J. Phys. Chem. C 2008, 112, 4061-4067
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A Prototype for Graphene Material Simulation: Structures and Interaction Potentials of Coronene Dimers Yan Zhao and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, UniVersity of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431 ReceiVed: NoVember 15, 2007; In Final Form: December 28, 2007
Graphene sheets are the building blocks of carbon nanotubes and a variety of functionalized nanomaterials. Methods to be used for computer-aided design of such materials or for the study of aromatic-aromatic interactions in biopolymers and other soft materials should be validated for smaller systems where reliable estimates of interaction energies are available. In this work, we first validated the M06-2X functional against the S22 database of noncovalent interaction energies of biological importance. We then applied the M06-2X functional to study aromatic-aromatic interactions in coronene dimers. We located six stationary points on the potential energy surface of coronene dimer, we calculated the potential energy curves for the sandwich, T-shaped, and parallel-displaced configurations of this prototype of aromatic-aromatic interactions, and we found that a parallel displaced configuration is the global minimum. The potential curves for the coronene dimers will aid the development of new force fields and potential energy functions that are computationally efficient and capable of modeling large graphene or aromatic clusters.
1. Introduction Aromatic-aromatic interactions play key roles in molecular recognition,1-4 protein folding,2,5-9 stacked nucleobases,10-12 drug13-16 and pesticide17 intercalation, nonlinear optical materials,18-20 crystal packing,21-23 self-assembly,24-28 soot,29-33 solvation,34-36 partitioningoforganicpollutantsintheenvironment,37-43 supramolecular chemistry,44-48 catalysis,49 and soft materials. Coronene (C24H12) is a highly symmetric polyaromatic hydrocarbon (PAH) consisting of seven perifused benzene rings.50 It is of particular interest in astrophysics because coronene has been identified as the most representative PAH that constitutes the carbon grains occurring in the interstellar medium,51 and it has also been identified in the atmosphere of Jupiter.52 Astronomical observations of bright PAH clusters suggest that up to 20% of the carbon in the universe could be tied up in PAHs.53 Moreover, coronene is of biological interest because most PAHs are carcinogenic and they can cause DNA damage in mammals. An emerging technological area is focused on the electrical properties54-58 of graphene sheets and similar carbon-based nanostructures, including hydrogen-terminated graphene nano ribbons with widths in the 1-3 nm range;56 these kinds of sheets and nanostructures provide the building-block materials for a myriad of anticipated new applications of functionalized nanomaterials. Planar PAHs with only six-membered rings, such as coronene, provide molecular models of graphene sheets, and, for this reason, planar PAHs should be of increasing interest. Carbon nanotubes can be self-assembled as rolled-up graphene,59 and multiwalled carbon nanotubes can self-assemble as rolledup graphene layers60,61 interacting by aromatic-aromatic noncovalent interactions. Electrical contacts between carbon nanotubes and graphene surfaces are also important for engineering electronic devices.62 Coronene dimers63-69 are a model system for studying the interactions between graphene sheets. * Corresponding author.
Because of the low vapor pressure of coronene and the anharmonicity of coronene-coronene interactions, it is difficult to study the coronene dimer in the laboratory. Therefore more information is available from electronic structure theory than from experiment. Both wave function theory (WFT) and density functional theory (DFT) may be considered. Coupled cluster theory70,71 with single and double excitations and a quasiperturbative treatment of connected triple excitations (CCSD(T)) and Møller-Plesset second-order perturbation theory72 (MP2) theory are two widely employed forms of WFT. Although the approach of directly estimating complete basis set (CBS) limit CCSD(T) has been successfully applied to study small van der Waals dimers,73-75 it is prohibitively expensive for the study of coronene dimers. Even for smaller systems (e.g., benzene dimers), a more indirect approach is required. One method to study π-π interactions is to combine the MP2 CBS limit with a correction (∆CCSD(T)) computed in a smaller basis set to estimate the difference between MP2 and CCSD(T).28,76-78 The importance of the ∆CCSD(T) correction for π-π interactions is illustrated by the demonstration79 that MP2/CBS is less accurate than the M05-2X80 density functional for this type of noncovalent interaction. The ∆CCSD(T) correction method has been used by Jurecka et al.81 to create the S22 database. The S22 database is a data set of 22 interaction energies for weakly bonded molecular complexes of biological importance. Jurecka et al. divided the S22 set into three subsets, namely, seven hydrogen-bonded complexes, eight dispersion-dominated complexes, and seven mixed complexes. The structures of these noncovalent complexes are shown in the Supporting Information. We will use these data in the present work to validate a density functional, M06-2X, that is affordable for coronene dimers. Although it is beyond our scope to test the wide variety of other density functionals that have been proposed, we note that, until recently, most density functionals were inaccurate for dispersiondominated weak interactions and aromatic-aromatic interac-
10.1021/jp710918f CCC: $40.75 © 2008 American Chemical Society Published on Web 02/28/2008
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tions. A succinct overview, with 20 references, of techniques suggested to overcome these deficiencies has recently been provided by Sato et al.82 In the literature, there are many theoretical studies of the interactioninbenzenedimers28,64,66,69,76-78,80,83-88 andPAHs,63-69,89-97 but studies of coronene dimers are scarcer.63-69 Furthermore, as summarized in a recent review,98 simulation studies of nanoscale carbon systems have usually been carried out with unreliable or low-level methods. In the present work, we investigate the energetics, geometries, and potential energy surfaces (PESs) of the coronene dimers with a validated density functional. The reliable results thus produced for coronene dimers will be helpful in the calibration of molecular mechanics63,67,92,99,100 and tight binding101 methods and potential energy functions for the simulations of large PAH clusters. The paper is organized as follows. Section 2 describes the computational methods used in the present work. Section 3 presents results and discussion, and Section 4 has concluding remarks. 2. Theoretical Methods and Computational Details In this work, two recently developed density functionals, namely M06-L102 and M06-2X,103 are employed for the study of coronene dimers. The details of these functionals are given in previous papers,102,103 and we simply note here that the work builds on previous work by Becke,104-106 Perdew,107,108 and Scuseria109 and their collaborators. M06-L is a local functional for main-group thermochemistry, thermochemical kinetics, noncovalent interactions, and transition metal chemistry, whereas M06-2X is a hybrid one with 54% Hartree-Fock (HF) exchange in the functional. Three basis sets, namely MIDI!, DIDZ, and MG3S, are employed in the present study. MIDI!110 is a well-balanced and economical double-ζ basis set that gives reasonably good molecular geometries and partial atomic charges. MIDI! is sometimes called MIDIX. DIDZ is a short name for an augmented polarized valence double-ζ basis set conventionally denoted as 6-31+G(d,p);111 we use the shorter name because it has seven less characters, which is helpful in both text and table headings. MG3S is an augmented polarized valence triple-ζ basis set,112 which is the same as the 6-311+G(2df,2p) basis set111 for molecules containing atoms no heavier than Si. We located six stationary points on the PES of coronene dimer (see Figure 1), including one sandwich (S) dimer, one twisted sandwich (TS) dimer, two T-shaped (T) dimers, and two parallel-displaced (PD) dimers. Potential energy curves (PECs) were computed as a function of the intermonomer distance R for the S and T-1 configurations and of R2 for the PD-1 configuration (see Figure 2). For the calculations of PECs, the coronene monomer is optimized at the M06-2X/DIDZ level, and the monomer geometry is frozen for the calculation of PECs. All DFT calculations were carried out using locally modified Gaussian 03113 and NWChem114 computer programs. We performed calculations both with and without counterpoise (Cp) corrections115,116 for basis set superposition error (BSSE). 3. Results and Discussion 3.1. Validation of the M06-2X Functional. In the BornOppenheimer approximation, the PES equals the fixed-nuclei electronic energy plus nuclear repulsion, and in Kohn-Sham DFT, the electronic energy has three components: noninteracting kinetic energy, classical electrostatic energy, and exchangecorrelation energy. Many studies80,117-121 have shown that conventional density functionals such as B3LYP are problematic
Figure 1. Sandwich (S and TS), T-shaped (T-1, T-2), and paralleldisplaced (PD-1, PD-2) conformers of coronene dimers.
for the description of aromatic-aromatic interactions. One way to improve the performance of DFT is to augment the DFT energy by a special dispersion term (in the functional or added to the energy) that yields the correct asymptotic form -C6R-6 (plus possibly higher order terms, if the multipole expansion is not truncated at the first term,).82,86,88,122-128 An example of such a DFT plus dispersion approach is the DFT-D approach (e.g., BLYP-D66 and B97-D69,88), which has been shown to be very successful in various applications.66,69,88,128-131 The M06-2X functional103 developed in our group is based on a different approach, namely, a hybrid meta exchange-correlation functional has been optimized to give good performance for a broad range of properties including noncovalent interactions. No molecular mechanics terms or special dispersion terms proportional to R-6 (nor any other molecular mechanics (MM) terms) are added; instead the attractive noncovalent interactions are produced by the hybrid meta exchange-correlation functional. Although the M06-2X exchange-correlation functional is not valid in the region where the potential curves vary as -C6R-6 (the longrange dispersion limit, which is important for elastic scattering in molecular beams), the M06-2X functional and its precursor M05-2X functional80 have been shown to provide good accuracy in the vicinity of van der Waals minima.79 Note that such minima occur where the gradient of the repulsive potential is exactly cancelled by the gradient of the attractive potential (for any method of apportioning the interaction energy into attractive and repulsive components), and the overlap-dependent exchange-
Structure/Interaction Potential of Coronene Dimers
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Figure 2. Definitions of the intermolecular distances in PD (R1, R2) and in S and T (R) conformers.
TABLE 1: Validation of the M06-2X Method against the S22 Noncovalent Interaction (kcal/mol) Database of Biological Importance WFT geometriesa M06-2X/DIDZ complex
best estimate
(NH3)2 (H2O)2 formic acid dimer formamide dimer uracil dimer 2-pyridoxine‚2-aminopyridine adenine‚thymine WC MSEb MUEb MMUE-HBb
-3.17 -5.02 -18.61 -15.96 -20.65 -16.71 -16.37
(CH4)2 (C2H4)2 benzene‚CH4 benzene dimer pyrazine dimer uracil dimer indole‚benzene adenine‚thymine stack MSEb MUEb MMUE-Db
-0.53 -1.51 -1.50 -2.73 -4.42 -10.12 -5.22 -12.23
ethene‚ethyne benzene‚H2O benzene‚NH3 benzene‚HCN benzene dimer bndole‚benzene T-shape phenol dimer MSEb MUEb MMUE-mixb AMUEc
-1.53 -3.28 -2.35 -4.46 -2.74 -5.73 -7.05
MP2/CBS
Cp
no Cp
M06-2X/DIDZ geometries M06-2X/MG3S Cp
no Cp
M06-2X/DIDZ Cp
no Cp
M06-2X/MG3S Cp
no Cp
Hydrogen Bonded Complexes -3.20 -3.98 -4.24 -3.30 -3.42 -5.03 -5.73 -6.54 -5.06 -5.52 -18.60 -18.18 -19.08 -18.57 -19.22 -15.86 -15.39 -15.98 -15.39 -15.79 -20.61 -19.36 -20.15 -19.23 -19.77 -17.37 -15.74 -16.42 -15.45 -15.93 -16.54 -15.17 -15.95 -14.90 -15.43 -0.10 0.42 -0.27 0.66 0.20 0.15 0.85 0.61 0.70 0.59 0.73 0.65
-4.07 -4.37 -5.81 -6.68 -19.25 -20.18 -15.48 -16.07 -19.47 -20.25 -16.38 -17.08 -15.80 -16.59 0.03 -0.67 0.70 0.79 0.74
-3.31 -3.45 -5.14 -5.60 -19.54 -20.22 -15.39 -15.78 -19.28 -19.81 -16.11 -16.61 -15.49 -16.04 0.32 -0.14 0.66 0.56 0.61
Dispersion-Dominated Complexes -0.51 -0.52 -0.53 -0.44 -0.47 -1.62 -1.37 -1.51 -1.55 -1.64 -1.86 -1.32 -1.44 -1.38 -1.56 -4.95 -2.51 -2.98 -2.71 -3.24 -6.90 -4.18 -4.85 -4.21 -4.84 -11.39 -10.06 -11.28 -9.95 -10.91 -8.12 -4.52 -5.26 -4.69 -5.48 -14.93 -12.63 -14.15 -12.40 -13.57 -1.50 0.15 -0.47 0.12 -0.43 1.51 0.24 0.48 0.17 0.45 0.36 0.31
-0.56 -0.58 -1.46 -1.62 -1.32 -1.46 -2.57 -3.03 -4.21 -4.88 -10.91 -12.35 -4.64 -5.37 -12.82 -14.41 -0.03 -0.68 0.32 0.69 0.51
-0.50 -0.53 -1.68 -1.79 -1.40 -1.59 -2.75 -3.27 -4.23 -4.85 -10.81 -11.94 -4.79 -5.56 -12.58 -13.80 -0.06 -0.63 0.25 0.63 0.44
-1.37 -1.54 -3.60 -4.13 -2.41 -2.74 -4.62 -4.93 -2.34 -2.67 -4.91 -5.41 -7.11 -8.07 0.11 -0.34 0.28 0.45 0.37 0.44 0.64
-1.31 -1.41 -3.73 -4.19 -2.42 -2.67 -4.79 -5.10 -2.32 -2.67 -4.98 -5.44 -6.75 -7.36 0.12 -0.24 0.36 0.38 0.37 0.42 0.52
-1.69 -3.61 -2.72 -5.16 -3.62 -7.03 -7.76 -0.64 0.64 0.76
Mixed Complexes -1.37 -1.52 -3.44 -3.90 -2.31 -2.65 -4.62 -4.91 -2.33 -2.67 -4.90 -5.42 -6.79 -7.76 0.20 -0.24 0.29 0.35 0.32 0.46 0.48
-1.32 -1.41 -3.49 -3.88 -2.34 -2.59 -4.78 -5.09 -2.32 -2.67 -4.99 -5.45 -6.53 -7.15 0.20 -0.15 0.35 0.29 0.32 0.41 0.44
The best estimates of WFT geometries from Jurecka. b MSE: mean signed error; MUE: mean unsigned error; MMUE ) 0.5(MUE-Cp + MUE-noCp). c AMUE is the average of the three MUEs, each weighted by one-third. a
repulsion terms are not negligible. The aromatic-aromatic noncovalent attraction in our calculations then arises from the correlation part of the exchange-correlation functional. In previous work,79,132 we have shown that the medium-range noncovalent interaction present in the M05-class and M06-class functionals that we have developed (and that require no multipole expansions, damping, or cutoffs) also results properly from the correlation functional, which is an important finding since some earlier functionals predicted a spurious binding due to the exchange functional. Thus, as compared to using classical force fields to study structural transformations of carbon-based
nanostructures,133 it is very appealing to use a formulation like the present one where the repulsion and attraction effects (beyond those due to noninteracting kinetic energy and classical electrostatics) are treated consistently with the same exchangecorrelation functional. Table 1 presents the results of M06-2X calculations for the S22 database. M06-2X gives good performance for all three types of noncovalent interactions both with WFT geometries and with M06-2X/DIDZ geometries. M06-2X/MG3S is slightly more accurate than M06-2X/DIDZ. Since M06-2X optimization with large basis sets is very expensive for the optimization of
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TABLE 2: Intermolecular Distancesa in Coronene Dimers (Å) for Six Configurations of the Coronene Dimer dimers
distance
M06-2X/DIDZ
M06-L/MIDI!
PD-1
R1 R2 R1 R2 R R R R
3.32 1.76 3.33 1.45 3.66 7.14 7.17 3.41
3.33 1.74 3.32 1.46 3.60 7.02 7.04 3.42
PD-2 S T-1 T-2 TS a
TABLE 3: Binding Energies (kcal/mol) for Six Configurations of the Coronene Dimer M06-2X/DIDZ// M06-L/MIDIX dimers PD-1 PD-2 S T-1 T-2 TS
M06-2X/DIDZ// M06-2X/DIDZ
M06-2X/MG3S// M06-2X/DIDZ
Cp
no Cp
Cp
no Cp
Cp
no Cp
17.91 17.31 9.04 3.09 3.06 17.19
21.13 20.54 12.17 4.35 4.31 20.57
18.35 17.64 9.27 3.41 3.39 17.31
21.70 20.87 12.32 4.59 4.56 20.76
18.55 17.88 9.77 3.30 3.29 17.59
21.73 20.98 12.64 4.36 4.34 20.92
See Figure 2 for the definitions of the intermolecular distances.
coronene dimers, we will use M06-2X/MG3S//M06-2X/DIDZ, where A/B//C/D denotes (as usual) geometry optimization with method C and basis set D followed by a single-point energy calculation with method A and basis set B. A close inspection of the M06-2X/MG3S//M06-2X/DIDZ results in Table 1 reveals that, for the dispersion-dominated complexes and mixed complexes, Cp-corrected results give better performance than uncorrected ones, whereas Cp-corrections slightly deteriorate the performance for the hydrogen-bonded complexes. Overall, Cp-corrected M06-2X/MG3S//M06-2X/DIDZ performs better than uncorrected calculations. An encouraging result is that the Cp-corrected M06-2X/DIDZ //M06-2X/DIDZ calculations give almost equally good performance as the Cp-corrected M06-2X/ MG3S//M06-2X/DIDZ calculation. For very large systems, M06-2X/DIDZ is a much more affordable level of theory than M06-2X/MG3S. Comparing MP2/CBS to M06-2X, we can see that, although MP2/CBS gives very accurate results for hydrogen bonds, it is less accurate than M06-2X for dispersion-dominated complexes and mixed complexes. Averaging over three types of interactions, as shown by the AMUE of Table 1, M06-2X is more accurate than MP2/CBS. Comparing M06-2X to other DFT-based approaches in the literature, we found that M06-2X has comparable accuracy to B97-D,129 TPSS-D,124 and LC-BOP+ALL.82 The B97-D and TPSS-D methods involve adding dispersion corrections with empirically determined coefficients and short-range damping, and LC-BOP-ALL involves a post-SCF van der Waals correlation functional based on a cutoff criterion and an empirical damping function. 3.2. Geometries of the Stationary Points. Six stationary points have been located by optimizations at the M06-2X/DIDZ and M06-L/MIDI! levels. The intermolecular distances are listed in Table 2. The values of 3.32-3.33 Å for the vertical separation between planes in the PD configurations and 3.41-3.42 Å in the TS configurations agree well with the observation134 that, in crystals, many aromatic molecules form stacks with approximately parallel molecular planes separated by 3.3-3.6 Å. For the PD-1 configuration, Ruuska and Pakkanen’s65 MP2/631G(d) calculation gave a vertical separation of 3.41 Å, which is slightly larger than the distances from our M06-L/MIDI! and M06-2X/DIDZ optimizations, and Grimme’s BLYP-D/TZV(2d,2p) calculation66 (without Cp) gave 3.40 Å. The consistency of our calculated value with previous WFT calculations and DFT with empirical C6 terms is encouraging. For each stationary point in Table 2, M06-L/MIDI! gives similar intermolecular distances to those calculations with M062X/DIDZ. This is a very important practical observation because M06-L/MIDI! is much more affordable than M06-2X/DIDZ for geometry optimization, since M06-L is a local functional (that is, it has no HF exchange), and MIDI! is a much smaller basis sets than DIDZ. For example, for coronene dimer, DIDZ involves 1032 contracted and 1704 primitive Gaussians, whereas MIDI! involves 480 contracted and 792 primitive Gaussians.
3.3. Energetics of the Stationary Points. Table 3 presents the energetics for the six stationary points obtained from three levels of theory, namely, M06-2X/DIDZ//M06-L/MIDI!, M062X/DIDZ//M06-2X/DIDZ, M06-2X/MG3S//M06-2X/DIDZ. The three approaches give consistent results. Among the six stationary points, PD-1 has the lowest energy, and the Cp-corrected M06-2X/MG3S//M06-2X/DIDZ calculation gives a binding energy of 18.55 kcal/mol, which compares well with the value of 18.66 kcal/mol from the Cp-corrected MP2/6-31G* calculations.65 The PD-2 and TS dimers have very similar binding energies, and they are about 0.8-1.0 kcal/mol less stable than the PD-1 dimer. It is surprising that the binding energy in the TS structure is about 8 kcal/mol greater than that in S, because this situation is very different from that in benzene dimers, where internal rotations give an energy difference of less than 0.1 kcal/ mol.76 The T-shaped dimers have the lowest binding energies; this is also different from the results in benzene dimers.76 The relative energies of various coronene dimer stationary points were also computed by Marsec64 and Rapacioli,67 who used less accurate semiempirical methods. Marsec found the sandwich to be lowest, and Rapacioli et al. found the twisted sandwich to be lowest, with a slightly stronger binding (22.7 kcal/mol) than PD-1 (22.4 kcal/mol). We too find that TS and PD-1 are close in energy (within ∼0.9 kcal/mol), but the sandwich is high enough in energy (see Table 3) to be excluded as a candidate for the global minimum. Table 3 also shows that the least expensive method, M062X/DIDZ//M06-L/MIDI!, gives results very similar to those of the most expensive method in Table 3. This is encouraging because M06-2X/DIDZ//M06-L/MIDI! can be employed for very large systems, as shown in a recent application47 to hostguest supramolecular chemistry involving C60, C70, nanotubes, and nanorings. Our nominally most accurate estimate of the coronene dimer binding energy would be the average of the largest-basis results with and without Cp (18.55 and 21.73 kcal/mol), which yields 20.1 kcal/mol. This may be compared to previous DFT-D estimates employing empirical corrections for dispersion interactions, which yield 21.6 kcal/mol66 (BLYP-D) and 21.5 kcal/ mol69 (B97-D at an empirical geometry). The difference of ∼7% is comparable to the typical error for smaller dispersiondominated complexes in Table 1, and it might be an indication that the present results underestimate the interaction energy, but it might also result from an overestimation of the interaction energy by the semiempirical DFT-D methods (e.g., B97-D overestimate the exfoliation energy of graphite by 27 ( 13%69). 3.4. Potential Energy Curves. The PECs for the S, T-1, and PD-1 configurations of the coronene dimer are plotted in Figures 3, 4, and 5, respectively. The shapes of these curves are very similar to the PECs of the S, T, and PD configurations of the benzene dimers, and hence the shape requires no discussion, but the magnitudes of the interaction strengths are much larger than those in benzene dimers due to the larger size of the system.
Structure/Interaction Potential of Coronene Dimers
Figure 3. PEC for the sandwich configuration of coronene dimer at the M06-2X/DIDZ level of theory as a function of interplane spacing R.
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Figure 5. PEC for the PD-1 configuration of coronene dimer at the M06-2X/DIDZ level of theory as a function of displacement distance R2 with interplane spacing R1 ) 3.35 Å (see Figure 2).
respectively. Thus, the in-plane charge polarization effect is expected to be negligible for misaligned graphene sheets. 4. Concluding Remarks
Figure 4. PEC for the T-1 configuration of coronene dimer at the M06-2X/DIDZ level of theory as a function of interplane spacing R.
3.5. Charge Polarization. Although the interaction energy between the coronene monomer is dominated by medium-range correlation energy, there is also an electrostatic component, including charge polarization due to dimer formation. To study this, we calculated partial atomic charges in the PD-1 structure by Charge Model 4135 (in particular, CM4/M06-2X/6-31G(d)// M06-2X/DIDZ). In the monomer, the charges are 0.000 on the six central carbons, -0.01 on the other carbons without attached hydrogens, -0.07 on carbons with attached hydrogens, and +0.07 on hydrogens. In PD-1, the average absolute value of the changes in these charges are 0.004, 0.004, 0.007, and 0.003,
At van der Waals minima, the attraction in polyaromatic carbon ring systems is due primarily to medium-range correlation energy,136,137 and the gradient of the attractive potential is equal and opposite to the gradient of short-range repulsive forces. Density functionals such as M06-2X that provide a good account of medium-range correlation energy are well suited to providing a realistic balance between medium-range correlation energy, short-range repulsion, electrostatics, and hydrogen bonding. The ability to balance all four kinds of features will be especially important when one considers carbonaceous and graphitic materials with heteronuclear substitution added for functionality. Here we consider only prototype hydrocarbons, so there is no hydrogen bonding, and we have shown that electrostatics play a reduced (but nonzero) role as compare to systems containing polar hydrogen or a heteroatom. This work first validated the M06-2X functional against the S22 database of noncovalent interaction energies of biological importance. It then applied the M06-2X functional to study the aromatic-aromatic interactions in the coronene dimers. We located six stationary points on the PES of coronene dimer, and we find that a parallel displaced configuration is the global minimum. We have calculated the PECs for the S, T-1, and PD-1 configurations of the prototype of aromatic-aromatic interactions. These curves for the coronene dimers will aid the development of new force field and potential energy functions that are computationally more affordable than direct electronic structure calculations and are capable of modeling large graphene structures or aromatic clusters. Acknowledgment. This work was supported in part by grant CHE07-04974 from the National Science Foundation (complex systems), by the Office of Naval Research under award number N00014-05-0538 (software tools), and by EMSL at PNNL (Computational Grand Challenge grant).
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