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Mechanisms for the growth of polycyclic aromatic hydrocarbon (PAH) cations
25 March 2002
Chemical Physics Letters 355 (2002) 159–163 www.elsevier.com/locate/cplett
Mechanisms for the growth of polycyclic aromatic hydrocarbon (PAH) cations Charles W. Bauschlicher b
a,*
, Alessandra Ricca
a,1
, Marzio Rosi
b
a NASA Ames Research Center, Mail Stop 230-3, Space Technology Division, Moffett Field, CA 94035, USA Department of Chemistry, CNR Center for High Performance Computing in Molecular Sciences, University of Perugia, Perugia 06123, Italy
Received 7 May 2001; in final form 17 September 2001
Abstract The barriers and heats of reaction for the conversion of benzene cation to naphthalene cation by acetylene additions using the B3LYP/6-31G* approach are reported. The barrierless path we previously reported, using the B3LYP/4-31G approach, is shown to be incorrect. New paths for the ring formation are shown to have low barriers. Ó 2002 Published by Elsevier Science B.V.
1. Introduction We recently studied [1] the growth of neutral polycyclic aromatic hydrocarbons (PAHs) by the Frenklach [2–4] and Bittner–Howard [5] mechanisms. Since PAH cations are found in flames [6] and interstellar space [7–9], we also studied a Bittner–Howard-like formation mechanism for the cations, and we found a barrierless cation ring growth mechanism. All of the calculations in our previous Letter [1] were carried out using the hybrid [10] B3LYP [11] approach in conjunction with a 4-31G basis set [12]. As described in [1], the choice of the small 4-31G basis set was based on the small difference between small and large basis *
Corresponding author. Fax: +1-650-604-0350. E-mail address: [email protected] (C.W. Bauschlicher). 1 ELORET, Mail Stop 230-3.
sets for two bond energies and two barriers typical of neutral ring growth reactions. The first step in the cation ring growth was the loss of an aromatic H. This was followed by the addition of C2 H2 , see reaction (A) in Fig. 1. A second C2 H2 added to the end of the first, and then the end of the C4 H4 chain reacted with the first ring to complete the second ring. Recently we found that the reaction (A) was an artifact of the small basis set, and that using a 6-31G* basis set yields reaction (B), where both H atoms are on the terminal carbon. It should be noted that C6 H5 – þ CCHþ 2 was more stable than C6 H5 –CCH even for the 4-31G basis set, but improving the basis set removes the barrier between the two species, so that only C6 H5 –CCHþ 2 is a minimum for the 6-31G* basis set. Since reaction (A) was critical to the barrierless cation growth mechanism [1], we have reinvestigated the cation growth mechanism using the larger 6-31G* basis.
0009-2614/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 2 0 2 - 6
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Fig. 1. The basis set dependence of the products of the reaction C6 Hþ 5 þ C2 H2 .
A similar H shift is possible for the neutral system, and at the B3LYP/6-31G* level of theory, C6 H5 –CCH2 is computed to be 10.8 kcal/mol below C6 H5 –CHCH. However there is a 47.0 kcal/ mol barrier for the conversion of C6 H5 –CHCH to C6 H5 –CCH2 . Thus improving the basis set does not affect the conclusions of the previous work for the neutral systems.
Fig. 2, the H adds to the ring and then H2 is eliminated in an overall exothermic reaction. For this addition/elimination reaction, the H approaches out of the plane of the ring and reacts þ with C6 Hþ 6 , without a barrier, to form C6 H7 ; at the B3LYP/6-31G* level we find that this reaction is exothermic by 75.3 kcal/mol. C6 Hþ 7 can loose an H2 in a process that has no barrier in excess of the endothermicity of the reaction. While some of the C 6 Hþ 7 species will be stabilized by collisions or IR emission, the fact that C6 Hþ 5 þ H2 is 11 kcal/mol below C6 Hþ þ H, suggests that some of the C6 Hþ 6 7 þ will dissociate to form C6 H5 . Thus C6 Hþ 5 can form in a barrierless process by way of the C6 Hþ 7 intermediate. We should note that if the extra hydrogen approaches the C6 Hþ 6 in the plane of the molecule retaining the C2v symmetry, the reactants and products have different symmetry, 1 B1 and 1 2 A1 , respectively, arising from C6 Hþ 6 ð B1 Þ þ þ 1 2 1 Hð A1 Þ and C6 H5 ð A1 Þ þ H2 ð A1 Þ. We find that these potentials intersect at about 12 kcal/mol above the reactants. Thus there is a barrier for the
2. Computational methods The geometries were optimized and the harmonic frequencies were computed using the B3LYP [11] hybrid [10] functional in conjunction with the 6-31G* basis sets [12]. The frequencies confirmed that we have either minima or transition states. We computed the zero-point energy as one half the sum of the B3LYP harmonic frequencies. The B3LYP calculations were performed using the GA U S S I A N 98 computer code [13].
3. Results and discussion We were unable to find any ring forming reactions for C6 Hþ 6 and C2 H2 , and therefore we assume that the first step in any growth mechanism is the loss of a hydrogen atom. In interstellar space this could occur as the result of the absorption of an ultraviolet photon, while in combustion this could occur by the abstraction of H by some reactive species like H or C2 H. In reaction (1) of
Fig. 2. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 1.
C.W. Bauschlicher et al. / Chemical Physics Letters 355 (2002) 159–163
H abstraction reaction even if the hopping between the surfaces occurs easily. Therefore we conclude that C6 Hþ 5 is formed mostly by the addition/elimination process. As noted in Section 1, C6 Hþ 5 plus C2 H2 yields C6 H5 CCHþ 2 (reaction (2) in Fig. 2). This reaction is quite exothermic and we could not find a barrier for this process. We were unable to find a path that adds a second C2 H2 to the end of the first C2 H2 , as found in the Bittner–Howard mechanism. That is, shifting the H atom to the terminal carbon appears to prevent a cation Bittner–Howard like mechanism. Therefore we conclude that our previously suggested mechanism is an artifact of the 4-31G basis set. While we did not find the end-to-end reaction, we found a reaction that forms a fourmembered ring. As shown in Fig. 2, reaction (3) has a very small barrier and is exothermic. Opening the four-membered ring, with or without an H transfer (reactions (4) and (5)), is an endothermic process. Since reaction (5) is only endothermic by 11 kcal/mol, we considered further reactions, but none led to a favorable route for the formation of the second ring. In reaction (6) of Fig. 3, a hydrogen adds to the four-membered ring. The reaction is exothermic and we do not expect a barrier. After the addition of the H, it is possible to open the four-membered ring with a small barrier, reaction (7). As shown in reaction (8), this radical can undergo ring closing, but this reaction has a high barrier. More likely reactions are shown in (9) and (10), where H addition followed by H2 elimination leads to two possible ring closing products depending on the orientation of the C4 H5 side chain. Removing the hydrogen at either carbon a or b followed by a geometry optimization yields one of the two products shown in Fig. 3, suggesting that there is no barrier in these two ring closing reactions. In Fig. 4 we consider other possible reactions for the C6 H5 CCHþ 2 molecule. Reaction (11) adds a hydrogen atom to the side group; this process is exothermic and expected to have no barrier. In reaction (12) the aromatic H adjacent to the side group is removed in a Frenklach-like mechanism. This reaction is slightly endothermic, and on the basis of (18) and previous work [1], we expect a barrier of about 10 kcal/mol. Reaction (13) adds a
161
Fig. 3. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 2.
C2 H2 in a very exothermic process that is not expected to have a barrier. We find two ring closing reactions; reaction (14) has the lower barrier of the two and leads to a five-membered ring, while reaction (15) yields the 2-hydronaphthalene cation. Both reactions are exothermic. The transition state for reaction (15) involves a transfer of a hydrogen from carbon ‘A’ to carbon ‘B’. The intrinsic reaction path shows that this transition state then undergoes a ring closing without any additional barriers. Since the addition of C2 H2 in reaction (13) involves a shift of the hydrogen in the opposite direction, we looked for a concerted process where the C2 H2 adds to C6 H4 CHCHþ 2 to yield the 2-hydronaphthalene cation directly, but were unsuccessful in finding such a process. However, despite the barriers for reactions (14) and (15), reaction (13) is sufficiently exothermic that after the C2 H2 additions there is more than enough energy for surmounting the barrier and, therefore, ring closure should be possible. In addition to this ring closing, there is a second route that first involves the exothermic addition of a H atom
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Fig. 4. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 3.
Fig. 5. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 4.
þ
(reaction (16)) to form a second C2 H3 side group. This product can close the ring in a reaction with only a small barrier (see reaction (17)), which could be overcome by the sizeable energy released in reaction (16). Thus reactions (16) and (17) appear to be more likely than (14) or (15). In Fig. 5, we consider a Frenklach-like mechanism starting from the product of reaction (2), where an aromatic H adjacent to the CCH2 group is abstracted – see reaction (18). This is found to be endothermic by about 10 kcal/mol with a 14 kcal/mol barrier. As shown in reaction (19), C2 H2 can add to the radical in a barrierless exothermic process. This can rearrange, see reaction (20), to CCH2 , which is also exothermic. We cannot find a transition state for the ring closure of þ C6 H4 ðCCH2 Þ2 , which is consistent with the two terminal carbons being so far apart. However, since reaction (20) has a sizable barrier, we consider the ring closure from C6 H4 ðCCH2 Þ
ðCHCHÞ , the product of reaction (19). As shown in reaction (21), this has a small barrier and is exothermic. While the hydrogen shift, i.e. reaction (22), has a barrier, the addition of a H atom, reaction (23), leads to the 2-hydronaphthalene cation; this process is very exothermic and is not expected to have a barrier. It should be noted that such protonated cations are more common in flames than cations without the extra hydrogen [6].
4. Conclusions We have reported three growth mechanisms that convert benzene cation into a derivative of the naphthalene cation. The reaction sequence (1), (2), (3), (6), (7), and (9) has a maximum barrier of 7 kcal/mol, the sequence (1), (2), (18), (19), (21), and (23) has a maximum barrier of 14 kcal/mol, while the sequence (1), (2), (11), (12), (13), (16),
C.W. Bauschlicher et al. / Chemical Physics Letters 355 (2002) 159–163
and (17) has a maximum barrier of 12 kal/mol. For the first and third sequences an exothermic step proceeds the ring closing and hence could supply the energy to drive the ring closing. Therefore the bottleneck in these sequences could be reaction (3), which has a barrier of 0.6 kcal/mol and reaction (12), which is estimated to have a barrier of 10 kcal/mol. Thus only the first sequence might be possible in very cold diffuse interstellar clouds. Clearly the second and third sequences are viable cation growth mechanism for flames, however, their barriers appear to be too large for this mechanism to occur in very cold diffuse interstellar clouds, unless the mechanism is somewhat different from that presented. For example photon absorption could supply the energy to remove the hydrogen in reactions (12) and (18), thus making the second and third mechanisms more likely at low temperature. It is also possible that absorption of a photon leads to an electronically excited molecule that converts to a vibrationally excited ground state molecule, and this vibrational energy drives reactions like reaction (21), the ring closure for the second mechanism. These pathways supersede our previously reported mechanism.
163
Acknowledgements A.R. was supported by NASA Contract No. NAS2-99092. M.R. thanks CNR for a short-term fellowship. References [1] C.W. Bauschlicher, A. Ricca, Chem. Phys. Lett. 326 (2000) 283. [2] H. Wang, M. Frenklach, J. Phys. Chem. 98 (1994) 11465. [3] H. Wang, M. Frenklach, Combust. Flame 110 (1997) 173. [4] J. Appel, H. Bockhron, M. Frenklach, Combust. Flame 121 (2000) 122. [5] J.D. Bittner, J.B. Howard, Symp. (Inter.) Combust. 18th (1981) 1105. [6] P. Weilm€ unster, A. Keller, K.-H. Homann, Combust. Flame 116 (1998) 62. [7] L.J. Allamandola, A.G.G.M. Tielens, J.R. Barker, Astrophys. J. Suppl. 71 (1989) 733. [8] J. Szczepanski, M. Vala, Nature 363 (1993) 699. [9] A. Leger, J.L. Puget, Astron. Astrophys. 137 (1984) L5. [10] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [11] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 (1994) 11623. [12] M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 (1984) 3265, and references therein. [13] M.J. Frisch et al., GA U S S I A N 98, Revision A.7, Gaussian, Inc., Pittsburgh PA, 1998.
Chemical Physics Letters 355 (2002) 159–163 www.elsevier.com/locate/cplett
Mechanisms for the growth of polycyclic aromatic hydrocarbon (PAH) cations Charles W. Bauschlicher b
a,*
, Alessandra Ricca
a,1
, Marzio Rosi
b
a NASA Ames Research Center, Mail Stop 230-3, Space Technology Division, Moffett Field, CA 94035, USA Department of Chemistry, CNR Center for High Performance Computing in Molecular Sciences, University of Perugia, Perugia 06123, Italy
Received 7 May 2001; in final form 17 September 2001
Abstract The barriers and heats of reaction for the conversion of benzene cation to naphthalene cation by acetylene additions using the B3LYP/6-31G* approach are reported. The barrierless path we previously reported, using the B3LYP/4-31G approach, is shown to be incorrect. New paths for the ring formation are shown to have low barriers. Ó 2002 Published by Elsevier Science B.V.
1. Introduction We recently studied [1] the growth of neutral polycyclic aromatic hydrocarbons (PAHs) by the Frenklach [2–4] and Bittner–Howard [5] mechanisms. Since PAH cations are found in flames [6] and interstellar space [7–9], we also studied a Bittner–Howard-like formation mechanism for the cations, and we found a barrierless cation ring growth mechanism. All of the calculations in our previous Letter [1] were carried out using the hybrid [10] B3LYP [11] approach in conjunction with a 4-31G basis set [12]. As described in [1], the choice of the small 4-31G basis set was based on the small difference between small and large basis *
Corresponding author. Fax: +1-650-604-0350. E-mail address: [email protected] (C.W. Bauschlicher). 1 ELORET, Mail Stop 230-3.
sets for two bond energies and two barriers typical of neutral ring growth reactions. The first step in the cation ring growth was the loss of an aromatic H. This was followed by the addition of C2 H2 , see reaction (A) in Fig. 1. A second C2 H2 added to the end of the first, and then the end of the C4 H4 chain reacted with the first ring to complete the second ring. Recently we found that the reaction (A) was an artifact of the small basis set, and that using a 6-31G* basis set yields reaction (B), where both H atoms are on the terminal carbon. It should be noted that C6 H5 – þ CCHþ 2 was more stable than C6 H5 –CCH even for the 4-31G basis set, but improving the basis set removes the barrier between the two species, so that only C6 H5 –CCHþ 2 is a minimum for the 6-31G* basis set. Since reaction (A) was critical to the barrierless cation growth mechanism [1], we have reinvestigated the cation growth mechanism using the larger 6-31G* basis.
0009-2614/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 2 0 2 - 6
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Fig. 1. The basis set dependence of the products of the reaction C6 Hþ 5 þ C2 H2 .
A similar H shift is possible for the neutral system, and at the B3LYP/6-31G* level of theory, C6 H5 –CCH2 is computed to be 10.8 kcal/mol below C6 H5 –CHCH. However there is a 47.0 kcal/ mol barrier for the conversion of C6 H5 –CHCH to C6 H5 –CCH2 . Thus improving the basis set does not affect the conclusions of the previous work for the neutral systems.
Fig. 2, the H adds to the ring and then H2 is eliminated in an overall exothermic reaction. For this addition/elimination reaction, the H approaches out of the plane of the ring and reacts þ with C6 Hþ 6 , without a barrier, to form C6 H7 ; at the B3LYP/6-31G* level we find that this reaction is exothermic by 75.3 kcal/mol. C6 Hþ 7 can loose an H2 in a process that has no barrier in excess of the endothermicity of the reaction. While some of the C 6 Hþ 7 species will be stabilized by collisions or IR emission, the fact that C6 Hþ 5 þ H2 is 11 kcal/mol below C6 Hþ þ H, suggests that some of the C6 Hþ 6 7 þ will dissociate to form C6 H5 . Thus C6 Hþ 5 can form in a barrierless process by way of the C6 Hþ 7 intermediate. We should note that if the extra hydrogen approaches the C6 Hþ 6 in the plane of the molecule retaining the C2v symmetry, the reactants and products have different symmetry, 1 B1 and 1 2 A1 , respectively, arising from C6 Hþ 6 ð B1 Þ þ þ 1 2 1 Hð A1 Þ and C6 H5 ð A1 Þ þ H2 ð A1 Þ. We find that these potentials intersect at about 12 kcal/mol above the reactants. Thus there is a barrier for the
2. Computational methods The geometries were optimized and the harmonic frequencies were computed using the B3LYP [11] hybrid [10] functional in conjunction with the 6-31G* basis sets [12]. The frequencies confirmed that we have either minima or transition states. We computed the zero-point energy as one half the sum of the B3LYP harmonic frequencies. The B3LYP calculations were performed using the GA U S S I A N 98 computer code [13].
3. Results and discussion We were unable to find any ring forming reactions for C6 Hþ 6 and C2 H2 , and therefore we assume that the first step in any growth mechanism is the loss of a hydrogen atom. In interstellar space this could occur as the result of the absorption of an ultraviolet photon, while in combustion this could occur by the abstraction of H by some reactive species like H or C2 H. In reaction (1) of
Fig. 2. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 1.
C.W. Bauschlicher et al. / Chemical Physics Letters 355 (2002) 159–163
H abstraction reaction even if the hopping between the surfaces occurs easily. Therefore we conclude that C6 Hþ 5 is formed mostly by the addition/elimination process. As noted in Section 1, C6 Hþ 5 plus C2 H2 yields C6 H5 CCHþ 2 (reaction (2) in Fig. 2). This reaction is quite exothermic and we could not find a barrier for this process. We were unable to find a path that adds a second C2 H2 to the end of the first C2 H2 , as found in the Bittner–Howard mechanism. That is, shifting the H atom to the terminal carbon appears to prevent a cation Bittner–Howard like mechanism. Therefore we conclude that our previously suggested mechanism is an artifact of the 4-31G basis set. While we did not find the end-to-end reaction, we found a reaction that forms a fourmembered ring. As shown in Fig. 2, reaction (3) has a very small barrier and is exothermic. Opening the four-membered ring, with or without an H transfer (reactions (4) and (5)), is an endothermic process. Since reaction (5) is only endothermic by 11 kcal/mol, we considered further reactions, but none led to a favorable route for the formation of the second ring. In reaction (6) of Fig. 3, a hydrogen adds to the four-membered ring. The reaction is exothermic and we do not expect a barrier. After the addition of the H, it is possible to open the four-membered ring with a small barrier, reaction (7). As shown in reaction (8), this radical can undergo ring closing, but this reaction has a high barrier. More likely reactions are shown in (9) and (10), where H addition followed by H2 elimination leads to two possible ring closing products depending on the orientation of the C4 H5 side chain. Removing the hydrogen at either carbon a or b followed by a geometry optimization yields one of the two products shown in Fig. 3, suggesting that there is no barrier in these two ring closing reactions. In Fig. 4 we consider other possible reactions for the C6 H5 CCHþ 2 molecule. Reaction (11) adds a hydrogen atom to the side group; this process is exothermic and expected to have no barrier. In reaction (12) the aromatic H adjacent to the side group is removed in a Frenklach-like mechanism. This reaction is slightly endothermic, and on the basis of (18) and previous work [1], we expect a barrier of about 10 kcal/mol. Reaction (13) adds a
161
Fig. 3. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 2.
C2 H2 in a very exothermic process that is not expected to have a barrier. We find two ring closing reactions; reaction (14) has the lower barrier of the two and leads to a five-membered ring, while reaction (15) yields the 2-hydronaphthalene cation. Both reactions are exothermic. The transition state for reaction (15) involves a transfer of a hydrogen from carbon ‘A’ to carbon ‘B’. The intrinsic reaction path shows that this transition state then undergoes a ring closing without any additional barriers. Since the addition of C2 H2 in reaction (13) involves a shift of the hydrogen in the opposite direction, we looked for a concerted process where the C2 H2 adds to C6 H4 CHCHþ 2 to yield the 2-hydronaphthalene cation directly, but were unsuccessful in finding such a process. However, despite the barriers for reactions (14) and (15), reaction (13) is sufficiently exothermic that after the C2 H2 additions there is more than enough energy for surmounting the barrier and, therefore, ring closure should be possible. In addition to this ring closing, there is a second route that first involves the exothermic addition of a H atom
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Fig. 4. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 3.
Fig. 5. Heats of reaction and barriers computed using the B3LYP/6-31G* approach: part 4.
þ
(reaction (16)) to form a second C2 H3 side group. This product can close the ring in a reaction with only a small barrier (see reaction (17)), which could be overcome by the sizeable energy released in reaction (16). Thus reactions (16) and (17) appear to be more likely than (14) or (15). In Fig. 5, we consider a Frenklach-like mechanism starting from the product of reaction (2), where an aromatic H adjacent to the CCH2 group is abstracted – see reaction (18). This is found to be endothermic by about 10 kcal/mol with a 14 kcal/mol barrier. As shown in reaction (19), C2 H2 can add to the radical in a barrierless exothermic process. This can rearrange, see reaction (20), to CCH2 , which is also exothermic. We cannot find a transition state for the ring closure of þ C6 H4 ðCCH2 Þ2 , which is consistent with the two terminal carbons being so far apart. However, since reaction (20) has a sizable barrier, we consider the ring closure from C6 H4 ðCCH2 Þ
ðCHCHÞ , the product of reaction (19). As shown in reaction (21), this has a small barrier and is exothermic. While the hydrogen shift, i.e. reaction (22), has a barrier, the addition of a H atom, reaction (23), leads to the 2-hydronaphthalene cation; this process is very exothermic and is not expected to have a barrier. It should be noted that such protonated cations are more common in flames than cations without the extra hydrogen [6].
4. Conclusions We have reported three growth mechanisms that convert benzene cation into a derivative of the naphthalene cation. The reaction sequence (1), (2), (3), (6), (7), and (9) has a maximum barrier of 7 kcal/mol, the sequence (1), (2), (18), (19), (21), and (23) has a maximum barrier of 14 kcal/mol, while the sequence (1), (2), (11), (12), (13), (16),
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and (17) has a maximum barrier of 12 kal/mol. For the first and third sequences an exothermic step proceeds the ring closing and hence could supply the energy to drive the ring closing. Therefore the bottleneck in these sequences could be reaction (3), which has a barrier of 0.6 kcal/mol and reaction (12), which is estimated to have a barrier of 10 kcal/mol. Thus only the first sequence might be possible in very cold diffuse interstellar clouds. Clearly the second and third sequences are viable cation growth mechanism for flames, however, their barriers appear to be too large for this mechanism to occur in very cold diffuse interstellar clouds, unless the mechanism is somewhat different from that presented. For example photon absorption could supply the energy to remove the hydrogen in reactions (12) and (18), thus making the second and third mechanisms more likely at low temperature. It is also possible that absorption of a photon leads to an electronically excited molecule that converts to a vibrationally excited ground state molecule, and this vibrational energy drives reactions like reaction (21), the ring closure for the second mechanism. These pathways supersede our previously reported mechanism.
163
Acknowledgements A.R. was supported by NASA Contract No. NAS2-99092. M.R. thanks CNR for a short-term fellowship. References [1] C.W. Bauschlicher, A. Ricca, Chem. Phys. Lett. 326 (2000) 283. [2] H. Wang, M. Frenklach, J. Phys. Chem. 98 (1994) 11465. [3] H. Wang, M. Frenklach, Combust. Flame 110 (1997) 173. [4] J. Appel, H. Bockhron, M. Frenklach, Combust. Flame 121 (2000) 122. [5] J.D. Bittner, J.B. Howard, Symp. (Inter.) Combust. 18th (1981) 1105. [6] P. Weilm€ unster, A. Keller, K.-H. Homann, Combust. Flame 116 (1998) 62. [7] L.J. Allamandola, A.G.G.M. Tielens, J.R. Barker, Astrophys. J. Suppl. 71 (1989) 733. [8] J. Szczepanski, M. Vala, Nature 363 (1993) 699. [9] A. Leger, J.L. Puget, Astron. Astrophys. 137 (1984) L5. [10] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [11] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 (1994) 11623. [12] M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 (1984) 3265, and references therein. [13] M.J. Frisch et al., GA U S S I A N 98, Revision A.7, Gaussian, Inc., Pittsburgh PA, 1998.