Chemistry Publications Chemistry

9-1998

Ab Initio Potential Energy Surface by ModifiedShepard Interpolation: Application to theCH3+H2→CH4+H ReactionTakeyuki TakataUniversity of Tokyo

Tetsuya TaketsuguUniversity of Tokyo

Kimihiko HiraoUniversity of Tokyo

Mark S. GordonIowa State University, mgordon@iastate.edu

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Ab Initio Potential Energy Surface by Modified Shepard Interpolation:Application to the CH3+H2→CH4+H Reaction

AbstractAn ab initiopotential energy surface for the six-atom reactionCH3+H2→CH4+H was constructed, within C3vsymmetry, by a modified Shepard interpolation method proposed recently by Collins et al. Selection of datapoints for the description of the potential energy surface was performed using both the Collins method andthe dynamic reaction path (DRP) method. Although the DRP method is computationally more expensive,additional data points can be determined by just one simulation. Analyses of distributions of the data points,reaction probability, and errors in energy and energy gradients determined by the two different methodssuggest a slight advantage for the DRP sampling in comparison with the iterative sampling.

KeywordsPotential energy surfaces, Surface reactions, Hydrogen reactions, Interpolation, Ab initio calculations

DisciplinesChemistry

CommentsThe following article appeared in Journal of Chemical Physics 109 (1998): 4281, and may be found atdoi:10.1063/1.477032.

RightsCopyright 1998 American Institute of Physics. This article may be downloaded for personal use only. Anyother use requires prior permission of the author and the American Institute of Physics.

This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/chem_pubs/346

Ab initio potential energy surface by modified Shepard interpolation: Application to theCH 3 +H 2 →CH 4 +H reactionTakeyuki Takata, Tetsuya Taketsugu, Kimihiko Hirao, and Mark S. Gordon Citation: The Journal of Chemical Physics 109, 4281 (1998); doi: 10.1063/1.477032 View online: http://dx.doi.org/10.1063/1.477032 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/109/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Gradient-based multiconfiguration Shepard interpolation for generating potential energy surfaces for polyatomicreactions J. Chem. Phys. 132, 084109 (2010); 10.1063/1.3310296 Interpolating moving least-squares methods for fitting potential energy surfaces: An application to the H 2 C Nunimolecular reaction J. Chem. Phys. 126, 104105 (2007); 10.1063/1.2698393 Ab initio potential-energy surfaces for the reactions OH+H 2 ↔ H 2 O+H J. Chem. Phys. 115, 174 (2001); 10.1063/1.1372335 Quantum dynamics on new potential energy surfaces for the H 2 +OH→H 2 O+H reaction J. Chem. Phys. 114, 4759 (2001); 10.1063/1.1354145 Automatic potential energy surface generation directly from ab initio calculations using Shepard interpolation: Atest calculation for the H 2 +H system J. Chem. Phys. 107, 3558 (1997); 10.1063/1.474695

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Ab initio potential energy surface by modified Shepard interpolation:Application to the CH 31H2˜CH41H reaction

Takeyuki Takata, Tetsuya Taketsugu, and Kimihiko HiraoDepartment of Applied Chemistry, Graduate School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

Mark S. GordonDepartment of Chemistry, Iowa State University, Ames, Iowa 50011

~Received 23 March 1998; accepted 12 June 1998!

An ab initio potential energy surface for the six-atom reaction CH31H2→CH41H was constructed,within C3v symmetry, by a modified Shepard interpolation method proposed recently by Collinset al.Selection of data points for the description of the potential energy surface was performed usingboth the Collins method and the dynamic reaction path~DRP! method. Although the DRP methodis computationally more expensive, additional data points can be determined by just one simulation.Analyses of distributions of the data points, reaction probability, and errors in energy and energygradients determined by the two different methods suggest a slight advantage for the DRP samplingin comparison with the iterative sampling. ©1998 American Institute of Physics.@S0021-9606~98!30835-1#

I. INTRODUCTION

The construction of an accurate potential energy surface~PES! is an essential step in the analysis of the dynamics ofa chemical reaction. The reliability of dynamic simulations isvery sensitive to the quality of the PES employed. The PES,as a function of nuclear coordinates, can be determined, inprinciple, by solving the electronic Schro¨dinger equation forfixed nuclear arrangements1 at some number of grid points inthe appropriate configuration space. However, computationallimitations have commonly necessitated the use of modelpotential energy functions~containing several fitting param-eters! in molecular dynamics simulations. When the processincludes a bond cleavage or bond formation, it is difficult todevise reliable potentials. As the number of atoms in thesystem of interest increases, a systematic determination ofthe PES becomes increasingly difficult, as does the subse-quent process of fitting the resulting points to an analyticrepresentation.

Recent developments in electronic structure theory,computational algorithms, and computer technology havemade it possible to integrate molecular dynamics simulationsinto ab initio electronic structure codes. For example, thedynamic reaction path~DRP! method,2–8 which was devel-oped originally as an efficient method to locate the intrinsicreaction coordinate~IRC!9–13 for elementary reactions, canbe used to obtain a classical ‘‘trajectory on-the-fly,’’ withoutprior knowledge of the PES. The DRP method uses the en-ergy gradient, obtained directly from electronic structure cal-culations, to determine atomic accelerations, velocities, andpositions. Thus, this method explores theab initio potentialenergy surface, although suchab initio trajectorycalculations14,15require considerably more computational ex-pense than traditional trajectory calculations.16 The DRPmethod has recently been used to extract the characteristicsof dynamical processes for several reactions.4–8

A promising method for incorporatingab initio potentialenergy surfaces into the dynamics of chemical reactions hasrecently been developed by Collins and co-workers,17–25andbeen applied to four-atom systems, includingNH1H2→NH21H;17 OH1H2→H2O1H;18–20,22 C2H2;

23

and C(3P)1H31(1A18).

24 In their method, which has recentlybeen extended to six-atom species,25 the PES is constructedby a modified Shepard interpolation26,27 in which the poten-tial energy at a given configuration,X, is determined byusing local Taylor expansions aboutNd known data points.At those data points, the energy and derivatives in terms ofnuclear coordinates are calculated byab initio methods inadvance. The total potential energy atX is evaluated as thesum over allNd data points of their weighted Taylor series,where the weighting function becomes larger for data pointscloser toX. Methods for iteratively improving the PES usingclassical trajectory simulations have also beendeveloped.17,21 With this interpolation methodology, severalgroups have attempted to develop more efficient methods toconstruct the PES.28–30

For a given elementary reaction, there are usually someregions of configuration space that are more important thanothers. The most important region may be that within a vi-brational amplitude of the minimum energy path~intrinsicreaction coordinate path, sometimes abbreviated as IRC!;that is, the region of a minimum energy path which connectsa reactant minimum, a transition state~TS!, and a productminimum on the PES. In the Collins interpolation scheme,the initial PES is usually constructed in terms of data pointsdistributed on the IRC17,21 ~referred to as the reaction pathpotential energy surface; RP-PES!. Once the IRC is definedon the PES, the reaction path Hamiltonian31 ~RPH!, whichdescribes the motion along the IRC in terms of harmonicdisplacements in directions orthogonal to the reaction path,can be determined with a reasonable cost.32 In addition to

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 11 15 SEPTEMBER 1998

42810021-9606/98/109(11)/4281/9/$15.00 © 1998 American Institute of Physics

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being a common starting point for the study of reaction dy-namics, the RPH provides important insights regarding thedynamics of a process of interest. If some amount of nuclearsymmetry is conserved along the reaction path, Collinset al.make use of their symmetry-invariant reaction pathpotentials.33

Although the one-dimensional reaction path is extremelyuseful, the IRC has two geometrical weaknesses that cancause problems in dynamics calculations. These occur forsharply curved regions5,34,35and for unstable regions36–40onthe reaction path. When a reaction path contains such re-gions, the significance of regions far from the IRC are ex-pected to increase. It then becomes necessary to estimate thepotential energy farther from the IRC than is ordinarilyneeded. Collinset al.employ classical trajectory calculationsto select those data points which are far from the IRC but areimportant in dynamics; they are added to the set of originaldata points to improve the interpolated PES.17,21

In this paper, we apply the interpolation method of Col-lins et al.17,21 to the construction of the PES for the six-atomreaction, CH31H2→CH41H, with 12 internal degrees offreedom. This reaction, including the nature of the IRC, hasbeen extensively studied.10,11,34,40–45The IRC conservesC3vsymmetry throughout, and there are four totally symmetric(A1) coordinates in thisC3v subspace. Since the reactioncoordinate couples only with the orthogonal totally symmet-ric vibrations along the reaction path, the totally symmetricconfiguration space may be particularly important. The en-ergy gradient on the IRC in orthogonal nontotally symmetricdirections is zero, and the energy variation in those direc-tions relative to the IRC can be described by an even func-tion. Thus, as a first step in developing a global potentialenergy surface, we construct the PES in the totally symmet-ric configuration space~referred to as the totally symmetricPES!, i.e., that part of the PES that may be expressed interms of four totally symmetric coordinates inC3v symme-try. The DRP method is employed for the selection of thedata points to represent the PES~DRP sampling!. Throughapplication of the interpolation scheme to larger molecularsystems, we investigate the applicability and limitations ofthe method.

II. INTERPOLATION SCHEME

We first summarize the modified Shepard interpolationmethod of Collinset al. for the construction and improve-ment of the PES,17,21with several additional modifications inthe present study. In this interpolation scheme, the potentialenergy at some nuclear arrangement,X, can be expressed asthe sum of contributions from several data points as

VMS~X!5(i 51

Nd

Wi~R!Vi~X!, ~1!

where X and R denote 3N Cartesian and 3N26 internalcoordinates, respectively, for anN atom system,Wi(R) de-notes a weight function for thei th data point,Vi(X) is aTaylor expansion function aboutX( i ), the Cartesian coordi-nates for thei th data point, andNd is the number of datapoints. Note that Cartesian coordinates are used to calculate

potential energy values,46 while the weight function is basedon internal coordinates. This modification of the originalscheme is made, because the potential energy can be moreeasily calculated in terms of Cartesian coordinates in thepresent case, and the weight function should depend on onlythe molecular internal structure. For the Taylor expansionabout data points, we employ the second order series as

Vi~X!5V~ i !1(j 51

3N

~Xj2Xj~ i !!gj

~ i !

11

2 (j ,k51

3N

f jk~ i !~Xj2Xj

~ i !!~Xk2Xk~ i !!, ~2!

where V( i ), gj( i ) , and f jk

( i ) are the potential energy, thej thcomponent of the gradient vector, and thejkth component ofthe Hessian matrix, respectively, calculated byab initio elec-tronic structure calculations atX( i ). The utility of higher or-der derivatives~third and fourth! was investigated by Collinset al.,19 and it was concluded that, without an efficient meansfor calculating analytic third~or fourth! derivatives, secondorder derivatives provide the most cost-effective means ofconstructing a global PES by interpolation. The weight func-tion Wi(R), which approaches 1 asR→R( i ) and approaches0 asuR2R( i )u→`, is given by

Wi~R!5wi~R!

(k51Nd wk~R!

, ~3!

wi~R!5H (j

3N26

~Rj2Rj~ i !!2J 2p

. ~4!

Equation~3! assures the normalization of the weight function(( i 51

Nd Wi(R)51). The parameter 2p in Eq. ~4! must begreater than the larger of 3N26 and the order of the Taylorseries expansion to guarantee convergence of the potentialVMS(X).18

The simplest PES for a given elementary reaction is thereaction path~RP!-PES. Within the context of the interpola-tion scheme, the RP-PES can be constructed in terms of datapoints that are evenly spaced along the IRC. The utility ofthe RP-PES is based on the significance of the~static! reac-tion path. Since a chemical reaction is a dynamical process,it is important to include points in dynamically significantregions in the interpolation set. For this purpose, Collinset al.17–21 devised the following procedures: First, the RP-PES is constructed in terms of data points on the IRC; thenclassical trajectory calculations are carried out on this PES.Weights are assigned for selected points on these classicaltrajectories. These weights are large in regions that are~1!sampled by many trajectories and~2! far from data pointsthat are already included. The former condition indicates thedynamical significance of that nuclear arrangement. The fol-lowing weight function satisfying these two conditions wasintroduced by Collinset al.,17,21

h@R~ j !#51

NT21

(n~Þ j !NT wn@R~ j !#

(kNdwk@R~ j !#

, ~5!

4282 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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whereR( j ) denotes a set of 3N26 internal coordinates atthe j th selected point from a trajectory,NT is the number ofselected points, andNd is the number of data points forinterpolation.47 The functional form ofwk(R) is given in Eq.~4!. The point giving the maximum weight is added to thegroup of data points. Then, the contribution from this newpoint moves from the numerator to the denominator in Eq.~5!, and the weights in Eq.~5! are recalculated for the re-maining trajectory points that have been selected. This selec-tion of data points is repeated untilnT new data points arechosen (nT is a parameter!. Finally, ab initio electronic struc-ture calculations are performed to determine the energy, en-ergy gradient, and Hessian matrix at these new data points,resulting in an improved interpolated PES. Classical trajec-tory calculations are again carried out on this revised PES,thennT additional points are selected again according to Eq.~5!, and implemented to a group of data points.

The procedure described above for improving the PES iscontinued until one or more physical quantities that are de-termined by the PES~e.g., the reaction probability or errorsin the potential energy or energy derivatives! have con-verged. The reaction probability can be calculated as thefraction of trajectories that reach the product side. The errorin the PES itself can be estimated by the averaged deviationsof the energy and energy gradient at several selected refer-ence points,

ERRV5(i

M

uVMS~X i !2Vab initio~X i !u/M , ~6!

ERRG5(i

M

uGMS~X i !2Gab initio~X i !u/M , ~7!

whereVMS andVab initio denote the potential energy atX i ( i threference point! estimated by the~modified Shepard! inter-polation method and theab initio electronic structure calcu-lation, respectively;GMS andGab initio are the correspondingmagnitudes of the energy gradient atX i , andM is the totalnumber of reference points used for the error analysis.

An alternative method is proposed in this work for theselection of dynamically important points to be used for theinterpolation process. The original Collins method17,21 im-proves the PES iteratively based on classical trajectories,which run on the updated interpolated PES. In the initialstages, the trajectories will contain some level of inaccuracy;as the PES converges to the correct result, the trajectoriesalso converge to the correct ones. This approach may bereferred to as iterative sampling. As an alternative for obtain-ing a correctab initio trajectory, one can use the dynamicreaction path~DRP! method,2–8 in which a trajectory is inte-grated according to Newton’s equation of motion by usingthe potential energy and energy gradient calculated directlyby ab initio electronic structure methods. In this method, theonly inherent approximation is the computational level ofabinitio theory employed. Of course, the tradeoff is the expenseof a directly calculatedab initio trajectory which limits thenumber of trajectories that one can practically calculate. TheDRP method can be used to obtain correct reference valuesfor physical quantities~e.g., the reaction probability; the

trace of a trajectory! to check the convergence of the inter-polated PES. This DRP method can be used to directly de-termine dynamically important points with just one trial~i.e.,DRP sampling!. In the present study, we carry out DRP sam-pling for the selection of the data points, and compare thePES obtained in this manner with that obtained by iterativesampling.

III. REACTION SYSTEM: CH31H2˜CH41H

The modified Shepard interpolation method17,21 is ap-plied to the six-atom reaction, CH31H2→CH41H. As de-scribed in the Introduction, this IRC retainsC3v symmetrythroughout, thus the totally symmetric PES for this reactioncan be described in terms of four totally symmetriccoordinates.34 In the present study, this four-dimensional to-tally symmetric PES is constructed as the first step in build-ing the entire 12-dimensional PES~the totally symmetricPES may be the most important region for the reaction dy-namics!. Note that a classical trajectory also retains the spa-tial symmetry determined by the initial conditions~i.e.,atomic positions and velocities!. Figure 1 shows the fourtotally symmetric coordinates,R1–R4 , employed in thepresent study. The positions of H3, H4, and H5 change so thatthe system retainsC3v symmetry, thusR2 andR4 each rep-resent three equivalent atomic distances inC3v symmetry.Then, molecular structures in theC3v subspace can be ex-pressed byR1–R4 . These internal coordinates are used inthe weight function in Eq.~4! for the interpolation. Since thismethod requires a large number ofab initio electronic struc-ture calculations, and the main intent of this work is to dem-onstrate the utility of the method, the unrestricted Hartree–Fock ~UHF! method is employed with the 6-31G(d,p) basisset48 using theGAMESS program.49

Figure 2 shows variations in~a! energy and~b! geo-metrical parameters along the IRC. The activation energy forthis reaction is estimated as 21.4 kcal/mol, and the energy ofproducts, CH41H, is calculated as22.4 kcal/mol relative tothat of reactants, CH31H2. Figure 2~b! indicates that thisreaction can be characterized by changes inR1 (C¯H1) andR3 (H1¯H2). R2 remains nearly constant~;1.08 Å!throughout, while changes inR4 reflect those inR1 , so R4

can be expressed by a linear function ofR1 along the IRC. Inthe following, we discuss the quality of the PES based ontwo-dimensional sections described in terms ofR1 and R3 ,with R2 fixed at 1.08 Å.R4 is determined by the aforemen-tioned linear function ofR1 . Figure 3 shows theab initioPES in the vicinity of the transition state, constructed in

FIG. 1. Four total symmetry coordinates,R1–R4 , which describe the reac-tion, CH31H2→CH41H, in C3v subspace.R2 and R4 represent threeequivalent atomic distances of CH and HH, respectively.

4283J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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terms of ab initio ~UHF! energies at 400 grid points. Theenergy difference between the contour lines is 5 and themaximum contour is set to 40 kcal/mol. Thisab initio PESwill be used as a reference to check the quality of the inter-polated PES.

To integrate a trajectory, we employ the algorithm pro-posed by Stewartet al. for the DRP method.2 The time stepis fixed at 0.1 fs. In the initial stages of trajectory calcula-tions, quantization of vibrational motions is taken into ac-count as in quasiclassical trajectory simulations. WithinC3vsubspace, there are three internal totally symmetric normalvibrational modes in the reactants (CH31H2): CH3 stretch-ing ~3260.0!; CH3 umbrella ~339.6!; HH stretching~4635.1cm21!. The remaining totally symmetric mode correspondsto the collisional motion of CH3 and H2 in which C3v sym-metry is retained. Zero-point energy is assigned to the threeinternal vibrational modes, and 35 kcal/mol is provided tothe collisional mode. The latter is based on preliminary cal-culations and the observation that too high a collision energyleads to a reaction probability of 100%, while too low anenergy leads to no reaction. Within the harmonic approxima-tion, thei th normal coordinateQi and its conjugate momen-tum Pi in the vibrational ground states are

Qi5Ahn i /l i cosv i , ~8!

Pi5Ahn i sin v i , ~9!

wheren i and l i denote the vibrational frequency and forceconstant of thei th normal mode, respectively,v i denotes aphase angle, andh is Planck’s constant. Four phase angles, 0,p/2, p, 3p/2, are considered. This leads to 64 (543) initial

conditions for the trajectory calculations. Trajectories are ini-tiated in the region ofR1;2.5 Å. To provide a range ofinitial conditions, the initial value ofR1 is taken as

R152.510.13sinS 2np

4 D , ~10!

where n refers to 1 of the 64 trajectories. The trajectorycalculation is continued for 60 fs or until H2 is scattered fromCH3; that is, if R1 decreases from;2.5 to below 1.6 Å andthen back to distances that are larger than 1.6 Å. The crite-rion for reaction to CH41H is taken to beR351.5 Å.

The unnormalized weight functionwi(R) in Eq. ~4! in-cludes a parameter,p, which must be greater than half thevalue of the larger of~the number of internal degrees offreedom/the order of the Taylor series expansion!.18 As p isincreased, the weight for the closest point in a normalizedweight function@Eq. ~3!# increases sinceRj2Rj

( i ),1. With asufficiently large value ofp, one obtains a smoothly varyingPES. An extensive preliminary survey suggested thatp>4.So, p is set to 4 in this study. In order to choose new datapoints and conserve CPU time, the weight given in Eq.~5! isassigned for one-seventh of all the points on trajectories. Inthe iterative sampling, the optimal number of data pointsadded in one cycle (nT) was determined to be 30 throughseveral trial and error calculations. By repeating this processten times, 300 data points are added into the group of datapoints for the interpolation.

IV. RESULTS AND DISCUSSION

In the interpolation scheme for the PES, the initial PES,i.e., the reaction path PES~RP-PES! constructed from datapoints evenly scattered along the IRC, may be the mostprimitive. With this RP-PES, the energy for a given pointcan be calculated by the weighted average of the secondorder Taylor series about the respective data points as indi-cated by Eqs.~1! and ~2!. In the present study, 39 points onthe IRC are chosen as data points: 1 corresponding to the TS,19 points from the reactant side, and 19 points from theproduct side. Figure 4 shows~a! positions of these datapoints along the IRC and~b! the calculated RP-PES in the(R1 ,R3) plane ~using interpolated energies at 3600 grid

FIG. 3. Ab initio PES around a TS region, constructed in terms of UHFenergies at 400 grid points.

FIG. 2. Variations of~a! energy and~b! geometrical parameters,R1–R4 ,along the IRC (CH31H2→CH41H).

4284 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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points!. In this figure, R2 is fixed at 1.08 Å whileR4 isdetermined as a linear function ofR1 ~see Sec. III!. One cansee from Fig. 4~a! that no significant molecular deformation~abstraction of H! occurs until~approaching from the right inthe figure! R1 decreases to;1.6 Å. The structural transfor-mation continues untilR3 exceeds about 1.3 Å@see also Fig.2~a!#. In Fig. 4~b!, the focus is on the PES around this im-portant region close to the transition state. The energy differ-ences between the contour lines is the same as that in Fig. 3.Since the harmonic approximation has been used for the di-rections orthogonal to the reaction path, the energy variationon both sides of the IRC looks symmetric relative to the IRC,in contrast with the PES in Fig. 3.

The energy variation in bond stretching directions ismore appropriately expressed by a Morse function than by aquadratic~harmonic! function. In the Morse function, theenergy increase along the positive coordinate~toward bonddissociation! is softened, while it becomes harder along thenegative coordinate. So, for a given energy, the system ismore easily displaced~i.e., is more anharmonic! in the posi-tive than in the negative direction. The reaction system in thepresent study has four totally symmetric coordinates, each ofwhich may be classified as a bondingtype coordinate. Sincethe reaction path is characterized explicitly by two of thesecoordinates,R1 and R3 , as illustrated in Fig. 2, 28 datapoints were added to the positive sections ofR1(51.3 Å).andR3(50.9 Å) relative to the IRC in preparing the initialPES@indicated by the black circles in Fig. 5~a!#. These ad-ditional data points are approximately on the reaction planedetermined by the reaction path tangent and curvature vec-

tors, which are frequently important in dynamics.5,34,35 Fig-ure 5 shows~a! positions of the initial data points~whitecircles for those on the IRC and black circles for those in theconcave side of the IRC; 67 in total! and~b! the interpolatedPES calculated with these data points. As illustrated in Fig.5~b!, the energy on the concave side of the IRC is loweredconsiderably in comparison with that in Fig. 4~b! ~RP-PES!.In the following, the PES in Fig. 5~b! is employed as theinitial PES to be improved.

Figure 6 shows~a! traces of 64 trajectories calculated onthe initial PES with 67 data points~white circles!, and ~b!traces of 64 DRP trajectories calculated with the same initialconditions. Note that the simulations presented here are ac-tually carried out on a four-dimensional PES, the four di-mensions corresponding to the four totally symmetric inter-nal coordinates. The DRP simulations are exact on theUHF/6-31G(d,p) PES. This computational level is also em-ployed for theab initio molecular orbital~MO! energies andderivatives used here to construct the interpolated PES.Thus, the DRP simulations can be used as a reference to testthe quality of the approximate PES employed. At firstglance, these two sets of trajectories exhibit similar behavior,but there are some interesting differences: DRP trajectoriescan deviate more extensively relative to the IRC in the prod-uct ~left! side. In addition, two trajectories return across theconcave side of the TS toward the reactant side in Fig. 6~a!,while only one trajectory returns in this manner in Fig. 6~b!.Overall, more trajectories seem to return to the reactant sidein Fig. 6~b!, judging from the density difference of trajecto-ries. This last point can be verified by estimating the reaction

FIG. 4. ~a! Positions of the initial data points evenly spaced along the IRCand ~b! the calculated RP-PES, in a (R1 ,R3) plane.

FIG. 5. ~a! Positions of the initial data points~white circles for those on theIRC and black circles for those in the concave side of the IRC! and~b! theinterpolated PES calculated with these data points.

4285J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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probability in the respective cases: 88% in Fig. 6~a!, 75% inFig. 6~b!.50 These differences may be due to differences ingeometric features of the PES near the TS region.

Next, we proceed to improved potential energy surfaces,in which the additional data points selected either by iterativesampling or by DRP sampling have been included. Usingiterative sampling, 30 data points are added by one simula-tion ~64 trajectory calculations on a given PES!; this proce-dure is repeated ten times, resulting in a total of 300 addi-tional data points. So, in this sampling, the PES is improvedstep by step through iterative trajectory calculations. On theother hand, in the DRP sampling, 300 data points for theinterpolation can be added through just one simulation, sincethe DRP trajectory runs on the exact PES for this level oftheory~of course, both sets of calculations are limited by theapproximations in theab initio MO method employed!. Theobvious disadvantage of the DRP sampling is the computa-tional cost.

The selection of data points was performed using theweight function in Eq.~5! for both methods. Figure 7 showsdistributions of initial ~white circle! and additional~blackcircle! data points selected by~a! the iterative sampling and~b! the DRP sampling. For both methods, many data pointsare observed on the convex side of the IRC around the TS.This is due to the centrifugal force acting on the reactionsystem in that region due to the curvature of the IRC. Com-

parison of Figs. 7~a! and 7~b! shows that the data points aredistributed more or less uniformly along the IRC in the it-erative sampling procedure, while the data points accumulatemore on the reactant~right! side in the DRP sampling. Thosedistributions reflect the distributions of trajectories shown inFig. 6.

Figure 8 shows the improved PES with the additionaldata points selected by~a! the iterative sampling and~b! theDRP sampling, corresponding to Figs. 7~a! and 7~b!, respec-tively. In comparison with theab initio PES in Fig. 3, theyhave somewhat ragged regions; the additional data points onthe convex side of the IRC lower the potential energy around(R1 ,R3)5(1.0,1.0). Figures 9~a! and 9~b! show traces of 64trajectories calculated on the improved PES with 367 datapoints, corresponding to Figs. 8~a! and 8~b!, respectively. Incomparison with Fig. 6~a!, they seem to approach more cor-rect behavior, especially on the product side.

To check the quality of the PES quantitatively, the reac-tion probability and errors in the potential energy@Eq. ~6!#and energy gradient@Eq. ~7!# were calculated using the twopotential energy surfaces. To determine how the interpolatedPES converges as additional data points are added, the reac-tion probability and errors for the DRP-sampling-PES werecalculated at every 20th additional data point. Figure 10shows variations in the reaction probability for the iterativesampling and the DRP sampling. The dotted line corresponds

FIG. 6. ~a! Traces of 64 trajectories calculated on the initial PES with 67data points of white circles and~b! DRP trajectories calculated with thesame initial conditions.

FIG. 7. Distributions of initial~white circle! and additional~black circle!data points for the interpolation:~a! the iterative sampling;~b! the DRPsampling.

4286 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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to the exact value~75%! for the UHF/6-31G(d,p) PES esti-mated by DRP simulations. Due to the small number of ad-ditional data points, the reaction probability decreases rap-idly from 88% to about 70% in both sampling methods. Theyshow fluctuations around 70% for a while, then convergenear 75% with more than 200 data points. As referencepoints for estimating errors in the energy and gradient fromab initio values @Eqs. ~6! and ~7!#, 448 data points wereselected from points on the DRP trajectories at random. Fig-ure 11 shows changes in these errors in~a! energy and~b!energy gradient against the number of additional data points.Both figures indicate a slight advantage of the DRP samplingin comparison with the iterative sampling. The errors de-crease rapidly for both methods after adding the first groupof data points, then undergo relatively small changes as thenumber of data points is increased further.

V. CONCLUSION

In the present paper, we have applied the modified Shep-ard interpolation method proposed by Collinset al. to theconstruction of the PES in a polyatomic reaction includingsix atoms, CH31H2→CH41H. As a starting point for aneventual study of the entire PES, the focus is given to thetotally symmetric~four-dimensional! PES inC3v symmetry,which is conserved along the IRC. The weight function isdescribed in terms of four totally symmetric~internal! coor-dinates, while the potential energy is calculated based on asecond order Taylor series in Cartesian coordinates. In this

methodology, starting from the reaction path PES, the PES isimproved by adding data points chosen using both the DRPsampling method, as well as the iterative method developedby Collins and co-workers. Although the DRP method re-quires a greater computational cost, the additional data points

FIG. 8. The improved PES interpolated by the initial data points plus addi-tional ones selected by~a! the iterative sampling and~b! the DRP sampling.

FIG. 9. Traces of 64 trajectories calculated on the improved PES with 671300 data points selected by~a! the iterative sampling and~b! the DRPsampling.

FIG. 10. Reaction probability as a function of number of additional datapoints by the iterative sampling and by the DRP sampling. The exact valueis calculated by the DRP simulation.

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can be determined by just one simulation. Comparison of thedata points determined by the two different methods suggeststhat, although their convergence of the reaction probability issimilar to each other, the distributions of data points and theconvergence in errors of energy and energy gradient seem tobe a little better in the DRP sampling than in the iterativesampling. Since the DRP method is generally applicable tomolecular systems of any size, the present study suggeststhat the DRP method should be an effective partner to theinterpolation method for the construction of potential energysurfaces.

ACKNOWLEDGMENTS

We are grateful to Dr. Nobuo Tajima, Mr. Kenichi Na-kayama, and Mr. Tomohiro Hashimoto for helpful sugges-tions on calculations of standard deviations of reaction prob-ability. T.T., T.T., and K.H. were supported by Grant-in-Aidfor Scientific Research on Priority Areas from Ministry ofEducation, Science and Culture in Japan, and M.S.G. wassupported by grants from the National Science Foundationand the Air Force Office of Scientific Research.

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4288 J. Chem. Phys., Vol. 109, No. 11, 15 September 1998 Takata et al.

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50Since these reaction probabilities~88% and 75%! are estimated based on asmall number~564! of trajectories, they should be viewed within thecontext of the statistical uncertainty. Their standard deviations were cal-culated as 0.041 and 0.054, respectively.

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