van Harrevelt R, Honkala K, Norskov JK, Manthe U. The effect of surface relaxation on the N2 dissociation rate on stepped Ru: A Transition State Theory Study. Journal of Chemical Physics. 2006;124(2):026102: 026102

Jens K. Nørskov

Karoliina Honkala

Rob van Harrevelt

Uwe Manthe

Wigner E.

van Harrevelt, R.

Honkala, K.

Norskov, J. K.

Manthe, Uwe

Publications at Bielefeld University

The effect of surface relaxation on the N2 dissociation rate on stepped Ru: A Transition State Theory Study

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The effect of surface relaxation on the N2 dissociation rate on stepped Ru: A transition state theory study

van Harrevelt, Rob

Honkala, Johanna Karoliina

Nørskov, Jens Kehlet

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017The effect of surface  relaxation on the N-2 dissociation rate  on stepped Ru: ATransition State Theory Studyvan Harrevelt, Rob; Honkala, Johanna Karoliina; Nørskov, Jens Kehlet; Manthe, UwePublished in:Journal of Chemical PhysicsLink to article, DOI:10.1063/.1.2150827Publication date:2006Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):van Harrevelt, R., Honkala, J. K., Nørskov, J. K., & Manthe, U. (2006). The effect of surface  relaxation on the N-2 dissociation rate  on stepped Ru: A Transition State Theory Study. Journal of Chemical Physics, 124(2),026102. DOI: 10.1063/.1.2150827The effect of surface relaxation on the N2 dissociation rate on stepped Ru:A transition state theory studyRob van HarreveltaTheoretische Chemie, Technische Universität München, Lichtenbergstraße 4, D-85747 Garching, GermanyKaroliina Honkalab and Jens K. NørskovCenter for Atomic-Scale Materials Physics, Department of Physics, Technical University of Denmark,DK-2800 Lyngby, DenmarkUwe MantheTheoretische Chemie, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, GermanyReceived 25 August 2005; accepted 18 November 2005; published online 10 January 2006DOI: 10.1063/1.2150827The dissociation of N2 on stepped Ru is the rate limitingstep in the industrial synthesis of ammonia on Ru catalystssee Refs. 1–4. In a preceding paper,5 we have examined theeffect of tunneling on the reaction rate for N2 on a steppedRu surface with 0001 terraces. The positions of the surfaceatoms were frozen, in order to reduce the computational ef-fort. The strong tunneling effect found in previous modelcalculations6–11 is not found. In contrast, the tunneling effectwas very small above room temperature. The thermal rateconstant obtained from harmonic transition state theoryTST therefore agrees very well with the accurate quantum-mechanical result. In this note we further examine the accu-racy of the harmonic TST rate constant and consider theeffect of surface motion. Different models to take the effectof surface motion in account are employed. The best theo-retical result for the thermal sticking coefficient, which isproportional to the thermal rate constant, is then comparedwith experiment.Density-functional theory DFT calculations are per-formed to localize the transition state and to determine ener-gies and vibrational frequencies at the transition state. Tech-nical details are the same as in Ref. 5, except that the presentwork also includes relaxations of the Ru atoms in the top-most two layers. The transition state of N2 dissociation islocalized by constraining the N–N distance and relaxing boththe other nitrogen degrees of freedom and the positions ofthe Ru atoms in the topmost two layers. By varying the N–Ndistance we localized the saddle point. The calculated barrierheight for a relaxed surface is 0.65 eV. It should be notedthat this barrier is higher than the one reported by Logadóttirand Nørskov.1 They used the same 42 unit cell and thesame relaxed surface model. The difference is mainly due tothe more accurate localization of the saddle point in thepresent work.The relaxation of the surface has a considerable effect onthe barrier height, lowering it from 1.0 to 0.65 eV. There-fore, it is obvious that surface degrees of freedom cannot beneglected. In principle, harmonic TST would require that allrelaxed degrees of freedom are included in to the dynamicaltreatment. This requires lots of computer time but does notnecessarily change results. Thus we include only selectedatoms in the dynamical treatment.The expression for the thermal rate constants k in tran-sition state theory is given ask =kBT2Q‡Qreactantse−E/kBT, 1where E is the vibrationally adiabatic barrier height and Q‡and Qreactants are the partition functions of the activated com-plex and reactants per unit volume, respectively. E, Q‡,and Qreactants are defined according to the specific reduceddimensional model employed. Here, the harmonic approxi-mation is used to evaluate E and the partition functions.The thermal rate constant k is related to the sticking coeffi-cient S according toS =2mkBTnactivek , 2where m is mass of a N2 molecule, and nactive the density ofactive sites on the surface. For nactive we use the value esti-mated by Logadóttir and Nørskov nactive1.571017 m−2.1The simplest approximation to include the motion of thesurface atoms into the rate constant is to calculate the barrierheight and the vibrational harmonic frequencies of N2 at thetransition state on a relaxed surface but neglect the vibrationsof the surface atoms. Here, we call this model the “relaxed/N2 model.” This type of modeling is frequently used in theliterature. To investigate the role of surface vibrations a moreadvanced model is needed. In a “relaxed/N2+2Ru” modelthe vibrations of two-step Ru atoms next to the upper N atomare included, see Fig. 1. Due to the interaction between adissociating N2 molecule and the surface, the Ru atoms at thestep relax upwards compared to the clean surface positions.In the relaxed/N2+2Ru model we consider the motion ofboth Ru atoms in all three dimensions. Thus, the vibrationalmotion at the transition state involves six nitrogen and sixruthenium degrees of freedom. The harmonic frequencies forthe different models are given in Table I. For the N2+2Rumodel, modes 1–5 and the mode corresponding to the imagi-nary frequency mainly involve motion of the nitrogen atoms,THE JOURNAL OF CHEMICAL PHYSICS 124, 026102 20060021-9606/2006/1242/026102/3/$23.00 © 2006 American Institute of Physics124, 026102-1Downloaded 16 Jun 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jspand modes 6–11 mainly involve motion of the Ru atoms. Bycomparing the frequencies of the relaxed/N2 and relaxed/N2+2Ru models we can conclude that the small differencesin frequencies are probably due to the coupling between ni-trogen and ruthenium degrees of freedom in the relaxed/N2+2Ru model.The surface vibrations do not change the vibrationallyadiabatic barrier height significantly, as can be seen in TableI. The coupling changes only the factor Q‡ /Qreactants andtherefore has only a small effect on the sticking probability.The thermal sticking coefficients are presented in Fig. 2. Theresults of the two models considered differ by less than 10%.This suggests that the relaxed/N2 model, which only takesrelaxation of the surface into account but neglects Ru fre-quencies explicitly, can be used to reliably predict the stick-ing probability or rate constant.Figure 2 shows that the sticking coefficient obtained us-ing the relaxed surface models with an activation energy of0.65 eV is two to four orders of magnitude smaller than theexperimental result of Dahl et al.3 Previous work shows5 thatthe tunneling is very small and can therefore not explain theobserved discrepancy between theory and experiment. How-ever, the dependence of the barrier height on the size of theunit cell partially explains the discrepancy. With a larger 52 unit cell the upper terrace is one atom row broader,2employing the relaxed/N2 model, we obtain a barrier of only0.49 eV.2 Using the 52 unit cell improves the agreementwith the experiment considerably see Fig. 2. If we lowerthe classical barrier further down to 0.35 eV, an absoluteagreement between theory and experiment is achieved, asshown in Fig. 2. This suggests that in reality the barrier isaround 0.35 eV, which agrees very well with the value mea-sured by Dahl et al.4 An error of 0.14 eV for the barrierheight is typical for DFT calculations on molecule-surfacesystems.The present results also provide an a posteriori justifica-tion of the frozen surface approximation in previous work,5where the contribution of the tunneling effect to the reactionrate was studied. Our justification is based on the followingtwo arguments. First, the normal-mode analysis for the re-laxed surfaces models suggests that the coupling betweennitrogen and ruthenium degrees of freedom is small at thetransition state. Second, the small differences between thetransition state harmonic frequencies for the frozen surfacemodel and the relaxed/N2 model see Table I indicate thatrelaxation of the surface atoms has a small effect on the localshape of the potential around the transition state. In particu-lar, the imaginary frequency changes by less than 20%. Asdiscussed in Ref. 5, the tunneling effect is mainly determinedby the imaginary frequency i and is adequately describedby the Wigner tunneling theory.12 According to this theory,the ratio between the rate constants with and without tunnel-ing is given by the tunneling factor =1+ 124i /kBT2.12FIG. 1. The transition state for dissociation of N2. The big and small ballsrepresent Ru and N atoms, respectively. The marked Ru atoms are the twosurface atoms whose vibrations are included in the relaxed/N2+2Ru model.TABLE I. DFT results: the classical and vibrationally adiabatic barrierheight and harmonic frequencies at the transition state and of the reactantsisolated N2g and 2 Ru atoms.Transition stateRelaxed/N2+2Ru Relaxed/N2 FrozenClassical barrier 0.648 eV 0.648 eV 0.999 eVE 0.641 eV 0.641 eV 0.992 eVImaginary frequency 484 i cm−1 483 i cm−1 409 i cm−11 mode 1 594 cm−1 564 cm−1 578 cm−12 mode 2 514 cm−1 512 cm−1 517 cm−13 mode 3 469 cm−1 466 cm−1 446 cm−14 mode 4 447 cm−1 437 cm−1 423 cm−15 mode 5 403 cm−1 402 cm−1 411 cm−16 mode 6 193 cm−17 mode 7 183 cm−18 mode 8 180 cm−19 mode 9 159 cm−110 mode 10 152 cm−111 mode 11 119 cm−1Isolated N2Geometry 2.11a0  2492 cm−1Isolated surface 2Ru atoms1…6 207, 192, 188, 148, 140, and 139 cm−1FIG. 2. Arrhenius plot of the thermal sticking coefficient obtained fromharmonic TST calculations employing different models. The circles are theexperimental results of Dahl et al. Ref. 3.026102-2 van Harrevelt et al. J. Chem. Phys. 124, 026102 2006Downloaded 16 Jun 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jspFor example, at room temperature 300 K, the tunnelingfactors are 1.16 and 1.22 for the frozen and relaxed surfacemodels, respectively. Although the tunneling effect is some-what stronger for the relaxed surface, it still has a marginaleffect on the rate constant. At temperatures relevant to am-monia synthesis, the difference is even less: at 750 K  is1.03 and 1.04 for the frozen and relaxed surface models,respectively. The effect of surface relaxation is thus predomi-nantly an energy shift. Due to this energy shift the frozensurface model cannot yield quantitative reaction rates, as isclearly demonstrated in Fig. 1. However, the quantum dy-namics at the transition state region is correctly described inthe frozen surface model.The present work shows that the effects of surface mo-tion are adequately accounted for in the relaxed/N2 model. Ina previous study we have shown that tunneling effects arenegligible.5 Thus, harmonic TST employing the relaxed/N2model is an accurate approach to calculate the rate constantsfrom DFT data. The applicability of this model in first-principles study of the ammonia synthesis rate2 is thereforeconfirmed by the present results. The main source of errorare the DFT calculations itself. Fortunately, the results of theoverall ammonia production are quite insensitive to the DFTerrors due to the compensation effect.13The work is financially supported by the European Com-mission through the RTN Program Predicting CatalysisHPRN-CT-2002-00170, the Deutsche Forschungsgemein-schaft, and by the Fond der Chemischen Industrie. One ofthe authors K.H. acknowledges the computer resourcesfrom the Danish Center for Scientific Computing.aPresent address: Instituut voor Theoretische Chemie, Radboud UniversiteitNijmegen, Toernooiveld 1, 6525 ED Nijmegen, the Netherlands. Electronicmail: r.vanharrevelt@science.ru.nlbPresent address: Department of Physics, Nanoscience Center, P.O. Box 35,FIN-40014 University of Jyväskylä, Finland.1 Á. Logadóttir and J. K. Nørskov, J. Catal. 220, 273 2003.2 K. Honkala, A. Hellman, I. N. Remediakis, Á. Logadóttir, A. Carlsson, S.Dahl, C. H. Christensen, and J. K. Nørskov, Science 307, 5709 2005.3 S. Dahl, E. Törnqvist, and I. Chorkendorff, J. Catal. 192, 381 2000.4 S. Dahl, Á. Logadóttir, R. C. Egeberg, J. H. Larsen, I. Chorkendorff, E.Törnqvist, and J. K. Nørskov, Phys. Rev. Lett. 83, 1814 1999.5 R. van Harrevelt, K. Honkala, J. K. Nørskov, and U. Manthe, J. Chem.Phys. 122, 234702 2005.6 S. Holloway, D. Halstead, and A. Hodgson, J. Electron Spectrosc. Relat.Phenom. 45, 207 1987.7 G. Haase, M. Asscher, and R. Kosloff, J. Chem. Phys. 90, 3346 1989.8 G. D. Billing, A. Guldberg, N. E. Henriksen, and F. Y. Hansen, Chem.Phys. 147, 1 1990.9 L. Romm, G. Katz, R. Kosloff, and M. Asscher, J. Phys. Chem. B 101,2213 1997.10 L. Romm, O. Citri, R. Kosloff, and M. Asscher, J. Chem. Phys. 112,8821 2000.11 N. E. Henriksen, F. Y. Hansen, and G. D. Billing, Chem. Phys. Lett. 330,139 2000.12 E. Wigner, Z. Phys. Chem. Abt. B 19, 203 1932.13 T. Bligaard, K. Honkala, Á. Logadóttir, J. K. Nørskov, S. Dahl, and C. J.H. Jacobsen, J. Phys. Chem. B 107, 9325 2003.026102-3 N2 dissociation of a stepped Ru surface J. Chem. Phys. 124, 026102 2006Downloaded 16 Jun 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

The effect of surface relaxation on the N-2 dissociation rate on stepped Ru: A Transition State Theory Study

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The effect of surface relaxation on the N-2 dissociation rate on stepped Ru: A Transition State Theory Study

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