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Theoretical investigation of the role of π–π interactions for the stability of phenylene ethynylene aggregates
Chemical Physics 270 (2001) 245±251
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Theoretical investigation of the role of p±p interactions for the stability of phenylene ethynylene aggregates M onica Pickholz *, Sven Stafstr om Department of Physics and Measurement Technology, Linkoping University, SE-581 83 Linkoping, Sweden Received 30 January 2001; in ®nal form 2 May 2001
Abstract A theoretical investigation of phenylene ethynylene macrocycles and oligomers aggregation was carried out. Two types of stacked supramolecular organization were investigated at the MP2 level: the intermolecular aggregation of macrocycles and the intramolecular conformational ordering of long oligomers. For the neutral monomer and orthoand meta-macrocycles bound states of p±p stacked pairs were found, having binding energies of 0.061, 0.378 and 0.81 eV, respectively. The relative stability between ortho- and meta-phenylene ethynylene chains and helices as function of the oligomer length n, in gas phase was also studied. The results show that p±p interactions stabilize the helical structures for long oligomers. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Macromolecules that show self-organization in the solid state and solution have attracted much interest recently [1,2]. In this kind of systems conformations are stabilized by noncovalent interactions such as hydrogen bonding, hydrophobic and van der Waals forces [3]. p-interactions between aromatic units play a signi®cant role in this context, although speci®c details on the nature of these interactions remain unclear [4]. The interaction between p systems can in¯uence the three dimensional structures of, for example proteins, DNA, polyaromatic molecules and crystal packing of aromatic molecules [5]. Phenylene ethynylene macromolecules constitute good examples of this
*
Corresponding author. Fax: +46-13-13-7568. E-mail addresses: [email protected] (M. Pickholz), svens@ ifm.liu.se (S. Stafstr om).
type of interaction. Macrocycles consisting of phenylene ethynylene units have been shown to associate in face-to-face p-stacks in some solvents and liquid crystals [3,6]. Furthermore, chains of phenylene ethynylene have been demonstrated to collapse into a p-stacked helical conformation, induced by both temperature and solvent eects [2]. The goal of this work is to investigate the role that p-interactions play in the stability of aggregates of phenylene ethynylene. The dispersion interaction, which has its origin in electron correlation [7], is an important interaction for these systems, therefore electron correlation is essential in this study and was taken into account using Mùller±Plesset (MP) perturbation theory [8]. The advantage of this method to take into account correlation eects is that MP calculations truncated at any order can be shown to be size consistent [9]. MP calculations are also computationally much faster than most of other methods that include correlation eects (e.g. con®guration interaction).
0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 8 7 - 1
246
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
As the ®rst step in this study we have investigated two phenylene ethynylene monomers in a parallel p-stack as function of the intermolecular distance. This system has been used as a test system to evaluate the eect of the size of the basis set and dierent orders in the perturbation theory. In the following sections we investigate at the MP2/ 3-21G level, the stability of larger aggregates. Well-de®ned phenylene ethynylene macrocycles (PEM) and ortho- and meta-phenylene ethynylene oligomers (linear as well as folded in helix conformations) were used to study interchain and intrachain p-interactions, respectively. Calculations were performed using the G A U S S I A N ' 9 8 package [10].
was then investigated using the 3-21G basis set. Finally, the basis set superposition error (BSSE) was estimated by the counterpoise method [11] at the MP2 level using two dierent basis sets. At the HF level the interaction between the monomers is found to be repulsive for all the basis sets used. Contrarily, at the MP2 level, a bound state is found in every case. The interaction energy, DE, of the pair is de®ned with respect to the system separated at in®nite distance, E R 1 as (E R 1 is equal to two times the energy of the monomer): DE R E R
Geometrical optimizations of the neutral phenylene ethynylene monomer were carried out at the ab initio HF/3-21G level. Optimized bond lengths of this monomer are shown in Section 3. The optimized monomers were arranged in parallel p-stack pairs, as is shown in Fig. 1. The evolution of the total energy as function of the intermolecular distance R was then calculated. Correlation eects were included using MP perturbation theory. As a ®rst step, the intermolecular interaction energies of the pairs were calculated at MP2 level, testing several dierent basis sets. The role of the higher correlation energy contributions
Fig. 1. Parallel p-stack of the phenylene ethynylene pairs. R is the intermolecular distance between the monomers.
1
and the binding energy (BE), is de®ned as, BE E Req
2. Short segments
E R 1
E R 1
2
where Req is the equilibrium intermolecular distance. In Fig. 2a is shown DEMP2 as a function of R for several dierent basis sets. Using the minimal STO-3G basis set, we found a very ¯at minimum with a BE 0:017 eV. For the basis at Req 4 A sets 3-21G and 3-21G** bound states were found with BE 0.061 and 0.079 eV, at Req 3:8 A, respectively. Increasing the basis set further we found that the minimum of the interaction energy is shifted to shorter distances, for basis sets 6 and the 31G** and cc-pVDZ, Req 3:7 and 3:6 A, BEs are 0.147 and 0.165 eV, respectively. A comparison between the calculations using the 3-21G and 3-21G** basis sets shows that including polarization functions lowers the interaction energy by about 30%, but does not aect the equilibrium distance. With these results we conclude that STO-3G is not at suitable basis set to describe these interactions, however with 3-21G it is possible to reproduce qualitatively the behavior obtained with bigger basis sets. In the next step of evaluation of the methodology, we compared the evolution of the interaction energy at HF, MP2, MP3, and MP4(SDTQ) level using the 3-21G basis set. These results are shown in Fig. 2b. From the repulsive interaction at ®rst order (HF level) we can see oscillations in the interaction energy at growing level in perturbation theory. The deepest minimum is found at the MP2 level, with a BEMP2 0:061 eV. The BEs at the MP3 and
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
247
Fig. 2. (a) Evolution of the MP2 interaction energy as function of the intermolecular distance R for dierent basis sets. (b) HF, MP2, MP3, and MP4(SDTQ) interaction energies using the 3-21G basis set. (c) Comparison between MP2 and MP2 BSSE for 3-21G and 631G** basis sets.
MP4 levels are 0.004 and 0.041 eV, respectively. Evidently, MP2 overestimates [12] and MP3 underestimates the interaction energy with respect to MP4. The BSSE was studied by the counterpoise method. MP2 calculations using the 3-21G and
6-31G** basis sets were carried out (the interaction is still repulsive for BSSE corrected HF calculations). In Fig. 2c is shown the results of the MP2 and MP2/corrected interaction energies as a function of the intermolecular distance. The corrected interaction energy is de®ned by:
248
DE Epair R
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251 dimer 2Emonomer R
3
where the sublabel indicates the molecules used in the calculations and the supralabel, the basis used. We can see in Fig. 2c that a bound state is found also after BSSE correction. However, the BE is much lower and the equilibrium distance longer than without correction. The MP2 intermolecular for equilibrium distances, Req , are 3.8 and 3.7 A the 3-21G and 6-31G** basis sets, respectively. With BSSE corrections these distances increase to respectively. 4.6 and 4.1 A, The dierence between the BSSE corrected and uncorrected diminish going from 3-21G to 631G** basis sets and it is expected to vanish in the limit of a complete basis set. The equilibrium dis tance is therefore expected to be lower than 3.7 A in this limit. The parallel p-stack of two benzene molecules was also investigated, using the same methodology. For this system there are extensive studies reported in the literature. Our MP2/3-21G results in benzene pair system show a bound state, with a although no BE of 0.03 eV and Req 3:8 A, bound state was found when BSSE correction was considered. Tsuzuki et al. [12] reported interaction energies of 0.03 eV (no bound state) to 0.11 eV going from 6-31G to aug(d)-6-311G* at the MP2 level taken into account BSSE correction (Experimental bonding enthalpy reported was 0.07 eV [13]). They also concluded that MP2 calculation overestimates the attractive interactions of benzene and naphthalene compared with more sophisticated methods. However using a relatively small basis set compensates for this eect. We can conclude from this study that MP2/3-21G gives a qualitative good picture of the interaction. This combination is therefore used in the following sections.
3. Macrocycles PEM are interesting systems for studies of p±p interactions because their planarity and the fact that they are free of end defects. Fig. 3 shows PEM built by linked phenylene ethynylene monomers in the ortho- (o-) and meta- (m-) position of the phe-
Fig. 3. Ortho- and meta-PEM.
Table 1 HF/3-21G optimized bond lengths of m-PEM and o-PEM, in A C1±C2 C2±C3 C3±C4 C4±C5 C5±C6 C6±C1 C2±C7 C7±C8
m-PEM
o-PEM
Monomer
1.395 1.397 1.384 1.384 1.397 1.395 1.455 1.176
1.402 1.390 1.380 1.384 1.380 1.390 1.429 1.190
1.391 1.391 1.382 1.384 1.384 1.382 1.434 1.189
nyl ring. Geometrical optimizations of both macrocycles have been performed at the HF level with the 3-21G basis set. In Table 1 we compare the geometries of the m-PEM and o-PEM and the phenylene ethynylene monomer used in last section. Because of the symmetry of the molecules it is enough to show the results for a monomer only. The numeration used in this table is shown in Fig. 4. Comparisons of the data presented in Table 1 show small dierences between the geometries of m-PEM, o-PEM and the monomer. The phenyl rings of the m-PEM and the monomer present a more pronounced aromatic character than for the o-PEM, in which their bond lengths go from 1.402 (C1±C2) to 1.380 A (C3±C4 and C5±C6). In the A ethylene group the single bond between atoms C2 and C7 is considerably longer in m-PEM (1.455 The triple bond between the carbons C7 and A). C8, that is also slightly shorter for the m-PEM as compared to o-PEM.
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
249
BE 3-21G BE STO-3G monomer-pair BE 3-21G 3:7 BE STO-3G rr-o-PEM
4
Due to the large size of the rr-m-PEM, it was only possible to study the pair at the MP2/STO-3G level, leading to a BE 0:221 eV and Req 3:9 However, using the last expression we can esA. timate the BE of this pair as BE 3-21G 3:7 BE STO-3G 0:81 eV. Fig. 4. Numeration of the monomer referred in Table 1. The arrows point at the connection with the next monomer, m and o indicates ortho- and meta-connections, respectively.
Neutral pairs of PEMs were arranged parallel to each other in a p-stack, with the aromatic rings on top of each other as illustrated in Fig. 5 for the o-PEM. MP2 calculations using STO-3G and 321G basis sets of o-PEM pairs were carried out as function of the intermolecular distance R. Their BEs were BE(STO-3G) 0.103 eV and BE(3 and Req 3:7 A, 21G) 0.378 eV at Req 3:9 A respectively. Comparisons with the results obtained in the last section show that for both basis sets the BE of the o-PEM pair is larger than three times the BE of the monomer pair. This shows that the delocalization of the p electrons around the macrocycle backbone further stabilizes the dimer. The ratio between both BEs of the same system using the two basis set is approximately the same,
Fig. 5. Two o-PEM in a parallel p±p stack.
4. Helices and chains In this section we discuss the relative stability between oligomers of ortho- (recently synthesized [14]) and meta- [15] phenylene ethynylene chains and helices as function of the oligomer length n, in gas phase. The oligomers, connected in meta- and ortho-positions (see Fig. 6), were optimized at HF/ 3-21G level for dierent chain lengths, n (n 3 to 10). It is not possible to optimize the helical structures at the ab initio quantum chemistry level since the size of the system is too large to take correlation eects into account. From the results presented above we know that at the HF level the stacking interaction is repulsive, which leads to the formation of chains instead of helices. The helices were therefore built by folding the optimized planar chains, without further optimizations. The chains were folded in a cisoid conformation varying the torsional angle between consecutive phenyl rings, / (see Fig. 7). The torsional angles were / 5:8° for the m-helices and / 32° for the o-helices. The p±p stacking between phenyl groups of dierent loops of the built helices imitate the parallel p-stack of PEM (in which the intermolecular [6]). Each loop involves six distance is 3.5 A phenyl rings in the m-helices and three in the ohelices. The total energy of chains and helices has been evaluated using the MP2 method and the 321G basis set. Their relative stability as a function of the chain length was investigated for oligomers having from n 3±10 phenyl rings. The relative
250
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
Fig. 6. Ortho- and meta-phenylene ethynylene oligomers (n 3±10).
Fig. 7. Examples of o- and m-helix.
stability between chains and helices is studied as function of the chain length. The dierence in MP2 energy between the helically ordered state and the unfolded (planar) chain, DEho , was calculated. The evolution of DEho as a function of the oligomer length n, for both systems is shown in Fig. 8. For the o-system, the helical structures are the most stable ones for n P 4. This oligomer lengths correspond to the o-helix having p overlap, i.e. pair of phenylene ethynylene units appear on top of each other as shown in Fig. 1. The energy difference DEho is 0.07 eV for n 4 and increase with n, but linearly only in each loop (see Fig. 8). The slope increase going from the second to the third loop, is consistent with the extension of the p-stack. For the m-system, the helical structure becomes the most stable ones for n 6. This structure has no overlaping phenylene ethynylene units. The n 6 m-helix is more stable than the mchain at both the HF and the MP2 levels, with the same energy dierence, DEho 0:02 eV. This suggests that this helical structure is not stabilized by
Fig. 8. MP2 energy dierence between helices and chains as a function of n for the m-system and the o-system. The onset for helical stability is given by DEho P 0. The continuous curves correspond to linear ®ttings of the data for each loop.
p±p interactions. Instead, hydrogen bonds with the aromatic end rings stabilize the six rings m-helix. For smaller structures the energy dierence between the chain and the helix is lower than 1 meV (favorable to the chains) which indicates that the small torsion (/ 5:8°) of the dihedral angles and cisoid conformations are practically unimportant in this context. Note that in the case of the o-system the energy cost due to torsion is much larger, as shown by the negative value of DEho for n 3. For n 7 the energy dierence is DEho 0:14 eV and increase linearly with n. The resulting stability of helical structures in which p±p interactions are present, here calculated at the MP2 level, (contrarily to the HF results)
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
shows that electron correlation eects play an important role for the intramolecular interactions. 5. Conclusions In this work, the role of the intramolecular and intermolecular interactions in phenylene ethynylene aggregates was studied. We have shown that aromatic p±p interactions could help the selfassociation of macrocycles in p-stacks and in the formation of helical structures. Electron±electron correlation eects were found to be responsible for the attractive interaction that lead to the aggregation in these systems. Because the size of the studied systems we had to compromise to include correlation eects at the MP2 level using in most of calculations the 3-21G basis set. In Section 1 we evaluate the eects of this approximation and we found that this combination leads to a qualitatively correct picture of the interaction. From the experimental point of view, the self organization of these systems involves a variety of noncovalent cohesive forces, including hydrogen bonds, electrostatic forces, steric packing and hydrophobic eects [2]. Besides, solvophobicity can be an eective way to drive the supramolecular organizations involving p-stacked structures [16]. However, in this study we focussed the attention to p±p stacked structures between equivalent units (i.e., phenyl ring phenyl ring stacking). The main motivation of this work has been, therefore, to point out the importance of the dispersive p±p interaction for the stability of three-dimensional structures of phenylene ethynylene. Acknowledgements Computational resources were provided by the Swedish Council for High performance Computing (NSC). Financial support from the Swedish Research Council for Engineering Science (TFR)
251
and the Swedish Natural Science Research Council (NFR) is gratefully acknowledged.
References [1] M.J. Mio, J. Moore, MRS bulletin (2000) 36. [2] J.C. Nelson, J.G. Saven, P. Wolynes, Science 277 (1997) 1793. [3] S. Lahiri, J.L. Thompson, J. Moore, J. Am. Chem. Soc. 122 (2000) 11315. [4] A.S. Shetty, J. Zhang, J. Moore, J. Am. Chem. Soc. 118 (1996) 1019. [5] S. Tsuzuki, T. Uchimaru, M. Mikami, K. Tanabe, Chem. Phys. Lett. 252 (1996) 206. [6] O.Y. Mindyuk, M.R. Stetzer, P.A. Heiney, J.C. Nelson, J. Moore, Adv. Mater. 10 (1998) 1363. [7] A.J. Stone, The Theory of Intermolecular Forces, ISBN 0-19-855884-8/, Clarendon, Oxford, 1996. [8] C. Mùller, M.S. Plesset, Phys. Rev. 46 (1934) 618. [9] A. Szabo, N. Ostlund, Modern Quantum Chemistry, Mineola, New York, 1982. [10] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery, R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Cliord, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B.G. Johnson, W. Chen, M.W. Wong, J.L. Andres, M. Head-Gordon, E.S. Replogle, J.A. Pople, G A U S S I A N 9 8 , Revision A.7, Gaussian, Inc., Pittsburgh PA, 1998. [11] F.B. van Duijneveldt, J.G.C.M. van Duijneveldt de Rijdt, J.H. van Lenthe, Chem. Rev. 94 (1994) 1873. [12] S. Tsuzuki, T. Uchimaru, M. Mikami, K. Tanabe, Chem. Phys. Lett. 319 (2000) 547. [13] H. Krause, B. Ernstberger, H.J. Neusser, Chem. Phys. Lett. 184 (1991) 411. [14] A. Orita, E. Alonso, J. Yaruva, J. Otera, Synlett 9 (2000) 1333. [15] K. Burguess (Ed.), Solid-Phase Organic Synthesis, Wiley, New York, 2000 (Chapter 4). [16] R.B. Prince, J.G. Saven, P. Wolynes, J. Moore, J. Am. Chem. Soc. 121 (1999) 3114.
www.elsevier.com/locate/chemphys
Theoretical investigation of the role of p±p interactions for the stability of phenylene ethynylene aggregates M onica Pickholz *, Sven Stafstr om Department of Physics and Measurement Technology, Linkoping University, SE-581 83 Linkoping, Sweden Received 30 January 2001; in ®nal form 2 May 2001
Abstract A theoretical investigation of phenylene ethynylene macrocycles and oligomers aggregation was carried out. Two types of stacked supramolecular organization were investigated at the MP2 level: the intermolecular aggregation of macrocycles and the intramolecular conformational ordering of long oligomers. For the neutral monomer and orthoand meta-macrocycles bound states of p±p stacked pairs were found, having binding energies of 0.061, 0.378 and 0.81 eV, respectively. The relative stability between ortho- and meta-phenylene ethynylene chains and helices as function of the oligomer length n, in gas phase was also studied. The results show that p±p interactions stabilize the helical structures for long oligomers. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Macromolecules that show self-organization in the solid state and solution have attracted much interest recently [1,2]. In this kind of systems conformations are stabilized by noncovalent interactions such as hydrogen bonding, hydrophobic and van der Waals forces [3]. p-interactions between aromatic units play a signi®cant role in this context, although speci®c details on the nature of these interactions remain unclear [4]. The interaction between p systems can in¯uence the three dimensional structures of, for example proteins, DNA, polyaromatic molecules and crystal packing of aromatic molecules [5]. Phenylene ethynylene macromolecules constitute good examples of this
*
Corresponding author. Fax: +46-13-13-7568. E-mail addresses: [email protected] (M. Pickholz), svens@ ifm.liu.se (S. Stafstr om).
type of interaction. Macrocycles consisting of phenylene ethynylene units have been shown to associate in face-to-face p-stacks in some solvents and liquid crystals [3,6]. Furthermore, chains of phenylene ethynylene have been demonstrated to collapse into a p-stacked helical conformation, induced by both temperature and solvent eects [2]. The goal of this work is to investigate the role that p-interactions play in the stability of aggregates of phenylene ethynylene. The dispersion interaction, which has its origin in electron correlation [7], is an important interaction for these systems, therefore electron correlation is essential in this study and was taken into account using Mùller±Plesset (MP) perturbation theory [8]. The advantage of this method to take into account correlation eects is that MP calculations truncated at any order can be shown to be size consistent [9]. MP calculations are also computationally much faster than most of other methods that include correlation eects (e.g. con®guration interaction).
0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 8 7 - 1
246
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
As the ®rst step in this study we have investigated two phenylene ethynylene monomers in a parallel p-stack as function of the intermolecular distance. This system has been used as a test system to evaluate the eect of the size of the basis set and dierent orders in the perturbation theory. In the following sections we investigate at the MP2/ 3-21G level, the stability of larger aggregates. Well-de®ned phenylene ethynylene macrocycles (PEM) and ortho- and meta-phenylene ethynylene oligomers (linear as well as folded in helix conformations) were used to study interchain and intrachain p-interactions, respectively. Calculations were performed using the G A U S S I A N ' 9 8 package [10].
was then investigated using the 3-21G basis set. Finally, the basis set superposition error (BSSE) was estimated by the counterpoise method [11] at the MP2 level using two dierent basis sets. At the HF level the interaction between the monomers is found to be repulsive for all the basis sets used. Contrarily, at the MP2 level, a bound state is found in every case. The interaction energy, DE, of the pair is de®ned with respect to the system separated at in®nite distance, E R 1 as (E R 1 is equal to two times the energy of the monomer): DE R E R
Geometrical optimizations of the neutral phenylene ethynylene monomer were carried out at the ab initio HF/3-21G level. Optimized bond lengths of this monomer are shown in Section 3. The optimized monomers were arranged in parallel p-stack pairs, as is shown in Fig. 1. The evolution of the total energy as function of the intermolecular distance R was then calculated. Correlation eects were included using MP perturbation theory. As a ®rst step, the intermolecular interaction energies of the pairs were calculated at MP2 level, testing several dierent basis sets. The role of the higher correlation energy contributions
Fig. 1. Parallel p-stack of the phenylene ethynylene pairs. R is the intermolecular distance between the monomers.
1
and the binding energy (BE), is de®ned as, BE E Req
2. Short segments
E R 1
E R 1
2
where Req is the equilibrium intermolecular distance. In Fig. 2a is shown DEMP2 as a function of R for several dierent basis sets. Using the minimal STO-3G basis set, we found a very ¯at minimum with a BE 0:017 eV. For the basis at Req 4 A sets 3-21G and 3-21G** bound states were found with BE 0.061 and 0.079 eV, at Req 3:8 A, respectively. Increasing the basis set further we found that the minimum of the interaction energy is shifted to shorter distances, for basis sets 6 and the 31G** and cc-pVDZ, Req 3:7 and 3:6 A, BEs are 0.147 and 0.165 eV, respectively. A comparison between the calculations using the 3-21G and 3-21G** basis sets shows that including polarization functions lowers the interaction energy by about 30%, but does not aect the equilibrium distance. With these results we conclude that STO-3G is not at suitable basis set to describe these interactions, however with 3-21G it is possible to reproduce qualitatively the behavior obtained with bigger basis sets. In the next step of evaluation of the methodology, we compared the evolution of the interaction energy at HF, MP2, MP3, and MP4(SDTQ) level using the 3-21G basis set. These results are shown in Fig. 2b. From the repulsive interaction at ®rst order (HF level) we can see oscillations in the interaction energy at growing level in perturbation theory. The deepest minimum is found at the MP2 level, with a BEMP2 0:061 eV. The BEs at the MP3 and
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
247
Fig. 2. (a) Evolution of the MP2 interaction energy as function of the intermolecular distance R for dierent basis sets. (b) HF, MP2, MP3, and MP4(SDTQ) interaction energies using the 3-21G basis set. (c) Comparison between MP2 and MP2 BSSE for 3-21G and 631G** basis sets.
MP4 levels are 0.004 and 0.041 eV, respectively. Evidently, MP2 overestimates [12] and MP3 underestimates the interaction energy with respect to MP4. The BSSE was studied by the counterpoise method. MP2 calculations using the 3-21G and
6-31G** basis sets were carried out (the interaction is still repulsive for BSSE corrected HF calculations). In Fig. 2c is shown the results of the MP2 and MP2/corrected interaction energies as a function of the intermolecular distance. The corrected interaction energy is de®ned by:
248
DE Epair R
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251 dimer 2Emonomer R
3
where the sublabel indicates the molecules used in the calculations and the supralabel, the basis used. We can see in Fig. 2c that a bound state is found also after BSSE correction. However, the BE is much lower and the equilibrium distance longer than without correction. The MP2 intermolecular for equilibrium distances, Req , are 3.8 and 3.7 A the 3-21G and 6-31G** basis sets, respectively. With BSSE corrections these distances increase to respectively. 4.6 and 4.1 A, The dierence between the BSSE corrected and uncorrected diminish going from 3-21G to 631G** basis sets and it is expected to vanish in the limit of a complete basis set. The equilibrium dis tance is therefore expected to be lower than 3.7 A in this limit. The parallel p-stack of two benzene molecules was also investigated, using the same methodology. For this system there are extensive studies reported in the literature. Our MP2/3-21G results in benzene pair system show a bound state, with a although no BE of 0.03 eV and Req 3:8 A, bound state was found when BSSE correction was considered. Tsuzuki et al. [12] reported interaction energies of 0.03 eV (no bound state) to 0.11 eV going from 6-31G to aug(d)-6-311G* at the MP2 level taken into account BSSE correction (Experimental bonding enthalpy reported was 0.07 eV [13]). They also concluded that MP2 calculation overestimates the attractive interactions of benzene and naphthalene compared with more sophisticated methods. However using a relatively small basis set compensates for this eect. We can conclude from this study that MP2/3-21G gives a qualitative good picture of the interaction. This combination is therefore used in the following sections.
3. Macrocycles PEM are interesting systems for studies of p±p interactions because their planarity and the fact that they are free of end defects. Fig. 3 shows PEM built by linked phenylene ethynylene monomers in the ortho- (o-) and meta- (m-) position of the phe-
Fig. 3. Ortho- and meta-PEM.
Table 1 HF/3-21G optimized bond lengths of m-PEM and o-PEM, in A C1±C2 C2±C3 C3±C4 C4±C5 C5±C6 C6±C1 C2±C7 C7±C8
m-PEM
o-PEM
Monomer
1.395 1.397 1.384 1.384 1.397 1.395 1.455 1.176
1.402 1.390 1.380 1.384 1.380 1.390 1.429 1.190
1.391 1.391 1.382 1.384 1.384 1.382 1.434 1.189
nyl ring. Geometrical optimizations of both macrocycles have been performed at the HF level with the 3-21G basis set. In Table 1 we compare the geometries of the m-PEM and o-PEM and the phenylene ethynylene monomer used in last section. Because of the symmetry of the molecules it is enough to show the results for a monomer only. The numeration used in this table is shown in Fig. 4. Comparisons of the data presented in Table 1 show small dierences between the geometries of m-PEM, o-PEM and the monomer. The phenyl rings of the m-PEM and the monomer present a more pronounced aromatic character than for the o-PEM, in which their bond lengths go from 1.402 (C1±C2) to 1.380 A (C3±C4 and C5±C6). In the A ethylene group the single bond between atoms C2 and C7 is considerably longer in m-PEM (1.455 The triple bond between the carbons C7 and A). C8, that is also slightly shorter for the m-PEM as compared to o-PEM.
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BE 3-21G BE STO-3G monomer-pair BE 3-21G 3:7 BE STO-3G rr-o-PEM
4
Due to the large size of the rr-m-PEM, it was only possible to study the pair at the MP2/STO-3G level, leading to a BE 0:221 eV and Req 3:9 However, using the last expression we can esA. timate the BE of this pair as BE 3-21G 3:7 BE STO-3G 0:81 eV. Fig. 4. Numeration of the monomer referred in Table 1. The arrows point at the connection with the next monomer, m and o indicates ortho- and meta-connections, respectively.
Neutral pairs of PEMs were arranged parallel to each other in a p-stack, with the aromatic rings on top of each other as illustrated in Fig. 5 for the o-PEM. MP2 calculations using STO-3G and 321G basis sets of o-PEM pairs were carried out as function of the intermolecular distance R. Their BEs were BE(STO-3G) 0.103 eV and BE(3 and Req 3:7 A, 21G) 0.378 eV at Req 3:9 A respectively. Comparisons with the results obtained in the last section show that for both basis sets the BE of the o-PEM pair is larger than three times the BE of the monomer pair. This shows that the delocalization of the p electrons around the macrocycle backbone further stabilizes the dimer. The ratio between both BEs of the same system using the two basis set is approximately the same,
Fig. 5. Two o-PEM in a parallel p±p stack.
4. Helices and chains In this section we discuss the relative stability between oligomers of ortho- (recently synthesized [14]) and meta- [15] phenylene ethynylene chains and helices as function of the oligomer length n, in gas phase. The oligomers, connected in meta- and ortho-positions (see Fig. 6), were optimized at HF/ 3-21G level for dierent chain lengths, n (n 3 to 10). It is not possible to optimize the helical structures at the ab initio quantum chemistry level since the size of the system is too large to take correlation eects into account. From the results presented above we know that at the HF level the stacking interaction is repulsive, which leads to the formation of chains instead of helices. The helices were therefore built by folding the optimized planar chains, without further optimizations. The chains were folded in a cisoid conformation varying the torsional angle between consecutive phenyl rings, / (see Fig. 7). The torsional angles were / 5:8° for the m-helices and / 32° for the o-helices. The p±p stacking between phenyl groups of dierent loops of the built helices imitate the parallel p-stack of PEM (in which the intermolecular [6]). Each loop involves six distance is 3.5 A phenyl rings in the m-helices and three in the ohelices. The total energy of chains and helices has been evaluated using the MP2 method and the 321G basis set. Their relative stability as a function of the chain length was investigated for oligomers having from n 3±10 phenyl rings. The relative
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Fig. 6. Ortho- and meta-phenylene ethynylene oligomers (n 3±10).
Fig. 7. Examples of o- and m-helix.
stability between chains and helices is studied as function of the chain length. The dierence in MP2 energy between the helically ordered state and the unfolded (planar) chain, DEho , was calculated. The evolution of DEho as a function of the oligomer length n, for both systems is shown in Fig. 8. For the o-system, the helical structures are the most stable ones for n P 4. This oligomer lengths correspond to the o-helix having p overlap, i.e. pair of phenylene ethynylene units appear on top of each other as shown in Fig. 1. The energy difference DEho is 0.07 eV for n 4 and increase with n, but linearly only in each loop (see Fig. 8). The slope increase going from the second to the third loop, is consistent with the extension of the p-stack. For the m-system, the helical structure becomes the most stable ones for n 6. This structure has no overlaping phenylene ethynylene units. The n 6 m-helix is more stable than the mchain at both the HF and the MP2 levels, with the same energy dierence, DEho 0:02 eV. This suggests that this helical structure is not stabilized by
Fig. 8. MP2 energy dierence between helices and chains as a function of n for the m-system and the o-system. The onset for helical stability is given by DEho P 0. The continuous curves correspond to linear ®ttings of the data for each loop.
p±p interactions. Instead, hydrogen bonds with the aromatic end rings stabilize the six rings m-helix. For smaller structures the energy dierence between the chain and the helix is lower than 1 meV (favorable to the chains) which indicates that the small torsion (/ 5:8°) of the dihedral angles and cisoid conformations are practically unimportant in this context. Note that in the case of the o-system the energy cost due to torsion is much larger, as shown by the negative value of DEho for n 3. For n 7 the energy dierence is DEho 0:14 eV and increase linearly with n. The resulting stability of helical structures in which p±p interactions are present, here calculated at the MP2 level, (contrarily to the HF results)
M. Pickholz, S. Stafstrom / Chemical Physics 270 (2001) 245±251
shows that electron correlation eects play an important role for the intramolecular interactions. 5. Conclusions In this work, the role of the intramolecular and intermolecular interactions in phenylene ethynylene aggregates was studied. We have shown that aromatic p±p interactions could help the selfassociation of macrocycles in p-stacks and in the formation of helical structures. Electron±electron correlation eects were found to be responsible for the attractive interaction that lead to the aggregation in these systems. Because the size of the studied systems we had to compromise to include correlation eects at the MP2 level using in most of calculations the 3-21G basis set. In Section 1 we evaluate the eects of this approximation and we found that this combination leads to a qualitatively correct picture of the interaction. From the experimental point of view, the self organization of these systems involves a variety of noncovalent cohesive forces, including hydrogen bonds, electrostatic forces, steric packing and hydrophobic eects [2]. Besides, solvophobicity can be an eective way to drive the supramolecular organizations involving p-stacked structures [16]. However, in this study we focussed the attention to p±p stacked structures between equivalent units (i.e., phenyl ring phenyl ring stacking). The main motivation of this work has been, therefore, to point out the importance of the dispersive p±p interaction for the stability of three-dimensional structures of phenylene ethynylene. Acknowledgements Computational resources were provided by the Swedish Council for High performance Computing (NSC). Financial support from the Swedish Research Council for Engineering Science (TFR)
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and the Swedish Natural Science Research Council (NFR) is gratefully acknowledged.
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