Optimal π-Stacking Interaction Energies in Parallel-Displaced Aryl

(10, 12) For instance, MP2 is known to routinely overestimate the strengths of these interactions. ... the strengths and weaknesses of different dimer...
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J. Phys. Chem. A 2010, 114, 9205–9211

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Optimal π-Stacking Interaction Energies in Parallel-Displaced Aryl/Aryl Dimers are Predicted by the Dimer Heavy Atom Count John M. Sanders* Department of Chemistry Modeling and Informatics, Merck Research Laboratories, West Point, PennsylVania 19486 ReceiVed: December 22, 2009; ReVised Manuscript ReceiVed: June 19, 2010

It is generally accepted that large basis sets are required for the accurate calculation of interaction energies for weakly bound aryl dimers, but it has recently been reported that MP2(full)/6-31G* energies, although inaccurate in absolute terms, are well-correlated with estimated CCSD(T)/CBS results. It is now shown that this correlation holds for MP2/aug-cc-pVDZ and SCS-MP2/aug-cc-pVDZ values. Linear regression of published CCSD(T)/CBS results with MP2 or SCS-MP2 results has been used to correct systematic errors observed with both MP2 theories, and these corrections are applied to 27 parallel-displaced aromatic dimers of interest in medicinal chemistry. The optimal computed interaction energies are found to be strongly correlated with the heavy atom counts of the aryl/aryl dimers. This relationship between heavy atom count and interaction energy also applies to a series of 12 aryl/non-aryl dimers such that a single linear regression equation accounts for all of the dimers studied. Introduction The calculation of aryl/aryl dimer interaction energies has been a subject of considerable interest for the last two decades.1-15 A number of theoretical studies have investigated the relative strengths of different geometries for the prototypical aryl dimer, the benzene dimer (Figure 1), since experimental results supported multiple gas phase geometries16-18 and early theoretical studies1,2 did not reach a consensus about the preferred geometry. The fact that articles published as recently as 2007 are devoted to the benzene dimer and its potential energy surface12 illustrates the subtleties of these systems, especially considering that early ab initio investigations were carried out in 1983.1 It is now thought that the C2V T-shaped and C2h parallel-displaced structures (Figure 1A,B) are quite close in energy and that the D6h sandwich geometry (Figure 1C) is less stable,10,12 but this understanding has only recently been reached through the application of increasingly sophisticated computational methods. Unfortunately, this has significant implications for the broad study of substituent effects since many more accessible methods are well-documented to have accuracy shortcomings for these systems.10,12 For instance, MP2 is known to routinely overestimate the strengths of these interactions.3-5,7,10,13 While recent progress with spin component-scaled MP219 (SCSMP2) has shown improvement in absolute accuracies,19,20 CCSD(T) and QCISD(T) are accepted as the benchmark methods.10,12 Of course, the importance of π-stacking events is well-known in the fields of medicinal chemistry21 and structural biology,22,23 but here the usage of CCSD(T) is beyond the reach of routine application. In practice, this makes understanding the strengths and weaknesses of different dimer interactions impossible in a field where such knowledge could reasonably be expected to influence the interpretation of biological data and the development of novel compounds. The recent observation that MP2 is able to reproduce the trends observed in CCSD(T)/CBS calcula* Phone: 215-652-7539. Fax: 215-652-4625. E-mail: john_sanders@ merck.com.

Figure 1. Structures of the benzene/benzene dimer: (A) T-shaped C2V point-face geometry; (B) parallel-displaced C2h atom-centered geometry; (C) sandwich D6h stacked geometry.

tions of substituted aromatic systems13 was viewed as a possible solution to this understanding gap despite the absolute errors of MP2. Indeed, it was hoped that raw MP2 values could be corrected by way of linear regression of published CCSD(T) values to provide guidance about the absolute strengths of different interactions, similar to the way that spin componentscaled MP2 uses individually rescaled parallel and antiparallel spin components of the MP2 energy to provide better agreement with experiment.19,20 Although the effects of solvation and other intermolecular interactions complicate the interpretation of biological phenomena, much can still be learned from such fundamental gas phase calculations. Results from the study of 27 parallel-displaced aromatic dimers are presented in terms of both raw and corrected MP2 and SCS-MP2 interaction energies. The set of dimers used in this study led to the observation that interaction energies are surprisingly well predicted by a simple molecular descriptor, the number of heavy atoms in the dimer, and extension of the study to include 12 aryl/non-aryl dimers suggests that this observation is somewhat general. This is consistent with recent findings that substituted aryl/aryl dimer interaction energies show evidence of additivity9,24 and benefit from direct interactions between the substituent of one monomer and the aryl ring of the other.14,15,25 Together, this supports the idea that optimal π-stacking interactions can be described simply by the dimer heavy atom count.

10.1021/jp912094q  2010 American Chemical Society Published on Web 08/03/2010

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TABLE 1: Interaction Energies for Reference Aryl/Aryl Dimers dimer

dimer geometry

point group

est-∆ECCSD(T)/CBS (kcal · mol-1)a

∆EMP2 (kcal · mol-1)b

∆ESCS-MP2 (kcal · mol-1)c

benzene/benzene benzene/benzene benzene/benzene toluene/benzene phenol/benzene fluorobenzene/benzene aniline/benzene cyanobenzene/benzene nitrobenzene/benzene

PDd Te Sf PD PD PD PD PD PD

C2h Cs D6h Cs C1 Cs Cs Cs Cs

-2.68 -2.86 -1.53 -4.00 -3.65 -3.44 -4.45 -4.35 -5.15

-4.21 -3.28 -2.89 -5.17 -5.33 -4.37 -5.77 -6.09 -6.97

-1.92 -2.02 -1.18 -2.47 -2.66 -1.92 -3.45 -3.19 -3.71

a Estimated CCSD(T)/CBS values are published elsewhere.13 energy. d Parallel-displaced (PD). e T-shaped (T). f Sandwich (S).

b

MP2/aug-cc-pVDZ interaction energy.

c

SCS-MP2/aug-cc-pVDZ interaction

Materials and Methods The use of MP2/aug-cc-pVDZ//MP2/cc-pVDZ to calculate interaction energies was motivated by Lee et al., who showed that MP2/6-31+G* interaction energies are highly correlated with estimated CCSD(T)/CBS energies for seven dimers (six substituted benzene dimers and the unsubstituted benzene dimer).13 In this work, Gaussian 0326 was used to carry out all electronic calculations. Initial geometries of low symmetry dimers were manually constructed whereas AMPAC AGUI version 8.16.727 was used to construct starting geometries for dimers with specific point group symmetries. Final gas phase geometries of the monomer and dimer systems were obtained through geometry optimization using MP2 with the cc-pVDZ basis set28 in the absence of any geometry restraints other than those imposed by the initial point group symmetry. Frequency calculations at MP2/cc-pVDZ were used to evaluate convergence and to characterize the stationary points found. Monomer energies were obtained at the MP2/cc-pVDZ geometries using MP2 and the aug-cc-pVDZ basis set.29 Dimer interaction energies were determined at optimized MP2/cc-pVDZ geometries using MP2/aug-cc-pVDZ theory with counterpoise correction30 and fragment relaxation; SCS-MP219 energies were calculated from the MP2 results. Zero point energies were not included in the calculation of the interaction energies. The dimer conformations with the lowest interaction energy identified (using MP2/aug-cc-pVDZ theory) are shown in Figures S1 and S2 of the Supporting Information. Details for all monomer and dimer geometries, frequency calculations, and energies are given in Tables S1-S4. Unless otherwise stated, the interaction energies described in the text were the lowest obtained for each dimer using MP2/aug-cc-pVDZ//MP2/cc-pVDZ theory. Results and Discussion Initially, the dimers reported by Lee et al.13 were investigated to determine the applicability of MP2/aug-cc-pVDZ interaction energies as approximations to the published estimated CCSD(T)/ CBS results. In making this comparison, interaction energies were determined using dimer configurations similar to those of Lee et al.,13 although the structures were optimized as described above (Figure S1). As shown in Table 1 and Figure 2, the MP2/ aug-cc-pVDZ and SCS-MP2/aug-cc-pVDZ results reported herein are well-correlated with the published CCSD(T)/CBS estimates13 (R2 ) 0.90 and 0.91 for MP2 and SCS-MP2 values, respectively), validating the abilities of both approaches to reproduce the trends in the published CCSD(T)/CBS estimates. In terms of absolute accuracy, the mean error of prediction for the SCS-MP2 energies was smaller than that of the traditional MP2 approach (1.07 and 1.33 kcal · mol-1, respectively) and the standard deviation of the mean error was tighter (0.39 vs 0.45

Figure 2. Comparison of estimated CCSD(T)/CBS interaction energies for published aryl/aryl dimers13 with MP2/aug-cc-pVDZ (() and SCSMP2/aug-cc-pVDZ (O) interaction energies. The 1:1 line is shown for reference.

kcal · mol-1), in agreement with prior reports.14,20,31 The improvement of SCS-MP2 with respect to MP2 is especially pronounced when comparing the Cs T-shaped and D6h sandwich geometries of the benzene dimer, Table 1. MP2 predicts that the Cs T-shaped and D6h sandwich geometries are similar in energy (∆EMP2 ) -3.28 and -2.89 kcal · mol-1, respectively), whereas SCS-MP2 and CCSD(T) both predict that the Cs T-shaped geometry is considerably more stable than the D6h sandwich structure (∆ESCS-MP2 ) -2.02 and -1.18 kcal · mol-1, respectively; est-∆ECCSD(T)/CBS ) -2.86 and -1.53 kcal · mol-1, respectively13). Of course, SCS-MP2 would be expected to perform better when comparing dimers of very different geometries, for example, T-shaped and parallel-displaced structures.20 Regardless, the performance of MP2 for paralleldisplaced geometries appears to be quite good in a relative sense (Table 1, Figure 2) hence MP2 should provide a reliable foundation for the study of additional parallel-displaced dimers. Having established that both methods reproduce the estimated CCSD(T)/CBS results published previously13 with a high correlation coefficient and some systematic error (all dimers are overbound with MP2 and under-bound with SCS-MP2), both were applied to a series of parallel-displaced aromatic dimers that one might expect to find in pharmaceutically relevant protein/ligand interactions. The investigation began by pairing aniline, benzene, chlorobenzene, fluorobenzene, furan, indole, naphthalene, nitrobenzene, phenol, pyrazine, pyridine, toluene, and trifluoromethylbenzene with benzene and indole to approximate the interactions of these monomers with the side

Interaction Energies for Aryl/Aryl Dimers

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TABLE 2: Optimal Interaction Energies and Heavy Atom Counts for Parallel-Displaced Aryl/Aryl Dimers dimer

point group

∆EMP2 (kcal · mol-1)a

∆ESCS-MP2 (kcal · mol-1)b

∆EMP2-corrected (kcal · mol-1)c

∆ESCS-MP2-corrected (kcal · mol-1)d

dimer HACe

aniline/benzene benzene/benzene chlorobenzene/benzene cyanobenzene/benzene fluorobenzene/benzene furan/benzene indole/benzene naphthalene/benzene nitrobenzene/benzene phenol/benzene pyrazine/benzene pyridine/benzene toluene/benzene trifluoromethylbenzene/benzene aniline/indole chlorobenzene/indole cyanobenzene/indole fluorobenzene/indole furan/indole indole/indole naphthalene/indole nitrobenzene/indole phenol/indole pyrazine/indole pyridine/indole toluene/indole trifluoromethylbenzene/indole

Cs C2h Cs Cs Cs Cs C1 C1 Cs Cs Cs Cs Cs Cs C1 C1 C1 C1 C1 Ci C1 C1 C1 C1 C1 C1 C1

-5.77 -4.21 -5.10 -6.09 -4.42 -3.14 -6.95 -7.25 -6.97 -5.33 -4.60 -4.81 -5.17 -5.03 -10.63 -9.53 -11.26 -7.92 -5.95 -10.87 -11.14 -11.58 -8.47 -8.77 -8.75 -8.49 -9.83

-3.45 -1.92 -2.39 -3.19 -2.14 -1.36 -3.40 -3.38 -3.71 -2.66 -2.26 -2.44 -2.47 -2.23 -6.86 -5.19 -6.78 -4.19 -3.53 -6.57 -5.95 -6.70 -4.96 -4.87 -4.98 -4.28 -5.56

-4.25 -3.03 -3.72 -4.50 -3.19 -2.19 -5.17 -5.41 -5.19 -3.91 -3.33 -3.50 -3.78 -3.67 -8.05 -7.19 -8.54 -5.93 -4.39 -8.23 -8.45 -8.79 -6.36 -6.59 -6.58 -6.38 -7.42

-4.76 -2.84 -3.42 -4.43 -3.11 -2.13 -4.69 -4.67 -5.08 -3.76 -3.26 -3.49 -3.53 -3.23 -9.03 -6.94 -8.93 -5.69 -4.85 -8.66 -7.89 -8.83 -6.64 -6.53 -6.68 -5.79 -7.39

13 12 13 14 13 11 15 16 15 13 12 12 13 16 16 16 17 16 14 18 19 18 16 15 15 16 19

a MP2/aug-cc-pVDZ interaction energy. b SCS-MP2/aug-cc-pVDZ interaction energy. c Corrected MP2/aug-cc-pVDZ interaction energy obtained with eq 1. d Corrected SCS-MP2/aug-cc-pVDZ interaction energy obtained with eq 2. e Dimer heavy atom count (HAC).

chains of phenylalanine and tryptophan (future work is planned to extend these studies to phenol- and imidazole-containing dimers to account for tyrosine and histidine). To gain some appreciation for the absolute magnitudes of these interaction energies, both MP2 and SCS-MP2 values have been corrected using the best-fit line parameters obtained from linear regression of the published, estimated CCSD(T)/CBS interactions energies against the MP2/aug-cc-pVDZ and SCS-MP2/aug-cc-pVDZ energies (Table 1):

∆EMP2-corrected ) 0.78 · ∆EMP2 + 0.26

(1)

∆ESCS-MP2-corrected ) 1.25 · ∆ESCS-MP2 - 0.43

(2)

The corrected MP2 and SCS-MP2 results obtained from eqs 1 and 2 are very highly correlated (R2 ) 0.97, Table 2 and Figure 3), so for simplicity the corrected MP2 energies will be the focus of the discussion. Using corrected MP2 interaction energies, it is clear that electron-donating and electron-withdrawing substituents both increase the interaction energy with respect to the benzene/ benzene dimer, consistent with prior investigations (Table 2).6,9,10,13-15 For instance, the addition of an electron-donating amine to form the aniline/benzene dimer yields ∆EMP2-corrected ) -4.25 kcal · mol-1, whereas addition of the electronwithdrawing cyano group to form the cyanobenzene/benzene dimer results in ∆EMP2-corrected ) -4.50 kcal · mol-1, both of which are stabilized by more than 40% with respect to the benzene/benzene dimer (∆EMP2-corrected ) -3.03 kcal · mol-1). This is also observed when comparing the aniline/indole and cyanobenzene/indole dimers with the benzene/indole dimer (∆EMP2-corrected ) -8.05, -8.54, and -5.17 kcal · mol-1, respectively), although here the stabilization provided by the substituents is ∼60% over the analogous benzene/indole complex.

Figure 3. Comparison of corrected MP2/aug-cc-pVDZ and corrected SCS-MP2/aug-cc-pVDZ interaction energies for parallel-displaced aryl/ aryl dimers.

Likewise, the toluene/benzene and trifluoromethylbenzene/ benzene dimers show similar stabilization with respect to the benzene/benzene dimer (∆EMP2-corrected ) -3.78, -3.67, and -3.03 kcal · mol-1, respectively) as do the toluene/indole and trifluoromethylbenzene/indole dimers compared to the benzene/ indole dimer (∆EMP2-corrected ) -6.38, -7.42, and -5.17 kcal · mol-1, respectively). Since electron donating and withdrawing groups both increase dimer interaction energies, what factors determine these values? To gain additional insight, dimers with extreme ∆EMP2-corrected values were considered. The dimer with the smallest interaction energy is the furan/benzene dimer, with ∆EMP2-corrected ) -2.19 kcal · mol-1, while the dimer with the largest interaction energy is the nitrobenzene/indole dimer, with ∆EMP2-corrected ) -8.79 kcal · mol-1. Given the ability of MP2 to reproduce more subtle trends in CCSD(T) values (Table 1), these estimates are quite

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Figure 4. Comparison of corrected MP2/aug-cc-pVDZ interaction energies and dimer heavy atom counts for parallel-displaced aryl/aryl dimers.

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Figure 5. Comparison of corrected MP2/aug-cc-pVDZ interaction energies for aryl/indole and aryl/benzene dimers.

TABLE 3: Optimal Interaction Energies and Heavy Atom Counts for Aryl/Non-aryl Dimers

distinct. In fact, the interaction energy of the nitrobenzene/indole dimer is so large that it exceeds BSSE-corrected estimates for the pyridine-water hydrogen bond interaction energy of -6.28 kcal · mol-1 calculated at the MP2/aug-cc-pVDZ level of theory.32 One obvious difference between these two dimers is that the furan/benzene dimer is the smallest of the aryl/aryl dimers studied (11 heavy atoms) whereas the nitrobenzene/ indole dimer is one of the largest (18 heavy atoms). This prompted the comparison of the dimer interaction energies with their respective dimer heavy atom counts (HAC), revealing that these two properties are correlated with R2 ) 0.80 for corrected MP2 interaction energies (Figure 4). Due to the composition of the data set, it is important to consider the possibility that the relationship between interaction energy and dimer size is spurious. For example, there is a systematic size difference between comparable benzenebased and indole-based dimers (indole-based dimers are three heavy atoms larger) that may be covariant with or even mask fundamental electronic differences between the two groups. Indeed, it is clear that the mean ∆EMP2-corrected values are much larger for the indole-based dimers than for the benzene-based dimers (-7.00 ( 1.33 kcal · mol-1 vs -3.92 ( 0.91 kcal · mol-1, respectively). However, when the two groups are treated independently the ∆EMP2-corrected/dimer HAC relationships are maintained with R2 values of 0.64 and 0.61 for the benzene-based and indole-based dimers, respectively (using ∆ESCS-MP2-corrected values instead, the R2 values are 0.43 and 0.44, respectively; R2 values are lowered in both subgroups due to a 38% contraction in HAC range). This suggests that the systematic size difference between the two groups is not the dominant explanation for these findings. Further, when the indole-based dimer interaction energies are plotted against the benzene-based dimer interaction energies for the 14 directly comparable dimers in the data set (e.g., comparing the aniline/indole dimer with the aniline/ benzene dimer, etc.), it is clear that the interaction energies are correlated (R2 ) 0.85 for ∆EMP2-corrected values, Figure 5). This suggests that the origins of the substituent effects observed in the two groups may be held in common and that competing group effects are unlikely to explain the overall correlation between dimer interaction energy and dimer HAC. The possibility that the optimal dimer interaction energy can be trivially estimated from the number of heavy atoms in the dimer

dimer

point group

∆EMP2 (kcal · mol-1)a

∆ESCS-MP2 (kcal · mol-1)b

dimer HACc

methane/benzene ethane/benzene cyclopropane/benzene cyclobutane/benzene cyclopentane/benzene cyclohexane/benzene methane/indole ethane/indole cyclopropane/indole cyclobutane/indole cyclopentane/indole cyclohexane/indole

C3V Cs Cs Cs Cs Cs C1 C1 C1 C1 C1 C1

-1.51 -2.40 -3.26 -3.08 -3.90 -3.70 -2.52 -3.83 -4.85 -5.21 -5.73 -5.85

-0.88 -1.37 -1.92 -1.78 -2.22 -2.08 -1.49 -2.25 -2.91 -3.06 -3.33 -3.37

7 8 9 10 11 12 10 11 12 13 14 15

a MP2/aug-cc-pVDZ interaction energy. b SCS-MP2/aug-cc-pVDZ interaction energy. c Dimer heavy atom count (HAC).

recalls earlier work describing additivity among substituents9,24 and direct substituent/ring interactions.14,15,25 Given the diversity of substituents investigated, it is therefore of interest to examine the generality of the observed dimer interaction energy/HAC relationship. To address this, and to specifically test the dimer size/ interaction energy correlation through the introduction of still smaller dimers, these methods were applied to a series of aryl/ non-aryl complexes. For this part of the discussion, it is appropriate to refer to raw ∆EMP2 values since it is not obvious that the best fit line relating MP2/aug-cc-pVDZ and estimated CCSD(T)/CBS aryl/ aryl dimer interaction energies should apply to aryl/non-aryl dimers. Beginning with one of the smallest possible dimers, the methane/ benzene dimer (which is not expected to form any π-stacking interactions with benzene in a classical sense), a raw ∆EMP2 ) -1.51 kcal · mol-1 (Table 3) is obtained s a value in good agreement with others obtained using similar methods33 and which is close to the CCSD(T)/CBS value of -1.45 kcal · mol-1.33 In comparison to the methane/benzene dimer, the raw ∆EMP2 values for the furan/benzene, benzene/benzene, and cyanobenzene/benzene complexes are all larger at -3.14, -4.21, and -6.09 kcal · mol-1, respectively. Of course, this is expected from chemical intuition, but it is also consistent with an interaction energy/HAC relationship. Similar conclusions are reached using raw SCS-MP2 values to make the same comparisons, as the methane/benzene ∆ESCS-MP2 value is -0.88 kcal · mol-1 and the furan/benzene, benzene/benzene, and cyanobenzene/benzene ∆ESCS-MP2 values are all larger at -1.36, -1.92, and -3.19 kcal · mol-1, respectively (Tables 2, 3). Extending the series with additional small alkane and cycloalkane molecules (ethane, cyclopropane, cyclobutane,

Interaction Energies for Aryl/Aryl Dimers

Figure 6. Comparison of dimer heavy atom count with (A) raw MP2/ aug-cc-pVDZ interaction energy and (B) raw SCS-MP2/aug-cc-pVDZ interaction energy. Aryl/non-aryl dimers are indicated by closed squares, aryl/aryl dimers by open circles.

cyclopentane, and cyclohexane), a size-dependence similar to that found with the aryl/aryl dimers is apparent (Table 3). This is in fact consistent with an investigation in which CCSD(T) was used for a series of alkane/benzene dimers,34 although this work did not explicitly consider the importance of dimer size (R2 ) 0.90 for the interaction energy/HAC correlation, data not shown). Interestingly, when the energies of all dimers (aryl/ aryl and aryl/non-aryl) are plotted together, it is clear that the interaction energies depend linearly on the dimer HAC (R2 ) 0.85 and 0.75 for raw MP2 and raw SCS-MP2 values, respectively, Figure 6). That is, there does not appear to be any clear distinction between the interaction energies of aryl/aryl and aryl/non-aryl dimers on the basis of the aromaticity or chemical functionality of the participating monomers. For example, the cyclohexane/benzene and benzene/benzene dimers each have twelve heavy atoms and roughly similar raw ∆EMP2 values (-3.70 and -4.21 kcal · mol-1, respectively). When the analogous raw ∆ESCS-MP2 values are considered, one again finds that the two dimers have similar energies (∆ESCS-MP2 values are -2.08 and -1.92, respectively, for the cyclohexane/benzene and benzene/benzene dimers). The same comparison can be made with the cyclohexane/indole and benzene/indole dimers of 15 heavy atoms each where ∆EMP2 values of -5.85 and -6.95 kcal · mol-1 and ∆ESCS-MP2 values of -3.37 and -3.40 kcal · mol-1 are respectively found. Considering now the cyclopentane/benzene and furan/benzene dimers, the aryl/non-aryl

J. Phys. Chem. A, Vol. 114, No. 34, 2010 9209 dimer is predicted to be more stable than the aryl/aryl dimer (∆EMP2 values are -3.90 and -3.14 kcal · mol-1, respectively; ∆ESCS-MP2 values are -2.22 and -1.36 kcal · mol-1, respectively), whereas the cyclopentane/indole and furan/indole dimers are similar in energy (∆EMP2 values are -5.73 and -5.95 kcal · mol-1, respectively; ∆ESCS-MP2 values are -3.33 and -3.53 kcal · mol-1, respectively). Taken together, these findings suggest that aryl/non-aryl dimers are similar in strength to aryl/aryl dimers of the same size despite the fact that there is no π-π contribution to their stability. To further explore the generality of the relationship between dimer strength and HAC, the results from a recent publication of Ringer and Sherrill35 were evaluated in the context of dimer HAC. In their paper, Ringer and Sherrill considered benzenecontaining sandwich geometry dimers which specifically included several monomers with extreme electrostatic potentials in order to probe how well Hammett parameters and electrostatic considerations would explain the strengths of the dimers they studied. This data set provides an interesting opportunity to examine the generality of a simple parameter such as the dimer HAC as an indicator of maximal dimer strength. In fact, the interaction energies (relative to the benzene/benzene dimer) of the dimers that Ringer and Sherrill studied35 are highly correlated with the dimer HAC (R2 ) 0.94, data not shown). While one very strongly bound dimer, the hexacyanobenzene/benzene dimer (10.46 kcal · mol-1 more strongly bound than the benzene/ benzene dimer), may exaggerate the magnitude of the correlation, exclusion of this point still leaves a very strong correlation (R2 ) 0.89, data not shown). This indicates that the relationship between HAC and interaction energy applies to sandwich geometries in addition to parallel-displaced geometries and further suggests that such simple descriptors may be useful. Of course, the fact that the dimer HAC is correlated with the dimer interaction energy immediately suggests that a number of other parameters may show similar trends. One particularly interesting example of such a parameter is the number of good intermolecular van der Waal’s contacts occurring within the dimers. To determine this, the MP2/cc-pVDZ geometries for the dimers examined in this work were read into Maestro36 and the number of good intermolecular contacts for each dimer was counted by using eq 3

C)

Dij Ri + Rj

(3)

where Dij is the distance between atoms i and j, Ri and Rj are the radii of atoms i and j (as defined by the OPLS_2005 force field), and contacts with C values between 0.89 and 1.3 were counted as “good.” As shown in Figure 7, the correlation between the number of good contacts and ∆EMP2 (R2 ) 0.90) is better than what is found when relating HAC to ∆EMP2 (R2 ) 0.85, Figure 6A). Together, these parameters suggest that increases in dimer interaction strength commonly result from increasing the number of beneficial intermolecular VDW interactions within the dimer and that increasing the HAC presents more opportunities to accomplish this. Put differently, larger dimers are expected to have the possibility of increased interaction energies through increased dispersion interactions. The above results should therefore allow for the estimation of optimal dimer interaction strength given a dimer’s HAC. Using linear regression to relate MP2 and SCS-MP2 interaction energies with dimer HAC, eqs 4 and 5 were obtained to allow

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Figure 8. Dimer geometries that minimize and maximize direct substituent/ring interactions. (A) Aniline/benzene dimer with minimal direct amine/ring interaction; (B) aniline/benzene dimer with maximal direct amine/ring interaction.

Figure 7. Comparison of the number of good intermolecular VDW contacts for the dimer with the raw MP2/aug-cc-pVDZ interaction energy. Aryl/non-aryl dimers are indicated by closed squares, aryl/aryl dimers by open circles.

for the estimation of the optimal MP2 and SCS-MP2 dimer interaction energies, respectively, from HAC:

∆EHAC-MP2 ) -0.88 · HAC + 5.80

(4)

∆EHAC-SCS-MP2 ) -0.50 · HAC + 3.40

(5)

Estimates obtained from eqs 4 and 5 do have some potential limitations. For instance, using a single linear regression to fit aryl/aryl and aryl/non-aryl dimers may not be the best way to treat the data (Figure 6). However, because of the limited size of the data set and the simplicity of the relationship, it is not clear that separate treatments for the two groups are justified. Also, note that estimates obtained with eqs 4 and 5 are based on fits to raw ∆EMP2 and ∆ESCS-MP2 values. Although it would be useful to estimate corrected MP2 and SCS-MP2 values instead, the lack of CCSD(T)/CBS results for aryl/non-aryl dimers prevents this. Given that SCS-MP2 is better than MP2 at reproducing CCSD(T)/CBS results for aryl/aryl dimers in an absolute sense, ∆EHAC-SCS-MP2 may provide better absolute interaction energy estimates though both should be useful in a relative sense. The unsigned errors of the estimates obtained with eqs 4 and 5 can be large (for instance, 3.25 kcal · mol-1 for the HAC-derived estimate of the trifluoromethylbenzene/ benzene dimer MP2 interaction energy), so some caution is advised. However, the average errors obtained from eqs 4 and 5 (0.80 ( 0.70 kcal · mol-1 and 0.64 ( 0.53 kcal · mol-1, respectively) do suggest that such simple calculations may be useful. Of course, in trying to interpret biological data it is important to remember that eqs 4 and 5 are based on calculations for isolated gas phase dimers and that solvation effects are not accounted for. Until this point, the discussion has pertained to the optimal interaction energies found for each dimer. There are, of course, higher energy stationary points for each of the dimers investigated (i.e., geometries not reported in Tables 2 and 3, see the Supporting Information for additional details). For the substituted benzene/benzene dimers, consideration of different stationary point geometries provides some insight into how the interactions of substituents with the unsubstituted aryl ring affect the interaction energies. The dimers that facilitate this comparison are those that have high-symmetry stationary points at

geometries that maximize the distance between the substituent and the ring (Figure 8A) or that minimize the distance between the substituent and the ring (Figure 8B). Benzene-based dimers containing aniline, chlorobenzene, cyanobenzene, fluorobenzene, nitrobenzene, toluene, and trifluoromethylbenzene readily allow these comparisons. With the exception of the fluorobenzene/ benzene dimer, the lowest energy geometries are the ones that orient the substituent over the unsubstituted benzene ring (Figure 8B). For fluorobenzene/benzene, the two geometries are very similar in energy (∆EMP2-corrected ) -3.19 and -3.16 kcal · mol-1 for the noninteracting and interacting geometries, respectively), and both are close to the optimal ∆EMP2-corrected value of the benzene/benzene dimer (-3.03 kcal · mol-1). For the other dimers, the ∆EMP2-corrected values for geometries that minimize the interaction between the substituent and the unsubstituted benzene are within a narrow range of 1.00 kcal · mol-1 from one another (Table S2) - smaller than the 2.03 kcal · mol-1 range found when the substituents interact with the ring (Table 2). It is also noted that some of these geometries are less stable than the benzene/benzene dimer (e.g., the noninteracting aniline/ benzene ∆EMP2-corrected ) -2.85 kcal · mol-1 vs benzene/benzene ∆EMP2-corrected ) -3.03 kcal · mol-1), whereas the noninteracting geometry of the nitrobenzene/benzene dimer is stabilized by 0.82 kcal · mol-1 relative to the benzene/benzene dimer (∆EMP2-corrected ) -3.85 and -3.03 kcal · mol-1, respectively). From this it is clear that different substituent effects are observed for different geometries, consistent with prior findings,14 and that the HAC does not appear to be critical for geometries with minimal ring/substituent interaction. Therefore, although the dimer HAC seems important for the optimal strength of the dimer (which usually involves interactions between substituent and ring), other mechanisms are at play and size-dependence is not observed when substituents are directed away from the partner ring. It is also worth noting that the dimers investigated thus far are likely stabilized primarily by dispersion interactions. For dimers that include strong electrostatic components, such as the water/benzene dimer,37 dimer size does not appear to be a dominant factor in the overall strength of the dimer. It has been reported that the water/benzene dimer (7 heavy atoms) is stronger than the benzene/benzene dimer (12 heavy atoms) despite its smaller size (-3.29 vs -2.62 kcal · mol-1 using CCSD(T)38), whereas the ammonia/benzene dimer (7 heavy atoms) is slightly less stable than the benzene/benzene dimer (-2.32 vs -2.62 kcal · mol-1 using CCSD(T)38). The difference between the water/benzene and ammonia/benzene dimer would not be expected on the basis of the HAC, and the fact that the water/benzene dimer is stronger than the benzene/benzene dimer clearly demonstrates that the relationship between HAC and interaction energy does not apply universally to all varieties of dimers containing aromatic monomers. Nevertheless, it appears from this work (at least when using MP2 theory) that the strength of π-stacked dimers depends

Interaction Energies for Aryl/Aryl Dimers heavily on the size of the dimer and not necessarily on the functionality of the individual monomers. At least one other recent publication39 supports this size-dependent conclusion for smaller systems, though it suggests a stronger role for π-mediated effects in larger systems. An alternative explanation for these results could be that MP2 theory is not equally appropriate for the treatment of aryl/aryl and aryl/non-aryl complexes, but this seems unlikely for several reasons. First, the estimate for the methane/benzene interaction energy of -1.51 kcal · mol-1 is in agreement with published CCSD(T)/CBS estimates33,38,40,41 in the range of -1.47 to -1.45 kcal · mol-1 as well as the experimental measurement of -1.13 to -1.03 kcal · mol-1.42 Additionally, the ∆EMP2 value of -2.40 kcal · mol-1 for the ethane/benzene complex is in agreement with published experimental and CCSD(T) values for the ethylene/benzene and acetylene/benzene dimers of the same HAC (D0 (expt1) ) -1.4 and -2.7 kcal · mol-1, respectively; ∆ECCSD(T) ) -2.17 and -2.75 kcal · mol-1, respectively40); although these results are not directly comparable, they do support the finding that the dimer interaction energy depends on the size of the dimer. Lastly, for the series of alkane/benzene dimers34 reported by Ran and Wong, CCSD(T)/aug-cc-pVTZ interaction energies are correlated with HAC (as they are with MP2). Together, these points argue against an inherent deficiency of MP2 for handling aryl/non-aryl systems. Since the MP2 results for aryl/aryl complexes are shown to scale nicely with published estimated CCSD(T)/CBS interaction energies, MP2 appears to be an appropriate tool for all of the dimers studied in this work. Conclusions MP2 and SCS-MP2 theories, which have previously been demonstrated to qualitatively reproduce CCSD(T) results for aryl/ aryl complexes, have been applied to a range of parallel-displaced aryl/aryl dimers of interest in medicinal chemistry and linear regression has been used to correct systematic error. The magnitudes of the interaction energies were found to depend less on the nature of aryl substituents than on the sizes of those substituents, and this is found to hold for a set of aryl/non-aryl dimers. This allows for dimer interaction energies to be simply estimated using only the dimer HAC and suggests that the highly detailed understandings afforded by sophisticated computational methods are not required to estimate optimal dimer interaction strengths. Acknowledgment. J. M. S. gratefully acknowledges Chris Culberson, Kate Holloway, Constantine Kreatsoulas, Georgia McGaughey, Edward Sherer, and Robert Sheridan for helpful discussion and Rich Bach, Gene Fluder, Joe Forbes, and Daniel McMasters for technical assistance. Supporting Information Available: The dimer structures, monomer and dimer coordinates, point group symmetries, numbers of imaginary frequencies, and energies. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Karlstro¨m, G.; Linse, P.; Wallqvist, A.; Jo¨nsson, B. J. Am. Chem. Soc. 1983, 105, 3777–3782. (2) Hobza, P.; Selzle, H. L.; Schlag, E. W. J. Am. Chem. Soc. 1994, 116, 3500–3506. (3) Hobza, P.; Selzle, H. L.; Schlag, E. W. J. Phys. Chem. 1996, 100, 18790–18794. (4) Tsuzuki, S.; Uchimaru, T.; Matsumura, K.; Mikami, M.; Tanabe, K. Chem. Phys. Lett. 2000, 319, 547–554. (5) Sinnokrot, M. O.; Valeev, E. F.; Sherrill, C. D. J. Am. Chem. Soc. 2002, 124, 10887–10893.

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