This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.16.69.49

This content was downloaded on 01/02/2015 at 10:21

Please note that terms and conditions apply.

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces

View the table of contents for this issue, or go to the journal homepage for more

1984 Phys. Scr. 29 360

(http://iopscience.iop.org/1402-4896/29/4/014)

Home Search Collections Journals About Contact us My IOPscience

Physica Scripta. Vol. 29. 360-371, 1984

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces

B. Hellsing and M. Persson

Institute of Theoretical Physics, Chalmers University of Technology. S 4 1 2 96 Goteborg, Sweden

Received June 23, 1983; accepted December 23, 1983

Abstract estimate the energy loss for an incoming atom. In particular we find that a typical thermal energy, 25meV. of an incoming atom is accommodated by the surface.

The electronic mechanism which causes lifetime broadening

First-principles calculations of the damping rate of vibrational and translational modes of single hydrogen atoms and molecules chemisorbed on metal surfaces are presented. m e decay into electron-hole pair excitations is considered; The metal is described in the so-called jellrum model, and the calculations are based on the Kohn-Sham density func- tional formalism extended to the quasi-static regime. For the actual evaluation of the damping an embedding scheme is used, which through

geneous limit, a comparison is made with an exact phase-shift formula. The local electron density is found to be a key parameter for the damp- ing rate, in particular in situations with no dramatic electron structure at the Fermi level. Adsorbates that induce states around the Fermi level should exhibit an enhanced damping rate. A direct relation between the friction coefficient of an adparticle and the vibrational damping rate is derived. The calculated rate values imply that the electronic mechanism

will be discussed primarily for vibrational modes with fre- quencies far outside the bulk phonon band. The case of a sur- face vibrational mode with the bulk phonon band,

not be discussed here [SI. The nature of the contributions from the anharmonic coupling to the phonons are elucidated t o some extent in the experimental and theoretical studies of local modes in insulating ionic crystals [61. Several different able effects have been suggested for distinguishing between these two different contributions to the broadening [ 2 , 71. The

explicit tests is shown to be appropriate, In particular in the homo- where the phonon mechanism is obvious and important,

is able to accommodate typical thermal energies of hydrogen atoms impinging on metal surfaces. From this result and comparisons with observed lifetime broadenings of vibrational spectral lines it is concluded that electron-hole pairs provide an important channel for energy transfer a t metal surfaces.

1 . Introduction

The relaxation of vibrationally excited adsorbates on a metal surface is an important dynamical process. The clarification of the physical mechanisms behind such a process from an exper- imental and theoretical point of view, is a first step towards the understanding of more complex dynamical phenomena, e.g., the sticking of atoms and molecules and surface reactions [ 11.

In this work we present results from ab initio calculations of the electron-hole pair mechanism for the broadening of vibra- tional modes of atomic and molecular hydrogen adsorbed on jellium surfaces. The calculational method is also described in detail. Preliminary results of these calculations have been presented in a recent letter [2] and in some conference contri- butions [3]. The calculations provide evidences that the elec- tronic mechanism is able t o account for the order of magnitude of observed values for the vibrational linewidth. In particular a direct tentative comparison can be made for H adsorbed on W(100) [4]. The calculated damping rates are shown to be influenced by the behaviour of the adsorbate induced electron structure in the surface region, which varies from one adsorbate- surface combination t o another. An enhanced damping rate is found, when the hydrogen molecule induces resonant states at the Fermi level. Further, a correlation is pointed out between the result for the hydrogen atom in the surface region and the corresponding result for the homogeneous electron gas with a density equal t o the local density.

These two features are expected t o be present for more general substrate and adsorbate combinations. Within a friction force description the calculated damping rates can be used t o

Physica Scripta 29

electron-hole pair mechanism gives a simple isotope depen- dence. The broadening is inversely proportional to the reduced mass of the vibrational mode, an effect which has been observed for CH30 on Cu [7]. The temperature dependence of the broadening differs markedly, being strong for the anharmonic mechanism [6] and negligible for the electronic mechanism.

The electron-hole pair mechanism has been treaded theoreti- cally earlier for the decay of phonons in metals. Lifetime broad- ening of phonon spectra peaks have been directly measured by neutraon scattering and indirectly with other techniques. For free-electron-like metals. such as Al, Li and K, the calculated broadening has been found t o give an important contribution to the measured linewidth at low temperature [SI. For a few transition metals, in particular for Nb [9] values for the broad- ening have shown some success in comparison with experimental results calculated in the rigid muffin tin approximation. For a more extensive discussion of this subject the reader is referred t o the literature [ 101.

Other calculations of the damping rate have been performed, both for physisorbed and chemisorbed species. In the case of physisorption, where the electrons of the adsorbate and the substrate are separated in space, the broadening has been calcu- lated in a few different models of the metal surface [ l 1, 121. It has been pointed out that the calculated value for the damp- ing rate depends sensitiviely on the description of the electronic motion in the surface region [13]. For instance, the infinite barrier model of the metal surface cannot account for the observed linewidth 0.5 meV of the stretch mode in CO adsorbed on Cu [12]. Hence, there is still some controversy about the order of magnitude of the damping rate in the case of physi- sorbed species.

In the case of chemisorption, an instructive calculation of the damping rate has been given for a hydrogen atom on a tight binding solid (cubium), giving a valuable first estimate [ 141. That calculation will be commented upon in Section 3. The

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces 361

observed linewidth of the stretch mode in CO has been accounted for in a semi-empirical model based on a simple Newns-Anderson Hamiltonian for the electronic structure of the chemisorbed molecule [ 151. In this work a simple expression was derived, which connects the damping rate to the strength of the charge fluctuations related to the vibrational motion. The damping rate can then be extracted approximately from other experimental observations, e.g., The change in dynamical dipole moment upon adsorption.

In Section 2 contact is made with the friction coefficient appearing in the context of stopping power for atoms in the bulk and surface region [16]. The friction coefficient is found to be directly related to the damping rate for the corresponding frustrated translation. Consequently; the phase shift formula for the friction coefficient of an atom immersed in a homo- geneous electron gas [17] is also applicable for calculating the damping rate in this case.

The theory for coupling between the electronic and vibra- tional motion has been treated in a vast number of papers and is well-developed [lo]. In technical terms, with details in Section 2, the broadening is determined from the imaginary part of the dynamical matrix. In this work we make a one- electron approximation for this imaginary part and relate it to ground state quantities of the density functional scheme [18, 191. A similar one-electron approximation has been made for the calculation of the photo absorption by atoms [20]. In our application, the Local Density Approximation (LDA) has been used for the exchange-correlation potential [21], This approximation has been successful in the calculations of static properties, such as binding energies, electron structure and vibrational frequencies of atoms, molecules and solids [22].

In order to be realistic, the electronic ground state properties that should be used as an input in the damping rate formula, should be calculated in a model, which retains the whole continuous spectrum of one-electron excitations. This is not the case for the commonly used cluster and finite-layer calcu- lations. On the other hand the embedding scheme, which we apply, has this feature [23]. It accounts for the extended nature of the substrate, while focussing on the local changes induced by the adsorbate. In addition, in the proper limit the results obtained in this scheme compare well with the results from accurate calculations based on the phase shift formula.

The necessary formulism for treating the electronic mechanism for vibrational damping is presented in Section 2. In Section 3 the calculational method is demonstrated to be appropriate for calculating the broadening of vibrational modes of adsorbates on metal surfaces. Finally, the results will be discussed in Section 4 and compared with experimental data in Section 5.

2 . Theoretical formulation 2.1. The electron-vibration coupling The large mass difference between the nuclei and the electrons forms the basis for the separation of the fast electronic and the slow nuclear motion, as utilized in the Born-Oppenheimer approximation [24]. The internuclear forces are then deter- mined by allowing the electronic motion to adiabatically follow the motion of the nuclei. In the harmonic approximation for the internuclear forces the nuclear motion can be described in terms of stationary vibrational modes, i.e., phonons in crystals or, as here, a localized vibrational mode with a fre- quency outside the range of the bulk phonon band. There are

two qualitatively different effects that can cause such a mode to be non-stationary: The fact that the electronic motion can not follow the nuclear motion instantaneously, and the presence of anharmonic contributions to the internuclear forces. This work considers the former effect in the case of alocalvibrational mode coupled to the electronic excitations of an extended system, in particular an adsorbate vibrational mode coupled to the electron-hole pairs of a metal substrate.

This kind of non-adiabatic effects has been considered for a long time in the field of electron-phonon interactions in metals [ lo ] , where the screening properties of the conduction electrons have to be accounted for as for the adsorbate-metal system. In this paper the formulation from the classical work by Baym [25], refined by others [28], is adopted.

To linear order in the nuclear displacements an equation for the nuclear motion in an ordinary metal can be derived [26]. In the case of adsorbed atoms/molecules on a surface we are faced with a system with different nuclear masses. For con- venience we therefore introduce a mass-weighted displacement vector g(R, t ) = m(R)”2u(k, t) , for each nucleus of mass m(R) and with equilibrium position R. The equation of motion for the correlation function of g then takes the form

ea d2 -@R, t)g(R’, 0)) = I d t ‘ x D(R, R”; t - t ‘ ) d t2 0 R”

x (C(R”, t ’)g(R’, 0))

where the sum is over all nuclei in the system and D is the generalized dynamical matrix. This matrix can be decomposed into two separate contributions

D(R, R’; t ) = Do(R, R’)6(t) + D’(R, R’; t ) (2)

where the first instantaneous part describes the static response of the motion of the electrons and the second dynamical part, the lossy response. The latter part can be expressed in terms of the dynamical part x‘ of the retarded electron density propagator x ( x , x’; t ) calculated for nuclei fixed in their mean positions

x x’(x, x’; t)V‘- 1

[x’ - R’I

where Z(R) is the charge of the nucleus at the position R. The prime interest in this work is the decay of a localized

vibrational eigenmode of Do. The matrix D’ has both diagonal and off-diagonal matrix elements with respect to this mode. As will be clarified further on, the imagonary part of the former elements corresponds to the decay of the mode into electronic excitations. The effects of the latter terms can be shown to be of second order in the non-adiabatic coupling between the electronic and nuclear motion and are neglected in our treatment.

To simplify the further treatment we consider a single vibrating nucleus with mass M and charge eZ with the normal coordinate Q.

The result can be generalized to an arbitrary mode. The equation of motion for the correlation function of the displace- ment coordinate Q is then

Physica Scripta 29

362 B. Hellsing and M. Persson

d2 - (Q(t)Q(O)) = - G(Q(t)Q(o)) d t2

+J’- 0 dt’A(t - t’)(Q(t’)Q(O)) (4)

The dynamic self-energy A(t) can be expressed in terms of the dynamical part of the density propagator

Iz(t) = - ! dx dx’@(x) ~ ‘ ( x , x‘; t)@(x’)

where

( 5 ) e’ - M

is the Coulomb field from the bare, displaced nucleus. The experimentally measurable energy loss spectrum is

related to the spectral function, S ( o ) , of the correlation func- tion in eq. (4) [27] :

S(W) = 2W {I - exp ( - F i ~ / t k B T ) ) - l Im xQ(w) ( 7 )

where xQ is the Fourier transform of the retarded displacement- displacement response function

1 X Q ( W ) = Ji d t exp (iut),([Q(t), Q(0)l) (8)

This function can be evaluated directly from eq. (4). In particu- lar for weak non-adiabaticity, i.e., when Im A(w)/wCLo < 1. the imaginary part of xQ, to be inserted into eq. ( 7 ) , takes a Lorenzian form

The mode is consequently lifetime broadened. The lifetime T is directly related to the full width at half maximum r = W Im A(no)/nO as 7 = W/r. There is also a small dynamical shift A, caused by the real part of A. which has been discussed earlier in the context of electron-phonon interactions in metals [ lo ] . The dynamical shift is of the order a0 /eF smaller than the broadening in most situations without any dramatic variation of the electron structure around the Fermi-level f F and can accordingly be ignored [ 2 8 , 291.

2.2. The one-electron approaimation In order to calculate the broadening one has to evaluate the density propagator for the electronic motion. To describe the electrons particularly in complex geometries, a theoretical method that is simple and yet accounts for exchange and correlation effects among the electrons is useful. The density functional theory provides a one-electron scheme for calculating ground state properties [ 181. In the Appendix this scheme is extended to slow time-dependent perturbations, and arguments in favour of this extension are presented. The main result of the Appendix is the following expression for the imaginary part of the dynamical self-energy a t non-zero temperatures:

x (f(fk) -f(fk’))6(fiW - f k + fk’) (10)

where $k(x) and f k are one-electron wavefunctions and energies.

Phj%sica Scvipta 29

respectively, using the density-functional-scheme in the descrip- tion of the groud state, f ( ~ ) is the Fermi-Dirac distribution function, and the factor 2 comes from spin degeneracy. The effective field Qeff has contributions from three terms: the bare Coulomb field from the displaced nuclei, the Coulomb field from the induced electronic charge distribution, and a local-field correction.

We will consider the quasi-static limit no -+ 0 of eq. (lo), which should be applicable, when the electronic structure varies slowly over the energy scale determined by the vibrational quantum energy h a o . In the situation described in this paper the matrix element in eq. (10) is slowly varying over the energy scale h a o as a function of the one-electron energies and from it one can derive the following expression in the quasi-static limit:

x 6 (p - fk)&(p - fk’) (1 1)

where p is the chemical potential and a V/aQ(x) the static limit of -eQeff(x, a,). The potential V is the full one-electron potential within the density functional scheme. In the tempera- ture range of interest in the laboratory the chemical potential differes neglible from its zero-point value f F .

2.3. The fomulas for the damping rate The linear dependence on 520 of the expression for Im A(a0) has its origin in the density of states of electron-hole pairs close to the Fermi level, and this factor disappears in the expression for the broadening

r = w ~m A ( S ~ ~ ) / S Z ~

(12)

l 2 2nW2 r = - 1 1 JIdx $2(x)+k(X)aV/aQ k , k ’

x 6 (fF - fk)&(fF - fk’)

This is the golden-rule like expression for the broadening which has appeared earlier in various forms [lo].

Conventionally, this formula is derived by applying the golden rule to the non-adiabatic interaction between the elec- tronic and nuclear motions, i.e., the interaction neglected in the Born-Oppenheimer approximation, which is linear in the nuclear displacement, with the nuclear motion subjected to the harmonic approximation. In the quasi-static limit the same expression, eq. (12), for the damping rate then appears. The main points of giving the above derivation of eq. (12) is that it considers the broadening of a localized vibrational mode and that an argument has been given in the Appendix for using the one-electron wavefunctions and potentials of the density functional scheme.

A physical interpretation can immediately be presented for the damping rate according to eq. (12): an electron incident with an energy close to the Fermi level can be scattered by the field from the displaced nuclei of the vibrational mode, screened by the conduction electrons, to an unoccupied state, and the corresponding transition amplitude is expressed by the matrix element in eq. (12). This transition can alternatively be described as an excitation of an electron-hole pair, It is note- worthy that the broadening is independent of the vibrational frequency n o in the quasi-static limit. This can be understood from eq. (12). On average, the nuclei are displaced with a

Electronic Damping of Atomic and Molecular Vibrations a t Metal Surfaces 363

root-mean square amplitude d m . The frequency Qo in this factor disappears when the matrix element is squared, as the available phase space depends linearly on no.

Two characteristic features are apparent in the expression for the broadening: (a) the isotope effect and (b) the negligible temperature dependence. Thus, r is sensitive for isotopic substi- tutions of those nuclei, which participate in the considered vibrational mode. One cannot rule out some weak indirect temperature effects on the broadening, e.g., due to the thermal expansion of the lattice.

2.4. The friction coefficient A simple description of the electronic contribution to the dissipation of translational energy of an atom in the surface region of a metal is provided by Langevin’s equation of motion or equivalently by a Fokker-Planck equation [16]. In this description the force acting on the nuclei from the surrounding electrons consists of a reactive part, derived from the adiabatic potential energy curves and a dissipative part, described by a friction coefficient and a stochastic part. In this section we show that there is a close relation between the friction coef- ficient and the damping rate.

The friction coefficient q is determined by the stochastic part Fst of the force acting on the nuclei fixed in space with charge Z [ 161,

q = - 1 Re /;dt(Fst(t)Fst(0))

kBT where the tensor character of the friction coefficient is indicated, the bracket denotes thermal average and

The damping rate re of a vibration of the atom along the direction e is determined by the self-energy A(t) and according to eq. (5) it can be expressed in terms of the corresonding component of the force Fst as,

1 ~ ( t ) = M x ( [ e . P t ( t ) , e .P (o ) ] ) e ( t ) (14)

where M is the mass of the nuclei. The fluctuation-dissipation theorem relates the imaginary

part of the Fourier transform of the retarded correlation function in eq. (14) to the Fourier transform of the correspond- ing correlation function [30]. In the quasi-static limit, o + 0, this relation reduces to,

Re l - ( e . Fst(t)e- Fst(0)> d t Im A(o) + - 0

MkgT O

Since the damping rate is given by, r = hA(Qo)/Q0, a simple identification of eqs. (13) and (14) yields directly,

M A

e - q - e = -re

This relation enables one to evaluate the friction coefficient tensor from calculated damping rates for vibrations of the atom in different directions in space.

3. The calculational method 3.1. The embedding scheme In the calculations the Kohn-Sham density functional scheme

[ 181 has been used with the local density approximation for the exchange-correlation potential [21], which allows the con- struction of effective numerical schemes for the calculation of the ground-state properties and electronic structure.

In order to calculate the damping rate from eq. (12), one needs, in particular, a numerical scheme to solve the Kohn- Sham equations, which retains the whole continuous spectrum of one-electron states of the metal substrate. The embedding scheme of Gunnarsson and Hjelmberg [23] has this important feature and it has been devised for the calculation of ground- state properties of adsorbates on metal surfaces. The scheme and its applications to different systems have been described elsewhere [31, 321 and we will only provide the necessary background for the presentation of the extension to the calcu- lation of the damping rate from eq. (12).

The scheme is based on the fact that the screening by the conduction electrons effectively limits the range of the pertur- bation caused by the adsorbate. The perturbed one-electron potential, AV is expanded in a localized finite basis set {In)},. In the Kohn-Sham scheme the one-electron potential consists of three terms: the bare Coulomb field from the nuclei, the Hartree field from the electrons and a field representing the effects of exchange and correlation. The latter two terms of the potential have to be determined self-consistently by an iteration procedure. The applied functional form of the exchange-correlation potential is the one suggested in [21].

In this scheme the perturbed one-electron potential is restricted to have the form AV = PAVP where the projection operator P is defined as P = Z, In)(n 1, This restriction forms the basis for the reformulation of the Schrodinger equation into a Dyson-like equation:

PC(E)P = PGO(E)P + PG~(E)PAVPG(E)P (17)

where PG(E)P and PGo(e)P are the Green functions for the coupled adsorbate and substrate system and the bare substrate, respectively and E is the energy.

The reduced Green functions appearing in eq. (17) contain all information about the one-electron wave functions and energies projected on the localized basis set. For instance, the induced charge by the adsorbate, Ap(x), is given by

E Im (xIP(G(E) - Go(E))PIx)

where the factor 2 comes from spin degeneracy. The charge density is a basic quantity in the local density approximation, as it determined the full one-electron potential. The Green function is then solved self-consistently from eq. (1 7) by iterating from an initial value for the density matrix.

The embedding scheme has earlier been deomonstrated to give a good description of the electronic ground-state for the hydrogen atom [31] and the molecule [32] respectively, adsorbed on different metal surfaces. These applications have used the semi-infinite jellium model, where the positive charge from the nuclei in the substrate is smeared out to a semi-infinite positive homogeneous background. Calculations have been performed for an adsorbed hydrogen aom [31], and a molecule that is oriented normal to the surface [32]. Both systems have axial symmetry. Hence, the components of the angular momen- tum normal to the surface are conserved. This fact reduces favourably the number of matrix elements both in the density and the potential matrices. One should note that the calculated equilibrium configuration for the hydrogen molecule on the

Physica Scripta 29

364 B. Hellsing and M. Persson

Mg(1000) surface is the one with the molecule lying flat on the surface [33], however.

3.2. The evaluation of the damping rate In the extension of the calculational scheme we make the additional assumption that the quasistatic field a V/aQ from the vibrating nuclei is also limited in range by the screening from the conduction electrons. Thus it should also be possible to represent this field by an expansion in a localized basis set. The validity of this assumption is investigated in the later subsections.

The sum over the wave functions in the matrix element appearing in eq. (12) can be replaced by the density of states matrix at the Fermi level. When introducing the imposed restriction on the quasistatic field, a V/aQ =Pa V/aQP, and invoking the resolution of the projection operator in the local- ized basis set, eq. (12) can be re-expressed as

r = (2nh2/M) Tr {WW) (19) where

( n I Wln') =

( n lp(+)Im) and ( n laV/aQlm) being the matrix elements of the density of states at the Fermi level, and in the field of the vibrating nuclei, respectively. The expression for the damping rate can thereby be evaluated directly through matrix multipli- cation and matrix contration.

In all our calculations of the matrix elements for the field aV/aQ, the center of the basis set is fixed in space. The one- electron potential A V has been calculated for three different amplitudes Q of the considered vibrational modes. In the following the A, B and C configurations denote the cases when (2 = 0. - AQi2 and AQi2, respectively. The field aV/aQ is

( n Ip(+)lm)(m la V/aQ la'). m

Table I. The vibrational lifetime broadening for HIJe (vibration normal to the surface) and H2/Je (stretch mode) as a function of the bulk electron density parameter r, and distance d from the jellium edge

H/ Je H,/Je

r , (a.u.j d (a.u.j r (meV) r , (a.u.) d (a.u.) r (meV) ~~~~~ ~~

3.99 -14.0 1.16 3.99 0.2 0.57 2.65 -14.0 2.50 2.65 0.5 0.81 3.00 - 2.0 1.55 3.00 -1.0 1 .oo 3.00 -0.5 0.83 3.00 0 .0 0.72 3.00 0.5 0.69 3.00 1 .o 0.70 3.00 1.5 0.75 3 .OO 2.0 0.83 2.07 - 2.0 4.06 2.07 -1.0 2.40 2.07 -0.3 1.78 2.07 0.3 1.48 2.07 0.3 1.36

3.99 -6.0 3.99 -4.0 3.99 - 3.0 3.99 - 2.5 3.99 - 2.0 3.99 -1.5 3.99 -1.0 3.99 0.0 3.99 1 .o 3.99 2 .o 2.65 -6.0 2.65 -4.0 2.65 -3.0 2.65 - 2.0 2.65 -1.0 2.65 0.0 2.65 1 .o 2.65 2.0

9.58 18.4 24.3 20.6 12.4 5.91 2.52 0.58 0.22 0.13

13.3 10.8 14.8 18.9 12.8

3.18 1.13 0.43

face a direct comparison can be made between the evaluated damping rate in this embedding scheme and the result from another numerical calculation of the friction experienced by the atom [34]. The latter calculation is based on phase shift formula for the friction coefficient valid for a homogeneous electron gas [ 171. From the direct relation between the friction coefficient and the damping rate stated in eq. (16), the same formula with the proper modification is also valid for this case,

simply evaluated from A V(C) and A V(B) as a simple difference quotient: ( I f l ) s in2 (61+ l -6,) (21) aV/aQ = (AV(C) - AV(B))/AQ

where r , is the electron gas density parameter and 6, is the phase shift experienced by the Ith partial wave of an incident electron at the Fermi level. The phase shifts have been caicu- lated within the density functional scheme, with the Local

(20)

The amplitudes used in the calculations are AQ = 0.2, 0.1 a.u. for the atom and the molecule, respectively. A measure of the linearity of the potentid A V in (2 is given by the quantity

r = ll(AV(C) + AV(B) - 2AV(A))/(AV(C) - AV(B))II

where I / W l / 2 = Tr (WWt) is a norm of the matrix W . For the hydrogen molecule r is of the order 0.1 for all considered configurations, which supports the chosen numerical evaluation of the matrix elements (nIaV/aQim). For the hydrogen atom at the distance d = 0.5 a.u. ourside the edge of a semi-infinite jellium with the electron gas density parameter Y, = 2.65. the dependence of the damping rate on the vibrational amplitude has been investigated and found to change less than 6% when decreasing the amplitude with a factor of 2.

The damping rate r has been evaluated for H and H2 on jellium with a self consistent solution for the matrix of the density of states at the Fermi energy for the configuration A. and the matrix of the field (eq. (20)). The used standard basis set is discribed in Section 3.5. The results are tabulated in Table I . The evaluated damping rate converges more slowly than the total energy for the equilibrium configuration as shown in Table 11.

3.3. A comparison with another calculation By forcing the hydrogen atom to be far inside the jellium sur-

Physica Scripta 29

Density Approximation for the exchange and correlation potential, by numerical integration of the self-consistent radial Schrodinger equation [34]. The results calculated with the two methods are compared in Table I11 for two different cases. The agreement is considered to be fair.

Table 11. The convergence of the chemisorption enera) A E and the broadening r (eq. (16 ) ) for H/Ag (rs = 3.00) and H2/Mg (r, = 2.65). The distance from the jellium edge is 1.Oa.u.

Iteration E (eVj r (meV)

9 - 1.75 1.076 10 - 1.76 0.707

HI Ag 11 - 1.77 0.698 12 -1.77 0.696 13 -1.77 0.695

9 - 3.75 1.14 10 -3.75 1.11 11 - 3.74 1.10 12 -3.74 1.11

H, /Mg

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces 365

Table 111. A comparison of the two results for H/Je (eq. (16)), far inside the jellium edge for two different values of the bulk electron density parameter r,. In the right column we give the result from eq. (18), described in Section 3.3

0“ rhom rs (ax.) d (am) r (mew

3.99 -14.0 1.16 1.45 2.65 - 14.0 2.50 2.88

3.4. The local basis set The results evaluated from eq. (17) are reliable only if the basis set represents accurately the ground state one-electron potential and its derivative. For practical reasons we have to work with a finite basis set. The sensitivity of the damping rate to the size of the basis set has been investigated with respect to the number of basis functions. The investigation is guided by qualitative considerations based on the nature of the electronic structure and the vibrational mode.

The functions $,(x) in the localized basis set consist of a radial part f p ( r ) multiplied with an angular part of spherical harmonics YL(X) with L = (1, m ) and $,(x) = fG(r)YL(X), i.e., they form a one center-basis set. The radial functions are constructed by a Gram-Schmidt orthogoalization procedure from Slater-like orbitals r” exp (- ar) truncated at r = R,, where R, = 8a.u. The parameters in the radial functions are chosen to be the same as those used in the earlier calculations for the ground state. The standard basis set used in our calcu- lations consists of 9 radial functions and spherical harmonics

For the hydrogen atom immersed in a homogeneous electron gas of a metallic density, the induced electronic structure for the H-atom is dominantly of s-character ( I = 0) [35]. When the atom is placed in the surface region of the semi-infinite jellium the dominance of the s-character prevails [31]. In the homo- generous situation the dominant contribution (99%) to the damping rate for the atom comes from s- and p-states and from the first erm of the sum in eq. (20) [34]. In the surface region the oneelectron potential is not necessarily spherically sym- metric. For the vibrating atom the potential can cause tran- sitions between states whose local angular momenta can differ by more than one unit. However, our calculated results, Table IV, indicate that these effects are small and that the chosen number of angular functions should be sufficient.

The electronic structure of thy hydrogen molecule involves states at the Fermi level with higher angular momenta than for the hydrogen atom. They are primarily introduced by the presence of the anti-bonding 2u* orbital of the free hydrogen molecule, and can be inferred to be dominantly of p-character.

up to 1 = 3.

Table IV. Test of the basis set. The vibrational lifetime broad- ening for H/Mg and H2/Mg a t the distance d = 1.Oa.u. from the jellium edge for different maximum 1-quantum numbers l , , of the basis set

H/Mg H, /Mg Jmax rImax/r4 rzmax/r4

2 1.01 0.63 3 1.01 1.02 4 1.00 1.00

Table V. Test of the basis set. The vibrational lifetime broad- ening for H2/Mg a t the distance d = 1.Oa.u. for the jellium edge, for different number of radial basis functions N

N r (meV)

I 8 9

1.18 1.15 1.10

The bare Coulomb fields from the vibrating protons in the molecule are of quadrupole character and should mainly cause transitions between states where the corresponding angular momentum values differ at most with A1 = 2. Hence, one needs at least s-, p - , d- and f-states for a meaningful result. That view is confirmed from Table IV, and it seems in practice to be sufficient to consider states up to 1 = 3.

The dependence of the calculated damping rate on the number of radial functions has also been investigated. As apparent from Table V the chosen radial basis set (N = 9) should be sufficient.

3.5. Localization of the field In the calculations (eq. (19)), the field aV/aQ coupling the elec- trons to the nuclear motion has been assumed to be relatively short ranged, to allow it to be expanded in the local basis set. This subsection investigates the consistency of this assumption. Close to the vibrating nuclei the electron experiences the bare Coulomb field which is of dipole character. The contribution from such a dipole field to the damping rate inside a sphere with radius R increases as R4 for small R . In this perspective the possible misrepresentation of the field close to the vibrating nuclei by the finite basis set should be unimportant.

20 16 - “H2 in Na”

Id = -3.0a.u.)

0 2 L 6 Rt r La.u.1

Fig. 1 . Illustration of the range of the field aV/aQ which couples the electronic and vibrational motion. The vibrational lifetime broadening r of the stretch mode of (a) “H, in Na” (jellium with rs = 3.99 and d = - 3.0 a.u.) and (b) H,/Mg (jellium with rs = 2.65 and d = 0.5 a.u.). The dashed and the solid line represents the contribution from the bare Coulomb field and the full field respectively. (c) The vibrational lifetime broadening of the mode normal to the surface. The dashed and solid line represents the bare Coulomb field and the full field respectively for H/Mg (jellium with rs = 2.65 and d = 0.5 ax.) . The dotted line represents the contribution from the full field for H/Ag (jellium with rs = 3.00 and d = 3.0 a.u.). The arrows indicates the results in theequilibrium positions.

Physica Scripta 29

366 B. Hellsing and M. Persson

The expansion of the field in the basis set gives a represen- tation only in a finite part of space. The possible contributions from long range components of the field to the damping rate have been investigated with the following method: The field aV/aQ is truncated in space at a radius R,. When invoking the matrix representation of the trucated field in the basis set the corresponding R,-dependent damping rate can be evaluated from eq. (19) (Fig. 1). The figure shows that in both cases for the molecule and the atom the field is short ranged, the excitation of the electrons taking place in the neighbour- hood of the vibrating species. This behaviour demonstrates that the radius R, = 8 a.u. chosen for the embedding sphere is large enough to yield a meaningful result.

In the case of the vibrating atom, Fig. l(c) shows that the field has a short range as the result of the effectiveness of the conduction electrons to screen out the field from the vibrating proton, which has dipole character. The contribution from the bare Coulomb field of the vibrating proton, on the other hand, increases steadily with the distance from the proton. In view of this result we argue that the neglect of screening in the other- wise irinstructive calculation in Ref. [ 141 constitutes the main deficiency of the latter. For the vibrating molecule, on the other hand, the short spatial range of the field a VfaQ is mainly due to the quadrupole character of the Coulomb field from the protons, as shown in Fig. l (a , b). This reflects the fact that the quadrupole field decays faster with the distance than the dipole field.

The fact that the excitation of the electron-hole pairs is localized in space around the atom and the molecule respect- ively, can also be found from the following consideration. The damping rate can be decomposed into separate contributions from states with a well-defined m-component of the angular momentum normal to the surface (Table VI). In general elec- trons in states with higher m-values will be found further out from the axis. Table VI shows the dominant contribution to come from m = 0, which is consistent with the assumption of a short-ranged field and with the result of Fig. 1.

In the next section, we will give a more detailed discussion of the correlation between the local electron structure and the calculated damping rates.

4. Results and discussion 4.1. General aspects Whether the electronic mechanism can provide an efficient channel for energy transfer between an adsorbate and a metal substrate or not is a question that is obviously determined by the magnitude of the lifetime broadening. A dimensional analysis of eq. (12) allows the damping rate to be expressed as r = f i ' / M l ~ ~ , where le, has a dimension of a length determined

TableVI. Analysis of the damping mechanism. The contri- bution to the vibrational lifetime braodening in meV, from different m-components of the angylar momentum. Results are given for three different configurations, H2/Na, H2 fMg and HIMg a t the indicated distances d from the jellium edge

m=O m = l m = 2

Fig. 2. The vibrational lifetime broadening r of an atom immersed in a homogeneous electron gas as a function of the atomic number 2. r is evaluated from eq. (18), where the sums over the phase shifts is taken from [33].

by the electronic structure. In the case of a hydrogen atom embedded in a homogeneous electron gas I,, is essentially given by the density parameter r, (see eq. (21)) as the r , dependence of the phase shifts is less dramatic. It is noteworthy that the magnitude is not determined by the ratio between the masses of the electron and the nuclei.

Some effects of the electronic structure on the damping rate are illustrated by the results from eq. (21) for atoms immersed in a homogeneous electron gas. The relevant sums over the phase shifts have been taken from [34]. From Fig. 2 it is apparent that the damping rate can be appreciable even for heavy atoms. such as C and K. The detailed explanation of the oscillation in the damping rate with the atomic number is described in [34]. Here it suffices to stress that the origin of these is the filling of atom induced resonance states in the conduction band.

For an inhomogeneous system. e.g.. an adsorbate on a metal surface the symmetry is lowered. and additional couplings are thereby introduced in the matrix element. For instance, while in a homogeneous system the filed from a single vibrating atom can only cause transitions between one-electron states, for which the angular momenta differ by unity, there is no such symmetry restriction in the surface region.

Already from eq. (12) it is possible to discuss the different electronic factors which can influence the magnitude of the damping rate r. The magnitude of the matrix element is deter- mined by the amplitude of the one-electron wavefunctions at the Fermi-level and the strength and spatial range of the field aV/aQ from the vibrating nuclei. An enhanced amplitude of the wavefunctions is expected in the neighbourhood of the adsorbate, when the latter induces many states at the Fermi- level. Whether those local or other more extended aspects of the electronic structure are important or not should be deter- mined by the spatial extension of the field.

These general questions about the different factors influenc- ing the magnitude of the broadening, will be answered to some extent by the results for the damping rate presented in the next section for vibrations of a hydrogen atom and molecule, chemi- sorbed on different jellium surfaces.

H,/Na (d = - 3.0 a.u.) 22.95 1.288 0.012 4.2. Adsorbate induced states o'Ool

H/Mg (d = 0.5 a.u.) 0.782 0.022 0.001 As already discussed in Section 3.5, (Fig. I) , the excitation of the delectron hole pairs takes place rather close to the vibrating

H ; / M ~ (d = 1.Oa.u.) 1.147 0.01 1

Physica Scripta 29

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces 367

nuclei. IN particular, it is more localized in space for the H atom than for the H2 molecule. As a consequece, the local electronic structure is a key factor for determining the magni- tude of the damping rate. It is therefore of immediate interest to review some of the results for the adsorbate induced elec- tronic structure obtained in earlier works within the same one-electron scheme for the atom and the molecule chemi- sorbed on jellium surfaces.

The electronic structure of the atom in the surface region can be interpreted in terms of the bound state following the local bottom of the conduction band, defined by the one- electron potential. The state will turn into a resonance when its energy overlaps with the bulk conduction band [36 ,37 ,38] .

The electronic structure of the hydrogen molecule is qualitatively different from that of the atom. When the molecule approaches the jellium surface the interaction with the substrate electrons gradually broadens and shifts the empty antibonding state 2a* down in energy and the formed reson- ance gets filled up to the Fermi level [32].

The similarities between the adsorbate induced electronic structure for the hydrogen atom in the surface region and that in the bulk region suggest a comparison of our calculated results with results from a local density approximation based on eq. (21). In such an approximation the values for the electron gas density parameter rs are determined from the electron density of the bare surface at the position of the atom. There is a certain correlation between this result in the inner surface region and that calculated from er?. (21), as demonstrated in Fig. 3. This result suggests that the bare surface electron density close to the adsorbate determines the magnitude of the damping rate, when there is no dramatic adsorbate induced electronic structure at the Fermi level. Further out, however, they exhibit different behaviours. The true surface result from eq. (19) is not decaying exponentially with the electron density but rather levelling off. This effect can be viewed as caused by the less efficient screening by the conduction electrons in the outer parts of the metal surface region. For instance, self-consistent calculations for adsorbed H have shown an increase separation in space between the negative excess charge around the proton and the positive screening charge residing on the substrate with increasing H-jellium distance. Similarly the screening charge induced by the bare Coulomb field from the vibrating proton outside the surface should reside on regions with high electron density. Increased charge separation with distance d

I

- 2 - 1 0 1 2 3 L 5 i

d (a. u.) Fig. 3. The vibrational lifetime broadening of the vibration normal to the surface for a H atom, as a function of the distance d from the edge of jellium representing Al (dots, rs = 2.07) and Ag (crosses, rs = 3.00), respectively. The solid line is the “local density approximation” results, i.e., the result extracted from eq. (18) for the damping rate of the atom immersed in a homogeneous electron gas with a density equal to the local density of the bare surface.

should also make the field to extend further and thus keep the magnitude of the damping rate high. This view is supported by the result that in the case of an H-atom forced to vibrate around a position far outside the jellium edge the main contribution comes from a region that is less localized in space than for the atom in the equilibrium position (Fig. l(c)).

For the hydrogen molecule, on the other hand, the bare Coulomb field from the individual protons is already screened out by the electrons in the bound orbital lo. This effect and the quadrupole character of the bare Coulomb field from the vibrating protons in the stretch mode cause the damping rate for this mode to decrease steadily with the decreasing electron density in the outer surface region, as shown in Fig. 4.

We have investigated the influence of adsorbate induced states at the Fermi level both for the atom and the molecule. In the latter case it has been made by forcing the molecule with a fixed internuclear distance to approach the bulk region. The results for the damping rate of the molecule in the region of the jellium surface, shown in Fig. 4, demonstrates that there is a strong correlation between the passage of an antibonding state 2o through the Fermi level. In particular, a maximum occurs when the resonance is half-filled. This fact can be under- stood simply from the behaviour of the Fermi-electron wave functions in the neighbourhood of the molecule. At the passage the amplitude of the one-electron wave functions is enhanced in the vicinity of the molecule. The resonance is rather broad, as shown schematically in Fig. 4. For a more narrow resonance an even greater enhancement of the damping rate is expected. Finally, the filling of antibonding states is also known to play a key role in other physical phenomena occurring, when molecules are chemisorbed on metal surfaces. Examples of such phenomena are bond breaking [32], shifts of vibrational frequencies [32] and changes in dynamic dipole moments [47].

10

- 5 t L I

-6 Li -3 -2 -1 -0 1 2 d (a.u)

Fig. 4. (a) The vibrational lifetime braodening r of the stretch mode of H, as a function of the distance d from the edge of jellium representing Na and Mg, respectively. (b) The bare surface electron density profiles of Na and Mg in the jellium model. (c) The position of the adsorbate induced resonance 20* for the molecule, as a function of the distance d from the jellium edge, and the corresponding width when it crosses the Fermi level 1311.

Physica Scripta 29

368 B. Hellsing and M. Persson

4.3. Damping of translational motion There are several different theoretical treatments of the damp- ing of translational motion of an adsorbate due to the excitation of electron-hole pairs of the substrate in the literature [ 14, 16. 40, 411. They can be grouped according to the region in space, where they are expected to occur. In the asymptotic region outside a metal, where there is no overlap between the elec- tronic wavefunctions of the adsorbate and the substrate, the dynamical corrections to the van der Waals interaction have been found to be negligible [39]. In the region closer to the surface, on the other hand, it is believed that appreciable energy loss can take place, if the adsorbate induces an electronic resonance that crosses the Fermi level. This is supported by model calculations using realistic parameters [40, 411.

In this section we calculate the energy loss in the region even closer to the surface for a case, when there is no dramatic changes in the density of states at the Fermi level. Under these circumstances arguments have been given for using a friction approach [42].

We consider a hydrogen atom moving in the direction normal to a jellium surface representing Ag. The atom is given an initial thermal energy of 25meV at the distance 4a.u. from the jellium edge. The friction force acting on the atom is given by

where v1 is the projection of the friction tensor, evaluated from eq. (16), in the direction normal to the surface. The mean velocity (v) is assumed to be given by the adiabatic potential energy curve, U@), as (v) = .\/21.3((E) - U). This approximation is valid, when the atom is far from thermal equilibrium with the surface. The results are shown in Fig. 5(b). Although the cal- culations are performed in a limited region of space, the thermal energy of the incoming hydrogen atom is seen to be effectively accommodated by the substrate electrons. This fact is mainly due to the enhanced velocity in the potential well.

The limitations of the assumption of a dissipative force, proportional to the adparticle velocity v, have been investigated in the context of stopping power of a charged particle moving in a homogeneous electron gas [ 171. In this case eq. (22) is obtained,

- 2 0 2 4 6 8 d (a.u.1

Fig. 5. The potential-energy curves for H and H- as a function of the distance d from the jellium edge. The curves are obtained from the effective medium theory [ 521 (dotted line) the image potential (dashed- dotted line) and the present calculation (solid line). The dahsed line is the result of interpolation by hand between the separate limits. (b) The total mean energy of the incoming H atom is integrated from eq. (19) and given as a function of the distance d from the jellium edge, solid lind. The incident atom has been given a typical thermal energy of 25 meV at the distance d = 4 a.u.

Physica Scripta 29

with 7) = (M/A)rhom and rhom given by eq. (21), when only the linear term in an expansion in the quantity (v/vF) is kept, v F being the Fermi velocity. In our calculation the velocity of the atom is small, (v/vF) < 2 x lo-*, and therefore a friction description, eq. ( 2 2 ) , should be applicable.

5. Comparison with experiments

The inherent linewidths of surface vibrational modes in adsorbed layers on metal surfaces have been resolved in a few instances by different surface sensitive spectroscopic methods with high resolution, such as Infrared Reflection Spectroscopy (IRS) [43-451 and the related technique Surface Electromagnetic Wave Spectroscopy (SEWS) [4] and Electron Energy Loss Spectroscopy (EELS) [46]. In the analysis of such recorded spectra one has to be aware of the fact that several qualitatively different mechanisms can contribute to the observed linewidth and lineshape.

Adsorbate overlayers exhibit a rich variety of different structures. For instance, the adsorbed species can form a liquid like phase, where they are rather free to move along the surface, or alternatively the individual species can be localized for a long time at different sites. In the cases of such disordered struc- tures there will be an inhomogeneous contribution to the observed linewidth and lineshape from fluctuations in the adsorbate-adsorbate or the adsorbate-substrate interactions. For a long range ordered adsorbate overlayer, on the other hand, these effects are absent and the broadening is homogeneous.

The effects of the adsorbate-adsorbate interactions on the inhomogeneous broadening has been investigated from a theor- etical and experimental point of view for the stretch mode of adsorbed CO on Cu (100) [47]. The theoretical analysis led to an asymmetric lineshape, and the calculated inhomogeneous contribution was found to be rather small in comparison with the observed homogeneous contribution. Such a small effect in this case is simply due to the relatively weak dipole-dipole interaction [47].

In the description of the homogeneous broadening of elemen- tary excitations concepts such as energy relaxation and phase relaxation are frequently used, e.g., in Nuclear Magnetic Resonance [48]. These two qualitatively different aspects of homogeneous broadening are possible to distinguish with different kinds of measurements [49]. In this work we calculate the electronic contribution to the homogeneous broadening due to the deexcitation of the vibrational mode, i.e., energy relax- ation. The corresponding lineshape is Lorentzian for weak non- adiabacity (cf. Section 2.1).

On the other hand, from studies of hydrogen impurities in alkali halides, anharmonic terms in the internuclear forces is known to cause relaxation, either through deexcitation into a coherent multi-phonon state or by the dephasing of the excited vibrational state, when scattered against phonons [6]. The former mechanism needs higher order of anharmonicity to be effective for increasing frequency of the localized mode. From that fact one expects this mechanism to be less important for higher ratios between the frequency of the localized vibrational mode and the maximum frequency of the bulk phonon band. Further, the latter mechanism is strongly temperature dependent and is even absent at zero temperature.

At present, the experimental data on inherent broadening of vibrational modes in ordered adsorbate layers of hydrogen atoms or molecules are scarce. In particular, the ordered struc-

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces 369

ture p ( 1 x l)H - W( 100) is so far the only one for which it is possible to make a direct comparison of the observed linewidth with the results from our calculation. For such an ordered structure the vibrations in the adsorbed overlayer form collective modes, which can be labelled by wavevectors kl, in the surface Brillouin zone. In this experiment one probes the collective mode k l l = 0 as the incident photon-field has zero momentum parallel to the surface. The lineshape of this mode is resolved and can be fitted by a Lorentzian, with a corresponding line- width (FWHM) of 1.7 meV [4]. Our model, with an H atom on ajellium surface is obviously far from ideal for H on W. It should give a good account for the coupling of the H vibration to the spelectrons, however, provided proper parameters are used. An estimate of the sp-electron density of bulk is given by the density of the selectrons derived from the constituent atoms (2 electrons from each atom) yielding an electron gas density parameter rs = 2.3. This value lies in between the electron gas density parameters of the jellia representing Al and Ag. The calculated values for the damping rate of the hydrogen atom in the equilibrium position on the surfaces of these jellia are 1.4 meV and 0.7 meV, respectively.

Such a jellium model of the tungsten surface is still in- complete, as it does not account for the effects from the delectron states of tungsten atoms. On the other hand, the results from the extended effective medium theory, which includes d-band hybridization for hydrogen on transition metals, show that a description based on a jellium model is basically sound as a starting point [ 5 0 ] . In the equilibrium position, calculated in this effective medium theory, the local electron gas density parameter of the bare tungsten surface is r, = 3.0.

This value is quite close to the value r, = 2.9 in the equilib- rium position of the atom on the jellium surface representing Al. The corresponding calculated damping rate agrees surprisingly well with the measured linewidth. This result implies that the electronic mechanism gives an important contribution to the measured broadening. It would therefore be of interest to test this conclusion by measuring the temperature and isotope dependenence of the observed broadening.

The electronic mechanism considered here is also conceivable for the hydrogen modes in CHBO chemisorbed on Cu(lOO), since the conduction electrons are shared between the molecule and the metal substrate. For this complex chemisorption system, we just note that the observed broadening for the different hydrogen vibrational modes [44], r = 2meV are of the same order of magnitude as the calculated ones.

So far there exists no experimental data for linewidths of vibrational modes in hydrogen molecules chemisorbed on metal surfaces. On the other hand, there are some data for carbon- monoxide and the isoelectronic nitrogren molecule chemi- sorbed on transition metals [45]. The electronic structures of these systems show some similarities with that of the hydrogen molecule, in particular the existence of absorbate induced states at the Fermi level accociated with the antibonding orbital of the molecule, 20* and 2n* for the hydrogen and carbon mon- oxide molecule, respectively. A prominent difference between the systems is the size of the masses. Simple mass scaling of the result, r = 1.3 meV, calculated for the hydrogen molecule oriented normal to the surface at the equilibrium distance deq = 1 a.u. for r, = 2.65 to account for the different reduced masses of CO and N2, yields r = 0.1 meV. This value is of the same order of magnitude as the observed broadening r = 0.5

meV for CO chemisorbed on Cu(100) [43]. A closer inspection of the differences in the electronic structure for these two systems makes evident several factors, which could conceivably influence the magnitude of the broadening. For instance, the calculated results show that the damping rate is sensitive to the position of the adsorbate induced resonances with respect to the Fermi level.

6 . Concluding remarks

In this section we would finally like to stress a few points about the electronic contribution to the broadening and also discuss some implications and possible future developments.

This work has demonstrated that the embedding scheme (Section 3) is well suited for the calculation of the damping rate. It would be of interest, however, to include in such calculations a more realistic description of the substrate than that of the semi-infinite jellium surface model. Such an exten- sion could provide results for an unambiguous comparison of both the damping rate and the bonding parameters with experiment. The comparison made in Section 5 is in this respect still tentative. On the other hand, the agrement found between the calculated lifetime broadening of the vibration of the hydrogen atom normal to the jellium surface and the measured linewidth for the hydrogen atom adsorbed on a tungsten surface indicates that the electronic mechanism is effective. This state- ment could be further tested in an experimental observation of the isotope and temperature dependences of the linewidth, since the electron-hole mechanism gives a definite behaviour for the broadening with respect to these quantities.

There are some key factors that influence the magnitude of the damping rate. It is obvious from eq. (12) that a light mass is favourable for a high damping rate. In addition the damping rate is found to be larger in magnitude when the adsorbate induces states at the Fermi level. This situation is expected to be encountered in many situations when chemisorption bonds are formed between an adsorbate and a metal substrate.

A close relation is also demonstrated between the vibrational broadening and the friction coefficient. In the application of this relation we find that a typical thermal energy of an incoming hydrogen atom is effectively accommodated by the electrons in the metal substrate. This result is of particular importance in the discussion of the relevance of “hot” atoms, i.e., atoms fare from thermal equilibrium, in promoting chemical reactions at metal surfaces.

In conclusion we have given rather strong evidence that the electron-hole pairs in a metal substrate provide an efficient channel for energy relaxation of excited adsorbates.

Acknowledgements

We would like to thank B. I. Lundqvist for stimulating discussions and most valuable suggestions, and also H. Hjelmberg for sharing his experiences with us. We acknowledge support from the Swedish Natural Science Research Council (NFR).

Appendix

Here we will discuss the extension of the density functional scheme to the regime of slow time dependent perturbations, which is applicable to the case of lifetime broadening of a vibrational mode.

Physica Scripta 29

370 B. Hellsing and M. Persson

The broadening r is directly related to the imaginary part of the self-energy for the mode and is determined from the lossy response of the electrons to the bare Coulomb field @ext

from the vibrating nuclei as,

Im A(w) = - e Im j d x 6n(x, w)@lxt(x, a) (AI)

where 6n is the induced electron density. The latter quantity is related to the external field through the retarded density propagator [51]

where a shorthand notation for the space-time points has been introduced, "1" denoting (x l , t l) . For easy reference the formula for this propagator is presented [51]:

(-43)

where ( , ) is the statistical average in thermal equilibrium, e ( t ) the Heaviside step function , and i ( l ) the particle density operator.

One way to construct a one-electron scheme for the evalu- ation of eq. (Al) is to replace the propagator in eq. (A2) by the one for an appropriate system of non-interacting electons, xo and to modify the external field in such a way that it gives rise to the proper induced density 6n.

It is possible to construct such an effective field @eff when adopting the assumption, put forward by Peuckert [52], that the propagators of both the interacting and non-interacting systems can be inverted,

where

A( l ,2) = xO'(1, 2) -x - ' ( l , 2) (-46)

Noteworthy is that this kernel is advanced in time, i.e., A( l , 2 ) = 0 when t l > t z .

In particular eq. (A5) is valid in the presence of a weak static external field. The main ingredient of the density-functional one-electron scheme is that it accounts for the equilibrium density, As a consequence the induced density to linear order in a static external field can be represented as in eq. (A5). The static effective field is in this case nothing else than the part of the full one-electron potential induced by the external field. Hence by such a choice for the non-interacting set of electrons the static limit of the effective field is known.

When neglecting the dynamical part of the kernel A( 1, 2 ) , the imaginary part of the self energy in eq. (Al) can be reformulated as.

x XO(X1, xz ; w > @ b (xz > (A7)

This approximation is expected to be approximation is expected to be appropriate for slow time dependent perturbations. The expression for the imaginary part of the self energy can be

evaluated from the representation of the propagator, XO, in terms of one-electon wave functions and energies of the ground state [51],

Xf(Ek)(l -f(Ek')) + C.C(W '-0) (A8)

where $&) is the wavefunction for the electron with wave- vector k in bulk and energy Ek, f ( e ) the Fermi-Dirac distri- bution function, and the factor 2 comes from spin degeneracy. These ground state quantities should be determined in the density functional scheme. Finally the results in eq. (A6, 7) yields,

(-49)

12 Im A(w) = h e 2 1 Jdx $t(X)$k(X)@eff(X, U ) k , k'

x (f(Ek) -f(Ek'))6(fia - f k + ck')

This expression for the imaginary part of the self energy Im A(a) is the one invoked in eq. (lo).

References

1.

2. 3.

4. 5.

6 .

7. 8.

9 .

10.

11.

12. 13. 14. 15.

16.

17.

18.

19. 20. 21. 22.

23. 24.

25, 26.

See, e.g., Lundqvist, B. I . or Metiu, H . and Gadzuk,J. W., in Vibrations a t Surfaces (Edited by R. Caudano e t al.). Plenum Press, New York (1 982) (and references therein). Person , M. and Hellsing, B., Phys. Rev. Lett. 4 9 , 6 6 2 (1982). Hellsing, B., Person, M. and Lundqvist, B. I., Surf. Sci. 126, 147 (1983); Person , M., Hellsing, B. and Lundqvist, B. I., J. Electr. Spectr. and Rel. Phen. 29, 119 (1983). Chabal, Y . J. and Sievers, A. J . , Phys. Rev. Lett. 44 , 944 (1980). See, e.g.. Localized Excitations in Solids (Edited by R. F. Wallis). Plenum Press, New York (1968). Barker, A. S. Jr. and Sievers. A. J., Rev. Mod. Phys. 47, S2, 74 (1975). Person , B. N. J. and Ryberg, R., Phys. Rev. Lett. 48 , 549 (1982). Bjorkman, G., Lundqvist, B. I. and Sjolander, A., Phys. Rev. 159, 551 (1967); Schneider, T. and Stoll, E., Phys. Kond. Mater. 9, 32 (1969); Price, D. L., Singwi, K. S. and Tosi, M. P., Phys. Rev. B2, 2983 (1970). Butler, W. H., Pinski, F. J . and Allen, P. B., Phys. Rev. B19, 3708 (1 979). Grimwall, G., Electron-Phonon Interactions in Metals. North- Holland, Amsterdam (1981). Person, B. N. J., J . Phys. C11, 4251 (1978); Kozhushner, M. A., Kustarev, V. G. and Shub, B. R., Surf. Sci. 81, 261 (1979). Persson, B. N. J. and Persson, M., Surf. Sci. 97, 609 (1980). Apell, P., Physica Scripta 24, 795 (1980). Brivio, G. P. and Grimley T. B., Surf. Sci. 89, 226 (1979). Person, B. N. J . and Persson, M., Solid State Commun. 36, 175 (1980). d'Agliano, E . G., Kumar. P., Schaich, W. and Suhl, H., Phys. Rev. B11, 22 (1975); Blandi, A., Nourtier, A. and Hone, P. W., J . Physique 37, 369 (1976). Finneman, J . , Dissertation, The Institute of Physics, Aarhus, 1968 (unpublished) and see also [ 161. Hohenberg, P. and Kohn, W., Phys. Rev. 136, B864 (1964); Kohn, W. and Sham, L. J., Phys. Rev. 140, A1133 (1965). Mermin,N. D., Phys. Rev. 137,A1441 (1965). Zangwill, A. and Soven, P., Phys. Rev. A21, 1561 (1980). Hedin, L. and Lundqvist, B. I., J. Phys. C4, 2064 (1971). See, e.g., Theory of the Inhomogeneous Electron Gas (Edited by S. Lundqvist and N. March), Plenum Press (to appear). Gunnarsson, 0. and Hjelmberg, H., Physica Scripta 11, 97 (1975). Born, M. and Huang, K., Dynamical Theory of Crystal Lattices. (Clarendon Press, Oxford, 1954). Baym. G., Ann. Phys. 14, l ( 1 9 6 1 ) . Sjolander, A. and Johnson, R., in Inelastic Scattering of Neutrons, p. 61. IAEA Proceeding Series, Vienna (1965).

Physica Scripta 29

Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces 37 1

27. Forster, D., Hydrodynamics Fluctuations, Broken Symmetry and Correlation Functions, Frontiers in Physics, Lecture Note Series 47. W. A. Benjamin, Inc. (1975).

28. Kagan, Y. and Browman, E. G., Sov. Phys.-JETP 25, 365 (1967). 29. The full shift including the static contribution has been discussed

by Gadzuk, J. W., Phys. Rev. B24,1651 (1981). 30. Kubo, R., Rep, Prog. Phys. 29,255 (1966). 31. Hjelmberg, H., Physica Scripta 18,481 (1978). 32. Hjelmberg, H., Lundqvist, B. I. and Nbrskov, J . K., Physica Scripta

20,192 (1979). 33. Nbrskov, Houmbller, A., Johansson, P. K. and Lundqvist, B. I.,

Phys. Rev. Lett. 46, 257 (1981); Johansson, P. K., Surf. Sci. 104, 510 (1981).

34. Puska, M. J. and Niemimn, R. M. (to be published). 35. Almbladh, C. O., von Barth, U., Popovic, Z. P. and Stot t , M. J.,

Phys. Rev. B14,2250 (1976). 36. Gunnarsson, O., Hjelmberg, H. and Lundqvist, B. I., Phys. Rev.

Lett. 37, 292 (1976). 37. Gunnarsson, O., Hjelmberg, H. and NBrskov, J . K., Physica Scripta

22, 165 (1980). 38. Lundqvist, B. I., Gunnarsson, 0. and Hjelmberg, H., Surf. Sci. 89,

196 (1979); Johansson, P. K., Lundqvist, B. I., Houmbller, A. and Nbrskov, J . K., in Recent Developments in Condensed Matter Physics (Edited by J. T. Devreese). Plenum Press, New York (1981); Lundqvist, B. I., in Vibrations a t Surfaces (Edited by R. Caudano e t al.), p. 541. Plenum Press, New York (1982).

39. Schaich, W. L. and Harris, J., J. Phys. R11,65 (1981). 40. Nbrskov, J . K. and Lundqvist, B. I., Surf. Sci. 89,251 (1979). 41. Schonhammer, K. and Gunnarsson, O., Phys. Rev. B24, 7084

(1981). 42. Schaich, W. L., J .Chem.Phys.60,1087 (1974). 43. Ryberg, R., Surf. Sci. 114,627 (1982). 44. Ryberg, R., Chem. Phys. Lett. 83,423 (1981). 45. Shigeishi, R. A. and King, D. A., Surf. Sci. 58, 379 (1976); Surf.

Sci 62, 379 (1977); Krebs, H. J. and Liith H., Appl. Phys. 14, 337 (1977); Bradshaw, A. M. and Hoffmann, F. H., Surf. Sci. 72, 513 (1978); Grunze, H., Driscoll, R. K., Burland, G . H., Cornish, J. C. L. Cornish, and Pritchard, J., Surf. Sci. 89, 381 (1979); Hollins, P. and Pritchard, J., Surf. Sci. 89, 486 (1979); Pfniir. H., Menzel, D., Hoffmann, F., Ortega, A. and Bradshaw, A. M., Surf. Sci. 95,131 (1980).

46. Demuth, J . E., Ibach, H. and Lehwald, S., Phys. Rev. Lett. 40, 1044 (1978); Anderson, S., Surf. Sci.79, 385 (1979).

47. Persson, B. N. J . and Ryberg, R., Phys. Rev. B24,6954 (1981). 48. Slichte, C. P., Principles of Magnetic Resonance. Harper and Row,

New York (1963). 49. Lauberau, A. and Kaiser, W., Rev. Mod. Phys. 50, 607 (1978). 50. Nordlander, P., Holloway, S. and Nbrskov, J. K., Surf. Sci. 136, 59

(1984). 51. Fetter, A. L. and Walecka, J . D., Quantum Theory of Many Particle

Systems. McGraw-Hill, New York (1971). 52. Peuckert, V., J. Phys. C11,4945 (1978). 53. NBrskov, and Lang, N. D., Phys. Rev. B21,2136 (1980).

Physica Scripta 29