crystals Article Stability-Ranking of Crystalline Ice Polymorphs Using Density-Functional Theory Pralok K. Samanta , Christian J. Burnham and Niall J. English * School of Chemical and Bioprocess Engineering University College Dublin, Belfield, Dublin 4, Ireland; [email protected] (P.K.S.); [email protected] (C.J.B.) * Correspondence: [email protected] Received: 18 December 2019; Accepted: 15 January 2020; Published: 16 January 2020 Abstract: In this work, we consider low-enthalpy polymorphs of ice, predicted previously using a modified basin-hopping algorithm for crystal-structure prediction with the TIP4P empirical potential at three pressures (0, 4 and 8 kbar). We compare and (re)-rank the reported ice polymorphs in order of energetic stability, using high-level quantum-chemical calculations, primarily in the guise of sophisticated Density-Functional Theory (DFT) approaches. In the absence of applied pressure, ice Ih is predicted to be energetically more stable than ice Ic, and TIP4P-predicted results and ranking compare well with the results obtained from DFT calculations. However, perhaps not unexpectedly, the deviation between TIP4P- and DFT-calculated results increases with applied external pressure. Keywords: crystal-structure prediction; basin-hopping methods; density-functional theory; many-body dispersion 1. Introduction The study of water’s properties, regardless of the particular condensed phase thereof, is of paramount practical and technical interest, in addition to constituting a fundamental and formidable scientific challenge [1]. One of the most intriguing properties of water is its large numbers of crystalline polymorphs (currently 18, although no doubt still set to grow in the future), which are stable over a broad range of temperature and pressure [2]. Ice Ih is one of the most frequently-observed phases of water on Earth. However, different forms of ice can be found at medium and high pressures [1,3,4]; Tammann and Bridgman discovered first high-pressure form of ice in the early 20th century [5,6]. Indeed, in this new, 21st century alone, six ice polymorphs have been found (ices XIII-XVIII) so far [7–11]. However, it has also been observed that most water present in the Universe is in its amorphous form [12]. Much of the detailed and technical characterization and properties-determination work done on both the crystalline and amorphous forms of water, has been, and will continue to be, experimental. However, theoretical and/or molecular-simulation studies can help to explain more insights into the rich behavior of solid (both crystalline and amorphous) water phases, especially for high-pressure ices at the electronic-structure level, where empirical models can often be found wanting (due to the inevitable electron-cloud overlap at high pressures being well outside the electronic experience of near-ambient-condition empirical-model parameterization) [13]. The theoretical pursuit, and leverage, of crystal-structure-prediction (CSP) methods can be used to explore the plausible different crystalline polymorphs, and indeed, polytypes, of ice [14]. Indeed, CSP may be a fruitful avenue for assessing the structural and energetic viability, in a prototyping or screening fashion, for potential novel, yet-to-be-discovered, ice structures. In general, CSP, done well, can provide de-novo predictions for various crystalline polymorphs of a material from a first-principles approach (either with accurate empirical force-fields or from electronic-structure approaches). Crystals 2020, 10, 40; doi:10.3390/cryst10010040 www.mdpi.com/journal/crystals Crystals 2020, 10, 40 2 of 8 Crystals 2020, 10, x FOR PEER REVIEW 2 of 8 CSPCSP cancan bebe deployed,deployed, interinter aliaalia,, forfor thethe developmentdevelopment ofof newnew pharmaceuticalpharmaceutical andand pigmentpigment compoundscompounds andand/or/or forfor thethe engineeringengineering ofof novelnovel molecularmolecular materialsmaterials [15[15–18].–18]. CSPCSP methodsmethods oftenoften predictpredict manymany possiblepossible polymorphspolymorphs withinwithin aa smallsmall relative-energy-direlative-energy-differencefference rangerange ((<10<10 kJkJ/mol)/mol) [[19].19]. Thus,Thus, in order to to gauge gauge and and rank rank polymorphs’ polymorphs’ relative relative energetic energetic stability, stability, it is itessential is essential to use to high- use high-levellevel quantum-chemical quantum-chemical methods methods to capture to capture as asaccura accuratelytely as as possible possible the the subtlety subtlety of intermolecularintermolecular interactions,interactions, and and to to refine refine the (free-)the (free-) energy energy/enth/enthalpy rankingalpy ranking from (potentially) from (potentially) less accurate less empirical accurate force-fieldsempirical orforce-fields more crude or electronic-structure more crude electronic representations-structure [ 20representations]. Intermolecular [20]. interactions, Intermolecular which areinteractions, relatively which weak, areare typicallyrelatively non-directional weak, are typically and long-rangenon-directional in nature; and long-range an accurate in description nature; an thereofaccurate is description a challenging thereof task foris a modelingchallenging molecules task for modeling and materials molecules [21]. Density-Functionaland materials [21]. Density- Theory (DFT),Functional combined Theory with (DFT), Grimme combined D3-BJ with methods Grimme (e.g., D3-BJ the D3methods method (e.g., with the Becke-Jonson D3 method with damping), Becke- andJonson more damping), sophisticated and more many-body sophisticated dispersion many-b (MBD)ody techniques,dispersion (MBD) are generally techniques, found are to generally be more suitablefound to for be a moremore accurate suitable description for a more of intermolecularaccurate description forces, especiallyof intermolecular in higher-pressure forces, especially crystalline in polymorphshigher-pressure [22, 23crystalline]. We also polymorphs have considered [22,23]. the We density-dependent also have considered dispersion the density-dependent correction (dDsC) developeddispersion bycorrection Steinmann (dDsC) et al. [24developed,25] for dispersion by Steinmann energy, et and al. compared [24,25] for the dispersion results with energy, D3-BJ andand MBDcompared results. thedDsC results is with used D3-BJ to calculate and MBD the results. interactions dDsC betweenis used to non-overlapping calculate the interactions densities, between which cannotnon-overlapping be described densities, accurately which by standard cannot be Density-Functional described accurately Theory. by standard Density-Functional Theory.In the present work, we consider the low-energy polymorphs of water at different pressures (0, 4In and the 8 present kbar), work, predicted we consider previously the low-energy by us using polymorphs a modified of basin-hoppingwater at different algorithm pressures for (0, crystal-structure4 and 8 kbar), predicted prediction; previously here, we by re-rank us using the a icemodified polymorphs basin-hopping using sophisticated algorithm for DFT crystal- and higher-levelstructure prediction; electronic-structure here, we re-rank calculations the ice [polymorphs14]. Some of using the polymorph sophisticated structures DFT and are higher-level shown in Figureelectronic-structure1. The structure calculations S12 /41620 8[14].10, for Some example, of the haspolymorph 12 (the numberstructures following are shown the in “S”) Figure molecules 1. The 1 20 10 perstructure unit cell. S12/4 The6 numbers8 , for example, following has the 12 (the forward number slash followin (“/”) giveg the the “S”) ring molecules count: this per structure unit cell. hasThe onenumbers four-membered following ring,the forward twenty-six-membered slash (“/”) give rings,the ring and count: ten eight-membered this structure has rings. one Infour-membered our previous workring, [twenty-six-membered14], basin hopping was rings, performed, and ten and eight-memb the ice polymorphsered rings. were In our ranked previous in order work of [14], enthalpic basin stabilityhopping at was these performed, three pressures and the using ice polymorphs the 4-site transferable were ranked interaction in order potential of enthalpic (TIP4P) stability water at model, these athree popular pressures empirical using model the for4-site water transferable developed inte by Jorgensenraction potential et al. [26 (TIP4P)]. In this water article, model, we compare a popular the resultsempirical of energetic-stability model for water developed ranking using by Jorgensen both TIP4P et wateral. [26]. and Inelectronic-structure this article, we compare calculations. the results of energetic-stability ranking using both TIP4P water and electronic-structure calculations. Figure 1. Crystal structures and unit cells for several selected structures. Figure 1. Crystal structures and unit cells for several selected structures. Crystals 2020, 10, 40 3 of 8 2. Computational Details We have considered our previously-reported unit-cell geometry and lattice parameters predicted using the basin-hopping algorithm with the TIP4P water force-field [27]. We re-optimized the geometry, including the relaxation of lattice parameters to the appropriate pressures (0, 4 and 8 kbar) using DFT, as implemented in the Vienna Ab-initio Simulation Package (VASP 5.4.4) [28–31]. Here, we have considered a plane-wave energy cutoff of 800 eV, within the framework of the Generalized Gradient Approximation (GGA), with the Perdew–Burke–Ernzerhof