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Electronic damping mechanism for vibrations, rotations, and translations of adsorbates on metal surfaces
147
Surface Science 126 (1983) 147-153 North-Holland Publishing Company
ELECTRONIC ROTATIONS, SURFACES B. HELLSING, Institute Received
DAMPING MECHANISM FOR VIBRATIONS, AND TRANSLATIONS OF ADSORBATES ON METAL
M. PERSSON
of Theoretical Physics. Chalmers 24 August
and B.I. LUNDQVIST University of Technology, S - 412 - 96 Gtiteborg, Sweden
1982
Further theoretical evidence for the importance of the electronic mechanism for the energy exchange between an adsorbate and a metallic substrate is provided by first-principle calculations of the damping of vibrational modes of chemisorbed H and H, on free-electron-like metal surfaces and of the friction coefficient of an approaching H atom. In reactive situations, i.e. when a resonance in the adsorbate-induced electron structure crosses the Fermi level of the substrate conduction electrons, enhanced damping is exhibited. With rough estimates for rotational lifetimes added, relevant relaxation times for molecules approaching metal surfaces are briefly discussed.
1. Introduction The dynamics of atoms and molecules at surfaces is fundamentally a complex quantum-mechanical problem, describable by, for instance, scattering theory, but is commonly described by the equations of irreversible thermodynamics. Such equations as the Pauli master equation, the Fokker-Planck equation and the generalized Langevin equation have been used in this context [l]. Whether their solutions give physically meaningful results depends very much on the quality of the input to the equations, that is the reactive and dissipative forces acting on the particles, whose motion is described [2]. The reactive forces can be derived from the potential-energy surfaces for the interaction between a particle and a surface. Recent progress in the calculation of potential-energy surfaces [3] and in providing efficient schemes for the extension of such results [4] have improved the prospect for a relatively realistic input of reactive forces. Energy exchange can occur from, for example, translational modes into internal excitations of the molecules, such as rotations [5], vibrations and electronic excitations [6], or to excitations of the substrate. The latter is a true dissipation effect, due to the many degrees of freedom of the substrate. Commonly, one has stressed direct energy exchange to phonons in this context [7]. Recently, electron-hole pairs have attracted a considerable attention as a 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
148
B. Hellsing et al. / Electronic damping mechanism
dissipation channel in metal substrates, both for translational modes, as in sticking or trapping [3,8], and for vibrational modes [9]. This paper is a progress report of an ongoing program for the calculation of the damping of translational, vibrational, and rotational modes of molecules and atoms at metal surfaces. It presents some new results and attempts to place these and already published ones [lo] in a perspective. In addition, some factors that affect the magnitude of energy exchange to electron-hole pairs, are pointed out. The damping of the motion of charged particles has been extensively studied [ 111. For fast particles, descriptions based on the dielectric response of the metal are common. The charged particle couples through the screened Coulomb field to the electron-hole pair excitations and, if it is energetic enough, by direct Coulomb interaction to the plasmons, the collective excitations build up by the electron-hole pairs [12]. When the charged particle moves slowly, the conduction electrons adjust, at least partially, to its presence, and a situation approaching that with a neutral object moving through the medium is obtained [ 131. Intuitively, one would expect an inert-gas atom to have a very weak coupling to the electron-hole pair excitations. No essential electronic redistribution occurs during the atom-substrate encounter. The conduction electrons have only to orthogonalize against the atom with its tightly bound electrons. In addition, basically for the same reason [14], inert-gas atoms do not penetrate the surface, but are kept away from the substrate atoms, making electronic overlap and coupling to the electron-hole pairs very small [ 151. Reactive atoms and molecules, on the other hand, seem likely to give a much stronger coupling to the electron-hole pairs. Per definition new bonds are formed, old ones are broken, and/or existing bonds are changed during the adsorption of such species. For example, a neutral alkali atom impinging on a metal surface creates charge transfer with a rapidness related to the speed of incidence and the strength of the adsorption forces. Such charge rearrangements, of course, stir up among the conduction electrons. Translational energy is thereby dissipated to electron-hole pairs. Adsorbed CO on, e.g., Ni has an electronic resonance that bounces up and down through the Fermi level, causing charge transfer between the molecule and the substrate, as the moleBoth the mentioned cases exhibit cule performs stretch oscillations. adsorption-induced resonances at the Fermi level, whose properties change with the motion of the adsorbate. The relation between the behaviour of adsorption-induced resonances [ 161 and damping has earlier been stressed in the contexts of trapping [8] and damping of vibrations [9].
B. Hellsing et al. / Electronic damping mechanism
149
2. Rough lifetime estimates Like for charged particles, it seems natural to simplify and first calculate the damping of neutral particle motion in a homogeneous electron system. There exist results for the damping of translation and vibration of an atom in a homogeneous electron gas. In this case the friction formula reads [ 131 l/3 ,=2
Q rs2
( 9a 1
m
,X0(/+ E
1) sin2(b+,
--4),
(1)
where 6, are the phaseshifts of an electron at the Fermi level due to the presence of the atom. The damping rate of the vibration of the atom against the electron gas is [lo] Lb=
(A/M)
17,
(2)
where M is the mass of the atom. Eq. (2) is valid with the motion of the atom described classically as well as quantum-mechanically. In the rotator case, eq. (1) has been used for an order of magnitude estimate of the damping, but so far only in a crude classical model. Describing the H, molecule as a rotating classical dumbbell of two neutral H atoms, we get the same expression for r as in eq. (2). Eqs. (1) and (2) need as input the density pa of the electron gas (or equivalently the parameter r,, defined by 4~(r,a~)~/3 = p; ‘) and the phaseshifts for the atom at this density. For rS values characteristic of metals and metal surfaces ( rS = 2-6) and with phase shifts for the H atom [ 131, the quotients between the relaxation time and the period lie in the range 7/T = 4-30 for the first excited vibration of an H atom and a similar order of magnitude for rotations of H, in the classical limit. These numbers indicate that a H atom or a H, molecule, impinging on a metal surface and reaching the region, where the chemisorption forces are effective, performs vibrations and rotations (for H2) that are significantly damped. In describing the dynamic adsorption process of H and H,, there is thus a need to properly account for these dissipative forces in the dynamic equations. 3. Calculations Our calculations of the electronic mechanism for lifetime broadening Qi, of vibrationally excited adsorbates [lo] and for friction n take their origin in the Born-Oppenheimer approximation with one electronic ground state and compute r,, to the lowest order in the screened ion-electron interaction. In the quasi-static regime, with ground-state quantities calculated in the embedding scheme of Gunnarsson and Hjelmberg [ 171, rVi, = (27rA/M)
Tr( WW),
(3)
150
B. Hellsing et al. / Electronic damping mechanism
where matrix elements between centred on the adsorbate,
the finite
number
of localized
functions
]n)
are summed over in the trace. In eq. (4), p(er) is the total one-electron density of states at the Fermi energy and W/aQ the derivative of the effective potential felt by each electron with respect to the displacement coordinate for the vibration considered. Self-consistent calculations have been performed in a model consisting of the adsorbates H and H,, respectively. kept in position by some external force, in interaction with a semi-infinite jellium, that is the planar-uniform background model for a metal surface, characterized by the density of r, value deep in the bulk. The electronic friction coefficient, 9, has been worked out in two different approaches starting from the force-force correlation function [ 1b] and the Keldysh propagator formalism [ 181. By assuming that the one-electron poten-
-6 -L
-3
-2
-1
-0
1
2
dfau) Fig. 1. {a) The vibrational damping rate r for a hydrogen molecule as a function of the distance d from the edge of a jellium surface modelling Na and Mg, respectively; the configuration of the molecule used in the calculation is also indicated (R = 1.4 au). (b) The free surface electron density profiles of Na and Mg in the jellium model. (c) The position of the adsorbate induced resonance 2a* for the hydrogen molecule as a function of the distance d from the jellium edge and the corresponding width when crossing the Fermi level [19].
B. Helking
et al. / Electronic damping mechanism
151
tial can be completely described by the local basis set, both expressions can be related to eq. (3) as in eq. (2). Thus, in this limit of near adiabaticity, both the vibrational damping and the friction can be obtained from the same computation.
4. Results Fig. 1 shows the results for the lifetime broadening r of the vibrational stretch mode of an H, molecule adsorbed on a jellium surface [lo], The clean jellium-surface electron density p0 is included in fig. lb to show that the strong variation of I’ with the distance d from the jellium edge is not a mere local-density-variation effect. Rather, the sensitivity of the damping rate r to the local electronic structure at the Fermi level is borne out by the correlations in location between the enhancement in the damping rate (fig. la) and the passage of the antibonding 2a* resonance through the Fermi level (fig. Ic) [19]. r
CmeV)
i
(meV)
H IAg 21 .,
. ieV)
f:
HlAg
o-2.-A--S--
c -2
0
2
1
6
dku)
Fig. 2. The damping rate for the vibration of a H atom perpendicular to a jellium surface as a function of the distance d from the edge of jellium modelling (a) Al and (b) Ag (dots). The result for a H atom immersed in a homogeneous electron gas with the density equal to the local density at the position of the atom, eq. (1) (solid line). (c) The density of states, Ap( c), induced by the H atom given at the distances d = 1.0 and 3.0 au from the edge of jellium modelling Ag.
152
B. Hellsing et al. / Electronic damping mechanism
L --
H;A’s /./‘-‘1 --
'. '.
0 _-
/.0‘ 0' '. / 0' ‘;' __=-_~s----
H+Ag
/-
_---l
a
A
-2
0
2
1
6
8
m
d(au)
Fig. 3. (a) The potential-energy curves for H and H- as a function of the distance d from the jellium edge. The curves are results from the effective medium theory [4] (dotted line) and the image potential (dash-dotted line) and the present calculation (solid line). The dashed line is the result from interpolation by hand between the separate limits. (b) The total energy, E, calculated from Langevin’s equation of motion for the H atom, as a function of the distance d from the jellium edge. The incoming atom has a typical thermal energy 25 meV.
The calculated damping rate for a hydrogen atom vibrating perpendicular to a jellium surface is presented in fig. 2. We see that the damping rate in a homogeneous electron gas (eq. (2)), evaluated at the local density, gives a good order-of-magnitude estimate but does not account fully for the variation with the distance d. The possibility that the levelling-off of the calculated r points (in the region d = O-4 au for H on Ag) should be due to the adsorbate-induced resonance being close to the Fermi level is indicated in the lower part of the figure. The observed vibrational linewidth 1.8 meV for H on W [20] compares favourably with the calculated damping rate, 1.6 meV in fig. 2a, at the position where the local electron density of the free surface is the same as proposed in the effective medium theory for describing static proeprties [4]. The same order of magnitude is also observed for the vibration of hydrogen in CH,O on Cu
1211. The friction coefficient is given by fig. 2b with the use of the relation 17= M/A&. For a slowly incoming H atom the total energy is calculated from the Langevin equation of motion, considering the adiabatic potential energy curve in the lower left part of fig. 3a and the friction. In fig. 3b it is indicated that this friction force is efficient in slowing down the atom. The trajectory is depicted only for d equal to 4 au and inwards, as the friction force description might not apply for the two-state problem, indicated by the crossing of the two diabatic potential-energy curves of fig. 3a. In conclusion, the electronic contribution to the damping of vibrational and translational motion is important. This is based on the following: (i) the
B. Hellsing et al. / Electronic damping mechanism
153
calculated vibrational damping rate for H on jellium compares favourably with experiment [20,21]; (ii) a slowly approaching H atom is predicted to be trapped on a Ag surface. In addition, in a reactive situation where an adsorbate-induced resonance crosses the Fermi level, the damping rate should be enhanced. Finally, we want to stress that eq. (2) can provide estimates of the vibrational damping rate for adsorbed atoms [22].
Acknowledgements We would like to thank H. Hjelmberg for sharing his experience with us. In addition we are grateful to J.K. Norskov and P. Nordlander for unpublished results and to A. Luntz for stimulating discussions. Financial support from the Swedish Natural Science Council is also gratefully acknowledged.
References [l] See, e.g.: (a) G.S. De, U. Landman and M. Rasolt, Phys. Rev. B21 (1980) 3256; (b) E.G. d’Agliano, P. Kumar, W. Schaich and H. Suhl, Phys. Rev. Bll (1975) 2122; (c) J.C. Tully, J. Chem. Phys. 73 (1980) 4. [2] J.C. Tully, Ann. Rev. Phys. Chem. 31 (1980) 319. [3] See, e.g., B.I. Lundqvist, in: Vibrations at Surfaces, Eds. Caudano et al. (Plenum, New York, 1982) p. 541, and references therein. [4] J.K. Norskov, Phys. Rev. Letters 48 (1982) 1620. [S] S. Andersson and J. Harris, Phys. Rev. Letters 48 (1982) 545; S. Andersson and J. Harris, to be published. [6] J.K. Norskov, D.M. Newns and B.I. Lundqvist, Surface Sci. 80 (1979) 179. [7] F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, New York, 1976). [8] J.K. Norskov and B.I. Lundqvist, Surface Sci. 89 (1979) 251. [9] B.N.J. Persson and M. Persson, Solid State Commun. 36 (1980) 175. [lo] M. Persson and B. Hellsing, Phys. Rev. Letters 49 (1982) 662. [l l] R.H. Ritchie, Phys. Rev. 114 (1959) 644. [ 121 D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963) p. 168. [13] P.M. Echenique, R.M. Nieminen and R.H. Ritchie, Solid State Commun. 37 (1981) 779. [14] J. Harris and A. Liebsch, J. Phys. Cl5 (1982) 2275. [ 151 0. Gunnarsson and K. Schiinhammer, Phys. Rev. B25 (1982) 2514. [ 161 See, e.g., B.I. Lundqvist, H. Hjelmberg and 0. Gunnarsson, in: Photoemission and the Electronic Properties of Surfaces, Eds. B. Feuerbacher, B. Fitton and R.F. Willis (Wiley-Interscience, New York, 1978) p. 227. [17] 0. Gunnarsson and H. Hjelmberg, Phys. Scripta 11 (1975) 97. [18] A. Nourtier, J. Physique 37 (1976) 369. [19] H. Hjelmberg, B.I. Lundqvist and J.K. Norskov, Phys. Scripta 20 (1979) 192. [20] Y.J. Chabal and A.J. Sievers, Phys. Rev. Letters 44 (1980) 944. [21] R. Ryberg, Chem. Phys. Letters 83 (1981) 423. [22] R.M. Nieminen, private communication.
Surface Science 126 (1983) 147-153 North-Holland Publishing Company
ELECTRONIC ROTATIONS, SURFACES B. HELLSING, Institute Received
DAMPING MECHANISM FOR VIBRATIONS, AND TRANSLATIONS OF ADSORBATES ON METAL
M. PERSSON
of Theoretical Physics. Chalmers 24 August
and B.I. LUNDQVIST University of Technology, S - 412 - 96 Gtiteborg, Sweden
1982
Further theoretical evidence for the importance of the electronic mechanism for the energy exchange between an adsorbate and a metallic substrate is provided by first-principle calculations of the damping of vibrational modes of chemisorbed H and H, on free-electron-like metal surfaces and of the friction coefficient of an approaching H atom. In reactive situations, i.e. when a resonance in the adsorbate-induced electron structure crosses the Fermi level of the substrate conduction electrons, enhanced damping is exhibited. With rough estimates for rotational lifetimes added, relevant relaxation times for molecules approaching metal surfaces are briefly discussed.
1. Introduction The dynamics of atoms and molecules at surfaces is fundamentally a complex quantum-mechanical problem, describable by, for instance, scattering theory, but is commonly described by the equations of irreversible thermodynamics. Such equations as the Pauli master equation, the Fokker-Planck equation and the generalized Langevin equation have been used in this context [l]. Whether their solutions give physically meaningful results depends very much on the quality of the input to the equations, that is the reactive and dissipative forces acting on the particles, whose motion is described [2]. The reactive forces can be derived from the potential-energy surfaces for the interaction between a particle and a surface. Recent progress in the calculation of potential-energy surfaces [3] and in providing efficient schemes for the extension of such results [4] have improved the prospect for a relatively realistic input of reactive forces. Energy exchange can occur from, for example, translational modes into internal excitations of the molecules, such as rotations [5], vibrations and electronic excitations [6], or to excitations of the substrate. The latter is a true dissipation effect, due to the many degrees of freedom of the substrate. Commonly, one has stressed direct energy exchange to phonons in this context [7]. Recently, electron-hole pairs have attracted a considerable attention as a 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
148
B. Hellsing et al. / Electronic damping mechanism
dissipation channel in metal substrates, both for translational modes, as in sticking or trapping [3,8], and for vibrational modes [9]. This paper is a progress report of an ongoing program for the calculation of the damping of translational, vibrational, and rotational modes of molecules and atoms at metal surfaces. It presents some new results and attempts to place these and already published ones [lo] in a perspective. In addition, some factors that affect the magnitude of energy exchange to electron-hole pairs, are pointed out. The damping of the motion of charged particles has been extensively studied [ 111. For fast particles, descriptions based on the dielectric response of the metal are common. The charged particle couples through the screened Coulomb field to the electron-hole pair excitations and, if it is energetic enough, by direct Coulomb interaction to the plasmons, the collective excitations build up by the electron-hole pairs [12]. When the charged particle moves slowly, the conduction electrons adjust, at least partially, to its presence, and a situation approaching that with a neutral object moving through the medium is obtained [ 131. Intuitively, one would expect an inert-gas atom to have a very weak coupling to the electron-hole pair excitations. No essential electronic redistribution occurs during the atom-substrate encounter. The conduction electrons have only to orthogonalize against the atom with its tightly bound electrons. In addition, basically for the same reason [14], inert-gas atoms do not penetrate the surface, but are kept away from the substrate atoms, making electronic overlap and coupling to the electron-hole pairs very small [ 151. Reactive atoms and molecules, on the other hand, seem likely to give a much stronger coupling to the electron-hole pairs. Per definition new bonds are formed, old ones are broken, and/or existing bonds are changed during the adsorption of such species. For example, a neutral alkali atom impinging on a metal surface creates charge transfer with a rapidness related to the speed of incidence and the strength of the adsorption forces. Such charge rearrangements, of course, stir up among the conduction electrons. Translational energy is thereby dissipated to electron-hole pairs. Adsorbed CO on, e.g., Ni has an electronic resonance that bounces up and down through the Fermi level, causing charge transfer between the molecule and the substrate, as the moleBoth the mentioned cases exhibit cule performs stretch oscillations. adsorption-induced resonances at the Fermi level, whose properties change with the motion of the adsorbate. The relation between the behaviour of adsorption-induced resonances [ 161 and damping has earlier been stressed in the contexts of trapping [8] and damping of vibrations [9].
B. Hellsing et al. / Electronic damping mechanism
149
2. Rough lifetime estimates Like for charged particles, it seems natural to simplify and first calculate the damping of neutral particle motion in a homogeneous electron system. There exist results for the damping of translation and vibration of an atom in a homogeneous electron gas. In this case the friction formula reads [ 131 l/3 ,=2
Q rs2
( 9a 1
m
,X0(/+ E
1) sin2(b+,
--4),
(1)
where 6, are the phaseshifts of an electron at the Fermi level due to the presence of the atom. The damping rate of the vibration of the atom against the electron gas is [lo] Lb=
(A/M)
17,
(2)
where M is the mass of the atom. Eq. (2) is valid with the motion of the atom described classically as well as quantum-mechanically. In the rotator case, eq. (1) has been used for an order of magnitude estimate of the damping, but so far only in a crude classical model. Describing the H, molecule as a rotating classical dumbbell of two neutral H atoms, we get the same expression for r as in eq. (2). Eqs. (1) and (2) need as input the density pa of the electron gas (or equivalently the parameter r,, defined by 4~(r,a~)~/3 = p; ‘) and the phaseshifts for the atom at this density. For rS values characteristic of metals and metal surfaces ( rS = 2-6) and with phase shifts for the H atom [ 131, the quotients between the relaxation time and the period lie in the range 7/T = 4-30 for the first excited vibration of an H atom and a similar order of magnitude for rotations of H, in the classical limit. These numbers indicate that a H atom or a H, molecule, impinging on a metal surface and reaching the region, where the chemisorption forces are effective, performs vibrations and rotations (for H2) that are significantly damped. In describing the dynamic adsorption process of H and H,, there is thus a need to properly account for these dissipative forces in the dynamic equations. 3. Calculations Our calculations of the electronic mechanism for lifetime broadening Qi, of vibrationally excited adsorbates [lo] and for friction n take their origin in the Born-Oppenheimer approximation with one electronic ground state and compute r,, to the lowest order in the screened ion-electron interaction. In the quasi-static regime, with ground-state quantities calculated in the embedding scheme of Gunnarsson and Hjelmberg [ 171, rVi, = (27rA/M)
Tr( WW),
(3)
150
B. Hellsing et al. / Electronic damping mechanism
where matrix elements between centred on the adsorbate,
the finite
number
of localized
functions
]n)
are summed over in the trace. In eq. (4), p(er) is the total one-electron density of states at the Fermi energy and W/aQ the derivative of the effective potential felt by each electron with respect to the displacement coordinate for the vibration considered. Self-consistent calculations have been performed in a model consisting of the adsorbates H and H,, respectively. kept in position by some external force, in interaction with a semi-infinite jellium, that is the planar-uniform background model for a metal surface, characterized by the density of r, value deep in the bulk. The electronic friction coefficient, 9, has been worked out in two different approaches starting from the force-force correlation function [ 1b] and the Keldysh propagator formalism [ 181. By assuming that the one-electron poten-
-6 -L
-3
-2
-1
-0
1
2
dfau) Fig. 1. {a) The vibrational damping rate r for a hydrogen molecule as a function of the distance d from the edge of a jellium surface modelling Na and Mg, respectively; the configuration of the molecule used in the calculation is also indicated (R = 1.4 au). (b) The free surface electron density profiles of Na and Mg in the jellium model. (c) The position of the adsorbate induced resonance 2a* for the hydrogen molecule as a function of the distance d from the jellium edge and the corresponding width when crossing the Fermi level [19].
B. Helking
et al. / Electronic damping mechanism
151
tial can be completely described by the local basis set, both expressions can be related to eq. (3) as in eq. (2). Thus, in this limit of near adiabaticity, both the vibrational damping and the friction can be obtained from the same computation.
4. Results Fig. 1 shows the results for the lifetime broadening r of the vibrational stretch mode of an H, molecule adsorbed on a jellium surface [lo], The clean jellium-surface electron density p0 is included in fig. lb to show that the strong variation of I’ with the distance d from the jellium edge is not a mere local-density-variation effect. Rather, the sensitivity of the damping rate r to the local electronic structure at the Fermi level is borne out by the correlations in location between the enhancement in the damping rate (fig. la) and the passage of the antibonding 2a* resonance through the Fermi level (fig. Ic) [19]. r
CmeV)
i
(meV)
H IAg 21 .,
. ieV)
f:
HlAg
o-2.-A--S--
c -2
0
2
1
6
dku)
Fig. 2. The damping rate for the vibration of a H atom perpendicular to a jellium surface as a function of the distance d from the edge of jellium modelling (a) Al and (b) Ag (dots). The result for a H atom immersed in a homogeneous electron gas with the density equal to the local density at the position of the atom, eq. (1) (solid line). (c) The density of states, Ap( c), induced by the H atom given at the distances d = 1.0 and 3.0 au from the edge of jellium modelling Ag.
152
B. Hellsing et al. / Electronic damping mechanism
L --
H;A’s /./‘-‘1 --
'. '.
0 _-
/.0‘ 0' '. / 0' ‘;' __=-_~s----
H+Ag
/-
_---l
a
A
-2
0
2
1
6
8
m
d(au)
Fig. 3. (a) The potential-energy curves for H and H- as a function of the distance d from the jellium edge. The curves are results from the effective medium theory [4] (dotted line) and the image potential (dash-dotted line) and the present calculation (solid line). The dashed line is the result from interpolation by hand between the separate limits. (b) The total energy, E, calculated from Langevin’s equation of motion for the H atom, as a function of the distance d from the jellium edge. The incoming atom has a typical thermal energy 25 meV.
The calculated damping rate for a hydrogen atom vibrating perpendicular to a jellium surface is presented in fig. 2. We see that the damping rate in a homogeneous electron gas (eq. (2)), evaluated at the local density, gives a good order-of-magnitude estimate but does not account fully for the variation with the distance d. The possibility that the levelling-off of the calculated r points (in the region d = O-4 au for H on Ag) should be due to the adsorbate-induced resonance being close to the Fermi level is indicated in the lower part of the figure. The observed vibrational linewidth 1.8 meV for H on W [20] compares favourably with the calculated damping rate, 1.6 meV in fig. 2a, at the position where the local electron density of the free surface is the same as proposed in the effective medium theory for describing static proeprties [4]. The same order of magnitude is also observed for the vibration of hydrogen in CH,O on Cu
1211. The friction coefficient is given by fig. 2b with the use of the relation 17= M/A&. For a slowly incoming H atom the total energy is calculated from the Langevin equation of motion, considering the adiabatic potential energy curve in the lower left part of fig. 3a and the friction. In fig. 3b it is indicated that this friction force is efficient in slowing down the atom. The trajectory is depicted only for d equal to 4 au and inwards, as the friction force description might not apply for the two-state problem, indicated by the crossing of the two diabatic potential-energy curves of fig. 3a. In conclusion, the electronic contribution to the damping of vibrational and translational motion is important. This is based on the following: (i) the
B. Hellsing et al. / Electronic damping mechanism
153
calculated vibrational damping rate for H on jellium compares favourably with experiment [20,21]; (ii) a slowly approaching H atom is predicted to be trapped on a Ag surface. In addition, in a reactive situation where an adsorbate-induced resonance crosses the Fermi level, the damping rate should be enhanced. Finally, we want to stress that eq. (2) can provide estimates of the vibrational damping rate for adsorbed atoms [22].
Acknowledgements We would like to thank H. Hjelmberg for sharing his experience with us. In addition we are grateful to J.K. Norskov and P. Nordlander for unpublished results and to A. Luntz for stimulating discussions. Financial support from the Swedish Natural Science Council is also gratefully acknowledged.
References [l] See, e.g.: (a) G.S. De, U. Landman and M. Rasolt, Phys. Rev. B21 (1980) 3256; (b) E.G. d’Agliano, P. Kumar, W. Schaich and H. Suhl, Phys. Rev. Bll (1975) 2122; (c) J.C. Tully, J. Chem. Phys. 73 (1980) 4. [2] J.C. Tully, Ann. Rev. Phys. Chem. 31 (1980) 319. [3] See, e.g., B.I. Lundqvist, in: Vibrations at Surfaces, Eds. Caudano et al. (Plenum, New York, 1982) p. 541, and references therein. [4] J.K. Norskov, Phys. Rev. Letters 48 (1982) 1620. [S] S. Andersson and J. Harris, Phys. Rev. Letters 48 (1982) 545; S. Andersson and J. Harris, to be published. [6] J.K. Norskov, D.M. Newns and B.I. Lundqvist, Surface Sci. 80 (1979) 179. [7] F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, New York, 1976). [8] J.K. Norskov and B.I. Lundqvist, Surface Sci. 89 (1979) 251. [9] B.N.J. Persson and M. Persson, Solid State Commun. 36 (1980) 175. [lo] M. Persson and B. Hellsing, Phys. Rev. Letters 49 (1982) 662. [l l] R.H. Ritchie, Phys. Rev. 114 (1959) 644. [ 121 D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963) p. 168. [13] P.M. Echenique, R.M. Nieminen and R.H. Ritchie, Solid State Commun. 37 (1981) 779. [14] J. Harris and A. Liebsch, J. Phys. Cl5 (1982) 2275. [ 151 0. Gunnarsson and K. Schiinhammer, Phys. Rev. B25 (1982) 2514. [ 161 See, e.g., B.I. Lundqvist, H. Hjelmberg and 0. Gunnarsson, in: Photoemission and the Electronic Properties of Surfaces, Eds. B. Feuerbacher, B. Fitton and R.F. Willis (Wiley-Interscience, New York, 1978) p. 227. [17] 0. Gunnarsson and H. Hjelmberg, Phys. Scripta 11 (1975) 97. [18] A. Nourtier, J. Physique 37 (1976) 369. [19] H. Hjelmberg, B.I. Lundqvist and J.K. Norskov, Phys. Scripta 20 (1979) 192. [20] Y.J. Chabal and A.J. Sievers, Phys. Rev. Letters 44 (1980) 944. [21] R. Ryberg, Chem. Phys. Letters 83 (1981) 423. [22] R.M. Nieminen, private communication.