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Determination of CNDO parameters for germanium and their application to simple germanium molecules
J Phys Chem. Sol& Vol Pnoted tn Great Entam
49. No
8. pp
883-886.
1988
0022.3697,/88 13 00 + 0 00 Pergamon Press
plc
DETERMINATION OF CNDO PARAMETERS FOR GERMANIUM AND THEIR APPLICATION TO SIMPLE GERMANIUM MOLECULES C. K. ONG and G. S. KHOO Department of Physics, National University of Singapore, Kent Ridge, Singapore 0511 (hceived
I June 1987; accepred 7 October 1987)
Abstract-We have obtained a set of CNDO parameters for crystalline germanium that can reproduce, to high accuracy, the experimental values for the cohesive energy, the valence bandwidth and the equilibrium internuclear separation for a perfect Ge crystal. The CNDO method is then used to calculate the equilibrium bond lengths of simple molecules of Ge. The results are compared with experimental data. This set of parameters may be useful for defect studies in Ge. Keywords: CNDO parameters, large unit cells.
1. INTRODUCTION
2. CNDO METHOD
In many defect calculations for semiconductors, information about the precise local atomic position associated with a particular charge state is crucial [l]. Therefore, any useful theoretical tool must be able to give the total energy of the system using selfconsistent methods. The complete neglect of the differential overlap (CNDO) method has the abovementioned properties. In fact, models of the solid as a large cluster of atoms with CNDO approximations have been used successfully in vacancies and interstitials in diamond [Z, 31, hydrogen in diamond and silicon [4], as well as self-interstitials in silicon [5,6]. We have also successfully studied hydrogen complexes in silicon [7]. The CNDO and other semi-empirical methods depend very much on their parameter sets, so it is very important that there exists a systematic procedure to choose these parameters. At present, there are no suitable CNDO parameters for GE. Therefore, before we can proceed further to investigate defects in Ge using the CNDO method, we have, first, to determine a suitable parameter set for Ge. In this work, we shall closely follow the approach of Harker and Larkins [8] by using a periodic I&atom large unit cell (LUC) to obtain a set of CNDO parameters for Ge. The Harker and Larkins parameters for diamond and silicon obtained by this approach have already heen used successfully to study vacancies, interstitials and hydrogen in diamond and silicon [2-71, as mentioned earlier. Hence, we expect that the set of parameters obtained for Ge in the present work may also be useful for defect studies on Ge. These parameters, once obtained, will then be applied to study the equilibrium bond length of Ge-X, in GeX,, where X = H, F and Cl.
The CNDO method is a semi-empirical selfconsistent molecular method. In CNDO, valence electrons, moving in fixed cores, are represented by a minimal Slater basis set. A further approximation is to approximate the Roothaan equation by neglecting the two electron overlap integrals on the different atoms. The elements of the Fock matrix in the order of overlap integrals are replaced by the semiempirical parameters, which depend only on the atomic species (for details, see Ref. 181). They are: (1) the orbital exponent, 5; (2) the electronegativity ~~and cP; (3) the bonding parameters, /I. Eacl. orbital of the valence electrons of an atom possesses these three parameters. In our case, the valence electrons of Ge atoms have 4s and 4p orbitals and hence Ge has five CNDO parameters, which are <,, r,, 5, 6Pand /I. The subscripts s, p are for s and p orbitals respectively. We have assumed here that s and p orbitals have the same orbital exponent, i.e. t;lPI fi depends only on the atomic species and not on the orbital type. For a given set of parameters, the CNDO method will perform self-consistent calculations similar to Hartree-Fock calculations, giving the energy eigenvalues and wavefunctions. We have used the Harwell MOSES codes [9] to perform the CNDO calculations. We use periodically repeated large unit cells (LUC), with each LUC containing 16 atoms to model the perfect Ge crystal. The k = 0 approximation and a cut-off distance equal to the equilibrium internuclear separation for Coulomb integrals have been applied (for details see Ref. 181). 883
C. K. ONG and G. S.
884 3. RESULTS Optimum
CNDO
parameters
In our parameterization procedure, we basically use three bulk Ge properties, namely the valence bandwidth, cohesive energy and equilibrium internuclear separation, to determine five Ge CNDO parameters. This is a tedious process since we have to vary the parameters independently, starting from the values obtained from the standard method as proposed by Pople and Beveridge [lo]. The dependence of the valence bandwidth and cohesive energy is illustrated in Fig. 1. For each case, only one CNDO parameter is varied independently, with the remaining parameters kept constant at their final values. The optimum set of CNDO parameters we have obtained for germanium is presented in Table 1. The results for the valence bandwidth, cohesive energy, and equilibrium internuclear separation from the CNDO calculations with these optimum CNDO parameters, together with the ex~~mental data, are given in Table 2. As we can see from the table, the calculated values fit the experimental values well. The resulting valence band-structure obtained with this parameter set is listed in Table 3. The energy eigenvalues of the valence band are calculated at the special symmetry points in the Brillouin zone. Our results are compared with various recent theoretical calculations and experimental data (see Table 3). Sahu et al. [l l] used the extended Htickel theory
expomt 3 W&r-‘1 lel
p Wllw (*VI IfI
KHOO
(EHT) with Messmer’s parameters [12]. The predictions of Newman and Dow [ 131 were obtained by using an empirical tight binding theory with the virtual-crystal approximation. They included secondneighbour interactions, while the results of Wang and Klein [14] were obtained by using the linear combination of Gaussian orbitab (LCGO) method with a local-density form of the exchange-correlation function. In general, the present calculated results agree reasonably we11with the experimental resufts and other theoretical calculations in the order and accuracy of the valence band energy levels, but the calculated band gap is rather large in comparison with the experimental value. This is largely due to the use of a limited basis set. We are aware that the CNDO method is not an accurate method for band-structure calculations compared with some of the other methods (e.g. local density method) listed in Table 3 and should not be treated as such. However, it is attractive and useful in defect studies
Ill. We have also performed CNDO calculations for the bulk modulus and found that its value is more than two times larger than the experimental result. There are three possible areas that may contribute to this bad bulk modulus result, namely, the k = 0 approximation in the LUC calculation, the CNDO approximations and the limited basis set. Recently, Smith et al. [ 151 commented that the k = 0 approximation with a cut-off distance for overlap and Coulomb integrals in the LUS-CNDO calculation
Is (*VI (9)
Ip WI IhI
Fig. 1. (a)-(h). Variation of the valence bandwidth and cohesive energy of germanium with the UNDO parameter set. For each diagram only one parameter is varied while the rest are kept constant. The
parameters when kept constant have the following values: <+,= 1.88 Bohr - ‘; fi = - 6.45 eV; C~= 7.7 eV; q, = 3.8 eV.
Determination
of CNDO parameters for germanium
885
Table 1. Ootimum CNDO narameter set Orbital exponent L (Bohr-‘)
Electronegativity t, (eV) 5 (eV)
1.88
7.1
Bonding parameter B (eV)
3.8
-6.45
Table 2. Bulk properties Present work Experiment Valence bandwidth (eV) Cohesive energy (eV/atom) Equilibrium internuclear separation (h;) Bulk modulus (tOi Pa) Hybridization state
13.5 3.85 2.43 20.49 S’ I6PZM
13.1t 3.853 2.45: 7.7% SPS
t Cardona and Pollak [18].
$ Kittel [19].
Table 3. Comparison of energy eigenvalues (eV) at special symmetry points in the Brillouin zone Symmetry points
tr;,
Present work
Ref. [II]
Ref. [14] Ref. (131
7.4 6.6 0.0 -1.6 -3.7 -8.5 -1.5 -8.7 -13.5
3.5 0.0 -1.3 -2.7 -8.1 -5.6 - 10.3 -12.3
2.8 0.0 -1.4 -3.1 -8.6 -7.6 - 10.7 - 12.7
3.3 0.0 - 1.7 -3.3 -8.7 -7.3 - 10.5 - 12.7
Experiment (Ref. 1181) 3.2 0.0 -1.2 -3.0 -9.0 -7.7 -11.3 -13.1
is set equal to zero.
by Harker and Larkins [S] was not satisfactory. They proposed a new formalism that included a real space modulating function in the LUC-CNDO equations to allow for the replacement of a Brillouin zone integral by finite samplings, and found good bulk modulus results for silicon and diamond. However, when their optimum CNDO parameters were used in defect calculations, problems of convergence in SCF calculations occurred [ 151. It should be noted that we intend to use the optimum CNDO parameter set we have obtained for germanium in defect studies where the defect clusters would be used instead of the LUC. Furthermore, we are interested in getting the total energy surface of a defect cluster calculation, and the volumedependent terms would not be used.
results for analogous Si molecules using the parameters of Harker and Larkins [8] are also presented for comparison. The parameters for H, Cl and F are those of Pople and Beveridge [IO]. The calculated
equilibrium bond length and bond energy for both GeH4 and SiH, are in excellent agreement with experiment [ 161.Thus, the parameters determined for Ge seem promising for defect studies in Ge, especially for hydrogen in Ge. However, there is some deviation between the calculated results and experimental data for the other molecules where F and Cl are involved. This could be due to the fact that the parameters for F and Cl are deduced from small molecule properties while the parameters for Si and Ge are from bulk properties of the crystalline solid. Thus, the parameters for F and Cl may have to be reparameterized in order to be used in the solid case.
4. APPLICATION TO SIMPLE Ce MOLECULES
5. CONCLUSION
We now apply the CNDO parameters obtained to calculate the equilibrium bond length for G-X, where X = F, Cl and H in GeX,. However, only the experimental value of the bond energy for Ge-H in GeH, is available for comparison. Therefore, calculations for the bond energy will be made only for Ge-H. The results of such CNDO calculations on GeH,, GeF, and GeCl., are illustrated in Table 4. The
We have obtained an optimum CNDO parameter set for Ge by using ldatom periodically repeated large unit cells. This set of optimum parameters is able to account for the electronic properties of the valence band but fails to give a good bulk modulus value. As mentioned in Section 3, Smith et al.% [IS] modulating function scheme was able to give good bulk moduli results for silicon and diamond. How-
886
C. K. ONG and G. S. KHOO Table 4. Predicted bond energy and equilibrium bond length for Ge-X and Si-X in GeX, and Six,
Molecules GeH,. Ge-H SiH,, Si-H GeF,, Ge-F SiF,, Si-F GeCl,, Ge-Cl SiCl, , Si-Cl
Bond energy (eV)
Equilibrium bond length (A)
Expt
Calc.
-
3.0 3.3
2.9 3.2 4.4 3.9 2.0 2.0
5.8 3.9
Expt.
Calc.
1.53
1.55 1.50 1.82 1.83 2.24 2.21
1.48 1.67 1.54 2.10 2.01
t Note: all experimental data are from Co~prehe~s~e Znorganic Chemistry [161. ever, when their optimum CNDO parameters were used in defect calculations, the problems of con-
vergence in SCF calculations occurred [15]. In short, we have reported the limitation and usefulness of the CNDO method in producing the bulk properties of germanium. We are using this present set of parameters to study the energy profiles of the different forms of hydrogen and other defect properties in germanium. Excellent rapid convergences have been obtained in SCF calculations and good agreement with experiment has been obtained [17], REFERENCES 1. Stoneham A. M., Phil. Mug. B, 51, 1611 (1985). 2. Mainwood A.. J. Phys. C, Solid St. Phys. 11, 2703 (1978). 3. Mainwood A., Larkins F. P. and Stoneham A. M.. So/id St. Electron. 21, 1431 (1978). 4. Mainwood A. and Stoneham A. M., Physica B. 116,101 (1983). 5. Masri P., Harker A. H. and Stoneham A. M., J. Phys. C, Solid St. Phys. 16, L613 (1983). 6. Khoo G. S. and Ong C. K., J. Phys. C, Solid St. Phys. M, SO37 (1987).
7. Ong C. K. and Khoo G. S., J. Phys. C, Solid SI. Phys. 20, 419 (1987). 8. Harker A. H. and Larkins F. P., J. Phys. C, Solid Sr. Phys. 12, 2487, 2497 (1979). 9. Harker A. H. and Lyon S. B., Hanvell Rep., AERE, R8S98 (1979). 10. Pople J. A. and Beveridge D.L., Approximate Molecrtiar Orbifal Theory, McGraw-Hill, New York (1970). 11. Sahu S. N., Borenstein J. T., Singh V. A. and Corbett J. W., Phys. Status Solidi B, 122, 661 (1984). 12. Messmer R. P., Chem. Phys. L&r. 11, 589 (1971). 13. Newman K. E. and Dow J. D., Phys. Rev. B, 30, 1929 (1984). 14. Wang C. S. and Klein B. M., Phys. Rev. B, 24, 3393 (1981). IS. Smith P. V., Szymanski J. E. and Matthew J. A. D., Phil. Mug. B, 51, 193 (1985). 16. Bailar J. C., Emeleus H. J., Nyholm Sir R. and Trotman-Dickenson A. F., Comprehensive Inorganic Chemistry, Vol. 1, p. 1381; Vol. 2, p. 23. Pergamon Press, Oxford. 17. Khoo G. S. and Ong C. K., J. Phys. C, Solid St. Phys. 20, 1385 (1987). 18. Cardona M. and Pollak F. H., Phys. Rev. 142, 530 (1966). 19. Kittel C., Introduction to Solid Stare Physics, 3 1, 74, 85 (1976).
49. No
8. pp
883-886.
1988
0022.3697,/88 13 00 + 0 00 Pergamon Press
plc
DETERMINATION OF CNDO PARAMETERS FOR GERMANIUM AND THEIR APPLICATION TO SIMPLE GERMANIUM MOLECULES C. K. ONG and G. S. KHOO Department of Physics, National University of Singapore, Kent Ridge, Singapore 0511 (hceived
I June 1987; accepred 7 October 1987)
Abstract-We have obtained a set of CNDO parameters for crystalline germanium that can reproduce, to high accuracy, the experimental values for the cohesive energy, the valence bandwidth and the equilibrium internuclear separation for a perfect Ge crystal. The CNDO method is then used to calculate the equilibrium bond lengths of simple molecules of Ge. The results are compared with experimental data. This set of parameters may be useful for defect studies in Ge. Keywords: CNDO parameters, large unit cells.
1. INTRODUCTION
2. CNDO METHOD
In many defect calculations for semiconductors, information about the precise local atomic position associated with a particular charge state is crucial [l]. Therefore, any useful theoretical tool must be able to give the total energy of the system using selfconsistent methods. The complete neglect of the differential overlap (CNDO) method has the abovementioned properties. In fact, models of the solid as a large cluster of atoms with CNDO approximations have been used successfully in vacancies and interstitials in diamond [Z, 31, hydrogen in diamond and silicon [4], as well as self-interstitials in silicon [5,6]. We have also successfully studied hydrogen complexes in silicon [7]. The CNDO and other semi-empirical methods depend very much on their parameter sets, so it is very important that there exists a systematic procedure to choose these parameters. At present, there are no suitable CNDO parameters for GE. Therefore, before we can proceed further to investigate defects in Ge using the CNDO method, we have, first, to determine a suitable parameter set for Ge. In this work, we shall closely follow the approach of Harker and Larkins [8] by using a periodic I&atom large unit cell (LUC) to obtain a set of CNDO parameters for Ge. The Harker and Larkins parameters for diamond and silicon obtained by this approach have already heen used successfully to study vacancies, interstitials and hydrogen in diamond and silicon [2-71, as mentioned earlier. Hence, we expect that the set of parameters obtained for Ge in the present work may also be useful for defect studies on Ge. These parameters, once obtained, will then be applied to study the equilibrium bond length of Ge-X, in GeX,, where X = H, F and Cl.
The CNDO method is a semi-empirical selfconsistent molecular method. In CNDO, valence electrons, moving in fixed cores, are represented by a minimal Slater basis set. A further approximation is to approximate the Roothaan equation by neglecting the two electron overlap integrals on the different atoms. The elements of the Fock matrix in the order of overlap integrals are replaced by the semiempirical parameters, which depend only on the atomic species (for details, see Ref. 181). They are: (1) the orbital exponent, 5; (2) the electronegativity ~~and cP; (3) the bonding parameters, /I. Eacl. orbital of the valence electrons of an atom possesses these three parameters. In our case, the valence electrons of Ge atoms have 4s and 4p orbitals and hence Ge has five CNDO parameters, which are <,, r,, 5, 6Pand /I. The subscripts s, p are for s and p orbitals respectively. We have assumed here that s and p orbitals have the same orbital exponent, i.e. t;lPI fi depends only on the atomic species and not on the orbital type. For a given set of parameters, the CNDO method will perform self-consistent calculations similar to Hartree-Fock calculations, giving the energy eigenvalues and wavefunctions. We have used the Harwell MOSES codes [9] to perform the CNDO calculations. We use periodically repeated large unit cells (LUC), with each LUC containing 16 atoms to model the perfect Ge crystal. The k = 0 approximation and a cut-off distance equal to the equilibrium internuclear separation for Coulomb integrals have been applied (for details see Ref. 181). 883
C. K. ONG and G. S.
884 3. RESULTS Optimum
CNDO
parameters
In our parameterization procedure, we basically use three bulk Ge properties, namely the valence bandwidth, cohesive energy and equilibrium internuclear separation, to determine five Ge CNDO parameters. This is a tedious process since we have to vary the parameters independently, starting from the values obtained from the standard method as proposed by Pople and Beveridge [lo]. The dependence of the valence bandwidth and cohesive energy is illustrated in Fig. 1. For each case, only one CNDO parameter is varied independently, with the remaining parameters kept constant at their final values. The optimum set of CNDO parameters we have obtained for germanium is presented in Table 1. The results for the valence bandwidth, cohesive energy, and equilibrium internuclear separation from the CNDO calculations with these optimum CNDO parameters, together with the ex~~mental data, are given in Table 2. As we can see from the table, the calculated values fit the experimental values well. The resulting valence band-structure obtained with this parameter set is listed in Table 3. The energy eigenvalues of the valence band are calculated at the special symmetry points in the Brillouin zone. Our results are compared with various recent theoretical calculations and experimental data (see Table 3). Sahu et al. [l l] used the extended Htickel theory
expomt 3 W&r-‘1 lel
p Wllw (*VI IfI
KHOO
(EHT) with Messmer’s parameters [12]. The predictions of Newman and Dow [ 131 were obtained by using an empirical tight binding theory with the virtual-crystal approximation. They included secondneighbour interactions, while the results of Wang and Klein [14] were obtained by using the linear combination of Gaussian orbitab (LCGO) method with a local-density form of the exchange-correlation function. In general, the present calculated results agree reasonably we11with the experimental resufts and other theoretical calculations in the order and accuracy of the valence band energy levels, but the calculated band gap is rather large in comparison with the experimental value. This is largely due to the use of a limited basis set. We are aware that the CNDO method is not an accurate method for band-structure calculations compared with some of the other methods (e.g. local density method) listed in Table 3 and should not be treated as such. However, it is attractive and useful in defect studies
Ill. We have also performed CNDO calculations for the bulk modulus and found that its value is more than two times larger than the experimental result. There are three possible areas that may contribute to this bad bulk modulus result, namely, the k = 0 approximation in the LUC calculation, the CNDO approximations and the limited basis set. Recently, Smith et al. [ 151 commented that the k = 0 approximation with a cut-off distance for overlap and Coulomb integrals in the LUS-CNDO calculation
Is (*VI (9)
Ip WI IhI
Fig. 1. (a)-(h). Variation of the valence bandwidth and cohesive energy of germanium with the UNDO parameter set. For each diagram only one parameter is varied while the rest are kept constant. The
parameters when kept constant have the following values: <+,= 1.88 Bohr - ‘; fi = - 6.45 eV; C~= 7.7 eV; q, = 3.8 eV.
Determination
of CNDO parameters for germanium
885
Table 1. Ootimum CNDO narameter set Orbital exponent L (Bohr-‘)
Electronegativity t, (eV) 5 (eV)
1.88
7.1
Bonding parameter B (eV)
3.8
-6.45
Table 2. Bulk properties Present work Experiment Valence bandwidth (eV) Cohesive energy (eV/atom) Equilibrium internuclear separation (h;) Bulk modulus (tOi Pa) Hybridization state
13.5 3.85 2.43 20.49 S’ I6PZM
13.1t 3.853 2.45: 7.7% SPS
t Cardona and Pollak [18].
$ Kittel [19].
Table 3. Comparison of energy eigenvalues (eV) at special symmetry points in the Brillouin zone Symmetry points
tr;,
Present work
Ref. [II]
Ref. [14] Ref. (131
7.4 6.6 0.0 -1.6 -3.7 -8.5 -1.5 -8.7 -13.5
3.5 0.0 -1.3 -2.7 -8.1 -5.6 - 10.3 -12.3
2.8 0.0 -1.4 -3.1 -8.6 -7.6 - 10.7 - 12.7
3.3 0.0 - 1.7 -3.3 -8.7 -7.3 - 10.5 - 12.7
Experiment (Ref. 1181) 3.2 0.0 -1.2 -3.0 -9.0 -7.7 -11.3 -13.1
is set equal to zero.
by Harker and Larkins [S] was not satisfactory. They proposed a new formalism that included a real space modulating function in the LUC-CNDO equations to allow for the replacement of a Brillouin zone integral by finite samplings, and found good bulk modulus results for silicon and diamond. However, when their optimum CNDO parameters were used in defect calculations, problems of convergence in SCF calculations occurred [ 151. It should be noted that we intend to use the optimum CNDO parameter set we have obtained for germanium in defect studies where the defect clusters would be used instead of the LUC. Furthermore, we are interested in getting the total energy surface of a defect cluster calculation, and the volumedependent terms would not be used.
results for analogous Si molecules using the parameters of Harker and Larkins [8] are also presented for comparison. The parameters for H, Cl and F are those of Pople and Beveridge [IO]. The calculated
equilibrium bond length and bond energy for both GeH4 and SiH, are in excellent agreement with experiment [ 161.Thus, the parameters determined for Ge seem promising for defect studies in Ge, especially for hydrogen in Ge. However, there is some deviation between the calculated results and experimental data for the other molecules where F and Cl are involved. This could be due to the fact that the parameters for F and Cl are deduced from small molecule properties while the parameters for Si and Ge are from bulk properties of the crystalline solid. Thus, the parameters for F and Cl may have to be reparameterized in order to be used in the solid case.
4. APPLICATION TO SIMPLE Ce MOLECULES
5. CONCLUSION
We now apply the CNDO parameters obtained to calculate the equilibrium bond length for G-X, where X = F, Cl and H in GeX,. However, only the experimental value of the bond energy for Ge-H in GeH, is available for comparison. Therefore, calculations for the bond energy will be made only for Ge-H. The results of such CNDO calculations on GeH,, GeF, and GeCl., are illustrated in Table 4. The
We have obtained an optimum CNDO parameter set for Ge by using ldatom periodically repeated large unit cells. This set of optimum parameters is able to account for the electronic properties of the valence band but fails to give a good bulk modulus value. As mentioned in Section 3, Smith et al.% [IS] modulating function scheme was able to give good bulk moduli results for silicon and diamond. How-
886
C. K. ONG and G. S. KHOO Table 4. Predicted bond energy and equilibrium bond length for Ge-X and Si-X in GeX, and Six,
Molecules GeH,. Ge-H SiH,, Si-H GeF,, Ge-F SiF,, Si-F GeCl,, Ge-Cl SiCl, , Si-Cl
Bond energy (eV)
Equilibrium bond length (A)
Expt
Calc.
-
3.0 3.3
2.9 3.2 4.4 3.9 2.0 2.0
5.8 3.9
Expt.
Calc.
1.53
1.55 1.50 1.82 1.83 2.24 2.21
1.48 1.67 1.54 2.10 2.01
t Note: all experimental data are from Co~prehe~s~e Znorganic Chemistry [161. ever, when their optimum CNDO parameters were used in defect calculations, the problems of con-
vergence in SCF calculations occurred [15]. In short, we have reported the limitation and usefulness of the CNDO method in producing the bulk properties of germanium. We are using this present set of parameters to study the energy profiles of the different forms of hydrogen and other defect properties in germanium. Excellent rapid convergences have been obtained in SCF calculations and good agreement with experiment has been obtained [17], REFERENCES 1. Stoneham A. M., Phil. Mug. B, 51, 1611 (1985). 2. Mainwood A.. J. Phys. C, Solid St. Phys. 11, 2703 (1978). 3. Mainwood A., Larkins F. P. and Stoneham A. M.. So/id St. Electron. 21, 1431 (1978). 4. Mainwood A. and Stoneham A. M., Physica B. 116,101 (1983). 5. Masri P., Harker A. H. and Stoneham A. M., J. Phys. C, Solid St. Phys. 16, L613 (1983). 6. Khoo G. S. and Ong C. K., J. Phys. C, Solid St. Phys. M, SO37 (1987).
7. Ong C. K. and Khoo G. S., J. Phys. C, Solid SI. Phys. 20, 419 (1987). 8. Harker A. H. and Larkins F. P., J. Phys. C, Solid Sr. Phys. 12, 2487, 2497 (1979). 9. Harker A. H. and Lyon S. B., Hanvell Rep., AERE, R8S98 (1979). 10. Pople J. A. and Beveridge D.L., Approximate Molecrtiar Orbifal Theory, McGraw-Hill, New York (1970). 11. Sahu S. N., Borenstein J. T., Singh V. A. and Corbett J. W., Phys. Status Solidi B, 122, 661 (1984). 12. Messmer R. P., Chem. Phys. L&r. 11, 589 (1971). 13. Newman K. E. and Dow J. D., Phys. Rev. B, 30, 1929 (1984). 14. Wang C. S. and Klein B. M., Phys. Rev. B, 24, 3393 (1981). IS. Smith P. V., Szymanski J. E. and Matthew J. A. D., Phil. Mug. B, 51, 193 (1985). 16. Bailar J. C., Emeleus H. J., Nyholm Sir R. and Trotman-Dickenson A. F., Comprehensive Inorganic Chemistry, Vol. 1, p. 1381; Vol. 2, p. 23. Pergamon Press, Oxford. 17. Khoo G. S. and Ong C. K., J. Phys. C, Solid St. Phys. 20, 1385 (1987). 18. Cardona M. and Pollak F. H., Phys. Rev. 142, 530 (1966). 19. Kittel C., Introduction to Solid Stare Physics, 3 1, 74, 85 (1976).