- Email: [email protected]
Molecular structure and dynamical properties of C36: a semi-empirical calculation
1 January 1999
Chemical Physics Letters 299 Ž1999. 64–68
Molecular structure and dynamical properties of C 36 : a semi-empirical calculation E. Halac, E. Burgos, H. Bonadeo
)
Departamento de Fısica, Comision AÕenida Gral. Paz 1499, (1650) San Martın, ´ ´ Nacional de Energıa ´ Atomica, ´ ´ Pcia. de Buenos Aires, Argentina Received 4 September 1998; in final form 27 October 1998
Abstract We have explored possible molecular structures and calculated the vibrational properties of the newly synthesized C 36 fullerene. We used a semi-empirical covalent potential that has been shown to reproduce these properties well in C 60 , C 70 , diamond and graphite. In agreement with ab initio calculations, we find that D6h and D 2d structures are the most stable. Using our semi-empirical potential we obtain infrared frequencies in the range of those observed for the crystal powder. We also calculate the Raman spectrum on the basis of the band polarizability model. q 1999 Elsevier Science B.V. All rights reserved.
The stability of different structures for molecules containing 36 carbon atoms was studied recently by Grossman et al. w1x, who used ab initio methods to obtain their total energies. They found that two carbon cages with 8 hexagonal and 12 pentagonal faces, one with D6h and the other with D 2d symmetry, were the most stable, with bond lengths of the order of those of the by now well studied C 60 and C 70 . Shortly after this, a C 36 sample was synthesized and isolated for the first time w2x. The available data on its physical properties are still scarce: NMR spectra favor a D6h symmetry for the molecule and a mid-infrared ŽIR. spectrum of the powdered crystal shows a number of broad features in the 400 Žthe instrumental lower limit. to 1800 cmy1 frequency range. Electron diffraction patterns indicate a hexag-
)
Corresponding author. E-mail: [email protected]
onal symmetry for the crystal, with very short Žf 1.7 ˚ . intermolecular carbon–carbon distances. Cote A ˆ ´ et al. w3x, in an ab initio calculation, also find that a trigonal C 36 crystal shows covalent bonding between carbon atoms belonging to different molecules, which ˚ apart. are only 1.56 A Although ab initio calculations are becoming more accessible, and the complexity of the systems which can be approached has greatly increased in the last years, the much simpler semi-empirical methods retain their usefulness for calculations on complex systems, especially in the case of the dynamic properties: ab initio phonon or molecular dynamics calculations are still confined to systems of relatively few atoms. One of the most widely used semi-empirical potentials for covalent carbon systems was proposed by Tersoff w4x; it was fitted to, and describes well, a series of structural and energetic properties of these materials. However, their dynamical properties are poorly reproduced. In a recent work, we have shown
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 2 2 6 - 3
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
65
and
°
ri j - R min ,
1,
~ 12 q 12 cos ŽpRŽ r
f c Ž ri j . s
i j y R min
.
ma x y R min .
¢
,
ri j ) R max .
0,
bisym j s
zi j s
R min - ri j - R max ,
1
½
bi j s 1 q Ž az i j .
Ž b q bji . , 2 ij
l
yn
5
,
f c Ž ri k .
Ý k/ Ž i , j .
½
= 1q
Fig. 1. Structure of D6h C 36 . The three groups of non-equivalent atoms are indicated by different fillings.
tor 2 E s Ý f c Ž ri j . VR Ž ri j . y bisym j VA Ž r i j . q Vi j 4
i/j
with VR Ž ri j . s VA Ž ri j . s
De
Ž S y 1. SDe
Ž S y 1.
exp yb'2 S Ž ri j y re .
½
½ '
5
exp yb 2rS Ž ri j y re .
5
d2
c2 q d 2 q h y cos u i jk 4
2
5
.
Here u i jk is the angle formed by bonds ij and ik; the torsional term is Vi tor j st
that, regardless of the parameters used, a potential of the form used by Tersoff – or a similar one proposed by Brenner w5x – cannot reproduce correctly the observed vibrational properties of graphite. We presented a modification to this form, by adding a torsional term, and refined the parameters to fit structural and dynamical properties of graphite and diamond. The potential thus obtained not only reproduces these properties well but predicts accurately the – few – observed properties of Lonsdaleite, gives values in reasonable agreement with ab initio calculations for hypothetical carbon structures, and, most important for the purposes of this work, yields structural and dynamical data for C 60 and C 70 which agree remarkably well with observed data w6x. In this work, we apply our potential to the calculation of the statical and dynamical properties of C 36 . The energy for each pair of bonded atoms i,j is written as
c2
f c Ž ri k . f c Ž r jl . Tk i jl
Ý Ž k ,l ./ Ž i , j .
with k / l and Tk i jl s
™ ™ i k = ri j
žr
™ ™ jl = r ji
/ Pžr
ri k ri j r jt r ji
/.
The parameters used in the present calculation, taken ˚ . s 1.3883; from Ref. w6x are: DeŽeV. s 6.362; reŽA ˚ . s 1.8; R max ŽA˚ . s 2.1; b ŽA˚ y1 . s 1.638; R minŽA S s 1.577; a s 1.5724 = 10y7 ; l s 0.891; n s 0.687276; c s 38049; d s 4.745; h s y0.7171; t ŽeV. s y0.208. There seem to be strong indications pointing to a D6h symmetry for the C 36 molecule w2x. Besides the NMR results, a simple hexagonal crystal is incompatible with a molecular structure like D 2d , which lacks a 3- or 6-fold axis. However, we have searched
Table 1 Comparison between the energy differences Žin eV. for two C 36 isomers relative to the D6h fullerene, calculated in this work, and using a density functional approach and the LDA and GGA approximations, taken from Ref. w1x
This work Ref. w1x
D6h
D 2d
D 3h
0.00 0.0
0.14 0.0
1.17 1.4 ŽLDA.; 1.8 ŽGGA.
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
66
Table 2 ˚ . in D6h C 36 , calculated for the totally relaxed Bond distances ŽA structures in this work and in Ref. w1x. Bonded atoms as defined in Fig. 1 Bonded atoms
C1–C1
C1–C2
C2–C3
C3–C3
This work Ref. w1x
1.46 1.41
1.49 1.48
1.47 1.43
1.43 1.43
different structures, as an application of our new interaction potential. Here we present calculations based on D6h , D 2d , and D 3h symmetries; the first two were found to have the lowest energy in ab initio calculations w1x, and the last one has a 3-fold axis. Fig. 1 shows the D6h structure; C1, C2 and C3 are three symmetry-independent atoms. We find an average energy per atom for the D6h isomer of y7.41 eV; this result is reasonable, compared with the y7.86 eV we obtained for C 60 , and y8.56 eV for graphite or diamond w6x, using the same potential and calculation method. Table 1 shows the energy differences, relative to the D6h structure, of the three isomers under study, calculated in this work and in Ref. w1x. It can be seen that the results are quite similar, although in our case D6h has a lower energy than D 2d . The four distinct bond distances in the D6h structure are also compared to those of Ref. w1x in Table 2; both patterns are similar, with differences in the details. Table 3 shows the coordinates of the independent atoms; we have chosen z 5C 6 ; the planes sÕ contain two C1 carbon atoms of the top and bottom hexagons of the structure Žsee Fig. 1.. Unfortunately comparison with Ref. w1x is not possible, since these data were not reported, and the four interatomic distances do not determine the seven independent coordinates. For the D6h structure, the normal mode classification of C 36 is: G s 6A 1g q 2A 2g q 4B1g q 5B 2g q 8E 1g q 9E 2g q 3A 1u q 5A 2u q 5B1u q 4B 2u q 8E 1u q9E 2u . The 13 IR active modes are A 2uŽ z . and E 1u Ž x, y ., and the 23 Raman active ones are A 1g Ž xx Table 3 ˚ . of independent atoms of D6h C 36 Coordinates Žin A Atom
x
y
z
C1 C2 C3
1.458 2.366 2.204
0 0 1.272
2.615 1.432 0.714
Fig. 2. Calculated vibrational frequencies of C 36 . Degenerate modes are indicated by bars of height 2.
q yy, zz ., E 1g Ž yz, xz . and E 2g Ž xx y yy, xy .. Fig. 2 shows the positions of the calculated normal mode frequencies; degenerate modes are indicated by lines of double height. In Table 4 we list the calculated optically active frequencies, and their symmetries. The only vibrational information presently available is a mid-IR transmission spectrum of the powder, apparently obtained under difficult conditions w2x, which is shown, together with the positions of our calculated IR-active peaks in Fig. 3: superimposed to a background typical of a granular sample, the spectrum shows a series of broad features. Our highest calculated frequency lies under 1500 cmy1 ; it should be pointed out that no other fullerene-like structure has been reported to show vibrational frequencies much above 1600 cmy1 . In any case, the spectra correspond to the solid phase; if the intermolecular bonding is of at least partially covalent character, as
Table 4 Calculated Raman ŽA 1g , E 1g and E 2g . and IR ŽA 2u and E 1u . active frequencies Žin cmy1 . of C 36 A 1g
E 1g
E 2g
A 2u
E 1u
1485 1160 766 714 498 332
1454 1307 1097 873 814 701 516 325
1433 1375 1210 1034 794 735 640 551 298
1320 1022 835 614 467
1460 1336 1147 972 737 583 457 435
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
has been suggested, this may result in large changes in the vibrational spectrum from the molecule to the crystal. The bond polarizability model has been applied with success to the calculation of Raman intensities of fullerenes w7,8x. This model assigns to each type of bond a longitudinal and a transverse polarizability, or, equivalently, a mean static polarizability and an anisotropy, and assumes that these depend only on the bond length. Since the mean static polarizability does not contribute to the Raman intensity, three parameters for each type of bond are necessary to complete the model, one for the anisotropy and two for the derivatives of the polarizability and the anisotropy with respect to the bond distance. C 60 has well-defined single and double bonds; for C 70 the situation is not so clear, since there are several bond lengths. For this molecule Guha et al. w8x find that a ˚ between single and double cut-off length of 1.425 A bonds gives the best agreement between predictions and experiment. In Table 2, we see that for C 36 there are two groups of bond lengths, above and below ˚ which we chose as our cut-off length, leav1.46 A, ing us with 36 single and 18 double bonds. In Fig. 4 we show the results of our calculation using three different parameter sets; the lines are broadened using Gaussians with s s 5 cmy1 . The three sets predict similar main features for the spectrum below 1000 cmy1 , where three groups of intense bands appear. Differences are larger in the higher-frequency region: although at least two bands are always present, other features show sizable variations.
67
Fig. 4. Calculated Raman spectrum of C 36 , with different parameters for the bond polarizability model: Ža. Ref. w8x; Žb. Ref. w7x; Žc. Refs. w9,10x.
The semi-empirical potential used in this calculation has been shown to yield good agreement with the experimental features of C 60 and C 70 ; in the present work, the main structural properties of C 36 are also well reproduced. In the absence of detailed vibrational data, we have predicted the dynamical behavior of the molecule. It is interesting that the nature of C 36 crystals – at least according to the available indications – is very peculiar in that it shows intermolecular covalent bonding, whereas all other known fullerenes form van der Waals crystals; the semi-empirical approach is well suited to explore the properties of such complex crystals, although more detailed structural data are probably necessary to allow a thorough study of the problem.
Acknowledgements This work was partially supported by CONICET Grant PMT-PICT0051.
References
Fig. 3. Observed mid-IR spectrum of C 36 w2x and calculated IR-active frequencies of C 36 .
w1x J.C. Grossman, M. Cote, ˆ ´ S.G. Louie, M.L. Cohen, Chem. Phys. Lett. 284 Ž1998. 344. w2x C. Piskoti, J. Yarger, A. Zettl, Nature 393 Ž1998. 771. w3x M. Cote, ˆ ´ J.C. Grossman, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 81 Ž1998. 697.
68
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
w4x J. Tersoff, Phys. Rev. B 39 Ž1989. 5566. w5x D.W. Brenner, Mater. Res. Soc. Symp. Proc. 141 Ž1989. 59. w6x E. Burgos, E. Halac, H. Bonadeo, Chem. Phys Lett. 298 Ž1998. 273. w7x D.W. Snoke, M. Cardona, Solid State Comm. 87 Ž1993. 121.
w8x S. Guha, J. Menendez, J.B. Page, G.B. Adams, Phys. Rev. B ´ 53 Ž1996. 13106. w9x J. Martin, S. Montero, J. Chem. Phys. 80 Ž1984. 4610. w10x M.F. Orduna, A. del Olmo, C. Domingo, S. Montero, J. Mol. Struc. 142 Ž1986. 201.
Chemical Physics Letters 299 Ž1999. 64–68
Molecular structure and dynamical properties of C 36 : a semi-empirical calculation E. Halac, E. Burgos, H. Bonadeo
)
Departamento de Fısica, Comision AÕenida Gral. Paz 1499, (1650) San Martın, ´ ´ Nacional de Energıa ´ Atomica, ´ ´ Pcia. de Buenos Aires, Argentina Received 4 September 1998; in final form 27 October 1998
Abstract We have explored possible molecular structures and calculated the vibrational properties of the newly synthesized C 36 fullerene. We used a semi-empirical covalent potential that has been shown to reproduce these properties well in C 60 , C 70 , diamond and graphite. In agreement with ab initio calculations, we find that D6h and D 2d structures are the most stable. Using our semi-empirical potential we obtain infrared frequencies in the range of those observed for the crystal powder. We also calculate the Raman spectrum on the basis of the band polarizability model. q 1999 Elsevier Science B.V. All rights reserved.
The stability of different structures for molecules containing 36 carbon atoms was studied recently by Grossman et al. w1x, who used ab initio methods to obtain their total energies. They found that two carbon cages with 8 hexagonal and 12 pentagonal faces, one with D6h and the other with D 2d symmetry, were the most stable, with bond lengths of the order of those of the by now well studied C 60 and C 70 . Shortly after this, a C 36 sample was synthesized and isolated for the first time w2x. The available data on its physical properties are still scarce: NMR spectra favor a D6h symmetry for the molecule and a mid-infrared ŽIR. spectrum of the powdered crystal shows a number of broad features in the 400 Žthe instrumental lower limit. to 1800 cmy1 frequency range. Electron diffraction patterns indicate a hexag-
)
Corresponding author. E-mail: [email protected]
onal symmetry for the crystal, with very short Žf 1.7 ˚ . intermolecular carbon–carbon distances. Cote A ˆ ´ et al. w3x, in an ab initio calculation, also find that a trigonal C 36 crystal shows covalent bonding between carbon atoms belonging to different molecules, which ˚ apart. are only 1.56 A Although ab initio calculations are becoming more accessible, and the complexity of the systems which can be approached has greatly increased in the last years, the much simpler semi-empirical methods retain their usefulness for calculations on complex systems, especially in the case of the dynamic properties: ab initio phonon or molecular dynamics calculations are still confined to systems of relatively few atoms. One of the most widely used semi-empirical potentials for covalent carbon systems was proposed by Tersoff w4x; it was fitted to, and describes well, a series of structural and energetic properties of these materials. However, their dynamical properties are poorly reproduced. In a recent work, we have shown
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 2 2 6 - 3
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
65
and
°
ri j - R min ,
1,
~ 12 q 12 cos ŽpRŽ r
f c Ž ri j . s
i j y R min
.
ma x y R min .
¢
,
ri j ) R max .
0,
bisym j s
zi j s
R min - ri j - R max ,
1
½
bi j s 1 q Ž az i j .
Ž b q bji . , 2 ij
l
yn
5
,
f c Ž ri k .
Ý k/ Ž i , j .
½
= 1q
Fig. 1. Structure of D6h C 36 . The three groups of non-equivalent atoms are indicated by different fillings.
tor 2 E s Ý f c Ž ri j . VR Ž ri j . y bisym j VA Ž r i j . q Vi j 4
i/j
with VR Ž ri j . s VA Ž ri j . s
De
Ž S y 1. SDe
Ž S y 1.
exp yb'2 S Ž ri j y re .
½
½ '
5
exp yb 2rS Ž ri j y re .
5
d2
c2 q d 2 q h y cos u i jk 4
2
5
.
Here u i jk is the angle formed by bonds ij and ik; the torsional term is Vi tor j st
that, regardless of the parameters used, a potential of the form used by Tersoff – or a similar one proposed by Brenner w5x – cannot reproduce correctly the observed vibrational properties of graphite. We presented a modification to this form, by adding a torsional term, and refined the parameters to fit structural and dynamical properties of graphite and diamond. The potential thus obtained not only reproduces these properties well but predicts accurately the – few – observed properties of Lonsdaleite, gives values in reasonable agreement with ab initio calculations for hypothetical carbon structures, and, most important for the purposes of this work, yields structural and dynamical data for C 60 and C 70 which agree remarkably well with observed data w6x. In this work, we apply our potential to the calculation of the statical and dynamical properties of C 36 . The energy for each pair of bonded atoms i,j is written as
c2
f c Ž ri k . f c Ž r jl . Tk i jl
Ý Ž k ,l ./ Ž i , j .
with k / l and Tk i jl s
™ ™ i k = ri j
žr
™ ™ jl = r ji
/ Pžr
ri k ri j r jt r ji
/.
The parameters used in the present calculation, taken ˚ . s 1.3883; from Ref. w6x are: DeŽeV. s 6.362; reŽA ˚ . s 1.8; R max ŽA˚ . s 2.1; b ŽA˚ y1 . s 1.638; R minŽA S s 1.577; a s 1.5724 = 10y7 ; l s 0.891; n s 0.687276; c s 38049; d s 4.745; h s y0.7171; t ŽeV. s y0.208. There seem to be strong indications pointing to a D6h symmetry for the C 36 molecule w2x. Besides the NMR results, a simple hexagonal crystal is incompatible with a molecular structure like D 2d , which lacks a 3- or 6-fold axis. However, we have searched
Table 1 Comparison between the energy differences Žin eV. for two C 36 isomers relative to the D6h fullerene, calculated in this work, and using a density functional approach and the LDA and GGA approximations, taken from Ref. w1x
This work Ref. w1x
D6h
D 2d
D 3h
0.00 0.0
0.14 0.0
1.17 1.4 ŽLDA.; 1.8 ŽGGA.
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
66
Table 2 ˚ . in D6h C 36 , calculated for the totally relaxed Bond distances ŽA structures in this work and in Ref. w1x. Bonded atoms as defined in Fig. 1 Bonded atoms
C1–C1
C1–C2
C2–C3
C3–C3
This work Ref. w1x
1.46 1.41
1.49 1.48
1.47 1.43
1.43 1.43
different structures, as an application of our new interaction potential. Here we present calculations based on D6h , D 2d , and D 3h symmetries; the first two were found to have the lowest energy in ab initio calculations w1x, and the last one has a 3-fold axis. Fig. 1 shows the D6h structure; C1, C2 and C3 are three symmetry-independent atoms. We find an average energy per atom for the D6h isomer of y7.41 eV; this result is reasonable, compared with the y7.86 eV we obtained for C 60 , and y8.56 eV for graphite or diamond w6x, using the same potential and calculation method. Table 1 shows the energy differences, relative to the D6h structure, of the three isomers under study, calculated in this work and in Ref. w1x. It can be seen that the results are quite similar, although in our case D6h has a lower energy than D 2d . The four distinct bond distances in the D6h structure are also compared to those of Ref. w1x in Table 2; both patterns are similar, with differences in the details. Table 3 shows the coordinates of the independent atoms; we have chosen z 5C 6 ; the planes sÕ contain two C1 carbon atoms of the top and bottom hexagons of the structure Žsee Fig. 1.. Unfortunately comparison with Ref. w1x is not possible, since these data were not reported, and the four interatomic distances do not determine the seven independent coordinates. For the D6h structure, the normal mode classification of C 36 is: G s 6A 1g q 2A 2g q 4B1g q 5B 2g q 8E 1g q 9E 2g q 3A 1u q 5A 2u q 5B1u q 4B 2u q 8E 1u q9E 2u . The 13 IR active modes are A 2uŽ z . and E 1u Ž x, y ., and the 23 Raman active ones are A 1g Ž xx Table 3 ˚ . of independent atoms of D6h C 36 Coordinates Žin A Atom
x
y
z
C1 C2 C3
1.458 2.366 2.204
0 0 1.272
2.615 1.432 0.714
Fig. 2. Calculated vibrational frequencies of C 36 . Degenerate modes are indicated by bars of height 2.
q yy, zz ., E 1g Ž yz, xz . and E 2g Ž xx y yy, xy .. Fig. 2 shows the positions of the calculated normal mode frequencies; degenerate modes are indicated by lines of double height. In Table 4 we list the calculated optically active frequencies, and their symmetries. The only vibrational information presently available is a mid-IR transmission spectrum of the powder, apparently obtained under difficult conditions w2x, which is shown, together with the positions of our calculated IR-active peaks in Fig. 3: superimposed to a background typical of a granular sample, the spectrum shows a series of broad features. Our highest calculated frequency lies under 1500 cmy1 ; it should be pointed out that no other fullerene-like structure has been reported to show vibrational frequencies much above 1600 cmy1 . In any case, the spectra correspond to the solid phase; if the intermolecular bonding is of at least partially covalent character, as
Table 4 Calculated Raman ŽA 1g , E 1g and E 2g . and IR ŽA 2u and E 1u . active frequencies Žin cmy1 . of C 36 A 1g
E 1g
E 2g
A 2u
E 1u
1485 1160 766 714 498 332
1454 1307 1097 873 814 701 516 325
1433 1375 1210 1034 794 735 640 551 298
1320 1022 835 614 467
1460 1336 1147 972 737 583 457 435
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
has been suggested, this may result in large changes in the vibrational spectrum from the molecule to the crystal. The bond polarizability model has been applied with success to the calculation of Raman intensities of fullerenes w7,8x. This model assigns to each type of bond a longitudinal and a transverse polarizability, or, equivalently, a mean static polarizability and an anisotropy, and assumes that these depend only on the bond length. Since the mean static polarizability does not contribute to the Raman intensity, three parameters for each type of bond are necessary to complete the model, one for the anisotropy and two for the derivatives of the polarizability and the anisotropy with respect to the bond distance. C 60 has well-defined single and double bonds; for C 70 the situation is not so clear, since there are several bond lengths. For this molecule Guha et al. w8x find that a ˚ between single and double cut-off length of 1.425 A bonds gives the best agreement between predictions and experiment. In Table 2, we see that for C 36 there are two groups of bond lengths, above and below ˚ which we chose as our cut-off length, leav1.46 A, ing us with 36 single and 18 double bonds. In Fig. 4 we show the results of our calculation using three different parameter sets; the lines are broadened using Gaussians with s s 5 cmy1 . The three sets predict similar main features for the spectrum below 1000 cmy1 , where three groups of intense bands appear. Differences are larger in the higher-frequency region: although at least two bands are always present, other features show sizable variations.
67
Fig. 4. Calculated Raman spectrum of C 36 , with different parameters for the bond polarizability model: Ža. Ref. w8x; Žb. Ref. w7x; Žc. Refs. w9,10x.
The semi-empirical potential used in this calculation has been shown to yield good agreement with the experimental features of C 60 and C 70 ; in the present work, the main structural properties of C 36 are also well reproduced. In the absence of detailed vibrational data, we have predicted the dynamical behavior of the molecule. It is interesting that the nature of C 36 crystals – at least according to the available indications – is very peculiar in that it shows intermolecular covalent bonding, whereas all other known fullerenes form van der Waals crystals; the semi-empirical approach is well suited to explore the properties of such complex crystals, although more detailed structural data are probably necessary to allow a thorough study of the problem.
Acknowledgements This work was partially supported by CONICET Grant PMT-PICT0051.
References
Fig. 3. Observed mid-IR spectrum of C 36 w2x and calculated IR-active frequencies of C 36 .
w1x J.C. Grossman, M. Cote, ˆ ´ S.G. Louie, M.L. Cohen, Chem. Phys. Lett. 284 Ž1998. 344. w2x C. Piskoti, J. Yarger, A. Zettl, Nature 393 Ž1998. 771. w3x M. Cote, ˆ ´ J.C. Grossman, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 81 Ž1998. 697.
68
E. Halac et al.r Chemical Physics Letters 299 (1999) 64–68
w4x J. Tersoff, Phys. Rev. B 39 Ž1989. 5566. w5x D.W. Brenner, Mater. Res. Soc. Symp. Proc. 141 Ž1989. 59. w6x E. Burgos, E. Halac, H. Bonadeo, Chem. Phys Lett. 298 Ž1998. 273. w7x D.W. Snoke, M. Cardona, Solid State Comm. 87 Ž1993. 121.
w8x S. Guha, J. Menendez, J.B. Page, G.B. Adams, Phys. Rev. B ´ 53 Ž1996. 13106. w9x J. Martin, S. Montero, J. Chem. Phys. 80 Ž1984. 4610. w10x M.F. Orduna, A. del Olmo, C. Domingo, S. Montero, J. Mol. Struc. 142 Ž1986. 201.