Atomic neon interactions with alkali ions: EGM potentials and collision integrals

Physica 93C (1978) 279-284 © North-Holland Publishing Company

ATOMIC NEON INTERACTIONS WITH A L K A L I IONS: EGM POTENTIALS AND COLLISION INTEGRALS F. A. GIANTURCO and U. T. LAMANNA lstituto di Chimica fisica dell'Universitit, Via Amendola 1 73, Bari, Italy*

Received 7 December 1976 Revised 5 May 1977

The electron gas model (EGM) is employed to compute potential curves for Na+, K+ and Li÷ interacting with neon atoms. A modified cut-off form is suggested for the induction energy contributions and its effect is shown to improve the agreement with previous results. Classical collision integrals are also obtained over a wide range of temperatures by numerical integration of the analytical fittings to the computed potential points. The results are finally discussed in comparison with a few experimental data and with other computations from mobility data given at various field strengths.

In view o f these difficulties, experimental data are generally analyzed by postulating a simple, parameterized potential model that is in turn employed within some tractable scattering formalism, with which the resulting rates are then obtained and improved upon by varying the empirical parameters [7]. Since the procedure is not unique, a potential obtained in this way may turn out to be adequate for only a limited set of data and one is therefore left without any lead for improving on the chosen empirical form in a more general way [8]. In this work we present instead a set o f ion-atom potentials that have been obtained from an entirely theoretical model and can thus be used to yield collision integrals over a fairly wide interval o f gas temperatures that allow us to test the relative effects o f both short-range repulsive regions and the long-range, attractive polarization tails.

1. Introduction It is known that a rather wide variety o f gas phase phenomena exist which appear to depend on intermolecular forces solely by binary collision dynamics, or at least which are observed under experimental conditions that make such an assumption fairly realistic. As examples one may cite gas kinetic properties such as viscosity and diffusion [1 ], general relaxation phenomena as those taking place in sound absorption and dispersion observations [2, 3 ] , NMR spin-lattice relaxation [4], and collision-induced pressure broadening of spectral lines [5]. It is therefore quite understandable that over the years the concerted efforts from many different approaches have been directed to correlate all these diverse phenomena most simply and fundamentally by the detailed features o f the relevant intermolecular potentials. It has, however, proved difficult to understand these effects in terms o f basic microscopic processes and virtually impossible to extract information about potentials by inverting any experimental result other than what has become available from simpler molecular beam experiments [6].

2. Description o f the model For the closed-shell systems discussed in this work only a central i o n - n e u t r a l interaction potential is examined, and therefore it is implicitly assumed that only elastic collisions take place between the ions and the neutrals and no thermal excitation or ionization

* Also: Quantum Chemistry Laboratory, CNR, Pisa, which partly supported this research. 279

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F. A. Gianturco and U. T. Lamanna/Interactions of neon atoms with alkali ions

of the gas occurs [9]. These assumptions are most realistically met in atomic gases like the neon case examined here, and at intermediate temperatures [9]. In molecular gases, in fact, non-isotropic orientationdependent forces are also present and the assumption concerning the elastic scattering mechanism breaks down at much lower temperatures [8]. For the present cases one can therefore write the interaction potential V(r) as previously outlined [10], i.e.

where ~(r) contains a "volume" factor related to tile ground state HF wavefunction of the neon target [10]. While the FAG potential has proved to be fairly efficient and accurate in dealing with heavy, manyelectron gases [10] it suggested also that for lighter systems, e.g. I_i+-Ne, the attractive tail penetrated too deeply within the core and therefore yielded larger wells and smaller a values than the ones expected [16]. A simple alternative can then be obtained by putting

V(r) = V~ionic(r ) + VKE(r) + Vex(r) + Vcorr(r)

ivan°D(r) = - ~ra04 F'(r),

"4- Vind(r)

(la)

= VHF(r) + Vcorr(r)+ Vind(r),

(lb)

where the first two terms on the r.h.s, ofeq. (lb) have been obtained within the uniform gas model theory [11, 12] and provide the undistorted contributions to the electron gas model (EGM) [13 ] potential at all relative distances. For the ionic systems, however, the need to describe the strong distortion arising at intermediate and short distances requires the inclusion of a further term, Vind(r), which will asymptotically exhibit the spherical dipole polarizability, r -4 dependence. Such an inclusion has been performed by Gordon and Kim [14] by employing the classical Drude model [15] as a replacement of the EGM scheme and by introducing an empirical parameter, Nf, as the effective electron number required to reproduce the correct C6 values (hereafter called GKD potential). An alternative approach has been suggested by us [10] (the FAG potential) whereby the induction term of the total potential is treated in an effective local way, within the overlap region, and correctly goe s to zero as r 2 for increasingly smaller distances, where the large repulsive interaction becomes the controlling factor of the potential v FAG(r) = "ind

~ ~

--

aO F(r), 2-~

(4)

with F'(r) = [1 - exp (-/3)] 6.

(5)

This last functional form provides a smoother cut-off to the asymptotic term and has been successfully tested for electron-molecule scattering at thermal energies in order to mimick the molecular charge distortion caused by the incoming projectiles [17]. Eq. (4) was then used to obtain the MODPOL potential functions of this work. Furthermore, a least-squares fitting for each of the computed sets of points provided analytic expressions that could be directly used [20] first to evaluate deflection functions and then the .corresponding transport cross sections at various energies:

Q(I)(E) : 27r 1

1 +(_)l[-1 2(1 + l) j

~ (1-cosl0)bdb, 0 (6)

which were in turn employed, after velocity averaging, to yield collision integrals as a function of temperature ~(l, s)(r ) = ((s + 1) ! (KT) s +2)-1

X ~ Q(I)(E) E s+ l e x p [ - E / k T ] dE. 0

(7)

(2) 3. Discussion

of results

with F(r) = [1 - exp (-/fi)],

(3)

The choice of concentrating on neon targets stems from several factors. First, experimental mobility data

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F. A. Gianturco and U. T. Lamanna/Interactions o f neon atoms with alkali ions

are more often found for lighter rare gases [19], while the EGM model does not fare too well for very light systems like He but becomes acceptable from Z = 10 upwards [ 13]. Moreover, the functional form suggested in eq. (2) has already given good results for the Ar through Xe targets [10], where the strong repulsive effect in the short range regions easily overshadows any detailed change in the smaller attraction part and therefore they appeared to be largely unaffected by changing from the ViFnAG to the VMind OD cutoff expressions. Finally, for alkali ions projectiles a recent use of calculated potentials to compute low temperature mobilities as a function of field strength [ 16] has indicated that the Li+-Ne FAG potential proposed by one of us [ 10] gives results that are in excellent agreement with experiments to the left of the mobility maximum, while becoming slightly more inaccurate near the maximum and beyond. The result seemed to suggest a repulsive branch of the potential that lies too low, thus leading to too deep a potential minimum [ 16]. This datum was also confirmed by high temperature calculations of classical collision integrals performed recently in our group [20]. A quick perusal through the figure of this work shows then that the here proposed functional cut-off of eq. (4) substantially changes the interaction behaviour with the lightest ion (fig. 1) both around its potential well and at the early onset of its repulsive branch which is now moved outwards to r ~ 3.02 a.u. (previous value ~ 2.76 a.u.). The same happens to its e value, while the attractive branch goes asymptotically with the expected r -4 behaviour [18]. However, when one moves to systems with a larger number of electrons, the classical coulombic contributions in eq. (1) become increasingly important, especially at short distances, thus reducing the relative effect of varying the description of the induction term within the region of non-local interaction. It therefore follows that Na + and K + ions are less noticeably affected by the different F ( r ) values, whereas the quality of the density functions chosen to represent the collision partners will play for them a greater role. For the sodium ion, in fact, all the computed potentials lay within fairly small ranges (e.g. ~0.0003 a.u. for the well depths) and the MODPOL correction exhibits the earliest onset for the repulsive branch,

i I

r (a. u . )

4

I

i

Ne -Li +

J

-2

GKD / /

0 /. / /. //" --4

• \

\

MODPOL' ~ \ ~

/"

/"

/ /

i

i/

--6

Fig. 1. Computed potential curves for the Li+-Ne system. The plots are only shown around their minima regions: = FAG results from ref. 10; . . . . . GKD results from ref. 14; . . . . . . present results (MODPOL).

even if the "strength" of it is slightly less than the value suggested by the GKD model, which employed a different basis set expansion for the partners [10]. The K + - N e interaction behaves very much the same way, while the well depth becomes smaller with respect to the previous cases: the new choice of eq. (4) for the inner region behaviour of the induction term still connects more smoothly with the repulsive branch of the potential, which again sets in at slightly larger radial values. To assess the relative effects of the above variations on quantities related to transport properties, we performed classical calculations for collision integrals of the mobility and viscosity types [7], and we obtained analytic forms of the potentials by least-square fits to the numerical data via the following piecewise expressions. They were also required to have first-order

282

F. A. Gianturco and U. T. Lamanna/lnteractions o f neon atoms with alkali ions

Table I Computed coefficients of the analytic fittings to the EGM potentials for neon-alkali ions interactions (all in a.u.). Li+

Na+

K+

(a)

(b)

(c)

(b)

(c)

(b)

(c)

A (d) B C

43.311025 2.7714 78.958587

316.953309 3.35 50.876711

155.71717 2.68 1920.8685

600.72428 2.99 283.64171

495.195289 2.59 486.04737

645.78392 2.566 718.20619

D (e) E F

67.935405 2.7734 2.5598

173.9364 2.95 2.8836

217.05987 2.69 2.9596

360.89274 2.74 3.4325

339.13642 2.43 3.6406

469.2201 2.43 4.3854

11.1814 0.40 2.55

22.7262 0.40 2.60

67.1868 0.4428 2.00

97.7657 0.4428 2.00

G. 104 (f) L M

7.093 0.3999 2.30

7.7434 0.40 2.60

(a) Coefficients as defined in eq. (8) of text. (b) Results from present work (MODPOL potential). (c) GKD potentials of ref. 14. (d) For radial values up to 2.0 for Li÷ (2.3 for GKD), 2.6 for Na÷ (2.8 for GKD) and 3.2 for K ÷. (e) Between the r values of (d) and 10.0 for Li÷, Na ÷ and 9.0 for K ÷. (f) For all radial values larger than those quoted in the above note (e).

continuity at the boundary regions: V(r)=Aexp(-Br)+C.r

-12,

r < ~ r 1,

= D exp ( - E r ) - ~ F r - 4 ,

r I < r ~< r 2, (8)

= - G exp ( - L r ) - ~ M r - 4 ,

r > r2.

This procedure had been previously applied to the F A G potential forms for the whole series of Ne, Ar, Kr and Xe targets interacting with alkali ions [20]. Here it is extended, for neon atoms only, to the GKD potentials given in the literature [14] and to the computed curves of this work (MODPOL potentials). The various coefficient values, with their ranges of validity, are listed in table I for all the six cases examined. The qualitative agreement between the two sets of values is rather close and it also comes fairly close to the previous values [20]. The Li + results, however, exhibit the largest differences between the F A G and MODPOL parameters of eq. (8). Collision integrals were then calculated within a classical framework whereby quantum interference is assumed to affect very little the mobility observables, an assumption generally accepted at higher temperatures or for heavy systems [21]. The relevant integrals

were thus expressed in the form o f triple integrals, which can be evaluated using Clenshaw-Curtis quadrature formulae for some chosen set of energy values. The other needed cross sections are then determined by Chebyshev interpolation. The used c o m p u t e d code was our modification of a well tested procedure that follows the above lines and which is already documented in the literature [22, 23]. The usual ~(1,1) and ~(2,2) integrals, in units o f ,8,2, are available on request for the MODPOL and GKD forms of potentials of the present table I. The general effect o f their magnitude falling with higher temperature appears over the whole large interval o f high T (from 1000 to 30,000 K) reported. On the whole, however, the changes from previous results [20] turn out to be rather small in these regions; the trajectories that contribute most to the cross sections are those samplingthe repulsive potential branches which exhibit here rather similar "hardness", roughly measured by the logarithmic derivatives o f the V(r) values. For the lightest Li + target a comparison with experiments at both low [16] and high [20] temperatures indicated the FAG potential to be too large. This is corrected here by the new MODPOL form which produces smaller ~ values for the whole range

F. A. Gianturco and U. T. Lamanna/Interactions o f neon atoms with alkali ions

+1

Z z

Z

+1 ,..1

-I-

+

+1

O

Z "4-

0

~6

+1 Z

Z

+1

283

284

R A. Gianturco and U. T. Lamanna/Interactions o f neon atoms with alkali ions

examined and also appears rather close to the GKD values, being larger than them at very high T ( > 5000 K) and slightly smaller below that temperature value. The Na + and K + ions, on the other hand, exhibit smaller e values and are all very close to one another, although their prepulsive branches are still being affected by the present changes, with the largest ~(1,1) values.

When going down with temperature, one is increasingly sampling the attractive branch of the potential and its long-range tail. The resulting collision integrals are presented in table II, down to T = 200 K. As expected, the differences among them increase for the lithium-neon case, while remain small for the other ions. The MODPOL potential yields figures that are in between the FAG and GKD results (columns 2, 8 and 14), whereas all the computed values come out very close to one another for the Na + and K + results, thus suggesting for them only a minor effect of eq. (4). References [ 1 ] J. H. Ferzinger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972). [2] K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New York, 1959). [3] L. Monchick, Phys. Fluids 7 (1964) 882.

[4] W. Nielsen and R. G. Gordon, J. Chem. Phys. 58 (1973) 414. [5] H. Rabitz, Ann. Rev. Phys. Chem. 25 (1974) 155. [6] U. Buck, Rev. Mod. Phys. 46 (1974) 369. [7] E. W. McDaniel and E. A. Mason, The Mobility and Diffusion of Ions in Gases (Wiley, New York, 1963). [8] L. Monchick and S. Green, J. Chem. Phys. 63 (1975) 2000. [9] L. A. Viehland and E. A. Mason, Ann. Phys. 91 (1975) 449. [10] F. A. Gianturco, J. Chem. Phys. 64 (1976) 1973. [11] V. I. Gaydaenko and V. K. Nikulin, Chem. Phys. Lett. 7 (1970) 360. [12] R. G. Gordon and Y. S. Kim, J. Chem. Phys. 60 (1974) 1842. [13] F. A. Gianturco and M. Dilonardo, J. Chim. Phys. 72 (1975) 315. [14] Y. S. Kim and R. G. Gordon, J. Chem. Phys. 61 (1974) 1 [ 15 ] P. K. L. Drude, The Theory of Optics (Longmans Green, London, 1933). [16] I. R. Gatland, L. A. Viehland and E. A. Mason, J. Chem. Phys. 66 (1977) 537. [17] F. A. Gianturco and D. G. Thompson, J. Phys. B 12 (1976) L383. [18] F. A. Gianturco, J. Chim. Phys. 73 (1976) 527. [19] E. A. Mason et al. Phys. Fluids 18 (1975) 1070. [20] F. A. Gainturco, U. T. Lamanna and M. Capitelli, to be published. [21] W. F. Morrison et al., J. Chem. Phys. 63 (1975) 2238. [22] H. O'Hara and F. J. Smith, Comput. Phys. Commun. 2 (1971) 47. [23] P. D. Neufeld and R. A. Aziz, Comput. Phys. Commun. 3 (1972) 269. [24] See, for example, P. Bertoncini and A. C. Wahl, Phys. Rev. Lett. 25 (1970) 991.