- Email: [email protected]
Photoemission spectrum for hydrogen chemisorbed on Ni
Solid State Communications,
Vol. 22, pp. 51-54,
PHOTOEMISSION
1977.
Pergamon Press.
SPECTRUM FOR HYDROGEN
Printed in Great Britain
CHEMISORBED ON Ni
K. Schonhammer* Department
of Physics, University (Received
of Pennsylvania
17 November
Philadelphia,
PA 19174, U.S.A.
1976 by E. But-stein)
Recent X,-SW cluster calculations, which indicate the relative importance of the sp-bands compared to the d-bands for hydrogen chemisorption on Ni are used to parameterize a model Hamiltonian. The adsorbate Green’s function is calculated and yields the experimental width of the extra peak seen in photoemission. It is shown that in the equilibrium geometry the qualitative changes in the spectrum due to many body effects are too small to be seen experimentally. _
IN THE LAST FEW YEARS three different kinds of theoretical approaches have been used to describe the chemisorption of hydrogen on metals but none of them is adequate to properly describe all aspects of the problem. In the work using the density functional formalism [ 1,2] a jelhum description of the semi-infinite metal has been used which is useful for simple metals. As the main experimental interest has been on the chemisorption on transition metals one has to go beyond the jellium model in this type of calculations. Newns, Grimley and others [4-81 have used the Anderson model [3] to describe hydrogen chemisorption in which the electron-electron interactiqn on the metal side is neglected. One of the main problems is then how to parameterize the matrix elements in the Hamiltonian. Recently cluster calculations have been performed [9 ] using the X,-scattered wave method with clusters of 4 or 6 Ni (Pd, Pt) atoms. These calculations yield useful information about the electronic structure, but obviously are unable to describe the width of the extra peaks seen in photoemission from hydrogen covered surfaces, which result from a broadening of sharp levels in the cluster calculation into resonances. In the following we will use the information on the relative importance of the s-p and d-bands in the chemisorption as obtained from these cluster calculations for a parameterization of the Anderson Hamiltonian. The Anderson Hamiltonian reads H =
c k. a
Eknka
+
C a
wba
+
+
c
cv,k
+iu+ka
lb) = V-’ 1
Vok Ik);
(1)
Vz = c
k
k
I&I*.
(2)
The only way the properties of the free metal enter the calculation of the adsorbate Green’s function is via the function F(z) l?(z) = p ‘bl-&
lb).
(3)
M
In the following we will use the bandstructure of Ni as obtained in a recent self-consistent calculation [lo], which shows a broad s-p band about one Rydberg wide and a d-band with a width of the order 4 eV. We have simulated this band structure by two semi-elliptical bands. For the sp-band this is not a bad approximation while for the d-band the detailed structure is not important, as the peaks we obtain in the spectral density are well below the bottom of the d-band. The relative importance of the sp- and d-bands enter via the state lb)
Unotno4
On leave of absence from Physikdepartment T IJ Munchen, West Germany.
h.c.1.
The first term describes the free metal surface with k labeling the wavevector k and the band index, the second and third term the free hydrogen atom with U the intra-atomic Coulomb repulsion and the last one the coupling between the metal and the adsorbate. The coupling term can be rewritten [6] using the creation and annihilation operator of a normalized state lb) localized at the metal surface in the vicinity of the hydrogen atom
lb) = ald)+&=-&p)
*
+
k.a
(4)
and cr is the measure how d-like lb) is. In almost all model calculations only the d-band has been considered
der
51
PHOTOEMISSION
52
(0~= 1). In a HF calculation for the adsorbate Green’s function [4] a sharp localized level has been found to split off the bottom of the d-band, which is essentially a linear combination of lb) with the hydrogen 1s state. The same kind of state appears in the X,-SW cluster calculation and we have used the s-p- and d contribution to the charge density of this state on the Ni atoms to obtain a value for 0~.For Ni 0~’= 0.4 while for Pd and Pt the d-bands give the larger contribution [9]. Melius et al. [ 111 have also performed calculations which indicate the importance of the s-band for hydrogen chemisorption on Ni. We use a value of V which reproduces the position of the peak seen in photoemission experiments (5.8 eV below the Fermi level for Ni [ 121). Newns [4] has used the experimental binding ener,v to obtain a value for V. As the binding energy is not a purely electronic property we prefer the first procedure. There has been much discussion [ 13, 141 about how large to chose the interatomic Coulomb repulsion (1. In a classical picture the ionisation level of the hydrogen is moved upwards by e2/4z (z distance from the surface) and the affinity level is pushed downwards by the same amount. As these image charge effects are not included in the Anderson Hamiltonian one has to use an effective value for U which is smaller than the 12.9 eV for z -+ 00. If one takes U,,, to be this difference between the affinity and the ionisation level one has U,,, = U e2/2z. As the hydrogen atom is adsorbed neutrally (A, = 0. le) the actual choice of U,,, only slightly effects the HF-result for the spectral density as e, + U. +I,) does not change very much by the replacements E, -) eg**,U -+Ueff. As we will discuss in the following also many body corrections to the HF-result, we chose a not too smaIl value for Ueff(8 eV) to obtain an upper bound on the importance of the correlation effects. We should point out that while it seems to be appropriate to use effective values for e, and U in the calculation of the photoelectron spectrum, we believe it is more reasonable to use the unmodified quantities in the calculation of the ground state ener,T. Using the parameters described above we have calculated the adsorbate Green’s function g,,(z) which gives the change in the photoemission current with adsorption [ 151 in two different approximations. We use Zubarev’s double time Green’s functions defined by
iI61 g&>
Vol. 22, No. 1
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
= --i1 e'*'({IC/,&), G,',P dt 0
with G,(t) = eiHf ic/, eeiHt, { } denoting the anticommutator and ( ) the expectation value in the exact ground state. The adsorbate Green’s function in the
(5)
i
I
p, I -15
I
l
-10
-5
n il
(e’
I-
-10
-15
5
(f
l
5
lf
-5
E [C”l
6ICVl Fig. 1. Adsorbate spectral density for weak coupling to the sp-band: HF result (upper curve) and result including many body corrections (lower curve).
‘J=E.OeV v =4.1
ev
a*=0.4
I
I
-15
-10
-I
-5
‘f
5
l
6 [e”l
“f
E
Ievl
Fig. 2. Adsorbate spectral density for strong coupling to the sp-band: HF result (upper curve) and result including many body corrections (lower curve). HF approximation
is given by [3,4]
HF = g*cJ
[z - e, - U(n,_,)
- r(z)]
-’ .
(6)
Figures 1 and 2 show the spectral density p::(e) = - Im g,“,“(e + iO)/n for V = 4.1 eV and the two values 01~= 0.8 and 0~’ = 0.4. In both cases pFF(e) shows a resonance well below the bottom of the d-band. The values of Vhas been chosen to reproduce the position of the experimental peak at E+ = - 5.8 eV below the Fermi level for Ni, after the inclusion of the many body effects. The width A of the peaks is approximately given by Cpsp density of states of the sp-band) A = k%r&,(E+)(l
- cz*>.
(7)
Using the parameter of OLas obtained in the cluster calculation leads to the experimental width of (3.0 k 0.5) eV [ 121. For Pd and Pt the sp-contribution is smaller [9] which is in agreement with the smaller experimental width of 1.5 eV. The effect of many body corrections for the Anderson model have been discussed previously [6-81. For the calculation of binding energies as a function of
Vol. 22, No. 1
PHOTOEMISSION
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
53
the distance from the surface variational methods have been proposed which are very efficient [B]. To obtain the correlation effects on the photoelectron spectrum one needs an improved calculation of g&z). Brenig and Schonhammer [6] have proposed an approximation which produces extra satellite peaks in the spectral function due to shake up processes, i.e. electronic excitations that accompany the emission of the photoelectron. As we are only interested in the spectral function in the equilibrium geometry we have calculated the shake up spectrum by considering self-energy corrections to second order in II using HF propagators as the unperturbed Green’s functions. In the Anderson model one has a non-vanishing self-energy M(z) only for the adsorbate Green’s function [ 171
Numerically we have evaluated M2(z) by first calculating the imaginary part Im M2(e + i0) and then obtained Re M2(e) as the Hilbert transform of Im M2(E + i0). Figures 1 and 2 show the effect of the many body corrections:
g,,(z)
A sharp peak due to the last process has been obtained previously [6] using the d-band only. In Fig. 1 (o’ = 0.8) the spectral density p,(c) shows rather narrow bonding and anti-bonding resonances and a well defined shake-up peak corresponding to this process can be seen. For the realistic value of the coupling to the s-band (02 = 0.4) this shake-up peak has almost completely disappeared, so it is no surprise that it has not been found in the experiments on Ni. In models which properly describe the electronelectron interaction on the metal one would also obtain shake-up lines due to the excitations of surface plasmons. Using a parameterization of the Anderson model as suggested by cluster calculations we have obtained the experimental width of the peak seen in photoemission from hydrogen covered Ni surfaces. This indicates that an interplay between the different theoretical approaches can be very useful. We have also shown that in the equilibrium geometry the qualitative changes in the spectrum due to many body effects are too small to be seen experimentally.
= [z - e, - UXn,_,) - r(z) -fif(z)]-‘.
(9)
M(z) is given to second order by
+ t1 -f(El)l
[I -f(E2)lf(E3%
(10)
There are several reasons, why this is a much better approximation than one might expect. To see this we first discuss the symmetric case (2e, + U = 2ef, half filled symmetric band): (a) M2(z) goes over to the exact expression [6] in the surface molecule limit (bandwidth/V + 0). (b) Using M2(z) the adsorbate spectral density goes over smoothly to the atomic limit (with a narrow third “Kondo-peak” at the Fermi level). (c) All odd order contributions vanish [ 1S] , i.e. the self-energy is correct up to third order. These statements are no longer exactly true if one goes away from the symmetric case. On the other hand it has been shown by Cederbaum et al. [19] that for molecules a finite order calculation of the self-energy leads to excellent improvements over Koopman’s theorem. Another important fact for the discussion of the shake-up spectrum is that M2(z) has the correct analytical properties.
(1) There is a shift of the bonding resonance of the order of 0.5 eV due to relaxation and correlation effects. (2) One obtains the shake-up spectrum due to the emission from the bonding resonance or the metal bands, accompanied by various kinds of particle hole excitations. The peak at - 15 eV in Fig. 1 for example corresponds to the emission from the bonding resonance + the excitation of another electron from the bonding to the antibonding resonance.
Acknowledgements - The author wishes to thank the members of the solid state group at the University of Pennsylvania for their hospitality. He also would like to thank Profs. J.R. Schrieffer and S. Lundqvist for interesting discussions. This work was supported by a research grant of the Deutsche Forsohungsgemeinschaft.
REFERENCES 1.
LANG N.D. & WILLIAMS A.R., Phys. Rev. Lett. 34,531
(1975).
2.
GUNNARSON
3.
ANDERSON P.W.,Phys. Rev. 124,41
4.
NEWNS D.M., Phys. Rev. 178, 1123 (1969).
5.
GRIMLEY T.B., int. School of Physics Enrico Fermi LVIII, 298, (1974).
O., HJELMBERG H. & LUNDQLJIST, B.I., Phys. Rev. Lett. 37,292 (1961).
(1976).
PHOTOEMISSION
54 6.
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
BRENIG W. & SCHONHAMMER K., Z. f’hys. 267,201 BRENIG W., Z. Phys. B22, 143 (1975). SCHUCK P., Phys. Rev. B13,5225
(1974); SCHijNHAMMER
Vol. 22, No. I
K., HARTUNG V. &
(1976).
8.
SCHONHAMMER K., Z. Phys. B21,389
9.
JOHNSON K.H. (private communication).
(1975); Phys. Rev. B13,4336
(1976).
10.
CALLAWAY T. &WANG C., Phys. Rev. B7, 1096 (1973).
11.
MELIUS C.F., MOSKOWITZ J.W., MARTALA A.P., BAILLE M.B. & RATNER M.A., Surf: Sci. 59,279 (1976).
12.
DEMUTH J.E. (to be published).
13.
SCHRIEFFER
14.
HEWSON A.C. & NEWNS D.M., Japan J. Appl. Phys. Suppl. 2, 121 (1974).
15.
PENN D.R., Phys. Rev. Lett. 28, 1041 (1972).
J.R., Int. School of Physics Enrico Fermi L VIII, 250, (1974).
16.
ZUBAREV D.N., Sov. Phys. Ushpekhi 3,320
17.
SCHRIEFFER
(1960).
18.
ICHE G. & ZAWADOWSKI A., Solid State Commun. 10, 1001 (1972).
19.
For a review see: CEDERBAUM L.S. & DOMCKE W., Adv. Chem. Phys. (to be published).
J.R. & MATTIS D., Phys. Rev. 140, 1412 (1965).
Vol. 22, pp. 51-54,
PHOTOEMISSION
1977.
Pergamon Press.
SPECTRUM FOR HYDROGEN
Printed in Great Britain
CHEMISORBED ON Ni
K. Schonhammer* Department
of Physics, University (Received
of Pennsylvania
17 November
Philadelphia,
PA 19174, U.S.A.
1976 by E. But-stein)
Recent X,-SW cluster calculations, which indicate the relative importance of the sp-bands compared to the d-bands for hydrogen chemisorption on Ni are used to parameterize a model Hamiltonian. The adsorbate Green’s function is calculated and yields the experimental width of the extra peak seen in photoemission. It is shown that in the equilibrium geometry the qualitative changes in the spectrum due to many body effects are too small to be seen experimentally. _
IN THE LAST FEW YEARS three different kinds of theoretical approaches have been used to describe the chemisorption of hydrogen on metals but none of them is adequate to properly describe all aspects of the problem. In the work using the density functional formalism [ 1,2] a jelhum description of the semi-infinite metal has been used which is useful for simple metals. As the main experimental interest has been on the chemisorption on transition metals one has to go beyond the jellium model in this type of calculations. Newns, Grimley and others [4-81 have used the Anderson model [3] to describe hydrogen chemisorption in which the electron-electron interactiqn on the metal side is neglected. One of the main problems is then how to parameterize the matrix elements in the Hamiltonian. Recently cluster calculations have been performed [9 ] using the X,-scattered wave method with clusters of 4 or 6 Ni (Pd, Pt) atoms. These calculations yield useful information about the electronic structure, but obviously are unable to describe the width of the extra peaks seen in photoemission from hydrogen covered surfaces, which result from a broadening of sharp levels in the cluster calculation into resonances. In the following we will use the information on the relative importance of the s-p and d-bands in the chemisorption as obtained from these cluster calculations for a parameterization of the Anderson Hamiltonian. The Anderson Hamiltonian reads H =
c k. a
Eknka
+
C a
wba
+
+
c
cv,k
+iu+ka
lb) = V-’ 1
Vok Ik);
(1)
Vz = c
k
k
I&I*.
(2)
The only way the properties of the free metal enter the calculation of the adsorbate Green’s function is via the function F(z) l?(z) = p ‘bl-&
lb).
(3)
M
In the following we will use the bandstructure of Ni as obtained in a recent self-consistent calculation [lo], which shows a broad s-p band about one Rydberg wide and a d-band with a width of the order 4 eV. We have simulated this band structure by two semi-elliptical bands. For the sp-band this is not a bad approximation while for the d-band the detailed structure is not important, as the peaks we obtain in the spectral density are well below the bottom of the d-band. The relative importance of the sp- and d-bands enter via the state lb)
Unotno4
On leave of absence from Physikdepartment T IJ Munchen, West Germany.
h.c.1.
The first term describes the free metal surface with k labeling the wavevector k and the band index, the second and third term the free hydrogen atom with U the intra-atomic Coulomb repulsion and the last one the coupling between the metal and the adsorbate. The coupling term can be rewritten [6] using the creation and annihilation operator of a normalized state lb) localized at the metal surface in the vicinity of the hydrogen atom
lb) = ald)+&=-&p)
*
+
k.a
(4)
and cr is the measure how d-like lb) is. In almost all model calculations only the d-band has been considered
der
51
PHOTOEMISSION
52
(0~= 1). In a HF calculation for the adsorbate Green’s function [4] a sharp localized level has been found to split off the bottom of the d-band, which is essentially a linear combination of lb) with the hydrogen 1s state. The same kind of state appears in the X,-SW cluster calculation and we have used the s-p- and d contribution to the charge density of this state on the Ni atoms to obtain a value for 0~.For Ni 0~’= 0.4 while for Pd and Pt the d-bands give the larger contribution [9]. Melius et al. [ 111 have also performed calculations which indicate the importance of the s-band for hydrogen chemisorption on Ni. We use a value of V which reproduces the position of the peak seen in photoemission experiments (5.8 eV below the Fermi level for Ni [ 121). Newns [4] has used the experimental binding ener,v to obtain a value for V. As the binding energy is not a purely electronic property we prefer the first procedure. There has been much discussion [ 13, 141 about how large to chose the interatomic Coulomb repulsion (1. In a classical picture the ionisation level of the hydrogen is moved upwards by e2/4z (z distance from the surface) and the affinity level is pushed downwards by the same amount. As these image charge effects are not included in the Anderson Hamiltonian one has to use an effective value for U which is smaller than the 12.9 eV for z -+ 00. If one takes U,,, to be this difference between the affinity and the ionisation level one has U,,, = U e2/2z. As the hydrogen atom is adsorbed neutrally (A, = 0. le) the actual choice of U,,, only slightly effects the HF-result for the spectral density as e, + U. +I,) does not change very much by the replacements E, -) eg**,U -+Ueff. As we will discuss in the following also many body corrections to the HF-result, we chose a not too smaIl value for Ueff(8 eV) to obtain an upper bound on the importance of the correlation effects. We should point out that while it seems to be appropriate to use effective values for e, and U in the calculation of the photoelectron spectrum, we believe it is more reasonable to use the unmodified quantities in the calculation of the ground state ener,T. Using the parameters described above we have calculated the adsorbate Green’s function g,,(z) which gives the change in the photoemission current with adsorption [ 151 in two different approximations. We use Zubarev’s double time Green’s functions defined by
iI61 g&>
Vol. 22, No. 1
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
= --i1 e'*'({IC/,&), G,',P dt 0
with G,(t) = eiHf ic/, eeiHt, { } denoting the anticommutator and ( ) the expectation value in the exact ground state. The adsorbate Green’s function in the
(5)
i
I
p, I -15
I
l
-10
-5
n il
(e’
I-
-10
-15
5
(f
l
5
lf
-5
E [C”l
6ICVl Fig. 1. Adsorbate spectral density for weak coupling to the sp-band: HF result (upper curve) and result including many body corrections (lower curve).
‘J=E.OeV v =4.1
ev
a*=0.4
I
I
-15
-10
-I
-5
‘f
5
l
6 [e”l
“f
E
Ievl
Fig. 2. Adsorbate spectral density for strong coupling to the sp-band: HF result (upper curve) and result including many body corrections (lower curve). HF approximation
is given by [3,4]
HF = g*cJ
[z - e, - U(n,_,)
- r(z)]
-’ .
(6)
Figures 1 and 2 show the spectral density p::(e) = - Im g,“,“(e + iO)/n for V = 4.1 eV and the two values 01~= 0.8 and 0~’ = 0.4. In both cases pFF(e) shows a resonance well below the bottom of the d-band. The values of Vhas been chosen to reproduce the position of the experimental peak at E+ = - 5.8 eV below the Fermi level for Ni, after the inclusion of the many body effects. The width A of the peaks is approximately given by Cpsp density of states of the sp-band) A = k%r&,(E+)(l
- cz*>.
(7)
Using the parameter of OLas obtained in the cluster calculation leads to the experimental width of (3.0 k 0.5) eV [ 121. For Pd and Pt the sp-contribution is smaller [9] which is in agreement with the smaller experimental width of 1.5 eV. The effect of many body corrections for the Anderson model have been discussed previously [6-81. For the calculation of binding energies as a function of
Vol. 22, No. 1
PHOTOEMISSION
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
53
the distance from the surface variational methods have been proposed which are very efficient [B]. To obtain the correlation effects on the photoelectron spectrum one needs an improved calculation of g&z). Brenig and Schonhammer [6] have proposed an approximation which produces extra satellite peaks in the spectral function due to shake up processes, i.e. electronic excitations that accompany the emission of the photoelectron. As we are only interested in the spectral function in the equilibrium geometry we have calculated the shake up spectrum by considering self-energy corrections to second order in II using HF propagators as the unperturbed Green’s functions. In the Anderson model one has a non-vanishing self-energy M(z) only for the adsorbate Green’s function [ 171
Numerically we have evaluated M2(z) by first calculating the imaginary part Im M2(e + i0) and then obtained Re M2(e) as the Hilbert transform of Im M2(E + i0). Figures 1 and 2 show the effect of the many body corrections:
g,,(z)
A sharp peak due to the last process has been obtained previously [6] using the d-band only. In Fig. 1 (o’ = 0.8) the spectral density p,(c) shows rather narrow bonding and anti-bonding resonances and a well defined shake-up peak corresponding to this process can be seen. For the realistic value of the coupling to the s-band (02 = 0.4) this shake-up peak has almost completely disappeared, so it is no surprise that it has not been found in the experiments on Ni. In models which properly describe the electronelectron interaction on the metal one would also obtain shake-up lines due to the excitations of surface plasmons. Using a parameterization of the Anderson model as suggested by cluster calculations we have obtained the experimental width of the peak seen in photoemission from hydrogen covered Ni surfaces. This indicates that an interplay between the different theoretical approaches can be very useful. We have also shown that in the equilibrium geometry the qualitative changes in the spectrum due to many body effects are too small to be seen experimentally.
= [z - e, - UXn,_,) - r(z) -fif(z)]-‘.
(9)
M(z) is given to second order by
+ t1 -f(El)l
[I -f(E2)lf(E3%
(10)
There are several reasons, why this is a much better approximation than one might expect. To see this we first discuss the symmetric case (2e, + U = 2ef, half filled symmetric band): (a) M2(z) goes over to the exact expression [6] in the surface molecule limit (bandwidth/V + 0). (b) Using M2(z) the adsorbate spectral density goes over smoothly to the atomic limit (with a narrow third “Kondo-peak” at the Fermi level). (c) All odd order contributions vanish [ 1S] , i.e. the self-energy is correct up to third order. These statements are no longer exactly true if one goes away from the symmetric case. On the other hand it has been shown by Cederbaum et al. [19] that for molecules a finite order calculation of the self-energy leads to excellent improvements over Koopman’s theorem. Another important fact for the discussion of the shake-up spectrum is that M2(z) has the correct analytical properties.
(1) There is a shift of the bonding resonance of the order of 0.5 eV due to relaxation and correlation effects. (2) One obtains the shake-up spectrum due to the emission from the bonding resonance or the metal bands, accompanied by various kinds of particle hole excitations. The peak at - 15 eV in Fig. 1 for example corresponds to the emission from the bonding resonance + the excitation of another electron from the bonding to the antibonding resonance.
Acknowledgements - The author wishes to thank the members of the solid state group at the University of Pennsylvania for their hospitality. He also would like to thank Profs. J.R. Schrieffer and S. Lundqvist for interesting discussions. This work was supported by a research grant of the Deutsche Forsohungsgemeinschaft.
REFERENCES 1.
LANG N.D. & WILLIAMS A.R., Phys. Rev. Lett. 34,531
(1975).
2.
GUNNARSON
3.
ANDERSON P.W.,Phys. Rev. 124,41
4.
NEWNS D.M., Phys. Rev. 178, 1123 (1969).
5.
GRIMLEY T.B., int. School of Physics Enrico Fermi LVIII, 298, (1974).
O., HJELMBERG H. & LUNDQLJIST, B.I., Phys. Rev. Lett. 37,292 (1961).
(1976).
PHOTOEMISSION
54 6.
SPECTRUM FOR HYDROGEN CHEMISORBED ON Ni
BRENIG W. & SCHONHAMMER K., Z. f’hys. 267,201 BRENIG W., Z. Phys. B22, 143 (1975). SCHUCK P., Phys. Rev. B13,5225
(1974); SCHijNHAMMER
Vol. 22, No. I
K., HARTUNG V. &
(1976).
8.
SCHONHAMMER K., Z. Phys. B21,389
9.
JOHNSON K.H. (private communication).
(1975); Phys. Rev. B13,4336
(1976).
10.
CALLAWAY T. &WANG C., Phys. Rev. B7, 1096 (1973).
11.
MELIUS C.F., MOSKOWITZ J.W., MARTALA A.P., BAILLE M.B. & RATNER M.A., Surf: Sci. 59,279 (1976).
12.
DEMUTH J.E. (to be published).
13.
SCHRIEFFER
14.
HEWSON A.C. & NEWNS D.M., Japan J. Appl. Phys. Suppl. 2, 121 (1974).
15.
PENN D.R., Phys. Rev. Lett. 28, 1041 (1972).
J.R., Int. School of Physics Enrico Fermi L VIII, 250, (1974).
16.
ZUBAREV D.N., Sov. Phys. Ushpekhi 3,320
17.
SCHRIEFFER
(1960).
18.
ICHE G. & ZAWADOWSKI A., Solid State Commun. 10, 1001 (1972).
19.
For a review see: CEDERBAUM L.S. & DOMCKE W., Adv. Chem. Phys. (to be published).
J.R. & MATTIS D., Phys. Rev. 140, 1412 (1965).