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Dynamical polarization effects in dissociative sticking of a diatomic molecule with energy loss to the substrate
Surface Science 449 (2000) L243–L247 www.elsevier.nl/locate/susc
Surface Science Letters
Dynamical polarization effects in dissociative sticking of a diatomic molecule with energy loss to the substrate G.P. Brivio a, *, T.B. Grimley b,1, M.I. Trioni a a Istituto Nazionale per la Fisica della Materia and Dipartimento di Scienza dei Materiali, Universita` di Milano ‘Bicocca’, via Cozzi 53, 20125 Milan, Italy b The Donnan Laboratories, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK Received 15 November 1999; accepted for publication 12 January 2000
Abstract We solve exactly and fully quantum mechanically a model Hamiltonian which describes energy loss to the substrate in dissociative sticking. The inelastic process is assumed as being due to electron–hole pairs. Results are calculated for the system H /Cu(111). It is shown that the main effect is a dynamical polarization of the electron system with 2 no electron–hole pairs in the final channel, which enhances the sticking coefficient with respect to that calculated only with the elastic molecule–solid potential. Substrate excitations in the final channel contribute little to the sticking coefficient. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Chemisorption; Energy dissipation; Models of surface chemical reactions; Sticking
Important information on the dissociative sticking of a diatomic molecule was recently obtained by computer simulations of the time evolution of a quantum mechanical wavepacket on the adiabatic molecule–surface multi-dimensional potential energy surfaces (PESs) [1–4]. Such PESs are derived from ab initio density functional based calculations which realistically describe the molecule–surface system. However, such first principle quantum methods cannot yet be used to treat the excitation system in reactive molecule–surface * Corresponding author. Fax: +39 02 6448 5403. E-mail address: [email protected] (G.P. Brivio) 1 Permanent address: 41 Beechways Drive, Neston, Cheshire CH64 6TF, UK.
dynamics. This problem needs to be addressed because for the dissociative sticking of molecules an energy of the order of an electronvolt is transferred to the substrate. Since phonon energies are 50 to 100 times smaller than this, either multi-phonon excitations, or perhaps single electron–hole pairs are obvious alternatives. The sticking of a reactive particle on a metal with the creation of phonons or electron–hole pairs has been the subject of several investigations over 20 years [5–12]. For dissociative sticking, this problem has been extensively investigated by Billing and coworkers [13–16 ]. They performed self-consistent semi-classical and quantum wavepacket calculations for H on copper, which 2 accounted for the multi-dimensional character of the PES, and used an effective dynamical coupling
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to describe the interaction both with the substrate electronic excitations and with the phonons. In this way they could obtain sticking coefficients within a framework capable of considering several phenomenological features of the system. But one would also like to ask a more general question on the relevance and the effects of energy loss mechanisms in dissociative sticking, because, whilst energy loss is essential for the sticking of a single atom, for the dissociative sticking of a diatomic molecule it is not. In fact, the exothermicity may be dissipated to the excitation system when the two atoms are far apart on the surface, in which case the sticking coefficient is uninfluenced by inelastic processes. But the energy loss may also occur at a more critical stage in the sticking dynamics, when the molecule bond is breaking, and the two surface bonds are forming. In this paper, in order to contribute to answering the above question, we present results for the dissociative sticking coefficient, s, obtained by a quantum analytically solvable model which contains an inelastic molecule–solid potential in addition to the simple static PES introduced originally by Grimley [17]. The inelastic potential we use can describe aspects of energy loss to phonons or electrons, and here, we parameterize the model for the latter. We show that the most important contribution to s from the inelastic potential is a dynamical polarization effect of the electronic system, non-adiabatically induced by the molecule. This effect does not imply electron–hole pairs in the final stuck channel, and increases the sticking coefficient with respect to that calculated on the adiabatic PES. We consider a diatomic molecule of mass M at normal incidence with its internuclear axis parallel to the solid surface. The Hamiltonian is: +H +H +U+W, trans rel s p2 H = z +V (z), f trans 2M
(1)
V =−gd(z)+ lim VH(z−d ), f V2 p2 H = x +V (x), i rel 2m
(3)
H=H
H =∑ |ne n|, s n n
(2)
(4) (5)
(6) U=ad(z) ∑ |nW1 W ∞ n∞|, n n n,n∞ W=h ∑ |mg1 Ug n|, h=c/a. (7) m n m≠n z is the molecule’s centre-of-mass coordinate, normal to the surface, x, the interatomic distance, is parallel to it, and m is the molecule’s reduced mass. The potential V in the final channel consists f of a d-function of strength −g at the minimum energy position z=0, and a hard wall at z=d representing the repulsive part of the atom–solid interaction. V supports one bound state of f energy E =−Mg2/2B2=−B2K2 /2M, where B B K =gM[1−exp(−2K d )]/B2. |n are eigenstates B B (bound and unbound) of the relative motion on V (x), and the molecule is initially in the bound i (vibrational ) state |i with energy e . H is the i s Hamiltonian for the excitation system, whose ground state is |0, and whose excited states are labelled |m, with m≠0. U is the elastic interchannel coupling with strength a, and W is the inelastic one with strength c. To obtain sticking coefficients in a closed form, both potentials U and W have to be separable, with details defined by the set of coefficients {W} and {g, W}, respectively. Projected onto the coordinate x, they can account for the structure of the elastic interchannel coupling, which depends on the local adiabatic electronic structure at z=0, and also determine the inelastic coupling to the excitation system. Still the Hamiltonian in Eqs. (1)–(7) provides a much simplified picture of dissociative sticking: the molecule does not rotate while approaching the surface, there is no surface corrugation, the elastic and inelastic potentials only act at a particular (close) distance from the surface. However, the model contains the essentials of dissociative sticking, i.e. energy conversion from the degrees of freedom of the impinging molecule to those parallel to the surface of the two dissociated atoms. Also the electron–hole pair coupling, due to an interaction rapidly decaying outside the metal, is described by the potential W in a limiting case. Since the potentials U and W are separable, the Lippmann–Schwinger equation for the relevant scattering state |Y+ can be solved exactly in an analytical way in terms of the amplitudes a+ (z)= z| m| m|Y+. By performing a calculamm
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tion, to be presented elsewhere [18], we obtain the sticking coefficient. Here, it is sufficient to record that: M
|RW |2|a+ (E )|2 ∑ A i i0 i m B2K m 0 W(E −e −E ) 2 i m B × n (E −e −E ), v i m B D(e ; E ) m i with: s=4p2
K
K
(8)
G
1 for m=0, (9) A = m |hg g1 |2 for m≠0. 0 m In Eq. (8) E =E +e is the initial energy i trans i of the molecule, whose translational one is E =B2K2 /2M. The constant R only depends trans 0 on K and d. The density n (e) refers both to B v discrete bound vibrational states of the molecule and to continuous unbound ones of the two dissociated atoms. The complex function D(e ; E ), m i which depends on all coupling constants, contains the details of the dynamics of sticking. It is important to note that the summation in Eq. (8) naturally separates into two terms, the former one, s 1 for m=0, and the latter one, s for m≠0, so that 2 s is the sum of two contributions, s=s +s , where 1 2 s involves only virtual excitations, whereas s 1 2 contains the real energy loss. To write s we have 2 to pass to the continuum for the excitation spectrum. After introducing the excitation probability P(e ) and density of states n (e ) per unit m e m energy, one can write:
K
K
W(E −E ) 2 i B n (E −E ), v i B D (E ) 0 i Ei−EB s = de P(e ) 2 m m
s 3 1
(10)
P
0 Ei−EB de n (e )A(e ) 3 m e m m
P 0
K
K
W(E −e −E ) 2 i m B n (E −e −E ). (11) v i m B D(e ; E ) m i The integral upper limit follows from energy conservation. Since s labels the dissociative sticking 0 ×
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coefficient on the static PES, s −s , which is the 1 0 result of virtual excitations, will be referred to as the dynamical polarization contribution. Up to this point no hypothesis on the excitation system has been made, only the density of states appears in our equations. Hereonwards, for sake of simplicity, we assume the inelastic potential W as being due to single electron–hole pair excitations with density of states per atom n˜ (e )3e (but see e m m the following). We also remark that the form of the inelastic potential in Eq. (7) is chosen so as to provide a model which is exactly solvable for multi-excitations, not simply for single excitations. And it is correct for linear coupling to a boson excitation system, such as the electron–hole pair system just introduced, as well as to a phonon one. The qualitative difference between the two systems does not mainly lie in the form of the density of states, which is quadratic for single vibrational excitations. It is related instead to the excitation energy cutoff. In fact that of the single electron–hole pairs is almost two orders of magnitude larger than the Debye energy of the phonon system. So the electron–hole pair cutoff energy is indeed comparable with those exchanged in dissociative sticking of light reactive molecules on metals. To obtain results, it is natural to parametrize our model to the bench mark system of dissociative sticking, i.e. to H /Cu(111). Spurious oscillations 2 in the sticking coefficients are avoided if K d≤p. B Accordingly we have chosen the parameter d equal to 0.07 Bohr. The distribution and the energy dependence of the coefficients {W} has been determined from an interchannel separable potential following Ref. [19]. The parameter a˜ =a/d=1.0 eV has been selected in order to give a reasonably realistic interchannel coupling. It is also multiplied by a transmission coefficient through a barrier of height E =0.5 eV to simulate an activated sticking b process such as H on Cu. Since we have no 2 information on separable potential couplings to electron–hole pairs, all the coefficients g , m≠0, m have been taken to be constant and equal to unity. The parameter |g |2h=20 is chosen so that s is a 0 good fit to the experimental results of Rettner et al. [20]. The results, for an impinging H 2 molecule in its vibrational ground state, and as a function of the translational kinetic energy E , trans
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are shown in Fig. 1, which displays s=s +s (solid 1 2 line), s (dashed line) and s (dot-dashed line). The 1 0 most important feature of these results is that the dynamical polarization contribution s −s is the 1 0 dominant effect of the inelastic potential; energy loss to electron–hole pairs has, by comparison, little effect on the dissociative sticking coefficient. To confirm this result, in Fig. 2 we plot s and s 1 as a function of |g |2h at E =1.0 eV with a˜ as 0 trans before, and we have also verified that such a
Fig. 1. The dissociative sticking coefficient of H on Cu(111) 2 versus E , the translational energy of the incident molecule. trans s (dot-dashed line) is the value calculated for the rigid potential 0 energy surface (inelastic potential set to zero), s (dashed line) 1 includes the dynamical polarization of the electron–hole pair system by the dissociating molecule, and s (solid line) is the total value which includes in addition s , the real energy loss to 2 electron–hole pairs. For parameters, see the text.
Fig. 2. The dissociative sticking coefficient of H on Cu(111) 2 versus |g |2h, the electron–hole pair coupling strength: s (solid 0 line) and s (dashed line). 1
behaviour of s is essentially independent of a˜ . Both the maximum and the asymptotic behaviour of s and s for large h are expected on general grounds. 1 For the latter observe that in the limit of a strong inelastic potential there is no energy loss, and s collapses to s . The minimum is instead due to the 1 particular form of the equations for this model. To find further support for our results, we have solved exactly a different Hamiltonian, that in Eq. (1) but with a=0. We have verified that for a wide and physically significant range of values of h, s %s , though here one cannot even define an elastic 2 1 sticking coefficient s . We have also checked that the 0 above mentioned result is not a consequence of the simple form used for the density of states n˜ (e ). e m Even if n˜ (e ) is quite different from that for single e m electron–hole pairs in the nearly-free electron spectrum, being instead a narrow peak at some energy E associated with a local adiabatic electronic p structure responsible for the chemisorption bond, this still does not lead to a larger contribution to s and 2 to any corresponding structure in s , because only 2 integrated quantities appear in s . This theoretical 2 conclusion agrees with experimental findings that dissociative sticking coefficients show a rather featureless dependence on the translational energy of the incoming molecule, and that no electron–hole pairs are detected in dissociative sticking of H on noble 2 metals [21]. Of course one could argue that the large ratio s /s which we found is somewhat a model depen1 2 dent result. But, in any case, the solution of the Hamiltonian [Eqs. (1)–(7)] points out that a static PES may not always be adequate for the dynamics, because we have shown that the inelasticity gives two contributions to dissociative sticking: the former, the dynamical polarization one, with no excitations in the stuck channel, and the latter one with exothermicity dissipated to the electron–hole pairs. Both effects contribute to increasing s with respect to that calculated on the adiabatic PES. We wish to observe that this result finds some support from the multi-dimensional ab initio simulations. In fact in Ref. [1] it is pointed out that the reported 6D simulation on the adiabatic PES underestimates the sticking coefficient of H on 2 Cu(100) with respect to the experimental result. It is also useful to make a connection between our
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results and the recent ones obtained by simulations where inelasticity is included. In Ref. [15] it is shown that inelasticity increases the dissociative sticking coefficient for H on copper, and in 2 Ref. [22] that it lowers the tunneling rate, hence hindering hydrogen resurfacing in associative desorption of H from Ni(111). Both results can 2 be attributed to a dissipative friction mechanism, in which, in agreement with our results, the energy transfer to electron–hole pair excitations is a small effect. The same finding is reported for the sticking coefficient of CO/Cu(100) [23]. But the dynamical polarization contribution discussed in this paper is not considered in all those works. In conclusion, the main result of this paper is that the inelastic coupling can provide, via the phenomenon of dynamical polarization, a new mechanism which favours dissociative sticking but still leaves the electron–hole pair system in its ground state. Of course, a full quantum mechanical calculation of the static PES already includes the static polarization effects due to substrate electron–hole pair excitations required to describe the molecule–surface bond. But the dynamical polarization is essentially a non-adiabatic effect, and hence only manifests itself in a dynamical approach such as that presented here.
Acknowledgements We are grateful to G. Guerra for his help in solving the equations, and to A.C. Levi and H. Nienhaus for useful discussions.
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References [1] G.J. Kroes, E.J. Baerends, R.C. Mowrey, Phys. Rev. Lett. 78 (1997) 3583, Phys. Rev. Lett. 81 (1998) 4781. [2] A. Gross, Surf. Sci. Rep. 32 (1998) 291. [3] A. Gross, M. Scheffler, Phys. Rev. B 57 (1998) 2493. [4] G.P. Brivio, M.I. Trioni, Rev. Mod. Phys. 71 (1999) 231. [5] J. Bo¨heim, W. Brenig, Z. Phys. B 41 (1981) 243. [6 ] B. Jackson, J. Chem. Phys. 90 (1989) 190. [7] G. Armand, J.R. Manson, Phys. Rev. B 43 (1991) 14 371. [8] J.K. Nørskov, B.I. Lundqvist, Surf. Sci. 89 (1979) 251. [9] K. Scho¨nhammer, O. Gunnarsson, Z. Phys. 838 (1980) 127, Phys. Rev. B 22 (1980) 1629. [10] R. Brako, D.M. Newns, Solid State Commun. 33 (1980) 713. [11] J.W. Gadzuk, H. Metiu, Phys. Rev. B 22 (1980) 2603. [12] G.P. Brivio, T.B. Grimley, Surf. Sci. 89 (1979) 226, Surf. Sci. 131 (1983) 475. [13] M. Cacciatore, G.D. Billing, Surf. Sci. 232 (1990) 35. [14] G.D. Billing, M. Cacciatore, Faraday Discuss. 96 (1993) 33. [15] G.D. Billing, J. Chem. Phys. 99 (1995) 15 378. [16 ] S.K. Mengen, G.D. Billing, J. Phys. Chem. 101 (1997) 10 781. [17] T.B. Grimley, Chem. Phys. Lett. 222 (1994) 551. [18] G.P. Brivio, T.B. Grimley, G. Guerra, M.I. Trioni, in preparation. [19] G.P. Brivio, M.L. Rossi, T.B. Grimley, Chem. Phys. Lett. 263 (1996) 597. [20] C.T. Rettner, H.A. Michelsen, D.J. Auerbach, J. Chem. Phys. 102 (1995) 4625. [21] H. Nienhaus et al., Phys. Rev. Lett. 82 (1999) 446. [22] R. Baer, Y. Zeiri, R. Kosloff, Phys. Rev. B 55 (1997) 10 952. [23] J.T. Kindt, J.C. Tully, M. Head-Gordon, M.A. Gomez, J. Chem. Phys. 109 (1998) 3629.
Surface Science Letters
Dynamical polarization effects in dissociative sticking of a diatomic molecule with energy loss to the substrate G.P. Brivio a, *, T.B. Grimley b,1, M.I. Trioni a a Istituto Nazionale per la Fisica della Materia and Dipartimento di Scienza dei Materiali, Universita` di Milano ‘Bicocca’, via Cozzi 53, 20125 Milan, Italy b The Donnan Laboratories, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK Received 15 November 1999; accepted for publication 12 January 2000
Abstract We solve exactly and fully quantum mechanically a model Hamiltonian which describes energy loss to the substrate in dissociative sticking. The inelastic process is assumed as being due to electron–hole pairs. Results are calculated for the system H /Cu(111). It is shown that the main effect is a dynamical polarization of the electron system with 2 no electron–hole pairs in the final channel, which enhances the sticking coefficient with respect to that calculated only with the elastic molecule–solid potential. Substrate excitations in the final channel contribute little to the sticking coefficient. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Chemisorption; Energy dissipation; Models of surface chemical reactions; Sticking
Important information on the dissociative sticking of a diatomic molecule was recently obtained by computer simulations of the time evolution of a quantum mechanical wavepacket on the adiabatic molecule–surface multi-dimensional potential energy surfaces (PESs) [1–4]. Such PESs are derived from ab initio density functional based calculations which realistically describe the molecule–surface system. However, such first principle quantum methods cannot yet be used to treat the excitation system in reactive molecule–surface * Corresponding author. Fax: +39 02 6448 5403. E-mail address: [email protected] (G.P. Brivio) 1 Permanent address: 41 Beechways Drive, Neston, Cheshire CH64 6TF, UK.
dynamics. This problem needs to be addressed because for the dissociative sticking of molecules an energy of the order of an electronvolt is transferred to the substrate. Since phonon energies are 50 to 100 times smaller than this, either multi-phonon excitations, or perhaps single electron–hole pairs are obvious alternatives. The sticking of a reactive particle on a metal with the creation of phonons or electron–hole pairs has been the subject of several investigations over 20 years [5–12]. For dissociative sticking, this problem has been extensively investigated by Billing and coworkers [13–16 ]. They performed self-consistent semi-classical and quantum wavepacket calculations for H on copper, which 2 accounted for the multi-dimensional character of the PES, and used an effective dynamical coupling
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to describe the interaction both with the substrate electronic excitations and with the phonons. In this way they could obtain sticking coefficients within a framework capable of considering several phenomenological features of the system. But one would also like to ask a more general question on the relevance and the effects of energy loss mechanisms in dissociative sticking, because, whilst energy loss is essential for the sticking of a single atom, for the dissociative sticking of a diatomic molecule it is not. In fact, the exothermicity may be dissipated to the excitation system when the two atoms are far apart on the surface, in which case the sticking coefficient is uninfluenced by inelastic processes. But the energy loss may also occur at a more critical stage in the sticking dynamics, when the molecule bond is breaking, and the two surface bonds are forming. In this paper, in order to contribute to answering the above question, we present results for the dissociative sticking coefficient, s, obtained by a quantum analytically solvable model which contains an inelastic molecule–solid potential in addition to the simple static PES introduced originally by Grimley [17]. The inelastic potential we use can describe aspects of energy loss to phonons or electrons, and here, we parameterize the model for the latter. We show that the most important contribution to s from the inelastic potential is a dynamical polarization effect of the electronic system, non-adiabatically induced by the molecule. This effect does not imply electron–hole pairs in the final stuck channel, and increases the sticking coefficient with respect to that calculated on the adiabatic PES. We consider a diatomic molecule of mass M at normal incidence with its internuclear axis parallel to the solid surface. The Hamiltonian is: +H +H +U+W, trans rel s p2 H = z +V (z), f trans 2M
(1)
V =−gd(z)+ lim VH(z−d ), f V2 p2 H = x +V (x), i rel 2m
(3)
H=H
H =∑ |ne n|, s n n
(2)
(4) (5)
(6) U=ad(z) ∑ |nW1 W ∞ n∞|, n n n,n∞ W=h ∑ |mg1 Ug n|, h=c/a. (7) m n m≠n z is the molecule’s centre-of-mass coordinate, normal to the surface, x, the interatomic distance, is parallel to it, and m is the molecule’s reduced mass. The potential V in the final channel consists f of a d-function of strength −g at the minimum energy position z=0, and a hard wall at z=d representing the repulsive part of the atom–solid interaction. V supports one bound state of f energy E =−Mg2/2B2=−B2K2 /2M, where B B K =gM[1−exp(−2K d )]/B2. |n are eigenstates B B (bound and unbound) of the relative motion on V (x), and the molecule is initially in the bound i (vibrational ) state |i with energy e . H is the i s Hamiltonian for the excitation system, whose ground state is |0, and whose excited states are labelled |m, with m≠0. U is the elastic interchannel coupling with strength a, and W is the inelastic one with strength c. To obtain sticking coefficients in a closed form, both potentials U and W have to be separable, with details defined by the set of coefficients {W} and {g, W}, respectively. Projected onto the coordinate x, they can account for the structure of the elastic interchannel coupling, which depends on the local adiabatic electronic structure at z=0, and also determine the inelastic coupling to the excitation system. Still the Hamiltonian in Eqs. (1)–(7) provides a much simplified picture of dissociative sticking: the molecule does not rotate while approaching the surface, there is no surface corrugation, the elastic and inelastic potentials only act at a particular (close) distance from the surface. However, the model contains the essentials of dissociative sticking, i.e. energy conversion from the degrees of freedom of the impinging molecule to those parallel to the surface of the two dissociated atoms. Also the electron–hole pair coupling, due to an interaction rapidly decaying outside the metal, is described by the potential W in a limiting case. Since the potentials U and W are separable, the Lippmann–Schwinger equation for the relevant scattering state |Y+ can be solved exactly in an analytical way in terms of the amplitudes a+ (z)= z| m| m|Y+. By performing a calculamm
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tion, to be presented elsewhere [18], we obtain the sticking coefficient. Here, it is sufficient to record that: M
|RW |2|a+ (E )|2 ∑ A i i0 i m B2K m 0 W(E −e −E ) 2 i m B × n (E −e −E ), v i m B D(e ; E ) m i with: s=4p2
K
K
(8)
G
1 for m=0, (9) A = m |hg g1 |2 for m≠0. 0 m In Eq. (8) E =E +e is the initial energy i trans i of the molecule, whose translational one is E =B2K2 /2M. The constant R only depends trans 0 on K and d. The density n (e) refers both to B v discrete bound vibrational states of the molecule and to continuous unbound ones of the two dissociated atoms. The complex function D(e ; E ), m i which depends on all coupling constants, contains the details of the dynamics of sticking. It is important to note that the summation in Eq. (8) naturally separates into two terms, the former one, s 1 for m=0, and the latter one, s for m≠0, so that 2 s is the sum of two contributions, s=s +s , where 1 2 s involves only virtual excitations, whereas s 1 2 contains the real energy loss. To write s we have 2 to pass to the continuum for the excitation spectrum. After introducing the excitation probability P(e ) and density of states n (e ) per unit m e m energy, one can write:
K
K
W(E −E ) 2 i B n (E −E ), v i B D (E ) 0 i Ei−EB s = de P(e ) 2 m m
s 3 1
(10)
P
0 Ei−EB de n (e )A(e ) 3 m e m m
P 0
K
K
W(E −e −E ) 2 i m B n (E −e −E ). (11) v i m B D(e ; E ) m i The integral upper limit follows from energy conservation. Since s labels the dissociative sticking 0 ×
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coefficient on the static PES, s −s , which is the 1 0 result of virtual excitations, will be referred to as the dynamical polarization contribution. Up to this point no hypothesis on the excitation system has been made, only the density of states appears in our equations. Hereonwards, for sake of simplicity, we assume the inelastic potential W as being due to single electron–hole pair excitations with density of states per atom n˜ (e )3e (but see e m m the following). We also remark that the form of the inelastic potential in Eq. (7) is chosen so as to provide a model which is exactly solvable for multi-excitations, not simply for single excitations. And it is correct for linear coupling to a boson excitation system, such as the electron–hole pair system just introduced, as well as to a phonon one. The qualitative difference between the two systems does not mainly lie in the form of the density of states, which is quadratic for single vibrational excitations. It is related instead to the excitation energy cutoff. In fact that of the single electron–hole pairs is almost two orders of magnitude larger than the Debye energy of the phonon system. So the electron–hole pair cutoff energy is indeed comparable with those exchanged in dissociative sticking of light reactive molecules on metals. To obtain results, it is natural to parametrize our model to the bench mark system of dissociative sticking, i.e. to H /Cu(111). Spurious oscillations 2 in the sticking coefficients are avoided if K d≤p. B Accordingly we have chosen the parameter d equal to 0.07 Bohr. The distribution and the energy dependence of the coefficients {W} has been determined from an interchannel separable potential following Ref. [19]. The parameter a˜ =a/d=1.0 eV has been selected in order to give a reasonably realistic interchannel coupling. It is also multiplied by a transmission coefficient through a barrier of height E =0.5 eV to simulate an activated sticking b process such as H on Cu. Since we have no 2 information on separable potential couplings to electron–hole pairs, all the coefficients g , m≠0, m have been taken to be constant and equal to unity. The parameter |g |2h=20 is chosen so that s is a 0 good fit to the experimental results of Rettner et al. [20]. The results, for an impinging H 2 molecule in its vibrational ground state, and as a function of the translational kinetic energy E , trans
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are shown in Fig. 1, which displays s=s +s (solid 1 2 line), s (dashed line) and s (dot-dashed line). The 1 0 most important feature of these results is that the dynamical polarization contribution s −s is the 1 0 dominant effect of the inelastic potential; energy loss to electron–hole pairs has, by comparison, little effect on the dissociative sticking coefficient. To confirm this result, in Fig. 2 we plot s and s 1 as a function of |g |2h at E =1.0 eV with a˜ as 0 trans before, and we have also verified that such a
Fig. 1. The dissociative sticking coefficient of H on Cu(111) 2 versus E , the translational energy of the incident molecule. trans s (dot-dashed line) is the value calculated for the rigid potential 0 energy surface (inelastic potential set to zero), s (dashed line) 1 includes the dynamical polarization of the electron–hole pair system by the dissociating molecule, and s (solid line) is the total value which includes in addition s , the real energy loss to 2 electron–hole pairs. For parameters, see the text.
Fig. 2. The dissociative sticking coefficient of H on Cu(111) 2 versus |g |2h, the electron–hole pair coupling strength: s (solid 0 line) and s (dashed line). 1
behaviour of s is essentially independent of a˜ . Both the maximum and the asymptotic behaviour of s and s for large h are expected on general grounds. 1 For the latter observe that in the limit of a strong inelastic potential there is no energy loss, and s collapses to s . The minimum is instead due to the 1 particular form of the equations for this model. To find further support for our results, we have solved exactly a different Hamiltonian, that in Eq. (1) but with a=0. We have verified that for a wide and physically significant range of values of h, s %s , though here one cannot even define an elastic 2 1 sticking coefficient s . We have also checked that the 0 above mentioned result is not a consequence of the simple form used for the density of states n˜ (e ). e m Even if n˜ (e ) is quite different from that for single e m electron–hole pairs in the nearly-free electron spectrum, being instead a narrow peak at some energy E associated with a local adiabatic electronic p structure responsible for the chemisorption bond, this still does not lead to a larger contribution to s and 2 to any corresponding structure in s , because only 2 integrated quantities appear in s . This theoretical 2 conclusion agrees with experimental findings that dissociative sticking coefficients show a rather featureless dependence on the translational energy of the incoming molecule, and that no electron–hole pairs are detected in dissociative sticking of H on noble 2 metals [21]. Of course one could argue that the large ratio s /s which we found is somewhat a model depen1 2 dent result. But, in any case, the solution of the Hamiltonian [Eqs. (1)–(7)] points out that a static PES may not always be adequate for the dynamics, because we have shown that the inelasticity gives two contributions to dissociative sticking: the former, the dynamical polarization one, with no excitations in the stuck channel, and the latter one with exothermicity dissipated to the electron–hole pairs. Both effects contribute to increasing s with respect to that calculated on the adiabatic PES. We wish to observe that this result finds some support from the multi-dimensional ab initio simulations. In fact in Ref. [1] it is pointed out that the reported 6D simulation on the adiabatic PES underestimates the sticking coefficient of H on 2 Cu(100) with respect to the experimental result. It is also useful to make a connection between our
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results and the recent ones obtained by simulations where inelasticity is included. In Ref. [15] it is shown that inelasticity increases the dissociative sticking coefficient for H on copper, and in 2 Ref. [22] that it lowers the tunneling rate, hence hindering hydrogen resurfacing in associative desorption of H from Ni(111). Both results can 2 be attributed to a dissipative friction mechanism, in which, in agreement with our results, the energy transfer to electron–hole pair excitations is a small effect. The same finding is reported for the sticking coefficient of CO/Cu(100) [23]. But the dynamical polarization contribution discussed in this paper is not considered in all those works. In conclusion, the main result of this paper is that the inelastic coupling can provide, via the phenomenon of dynamical polarization, a new mechanism which favours dissociative sticking but still leaves the electron–hole pair system in its ground state. Of course, a full quantum mechanical calculation of the static PES already includes the static polarization effects due to substrate electron–hole pair excitations required to describe the molecule–surface bond. But the dynamical polarization is essentially a non-adiabatic effect, and hence only manifests itself in a dynamical approach such as that presented here.
Acknowledgements We are grateful to G. Guerra for his help in solving the equations, and to A.C. Levi and H. Nienhaus for useful discussions.
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