Vibrational damping of adsorbed molecules: Effects of a realistic metal surface

Solid'State Communications, Vol. 47, No. 8, pp. 615-618, 1983. Printed in Great Britain.

0038-1098/83 $3.00 + .00 Pergamon Press Ltd

VIBRATIONAL DAMPING OF ADSORBED MOLECULES: EFFECTS OF A REALISTIC METAL SURFACE P. Apell* Physics Departmentt, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.

(Received 14 September 1982 by H. Suhl) The damping of a vibrationally excited CO molecule adsorbed on copper can be fully accounted for within an image-dipole interaction provided the effects of nonlocality and particularly a smooth electron density profile of the metal are taken into account. 1. INTRODUCTION A CONSIDERABLE EFFORT is devoted to the study of molecular vibrations at surfaces, particularly carbon monoxide, using techniques such as infrared reflection absorption and electron energy loss spectroscopies. The focus has mainly been on the understanding of phenomena characteristic of the adsorption, for instance adsorption sites and chemisorption bonds. However the presence of a vibrational excited molecule outside a metal surface is also a probe of the metal's response, in this case to an electromagnetic field set up by the vibrational motion. This image interaction will in general lead to both a shift of the vibration frequency and a lifetime broadening, which both contain important information about the way the metal responds to the perturbing electromagnetic field. We will in this communication use the lifetime of the molecule for discussing the response, since this is a convenient measure of the metal's ability to deplete the molecule of excess energy. We are especially interested in a spatial region very close to the metal surface where the coupling to single-particle excitations is very strong, and of particular relevance to us is to incorporate a realistic surface profile for the metal, since this will turn out to have a profound influence on the response. The damping of an excited molecule outside a metal surface has been calculated previously [ 1-3] for a couple of different models of the metal and the results show a clear trend: the smoother or softer you make the electron density profile at the surface the larger is the damping. In [2] Persson and Persson calculated the lifetime of CO on copper (r) using the infinite barrier model and found ~-~ 1 x 10 -l° sec which should be compared with the experimental value of T ~-- 1.3 +

0.1 x 10 -12 sec [4]. Using a finite step barrier model Kozhushner et al. [3] found ~"to be = 2 x 10-11 sec, i.e. one order of magnitude smaller. Based on a similar trend for the damping of the surface plasmon at a vacuum-metal interface, it is also a localized electromagnetic disturbance in the surface region of a metal, the conjecture was made in [5] that using still a better electron density profile (preferably self-consistent) the lifetime should be even smaller and it was estimated to be ~-- 5 x 10 -12 sec. It is the purpose of this communication to put this estimate on a more firm basis as well as pointing out the relevant features in characterizing the metal's response. The resulting lifetime is found to be in fair agreement with the experimental one. However it should be pointed out that this damping due to the oscillating dipole field of the molecule is obviously not the only mechanism possible, for instance a model based on charge transfer between the molecule and the metal also give comparable results [6] : Our results are thus an indication of that the order of magnitude of damping which one gets from the dipole field mechanism, using gas phase values for the CO parameters, in conjunction with a realistic surface profde for the metal is larger than previously estimated [ 1-3 ] but do not exclude the other possible mechanisms since they are of comparable size. A calculation has therefore to be done for every separate molecule-metal system in order to find their relative contributions to the total damping. An experimental set-up which could monitor the damping of molecules, say up to 10 A outside of the metal surface could thus be of value in judging the significance of the various contributions, since so far outside the charge transfer mechanism is inhibited and the long-range dipole field dominates, if ever.

* IBM post-doctoral fellowship holder. 2. THEORY t Research supported in part by the Office of Naval Research.

Consider a molecule with polarizability ao(eO) 615

616

VIBRATIONAL DAMPING OF ADSORBED MOLECULES ~co~

ao(co) = ae + oo2 60---------_ 5 ,

(1)

where ae is the electronic part (assumed constant in this frequency range) and av is the vibrational part associated with the normal mode with frequency coy. We stay in the extreme low coverage limit to avoid all local field effects from other molecules and their images, and the actual molecule is represented by a point dipole situated a distance d outside of the jeUium background of the metal. In the presence of a metal surface this polarizability ao(co) is renormalized according to ao(co) -~ ao(co)/(1 - ao(co)G),

(2)

(from the self-consistency criterion for the local field) where the feedback function G characterizing the metal's response is given by (for a dipole perpendicular to the surface) G = f

d k k 2 e-2kdpp(k, co),

(3)

0

where we describe the metals response by the coefficient p , for a photon of frequency co with momentum k parallel to the surface, which is possible if there is no charge overlap between the molecule and the metal. Since we are interested in distances close to the surface the finite velocity of light is of no concern to us and G is therefore given in its quasistatic form. This form of G becomes evident if we decompose the electric field associated with the point dipole in propagating and evanescent plane waves. Thereby the molecule is represented by a set of photons with frequency coy with all momentas parallel to the surface allowed and each one is reflected by the reflection coefficient pp. Thus G is the feedback from the metal and will contain both reactive and lossy parts and therefore will be in general complex. Poles to equation (2) ( ~ ) give the molecule a frequency shift Aco -- Re ( ~ ) -- coy and a life time broadening 1 / r - - - 2 I m (c~), viz: Aco/cov ~-- --½av Re G(1 --O/e Re G) -1 ,

(4)

1/rcov "" 0% Im G (1

(5)

--R

e

Re G) -a .

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estimates the single-particle strength [8], and it therefore necessary to include a better description of the interface. This can be done in simple terms via the reflection coefficient &o [5]. If d±(co) (a length) denotes the center of gravity of the induced density in a metal subjected to an external electromagnetic disturbance one finds in the quasi static limit that pp is given by

PP

(e -- 1)(1 -- kd±) e + l + (e--1)kd±'

(6)

where e = 1 -- w~/co 2 is the Drude dielectric function. For a classical sharp interface all the induced charge resides right at the boundary and d± -= 0. This gives pp = e -- 1/e + 1 and we retrieve the classical image interaction G = e -- 1/(e + 1)" 1/4d 3 from equation (3), but as soon as the induced density can relax as it does in a non-local description of the metals response and/or with a smooth variation in the electron density profile at the surface d± is non-zero, having both real and imaginary parts. For the vibration frequencies in question coy ~ coy (e -+ oo) and Pv can thus be approximated by

(7)

1 -- kd±

Op -- 1 + kd±

This means that the metals response is largely determined by where the induced density resides (d±). In order to take into account the higher spatial fourier components in the dipole field when the molecule is close to the surface we will for d± use d± = d±(k = 0)/ 1 + kd±(k = 0), thereby pp tends to zero for very high momentum transfers as it should (compare high energy transfers when e -+ 1 and thus pp ~ 0). We introduce a - -- d/d±(k = 0) and we can write

acolco~

-- - ½- go" Re F(a),

for the shift and

(8) .

1/rcov = 6v" Im F(a),

for the damping [9], where a v - av[ 8da and we have neglected the (1 - ae Re G)-factors since in general, for physically reasonable d-values, ae '~ 4d a. F ( a ) = -- a -- a 2 -- a a e-aE1 (-- a),

If we have several molecules present G -+ G - - S ( O ) where S ( 0 ) is a coverage dependent local field factor [15]. Replacing a v by I.t2/2hcov in equation (5), where/a is the classical dynamical dipole moment [7] Im G is seen to be a generalization of the damping function which was originally introduced in [ 1 ] for a sharp interface between vacuum and the metal, with the possibility of exciting electron-hole pairs included via the Lindhard dielectric function appropriate for an infinite homogeneous medium (SCIB-model). It is known from photoyield measurements on A1 that this particular model under-

(9)

(10)

where Ex is the exponential integral [ 10]. We proceed next to discuss an application of these results to the system CO on copper. 3. RESULTS AND DISCUSSION The present theory is applicable to molecules without any appreciably charge overlap with the metal, i.e. situated at least 3--4 A outside o f the jellium edge where the electron density is effectively zero. In Fig. 1 the

Vol. 47, No. 8

VIBRATIONAL DAMPING OF ADSORBED MOLECULES .,jlrnF

/#.

:1

3. 2 --I

'2'0

do(O do~O

3'0 " d4d,~

,/I Fig. 1. A molecule is located a distance d outside a metal surface with the jellium background terminated at z = 0. do is the position of the induced density. The shift and lifetime broadening for the molecule because of its interaction with the metal is proportional to a function F. The real part o f f (shift) is shown for do < 0 corresponding to a smooth electron density profile and for do > 0 which is characteristic of a stepprofile. The imaginary part of F (damping) is on the scale in the figure only visible for the former model. Indicated is also the electron density profile for a jeUium metal (r s = 2) and F is shown dotted in the region where the electron density is tailing out into the vacuum and a charge overlap with molecular wave functions is expected. electron density for an r s = 2 jellium metal is shown as a function of d/Idol where do = Re d±(0) ( ~ 0.85 )~ for r 8 = 2 [11]) is the static value of d± which should be a reasonable approximation to use for do when dealing with vibration frequencies. This figure also shows the function F ( a ) determining the shift and damping in equations (8) and (9). The curve labelled do < 0 is the real part of F for a L a n g - K o h n electron density profile [11 ] where the induced density is at a point (Re d j_) outside the jellium edge while the curve "do > 0" corresponds to a calculation within the semi-classical infinite barrier (SCIB) model where the induced density is forced to stay within the jellium. The peak curve is the imaginary part o f F for the smooth density profile while a SCIBcalculation gives values for Im F ~-- 10 -2 [1] which is not visible on the scale of the figure and thus indicates the intuitatively qualitative larger influence on the damping a softer electron density has. However no experiments are performed in what we have indicated as the acceptable region for the theory in Fig. 1 so in order to compare the theory with presently available experiments we have to go beyond its actual range of validity. We then have to do this in the spirit of getting an order of estimate for the magnitude of the damping due to the dipole mechanism, where we put the molecule that close to the surface and neglecting all other effects. This has to be viewed also in the light of the point-dipole approximation

617

utilized, truly both the extension of the molecule and higher moments ought to play a role when we are as close to the jellium background as the molecules own extension [ 18]. Furthermore other calculations available either use the damping as a parameter to adjust the distance between the pointqike molecule and the metal or use a model without any electron density profile and then calculates the damping right up to the jellium edge. To proceed we thus need values for a v, cov, d and d± in order to evaluate F. The gas phase values for ct v and w v are 0.05 A 3 [12] and 2143 cm -1 respectively. If we put the point-dipole at the center of mass of the CO molecule, which is a natural choice since we are dealing with the stretch vibration, LEED measurements [13] indicates that this is about 2.6 A outside of the first copper lattice plane. This lattice plane is 0.9 A behind the jellium edge and thus d ~-- 1.7/k. In order to stress that the present model does not need ad hoc small values of d to artificially increase the coupling strength we will use d = 2 ,~ in our rough estimate. For do we use -- 0.7 A (r s = 3 [ 11 ]) for the real part and from an approximate expression Im d l ( w v ) ~ ~r/2. Re d±(0)" Wv[COp [14] we get Im d± ~----2.5 x 10-2A. Inserting these values we get for the shift Aco ~ - - 2 c m

-1,

(I1)

and for the lifetime r ~- 1 x 10 -12sec,

(12)

which is slightly shorter than the experimental value (1.3 x 10 -12 sec) but certainly is of the right magnitude. The magnitude of the shift clearly indicates that the total shift of about 65 cm -1 (from 2143 to 2078 cm -t) when a single CO molecule is adsorbed on a copper surface is not of electrodynamical origin as suggested in [15]. There the metal's response was incorporated by letting the image interaction have its classical form but measuring d from the image plane. This is only an asymptotic formula which does not include the saturation which is inherent in our formulation, with a drastic reduction in the strength of the interaction as the result for the shift shows. This shift is also much less than the mode coupling shift of 39 cm -t [16] from the coupling between the CO stretch vibration and the vibration of the whole molecule against the surface. A weakening of the CO-bond due to backbonding leading to an increase in the equilibrium distance of ~ 10 -2 )k is also capable of giving shifts = 5 0 c m -1 [17]. It thus seems that the "mechanical" shifts due to the coupled modes and the weakening of the bond are far larger than the electrodynamical one. We have in this communication indicated that a

618

VIBRATIONAL DAMPING OF ADSORBED MOLECULES

reasonable description of the electron density profile at a metal surface changes the damping rate of an adsorbed molecule considerably compared to more stiff profiles, leading to results for the damping of the stretch vibration of CO on copper which are in fair agreement with experirnental findings. The frequency shift on the other hand is very small especially compared to expected chemical shifts.

7. 8. 9. 10. 11. 12.

REFERENCES 1. 2. 3. 4. 5. 6.

B.N.J. Persson, J. Phys. C11,4251 (1978). B.N.I. Persson & M. Persson, Surf. Sci. 97,609 (1980). M.A. Kozhushner, V.G. Kustarev & B.R. Shub, Surf. Sci. 81,261 (1979). R. Ryberg, Surf. Sci. 114,627 (1982). P. Apell, Physica Scripta 24,795 (1981). B.NJ. Persson & M. Persson, Solid State Commun.

13. 14. 15. 16. 17. 18.

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36,175 (1980). bt = 2(fl) where/2 is the quantum mechanical dipole operator. P. Apell, Physica Scripta 25, 57 (1982). 1/r ~- FWHM for a Lorentzian. Tables of the exponential integral for complex arguments. N.B.S. Appl. Math. Series 51 (1958).

El(z) = f z du e-U/u. N.D. Lang & W. Kohn, Phys. Rev. B7, 3541 (1973). B.N.J. Persson & A. Liebsch, Surf. Sci. 110, 356 (1981). S. Andersson & J.B. Pendry, Phys. Rev. Lett. 43, 363 (1979). P. Apell, submitted to SolidState Commun. M. Scheffler, Surf. Sei. 81,562 (1979). G.W. Ford & W.H. Weber, Bull. of Am. Phys. Soc. 27, March MK11, p. 410 (1982). S. Efrima & H. Metiu, Surf. Sci. 92,433 (1980). G.W. Ford & W.H. Weber, Surf. Sei. 109, 451 (1981).