Are Crystal Phase Diagrams Predictable with Force Fields? Case of

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Are crystal phase diagrams predictable with force fields? Case of benzene polymorphs. Detlef Hofmann, and Liudmila Kuleshova Cryst. Growth Des., Just Accepted Manuscript • Publication Date (Web): 11 Jul 2014 Downloaded from http://pubs.acs.org on July 15, 2014

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Crystal Growth & Design

Are crystal phase diagrams predictable with force fields? Case of benzene polymorphs D.W.M. Hofmann∗ and L.N. Kuleshova∗ CRS4, Parco Scientifico e Technologico, Sardegna Ricerca, Edificio 1, Loc. Piscina Mana, Pula, Italy 09010 E-mail: [email protected]; [email protected]

Abstract The accurate prediction of the properties of materials under non-standard or extreme conditions, such as the high pressure of ocean depths and the near-absolute zero (0 K) temperatures of space has become increasingly important. The ability to predict crystal polymorphs that are realized at a given pressure P and temperature T would allow the calculation of a crystal phase diagram for any given substance, as well as the ability to forecast the properties of different phases under certain conditions, even prior to their synthesis. This would have a large effect on several fields of science and technology. We present an efficient approach to parameterizing and/or optimizing the parameters of force fields, which takes into account external conditions.

Introduction The formation of crystal structures is affected by external conditions. Depending on temperature and pressure, substances can crystallize into different crystal structures, thus showing polymor∗ To

whom correspondence should be addressed

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phism. The prediction of the crystal structure of a given molecule under certain conditions is a very challenging task. However, the successful prediction of crystal structures is problematic because of the immense number of reasonable crystal structures that are generated during prediction, which possess tiny differences in the lattice energy. Ten years ago, when Jack Dunitz 1 published the article, "Are crystal structures predictable?" the answer to the question was still "no". Similarly, the blind tests of crystal structure prediction conducted by the Cambridge Crystallography Data Centre (CCDC) 2–4 showed a very low success rate. However, the results of subsequent blind tests 5,6 contributed significantly to this field, and several research groups are now able to predict crystal structures with high reliability. The fundamental approach to molecular simulations is based on the following simple thermodynamics equation Eq. (1): G = U + PV − T S

(1)

where G is Gibbs free energy, U is the internal energy of the system, P,V, T, S is the pressure, volume, temperature, and entropy, respectively. Because the calculation of entropy is well known to be difficult, instead of Gibbs free energy, the enthalpy (H = U + PV ) is minimized, which effectively corresponds to the T = 0(K) case. Nevertheless, in molecular modelling, some force fields are parameterized to reproduce free energy G. This substitution was not often stated explicitly because molecular modelling was not accurate enough to distinguish the difference. However, with modern, improved force fields, it has become important. Moreover, different methods of molecular modelling require different parameterization. Molecular mechanics, which tries to obtain experimental molecular or crystal structures by minimization, requires Gibbs energy G as the parameter. In contrast, molecular dynamics requires the internal energy U as the parameter because the effects of pressure and temperature are intrinsic to this method and are added during the integration of the equations of motion. The parameters for Gibbs free energy are optimized to reproduce experimental molecular structures 7 or crystal structures. 8–10 The parameters for internal energy are optimized to fit experimental, measured radial distribution functions 11,12 or to reproduce quantum-chemical calculations. 13,14 2 ACS Paragon Plus Environment

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However, in quantum chemistry, enthalpy is the primary result. The effects of temperature, pressure, and mass have to be added in subsequent and tedious calculations of molecular dynamics. Moreover, in the third step, the results have to be transformed to effective potentials to allow a plausible interpretation of the result. Parinello‘s metadynamics 15 or its variations, as implemented in the VAMP/VASP program, 16 are used the most often. However, another possibility of relating the thermodynamic parameters of a system, such as temperature and pressure, with the microscopic behaviour of atoms and molecules could be provided by using statistical mechanics. Although this approach was applied in 1975 10 to predict the thermodynamics of crystalline benzene, it was not broadly applied in the field of crystal structure prediction. The reason is that by increasing the degrees of freedom in crystalline systems, the number of points needed for screening the energy surfaces increases exponentially, which makes the calculation process extremely time-consuming. However, the growth of experimental databases and the development of the data mining technique have made possible an alternative calculation of effective potentials. In this paper, we present an efficient data mining approach to parameterising and/or optimising the parameters of a force field, which takes external conditions into account. To illustrate the effects of temperature and pressure on atom-pair potentials, we use the example of the crystal polymorphs of benzene, which are unstable under standard conditions and, consequently, could not be predicted correctly using standard force fields. The pressure- and temperature-induced polymorphs of benzene are also studied, both experimentally 17,18 and theoretically. 19

Data Mining on crystal structures. Data mining was created to obtain the optimal parameters of any model, by which sets of measured data are connected with the desired properties. The enormous amount of information about crystal structures is currently stored in crystal structural databases, such as the Cambridge structural database (CSD), which provides an excellent opportunity to apply the advanced technique of data mining to derive the potentials in crystal and molecular modelling.



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In thermodynamics, it is known that any system observed in nature is characterized by a local minimum of free energy. Similarly, any experimentally observed (and stable) crystal structure is characterized by a local minimum with negative values of free energy according to the position of the atoms in the crystal structure, as shown by Eq. (2). The data mining structural information is conducted to exploit this property in order to derive the free energy from a set of experimental crystal structures:

∂G =0 ∂ ri

(2)

A support vector machine with associated algorithms can be used to solve the problem. In this approach, when an arbitrary energy function (model) is imposed, the parameters of this model should be optimized by learning algorithms. For this purpose, two sets (classes) of structures are imposed, using a classification tool. The first class, "correct", comprises experimental crystal structures, which possess a minimum of negative values of the free energy. The second class of crystal structures, "virtual", is generated by small distortions in the cell parameters and/or atom positions in the experimental structures. In this class, the structures are not set at the minimum, and they can possess both positive and negative values of free energy: virtual } set experimental = {crystalkexperimental } → set virtual = {crystalk,m

(3)

From these two sets, we can set up two systems of inequations: margin Gvirtual − Gexperimental km k

(7)

This separation can be achieved only approximately because some experimental structures can be erroneous, and/or the model is not perfect and contains some approximations. The achieved outliers, nevertheless, are very important for further refinement of the model. They also help to clean the training set and the used data base of erroneous entries. The solution to this system of inequalities allows the determination of the supposed parameter for the function G. Commonly, the energy G is supposed as linear in the atomic pair of interactions g.

Gcrystal = ∑ f requencymnr ginteraction (r) mn

(8)

The parameters obtained during training, as well as the definition of the model, provide our force field for molecular modelling. The special advantage of data mining force fields in molecular mechanics is that its effective potentials directly calculate the free energy of a system. The fact that the method produces the free energy instead of the enthalpy can be illustrated by a short example. The method trains the interaction potentials until the energy function assigns the lowest energy to the experimental structure, among all possible polymorphs. For example, if we use high pressure-induced crystal structures in training, then after training, the energy function assigns the lowest energy to the high-pressure polymorph. If the second training set contains crystal structures that are stable at low temperature, after the training the energy function assigns the lowest energy to the low temperature-induced polymorph. Obviously, this energy function includes the effects of temperature and pressure and corresponds to G(p, T ) instead of enthalpy, which by definition is not adequate to predict the relative stability of polymorphs that depend on external conditions. 5 ACS Paragon Plus Environment


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In this context, it is important to keep in mind that the interaction potentials are inherently dependent on pressure and temperature. Hence, we denote the atom pair potentials as gi j instead of ei j , which is used in common force fields. A general description of the method can be found in the textbook "Data Mining in Crystallography". 20 In earlier studies, 21 similar consideration was successfully applied to distinguish the parameters of H and D atoms, which allowed the reproduction and explanation of the effect of isotopic substitution on the formation of crystalline polymorphs of acridine.

Data Mining Force Field for benzene In this section, we will illustrate the procedure of deriving fine-tuned parameters for the specific case of pressure- and temperature-induced polymorphs of benzene. The energy function g is assumed as the sum of atom pair interactions, which is the case of the majority of force fields in molecular mechanics and molecular dynamics. The atom-pair interaction gi j (r) depends on the type of atom and the distance. In the case of benzene, two types of atoms have to be defined: C and H. The dependence on the distance is developed in a series of the expansion of inverse power. In this special case, we restricted the expansion to the powers of 1, 6, and 12. Hence, the model of potential coincides with common van der Waals potentials.

g(r) = ∑

ai with i ∈ {1, 6, 12} ri

(9)

For the initial potential, we used the standard potentials 9 implemented in the FlexCryst program suite.


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Data sets for polymorphs of benzene Class of "correct" structures of benzene The set of "correct" structures required for training was obtained by extracting the experimental crystal structures from the CCDC. The keyword "benzene" resulted in 19 structures (BENZEN 0018). From this structural set, we excluded four entries without 3-D coordinates (BENZEN 05, 08, 09, 10). The retained crystal structures were clustered according to the procedure 22 implemented in the FlexCryst program suite Figure 2. The clustering clearly revealed three clusters of structures: polymorphs II (16, 17); III (03, 04); and polymorph I (00-02, 06-07, 11-15, 18-19). Structure 12, determined at extremely high pressure with Rw =14,2%, was defined as intermediate one. The set of structures of polymorph I was subdivided into two sub-clusters: low-temperature and high-pressure modifications with considerable variation in crystal density Table 1. The density of high-pressure structures varied from 1.14 to 1.26 g/cm3 , and the density of low temperature structures varied from 1.02 to 1.12 g/cm3 . Table 1: Experimental crystal structures of benzene in the Cambridge Structure Database. The double line separates high pressure and low temperature modifications (N - neutron scattering, D hexadeutero-benzene). structure p, GPa BENZEN 03 >2.0 BENZEN 04 >2.0 BENZEN 16 0.91 BENZEN 17 0.97 BENZEN 12 1.1 BENZEN 11 0.7 BENZEN 13 0.3 BENZEN 15 0.15 BENZEN 00 0.1 BENZEN 18 0.1 BENZEN 02 0.1 BENZEN 01 0.1 BENZEN 07 0.1 BENZEN 06 0.1 BENZEN 14 0.1

T, K 298 298 295 295 296 296 296 295 218 150 270 138 123 15 4.2

d, g/cm3 1.26 1.26 1.19 1.20 1.20 1.16 1.14 1.07 1.06 1.05 1.02 1.09 1.10 1.12 1.12

polymorph III III II II I I I I I I I I I I I


remark X-Ray X-Ray X-Ray X-Ray X-Ray X-Ray X-Ray X-Ray N X-Ray X-Ray N N, D N, D N, D


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Class of "wrong" structures of benzene The required virtual crystal structures of the class "wrong" were generated by certain distortions of the cell parameters (a, b, c, α , β , γ ) and by translations and rotations of the molecules in elementary cells.

Refining of the potential for benzene After defining the two classes, we refined the initial potentials by introducing an error function, which weighted the separation of these two classes according to the potential function. The refinement was performed by an iterative minimization of the error function using the simplex method. If the energy of the minimized structure lower than the energy of the experimental structure and violates the principle of Eq. (5), then the relative error is added to the total error. If the minimized structure is above the energy of the experimental structure, the relative error function does not contribute to the total error. Using this procedure, we performed two data sets of benzene structures: polymorphs obtained at high pressure and at low temperature. These sets of structures possessed significantly different structural characteristics, including density and system of short intramolecular contacts, which were carefully analysed in. 17–19 The energy functions and parameters of the potentials (see Eq. (9)) obtained in this way are listed in Table 2 and shown in Figure 3. The "high pressure" potential is characterized as follows: mean error 2.48; largest error 6.11 for BENZEN 04; outliers 9.23%. The "low temperature" potential is characterized as follows: mean error 0.77; largest error 1.62 for BENZEN 02; outliers 4.01%. It is necessary to keep in mind two particularities of effective potentials. First, effective potentials are naturally dimensionless. The set of effective potentials is trained to assign a lower energy to the experimental structure than to any virtual structure. Thus, the obtained values should be scaled with some arbitrary factor to be compatible with commonly used numbers. Moreover, the multiplication of the energies by an arbitrary factor does not influence the property of the potentials to assign the lowest energy to the experimental structures. In this study, we scaled our potentials 8 ACS Paragon Plus Environment

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Table 2: The parameters of fine-tuned potentials Eq. (9) for two sets of benzene structures data set atoms a1 a6 a12 high pressure H,H -0.219458E+00 0.226153E+02 0.608363E+01 low temperature H,H -0.177138E+00 0.900505E+02 0.287161E+01 high pressure H,C -0.741639E-01 0.556881E+02 0.102662E+02 low temperature H,C -0.668084E-01 0.891746E+02 0.716776E+01 high pressure C,C -0.466172E-02 0.456968E+02 0.372193E+06 low temperature C,C -0.204315E+00 0.333451E+01 0.372193E+06 so that the carbon-carbon interaction had a minimum value of -0.266. This coincides approximately with the calibration to kJ/mol. Second, the physical interpretation of the terms of effective potentials does not necessarily hold. In particular, this concerns, the first (Coulomb) term, which is interpreted as the charge-charge interaction in the Van der Waals potentials. The parameter a1 can assume arbitrary values in effective potentials and does not have to coincide with a reasonable charge in a specific atom pair. Nevertheless, the interpretation of potentials is possible. As Figure 3 shows, the high pressure (HP) and low temperature (LT) potentials differ in their interactions with hydrogen-hydrogen and hydrogen-carbon. It can be concluded that low-temperature structures are governed mainly by H...H interactions, whereas the determining factor in high-pressure structures is C...H interactions. The importance of H...H interactions in the formation of low temperature polymorph was reflected in a deeper minimum and in the significant shift of this minimum to shorter distances. Reciprocally, the deeper minimum of the C...H potential was accompanied by the shift in the minimum position to shorter distances in high-pressure polymorphs. C...C interactions did not differ significantly, and in the case of low-temperature structures, they had extremely low attractive values. This data confirmed the analysis of crystal packages performed in experimental works. 17–19

Prediction of benzene structures with the refined potentials Predictions of crystal structure of benzene performed with standard (initial) potentials (CPU-time 10 hours, 700 structures) were able to generate the reasonable structure of polymorph II in rank 9 ACS Paragon Plus Environment


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110 (with energy -88.68 kJ/mol, density 1.20 g/cm3 ) ; the polymorphs I was not found at all. Moreover, the analysis of the all predicted densities, Figure 4, has reveal that they correspond to an averaged density of experimentally known structures and do not reflect the peculiarities, induced by external conditions. The predicted densities vary between 1.1-1.2 g/cm3 . Experimental data varies between 1.06 and 1.26 g/cm3 . Obviously, a successful prediction would require different potentials.for different cases. Instead, a control prediction already performed with fine-tuned HP- and LT-potentials resulted in significantly different structures. The fined-tuned HP-potential during the prediction of crystal structures produced high densities between 1.15 and 1.27 g/cm3 . This corresponded exactly to the experimental values found in high-pressure modifications (Table 1). Clustering of the structures generated with HP-potential (Figure 5) revealed the correctly predicted crystal structures within the first twelve ranks. Both the experimental crystal structures (Figure 2) and predicted crystal structures were clearly subdivided into two groups: cluster of polymorphs I and cluster of polymorphs II and III. The precise structure of polymorph III (molecular rms = 0.178)) had already been found in rank 1. The superposition of predicted and experimental crystal structures is shown in Figure 6. The fine-tuned HP potential also allowed the prediction of peculiarities in polymorph I under high pressure. For example, the structure of BENZEN 12 (see Table 1) was very well reproduced in rank 12 of the prediction (with molecular rms = 0.506). The predicted structures in ranks 11 and 13 were also placed in one cluster with the structures BENZEN 11 and 13 (numbers fit accidentally) with rms = 0.588 and 0.545, respectively. The fine-tuned LT potential produced structures with densities between 1.08 and 1.12 g/cm3 , which corresponded to the experimental values of low temperature modifications (Table 1). The correctly predicted structures were placed in ranks 13-22 (Figure 7). Even if the range of predictability was slightly lower than in the case of the HP-potentials, the fit of the experimental structure BENZEN 18 and predicted rank 22 resulted in molecular rms = 0.547 (Figure 8). The lower ranks of prediction and values of rms in the case of low temperature modifications


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could have been caused by the dataset (mixed X-ray and neutron scattering data) used in this study. However, although the dataset was sparse, is allowed the accurate reproduction of the experimental data. Moreover, it is important to note that in our approach, the relative stabilities of the polymorphs were reproduced correctly. The free energy of low-temperature modifications varied from 39.85 kJ/mol to 40.11 Kj/mol. The stability of the high-pressure polymorphs II and III varied from 33.64 kJ/mol to 33.73 kJ/mol. The stability of polymorph I under the high pressure was remarkably decreased to 33.15-33.21 kJ/mol, which could reflect the possibility of polymorphic transition from I to II under the condition of high-pressure.

Conclusions We demonstrated that an alternative method that takes into account external conditions during crystal structure prediction allowed the reproduction of temperature- and pressure-induced polymorph modification of benzene, with the correct order of stability polymorphs. The presented approach to deriving/refining the parameters of the potentials for molecular mechanics could be applied to a number of structural tasks. Depending on the data used in training, the effective potentials could include isotopic effects, temperature and pressure conditions, and specific classes of chemical compounds, such as organometallic and mixed compounds. The constant growth of the existing databases opens several possibilities for further investigations.

References (1) Dunitz, J. D. Chem. Commun 2003, 545. (2) Lommerse, J.; Motherwell, W.; Ammon, H.; Dunitz, J.; Gavezzotti, A.; Hofmann, D.; Leusen, F.; Mooij, W.; Price, S.; Schweizer, B. Acta Crystallographica Section B: Structural Science 2000, 56, 697. (3) Motherwell, W.; Ammon, H.; Dunitz, J.; Dzyabchenko, A.; Erk, P.; Gavezzotti, A.; Hof-



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mann, D.; Leusen, F.; Lommerse, J.; Mooij, W. Acta Crystallographica Section B: Structural Science 2002, 58, 647. (4) Day, G.; Motherwell, W.; Ammon, H.; Boerrigter, S.; Della Valle, R.; Venuti, E.; Dzyabchenko, A.; Dunitz, J.; Schweizer, B.; Van Eijck, B.; Hofmann, D. Acta Crystallographica Section B: Structural Science 2005, 61, 511. (5) Day, G.; Cooper, T.; Cruz-Cabeza, A.; Hejczyk, K.; Ammon, H.; Boerrigter, S.; Tan, J.; Della Valle, R.; Venuti, E.; Jose, J.; Hofmann, D. Acta Crystallographica Section B: Structural Science 2009, 65, 107. (6) Bardwell, D.; Adjiman, C.; Arnautova, Y.; Bartashevich, E.; Boerrigter, S.; Braun, D.; CruzCabeza, A.; Day, G.; Della Valle, R.; Desiraju, G.; Hofmann, D.; Hofmann, F.; Kuleshova, L. Acta Crystallographica Section B: Structural Science 2011, 67, 535. (7) Weiner, S.; Kollman, P.; Case, D.; Singh, U.; Ghio, C.; Alagona, G.; Profeta, S.; Weiner, P. Journal of the American Chemical Society 1984, 106, 765. (8) Filippini, G.; Gavezzotti, A. Acta Crystallographica Section B: Structural Science 1993, 49, 868. (9) Hofmann, D.; Apostolakis, J. Journal of Molecular Structure 2003, 647, 17. (10) Pertsin, A.; Nauchitel, V.; Kitaigorodsky, A. Mol. Cryst.Liq. Cryst. 1975, 31, 205. (11) Lyubartsev, A.; Laaksonen, A. Physical Review E 1995, 52, 3730. (12) Hofmann, D. W. M.; Kuleshova, L. N.; D’Aguanno, B. Chem.Phys.Lett. 2007, 448, 138. (13) Hofmann, D.; Förner, W.; Ladik, J. Physical Review A 1988, 37, 4429. (14) Kaminski, G.; Friesner, R.; Tirado-Rives, J.; Jorgensen, W. The Journal of Physical Chemistry B 2001, 105, 6474. (15) Car, R.; Parrinello, M. Physical Review Letters 1985, 55, 2471. 12 ACS Paragon Plus Environment

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Figure 4: Density versus energy in the crystal structures of benzene, predicted with standard potential in comparison with the densities of the experimental structures.




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Are Crystal Phase Diagrams Predictable with Force Fields? Case of

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