Exchange-Hole Dipole Dispersion Model for Accurate Energy Ranking in Molecular Crystal Structure Prediction II: Nonplanar Molecules

Frontiers of molecular crystal structure prediction for pharmaceuticals and functional organic materials

The reliability of organic molecular crystal structure prediction has improved tremendously in recent years. Crystal structure predictions for small, mostly rigid molecules are quickly becoming routine. Structure predictions for larger, highly flexible molecules are more challenging, but their crystal structures can also now be predicted with increasing rates of success. These advances are ushering in a new era where crystal structure prediction drives the experimental discovery of new solid forms. After briefly discussing the computational methods that enable successful crystal structure prediction, this perspective presents case studies from the literature that demonstrate how state-of-the-art crystal structure prediction can transform how scientists approach problems involving the organic solid state. Applications to pharmaceuticals, porous organic materials, photomechanical crystals, organic semi-conductors, and nuclear magnetic resonance crystallography are included. Finally, efforts to improve our understanding of which predicted crystal structures can actually be produced experimentally and other outstanding challenges are discussed.

Organic Crystal Packing Is Key to Determining the Photomechanical Response

Organic photomechanical crystals have great promise as molecular machines, but their development has been hindered by a lack of clear theoretical design principles. While much research has focused on the choice of the molecular photochrome, density functional theory calculations here demonstrate that crystal packing has a major impact on the work densities that can be produced by a photochrome. Examination of two diarylethene molecules reveals that the predicted work densities can vary by an order of magnitude across different experimentally known crystal structures of the same species. The highest work densities occur when molecules are aligned in parallel, thereby producing a highly anisotropic photomechanical response. These results suggest that a greater emphasis on polymorph screening and/or crystal engineering could improve the work densities achieved by photomechanical engines. Finally, an inherent thermodynamic asymmetry is identified that biases photomechanical engines to exhibit higher work densities in the forward stroke direction.

XDM-corrected hybrid DFT with numerical atomic orbitals predicts molecular crystal lattice energies with unprecedented accuracy

Molecular crystals are important for many applications, including energetic materials, organic semiconductors, and the development and commercialization of pharmaceuticals. The exchange-hole dipole moment (XDM) dispersion model has shown good performance in the calculation of relative and absolute lattice energies of molecular crystals, although it has traditionally been applied in combination with plane-wave/pseudopotential approaches. This has limited XDM to use with semilocal functional approximations, which suffer from delocalization error and poor quality conformational energies, and to systems with a few hundreds of atoms at most due to unfavorable scaling. In this work, we combine XDM with numerical atomic orbitals, which enable the efficient use of XDM-corrected hybrid functionals for molecular crystals. We test the new XDM-corrected functionals for their ability to predict the lattice energies of molecular crystals for the X23 set and 13 ice phases, the latter being a particularly stringent test. A composite approach using a XDM-corrected, 25% hybrid functional based on B86bPBE achieves a mean absolute error of 0.48 kcal mol^-1 per molecule for the X23 set and 0.19 kcal mol^-1 for the total lattice energies of the ice phases, compared to recent diffusion Monte-Carlo data. These results make the new XDM-corrected hybrids not only far more computationally efficient than previous XDM implementations, but also the most accurate density-functional methods for molecular crystal lattice energies to date.

A theoretical framework for the design of molecular crystal engines

Photomechanical molecular crystals have garnered attention for their ability to transform light into mechanical work, but difficulties in characterizing the structural changes and mechanical responses experimentally have hindered the development of practical organic crystal engines. This study proposes a new computational framework for predicting the solid-state crystal-to-crystal photochemical transformations entirely from first principles, and it establishes a photomechanical engine cycle that quantifies the anisotropic mechanical performance resulting from the transformation. The approach relies on crystal structure prediction, solid-state topochemical principles, and high-quality electronic structure methods. After validating the framework on the well-studied [4 + 4] cycloadditions in 9-methyl anthracene and 9-tert-butyl anthracene ester, the experimentally-unknown solid-state transformation of 9-carboxylic acid anthracene is predicted for the first time. The results illustrate how the mechanical work is done by relaxation of the crystal lattice to accommodate the photoproduct, rather than by the photochemistry itself. The large ∼10⁷ J m^-3 work densities computed for all three systems highlight the promise of photomechanical crystal engines. This study demonstrates the importance of crystal packing in determining molecular crystal engine performance and provides tools and insights to design improved materials in silico.

Minimal Basis Set Hartree-Fock Corrected with Atom-Centered Potentials for Molecular Crystal Modeling and Crystal Structure Prediction

Crystal structure prediction (CSP), determining the experimentally observable structure of a molecular crystal from the molecular diagram, is an important challenge with technologically relevant applications in materials manufacturing and drug design. For the purpose of screening the randomly generated candidate crystal structures, CSP protocols require energy ranking methods that are fast and can accurately capture the small energy differences between molecular crystals. In addition, a good ranking method should also produce accurate equilibrium geometries, both intramolecular and intermolecular. In this article, we explore the combination of minimal-basis-set Hartree-Fock (HF) with atom-centered potentials (ACPs) as a method for modeling the structure and energetics of molecular crystals. The ACPs are developed for the H, C, N, and O atoms and fitted to a set of reference data at the B86bPBE-XDM level in order to mitigate basis-set incompleteness and missing correlation. In particular, ACPs are developed in combination with two methods: HF-D3/MINIs and HF-3c. The application of ACPs greatly improves the performance of HF-D3/MINIs for lattice energies, crystal energy differences, energy-volume and energy-strain relations, and crystal geometries. In the case of HF-3c, the improvement in the crystal energy differences is much smaller than in HF-D3/MINIs, but lattice energies and particularly crystal geometries are considerably better when ACPs are used. The resulting methods may be useful for CSP but also for quick calculation of molecular crystal lattice energies and geometries.

A density-functional benchmark of vibrational free-energy corrections for molecular crystal polymorphism

Many crystal structure prediction protocols only concern themselves with the electronic energy of molecular crystals. However, vibrational contributions to the free energy ( Fvib) can be significant in determining accurate stability rankings for crystal candidates. While force-field studies have been conducted to gauge the magnitude of these free-energy corrections, highly accurate results from quantum mechanical methods, such as density-functional theory (DFT), are desirable. Here, we introduce the PV17 set of 17 polymorphic pairs of organic molecular crystals, for which plane wave DFT is used to calculate the vibrational free energies and free-energy differences (Δ Fvib) between each pair. Our DFT results confirm that the vibrational free-energy corrections are small, having a mean value of 1.0 kJ/mol and a maximum value of 2.3 kJ/mol for the PV17 set. Furthermore, we assess the accuracy of a series of lower-cost DFT, semi-empirical, and force-field models for computing Δ Fvib that have been proposed in the literature. It is found that calculating Fvib using the Γ-point frequencies does not provide Δ Fvib values of sufficiently high quality. In addition, Δ Fvib values calculated using various approximate methods have mean absolute errors relative to our converged DFT results of equivalent or larger magnitude than the vibrational free-energy corrections themselves. Thus, we conclude that, in a crystal structure prediction protocol, it is preferable to forego the inclusion of vibrational free-energy corrections than to estimate them with any of the approximate methods considered here.

The interplay of intra- and intermolecular errors in modeling conformational polymorphs

Conformational polymorphs of organic molecular crystals represent a challenging test for quantum chemistry because they require careful balancing of the intra- and intermolecular interactions. This study examines 54 molecular conformations from 20 sets of conformational polymorphs, along with the relative lattice energies and 173 dimer interactions taken from six of the polymorph sets. These systems are studied with a variety of van der Waals-inclusive density functionals theory models; dispersion-corrected spin-component-scaled second-order Møller–Plesset perturbation theory (SCS-MP2D); and domain local pair natural orbital coupled cluster singles, doubles, and perturbative triples [DLPNO-CCSD(T)]. We investigate how delocalization error in conventional density functionals impacts monomer conformational energies, systematic errors in the intermolecular interactions, and the nature of error cancellation that occurs in the overall crystal. The density functionals B86bPBE-XDM, PBE-D4, PBE-MBD, PBE0-D4, and PBE0-MBD are found to exhibit sizable one-body and two-body errors vs DLPNO-CCSD(T) benchmarks, and the level of success in predicting the relative polymorph energies relies heavily on error cancellation between different types of intermolecular interactions or between intra- and intermolecular interactions. The SCS-MP2D and, to a lesser extent, ωB97M-V models exhibit smaller errors and rely less on error cancellation. Implications for crystal structure prediction of flexible compounds are discussed. Finally, the one-body and two-body DLPNO-CCSD(T) energies taken from these conformational polymorphs establish the CP1b and CP2b benchmark datasets that could be useful for testing quantum chemistry models in challenging real-world systems with complex interplay between intra- and intermolecular interactions, a number of which are significantly impacted by delocalization error.

How many more polymorphs of ROY remain undiscovered

With 12 crystal forms, 5-methyl-2-[(2-nitrophenyl)amino]-3-thiophenecabonitrile (a.k.a. ROY) holds the current record for the largest number of fully characterized organic crystal polymorphs. Four of these polymorph structures have been reported since 2019, raising the question of how many more ROY polymorphs await future discovery. Employing crystal structure prediction and accurate energy rankings derived from conformational energy-corrected density functional theory, this study presents the first crystal energy landscape for ROY that agrees well with experiment. The lattice energies suggest that the seven most stable ROY polymorphs (and nine of the twelve lowest-energy forms) on the Z′ = 1 landscape have already been discovered experimentally. Discovering any new polymorphs at ambient pressure will likely require specialized crystallization techniques capable of trapping metastable forms. At pressures above 10 GPa, however, a new crystal form is predicted to become enthalpically more stable than all known polymorphs, suggesting that further high-pressure experiments on ROY may be warranted. This work highlights the value of high-accuracy crystal structure prediction for solid-form screening and demonstrates how pragmatic conformational energy corrections can overcome the limitations of conventional density functionals for conformational polymorphs.

Crystal structure prediction suggests that the low-energy polymorphs of ROY have already been found, but a new high-pressure form is predicted.

Modeling the α- and β-resorcinol phase boundary via combination of density functional theory and density functional tight-binding

The ability to predict not only what organic crystal structures might occur but also the thermodynamic conditions under which they are the most stable would be extremely useful for discovering and designing new organic materials. The present study takes a step in that direction by predicting the temperature- and pressure-dependent phase boundary between the α and β polymorphs of resorcinol using density functional theory (DFT) and the quasi-harmonic approximation. To circumvent the major computational bottleneck associated with computing a well-converged phonon density of states via the supercell approach, a recently developed approximation is employed, which combines a supercell phonon density of states from dispersion-corrected third-order density functional tight binding [DFTB3-D3(BJ)] with frequency corrections derived from a smaller B86bPBE-XDM functional DFT phonon calculation on the crystallographic unit cell. This mixed DFT/DFTB quasi-harmonic approach predicts the lattice constants and unit cell volumes to within 1%-2% at lower pressures. It predicts the thermodynamic phase boundary in almost perfect agreement with the experiment, although this excellent agreement does reflect fortuitous cancellation of errors between the enthalpy and entropy of transition.

Theoretical investigation of polymorph- and coformer-dependent photoluminescence in molecular crystals

A novel density-functional approach provides accurate predictions for the colour zoning of ROY polymorphs and the fluorescence energies of a family of 9-acetylanthracene cocrystals.

Reduced-cost supercell approach for computing accurate phonon density of states in organic crystals

Phonon contributions to organic crystal structures and thermochemical properties can be significant, but computing a well-converged phonon density of states with lattice dynamics and periodic density functional theory (DFT) is often computationally expensive due to the need for large supercells. Using semi-empirical methods like density functional tight binding (DFTB) instead of DFT can reduce the computational costs dramatically, albeit with noticeable reductions in accuracy. This work proposes approximating the phonon density of states via a relatively inexpensive DFTB supercell treatment of the phonon dispersion that is then corrected by shifting the individual phonon modes according to the difference between the DFT and DFTB phonon frequencies at the Γ-point. The acoustic modes are then computed at the DFT level from the elastic constants. In several small-molecule crystal test cases, this combined approach reproduces DFT thermochemistry with kJ/mol accuracy and 1-2 orders of magnitude less computational effort. Finally, this approach is applied to computing the free energy differences between the five crystal polymorphs of oxalyl dihydrazide.

Application of XDM to ionic solids: The importance of dispersion for bulk moduli and crystal geometries

Dispersion corrections are essential in the description of intermolecular interactions; however, dispersion-corrected functionals must also be transferrable to hard solids. The exchange-hole dipole moment (XDM) model has demonstrated excellent performance for non-covalent interactions. In this article, we examine its ability to describe the relative stability, geometry, and compressibility of simple ionic solids. For the specific cases of the cesium halides, XDM-corrected functionals correctly predict the energy ranking of the B1 and B2 forms, and a dispersion contribution is required to obtain this result. Furthermore, for the lattice constants of the 20 alkali halides, the performance of XDM-corrected functionals is excellent, provided that the base functional's exchange enhancement factor properly captures non-bonded repulsion. The mean absolute errors in lattice constants obtained with B86bPBE-XDM and B86bPBE-25X-XDM are 0.060 Å and 0.039 Å, respectively, suggesting that delocalization error also plays a minor role in these systems. Finally, we considered the calculation of bulk moduli for alkali halides and alkaline-earth oxides. Previous claims in the literature that simple generalized gradient approximations, such as PBE, can reliably predict experimental bulk moduli have benefited from large error cancellations between neglecting both dispersion and vibrational effects. If vibrational effects are taken into account, dispersion-corrected functionals are quite accurate (4 GPa-5 GPa average error), again, if non-bonded repulsion is correctly represented. Careful comparisons of the calculated bulk moduli with experimental data are needed to avoid systematic biases and misleading conclusions.

Overcoming the difficulties of predicting conformational polymorph energetics in molecular crystals via correlated wavefunction methods

Molecular crystal structure prediction is increasingly being applied to study the solid form landscapes of larger, more flexible pharmaceutical molecules. Despite many successes in crystal structure prediction, van der Waals-inclusive density functional theory (DFT) methods exhibit serious failures predicting the polymorph stabilities for a number of systems exhibiting conformational polymorphism, where changes in intramolecular conformation lead to different intermolecular crystal packings. Here, the stabilities of the conformational polymorphs of o-acetamidobenzamide, ROY, and oxalyl dihydrazide are examined in detail. DFT functionals that have previously been very successful in crystal structure prediction perform poorly in all three systems, due primarily to the poor intramolecular conformational energies, but also due to the intermolecular description in oxalyl dihydrazide. In all three cases, a fragment-based dispersion-corrected second-order Møller-Plesset perturbation theory (MP2D) treatment of the crystals overcomes these difficulties and predicts conformational polymorph stabilities in good agreement with experiment. These results highlight the need for methods which go beyond current-generation DFT functionals to make crystal polymorph stability predictions truly reliable.

What is “many-body” dispersion and should I worry about it?

“Many-body” dispersion can refer to two distinct phenomena, here termed electronic and atomic many-body effects, both of which cause the dispersion energy to be non-additive.

Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters

Crystal structure prediction is based on the assumption that the most thermodynamically stable structure will crystallize first. The existence of other structures such as polymorphs or from counterenantiomers requires an accurate calculation of the electronic energy. Using atom-centered Gaussian basis functions in periodic Density Functional Theory (DFT) calculations in Turbomole, the performance of two dispersion-corrected functionals, PBE-D3 and B97-D, is assessed for molecular organic crystals of the X23 benchmark set. B97-D shows a MAE (mean absolute error) of 4 kJ/mol, compared to 9 kJ/mol for PBE-D3. A strategy for the convergence of lattice energies towards the basis set limit is outlined. A simultaneous minimization of molecular structures and lattice parameters shows that both methods are able to reproduce experimental unit cell parameters to within 4–5%. Calculated lattice energies, however, deviate slightly more from the experiment, i.e., by 0.4 kJ/mol after unit cell optimization for PBE-D3 and 0.5 kJ/mol for B97-D. The accuracy of the calculated lattice energies compared to the experimental values demonstrates the ability of current DFT methods to assist in the quest for possible polymorphs and enantioselective crystallization processes.

Crystal-energy landscapes of active pharmaceutical ingredients using composite approaches

Composite methods employing dispersion-corrected DFT consistently identify experimentally isolated polymorphs as the lowest-energy crystal structures of common APIs.

Solid state photodimerization of 9-tert-butyl anthracene ester produces an exceptionally metastable polymorph according to first-principles calculations

Molecular crystal engineering seeks to tune the material properties by controlling the crystal packing.

Density-functional description of alkalides: introducing the alkalide state

Alkalides are crystalline salts in which the anion is a negatively charged alkali metal. A systematic investigation of the electronic structure of thirteen alkalides, with known crystal structures, is conducted using density-functional theory. For each alkalide, a high-lying valence state is identified that is localised on the alkali anions and is consistent with the low band gap and strong reducing power characteristic of these materials. This 'alkalide state' is compared to a similar state in the related class of electride materials, where the alkali anions are replaced by crystal voids occupied by localised, interstitial electrons. Finally, a thermodynamic cycle is constructed to examine the energy differences between the alkalides and electrides, revealing that the alkali-metal anion significantly stabilises the crystal.

Pervasive Delocalisation Error Causes Spurious Proton Transfer in Organic Acid-Base Co-Crystals

Dispersion-corrected density-functional theory (DFT-D) methods have become the workhorse of many computational protocols for molecular crystal structure prediction due to their efficiency and convenience. However, certain limitations of DFT, such as delocalisation error, are often overlooked or are too expensive to remedy in solid-state applications. This error can lead to artificial stabilisation of charge transfer and, in this work, it is found to affect the correct identification of the protonation site in multicomponent acid-base crystals. As such, commonly used DFT-D methods cannot be applied with any reliability to the study of acid-base co-crystals or salts, while hybrid functionals remain too restrictive for routine use. This presents an impetus for the development of new functionals with reduced delocalisation error for solid-state applications; the structures studied herein constitute an excellent benchmark for this purpose.

Composite and Low-Cost Approaches for Molecular Crystal Structure Prediction

Molecular crystal structure prediction (CSP) requires evaluating differences in lattice energy between candidate crystal structures accurately and efficiently. In this work, we explore and compare several low-cost alternatives to dispersion-corrected density-functional theory (DFT) in the plane-waves/pseudopotential approximation, the most accurate and general approach used for CSP at present. Three types of low-cost methods are considered: DFT with a small basis set of finite-support numerical orbitals (the SIESTA method), dispersion-corrected Gaussian small or minimal-basis-set Hartree-Fock and DFT with additional empirical corrections (HF-3c and PBEh-3c), and self-consistent-charge dispersion-corrected density-functional tight binding (SCC-DFTB3-D3). In addition, we study the performance of composite methods that comprise a geometry optimization using a low-cost approach followed by a single-point calculation using the accurate but comparatively expensive B86bPBE-XDM functional. All methods were tested for their abilities to produce absolute lattice energies, relative lattice energies, and crystal geometries. We show that assessing various methods by their ability to produce absolute lattice energies can be misleading and that relative lattice energies are a much better indicator of performance in CSP. The EE14 set of relative solubilities of homochiral and heterochiral chiral crystals is proposed for relative lattice-energy benchmarking. Our results show that PBE-D2 plus a DZP basis set of numerical orbitals coupled with a final B86bPBE-XDM single-point calculation gives excellent performance at a fraction of the cost of a full B86bPBE-XDM calculation, although the results are sensitive to the particular details of the computational protocol. The B86bPBE-XDM//PBE-D2/DZP method was subsequently tested in a practical CSP application from our recent work on the crystal structure of the enantiopure and racemate forms of 1-aza[6]helicene, a chiral organic semiconductor. Our results show that this multilevel method is able to correctly reproduce the energy ranking of both crystal forms.

Crystal structure prediction of flexible pharmaceutical-like molecules: density functional tight-binding as an intermediate optimisation method and for free energy estimation

Successful methodologies for theoretical crystal structure prediction (CSP) on flexible pharmaceutical-like organic molecules explore the lattice energy surface to find a set of plausible crystal structures. The initial search stages of CSP studies use relatively simple lattice energy approximations as hundreds of thousands of minima have to be considered. These generated crystal structures often have poor molecular geometries, as well as inaccurate lattice energy rankings, and performing reasonably accurate but computationally affordable optimisations of the crystal structures generated in a search would be highly desirable. Here, we seek to explore whether semi-empirical quantum-mechanical methods can perform this task. We employed the dispersion-corrected tight-binding Hamiltonian (DFTB3-D3) to relax all the inter- and intra-molecular degrees of freedom of several thousands of generated crystal structures of five pharmaceutical-like molecules, saving a large amount of computational effort compared to earlier studies. The computational cost scales better with molecular size and flexibility than other CSP methods, suggesting that it could be extended to even larger and more flexible molecules. On average, this optimisation improved the average reproduction of the eight experimental crystal structures (RMSD15) and experimental conformers (RMSD1) by 4% and 23%, respectively. The intermolecular interactions were then further optimised using distributed multipoles, derived from the molecular wave-functions, to accurately describe the electrostatic components of the intermolecular energies. In all cases, the experimental crystal structures are close to the top of the lattice energy ranking. Phonon calculations on some of the lowest energy structures were also performed with DFTB3-D3 methods to calculate the vibrational component of the Helmholtz free energy, providing further insights into the solid-state behaviour of the target molecules. We conclude that DFTB3-D3 is a cost-effective method for optimising flexible molecules, bridging the gap between the approximate methods used in CSP searches for generating crystal structures and more accurate methods required in the final energy ranking.

Identifying pragmatic quasi-harmonic electronic structure approaches for modeling molecular crystal thermal expansion

Quasi-harmonic approaches provide an economical route to modeling the temperature dependence of molecular crystal structures and properties. Several studies have demonstrated good performance of these models, at least for rigid molecules, when using fragment-based approaches with correlated wavefunction techniques. Many others have found success employing dispersion-corrected density functional theory (DFT). Here, a hierarchy of models in which the energies, geometries, and phonons are computed either with correlated methods or DFT are examined to identify which combinations produce useful predictions for properties such as the molar volume, enthalpy, and entropy as a function of temperature. The results demonstrate that refining DFT geometries and phonons with single-point energies based on dispersion-corrected second-order Møller-Plesset perturbation theory can provide clear improvements in the molar volumes and enthalpies compared to those obtained from DFT alone. Predicted entropies, which are governed by vibrational contributions, benefit less clearly from the hybrid schemes. Using these hybrid techniques, the room-temperature thermochemistry of acetaminophen (paracetamol) is predicted to address the discrepancy between two experimental sublimation enthalpy measurements.