first-principles modeling of molecular crystals ... · of molecular crystals: structures and...

Advanced Review

First-principles modelingof molecular crystals: structuresand stabilities, temperatureand pressureJohannes Hoja,1,2 Anthony M. Reilly3 and Alexandre Tkatchenko1,2*

The understanding of the structure, stability, and response properties of molecu-lar crystals at finite temperature and pressure is crucial for the field of crystalengineering and their application. For a long time, the field of crystal-structureprediction and modeling of molecular crystals has been dominated by classicalmechanistic force-field methods. However, due to increasing computationalpower and the development of more sophisticated quantum-mechanical approxi-mations, first-principles approaches based on density functional theory can nowbe applied to practically relevant molecular crystals. The broad transferability offirst-principles methods is especially imperative for polymorphic molecular crys-tals. This review highlights the current status of modeling molecular crystalsfrom first principles. We give an overview of current state-of-the-art approachesand discuss in detail the main challenges and necessary approximations. So far,the main focus in this field has been on calculating stabilities and structureswithout considering thermal contributions. We discuss techniques that allow oneto include thermal effects at a first-principles level in the harmonic or quasi-harmonic approximation, and that are already applicable to realistic systems, orwill be in the near future. Furthermore, this review also discusses how to calcu-late vibrational and elastic properties. Finally, we present a perspective on futureuses of first-principles calculations for modeling molecular crystals and summa-rize the many remaining challenges in this field. © 2016 John Wiley & Sons, Ltd

How to cite this article:WIREs Comput Mol Sci 2016. doi: 10.1002/wcms.1294

INTRODUCTION

Molecular crystals are versatile materials thatfind applications as pharmaceuticals, explo-

sives, organic semiconductors, in solid-state reac-tions, and plastic materials.1–8 Molecular crystals aresolids composed of well-defined molecular moieties

that are held together by noncovalent forces (inter-molecular interactions). These interactions betweenmolecules are typically weaker than intramolecularcovalent bonds or bonds in hard solid materials.However, an understanding of intermolecular inter-actions is crucial for the fast-growing field of crystalengineering, where the main goal is to design andultimately synthesize new crystalline materials with apredefined arrangement of molecules that exhibit cer-tain properties and functionalities.9,10 This involvesspecific self-organization and self-recognition ofmolecules, which is governed by intermolecular inter-actions. In this context, recognition events are oftendiscussed in terms of specific intermolecular interac-tions such as hydrogen bonds, π-π stacking, C–H! ! !π interactions, halogen bonds, dipole–dipole

*Correspondence to: [email protected] der Max-Planck-Gesellschaft, Berlin,Germany2Physics and Materials Science Research Unit, University of Lux-embourg, Luxembourg City, Luxembourg3The Cambridge Crystallographic Data Centre, Cambridge, UK

Conflict of interest: The authors have declared no conflicts of inter-est for this article.

© 2016 John Wiley & Sons, Ltd

interactions, and van der Waals (vdW) interactions.10

In order to model all these interactions correctly andachieve predictive power, classical force fields areoften insufficient and the use of first-principles meth-ods is required.

For small molecular dimers, the intermolecularinteraction energy can be calculated rather accuratelywith quantum-chemical coupled cluster techniques11

or the quantum Monte Carlo (QMC) method.12 Suchexplicitly correlated approaches have been used asbenchmarks for several datasets of noncovalentinteractions.13–15 However, these methods are onlyapplicable to relatively small systems due to theirrather high computational cost. Therefore, in recentyears density functional theory (DFT) has emerged asthe first-principles method of choice for larger sys-tems, especially with the continuing development ofimproved density functional approximations (DFA)and the incorporation of the long-range vdW energyvia several models for dispersion interactions.16–22

However, the quality of interaction energies isnot always optimal with vdW-inclusive DFAapproaches. For example, the accuracy of DFA+vdWfor strong hydrogen bonds and charge transfer canbe sometimes inadequate.23 For periodic molecularcrystals, one requires a DFA that is computationallyaffordable and that provides a satisfactory descrip-tion of all relevant intra- and intermolecular interac-tions. Since reliable quantum-chemical reference datais hard to generate for molecular crystals, the perfor-mance of DFA+vdW methods is currently evaluatedby comparing with experimental structures andextrapolated lattice energies.24,25

A molecular crystal composed of the same moi-eties can have several different crystal-packing motifsor polymorphs. This has far-reaching consequences,as polymorphs can exhibit completely different solu-bilities, kinetic stabilities, densities, vibrational spec-tra, nuclear magnetic resonance (NMR) chemicalshifts, melting points, conductivities, refractive indi-ces, vapor pressures, elastic constants, heat capaci-ties, etc.9 It was believed that polymorphism mostlyoccurred for large and flexible molecules but recentlyit was suggested by Cruz-Cabeza et al. that polymor-phism may be much more common.26 In the set ofcrystals they studied, at least one in two moleculeshad multiple polymorphs. The energy differencesbetween experimentally observed polymorphs areusually less than 4.2 kJ/mol (1 kcal/mol) per mole-cule but can even be lower than 1 kJ/mol.26 As aresult, to ensure an accurate stability ranking of low-energy polymorphs, calculations with sub-kJ/molaccuracy are necessary. This is already challengingenough, but in addition, experimental crystal

structures are always grown and studied at finitetemperatures. Thermal effects are usually neglectedin state-of-the-art first-principles calculations but cancompletely change the relative stability and orderingof low-energy polymorphs.27,28 Furthermore, even ifthe thermodynamically most stable polymorph at afinite temperature is modeled correctly, it might notcrystallize in experiment due to kinetic effects.29 Forexample, if the molecular conformation in the ther-modynamically stable polymorph is very differentfrom the stable conformation of the isolated mole-cule, it may not be accessible under the conditionsemployed in the crystallization experiment. In addi-tion, a predicted polymorph might not be found inexperiment because the necessary crystallizationexperiment has not been performed yet or cannot beperformed.30 Beyond polymorphs of a single-component molecular crystal, we have to deal alsowith multicomponent systems such as cocrystals,salts, hydrates, and solvates, which will add an addi-tional layer of complexity. The description of allthese subtle effects in condensed molecular systemsdemands a quantum-mechanical first-principlesapproach if predictive power is desired.

The modeling of response properties of molecu-lar crystals such as vibrational spectra, optical, andelastic properties has also proven to be quite chal-lenging. All of these properties are highly structuredependent and hence also temperature dependentdue to thermal expansion. Therefore, accurate THzspectra and elastic constants would require theknowledge of the coordinates and cell parameterscorresponding to the desired temperature. Further-more, these properties are usually calculated by usingthe (quasi-)harmonic approximation, which meansthat anharmonic effects due to atomic motion areneglected. However, these effects can be very signifi-cant at temperatures close to room temperature.

From this discussion, it is obvious that accuratemodeling of molecular crystals from first principles isan intricate task and involves incorporating severaleffects usually neglected in state-of-the-art calcula-tions. Therefore, we will not only discuss the mostimportant concepts for modeling molecular crystalsbut also consider the limitations of the approxima-tions employed. First, we will give an overview of theultimate goal of modeling of molecular crystals, fol-lowed by the hard truth of reality. We then discusshow we can predict crystal structures without anyexperimental information and which first-principlesmethods can be reasonably applied to realistic sys-tems given current computational power, with a spe-cial focus on vdW-inclusive DFT. In the second partof this review, we discuss how we can include

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thermal and pressure effects in the calculation ofstructures, energetics, and response properties.

WHERE WE WOULD LIKE TO BE—ADAY DREAMIdeally, we would like to predict the three-dimensional (3D) crystal structure and correspondingsolid-state response properties of a given moleculefor desired experimental conditions, which couldthen be used to guide or even predict experiments.For example, in the context of crystal engineering wewould then be able to propose molecular crystalsthat show certain desirable properties under specificthermodynamic conditions. As a result, we wouldlike to know the complete crystal energy landscapefor any temperature and pressure, i.e., the completecrystalline phase diagram calculated from first princi-ples. This would provide information about the ther-modynamic stability of different polymorphs for agiven molecular crystal.

In principle, given sufficient computationalresources, the exact energy of molecular crystalscould be calculated in the nonrelativistic limit byusing the full configuration interaction (full CI)method,31,32 which considers Slater determinants ofthe ground state and all possible excitations at 0 K.Obviously, it is imperative to employ an electronic-structure method that is size extensive and capable ofyielding analytical expressions for the atomic forces(such as full CI, coupled cluster, or infinite-order per-turbation theories). Then, temperature and pressureeffects are accessible via molecular dynamics(MD) simulations.33 As most experiments are per-formed at constant temperature and pressure, theisothermal–isobaric (NPT) statistical ensemble shouldprovide a picture close to reality. However, thisensemble is computationally demanding, since thevolume of the cell is not constant during the simula-tion. Ideally, we would perform the MD simulationwith a supercell large enough to capture all long-range effects and potentially to model small defectsin the crystal. In order to correctly capture the quan-tum nature of the nuclei, it is also necessary to usepath-integral MD, since standard MD describes thedynamics of the nuclei only on a classical level. Inaddition, we would like to explicitly include crystalli-zation conditions in crystal-structure prediction cal-culations and explore kinetic effects. Kinetics couldbe studied by simulating crystal growth by using akinetic Monte Carlo algorithm.34–36 For a detaileddiscussion on kinetic effects, the reader is referred toRefs 30,37–39.

These simulations would provide us with thestructure of the most stable polymorph of a certainmolecule under specific thermodynamic conditions.Vibrational spectra could be obtained from the MDsimulations via a Fourier transform of the velocityautocorrelation function. The same is true for elasticproperties. Optical and dielectric properties, as wellas NMR chemical shifts and other properties, can becalculated based on a time-averaged static structure.

WHERE WE ARE—BACK TO REALITYUnfortunately, the previously described MD simula-tions are currently impossible to perform with first-principles methods for realistic systems, especially forcrystals of flexible molecules. The necessary size ofthe simulation cell and the required timescale formeaningful MD simulations is currently far beyondavailable computational resources. We expect thatthis kind of simulations will not be possible for manyyears for the majority of realistic systems.

Therefore, geometry optimizations are usuallydone without explicitly including thermal or zero-point effects. The thermal expansion of a molecularcrystal can be approximated by using the so-calledquasi-harmonic approximation (QHA). With thisapproach, which will be discussed in detail below,one can capture about 80% of the actual thermalexpansion. To our knowledge, kinetic effects havenot been studied so far in crystal-structure predictioncalculations at a fully first-principles level. However,this is only due to the lack of the necessary computa-tional power to perform these calculations. In princi-ple, the kinetic Monte Carlo method would beappropriate for modeling these effects.34–36 In addi-tion, most lattice-energy calculations involve onlyelectronic energies and nuclear repulsion, withfree-energy contributions commonly neglected.Vibrational free energies can be approximated in theharmonic limit, and with this approach we canexpect an accuracy for lattice energies up to 4.2 kJ/mol.25 Part of the anharmonic contributions can beincluded by using the QHA, where thermal expan-sion is considered.40,41 Recently, a so-called vibra-tional self-consistent field (VSCF) method wasdeveloped, in which anharmonicity is explored bycalculating several points along normal-mode displa-cements.42 This method has already been applied tosolid molecular hydrogen43 and ice.44

Furthermore, the calculation of first principlesvibrational spectra is typically limited to the har-monic approximation (HA), but the volume depend-ence of vibrational modes can at least be

WIREs Computational Molecular Science First-principles modeling of molecular crystals


approximated by the QHA. While this provides rea-sonable vibrational frequencies at very low tempera-tures, the low-frequency modes differ significantlyfrom experimental measurements at higher tempera-tures due to large anharmonicities.45 However, westress that this approach is still much more accuratethan performing molecular-dynamics simulationswith traditional force fields because the accuracy offorce-field forces is in this case usually not sufficientfor modeling response properties. Mechanical andoptical properties are also usually modeled at theminimum of the electronic potential-energy surface(PES) or with the quasi-static approach, i.e., fixingthe unit cell to the experimental volume for a certaintemperature.46 Note that the calculation of opticalproperties is essentially only possible with first-principles methods because this requires considera-tion of excited electronic states.

Despite all these remaining challenges, first-principles methods have already yielded significantnew insights into molecular crystals. The most recentcrystal-structure prediction blind test47 has shownthat first-principles methods can now be successfullyapplied for pharmaceutically relevant flexible molec-ular crystals, salts, and hydrates, and that they pro-vide better structures and stabilities than force-fieldapproaches. Furthermore, it was shown by Reillyand Tkatchenko that a high-level vdW-inclusive DFTmethod is able to describe the relative enthalpy andentropy of the two known polymorphs of aspirincorrectly, offering a resolution for a long-standingcontroversy regarding the metastability of form-IIaspirin.27 In addition, Schatschneider et al. haveshown that vdW-inclusive DFT is able to describetemperature-induced phase transitions betweenmolecular crystal polymorphs of tetracyanoethylenein excellent agreement with experimental data.48

INTERMOLECULAR INTERACTIONSIN MOLECULAR CRYSTALSAs molecular crystals consist of molecular moieties,the cohesion within the crystal arises from noncova-lent or intermolecular interactions, with the majorityof their properties also governed by intermolecularinteractions. Therefore, it is necessary to understandand properly model these many different types ofinteractions in an accurate and balanced manner. Ingeneral, we can loosely distinguish between four dif-ferent types of noncovalent interactions betweenclosed-shell molecules: electrostatics, exchange repul-sion, induction, and dispersion.49 We can understandthese interactions in a textbook picture based on

perturbation theory50 as follows: electrostaticsdescribes the static Coulomb interaction betweenfixed molecular charge distributions, i.e., the interac-tion between permanent multipoles with the leadingterms being dipole–dipole, dipole–quadrupole,quadrupole–quadrupole, etc. The exchange repulsionis comprised of two effects. The first one is an attrac-tive interaction, which arises due to the fact that elec-trons can move over both molecules at shortintermolecular distances, and the second effect is aresult of the Pauli exclusion principle, which leads tothe existence of a repulsive force between electrons ofthe same spin. Overall, the exchange repulsion isdominated by the second effect and is thereforerepulsive. Induction describes the interaction betweenpermanent and induced multipoles. Finally, the dis-persion term describes the interaction betweeninstantaneously created multipoles and induced mul-tipoles, i.e., zero-point fluctuations of electrons inone molecule create an instantaneous multipolemoment, which can then induce a multipole momentin another molecule. The dispersion interaction isubiquitous because it does not require the existenceof permanent multipole moments.

As we wish to discuss in this review the model-ing of molecular crystals with density functionalmethods, we also provide a slightly different inter-pretation of intermolecular interactions based onelectron densities (see Figure 1). In all first-principlescalculations, the electron density of a system isobtained via an iterative self-consistent field (SCF)approach.31 At the beginning of the SCF procedure,a static density is initialized, typically using atomic-charge information. This initial stage can be viewedas electrostatics. When electron clouds overlap, theexchange of electrons becomes possible. The Pauliprinciple requires that the total wave function isantisymmetric with respect to exchange of electronsof the same spin. The adaptation of the wave func-tion is connected with an energy loss and therefore

Electrostatics

Dispersion Exchange repulsion

Polarization

BA

ΨAB(r1,r2) = -ΨAB(r2,r1)

Static density Relaxed density

Fluctuating density Pauli exclusion principle

FIGURE 1 | Schematic representation of intermolecularinteractions.



repulsive interaction. As the SCF cycle progresses,the electron densities of the atoms/molecules relax inresponse to the surrounding charge distributions.This effect can be understood as polarization orinduction. The three effects discussed so far woulddescribe the picture of intermolecular interactionswithin Hartree–Fock theory and semi-local DFAs.For any method accounting for long-range electroncorrelation energy, the instantaneous electronic fluc-tuations lead to an additional modification of elec-tron densities compared to electronic mean-fieldmodels. The intermolecular interaction arising dueto this effect can be loosely interpreted as dispersion.It has been shown by Ferri et al.51 that long-rangevdW (dispersion) interactions can significantly mod-ify the electron density and related electronic proper-ties. In general, all of the discussed effects(repulsion, electrostatics, polarization, and disper-sion) are intimately linked together and there is nounambiguous way to separate them in practice, espe-cially for extended crystals.

The complexity of describing all of the dis-cussed interactions on equal footing makes the use offirst-principles methods imperative for an accuratemodeling of molecular crystals. However, most cal-culations are still carried out with empirical forcefields that often only describe electrostatics andemploy rather crude models for Pauli repulsion anddispersion interactions. Typically, polarization istreated only poorly or neglected completely by theseclassical models, although polarizable force fieldshave been developed. While fully self-consistent first-principles methods can also capture complex nonad-ditive effects between exchange-repulsion, electrostat-ics, polarization, and dispersion, force fields treatthese terms only in an additive fashion.

As discussed by Dobson and Gould,52 the termvan der Waals (vdW) interactions often has varyingmeaning in different communities. In chemistry, thesum of all intermolecular interactions is often termedvdW interactions, whereas in solid-state physics, itusually describes effects due to correlations betweenelectronic fluctuations, which correspond to the dis-persion interactions defined above. We will adopt thelatter definition throughout this review. Withinmolecular crystals, vdW interactions can dominatethe cohesive energy given their long-range nature andubiquity. However, hydrogen bonds53 and also halo-gen bonds play a very important role in crystal engi-neering because of their directional-dependentattraction at short range. This can be used to controlstructural motifs in the formation of molecular crys-tals. The nature of intermolecular interactions withinhydrogen- and halogen-bonded systems was recently

analyzed in Refs 54–56 using symmetry-adapted per-turbation theory.57,58

We now proceed to discuss how one can pre-dict crystal structures and properties of molecularcrystals by combining a hierarchy of modelingapproaches for the increasingly accurate descriptionof intermolecular interactions.

PREDICTING CRYSTAL STRUCTURESAND PROPERTIESFor the modeling of experimentally known molecularcrystals, the determined crystal structure is often usedas starting point for theoretical calculations. Usually,this experimental structure is reoptimized with thefirst-principles method of choice and all propertiesare calculated from that structure. However, as wewant to predict certain properties of molecular crys-tals prior to experiment, we need to determine themost stable crystal structure of a certain moleculefrom scratch. Since 1999, the Cambridge Crystallo-graphic Data Centre (CCDC) has organized severalcrystal-structure prediction (CSP) blind tests.47,59–63

In these blind tests, the participants were required topredict the experimentally determined crystal struc-tures of several molecules based only on the knowl-edge of the two-dimensional (2D) chemical diagram.Most groups used force-field approaches but thenumber of first-principles contributions (especiallyDFA+vdW) is increasing. Since 2009, one methodhas correctly predicted most of the target crystalstructures.47,62,63 However, the accurate stabilityranking of polymorphs still remains a challenge, withonly the most sophisticated many-body dispersion(MBD) method leading to a significantly improvedrank order prediction.47 In addition, thermal effectsand kinetic effects are still mostly neglected in currentcrystal-structure prediction calculations. Kineticeffects due to crystallization conditions in experi-ments can restrict the diversity of accessible poly-morphs, while thermal effects can radically changethe relative stabilities of polymorphs.

Whenever the crystal structure is not knownexperimentally, we have to start with a 2D chemicaldiagram of the target molecule. The procedure forfinding the lowest-energy polymorph can be dividedinto three basic steps, which we will briefly mentionin the following sections. For a more detailed descrip-tion of this process, the reader is referred to Ref 37.All properties of molecular crystals are highlydependent on the crystal structure. Therefore, prop-erties can be calculated after we have obtained themost stable crystal structure. The calculation of



vibrational and elastic properties will be discussed inthe Addressing the Temperature and Pressure Gapsection.

Step 1: Generating MolecularConformationsAs molecular crystals consist of noncovalently boundmolecular entities, a reasonable starting point is tocalculate the most stable conformer of a molecule inthe gas phase. In many instances, this conformer willalready be a good estimate of the conformationadopted within the crystal, especially for rigid mole-cules. However, when a molecule can form intermo-lecular hydrogen bonds, the additional stabilizationof these can lead to significant changes in conforma-tion to accommodate their formation. This is espe-cially an issue for molecules that form intramolecularhydrogen bonds. Furthermore, several rotationsabout single bonds can occur in flexible moleculesthat do not dramatically change the conformationalenergy, e.g., rotation of alkyl linkers or phenylgroups.

It is also possible that the most stable geometryin the crystal is a different tautomer compared to themost stable geometry in vacuum. For example, barbi-turic acid was believed to occur only in the ketoform. Recently, a polymorph featuring the enol formwas found and is more stable at room temperaturethan all previously known polymorphs.64

Step 2: Crystal-Packing ArrangementIn the second step, a large number of different puta-tive crystal-packing arrangements must be calculated.This task is highly nontrivial, as we do not knowhow many molecules are in the unit cell or whichspace group the molecule will crystallize in. Even ifwe have only one kind of molecule in the crystal andif the conformation of all molecules is the same, thesize of the crystallographic space to be covered is tre-mendous. One important parameter in this search isthe number of molecules in the asymmetric unit ofthe unit cell (Z0), i.e., the number of molecules thatare necessary in order to generate all atomic posi-tions within the unit cell through symmetry opera-tions. In most crystal-structure searches, only Z0 = 1structures are considered but for several systems, itcan easily be greater than 1. However, the samplingfor these less symmetric structures vastly increasesthe number of possible crystal structures that need tobe considered and therefore significantly increases thecomputation time. Hence, for every crystal-structureprediction, one has to find a balance between

accuracy and cost of the search. In a first approxima-tion, gas-phase molecular structures are used to cre-ate initial crystal-packing geometries. For flexiblemolecules, several torsion angles must be consideredfor rotatable bonds, which vastly increase the com-plexity of the crystal-structure prediction.65,66 Inaddition, one may have to consider also the possibil-ity of hydrates, solvates, and different stoichiometry.In some cases, it can also be difficult to distinguishbetween salts and cocrystals, which often only differin the position of a proton. Such systems might there-fore require considering both possibilities.

Step 3: Stability RankingOnce a sufficiently large set of crystal-packingarrangements has been created, one has to rank all ofthem according to a certain measure of stability. Usu-ally, this is done by calculating the lattice energyElatt, which is given by

Elatt =Ecryst

Z−Egas ð1Þ

where Ecryst is the energy of the crystal unit cell, Egas

is the energy of one isolated molecule in the lowest-energy gas-phase conformation, and Z is the numberof molecules in the unit cell. It describes the energy offormation of a molecular crystal from infinitely sepa-rated molecules in their gas-phase minimum-energyconformation, or in other words, the molecularenergy gained by forming a crystal. If a crystal is sta-ble, the lattice energy has a negative sign. Typically,lattice-energy calculations do not include any thermaleffects, i.e., only the electronic energy and thenuclear-repulsion energy are considered.

Ideally, all calculations should be done by first-principles methods to ensure accurate modeling ofconformations and crystal structures through thewhole CSP process. However, the proper sampling ofthe crystal-structure search space requires a largenumber of calculations. Therefore, it is common touse the following approach: at first, the molecularconformation (step 1) is optimized with a first-principles method, such as DFT, followed by the gen-eration of parameters for a first-principles-derivedforce field. Next, a large number of possible crystal-packing arrangements are created based on theDFT-optimized molecular structures (step 2) and theyare ranked by their stability according to the force-field lattice energy (step 3). After this initial ranking,an appropriate number of low-energy structures canthen be reoptimized and reranked with a hierarchy ofincreasingly accurate methods, starting with a first-



principles-derived force field up to high-level first-principles methods. If done properly, this eventuallyleads to the most-stable crystal structure with thefinal method of choice.

Step 4: Calculation of PropertiesThe stable structure is merely a proxy for calculatinga range of properties desirable in actual solid-stateapplications. With the structure determined inStep 3, we are now able to calculate a variety ofresponse properties. Vibrational properties can becalculated within the HA by determining the forcesdue to finite atomic displacements. This providesinformation about vibrational modes of the molecu-lar crystal, including internal modes and phononmodes. With that, one can obtain THz fingerprintspectra [low-frequency infrared (IR) spectra], freeenergies at finite temperatures, and heat capacities atconstant volume. The mechanical stability of amolecular crystal can be assessed via elastic constantsand moduli. These quantities are calculated by deter-mining the stress tensor on several strained unit cells.All these properties will be discussed later on in thisreview. In addition, we can also calculate charge-transport properties of molecular crystals. Recently,Motta and Sanvito67 have proposed a model for cal-culating the charge mobility of a molecular crystal atspecific temperatures using data from lattice-dynamics calculations. This model constructs aneffective tight-binding Hamiltonian that describes theelectron–phonon coupling. The NMR chemical shiftsof molecular crystals can be calculated from thechemical-shielding tensor, the components of whichare the second derivatives of the energy with respectto an external magnetic field and the nuclear mag-netic moment. Most calculations use the so-calledgauge-including projector augmented wave (GIPAW)method.68,69 Recently, a fragment-based method hasalso been developed by Hartman et al.70 Chemicalshifts can be used to distinguish between differentpolymorphs or to assign peaks in experimental NMRspectra.70,71 However, the differences in chemicalshifts between polymorphs are small and sensitive tothe crystal-structure geometry, which makes accuratemodeling especially challenging.

It is important to remember, however, that thefinal structures from a crystal-structure predictioncalculation typically do not include any contributionfrom thermal expansion or motion. To incorporatethese effects the QHA can be used to determinefinite-temperature lattice constants, which will be dis-cussed in the Addressing the Temperature and Pres-sure Gap section. It can also be used to model an

approximate temperature dependence in vibrationalmodes, elastic constants, and NMR shifts. In addi-tion, it enables the calculation of thermal expansioncoefficients or heat capacities at constant pressure.

CALCULATING CRYSTALSTRUCTURES AND LATTICEENERGIES

Method OverviewThere is a range of methods that can be used to cal-culate energies, geometries, and response propertiesof molecular crystals. Before considering first-principles methods, it is worth noting that force-fieldmethods are indispensable for crystal-structure pre-diction calculations. Owing to their low computa-tional cost, they can be used to rapidly evaluate allof the generated crystal-packing arrangements andprovide a preselection for first-principles methods.However, even the most recent force fields do notprovide the reliability and accuracy of high-qualityfirst-principles calculations. For molecular crystals,first-principles-derived force fields are more success-ful than conventional force fields because they arefitted especially for the molecule being studied andtherefore usually describe lattice energies far moreaccurately than conventional force fields.72,73 Semi-empirical density functional tight-binding methodscan also be used for preselection of structures,which are then subsequently investigated with first-principles methods.74,75 However, the parametriza-tion of these methods can lead to problems forflexible molecular crystals with a large number offreely rotatable functional groups. The relative sta-bility ranking of conformations with these methodscan also be poor, giving spurious results in CSPcalculations.47

In recent years, various first-principles methodshave been developed for modeling extended molecu-lar crystals. We will briefly mention important classesof methods. For a more in-depth discussion, thereader is referred to Ref 71. We begin with periodicDFT. Most traditional DFAs lack the ability todescribe long-range correlation energy, which canlead to completely wrong lattice energies and geome-tries for molecular crystals. In recent years, this defi-ciency has been addressed by developing models forvdW interactions that are added to the total DFTenergy16–20 (which are then termed vdW-inclusivemethods) and also by the development of nonlocalfunctionals.76 These methods will be discussed inmore detail in the next section. As a result of theserecent developments, DFT is becoming the method of



choice for molecular crystals, as it provides a goodbalance between accuracy and computational cost.

Periodic post Hartree–Fock calculations havealso recently become possible for small molecularcrystals. Periodic local second-order Møller–Plessetperturbation theory (MP2) has successfully beenapplied to several small molecular crystals includingcarbon dioxide,77 urea,78 and ammonia.77 Further-more, Grüneis et al. developed a periodic MP2method based on a plane-wave basis set.79 Thisdevelopment is encouraging, however, these calcula-tions are much more time consuming than DFT cal-culations and require much larger basis sets forconverged results. Another high-level first-principlesmethod is diffusion QMC, which is mainly used forbenchmarking, as the required computation timescales rapidly with system size. The largest reportedcalculation so far is that of the energy differencebetween two polymorphs of diiodobenzene.80,81

However, converging small energy differencesbetween polymorphs with QMC is challenging dueto finite-size effects and inherent statistical samplingerrors.

Finally, the last important class of methods arethe so-called fragment methods. In these methods,the total energy of a molecular crystal is decomposedinto molecular energies Ei (monomers), the interac-tion between two molecules ΔEij, three-moleculeinteractions ΔEijk, etc.

Etot =X

i

Ei +X

ij

ΔEij +X

ijk

ΔEijk +X

ijkl

ΔEijkl + ! ! ! ð2Þ

This has the advantage that the overall computa-tional cost is lowered substantially, which enablescalculations with high-level wave function-basedmethods. For example, Yang et al. were able to cal-culate the lattice energy of the benzene crystal in thisway with a coupled cluster method with perturbativetriple excitations.82 Interactions up to four-moleculeterms are necessary to converge the lattice energy.83

Heit et al. applied the fragment-based hybrid many-body interaction (HMBI) model to crystalline carbondioxide.40,84 This method calculates the intramolecu-lar energy and the short-range two-body energy witheither MP2 or coupled cluster singles and doubleswith perturbative triples [CCSD(T)], and the long-range two-body and remaining many-body interac-tions by using the AMOEBA polarizable force field.However, MP2 suffers from the problem that it over-estimates lattice energies, especially for dispersion-bound systems with π-stacking.85 Another importantfragment method is symmetry-adapted perturbationtheory.50,57 With this method, one can decompose

interaction energies between molecules into electro-statics, exchange, induction, and dispersion contribu-tions. For a detailed description of its application tomolecular crystals, the reader is referred to Ref 86.

vdW-Inclusive DFTIn vdW-inclusive DFT, the missing dispersion energyEdisp is calculated a posteriori and then added to thetotal DFT energy of the DFA used. Most pairwiseschemes have the following general form

Edisp = −X

i > j

fdmpCij

6

R6ij; ð3Þ

where Rij is the interatomic distance, Cij6 describes the

dipole–dipole dispersion coefficient between atomsi and j, and fdmp is a damping function. For a detaileddiscussion of the importance of the used dampingfunction, the reader is referred to Ref 87. The disper-sion coefficients are related to the atomic polarizabil-ities. This approach is used, for example, for theDFT-D2 method16 and the Tkatchenko–Scheffler(TS) method.19 In the D2 scheme fixed empirical dis-persion coefficients are used, while in the TS scheme,they are directly derived ‘on the fly’ from the calcu-lated electron density. Furthermore, the TS and therelated MBD method are explicit density functionalsand hence permit investigation of the effect of vdWinteractions on all properties of interest beyond struc-tures and stabilities of molecular crystals.51,88

The dispersion interaction can be expressed in amultipole expansion. The D2 and TS schemes includeonly dipole–dipole interactions. Therefore, one possi-bility is to augment them by dipole–quadrupole inter-actions, adding a Cij

8=R8ij term. This is done in the

DFT-D3 method,17 where the fixed dispersion coeffi-cients were replaced by coefficients dependent onatomic coordination. Another important dispersionmodel is the exchange-dipole moment (XDM)model,18,89,90 which is also density dependent andincludes pairwise interactions up to quadrupole–quadrupole interactions.

However, dispersion interactions are not pair-wise additive and therefore many-body interactionshave to be considered as well.91 In the case of DFT-D3, the many-body interactions are approximated bythe lowest-order three-body term (Axilrod-Teller-Muto), which describes the interaction between threeinstantaneous dipoles. In 2012, Tkatchenkoet al. developed the so-called many-body dispersion(MBD) method, which considers many-body dipolarinteractions up to infinite order.20 The MBD method



also includes electrodynamic response (dielectricscreening) effects by a self-consistent screening ofatomic polarizabilities. These effects are especiallyimportant for extended systems like molecular crys-tals because the presence of the crystal field changespolarizabilities considerably compared to isolatedmolecules. The MBD dispersion energy (long-rangecorrelation energy) is given by

EMBD =12π

ð∞

0dωTr ln 1−ATð Þ½ % ð4Þ

where A is a diagonal 3n × 3n matrix containing theisotropic atomic polarizabilities and T is the dipole–dipole coupling tensor. This expression is equivalentto diagonalizing a model Hamiltonian of coupled iso-tropic quantum harmonic oscillators. For a more in-depth discussion of the MBD method, the reader isreferred to Refs 91–93.

In contrast to small gas-phase molecules, dis-persion interactions can extend over large distancesin molecular crystals. This is illustrated in Figure 2

for a cubic deutero ammonia (ND3) and a hexam-ethylbenzene (HMB) crystal. The figure shows theconvergence of the vdW lattice energy (per molecule)with respect to a cut off radius rcut, i.e., the com-pletely converged energy is set to zero. In the case ofthe pairwise dispersion approach (TS), interactionsbetween atoms are only considered if the interatomicdistance is less than or equal to rcut. In the MBDcase, interactions are only included in the dipole–dipole tensor if the interatomic distance between twoatoms is less than or equal to rcut. In addition, thecoupling between the harmonic oscillators is consid-ered within a cubic supercell of lattice constant a =rcut. It can be seen that the coupling of the harmonicoscillators (MBD) leads to a significantly slower con-vergence for both systems than the pairwise disper-sion energy (TS). In order to converge the latticeenergy of HMB within 0.5 kJ/mol, one needs to con-sider distances of roughly 22 Å for TS and alreadyaround 35 Å for MBD. This shows immediately thatwe are missing important collective effects by onlyconsidering pairwise interactions. In contrast, MBDconsiders many-body interactions up to infiniteorder. We mention in passing that molecular-crystalpolymorphs often have similar short-range order andinteractions, and are therefore sometimes only distin-guishable at relatively large distances between mole-cules. It is precisely at these distances that many-body screening effects are largest; hence, pairwise dis-persion corrections are frequently unable to capturethe subtle energetic difference between polymorphs.Both ND3 and HMB are rather symmetric (cubic)molecular crystals. For more complex geometrieswith lower symmetry, we expect the range of many-body vdW interactions to be even longer, as forexample exemplified by the analysis of vdW interac-tions in nanostructured materials.94,95

Lattice-Energy BenchmarksFor molecular dimers, benchmarks of interaction ener-gies can be easily made by comparing with CCSD(T)calculations at the basis-set limit using several data-bases. However, such theoretical benchmark sets areunfortunately not available for molecular crystals. Thebest solution is to compare calculations to experimen-tal sublimation enthalpies. One very well-studied sys-tem is the benzene crystal. The experimental latticeenergy is −55.3 & 2.2 kJ/mol.82 The highest-level first-principles result available was calculated with a frag-ment coupled cluster approach by including two,three, and four-molecule interactions, yielding a latticeenergy of −55.9 & 0.76 & 0.1 kJ/mol.82 The fragmentHMBI model based on CCSD(T) provides a lattice

ND3

HMB

0

1

2

3

4

5

0 10 20 30 40 50

Rel

ativ

e en

ergy

/ kJ

/mol

rcut / Å

ND3-TS

ND3-MBD

HMB-TS

HMB-MBD

FIGURE 2 | Convergence of the pairwise (TS) and MBD vdWlattice energy with respect to the dipole–dipole cut off radius andMBD supercell cut off radius for ND3 and HMB. HMB,hexamethylbenzene; MBD, many-body dispersion; ND3, cubic deuteroammonia; TS, Tkatchenko–Scheffler; vdW, van der Waals.



energy of −53.0 kJ/mol96,97 and the periodic localMP2 approach leads to a lattice energy of −56.6 kJ/mol.98 This shows that all calculations using post-Har-tree-Fock methods provide excellent agreement withexperiment for the benzene crystal. In contrast, thewidely used density functional B3LYP without any dis-persion correction yields a lattice energy of only−15.9 kJ/mol, which corresponds to an error of almost40 kJ/mol.99 Therefore, it is imperative to includeproper vdW interactions in DFT calculations. ThePerdew-Burke-Ernzerhof (PBE) DFA100 supplementedby the MBD and the XDM methods yield latticeenergies of −55.0 and −49.5 kJ/mol, respectively.24,25

In addition, the lattice energy calculated with PBE+D3with and without three-body terms amounts to −51.0and −54.8 kJ/mol, respectively.101 It can be seen thatmost vdW-inclusive DFT methods provide lattice ener-gies for the benzene crystal that are in excellent agree-ment with the experimental value.

In 2012, Otero-de-la-Roza and Johnsonassembled the C21 benchmark set containing21 molecular crystals ranging from dispersion-boundto hydrogen-bonded systems.24 The experimentalsublimation enthalpies were back corrected for vibra-tional contributions, yielding benchmark values for0 K lattice energies. This benchmark set wasextended and refined by Reilly and Tkatchenko,yielding the X23 benchmark set.25 The relative errorin the lattice energies compared to the benchmarkvalues is shown in Figure 3 for the PBE and PBE0DFAs supplemented by the TS and MBD methods.

It can be seen that MBD produces systemati-cally more accurate lattice energies than TS. The

largest difference between TS and MBD is found forthe vdW-bound systems. In addition, switching fromthe generalized gradient approximation (GGA) func-tional PBE to the PBE0 hybrid functional leads to aconsistently better description of the lattice energy. TheTS method consistently overestimates lattice energies,giving a mean absolute error (MAE) of 10.0 kJ/mol forPBE0+TS. For PBE0+MBD, the MAE amounts to only3.9 kJ/mol, placing it within the 4.2 kJ/mol (1 kcal/mol)window of chemical accuracy. The D3 dispersion cor-rection has also been tested for several functionals withthe X23 benchmark set.101 The best performance wasachieved for TPSS-D3 without three-body terms, alsoyielding an MAE of 3.9 kJ/mol.

Nyman et al.102 have recently studied theaccuracy of several force fields widely used incrystal-structure prediction calculations for the X23benchmark set. The resulting deviations in latticeenergies are around 2–3 times larger than the bestvdW-inclusive first-principles methods. These devia-tions are reasonable in the context of using suchforce fields in the intermediate stages of crystal-structure prediction calculations but not sufficientfor correctly predicting the rank ordering of poly-morphs and it should also be noted that they areparameterized for a limited number of atom typesand environments and are therefore not verytransferable.

All of the above benchmarks are for absolutelattice energies. However, for crystal-structure pre-diction calculations, relative energies are required.The C21 and X23 test sets include two polymorphsof oxalic acid. Experimentally,24,25,103,104 the α poly-morph is slightly more stable than the β form, by0.2 kJ/mol. The DFA methods discussed above yieldenergy differences ranging from about −4 kJ/mol to4 kJ/mol. For PBE0+TS, the β form is more stable byabout 1.5 kJ/mol, while PBE0+MBD predicts thatthe α form is more stable by about 1 kJ/mol, whichis consistent with experiment. In addition, the rela-tive stability of three glycine polymorphs has beenstudied for several vdW-inclusive DFAs by Maromet al.105 Only PBE0+MBD was able to capture cor-rectly the qualitative stability ranking, while PBE+TS, PBE0+TS, and PBE+MBD yielded a differentqualitative picture. The error in the calculated rela-tive energies amounts to about 1 kJ/mol for PBE0+MBD. All these benchmarks suggest that high-levelvdW-inclusive DFAs are necessary for accurate rela-tive stabilities and that the stability ranking of poly-morphs within an energy window of 1–2 kJ/molremains a challenge. It also is important to rememberthough that the experimental reference values forpolymorph and absolute stabilities are associated

vdW bonding

60

PBE+TS PBEO+TS

PBEO+MBDPBE+MBD

50

40

30

20

Rel

ativ

e er

ror

in E

lat /

%

10

0

–10

–20

Hydrogen bonding

Urea

Succinic acid

Oxalic acid (!)

Oxalic acid (")

Foram

ide

Ethylcarbam

ate

Cyanam

ide

Am

monia

Acetic acid

Uracil

Imidazole

Cytosine

Trioxane

Triazine

Pyrazole

Pyrazine

Naphthalene

Hexam

ine

Benzene

Anthracene

Adam

antane

1,4-cyclohexanedione

FIGURE 3 | Relative error in the calculation of lattice energies for22 molecular crystals for PBE+TS, PBE+MBD, PBE0+TS, and PBE0+MBD calculations. (Reprinted with permission from Ref 25. with thepermission of AIP Publishing) MBD, many-body dispersion; TS,Tkatchenko–Scheffler.



with an uncertainty, and that the often-used backcorrection of experimental sublimation enthalpiesalso introduces an additional uncertainty in thesebenchmarks.

Geometry BenchmarksAs with lattice energies, there is no high-level first-principles benchmark set available for geometries ofmolecular crystals. Therefore, we have to comparethe theoretically optimized structures with experi-mental crystal structures, usually measured by X-raydiffraction. However, all experimental structures aremeasured at finite temperatures, whereas theoreticalgeometry optimizations correspond to 0 K and alsodo not include zero-point effects. Most molecularcrystals expand with increasing temperature and eventhe zero-point vibrations alone can lead to a volumeexpansion of about 3%, which will be shown lateron for phase-I ammonia. One approach to combatthis is to benchmark theoretical structures against thelowest-possible temperature structure available inexperiment, minimizing the influence of thermaleffects. The lattice vectors obtained with PBE+TS andPBE+MBD optimizations for a subset of the X23 testset show mean relative errors of −0.55 and −0.75%,respectively.25 Moellmann and Grimme101 studiedthe structures of the C21 set by using PBE+D3 andTPSS+D3, yielding mean relative errors in the cellvolume of −1.1 and −2.3%, respectively. Calculatedstructures usually show a smaller volume comparedto experiment, most likely due to neglecting thermalexpansion.25,106 Schatschneider et al.106 studied the

structures of a large set of crystalline polycyclic aro-matic hydrocarbons with PBE+TS, yielding on aver-age an error of about &2% for the lattice vectors. Acomparison of the calculated densities with experi-mental values is shown in Figure 4. Most of the roomtemperature experimental densities agree within 5%,with the calculated densities, while for lower tem-peratures, the agreement improves to an averagedeviation of 2.3%.106 The main reason for the over-estimation of the densities is again the neglect of anytemperature or zero-point effect in the geometry opti-mization. The most-recent blind test of crystal-structure prediction methods47 has also shown thatseveral first-principles methods are able to predictstructures in very good agreement with experimentalgeometries and, in general, with a higher accuracythan force-field approaches.

ADDRESSING THE TEMPERATUREAND PRESSURE GAPThe lattice energies discussed so far contain onlytotal energies (electronic and nuclear repulsion) cal-culated with a first principle method; from now onabbreviated with Etot. These energies do not dependon temperature and pressure and are therefore onlyvalid at a temperature of 0 K and a pressure of0 bar. Strictly speaking, we are not even describing0 K correctly, as vibrational zero-point energies aremissing. While this might sometimes be a reasonableapproximation for modeling small isolated moleculesin vacuum, temperature and pressure effects can becrucial for the accurate modeling of molecularcrystals.

In order to include temperature effects, we needto calculate the Helmholtz free energy F(T, V),given by

F T,Vð Þ=Etot Vð Þ + Fvib T,Vð Þ ð5Þ

where Fvib(T, V) is the vibrational free energy due tothe nuclear motion on the Born–Oppenheimer energysurface, T is the temperature, and V is the unit-cellvolume of the molecular crystal. There are additionalcontributions to F(T, V), such as the electronic freeenergy107 and magnetic contributions. However,these effects can normally be neglected for insulatingmolecular crystals. Furthermore, we assume that weare dealing with a perfectly periodic molecular crys-tal. The existence of defects or disorder would alsointroduce additional terms in F(T, V). The effect ofpressure can be readily included by calculating theGibbs free energy of the crystal:

1.7

1.6

1.5

1.4

1.3

1.2

1.1

11 1.1 1.2 1.3

Experimental density (g cm–3)

Cal

cula

ted

dens

ity (g

cm

–3)

1.4 1.5

Low temperatureRoom temperature

1.6

FIGURE 4 | Calculated versus experimental densities for91 polycyclic aromatic hydrocarbons. Structures obtained at roomtemperature are shown in red and structures obtained below roomtemperature are shown in blue. The black line marks perfectagreement, while the green dotted lines mark a deviation of &5%.(Reprinted with permission from Ref 106. Copyright 2014 AmericanChemical Society).



G p,T,Vð Þ= F T,Vð Þ+ pV ð6Þ

where p is an external hydrostatic pressure acting onthe unit cell.

In the following subsections, we will discuss thebasics of approximations for including thermaleffects with the aid of a simple model system phase-Ideutero ammonia (ND3). In addition, we also discusshow vdW interactions influence the results. All calcu-lations for the model system were performed withinthe all-electron DFT code FHI-aims,108–111 using thetight species default settings and phonopy, a code forphonon calculations.112

The Harmonic ApproximationThe simplest way to calculate the Helmholtz freeenergy is the so-called harmonic approximation(HA). In principle, the calculation of Fvib requires theknowledge of the total PES, which is a 3n-dimensional object, where n is the number of atomsin the unit cell. In the HA, the PES is expressedaround the equilibrium geometry with coordinatesReq by a second-order Taylor expansion

E Req +ΔR" #

=E Req" #

+X

i

∂E∂Ri

ΔRi

+12

X

i, j

∂2E∂Ri∂Rj

ΔRiΔRj +OðΔR3Þ ð7Þ

where E(Req) is the total energy of the equilibriumgeometry. The first-order term vanishes since theforces are zero at a local minimum of the PES. Thesecond-order term will provide us with the harmonicforce constants Φij

∂2E∂Ri∂Rj

= −∂Fj∂Ri

=Φij ð8Þ

which build up the Hessian matrix, or for periodicsystems the so-called dynamical matrix.113 All effectsarising from higher-order terms in the Taylor expan-sion in Eq. (7) are called anharmonic effects and arecompletely neglected in the HA.

The harmonic force constant Φij describes thechange of the force on atom j when displacing atomi. The dynamical matrix is typically calculatednumerically by finite differences, but is also accessibleby using density functional perturbation theory.114

The eigenvalues of the dynamical matrix are the pho-non frequencies ω (vibrational frequencies in nonper-iodic systems) and the eigenvectors describe the

corresponding phonon modes. The complete dynam-ics of the harmonic system is described by 3n-independent quantum harmonic oscillators; one foreach phonon mode.

As the HA only approximates the PES aroundthe equilibrium structure with a corresponding unit-cell volume Veq, the harmonic Helmholtz free energyFHA does not have an explicit volume dependenceand can be expressed as

FHA Tð Þ =Etot +FHAvib Tð Þ ð9Þ

with

FHAvib Tð Þ=

ðdωg ωð Þℏω

2+ðdωg ωð ÞkBT ln 1−exp −

ℏωkBT

$ %& '

ð10Þ

where g(ω) is the phonon density of states (pDOS),i.e., the number of vibrational states at a certain fre-quency. The first integral in Eq. (10) describes zero-point vibrations, which are present in every quantumsystem even at 0 K, while the second integraldescribes thermally-induced vibrations and accountsfor vibrational entropy. Note that this term is partic-ularly important for low-frequency vibrations, as theright part of the second integral is largest in magni-tude for low frequencies (see Figure 5).

If a phonon calculation is performed by usingfinite differences, there are several technical aspectsone must consider. First, the size of the atomic dis-placements used must be tested carefully, as it is sys-tem dependent and also depends on the accuracy ofthe forces from the electronic-structure code. Thedisplacements must be large enough not to causenumerical errors in the forces but small enough stillto be in the harmonic regime. In our ND3 case, dis-placements between 0.001 and 0.01 Å yielded con-sistent results with a force accuracy of 10−4 eV/Å,therefore, we have used 0.005 Å for all calculations.Second, a large enough cell has to be used so thatthe effect of one atomic displacement does not pro-duce artifacts between periodic images. This was dis-cussed for the X23 test set by Reilly andTkatchenko25 with the conclusion that for these sys-tems, the cell should extend at least about 9–10 Å ineach direction. However, this should be evaluatedcarefully for the studied system, especially for saltsor molecular crystals involving heavier halogens. Allof the calculations presented here were performedby using a 2×2×2 supercell. For numerical stabilityof the calculation, it is important to ensure that thereciprocal-space sampling in the supercell is exactly



the same as in the optimized unit cell, i.e., if the unitcell was optimized with a n × n × n k-point grid, a2×2×2 supercell should have n

2 ×n2 × n

2 k points. Fur-thermore, the pDOS has to be evaluated with a densek-point grid in reciprocal space (or by calculatinglarge supercells) in order to get accurate free energies.We have used the program phonopy,112 which hasinterfaces to most of the popular periodic electronic-structure codes. Finally, the results have to be care-fully analyzed for ‘imaginary’ frequencies. Theappearance of an imaginary frequency at the Γ pointindicates that the crystal is not in a local minimum ofits PES and this will have a large impact on free ener-gies and low-frequency phonon modes. Owing to theacoustic sum rule, the three acoustic modes have tobe zero at the Γ point. However, small deviations arecommon due to the numeric nature of these calcula-tions. Therefore, these three modes often have smallimaginary frequencies, typically less than 1 cm−1 inmagnitude.

Harmonic Helmholtz Free EnergiesUsing the harmonic Helmholtz free energies, the rela-tive stability of polymorphs can be determined fordesired thermodynamic conditions. It was shown byMarom et al.105 and Rivera et al.115 that zero-point

energies can influence the calculated relative stabili-ties of glycine polymorphs. The aspirin crystal hastwo polymorphs with degenerated lattice energies,form I and form II, but form I is much more abun-dant experimentally. Considering only harmoniczero-point energies, the lattice-energy differencebetween the two polymorphs remains below 1 kJ/mol.27 However, when considering FHA

vib at 300 K,form I becomes more stable by 2.6 kJ/mol when cal-culated with PBE+MBD. In the case of PBE+TS, formII is more stable by 0.7 kJ/mol. This shows theimportance of both MBD interactions and Helmholtzfree energies for the prediction of polymorphstabilities.

Harmonic Vibrational SpectraThe pDOS provides information about all vibrationalstates within the molecular crystal, i.e., it containsnot only modes accessible by IR and Raman spec-troscopy (modes at the Γ point) but also out-of-phasephonon modes, which are long-range intermolecularmodes. Figure 6(a) shows the calculated pDOS forthe ND3 crystal at the respective optimized geometryof PBE, PBE+TS, and PBE+MBD. The four peaksabove 800 cm−1 correspond to the internal vibrationsof the ammonia molecules inside the crystal,i.e., inversions, angle bends, symmetric, and asym-metric stretching vibrations. The experimentallydetermined internal frequencies116 for ND3 areshown as gray lines in Figure 6(a). It can be seen thatthese frequencies are nicely captured by all threemethods. In this frequency range, all the vibrationsinvolve large energy changes, and therefore disper-sion interactions play only a minor role for thesevibrations. The peaks below 500 cm−1 correspond tophonons (or lattice vibrations), i.e., vibrations invol-ving intermolecular motion. Most of these low-frequency modes correspond to intermolecular trans-lations and librations. If the molecules in a crystalhave freely-rotating functional groups, like methylgroups, these low-energy intramolecular rotationscan also occur in this frequency range. The low-frequency vibrations of ND3 are shown in detail inFigure 6(b). It can be seen that there are qualitativedifferences, as well as shifts, in the peak positions ofthe pDOS between PBE and the vdW-inclusive meth-ods. However, as our test system is a highly symmet-ric molecular crystal of a small, rigid molecule, thedifferences are small and stem mostly from differ-ences in the optimized unit cells. For example, theunit-cell volume calculated with PBE is about 15%larger than with PBE+MBD, with the latter being clo-sest to experiment.

–10

–8

–6

–4

–2

0

0 100 200 300 400 500 600

k BT

ln[1

-exp

(-hω

/kBT

)] /

kJ/m

ol

Wave number / cm–1

T = 50 K

T = 100 K

T = 200 K

T = 300 K

FIGURE 5 | The temperature-dependent factor in the secondintegral of Eq. (10).



In contrast, Figure 6(c) shows the low-frequency pDOS for a cubic HMB crystal calculatedfor the three methods at the experimental unit-cellvolume. HMB has more flexibility than ammoniadue to its methyl groups and the crystal is bound bydispersion interactions. In this case, we observe sub-stantial differences in the pDOS between the differentmethods, even at the same volume. To understandhow important these differences are, Figure 6(c) alsoshows an experimental inelastic neutron scattering(INS) spectrum, measured at 15 K.117 As an INSspectrum does not represent a generalized pDOS, wecannot directly compare the intensities but be candirectly compare the position of the peaks (see thegray lines). It can be seen that PBE+MBD reproducesthe experimental peak positions very well, whereasthe peaks of PBE and PBE+TS appear to be slightlyshifted to higher frequencies. These spectra are com-pared at the experimental volume to allow for directcomparison in the absence of structural effects. Theresults are quite different if calculated at the respec-tive minima of Etot for each method. In this case, thePBE structure has a significantly larger volume andtherefore all peaks in the pDOS are shifted. Compar-ing different methods and experiment is thereforelikely to be very sensitive to the temperature of theexperiment and the unit cell used in calculating thespectra.

Although differences between the pDOS calcu-lated with different methods can appear subtle, theycan be very significant. Motta et al. recently calcu-lated the pDOS of a durene crystal and found thatincluding dispersion (using PBE+TS) was essential for

getting good agreement with a low-temperature INSspectrum.67 As noted above, the relative ordering ofthe two known polymorphs of aspirin changes signif-icantly when calculating Helmholtz free energies withPBE+MBD or PBE+TS.27 This can be traced to thepDOS calculated with the two methods. For PBE+TS, the two forms have comparable spectra but inthe MBD case, a peak is found at about 30 cm−1 forform I that is completely missing in the TS case andwas also not found in form II. It is this difference inthe pDOS of the two forms that leads to the changesin the calculated relative Helmholtz free energies andleads to PBE+MBD rationalizing the experimentalobservation of form I being more abundant.

In recent years, THz spectroscopy has emergedas very successful tool to detect drugs and explosives,as well as distinguish between different polymorphs ofmolecular crystals.118,119 THz spectroscopy detectsonly vibrational modes at the Γ point, possesses thesame selection rules as IR spectroscopy and is used tostudy only the low-frequency phonon modes, whichare highly correlated with the packing arrangement ofmolecules inside a molecular crystal.

When we compare the intermolecular modes atthe Γ point for our ND3 example with experimentalmeasurements, we observe, in general, deviationsbetween 10 and 70 cm−1. PBE always yields lowerfrequencies than the vdW-inclusive methods. The rea-son is mostly due to the overestimation of the unit-cell volume in the case of PBE. Furthermore, we arecomparing calculated results without any considera-tion of thermal effects, while the experimental mea-surements correspond to temperatures between

0 500 1000 1500 2000 2500

pDO

S /

arb.

u.

Wave number / cm–1 Wave number / cm–1 Wave number / cm–1

PBE

(a) (b) (c)

PBE+TS PBE+MBD

0 100 200 300 400 500

pDO

S /

arb.

u.

PBEPBE+TS

PBE+MBD

0 50 100 150 200 250

pDO

S /

arb.

u.

PBEPBE+TS

PBE+MBDINS-Exp.

FIGURE 6 | Full pDOS of ND3 (a), the low-frequency region of the ND3 pDOS (b), and the low-frequency region of the HMB pDOS (c). The graylines in (a) show the location of the experimental internal modes as determined by Holt et al.,116 and the gray lines in (c) mark the peak maximaof the experimental INS spectrum measured at 15 K by Ciezak et al.117 HMB, hexamethylbenzene; INS, inelastic neutron scattering; ND3, phase-Ideutero ammonia; pDOS, phonon density of states.



18 and 61 K. Already at these temperatures, anhar-monic effects due to thermal expansion and atomicmotion lead to significant frequency shifts. Reillyet al.120 extensively studied the lattice modes ofammonia and deutero ammonia by using the har-monic approximation as well as molecular-dynamicssimulations. The observed difference between thosemethods amounts to 15–30 cm−1 at 77 K.

The intensities of IR and THz spectra are pro-portional to the change of the dipole moment μ

IIR / ∂μ∂Q

$ %2

ð11Þ

where Q is a normal-mode coordinate. Calculationof IR intensities is much more involved for periodicsystems compared to molecules in vacuum becausethere is no unique definition of the dipole momentfor periodic systems and therefore the calculationrequires in principle a Berry-phase approach.121

However, this option is not available in a large num-ber of electronic-structure codes. The intensity ofeach mode can be calculated indirectly by calculatingthe difference between dipole moments of the unitcell calculated from atomic charges, e.g., Hirshfeld orMulliken charges. This unit-cell dipole is then calcu-lated for the ground state and for a geometry that isdisplaced along the normal-mode coordinate. Usu-ally, this method is referred to as the difference-dipole method. Allis et al. have applied this approachfor a variety of DFAs in order to calculate the THzspectrum of the explosive HMX.122 This study showsthat the THz modes highly depend on the DFA used.In recent years, vdW-inclusive DFT has been used tostudy the THz spectra of a number of molecular

crystals including naproxen,123 naphthalene,124

durene,124 enantiomers of ibuprofen,2 isomers ofbenzenediols,125 polymorphs of diclofenac acid,126

and α-D-glucose.45 In general, vdW-inclusive DFTmethods provide better THz spectra than traditionalDFAs but major differences can be found betweenseveral vdW-inclusive methods. The best agreementbetween calculated THz spectra in the HA andexperimental measurements is found at very low tem-peratures (<10 K), where anharmonic effects arelikely to be minimized.

What Is Missing in the HarmonicApproximation?The harmonic approximation enables us to calculatefree energies and obtain vibrational spectra. But whatexactly are we missing in this approximation? As dis-cussed before, we have omitted all higher-order termsin the Taylor expansion of the PES [see Eq. (7)]. Sohow valid is this approximation? To answer thatquestion, we have taken two modes of ND3 and cal-culated the energy with PBE+MBD for several displa-cements along the normal-mode coordinates (seeFigure 7).

For a unitary displacement, the norm of theeigenvectors is equal to one. The mode in (a) showsan intermolecular translation and the mode in(b) corresponds to an internal wagging motion. Itcan be seen for (a) that the HA provides good resultsfor small displacements from the equilibrium geome-try but there are significant differences from the exactresult at larger displacements. We can see from theenergy scale in (a) that only a few kJ/mol arerequired to get into the anharmonic regime of that

0

5

10

15

20(a) (b)

–10 –5 0 5 10

Rel

ativ

e en

ergy

/ kJ

/mol

Displacement

HarmonicExact

0

10

20

30

40

–2 –1 0 1 2

Rel

ativ

e en

ergy

/ kJ

/mol

Displacement

HarmonicExact

FIGURE 7 | Harmonicity of two phonon modes in ND3 calculated with PBE+MBD. The displacements are shown at a relative scale in which adisplacement of 1 means that the norm of the eigenvectors is equal to one. MBD, many-body dispersion; ND3, phase-I deutero ammonia.



phonon mode. This suggests that anharmonicity willbe a serious issue for low-frequency phonon modesat high temperatures. Displacing along mode(b) requires much more energy compared to (a). Inthe case of (b), it can be clearly seen that the mini-mum of the harmonic curve does not correspond tothe minimum of the exact curve. Considering theseanharmonic effects would lead to a shift in thephonon-mode frequency.

Furthermore, the modes are described in theHA as independent oscillators, which means thatthere are no interactions or coupling betweenthem. In a system at finite temperature, the motionof the atoms inside the crystal will certainly not beconstrained to movements along normal-modecoordinates. Hence, in order to describe theatomic motion correctly, one would need at leasta superposition of several normal modes. Anotherserious problem is that we have only approxi-mated our PES around the minimum of the totalenergy Etot. As a result, the pDOS corresponds toexactly this geometry and the free energy at finitetemperatures also assumes that the structure ofthe system does not change at all with tempera-ture. Therefore, it is not possible to observe or

model phase transitions in the HA. One effectoften neglected is the thermal expansion of molec-ular crystals. The volume of the unit cell will typi-cally increase with increasing temperature. Thiswill be discussed in detail in the next section.Finally, it is not possible to describe charge trans-port with the HA since phonon lifetimes and ther-mal conductivity are infinite.

The Quasi-Harmonic ApproximationA straightforward way to improve upon the HA isthe so-called quasi-harmonic approximation(QHA).127 In this approximation, the HA is stillapplied but at several different unit-cell volumes.We start with the HA at the minimum of Etot. Sub-sequently, the geometry of the unit cell is optimizedfor several fixed unit-cell volumes V around theequilibrium unit-cell volume and the HA is appliedto each of these structures. In the QHA, the freeenergy now depends explicitly on the unit-cellvolume

FQHA T,Vð Þ =Etot Vð Þ+ FHAvib T,Vð Þ ð12Þ

and therefore, the pDOS also depends on the cell vol-ume. Figure 8 shows the dependence of the low-frequency pDOS on the unit-cell volume for the ND3

crystal calculated at the PBE level of theory. It can beclearly seen that, in general, peaks shift to smallerfrequencies with increasing volume and that thepDOS below 200 cm−1 changes dramatically withincreasing volume.

The result of applying the QHA is that we nowknow for several unit-cell volumes V, the harmonicfree energy as a function of temperature T. There-fore, we can calculate the unit-cell volume corre-sponding to a specific temperature by fitting our datato an equation of state (EOS). The MurnaghanEOS128 is often used for this purpose:

FQHA Vð Þ= F0 +B0VB00

V0=Vð ÞB00

B00−1

+ 1

" #

−B0V0

B00−1

ð13Þ

where F0 is the equilibrium free energy for a certaintemperature, V0 is the corresponding equilibrium vol-ume, B0 is the bulk modulus at equilibrium volumeV0, and B0

0 is its pressure derivative.

Thermal ExpansionFigure 9 shows the EOS fits for the model ND3 crys-tal, while the resulting unit-cell volumes are

0 100 200 300 400 500

pDO

S /

arb.

u.

Wave number / cm–1

V = 180 Å3

V = 170 Å3 V = 160 Å3 V = 150 Å3

V = 140 Å3 V = 130 Å3 V = 120 Å3

FIGURE 8 | Phonon density of states of solid ND3 calculated withPBE at several unit-cell volumes. ND3, phase-I deutero ammonia.



compared with the experimental data in Table 1. At2 K, the unit-cell volume was measured experimen-tally as 128.6 Å3. We can immediately see that a sim-ple lattice optimization does not provide satisfactoryresults in terms of the unit-cell volume. The PBEfunctional overestimates V by about 11% due tomissing attractive interactions between the moleculesand the two vdW-inclusive methods underestimateV by about 4%. As discussed before, lattice optimiza-tions do not take into account zero-point vibrations,which results in most cases in slightly too small unit-cell volumes for vdW-inclusive DFT.25 The QHAincludes zero-point vibrations and the DFA+vdWresults agree at 2 K very well with experimentalresults; the error for PBE+TS and PBE+MBDamounts to 1.9 and 0.5%, respectively. Note, thatPBE alone shows now an error of 17%, which illus-trates how neglecting thermal and zero-point contri-butions can lead to spurious cancelation of someerrors. It also reinforces the importance of dispersioninteractions in modeling even a largely hydrogen-bonded system such as ammonia. Increasing the tem-perature to 180 K results in thermal expansion ofabout 5%.129 PBE without dispersion now shows anerror of about 22% in predicting the unit-cell vol-ume, while PBE+TS and PBE+MBD provide a veryreasonable description with errors being 2.5 and1.5%, respectively. The PBE+MBD results agree con-sistently better with the experimental results thanPBE+TS.

It is also possible to include pressure effects bysimply adding the pV term to FQHA, which thenbecomes GQHA.

GQHA p,T,Vð Þ=Etot Vð Þ + FHAvib T,Vð Þ+ pV ð14Þ

At ambient conditions pV is very small and is there-fore neglected in our QHA calculations. If one wishesonly to account for pressure effects without consider-ing thermal effects, an external hydrostatic pressurecan simply be added to the stress tensor. For exam-ple, Schatschneider et al. studied oligoacenes up to apressure of 25 GPa using PBE+TS.130 This hydro-static external pressure also enables one to approxi-mate thermal expansion. Describing the effect ofFHAvib T,Vð Þ using just the pV term leads to

pth =∂FHA

vib T,Vð Þ∂V

ð15Þ

where pth is a negative thermal pressure, which canbe applied to the stress tensor during optimization tomimic thermal effects. In a very crude approxima-tion, pth can be obtained by finite differences usingthe free energies for a certain temperature of the opti-mized unit cell and a slightly larger or smaller unitcell. In order to demonstrate this, we have calculatedthe thermal pressure for the ND3 model system for180 K by considering only the optimized unit cell

0

5

10

15

20

130 140 150 160 170 180

Rel

ativ

e fr

ee e

nerg

y / k

J/m

ol

Cell volume / Å3 Cell volume / Å3 Cell volume / Å3

2 K

77 K

180 K

PBE PBE+TS PBE+MBD

0

5

10

15

20

120 125 130 135 140 145

Rel

ativ

e fr

ee e

nerg

y / k

J/m

ol

2 K

77 K

180 K

0

5

10

15

20

120 125 130 135 140 145

Rel

ativ

e fr

ee e

nerg

y / k

J/m

ol

2 K

77 K

180 K

FIGURE 9 | Quasi-harmonic approximation for solid ND3 calculated with different methods at three different temperatures. The solid linesrepresent Murnaghan EOS fits and the red triangles mark the corresponding minima. EOS, equation of state; ND3, phase-I deutero ammonia.



and the closest available cell from the QHA, whichhas a larger volume than the optimized cell. Thisthermal pressure was then applied during a latticerelaxation. The obtained values for pth and the result-ing volumes are presented in Table 1. It can be seenthat the obtained volumes agree with the values fromthe QHA at 180 K within 2%. Using this thermalpressure approach, Otero-de-la-Roza and Johnsonfound that for the C21 dataset PBE+D2, PBE+TS,PBE+XDM, and vdW-DF2 yield mean absolute per-centage deviations between 1.3 and 2.8% for celllengths and between 0.1 and 0.3% for angles.24

Recently, Heit et al. studied the thermal expan-sion of crystalline carbon dioxide with the QHAusing MP2.40 Up to a temperature of 195 K, theyunderestimated the unit-cell volume by only about2–3% compared to experiment. Erba et al. haverecently reported the directional-dependent thermalexpansion of the urea crystal studied within theQHA by using several density functional approxima-tions and different dispersion corrections.41

The unit-cell volume for a specific temperatureis determined by an interplay between the 0 K totalenergies and the vibrational free energies. A measurefor the actual expansion of the crystal with tempera-ture is the volumetric thermal expansion coefficientαV, which can be written as

αV =1V

∂V∂T

$ %

pð16Þ

The knowledge of αV enables us to calculate also theheat capacity at constant pressure (Cp), which can bedirectly compared with experimental calorimetricmeasurements. Cp is given by

Cp Tð Þ =CV Tð Þ + αV Tð Þ2B Tð ÞV Tð ÞT ð17Þ

where B is the bulk modulus and CV can be calcu-lated from

CV = kBðdωg ωð Þ ℏω=kBTð Þ2exp ℏω=kBTð Þ

exp ℏω=kBTð Þ−1½ %2ð18Þ

Figure 10 shows the calculated values for Cp and thelinear thermal expansion coefficient α = αV/3 for theND3 example compared with experimental values.131

It can be seen that PBE overestimates Cp at low tem-peratures and underestimates it at higher tempera-tures, whereas α is constantly overestimated by about100%. In contrast, the vdW-inclusive methodsunderestimate Cp and α by about 20%. The onlyexception is PBE+TS, which follows the experimentalvalues at low temperature but starts underestimatingα at about 100 K. This underestimation of thermalexpansion is to be expected, since we are neglectinganharmonic effects due to internal atomic motion.

Elastic PropertiesAnother type of response property that is highly tem-perature dependent is the elastic constants, whichquantify a crystal’s response to elastic deformation.The elastic constant matrix C of a molecular crystalcan be obtained by a Taylor expansion around theequilibrium geometry:132

E V,εð Þ =E0 +V0

X6

i =1

σiεi +12

X6

ij = 1

Cijεiεj

!

ð19Þ

where ε is the strain applied to the unit cell and σ isthe corresponding stress of the unit cell. The elasticconstants can then be approximated by the second-order derivatives of the energy with respect to theapplied strain:

Cij =1V0

∂2E∂εi∂εj

ð20Þ

Another approach to calculate the elastic constants isvia the stress-strain relation

TABLE 1 | Unit-Cell Volumes for ND3 Phase I for Several Methods Calculated via Optimization (opt), Extracted from a Quasi-HarmonicApproximation (n K) and Optimization under Thermal Pressure (pth)

Method V(opt) [Å3] V(2 K) [Å3] V(77 K) [Å3] V(180 K) [Å3] V(pth) [Å3] pth [GPa]

PBE 142.6 149.9 152.2 163.9 162.6 −0.52

PBE+TS 123.0 126.2 127.2 131.2 133.3 −0.58

PBE+MBD 123.8 127.9 128.6 132.5 131.8 −0.39

Exp.129 — 128.6 130.6 134.6 — —

MBD, many-body dispersion; ND3, phase-I deutero ammonia; TS, Tkatchenko–Scheffler.



σ =Cε ð21Þ

where the elastic constant matrix is obtained by cal-culating the stress for several strained unit cells.Often, the elastic constants are calculated based onthe equilibrium geometry and correspond thereforeto 0 K. Thermal effects can also be included in a sim-ple quasi-harmonic way by obtaining the unit-cellvolume for the desired temperature from the QHAand calculating the elastic constants for this struc-ture. We will illustrate this temperature dependenceby using our ND3 model crystal. Our example has acubic unit cell and therefore possesses only threeunique elastic constants: C11, C12, and C44. The firsttwo constants are related to volumetric elasticity andC44 is related to shear deformation. The bulk modu-lus B is an inverse measurement for the compressibil-ity of the molecular crystal and can be calculated fora cubic crystal as

B =C11 + 2C12

3ð22Þ

Calculated elastic constants for PBE, PBE+TS, andPBE+MBD are given in Table 2 and compared withexperiment.

First, we compare the results obtained for theoptimized geometries with the experimental values at95 K. It can be seen that PBE underestimates the bulkmodulus and all elastic constants while both vdW-inclusive methods overestimate the experimental

values. Comparing the results obtained at the esti-mated unit-cell volumes at 194 K with the experimen-tal results at 194 K, we can see that all of the elasticconstants decrease with increasing temperature (andhence volume). The PBE bulk modulus is now almostthree times smaller than the experimental value, whilethe vdW-inclusive methods agree very well with exper-iment. The deviation in the bulk modulus is 1.0 GPafor PBE+TS and only 0.1 GPa for PBE+MBD.

Another important elastic property of a crystalis the Young’s modulus, which describes the tendencyof deformation along an axis and can also be calcu-lated directly from the elastic constants. Figure 11shows spherical polar plots of the Young’s modulus ofammonia. The plot for PBE is very isotropic, whilePBE+MBD is more anisotropic. If we compare theplots of the optimized geometries with the experimen-tal plot, one might think that PBE is describing theYoung’s modulus better than PBE+MBD, but whenwe compare the values for 194 K, we can immediatelysee that vdW interactions have a large influence onelastic properties. We can also measure the anisotropyof the elastic constants with the so-called anisotropyfactor A, which for cubic crystals is given by

A =2C44

C11−C12ð23Þ

It can be seen that PBE+MBD shows the largestanisotropy among the theoretical values, but stills

0

10

20

30

40

50(a) (b)

0 50 100 150 200

Cp

/ J/m

ol/K

Temperature / K

PBEPBE+TS

PBE+MBDExp

0

5

10

15

20

25

30

0 30 60 90 120 150

α x

105

K–1

Temperature / K

PBEPBE+TS

PBE+MBDExp

FIGURE 10 | Heat capacity at constant pressure (a) and linear thermal expansion coefficient (b) of solid ND3 obtained from the QHAcompared to experimental values from Ref 131. ND3, phase-I deutero ammonia; QHA, quasi-harmonic approximation.



underestimates the degree of anisotropy compared toexperimental measurements, likely due to the lack ofanharmonicity in calculating the elastic constants.

The use of DFAs for studying mechanical prop-erties has grown in recent years. The elastic proper-ties of urea have been studied by Erba et al. in aquasi-harmonic fashion yielding encouraging resultsfor PBE, PBE0, and B3LYP when paired with the D3dispersion correction.41 The mechanical properties ofthe two aspirin polymorphs have also been studiedusing PBE+TS and PBE+MBD.27 Despite their neardegenerate lattice energies and a free-energy differ-ence of 2.6 kJ/mol, the two forms show remarkably

different elastic properties. As in the case of ammo-nia, PBE+MBD also yields more anisotropic elasticresponse for the two polymorphs of aspirin andsmaller elastic constants, agreeing better with experi-mental values.

CONCLUSIONSThe prediction and modeling of molecular crystalsfrom first principles has evolved considerably inrecent years. With the development of accurate vdW-inclusive DFT methods, in particular, first-principles

TABLE 2 | Elastic Constants, Bulk Modulus, and Anisotropy Ratio for Solid ND3 Calculated at the Minima of Etot (opt) and at VolumesCorresponding to 194 K According to Respective QHAs, Compared to Experimental Values

Method C11 [GPa] C12 [GPa] C44 [GPa] B [GPa] A

PBE(opt) 9.8&0.3 3.7&0.1 4.6&0.1 5.7 1.5

PBE+TS(opt) 15.6&0.4 6.8&0.3 8.3&0.1 9.8 1.9

PBE+MBD(opt) 15.6&0.2 6.4&0.1 8.7&0.1 9.5 1.9

PBE(194 K) 3.5&0.2 1.6&0.1 1.3&0.1 2.3 1.4

PBE+TS(194 K) 9.8&0.3 3.7&0.2 5.0&0.1 5.8 1.6

PBE+MBD(194 K) 11.0&0.3 4.6&0.2 5.8&0.1 6.7 1.8

Exp(95 K)134 10.0&0.5 5.6&1.0 5.6&0.4 7.1 2.5

Exp(194 K)135 9.5&0.8 5.5&0.9 4.9&0.4 6.8 2.4

MBD, many-body dispersion; ND3, phase-I deutero ammonia; QHA, quasi-harmonic approximation; TS, Tkatchenko–Scheffler.

10

PBE

Opt.

194 K

PBE+MBD Exp.

0c

–10

–10

–10

10

0 b

10

0a

10

0c

–10

–10

–10

10

0 b

10

0a

10

0c

–10

–10

–10

10

0 b

10

0a

10

0c

–10

–10

–10

10

0 b

10

0a

10

0c

–10

–10

–10

10

0 b

10

0a

FIGURE 11 | Spherical plots of the Young’s modulus of ND3 (in GPa) obtained at the minima of Etot (opt) and at volumes corresponding to194 K from the QHA for PBE and PBE+MBD, as well as experimental values at 194 K of Ref 135. MBD, many-body dispersion; ND3, phase-Ideutero ammonia; QHA, quasi-harmonic approximation.



methods can now be applied to realistic molecularcrystals, leading to new insights and understandingof structure, stabilities, polymorphism, and responseproperties such as phonons and elastic moduli. Thisability of first-principles methods to model these prop-erties in an accurate, balanced, and transferable fash-ion will make methods such as vdW-inclusive DFTcentral tools for predicting and engineering molecularcrystals in future. High-level first-principles methodssuch as MP2 and beyond can already be applied toseveral molecular crystals and encouraging progress isbeing made to enable the applications for realistic sys-tems in the future.71 Also, the utilization of symmetry-adapted perturbation theory for periodic systems willenable new insights into the nature of intermolecularinteractions in molecular crystals.86

The current state of the art in first-principlesmethods enables us to calculate lattice energies witha accuracy better than 4.2 kJ/mol. Key to achievingthis accuracy is to go beyond a pairwise model of dis-persion and include nonadditive many-body contri-butions, for example, by using the MBDmethod.22,91 Truly predicting and understandingphenomena such as polymorphism requires greateraccuracy still, with a number of studies of a range ofsystems having shown the importance of thermal andpressure contributions to the relative stabilities ofpolymorphs.133

Another key aspect of modeling molecularcrystals is in understanding that all of their

properties (including their structure) can be highlytemperature dependent. Many first-principles stud-ies of molecular crystals treat them in the harmoniclimit. While this can give powerful insights, the har-monic approximation neglects anharmonic effectsdue to the expansion of the unit cell and thermalmotion. Omitting these effects when calculatingderived properties and quantities such as low-frequency vibrational spectra (THz spectra) andelastic constants, can lead to large deviationsbetween experiment and theory, as we have shownusing our model system of phase-I deutero ammo-nia. An efficient way of approximating anharmoniccontributions with first-principles methods is thequasi-harmonic approximation, which captures themajority of the thermal expansion.

Perhaps one of the most important directions infirst-principles modeling of molecular crystals is thedevelopment of accurate force fields derived fromfirst-principles data. Recent results already showencouraging accuracy for molecular crystals.102 Suchforce fields would enable the types of molecular-dynamics simulations and multiscale models needed tofully understand the thermodynamics and kinetics ofmolecular crystals as a function of temperature andother variables. At the same time, fully first-principlesmethods can be used to determine the underlying elec-tronic energies and key response properties accurately,providing a full picture of the formation, stability, andproperties of molecular crystals.

ACKNOWLEDGMENTJ.H. and A.T. acknowledge the support from the Deutsche Forschungsgemeinschaft under the program DFG-SPP 1807.

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