Volume 87. number 5 CHEMICAL PHYSICS LETTERS 9 April 1982

BOUNDLEVEL RESONANCES TN DIATOM-(RIGID) SURFACE SCATTERING

Reinhard SCHINKE

Rccck_’ 20 January 1987. m fmaJ form 10 March 1981

Model calculations are presented which show pronounced bound-level resonances in ro13lion3Uy in&&tic sallering oi HD molecules from weaNy cormgatcd crystal surfaces. The probabilities are obtained Mthin the difiraction sudden ap pro\lmation treating the rotational dcgrce of ireedom exactly.

1. Introduction

There is much esperimental Interest in understand- ing the microscopic dynanucs of melastic atom/surface [I 1 and molecule/surface collisions [2-S]. While in ref. [I] the inelastic energy exchange between the pro-

jectile and the surface is investigated, other experi- ments [Z-S] are primarily aimed at studying the en- ergy transfer between the translational and the rota- tional degrees of freedom of the molecule. At the same time there is much theoretical activity [6-121 to es- tablish accurate but feasible approximations with the goal to Interpret the detaiIed eaperiments and to ob- tain, if posstble, information about the Interaction per tential.

Recently Cowin et al. [s] reported rotational tran- sition probabilities of HD molecules scattered from a clean Pt crystal. Changing the incident angle from the surface normal f?,, but keeping the initial kinetic en- ergy at the constant value E = 109 meV and monitor- ing the heights of the various peaks within the specular channel, these authors observed distinct, closely spaced structures in the elastic, 0 + 0, as well as in the inelastic 0 + I and 0 + 2 rotational channels. Since

the corrugation of the HD/Pt potential energy surface (PES) is assumed to be very weak, momentum ex- changz between the normal and the parallel compo- nents is unfavourable and rotational excitation pre- dominantly depends on the normal incident energy, En = E COS’Biv Thus, the sharp modulations of the various transition probabdities versus Bi were inter-

preted as resonances between bound levels of the PES and negative kinetic energies correspondmg to

closed rotational channels [13]. Similar effects are well known In scattering of atoms from corrugated surfaces [ 141. The esperimental resolution of these resonances in many cases has led to the determination of the corresponding atom/surface Interaction poten- tiaI [IS], especially the weII region, and thus Cowin et al. [S] anticipate that analysis of experimental re- sults such as their HD/Pt data should determine the short-range interaction potential of a molecule on a smooth metallic surface.

Stimulated by the work of CowIn et al. [5] we perform model calculations which computationally verify the existence of bound-level resonances in ro- tationaIly elastic and Inelastic molecule/surface scat- tering. Such calculations should give important de- tails about the position, shape and width of the resO- nances and their dependence on the PES.

2. Theory and calculalions

The various rotational transition probabilities withing the specular diffraction channel are calculated within an approximation to the exact close-coupling treatment of diatom-(rigid) surface scattering [16]. The rotational degree of freedom is treated exactly whereas the diffraction channels are decoupled in the so-called coordinate-representation-sudden appro.xi- mation established by Gerber et al. [6]. Within this

438 0 009-2614/82/0000-OOOO/% 02.75 0 1982 North-Holland

Volume 87. number 5 CHEMICAL PHYSICS LETTERS 9 Apnl1982

model the rotationally coupled equations are solved for a fNed two-dimensional vector p = (X,JJ, 0) with x andy being coordinates along the lattice ax2s, i.e.

(d’/k2 + k,%~nl,-(z IP)

In 2q. (1) z is the distance of the molecular center of mass from the surface; the channel wavenumber for initial stat2 j and final state]’ is defined in au

k; = 2&5” + Ej - E,.) , (7-)

wHh P the reduced mass of the molscule, E,, the nor- mal initial kinetic energy and E,, =Bj’ (J’+ 1) the ro- tational energy in channelj. The coupling matrk is

defined by 2n

v ~mj,~,,i”(zl~) = s d0 j dy sin Y 0 0

x T:m,,(Y, @) %,Y, =, Y, $9 ~#‘~#X 4) I (3)

with Ymmj(y, 0) the wavefunction descnbing the mo- tion o the molecule in its center-of-mass frame in ro- t’ tational state jr?z,. Applying the appropriate boundary conditions of the scattering wavefunctions ~j~,(ZlP) [ 161 yields the p-dependent S matrix for a rotational transition, i.e. S(imi +j’m,Jp) from which the full S matrix for a rotation-diffractlon transition is evalu- ated as

S(inrj; ttm + j’n$.; tim’)

=A-’ / dp exp[ip(G,, -G,.,.)]

X S(imi ?j’ml. I p) . (4)

In 2q. (4) G,, = (h/a, 2rm/a, 0). a, and A are a reciprocal space vrctor, the unit cell length and the unit cell area, respectively. The diffraction sudden approximation is assumed to be valid if the incident momentum along the z direction is large compared to

,I [6]. If the PES is only weakly corrugated Fimh. th w c IS 2 case of current interest for us) transi-

tions into diffraction channels with n’ #n or m’ # m are very unlikely. Thus, w2 ar2 confident that in this case the diffraction sudden approximation gives an (at least qualitatively) accurate description of the

various rotational transition probabilities within the specular channel even if the above accuracy criterion is only approximately fulfiiled.

The enormous time saving in this hybrid approw mation compared to the exact theory is due to the drastic reduction of the total number of coupled rotation-diffraction channels. In the present calcula- tlon aU,open channels (k; > 0) and three closed chan- nels (k,; > 0) are included m the rotational basis. The two-dImensional integral in eq. (1) is conveniently evaluated if S(irnj +j’mflp) is expanded into a Fourier series. Detads of the calculations will be re- ported m a forthcoming paper.

The model intzraction potential used in the prss- ent study is of the form

T’,(z)=D{I -exp[-o(:-i)]}2-D, (6)

V,(z) = exp [-2a(r - P)] [%P~(cos 7) + 2fz(cos T)] ,

(7)

Q(x,y) = cos(27xr/a) cos(2qfa). (8)

The well depth of the isotropic (I) part of the poten- tial is taken as 55 meV according to experimental estimates [SJ. The steepness parameter (Y and the well distance f are set to 0.6245 au and 7 au, rcspec- tively. The parameters S and E, which control the strength of the rotational coupling, are chosen to re- produce roughly the experimental transition proba- bdities in fig. 1 of ref. [S]. They are 6 = 8.8 meV and E = 1. I meV. The potential is independent of the azimuthal angle @ and thus one has the selection rule h=mf-mi=O.

3. Results and discussion

We show in fig. I j = 0 + 1 and j = 0 + 2 rotational transition probabilities versus normal incident kinetic energy En for a flat surface, i.e. fl= 0. The spacing between the computed energy points is 0.2 meV. The 0 + 1 probability rises steeply at the threshold ener- gy, Et& = I) = II.06 meV, from zero to one and e.xhlbits several deep and broad dips in the energy

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Volume 87. number 5 CHEMICAL PHYSICS L?Zl-ERS 9 Apxil1982

0. 0 20 LO 60 SO

Normat mcldent energy fZ,, Cm&l

-I

1

rig. 1. 0 - 1 and 0 - 2 rotationti kmsitlon probabtities within the spccula~ diffractton chzmncl versus normal in&

dent energy ER- Flat surface restits 03 = 0)

range below the]’ = 2 threshold at 33.18 meV. It drops suddenly to inte~edjate values where the 0 + 2 channel opens and apart from the narrow peaks below Et& = 3) and the even sharper dips above that ~resho~d it declines slowly to higher energies. The 0 + 2 probability rises instantly at the threshold and increases on the average slowly with En. It e.xhibtts sharp structures at esactly the same energies as the 0 -+ 1 probabihty. A sudden change occurs at the opening of the 0 --F 3 channel.

Within the diffraction sudden limit bound-level resonances are expected to occur whenever a channel wavenumber kj, defmed in eq. (2) is imaginary (i.e. j, is a closed channel) and matches that of a bound level g, of the ha~toni~. In that case the molecule is temporarily trapped at the surface and the transi- tion probabilities show sharp structures in the vicinity of this resonance energy ER(jc, n) defined by

ERfjc, n) =?,, + ejc . (9) Assuming in zeroth order a free rater bound by the Morse potential in eq. (61, V&z), the resonance ener- gies are readily equated as

- 0.5&p(n + l/2)’ - D + Bj,(j, + 1) (10)

and given in table 1 for 0 G II Q 7 and 1 dj, Q 4. The Morse potential with the parameters quoted just after eq. (8) supports eight bound states for HD.

Some oi the free-rotor resonance energies of table 1 are indtcated in fig. 1. It is apparent that the sharp dips in the 0 + 1 transition probability at En r= 17.8, 24.6,7,9.4 and 32.4 meV correspond to the series with j, = 2 and 3 Qn G 6. The first three members of this series are below the 0 -+ 1 threshold and therefore not observable. Simiiardy, the sharp structures in the 0 + 1 as well as in the 0 + 2 probability below the 0 + 3 threshold correspond to the series ERGc = 3, 2 G n Q 6). The jc = 3, or= 0 and 1 resonances are below the O-2 threshold and only obse~ab~e in the 0 + 1 probabihty. They occur at En w 18.9 meV and En e 31.1 meV as very narrow peaks. A third series co~esponding to jc = 4 is partly seen above the 0 -+ 3 threshold. We note the following general observations from fig. 1: (a) A 0 -+j’ transition probability shows a dip at a resonance if the corresponding closed chan- nel isj’ 4 1. (b) Due to weaker coupling between high- er rotational states the resonance structures become narrower with in~reasingi~ (c)Within a particularj~ ladder the free-rotor approximation, which neglects hindering of the molecular rotation due to the cou-

Table 1 Free-roror resonance energies ER&, n) (iin mcV) defied m

eq. (10)

n JC

1 2 3 4

0 -36.9 -14.8 18.4 62.6 1 -24.3 -2.2 31.0 75.2 2 -13.6 8.5 41.7 85.9 3 -4.9 il.2 SO.4 94.6 4 1.9 24.1 57.2 101.5 5 6.8 28.9 62.1 106.4

6 9.8 31.9 65.1 109.3 7 10.8 33.0 66.1 110.4

Volume 87. number 5 CHEMICAL PHYSICS LEl-l-ERS 9 Apni 1982

pling term V,(r, 7) in eq. (7), provides more accurate estimates of the resonances for the higher bound states. (d) The free-rotor appro.ximation becomes more reliable asj, increases.

The data of fig. I are all obtained assuming an un- realistic flat surface. There is of course the question to which extent the narrow structures might be smeared out by the inclusion of surface corrugation,

i.e. the integration over p in eq. (1) in the case /I # 0. To study the influence of surface corrugation we show

in fig. 2j = 0 + 1 probabilities within the specular dif- fraction channel (n = II’, nz = rn’) around ER(jc = 2, II = 4) for various values of p_ Within the consldered range of the strength parameter fl the positions of the two resonances remain almost unchanged. This could be surmised from eqs. (S)-(8) because the maximum change of the well depth is only ?5% for p = 0.025 if x and y are varied in the accessible range of [0, n]. The variation of ER is of the same order and thus the net effect of including weak surface corrugation is negligible when the (.r,y) integration m eq. (1) is performed. The main effect is a decrease of the 0 + 1 transition probabihty in the various maxima ifP is en- larged. This is not due to reshuffhng of the probabdi- ty among the 0 + 0 and 0 + 1 channels but it is due to the lost of probability to non-specular dlffraction channels, i.e. 11’ f n and m’ #RI.

Nthough the present calculations are based on a model PES at least qualitative agreement with the experimental results of Cowin et al. [S] is partly achieved. For example, four dips in rhe 0 --t I transi-

10

0 20 22 2L 26 28 30

Normal uwdenl emqy E. tmeVJ

Fig 2. 0 --t 1 rotational transition probabilities witbln the specuku diffrxtion chwmel wrsus normal incident energy En for various corruption panmetcrs 8.

tion probabihty are clearly resolved experimentally. They occur at normal incident kinetic energies of

En =Z 17.3, 232,272 and 3 1.1 meV and are in quite good agreement with the theoretical ones quoted above. Three slightly indicated dips in the 0 + 7- prob-

ability are observed at En = 45.4.Q.6 and 59.0 meV which are also in fair agreement with the theoretical results, i.e. En - -+ 42.4,s 1.0 and 57.6 meV. Two ad- ditional resonances predicted at En = 62.0 and 65.2 meV, which correspond to j, = 3 and II = 5 and 6, respectively, are not observed eupcrimcntaliy. Indeed, these resonances are very narrow, as can be seen in fig. I.

The agreement between the present calculation and experiment, if based solely on the resonance positions, could easily be improved by varying the potent131 pa- rameters, especially the well depth and the steepness parameter Q m eq. (6). In our opinion, however, it is also necessary for a potential determination to in- clude the individual rotational transition probabilities, preferably for several collision energies. Unfortunately the data presented in ref. [S] arc scattering intensities rather than transition probabilitres and cannot be com- pared directly with theoretical results. Thus full de- convolution of the esperimentd data has to be await- ed before one can attempt to extract a reliable HD/Pt potential energy surface from the experiment.

4. Conclusion

It has been comput3tlonally demonstrated that resonances between bound levels of the interaction potential and closed rotational channels are distinct features of molecular scattering from weakly corrugat- ed crystal surfaces. Some of these rcsonanccs are suf- ficiently broad to be resolved wth present expenmcn- tal technology [.5]. They are analogous to the well- known resonances in atom/surface scattering which, however, are due to coupling to closed diffraction channels.

Analysis of the resonances in rotation3lly elastic and inelasrlc transition probabilities in molecule/ surface scattering should yield informatron about the interaction potential, especially in the well region. Before this problem can be attacked we feel that ad- ditional theoretical investigations are necessary. Such future work should consider: (3) The position of the

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\rolume 87, number 5 CHEMICAL PHYSICS LEl-l-ERS 9 April 1982

resonances and the proper relation to bound levels, especially the influence of the rotor hindering due to the anisotropic part of the potential. (b) The analysis of the shape and the width of the resonance structures. (c)The dependence on the interactron potential. (d) The inclusion of surface corrugation and the test of the diffraction sudden approsimation by comparison with exact calculations. Work in these directions is in progress and will be reported later.

Note added in proof

Simdar model calculations have been performed independently by Whaley,et al. [ 171.

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