Article pubs.acs.org/crystal
How Reliable Are Intermolecular Interaction Energies Estimated from Topological Analysis of Experimental Electron Densities? Mark A. Spackman*
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School of Chemistry and Biochemistry, University of Western Australia, 35 Stirling Highway, Perth, Western Australia 6009, Australia ABSTRACT: We examine the reliability of intermolecular interaction energies estimated from intermolecular bond critical point properties of experimental electron densities and show that they are inherently unreliable, typically underestimating but sometimes overestimating more reliable values. Based on these estimates, conclusions regarding the energetic importance of specific intermolecular interactions, especially in the context of crystal packing and crystal engineering, can be misleading.
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INTRODUCTION
The Espinosa−Molins−Lecomte (EML) relationship has been discussed in considerable detail by Gatti,7 who proposed several important caveats associated with its use that need to be borne in mind. The potential energy density at the bcp calculated from experimental charge density analyses relies on Abramov’s approach,8 which relates the kinetic energy density at the bcp, Gb, to values of the experimental electron density and its Laplacian at the bcp:
It has become increasingly common in modern experimental charge density studies to use the potential energy density, Vb, at the bond critical point (bcp) identified using Bader’s quantum theory of atoms in molecules (QTAIM)1 to estimate energies associated with specific interatomic interactions between molecules in crystals. These energies in turn are used to assess the relative importance of different intermolecular interactions identified in a crystal, and indeed whether they may be considered “structure determining”, a consideration of some importance in crystal engineering. These energy estimates use the relationship Eint = 0.5Vb, proposed by Espinosa, Molins, and Lecomte2 on the basis of topological analyses of experimental electron densities for a large number of X−H···O (X = O, N or C) hydrogen bonds. Although there is no reason to believe that this relationship will be universal and the coefficient 0.5 will apply to a wide range of intermolecular interactions, in the absence of substantial evidence to the contrary that has become a widespread assumption. For example, Chen et al. reported3 energy estimates based on Eint = 0.5Vb, arguing that “Although this expression was derived for hydrogen bonds, it should also provide a reasonable estimate for other weak closed-shell interactions”. More recently, many charge density studies have reported energy estimates using this expression for a range of noncanonical intermolecular interactions, while also acknowledging that its applicability has not been demonstrated.4 Different values for the proportionality constant have also been proposed. For example, Espinosa and co-workers5 analyzed theoretical electron densities for a range of neutral hydrogen bonded dimers involving fluorine and suggested Eint ≈ 0.42(2) Vb for those interactions, while a recent study on solid I2 has proposed that the relationship Eint = 0.68Vb be used for interactions involving iodine atoms.6 © 2015 American Chemical Society
Gb =
1 3 (3π 2)2/3 ρb5/3 + ∇2 ρb 10 6
(1)
Assuming that multipole-refined electron distributions obey the local viral theorem, Espinosa et al. used this value of Gb to determine Vb: Vb =
ℏ2 2 ∇ ρb − 2G b 4m
(2)
For a set of 83 O···H hydrogen bonds Espinosa et al. analyzed the behavior of these experimental estimates of Gb and Vb as functions of the O···H distance and observed that both display a similar exponential dependence on the O···H distance over the range 1.6−3.0 Å. Literature values of hydrogen bond dissociation energies, obtained from a variety of theoretical methods, were then found to display essentially the same exponential dependence. Observing a close similarity between the fitting functions led Espinosa et al. to propose the equality E int = 0.5Vb
(3)
where the proportionality factor 0.5 has units of volume. Although the authors anticipated this to be used to estimate Received: September 14, 2015 Revised: October 5, 2015 Published: October 7, 2015 5624
DOI: 10.1021/acs.cgd.5b01332 Cryst. Growth Des. 2015, 15, 5624−5628
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Table 1. Details of Experimental Charge Density Analyses
a
compound
ref
T, K
atom···atom contacts
austdiol 1,3,4-trinitro-7,8-diazapentalene L-alanine DL-alanine 2,5-dichloro-1,4-benzoquinone 2-chloro-4-fluorobenzoic acid 4-fluorobenzamide hexachlorobenzene pentachlorophenol hexabromobenzene pentabromophenol ferulic acid selenophthalic anhydride 6-methyl-2-thiouracil pentafluorophenyl-2,2′-bisthiazole picolinic acid N-oxide croconic acid 7,9-di(thiophen-2-yl)-8H-cyclopenta[a]acenaphthylen-8-one β-pyrazinamide α-nicotinamide 2,3-dibromo-2,3-dihydroinden-1-one phthalic acid 2,6-dinitrophenol
22 3 23 23 24 25 25 26 27 27 27 28 29 30 31 32 33 4b 34 34 4a 35 36
70 90 23 19 90 100 100 100 100 a 100 100 100 90 110 100 20 100 90 90 100 20 20
O···H O···O, O···N, O···C, O···H O···H O···H Cl···Cl, Cl···O, Cl···H, O···C, O···H Cl···F, F···H, O···H F···F, F···H, O···H Cl···Cl Cl···Cl, O···H Br···Br Br···Br, O···H O···H Se···Se, Se···O, Se···H, O···O, O···H S···N, O···N, O···H, N···C, C···H, H···H S···F, F···F, F···C, C···C O···H O···O, O···C, O···H S···S, O···H O···N, O···C, O···H, N···N, N···C, N···H, H···H O···N, O···H, N···N, N···C, N···H, C···C, H···H Br···O, Br···H, O···H, C···C O···O, O···C, O···H, C···C, H···H O···O, O···N, O···C, O···H, C···C
Results for hexabromobenzene come from a theoretical electron density.
more importantly, the EML estimate depends on identifying all bcps between the molecules of interest; failure to locate all bcps will lead to a systematic underestimate of the interaction energy. Because the crystalline electron density between molecules is typically extremely flat, locating and characterizing all intermolecular bcps is nontrivial and arguably more challenging for experimental electron distributions. As discussed by Gatti,7 the number and type of critical points in a crystal must satisfy the Morse equation, and failure to do so implies that one or more critical points has been missed. Although this is a necessary condition that must be satisfied for a set of critical points in a crystal, it is not sufficient to guarantee that all critical points have been located. For theoretical electron distributions, correlations that have been observed between intermolecular binding energies and the sum of bcp electron densities (or EML energy estimates) suggest that sensible intermolecular interaction energies can be estimated for hydrogen bond and stacking interactions using an EML-type approach, provided that all bcps are located.15 This comparison of EML vs PIXEL energies suggested that meaningful intermolecular interaction energies may be based on experimental estimates of Vb, provided that sufficient attention is paid to the reduction of systematic errors that might compromise experimental charge densities. However, it was also clear from that work that more careful studies and comparisons would be necessary before this could become widely accepted and exploited. The present study, based on QTAIM analyses from a large number of recent experimental charge densities, represents a more detailed attempt at such a careful study and comparison, and it is presented in two ways. The first is an exploration of trends evident from the dependence of experimental EML energies on interatomic separation. The second compares estimates of pairwise intermolecular energies obtained by summing over all interatomic bcps identified between the two molecules, as
hydrogen bond energies from experimental charge densities, the use of an equality without any useful estimate of an error range has encouraged its widespread application and, in particular, far beyond the bounds of its derivation (i.e., O···H hydrogen bonds). To our knowledge, the EML relationship has not been rigorously tested at all, but in a recent article,9 we attempted a modest examination of its application to halogen bonded systems using data from ab initio QTAIM analyses of Alkorta et al.10 on complexes of hypohalous acids and nitrogen bases and Lu et al.11 on dimers involving bromobenzene. Ab initio interaction energies and EML estimates for 14 neutral complexes were found to exhibit the same basic trends, although differences were as large as 7 kJ mol−1 (for HOI··· NH3) and the rms deviation was 3.3 kJ mol−1. With this comparison in mind, we estimated EML intermolecular interaction energies, based on experimental bcp properties, for seven pairs of adjacent molecules in each of the co-crystals of (E)-1,2-bis(4-pyridyl)ethylene with 1,4-dibromotetrafluorobenzene (bpe·F4DBB12) and 1,4-diiodotetrafluorobenzene (bpe·F4DIB13), by summing over all interatomic bcps identified between the two molecules. This summation implicitly assumes that for molecular pairs linked by more than one bcp the total interaction energy can be estimated by simply summing over the Vb values for individual bcps. The resulting energy estimates were compared with intermolecular energies reported for the same molecular pairs based on Gavezzotti’s PIXEL approach.14 For most molecular pairs, the EML result was found to be well below the PIXEL value, especially for bpe·F4DBB, but for both complexes the EML estimate of the energy for the halogenbonded pair exceeded the PIXEL estimate by as much as 18 kJ mol−1. We attributed this behavior in part to the inherent difficulties associated with multipole refinements involving heavy atoms such as Br and I, coupled with the possible need to model anharmonic thermal motion for these atoms. Perhaps 5625
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Figure 1. (a) Log−linear plot of EML energies from experimental charge density analyses vs internuclear distance; (b) Log−linear plot of EML energies vs internuclear distance minus the sum of van der Waals radii.19 The line of best fit to the O···H data points is given in both plots.
plot of all 166 EML energies (here we actually plot EEML = −Eint in eq 3) vs interatomic separation in Figure 1a shows a number of interesting trends: (1) The O···H hydrogen bonds dominate the data and are the only interactions with very short internuclear separations. As expected, these data exhibit reasonably good linear behavior, with a line of best fit Eint = −4.69 × 103 exp(−2.669dO···H/Å) kJ mol−1. This relationship differs markedly from that originally proposed by Espinosa et al., Eint = −25.0 × 103 exp(−3.6dO···H/Å) kJ mol−1. We note that the present data include many more interactions at separations greater than 2.6 Å, while Figure 2 in ref 2 indicates that the original fit was dominated by data points below 2.0 Å. Data for N···H hydrogen bonds closely follow the trend for O··· H contacts in Figure 1a. (2) Apart from strong O···H hydrogen bonds, a few N···H hydrogen bonds, an S···H interaction, and one Br···O interaction, the remaining EML energies are all less than 10 kJ mol−1 in magnitude. (3) Bond critical points between atoms involved in stacking interactions (triangles in Figure 1, for example, C···C, N···C, O···C, O···N) are frequently unobserved in many experimental studies, and often difficult to characterize, but taken as a group, they display the same behavior as all other atom···atom interactions in Figure 1a. (4) Perhaps surprisingly, to a very good approximation all families of points for particular atom···atom interactions in Figure 1a display essentially the same linear trend as observed for O···H hydrogen bonds. This cannot be quantified with the limited number of data points available, but it is an intriguing result. This similarity can be seen much more clearly in Figure 1b, where the EML energies are plotted against their respective internuclear separations minus the sum of the van der Waals radii (from Bondi19) for the two atoms. All data points are now seen to closely follow the line of best fit to the O···H hydrogen bond data points, and where the atom···atom distance equals the sum of van der Waals radii, the points cluster around an energy of 2−3 kJ mol−1. (5) Figure 1b suggests the existence of an approximate universal relationship between Eint, and hence Vb, and the atom···atom separation for intermolecular bond critical points, namely, Eint = −3.30 −1 exp(−2.669[dA···B − dvdW A···B]/Å) kJ mol . The rms deviation
described above, with CE-B3LYP model energies described in our recent work.16
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METHODS
Database of Interatomic QTAIM Properties. The present analyses are based on results of experimental charge density studies for 23 molecular crystals, published in the past decade. Table 1 summarizes these studies, along with some relevant statistics; they include the atoms H, C, N, O, F, S, Cl, Se, and Br. QTAIM topological analysis of the multipole-refined electron densities provides data for 166 separate atom···atom interactions spanning 27 different atom··· atom pairs; O···H hydrogen bonds comprise 37% of the pairs. Although all of these studies report values of ρb and ∇2ρb at intermolecular bcps, not all have used these data to derive EEML, but the calculation using eqs 1−3 is straightforward. Model Energies for Intermolecular Interactions. We have recently described an efficient procedure for obtaining accurate model energies for intermolecular interactions in molecular crystals.16 The approach uses electron densities of unperturbed monomers to obtain accurate estimates of electrostatic, polarization, and repulsion energies, and these are combined with Grimme’s D2 dispersion corrections17 to derive a family of energy models by scaling the separate energy components to fit dispersion-corrected DFT energies for a large number of molecular pairs extracted from organic and inorganic molecular crystals. Here we use the best performing model, denoted CE-B3LYP, which was shown to reproduce B3LYP-D2/6-31G(d,p) counterpoise-corrected energies with a mean absolute deviation (MAD) of just over 1 kJ mol−1 but in considerably less computation time. It also performed surprisingly well (MAD of 2.5 kJ mol−1) for a combined data set of CCSD(T)/CBS benchmark energies for 152 molecular dimers from the literature. CrystalExplorer18 was used to obtain CE-B3LYP energies using the crystal structures from the experimental charge density studies but with bond lengths to hydrogen atoms normalized to standard neutron values.
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RESULTS AND DISCUSSION Trends Evident in the EML Energies. Because EML estimates of interaction energies for such a wide range of atom···atom interactions have not been examined in detail previously, we thought it would be useful and potentially informative to examine the behavior of these energies as a function of interatomic separation. The resulting log−linear 5626
DOI: 10.1021/acs.cgd.5b01332 Cryst. Growth Des. 2015, 15, 5624−5628
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Figure 2. EML estimates of interaction energies vs CE-B3LYP model energies for molecular pairs: (a) full data range; (b) data for which either energy lies between −30 and +10 kJ mol−1. The dashed line has a slope of unity.
Table 2. Intermolecular Energies (kJ mol−1) for Selected Molecular Pairs in Figure 2a
from this equation is remarkably small; for all points in Figure 1b, it is only 3.1 kJ mol−1. Comparison between EML-based Intermolecular Energies and CE-B3LYP Results. As described in the Introduction, EML atom···atom energies are commonly summed over all intermolecular bcps identified with pairs of neighboring molecules in a crystal to obtain an estimate for their energy of interaction. These energies are then used to discuss the relative importance of various intermolecular interactions in the crystal. Using the database of 166 atom··· atom interactions, we have calculated EML estimates of intermolecular energies for 106 different nearest neighbor molecular pairs in the crystals listed in Table 1. These energies are compared with CE-B3LYP energies for the same molecular pairs in Figure 2a (over the energy range −160 to +20 kJ mol−1) and Figure 2b (energies restricted to −30 to +10 kJ mol−1; 81 data points fall within this range), revealing the following: (1) EML estimates based on bcp data typically underestimate accurate molecule···molecule energies and by a substantial amount. However, there are some significant exceptions where the EML result far exceeds the model energy. Results for the largest outliers are given in Table 2, where EML and CE-B3LYP energies are compared with benchmark DFT energies (B3LYP-D2/6-31G(d,p) counterpoise-corrected) computed using the same molecular geometries. With the exception of the stacked molecular pair in pentafluorophenyl2,2′-bithiazole, all outlying molecular pairs incorporate strong hydrogen bonds, and it is readily seen that, as expected, the CEB3LYP model energies agree well with the benchmark DFT results. (2) The typical underestimation of the intermolecular energy by summing EML atom···atom energies can be rationalized by assuming that the topological analysis of multipole-refined electron distributions frequently fails to locate all intermolecular bond critical points, as discussed in the Introduction. However, the instances of large overestimates (for molecular pairs in ferulic acid, croconic acid, and phthalic acid) are more difficult to explain. The molecular pairs in ferulic acid and phthalic acid both involve a cyclic carboxylic acid hydrogen bond synthon, while in croconic acid, the two molecular pairs in question feature a single carboxylic acid O− H···O interaction. In all cases the experimental electron
L-alanine L-alanine DL-alanine pentafluorophenyl-2,2′-bisthiazole 4-fluorobenzamide ferulic acid phthalic acid croconic acid croconic acid croconic acid 6-methyl-2-thiouracil phthalic acid
EML
CEB3LYP
DFT benchmark
−52.5 −46.8 −46.1 −24.6 −22.5 −138.7 −130.2 −77.6 −68.2 −16.2 −2.3 −2.3
−119.5 −150.1 −119.1 −82.8 −67.1 −75.7 −78.0 −45.9 −32.6 +0.7 +3.3 +6.9
−116.7 −157.7 −116.8 −86.7 −73.1 −82.4 −89.8 −47.1 −35.8 +1.9 +2.7 +6.6
a
DFT benchmark energies are counterpoise-corrected B3LYP-D2/631G(d,p) using the same geometry as the CE-B3LYP model energy calculation.
densities at the intermolecular bcps, ρb, are between 0.33 and 0.39 e Å−3 and are unusually large compared with those in the original EML data set (for which all values are plotted in Figure 4 of ref 20). These in turn dominate the EML estimates, contributing approximately 90% of the Abramov estimate of Vb, and it is notable that in all cases the experimental ρb values exceed those reported from theory in those charge density studies. (3) EML estimates are always negative (i.e., binding), a direct consequence of eq 3 and the fact that the potential energy density is negative everywhere.21 But Figure 2b shows that there are three molecular pairs in the present analysis for which the CE-B3LYP energy is actually positive. For example, the energy for the molecular pair in phthalic acid derives from the bcp between O(1) atoms separated by 3.291 Å and for which bcp properties are faithfully reproduced in a theoretical electron density. For phthalic acid, the large overestimate of the energy is due to three well-characterized intermolecular bcps (O(1)···O(2) at 3.006 Å, O(5)···O(4) at 3.072 Å, and O(5)··· C(4) at 3.012 Å). Benchmark DFT results for these three molecular pairs are all positive and agree well with CE-B3LYP model energies (Table 2), strongly supporting the conclusion 5627
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(11) Lu, Y. X.; Zou, J. W.; Wang, Y. H.; Jiang, Y. J.; Yu, Q. S. J. Phys. Chem. A 2007, 111, 10781−10788. (12) Forni, A. J. Phys. Chem. A 2009, 113, 3403−3412. (13) Bianchi, R.; Forni, A.; Pilati, W. Chem. - Eur. J. 2003, 9, 1631− 1638. (14) Gavezzotti, A. Mol. Phys. 2008, 106, 1473−1485. (15) (a) Zhikol, O. A.; Shishkin, O. V.; Lyssenko, K. A.; Leszczynski, J. J. Chem. Phys. 2005, 122, 144104. (b) Waller, M. P.; Robertazzi, A.; Platts, J. A.; Hibbs, D. E.; Williams, P. A. J. Comput. Chem. 2006, 27, 491−504. (c) Matta, C. F.; Castillo, N.; Boyd, R. J. J. Phys. Chem. B 2006, 110, 563−578. (16) Turner, M. J.; Grabowsky, S.; Jayatilaka, D.; Spackman, M. A. J. Phys. Chem. Lett. 2014, 5, 4249−4255. (17) Grimme, S. J. Comput. Chem. 2006, 27, 1787−1799. (18) Wolff, S. K.; Grimwood, D. J.; McKinnon, J. J.; Turner, M. J.; Jayatilaka, D.; Spackman, M. A. CrystalExplorer 3.2, University of Western Australia: Perth, 2015, http://hirshfeldsurface.net. (19) Bondi, A. J. Phys. Chem. 1964, 68, 441−447. (20) Spackman, M. A. Chem. Phys. Lett. 1999, 301, 425−429. (21) Matta, C. F.; Boyd, R. J. An Introduction to the Quantum Theory of Atoms in Molecules. In The Quantum Theory of Atoms in Molecules. From Solid State to DNA and Drug Design; Matta, C. F., Boyd, R. J., Eds.; Wiley-VCH: Weinheim, Germany, 2007. (22) Lo Presti, L.; Soave, R.; Destro, R. J. Phys. Chem. B 2006, 110, 6405−6414. (23) Destro, R.; Soave, R.; Barzaghi, M. J. Phys. Chem. B 2008, 112, 5163−5174. (24) Hathwar, V. R.; Gonnade, R. G.; Munshi, P.; Bhadbhade, M. M.; Row, T. N. G. Cryst. Growth Des. 2011, 11, 1855−1862. (25) Hathwar, V. R.; Row, T. N. G. Cryst. Growth Des. 2011, 11, 1338−1346. (26) Bui, T. T. T.; Dahaoui, S.; Lecomte, C.; Desiraju, G. R.; Espinosa, E. Angew. Chem., Int. Ed. 2009, 48, 3838−3841. (27) Brezgunova, M. E.; Aubert, E.; Dahaoui, S.; Fertey, P.; Lebegue, S.; Jelsch, C.; Angyan, J. G.; Espinosa, E. Cryst. Growth Des. 2012, 12, 5373−5386. (28) Thomas, S. P.; Pavan, M. S.; Row, T. N. G. Cryst. Growth Des. 2012, 12, 6083−6091. (29) Brezgunova, M. E.; Lieffrig, J.; Aubert, E.; Dahaoui, S.; Fertey, P.; Lebegue, S.; Angyan, J. G.; Fourmigue, M.; Espinosa, E. Cryst. Growth Des. 2013, 13, 3283−3289. (30) Jarzembska, K. N.; Kaminski, R.; Wenger, E.; Lecomte, C.; Dominiak, P. M. J. Phys. Chem. C 2013, 117, 7764−7775. (31) Pavan, M. S.; Prasad, K. D.; Row, T. N. G. Chem. Commun. 2013, 49, 7558−7560. (32) Shishkina, A. V.; Zhurov, V. V.; Stash, A. I.; Vener, M. V.; Pinkerton, A. A.; Tsirelson, V. G. Cryst. Growth Des. 2013, 13, 816− 828. (33) Zhurov, V. V.; Pinkerton, A. A. Z. Anorg. Allg. Chem. 2013, 639, 1969−1978. (34) Jarzembska, K. N.; Hoser, A. A.; Kaminski, R.; Madsen, A. O.; Durka, K.; Wozniak, K. Cryst. Growth Des. 2014, 14, 3453−3465. (35) Zhurov, V. V.; Pinkerton, A. A. Cryst. Growth Des. 2014, 14, 5685−5691. (36) Cenedese, S.; Zhurov, V. V.; Pinkerton, A. A. Cryst. Growth Des. 2015, 15, 875−883.
that the EML approach cannot be applied to these interactions. (4) EML energy estimates for hydrogen bonds are no more reliable (or unreliable) than those for other interactions. (5) For the subset of 81 molecular pairs in Figure 2b, the mean deviation of EML estimates from CE-B3LYP model energies is −3.1 kJ mol−1, the mean absolute deviation is 5.3 kJ mol−1, and the rms difference is 6.9 kJ mol−1.
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CONCLUSIONS The results presented here, especially Figures 1 and 2, speak for themselves and need little emphasis. It seems quite clear that EML estimates of atom···atom energies, based on modern experimental charge density analyses and eqs 1−3, provide little more than a very crude estimate of an interaction energy. Moreover, Figure 1b suggests that this crude estimate can be surprisingly well approximated by a universal equation that depends only on the atom···atom separation relative to the sum of van der Waals radii. Intermolecular energies based on sums of EML atom···atom energies are seldom reliable, typically underestimating and sometimes overestimating more reliable values. Based on these estimates, conclusions regarding the energetic importance of specific intermolecular interactions, especially in the context of crystal packing and crystal engineering, can often be quite misleading, and the continued use of EML energy estimates for this purpose should be discouraged.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The author acknowledges the hospitality of the Department of Chemistry, Aarhus University, during a short and productive visit. This work was supported in part by the Australian Research Council (Grant DP130103488) and the Danish National Research Foundation (DNRF93) Center for Materials Crystallography.
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REFERENCES
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