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Conference Overview, Directions for Future Work and Critical Issues: Theory*
Journal of Electron Spectroscopy and Related Phenomena, 30 (1983) 229-235 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
229
CONFERENCE OVERVIEW, DIRECTIONS FOR FUTURE WORK AND CRITICAL ISSUES: THEORY* T,
B. GRIMLEY
Donnan Laboratories,University of Liverpool, P.O. Box 147,Liverpool L69 3BX UK
ABSTRACT This paper describes the goals of theoretical work on vibrations at surfaces, and comments on some of the contributions to VAS-82. Its main purpose however is to specify areas where theoretical work is urgently needed, and to identify some critical issues. Among the latter, the most important is the use of clusters to model atomic and molecular processes on solid surfaces. I NTRODUCTI ON The goals of theory are to provide fully microscopic descriptions of all measurable phenomena associated with vibrations at surfaces, and to furnish a comprehensive back-up for experimental workers in the field.
Consequently we
need: A) Theoretical models for ab initio numerical work. B) Qualitative theoretical models with usable concepts, and with correlating (and if possible, predictive) power. C) Fundamental theoretical work on imperfectly understood phenomena. In addition we need to produce well-documented software, implementing quantitative theoretical models, for experimental workers to use to extract fundamental structural, and dynamical parameters from their data.
There are consi-
derable achievements in this area in LEED, less in photoemission, EELS, and diffractive scattering, where much more could, and should, be done. Including all papers at the conference with a substantial theoretical content, the numbers in the categories A, Band C above are as follows: A
B
C
6
12
4
so most contributions fall into category B.
Of course, theoretical work in any
category may be useless because some essential feature of the chemistry or *
Written on the basis of oral presentations at the conference; I have not seen any manuscripts.
0368-2048/83/0000-0000/$03.00 © 1983 Elsevier Scientific Publishing Company
230
physics has been overlooked, but any such deficiency in category A seems to me particularly serious; ab initio numerical work must in the end output numbers of chemical accuracy to be useful. Theoretical models for equilibrium properties, probe dynamics, and process dynamics, whether for clean surfaces, or for adsorbates, can be made for: a) clusters b) embedded clusters c) slabs d) embedded surfaces e) semi-infinite systems. Only work with (a) and (e) was presented at the conference, although work with (b) and (c) was referred to briefly in the discussions. CLEAN SURFACES Clean surfaces are studied to provide the static input to probe dynamics, for example to atomic beam-surface inelastic scattering.
Thus, if we treat the
beam atom in the classical trajectory approximation so that it only provides a time-dependent addition Vet) : V(±oo)
= 0, to the Hamiltonian Ho describing the
loss modes of the clean substrate, then the probability ~ as t + 00
for the substrate initially (t
+ _00)
where 1'I'(t»
-i(H -E )t 0
0
to observe a loss o
with energy Eo is
P(~) = ~1I f:dt<'I'(t) Ie
P(~)
in its ground state I~ >
.
1'I'(t»el~t
is the time-evolved substrate state
H
O
+
Vet).
Evidently the substrate eigenstates, which define H are the static input to o' this dynamical problem. Experimentally, the substrate excitations are detected as energy losses in the reflected beams, and ultimately therefore, we need a full quantum theory of the angle-dependence of the loss process, for example of the one-phonon loss. This shifts attention to the T-matrix element for the process, which, keeping only the so-called "direct" term, has the form
for excitation of the mode 0 with parallel wavevector
K,
polarization vector
231 +
-+
+
Ea(K) , and frequency na(K) in a lattice of atoms with mass M.
+
V+V(R,O) is the u
gradient with respect to the position of an arbitrary surface atom of the beam ++
atom-substrate interaction potential V(R,u) when all other substrate atoms are f+ is the wavein their equilibrium positions, and the beam atom is at
R.
+
p
function for the diffractive scattering off the rigid substrate of an incident +
+-
beam atom with momentum p (parallel momentum P), and f+
is the time-reversed
q
+
diffractive scattering wavefunction for a scattered beam atom with momentum q (parallel component
Q).
~ conserves parallel momentum to within a reciprocal
vector of the substrate surface net; energy is conserved when we use this Tmatrix element to calculate the transition rate.
I have written down this for-
mula because it shows us exactly what we need to calculate the dynamics of the one-phonon loss in beam-surface scattering (see Benedek's paper). At the conference there were contributions to probe, and process dynamics by Dietz (electron-surface scattering),
Sch~nhammer,
Garcia and Benedek (atom-
surface scattering), and M. Persson (an H atom moving near a jellium). For clean surfaces we need: ++
i)
Ab initio rare gas-solid interaction potentials V(R,u) with the gas atom at
R,
and the substrate atoms in configurations ~ near the equilibrium confi+
+
guration u = 0 to calculate the linear coupling V+(R,O) to phonons. u
ii)
Realistic microscopic models to calculate the non-adiabatic coupling terms
=
=(R) between electronically adiabatic states in the Born-
A (R) and B
+
Oppenheimer coupled equations for an atom at R near a metal surface so as to compute the inelasticity due to electronic excitations in beam-surface scattering (see
Sch~nhammer's
paper for more details).
ADSORBATES Most conference papers were concerned with adsorbates, and here, the present tasks of theory are to: a) Calculate the vibration frequencies of adsorbed species. b) Work out the probe dynamics of, for example, EELS and SERS. c) Study lateral interactions between adsorbate vibrations. Vibration frequencies The papers presented by Batra and by Goddard are good examples of what is being done theoretically.
Both use cluster models of chemisorption, and the
advantage of this approach is that rather accurate theoretical models can be used.
Goddard uses the Generalized Valence Bond (GVB) model, although many
results reported at the conference were only at the Hartree-Fock (HF) level.
232
It is the task of theory to calculate vibration frequencies for all relevant surface species, and assumed geometries, so that experimental measurements of actual frequencies become a structural tool without recourse to qualitative arguments, or dubious comparisons.
To achieve this we must be able to calculate
adiabatic potential energy surfaces, or Hellman-Feynman forces, with chemical accuracy around the equilibrium configuration.
Chemical accuracy in this con-
text means vibration frequencies to about 10 cm-l since shifts of this amount
can be chemically (or physically) significant. achieve for strong bonds than for weak ones.
Such accuracy is easier to For the latter, electron correla-
tions are important, and something better than single configuration IIF is
neede~
Either GVB, or configuration interaction (CI), or perhaps the local spin density functional approach might be used.
At the conference, only cluster work using
either HF or GVB was reported; no work was reported for extended substrates, or embedded clusters. Probe dynamics An important paper under this heading was presented by Schatz in connection with SERS.
Again a small substrate cluster was used to model H chemisorption 2
by lithium, and again as a consequence, a rather elaborate computation could be made (coupled TDIIF).
There was no theoretical contribution to EELS, although
Bare showed a detailed comparison between muffin-tin EELS theory (ref. l),and experiment, for a p(lxl)H
overlayer on (lOO)W.
Aspects of the electronic
damping mechanism for adsorbate motions near jelliums were treated by B.Persson, and by M. Persson. Lateral interactions B. Persson's paper gives a very clear account of some of the physics for systems where overlayer phonon band structure effects are not important. Allan's contribution shows how some aspects of the band structure effects can be included by parameterizing a simple model.
What is now needed is a comprehensive
ab initio study of an adsorbate on a real (deformable) substrate so that all effects contributing to the lateral interaction between adsorbate vibrations are included from the beginning. For adsorbates we need: i)
77
Adiabatic potential energy surfaces V(R,u) (or lIellman-Feynman forces) where
R stands
for the adsorbate configuration, and ~ the substrate atom
displacements, round the equilibrium configuration adsorbate vibration frequencies. ii)
(Ro ,0)
so as to compute
Detailed theoretical models for computing the non-adiabatic coupling terms in the Born-Oppenheimer system of equations in the region of space around
233
(Ro ,0)
so as to compute the damping of adsorbate vibrations due to energy
loss to the electron system. iii) A computational scheme for handling non-muffin-tin potentials in the theory of EELS so as to include impact and dipole scattering in the same model. CRITICAL ISSUES i)
What is the role of chemical bond formation in energy dissipation to electrons at surfaces? It is not easy to answer this question when, for the purpose of handling the
dynamics, Anderson- or HUckel-type Hamiltonians are used to model the static (chemisorption) results obtained with jellium substrates.
With jelliums, the
prominent effects are changes in the position, width and occupancy of an adsorbate affinity resonance with adsorbate motion near the surface.
However reso-
nances in the adsorbate-induced density of states with Anderson's Hamiltonian are quantitatively different from those found with adsorbates on jelliums. With the former, a resonance wholly below the Fermi level holds one electron per spin, with the latter, depending on the electronic structure of the free adsorbate, such a resonance might hold only
~
electron in total.
This difference
arises because with jelliums, Coulomb interactions between electrons in the substrate are explicitly handled.
Consequently, the adsorbate-jellium surface
bond cannot always be modelled with an Anderson- or HUckel-type Hamiltonian (but see ref. 2).
Furthermore, the dynamics of the time-dependent Kohn-Sham
(TDKS) approximation are different from those of time-dependent Anderson-type (TDA-T) models.
To see this we only need to implement TDKS in the (over-
complete) orbital basis consisting of the important adsorbate orbital la>, and the clean surface Kohn-Sham orbitals {!k>}.
Then the TDKS approximation will
have k-state scattering, but the TDA-T model does not.
I remark also that, if
we move from jelliums, and put the substrate chemistry into the problem, and then use the LCAO-MO approach to either the Kohn-Sham, or the Hartree-Fock equations, it can be seen that, changes in an adsorbate affinity level with adsorbate motion do not exhaust the possibilities for non-adiabatic behaviour. +
All elements P.. (R) of the charge- and bond-order matrix in the orbital basis 1J
+
{Ii>} (the P-matrix of quantum chemistry) at the adsorbate configuration R enter into the non-adiabatic coupling terms, whereas the affinity orbital is (or can be chosen to be) only one member of {Ii>}. ii)
Are cluster models adequate for the ab initio calculation of adsorption geometries, and vibration frequencies?
234
This is an extremely important question.
With present-day computing re-
sources, we can obtain information on wavefunctions, and potential energy surfaces including electron correlations, for adsorbates on small substrate clusters so as to extract optimum geometries, and vibration frequencies with chemical accuracy (or very near it).
We cannot achieve this with computations on
extended substrates (slabs or semi-infinite), and since most substrate atoms in a small cluster do not have the number of nearest neighbours appropriate to the extended substrate, it is of the utmost importance to know whether an ab initio cluster calculation of, for example, an adsorbate vibration frequency is converged with chemical accuracy with respect to cluster size.
Goddard has
addressed this point, and it seems that ZO nickel atoms are needed to model (IOO)Ni for CH, CH
and CH 3 adsorption. I emphasize again that ab initio calZ culations are undertaken to produce chemically accurate information, and if they do not do so, they are useless.
We ought therefore to be slightly wary of
any such computations using less than ZO nickel atoms to model the (100) surfaca iii)
Can we model probe and process dynamics with clusters?
Again the point is that, if there are dynamical processes on clusters which are insensitive to the fact that most substrate atoms in the cluster do not have the environment of the extended substrate, then cluster calculations would provide a way forward with chemical accuracy.
Much preliminary work is needed
to explore such an approach to dynamical problems at surfaces, and it is not appropriate to begin it here.
I remark only that quite simple models contain
the essential problems, and that the question can probably be answered fairly easily for specific probes (e.g. He atoms), and specific processes (e.g. vibrational damping). iv)
What aspects of SERS can be studied with cluster models? This question is really included in (iii).
However, Schatz has already cal-
culated Raman intensities from a cluster using, in effect, a microscopic model of coupled photons, electrons and nuclei, and consequently the above question should now be asked. CONCLUSION Only ab initio work using either jelliums, or small substrate clusters was presented at the conference, and because of the lack of substrate chemistry with jelliums, it is important to consider the limits of clusters as models of semi-infinite substrates.
It is fairly well established that small substrate
clusters are adequate for the calculation of adsorbate, and surface bond
235
lengths, that larger clusters are needed for vibration frequencies, and that quite large clusters must be used to obtain the adsorbate-induced electronic structure.
But "small", "larger", and "quite large" are vague terms whose pre-
cise meaning depends on the chemisorption system under investigation, and much more theoretical work with good theoretical models, which include electron correlation effects, is needed to widen our experience, and to improve our knowledge in this area.
Finally, the possible use of cluster models for probe, and
process dynamics, about which little is known, should be systematically explored; cluster embedding is computationally rather expensive, and we need to know much more about the circumstances in which we can avoid it.
REFERENCES 1 2
G.C. Aers, T.B. Grimley, J.B. Pendry and K.L. Sebastian, J. Phys. C: Solid St. Phys., 14 (1981) 3995; also unpublished computations. J.P. Muscat and n.M. Newns, Phys. Letters,60A (1977) 348, 6lA (1977) 481.
229
CONFERENCE OVERVIEW, DIRECTIONS FOR FUTURE WORK AND CRITICAL ISSUES: THEORY* T,
B. GRIMLEY
Donnan Laboratories,University of Liverpool, P.O. Box 147,Liverpool L69 3BX UK
ABSTRACT This paper describes the goals of theoretical work on vibrations at surfaces, and comments on some of the contributions to VAS-82. Its main purpose however is to specify areas where theoretical work is urgently needed, and to identify some critical issues. Among the latter, the most important is the use of clusters to model atomic and molecular processes on solid surfaces. I NTRODUCTI ON The goals of theory are to provide fully microscopic descriptions of all measurable phenomena associated with vibrations at surfaces, and to furnish a comprehensive back-up for experimental workers in the field.
Consequently we
need: A) Theoretical models for ab initio numerical work. B) Qualitative theoretical models with usable concepts, and with correlating (and if possible, predictive) power. C) Fundamental theoretical work on imperfectly understood phenomena. In addition we need to produce well-documented software, implementing quantitative theoretical models, for experimental workers to use to extract fundamental structural, and dynamical parameters from their data.
There are consi-
derable achievements in this area in LEED, less in photoemission, EELS, and diffractive scattering, where much more could, and should, be done. Including all papers at the conference with a substantial theoretical content, the numbers in the categories A, Band C above are as follows: A
B
C
6
12
4
so most contributions fall into category B.
Of course, theoretical work in any
category may be useless because some essential feature of the chemistry or *
Written on the basis of oral presentations at the conference; I have not seen any manuscripts.
0368-2048/83/0000-0000/$03.00 © 1983 Elsevier Scientific Publishing Company
230
physics has been overlooked, but any such deficiency in category A seems to me particularly serious; ab initio numerical work must in the end output numbers of chemical accuracy to be useful. Theoretical models for equilibrium properties, probe dynamics, and process dynamics, whether for clean surfaces, or for adsorbates, can be made for: a) clusters b) embedded clusters c) slabs d) embedded surfaces e) semi-infinite systems. Only work with (a) and (e) was presented at the conference, although work with (b) and (c) was referred to briefly in the discussions. CLEAN SURFACES Clean surfaces are studied to provide the static input to probe dynamics, for example to atomic beam-surface inelastic scattering.
Thus, if we treat the
beam atom in the classical trajectory approximation so that it only provides a time-dependent addition Vet) : V(±oo)
= 0, to the Hamiltonian Ho describing the
loss modes of the clean substrate, then the probability ~ as t + 00
for the substrate initially (t
+ _00)
where 1'I'(t»
-i(H -E )t 0
0
to observe a loss o
with energy Eo is
P(~) = ~1I f:dt<'I'(t) Ie
P(~)
in its ground state I~ >
.
1'I'(t»el~t
is the time-evolved substrate state
H
O
+
Vet).
Evidently the substrate eigenstates, which define H are the static input to o' this dynamical problem. Experimentally, the substrate excitations are detected as energy losses in the reflected beams, and ultimately therefore, we need a full quantum theory of the angle-dependence of the loss process, for example of the one-phonon loss. This shifts attention to the T-matrix element for the process, which, keeping only the so-called "direct" term, has the form
for excitation of the mode 0 with parallel wavevector
K,
polarization vector
231 +
-+
+
Ea(K) , and frequency na(K) in a lattice of atoms with mass M.
+
V+V(R,O) is the u
gradient with respect to the position of an arbitrary surface atom of the beam ++
atom-substrate interaction potential V(R,u) when all other substrate atoms are f+ is the wavein their equilibrium positions, and the beam atom is at
R.
+
p
function for the diffractive scattering off the rigid substrate of an incident +
+-
beam atom with momentum p (parallel momentum P), and f+
is the time-reversed
q
+
diffractive scattering wavefunction for a scattered beam atom with momentum q (parallel component
Q).
~ conserves parallel momentum to within a reciprocal
vector of the substrate surface net; energy is conserved when we use this Tmatrix element to calculate the transition rate.
I have written down this for-
mula because it shows us exactly what we need to calculate the dynamics of the one-phonon loss in beam-surface scattering (see Benedek's paper). At the conference there were contributions to probe, and process dynamics by Dietz (electron-surface scattering),
Sch~nhammer,
Garcia and Benedek (atom-
surface scattering), and M. Persson (an H atom moving near a jellium). For clean surfaces we need: ++
i)
Ab initio rare gas-solid interaction potentials V(R,u) with the gas atom at
R,
and the substrate atoms in configurations ~ near the equilibrium confi+
+
guration u = 0 to calculate the linear coupling V+(R,O) to phonons. u
ii)
Realistic microscopic models to calculate the non-adiabatic coupling terms
=
=(R) between electronically adiabatic states in the Born-
A (R) and B
+
Oppenheimer coupled equations for an atom at R near a metal surface so as to compute the inelasticity due to electronic excitations in beam-surface scattering (see
Sch~nhammer's
paper for more details).
ADSORBATES Most conference papers were concerned with adsorbates, and here, the present tasks of theory are to: a) Calculate the vibration frequencies of adsorbed species. b) Work out the probe dynamics of, for example, EELS and SERS. c) Study lateral interactions between adsorbate vibrations. Vibration frequencies The papers presented by Batra and by Goddard are good examples of what is being done theoretically.
Both use cluster models of chemisorption, and the
advantage of this approach is that rather accurate theoretical models can be used.
Goddard uses the Generalized Valence Bond (GVB) model, although many
results reported at the conference were only at the Hartree-Fock (HF) level.
232
It is the task of theory to calculate vibration frequencies for all relevant surface species, and assumed geometries, so that experimental measurements of actual frequencies become a structural tool without recourse to qualitative arguments, or dubious comparisons.
To achieve this we must be able to calculate
adiabatic potential energy surfaces, or Hellman-Feynman forces, with chemical accuracy around the equilibrium configuration.
Chemical accuracy in this con-
text means vibration frequencies to about 10 cm-l since shifts of this amount
can be chemically (or physically) significant. achieve for strong bonds than for weak ones.
Such accuracy is easier to For the latter, electron correla-
tions are important, and something better than single configuration IIF is
neede~
Either GVB, or configuration interaction (CI), or perhaps the local spin density functional approach might be used.
At the conference, only cluster work using
either HF or GVB was reported; no work was reported for extended substrates, or embedded clusters. Probe dynamics An important paper under this heading was presented by Schatz in connection with SERS.
Again a small substrate cluster was used to model H chemisorption 2
by lithium, and again as a consequence, a rather elaborate computation could be made (coupled TDIIF).
There was no theoretical contribution to EELS, although
Bare showed a detailed comparison between muffin-tin EELS theory (ref. l),and experiment, for a p(lxl)H
overlayer on (lOO)W.
Aspects of the electronic
damping mechanism for adsorbate motions near jelliums were treated by B.Persson, and by M. Persson. Lateral interactions B. Persson's paper gives a very clear account of some of the physics for systems where overlayer phonon band structure effects are not important. Allan's contribution shows how some aspects of the band structure effects can be included by parameterizing a simple model.
What is now needed is a comprehensive
ab initio study of an adsorbate on a real (deformable) substrate so that all effects contributing to the lateral interaction between adsorbate vibrations are included from the beginning. For adsorbates we need: i)
77
Adiabatic potential energy surfaces V(R,u) (or lIellman-Feynman forces) where
R stands
for the adsorbate configuration, and ~ the substrate atom
displacements, round the equilibrium configuration adsorbate vibration frequencies. ii)
(Ro ,0)
so as to compute
Detailed theoretical models for computing the non-adiabatic coupling terms in the Born-Oppenheimer system of equations in the region of space around
233
(Ro ,0)
so as to compute the damping of adsorbate vibrations due to energy
loss to the electron system. iii) A computational scheme for handling non-muffin-tin potentials in the theory of EELS so as to include impact and dipole scattering in the same model. CRITICAL ISSUES i)
What is the role of chemical bond formation in energy dissipation to electrons at surfaces? It is not easy to answer this question when, for the purpose of handling the
dynamics, Anderson- or HUckel-type Hamiltonians are used to model the static (chemisorption) results obtained with jellium substrates.
With jelliums, the
prominent effects are changes in the position, width and occupancy of an adsorbate affinity resonance with adsorbate motion near the surface.
However reso-
nances in the adsorbate-induced density of states with Anderson's Hamiltonian are quantitatively different from those found with adsorbates on jelliums. With the former, a resonance wholly below the Fermi level holds one electron per spin, with the latter, depending on the electronic structure of the free adsorbate, such a resonance might hold only
~
electron in total.
This difference
arises because with jelliums, Coulomb interactions between electrons in the substrate are explicitly handled.
Consequently, the adsorbate-jellium surface
bond cannot always be modelled with an Anderson- or HUckel-type Hamiltonian (but see ref. 2).
Furthermore, the dynamics of the time-dependent Kohn-Sham
(TDKS) approximation are different from those of time-dependent Anderson-type (TDA-T) models.
To see this we only need to implement TDKS in the (over-
complete) orbital basis consisting of the important adsorbate orbital la>, and the clean surface Kohn-Sham orbitals {!k>}.
Then the TDKS approximation will
have k-state scattering, but the TDA-T model does not.
I remark also that, if
we move from jelliums, and put the substrate chemistry into the problem, and then use the LCAO-MO approach to either the Kohn-Sham, or the Hartree-Fock equations, it can be seen that, changes in an adsorbate affinity level with adsorbate motion do not exhaust the possibilities for non-adiabatic behaviour. +
All elements P.. (R) of the charge- and bond-order matrix in the orbital basis 1J
+
{Ii>} (the P-matrix of quantum chemistry) at the adsorbate configuration R enter into the non-adiabatic coupling terms, whereas the affinity orbital is (or can be chosen to be) only one member of {Ii>}. ii)
Are cluster models adequate for the ab initio calculation of adsorption geometries, and vibration frequencies?
234
This is an extremely important question.
With present-day computing re-
sources, we can obtain information on wavefunctions, and potential energy surfaces including electron correlations, for adsorbates on small substrate clusters so as to extract optimum geometries, and vibration frequencies with chemical accuracy (or very near it).
We cannot achieve this with computations on
extended substrates (slabs or semi-infinite), and since most substrate atoms in a small cluster do not have the number of nearest neighbours appropriate to the extended substrate, it is of the utmost importance to know whether an ab initio cluster calculation of, for example, an adsorbate vibration frequency is converged with chemical accuracy with respect to cluster size.
Goddard has
addressed this point, and it seems that ZO nickel atoms are needed to model (IOO)Ni for CH, CH
and CH 3 adsorption. I emphasize again that ab initio calZ culations are undertaken to produce chemically accurate information, and if they do not do so, they are useless.
We ought therefore to be slightly wary of
any such computations using less than ZO nickel atoms to model the (100) surfaca iii)
Can we model probe and process dynamics with clusters?
Again the point is that, if there are dynamical processes on clusters which are insensitive to the fact that most substrate atoms in the cluster do not have the environment of the extended substrate, then cluster calculations would provide a way forward with chemical accuracy.
Much preliminary work is needed
to explore such an approach to dynamical problems at surfaces, and it is not appropriate to begin it here.
I remark only that quite simple models contain
the essential problems, and that the question can probably be answered fairly easily for specific probes (e.g. He atoms), and specific processes (e.g. vibrational damping). iv)
What aspects of SERS can be studied with cluster models? This question is really included in (iii).
However, Schatz has already cal-
culated Raman intensities from a cluster using, in effect, a microscopic model of coupled photons, electrons and nuclei, and consequently the above question should now be asked. CONCLUSION Only ab initio work using either jelliums, or small substrate clusters was presented at the conference, and because of the lack of substrate chemistry with jelliums, it is important to consider the limits of clusters as models of semi-infinite substrates.
It is fairly well established that small substrate
clusters are adequate for the calculation of adsorbate, and surface bond
235
lengths, that larger clusters are needed for vibration frequencies, and that quite large clusters must be used to obtain the adsorbate-induced electronic structure.
But "small", "larger", and "quite large" are vague terms whose pre-
cise meaning depends on the chemisorption system under investigation, and much more theoretical work with good theoretical models, which include electron correlation effects, is needed to widen our experience, and to improve our knowledge in this area.
Finally, the possible use of cluster models for probe, and
process dynamics, about which little is known, should be systematically explored; cluster embedding is computationally rather expensive, and we need to know much more about the circumstances in which we can avoid it.
REFERENCES 1 2
G.C. Aers, T.B. Grimley, J.B. Pendry and K.L. Sebastian, J. Phys. C: Solid St. Phys., 14 (1981) 3995; also unpublished computations. J.P. Muscat and n.M. Newns, Phys. Letters,60A (1977) 348, 6lA (1977) 481.