- Email: [email protected]
Structural and electronic properties of molten semimetals: An ab initio study for liquid antimony
,OURNAL
OF
NON-CRYSTALIdNESO Journal
of Non-Crystalline
Solids 205-207
(1996)
871-874
Structural and electronic properties of molten semimetals: An ab initio study for liquid antimony L
K. Seifert, J. Hafner *, G. Kresse Institutfiir
Theoretische
Physik,
Technische
Universitiit
Wien, Wiednev HauptstraPe
S-10, A-1040
Vienna, Austria
Abstract The structural and electronic properties of the crystalline semimetals As and Sb may be understood in terms of a Peierls distortion from the simple cubic structure bonds: as the (pp~)-band is exactly half-filled, the cubic structure is unstable against dimerization dividing the six nearest-neighbour bonds into three strong and three weak bonds and opening a deep gap in the electronic density of states at the Fermi level. On the basis of neutron-diffraction experiments it has been suggested that some of the anomalous structural and electronic properties survive in the liquid phase. Here, an ab initio investigation of the structural, electronic, and dynamic properties of molten Sb that is based on the full set of quantum-many-body forces is presented. The results confirm the picture of a Peierls-distortion in the liquid.
1. Introduction The analysis of the crystal structures of the groupV elements shows an interesting trend from insulating molecular (N,) structures in nitrogen, over coexisting molecular (P,), ‘polymeric’ and semimetallic structures in P, to rhombohedral structures in As, Sb, and Bi with a decreasing deviation from the simple cubic limit as the atomic number increases. The correct description of the transition from the molecular to semimetallic phases goes to the limits of local-density-functional (LDF) theory, because it is less accurate in the atomic and molecular limits than for condensed phases [1,2]. The structural variations of the heavier elements, however, such be well describable by LDF methods. It has been suggested that the rhombohedral A7-structures of As, Sb, and Bi and the orthorhombic structures of P and As
* Corresponding author. Tel.: +43-l 5880 15676; 586 7760; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright PZI SOO22-3093(96)00473-5
0 1996 Elsevier
fax:
Science
+43-l
originate from a Peierls-distortion of a simple cubic lattice [3]. This has been confirmed by recent ab initio LDF calculations that demonstrate that the precise amount of the distortion, its variation under pressure, and the pressure-induced phase transitions are correctly predicted [4]. There is also good evidence from diffraction experiments [5,6,8] that the liquid structures follow a trend similar to that in the crystal structures. The classical argument for the occurrence of a Peierls distortion requires periodicity. On the basis of semiempirical tight-binding calculations in combination with Monte Carlo simulations [9,7] it has been argued that lattice periodicity is not a necessary conditions for the occurrence of a Peierls distortion, but that purely local considerations lead to the same result. On the other hand, it has been shown that the structures of the polyvalent liquid elements may be explained in terms of the modulation of the denserandom-packing (DRP) structure by the mediumrange Friedel oscillations in the effective interatomic
B.V. All rights reserved.
872
K.
Seifert et al,/Joumal
of Non-Crystalline
potential [lo,1 11. It turns out that the Peierls-distortion argument and the Friedel-modulation picture are different descriptions of the same physical effect. The pair-potential modelling has also been extended to treat dynamical properties and it has been suggested that electronically driven soft modes might exist in the vibrational spectrum [12]. However, quantitative results depend quite sensitively on the parametrization of the TB- or pseudopotentialHamiltonians. Therefore detailed ab initio studies are of great interest. For liquid phosphorus[13] and arsenic [14] ab initio calculations using the Car-Parrinello approach have essentially confirmed the Peierls-distortion-Friedel-modulation scenario, except for details depending on the true covalent nature of the bond. These materials remain narrow-gap semiconductors even in the liquid state. Interesting questions remain for the heavier elementsSb and Bi where the distortions from the simplecubic structure are smaller and which are close to a metal/non-metal transition.
2. Ab initio LDF molecular dynamics on the Born-Oppenheimer surface In our calculations we have used the Vienna ab initio simulation program VASP based on the following principles [151. (1) We use the finite-temperature version of the LDF theory. At finite temperature the free energy F[ n(r), fi, ~1 depending on the electron density n(r), the Fermi-Dirac occupation probability J of the one-electron states $i(r) (n(r) = l&ifil$f(r>/2>, and the chemical potential y, is the proper variational functional. However, even at finite temperature, the proper LDF force is equal to the Hellmann-Feynman force. Instead of the Fermi-Dirat broadening of the one-electron energiesit may be computationally convenient to use a Gaussianbroadening instead. (2) The minimization of the total energy (respectively, the total free energy) is performed using an efficient matrix diagonalization scheme based on conjugate-gradient or residual-minimization techniques [15] and efficient proceduresfor charge-density mixing. (3) The atomic motion is described using NosCdynamics generating a canonical ensemble.
Solids 205-207
(1996)
871-874
(4) After moving the atoms, the new wavefunctions are estimated using a subspace alignment scheme. (5) The electron-ion interaction is describedby optimized ultrasoft pseudopotentials[ 161.For all further details we refer to Refs. [14,15].
3. Properties of liquid antimony Our ab initio simulation for 1-Sb were performed for an ensemble of N = 64 atoms at a density of p = 6.4! g crnm3(th e ed,De of the MD cell measures 12.59 A) at a temperature of T= 1073 K. The ultrasoft pseudopotential for Sb has been described in our paper on the crystalline phases[4]. Forces are calculated at the r-point only, with a Gaussian smearing of CT= 0.1 eV. For the calculation of the electronic density of states a finer grid of special k-points has been used. The time-step for the MD simulationswas A.t = 3 fs. The simulationwas started from an initial configuration generated via classical MD and effective pair potentials. The simulation run was extended over 8.1 ps. 3.1. Liquid
structure
Fig. 1 and Fig. 2 present our results for the pair correlation function g(R) and the static structure factor S(Q), comparedto the experimental data. The characteristic features of the structure factor are the weak side-maximum to the right of the main peak
1.5 ci m
1,o 0.5
0.0s 0
5
10
Q (A-‘, Fig. 1. Static structure factor S(Q) for liquid Sb at T= 1072 K. Histogram - ab initio MD, squares - neutron diffraction data of Gaspard et al. [S].
K. Seifert et al. / Journal
of Non-Crystalline
(which is much stronger in I-As), and the ratio of the positions of the first two principal peaks, Q2/Ql - 2, which is larger than the value characteristic for dense-random-packing fluids, but smaller than for l-As. The calculated structure factor is in excellent agreement with the neutron-diffraction data of Gaspard et al. 181, even for the details of the higher-order oscillations. We also note a good agreement with the results of the pair-potential $mulations [ll]. The side-peak occurs at $ = 3.2 A-‘, i.e., close to the diameter 2k, = 3.36 A-’ of the Fermi-sphere. This is a clear indication that the structural anomaly is electronically driven. In the pair correlation function we find a small shoulder at the right-hand side of the main peak (which is a remnant of a true peak in I-As), and again a ratio RJR, = 1.98, smaller than in l-As (RJR1 = 2.281, but larger than in DRP liquids (RJR1 = 1.75). Again the agreement with the experimental data of Waseda [5] is good, except for a small difference in the position of the first peak. We think that the discrepancy is due partly to the notorious tendency of the LDF theory to slightly underestimate the bond-length [4], but partly also to truncation errors in the experimental g(R) arising from the limited Q-range of Waseda’s diffraction experiment. With the given form of the pair correlation function, it is difficult to determine a meaningful value for the coordination number. The best approach is to study the variation of the coordination number NC
Fig. 2. Pair correlation function g(R) for liquid Sb. Full line - ab initio MD at T = 1072 K, circles - X-ray diffraction data of Waseda [5] at T = 932 K.
Solids 205-207
(1996)
871-874
0
50
873
100 0 [deg)
150
Fig. 3. Bond-angle distribution function f(8) = ga(B, R,,,) for varying maximum bond-length: (a) R,,, = 3.40 A, (b) R,,, = 4.25 A, (c) R,,, = 5.10 A. (cf. text.)
(defined as the integral over g(R) up to R,,,) and of the distribution of the bond-angles in nearestneigbbour triplets as a function of R,,,. For R,,, = 3.4 A (i.e., including essentially the symmetric part of the first peak), we find NC = 4.73 and a bond-angle distribution with broad peaks at 9 N 90” and 8 N 180” (see Fig. 3). This means that the nearestneighbour configuration is that corresponding to a weak Peierls distortion with only a gradual difference between the short and the long (pp~) bonds, a coordination number halfway between 3 (A7 structure) and 6 (simple cubic), and bond angles representing almost rectangular atd collinear triplet-bonds. Increasing R,,, to 4.25 A (i.e., just beyond the shoulder in g(R)) leads to NC N 10 and peaks in f(9) close to %- 60” and 9 N 90”. The coordination number is roughly equal to the sum of the nearestand next-nearest-neighbour coordination numbers in the A7 structure NC = 3 + 6, the N 60” bond-angles correspond to the angle between next-nearestneighbour bonds in a weakly Peierls-distorted simple-cubic structure. Extending R,,, to 5.10 A brings the coordination number to NC = 16.8, close to the sum over the first two coordination shells in the simple cubic structure. Our interpretation of the structure of liquid Sb differs slightly from that given by Bichara et al. [7] on the basis of their semi-empirical tight-binding simulations. They concluded that no Peierls distortion exists in 1-Sb. However, our calculations show clearly, that although these distortions are distinctly smaller than in I-As, they are still present in the case of 1-Sb.
874
K. Seifert et al./Joumal
3.2. Liquid
of Non-Crystalline
Preliminary results for the dynamical propertiesof 1-Sb (velocity auto-correlation function, vibrational frequency spectrum,dynamical structure factor) indicate substantial differences to those of ‘normal’ liquid metals: lesspronouncedoscillations in the velocity auto-correlation function (i.e., a weaker ‘cage effect’), only a weak indication of an inelastic peak in the frequency spectrum associatedwith existence of collective excitations, and ‘soft modes’ in the dynamical structure factor induced by electronic effects (similar to those predicted by calculations on the basis of effective pair interactions [12]). Details will be reported elsewhere after extending the MD runs. structure
The electronic density of statesfor I-Sb shown in Fig. 4 is very similar to that of the crystalline phase [4] and to the earlier results obtained for the pairpotential-generated structure [ll] (we also refer to this work for a detailed comparison with photoelectron spectroscopy). The lower part of the valence-band is of s-character, the Peierls-splitting of the p-bandsat the Fermi level is indicated by a weak DOS minimum. The most important difference is the filling of the narrow (about 0.7 eV wide at half-depth) pseudogapat the Fermi-level. In the crystal, the DOS is almost zero in the pseudogap,whereasin the liquid, only a slight depressionsubsistsat the Fermilevel. However, the negative slope of the DOS is sufficient to causeelectronic transport properties that
G Ir E IeVl Fig. 4. Electronic initio LDF-MD.
density
(1996)
871-874
are characteristically different from normal polyvalent metals.
dynamics
3.3. Electrorzic
Solids 205-207
of states &!Z) for I-Sb calculated
via ab
4. Conclusions We have analyzed the structural, dynamical and electronic properties of the molten semimetal Sb using ab initio LDF molecular dynamics on the Born-Oppenheimer surface. Our results show that the structural anomaliesof 1-Sbmay be interpreted in terms of a weak Peierls distortion of the (ppa)bonded network and confirm earlier interpretations given on the basisof effective pair-forces with shortand medium-rangeFriedel oscillations.
Acknowledgements This work has been supported by the Austrian Science Foundation under project No. P10445-PHYS within the framework of COST-Project D6. References Ill L. 121G.
Mitas and R.M. Martin, Phys. Rev. Lett. 72 (1994) 2438. Kern, J. Hafner and G. Kresse, to be published. Crit. Rev. Solid State Mater. Sci. 11 (1983) E33 P.B. Littlewood, 229. Nl K. Seifert. J. Hafner, G. Kresse and J. Furthmiiller, J. Phys.: Condens. Matter 7 (1995) 3683. Materials I51Y. Waseda, The Structures on Non-Crystalline Liquids and Amorphous Solids (McGraw-Hill, New York, 1981). 161R. Bellissent, C. Bergman, R. Ceolin ‘and J.P. Gaspard, Phys. Rev. Lett. 59 (1987) 661. t71 C. Bichara, A. Pellegatti and J.P. Gaspard, Phys. Rev. B47 (1993) 5002. IS1 J.P. Gaspard, R. Bellissent, A. Menelle, C. Bergman and R. Ceolin, Nuovo Cim. D12 (1990) 650. and A. Pellegatti, Europhys. Lett. 191 J.P. Gaspard, F. Marrinelli 3 (1987) 1095. DOI J. Hafner, Phys. Rev. Lett. 62 (1989) 784. J. Hafner and W. Jank, Phys. Rev. B45 (1992) 2739. 1111 1121Ya. Chushak, J. Hafner and G. Kahl, Phys, Chem. Liq. 29 (1995) 159. [I31 D. Hohl and R.O. Jones, Phys. Rev. B4.5 (1992) 8995; Phys. Rev. BSO (1994) 17047. Cl41 X.P. Li, P.B. Allen, R. Car, M. Parrinello and J.Q. Broughton, Phys. Rev. B41 (1990) 8392. IN G. Kresse and J. Hafner, Phys. Rev. B49 (1994) 14251. 1161G. Kresse and J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245.
OF
NON-CRYSTALIdNESO Journal
of Non-Crystalline
Solids 205-207
(1996)
871-874
Structural and electronic properties of molten semimetals: An ab initio study for liquid antimony L
K. Seifert, J. Hafner *, G. Kresse Institutfiir
Theoretische
Physik,
Technische
Universitiit
Wien, Wiednev HauptstraPe
S-10, A-1040
Vienna, Austria
Abstract The structural and electronic properties of the crystalline semimetals As and Sb may be understood in terms of a Peierls distortion from the simple cubic structure bonds: as the (pp~)-band is exactly half-filled, the cubic structure is unstable against dimerization dividing the six nearest-neighbour bonds into three strong and three weak bonds and opening a deep gap in the electronic density of states at the Fermi level. On the basis of neutron-diffraction experiments it has been suggested that some of the anomalous structural and electronic properties survive in the liquid phase. Here, an ab initio investigation of the structural, electronic, and dynamic properties of molten Sb that is based on the full set of quantum-many-body forces is presented. The results confirm the picture of a Peierls-distortion in the liquid.
1. Introduction The analysis of the crystal structures of the groupV elements shows an interesting trend from insulating molecular (N,) structures in nitrogen, over coexisting molecular (P,), ‘polymeric’ and semimetallic structures in P, to rhombohedral structures in As, Sb, and Bi with a decreasing deviation from the simple cubic limit as the atomic number increases. The correct description of the transition from the molecular to semimetallic phases goes to the limits of local-density-functional (LDF) theory, because it is less accurate in the atomic and molecular limits than for condensed phases [1,2]. The structural variations of the heavier elements, however, such be well describable by LDF methods. It has been suggested that the rhombohedral A7-structures of As, Sb, and Bi and the orthorhombic structures of P and As
* Corresponding author. Tel.: +43-l 5880 15676; 586 7760; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright PZI SOO22-3093(96)00473-5
0 1996 Elsevier
fax:
Science
+43-l
originate from a Peierls-distortion of a simple cubic lattice [3]. This has been confirmed by recent ab initio LDF calculations that demonstrate that the precise amount of the distortion, its variation under pressure, and the pressure-induced phase transitions are correctly predicted [4]. There is also good evidence from diffraction experiments [5,6,8] that the liquid structures follow a trend similar to that in the crystal structures. The classical argument for the occurrence of a Peierls distortion requires periodicity. On the basis of semiempirical tight-binding calculations in combination with Monte Carlo simulations [9,7] it has been argued that lattice periodicity is not a necessary conditions for the occurrence of a Peierls distortion, but that purely local considerations lead to the same result. On the other hand, it has been shown that the structures of the polyvalent liquid elements may be explained in terms of the modulation of the denserandom-packing (DRP) structure by the mediumrange Friedel oscillations in the effective interatomic
B.V. All rights reserved.
872
K.
Seifert et al,/Joumal
of Non-Crystalline
potential [lo,1 11. It turns out that the Peierls-distortion argument and the Friedel-modulation picture are different descriptions of the same physical effect. The pair-potential modelling has also been extended to treat dynamical properties and it has been suggested that electronically driven soft modes might exist in the vibrational spectrum [12]. However, quantitative results depend quite sensitively on the parametrization of the TB- or pseudopotentialHamiltonians. Therefore detailed ab initio studies are of great interest. For liquid phosphorus[13] and arsenic [14] ab initio calculations using the Car-Parrinello approach have essentially confirmed the Peierls-distortion-Friedel-modulation scenario, except for details depending on the true covalent nature of the bond. These materials remain narrow-gap semiconductors even in the liquid state. Interesting questions remain for the heavier elementsSb and Bi where the distortions from the simplecubic structure are smaller and which are close to a metal/non-metal transition.
2. Ab initio LDF molecular dynamics on the Born-Oppenheimer surface In our calculations we have used the Vienna ab initio simulation program VASP based on the following principles [151. (1) We use the finite-temperature version of the LDF theory. At finite temperature the free energy F[ n(r), fi, ~1 depending on the electron density n(r), the Fermi-Dirac occupation probability J of the one-electron states $i(r) (n(r) = l&ifil$f(r>/2>, and the chemical potential y, is the proper variational functional. However, even at finite temperature, the proper LDF force is equal to the Hellmann-Feynman force. Instead of the Fermi-Dirat broadening of the one-electron energiesit may be computationally convenient to use a Gaussianbroadening instead. (2) The minimization of the total energy (respectively, the total free energy) is performed using an efficient matrix diagonalization scheme based on conjugate-gradient or residual-minimization techniques [15] and efficient proceduresfor charge-density mixing. (3) The atomic motion is described using NosCdynamics generating a canonical ensemble.
Solids 205-207
(1996)
871-874
(4) After moving the atoms, the new wavefunctions are estimated using a subspace alignment scheme. (5) The electron-ion interaction is describedby optimized ultrasoft pseudopotentials[ 161.For all further details we refer to Refs. [14,15].
3. Properties of liquid antimony Our ab initio simulation for 1-Sb were performed for an ensemble of N = 64 atoms at a density of p = 6.4! g crnm3(th e ed,De of the MD cell measures 12.59 A) at a temperature of T= 1073 K. The ultrasoft pseudopotential for Sb has been described in our paper on the crystalline phases[4]. Forces are calculated at the r-point only, with a Gaussian smearing of CT= 0.1 eV. For the calculation of the electronic density of states a finer grid of special k-points has been used. The time-step for the MD simulationswas A.t = 3 fs. The simulationwas started from an initial configuration generated via classical MD and effective pair potentials. The simulation run was extended over 8.1 ps. 3.1. Liquid
structure
Fig. 1 and Fig. 2 present our results for the pair correlation function g(R) and the static structure factor S(Q), comparedto the experimental data. The characteristic features of the structure factor are the weak side-maximum to the right of the main peak
1.5 ci m
1,o 0.5
0.0s 0
5
10
Q (A-‘, Fig. 1. Static structure factor S(Q) for liquid Sb at T= 1072 K. Histogram - ab initio MD, squares - neutron diffraction data of Gaspard et al. [S].
K. Seifert et al. / Journal
of Non-Crystalline
(which is much stronger in I-As), and the ratio of the positions of the first two principal peaks, Q2/Ql - 2, which is larger than the value characteristic for dense-random-packing fluids, but smaller than for l-As. The calculated structure factor is in excellent agreement with the neutron-diffraction data of Gaspard et al. 181, even for the details of the higher-order oscillations. We also note a good agreement with the results of the pair-potential $mulations [ll]. The side-peak occurs at $ = 3.2 A-‘, i.e., close to the diameter 2k, = 3.36 A-’ of the Fermi-sphere. This is a clear indication that the structural anomaly is electronically driven. In the pair correlation function we find a small shoulder at the right-hand side of the main peak (which is a remnant of a true peak in I-As), and again a ratio RJR, = 1.98, smaller than in l-As (RJR1 = 2.281, but larger than in DRP liquids (RJR1 = 1.75). Again the agreement with the experimental data of Waseda [5] is good, except for a small difference in the position of the first peak. We think that the discrepancy is due partly to the notorious tendency of the LDF theory to slightly underestimate the bond-length [4], but partly also to truncation errors in the experimental g(R) arising from the limited Q-range of Waseda’s diffraction experiment. With the given form of the pair correlation function, it is difficult to determine a meaningful value for the coordination number. The best approach is to study the variation of the coordination number NC
Fig. 2. Pair correlation function g(R) for liquid Sb. Full line - ab initio MD at T = 1072 K, circles - X-ray diffraction data of Waseda [5] at T = 932 K.
Solids 205-207
(1996)
871-874
0
50
873
100 0 [deg)
150
Fig. 3. Bond-angle distribution function f(8) = ga(B, R,,,) for varying maximum bond-length: (a) R,,, = 3.40 A, (b) R,,, = 4.25 A, (c) R,,, = 5.10 A. (cf. text.)
(defined as the integral over g(R) up to R,,,) and of the distribution of the bond-angles in nearestneigbbour triplets as a function of R,,,. For R,,, = 3.4 A (i.e., including essentially the symmetric part of the first peak), we find NC = 4.73 and a bond-angle distribution with broad peaks at 9 N 90” and 8 N 180” (see Fig. 3). This means that the nearestneighbour configuration is that corresponding to a weak Peierls distortion with only a gradual difference between the short and the long (pp~) bonds, a coordination number halfway between 3 (A7 structure) and 6 (simple cubic), and bond angles representing almost rectangular atd collinear triplet-bonds. Increasing R,,, to 4.25 A (i.e., just beyond the shoulder in g(R)) leads to NC N 10 and peaks in f(9) close to %- 60” and 9 N 90”. The coordination number is roughly equal to the sum of the nearestand next-nearest-neighbour coordination numbers in the A7 structure NC = 3 + 6, the N 60” bond-angles correspond to the angle between next-nearestneighbour bonds in a weakly Peierls-distorted simple-cubic structure. Extending R,,, to 5.10 A brings the coordination number to NC = 16.8, close to the sum over the first two coordination shells in the simple cubic structure. Our interpretation of the structure of liquid Sb differs slightly from that given by Bichara et al. [7] on the basis of their semi-empirical tight-binding simulations. They concluded that no Peierls distortion exists in 1-Sb. However, our calculations show clearly, that although these distortions are distinctly smaller than in I-As, they are still present in the case of 1-Sb.
874
K. Seifert et al./Joumal
3.2. Liquid
of Non-Crystalline
Preliminary results for the dynamical propertiesof 1-Sb (velocity auto-correlation function, vibrational frequency spectrum,dynamical structure factor) indicate substantial differences to those of ‘normal’ liquid metals: lesspronouncedoscillations in the velocity auto-correlation function (i.e., a weaker ‘cage effect’), only a weak indication of an inelastic peak in the frequency spectrum associatedwith existence of collective excitations, and ‘soft modes’ in the dynamical structure factor induced by electronic effects (similar to those predicted by calculations on the basis of effective pair interactions [12]). Details will be reported elsewhere after extending the MD runs. structure
The electronic density of statesfor I-Sb shown in Fig. 4 is very similar to that of the crystalline phase [4] and to the earlier results obtained for the pairpotential-generated structure [ll] (we also refer to this work for a detailed comparison with photoelectron spectroscopy). The lower part of the valence-band is of s-character, the Peierls-splitting of the p-bandsat the Fermi level is indicated by a weak DOS minimum. The most important difference is the filling of the narrow (about 0.7 eV wide at half-depth) pseudogapat the Fermi-level. In the crystal, the DOS is almost zero in the pseudogap,whereasin the liquid, only a slight depressionsubsistsat the Fermilevel. However, the negative slope of the DOS is sufficient to causeelectronic transport properties that
G Ir E IeVl Fig. 4. Electronic initio LDF-MD.
density
(1996)
871-874
are characteristically different from normal polyvalent metals.
dynamics
3.3. Electrorzic
Solids 205-207
of states &!Z) for I-Sb calculated
via ab
4. Conclusions We have analyzed the structural, dynamical and electronic properties of the molten semimetal Sb using ab initio LDF molecular dynamics on the Born-Oppenheimer surface. Our results show that the structural anomaliesof 1-Sbmay be interpreted in terms of a weak Peierls distortion of the (ppa)bonded network and confirm earlier interpretations given on the basisof effective pair-forces with shortand medium-rangeFriedel oscillations.
Acknowledgements This work has been supported by the Austrian Science Foundation under project No. P10445-PHYS within the framework of COST-Project D6. References Ill L. 121G.
Mitas and R.M. Martin, Phys. Rev. Lett. 72 (1994) 2438. Kern, J. Hafner and G. Kresse, to be published. Crit. Rev. Solid State Mater. Sci. 11 (1983) E33 P.B. Littlewood, 229. Nl K. Seifert. J. Hafner, G. Kresse and J. Furthmiiller, J. Phys.: Condens. Matter 7 (1995) 3683. Materials I51Y. Waseda, The Structures on Non-Crystalline Liquids and Amorphous Solids (McGraw-Hill, New York, 1981). 161R. Bellissent, C. Bergman, R. Ceolin ‘and J.P. Gaspard, Phys. Rev. Lett. 59 (1987) 661. t71 C. Bichara, A. Pellegatti and J.P. Gaspard, Phys. Rev. B47 (1993) 5002. IS1 J.P. Gaspard, R. Bellissent, A. Menelle, C. Bergman and R. Ceolin, Nuovo Cim. D12 (1990) 650. and A. Pellegatti, Europhys. Lett. 191 J.P. Gaspard, F. Marrinelli 3 (1987) 1095. DOI J. Hafner, Phys. Rev. Lett. 62 (1989) 784. J. Hafner and W. Jank, Phys. Rev. B45 (1992) 2739. 1111 1121Ya. Chushak, J. Hafner and G. Kahl, Phys, Chem. Liq. 29 (1995) 159. [I31 D. Hohl and R.O. Jones, Phys. Rev. B4.5 (1992) 8995; Phys. Rev. BSO (1994) 17047. Cl41 X.P. Li, P.B. Allen, R. Car, M. Parrinello and J.Q. Broughton, Phys. Rev. B41 (1990) 8392. IN G. Kresse and J. Hafner, Phys. Rev. B49 (1994) 14251. 1161G. Kresse and J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245.