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the quantitative understanding of the dynamics of electronic excitations in the shockley state band at the ( 111 ) face of ag has been the subject of a considerable number of research reports ( for a review see ref .
recent experimental studies used photoelectron spectroscopy and the scanning tunnelling microscope ( stm ) @xcite .    theoretically , many - body lifetime calculations using the gw approximation @xcite have clarified the role of electron - electron ( _ e - e _ ) interactions as well as the relative importance of intraband and interband scattering and screening . the damping rate due to electron - phonon scattering is also quantitatively understood @xcite . however , comparing theoretical with stm - derived experimental damping rates of electronic excitations with energies above the fermi level @xmath2 show that the experimental decay rates increase more rapidly with increasing energy @xmath1 . recently a similar discrepancy between image state lifetimes measured by means of the stm and two - photon - photoemission spectroscopy was lifted @xcite .
it was shown that the applied electric field during the stm - measurement causes an increase in both the efficiency of the image state decay channels as well as their number . in the light of this work , becker _
et al . _ theoretically investigated the impact of the stm - induced electric field on the inelastic _ e - e _ damping rate of the electron- and hole - like excitations in the shockley surface state band at ag(111 ) @xcite .
their results indicate that under typical tunnelling conditions the stm does not significantly alter the surface state wave function and so previous stm - derived shockley state lifetimes need not be corrected .
nevertheless , while experimental and calculated lifetimes agree quite well over a range of energies , a quantitative difference remains for unoccupied states of energies @xmath3  mev .
the origin of this different energy dependence , which is still an open question , is addressed in this report .
the calculations presented here are based upon the approach developed by chulkov _ et al . _
@xcite , and used widely in calculations of surface state dynamics @xcite . within this approach the damping rate or inverse lifetime of an excitation in the state @xmath4 with energy @xmath1
is obtained from the expectation value of the imaginary part of the electron self - energy , @xmath5 : @xmath6 unless stated explicitly , atomic units are used throughout the text , i.e. , @xmath7 .    in the gw approximation @xcite of many - body theory the imaginary part of the self energy
is calculated to first order in terms of the screened coulomb interaction @xmath8 and the green function @xmath9 @xmath10 here we apply the widely used zeroth order approximation to the green function in the spectral representation @xmath11 where the @xmath12 are one - electron eigenfunctions with energies @xmath13 and @xmath14 a positive infinitesimal .
the screened coulomb interaction is evaluated in the random phase approximation ( rpa ) @xmath15 where @xmath16 is the bare coulomb interaction and @xmath17 is the density - density response function of the non - interacting electron system . an explicit expression for @xmath18
can be found in ref .
@xcite .
the single particle states have been obtained by solving the schrdinger equation using a one - dimensional pseudopotential varying in the direction perpendicular to the surface @xcite .
the pseudopotential does not describe the @xmath19-electrons of the substrate , which are anyways too low in energy to play a significant role as final states for the decay , but their contribution to the screening is included via a polarisable background @xcite .
( color online ) ag(111 ) surface band structure .
shaded area and solid line are the projected bulk and surface state dispersion from _ ab - initio _ calculations which we use for calculating the _ e - e _ decay rate .
the dashed lines are parabolic dispersions with effective masses @xmath20 ( surface state band ) and @xmath21 ( band edge).,width=264,height=245 ]    the electron - phonon contribution to the linewidth is taken to be constant @xmath22  mev for energies in excess of @xmath23  mev above the fermi level @xcite .
previous calculations of the decay rate of the ag(111 ) shockley state have assumed parabolic dispersion with effective masses of @xmath20 and @xmath24 for the intrinsic surface state and the lower band edge respectively @xcite .
however , over the extended energy range of interest here , this approximation needs to be improved as evident from fig .
[ fig : disp ] . figure [ fig : disp ] shows the parabolic dispersions ( dashed lines ) for the surface state band and the band edge with effective masses as mentioned above .
the grey shaded area and the solid line are the projected bulk and surface state dispersion , respectively , that are found from _ ab - initio _ calculations .
the parabolic dispersion underestimates the gap between surface state band and band edge .
we have therefore used the _ ab - initio _
dispersions for our lifetime calculations .
as previously noted @xcite there are important changes in the shape of the surface state wave function with @xmath25 , the surface wave vector , and we take these into account by recalculating the wave function for different @xmath25 with the pseudopotential parameters changed to take into account the appropriate _ ab - initio _ band edges and surface state energy .
we focus on the energy range between @xmath26 and @xmath27  ev above the fermi level where the surface state is energetically separated from the bulk states and thus is well defined .
this energy range is of interest as according to ref .
@xcite there is a cross - over in the primary decay channel from intraband to interband dominated decay .
our calculated results for the total damping rate @xmath28 are shown in fig .
[ fig : damp ] as a solid line .    for energies below @xmath29
ev we find that our results for the inelastic _ e - e _ damping rate of excitations in the shockley state band agree well with the inverse lifetime estimation from echenique _
( dashed line ) @xcite using similar methods but with an effective mass approximation .
this agreement is due to the fact that the effective mass approximation gives a very good description of the projected _ ab - inito _ surface band structure for energies with @xmath30  ev as evident from fig .
[ fig : disp ] .    for larger energies
we find that the _ e - e _ damping rate as a function of energy increases more rapidly with the _ ab - initio _ dispersion than calculated with parabolic dispersions .
this result can be understood as follows : for energies @xmath31  ev the parabolic dispersions underestimate the energy gap between surface state band and the bulk band edge .
consequently , the _ ab - initio _ dispersions reveal a reduced penetration of the surface state wave function into the interior of the crystal and an increased probability amplitude in the proximity of the metal surface .
this modification leads to two competing effects .
since screening is reduced at the surface @xcite intraband decay becomes more efficient .
on the other hand , interband decay becomes less probable owing to the reduced overlap with bulk states .
it turns out that the increase of intraband decay outweighs the decrease in interband transitions . the net result is a more steep increase of @xmath0 than obtained from parabolic dispersions .
we thus find the intraband contribution to dominate up to @xmath32  ev where the interband and intraband contributions become similar in magnitude .
comparing to published experimental data ( symbols in fig .
[ fig : damp ] ) the agreement is improved .    in summary , we have presented a theoretical many - body analysis of the inelastic _ e - e _ lifetimes of electronic excitations in the shockley surface state band of ag(111 ) .
the calculations are based on the gw approximation using a one - dimensional pseudopotential together with an _ ab - initio _ description of the projected surface band structure .
we have shown that the consistency of experimental and theoretical derived damping rates is significantly improved when using _ ab - initio _ band structures instead of an effective mass approximation for calculating the decay rate .
m.b . and r.b .
thank the deutsche forschungsgemeinschaft for financial support through _ spp 1093_.    nnnyy p.
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we present a theoretical many - body analysis of the electron - electron ( _ e - e _ ) inelastic damping rate @xmath0 of electron - like excitations in the shockley surface state band of ag(111 ) . 
 it takes into account _ ab - initio _ band structures for both bulk and surface states . 
 @xmath0 is found to increase more rapidly as a function of surface state energy @xmath1 than previously reported , thus leading to an improved agreement with experimental data .