Classical Wigner theory of gas surface scatteringdigital.csic.es/bitstream/10261/74060/1/Pollak.pdf · 2016-02-17 · Classical Wigner theory of gas surface scattering Eli Pollak,1,a

Classical Wigner theory of gas surface scatteringEli Pollak, Santanu Sengupta, and Salvador Miret-Artés Citation: J. Chem. Phys. 129, 054107 (2008); doi: 10.1063/1.2954020 View online: http://dx.doi.org/10.1063/1.2954020 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v129/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Classical Wigner theory of gas surface scatteringEli Pollak,1,a� Santanu Sengupta,1 and Salvador Miret-Artés2

1Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel2Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123,28006 Madrid, Spain

�Received 7 May 2008; accepted 12 June 2008; published online 5 August 2008�

The scattering of atoms from surfaces is studied within the classical Wigner formalism. A newanalytical expression is derived for the angular distribution and its surface temperature dependence.The expression is valid in the limit of weak coupling between the vertical motion with respect to thesurface and the horizontal motion of the atom along the periodic surface. The surface temperaturedependence is obtained in the limit of weak coupling between the horizontal atomic motion and thesurface phonons. The resulting expression, which takes into account the surface corrugation, leadsto an almost symmetric double peaked angular distribution, with peaks at the rainbow angles. Theanalytic expression agrees with model numerical computations. It provides a good qualitativedescription for the experimentally measured angular distribution of Ne and Ar scattered from a Cusurface. © 2008 American Institute of Physics. �DOI: 10.1063/1.2954020�

I. INTRODUCTION

The scattering of atoms by metal surfaces has been em-ployed as a probe of surface properties such as diffusion,vibrational relaxation, and structure.1 Rare gas atoms such asHe, Ne, and Ar typically do not penetrate into the bulk andso the information obtained from the scattering process re-lates mainly to the surface layer. Since these atoms are rela-tively inert, they act as good probes for a large variety ofatoms and molecules adsorbed on the surface. Experimentalstudies of the scattering of He, Ne, and Ar from copper vici-nal surfaces have been reported in a series of papers.2–6 Fur-ther studies of Ne and Ar atoms scattered from a variety ofsurfaces have been reported in Refs. 7–13.

The initial theoretical studies of helium atom scatteringby metal surfaces predated the experiments.14 Due to thelight mass of the He atom, it necessitates a quantum me-chanical treatment. A general coupled channel method of cal-culating diffraction intensities for a given geometry was de-veloped by Wolken.15 More approximate methods have beenused to invert measured angular distributions into force fieldsdescribing the interaction of the elastically16 andinelastically17 scattered atoms from copper vicinal surfaces.

Miller and co-workers,18–21 Doll,22 and Guantes et al.23

utilized semiclassical S-matrix theory to study He atom scat-tering from surfaces. In contrast to a purely classical dynam-ics approach, the semiclassical theory could account for theimportant features of the diffraction pattern. Two recent re-views of classical, semiclassical, and quantum methods maybe found in Refs. 24 and 25. For noble gas atom scatteringthe corrugated Morse potential �CMP� proposed by Armandand Manson4 has been found26 to be in satisfactory agree-ment with experimental results. Further work by Gorse et al.5

has led to the introduction of a modified corrugated Morsepotential �MCMP� for the low corrugated �110� and �113�faces.

Due to their heavier masses, the scattering of Ne andespecially Ar is readily described in terms of classical mod-els. Brako27 developed a classical model in which the verti-cal motion �z direction� is coupled linearly to the phononbath of the surface. He derived an analytical expression forthe angular distribution, as well as for the energy and mo-mentum transfer. However, he did not consider the effect ofcorrugation on the scattering. As a result he also ignored thecoupling between the phonon bath and the horizontal motionof the atom along the surface. The classical rainbow effectwhich is caused by surface corrugation was then studied byHorn et al.28 The effect of surface corrugation was furtherstudied through the introduction of the so called “washboardmodel” of Tully.29 However, as in the paper of Horn et al.28

the classical collision dynamics was simplified by assumingan impulsive collision model. A further refinement of thewashboard model with applications to the scattering of Neand Ar atoms may be found in Ref. 30. Most recently, Man-son and co-workers31,32 have employed the theory of Brako27

to analyze the experimental results for the scattering of Aratoms from Ru�0001�. Detailed molecular dynamics investi-gation of Ar atom scattering on Ni�001� may be found inRef. 33 and Ar on Pt�111� in Ref. 34.

The purpose of the present study is to derive and apply aclassical theory for the angular distribution that takes intoaccount surface corrugation and surface temperature anddoes not use an impulsive approximation for the dynamics.The formalism we will use is known as the classical Wignerapproximation,35–38 in which the initial conditions of thephonon bath and the incident wavepacket are treated quan-tum mechanically while the ensuing dynamics is obtainedfrom the classical equations of motion. The same approachhas been called “mixed quantum classical theory”39,40 or thea�Electronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 129, 054107 �2008�

0021-9606/2008/129�5�/054107/13/$23.00 © 2008 American Institute of Physics129, 054107-1

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“linearization approximation”41 since it is a linearized limitof a semiclassical initial value representation treatment of thesame problem.42

The central result of this paper is that the angular distri-bution is given by the expression

P��� =1

��

−�

�

dZ� 1

2��2

�exp�−Z2

2�2�H�K2 − �� + �i0 + ��1 + Z�2��K2 − �� + �i0 + �� + Z�2

,

�1.1�

where H�x� is the Heaviside function and �i0 is the incidentscattering angle. The distribution is determined by threeparameters—a shift parameter ��, the rainbow angle K �rela-tive to the specular scattering angle −�i0�, and the thermalwidth as given in terms of the variance �2 that �in the hightemperature classical limit� is linearly proportional to the sur-face temperature. This expression is valid provided that thefollowing conditions hold: �a� the incident wavepacket is lo-calized in momentum space, �b� the scattering is classical innature, �c� the coupling between the vertical and the horizon-tal motion of the scattered particle is weak, and �d� the fric-tional force exerted on the particle by the surface is weak.

In Sec. II we provide the framework and the workingmodel underlying the theory. We then provide in Sec. III theexpression for the angular distribution first in the absenceand then in the presence of interaction with the phonon bath.The detailed derivations are given in Appendixes A thru C.The analytic theory is then applied in Sec. IV to a model inwhich the potential in the vertical direction is taken to be theMorse potential. The results are compared with numericallyexact simulations. We find excellent agreement, further jus-tifying the analytic theory. We then apply the theory to thescattering of Ne and Ar from a Cu surface. Good qualitativeagreement is found for both systems.

We end with a discussion of further extensions of thetheory, as well as the possibility of solving the same problembut within a semiclassical initial value representation treat-ment of the scattering dynamics. Such a treatment wouldalso be valid for the scattering of He where due to the smallmass, quantum interference effects cannot be ignored.

II. THEORETICAL FRAMEWORK

A. The thermal angular distribution

We consider the scattering of a particle with mass Mfrom a surface. In the vertical z direction the particle inter-acts with the surface via a potential V�z�, which would typi-cally have the form of a Morse potential. The potential alongthe x direction is periodic. For simplicity we ignore the ycoordinate. When the particle is sufficiently distant from thesurface, it is a free particle. When it comes closer to thesurface, there is an interaction between the vertical and hori-zontal motion. We also assume that the interaction with thephonon bath comes from coupling between the horizontal

motion and the phonons of the surface; the vertical motion isnot directly coupled to the bath. We then use the followingmodel Hamiltonian:

H =px

2 + pz2

2M+ V�z� + V1f�z�cos�2�x

l�

+1

2j=1

N pj2 + � j

2�xj −cj

�M

� j2 xg�z��2� , �2.1�

where l is the periodic lattice length and pj ,xj , j=1, . . . ,Nare the mass weighted momentum and position operators ofthe bath degrees of freedom, which are linearly coupled tothe system motion. The function f�z� gives the coupling dueto the corrugation and g�z� couples the harmonic bath to thehorizontal motion when the particle is close to the surface.

The bath Hamiltonian �in mass weighted coordinates andmomenta� is defined to be

HB =1

2j=1

N

�pj2 + � j

2xj2� . �2.2�

Initially, the atom is sufficiently distant from the surface sothat it does not interact with it. It is then described by aGaussian wave function �� . The exact quantum mechanicalexpression for the angular distribution is written in terms ofa correlation function with factorized initial conditions as

P��� = limt→�

P��,t�

= limt→�

Tr� e−HB

ZB�� ���K†�t��� − ��K�t�� . �2.3�

Here, K�t� is the exact quantum mechanical propagator

�K�t�=exp�−iHt /���, �x� is the Dirac “delta” function, andthe angular operator is defined as the angle with respect tothe normal to the surface,

� = tan−1� px

pz� . �2.4�

B. The classical Wigner approximation

The Wigner representation of the thermal harmonic bathis

�B,W�p,x� � � e−HB

ZB�

W

= �j=1

N � j/����exp�− j

�� j�pj

2 + � j2xj

2��� , �2.5�

with

j � tanh��� j

2� , �2.6�

and the distribution is normalized.The initial wave function is chosen to have the Gaussian

form

054107-2 Pollak, Sengupta, and Miret-Artés J. Chem. Phys. 129, 054107 �2008�

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�x,z�� = � �

��1/2

exp�−�

2��x − x0�2 + �z − z0�2�

+ı

�px0

�x − x0� +ı

�pz0

�z − z0�� . �2.7�

The momenta px0and pz0

define the incident scattering angle

�i0 = tan−1� px0

pz0

� �2.8�

and the radial initial momentum

p02 = px0

2 + pz0

2 . �2.9�

The Wigner representation of �� ��� is

�S�px,pz,x,z� = � 1

���2

exp�− ���x − x0�2 + �z − z0�2�

−�px − px0

�2 + �pz − pz0�2

�2�� , �2.10�

and in this representation the angular distribution thenbecomes

P��� = limt→�

�−�

�

�j=1

Ndpjdxj

2���

−�

�

dpxdpzdxdz�B,W�p,x�

��S�px,pz,x,z��� − tan−1� px�t�pz�t�

�� , �2.11�

where the notation pz�t� , px�t� stands for the time evolution.In the classical Wigner approximation which will be consid-ered henceforth in this paper, this time evolution is given bythe classical equations of motion.

C. The classical dynamics

Using the shorthand notation V�x ,z�=V�x�+V1f�z�cos�2�x / l�, defining a spectral density

J��� =�

2 j=1

Ncj

2

� j�� − � j� �2.12�

and an associated friction function

��t� =2

��

0

�

d�J���

�cos��t� , �2.13�

one finds that the equations of motion for the system vari-ables are the generalized Langevin equations �GLEs�

Mxt +�V�xt,zt�

�xt+ Mg�zt��

−t0

t

dt���t − t��d�xt�g�zt���

dt��

= �MF�t�g�zt� , �2.14�

Mzt +�V�xt,zt�

�zt+ Mg��zt�xt�

−t0

t

dt���t − t��

�d�xt�g�zt���

dt�� = �MF�t�g��zt�xt, �2.15�

where the trajectory is initiated at the time −t0. The initialwavepacket is chosen such that z0 is sufficiently far awayfrom the surface. For the chosen value of the width �, we cansafely assume that for all initial conditions the phonon bathis not coupled to the atomic motion, that is, g�z�=0. TheGaussian noise function is therefore dependent only on theinitial conditions of the phonon bath,

F�t� = j=1

N

cj�xj cos�� j�t + t0�� +pj

� jsin�� j�t + t0��� .

�2.16�

It has zero mean and its correlation function is proportionalto the time dependent friction function.

III. CLASSICAL WIGNER THEORY FOR THEANGULAR DISTRIBUTION

A. Scattering in the absence of dissipation

In this subsection we will present an analytical expres-sion for the angular distribution in the absence of interactionwith the bath and in the limit of weak corrugation—the po-tential parameter V1 that couples the vertical to the horizontalmotion will be considered to be small. The dynamics issolved to leading order in V1. In this way we obtain an ex-pression for the momentum added �or subtracted� to the hori-zontal motion as a result of this coupling. Through energyconservation this also leads to a shift in the vertical momen-tum. We then use these results to derive an expression for theangular distribution, in the limit that the initial wavepacket isvery narrow in the momentum space or, more specifically, inthe limit that the width parameter in the incident wavepacket�→0.

As shown in Appendix A, using a perturbation theory forthe dynamics, in which the small parameter is the corruga-tion strength V1, which is assumed to be much smaller thanthe well depth V0, one finds that for a given trajectory, theshift in the scattering angle induced by the corrugation isgiven by the expression

�i =px�x,px,pz�

pz� K�pz,px�sin�2�

lx +

pxt0

M�� . �3.1�

As shown below, the rainbow angles are found to be −�i�K,where K�pz , px� is given in terms of the potential parametersand the known unperturbed trajectory in the vertical direc-tion, which depends on the initial momentum in the verticaldirection,

K�pz,px� =4�V1

pzl�

0

�

dt� cos�2�px

lMt�� f�z�t��� . �3.2�

To obtain the angular distribution we must integrate overthe initial coordinates and momenta. The details of the inte-gration are presented in Appendix B. The only additional

054107-3 Wigner theory of gas surface scattering J. Chem. Phys. 129, 054107 �2008�

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approximation introduced is that the width parameter � ofthe initial wavepacket is assumed to be small as compared tothe lattice length l, such that �l2�1. One then finds that

P��� =1

�

H�K2 − �� + �i0�2��K2�p0,�i0� − �� + �i0�2

, �3.3�

where H�x� is the Heaviside function, H�x�=1 if x�0 and is0 otherwise. The scattered angular distribution is a functionthat has maxima at the rainbow angles �+�i0= �K�p0 ,�i0�and a minimum at the specular angle for which

P�− �i0� =1

��K�p0,�i0��. �3.4�

Outside of the range −�K � ��+�i0� �K� the angular distribu-tion vanishes.

B. Scattering with weak dissipation

Typically, the friction exerted on the incident atom willhave a phonon component that can be characterized with twoparameters—a friction strength parameter ��� and a cutofffrequency �c that reflects the Debye frequency of the crystal.The spectral density would be thus typically Ohmic with anexponential cutoff frequency. This gives rise to memory inthe time dependent friction. However, one may expect thatthe memory effect on the dynamics will be negligible sincethe cutoff frequency is typically much larger than the atomsurface frequencies. We therefore simplify the considerationof the dynamics and specify it for purely Ohmic friction suchthat

��t� = 2��t� . �3.5�

As in the dissipationless case, we proceed in Appendix Cto use perturbation theory to derive an expression for thechange in final scattering angle for a trajectory with specifiedinitial conditions for the system variables and a given real-ization of the noise. In addition to considering the corruga-tion parameter V1 as small relative to V0, here we also as-sume that the friction is weak, allowing for a first orderperturbation theory in these parameters.

The straightforward application of perturbation theorythen leads to the following result for the final momentum inthe horizontal �x� direction:

�px = pzK�px,pz�sin�2�

lx +

px

Mt0�� + �px,1 + �px,2.

�3.6�

Here, the rainbow angle function K is defined as in the dis-sipationless case �see Eq. �3.2��. In addition one finds a fric-tion induced change in the momentum

�px,1 = − �px�0

�

dtg2�zt� �3.7�

and a noise induced change

�px,2 = 2�Mj=1

N

cjhj�xj cos�� jt0� +pj

� jsin�� jt0�� , �3.8�

where hj is the cosine Fourier transform of the vertical mo-tion dependent coupling function

hj � �0

�

dtg�zt�cos�� jt� . �3.9�

As in the dissipationless case, the shift in the verticalmomentum is determined by the energy balance. However,here one has to take also into account the energy gained bythe bath during the collision,

�EB = ��EB + EB. �3.10�

The energy loss to the bath has two parts—the average en-ergy loss to the bath ��EB and the fluctuational energy lossEB. Explicit expressions for the energy losses in terms ofthe system and bath dynamics are given in Appendix C. Theresulting change in the vertical momentum after the scatter-ing event is then found to be

�pz =px

pz�px +

M�EB

pz. �3.11�

Expanding the final scattering angle to first order with re-spect to the changes in the horizontal and vertical momenta,one finds that the change in the scattering angle is

�i � K�px,pz�sin�2�

lx +

px

Mt0�� + ��1 + ��2, �3.12�

where the angular shift is

��1 =�px,1

pzcos2 �i �3.13�

and the angular fluctuation is

��2 =�px,2

pzcos2 �i. �3.14�

To obtain the angular distribution it remains to integrateover the system phase space variables and to average overthe bath. The detailed derivation is provided in Appendix C;here we provide only the essential results. One finds that theangular distribution has a form that is similar to the angulardistribution in the absence of dissipation, except that now itis broadened by a Gaussian distribution that reflects the in-teraction of the particle with the phonon bath of the surface.The expression for the angular distribution is

P��� =1

��

−�

�

dZH�K2 − �� + �i0 + ��1 + Z�2�

�K2�p0,�i0� − �� + �i0 + ��1 + Z�2

�� 1

2��2 exp�−Z2

2�2� , �3.15�

where the temperature and friction dependent variance of thedistribution is expressed in the continuum limit as

054107-4 Pollak, Sengupta, and Miret-Artés J. Chem. Phys. 129, 054107 �2008�

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�2 =4�M

�p02 �

0

�

d�J�����

0

�

dtg�zt�cos��t��2

tanh��

2

. �3.16�

Equations �3.15� and �3.16� are the central results of thispaper. The coupling to the thermal bath induces a shift in thepeaks through the term ��1 �cf. Eq. �3.13�� and a broadeningfrom the Gaussian averaging. The variance of the distribu-tion is readily obtained in the continuum limit through thespectral density. Note that the variance is quantum mechani-cal; it does not vanish in the limit of zero temperature butincludes the zero point fluctuations of the bath. This reflectsthe classical Wigner approximation, in which the initial ther-mal conditions are treated quantum mechanically and it isonly the dynamics that is treated classically.

IV. APPLICATIONS

A. Analytic results for a Morse oscillator potential inthe absence of dissipation

In the following we will use the known analytic dynam-ics of the Morse oscillator to obtain an explicit expressionfor the rainbow angle function K�p0 ,�i0�. The model poten-tial we use is

V�x,z� = V0�1 − exp�− �z��2 − V0 + V1

�exp�− 2�z�cos�2�x

l� , �4.1�

where the function coupling the vertical motion to the hori-zontal motion has been specified to be f�z�=exp�−2�z�. Theinitial vertical energy of the particle when it is far from thesurface is denoted as Ez and the initial vertical momentum ispz. As already noted above, the origin of time was chosen asthe point at which the trajectory reaches the turning point�ztp� so that at time t=0 the vertical velocity vanishes. There-fore

Ez = V�ztp� =pz

2

2M. �4.2�

In the absence of coupling, the equation of motion for the zcoordinate is

Mz = − 2�V0�1 − exp�− �z��exp�− �z� . �4.3�

One can then find by inspection that the time dependence ofthe classical trajectory is given by the relation

exp��zt� =V0

Ez���1 +

Ez

V0�cosh��t� − 1� , �4.4�

where the energy dependent frequency is given in terms ofthe energy and potential parameters as

�2 =2�2Ez

M. �4.5�

The resulting expression for the rainbow angle function is

K�p0,�i0� =− 4�V1Ez

2

p0l cos �i0�Ez + V0�V0�

0

�

dt�

�

cos�2�px

lMt��

�cosh��t�� −� V0

�Ez + V0��2 . �4.6�

In the limit of high incident energy as compared to the welldepth of the Morse potential �Ez�V0�, one obtains the result,

K�px,pz� � −2�V1

�lV0�

0

�

dt�

cos�2� tan��i0�l�

t��cosh2�t��

=2�V1

�lV0

�2�tan �i0�

l� sinh��2�tan��i0��l�

� . �4.7�

B. Analytic results for a Morse oscillator potential inthe presence of dissipation

The horizontal motion will typically be coupled to thesurface phonon bath when the approaching atom is close tothe surface. For the Morse potential model used above, it isthus reasonable to choose the coupling function as a Gauss-ian centered at the minimum of the Morse potential

g�z� = exp�− z2/�2� . �4.8�

As can be inferred from Eq. �4.4� the time t� at which theatom goes through the minimum �z=0� is

TABLE I. Parameters for helium scattering.

�i0

�deg�E0

�meV�Mass�a.u.�

Beam parameters −32.5 63.0 7296.12

Cu�110� potential V0 �meV� V1 /V0 � �Å−1� l �Å�Surface parameters 6.35 0.03, 0.0885 1.05 3.6Bath parameters �c �a.u.� � �a.u.� � �Å�

0.001 061 17 0.0002 �c 2.645Wavepacket parameters � �a.u.� x0 �a.u.� z0 �a.u.� px0 �a.u.� pz0 �a.u.�

0.020 933 9 −25.4828 40.0 3.1249 −4.9051

054107-5 Wigner theory of gas surface scattering J. Chem. Phys. 129, 054107 �2008�

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cosh��t�� =�1 +Ez

V0. �4.9�

At this time,

��dz

dt�2�

t=t�=

�2

�2 �1 +V0

Ez� , �4.10�

so that we may approximate

exp�− z2/�2� � exp�−�2

�2�2�1 +V0

Ez��t − t��2� . �4.11�

It then follows that

hj = �0

�

dtg�zt�cos�� jt�

� �−�

�

dt cos�� j�t + t���exp�−�2

�2�2�1 +V0

Ez�t2�

=���� cos�� jt

��

��1 +V0

Ez

exp�−�2�2� j

2

4�2�1 +V0

Ez�� . �4.12�

and

�px,1 = − �px�0

�

dtg2�zt� � −�px����

��2�1 +V0

Ez� . �4.13�

We thus have that

��1 =�px,1

pzcos2��i0� = −

����� sin��i0�cos��i0�

��2�1 +V0

Ez� .

�4.14�

Using the definition of the spectral density, specifying toOhmic friction with an exponential cutoff

�J��� = �� exp�− �/�c��

and taking the classical limit for the thermal bath distribution��→0�, we then have

�2 =4M cos4��i0�

pz2

j

cj2hj

2

� j2

=2M��2 cos4��i0�

Ez�Ez + V0� �0

�

d� cos2��t��exp�−�2�2�2

2�2�1 +V0

Ez�

−�

�c�

=1

Ez

��M�� cos4��i0��Ez + V0

exp�Ez + V0

M�c2�2 ��1 − erf��Ez + V0

M�c2�2�� +

��M�� cos4��i0�Ez

�Ez + V0

�Reexp�Ez + V0

M�c2�2 �1 − 2i�ct

��2��1 − erf��1 − 2i�ct���Ez + V0

M�c2�2��� . �4.15�

For Ohmic friction the result is simpler,

�2 =1

Ez

��M�� cos4��i0��Ez + V0

��1 + exp�−

2t�2�2�1 +

V0

Ez�

�2�2 �� . �4.16�

C. Numerical verification

In this subsection we will compare the analytic expres-sion for the angular distribution with numerical simulations,in which the dissipative bath is represented in terms of afinite number of oscillators, which closely mimic the con-tinuum limit. The resulting friction kernel in the simulation isthus not strictly Ohmic; however, as already noted, the

memory time is sufficiently short, so it does not really makea big difference. The parameters used for the comparisonwith simulation roughly describe the scattering of He fromthe Cu�110� surface. The relevant scattering, potential, initialwavepacket, and frictional parameters are provided in TableI.

We first present in Fig. 1 results obtained for scatteringwithout dissipation and two values of the corrugationstrength parameter V1=0.03V0 ,0.0885V0. The initial distanceof the wave packet center from the surface is sufficientlylarge to ensure that the wavepacket is in the asymptotic re-gion where the interaction with the surface is negligiblewhile the impact parameter is taken to be zero. The analyticresults are obtained by averaging the expression for the an-gular distribution given in Eq. �B5� over the Gaussian distri-bution of the initial wavepacket. As can be seen from the

054107-6 Pollak, Sengupta, and Miret-Artés J. Chem. Phys. 129, 054107 �2008�

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figure, for the weaker corrugation, the agreement is nearlyperfect; it worsens slightly as the corrugation strength in-creases. We note especially that for the stronger corrugation,the angular distribution is not perfectly symmetric about thespecular angle. This asymmetry, already noted by Tully,29

which appears both in the simulation and in the analytictheory is a result of the initial Gaussian distribution, which ischosen to be Gaussian in the Cartesian momenta and so isnot symmetrically distributed in the angular space. Theasymmetry grows as the corrugation strength increases, in-creasing the magnitude of the difference between the rain-bow angle and the specular angle.

The numerical results with dissipation were obtained byassuming an Ohmic spectral density with a cutoff frequency.For the parameters used, it was sufficient to use only 20 bathmodes. The discretization was carried out as in Ref. 44 byassuming that there are Nb+1 bath oscillators and that themaximal bath frequency is infinity for the Nb+1th mode. TheNb bath oscillator frequencies and coupling coefficients arethen determined from the relations

� j = − �c ln�1 −j

Nb + 1� , �4.17�

cj =� 2� j2��c

��Nb + 1�. �4.18�

We first plot in Fig. 2 the temperature dependence of thevariance �cf. Eqs. �C26�, �C27�, and �4.15�� for V1 /V0

=0.03 at three different levels of treatment. The dashed linedenotes the values obtained for Ohmic friction with the ex-ponential cutoff �the friction strength and cutoff frequencyare as given in Table I�. The solid line represents the valuesfrom the classical limit ��→0� approximation �cf. Eq.�4.15��. The points denote the values evaluated by using thesimpler expression �Eq. �4.16�� for purely Ohmic friction.The quantum variance has a finite nonzero value at zero tem-perature while the classical approximation vanishes at T=0 K. However, this difference is noticeable only at verylow temperature. It is also evident that the differences be-tween the Ohmic and cutoff spectral densities are not verylarge.

In Figs. 3 and 4 we present results for the angular dis-tribution as a function of temperature. Figure 3 shows theresults for the weak corrugation case. The analytic resultshere include averaging over the initial wavepacket. The vari-

FIG. 1. Comparison between the angular distributions for classical heliumatom scattering obtained numerically �solid line� and using the analyticalexpression �dashed line� in the absence of dissipation for two different cor-rugation amplitudes. The inset gives a schematic profile of the scattering. FIG. 2. Temperature dependence of the variance of the thermal bath ��2� for

the parameters of He atom scattering as given in Table I in units of deg2.

FIG. 3. Comparison between the angular distributionsfor classical He atom scattering obtained numerically�solid lines� and using the analytical expression �dashedlines� at different temperatures for very weak corruga-tion V1 /V0=0.03.

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ance has been computed for the spectral density of the dis-cretized harmonic bath of 20 oscillators, rather than the con-tinuum limit so as to make the comparison with thenumerical simulations as close as possible. The simulatedresults were obtained by Monte Carlo averaging over initialconditions; the sample size used was 2.5�108 points. As isevident from the figure, the analytical expression is indeed avery good description for the classical angular distribution.Increasing the temperature causes a smearing out of the rain-bow peaks, ultimately leading to a unimodal function.

The same results are shown also in Fig. 4, but this timefor the larger value of the corrugation parameter. Again, theanalytic expression provides a good representation of thetrue distribution; however the differences are more notice-able than for the weaker corrugation shown in Fig. 3. Thelarger corrugation causes the distance between the rainbowpeaks to grow. As a result, even at high temperature oneremains with a bimodal distribution.

D. Experimental application

The He atom is very light. Its angular distribution isdistinctly nonclassical; even at low temperatures, the maxi-

mum of the distribution is found at the specular angle.3 How-ever the angular distributions for both Ne and Ar scatteredfrom the Cu surface and observed in the scattering plane aremuch more classical in nature. Already for Ne, one observestwo peaks located at the rainbow angles, although these aremodulated by nonclassical oscillations due to quantum dif-fraction. For Ar, there are no oscillations; one distinctly ob-serves the double peaked structure as derived from thepresent classical theory.

In Fig. 5 we show fits of the analytic expression �Eq.�1.1�� to the experimental results for both Ne and Ar. Theparameters used for the analytical fit are given in Table II. Asis evident from the figure, the functional form provides areasonable fit to the experimental results. The fact that thefits are not perfect reflects a number of approximations usedin our derivation. Especially for Ne scattering, the classicaldynamics cannot capture the diffraction oscillations, but onlythe overall envelope of the angular distribution. The low am-plitude of the experimental high angle shoulder in the case ofAr scattering cannot be explained at the present level of thetheory; we do note that we have ignored the third degree offreedom �the motion along the y direction� and this mighthave some influence on the resulting distribution, especiallywhen the scattering angle is large.

FIG. 5. Analytical curve �dashed line� fitted to the experimentally obtainedscattering distribution �solid line� of �a� Ar and �b� neon atoms. The experi-mental data are adapted from Ref. 3.

FIG. 4. Comparison between the angular distributionsfor classical He atom scattering obtained numerically�solid lines� and using the analytical expression �dashedlines� at different temperatures for a stronger corruga-tion V1 /V0=0.0885.

TABLE II. Parameters for experimental fits.

Parameters Argon Neon

Expt. Beam E �meV� 63.75 63.38�i0 �deg� −28.5 −32.5M �a.u.� 72 820.41 36 785.7

Surfacecharacteristicsa

V0 �meV� 50.8 12.70V1 /V0 0.03 0.03T �K� 105 105

Fit parametersfor analytic curve

���deg� −0.5 −0.1�2 �deg2� 11.3 7.0K �deg� 15.3 18.0

aVide Ref. 3.

054107-8 Pollak, Sengupta, and Miret-Artés J. Chem. Phys. 129, 054107 �2008�

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V. DISCUSSION

We have presented an analytical treatment of the classi-cal Wigner theory of scattering from a corrugated surface.The derived expression is valid in the limit of weak couplingbetween the vertical and horizontal motions of the scatteredparticle as well as weak coupling with the surface phononbath. Although we did not include any coupling between thevertical motion and the phonon bath, this is not essential.Such a coupling would only serve to increase the energy lossto the bath, leading to a further broadening of the distributiondue to the thermal motion. The theory presented has beenderived for Ohmic friction; however extension to a morerealistic form of the friction which would include a cutofffrequency determined by the Debye frequency of the surfaceis straightforward and would not cause a significant changein the results.

The resulting angular distribution has two peaks, reflect-ing the rainbow scattering angles. The distribution is broad-ened and smeared out by interaction with the surfacephonons. Increasing the temperature tends to turn the distri-bution into a unimodular one. Direct comparison of thetheory with numerical simulations shows excellent agree-ment provided that the parameters are indeed in the weakinteraction limit specified by the theory. As the coupling isincreased, one finds that the qualitative form of the distribu-tion is still well described by the analytic expression.

As already mentioned in Sec. I, there have been previousattempts at deriving a theory for the classical scattering. Themajor improvements of our theory are that �a� we include thesurface corrugation; �b� coupling to the phonons is includedvia the horizontal motion; �c� we do not invoke impulsivecollisions, but expressly use the potential of interaction andclassical mechanics to derive the rainbow angles, energyloss, etc.; �d� when fitting to experimental results, the param-eters may be inverted to provide an estimate of the frictionalforce felt by the scattered atom.

The analytic theory has been derived for in plane scat-tering. One may extend it also to include the coupling of thevertical motion to the phonon bath as well as inclusion of thesecond dimension of the surface, that is, to provide a threedimensional theory. These are topics for ongoing and futurework. The present theory could also include the classicaldynamics of a diatom scattered from a metal surface such asN2, for which experimental results are available.3 One mayuse the present theory also to determine the differential an-gular and energy distribution as well as sticking probabili-ties.

The fact that the analytical treatment for the angular dis-tribution is in excellent agreement with the numerically exactclassical Wigner computation of the distribution suggeststhat one should extend the present treatment to also study thequantum diffraction, using the semiclassical initial value rep-resentation �SCIVR� formalism for the angular scattering. Itwould be of interest to understand whether the SCIVR ap-proach can account for the experimentally observed diffrac-tion pattern especially in the case of the scattering of the Heatom. Preliminary computations show that indeed the SCIVRangular distribution for He is peaked at the specular angle.

Deriving an analytic expression within the SCIVR formalismis a much more formidable challenge, as it is necessary toderive explicit expressions also for the action of the trajec-tories.

ACKNOWLEDGMENTS

We gratefully acknowledge support of this work by agrant of the Israel Science Foundation. S.M.-A. would like tothank the Ministry of Science and Innovation of Spain for aproject with Reference No. FIS2007–62006.

APPENDIX A: THE CHANGE IN THE SCATTERINGANGLE

The origin of time is chosen as the point at which theunperturbed trajectory in the vertical direction reaches theturning point �ztp� so that at time t=0 the vertical velocityvanishes. From Hamilton’s equations we have that

px�t� = px +2�V1

l�

−t0

t

dt� sin�2�

lx�t��� f�z�t��� . �A1�

We assume that the corrugation parameter V1 is small so thatto leading order in the coupling constant V1, the motion inthe horizontal direction is that of a free particle,

x�t� � x +px

M�t + t0� . �A2�

The coupling function f�z�t�� is by construction symmetric intime. The momentum at the time t0 is then given within thisperturbation theory by the expression

px�t0� � px +4�V1

lsin�2�

lx +

px

Mt0��

��0

�

dt� cos�2�px

lMt�� f�z�t���

� px + px. �A3�

Using the conservation of energy and the known shift in thefinal momentum in the horizontal direction, we deduce tofirst order the shift in the vertical momentum,

pz�t0� = − pz +px

pzpx. �A4�

The incident scattering angle is by definition

�i = tan−1� px

pz� . �A5�

The final scattering angle may now be written as

��t0� = − �i − �i

� tan−1� px + px�x,px,pz�

− pz +px

pzpx

�� − �i −

px�x,px,pz�pz

. �A6�

We thus have that the change induced in the scattering angle

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due to the interaction of the vertical and horizontal motionsis as given in Eq. �3.1�.

APPENDIX B: DERIVATION OF THE ANGULARDISTRIBUTION IN THE ABSENCE OF DISSIPATION

In this appendix we provide in some detail the manipu-lations that lead to the expression for the angular distribu-tion, as given in Eq. �3.3�. The angular distribution is ob-tained by averaging the angular shift �i as given in Eq. �3.1�over the initial conditions. For this purpose, we first integrateover the horizontal coordinate. Changing variables from x toy=x+� �using the notation �= pxt0 /M� we find that

�−�

�

dxe−��x − x0�2�� + �i + �i�

= �−�

�

dye−��y − � − x0�2�� + �i + K sin�2�y

l��

=1

2��

−�

�

d��−�

�

dxe−��x − � − x0�2+i���+�i+K sin�2�x/l��.

�B1�

The width � is now chosen such that ��l�1. This thenmeans that ��x−�−x0�2 can be considered to be a constant asx changes over the interval l. We can then use the followingapproximation:

�−�

�

dxe−��x − y�2+i�K sin�2�x/l�

� j=−�

�

exp�− �l� j +1

2� − y�2�

��0

l

dx exp�i�K sin�2�

lx��

� J0��K��−�

�

dx exp�− ��x − y�2�

=��

�J0��K� , �B2�

where in the second line, the summation has been approxi-mated by an integral. We thus have that

�−�

�

dx exp�− ��x − x0�2��� + �i + �i�

�1

2���

��

−�

�

d� exp�i��� + �i��J0��K�

=� 1

��

H�K2 − �� + �i�2��K2 − �� + �i�2

, �B3�

where H�x� is the Heaviside function, H�x�=1 if x�0 and 0otherwise.

Since this result is independent of the vertical coordinatez, we can readily integrate also over the vertical coordinateso that

P��� = limt→�

� 1

�����2�

−�

�

dpxdpz

�exp�−�px − px0

�2 + �pz − pz0�2

�2��

�H�K2 − �� + �i�2��K2 − �� + �i�2

. �B4�

In the limit that �→0 we then find �changing to radial mo-menta p2= px

2+ pz2, p0

2= px0

2 + pz0

2 , �i0=tan−1�px0/ pz0

�� the de-sired result as also given in Eq. �3.3�,

P��� = �0

� pdp

��

0

2�

d�i�p − p0 cos��i − �i0��

��p0 sin��i − �i0��H�K2 − �� + �i�2�

�K2�p,�i� − �� + �i�2

=1

�

H�K2 − �� + �i0�2��K2�p0,�i0� − �� + �i0�2

. �B5�

APPENDIX C: DERIVATION OF THE ANGULARDISTRIBUTION IN THE PRESENCE OF DISSIPATION

1. The change in the horizontal momentum

For Ohmic friction, the motion in the horizontal x direc-tion is governed by the Langevin equation

Mxt −2�V1

lf�zt�sin�2�xt

l� + Mg�zt���xtg�zt�

+ xtztg��zt�� = �MF�t�g�zt� , �C1�

from which it follows that

px�t0� = px +2�V1

l�

−t0

t0

dt sin�2�xt

l� f�zt�

− M��−t0

t0

dtg�zt�d

dt�xtg�zt��

+ �−t0

t0

dt�MF�t�g�zt� . �C2�

Therefore, to first order in the corrugation strength V1 andthe noise strength, using the fact that initially the particle isfar from the surface so that the coupling function f�z�=0 andthat it is symmetric with respect to time, we have that

px�t0� � px +2�V1

l�

−�

�

dt sin�2�

lx +

px

M�t + t0��� f�zt�

− �px�0

�

dtg2�zt� + �M�−t0

t0

dtF�t�g�zt� . �C3�

Given the unperturbed trajectory in the vertical direction,one can readily evaluate both the frictional shift

054107-10 Pollak, Sengupta, and Miret-Artés J. Chem. Phys. 129, 054107 �2008�

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�px,1 = − �px�0

�

dtg2�zt� �C4�

as well as the noise induced shift

�px,2 = 2�Mj=1

N

cjhj�xj cos�� jt0� +pj

� jsin�� jt0�� , �C5�

with

hj � �0

t0

dtg�zt�cos�� jt� . �C6�

This allows us to write down that

px�t0� � px + pzK�px,pz�sin�2�

lx +

px

Mt0��

+ �px,1 + �px,2

� px + �px, �C7�

where K�px , pz� is as defined in Eq. �3.1�.

2. The change in the vertical momentum

As noted above, the shift in the vertical momentum isdetermined by the energy balance. In addition to the changein momentum in the horizontal direction, one must also takeinto consideration the loss of energy to the bath.

The average energy loss to the bath is given by theexpression43

��EB =M

2�

−t0

t0

dt1�−t0

t0

dt2

d�xt1g�zt1

��

dt1

���t1 − t2�d�xt2

g�zt2��

dt2

= M��−t0

t0

dt1�d�xt1g�zt1

��

dt1�2

, �C8�

where the second equality is specified for Ohmic friction.The fluctuational energy loss is given by the expression

EB = − �M�−t0

t0

dtd�xtg�zt��

dtF�t� . �C9�

Since the function g�z� that couples the horizontal motion tothe bath is typically localized around the minimum of thepotential energy surface or at the turning point for the verti-cal motion, the term with dg /dt may be neglected in Eq.�C9�, so that

��EB ��px

2

M�

−t0

t0

dtg2�zt� = −px�px,1

M�C10�

and

EB � −px

�M�

−t0

t0

dtg�zt�F�t� = −px�px,2

M. �C11�

Energy conservation then implies that

px2�t0� + pz

2�t0�2M

=px

2 + pz2

2M− �EB, �C12�

so that the shift in the vertical momentum is given by

�pz =px

pz�px +

M�EB

pz. �C13�

3. The change in the scattering angle

Given the shifts in the horizontal and vertical momenta,it is then straightforward to determine the change in the finalscattering angle

��t0� = − �i − �i

� tan−1� px + �px

− pz +px�px + M�EB

pz�

� − �i −cos2��i�

pz��px�1 +

px2

pz2� +

Mpx�EB

pz2 � ,

�C14�

so that

�i � K�px,pz�sin�2�

lx +

px

Mt0�� + ��1 + ��2. �C15�

The angular shift is then

��1 =�px,1

pzcos2 �i, �C16�

while the angular fluctuation is

��2 =�px,2

pzcos2 �i. �C17�

4. The temperature dependent angular distribution

We now carry out the integration over the horizontalvariable. The procedure is the same as in the case withoutdissipation and we find

�−�

�

dx exp�− ��x − x0�2��� + �i + �i�

�� 1

��

H�K2 − �� + �i + ��1 + ��2�2��K2 − �� + �i + ��1 + ��2�2

. �C18�

Integrating over the vertical variable, letting the width pa-rameter of the Gaussian wavepacket �→0, changing to ra-

dial momenta p2= px2+ pz

2, p02= px0

2 + pz0

2 , �i0=tan−1� px0

pz0

�, we

find that

P��� =1

�� �

j=1

Ndpjdxj

2���B,W�p,x�

�1

�K2�p0,�i0� − �� + �i0 + ��1 + ��2�2, �C19�

where it is understood that all variables in the expression are

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now functions of the incident energy and scattering angleonly and we have omitted the Heaviside function for the sakeof brevity.

To carry out the integration over the bath coordinates,we change variables from pj ,xj to

pj�0� = pj cos�� jt0� − � jxj sin�� jt0� , �C20�

xj�0� = xj cos�� jt0� +pj

� jsin�� jt0� . �C21�

Since

dpj�0�dxj�0� = dpjdxj �C22�

and

pj2�0� + � j

2xj2�0� = pj

2 + � j2xj

2, �C23�

the change of variables simplifies the integration over thebath. The dependence on the bath comes from the term ��2,which is only dependent on the xj�0�’s. Defining a “general-ized bath angle,”

Z � ��2 =2�M

pzcos2��i0�

j=1

N

cjhjxj�0� , �C24�

we note that

P��� =1

�� �

j=1

Ndpjdxj

� j exp�−

j

�� j�pj

2 + � j2xj

2�� H�K2 − �� + �i0 + ��1 + ��2�2��K2�p0,�i0� − �� + �i0 + ��1 + ��2�2

=1

��

−�

�

dZH�K2 − �� + �i0 + ��1 + Z�2�

�K2�p0,�i0� − �� + �i0 + ��1 + Z�2� 1

2��2 exp�−Z2

2�2� , �C25�

where the temperature and friction dependent variance of thedistribution is

�2 =2�M

p02 cos2��i0�

j

cj2hj

2

j� j. �C26�

One finds that the variance is readily expressed in the con-tinuum limit by noting that

j

cj2hj

2

j� j=

2

��

0

�

d�J�����

0

�

dtg�zt�cos��t��2

tanh��

2

. �C27�

Equations �C25�–�C27� are the central result of thisAppendix.

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