Plane-Wave Density Functional Theory Meets Molecular Crystals

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Plane-Wave Density Functional Theory Meets Molecular Crystals: Thermal Ellipsoids and Intermolecular Interactions Volker L. Deringer,*,†,§ Janine George,† Richard Dronskowski,†,‡ and Ulli Englert† †

Institute of Inorganic Chemistry and ‡Jülich−Aachen Research Alliance (JARA-HPC), RWTH Aachen University, Landoltweg 1, 52056 Aachen, Germany

CONSPECTUS: Molecular compounds, organic and inorganic, crystallize in diverse and complex structures. They continue to inspire synthetic efforts and “crystal engineering”, with implications ranging from fundamental questions to pharmaceutical research. The structural complexity of molecular solids is linked with diverse intermolecular interactions: hydrogen bonding with all its facets, halogen bonding, and other secondary bonding mechanisms of recent interest (and debate). Today, high-resolution diffraction experiments allow unprecedented insight into the structures of molecular crystals. Despite their usefulness, however, these experiments also face problems: hydrogen atoms are challenging to locate, and thermal effects may complicate matters. Moreover, even if the structure of a crystal is precisely known, this does not yet reveal the nature and strength of the intermolecular forces that hold it together. In this Account, we show that periodic plane-wave-based density functional theory (DFT) can be a useful, and sometimes unexpected, complement to molecular crystallography. Initially developed in the solid-state physics communities to treat inorganic solids, periodic DFT can be applied to molecular crystals just as well: theoretical structural optimizations “help out” by accurately localizing the elusive hydrogen atoms, reaching neutron-diffraction quality with much less expensive measurement equipment. In addition, phonon computations, again developed by physicists, can quantify the thermal motion of atoms and thus predict anisotropic displacement parameters and ORTEP ellipsoids “from scratch”. But the synergy between experiment and theory goes much further than that. Once a structure has been accurately determined, computations give new and detailed insights into the aforementioned intermolecular interactions. For example, it has been debated whether short hydrogen bonds in solids have covalent character, and we have added a new twist to this discussion using an orbital-based theory that once more had been developed for inorganic solids. However, there is more to a crystal structure than a handful of short contacts between neighboring residues. We hence have used dimensionally resolved analyses to dissect crystalline networks in a systematic fashion, one spatial direction at a time. Initially applied to hydrogen bonding, these techniques can be seamlessly extended to halogen, chalcogen, and pnictogen bonding, quantifying bond strength and cooperativity in truly infinite networks. Finally, these methods promise to be useful for (bio)polymers, as we have recently exemplified for α-chitin. At the interface of increasingly accurate and popular DFT methods, ever-improving crystallographic expertise, and new challenging, chemical questions, we believe that combined experimental and theoretical studies of molecular crystals are just beginning to pick up speed.

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INTRODUCTION Knowing and understanding molecular structure is a central quest for all of chemical research. Diffraction experiments have played important parts in this: from early, groundbreaking work on the crystal structures of benzene1 or insulin2 to the massive body of chemical and crystallographic knowledge that is today collected in the Cambridge Structural Database (CSD).3 Increasingly fast and reliable measurements are now routinely © 2017 American Chemical Society

performed on known and new molecular crystals. At the same time, cutting-edge experiments provide unprecedented insight: the electron density can be obtained with subatomic resolution, yielding detailed information about the nature of intermolecular interactions;4−6 neutron diffraction provides atom-resolved Received: February 2, 2017 Published: May 3, 2017 1231

DOI: 10.1021/acs.accounts.7b00067 Acc. Chem. Res. 2017, 50, 1231−1239

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Accounts of Chemical Research

Figure 1. Overview of topics in this Account. Starting from the periodic Kohn−Sham equations in reciprocal (“k”) space, DFT-based tools are commonly applied to inorganic materials (gray) but can likewise be useful for studying molecular crystals (blue/green).

structures of biomolecules.7 X-ray diffraction (XRD) techniques have seen major advances as well, regarding both instrumentation8 and interpretation.9 Nonetheless, important challenges remain. Many of the above experiments require highly advanced apparatuses at large research facilities, such as neutron sources or synchrotrons, and also samples of a certain quality, stability, or limited complexity. More fundamental issues arise with the very definition of intermolecular interactions: even if a neutron diffraction experiment yields accurate hydrogen···acceptor distances in an organic crystal, this does not yet reveal the nature and energetic strength of the hydrogen bonds involved. Similarly, other interactions such as “halogen bonds” are often assigned on structural grounds, which has led to heated and ongoing debates about the entire concept of “crystal engineering”.10−12 In principle, quantitative information about structure and bonding of solids can be extracted from first-principles quantumchemical computations,13 but their application to molecular crystals is far from trivial. On the one hand, organic molecules have been studied by so-called “wavefunction-based” electronicstructure methods for decades. Usually applied in the gas phase (for isolated molecules or small n-mers), these methods have recently been transferred to the solid state using extended clusters;14−16 this emerging approach is particularly promising for computing very accurate lattice energies17 and thermodynamic properties18 and has been summarized in two recent articles by Beran and co-workers.15,16 On the other handand in a largely complementary way, as we will see in a moment periodic density functional theory (DFT) has become the workhorse of choice for solid-state inorganic chemistry, physics, and materials science.13 Our Account focuses on periodic DFT with delocalized plane-wave basis sets rather than localized atomic orbitals, but both are viable choices.19 Combined with suitable and economic dispersion corrections,20,21 periodic DFT is now becoming a powerful tool for treating inorganic and particularly organic molecular solids.15,22,23 An important application (however beyond the scope of this work) is crystal structure prediction in the absence of experimental information.24 In this Account, we show that combinations of periodic DFT and molecular crystallography can go even further, jointly making new and useful contributions to chemical research. As is customary in this journal, we focus on recent joint ventures of ours (Figure 1). We begin by showing how theory can support

and complement experiments but focus on gaining new chemical insight, using periodic DFT to explore intermolecular interactions in crystals. We aim to provide a “pocket guide” rather than a comprehensive review and hope in particular to encourage future synergies between experiments and theory.

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QUANTUM-CHEMICAL “LITTLE HELPERS” High-resolution XRD and neutron diffraction can accurately locate all atoms in a crystal structure, but it is not that simple in many “household” experiments. All too often, a crystal structure is essentially known but details are notsuch as the precise location of hydrogen atoms, because of their low X-ray scattering power. Theory can provide a way out: periodic DFT readily describes the forces on atoms in a candidate structure, and these can be used to guide the hydrogen atoms into their respective local minima. The technical details are not the topic of this Account; they are described in ref 25 and in earlier work cited there. In a nutshell, to validate the robustness of plane-wave DFT approaches, we assembled a set of crystals with various X−H bonds for which previous XRD and also reference neutron diffraction experiments are available. While X−H distances from standard XRD are often underestimated, selective computational optimization brings them into very close agreement with the neutron benchmark (“selective” here indicates that cell parameters, heavy-atom positions, and symmetry are locked to their experimental values).25 We stress that these are “static” computations: no temperature enters the Kohn−Sham equations, and aside from the use of experimental lattice parameters, the calculations do not account for thermal effects. Take, for example, a hydrogen atom in a very short, almost symmetric intramolecular O··H··O bond: during DFT optimization, it will be “pulled back” into an asymmetric O−H···O configuration, in agreement with neutron data at very low temperature (see ref 25 and references therein). Indeed, going beyond zero Kelvin is a current important frontier, and again one where concepts from the solid-state physics and chemistry communities can come in helpful. To quantify the thermal motion of atoms in molecular solids and to solve complex crystal structures, crystallographers use anisotropic displacement parameters (ADPs), which can be visualized using the famous Oak Ridge Thermal Ellipsoid Plot (ORTEP) technique. The experimental determination of reliable ADPs, however, requires crystals of good quality and can be severely hampered by effects such as parameter correlation or 1232


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nitrogen temperature and thus routinely used in diffraction experiments. Our computations also allowed comparison of different dispersion-correction methods;29 however, a choice of “best” method among the many high-quality ones available (both for the correction strategy and the underlying DFT functional) will likely depend on the system at hand. More recently, we validated predicted ADPs for transition-metal carbonyls as model systems for organometallic compounds.30 The latter field in particular, we are convinced, holds promise because next to heavy-metal atoms, hydrogens are even more challenging to locate experimentally. By contrast, theory can provide highquality ADPs for the light H atoms just as well (Figure 2b).30 Quantum-chemical “little helpers” as discussed here are going through three stages. First, computational results can be checked against existing literature,25,27 as there is a wealth of crystallographic experience already collected in the CSD.3 Second, one can design new, targeted experiments to address specific questionssuch as for pentachloropyridine above.29 Ultimately, once thus validated, periodic DFT can help with new experiments and structures. Chemical crystallographers often use pseudoobservations, so-called “restraints”, to improve the ratio of observables to variables and to ensure convergence to physically meaningful structure models. Currently, these restraints are chosen from intuition and experience or based on database searches such as in the CSD,3 but DFT could provide individual, structure-specif ic restraints in the near future.31 In particular, a handful of tailor-made high-quality experiments can be used to validate periodic DFT methods and their application in molecular crystallography. Subsequently, this can boost the outcome of experiments with limited resolution, sample quality, or commitmenti.e., the vast majority! We provide an online tutorial at http://www.ellipsoids.de, alongside further historical and technical information about ADPs from experiment and theory.

pseudosymmetry. In such cases, phonon computations based on DFT can be used to compute, for each atom in a crystal, the direction (phonon eigenvector) and the extent of its movement (which increases with temperature according to statistical thermodynamics): one can therefore predict ADPs from periodic DFT, at least in principle.26,27 We exemplify this in Figure 2a for

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Figure 2. (a) Anisotropic displacement parameters (ADPs) in crystalline guanidine at 100 K, visualized using ORTEP-style drawings. Ellipsoids correspond to 90% probability of finding an atom within them. Adapted with permission from ref 27. Copyright 2014 Royal Society of Chemistry. (b) Scatterplot of reliable experimental vs computed ADPs, compiling data for urea and guanidine (ref 27), pentachloropyridine (ref 29), and two chromium carbonyl complexes (ref 30); technical details are given in the original references.

QUANTIFYING HYDROGEN BONDING IN SOLIDS We now move from a mainly “supportive” role of DFT to one where it gives new and complementary chemical insight into molecular crystals: namely, helping to understand the microscopic forces that hold them together, from the well-known hydrogen bonds (HBs) to the ubiquitous dispersion forces. HBs in solids have been prominently studied for decades, mainly from a structural perspective:32 for example, Jeffrey’s scheme classifies them as “strong”, “moderate”, and “weak” according to the H··· acceptor distance.33 To gauge the nature and strength of HBs, the scaling behavior of vibrational frequencies34 and (where highquality crystals are available) the charge density at the bond critical point4,5,35−37 are widely established techniques based on experimental observables; from the theoretical side, symmetryadapted perturbation theory (SAPT) has been used with success.38 This section discusses an alternative approach based on periodic DFT and delocalized plane-wave basis sets. It has long been suggested that short and strong HBs have a (partly) covalent nature.39 Some of us have been developing tools to analyze covalent bonds in solids using plane-wave-based DFT,40 initially for materials chemistry applications (see references in ref 40), but we were now eager to try these tools out for molecular solids. Our starting point was a peculiar guanidinium salt of squaric acid (Figure 3a): its structure contains HBs of very different lengths and strengths (affording a comprehensive and consistent experimental data set),5 and the compound forms large single crystals suitable for neutron and high-resolution X-ray diffraction. The first method allows the

guanidine, urea’s all-nitrogen analogue, for which single-crystal neutron data and thus high-quality ADPs are available.28 The results hold up favorably against experiment: almost no difference between the neutron and DFT results is seen with the naked eye. Following initial proofs of concept on ADP prediction,26,27 it next became important to develop systematic benchmarks. Phonon computations are today routinely done in the harmonic approximation, that is, at the equilibrium geometry: very simply put, the atoms go beyond zero Kelvin, but the lattice vectors do not. This approximation is computationally feasible and pays off at low temperature, where the lattice expansion is small. To explore what “low temperature” means in practice, we next designed experiments on pentachloropyridine (a simple system with halogen-bonding and dispersive interactions but no hydrogen atoms).29 Taking high-resolution XRD measurements in small temperature steps, refining the ADPs at each, and contrasting them with theory output showed that this method is indeed well applicable (giving results largely within the threefold experimental standard deviation) at 100 K: this is above liquid1233


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Figure 3. Hydrogen bonds in solids and ways of looking at them. (a) Molecular structure of N,N-dimethylbiguanidinium bis(hydrogen squarate).5 Distances for the three shortest HBs and atom labels are given as in the initial publication.5 (b) Laplacian of the experimental charge density for the anion dimer in the crystal. Reproduced with permission from ref 5. Copyright 2011 International Union of Crystallography. (c) Electronic densities of states (DOS) and projected crystal orbital Hamilton population (pCOHP) analysis.39 (d) A combined picture from both methods. Panels (c) and (d) adapted with permission from ref 42. Copyright 2014 Royal Society of Chemistry.

integrated pCOHPs for each contact against the respective H··· acceptor distance then led to an instructive picture (Figure 3d): the short HBs in the hydrogen squarate dimer show strong covalent contributions; this is less pronounced in the N−H···O bonds, and finally the weak “nonclassical” C−H···O HBs exhibit no covalency. This behavior is very similar to what is found from experimental charge densities at the bond critical points (ρbcp),5 even though the two approaches are fundamentally different; it is not limited to the results in Figure 3 but is more general and observed for nonionic compounds as well.42 Such combined perspectives will be useful for further studies of crystals with strong HBsin particular as pCOHPs can reliably distinguish between hydrogen bonding and mere vicinity,42 even in the absence of highly accurate diffraction data.

hydrogen atoms to be localized, and the second represents an established approach for quantitative experimental studies of HBs (Figure 3b).35 Solid-state computations can yield bonding information from a different angle by analyzing the self-consistent wave function ψk in the language of orbitals and bonds.40 The underlying idea is already described in undergraduate textbooks: invariably, they contain a molecular orbital (MO) diagram for the H2 molecule, with a bonding (stabilizing) and an antibonding (destabilizing) linear combination of the 1s orbitals. In solids, an analogue of the MO scheme is given by the densities of states (DOS), but this does not yet tell bonding from antibonding interactions. Instead, DOS contributions relating to pairs of neighboring orbitals can be singled out and weighted with the corresponding Hamiltonian matrix elements; this is called crystal orbital Hamilton population (COHP) analysis.41 The LOBSTER software performs such analyses routinely for the output of plane-wave DFT (ref 40 and references therein): it takes the self-consistent wave functions and projects them onto local orbitalshence “projected COHP”, or pCOHP. Figure 3c shows results for the two shortest HBs in the aforementioned guanidinium compound.42 Alongside the DOS, pCOHP curves provide chemical information by placing bonding and antibonding orbital interactions on either side of the energy axis. Finally, they can be integrated along the energy axis (from the lowest bands up to the Fermi level εF), yielding a measure of covalent bond strength (Figure 3c).41 Plotting

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INTERMOLECULAR INTERACTIONS IN EXTENDED NETWORKS The previous section may have left the impression that short HBs are what determines a molecular crystal’s properties and that everything beyond those is irrelevant. This is, of course, a dangerous simplification.43 We have therefore developed approaches that give a more comprehensive picture of bonding in molecular solids; in other words, we now look beyond covalent HBs (and beyond HBs in general) onward to the overall interaction energy, which we seek not for a gas-phase dimer but for an entire crystal. In so doing, we can gain new information about the dimensionality of the different interactions. Figure 4a 1234


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Figure 4. (a) Crystal structure of bromomalonic aldehyde as determined from XRD. (b) Schematic drawing of a supercell model as used in DFT energy analyses, here, for a one-dimensionally infinite H-bonded chain. In computational practice, the vacuum regions are usually larger. (c) DFT-computed interaction energies (relative to the free molecule) when tearing the crystal apart, one direction at a time. Adapted with permission from ref 46. Copyright 2013 Royal Society of Chemistry.

shows an examplethe structure of bromomalonic aldehyde, which seems straightforward: there are strong HBs in one direction (red) and somewhat weaker halogen bonds in another (orange), and then the so-assembled sheets stack to form the three-dimensional crystal. But is it really that simple? The question of interaction energies in solids is a fundamental one. To compute the cohesive energy of, say, diamond, a solidstate physicist will obtain from DFT the per-atom energy of the crystal and of a free carbon atom and then subtract one from the other. With the current machinery, the same can be done for a molecular crystal, first computing the DFT energy of the extended structure and second that of an isolated molecule (which is therefore put into a box, or “supercell”, with plenty of vacuum around it). Again, the interaction strength is given by the properly normalized energy difference between the two systems. This approach has been pioneered by Morrison and Siddick,44 who used periodic DFT to compute the hydrogen-bonding energy in crystalline urea. Indeed, the supercell approach can be taken further: not only simulating isolated molecules in vacuo but cleaving a crystal systematically in different spatial dimensions and thus computing energies for isolated fragments or “building blocks” (Figure 4b). As in the ORTEP case, we began our studies with guanidine,45 which contains several different HBs. However, the true strength of this approach is that it automatically and quantitatively captures all relevant interactions, HBs and others, without empirical preconceptions. We illustrate such an analysis in Figure 4c. There, DFTcomputed interaction energies are shown as we progressively disassemble the bromomalonic aldehyde crystal structure into 2D layers and then into 1D chains (cf. Figure 4b), which are broken apart in the last step.46 All of the energies are normalized and relative to one molecule, allowing comparison of fragments of different sizes. Somewhat unexpectedly at first, the interaction between the layers (3D → 2D) is larger than that in the putative halogen-bonded direction between the chains (2D → 1D). Subsequent analyses of electrostatic potentials resolved this puzzle: there is strong Coulombic attraction between bromine (δ+) and oxygen (δ−) across the layers, making the 3D → 2D cleavage comparatively expensive. These interactions, however, are nondirectional and thus are easily overlooked when assigning interactions purely on structural grounds. As said above, such DFT analyses capture all relevant interactions, directional or not. It goes without saying that appropriate dispersion corrections to DFT are crucial for the success of this,20,21 especially when interlayer interactions or

Figure 5. (a) A dimer, a trimer, and an infinite chain of ICN molecules, which count among the simplest prototypes for halogen-bonded systems. Using periodic DFT, these are easily modeled with the supercell technique (cf. Figure 4b). Interaction energies are given per bond to directly assess the “cooperative” amplification of halogen bonding. (b) A systematic survey of cooperativity in pnictogen-, chalcogen-, and halogen-bonded systems. For the respective heaviest homologue in particular, there is a pronounced increase in stabilization when going from trimers to chains, indicating that the bonds amplify each other. Adapted from ref 47. Copyright 2014 American Chemical Society.

even systems without any HBs are investigated. In principle, charge-density studies based on high-resolution XRD may provide an experimental counterpart;35−37 for bromomalonic aldehyde, this has not (yet) been done. Approaches as sketched in Figure 4b then allowed for systematic studies of halogen bonding (“XB”) and related mechanisms involving chalcogen and pnictogen atoms.47,48 We have thereby taken a step into the world of “σ-hole” interactions 1235


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What about applications? “Molecular solids” is a more diverse term today than it used to be, creating exciting new prospects for combined experimental and theoretical work. From ORTEPs in organometallics,30 the conceptual road leads onward to metal− organic frameworks61,62 and organic−inorganic hybrid materials. The role of hydrogen bonding in hybrid perovskites, as but one example, was recently explored by computations.63 Another vast field is given by biomaterials: we recently studied α-chitin with several tools discussed herein (Figure 6),64 and there are many

(see ref 49 for an exemplary, early plane-wave DFT study and ref 50 for introductory references and a survey of recent computational efforts). We set out to study energetic cooperativity (bonds amplifying each other) in several types of σ-hole bonds,47 using as model systems infinite chains of X(CN)n molecules (Figure 5). This avoids finite-size effects (look at Figure 5a: the trimer behavior is clearly different from that of the infinite chain), better reflecting the “real-life” periodic network. All of the chains exhibit large energetic cooperativity (Figure 5b). On a side note, the optimized bond lengths in the isolated halogen cyanide chains agree well with experimental values in the crystal structures (in which chains are packed together). Chalcogen cyanides, by contrast, form characteristic 2D networks, which we investigated in subsequent work:48 throughout the literature, crystals are described as “layered”, but when is a structure truly layered (as in graphite) and when is it not? The energetic results for homologous, structurally similar series may vary: from S(CN)2 to Se(CN)2 to Te(CN)2, the ratios of intra- to intersheet energies differ strongly.48 Ultimately, these analyses must be corroborated by experimentfor example, it would be interesting to probe the anisotropic thermal expansion of the above X(CN)2 crystals.

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Figure 6. α-Chitin, one of nature’s most abundant building materials, as an example of how the techniques discussed herein may be transferred to biomolecules. The disordered structure67,68 has been approximated using (necessarily small) discrete models and periodic DFT optimization. Here a polymer is shown with alternating configurations for the disordered hydroxymethyl groups (labeled “A/B”) and HBs within (blue) and between the chains (green). Subsequent energy analyses, similar in spirit to Figure 4c, allowed us to explore the bonding nature of the three-dimensional structure.64 Adapted from ref 64. Copyright 2016 American Chemical Society.

FUTURE DIRECTIONS Solid-state DFT methods, as we hope to have convinced the reader, are useful additions to the toolkit of chemical crystallography and by and large are ready for use today. Still, the theory communities are very active in further improving the computational machinery. One such aspect is the treatment of dispersion forces: pairwise corrections to DFT are popular for good reason,20−23 but new developments beyond these may describe molecular crystals even better.51,52 Recently, for example, many-body dispersion corrections resolved the puzzle of why one aspirin polymorph is more stable than the other.53 Furthermore, we believe that it is very important to improve ORTEP prediction at “high” (i.e., ambient) temperature27,29 and finite-temperature modeling of molecular solids in general; a more detailed discussion of this than is possible here and a survey of interesting work are provided in refs 15 and 54. Again, many such approaches were initially developed by solid-state physicists, using phonon computations that are ubiquitous for inorganic materials nowadays, and transferred to molecular crystals in initial case studies.54,55 Very recently, we showed how quasiharmonic phonon computations yield high-quality ORTEPs for crystalline sulfur up to 200 K.56 With all due respect for theory, none of this Account would have been written if there had not been careful experiments before. We studied guanidine (Figure 2a) a number of times, and all of these analyses built on the initial experimental crystal structure determination which, frankly, came almost 150 years after the compound was first described by Strecker!57 In return, periodic plane-wave-based DFT now seems well-poised to complement routine as well as high-resolution experiments. In the first case, theory may readily provide reasonable starting values,58 structure-specific geometry and ADP restraints during refinement, and much-needed “sanity checks”;59 in the second case, it can help to deconvolute ADPs and population parameters of atom-centered multipoles60 for improved charge-density studies. In the near future, charge-density analyses may even be enabled by computed ADPs in the first place. This promises new, valuable, and more abundant experimental insight into the intermolecular interactions discussed above.

other structural “building blocks” in biology.65 In all of these cases, synergies between periodic DFT and (fiber) diffraction as well as local probes such as NMR16,66 seem very much worth exploring. In the long run, theory must help to develop new and optimized molecular materials. Initial steps have been taken: say, searching for new substrates for organic molecular beam epitaxy, which again requires dimensionally resolved understanding of crystals.69 To enable further developments, streamlined procedures and file formats would help to exchange data between the communities. Even more important, however, is the exchange of ideas: initiating dialogue between crystallographers and solid-state theorists; constantly challenging the respective “other side”; getting to know their possibilities, their limitations, and their way of saying things. For the present authors, the relationship between theory and experiment has evolved from peaceful coexistence to productive synergy. We hope that this Account will inspire more of this in the future.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Volker L. Deringer: 0000-0001-6873-0278 Janine George: 0000-0001-8907-0336 Richard Dronskowski: 0000-0002-1925-9624 Ulli Englert: 0000-0002-2623-0061 1236


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Accounts of Chemical Research Present Address

Structure of Rhombohedral 2 Zinc Insulin Crystals. Nature 1969, 224, 491−495. (3) Groom, C. R.; Allen, F. H. The Cambridge Structural Database in Retrospect and Prospect. Angew. Chem., Int. Ed. 2014, 53, 662−671. (4) Coppens, P. X-Ray Charge Densities and Chemical Bonding; Oxford University Press: New York, 1997. (5) Serb, M.-D.; Wang, R.; Meven, M.; Englert, U. The Whole Range of Hydrogen Bonds in One Crystal Structure: Neutron Diffraction and Charge-Density Studies of N,N-Dimethylbiguanidinium Bis(hydrogensquarate). Acta Crystallogr., Sect. B: Struct. Sci. 2011, 67, 552−559. (6) Wang, R.; Dols, T. S.; Lehmann, C. W.; Englert, U. The Halogen Bond Made Visible: Experimental Charge Density of a Very Short Intermolecular Cl···Cl Donor-Acceptor Contact. Chem. Commun. 2012, 48, 6830−6832. (7) Langan, P.; Chen, J. C.-H. Seeing the Chemistry in Biology with Neutron Crystallography. Phys. Chem. Chem. Phys. 2013, 15, 13705− 13712. (8) Thompson, S. P.; Parker, J. E.; Marchal, J.; Potter, J.; Birt, A.; Yuan, F.; Fearn, R. D.; Lennie, A. R.; Street, S. R.; Tang, C. C. Fast X-Ray Powder Diffraction on I11 at Diamond. J. Synchrotron Radiat. 2011, 18, 637−648. (9) Capelli, S. C.; Bürgi, H.-B.; Dittrich, B.; Grabowsky, S.; Jayatilaka, D. Hirshfeld Atom Refinement. IUCrJ 2014, 1, 361−379. (10) Dunitz, J. D. Intermolecular Atom−Atom Bonds in Crystals? IUCrJ 2015, 2, 157−158. (11) Thakur, T. S.; Dubey, R.; Desiraju, G. R. Intermolecular Atom− Atom Bonds in Crystals − a Chemical Perspective. IUCrJ 2015, 2, 159− 160. (12) Lecomte, C.; Espinosa, E.; Matta, C. F. On Atom−Atom “Short Contact” Bonding Interactions in Crystals. IUCrJ 2015, 2, 161−163. (13) Dronskowski, R. Computational Chemistry of Solid State Materials; Wiley-VCH: Weinheim, Germany, 2005. (14) Č ervinka, C.; Fulem, M.; Růzǐ čka, K. CCSD(T)/CBS FragmentBased Calculations of Lattice Energy of Molecular Crystals. J. Chem. Phys. 2016, 144, 064505. (15) Beran, G. J. O. Modeling Polymorphic Molecular Crystals with Electronic Structure Theory. Chem. Rev. 2016, 116, 5567−5613. (16) Beran, G. J. O.; Hartman, J. D.; Heit, Y. N. Predicting Molecular Crystal Properties from First Principles: Finite-Temperature Thermochemistry to NMR Crystallography. Acc. Chem. Res. 2016, 49, 2501− 2508. (17) Yang, J.; Hu, W.; Usvyat, D.; Matthews, D.; Schütz, M.; Chan, G. K.-L. Ab Initio Determination of the Crystalline Benzene Lattice Energy to Sub-Kilojoule/mol Accuracy. Science 2014, 345, 640−643. (18) Heit, Y. N.; Nanda, K. D.; Beran, G. J. O. Predicting FiniteTemperature Properties of Crystalline Carbon Dioxide from First Principles with Quantitative Accuracy. Chem. Sci. 2016, 7, 246−255. (19) Evarestov, R. A. Quantum Chemistry of Solids; Springer: Berlin, 2012. (20) Kronik, L.; Tkatchenko, A. Understanding Molecular Crystals with Dispersion-Inclusive Density Functional Theory: Pairwise Corrections and Beyond. Acc. Chem. Res. 2014, 47, 3208−3216. (21) Grimme, S.; Hansen, A.; Brandenburg, J. G.; Bannwarth, C. Dispersion-Corrected Mean-Field Electronic Structure Methods. Chem. Rev. 2016, 116, 5105−5154. (22) Moellmann, J.; Grimme, S. DFT-D3 Study of Some Molecular Crystals. J. Phys. Chem. C 2014, 118, 7615−7621. (23) Cutini, M.; Civalleri, B.; Corno, M.; Orlando, R.; Brandenburg, J. G.; Maschio, L.; Ugliengo, P. Assessment of Different Quantum Mechanical Methods for the Prediction of Structure and Cohesive Energy of Molecular Crystals. J. Chem. Theory Comput. 2016, 12, 3340− 3352. (24) Price, S. L. Computed Crystal Energy Landscapes for Understanding and Predicting Organic Crystal Structures and Polymorphism. Acc. Chem. Res. 2009, 42, 117−126. (25) Deringer, V. L.; Hoepfner, V.; Dronskowski, R. Accurate Hydrogen Positions in Organic Crystals: Assessing a QuantumChemical Aide. Cryst. Growth Des. 2012, 12, 1014−1021.

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V.L.D.: Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom. Notes

The authors declare no competing financial interest. Biographies Volker L. Deringer (born 1987) studied chemistry at RWTH Aachen University, where he received his diploma (2010) and doctorate (2014) under the guidance of Richard Dronskowski. Currently he is a postdoctoral researcher at the University of Cambridge, supported by a fellowship from the Alexander von Humboldt Foundation. He is a computational solid-state chemist, curious about the atomistic modeling and bonding theory of complex solids. Janine George (born 1989) received her B.Sc. (2011) and M.Sc. (2013) from RWTH Aachen University. She is currently a doctoral candidate under the direction of Richard Dronskowski and has been supported by a Chemiefonds Fellowship from the Fonds der Chemischen Industrie. She is a solid-state chemist, interested in intermolecular interactions and vibrational properties of molecular crystals. Richard Dronskowski (born 1961) studied chemistry and physics in Münster and obtained his doctorate under the guidance of Arndt Simon (Stuttgart) in 1990. After a stay with Roald Hoffmann (Cornell) and receiving his habilitation in Dortmund, he moved to RWTH Aachen University in 1996, where he holds the Chair of Solid-State and Quantum Chemistry. His research spans experimental solid-state chemistry (carbodiimides, guanidinates, nitrides, and metastable phases), neutron diffraction (including the development of the highresolution time-of-flight neutron diffractometer POWTEX), and the quantum chemistry of solids (electronic structure, bonding theory, magnetism, and thermochemistry). Ulli Englert (born 1957) studied chemistry and biology in Tübingen; Joachim Strähle was the supervisor of his doctoral dissertation. After a postdoc with Fausto Calderazzo (Pisa), he moved to Aachen in 1989, where he has been a professor since 2003. He regularly teaches crystallography at a German summer school (since 2000) and serves as co-editor for Acta Crystallographica, Section C (since 2008). His research interests center around molecular crystals, mostly from an experimental point of view. Activities of his group include molecular recognition, solid-state reactivity, structural phase transitions, and charge density; the last of these is a challenging meeting ground with theory.

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ACKNOWLEDGMENTS We are grateful to our experimental collaborators, past and present, for sparking our interest in molecular crystals and for many joint projects. Philipp Jacobs is thanked for a critical reading of the manuscript. Our research has been supported by the Studienstiftung des deutschen Volkes (V.L.D.), the Fonds der Chemischen Industrie (J.G.), the Jülich−Aachen Research Alliance (JARA-HPC Project jara0069), and the Deutsche Forschungsgemeinschaft (DFG).

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DEDICATION This Account is dedicated to the memory of Walter Kohn (1923−2016) and Hugo Rietveld (1932−2016). REFERENCES

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Plane-Wave Density Functional Theory Meets Molecular Crystals

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