9206
J. Phys. Chem. 1996, 100, 9206-9215
ARTICLES The Weakest Bond: Collisions of Helium Dimers with Xenon Atoms E. Buonomo, F. A. Gianturco,* and F. Ragnetti Department of Chemistry, The UniVersity of Rome, Citta` UniVersitaria, 00185 Rome, Italy ReceiVed: NoVember 17, 1995; In Final Form: February 23, 1996X
The existence of an He2 bound state is employed in an analysis of the possible outcomes from the lowenergy collisions between helium dimers and xenon atoms that are currently being observed in molecular beam experiments. Existing interaction potentials between helium atoms and between He and Xe atoms are used to construct the necessary potential energy surface (PES). Various dynamical approximations for the evaluation of elastic and total cross sections are reviewed, and the results are analyzed in detail, helping us to clarify the specific physical meanings of the various decoupling schemes used in the calculations. The special nature of this unusual diatomic target turns out to play, as expected, a crucial role in the cross section behavior at low collision energies.
1. Introduction For a long time the helium dimer has been the only rare gas dimer which had not been experimentally observed. The corresponding theoretical results have yielded mixed conclusions on the question of whether a bound state of He2 should exist, since even the best calculations predicted a zero-point energy almost exactly equal to the well depth.1 This means that small errors can shift the computed Born-Oppenheimer (BO) energy of the dimer from a value which is slightly negative with respect to the separated atoms to one which is slightly positive, or Vice Versa, thus making tremendous demands on the absolute accuracy of the ab initio calculations in order to achieve with them any conclusive result.2 Furthermore, the fact that this possible bound state of He2 is found nearly resonant with the state of the two separate atoms with zero kinetic energy is taken to be the physical cause for the unusual cooling which occurs through long-range interactions in extreme He expansion experiments3 and provides additional clues as to its existence. The theoretical and computational interest in getting quantitative details on the dimer geometry and in the number and positions of its possible rovibrational states started very early on with work by Slater4 and progressed through a long series of subsequent calculations and models which yielded well depths ranging from 0 to 13 K. Recent ab initio work2,5 seems to be converging to a well depth of about 10.9 K at a separation of 2.97 Å, supporting only one bound state with binding energy between 0.8 and 1.6 × 10-3 K. The recent empirical potential suggested by Aziz et al.6,7 was constructed to fit some calculated potential points and also fitted the experimental viscosity and second virial coefficients data, together with various integral cross section measurements.8 It was recently modified by the inclusion of retardation effects in the calculation of its longrange part,9 and the bound state position was only very slightly perturbed by such an alteration. Recently, new experimental observations10 have detected helium dimer ions after electron impact ionization of a superX
Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)03386-7 CCC: $12.00
sonic expansion of helium with a translational temperature near 1 × 10-3 K. The dependence of the ion signal on various experimental features was measured, and the signal was finally determined to arise from ionization of neutral helium dimers. The above results, however, are still challenged by further experiments11 and are still subject to discussion in the current literature.12 An entirely different set of experiments has been carried out very recently,13 whereby molecular beams that contain various helium clusters produced in a conventional nozzle beam expansion are diffracted from a transmission grating. The cluster size selectivity was improved in the above experiments by producing clusters with nearly all the same velocities and narrow distributions ∆V/V of typically ∼5%. The employed technique was also entirely nondestructive since only those particles which do not strike the bars of the grating can contribute coherently to the diffraction peak intensity. As reported in that recent publication,13 it is of great current interest to select and characterize clusters of different sizes using various deflection scattering techniques, which are, however, restricted either to large clusters (N > 500)14 or to small, tightly bound clusters (N < 10).15 To circumvent these problems, a new method was used in which molecular beams that contain clusters from a conventional nozzle beam expansion are diffracted from a transmission grating. Such a technique could also be used in a further group of experiments where the individual clusters formed after the grating are made to interact with other atoms or molecules.16 In this new situation, then, one has to deal both with the description of the cluster intramolecular interaction and with the additional intermolecular potential with the projectile. Given the special nature of the diatomic helium, and the inevitably greater strength of any other interaction of it with different species, it becomes of interest to ask what would be the outcomes of such collisional events as one examines some of the contributing processes to the total cross sections, beginning with the simplest case of collisions with an additional rare gas atom. As new experiments are currently being attempted with xenon atoms,16,17 we will start here by examining this specific case. © 1996 American Chemical Society
He2-Xe Collisions
J. Phys. Chem., Vol. 100, No. 22, 1996 9207
2. The Interaction Forces The intramolecular potential for the helium dimer has been accurately estimated by a multiproperty fit for that system,6,7 and recent calculations9 indicate its equilibrium geometry to be given by req ) 2.9695 Å and the binding energy of its only bound rovibrational state (ν ) j ) 0) to be 1.31 × 10-3 K. These results were also recently reproduced by a new theoretical approach.18 Such a system turns out to be the world’s largest diatomic molecule since the expectation value of the relative distance between the two atoms is found to be of 51.9 Å, with a retardation correction that is rather remarkably more than 5%, hence giving a modified expectation value of 54.6 Å.9 The corresponding scattering length is on the order of ∼100 Å; hence, the low-energy limit of the scattering cross section for the He-He collisions is ∼250 000 Å2! On the other hand, the He-Xe interaction is a stronger potential. It is also fairly well known and has been quite accurately given by the Tang-Toennies modeling which reproduces several properties of this heteronuclear dimer.19 In what follows we will therefore attempt to describe the threeatom interaction by simply using a direct superposition of atomatom potentials,
V(R,r,γ) ) VHe-He(r) + VHe-Xe(R1) + VHe-Xe(R2)
(1)
where the variables (R,r,γ) represent the full set of Jacobi coordinates for the atom-diatomics system (with r being the He2 internuclear distance), while the variables on the right-hand side (rhs) of the above equation describe the internal coordinates of the atom-atom interactions. It is important to note here, and we shall show this point further on, the importance of the fact that the atom-diatomic full potential energy surface (PES) originates from an intermolecular interaction, which is stronger than the intramolecular binding interaction. In more conventional van der Waals (vdW) systems, on the other hand, the rare gas interaction with a molecular partner is by far the weaker interaction with respect to the usual molecular binding potentials. One should notice at this point that, given the unusual size of the He2 wave function and the relative strengths of the two atom-atom interactions, in the scattering process the dimer potential acts chiefly as a “perturbation” on the He-Xe interaction. Thus, is it reasonable to assume that the retardation effect plays no role during collisions, and therefore we can make use of the dimer potential of ref 18. Just to give a pictorial idea of the relative importance of the interaction terms involved in the present system, we report in Figure 1 both the atom-atom potentials and the full atomdiatomic potential energy surface that follows from the former. The top part of Figure 1 shows, on the same scale, the He2 potential from ref 18 (solid line) which reproduces the empirical potential of ref 7 and the He-Xe potential from ref 19 (dashed line). One clearly sees there that the equilibrium value of the xenon distance from He is much smaller than the expectation value of r in He2. On the other hand, the well depth is much deeper in the former system than in the latter. It therefore supports a larger number of bound states and also produces a much more localized nuclear wave function. It thus follows that the interaction between the xenon atom and each helium atom is stronger and shorter ranged than that between two helium atoms. The combined effect of such interactions within the full potential of eq 1 is seen in the two geometries of the potential energy surface (PES) reported in the lower part of Figure 1. One sees that the C2V geometry becomes increasingly more attractive as the two helium atoms are pulled apart, while the
Figure 1. Top: interaction potentials for the He-Xe (dashed line) and He-He (solid line) systems. Bottom: two-dimensional representations of the radial Jacobi coordinates for the atom-diatom system at two different orientations.
collinear arrangement remains obviously repulsive as the farther helium atom is moved away. In either case, one clearly notices the dominance of the He-Xe interaction in setting up the main features of the PES. The general behavior of the nuclear wave function for the weakly bound state sitting nearly at the top of the well shows the radial density of the “target” molecular state to be extended well outside the range of the actual interaction potential: to include ∼99% of the density, one needs, in fact, to extend the relative distance between the nuclei up to ∼500 a0! Furthermore, the radial density, r2F0(r), peaks at ∼14 a0, and the expectation value of r is found to be larger than 100 a0, as mentioned before. Figure 2 reports the shape of the bound wave function for
9208 J. Phys. Chem., Vol. 100, No. 22, 1996
Buonomo et al.
Figure 2. Radial dependence of the density for the bound state of the helium dimer. All quantities in atomic units.
the radial part of the dimer and over the range of distances which includes more than 95% of the total density of that state. One could therefore imagine the He2 molecule as a very floppy diatomic where the two partners are quantal objects which are strongly delocalized within the very shallow potential well. They therefore “react” to the xenon projectile rather differently depending on which angle of approach dominates during each specific trajectory. Furthermore, contrary to what one usually finds in conventional chemical bonds, the internuclear distance contains very little charge density, and therefore one should view such a molecular species more like a large “dumbbell” than like an ellipsoidal disc, as often done in modeling atom-diatomic scattering observables. In the following we will therefore describe, in the light of the above findings, possible ways in which the relative couplings between collisional and internal motions could be taken into consideration within a quantum treatment of the scattering process. 3. The Dynamical Models To begin with, one needs to find out in qualitative terms what processes could take place during the experiments and what are the most likely final channels which define the total cross sections as the sum of the elastic processes and of all possible inelastic processes,
σtot ) σel + σinel
(2)
One should keep in mind that it is the above quantity, σtot, which is being measured in the current experiments,16 and therefore it becomes useful to try to qualitatively identify different possible contributions to the total cross section,
σtotal ) σel + σexch + σbreakup
(3)
Xe + He2(V)0, j)0) f Xe + H2(0,0)
(4a)
σexch:
Xe + He2(0,0) f HeXe(V,j) + He
(4b)
σbreakup:
Xe + He2(0,0) f Xe + He + He
(4c)
where
σel:
Given the relative collision energies considered in the experiments, no electronic excitation processes have been included. Qualitatively, one further expects that channels 4b and 4c should be the dominant processes, with the breakup channel likely to be the most important.
If one begins by considering the subreactive collisions only, one should keep in mind that the conventional expansion20 of the total wave function over asymptotic (diabatic) target rovibrational states cannot be applied effectively here since only one vibrational (V ) 0) and one rotational (j ) 0) level can exist as bound states of He2. Furthermore, in view of the small amount of energy involved in the bound state, the relative motion of the helium atoms can be classically considered as very slow, and therefore the target molecule can be thought of as being largely still during the collision times that are relevant to the present experiments. Such a condition possibly corresponds to the best actual occurrence of the dynamical decoupling usually invoked within the impulse approximation (IOS)21 or the adiabatic nuclei approximation (AN),22 whereby the inelastic scattering process is treated by keeping either the internal coordinate vector jr or the relative orientation γ as a parametric index during the dynamics. In the same spirit, earlier on23-25 a reductive scheme was introduced for inelastic collisions by treating both the vibrational and rotational degrees of freedom within the infinite order sudden approximation. This scheme was called the (VRIOS) decoupling scheme. At any rate, the most common types of impulse approximations, be it the IOS or the AN decoupling scheme, would tend to look at the scattering of Xe atoms as essentially occurring from an He2 molecule considered to be a “conventional” diatomic molecule and treat it as being held fixed in space during the dynamics. Thus, although we shall show below that the above treatments handle coherence between the various “trajectories” that hit the target with different impact parameters in different ways, both consider an important feature of the process to keep a “permanent” molecular partner during the collisions with xenon. The present system is, however, very different from any previous one in the sense that the target molecule is much “floppier” than a conventional diatomic, and the most likely outcome of any low-energy collision with xenon atoms is that of breaking the bond rather than of exciting possible molecular internal degrees of freedom. Thus, one may qualitatively expect that most of the coherent interference between trajectories caused by molecular interactions in conventional atom-molecule collisions will be lost here because the “hit” of He2 by xenon projectiles will separate the two atoms and will cancel any existing correlation between their relative motions. A totally different approach, therefore, could be had by considering the He2 target as given by such weakly interacting pairs of atoms that its scattering cross section with Xe is simply obtained from collisions occurring independently with each of the two He atoms (IA model). In other words, one could consider the “molecular” cross section as being given by the sum of two separate “atomic” cross sections from the scattering of Xe by He monomers, as we shall further discuss below. To acquire some preliminary confidence on the overall reliability of the He-Xe interaction employed here, we first carried out calculations for the elastic cross sections of this system over an energy range between 2 and 30 meV. The experimental measurements at 8.5 meV provide a value of the σel of 275 Å2,17 and our calculated value at the same energy is 273 Å2. Its general energy dependence over the whole energy interval turns out to be rather smooth, and the agreement of our calculations with experiments is rather good: they indicate17 a value of 391.72 Å2 at an energy of 3.15 meV, where our calculations give a value of about 390 Å2. We could therefore accept both interaction components as describing rather realistically the three-particle system under study here.
He2-Xe Collisions
J. Phys. Chem., Vol. 100, No. 22, 1996 9209
3.1. The Independent-Atoms (IA) Model. The simplest possible estimate of the elastic part for the triatomic σtot could be obtained by considering the collisions of the Xe projectile with each of the helium atoms in the dimer as mainly separate processes.
provides some sort of approximate “weighting factor” for the target shape. We will come back to this point in the next section. One can further say, however, that for the “molecular” case the above equation modifies into
tot tot tot σHe ≈ σXe-He + σXe-He ) 2σHe-Xe 2-Xe 1 2
tot el inel σHe ∝ Ixe/NHe2 ) σHe + σHe 2-Xe 2-Xe 2-Xe
(5)
Such simple estimates essentially assume that the heteronuclear interaction dominates the elastic scattering. The results of this simple model suggest an elastic contribution, at E ) 18.4 meV, of about 1500 a02 to the total cross section of eq 3. This is an interesting result which we could further analyze by remembering that, in the simple case of scattering from a spherical potential,26a,b one could define a partial wave phase shift, ηl, as given by the sum of a real and an imaginary part,
ηl ) ξl + iχl
l e σell ≈ σinel
π (2l + 1) k2
(7a)
the equality on the rhs being valid whenever the opacity function, a direct measure of the attenuation of elastic scattering,
Pl ) 1 - exp(-4χl)
(7b)
becomes unity, i.e. inelasticity reaches its maximum value. In the present case, the simple IA model of incoherently summing the atom-atom cross sections provides some sort of estimate also for the inelastic cross section of the atom-dimer collisions. In particular, if one considers a system made up of isolated, structureless scattering centers, then the total cross section only has elastic contributions and is given by the ratio between the overall flux Ixe of projectiles and the number n of scattering particles:26 tot el ) σHeXe ∝ Ixe/nHe σHeXe
(8)
The corresponding total differential cross section within an IA model is given by
dσ n′ ) dΩ IN
(9)
∑i n′i
N
I
tot tot ∝ Ixe/0.5nHe ≈ 2σHeXe σHe 2Xe
( )
(12)
Using now the result of eqs 7 and 12 for strongly inelastic processes, which we expect to be valid in the present case, we can further write el inel tot ≈ σHe ) 1/2σHe σHe 2Xe 2Xe 2Xe
(13a)
and therefore, from eq 12 one further gets el el ≈ σHeXe σHe 2Xe
(13b)
Within the IA approximation the elastic and inelastic components of the present collisions could then be estimated from the elastic atom-atom cross sections only. What we are really doing is considering the helium dimer as a “nearly” dissociated molecule where the bound state is so close to the continuum that the two component atoms can be treated independently. Let us now go back to the more conventional “molecular” modeling of the xenon dynamics with helium dimers. 3.2. The Adiabatic Nuclei (AN) Approximations. When one employs Jacobi coordinates, the standard Hamiltonian for the relative motion of the A + BC(Vi,ji) system is given by20
{
ˆl 2 p2 ∂2 + + V1(R,r,γ) H) 2µ ∂R2 2µR2
}
ˆj 2 p2 ∂2 + + V2(r) 2m ∂r2 2mr2
(14)
where R is the atom-molecule distance, r is the internuclear molecular coordinate, and γ ) arcos(Rˆ ‚rˆ). The reduced mass of the (A + BC) system is given by µ, while m is that for the isolated diatomic target. The full potential has been written as a sum of two separate contributions:
V(R B,b) r ) V1(R,r,γ) + V2(r)
(15)
Following the well-known approximation introduced by Chase,22 one seeks an approximate solution for the total wavefunction as given by
B,b) r = φ(R B;b) r χn(b) r Ψtot(R
where n′ is the number of scattered particles, I is the incoming flux, and N is the number of scattering centers. If one writes n′ ) ∑in′i, considering the scattering centers as independent from each other, then one can write
1
If we now realize that NHe2 ) 1/2nHe, then the IA model gives us
(6)
that would contribute to elastic and inelastic processes, respectively. This allows one to define two different cross sections that correspond in turn to the elastic and to inelastic contributions and to note that when χl becomes very large, i.e. for strongly inelastic processes, the two sets of processes, elastic and inelastic, tend to become equal.27 This means that the maximum possible contribution to inelastic processes should be given, for each partial wave, as
(11)
(16)
where the continuum function φ depends only parametrically b) are on the molecular internal coordinate b r and the χn(r asymptotic (V ≈ V2) bound functions of the isolated BC target. The adiabaticity condition can be written as
(10)
1 ∂2 φ(R B;b) r ≈0 2m ∂r2
(17a)
For a large number of scattering centers we can see that the sum can be replaced by an integral and that the factor 1/N
1 2 ˆj φ(R B;b) r ≈0 2mr2
(17b)
dσ dΩ
)
)
1
N
∑
dσ
N i)1 dΩ
i
9210 J. Phys. Chem., Vol. 100, No. 22, 1996
Buonomo et al.
Since we further know that
{
l φl′m ∼ l′
}
ˆj 2 1 ∂2 + V(r) + χn(b) r ) nχn(b) r 2m ∂r2 2µr2
rf∞
(18)
we can rewrite the Schro¨dinger equation for the Ψtot of eq 14 as
[
]
1 ∂2 l2 + V(R,r,γ) + φ(R B;b) r ) φ(R B;b) r (19) 2µ ∂R2 2mR2
where ) Etot - n is the collision energy and k2 ) 2µ in the usual expression
e-ikR R
ik B‚R B B;b) r R∼ + f(Ω;b) r φ(R f∞e
(20)
1 {δll′ exp[-i(kR - lπ/2)] k1/2 S(l′ml′|lml′|rˆ) exp[i(kR - lπ/2)]} (26)
which, after standard manipulations20 yields the final expression for the AN scattering amplitude,
r ) fAN(Rˆ ;b)
π1/2 k
il-l′+1 × ∑l (2l + 1) ∑ l′m l′
l′ [δll′ δml′0 - Sl′m (rj) Y ml′ l′(Rˆ )] (27) l′
The above result requires a final quadrature over molecular coordinates, as in eq 22, to yield the scattering amplitude at the fixed energy ) k2/2µ and for the (n f n′) inelastic process. It is important to note here that the corresponding cross sections are given by
Let us recall here that the exact form of the scattering amplitude is given by26
fnn′ )
µ 〈Ψ |V |Ψ(+) kn 〉 2π k′n′
(21)
where Ψk′n′ ) eik′Rχn′( b) r and Ψ(+) kn is the exact scattering solution. Through the approximation discussed before, we can now rewrite the scattering amplitude as
fnn′(Ω) = )
µ 〈Ψ |V |Ψtot〉 2π k′n′ µ 〈χ |〈eik′R|V |φ〉Rh |χn〉jr 2π n′
(22)
r n〉jr = 〈χn′|f(Ω;b)|χ where the variable over which the integration is performed is explicitly indicated. To perform the integration over B R we have taken k′ ) k, which would be valid if
(∆k)R )
n′ - n 1 (kn′ + kn)Rm 2µRm2
,1
(23)
(24)
We therefore have introduced both the presence of a slow variable during the scattering process and the impulse approximation in terms of energy transfer. The latter simplification will make the model give the wrong solution for inelastic cross sections near their threshold and will yield the wrong behavior of them as k′ ≈ 0, as recently discussed by us in another context.29 When solving the scattering problem using the AN simplification, one could also expand φ over eigenfunctions of ˆl2 and write:
B;b) r ) φ(R where the
l φl′m′
1 R
l (R B;b)Y r ml′ l′(Rˆ ) ∑ Al ∑ φl′m l′ l
l′ml′
satisfies the following conditions:
(25)
(28)
which implies interference effects between the double sums coming from expression (24) after having carried out the quadrature over b. r An equivalent derivation of the above AN approximation was also introduced a while ago as the “lz-conserving” approximation for purely rotational internal degrees of freedom.30,31 In spite of the apparent differences between the final expression for that scattering amplitude and the one given above, the rotational degree of freedom of the target is decoupled, during the scattering process, from the relative motion of the partners in both cases and the scattering amplitude is obtained at a fixed orientation in both instances.20 The Vibrational degrees of freedom, however, are not involved in the “lz-conserving” decoupling scheme and have to be treated later on by using the AN approximation. More specifically, one can use the AN amplitude of eq 27 to obtain state-to-state scattering amplitudes,
r fAN(j′mj′V′|jmjV|Ω) ) 〈j′mj′V′|fAN(Rˆ ;b)|jm jV〉
(29)
and then the corresponding differential cross sections,
dσ ) |fAN(j′mj′V′|jmjV|Ω)|2 dΩ
where Rm is the effective range of action of the interaction V1 in eq 15. If such a relation is valid, then we can further write
µ ik′R µ 〈e |V|φ〉BR = 〈eikR|V|φ〉 ) f(Ω;rj) 2π 2π
dσ (n f n′|Ω) ∝ |fnn(Ω)|2 dΩ
(30)
where the interference effects of the contributing partial waves are clearly seen. In many experiments, however, one often finds that only state-unresolved total cross sections are determined. One could thus obtain them by summing over all possible open channels from one selected initial state,
(
)
dσ(jmjV) dΩ
) tot
2 |〈j′mj′V′|f(Rˆ ;b)|jm r ∑ jV〉| j′m V′ j′
j ) ∫|fAN(Rˆ ;b)| r 2|Y mj χV|2 db r
(31)
dσ j χV|2 db r ) ∫ (b)|Y r mj dΩ b)|2. where (dσ/dΩ)(r b) ) |fAN(Rˆ ;r The corresponding integral cross section is then given by
dσ σ(jmjV) ) ∫ (jmjV) dRˆ dΩ
He2-Xe Collisions
J. Phys. Chem., Vol. 100, No. 22, 1996 9211
dσ j ) ∫ (b)|Y χV|2 db r dRˆ r mj dΩ
(32)
j ) ∫σ(b)|Y r mj χV|2 db r
It is interesting to compare the above results with the classical expression of total scattering cross section from several, independent centers of scattering. By remembering the result of eq 10, in fact, and by considering that a more realistic 1/N factor is now given by the amplitudes of the target rovibrational states, |Y mj jχV|2, we could recover eq 31. It represents a physical situation where the earlier IA approximation is modified by the presence of a “molecular” form factor introduced by the AN approximation discussed here. The definition of an elastic, integral cross section within the AN approximation involves the quadrature in eq 29 over the initial state only:
r 0mj0V0〉 f(j0mj0V0|j0mj0V0|Ω) ) 〈 j0mj0V0|fAN(Rˆ ;b)|j el (Ω) ) f AN
(33)
)∫
el |f AN (Ω|k2)|2
dΩ
(34)
On the other hand, the total cross section within the same decoupling scheme should be obtained directly from eq 32 by performing a double integration over molecular coordinates and over space-fixed orientations: tot (k2) σAN
) ∫|fAN(Rˆ ;b)| r
2
|Y mj j(rˆ)
χV(r)| db r dΩ 2
(36)
and |Y mj j(rˆ) χV(r)|2 is a weighting factor that is also a positive quantity. If one insert now eq 25 into the corresponding equation of motion (19), the potential term gives rise to a sum over l that contains a large number of (ll′) coupling matrix elements. According to many authors,20 this point has been recognized as the main drawback in the application of the AN approximation in heavy-particle scattering problems, where one usually deals with thousands of such matrix elements. One can, however, avoid the difficulty by further applying the centrifugal sudden approximation (CS):
ˆl ) hl (lh + 1) 2
(37)
One now easily sees that the new equation depends parametrically on γ, and we can define a new solution, φhl, from such an equation.
[
]
(l + 1) d2 - hl + k2 φhl (R;r,γ) ) 2µV(R;r,γ) φhl (R;r,γ) 2 dR R2 (38)
The above approximation can also be obtained in a different way. In fact, following Khare,32 we introduce the transformation lml (R) Y lm * l(Rˆ ) φ(R,Rˆ ,Rˆ ′) ) ∑∑Yl′ml′(Rˆ ′) φl′m l′ l′ml′ lml
rf∞
{ [ (
)]}
1 π π exp -i kR - hl - Shl (γ,r) exp i kR - hl 1/2 2 2 k
)]
[(
(40) The inverse transformation is given by lml ) (i)l-lh〈l′ml′|φhl |lml〉 φl′m l′
(41)
and the ensuing S matrix, by comparing eq 26 with 41, is given by
r ) il+l′-2lh〈l′ml|Shl (lml)〉 S(l′ml′|lml|b)
(42)
r ) il′-l〈l′ml′|Tl|lml〉 (43) T(l′ml′|lml|rˆ) ≡ I - S(l′ml′|lml;b) Introducing now the above quantity into eq 27, with hl ) l, one obtains
fAN-CS(R;r,γ,Rˆ ) ) iπ1/2 k
〈l′ml′|Thl |l0〉 × Yl′m (Rˆ ) ) ∑l (2l + 1) ∑ l′m
(39)
l′
l′
iπ1/2
(35)
which implies a quadrature over real positive quantities since
r 2 dΩ ) σ(b|k r 2) ∫|fAN(Rˆ ;b)|
φhl ∼
By choosing hl ) l, where l is the initial orbital angular momentum,
hence el (k2) σAN
lml in which φl′m can be considered as the same function that l′ appears in the summation of the rhs of eq 25. Equation 38 can then be obtained by replacing ˆl2 with hl(lh + 1) in eq 19 and by applying the transformation (39) to the resulting equation. The φhl defined in eq 38 verifies the following boundary condition:
k
∑l (2l + 1)Tl(r,γ) Yl0(Rˆ )
(44)
The above quantity is then a sort of potential scattering amplitude for the fixed-orientation problem with respect to both rˆ and r, in terms of r and γ. Finally, we note that eq 38 is to be solved on a twodimensional grid: since γ, however, represents the body-fixed (BF) orientation of the target (in a frame defined by Rˆ ) zˆ), we need to further rotate the target wave function of eq 29 to the BF frame. Thus, we can rewrite eq 29 in the following way:
fAN(j′mj′V′|jmjV|Ω) )
j* j′ D mj′λ′ × 〈j′λ′V′|fAN-CS|jλV〉BF ) ∑j ∑j′ D mjλ j* j′ D mj′λ 〈j′λV′|fAN-CS|jλV〉BF ∑j D mjλ
(45)
in which the last term follows from the expression of fAN-CS (eq 44) that depends only on γ. It should be noted now that the AN-CS approximation discussed above is really equivalent to the (VR-IOS) approximation introduced earlier by us and tested on the He-Li2 system.23 In the case of the helium dimer, however, such a decoupling scheme should be particularly valid given the slow rotational and vibrational motions of this unusually large “diatomic” with respect to its interaction time with the impinging projectile of the present case. 3.3. The Vibrational Close-Coupling-RIOS Approximation (VCC-RIOS). This decoupling scheme has been discussed several times before20,23,33 and will only be briefly derived here. The basic assumptions involve the parametric dependence of the total wave function on the relative orientation angle γ, while the internal, vibrational coordinate is treated within a conven-
9212 J. Phys. Chem., Vol. 100, No. 22, 1996
Buonomo et al.
tional close-coupling (CC) expansion, VCC-RIOS hj hl V Ψtot (R,r,γ) ) Ψtot (R,r,γ) ) ∑g V′hj hl V(R;γ) χ V′hj (r) (46) V′
and the g coefficients are obtained from the set of CC equations where ˆl2 ) hl(lh + 1) and ˆj2 ) hj(jh + 1),33
{
d2 2
+ kh2j V -
}
hl (lh + 1)
dR
R
2
g V′hj hl V(R;γ) ) hj hj hl V 2µ ∑〈χ V′hj |V|χ V′′ 〉g V′′ (R; γ) (47) V′′
The usual search of the scattering solutions is carried out by propagating the coefficients of eq 46 through eq 47 and also by checking convergence on the sum of potential coupling terms on the rhs of eq 47. The orientation-dependent scattering amplitude is given by, with hl ) l initial, hj (V,V′;γ|Ω) ) f VCC-RIOS
i 2
∑(2l + 1)[δVV′ - ShjVV′l (γ)]Pl(cos ϑ) l
(48)
which yields in turn the degeneracy-averaged differential cross section:
( ) dσ
dΩ
) VCC-RIOS
hj (V,V′;γ|Ω|)2|jmj〉 (49) (2j + 1)-1 ∑ 〈 jmj|f VCC-RIOS mj
One should note that the double sum over angular momenta in the amplitude of eq 49 reduces to a single sum because of the products of spherical harmonics if the Ω-integration implied by eq 49 is carried out before the orientational quadrature over γ.
( ) dσ
π
hj l (γ)]|2 ) ∫ dΩ VCC-RIOS dΩ ) 2 ∑ (2l + 1)[δVV′ - S VV′ l
khj V
σVCC-RIOS(V,V′|γ) (50) Hence,
σVCC-RIOS(V,V′|kh2j V) )
1 1 VCC-RIOS (V,V′|γ) d cos γ (51) ∫σ 2 -1
A further simplification is obtained when in eq 47 only one vibrational target state is considered. This is usually called the vibrational-diabatic approach,23 the VD-RIOS simplification. It implies that the scattering process is carried out under the action of an anisotropic potential averaged over a single vibrational asymptotic state and left unchanged during the collision process. In the case of the ground vibrational level of the target molecule one has that
〈χ V′hj |V(R,r,γ)|χhV′j 〉 w 〈χ Vj′ |V|χ Vhj ′〉 ) V00(R,γ) ∑ V′′ 0
0
(52)
and the scattering is thought of as occurring via a single, asymptotic vibrational state of the molceule: its initial ground state |V0〉. For the present system only one vibrational state exists. Therefore, even the expansion of eq 46 implies a sum over one target state only unless one further expands on the continuum states of the He dimer which belong to the reactive channels in eq 5. This is a rather crucial difference between
Figure 3. Computed integral cross sections, elastic and total, for the He2-Xe system. All the curves refer to the different coupling schemes discussed in the main text. See text for the meaning of the acronyms.
VCC-RIOS results and those implied by the AN model, as we shall further discuss in the next section. 4. Discussion of Results The different approximations introduced in the last section are now applied to evaluate the elastic and total cross sections. The collision energies are chosen in the range between 3 and 30 meV, corresponding to typical translational energies of Xe atoms, following the supersonic expansion.16 Since the collision energies are larger than the bound state energy of the helium dimer, the collision has an impulsive character. This is a kinematic requirement for both the AN and the VCC-RIOS approximations, which we discussed before. One should also remember that the helium dimer has only one bound state, and we shall see that this will affect with specific modifications the approximations already described. The integral cross sections resulting from the methods of the previous section are shown in Figure 3. In that figure we also reported the behavior of the IA approximation: the elastic cross section estimated by this method is equal to the cross section for He-Xe scattering, and it is also taken to be equal to the cross section for the total inelastic processes given by the IA approximation. In particular, one gets an estimate for the IA inelastic cross sections which in the limit of infinite inelasticity can be considered as an upper bound to the exact quantity. It is also interesting to note that all the curves reported in Figure 3 are smooth, decreasing functions of the energy, similarity to the shape of the experimental cross sections for the scattering of helium atoms by xenon atoms.17 We have calculated the cross sections in the AN-CS approximation using eqs 29 and 32 for the elastic and total cross sections, respectively. Those fixed-orientation calculations were done in a two-dimensional grid composed of 40 angles, whose distributions will be discussed later, and 100 values of the helium dimer distance, from r ) 2.6 a0 to r ) 300 a0; this range contains about 96% of the vibrational wave function. Since the helium dimer has only one bound state, we need to justify the use of eq 32. Given the high collision energies of the present calculations, one should, in fact, include some additional functions of b r describing the dissociating states in order to obtain the closure relation required in the derivation of eqs 31 and 32. Our exploratory classical calculations in the collinear arrangement35 indicate that the reactive process of eq 4b is expected to be negligible in this system. We could therefore start by disregarding the exchange functions (i.e. the HeXe
He2-Xe Collisions
J. Phys. Chem., Vol. 100, No. 22, 1996 9213
Figure 4. Angular-dependent AN-CS opacity functions at fixed target geometries (r values) and over the range of relative orientation γ. The collision energy is 16.2 meV.
bound states), which depend on a different coordinate, in the exact closure relation. In this way, the additional functions we need in order to derive eq 32 coincide with the dissociative wave functions of the breakup process of eq 4c. Thus, in some of the cross sections given by eq 29, the breakup process could be included if one further adds an arbitrary set of functions of b r that, together with the bound state, form a complete set for the present process. One can explicitly calculate the continuum wave functions by then solving eq 18 for positive energy values, applying finite box boundary conditions. On the other hand, it is interesting to note that using instead the AN summation of eq 31, one doesn’t need to actually include those functions in the calculation because of the required completeness in the sum of the rhs of that equation, and the dissociation cross sections can be obtained within the computational effort required by eq 32 without explicitly considering the coupling between the bound state and the continuum wave functions.34 In Figure 4, the total AN-CS opacity function at some fixed r values is reported as a function of the Jacobi coordinate γ. This quantity is the cross section obtained from the potential scattering calculations at each fixed γ. At small r values (r ) 5 and 20 a0) the opacity function has a broad distribution, over the whole range of orientations. At very large r values, on the other hand, one can identify four different angular regions according to the projection of one of the helium atom coordinates along the line defined by the fixed orientation
r d ) sin γ 2
(53)
The detailed behavior of such regions is quickly summarized as follows. (i) For d < rt, where rt is the turning point of the He-Xe interaction, the potential along the line with fixed γ is similar to the potential for γ ) 0 (slightly enlarged for γ * 0) and the opacity function is a smooth function of orientation. (ii) For rt e d < r0, where r0 is defined by VHeXe (r0) ) 0, a barrier appears in the potential (whose maximum is at r ) d) and the opacity function has strong oscillations (≈7 for r ) 100 a0, ≈4 for r ) 200 a0). (iii) For r0 < d < a, where a is the full range of the potential (≈13 a0), the potential along the line is attractive and the opacity function is smooth and reaches its maximum for a specific orientational value. (iv) For d > a, the potential is negligible and the opacity function decreases exponentially for increasing γ.
Figure 5. Radial dependence of the AN-CS opacity functions for different collision energies. All values in atomic units.
TABLE 1: He2-Xe Cross Sections Using the AN-CS Approximation E/meV
elastic/a02
total/a02
Pdiss
3.0 10.0 16.4 25.0
110.6 66.31 58.59 54.00
1230.3 931.5 863.6 792.0
0.910 0.929 0.932 0.932
According to these observations, the γ distributions presented in Figure 4 were chosen in order to describe the features of each region of interaction that are clearly displayed by our calculations. In Figure 5 we further show the total integral cross sections, at fixed helium dimer distances; this is the AN result of the integration over γ of the opacity functions discussed above for the relevant range of collision energies. It is the radial integral in the last line of eq 32. One can observe that the relative maximum around 20 a0 is also close to the maximum of the density distribution of the He2 bound state. As the value of r increases, the cross sections tend to an asymptote that is close to the cross section of the He-Xe scattering for each of the collision energies examined. This is in contradiction with the IA approximation discussed before, which predicts a value for the total cross section equal to twice the He-Xe cross section. Obviously, the IA approximation of eq 34 is now replaced by a more realistic choice of modulating factors, as is done in eq 32. In Table 1 we have reported the elastic and total cross sections, together with the dissociation probabilities defined as
Pdiss ) 1 -
σel σtot
(54)
The computed cross sections as a function of collision energy are also shown in Figure 3. The AN-CS elastic cross section comes out to be less than 10% of the total cross section, due to the interference effect implied by eq 31. It is interesting to note that the AN total cross section is close to the IA cross section that uses the He-Xe atomic scattering model and that the difference between these two cross sections is on the order of magnitude of the elastic cross section in the AN-CS approximation (bottom curve of Figure 3). As the He-Xe elastic cross section gives an upper bound to the molecular inelastic cross section, we can conclude that, in this process, the inelasticity tends to its maximum value and the AN-CS total cross section is made up mostly of inelastic contributions and of a rather small elastic contribution which is around 10% of the total. We have also carried out further test calculations using the VCC-RIOS described in section 3.3. In this approximation,
9214 J. Phys. Chem., Vol. 100, No. 22, 1996
Buonomo et al.
Figure 7. Angular distributions of total scattered xenon atoms for He2 targets (AN-CS approximation) at two different collision energies. ϑcm is in degrees, and the amplitude is in atomic units.
TABLE 2: Computed Cross Sections, Total and Partial, Using the Other Approximations Described in the Main Text (All Quantities in au)
Figure 6. Three-dimensional representation of the diabatic potential for the ground rovibrational level of He2 interacting with xenon. Top: full view of the interaction in atomic units of energy at intervals shown. The x/y axes follow the molecular geometry: the y axis is perpendicular to the bond distance, which is in turn given along the x axis. Both quantities are in angstroms. Bottom: projection on the molecular plane, where x describes the coordinate along the bond length of the dimer. The energy spacing of the level curves is the same as in the top diagram, with the zero given by the outermost solid line.
the diabatic expansion of eq 46 is carried over the discrete state of the target. Due to the high collision energies and the smallness of the He2 interaction, it is very difficult to include pseudocontinuum states of He2 which could be obtained using finite box boundary conditions, as discussed in ref 34. We have therefore restricted our present expansion to just one state and named this approximation the “vibrationaldiabatic-RIOS” (VD-RIOS). In Figure 6 we report the potential matrix element defined by eq 52. One sees that, because of the spatial extension of the He2 bound state, the Coulomb cusp is present in the interaction even at large R distances (for small γ values) and the potential has a peculiar “rectangular” shape. The physical picture implied by this dynamical model, given the dimension of the He2 target and the assumption of fast vibration during collisions, is one where the scattering of Xe atoms occurs from a “molecular bar” which is about 100 a0 long and rather large in width (≈25 a0). The final result is given by the elastic cross sections reported in Figure 3, which turn out to be even larger than the rather large IA total cross sections also shown the figure. As a final example, we have reported in Figure 7 the elastic differential cross sections computed at two different collision energies from the AN-CS calculations. One can easily see that the forward scattering turns out to be strongly favorable, while for the other directions the scattering flux is exponentially small and markedly decreasing, due to the interference effects already
E/meV
el σHeXe
IA σtot
σelIA
VCC-RIOS σtot
4.0 8.0 12.0 16.2 20.0 25.0 30.0
1125.9 962.9 814.0 706.5 637.6 572.1 524.7
2251.8 1925.8 1628.0 1413.1 1275.2 1144.2 1049.5
1125.9 962.9 814.0 706.5 637.6 572.1 524.7
2632.8 2163.5 1922.5 1763.5 1662.7 1567.2 1498.5
mentioned. As the energy increases, however, the forward scattering becomes more important, possibly showing the consequences of physical processes in which the Xe atom “crosses” the helium dimer without being deflected by the interaction potential. 5. Conclusions We report in Tables 1 and 2 a full list of all the computed cross sections, at some of the relevant energies, in order to show more specifically the quantitative differences between the computational schemes adopted here to evaluate total and elastic cross sections for Xe scattering from helium dimers at meV collision energies. The three different dynamical schemes that we have put to work in the present computational study show some interesting differences in the case of our title system, precisely because of the unusual nature of the intramolecular potential Vis a` Vis the intermolecular interaction with another rare gas atom. (i) The VCC-RIOS scheme, given the smallness of the vibrational basis provided by the system, describes unrealistically the coupling between target vibrational motion and impinging projectile translational motion. As a consequence, interference structures originating from the very large size of the target dumbbell are underestimated by the above scheme, thereby yielding the largest total cross sections of all the models employed. (ii) The IA simple model indicates again that its total cross sections contain a rather large elastic component because of the approximations used by such a model in describing the effects of “molecular” form factors during the scattering. As a result, the estimated total cross sections are also fairly large, albeit smaller than the corresponding VCC-RIOS quantities. It appears, however, that the limit of maximum inelasticity contributions during collisions is likely to be reached by the present system and therefore that the corresponding estimates of inelastic contributions given by the IA model are better than its elastic cross section estimates.
He2-Xe Collisions (iii) The AN-CS decoupling scheme seems to include most aspects of the collision dynamics and to also treat, using its closure relations over the r-dependent expansion functions, rather realistically the large inelastic contributions coming from the breakup process of eq 4c. It does not include, however, effects from reactive exchange processes. The latter contributions, on the other hand, may be rather small at the collision energies considered here, as our classical calculations for the collinear geometry already indicate.35 (iv) The elastic processes described by the adiabatic nuclei scheme are contributing rather little to the total cross sections, which therefore appear to be dominated by the breakup effects during collisions. The interaction forces employed to describe the present triatomic system also turn out to be rather realistic in the sense that our Xe-He potential allows us to reproduce well the measured elastic cross sections for this subsystem.17 Furthermore, the analysis of the orientation-dependent and bonddependent opacity contributions to the AN-CS cross sections allows us to shed more light on the microscopic mechanisms which preside over the breakup process during collisions and help us to better understand the scattering dynamics of this system. Given the current experimental interest on such cluster interactions, we feel that a detailed computational study like the present one can usefully direct the experimental search for special cross section features of the helium cluster collisions with any of the other rare gas partners. It could also help to extend the present understanding of such dynamical processes down to the much lower collision energies of relevance in quantum condensation studies. Acknowledgment. We are grateful to Professor Peter Toennies for having suggested the present problem to us and for his encouragement in carrying out this study. One of us (F.A.G.) also thanks the Von-Humboldt-Stiftung for supporting his stay in Go¨ttingen during the summer of 1994, when the work was begun. The financial support of the Italian National Research Council (CNR) and of the Italian Ministry for University and for Scientific Research (MURST) is also acknowledged. References and Notes (1) Chalasinsky, G.; Gutowski M. Chem. ReV. 1988, 88, 943.
J. Phys. Chem., Vol. 100, No. 22, 1996 9215 (2) For example see: Lin, B.; Mc Lean, A. D. J. Chem. Phys. 1989, 91, 2348. (3) Toennies, J. P.; Winkelmann, K. J. Chem. Phys. 1977, 66, 3965. (4) Slater, J. C. Phys. ReV. 1928, 32, 349. (5) Vos, R. J.; Van Lenthe, J. H.; van Duijneveldt, R. B. J. Chem. Phys. 1990, 93, 643. (6) Aziz, R. A.; Slaman, M. J. J. Chem. Phys. 1991, 94, 8047. (7) Aziz, R. A.; Slaman, M. J. Metrologia 1990, 27, 211. (8) Feltgen, R.; Kirst, H.; Kohler, K. A.; Pauly, M.; Torello, F. J. Chem. Phys. 1982, 76, 2360. (9) Luo, F.; Kim, G.; Mc Bane, G. C.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1993, 98, 9687. (10) Luo, F.; Mc Bane, G. C.; Kim, G.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1993, 98, 3564. (11) Meyer, E. S.; Mester, J. C.; Silvera, I. F. J. Chem. Phys. 1994, 100, 4021. (12) Luo, F.; Mc Bane, G. C.; Kim, G.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1994, 100, 4023. (13) Scho¨llkopf, W.; Toennies, J. P. Science 1994, 266, 1345. (14) Lewerenz, M.; Schilling, B.; Toennies, J. P. Chem. Phys. Lett. 1993, 206, 381. (15) Buck, U.; Meyer, H. J. Chem. Phys. 1986, 84, 4854. (16) Scho¨llkoopf, W.; Toennies, J. P. Private communication. (17) Scho¨llkopf, W. Diplomarbeit, University of Go¨ttingen, 1994. (18) Tang, K. T.; Toennies, J. P.; Yiu, C. L. Phys. ReV. Lett. 1995, 74, 1546. (19) Tang, K. T.; Toennies, J. P. J. Chem. Phys. 1977, 76, 2424. (20) For example, see: Gianturco, F. A. The Transfer of Molecular Energy by Collisions; Springer: Berlin, 1979. (21) Rapp, D.; Kassal, T. Chem. ReV. 1969, 69, 61. (22) Chase, D. M. Phys. ReV. 1956, 104, 838. (23) Gianturco, F. A.; Serna, S.; Delgado-Barrio, G.; Villarreal, P. J. Chem. Phys. 1991, 95, 5024. (24) Pack, R. T. J. Chem. Phys. 1974, 60, 633. (25) Mc Guire, P.; Kouri, D. J. J. Chem. Phys. 1974, 60, 2488. Mc Guire, P. Chem. Phys. 1974, 4, 483. (26) (a) Newton, R. G. Scattering Theory of WaVes and Particles; McGraw-Hill: New York, 1966. (b) Levine, R. D. Quantum Mechanics of Molecular Rate Processes; Clarendon Press: Oxford, 1969. (27) Fuchs, M.; Toennies, J. P. J. Chem. Phys. 1986, 85, 7062. (28) Buonomo, E.; Gianturco, F. A. In preparation. (29) Althorpe, S.; Gianturco, F. A.; Sanna, N. J. Phys. B 1995, 28, 4165. (30) Khare, V. J. Chem. Phys. 1977, 67, 3897. (31) Khare, V.; Kouri, D. J. J. Chem. Phys. 1978, 68, 1237. (32) Khare, V. J. Chem. Phys. 1977, 67, 387. (33) Shimoni, Y.; Kouri, D. J. J. Chem. Phys. 1977, 66, 2841. Goldflam, R.; Kouri, D. J.; Green, S. J. Chem. Phys. 1977, 67, 5661. Parker, R.; Pack, R. T. J. Chem. Phys. 1978, 6, 1585. (34) Nobusada, K.; Sakimoto, K. Chem. Phys. 1995, 197, 157. (35) Buonomo, E.; Gianturco, F. A. In preparation.
JP953386M
