32
Surface Science 127 (1983) 283-314 North-Holland Publishing Company 283 ROTATIONALLY MEDIATED SELECTIVE ADSORPTION IN RIGID ROTOR/RIGID SURFACE SCATTERING Reinhard SCHINKE Max - Planck - Institut fiir Strlimungsforschwg, D - 3400 Gdttingen, Fed. Rep. of Germany Received 21 October 1982; accepted for publication 16 December 1982 We present a detailed investigation of a new effect in molecule/surface scattering, rotationally mediated selective adsorption, which has been first observed by Cowin et al. (J. Chem. Phys. 75 (1981) 1033) in rotationally inelastic HD/Pt (111) scattering. It is due to resonance between an asymptotically closed rotational channel and a vibrational bound state of the molecule/surface system. Exact close-coupling and diffractionally sudden calculations are performed for HD scattered from a rigid, flat and weakly corrugated surface. Both variation of the collision energy and variation of the incident angle are considered. In the latter case we include the averaging over the collision energy of the initial beam and compare qualitatively our results with the experiment. Taking into account the resonances below the j = 0 + 1 threshold we are able to explain all of the experimentally observed resonances and to assign them with the appropriate rotational-vibrational quantum numbers. The largest deviation between experimental and theoretical resonance angles is - 1.6’ at a collision energy of E = 109 meV. An interesting interference effect is found in the case of weak surface corrugation. Although the diffractionally sudden approximation generally gives only poor results, it also reproduces this effect and provides a simple explanation of it in terms of the Breit-Wigner representation of the S-matrix. 1. Introduction There is currently growing interest in understanding the microscopic dy- namics of rotationally inelastic molecule/surface scattering. Most detailed information is obviously obtained from state selective experiments which - in principle - measure the probability associated with a single j +j’ rotational transition. For light molecules such as H 2, HD or D2, the diffractive peaks can be resolved to extract rotational transition probabilities [l-6]. Since the final angular spacing between two rotational peaks becomes narrower as the energy transferred from the translational motion to molecular rotation decreases, this method is not suitable for heavier systems with small rotational constants. In some cases (NO [7,9], HF, CO [lo]), the powerful laser-induced-fluorescence technique is applied to determine rotational transition probabilities. The availability of these new and detailed experimental data has attracted a number of theoretical studies which attempt to understand the dynamics of 0039-6028/83/0000-0000/$03.00 0 1983 North-Holland

Rotationally mediated selective adsorption in rigid rotor/rigid surface scattering

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Surface Science 127 (1983) 283-314 North-Holland Publishing Company

283

ROTATIONALLY MEDIATED SELECTIVE ADSORPTION IN RIGID ROTOR/RIGID SURFACE SCATTERING

Reinhard SCHINKE

Max - Planck - Institut fiir Strlimungsforschwg, D - 3400 Gdttingen, Fed. Rep. of Germany

Received 21 October 1982; accepted for publication 16 December 1982

We present a detailed investigation of a new effect in molecule/surface scattering, rotationally mediated selective adsorption, which has been first observed by Cowin et al. (J. Chem. Phys. 75 (1981) 1033) in rotationally inelastic HD/Pt (111) scattering. It is due to resonance between an asymptotically closed rotational channel and a vibrational bound state of the molecule/surface system. Exact close-coupling and diffractionally sudden calculations are performed for HD scattered from a rigid, flat and weakly corrugated surface. Both variation of the collision energy and variation of the incident angle are considered. In the latter case we include the averaging over the collision energy of the initial beam and compare qualitatively our results with the experiment. Taking into account the resonances below the j = 0 + 1 threshold we are able to explain all of the experimentally observed resonances and to assign them with the appropriate rotational-vibrational quantum numbers. The largest deviation between experimental and theoretical resonance angles is - 1.6’ at a collision energy of E = 109 meV. An interesting interference effect is found in the case of weak surface corrugation. Although the diffractionally sudden approximation generally gives only poor results, it also reproduces this effect and provides a simple explanation of it in terms of the Breit-Wigner representation of the S-matrix.

1. Introduction

There is currently growing interest in understanding the microscopic dy- namics of rotationally inelastic molecule/surface scattering. Most detailed information is obviously obtained from state selective experiments which - in principle - measure the probability associated with a single j +j’ rotational transition. For light molecules such as H 2, HD or D2, the diffractive peaks can be resolved to extract rotational transition probabilities [l-6]. Since the final angular spacing between two rotational peaks becomes narrower as the energy transferred from the translational motion to molecular rotation decreases, this method is not suitable for heavier systems with small rotational constants. In some cases (NO [7,9], HF, CO [lo]), the powerful laser-induced-fluorescence technique is applied to determine rotational transition probabilities.

The availability of these new and detailed experimental data has attracted a number of theoretical studies which attempt to understand the dynamics of

0039-6028/83/0000-0000/$03.00 0 1983 North-Holland

284 R. Schinke / Rotationally mediated selective adsorption

molecule/surface collisions on a microscopical scale and to obtain important

information on the interaction potential by comparison between theory and experiment. These model calculations are performed on basically three levels of accuracy, i.e., the exact close-coupling (CC) approach [ 1 l-141, the infinite order sudden (10s) approximation [ 11,15,16] and the quasi-classical trajectory (QCT) approach [17-201. The CC method is exact but it requires usually very large computation times when many rotational states have to be included in the expansion basis. Moreover, the resulting distributions normally cannot be explained in simple physical pictures and an understanding of the collision dynamics is often obscured by the considerable numerical efforts. The great

asset of the 10s approximation is its interpretative power, especially if its semiclassical limit is explored [21-231. However, it is only applicable in cases

when the transferred energy is small compared to the translational energy and

might lead to serious errors otherwise [ 11,24,25]. The QCT method does not have this shortcoming but it also can yield spurious results in cases when quantum effects are prominent [25]. It is extensively used to study multiple collision effects which are due to reflection at the repulsive wall of the interaction potential [ 19,201. Such effects seem to be relatively unimportant in gas phasse collisions but might be significant in gas/surface scattering. An overview article on theoretical models in molecule/surface scattering is re- cently written by Gerber and Kouri [26].

Two of the recent experiments are particularly interesting because it seems possible to attribute certain regions of the interaction potential with distinct features of the experimental results. Auerbach, Kleyn and Luntz [9] reported highly non-statistical final state distributions of NO molecules inelastically scattered from the Ag(ll1) surface. Because of the relatively high collision energies of about 1 eV, it is believed that mainly the strongly repulsive part of the interaction potential is probed. The high Aj portion of the experimental distributions is interpreted in terms of rotational rainbows [14,16], an effect which is well known in atom/molecule scattering [27]. Thus, these measure- ments should yield information on the anisotropy of the interaction potential in the repulsive region. Model calculations in this direction are presented by

Barker et al. 1141. Quite different results are obtained by Cowin et al. [5]. These authors

scattered HD molecules from the Pt( 111) surface and resolved for the first time rotationally mediated selective adsorption structures which are due to reso- nance between an asymptotically closed rotational channel and a bound state energy of the interaction potential. Selective adsorption is well known in atom/surface scattering and is now routinely exploited to accurately determine the attractive part of the interaction potential [28]. Similar information is expected from molecular scattering experiments, especially in the case of very weakly corrugated surfaces when the “ usual” selective adsorption structures (which are due to an asymptotically closed diffractional channel and a bound

R. Schinke / Rotationally mediated selective adsorption 285

state energy) are very narrow and thus impossible to be resolved experimen-

tally. Two groups [ 12,131 recently published preliminary calculations which

show how rotationally mediated resonances can be modeled by exact CC calculations.

With the present article we continue our earlier investigations 1121 and analyse the resonance effect in more detail. It is organized as follows: The

theoretical methods used in this study are summarized in section 2. The interaction potential is decribed in section 3. The results for a flat, i.e. uncorrugated, surface are presented and discussed in section 4. Finally, modifi- cations in the case of weak surface corrugation are investigated in section 5. A

summary and a prognosis for future studies are given in section 6. All calculations ‘exclude the interaction with surface phonons as well as the thermal motion of the surface atoms, i.e. we assume that the molecules are scattered from a rigid surface. In addition we assume that vibrational excita- tion is negligibly small and thus the molecule can be treated as a rigid rotor.

2. Theory

We will present calculations performed on two levels of accuracy: (a) exact close-coupling (CC) calculations including both rotational and diffractional states in the expansion of the total scattering wavefunction, and (b) diffraction-

ally sudden (DS) calculations, however, treating the rotational degree of freedom exactly. The DS approximation is valid for small changes in the momentum parallel to the surface.

2. I. Close-coupling approach

The exact formulation of collisions between a diatomic molecule and a rigid surface has been given by Wolken [29] as an extension of his earlier work on atom/surface collisions [30]. The Hamiltonian for a rigid rotor colliding with a rigid surface is (units are such that tt = 1)

H(R, i) = - (2M)-1 v,z + h,,,(P) + V(R, i), (1)

with the first term being the relative translational energy and M = ml + m, the total mass of the molecule. The term h,,,(i) is the internal Hamiltonian describing the rotational degree of freedom of the molecule. The corresponding eigenstates and energies are I;,,(i) and E, = B,,,j(j + l), respectively. V(R, i) is the interaction potential depending on R = (x, y, z), the vector from the origin to the centre of mass of the molecule, and i, the internuclear vector of the molecule with r = req. The vector p = (x, y) is parallel to the surface and the coordinate z is perpendicular to the surface. Since for each internal molecular orientation i = (y, ‘p) the potential is periodic in the (x, y) plane it

286 R. Schinke / Rolaiia~al~y mediated selective adsorption

can be described in terms of the two-dimensional reciprocal lattice vectors G,,(k, I= 0, rfr 1, k 2,. . . ) with G,, = 0. For a rectangular lattice with peri- odicities a and b (the case which will be considered in this study) they are given

by

Thus, V(R, i) can be expanded in the form

V/(R, i) = c+,(z, Y, ‘P) exp(iG,,+p). k.1

(2)

Since the potential is real G,, must appear with G_, _,, and v_~ _, = u:,. The total wavefunction describing a molecule impinging on the periodic

surface with initial wave vector k = (k,, k,, k,) is expanded as

44R F) = x C@jmj:p4(L) Tmj(f) exp[i@+ G&P]~ (3) 9, Pq

with K = (k,, k,, 0) a two-dimensional vector consisting of the (x, y) compo- nents of k. The unknown functions u~~,;~~(z) satisfy the following set of coupled second order differential equations:

= 2M c c qn,pq:j’m,.p’q’f4 =jhj,;p,q44. j'm,. p’q’

The channel wave numbers are defined by

d&=2M(E+c, -c~)-(K+$~)~,

where E is the initial relative translational energy andk the initial state of the molecule. The coupling matrix elements are defined by

(4

(5)

rotational

yrn / pq; j’m,,p’q’ (z) =A-‘j-dildp

x I;*,,(f) exp( -iGP4.p ) V(X,.Y, 2, Y, 91) exp(iG,,,,.p) q,,,.(i)? (6)

where the integration dp is over the unit cell with area A. Utilizing the hu-ier expansion (2), the coupling matrix element reduces to

y-m,pq; j’m,.p’q’ cd -I di I$(:) u~+~_~&, Y, ‘P> y;.,,,,(;). (7)

In the present study we assume a q-independent interaction potential such that eq. (7) is diagonal in the magnetic quantum number, i.e., mj = mj,. This assumption is striclty valid only for a flat surface when a rotation of the molecule about an axis normal to the surface does not change the potential, It

R. Schinke / Rotationally mediated selective aa!sorption 287

is unphysical, however, for a corrugated surface even if the corrugation is weak. In this case, which will be discussed in section 5, one might first average the potential over the azimuthal angle 91 before performing the scattering calculations. Such a strategy is believed adequate if only degeneracy averaged quantities are of interest. Expanding - as usual - the Fourier coefficients in terms of Legendre polynomials

u/&7 Y) = cu;,w PA(COS Y) (8) A

finally gives for the coupling element eq. (6)

V ,m,pq;j’m,~p’q’w = 4?2,m,, (- 0”’ K2j + 1)(2j’ + N”*

x C(?.h + l)_’ C(jj’AP00) C(jj’h(rnj -mjO)~U;_p+r’(Z),

where C( . . - ) . . . ) is a Clebsch-Gordan coefficient. The coupled equations (4) are to be solved subject to the standard boundary

conditions [29]. The resulting S-matrix S( jmi; pq + j’mj,; p’q’) contains all the scattering information. The probability for a jm,; pq + j’m,,; p’q’ transition is

]S(*.* )I’. Physically, we are interested in scattering from the specular channel, i.e., p = q = 0. A very comprehensive overview and comparison of all the available numerical algorithms to solve the set of coupled differential equa- tions as they occur in scattering problems is given by Thomas et al. [3 11.

The CC treatment is exact but has the serious limitation that the number of coupled equations N = P X Q x J rapidly exceeds a size which can be handled with standard computer’codes, i.e. N - 100 (P X Q is the number of diffrac- tional states and J is the number of rotational states included in (3)). Therefore it is advisable to search for methods which treat one or the other degree of freedom approximately. Since the main purpose of this study is the investiga-

tion of resonances due to the temporary excitation of asymptotically closed rotational states the rotational degree of freedom must be treated exactly.

However, we already assumed weak surface corrugation which suggests an approximate description of the diffractional degree of freedom.

2.2. Diffractionally sudden approximation

The diffractionally sudden (DS) approximation has been first derived by Gerber et al. [32] for atom/surface scattering. Its basic assumption is that the channel wave numbers dp4 in eq. (5) (the rotational index j is omitted) can be approximated by a single value dFQ for all channels which significantly contribute to the scattering. Choosingp = 4 = 0 gives the validity criterion [32]

288 R. Schinke / Rotationally medrared selectroe adForprron

According to eq. (10) the DS approximation is expected to give reliable

transition probabilities for (a) large unit cell lengths, (b) large energies, (c) heavy masses, and (d) small angles of incidence (measured with respect to the

surface normal). The accuracy of the DS approximation has been tested by comparison with exact CC calculations (33-351.

The DS approximation is most conveniently derived by replacing the kinetic

energy operators -(2M)-‘i32/&2 and -(2M)-‘i!12/ay2 in eq. (1) by some constant values TX and r,,, respectively. The resulting Schrodinger equation contains x and y only as parameters through the potential. Expanding the total wavefunction for each point (x, y) on the surface in terms of rotor eigenstates,

&KY, z, Y, ‘PI = c t&,(%Y. z) y,,,(Y, cp), (11) Jm,

leads to a set of equations which are coupled only in the rotational quantum

number, i.e.,

($+;I:) G, (~3~3 z) = 2M c v/,,,:J,,,,.(x,~. z) fi,,,,,(w. z), (12) I'",.

with

and JJ2 = 2 M( E + cJ, - rK - T,. - c,). Applying the appropriate boundary con-

ditions yields a p-dependent S-matrix for a rotational transition, i.e. S( jm, + j’m,,]P). Now it is easy to show that

iiJ.,,,,,:p.4.( Z) = A -- ‘/dp exp( - iGP,y,. P) cJ.“,, (x9 Y, z) exP(iG,, . P) (14)

is a solution of the exact set of coupled equations (4) with the exact value d,$,, replaced by (il’. Analyzing the z 4 30 behaviour of the approximate solution eq. (14) gives for the full S-matrix

S( jmJ; p4 + j’m,.; p’4’)

= A - ’ /

dp exp( - iGPz,,, VP) S(jm, +j’m,b) exp(iG,;p). (15)

Note the similarity between this equation and the definition for the coupling matrix element in eq. (6).

This hybrid approximation yields a drastic reduction in computation time because the rotational channels are decoupled from the diffractional channels and the number of coupled equations is equal to the number of rotational states included in the basis expansion. However, this reduced set has to be solved for various values of (x,y) in order to perform the integration in eq. (15). Gerber et al. [ 151 introduced an approximation which also decouples the

R. Schinke / Rotationally mediated selective adsorption 289

rotational channels and which is valid if in addition to eq. (10) the energy transferred to molecular rotation is small compared to the collision energy.

3. The interaction potential

The calculations are performed for HD as gas molecule using the interaction

potential

where PA is a Legendre polynomial of order A. Since HD is a heteronuclear molecule even as well as odd polynomials must be included in eq. (16). The

zeroth order term is

V,(x,y,z)=D{l -exp[-cr(z-z,)]}‘-D

+PD Q(x,Y) exp[-2a(z -z,)], (17)

with the dissociation energy chosen as D = 57.75 meV which is close to the experimentally estimated value of 55 meV [5]. The higher order terms are purely repulsive and given by (A = 1 and 2)

v,(x,~,z)=x&Cl +PQ<x,Y)I exp[-24z--e)], (18) with S, = 9.23983 meV and 13, = 1.15498 meV. The parameter x is introduced to vary the rotational coupling strength. It is usually one if not stated otherwise. The exponent in eqs. (17) and (18) is the same and chosen as a = 0.9448 A-‘. The equilibrium distance z, is arbitrary.

The corrugation of the surface is described by

2a 2n Q(x,v) = cosclx + cos-y,

b (19)

with a and b the length of the unit cell which in the present study is assumed to be rectangular. The actual values are a = b = 1.385 A. The strength of the corrugation is controlled by the parameter p which is set to p = 0 (flat surface, section 4), B = 0.01 and j3 = 0.02 (section 5).

The parameters D and (Y are chosen to obtain a rough fit of the experimen- tally observed resonance positions as we will show in the next section. The coupling parameters S, and 6, are set such that the overall energy dependence of the rotational transition probabilities qualitatively reflects the experimental result. The present potential form and the parameters are slightly different from ref. [ 121. The potentials V,, V, and V, are displayed in fig. 1 for /3 = 0. The average potential, VO, supports ten bound states for the HD molecule.

290 R. Schinke / Rotationally mediated selective adsorption

-1 0 1 2 3 L 5 6

2 [AI Fig. 1. Legendre expansion coefficients V,, eqs. (17) and (18) for fi = 0. The bound state energies

?, (n = 0,. . ,8) for HD are marked by the horizontal lines.

4. Flat surface

In this section we present and analyze results for HD molecules scattered from a flat surface, i.e. p = 0 in eqs. (17) and (18). Thus, only the specular channel p = q = 0 has to be included in the expansion of the total wavefunc- tion (3) and the total number of coupled equations is equal to the number of rotational states. Then the channel wave numbers in eq. (5) reduce to

dj’ = d&) = k; + 2M(c, - Zj), (20)

with k, = k cos 8, where 8 is the incident angle, and ji is the initial rotational state. For each pair of collision energy E and incident angle 8 all open channels (d,? > 0) and the lowest three closed channels (d,: < 0) are included in the rotational basis. This ensures that all transition probabilities reported below are converged to a sufficient degree of accuracy, usually three or four digits.

4.1. Variation of the collision energy

In this subsection we discuss the energy dependence of the various rota- tional transition probabilities for a fixed incident angle which in all cases is 0 = 0”. In fig. 2 the probabilities for j = 0 -j’ = 0, 1, 2 and 3 are plotted

R. Schinke / Rotationally mediated selective adsorption 291

opt: 1 Et,, iO--3)

/ I I I I

0 20 LO 60 60 100

collision energy E lmeV I

Fig. 2. Rotational transition probabilities versus collision energy E for a flat surface, p = 0. The angle of incidence is 0 = 0’. The threshold energies Et,,(j) are marked by the arrows.

versus collision energy in the range 0 < E < 100 meV. We note the following observations: (a) Each transition probability rises steeply at its threshold Eth(O -j') = B&'(j' + 1) and because of the unitary of the S-matrix all the other probabilities for the lower states also show sudden changes at exactly the same energy. (b) Each probability shows pronounced structures which for all transitions occur at exactly the same energies. However, the shape of the structures can be very different for the various j’. (c) The structures for a

292 R. Schinke / Rotationa& medtated selective adsorptron

particular transition are different in the various energy intervals between two thresholds.

These structures are so-called Feshbach resonances [36] and occur whenever a channel wave number d,‘ defined in eq. (20) is imaginary (i.e. j, is a closed channel) and matches a wave number of a bound vibrational molecule/surface

state with energy c’,,. In view of eq. (20) and assuming a free rotor. i.e.. zeroth order perturbation theory, the resonance energies are given by

ER( j,, PI) = (t, + c,, - r,,)/cos28. (21) For collision energies close to a resonance the molecule is temporarily trapped at the surface in a quasi-bound state and the time during which the rotational-translational coupling is active is greatly enhanced compared to an off-resonance energy. Since rotational momentum and energy transfer depend

strongly on the collision time, the individual transition probabilities show sharp structures in the vicinity of a resonance.

The analogous effect in atom/surface scattering is known as selective adsorption and was first observed experimentally by Frisch and Stern [37] and

Table 1

Resonance energies ‘)

1 2 3 4

0 - 40.873 b, - 18.753 14.427

1 - 30.165 - 8.045 25.135

2 - 20.692 1.428

3 - 12.454 9.666

4

5 0.318

6

7

8 10.213

9 11.042 33.162

- 5.45 1

4.85 1

8.150

16.669

18.0”

22.438

23.4

26.97 I 21.8

30.270

30.8

32.333

34.608

35.2

42.846 43.4

49.849

50.4

55.617

56.0

60.151

60.6

63.449

63.8

65.5 13

65.6

66.342

58.667

69.375

69.8

78.848

19.2

87.086 87.4

94.089

94.4

99.857

104.391

107.689

109.753

110.582

‘) All energies are in meV.

h’ First entry: free-rotor approximation, eq. (21) with 8 = 0”.

‘) Second entry: from scattering calculations with A E = 0.2 meV.

R. Schinke / Rotationally mediated selective adsorption 293

1.0 I I j,=2.n= 4 5 6 789 ’

I I I I

It , I I I I I I 0 20 LO 60 80 100

1.0

0.8

0.6

0.4

0.2 +---I 0

0 20 LO 60 80 100

0.8

0.6 C

I I , I I

j,=&,n=O 1 2 3 4

I IIII

20 10 60 80 1 1 oc

collision energy E tmeV 1 Fig. 3. j = 0 * 1 transition probability versus collision energy E for a flat surface, p = 0. The

free-rotor resonance energies /?a( j,, n), eq. (21) with 8 = O”, are qarked by the vertical lines and

labeled with the appropriate rotational, vibrational quantum numbers. The strength parameter

controlling the rotational coupling is (a) x = 0.5, (b) x = 1.0, (c) x = 1.5.

explained theoretically by Lennard-Jones and Devonshire [38]. Both effects are very similar. The only difference is the “internal” degree of freedom which assists the trapping mechanism. In the atom/surface case the translational

294 R. Schinke / Ro~attonal!v mediated selectroe ndsorp:ion

energy for the motion perpendicular to the surface is temporarily transferred to the motion parallel to the surface and the closed channel necessary to form a resonance is a diffractional state GP y,. In order to distinguish both effects we

suggest to term the first “rotational& mediated selective adsorption” [ 131 and the second “diffractionally mediated selective adsorption”. Obviously, both

resonance types can occur simultaneously in the case of a molecule scattered from a corrugated surface.

The free-rotor resonance energies EK( j,, n), eq. (21) for 8 = 0”. are listed in table I for 1 <j, < 4 and 0 < n < 9. At or in the vicinity of these energies resonances are expected to occur. In order to demonstrate that the observed structures in fig. I correlate well with these zeroth order predictions, we plot in fig. 3 the j = 0 * I transition probability for three values of the strength parameter x in eq. (18) and mark the energies ER( j,, n). First, we observe that the structures become significantly broader and more pronounced as the rotational coupling is increased. This effect has been noticed before in various calculations for atom/surface scattering (301 as well as in gas phase scattering

1391. In fig. 3a the five narrow dips below the E,,,( j’ = 2) threshold at 33.18 meV

belong to the series j, = 2 and n = 4, 5, 6. 7 and 8. The last member of this

series, n = 9, is not resolved; its zeroth order prediction is 33.16 meV and coincides almost exactly with the j’ = 2 threshold (C,, _9 = - 0.018 meV). The resonances with n < 4 are all below the j’ = I threshold and therefore not observable in the 0 + I transition probability. We will discuss the appearance of resonances below the j = I threshold in section 4.3. Increasing the rotational coupling increases the width of the resonances and their shift away from the predicted ones such that an unambiguous correlation with the energies of table 1 becomes questionable. For example, the minimum at E - 13 meV for x = I .5 (fig. 3c) could correspond tojc = 2, n = 4 or alternatively tojc = 2, n = 3. In the latter case the free-rotor prediction is below the 0 + I threshold but due to the strong coupling the actual resonance could be shifted by 3-4 meV to larger energies and so become observable.

The series belonging to j, = 3 starts at E - 14.5 meV below the 0 + 2 threshold and the first two members cannot be observed in the 0 -+ 2 transition probability. However, they are clearly visible in fig. 3c at E - 15 meV and E - 26.5 meV, respectively. They are slightly indicated in fig. 3b for x = 1.0 but too narrow to be observable for x = 0.5 with a step size of AE = 0.2 meV. The seven distinct peaks or dips above the 0 --, 2 and below the 0 + 3 threshold correspond to the series j, = 3 and 2 Q n Q 8. Note the dramatic change of the shape as the coupling parameter is increased from x = 0.5 through x = 1.5. A third ladder with j, = 4 starts at E - 58.7 meV. The first member, n = 0, is below the 0 4 4 threshold and is unresolved even for the highest coupling strength. The higher terms with n > 1 are only slightly

indicated.

R. Schinke / Roiotionally mediated selective adsorption 295

o/L-J---- 0

I j= l-2

m=O

I I 75 100

0.6 -

0.4 -

,! 0 ._ ,E 0.2-

z

1 2 3 G 5 6761

j=l-2

m=l

collision energy E [meVl

Fig. 4. j = I --) 2 rotational transition probability versus collision energy for magnetic quantum numbers m = 0 and + I for a flat surface, /3 = 0. The results for m = 1 and M = - 1 are identical. The free-rotor resonance energies ER(jC, n), eq. 21 with B = O’, are marked by the vertical lines

and labelled with the appropriate rotational-vibrational quantum number.

Some of the resonance energies as obtained from the scattering calculation (AE = 0.2 meV, x = 1.0) are compared in table 1 with the zeroth order predictions. The shift can be as large as - 1.3 meV (j, = 2, n = 4). The conclusions from figs. 2 and 3 and table 1 can be summarized as follows: (a) If

j, is the resonant channel the probability of the next lower open state, i.e. j’ =j, - 1, shows a dip. (b) Due to the gradually weaker coupling into higher rotational states the resonance structures become narrower with increasing j,. (c) Within a particularj, ladder the accuracy of the free-rotor approximation, which neglects hindering of the molecular rotation, increases with the vibra-

296 R. Schrnke / Rorarionally mediated selecrive udsorptron

tional quantum number. (d) The free-rotor approximation becomes more

reliable as j, increases, i.e. when the coupling between the open and the closed channels decreases.

In fig. 4 we show j = 1 --*j’ = 2 rotational transition probabilities for m, =

m,. = 0 and 1. Since the interaction potential is independent of the azimuthal

molecular orientation angle cp the magnetic quantum number is conserved, i.e., m, = m,.. The probabilities are independent of the sign of m. This follows

directly from the symmetry of the coupling elements in eq. (9) with respect to +m. According to eq. (21) the free-rotor resonance energies are shifted by

<,‘I = 11.06 meV towards lower energies and they agree very well with those obtained from the scattering calculations. Comparing the 0 + 1 probabilities in fig. 1 with the 1 + 2 results in fig. 4 we note a considerable change in the shape and the width of the resonances although both are A j = 2 transitions. This could be expected, however, from the potential coupling matrix elements in eq. (9) which depend on both the initial and the final state rather than the momentum transfer. They also depend on m which explains the different variation with the energy for m = 0 and m = f 1.

Considerable efforts have been made to derive analytical expressions for the S-matrix in the neighbourhood of a resonance. The first rigorous approach has

been given by Feshbach [40] using projection operators which project either onto the subspace spanned by the open channels or onto the subspace spanned

1

j= O-2 1

I I 0 1

Re S(OO;OO- j’O;OOl

Fig. 5. Argand plot of thej = 0 + 2 S-matrix around the,, = 3. n = 3 resonance. The energy step

size is 0.05 meV for 43 < E $44 meV and 0.1 otherwise.

R. Schinke / ~otatjo~~i~ mediated selective adsorption 297

Fig. 6. Argand plot of thej = 0 -+ I S-matrix around the& = 2, n = 4 and n = 5 resonances. The energy step size is 0.1 meV.

by the closed channels. The resulting expressions for the S-matrix and other quantities such as the resonance energy and its width are very complicated and extremely difficult to handle. In many situations the S-matrix elements for a transition a -+ 0~’ in the neighbourhood of a single, non-degenerate resonance can be represented by the generalized Breit-Wigner expression [36]

where Sbg is the so-called background contribution which is assumed to vary

smoothly with the energy. The numbers y, can be chosen to be real and r = C,ly,12 is the total width. In fig, 5 we show an Argand plot of the 0 + 2 S-matrix around the j, = 3, n = 3 resonance. Plotted is the imaginary part of the S-matrix versus its real part with the energy as parameter. Except for the resonance region between E - 43 meV and E - 44 me%‘, the S-matrix describes an almost perfect circle around the origin. This suggests that the background term can be written in the form

Sbg(E)=Aexp[i(a+bE)], (23) with A, a, and b being constants. The resonance part, i.e., the second term in eq. (22), would be an approximate, counterclockwise circle in the lower halfplane. A quite complicated resonance behaviour is shown in fig. 6 for the j = 0 + 1 S-matrix in the neighbourhood of the two resonances corresponding

298 R. Schinke / Rotationally mediated selective adsorption

to j, = 2, n = 4 and 5, respectively. The two resonances are clearly visible at E - 18 meV and E = - 23.4 meV when the curve goes through the origin. Away from the resonance the S-matrix again describes a circle with almost energy independent radius and angular velocity. This is seen for the intervals 16 < E < 17 meV and 21 < E < 22 meV. Plots like those in figs. 5 and 6 can be helpful to separate the contributions from direct and resonance scattering.

4.2. Variation of the incident angle

Since the resonance widths are typically - 1 meV (see figs. 5 and 6), it would be extremely difficult, if possible at all, to resolve resonances in an

experiment which scans the collision energy at a fixed angle of incidence, 8. However, according to eq. (20) the component of the incident wave vector

normal to the surface, k, = &%% cos 8, is the quantity which is responsible for the formation of a resonance. Thus it is experimentally much more

1.0

0.6

‘t III I I I j,=2.n=987 6 5 L

1.0 [ I I I I I

0.8

0.6

30 60

incident angle 0 [deg

1 9(

Fig. 7. j = 0 + 1 transition probability versus angle of incidence 0 for a flat surface, p = 0, and collision energies of E = 40.8 meV and E = 109 meV. The free-rotor resonance angles 8,(j,. n).

eq. (24), are marked in the upper pane1 and the threshold angles t’,,(j) are marked by arrows in

the lower panel.

R. Schinke / Rotationally mediated selective adsorption 299

convenient to resolve resonance structures by varying 8 at a fixed energy. The

angular resonance positions in the free-rotor approximation are simply given by inverting eq. (21), i.e.

cos28,( j,, n) = (5” + Ejc - E,,)/E.

Fig. 7 shows two examples for the B-dependence of thej = 0 + 1 transition probability. The energies, E = 40.8 meV and E = 109 meV, are those of the experiment [5]. The resonance structure is equivalent to that observed for j = 0 + 1 in figs. 2 and 3. As predicted from eq. (24) the resonances become narrower as the collision energy is increased. The free-rotor predictions for

j, = 2 which are marked for the lower energy can be off from the actual resonances by 2”-3”.

Sharp structures as displayed in fig. 7 are obviously not resolved in the experiment because several factors can lead to significant averaging. These

factors which are not included in the present simple calculations are: (a) Interaction with surface phonons; (b) thermal motion of the surface atoms (the surface temperature in the experiment is T = 500 K [5]); (c) surface impurities; (d) final angular resolution (1.2’ in the experiment [5]); (e) energy spread in the initial HD beam; (f) influence of surface corrugation, i.e., diffraction scattering. The inclusion of coupling to surface phonons is obviously very difficult and we are not aware of any practical way to include this effect in a scattering calculation. The averaging due to thermal motion of the surface atoms in principle could be incorporated using Debye-Waller factors. How- ever, the appropriate form of Debye-Waller factors for molecule scattering is not known, at least to our knowledge. Impurities of the surface could be modeled by introducing optical, i.e., complex potentials, however, the physical inside gained would be questionable in our opinion. The influence of surface corrugation on rotationally mediated selective adsorption will be discussed in section 5.

The effect of the energy spread in the initial beam can quite easily be incorporated by averaging the transition probability at a fixed angle 0 over the energy weighted with a gaussian function, i.e.,

(25)

with N = G/a the normalization constant. E(O) is the mean energy and a2 is chosen to give the experimental FWHM. Using E, = E cos2 B (where E, is the translational energy of the motion normal to the surface), eq. (25) can be rewritten as

~ave(j -_f> = (N cos2B) - ’ jd E,

Xexp[ -a’(E, - E,‘“))2/cos48] IS(j-j’lE,)12, (26)

3ocl R. Schinke / Rotationally mediated selective aa%orption

.-_)

t

0.8

0.6

Oh

0.2

0

01.

0.2

0

-AE=l2meV

--- AE= 6meV

0 -1

0 -2

o-3

E = 109 meV

I I I

0 30 60 90

incident angle 0 [deg I

Fig. 8. Energy averaged rotational transition probabilities P,,, (j * j’), eq. (26), versus angle of

incidence for E = 109 meV and two FWHM, A E = 6 meV and A E = 12 meV. The arrows mark the

experimental resonance angles (51.

where we used the fact that the transition probabilities for a flat surface depend on E, rather than on E. Once the probabilities are calculated at a sufficient number of energies E,, the integration in eq. (26) can be easily performed for any angle 8.

The effect of energy averaging is demonstrated in fig. 8 for EC*) = 109 meV and for two FWHM, AE = 6 meV and AE = 12 meV. Let us first consider the latter case, A E = 12 meV, which is the one quoted in the experiment [5]. Above the threshold angle for the 0 -+ 2 transition we see three distinct minima in the 0 -+ 1 probability corresponding to thejc = 2, n = 4, 5 and 6 resonances. They occur at angles of 65.8O, 62.5O and 59.5O. For comparison distinct minima are resolved experimentally at angles of 66.5”, 62.5” and 60”. The experimental resonance angles are marked by arrows in fig. 8. A fourth minimum in the experimental distribution is slightly indicated at 8 = 57.7”. It is not observed in the theoretical curve with A E = 12 meV but appears in the A E = 6 meV curve at B = 57.8O. All experimental resonance angles are read off fig. 2 of ref. [S].

Three minima are vaguely indicated in the experimental 0 -+ 2 transition

probability at angles of 50. I “, 46.1” and 43.1’. Hardly any structure is observed in the theoretical curve with AE = 12 meV, but five minima are exhibited in the AE = 6 meV distribution. They occur at B = 55.3”. 50.8”, 47.2O. 43.8’ and 41.5” and correspond to the resonances with j, = 3 and n = 2, 3, 4, 5 and 6, respectively. The resonances be~on~ng to M = 3,4 and 5 correlate with those observed experimentally. The 0 --, 1 probability shows maxima at those angles where the 0 + 2 probabii~ty exhibits ~~irna. No structure is expe~men&aliy observed in the 0 --+ 1 probability below the 0 -+ 2 threshold angle. All the other resonances, especially those belonging to thej, = 4 ladder, are completely smeared out by the averaging and a smooth angular depen- dence is obtained for all probabilities beIow 8 - 40’. This is in accordance with the expe~ment.

Cowin et al. [S] also observed four distinct minima in the rotationally elastic 0 -+ 0 transition probability below the 0 + 1 threshold at 11.06 meV “‘which cannot be put into agreement with the levels” (51 obtained from the 0 + 1 and 0 + 2 resonance minima. These 0 -+ 0 minima are as pronounced and wide as the minima in the 0 --* 1 probability. The free-rotor ~ppro~mation. eq. (21), predicts seven resonances below the 0 + 1 threshold corresponding CO j, = 1, 5~n~9 and j, = 2, n = 2, 3. If only rotational channels are included in the expansion of the total scattering wave function, reso~aflces below the first threshold cannot be observed in the elastic transition probability because of the unitarity of the S-matrix. Regardless of how long the molecule is trapped in a quasi-bound state and of how far it moves along the surface, it finally has co emerge into the gas phase without being excited. The final angle must be equal to the initial angle of incidence. Not so in reality where surface impurities or phonon excitation can lead to a loss of particies from the specular channel. Another explanation is a phonon assisted direct resection into the open j’ = 1 channel, the missing energy being provided by the de-excited phonons. These processes are expected to be most prominent at glancing angles, i.e. the angular region of interest, B 2 70”. The opening of diffraction channels would be another possible explanation for the dips in the 0 ---f 0 probability. However the ~r~gation of the HD/Pt(l 1 I) surface seems to be too weak (see fig. 1 of ref. [5]) to explain such a strong modulation in the specular beam.

Scattering re~nances in the 0 --) 0 transition probability below the 0 -+ I threshold can be visualized by looking at the delay time r = d#( E )/dE where 9 is the phase of the 0 -+ 0 S-matrix element written in the form S = A exp(i+). An example is shown in fig. 9b in the vicinity of the j, = 1, n = 6 resonance. The incident angle is 6 = 0” and the two lines correspond to different coupling parameters x in eq, (18). For energies which are not too close to the resonance

302 R. Schinke / Rotationally mediated selective adsorption

-5b- 7 collision energy ECmeVl

Fig. 9. (a) Diffractionally sudenj = 0 - 0 transition probability within the specular beam. p = q = p’= q’= 0, below thej = 0 4 1, threshold. The corrugation parameter is fl= 0.01 and the angle of

incidence is 8 = 0”. (b) Delay time d+/d E for the elastic j = 0 + 0 transition below the j = 0 ---) I threshold. Flat surface and 0 = 0’. The vertical line marks the j, = 1. n = 6 free-rotor resonance

energy. The rotational coupling parameter is x = 1.0 (- )andx=0.5(----_).

the delay time is a smooth function of E. Here the resonance part of the S-matrix in eq. (22) ‘is negligible and T - 7bg = dGbg/dE. Near the resonance, however, T exhibits a rather narrow peak which shifts towards the free-rotor prediction as the coupling is decreased. This peak in the delay time manifests the appearance of a dynamical resonance. With 7res - 7 - 7bg we obtain roughly a delay time caused by the resonance of lo-” s.

We will introduce another possibility to locate resonances below the first threshold. As we will show in section 5 the diffractionally sudden (DS) approximation yields non-specular transition probabilities which are artifi- cially too large as compared to exact close-coupling calculations, even for a very small corrugation. This artificat is most drastic at very low energies. In fig. 9a we show the specular transition probability IS(O0; 00 + 00; OO)j2 calculated in the DS approximation versus energy. The corrugation parameter in eqs. (17) and (18) is p = 0.01. The pronounced minima occur at exactly the same

15

IO

5

0

-5

r

1 2

R. Schinke / Rotationally mediated selective aa!wrption

I I I I

2.3 I.9

' 1.8 ' I o--o

1.6 1.7

th(”

12

303

collision energy E ImeVl

Fig. 10. Delay time dg/dE for the elastic j = 0 -+ 0 transition below the j = 0 --) 1 threshold for a

flat surface. The free-rotor resonance energies E, (j,, n), eq. (21) with 8 = O’, are marked by the vertical lines.

energies where the delay time peaks. At resonance almost all outgoing flux is distributed among non-specular diffraction channels although many of them are closed at these energies. The DS approximation acts like an optical potential, Although the resulting transition probabilities are physically meaningless the DS approximation provides an easy way to detect resonances below the first rotational threshold.

Fig. 10 shows the delay time in the energy region 2 G E G 12 meV. The 0 + 1 threshold at E = 11.06 meV is marked. Five resonances are clearly observable. The free-rotor predictions in this range are also marked and labeled with the corresponding quantum numbersj,, n. The peaks at E - 5.7, 8.45 and 10.0 meV obviously belong to j, = 1, n = 6, 7 and 8, respectively. A separate calculation with only two channels, j = 0 and j, = 1, produces a peak at E - 10.5 meV and thus we can exclude that the peak at E = 10.0 meV correlates with thejc = 2, n = 4 resonance. Probably because of the appearance of three resonances which interfere with each other and the 0 -+ 1 threshold within a narrow energy interval it happens that the actual (1, 8) resonance occurs below the corresponding free-rotor prediction although the other mem-

304 R. Schinke / Rotationally mediated selective adsorption

bers of this series show the opposite behaviour. One clearly sees that the free-rotor approximation becomes gradually better as the vibrational quantum number increases. The lowest resonance at E - 2.65 meV very likely correlates with thej, = 2, n = 2 rigid-rotor prediction at E = 1.4 meV. A fifth resonance,

j, = 1, n = 9, occurs just at the opening of the 0 + 1 channel. Cowin et al. [5] resolved strong dips in the 0 -+ 0 transition probability at

angles of 8 = 80.4”, 78.3’, 75.4” and 73.5’. Using the resonance energies of fig. 10 we would obtain angles of 0 = 81.0’ (j, = 2, n = 2), 76.8” (1, 6) 73.8” (1, 7) and 72.4’ (1, 8) for the experimental collision energy of 109 meV. Although the

deviations between experimental and theoretical resonance angles are consider- able, 1.6’ at its maximum, we believe that our explanation and correlation with the free-rotor approximation is correct. The j, = 1, n = 9 resonance occurs just at the 0 -+ 1 threshold and is probably too narrow to be experimentally resolved. A concluding discussion of the comparison between theory and experiment is given in section 6.

5. Weakly corrugated surface

We will present diffractionally sudden (DS) and exact close-coupling (CC) results to investigate the influence of surface corrugation on rotationally mediated selective adsorption. Because of the numerical difficulties which are considerable in both cases, we must restrict ourselfs to a very weak corruga- tion. This is in accordance with our main purpose, i.e., model calculations for HD molecules scattered from the Pt(ll1) surface which seems to be only weakly corrugated [5]. We will present results for p = 0.01 and p = 0.02 as strength parameter in eqs. (18) and (19). Only specular transitions, p = q =p’

= q’ = 0, will be analyzed. In all cases the direction of the incident beam is (0 = O”, Qi = 0’) such that k = k,.

5.1. Diffractionally sudden calculations

For p = q = p’ = q’ = 0, eqs. (15) becomes

S( jmj; 00 + j'rn,; 00) = K’/dp S( jmi -+ j’m,lp), (27)

where the p-dependent S-matrix for the rotational transition is obtained from the numerical solution of the rotationally coupled equations in (12). The evaluation of the integral in eq. (27) does not cause numerical problems for collision energies well separated from a rotationally mediated resonance. Then the S-matrix as the interaction potential is a smooth function of p. However, the p-dependence of S becomes much more complicated in the vicinity of a resonance. Examples for ]S(OO; 00 -+ 20; OO]P)]~ are shown in fig. 11. The

R. Schrnke / Rotationally mediated selective aatsorprion 305

c-4 -

-2 X‘

04 , 1 I I ---__ --‘; - y=o

_- ___---- --. ‘\

\ -\ --- y= ok ,/’ /--

\ xI \ ‘:.,

_..._.. y = a/a

f

,’ // !’

0.3 1 /

\ ‘t !’

/

02-

I

1

J a

Fig. I I. (x, ))-dependence of the j = 0 -+ 2 transition probability at E = 43.4 meV for normal

incidence, 6’ = 0’. The corrugation parameter is p = 0.01.

collision energy is E = 43.4 meV, i.e. the energy at which the j, = 3, n = 3 resonance occurs for p = 0. Since the position of the resonance depends on p,

the energy dependence of JSl2 is reflected in this figure. The most convenient way to perform the averaging in eq. (27) is to expand the S-matrix into a Fourier series and then to calculate the integral analytically. The order of the expansion is usually small for off-resonance conditions but it must be signifi- cantly larger in the vicinity of a resonance.

Diffractionally sudden results for rotational transitions j = 0 + I and 2 are shown in figs. 12 and 13. The corrugation parameter is /3 = 0.01 and p = 0.02, respectively. The appearance of the resonance structure for /3 = 0.01 is qualita- tively similar to the flat surface results which for comparison are also shown in the figures. The main effect of the surface corrugation within the DS ap- proximation is a broadening of the peaks and dips which, of course, was surmised from eq. (27).

Drastic changes, however, are obseved in fig. 13 for p = 0.02, a value which

is still quite small. The average probabilities are smaller by roughly a factor of two. This decrease in probability is not due to a reshuffling among the various

rotational channels (the only additional channel is the elastic one) but due to the lost of probability to non-specular channels, i.e. p * p’ and 4 * 4’. Since the DS approximation neglects IG,,I’ in the channel wave number d&, eq. (5),

306 R. Schinke / Rotationally mediated selective adsorption

w 0.5 I \ 1 /

s o-2 * 0.4 -

-3 p = %I., 0.01 -

t 0 0.2 - 0

go.1 -

m - 0 I I I /

10 20 30 LO 50 60 1.0 ) I I , I I

0.9

0.6

‘v, 0.7

z d’ 0.6 -

0.2

0.1

0

---- flat surface - OS ***a CC (exact)

-I

O-l

10 20 30 40 50

collision energy E lmeV1

60

Fig. 12. j = 0 + 1 and 2 rotational transition probabilities within the specular beam, p = q = p’ =

q’ = 0, versus collision energy E. The incident angle is i3 = 0’. The surface corrugation is p = 0.01.

Comparison between diffractionally sudden ( -) and exact close coupling (. . . .) calcu-

lations and the flat surface results (- - -). The arrows mark structures discussed in the text.

even asymptotically closed diffractional channels (i.e., d/‘p4 < 0) have non-zero probabilities. This is of course a serious breakdown of the DS approximation. Although we do not show non-specular transition probabilities, we incidentally note that they exhibit the same resonance structures as the specular beams.

Besides the decrease of the overall transition probability one also observes qualitative changes of the resonance structures. They are significantly broader compared to the j3 = 0 case and exhibit a secondary peak/dip on each side of

R. Schinke / Rotationally mediated selective adsorption 307

Fig. 13.

. . . . CC(exact)

10 20 30 LO 50 60

collision energy E lmeV3 Same as for fig. 12 but with /? = 0.02.

the original resonance. This is most pronounced in the j = 0 + 2 probability for the& = 3 resonances. These additional structures are not an artifact of the numerical integration. This has been tested by increasing the order of the Fourier expansion of the p-dependent S-matrix. However, at this point we cannot exclude it to be an artifact of the DS approximation.

5.2. Exact calculations

In view of the results in fig. 12 we do not have confidence in the DS approximation even if the corrugation is as weak as in the present case. Thus,

308 R. Schinke / Rotationally mediated selective adsorption

we perform exact close-coupling calculations in the same energy range and for the same values of p. The rotational basis consists of all open channels and three closed channels as in the case of the flat surface. Each rotational state is extended with nine diffractional states withp, q = 0 and + 1 leading to a total number of 54 coupled equations at an energy of E = 60 meV. The restriction to a one-dimensional surface corrugation allows for an extension of the diffrac- tional basis to include higher GP,O ( p = 0, + 1, + 2, + 3) states. Such calcula- tions have been performed and compared to the results obtained with the p = 0, + 1 basis set. These test calculations assured that the specular diffraction probabilities, p’ = q’ = 0, shown in figs. 12 and 13 are converged to at least three figures. The exact calculations are very time consuming, such that the

number of considered energies must be limited. The step size is AE = 0.5 meV for E -C Eth( j = 2) and A E = 1 meV otherwise compared to 0.2 meV for the

sudden and the flat surface calculations. For energies below the j = 2 threshold the /3 = 0.01 and even the p = 0.02 results agree quite well with those obtained from the flat surface. This is in contrast to the DS calculations. Although the dominant features, the rotationally mediated resonances, remain almost un- changed as the surface corrugation is included, finer details can change, however. For example, the j, = 3, n = 0 resonance appears as a sharp peak at E - 15 meV for p = 0, but appears as a sharp dip at the same energy for /3 = 0.01 and fi = 0.02. Note that thej = 0, p, q = + 1 diffraction states open at the same energy. Another sharp dip in the 0 + 1 probability is seen in fig. 13 at E - 20 meV. It is only slightly indicated in fig. 12 for /3 = 0.01. An analogous structure is not observed for the flat surface and there is also no threshold present in this energy region. Thus, this structure might be due to the resonance between a closed rotational-diffractional channel and a bound state of the system. For example, zeroth order perturbation theory (d,tP,g, = 2Mc,,) predicts a j, = 1, p, = + 1, q, = 0 (or p, = 0, q, = + l), n = 6 resonance at 19.1

meV and a j; = 2, p, = qC = + 1, n = 1 resonance at 20.4 meV. The influence of surface corrugation is more prominent at energies above

the j = 0 + 2 threshold, where - on the average - the probabilities of the p’ = q’ = 0 beams are clearly smaller than those for the flat surface. Comparing the CC and the DS results we have to conclude that the DS approximation spuriously overestimates the transfer of flux from the specular to the non-spec- ular channels. The scattering in the CC data, especially for j = 0 + 1, indicates that additional structures other than rotationally mediated resonances might be hidden in the spectra. In principle, they could be due to the opening of new channels or to combined rotationally-diffractionally mediated resonances.

Fig. 14 shows an enlargement of the energy range around the j, = 3, n = 2 and n = 3 resonances with a step size of A E = 0.25 meV. One clearly notices an additional peak (dip) at each side of the j, = 3, n = 3 resonance forj = 0 + l(2). These structures are only slightly indicated for p = 0.01 but prominent for /3 = 0.02. A plausible assignment with thresholds or combined rotationally-dif-

R. Schinke / Rotationally mediated selective a&orption

.

o-1

32 34 36 38 LO 42 44 16

collision energy E ImeVl

Fig. 14. Exact close-coupling j = 0 + 1 and 0 -P 2 rotational transition probabilities within the

specular beam, p = q = p’= q’= 0, versus collision energy. The corrugation parameter is: p = 0

( -), /3 = 0.01 (00 0) and B = 0.02 (. . . .). The free-rotor resonance energies ER( j,, n),

eq. (21) with B = O’, are marked by the vertical lines. The arrows in the upper panel mark the

positions of the secondary minima of the DS probabilities in fig. 13.

fractionally mediated resonances is not possible. However, they are astonish-

ingly similar to those obtained within the DS approximation. Moreover, the positions of the secondary dips in thej = 0 + 2 DS probability from fig. 13 are marked in fig. 14 and even quantitative agreement (within 0.3 meV) is achieved. Since the DS approximation qualitatively reproduces these additional structures, it should be possible to find an explanation by analyzing the DS expression for the S-matrix.

310 R. Schinke / Rotarionally mediated selective arisorpiion

5.3. interference effects

We will discuss an approximate evaluation of the diffractionally sudden integral in eq. (27) which leads to a simple explanation of the structures observed in figs. 13 and 14. We will concentrate on the j = 0 --, 2 transition, especially the jC = 3, n = 3 resonance region at E - 43.4 meV. In order to simplify the notation we will omit all indices. In fig. 5 we have demonstrated that the 0 -+ 2 S-matrix in the case of a flat surface can be well represented by the Breit-Wigner expression given in eq. (22) with eq. (23) as the background contribution. We will base our simple approximation on this representation.

Inserting eq. (22) into eq. (27) gives

+~(x,Y) CE-E,(~,Y)+~~(X,Y)/~]-'} (28) for the full S-matrix where all the quantitaties, i.e., Sbs, 6 = - iy,v,,, E, and r, depend on the vector p = (x, y). The background contribution is assumed to be a smooth function of (x, y) yielding an average Sbg( E) which determines the overall magnitude of the specular and non-specular peaks for the considered rotational transition in the energy range between two resonances. In view of fig. 13, the (x, ~)-dependence of Sbs must be quite large to explain the significant decrease of the specular transition probability. In the following we will only consider the resonance part, i.e. the second term in eq. (28). Thus, interferences between resonance and background contribution are neglected. We further assume that the main contribution to the (x, y)-dependence of SreS( E) is due to the (x, y)-dependence of the resonance energy E,. Although it is generally weak, its consequence for S’“” can be large because En appears in the denominator of eq. (28). We find that E, is well approximated by

ER(X,Y) = Ep 1 + p co+ + co?+ i i ii

, (29)

with Eke’ the flat surface result, E, (‘I= 43.4 meV, and B - 0.014 for a corruga-

tion parameter /3 = 0.02. Thus, we obtain for the resonance part of the

S-matrix

S=*(E) = SA-tiQdxibdy

X{ E-EkO’[l+B(cos~x+cos~y)]+i~,2}-~.

Next, we expand the integrand in powers of p up to second order and perform the integration analytically. We find the interesting result that the contribution of the term linear in fi is exactly zero. Thus we obtain

S”‘(E) = S(r eib + 6x3 e3’*), (31)

R. Schinke / R~tat~onaliy mediated seiectiw adforptian 311

with the abbreviations

r= [(E-Ep)2+(r/2)2]-“2, (32)

c/J = tan-‘[(r/2)/( E- Ek”))], (33)

w = j’E$_“=/2. (34)

After some simple algebra we finaliy get

IS’““( E)(Z = [S/2 [ ( r* + id2r6 + 2w4 cos 2+)]. (35)

The first term in eq. (31) is the flat surface result and the second term represents the influence of the corrugation. Both contributions interfere with each other and thus might lead to the structures discussed in figs. 13 and 14. The first two terms in eq. (35), r2 and w2r6, are positive and peak at E = EL’). However, the third term can be positive and negative because of the factor cos 24, which has the following behaviour:

i

1 for JE - E,$“I large,

cos 24 = 0 for E=Ei”+r/2,

- 1 for E=Ei”.

Eq. (35) predicts: (a) A s~rne~c behaviour of fSresj2 with respect to EioP’. (b) Since (p is independent of ,!? the energy positions of the secondary structures should be independent on the corrugation parameter p. (c) Since this is an effect of second order in fi, small changes of p should lead to a strong amplification of the interference structures. All of these predictions are clearly confirmed by the exact results in fig. 14.

6. Conchsions and prognosis

We presented a detailed investigation of a new effect, rotationally mediated selective adsorption, which has recently been observed experimentally by Cowin et al. [5] resolving state-to-state rotational excitation probabilities in HD/Pt( 111) scattering. It is caused by resonance between a closed rotational channel and a bound state of the interaction potential and is analogous to the effect of selective adsorption in atom/surface scattering. As in the latter case, the resolution of rotationally mediated resonances leads to the dete~nation of the bound state energies of the system and therefore to an accurate estimation of the attractive part of the interaction potential. If the coupling between the rotational and the translational degree of freedom is large (this is definitely the case for HD scattering), the free-rotor approximation can lead to significant deviations from the actual resonances which - in the present model

312 R. Schinke / Rotationally mediated selectiue adsorption

Table 2

Comparison of theoretical and experimental resonance angles 8, (in degrees) a)

Transition Theory Experiment Difference

2, 2 o+o 81.0 b,

1, 6 76.8 b’

1, 7 73.8 b,

1, 8 72.4 b’

1, 9 71.7 b’

2, 4 o-t1 65.8 d,

2, 5 62.5 *)

2, 6 59.5 *)

2, 7 57.8 e)

3, 2 O-2 55.3 e,

3, 3 50.8 e,

3, 4 41.2 e,

3, 5 43.8 e)

3, 6 41.5 e)

80.4 ‘) -0.6

78.3 1.5

15.4 1.6

13.5 1.1

66.5 0.7

62.5 0.0

60.0 0.5

51.1 -0.1 _

50.1

46.1

43.1 _

-0.7

- 1.1

-0.7

a) E = 109 meV, flat surface.

b, Obtained from the analysis of the delay time.

‘) Taken from fig. 2 of Cowin et al. 151.

d, Taken from fig. 8 with A E = 12 meV.

e, Taken from fig. 8 with AE = 6 meV.

study - are as large as - 1 meV. Thus, it seems necessary to determine the potential by fitting exact close-coupling calculations to experimental results rather than using the free-rotor predictions. Such a procedure should also include variations of the anisotropical part of the interaction potential and an averaging over the collision energy. Taking into account the spread of the initial beam energy, we find at least qualitative agreement with the experiment: Only the most prominent resonances survive the energy averaging while finer details are completely smeared out. Another possibility to fit the resonance positions would be the use of first order perturbation theory 1131. However, such a procedure is only applicable for a weakly anisotropical potential

surface. Including in our analysis the resonances in the elastic j = 0 + 0 probability

below the j = 0 + 1 threshold by analyzing the resonance delay time, we are able to assign all the experimentally observed resonances with a set of rotational-vibrational quantum numbers. The theoretical and experimental resonance angles for the collision energy of E = 109 meV are compared in table 2. The largest deviation is for the j’ = 1, n = 7 resonance. The agreement could be improved by varying the potential parameters or the form of the interaction potential. It is well known that the Morse potential is not adequate to describe the long range attraction which usually goes as zP3. This would be most important for the bound state energies close to dissociation. A quantita-

R. Schrnke / Rotorionally mediated selectice adwrptron 313

tive comparison is not attempted in the present model study but will be attacked in the future. The purpose of this article is the investigation of the general structures of rotationally mediated selective adsorption. If the assign- ment in table 2 is correct, then n = 2 in the lowest vibrational state seen in the experiment. This in turn raises the question about the uniqueness of the potential derived from the resonances. One could of course speculate whether a

significantly flatter potential would also fit the observed resonances. These

questions must also be considered in future work. Most of the calculations presented in this article are for an uncorrugated

surface. Although the corrugation of the HD/Pt( 111) potential is weak accord- ing to the smallness of the non-specular beams (51, a flat surface is yet quite unrealistic. The influence of weak surface corrugation on rotationally mediated selective adsorption resonances is investigated in exact close-coupling calcula- tions. While finer details such as the shape and the width of the resonances can be affected, the resonance positions remain unchanged at least in the range of corrugation probed here. An interesting interference effect is found which leads to secondary minima/maxima on each side of the original resonance.

Although the diffractionally sudden approximation is generally poor, it allows a simple explanation when the Breit-Wigner representation is used for the S-matrix. It is questionable, of course, whether such fine structures can be resolved experimentally provided they exist for realistic systems. If so, how-

ever, they might lead to an estimation of the surface corrugation without measuring non-specular beams. A somewhat related effect is known in

atom/surface scattering, i.e., the splitting of two degenerate resonances [41). We are quite confident that a variety of interesting new structures might be found for cases with both strong rotational coupling and significant surface corrugation. Although such calculations are very difficult, they should be attacked in future studies.

In this article we (qualitatively) compared only the resonance spectrum as obtained from experiment and theory. Much more information on the interac- tion potential, especially the anisotropy of the interaction potential, will be obtained if also the magnitudes of the various transition probabilities are considered over a wide range of collision energies. However, a comparison of theory and experiment would require either a careful extrapolation of the experimental results to zero surface temperature or an appropriate Debye-Waller correction of the calculations. Efforts in both directions are necessary and important for the study of the molecule/surface dynamics.

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