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Generalized valence bond orbital interactions (GVB-OI) and stabilization (resonance) energies: Part II — Buckminster fullerene (C60)
THEO CHEM Journal of Molecular
Structure (Theochem)
424 (1998) 28 l-283
Generalized valence bond orbital interactions (GVB-01) and stabilization (resonance) energies: Part I&Buckminster hllerene (C,,-J SK. Pal Department of Chemistq
Bidhannagar College, Sector-I, 142 BF-Block, Salt Lake, Calcutta-700064, India Received
10 April 1997; accepted
17 April 1997
Abstract In a generalized valence bond (GVB) set up, the stabilization (resonance) energy of buckminster fnllerene, BF, (C&, has been determined from the minimization energies associated with Pauli’s orbital interactions (POI) involving all its sixty 2p, (GVB) carbon orbitals. As an useful guide, BFi has been considered to contain six independent naphthalene skeletons at a time on its spheroidal surface and on that basis the stabilization (resonance) energy OfBFi has been calculated by considering PO1 in all the independent naphthalene skeletons obtainable in the best possible way from the twenty hexagonal carbon rings present in BF,. 0 1998 Elsevier Science B.V. Keywords:
Pauli’s orbital interaction (POI); Minimization energy; Stabilization (resonance) energy; Bucluninster fullerene
@Pi)
2. Determination of stabilization energy in terms of PO1
1. Introduction Considering BF, in conventional aromatic terms, the stabilization energy of buckminster fullerene (BFJ has been calculated by the Huckel MO method [l]. However, no attempt has so far been made to calculate the stabilization (resonance) energy of BFi in terms of Pauli’s orbital interaction (POI) [2] in a set up of generalized valence bond (GVB) orbitals [3]. Therefore, an attempt has been made here in this direction.
For
I see ref. [4].
0166-1280/98/$19.00
0 1998 Elsevier
PII SO166-1280(97)00177-2
Science
(resonance)
The molecule buckminster fullerene having icosahedral symmetry (BFi) has been considered here to be formed of generalized valence bond (GVB) orbitals. Now, the orbital interactions involving all the 2p, (GVB) carbon orbitals, GVB-01, present in BFi have been considered in the following way. Twelve pentagonal carbon rings and twenty hexagonal carbon rings present on the spheroidal surface of BFi are arranged in such a way that BFi molecule can be imagined to contain six independent carbon skeletons of six naphthalenes (one unit) at any moment. Such an unit of six independent naphthalene
B.V. All rights reserved.
skeletons can be picked up out of the twenty hexagonal carbon rings (marked 1 to 20) (Fig. 1) present on the spheroidal surface of BF, in the following five ways: [1,5]-[3,13]-~18,19]-[7,20]-[9,10]--[11,15] [2,3]-[12,20]-f5,6]-[14,15]-[8,9]-[17,19] [14,lS]-[3,4]-[1.17]--[I 1,12]-[9,16]-[6,7] [1,2]-[4,20]-[13,14]-[10,6]-[&Ill-[16,19] [2,18]-(4,5]-[12,13]-[7,8]-[10,17]-[l&16]
(if (ii) (iii) (iv) (v)
Twenty hexagonal carbon rings present in BFi have been marked in arabic numerals, and a pair of numbers shown in a bracket represents two six-membered carbon rings fused to each other. This means that each pair of numbers shown in each bracket above stands for an independent naphthalene skeleton. These naphthalene skeletons have been picked up out of the twenty hexagonal carbon rings in such a way that each naphthalene skeleton is considered once, and no naphthalene skeleton is repeated in all the five units [(i)-(v)]. An important feature of the units [(i)-(v)] is that each hexagonal carbon ring would appear thrice when all the five units [(i)-(v)] are considered. So, in a generalized valence bond set up when (POI) [2], each of which involves two Zp, (GVB)bonding orbitals and one 2p, (GVB)-nonbonding orbital, are considered in all the independent naphthalene skeletons shown in the units [(i)-(v)], each of the sixty 2p, (GVB) carbon orbitals present in BFi would then participate thrice in the overall process of PO1 involving all the naphthalene skeletons shown in the units [(i)-(v)], because, as mentioned above, each hexagonal carbon skeleton would participate thrice in the overall process. Remember, however, at a time PO1 involving six naphthalene skeletons shown
(4
(b)
Fig. 1. The (a) front and (b) back surfaces of BF;.
in any one of the five units [(i)-(v)] should be considered. Now, in my previous paper [4] it has been shown how in a generalized valence bond set up the stabilization energy @Q/resonance energy (RE) of naphthalene can be calculated in terms of the minimization energies (e,) associated with the PO1 in the C3-fragment parts obtainable from the molecule, where SE/RE=
1 -x(2n+1)2xem=(2n+1)xfxm (2n + 1)
(1) when n = non-negative integer (n = 2 for naphthalene), ,f= stabilizing factor ( = 0. ‘6 for hexagonal ring system), pn = resonance integral = 75.312 kJ mol-I (for CA-FP) and e, = f x m. Since both the naphthalene hydrocarbon and the assumed naphthalene skeleton present in BFi contain an assembly of ten 2p, (GVB)-orbitals in a ten membered ring (the ring fusion bond being omitted), it is logical to suppose that their resonance energies may be determined by the same Eq. (1). So, when PO1 is considered in each independent naphthalene skeleton present in BF,, the stabilization (resonance) energy associated with a naphthalene skeleton = (2n + 1) x .f’x )11= 5 x f x m (here n = 2). Hence, if POI are considered in each independent naphthalene skeleton shown in each of the units [(i)(v)] above, a total of 6 x 5 naphthalene skeletons (six naphthalene skeletons in each unit) would participate in the POI, which would be associated with the stabilization (resonance) energy of 5 x 6 x 5fm. But it has been stated above that in the overall process of PO1 involving these 6 x 5 independent naphthalene skeletons in the five units ](i)-(v)], each of the sixty 2p, (GVB)-carbon orbitals of BF, would participate thrice. So, the actual stabilization (resonance) energy of BFi = 113 x 5 x 6 x 5 x f x m = 2510.1 kJ mol-’ (using the values offand nz mentioned above). Hence, SEiRE per carbon = l/60 x 25 10.1 kJ mol-‘, which equals 41.835 kJ mall’, which equals 9.9988048 kcal mall’, which equals 0.5554891& which equals approximately 0.555/3, where @ = 18 kcal mol-‘. The calculated value (0.5558) is very close to the value (0.5538) obtained by the Huckel MO (Molecular Orbital) method [ 11.
S.K. Pal/Journal of Molecular Structure (Theochem) 424 (1998) 281-283
3. Conclusion BFi attains aromatic stabilization due to PO1 involving all the sixty 2p, (GVB)-carbon orbitals present in it. Stabilization (resonance) energy associated with the PO1 involving these sixty 2p, (GVB)-carbon orbitals can be derived from the total stabilization (resonance) energy associated with all the independent naphthalene skeletons obtainable on the spheroidal surface of BF,. The stabilization (resonance) energy associated with an independent naphthalene skeleton can be calculated in terms of the minimization energies associated with the PO1 occurring in the C3-FP (C3-fragment parts, viz. >C=C-b< ) of naphthalene skeletons. Since all the sixty 2p, (GVB)carbon orbitals would participate thrice in the overall process of PO1 involving all the five units [(i)-(v)] (because all the twenty hexagonal carbon rings present in BF, would occur thrice), the SE(RE) of BF, can be obtained by dividing the total stabilization (resonance) energy associated with the PO1 involving all the five units [(i)-(v)] by three.
4. Discussion Since all the twenty hexagonal carbon rings (and hence all the sixty 2p,-GVB carbon orbitals) present in BFi should be considered equally in the overall process of POI, so PO1 involving all the five units [(i)-(v)] have been taken into account, because these five units [(i)-(v)] are the minimum number of units that can involve all the hexagonal carbon rings [and hence all the sixty 2p, (GVB)-carbon orbitals] equally (thrice) in the overall process of POI. PO1 involving only one set of six independent naphthalene skeleton (one unit) can be considered at a time on the spheroidal surface of BFi. So, once one such unit has been considered, another different unit, i.e. a different set of six independent naphthalene skeletons, should be considered the next time. In this way PO1 involving all the units [(i)-(v)] should be considered. There appears a continuous ‘rock-‘n-roll’
283
process of PO1 involving all the sixty 2p, (GVB)-electrons going on the spheroidal surface of BFi for which all the sixty sp2-carbon present in BFi become equivalent, as is evident from the 13C n.m.r. spectra (single line at 143 ppm at 295 K) [5]. Besides the ring strain due to the presence of twelve pentagonal rings on its surface, BFi possesses angular strain due to its spheroidal surface. The effect of this angular strain on the process of PO1 has been avoided by considering PO1 in smaller skeletons (viz. naphthalene skeletons) present on its surface. The surface of the naphthalene skeleton would appear almost planar, although the total surface of BF, is curved (spheroidal). Hence, in the calculation of minimization energy associated with PO1 in the C3-FP of naphthalene skeleton, the values off and m have been considered to be equal to 0.6, i.e. 0.666.. . and 75.3 12 kJ mol-’ respectively, as have been considered for aromatic hydrocarbons [4]. The idea of consideration of naphthalene skeletons as building blocks is due to Li and Yen [6], although Babic and Trinajstic have advocated in favour of more simpler building blocks [7]. However, for the purpose of calculating the SE(RE) of BFi avoiding angular strain, it has been found very useful to calculate the SE(RE) of BFi in terms of the minimization energies associated with PO1 in the Cj-FP of naphthalene skeletons present in BFi.
References [l] M.D. Newton, R.E. Stanton, J. Am. Chem. Sot. 108 (1986) 2469. [2] S.K. Pal, J. Mol. Struct. (Theochem), 337 (1995) 151-I 54; W.A. Goddard, J. Am. Chem. Sot., 94 (1972) 793. [3] W.A. Goddard, Phys. Rev. 157 (1967) 8; R.C. Ladner. W.A. Goddard, J. Chem. Phys., 51 (1969) 1073. [4] S.K. Pal, J. Mol. Struct. (Theochem) 389 (1997) 285-289. [5] C.S. Yannoni, R.D. Johnson, G. Meijer, D.S. Bethune, J.R. Salem, J. Phys. Chem. 95 (1991) 9-10. [6] Lin-Feng Li, Xiao-zeng Yen, J. Mol. Struct. (Theochem) 280 (1993) 147-149. [7] Darke Babic, Nenad Trinajstic, J. Mol. Struct. (Theochem) 303 (1994) 283.
Structure (Theochem)
424 (1998) 28 l-283
Generalized valence bond orbital interactions (GVB-01) and stabilization (resonance) energies: Part I&Buckminster hllerene (C,,-J SK. Pal Department of Chemistq
Bidhannagar College, Sector-I, 142 BF-Block, Salt Lake, Calcutta-700064, India Received
10 April 1997; accepted
17 April 1997
Abstract In a generalized valence bond (GVB) set up, the stabilization (resonance) energy of buckminster fnllerene, BF, (C&, has been determined from the minimization energies associated with Pauli’s orbital interactions (POI) involving all its sixty 2p, (GVB) carbon orbitals. As an useful guide, BFi has been considered to contain six independent naphthalene skeletons at a time on its spheroidal surface and on that basis the stabilization (resonance) energy OfBFi has been calculated by considering PO1 in all the independent naphthalene skeletons obtainable in the best possible way from the twenty hexagonal carbon rings present in BF,. 0 1998 Elsevier Science B.V. Keywords:
Pauli’s orbital interaction (POI); Minimization energy; Stabilization (resonance) energy; Bucluninster fullerene
@Pi)
2. Determination of stabilization energy in terms of PO1
1. Introduction Considering BF, in conventional aromatic terms, the stabilization energy of buckminster fullerene (BFJ has been calculated by the Huckel MO method [l]. However, no attempt has so far been made to calculate the stabilization (resonance) energy of BFi in terms of Pauli’s orbital interaction (POI) [2] in a set up of generalized valence bond (GVB) orbitals [3]. Therefore, an attempt has been made here in this direction.
For
I see ref. [4].
0166-1280/98/$19.00
0 1998 Elsevier
PII SO166-1280(97)00177-2
Science
(resonance)
The molecule buckminster fullerene having icosahedral symmetry (BFi) has been considered here to be formed of generalized valence bond (GVB) orbitals. Now, the orbital interactions involving all the 2p, (GVB) carbon orbitals, GVB-01, present in BFi have been considered in the following way. Twelve pentagonal carbon rings and twenty hexagonal carbon rings present on the spheroidal surface of BFi are arranged in such a way that BFi molecule can be imagined to contain six independent carbon skeletons of six naphthalenes (one unit) at any moment. Such an unit of six independent naphthalene
B.V. All rights reserved.
skeletons can be picked up out of the twenty hexagonal carbon rings (marked 1 to 20) (Fig. 1) present on the spheroidal surface of BF, in the following five ways: [1,5]-[3,13]-~18,19]-[7,20]-[9,10]--[11,15] [2,3]-[12,20]-f5,6]-[14,15]-[8,9]-[17,19] [14,lS]-[3,4]-[1.17]--[I 1,12]-[9,16]-[6,7] [1,2]-[4,20]-[13,14]-[10,6]-[&Ill-[16,19] [2,18]-(4,5]-[12,13]-[7,8]-[10,17]-[l&16]
(if (ii) (iii) (iv) (v)
Twenty hexagonal carbon rings present in BFi have been marked in arabic numerals, and a pair of numbers shown in a bracket represents two six-membered carbon rings fused to each other. This means that each pair of numbers shown in each bracket above stands for an independent naphthalene skeleton. These naphthalene skeletons have been picked up out of the twenty hexagonal carbon rings in such a way that each naphthalene skeleton is considered once, and no naphthalene skeleton is repeated in all the five units [(i)-(v)]. An important feature of the units [(i)-(v)] is that each hexagonal carbon ring would appear thrice when all the five units [(i)-(v)] are considered. So, in a generalized valence bond set up when (POI) [2], each of which involves two Zp, (GVB)bonding orbitals and one 2p, (GVB)-nonbonding orbital, are considered in all the independent naphthalene skeletons shown in the units [(i)-(v)], each of the sixty 2p, (GVB) carbon orbitals present in BFi would then participate thrice in the overall process of PO1 involving all the naphthalene skeletons shown in the units [(i)-(v)], because, as mentioned above, each hexagonal carbon skeleton would participate thrice in the overall process. Remember, however, at a time PO1 involving six naphthalene skeletons shown
(4
(b)
Fig. 1. The (a) front and (b) back surfaces of BF;.
in any one of the five units [(i)-(v)] should be considered. Now, in my previous paper [4] it has been shown how in a generalized valence bond set up the stabilization energy @Q/resonance energy (RE) of naphthalene can be calculated in terms of the minimization energies (e,) associated with the PO1 in the C3-fragment parts obtainable from the molecule, where SE/RE=
1 -x(2n+1)2xem=(2n+1)xfxm (2n + 1)
(1) when n = non-negative integer (n = 2 for naphthalene), ,f= stabilizing factor ( = 0. ‘6 for hexagonal ring system), pn = resonance integral = 75.312 kJ mol-I (for CA-FP) and e, = f x m. Since both the naphthalene hydrocarbon and the assumed naphthalene skeleton present in BFi contain an assembly of ten 2p, (GVB)-orbitals in a ten membered ring (the ring fusion bond being omitted), it is logical to suppose that their resonance energies may be determined by the same Eq. (1). So, when PO1 is considered in each independent naphthalene skeleton present in BF,, the stabilization (resonance) energy associated with a naphthalene skeleton = (2n + 1) x .f’x )11= 5 x f x m (here n = 2). Hence, if POI are considered in each independent naphthalene skeleton shown in each of the units [(i)(v)] above, a total of 6 x 5 naphthalene skeletons (six naphthalene skeletons in each unit) would participate in the POI, which would be associated with the stabilization (resonance) energy of 5 x 6 x 5fm. But it has been stated above that in the overall process of PO1 involving these 6 x 5 independent naphthalene skeletons in the five units ](i)-(v)], each of the sixty 2p, (GVB)-carbon orbitals of BF, would participate thrice. So, the actual stabilization (resonance) energy of BFi = 113 x 5 x 6 x 5 x f x m = 2510.1 kJ mol-’ (using the values offand nz mentioned above). Hence, SEiRE per carbon = l/60 x 25 10.1 kJ mol-‘, which equals 41.835 kJ mall’, which equals 9.9988048 kcal mall’, which equals 0.5554891& which equals approximately 0.555/3, where @ = 18 kcal mol-‘. The calculated value (0.5558) is very close to the value (0.5538) obtained by the Huckel MO (Molecular Orbital) method [ 11.
S.K. Pal/Journal of Molecular Structure (Theochem) 424 (1998) 281-283
3. Conclusion BFi attains aromatic stabilization due to PO1 involving all the sixty 2p, (GVB)-carbon orbitals present in it. Stabilization (resonance) energy associated with the PO1 involving these sixty 2p, (GVB)-carbon orbitals can be derived from the total stabilization (resonance) energy associated with all the independent naphthalene skeletons obtainable on the spheroidal surface of BF,. The stabilization (resonance) energy associated with an independent naphthalene skeleton can be calculated in terms of the minimization energies associated with the PO1 occurring in the C3-FP (C3-fragment parts, viz. >C=C-b< ) of naphthalene skeletons. Since all the sixty 2p, (GVB)carbon orbitals would participate thrice in the overall process of PO1 involving all the five units [(i)-(v)] (because all the twenty hexagonal carbon rings present in BF, would occur thrice), the SE(RE) of BF, can be obtained by dividing the total stabilization (resonance) energy associated with the PO1 involving all the five units [(i)-(v)] by three.
4. Discussion Since all the twenty hexagonal carbon rings (and hence all the sixty 2p,-GVB carbon orbitals) present in BFi should be considered equally in the overall process of POI, so PO1 involving all the five units [(i)-(v)] have been taken into account, because these five units [(i)-(v)] are the minimum number of units that can involve all the hexagonal carbon rings [and hence all the sixty 2p, (GVB)-carbon orbitals] equally (thrice) in the overall process of POI. PO1 involving only one set of six independent naphthalene skeleton (one unit) can be considered at a time on the spheroidal surface of BFi. So, once one such unit has been considered, another different unit, i.e. a different set of six independent naphthalene skeletons, should be considered the next time. In this way PO1 involving all the units [(i)-(v)] should be considered. There appears a continuous ‘rock-‘n-roll’
283
process of PO1 involving all the sixty 2p, (GVB)-electrons going on the spheroidal surface of BFi for which all the sixty sp2-carbon present in BFi become equivalent, as is evident from the 13C n.m.r. spectra (single line at 143 ppm at 295 K) [5]. Besides the ring strain due to the presence of twelve pentagonal rings on its surface, BFi possesses angular strain due to its spheroidal surface. The effect of this angular strain on the process of PO1 has been avoided by considering PO1 in smaller skeletons (viz. naphthalene skeletons) present on its surface. The surface of the naphthalene skeleton would appear almost planar, although the total surface of BF, is curved (spheroidal). Hence, in the calculation of minimization energy associated with PO1 in the C3-FP of naphthalene skeleton, the values off and m have been considered to be equal to 0.6, i.e. 0.666.. . and 75.3 12 kJ mol-’ respectively, as have been considered for aromatic hydrocarbons [4]. The idea of consideration of naphthalene skeletons as building blocks is due to Li and Yen [6], although Babic and Trinajstic have advocated in favour of more simpler building blocks [7]. However, for the purpose of calculating the SE(RE) of BFi avoiding angular strain, it has been found very useful to calculate the SE(RE) of BFi in terms of the minimization energies associated with PO1 in the Cj-FP of naphthalene skeletons present in BFi.
References [l] M.D. Newton, R.E. Stanton, J. Am. Chem. Sot. 108 (1986) 2469. [2] S.K. Pal, J. Mol. Struct. (Theochem), 337 (1995) 151-I 54; W.A. Goddard, J. Am. Chem. Sot., 94 (1972) 793. [3] W.A. Goddard, Phys. Rev. 157 (1967) 8; R.C. Ladner. W.A. Goddard, J. Chem. Phys., 51 (1969) 1073. [4] S.K. Pal, J. Mol. Struct. (Theochem) 389 (1997) 285-289. [5] C.S. Yannoni, R.D. Johnson, G. Meijer, D.S. Bethune, J.R. Salem, J. Phys. Chem. 95 (1991) 9-10. [6] Lin-Feng Li, Xiao-zeng Yen, J. Mol. Struct. (Theochem) 280 (1993) 147-149. [7] Darke Babic, Nenad Trinajstic, J. Mol. Struct. (Theochem) 303 (1994) 283.