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Fullerene-like cages versus alternant cages: isomer stability of B13N13, B14N14, and B16N16
Chemical Physics Letters 383 (2004) 95–98 www.elsevier.com/locate/cplett
Fullerene-like cages versus alternant cages: isomer stability of B13N13, B14N14, and B16N16 Douglas L. Strout
*
Department of Physical Sciences, Alabama State University, 915 S. Jackson, Montgomery, AL 36101, USA Received 19 July 2003; in final form 22 October 2003 Published online: 2 December 2003
Abstract The production of boron nitride nanotubes has prompted numerous investigations into the structural properties of intermediatesized boron nitride molecules. For molecules with more than twenty atoms, cage isomers are the most stable. In this study, two classes of boron nitrides are compared: fullerene-like structures consisting of pentagons and hexagons, and alternant structures consisting of squares and hexagons. These two classes are compared for B13 N13 , B14 N14 , and B16 N16 by theoretical calculations using Hartree–Fock theory and density functional theory (B3LYP and LDA). The major result is that the alternant structures are more stable than the molecules based on fullerenes. Ó 2003 Published by Elsevier B.V.
1. Introduction Boron nitrides (BN)x , the isoelectronic analogues to the fullerenes, have been extensively studied, as both boron nitride molecules [1,2] and nanotubes [3–6] have been synthesized and/or characterized. For small molecules (x ¼ 3–10), ring isomers are the most stable form of boron nitride [7,8]. For x > 10, cages consisting of threecoordinate networks of boron and nitrogen are the most stable species [9,10]. However, there exist two major classes of boron nitride cages with very different factors determining their stability properties. A class of molecules based on fullerene structures with networks of pentagons and hexagons has the energetic advantage of relatively low structural strain. However, since the pentagons have an odd number of atoms, boron–boron bonds and nitrogen–nitrogen bonds in the network are unavoidable, and such bonds are an energetic disadvantage. A second class of boron nitride networks, namely the alternants, consist entirely of four-membered and six-membered rings. This provides for full alternation of boron and nitrogen atoms, with the result that all bonds are B–N bonds, which are energetically favorable. *
Fax: +3342291088. E-mail address: [email protected] (D.L. Strout).
0009-2614/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.cplett.2003.10.141
The four-membered rings, however, result in an increase in angular strain in the network. Rogers et al. [11] described these two competing forces as Ôchemical frustrationÕ (the presence of B–B and N–N bonds in the fullerene class) and Ôsteric frustrationÕ (the presence of strained four-membered rings in the alternant class). Chemical frustration and steric frustration are the two major competing forces that determine the relative stability of fullerene versus alternant cage isomers of boron nitride. For the 12 pentagons of a fullerene-class structure, the minimum number of B–B and N–N bonds is three of each per molecule. Rogers et al. applied the density-functional tight-binding (DFTB) method to a wide range (x ¼ 13–35) of boron nitrides (BN)x . The conclusion of that study is that the fullerene class is more stable than the alternant class over the entire range of molecule sizes. In the current study, ab initio theoretical calculations are carried out to quantify the relative stability of fullerene and alternant isomers of B13 N13 , B14 N14 , and B16 N16 . Rogers et al. determined that three fullereneclass B16 N16 isomers exist with the minimum number of B–B and N–N bonds, three of each type of homonuclear bond. Only one such isomer exists for B13 N13 and B14 N14 , and none exist for B15 N15 [11]. These fullereneclass isomers are compared to alternant structures with
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D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
squares and hexagons. The representative alternant B16 N16 is chosen with all six squares isolated by hexagons, since edge-sharing pairs of four-membered rings are even more strained and more destabilizing than four-membered rings are individually. However, no such Ôisolated squareÕ isomer exists for B13 N13 and B14 N14 , so alternants at those molecule sizes are chosen to minimize the number of edges shared between squares. At each molecule size, the relative energetics of the alternants and fullerene-class isomers are calculated to determine which one is the most stable.
2. Computational details Geometries of the eight molecules are optimized with Hartree–Fock theory, the B3LYP density functional method [12,13], and the local density approximation (LDA) [14,15]. The basis set [16] is the correlationconsistent double-zeta (CC-PVDZ) of Dunning. All calculations are carried out with the GA U S S I A N 98 quantum chemistry software package [17].
Fig. 1. B13 N13 and B14 N14 fullerene-class and alternant isomers. (a) B13 N13 fullerene-class isomer (C3 symmetry from D3h fullerene topological parent), (b) B13 N13 alternant (C3v symmetry with three edges shared by squares), (c) B14 N14 fullerene-class isomer (C1 symmetry from D2 fullerene topological parent) and (d) B14 N14 alternant structure (Cs symmetry with one edge shared by squares).
3. Results and discussion The B13 N13 and B14 N14 molecules in this study are shown in Fig. 1. The B13 N13 fullerene-class isomer has C3 symmetry and derives from a parent C26 cage with D3h symmetry. The B13 N13 alternant has C1 symmetry and has two B–N bonds that are edges shared between four-membered rings. The B14 N14 fullerene-class isomer has C1 symmetry and is derived from the C28 topological parent with D2 symmetry. The B14 N14 alternant has Cs symmetry and has one B–N bond shared between fourmembered rings. The relative energies of these molecules are shown in Table 1. The results for B13 N13 and B14 N14 indicate that, contrary to the previous report [11], it is in fact the alternants that are the most stable isomers, by more than 500–600 kJ/mol relative to the corresponding fullerene-class isomers. HF, B3LYP, and LDA all agree on this point, although the density functional (B3LYP and LDA) energy differences are 80–120 kJ/mol less than the corresponding HF results. The three fullerene-class B16 N16 isomers are shown in Figs. 2–4. Fig. 2 shows a B16 N16 molecule that has C3 symmetry and is topologically derived from the parent C32 fullerene with D3d symmetry. In this molecule, the pentagons are clustered around the C3 axis at the top and bottom of the molecule, and therefore the homonuclear bonds are also relatively near to the C3 axis. Fig. 3 shows a B16 N16 molecule, also with C3 symmetry
Fig. 2. B16 N16 isomer with C3 symmetry (from D3d symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. Three B–B bonds are visible in the foreground. The three N–N bonds are distributed similarly on the far side of the molecule.
Table 1 Relative energies of fullerene-class and alternant isomers of B13 N13 and B14 N14 molecules (energies in kJ/mol) B13 N13 energies
HF/DZ B3LYP/DZ LDA/DZ a b
B14 N14 energies
Alternant (C1 )
Fullerene (C3 )a
Alternant (Cs )
Fullerene (C1 )b
0.0 0.0 0.0
+637.6 +568.2 +559.0
0.0 0.0 0.0
+759.4 +666.9 +641.0
Fullerene-class isomer has D3h symmetry C26 cage as topological parent. Fullerene-class isomer has D2 symmetry C28 cage as topological parent.
D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
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Table 2 Relative energies of fullerene-class and alternant isomers of B16 N16 molecules (energies in kJ/mol) Alternant (D2d )
HF/DZ B3LYP/DZ LDA/DZ
0.0 0.0 0.0
Fullerene-class isomersa C1 –D2
C3 –D3d
C3 –D3h
+949.8 +836.4 +783.2
+1006.7 +900.4 +859.8
+1056.0 +939.7 +893.3
a
Fullerene-class isomers are listed by symmetry of the B16 N16 molecule, followed by the symmetry of the topological parent fullerene. Fig. 3. B16 N16 isomer with C3 symmetry (from D3h symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. Three N–N bonds are visible on the edges of the foreground. The three B–B bonds are distributed similarly on the other side of the molecule.
Fig. 4. B16 N16 isomer with C1 symmetry (from D2 symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. A B–B bond is visible on the left edge, and both boron atoms are bonded to nitrogen atoms participating in N–N bonds. An N–N bond is visible on the right edge, and both nitrogens are bonded to boron atoms participating in B–B bonds.
but derived from the parent C32 with D3h symmetry. In this molecule, both the pentagons and the homonuclear bonds are distributed around the equator of the molecule. Fig. 4 shows the third fullerene-class B16 N16 , a molecule derived from a topological parent fullerene with D2 symmetry, and this B16 N16 itself belongs to the C1 point group. Fig. 5 shows the isolated-square alternant composed of squares and hexagons; this molecule has D2d symmetry. Relative energies of the four B16 N16 molecules from calculations with the CC-PVDZ basis set are shown in Table 2. The primary feature of the data is the large
stability advantage for the alternant structure. All of the fullerene-class structures are much higher in energy than the D2d alternant, which is the most stable by a margin of approximately 800–1000 kJ/mol, with B3LYP and LDA lowering the energy gap by 150–170 kJ/mol relative to the HF results. As with B13 N13 and B14 N14 , the results contradict the previous DFTB results [11]. The energetic advantage for the alternant is greater for B16 N16 than for B13 N13 and B14 N14 , because the B16 N16 alternant has all six squares isolated by hexagons. For all molecule sizes in this study, it seems that the Ôchemical frustrationÕ of the fullerene-class isomers is still the dominant energetic driving force. For the molecules in this study, basis set effects may be significant and should be explored. Calculations with a larger basis set, such as the Dunning triple-zeta (CC-PVTZ) or augmented double-zeta (AUG-CC-PVDZ), would provide more accurate results. Such calculations are beyond the computational resources available for the current study. However, considering the magnitude of the energy differences between the alternant and the fullerene-class isomers, it is unlikely that basis set effects would overturn the result that the alternant is the most stable molecule. 4. Conclusion At a molecule size of 26–32 atoms, (BN)x cage stability still favors the fully alternant isomer composed of hexagons and squares. The fullerene-class isomers are higher in energy because of B–B and N–N homonuclear bonds in the cage network. For molecules of these sizes, the issue of Ôchemical frustrationÕ outweighs Ôsteric frustrationÕ in determining molecule stability. For 32 atoms, the energetic difference is still substantial, which probably indicates that alternants will be preferred for molecules larger than B16 N16: The fullerene-class molecules may at some much larger molecule size become energetically preferred over the alternants, but further analysis of larger molecules is required to determine if this is so. Acknowledgements
Fig. 5. B16 N16 isomer with D2d symmetry (fully alternant structure with squares and hexagons). Boron atoms are shown in white, and nitrogen atoms are shown in black.
The Alabama Supercomputer Authority is gratefully acknowledged for a grant of computer time on the Cray
98
D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
SV1 operated in Huntsville, AL. The taxpayers of the state of Alabama are also gratefully acknowledged.
References [1] F. Jensen, Chem. Phys. Lett. 209 (1993) 417. [2] I. Silaghi-Dumitrescu, F. Lara-Ochoa, P. Bishof, I. Haiduc, J. Mol. Struct. 367 (1996) 47. [3] N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Science 269 (1995) 966. [4] M. Terrones, W.K. Hsu, H. Terrones, J.P. Zhang, S. Ramos, J.P. Hare, R. Castillo, K. Prassides, A.K. Cheetham, H.W. Kroto, D.R.M. Walton, Chem. Phys. Lett. 259 (1996) 568. [5] A. Loiseau, F. Williame, N. Demoncy, G. Hug, H. Pascard, Phys. Rev. Lett. 76 (1996) 568.
[6] Y. Saito, M. Maida, J. Phys. Chem. A 103 (1999) 1291. [7] J.M.L. Martin, J. El-Yazal, J.-P. Francois, R. Gijbels, Chem. Phys. Lett. 232 (1995) 289. [8] J.M.L. Martin, J. El-Yazal, J.-P. Francois, Chem. Phys. Lett. 248 (1996) 95. [9] D.L. Strout, J. Phys. Chem. A 104 (2000) 3364. [10] D.L. Strout, J. Phys. Chem. A 105 (2001) 261. [11] K.M. Rogers, P.W. Fowler, G. Seifert, Chem. Phys. Lett. 332 (2000) 43. [12] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [13] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [14] J.C. Slater, Quantum Theory for Molecular Solids, in: The SelfConsistent Field for Molecular Solids, vol. 4, McGraw-Hill, New York, 1974. [15] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. [16] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [17] M.J. Frisch et al., GA U S S I A N 98, Revision A.7, Gaussian, Inc., Pittsburgh, PA, 1998.
Fullerene-like cages versus alternant cages: isomer stability of B13N13, B14N14, and B16N16 Douglas L. Strout
*
Department of Physical Sciences, Alabama State University, 915 S. Jackson, Montgomery, AL 36101, USA Received 19 July 2003; in final form 22 October 2003 Published online: 2 December 2003
Abstract The production of boron nitride nanotubes has prompted numerous investigations into the structural properties of intermediatesized boron nitride molecules. For molecules with more than twenty atoms, cage isomers are the most stable. In this study, two classes of boron nitrides are compared: fullerene-like structures consisting of pentagons and hexagons, and alternant structures consisting of squares and hexagons. These two classes are compared for B13 N13 , B14 N14 , and B16 N16 by theoretical calculations using Hartree–Fock theory and density functional theory (B3LYP and LDA). The major result is that the alternant structures are more stable than the molecules based on fullerenes. Ó 2003 Published by Elsevier B.V.
1. Introduction Boron nitrides (BN)x , the isoelectronic analogues to the fullerenes, have been extensively studied, as both boron nitride molecules [1,2] and nanotubes [3–6] have been synthesized and/or characterized. For small molecules (x ¼ 3–10), ring isomers are the most stable form of boron nitride [7,8]. For x > 10, cages consisting of threecoordinate networks of boron and nitrogen are the most stable species [9,10]. However, there exist two major classes of boron nitride cages with very different factors determining their stability properties. A class of molecules based on fullerene structures with networks of pentagons and hexagons has the energetic advantage of relatively low structural strain. However, since the pentagons have an odd number of atoms, boron–boron bonds and nitrogen–nitrogen bonds in the network are unavoidable, and such bonds are an energetic disadvantage. A second class of boron nitride networks, namely the alternants, consist entirely of four-membered and six-membered rings. This provides for full alternation of boron and nitrogen atoms, with the result that all bonds are B–N bonds, which are energetically favorable. *
Fax: +3342291088. E-mail address: [email protected] (D.L. Strout).
0009-2614/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.cplett.2003.10.141
The four-membered rings, however, result in an increase in angular strain in the network. Rogers et al. [11] described these two competing forces as Ôchemical frustrationÕ (the presence of B–B and N–N bonds in the fullerene class) and Ôsteric frustrationÕ (the presence of strained four-membered rings in the alternant class). Chemical frustration and steric frustration are the two major competing forces that determine the relative stability of fullerene versus alternant cage isomers of boron nitride. For the 12 pentagons of a fullerene-class structure, the minimum number of B–B and N–N bonds is three of each per molecule. Rogers et al. applied the density-functional tight-binding (DFTB) method to a wide range (x ¼ 13–35) of boron nitrides (BN)x . The conclusion of that study is that the fullerene class is more stable than the alternant class over the entire range of molecule sizes. In the current study, ab initio theoretical calculations are carried out to quantify the relative stability of fullerene and alternant isomers of B13 N13 , B14 N14 , and B16 N16 . Rogers et al. determined that three fullereneclass B16 N16 isomers exist with the minimum number of B–B and N–N bonds, three of each type of homonuclear bond. Only one such isomer exists for B13 N13 and B14 N14 , and none exist for B15 N15 [11]. These fullereneclass isomers are compared to alternant structures with
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D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
squares and hexagons. The representative alternant B16 N16 is chosen with all six squares isolated by hexagons, since edge-sharing pairs of four-membered rings are even more strained and more destabilizing than four-membered rings are individually. However, no such Ôisolated squareÕ isomer exists for B13 N13 and B14 N14 , so alternants at those molecule sizes are chosen to minimize the number of edges shared between squares. At each molecule size, the relative energetics of the alternants and fullerene-class isomers are calculated to determine which one is the most stable.
2. Computational details Geometries of the eight molecules are optimized with Hartree–Fock theory, the B3LYP density functional method [12,13], and the local density approximation (LDA) [14,15]. The basis set [16] is the correlationconsistent double-zeta (CC-PVDZ) of Dunning. All calculations are carried out with the GA U S S I A N 98 quantum chemistry software package [17].
Fig. 1. B13 N13 and B14 N14 fullerene-class and alternant isomers. (a) B13 N13 fullerene-class isomer (C3 symmetry from D3h fullerene topological parent), (b) B13 N13 alternant (C3v symmetry with three edges shared by squares), (c) B14 N14 fullerene-class isomer (C1 symmetry from D2 fullerene topological parent) and (d) B14 N14 alternant structure (Cs symmetry with one edge shared by squares).
3. Results and discussion The B13 N13 and B14 N14 molecules in this study are shown in Fig. 1. The B13 N13 fullerene-class isomer has C3 symmetry and derives from a parent C26 cage with D3h symmetry. The B13 N13 alternant has C1 symmetry and has two B–N bonds that are edges shared between four-membered rings. The B14 N14 fullerene-class isomer has C1 symmetry and is derived from the C28 topological parent with D2 symmetry. The B14 N14 alternant has Cs symmetry and has one B–N bond shared between fourmembered rings. The relative energies of these molecules are shown in Table 1. The results for B13 N13 and B14 N14 indicate that, contrary to the previous report [11], it is in fact the alternants that are the most stable isomers, by more than 500–600 kJ/mol relative to the corresponding fullerene-class isomers. HF, B3LYP, and LDA all agree on this point, although the density functional (B3LYP and LDA) energy differences are 80–120 kJ/mol less than the corresponding HF results. The three fullerene-class B16 N16 isomers are shown in Figs. 2–4. Fig. 2 shows a B16 N16 molecule that has C3 symmetry and is topologically derived from the parent C32 fullerene with D3d symmetry. In this molecule, the pentagons are clustered around the C3 axis at the top and bottom of the molecule, and therefore the homonuclear bonds are also relatively near to the C3 axis. Fig. 3 shows a B16 N16 molecule, also with C3 symmetry
Fig. 2. B16 N16 isomer with C3 symmetry (from D3d symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. Three B–B bonds are visible in the foreground. The three N–N bonds are distributed similarly on the far side of the molecule.
Table 1 Relative energies of fullerene-class and alternant isomers of B13 N13 and B14 N14 molecules (energies in kJ/mol) B13 N13 energies
HF/DZ B3LYP/DZ LDA/DZ a b
B14 N14 energies
Alternant (C1 )
Fullerene (C3 )a
Alternant (Cs )
Fullerene (C1 )b
0.0 0.0 0.0
+637.6 +568.2 +559.0
0.0 0.0 0.0
+759.4 +666.9 +641.0
Fullerene-class isomer has D3h symmetry C26 cage as topological parent. Fullerene-class isomer has D2 symmetry C28 cage as topological parent.
D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
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Table 2 Relative energies of fullerene-class and alternant isomers of B16 N16 molecules (energies in kJ/mol) Alternant (D2d )
HF/DZ B3LYP/DZ LDA/DZ
0.0 0.0 0.0
Fullerene-class isomersa C1 –D2
C3 –D3d
C3 –D3h
+949.8 +836.4 +783.2
+1006.7 +900.4 +859.8
+1056.0 +939.7 +893.3
a
Fullerene-class isomers are listed by symmetry of the B16 N16 molecule, followed by the symmetry of the topological parent fullerene. Fig. 3. B16 N16 isomer with C3 symmetry (from D3h symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. Three N–N bonds are visible on the edges of the foreground. The three B–B bonds are distributed similarly on the other side of the molecule.
Fig. 4. B16 N16 isomer with C1 symmetry (from D2 symmetry C32 fullerene parent). Boron atoms are shown in white, and nitrogen atoms are shown in black. A B–B bond is visible on the left edge, and both boron atoms are bonded to nitrogen atoms participating in N–N bonds. An N–N bond is visible on the right edge, and both nitrogens are bonded to boron atoms participating in B–B bonds.
but derived from the parent C32 with D3h symmetry. In this molecule, both the pentagons and the homonuclear bonds are distributed around the equator of the molecule. Fig. 4 shows the third fullerene-class B16 N16 , a molecule derived from a topological parent fullerene with D2 symmetry, and this B16 N16 itself belongs to the C1 point group. Fig. 5 shows the isolated-square alternant composed of squares and hexagons; this molecule has D2d symmetry. Relative energies of the four B16 N16 molecules from calculations with the CC-PVDZ basis set are shown in Table 2. The primary feature of the data is the large
stability advantage for the alternant structure. All of the fullerene-class structures are much higher in energy than the D2d alternant, which is the most stable by a margin of approximately 800–1000 kJ/mol, with B3LYP and LDA lowering the energy gap by 150–170 kJ/mol relative to the HF results. As with B13 N13 and B14 N14 , the results contradict the previous DFTB results [11]. The energetic advantage for the alternant is greater for B16 N16 than for B13 N13 and B14 N14 , because the B16 N16 alternant has all six squares isolated by hexagons. For all molecule sizes in this study, it seems that the Ôchemical frustrationÕ of the fullerene-class isomers is still the dominant energetic driving force. For the molecules in this study, basis set effects may be significant and should be explored. Calculations with a larger basis set, such as the Dunning triple-zeta (CC-PVTZ) or augmented double-zeta (AUG-CC-PVDZ), would provide more accurate results. Such calculations are beyond the computational resources available for the current study. However, considering the magnitude of the energy differences between the alternant and the fullerene-class isomers, it is unlikely that basis set effects would overturn the result that the alternant is the most stable molecule. 4. Conclusion At a molecule size of 26–32 atoms, (BN)x cage stability still favors the fully alternant isomer composed of hexagons and squares. The fullerene-class isomers are higher in energy because of B–B and N–N homonuclear bonds in the cage network. For molecules of these sizes, the issue of Ôchemical frustrationÕ outweighs Ôsteric frustrationÕ in determining molecule stability. For 32 atoms, the energetic difference is still substantial, which probably indicates that alternants will be preferred for molecules larger than B16 N16: The fullerene-class molecules may at some much larger molecule size become energetically preferred over the alternants, but further analysis of larger molecules is required to determine if this is so. Acknowledgements
Fig. 5. B16 N16 isomer with D2d symmetry (fully alternant structure with squares and hexagons). Boron atoms are shown in white, and nitrogen atoms are shown in black.
The Alabama Supercomputer Authority is gratefully acknowledged for a grant of computer time on the Cray
98
D.L. Strout / Chemical Physics Letters 383 (2004) 95–98
SV1 operated in Huntsville, AL. The taxpayers of the state of Alabama are also gratefully acknowledged.
References [1] F. Jensen, Chem. Phys. Lett. 209 (1993) 417. [2] I. Silaghi-Dumitrescu, F. Lara-Ochoa, P. Bishof, I. Haiduc, J. Mol. Struct. 367 (1996) 47. [3] N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Science 269 (1995) 966. [4] M. Terrones, W.K. Hsu, H. Terrones, J.P. Zhang, S. Ramos, J.P. Hare, R. Castillo, K. Prassides, A.K. Cheetham, H.W. Kroto, D.R.M. Walton, Chem. Phys. Lett. 259 (1996) 568. [5] A. Loiseau, F. Williame, N. Demoncy, G. Hug, H. Pascard, Phys. Rev. Lett. 76 (1996) 568.
[6] Y. Saito, M. Maida, J. Phys. Chem. A 103 (1999) 1291. [7] J.M.L. Martin, J. El-Yazal, J.-P. Francois, R. Gijbels, Chem. Phys. Lett. 232 (1995) 289. [8] J.M.L. Martin, J. El-Yazal, J.-P. Francois, Chem. Phys. Lett. 248 (1996) 95. [9] D.L. Strout, J. Phys. Chem. A 104 (2000) 3364. [10] D.L. Strout, J. Phys. Chem. A 105 (2001) 261. [11] K.M. Rogers, P.W. Fowler, G. Seifert, Chem. Phys. Lett. 332 (2000) 43. [12] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [13] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [14] J.C. Slater, Quantum Theory for Molecular Solids, in: The SelfConsistent Field for Molecular Solids, vol. 4, McGraw-Hill, New York, 1974. [15] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. [16] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [17] M.J. Frisch et al., GA U S S I A N 98, Revision A.7, Gaussian, Inc., Pittsburgh, PA, 1998.