I . Phys. B: At. Mol. Opt. Phys. 28 (1995) 1287-1300. Printed in the UK

The Ramsauer minimum of methane

F A Gianturco?, J A Rodrigues-Ruizf: and N Sannag t Department of Chemistry. The University of Rome, Citti Universitaria, 00185 Rome, Italy t Departamento de Quimica Fisica, Universidad de Malaga, Campus de Teatinos, 29071 Malaga, Spain 6 Center for Suoercomoutine Aoolications to Universitv Research. CASPUR. Citti Univer- . I .. sitaria, 00185 Rome, Italy

Received 22 November 1994

Abstract. Vibrationally elastic, rotationally-summed cross Sections for electron collisions with CH, are calculated with ob irtrlio static+exchange (SE) interactions and using a sym- metry-adapted, single-centre expansion (SCE) representation for the close-mupled (cc) equations. The dynamical correlation forces are included through a local density-functional theory (DFT) approach. Both integral and differential cross sections are calculated at the Ramsauer-Townsend (RT) minimum and at energies close to it, Comparisons with experi- ments and with previous calculations show that the present approach exhibits one of the best overall accords with measurements while still keeping the comoutational effort within modest limits,

1. Introduction

In recent years considerable progress has been made in both the theoretical and experi- mental aspects of lowenergy collisions of electron beams with polyatomic molecules. Such developments have been particularly remarkable for the CH4 molecule, which has been studied by several groups for a long time, becoming in a sense the principal testing system of both theories and experiments involving polyatomic targets. Methane is, of course, an important constituent of the atmospheres of the outer planets and is one of the key trace elements which populate the upper atmosphere on earth, hence its relevance in molecular astrophysics. Furthermore, it has become of interest for plasma processing, particularly for deposition processes, and also plays a relevant role in edge plasmas for fusion devices (Morgan, 1992).

The observed cross sections in electron-methane scattering present a very distinct Ramsauer-Townsend (RT) minimum around 0.4 eV and a markeq increase at higher energies with a maximum at about 8 eV. Both of those structures have been well exam- ined by several experiments in terms of integral cross sections, total, partial and elastic, and in terms of differential cross sections at several collision energies and for a broad range of angles (e.g. see Tanaka er a/ 1982, Ferch et Q/ 1985, Sohn et ai 1986, Lohmann and Buckman 1986).

Calculations have also been carried out by several authors, who tested their theor- etical models against the above experimental findings. Very extensive studies of electron- methane scattering were, in fact, published by Jain (1986) using model potentials.'

0953-4075/95/071287 t 14S19.50 0 1995 IOP Publishing Ltd 1287

1288 F A Gianrurco et al

Gianturco and Scialla (1987) reproduced the Ramsauer-Townsend (RT) minimum in the integral cross section close to the experimental values, using a modified semiclassical exchange (MSCE) and a correlation-polarization potential from a free-electron-gas (FEG) model. Ab initio results w’ere obtained by Lima et al(1989) with the Schwinger multichannel method (SMC) and by McCurdy and Rescigno (1989) using the complex Kohn variational (CKV) method. The calculations of Lima et a/ provided differential cross sections at the static-exchange (SE) level and the static-exchange-plus-polarization (SEP) level of computation. The SE level failed to produce any minimum feature, while at the SEP level they found the RT minimum to located a t 0.1 eV, i.e. shifted to lower energy with respect to the experiments.

Similarly, the CKV calculations of McCurdy and Rescigno (1989) obtained partial cross sections in the A,, T2 and E symmetries at the SE level and their resulting cross sections were in good agreement with the experimental results at higher energies. Further calculations (Lengsfield el al 1991) using the same approach but at the SEP level, however, found the RT minimum at 0.4 eV, as the experiments suggest, and obtained computed differential cross sections for low collision energies (0.5-1.0 eV) which were also in good accord with the measurements.

Even more recently (Nestmann et al 1994), further ab initio calculations which employed the R-matrix (RM) treatment were also published, reporting the integral and partial elastic cross sections below the ionization limit and in the energy range of the experimental RT minimum. The corresponding DCS were also computed in that region and found to be only in qualitative agreement with the measurements. It is interesting to note here that even the very sophisticated computational methods reported above still show discrepancies with the experiments and indicate that a fully converged SEP calculation for the cross sections of this test molecule still needs to be done satisfactorily with balanced treatment of all interaction forces.

In the present work we extend our earlier calculations on the methane molecule by treating more accurately both the target wavefunction and the exchange interaction between the scattering electron and the bound electrons. We further test various types of correlation-polarization potential functions ( Vcp) by employing different forms of density functional theory (DFT) recently proposed by us for electron and positron scattering from atomic targets (Gianturco and Rodriguez-Ruiz 1993a, b). The aims of the calculations which we will report in this paper are therefore as follows:

(i) to employ the symmetry-adapted, single centre expansion (SCE) for the total wavefunction within a close-coupled (cc) set of scattering equations starting with a multicentre description of the bound target electrons. This approach has been recently tested by us for other polyatomic molecules like SiH4 and CF, (Gianturco et al I994a, b) and found to yield very good agreement with experiments;

(ii) to show that the inclusion of polarization effects via a DFT description of such forces is capable of giving final cross sections in quantitative accord with existing data but with a substantial reduction of computafional effort with respect to more traditional multi-configurational (MC) approaches;

(iii) to show that the use of an exact exchange treatment, i.e. the improved iterative exchange approach discussed earlier by us (Gianturco et al 1994a, b), allows one here to obtain integral and differential cross sections that provide remarkably good agree- ment with measurements around the RT minimum region. They also show good accord with measurements at higher energies (Gianturco et al 1995a, b);

(iv) to compare our computed DCS, over a broad range of collision energies, with existing experiments and with the most recent of the computed values available in the literature.

The Ramsauer minimum of methane I289

In the following section we describe our present computational approach and com- pare our new results with those obtained with previous treatments, while section 3 reports our integral cross sections for the rotationally-summed, vibrationally elastic scattering processes and compares them with experiments and with other calculations at the RT minimum.

Section 4 presents our calculated angular distributions and carries out the same sort of comparison as before with experimental DCS, while our final conclusions are presented in section 5.

2. The single-centre expanded (SCE) equations

The initial step in our treatment is to generate the full electron-molecule interaction potential as a local and non-local function of the electronic density of the target molecule as given by its SCF-HF molecular orbitals (MOS) produced via a multicentre expansion over Gaussian-type analytic functions (GTOS) (Gianturco et af 1994a, b).

We describe the collisional process in terms of the solutions of the Schrodinger equation written in the form:

I?Y(r, x) =EY(r. x) (1)

where

with ?being the kinetic en:rgy operator for the scattering electron, is the electron- molecule interaction and H , is the Hamiltonian of the molecular target. We let x represent collectively the coordinates of the bound electrons and of the molecular nuclei and intend to refer all particles to a frame of reference fixed to the molecule (BF frame).

The many-body problem in the scattering, electronic coordinate r is now converted into an effective single-particle problem by expanding:

where A is the antisymmetrization operator for the electronic coordinates, while the molecular nuclei are considered as being fixed in space during the scattering event (FN approximation).

By inserting equation (3) into equation (1) . multiplying to the left by the conjugate of a representative state in the expansion (3) one obtains the familiar set of coupled integro-differential equations (IDE):

In the above set of close-coupled (cc) equations for the scattering problem the following meaning of the symbols applies:

Ha =V', + k', ( 5 )

1290 F A Gianfurco et a1

Here lc: is given by 2(E-E.), with E being the total collision energy and E. the molecular internal energy in the target state la). The local icteraction involves, in our treatment, the exact electrostatic interaction with the target, VSl, and the linear response function of its electrons to the impinging projectile, $e correlation-polarization poten- tial I f q , described below. The non-local interaction, WaP, describes the exchange poten- tial between the bound and continuum orbitals, obtained exactly via energy-optimized, iterative schemes. In the case of only one single term in the expansion (3), the elastic scattering problem is then dealt with for a specific electronic state la ) of the target molecule within the M approximation (Gianturco et a1 1994a, b).

In order to solve the numerical problem as that ofaset ofcoupled integro-differential radial equations, one needs now to expand both the bound (Q i (x i ) ) , multicentre MOS and the continuum electron function (F(r) ) over a set of single-centre-expanded (SCE),

symmetry adapted partial waves X: ~ ~ ( x ~ ) = c r - ' u i , ~ ( ) . ) x ~ ( e , 4 ) ( 8 4

hl

F,(v)=C r-'ffYWXV(W ). (8b)

Here li) labels a specific, multicentre, occupied MO within the single-determinant (SD) description of the SCF-HF wavefunction of target molecule. The indices of the continuum function and ofeach contribution, I p p ) , label one of the relevant irreducible representation (IR), p , and one of its component p. The index h labels a specific basis. for the given partial-wave I, used within thepth IR one is considering. The generalized, symmetry adapted harmonics have been given before many times [5 ,6] and will not be discussed here again. The corresponding coefficients U;,,= u$f"'(r) are the essential ingred- ients for computing the interaction potentials of equation (7) and were here obtained by numerical quadrature of the multicentre GTOS given as artesian Gaussian functions:

( 9 4 labelled by the kth atomic centre, of which g is the j t h function, and by the contraction index v of the primitive Gaussian within the subgroup that belongs to a given set of contraction coefficients d,

hI

&'(xk) = N(a, 6, c, a)x" )p z' exp( -ax2)

"ma"

G"(xiJ= C d?g?(xk). (96) " - 1

The remaining parameters are included in the normalization constant Nand will not be further discussed here.

The quadratures were carried out via Gauss-Laguerre grids using a discrete, variable radial grid, for each point of which the spherical grid in the (0,b ) points was evaluated. Convergence was achieved with grids of 46 x 46 sets of angular points, while up to 500 radial values were generated in the centre of mass (COM) of the molecular target (BF frame). Several numerical tests were carried out using different grids and the final results can be considered converged (on the wave function representation) within less than 1%. A further discussion on the accelerated convergence for the iterative exchange approach has already been given in our previous work (Gianturco et a1 1994a, b).

3. Integral elastic cross sections

As mentioned in the introduction, because of its popularity both for theory and experi- ments, the methane molecule has become a very useful system on which to test the

The Rmnsauer minitiiunz of methane 1291

quality of the computational models employed to describe the dynamics of the interaction.

The GTO multicentre wavefunction was obtained using a quantum chemistry code (Gianturco et ai 1992) and was given by a triple-zeta expansion plus d-type polarizatioq functions on C and ptype on the H atoms. The bond distance was kept fixed at 2.063 au.

The SCE implementation was carried out, in a symmetry-adapted form, for all the potential contributions of equation (7), and for both the bound and continuum orbitals of equation (8), up to /ma= 12. The IR considered in the calculations were the A I , the T2 and the E symmetries for the continuum orbitals, while only the a] and tz are occupied inthe 'AI ground electronic state of the target molecule. Convergence tests on the partial-wave expansions were carried out in all the symmetries and 1,,,=7 was found to provide fully converged K-matrix elements for each considered symmetry.

As often discussed in the literature (e.g. see Lohmann and Buckman 1986) the presence of a marked minimum in the cross section at low collision energies has been detected for a wide variety of polyatomic targets and the recent growth of synchrotron radiation experiments (Randell et ai 1994) has shown quite definitely that such features are rather common in the very-low-energy region of electron scattering from complex polyatomics. They are also a very stringent test of possible theoretical models since the fact that the target molecules become 'transparent' to the electron beam over a narrow energy range is obviously related to a delicate interplay of the interaction forces. Qualita- tively, one can say that the electron-electron local and non-local repulsion effects are balanced by attractive nuclear Coulomb forces and by the local medium polarization effects in such a way as to minimize the classical deflection function at the end of the contributing trajectories and after their interference. This simply means, as mentioned in earlier work (Gianturco and Scialla 1987), that the presence of such a minimum is only reproduced with the inclusion of polarization effects and that its location at the correct energy values is the result of a delicate balance, in the theoretical model, of repulsive and attractive contributions to the full electronuclear interaction forces.

In the present work we have treated the Coulomb static interaction correctly within the chosen target basis set and we have also treated the exchange interaction accurately, through a converged iterative procedure. Furthermore, we have introduced polarization forces with two different, parameter-free models.

(i) We treated the short-range, dynamical correlation using a density functional modelling of this quantity (Gianturco and Rodriguez-Ruiz 1993a, b) and connecting the ensuing local potential to the long-range polarizability term. The details of the model are given in the above references and will not be repeated here. Results for the CH4 molecule in the resonance region were given in another recent publication (Gianturco er a1 1995a, b).

(ii) We also treated the short-range correlation by using the simpler, free-electron- gas (FEG) modelling of the local functional form, as introduced earlier by O'Connel and Lane (1983), Padial and Norcross (1984) and Gianturco el a1 (1987).

The various calculations for the RT region of the elastic, rotationally summed cross section are shown in figure I . The experimental data are given by the open squares and are the elastic cross sections from Ferch et al ( l985) . The full curve shows the present calculations using the DFT modelling of correlation forces, while the FEG model was used for the calculations given by the broken curve. It is also interesting to note that one can use a different description of the target wavefunction, i.e. a direct expansion over single-centre Slater-type orbitals (STOS) as we have previously done for this system

1292 F A Gianturco el a1

0 a2 0.4 0.6 0.8 IO Electron Energy le VI

Figure 1. Computed and measured integral elastic cross secticns in the region of the Ramsauer-Townsend (RT) minimum of methane. Squares, Ferch el a/ (1985): full curve, present calculations using D F r model; broken curve, present calculations using the FEG model; chain curve, present calculalions using the FFG model and Slater-type orbitals to describe the target molecule.

(Gianturco and Scialla 1987). Once the same descriptio11 of the scattering dynamics is employed, however, one finds that the results are affected very little, as shown by the chain curve reported in figure I . This means that, in both cases, the chosen target wavefunctims provide essentially the same description of the charge density distribu- tions of the bound electrons.

All the calculations shown in the figure manage to represent the presence of the RT minimum and the FEG model for correlation forces also locates it at the right position. The fact that the calculated cross sections are smaller than the experiments is not entirely surprising since one usually expects (Randell er al 1994) that vibrationally inelastic processes play an important role at these energies, while our calculations were performed within the FN approximation.

A further comparison with other calculations and with other experimental data is reported in figure 2. The experiments given by open squares are from Ferch er ai (1985), while those given by open triangles are from Sohn er al (1986). The crosses are the measurements of Lohmann and Buckman (1986). The results of Sohn et alare obtained from differential cross sections by integration over the scattering angle, the other two experimental results are obtained directly from time-of-flight transmission experiments.

One sees here that the present calculations are among the most extensive calculations available as functions of energy and describe very well, with both models for correlation forces, the steep rise of cross sections at very low collision energies. Furthermore, the location of the RT is given correctly by our calculations which use the FEC modelling of correlation forces (broken curve), while the DFT calculations give a minimum at lower energy and rise too rapidly as the energy increases. By comparison, the recent R-matrix (RM) calculations of Nestmann et al (1994) give a much deeper minimum and are in generai much smaller than the experiments over the whole range of energy covered by the RT minimum. The same can be said of the earlier calculations which used thecomplex Khon variational (CKV) method (Lengsfield et al1991) and produced much smaller cross sections around the RT minimum.

The Ramsauer ntininiuni of melhane 1293

DFT ---FEG

RM O O O C K V

-

I n

Electron Energy leu

Fiyre 2. Same as figure I but over a broader range of collision energy. Squares, experi- mental from Perch el al(1985) ; open triangles, experiments from Sohn er 01 (1986); crosses, experiments from Lohmann and Buckman (1986). Full curve, present calculations using DFT; broken curve present calculations using FEC; chain curve, calculations from Nestmann er a/ (1994); open circles, calculations from Lengsfield et a1 (1991).

4. The differential cross sections

The study of the electron angular distributions as a function of collision energy can further provide a rather close test on the quality of a given computational model. In the present instance, we have shown before that the modelling of correlation forces via a local density functional yields a rather good description of the correlation forces via a local density functional yields a rather good description of the interaction of the continuum electron with the many-electron target molecule and provides quantitative agreement with the existing measurements in the region of the RT minimum. Earlier calculations in the higher energy regimes (Gianturco et U/ 1995a, b) also showed very good accord with existing experiments. We now turn to the analysis of the angular distributions within the same, low-energy region or scattering experiments and we will try to answer the following questions.

(i) Can the present treatment establish the importance of rotational inelasticity at low energies?

(ii) Do we obtain the rapid changes of angular distributions, as the collision energy sweeps across the RT minimum, that have been seen in the experiments?

The general, qualitative interpretation of the RT minimum, as discussed earlier, has been related to an overall potential with an attractive component for 1 =0, the s-wave component, at a collision energy very close to Eco~~--O. That component tends to become repulsive if the energy is increased and therefore is likely to vanish at some very low E,,, value, thereby causing the scattering phaseshift due to the s-wave also to vanish. As it is often assumed that the higher partialwave contributions are rather small at such low energies because of their centrifugal barriers that keep the electron outside the molecular ‘core’, then it follows that the integral cross sections also become vanish- ingly small. Recent sensitivity studies (Nestmann el a/ 1994) have indeed shown a strong connection between the s-phaseshift threshold behaviour and the position of the

I294 F A Gianturco et ai

I a I

Figure 3. Computed and measured elastic differentia! cross sections. Full circles, experi- ments from Sohn et a/ (1986). Full curve, present calculation using DtT: broken curve, present calculations using FEG; chain curve, calculations from Nestmann et al. The DCS are evaluated at 0.2 eV of collision energy.

computed RT minimum in the cross sections, indicating a high sensitivity of these quantities to the actual potential employed to treat the scattering. That the present results show such good accord with measurements is therefore a good indicator of the overall reliability of our computed interaction forces.

Figure 3 presents now the elastic differential cross sections (DCS) at one of the lowest collision energies: 0.2 eV. Our calculated values are given by the full and the broken curves, which represent the use of the DFT and the FEG models described before for the short-range correlation effects. The experiments are from Sohn et a/ (1983, 1986) and the chain curves refer to the recent R-matrix calculations of Nestmann et a/ (1994). One clearly sees that, at this energy, our calculations follow very closely the experimental findings and that the FEG model seems to reproduce them more accurately than the DFT one. The R-matrix results, on the other hand, go down too rapidly with increasing scattering angles and appear to have the wrong behaviour in the large-angle, back- scattering region. This result will definitely affect the estimates of the momentum trans- fer cross sections in comparison with the swarm data, as we shall discuss below.

Further calculations of DCS values at higher collision energies are reported in figures 4, 5 and 6. The first of the figures refers to an EWll of 0.5 eV, while figures 5 and 6 show further results at 1.5 eV and at 2.5 eV. One clearly sees that a maximum begins to appear at all energies in the experiments and that i t shifts to lower angles as the E,ll increases. Our calculations reproduce such behaviour very well, while the RM calcula- tions get them at the wrong scattering angles. Furthermore, we see that the two different models for correlation forces are begining to yield DCS values which are very close to each other: as the collision energy increases, in fact, we have already found that the intermediate range of the full interaction is dominated by Coulomb and exchange forces

The Ramsauer minimum of"mthnne 1295

h B 5 0.8 0

Figure 4. Same as figure 3 but for R collision energy of0.5 eV. The meaning of the symbols is the same as in the previous figure.

1.0

- t 1.5 eV

Scattering Angle (deg)

Figure 5. Same as in figure 3 but for a collision energy of 1.5 eV. The elastic differential cross sections are reported following the symbols of figure 3. The open circles are now the experiments from Tanaka et ai (1982).

1296 F A Gianturco et a1

2.0 , 1 I

0 30 1 20 180 Scattering Angle (deg)

Figure 6. Same as in figure 3 bul for a collision energy of 2.5 eV. The meaning of symbols is the same as in that figure.

(Gianturco et a1 1995a, b) thereby making differences in the short-range modelling of the Vcn interaction less capable of affecting the final cross sections.

The results of figure 7 now show both the elastic and the rotationally inelastic DCS at 5 eV of collision energy. The top part of the figure reports our calculations, the RM calculations of Nestmann et ai (1994) and the experiments of Sohn ef al (1983). All the calculated quantities are now closer to each other in size and shape than they were at lower collision energies. Our calculations, however, are clearly closer to the experiments and give correctly the DCS maximum around 60". I1 should be noted here that our calculations involve a much larger number of partial waves than those included in the R M calculations: 12 in our case, versus 1,,,,.=2 in RM results. Such reduced interference effects in the latter case may be responsible for the discrepancies observed in the angular distributions, especially at larger scattering angles.

The state-to-state differential cross sections for rotationally inelaslic processes are shown at the bottom of figure 7, together with the experiments. All quantities are now shown on a log scale. One clearly sees there that, at this higher Ec0,,, the inelastic component of the cross sections becomes more important and that, above 5 eV, one should really compare with rotationally summed cross sections if such DCS are available experimentally. On the other hand, the data shown from experiments clearly indicate that the quantity being measured is indeed the elastic cross section and that the calcu- lated inelastic contributions would greatly modify the shape of the angular distribution, as seen from the chain curve reported in the figure.

The role played by inelastic processes at low ,collision energies is pictorially repre- sented by our calculated results at two different collision energies and reported in figure 8. Both sets of data are again shown on a log scale. The top part refers to a collision energy of 0.2 eV, while the bottom part refers to 1.5 eV. One clearly sees that at the lower energy the shape of the total DCS is modified by inelastic processes around the

The Rmnsauer miniinurn of methane 1297

\ :.

0

Figure I. Top: elastic differential cross sections at 5 eV. The three curves correspond to calculated values as labelled in figure 3. The open and full circles are the experiments already quoted in that figure. Bottom: calculated rotationally inelastic differential cross sections. Broken curve, (0-4) excitation; dotted curve, (0-3) excilation; full curve (0-0) process; chain curve, rotationally summed cross section.

1298 F A Gianlurco et ai

7

Scattering Angle (deg)

I I20 150 180 30 60 90 0

to' Scattering Angle (deg)

Figure 8. Computed and measured rotationally summed, elastic and state-to-state inelastic diferential cross sections. Top: results at 0.2eV, with the same meaning of symbols as in figure I. Bottom: results at 1.5eV. T h e higher full curve reports the rotationally summed computed DCS, while the lower one shows the (0-0) elastic cross section. The broken curve and thc dotted curve have the same meaning as in figure I.

The Ramsauer minimum of methane 1299

deep minimum, thus suggesting that the experiments may partly include such effects at thos angles. Inelasticity is, in fact, fairly small at low collision energies as one sees from our calculations in the figure: the slow, light projectile mainly contributes with the spherical part of its interaction and therefore the latter couples very inefficiently with the rotor states of the target. Since the mass factor is also unfavourable in this case, very little dynamical torque can be applied to the rotating target by the short-range potential due to the travelling electron. On the other hand, as the energy increases higher partial waves, i.e. more anisotropic contributions, begin to play a greater role around some specific scattering angles and therefore the extent of the rotational inelas- ticity increases, becoming comparable to the elastic component as shown by our calcula- tions at the bottom part of the figure. As a result, one could see modifications in the angular distribution once the rotationally-summed cross section is considered and therefore better assess their importance in the experimental data.

5. Conclusions

Our recently developed numerical method for treating electron scattering from non- linear, polyatomic targets has been in this work applied to the analysis of the Ramsauer- Townsend feature of the methane molecule. We employed a single-centre expansion of a multicentre description of the target electronic wavefunction and treated the continuum problem within a close-coupled (cc) approach, via symmetry-adapted angular functions and the solutions of integro-differential equations, to obtain the partial wave compo- nents for the scattered electron. The static and exchange (SE) interaction was treated exactly while the short-range correlation and the long-range polarization effects were treated using a parameter-free modelling through local density functionals. The present method has been used by as successfully to treat SiH4 (Gianturco et al1994b) and CFo (Gianturco et al1994a) as molecular targets and the details of the implementation have been given there.

The results of the present work indicate very clearly that our method can give results of quantitative accuracy for integral and differential eleastic cross sections, can provide estimates of rotationally inelastic cross sections and can reproduce well the RT features of the CH4 molecule.

Given the rather contained computational effort required to develop the DFT forms of correlation forces, as opposed to more complex CI methods, the present approach appears to be very suitable for treating large polyatomic molecules and for studying the rather complex structure of their low-energy shape resonances. Meanwhile, i t has provided for the present tetrahedral target, the most popular one in experimental and theoretical studies, among the best existing accords with the experimental findings around the RT feature, as well as at larger values of Eco,, (Gianturco et a/ 1995b). Its current implementation for the study of more complicated molecules such as sulphur hexafluoride will soon be reported by our group in forthcoming publications (Gianturco et a1 1995a).

Acknowledgements

The financial support of the Italian National Research Council (CNR) and of the Italian Ministry for University and Research (MURST) is here gratefully acknowledged. We

1300 F A Gianluvco er al

also thank the European Union for support through its Human Capital and Mobility (HCM) programme of a Research Network. One of us (JARR) also thanks the EU for funding his stay at the University of Rome during the summer of 1994.

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