π-π stacking of curved carbon networks: the corannulene dimer

�-� Stacking of Curved CarbonNetworks: The Corannulene Dimer

ANDRZEJ SYGULA, SVEIN SAEBØDepartment of Chemistry, P. O. Box 9573, Mississippi State University,Mississippi State, MS 39762

Received 17 April 2008; accepted 22 April 2008Published online 6 August 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21794

ABSTRACT: Dimers of corannulene, a curved, saucer shaped molecule, werestudied by theoretical calculations using second order Møller-Plesset perturbationtheory and a large polarized triple zeta basis set. Three dimer motifs were investigated:the “native” dimer is the concave-convex stacking of two monomers with thegeometries of both monomers conserved; the “planar” motif with both monomersforced to be planar; and the “C60-like” dimer where the outer monomer has the nativegeometry while the inner one has the curvature of buckminsterfullerene C60. Bothstaggered and eclipsed conformations of the dimers were investigated. Our calculationsshow that the binding energy of the native concave-convex corannulene dimer is quitesubstantial (17.2 kcal/mole at the “best” SCS-MP2/cc-pvtz level of theory) with anequilibrium distance of about 3.64 Å. Surprisingly, there are only minor differences inboth binding energies and equilibrium distances between the three different dimermotifs. This suggests that the curvature of the conjugated carbon networks does notdisable their ability to form �-� stacked assemblies similar to the planar systems.However, in contrast to the planar systems, at least part of the binding energies in thestacked curved systems can be attributed to attractive electrostatic dipole-dipolecontributions since buckybowls exhibit significant dipole moments. For the “planar”dimer, a staggered arrangement of the two monomers is preferred, while eclipsedconformations are the most stable for all curved dimers. For all systems, the basis setsuperposition errors are large (ca. 7 kcal/mol) at the equilibrium distance even with ourlargest basis sets. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem 109: 65–72, 2009

Key words: dispersion interactions; �-� stacking; MP2; corannulene; basis setsuperposition errors

Correspondence to: A. Sygula; e-mail: [email protected] or S. Saebø; e-mail: [email protected]

Contract grant sponsor: Chemical Sciences, Geosciences andBiosciences Division, Office of Basic Energy Sciences, Office ofScience, U.S. Department of Energy.

Contract grant number: DE-FG02–04ER15514.

International Journal of Quantum Chemistry, Vol 109, 65–72 (2009)© 2008 Wiley Periodicals, Inc.

Introduction

D ispersion interactions are important for awide range of chemical and physical proper-

ties. According to London’s classical formula, theinteraction energy is proportional to the product ofthe polarizabilities of the two interacting systems.Since �-electrons in extended conjugated systemsare easily polarizable, the dispersion interactionsbetween �-systems are relatively strong, and �-�interactions constitute an important part in manyphenomena including chiral chromatography [1],nucleic acid structure [2, 3], stabilities and tertiarystructures of proteins, protein crystallization [4–6],and molecular recognition [7–9].

The binding energy for the simplest �-stackedsystem, the benzene dimer, is about 2 kcal/mol [10,11] and potentially the interaction between largerstacked �-systems could be quite strong. A largenumber of both theoretical and experimental re-sults illustrating the importance of face-to-facestacking of planar aromatic and heteroaromaticmolecules in condensed phases have been reported[12–14].

Recent progress in fullerene chemistry intro-duced a novel aspect to the �-� stacking interac-tions, i.e., interactions of curved conjugated molecularnetworks [15–17] Indeed, several supramolecularcomplexes of fullerenes with aromatic systems likeporfirynes, calixarenes, etc. have been characterizedby X-ray crystallography and showed close contactsof significantly curved fullerene convex surfaceswith planar or bent aromatic and heteroaromaticmolecules. In this context buckybowls, i.e., bowl-shaped polycyclic aromatic hydrocarbons withcurved carbon networks representing fragments offullerenes offer an attractive possibility for forma-tion of stacking assemblies with fullerenes and witheach other since buckybowls posses both convexand concave surfaces as demonstrated below forthe smallest buckybowl, corannulene (1).

The potential of the concave–convex stackingmotif was recognized early and proposed as a ma-jor factor of gas-phase formation of dimeric species

like [12].� and [1*C60].� detected by Mass Spectrom-etry [18]. On the other hand, crystal structure stud-ies of buckybowls provided mixed results sincesome of the buckybowls crystallize in a concave-convex fashion while the others (including coran-nulene) do not exhibit any amount of that type ofstacking [17]. In addition, lack of experimental ev-idence for a strong complexation of fullerenes withbowl-shaped conjugated systems in solutions led tothe conclusion that “the attractive force of the con-cave-convex �-� interaction is not so significant, ifat all” [19]. Therefore, the question arises whetherthe curvature of a curved extended �-system wouldto some extent disable the potential for effectivestacking as compared with the planar conjugatedsystems.

Computational studies of �-� interactions in pla-nar aromatic systems have been well documentedin literature [20, 21]. In contrast, very few theoreti-cal studies addressing �-� interactions in curvedconjugated systems have been published. The onlycomputational study concerning corannulenedimer formation has been reported by Tsuzuki et al.who studied convex–convex dimer formation as amodel for �-� interaction between two C60 mole-cules [22].

To answer some of the questions concerning �-�stacking interactions between curved systems wecarried out ab intio calculations on several coran-nulene dimer motifs with a major emphasis on therelative importance of �-� interactions betweenbowl-shaped systems and those in planar aromaticsystems.

Computational Details

The corannulene monomer was optimized at theDFT level using the PBE1 [23] functional and the6-31G* basis set. As demonstrated earlier, this ge-ometry is virtually identical to the reported X-raystructure of corannulene [24] so further refinementsof the monomer geometry using more accuratemethods were not attempted. The same computa-tional model was used for geometry optimizationof the planar corannulene monomer. This structure,higher in energy by 9.3 kcal/mol, than the mini-mum energy bowl-shaped conformer, representsthe transition state for the bowl-to-bowl inversion.The “C60-like” corannulene (see later) was con-structed by cutting of the appropriate C20 fragmentfrom the PBE1/6-31G* optimized buckminster-fullerene, C60, structure and attaching 10 hydrogen

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66 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 1

atoms to the dangling bonds. The positions of thesehydrogen atoms were optimized keeping the car-bon network frozen. This highly distorted structureis less stable than the minimum energy form ofcorannulene by 59 kcal/mol at PBE1/6-31G* levelof theory.

Dispersion type interactions contribute signifi-cantly to the intermolecular forces in our targetsystems and a correlated method must be used fortheir proper description. Because of the size of thecorannulene dimer the MP2 method was initiallychosen. However, this method is known to overes-timate the correlation effect [25, 26] and we alsoemployed Grimme’s spin-component-scaled MP2(SCS-MP2) method [27]. The SCS-MP2 energy canbe easily extracted from MP2 calculations at essen-tially no extra cost. It has been demonstrated thatthe SCS-MP2 combined with the cc-pvtz basis setgives reliable results for anthracene dimers forwhich an experimental binding energy is available[28]. For the extensively studied benzene dimer, theMP2 results are both quantitatively and qualita-tively incorrect, while the results are significantlyimproved when the SCS-MP2 methods were used[29].

Initially, the calculations were carried out usingcc-pvdz and cc-pvtz basis sets [30]. Studies of ben-zene dimers clearly demonstrate the importance ofincluding diffuse functions in the basis set for thistype of system [20, 29]. However, both the aug-cc-pvdz and the aug-cc-pvtz were severely redundantfor the corannulene dimers and this prevented usfrom using these basis sets in our study. We werehowever able to use the aug-cc-pvdz basis set witha set of diffuse p-functions and a set of diffused-functions removed from the carbon and hydro-gen atoms, respectively. Throughout this article,this basis set will be referred to as aug(1)-cc-pvdz todistinguish it from the full aug-cc-pvdz basis. Inaddition, calculations were also carried out with theaug-cc-pvdz basis with only the set of diffuse p-functions removed from hydrogen atoms. The latterbasis set will be referred to as aug(2)-cc-pvdz. Thenumber of contracted basis functions (cbf) for thecorannulene dimer were 660, 840, 1,040, and 1,480for the cc-pvdz, aug(1)-cc-pvdz, aug(2)-cc-pvdz,and the cc-pvtz basis sets, respectively. The aug(2)-cc-pvdz (1,040 cbf) and the cc-pvtz (1,480 cbf) basisset gave similar results, however, in spite of thelarger number of basis functions, the cc-pvtz basisset was significantly more economical for the coran-nulene dimers since the augmented basis set exhib-ited very slow SCF convergence. The poor SCF

convergence with augmented basis sets has alsobeen noted by others [31].

All calculations of weak interactions are plaguedby basis set superposition errors (BSSE). We evalu-ated the BSSE by standard counterpoise calcula-tions [32]. The required BSSE corrections preventedthe use of gradient methods for the dimer calcula-tions, and a series of single point calculations withfrozen monomer geometries and the intermolecularseparations systematically varied between 3 and 4Å were carried out. The intermolecular separationswere defined as the distance between the centralfive membered rings. The results were fitted to aquartic function to determine the equilibrium dis-tance and binding energy.

The monomer calculations were carried out us-ing the Gaussian-03 program suite [33], while thedimer as well as the phantom basis calculationswere carried out using the PQS [34] program. Allcalculations were carried out on small (four CPUs)PQS Linux clusters.

Results and Discussion

CORANNULENE MONOMER

Corannulene, the smallest buckybowl, has C5vsymmetry and can be considered a slice of the top20 carbon atoms of buckminsterfullerene, C60, withthe 10 peripheral carbon atoms capped with hydro-gen atoms. The PBE1/6-31G* calculated bondlengths for the bonds denoted a through d beloware 1.4127, 1.3812, 1.4427, and 1.3853 Å, respec-tively, in perfect agreement with the X-ray data.These bond lengths suggest, as expected, significant�-delocalization in the corannulene system.

For the C60-like monomer, the bonds a through dwere 1.4473, 1.3908, 1.4473, and 1.3908 Å, respec-tively, and for the planar from 1.3936, 1.3644,1.4563, and 1.3995 Å. The curved monomers haveC5v symmetry while the planar from exhibit D2hsymmetry.

�-� STACKING OF CURVED CARBON NETWORKS

VOL. 109, NO. 1 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 67

Corannulene has a bowl-shaped geometry but itscurvature is significantly reduced in comparisonwith C60. The local curvature of a conjugated sys-tem can be conveniently described by the POAV(�-orbital axis vector) pyramidalization angles [35].The pyramidalization of carbon atoms in 1 is high-est in the central five-membered ring (Ch) with thePOAV of 8.4° while the five equivalent rim carbons(Cr) are much less pyramidalized and have POAVvalues of 3.8° only. Pyramidalization of the 10 CHrim carbon atoms (CrH) is quite marginal. In con-trast, the POAV angles for all 60 carbon atoms inbuckminsterfullerene C60 is 11.6°. The bowl depthof 1, defined by the distance between the plane ofthe hub five-membered ring and the plane of the 10rim carbon atoms, is 0.871 Å.

One of the consequences of the curvature ofcorannulene is its substantial dipole moment. Anexperimental value of 2.07 D has been reported [36]and our calculated values of 2.11 and 2.07 D atMP2/cc-pVTZ and SCS-MP2/cc-pVTZ levels, re-spectively, reproduce this value quite well. A di-pole moment of �2 D is rather large for a hydro-carbon, and it may explain some of the bindingproperties the corannulene dimers discussed later.The calculated dipole moments for the C60-likemonomer were 3.77, and 3.69 D at the MP2 andSCS-MP2 levels, respectively. The dipole momentvectors are aligned along the C5 symmetry axis andpoint toward the hub five-membered ring.

CORANNULENE DIMERS.

A series of single-point calculations at the MP2and SCS-MP2 level with cc-pvtz basis set and fro-zen monomer geometries were performed for threedifferent dimer motifs. Stacked concave–convexdimers of corannulene in the optimized bowl-shaped monomer geometry discussed above will bereferred to as the “native” dimers. To assess theeffect of the curvature of the conjugated systems ontheir ability to form �-� stacked assemblies we alsoconsidered the “planar” dimers, with both mono-mers forced to be planar. The planar corannulenemonomer represents in fact the transition state forthe bowl-to-bowl inversion of the system. Finally, athird type of the dimer with the inner corannuleneforced to adopt the curvature of the buckminster-fullerene molecule (C60) while the outer coran-nulene geometry was left unchanged (i.e., at theminimum energy of the monomer) will be referredto as “C60-like” dimer. This was included into ourstudy to mimic the concave-convex interaction be-

tween corannulene and C60. Since both “planar”and “C60-like” monomers do not represent mini-mum energy structures and have significantlyhigher energies than the minimum energy “native”form, the “planar” and “C60-like” dimers havemuch higher total energies than the “native”dimers. Side views of all three dimer motifs areshown in Figure 1.

For each of the three types of dimers we consid-ered two conformations, i.e., the eclipsed one, withall bonds in one monomer eclipsing the analogousbonds in the other monomer, and the staggeredform, in which one of the monomers is rotated by36° along the axis defined by the centers of both“hub” five-membered rings (see Fig. 1). The resultsof our calculations on the corannulene dimers aresummarized in Table I. The BSSE for these systemswere significant, about 7 kcal/mole at the equilib-rium distances with our largest basis set (cc-pvtz).All the results given in Table I have been correctedfor BSSE by standard counterpoise calculations.

Figure 2 shows the SCS-MP2/cc-pvtz calculatedinteraction energies of the eclipsed “native,” stag-gered “planar” and eclipsed “C60-like” corannulenedimers as a function of the separation distance be-tween the monomers. The calculated potentials arerather flat around the minimum energy distanceswhich vary between 3.4 Å for the staggered “pla-nar” dimer to 3.64 Å for the eclipsed “native”dimer. It should be pointed out that Figure 2 dis-plays interaction energies of the corannulene units

FIGURE 1. Graphical representations of the studiedmotifs and conformations of corannulene dimers.

SYGULA AND SAEBØ


and not the total energies of the dimers. The nativedimer which represents a minimum on the poten-tial energy surface, has the lowest total energywhile the planar dimer and the C60-like dimer aresignificantly less stable.

It can be clearly seen from Table I that the bind-ing energies are significantly underestimated when

the smaller, cc-pvdz, basis was used in comparisonwith the larger cc-pvtz results by both MP2 andSCS-MP2 methods. On average, the cc-pvtz bindingenergies are higher by ca. 45% than the cc-pvdzresults. As a result of the bonding underestimationby the smaller basis set, the equilibrium monomer–monomer distances are also systematically calcu-lated longer with cc-pvdz as compared with cc-pvtz.

Incidentally, cc-pvtz was the largest basis set thatwe were able to use for this system as addingadditional functions (e.g., diffuse functions) to thisbasis set resulted in a severely redundant basis setand numerical problems. However, as mentionedabove, we were able to perform calculations on thecorannulene dimers with the cc-pvdz basis set aug-mented with diffuse functions. Results using theaug(1)-cc-pvdz and aug(2)-cc-pvdz basis sets, de-scribed above, have been included in Table I. It canbe seen that the results with the larger aug(2)-cc-pvdz basis set are quite similar to those obtainedwith the cc-pvtz basis. The redundancy problems,discussed earlier, prevented us from making ex-trapolations and predictions about infinite basis re-sults in this study. However, the results in Table Isuggest that the effect of adding diffuse functions tois about the same as expanding the basis set fromdouble zeta to triple zeta quality.

There is strong evidence in the literature that �-�interaction energies are overestimated at the MP2level [25, 26]. In a recent benchmark study of the

TABLE I ______________________________________________________________________________________________Equilibrium distances (in Å) and counterpoise corrected binding energies (in kcal/mol) calculated using thecc-pvtz basis set (cc-pvdz results in parentheses).

Method Distance Binding energy Aug1a Aug2a

Native staggered MP2 3.50 (3.61) 25.7 (17.8) 22.6 26.1Native eclipsed MP2 3.45 (3.56) 27.7 (19.5) 24.8 28.0Planar staggered MP2 3.22 (3.33) 29.7 (20.6) 25.7 30.2Planar eclipsed MP2 3.51 (3.59) 21.1 (14.3) 18.5 21.8C60 like staggered MP2 3.41 (3.51) 23.4 (16.7) 20.5 23.5C60 like eclipsed MP2 3.26 (3.37) 27.8 (20.1) 24.4 28.0Native staggered SCS-MP2 3.67 (3.73) 15.7 (10.3) 14.0 16.8Native eclipsed SCS-MP2 3.64 (3.68) 17.2 (11.7) 15.9 18.4Planar staggered SCS-MP2 3.40 (3.48) 18.4 (12.4) 16.3 19.8Planar eclipsed SCS-MP2 3.67 (3.74) 13.0 (8.3) 11.6 14.2C60 like staggered SCS-MP2 3.61 (3.68) 14.1 (9.8) 12.8 15.2C60 like eclipsed SCS-MP2 3.45 (3.54) 17.0 (12.1) 15.3 18.2

aAug1 and Aug2 are binding energies calculated at the SCS-MP2/cc-pvtz equilibrium distances using the aug(1)-cc-pvdz andaug(2)-cc-pvdz basis sets, respectively.

FIGURE 2. BSSE corrected SCS-MP2/cc-pvtz inter-action energies of the staggered planar (circles),eclipsed native (squares), and eclipsed C60-like (trian-gles) corannulene dimers. [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com.]



benzene dimer it was demonstrated that MP2 didnot only overestimate the interaction energy but theresults were qualitatively wrong as MP2 failed topredict the correct order of the three studied formsof the benzene dimer [26, 29]. SCS-MP2 was muchbetter and when the results were corrected forBSSE, the method used in the present study (i.e.,SCS-MP2/cc-pvtz) appeared to perform quite sat-isfactorily, with slight underestimation of the inter-action energy [29]. Similar results were also ob-tained for the anthracene photo-dimer [28]. Inaccordance with the previous findings, the MP2calculated binding energies for all corannulenedimers are significantly larger than those obtainedusing SCS-MP2. Similarly, the minimum energyseparations of the corannulene subunits calculatedat the MP2 level are consistently shorter by 0.16–0.20 Å than the SCS-MP2 estimates. We considerthe SCS-MP2 calculated values as a more reason-able estimate of van der Waals interactions in thissystem.

In accordance to the earlier observation byHobza et al. [37], the MP2 binding energies calcu-lated with the smaller cc-pvdz basis set are quitecomparable to the SCS-MP2 energies obtained withthe larger cc-pvtz basis set due to the cancellation ofthe two errors—the overestimation of the dispersionforces by MP2 method and the their underestimationdue to the smaller basis set.

The most striking result of our study is that thecalculated binding energies of the three differentdimer motifs are rather similar (Table I). Using ourCP corrected SCS-MP2/cc-pvtz results, the stag-gered dimer of two planar corannulene subunits ispredicted to exhibit the strongest binding (18.4kcal/mol) but both the “native” and “C60-like”concave-convex dimers appear to be only slightlyless bonded (17.2 and 17.0 kcal/mol, respectively).It has been suggested that distortion of the ex-tended �-system in bowl-shaped molecules resultsin less favorable �-� interactions compared withplanar systems. Our results show that the curvedsystems stacked in a convex–concave fashion doindeed bind with energies quite comparable to theplanar conjugated carbon network of similar size.On the other hand, a significant stabilizing dipole–dipole component in the nonplanar dimers shouldbe present since the bowl-shaped corannulene ex-hibits a dipole moment of about 2 D. The planarcorannulene is nonpolar so the major electrostaticcontribution to the binding energy in the planardimer should come only from quadrupole–quadru-pole interactions while dipole–dipole interactions

may contribute significantly to the binding incurved concave–convex dimers. Interaction of two2.07 D dipoles aligned like in the convex–concave“native” dimer with the distance of 3.64 Å results inca. 2.6 kcal/mol binding due to dipole–dipole in-teraction. This accounts for ca. 15% of the bindingenergy calculated at SCS-MP2/cc-pvtz level (TableI). These results suggest that the dispersion interac-tions of the curved conjugated carbon networks inthe concave–convex stacked corannulene dimersare slightly weaker than in the planar systems ofthe same size, but because of non-zero dipole mo-ments of the curved components their actual dimerbinding energies are quite comparable with the pla-nar analogs. Therefore, the statement that “the at-tractive force of the concave-convex �-� interactionis not so significant, if at all” cannot be supportedby the results of our study [19]. Quite recently wehave reported a synthesis of a molecular clip whichcontains two corannulene subunits which make astrong inclusion complex with C60 both in the solidstate and in solution providing an experimentalevidence that the dispersion interaction betweenthe curved conjugated carbon surfaces are indeedimportant and can play a significant role in su-pramolecular chemistry of these systems [38]. Inter-action energies calculated at the Hartree–Fock andDFT levels were strongly repulsive (�15 kcal/mol)in spite of the large dipole moments for the curvedsystems. The electron correlation effects were largeand attractive leading to significantly attractivebinding energies for all dimers.

Interestingly, we found rather significant bind-ing energy differences between eclipsed and stag-gered dimers of corannulene. This is in contrast tothe results of the previous studies of the benzenedimer which concluded that for the “sandwich”dimer the eclipsed and staggered conformationshad virtually the same energies [20]. We found themost significant difference between the staggeredand eclipsed form in the “planar” dimer, where thestaggered conformation is calculated to be 5.4 kcal/mol more stable than the eclipsed conformer. Theequilibrium distances defined by the separation ofthe planes of the two “hub” five-membered rings ofthe monomers are also significantly different for thetwo conformers of the “planar” dimer (Table I). Astraightforward explanation for this is that while inthe eclipsed conformation there are only 20 pairs ofcarbon atoms in the attracting van der Waals range(3.67 Å, the optimum distance calculated for theeclipsed “planar” dimer) the number of such COCcontacts is significantly increased in the staggered

SYGULA AND SAEBØ


conformation. For example, at the equilibrium dis-tance of 3.40 Å each of the five “hub” carbon atoms(Ch, see above) is now in a close contact (3.48 Å)with two “hub” carbon atoms of the second mono-mer. Also, the staggered conformation brings eachof the five Cr carbon atoms into van der Waalscontact of 3.55 Å with two rim CrH carbon atoms inthe opposite monomer. The analogous distance inthe eclipsed conformer is 3.95 Å. As a result, thedispersion attraction is stronger in the staggered“planar” dimer.

Surprisingly, the staggered vs. eclipsed prefer-ence is reversed in the concave-convex “native” and“C60-like” dimers. The eclipsed conformations aremore stable by 1.5 and 2.9 kcal/mol, respectively. Acloser examination of the minimum energy struc-tures of the staggered dimers provides the explana-tion for this behavior. Curvature of the corannulenesubunits brings Ch and Cr carbon atoms of the inner(“bottom,” Fig. 1) monomer closer to Cr and CrH

carbons of the outer (top) corannulene. In the caseof the staggered “native” dimer, five Cr carbonatoms of the “bottom” corannulene subunit get toclose to the 10 CrH carbons of the “top” monomer(only 3.47 Å at the optimized separation of 3.67 Å)and prevent the optimal intermolecular contacts tobe reached by the remaining carbon atoms in thedimer. For example, Ch-Ch, Cr-Cr and CrH-CrH car-bon-carbon distances (3.74, 3.98, and 3.73 Å, respec-tively) are significantly longer in the staggered “na-tive” dimer than in its eclipsed conformation (3.64Å) due to this effect.

Finally, we are surprised to find that the calcu-lated binding energies of the “native” and “C60-like” dimers are virtually identical despite the dras-tic difference of the curvature of their “bottom”portion. Both these dimers prefer eclipsed confor-mations and the separation distances of the mono-mers differ significantly. Not surprisingly, the sep-aration of the monomers in the “C60-like”dimer issmaller since the increased curvature of the “bot-tom” component allows the “top” monomer to getcloser. However, the misfit of the curvatures doesnot allow for an efficient van deer Waals attraction.The shortest Ch-Ch, Cr-Cr, and CrH-CrH intermolec-ular carbon-carbon distances in the eclipsed “C60-like”dimer are 3.45, 3.66, and 4.07 Å, respectively,while they are uniformly 3.64 Å in the eclipsed“native” dimer. Anyway, the very similar bindingenergies of the “native” and “C60-like”dimers sug-gests that the curved-surfaced conjugated systemslike corannulene could form relatively strong �-�

stacked assemblies with fullerenes of various sizesand curvatures.

Conclusions

Our calculations demonstrate that the interac-tions between curved �-systems are of comparablemagnitude as for planar systems of the same size.Our best calculations suggest that the interactionenergy for the corannulene dimer is about 17 kcal/mole and an equilibrium distance of a about 3.6 Åand that �-� stacking of curved surfaces appears bean important factor in the supramolecular chemis-try of curved conjugated systems. However, a sig-nificant part of binding energies of the bowl-shapedsystems can attributed to the electrostatic dipole–dipole attraction which is absent in “planar” dimers.For the curved corannulene dimers eclipsed confor-mations are preferred while a staggered conforma-tion is preferred for planar dimers. The basis setsuperposition errors are large, (about 7 kcal/mol atthe equilibrium distances), even with the cc-pvtzbasis set.

ACKNOWLEDGMENTS

We thank Dr. Debbie Beard, Mississippi StateUniversity, for insightful comments.

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π-π stacking of curved carbon networks: the corannulene dimer

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