consumed GJ three times a day for three days (25, 26, and 27) at steady-state sunitinib PK. On the last sunitinib treatment day (day 28)

in the six week treatment cycle, the sunitinib PK was determined and compared to the data obtained without consumption of GJ. The effect of GJ was estimated on the relative bioavailability of sunitinib, since GJ is a potent intestinal CYP3A4 inhibitor and therefore, only an effect on the sunitinib uptake is expected rather than on sunitinib clearance, volume of distribution, absorption rate constant and absorption lag time. "In our study we have focused on the effect of GJ, an intestinal CYP3A4 inhibitor, on the pharmacokinetics of sunitinib. A limited if any effect was expected on the SU12662 exposure since hepatic CYP3A4 activity is not interfered by GJ consumption and therefore only sunitinib was quantified".

Indeed the concomitant use of grapefruit juice results in a significant increase of 11% in sunitinib exposure. However, since the reported interpatient variability in sunitinib clearance is large 40% the effect of GJ on sunitinib exposure is negligible and should not be regarded as clinically relevant. [START_REF] Houk | A population pharmacokinetic meta-analysis of sunitinib malate (SU11248) and its primary metabolite (SU12662) in healthy volunteers and oncology patients[END_REF] Moreover, the marginal 11% increase in sunitinib exposure is unlikely to result in an increased toxicity or different anticancer efficacy, although data on the pharmacokineticpharmacodynamic relationship are not available yet.

GJ irreversibly inhibits CYP3A4 and therefore it takes time to restore CYP3A4 functionality since it is dependent on formation of new enzymes. The recovery half-life of CYP3A4 activity after consuming GJ was set at 23 hours according to the study of Greenblatt et al. [START_REF] Paine | Two major grapefruit juice components differ in intestinal CYP3A4 inhibition kinetic and binding properties[END_REF] The recovery half-life was confirmed by several interaction studies between midazolam and GJ over different time intervals [31;42;43].

The half-life of sunitinib is relatively long ( 50 hours) and therefore steady-state PK of sunitinib is reached after 8 days. Similarily, it takes 8 days to achieve steady-state after coadministration of GJ. Therefore, at the second PK day, after three days co-administration of GJ, steady-state sunitinib was not reached. Since a large increase of sunitinib exposure was expected upon co-administration of GJ and consequently more toxicity, it was considered unethical to continue coadministration until sunitinib steady-state was reached. Therefore, we chose to analyse the effect of GJ on sunitinib PK by a compartmental PK approach. The estimated apparent clearance and volume of distribution found in our study are similar to those described earlier by compartimental approach [START_REF] Houk | A population pharmacokinetic meta-analysis of sunitinib malate (SU11248) and its primary metabolite (SU12662) in healthy volunteers and oncology patients[END_REF]. Conversely, a non-compartmental approach was used for determining midazolam exposure used as a phenotypic probe. Indeed, we could adequately determine the exposure to midazolam with and without GJ since extensive sampling was done from start of midazolam administration until plasmalevels of midazolam were undetectable.

The lack of a clinically relevant effect of GJ on sunitinib exposure was not related to the batch of GJ that was used in this study. First, the GJ selected had a sufficient content of BG (2.2mg/ 1.6mg) to induce a significant effect on CYP3A4

activity. [START_REF] Goosen | Bergamottin contribution to the grapefruit juice-felodipine interaction and disposition in humans[END_REF] Secondly, even after the recovery of a proportion of the intestinal CYP3A4 enzymes on the second PK day, a significant effect 50% was observed on the phenotypic drug midazolam, which is comparable to the effect of GJ on midazolam exposure explored in earlier interaction studies [27;42]. No effect of sunitinib on midazolam exposure was observed since midazolam exposure was similar to the exposures reported in earlier studies [42;44]. The increase in midazolam exposure due to GJ co-administration confirms the significant effect of GJ on intestinal CYP3A4 activity. Hence, the marginal effect observed on sunitinib bioavailability is likely to be the result of the limited efficiency of sunitinib metabolism by intestinal CYP3A4. The limited effect of GJ is in contrast with the large effect (51% increase) observed after co-administration of the CYP3A4 inhibitor ketoconazol. [START_REF] Goodman | Approval summary: sunitinib for the treatment of imatinib refractory or intolerant gastrointestinal stromal tumors and advanced renal cell carcinoma[END_REF] Notably, the interaction study with ketoconazol was performed after a single sunitinib dose while in the current study the interaction with GJ was determined after multiple dosing. Inhibition of a major metabolic pathway may lead to shunting of the metabolism to alternative metabolic pathways at steady state PK that remains unnoticed after single dose administration. Indeed, we have reported that alternative metabolic pathways can play a dominant role after prolonged exposure to imatinib. [START_REF] Van Erp | Influence of CYP3A4 inhibition on the steadystate pharmacokinetics of imatinib[END_REF] Another explanation could be that ketoconazol is a strong intestinal and hepatic CYP3A4 inhibitor while GJ is only capable of inhibiting intestinal CYP3A4.

In conclusion, upon co-administration of GJ only marginally increases in sunitinib exposure were observed which is not regarded clinically relevant. The population prediction (solid line), measured sunitinib concentration (grey dots) and 50% prediction interval (greyed area) on the first PK day (without the coadministration of sunitinib) and second PK day (with the coadministration of sunitinib).

( 4 .

 4 6 x 100 mm, i.d 3 m) using the following gradient [time scale (minutesminutes)/ percentage of solvent A (water 2500/phosphoric acid 1.25)/ percentage of solvent B (acetonitril)]: 0-7/70/30, 7-17 70/30 0/100, 17-18/0/100, 18-19 0/100 70/30, 19-22/70/30. DHB, fenprocoumon and BG eluted at 10.9, 12.8 and 16.5

  d 3 m) and eluted with the following gradient [flow rate (ml/min)/ time (minutes)/ percentage of solvent A (formic acid 0.1% in water)/ percentage of solvent B (formic acid 0.1% in acetonitril)]: 0.3/0.5/85/15/, 0.3/1/10/90, 0.3/4.3/10/90, 0.5/0.01/10/90, 0.5/0.39/85/15, 0.5/3.3/85/15, 0.3/0.05/85/15, 0.3/0.05/85/15. The effluent was monitored with a Micromass Quattro LC triple-quadruple mass-spectrometric detector (Waters, Milford, MA, USA) using the electrospray positive ionization mode. The calibration curve of midazolam was linear over the range of 1 -ng/mL. The within day and between day precision and accuracy were < 5%. The LLQ of the midazolam assay was 0.3 ng/mL. Pharmacokinetic analysis of midazolam Midazolam plasma concentrations were analyzed by non-compartimental methods using WinNonlin (version 5.2.1) (Pharsight Corporation, Mountain View, CA, USA). The midazolam area under the concentration time curve (AUC 0-24hr ) was calculated and was compared for each patient between the first and second PK day. Statistical analysis included the two-sided paired Student's t-test, and P values < .05 were considered statistically significant. The statistical calculations were performed using SPSS 16.0 (SPSS Inc. headquarters, Chicago, Illinois, USA)

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 4A :

 12344A Figure 1: study design Abbreviations: GFJ = grapefruit juice; PK = pharmacokinetics; od = once daily

Figure 5 :

 5 Figure 5: Effect GJ on sunitinib PK

Table 1 : Patient characteristics Characteristic Value

 1

	Number of patients	8
	Sex (female / male)	1/7
	Age, years *	54 (41 -78)
	Baseline serum renal and liver function parameters
	Creatinine, M *	77 (56 -122)
	Total bilirubin, M *	9 (6 -15)
	ALT, units/L *	39 (18 -68)
	Baseline bone marrow function parameters
	Hb, mM *	8.7 (7 -9.4)
	WBC, x 10 9 /L *	5.5 (3.5 -38.2)
	Thrombocytes, x 10 9 /L *	196 (149 -318)
	* median values (range)

Abbreviations: RSE = relative standard error; Cl/F = apparent clearance; Vd/F = apparent volume of distribution; ka = absorption rate constant; F = bioavailability; nd = not determined; AUC 0-24hr = area under the plasma concentration-time curve over the dose interval 0-24hr at steady-state pharmacokinetics; t 1/2 = elimination half-life; T max = time to reach peak plasma concentration. Between-subject variability was assessed using exponential models. * Derived parameters are calculated from estimated parameters and are demonstrated as mean values (range).