Multidimensional potential energy surface determination by modified Shepard interpolation for a...

Chemical Physics Letters 376 (2003) 566–575

www.elsevier.com/locate/cplett

Multi-dimensional potential energy surface determinationby modified Shepard interpolation for amolecule–surface reaction: H2 + Pt(1 1 1)

C. Crespos a, M.A. Collins b, E. Pijper a, G.J. Kroes a,*

a Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlandsb Research School of Chemistry, Australian National University, ACT 0200, Australia

Received 20 March 2003; in final form 4 June 2003

Published online: 9 July 2003

Abstract

A modified Shepard interpolation method, developed for constructing potential energy surfaces (PESs) for gas phase

reactions, has been adapted to generate PESs for molecule–surface reactions, and applied to the dissociative chemi-

sorption of H2 on Pt(1 1 1). To provide a test of the method, the input data were taken from an existing PES. Reaction

probabilities computed using classical and quantum dynamics on the new PES are in excellent agreement with results

for the old PES, the construction of which required twice as many points. This shows that the modified Shepard in-

terpolation method can be used efficiently to build PESs which yield accurate dynamics results for molecule–surface

reactions.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

A key step in theoretical studies of chemical re-

actions dynamics is the determination of the po-

tential energy surface (PES). Many fitting andinterpolation methods have been proposed to con-

struct continuous representations of PESs using ab

initio calculated points. Exact dynamics calcula-

tions of, for instance, initial-state selected reaction

probabilities require the use of a sufficiently accu-

rate PES, defined on a large area of configuration

* Corresponding author. Fax: +31-71-527-4488.

E-mail address: [email protected] (G.J. Kroes).

0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0009-2614(03)01033-9

space. Although the development of low-dimen-

sional models [1–3] to understand and interpret the

dynamics is very important, the validation of elec-

tronic structure methods through comparisons with

experiment often requires accurate high-dimen-sional simulations [4–7]. The study of gas–surface

reactions implies many degrees of freedom and the

development of an efficient scheme to produce ac-

curate high-dimensional PESs is a crucial task.

In recent years, significant advances have been

made in the evaluation of the energy of a molecule

interacting with a metal surface. In particular,

density functional theory (DFT) [8,9] within thegeneralized gradient approximation (GGA) [10,11]

is a convenient tool to compute molecule–surface

ed.

mail to: [email protected]

C. Crespos et al. / Chemical Physics Letters 376 (2003) 566–575 567

interactions involving many degrees of freedom.

Thus, comparable advances in interpolation and

fitting techniques are highly desirable.

The first PES models used in gas–surface dy-

namics simulations were derived from the general

London–Eyring–Polanyi–Sato (LEPS) analyticalform fitted on ab initio data [12–14]. Other authors

have expanded the potential of a diatomic mole-

cule interacting with a surface in functions which

are adapted to the symmetry associated with the

surface unit cell [15,16]. A new idea has been to

first remove the greater part of the corrugation

and the anisotropy from the molecule–surface in-

teraction, by taking sums of atom–surface inter-actions (for the atoms constituting the molecule)

as a zero-order expression for the potential, and

then using symmetry-adapted functions to inter-

polate the remainder of the interaction. This

method has been called the corrugation reducing

procedure (CRP) [17]. This new interpolation

scheme [17] has proved its efficiency in many

studies of diatomic molecules reacting or scatter-ing on metal surfaces, for instance, in six-dimen-

sional quantum and classical calculations on the

reaction of H2 on Pt(1 1 1) [4]. However, it is not

easy to extend this method to problems involving

more than six degrees of freedom.

A major goal in theoretical surface science is to

be able to study the dynamics of a polyatomic

molecule reacting on a surface. The achievement ofthis goal requires the availability of accurate, glo-

bal high-dimensional PESs for such systems. As a

step towards this goal, we have therefore tested

and adapted a method for generating PESs based

on modified Shepard interpolation, that has pre-

viously been applied to gas phase reactions [18–

22]. This method does not require the knowledge

of an a priori analytical form of the PES, and usesclassical trajectory simulations to provide an iter-

ative scheme for successively improving the PES.

The PES is given by an interpolation of local

Taylor expansions, and an iterative procedure

places new points only in regions of configuration

space that are important for dynamics. So instead

of computing a regular grid of ab initio points and

performing interpolation on it, the method focuseson �dynamically interesting� parts of the PES. One

of the great advantages of this method is its

economy in terms of the number of ab initio points

needed and, as a consequence, its economy in CPU

time. For example, only about 200 ab initio points

were needed for the reaction OH +H2 !H2O + H

[21].

The main goals of this work are: (i) to pro-duce a modified version of this interpolation

method that allows the construction of PESs for

gas–surface reactions and (ii) to test the method

by applying it to the system H2 + Pt(1 1 1). The

diffraction and dissociation processes of H2 on

Pt(1 1 1) surface have already been studied [4],

using an accurate representation of the PES [23]

based on the corrugation reducing procedure[17]. The latter PES (designated as �CRP-PES� for

corrugation reducing procedure PES) has been

used here to generate a new PES via the modified

Shepard interpolation method (named �MS-PES�for modified Shepard PES). Thus, the CRP-PES

plays the role of an ab initio calculation code

and gives for any geometry a value of the energy

and the first derivatives, considered as the exactones. The efficiency and the accuracy of the

modified Shepard interpolation method is tested

by comparing the results of dynamics simulations

on both the CRP-PES and the MS-PES (which

should be identical if the interpolation were

exact).

Section 2 gives a brief overview of the modified

Shepard interpolation method as revised for mol-ecule–surface interactions, and describes the way

we have prepared the MS-PES from the CRP-PES.

In Section 3, a discussion of the accuracy of the

interpolation method is provided by comparing

reaction probabilities obtained from classical and

quantum dynamics calculations that were ob-

tained with both PESs.

2. Method

In this section, the method for iteratively de-

veloping an interpolated PES, by modified Shep-

ard interpolation, is briefly presented. The

complete methodology will be detailed elsewhere

[24]. Here, the method will be applied, for the firsttime, to the system of a diatomic molecule reacting

on a surface.

568 C. Crespos et al. / Chemical Physics Letters 376 (2003) 566–575

2.1. Interpolated PES

The interpolated PES is given by a weighted

series of Taylor expansions centered at ab initio

data points, sampled throughout the configurationspace of the system. This sample of data points is

non-uniform and describes the regions of the

configuration space that are most important for

reaction with the highest accuracy. This sample of

points will be called the �PES data set� in the fol-

lowing. The method to determine the location of

these ab initio data points will be detailed below.

The H2 +Pt(1 1 1) system is described using in-verse inter-atomic distances, Q ¼ f1=R1; 1=R2; . . . ;1=RNðN�1Þ=2g with inter-atomic distances Ri and Nbeing the number of atoms. The model and the

system of coordinates to be used in the interpola-

tion method is depicted in Fig. 1. The impinging

molecule is a diatomic and the frozen surface is

represented by three atoms which are kept fixed

(these three atoms, defining the surface unit cellvectors, are required to represent the motion of the

molecule center of mass parallel to the surface).

Thus N ¼ 5 atoms are required to fully represent

the problem. The inter-atomic distances are kept

fixed for the surface atoms (the three Pt–Pt inter-

atomic distances). Only six coordinates are needed

to describe the reaction of a diatomic molecule

Fig. 1. Representation of the H2 +Pt(1 1 1) system, used in the

modified Shepard interpolation scheme.

with a frozen surface, so that any given geometry

nðiÞ ¼ n½QðiÞ is expressed in terms of 6 linear

combinations n ¼ ðn1; . . . . . . :n6Þ of the ½NðN � 1Þ=2 � 3 ¼ 7 remaining inter-atomic distances (one

of them being redundant).

The potential energy V , at a given configurationn, in the vicinity of an ab initio data point nðiÞ, can

be expanded as a second-order Taylor series, TiðnÞ:

TiðnÞ ¼ V ½nðiÞ þX6

k¼1

½nk � nkðiÞoVonk

��n¼nðiÞ

þ 1

2!

X6

k¼1

X6

j¼1

½nk � nkðiÞ

� ½nj � njðiÞo2V

onkonj

��n¼nðiÞ

þ � � � ð1Þ

An accurate evaluation of V can be obtained by a

modified Shepard interpolation [25,26], where the

potential energy is obtained as a weighted average

of the different Taylor series Ti (i ¼ 1; . . . ;Ndata)

calculated from each data point of our configura-

tion space sample (the number of data points in

the PES data set being Ndata) and their symmetry

equivalent points:

V ðnÞ ¼X

g2G

XNdata

i¼1

wg�iðnÞTg�iðnÞ; ð2Þ

where G denotes the symmetry group of the system

and g � i express the fact that the ith data point is

transformed by the group element g (in the origi-

nal study of the H2 + Pt(1 1 1) system, the differ-

ence between the fcc and hcp sites is neglected and

the symmetry group of the surface unit cell is C6v

[23]). The sum over the group elements in Eq. (2)

ensures that the PES has the correct symmetry

with respect to the permutation of indistinguish-

able particles. The symmetry of the PES with re-

spect to translation on the surface is ensured by

translating (in X and Y ) the center of mass of the

impinging molecule to the primary unit cell before

the PES of Eq. (2) is evaluated. The weight wg�i ofeach Taylor series in Eq. (2) is evaluated in a

similar manner to that used previously [22,27].

Further technical details concerning the weights

and other aspects of the PES interpolation method

will be presented elsewhere [24].


2.2. Iterative development of the PES

In order to build and optimize the PES data set,

an iterative scheme has been developed [18]. The

idea is to start with only a few calculated pointsand compute classical trajectories using this initial

version of the PES sample. The initial guess con-

sists of calculated data points mainly located close

to some hypothetic reaction pathways, assuming a

good accuracy in these regions is required. The

configurations visited by each of the classical tra-

jectories are stored in files and a choice of relevant

data points to be added to the initial sample ismade according to two different criteria, with the

aim of computing accurate observables (reaction

probabilities). For this purpose, one argument

suggests that the best location for a new data point

would be in the region most frequently visited by

the classical trajectories (�h-weight� criterion). A

second argument suggests that the accuracy of the

PES could be improved if a new data point isadded in the region where the interpolated poten-

tial is expected to be least accurate (�variance

sampling� criterion). The least accurate regions are

the ones for which the variance of the weighted

average in Eq. (2) is largest.

The distribution of data points chosen to be

added to the PES data set is driven by the con-

figurations visited by the classical trajectories. Theselection criteria for growing the sample of data

points ensure an efficient improvement of the PES

and a convergence of observables like reaction

probabilities. Indeed, the observation of such

quantities gives us a criterion to control the gain in

accuracy during the growing process of the PES

data set. In this study we have chosen to check the

convergence of the initial-state selected moleculardissociation probability as the size of the data set

increases. Many past studies show a good con-

vergence of reaction probability with the growth of

the PES data set for gas phase reactions [21,28–

33].

In summary, an initial PES data set is grown by

iteratively adding points corresponding to config-

uration space regions where accurate determina-tion of the potential is required. The growth of the

PES is stopped when a computed observable (in

our case the reaction probability) is considered as

converged, within a given tolerance. The key point

of the modified Shepard interpolation method is

that classical trajectories are used to scan the

configuration space of the system and locate the

dynamically important regions.

2.3. H2 +Pt(1 1 1) system as a benchmark

The aim of the present study is to test the re-

vised interpolation method for the case of gas–

surface reactions. We have modified the original

code to be able to treat the translational symmetry

of the problem (motion of H2 center of mass

parallel to the surface).The diffraction and dissociation processes of H2

on Pt(1 1 1) surface have already been extensively

studied [4], using an accurate representation of the

PES based on the corrugation reducing procedure

[17]. We use this accurate PES (CRP-PES) to

produce data points required in the modified

Shepard interpolation method. At the end of the

interpolated PES (MS-PES) growing process, weare left with two versions of the H2 + Pt(1 1 1)

potential, one made from the other. Quantum and

classical dynamics simulations are then carried out

on both PESs to test the efficiency and accuracy of

the modified Shepard interpolation method.

In the following, we will use the system of co-

ordinates depicted in Fig. 2a to discuss the results

of the classical and quantum dynamics calcula-tions. This system of coordinates is the one com-

monly used to discuss diatomic molecules reacting

on surfaces. The motion of the H2 center of mass is

represented by the set of coordinates (X , Y , Z), the

orientation of the molecular axis by the angles (h,

/) and the stretching of the molecular bond by the

coordinate r. To aid the discussion in Section 3,

some two-dimensional cuts through the CRP-po-tential [23] are shown in Fig. 3. Fig. 3a shows the

dependence of the potential on r and Z, for a ge-

ometry in which the molecule is located at the

threefold hollow site, the molecular axis being

parallel to the surface. Fig. 3a shows where the

reactants valley, the barrier region, and the prod-

ucts valley are approximately located for the sys-

tem considered here. Fig. 3b shows how thereaction barrier height varies across the surface,

for the molecule lying parallel to the surface (the

Fig. 2. Coordinate system used in the dynamics study of the

H2 +Pt(1 1 1) reaction. In (a) the H2 center of mass is fixed at

(X , Y , Z), the molecular orientation is described by (h, /), and

the bond elongation by r. In (b) the Pt(1 1 1) surface is repre-

sented, the surface unit cell being defined by three surface at-

oms. The high symmetry sites (top, bridge, threefold) of the

surface are also depicted.

Fig. 3. Contour plots of 2D-cuts through the CRP-PES. In (a)

the PES is plotted as a function of Z and r, for the molecule

being above threefold hollow site with h ¼ 90� (the level spacing

is 0.1 eV), with specification of products and reactants valleys

separated by the barrier region. In (b) contour plots show the

reaction barrier height (eV) as a function of the coordinates Xand Y . The barrier height has been minimized with respect to hand / for each combination of X and Y . Note that the coor-

dinate axes are taken orthogonal even though the (1 1 1) surface

unit cell is diamond shaped [4].


barrier height being minimized with respect to hand / for each X and Y value). As can be seen,

the transition state is �loose�, in the sense that thebarrier height is only weakly dependent on the

X - and Y -coordinates.

As previously mentioned the PES growth stop

criterion is the convergence of the computed mo-

lecular dissociation probability. In the classical

trajectories, dissociation is defined to occur once rbecomes three times the equilibrium distance of

H2, with the conjugated moment pr being positive.

2.4. Modified Shepard interpolation method applied

to gas–surface reactions

The modified Shepard interpolation method has

been developed and applied to several gas phase

reactions. Our purpose is to use this method to

deal with the problem of a molecule reacting on asurface, and in order to achieve this aim we have

implemented a few modifications of the interpo-

lation strategy.

First, as already discussed for H2 + Pt(1 1 1), ingas–surface reactions the transition state is often

�loose� in that the dependence of the reaction bar-

rier height on the X and Y coordinates is weak.

The direct consequence is that many reaction

pathways may have to be considered as initial

guess for starting the modified Shepard interpo-

lation (the dissociation can occur at many posi-

tions of the center of mass of the molecule over thesurface). Our strategy has therefore been to com-

pute ab initio points for not just one but three

initial reaction pathways, which correspond to the

high symmetry sites of the surface (top, threefold

hollow, and bridge) depicted in Fig. 2b. Thus, the

initial PES data set is composed of data for three

reaction pathways (75 points in total, 25 points per

reaction path).


Second, previous studies in dynamics of gas–

surface reactions exhibit quite a large complexity of

mechanisms leading to dissociation. In some cases,

like H2 reacting on Pd(1 1 1) [34], the molecular

dissociation mechanisms are different at low and

high value of the translational energy of the H2

molecule (collision energy), and the low energy

mechanism is sensitive to small structural features

of the PES. As a consequence, itmay be necessary to

grow the potential at different values of the trans-

lational energy of the impinging molecule. Indeed,

the scan of the PES could be dependent of the choice

of the collision energy at which the trajectories are

performed. In order to obtain a good level of ac-curacy for the whole range of energies implied by

our dynamics study, we have decided, in this work,

to grow the PES data set at three different collision

energies simultaneously. For example, in the case of

H2 + Pt(1 1 1), the dissociation dynamics is studied

for collision energies between E ¼ 0:05 and 0.5 eV.

Therefore, we have decided to grow the PES at

E ¼ 0:1, 0.3 and 0.5 eV simultaneously.From a more technical point of view, the inter-

polation of the PES, in the approximation of frozen

surface, is performed by using fixed surface atoms

and periodic boundary conditions. In order to re-

produce the symmetry of the potential with respect

to the motion of the molecular center of mass par-

allel to the surface, we have chosen to use the pri-

mary unit cell of the surface (the vectors ofwhich aredefined by three atoms) as a periodic box. Appro-

priate constraints on the surface atoms have been

used to keep them in fixed positions during the

classical trajectories calculations. Additional details

concerning, for instance, the need of three surface

atoms will be presented elsewhere [24]. Keeping all

the surface atoms fixed is not a restriction imposed

by the MS-method and we could in principle allow afew surface atoms to move, in order to build a PES

enabling a study of the energy exchange between the

molecule and the surface atoms. However, this was

not the goal of our research.

3. Results and discussion

We present here the results of the interpolation

and results of quantum and classical dynamics

calculations performed using the CRP-PES and

the MS-PES. We focus on initial-state selected

reaction probabilities, where the initial rovibra-

tional state of the impinging molecules is fixed at

(v ¼ 0, j ¼ 0), v being the vibrational and j the

rotational angular momentum quantum number.

3.1. Sampling of the configuration space

One very important aspect of the interpolation

method is that it avoids the calculation of a uni-

form grid of data points over configuration space

but rather focuses on computing points in the re-

gions relevant for the dynamics. The modifiedShepard interpolation method should locate the

so-called dynamically important region of the PES

and in that way give essential information on

where the reaction takes place.

In Fig. 4, different representations of the PES

data set are plotted. First, 2D representations in

(Z, r) are depicted for two different molecular

translational energies at which the PES was grown,E ¼ 0:1 eV (Fig. 4a) and E ¼ 0:5 eV (Fig. 4b).

Thus, the regions visited by the classical trajecto-

ries, relevant in terms of dynamics, are repre-

sented. As the figures show, the dynamically

relevant region is located in the region of high Zvalues and r values close to the H2 equilibrium

bond distance. This means that the reactivity is

mainly determined by the region corresponding tothe approach towards the surface and the region of

the barrier, as expected (see also Fig. 3a). For

obtaining initial-state selected reaction probabili-

ties, the accuracy of the potential beyond the dis-

sociation barrier (products valley) is not as

important as the accuracy of the potential in the

incoming part (reactants valley and barrier re-

gion). It is interesting to note that the sampledregion is slightly different for the lower and higher

value of the energy during the growing process.

When the molecules approach the surface with a

relatively low energy (E ¼ 0:1 eV), the dynamics is

sensitive to the structure of the potential in the

incoming region, and the proportion of reflected

back trajectories is important. For the high energy

case (E ¼ 0:5 eV) the region of the barrier(2:5 a:u: < Z < 4 a.u.; r � 1:5 a.u.) and the region

behind it, are more extensively visited because

Fig. 4. PES data set in (Z, r) representation (a and b), and in (X , Y ) representation (c and d). In each plot the initial data points are

depicted using filled diamonds and the ab initio points added during the growth of the PES using open circles. The plots (a) and (c) are

obtained for low energy growth of the PES (E ¼ 0:1 eV) and the plots (b) and (d) for high energy growth (E ¼ 0:5 eV).


of the higher probability of reaction, but never-

theless the points added in the incoming region are

still the most important.

In Figs. 4c and d, the X and Y coordinates of the

data points of the same PES data set are plotted.

This 2D view shows where on the surface the dy-namics takes place. Clearly, the dynamics is not

particularly driven by forces along X and Y , and the

distribution of the PES data points is almost uni-

form. Nevertheless, a slightly larger concentration

of points is seen close to the top sites for low energy

(see Figs. 2b and 4c). This last result is not surprising

because the top site presents the lowest barrier for

dissociation, and the neighborhoods of these sitesare the only dissociation windows accessible at

E ¼ 0:1 eV (Fig. 3b). At higher energy there are no

notable differences of density of points along the

unit cell and an almost uniform coverage of the

surface unit cell is obtained in the PES data set.

These different observations show two points:

first, the strategy of setting up three reaction paths

for three different surface sites as initial guess

could be expected to be efficient in providing a

uniform sampling of the surface unit cell. Second,

the differences in the sampling of data points for

E ¼ 0:1 and 0.5 eV suggest that sampling the PES

data set using three different values of the energysimultaneously may be a good strategy for ob-

taining a PES which yields reliable results on a

large range of energies. Furthermore, the differ-

ences between (Z, r) cuts (Figs. 4a and b) suggest

the importance of determining the dynamically

important region for interpolation.

3.2. Convergence control of the growing process

During the PES growing process, new data

points are iteratively added to the PES data set in

order to increase its accuracy. The accuracy of the

PES is tested by regularly performing classical

trajectories calculations to compute the reaction

probability (after each hundred of points added to

Fig. 5. The dissociation probability (v ¼ 0, j ¼ 0) H2 on

Pt(1 1 1) is compared for the CRP-PES and the MS-PES. The

results of classical trajectories simulations are plotted in (a) and

results of quantum wave-packet simulations in (b).


the data set). The convergence of the reaction

probability within a given tolerance is then used as

a criterion for stopping the growing process of the

PES. Full details of the criterion we used will be

presented elsewhere [24].

For the results presented below, we have chosento grow the PES at three different energies simul-

taneously, using as criterion for stopping the PES

growth the simultaneous convergence of the reac-

tion probability at the three growing energies.

Using this procedure, convergence was achieved

for a sample of 1300 data points, whereas the

corrugation reducing procedure [17] required more

than twice as many points for the development ofthe original CRP-PES.

The results presented in Section 3.1 were ob-

tained by growing different PES data sets for dif-

ferent collision energies. As will be discussed in a

more detailed paper [24], using this procedure re-

sults in a PES that yields reaction probabilities

which are accurate for energies which are very

different from the energy at which the PES wasgrown.

3.3. Classical and quantum results for (v ¼ 0,

j ¼ 0)

In order to test the Taylor-based interpolation

method we have performed accurate dynamics

simulations on both the CRP-PES and the MS-PES. Results of six-dimensional classical trajectory

calculations for the (v ¼ 0, j ¼ 0) initial state and

normal incidence are represented in Fig. 5a. For

each calculation we have used between 5000 and

10 000 trajectories to reduce the statistical error in

the reaction probability determination to about

0.5%. In our calculations for either PES, molecule-

phonon energy exchange and electron–hole pairexcitation are not taken into account. The validity

of these approximations has already been dis-

cussed elsewhere [35].

As one can see, very good agreement is ob-

tained between the reaction probability computed

for both PESs. This result confirms the accuracy

and the efficiency of the modified Shepard inter-

polation method.The other question we considered was whether

the growing process, which is based on classical

dynamics, is able to yield accurate quantum dy-

namical reaction probabilities. To answer this

question, we have performed quantum calculations

on both PESs. The method used is the time-de-

pendent wave-packet (TDWP) method, as imple-mented in [4]. In Fig. 5b, the results of the quantum

calculation for (v ¼ 0, j ¼ 0) show again very good

agreement between the reaction probabilities ob-

tained with the two PESs. The good agreement

between the quantum results shows that the mod-

ified Shepard interpolation procedure can yield a

PES which gives accurate quantum dynamics re-

sults for reaction, even though the growing of thePES is based on classical dynamics. This observa-

tion suggests that, for the H2 + Pt(1 1 1) system and

the range of energies considered, the dynamically


relevant regions for reaction in the quantum regime

do not really differ from the regions relevant for the

classical reaction dynamics.

4. Summary

The modified Shepard interpolation method,

previously developed for the construction of high-

dimensional potential energy surfaces for gas

phase reactions, has been adapted to molecule–

surface reactions and applied to the H2 +Pt(1 1 1)

system. Modifications, implemented to treat mol-

ecule–surface reactions, include the use of periodicboundary conditions for motion along the surface

and the freezing of the atoms representing the

surface. Furthermore, the initial data set used to

start the growing process contained data for three

reaction paths rather than one, the three reaction

paths describing dissociation over three different

surface sites. This strategy was adopted to pro-

mote a uniform sampling of the potential along themetal surface, because the dependence of the re-

action barrier height on the coordinates for mo-

tion along the surface is weak. Finally, a new

element has been that the PES was grown for three

different collision energies simultaneously, to allow

accurate reaction probabilities to be extracted

from its use for a large range of collision energies.

The accuracy and the efficiency of the modifiedShepard interpolation method was tested by ap-

plying the method to an initially known accurate

potential energy surface, thereby avoiding the

problem of errors related to ab initio calculations.

In a comparison with results obtained for the

original PES, classical and quantum dynamics

calculations showed that the grown PES yields

accurate reaction probabilities for the collisionenergy range 0.05–0.5 eV. This result was achieved

using a data set which consists of a number of

points (1300) that was less than half the amount of

(ab initio) points required to construct the original

PES, which was constructed using a method (the

corrugation reducing procedure) that employs in-

terpolation of a uniform grid of points.

The results show that the modified Shepardinterpolation method, which does not require a

priori knowledge concerning potentially useful

analytical fitting forms, can be efficiently used to

construct PESs for molecule–surface reactions

from which accurate reaction probabilities can be

derived. Generalization of the method to reactions

of polyatomic molecules with surfaces is straight-

forward and we expect that the method can beeasily applied to, for instance, the dissociative

chemisorption of methane on metal surfaces.

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