Abstract
This work presents a periodic extension of the GFN-FF force field for molecular crystals named mcGFN-FF. Non-covalent interactions in the force field are adjusted to reduce the systematic overbinding of the original, molecular version for molecular crystals. A diverse set of molecular crystal benchmarks for lattice energies and unit cell volumes is studied. The modified force field shows good results with a mean absolute relative deviation (MARD) of 19.9 % for lattice energies and 10.0 % for unit cell volumes. In many cases, mcGFN-FF approaches the accuracy of the GFN1-xTB quantum chemistry method which has an MARD of 18.7 % for lattice energies and 6.2 % for unit cell volumes. Further, the newly compiled mcVOL22 benchmark set is presented which features r2SCAN-D4/900 eV DFT reference volumes for molecular crystals with phosphorus-, sulfur-, and chlorine-containing compounds of various sizes. Overall, the mcGFN-FF poses an efficient tool for the optimization and energetic screening of molecular crystals containing elements up to radon.
1 Introduction
Due to the substantial growth of the synthetically accessible chemical space in recent decades [1], [2], [3], experimental exploration has become more time-consuming and expensive. Computational chemistry assists these efforts by identifying and ranking energetically favorable candidates. A prominent application in drug development is the mandatory screening for polymorphs. Crystal structure prediction software commonly uses atomistic force fields (FFs) to generate an initial ensemble of polymorphs. Therefore, accurate FF methods are essential to ensure that all relevant structures are included in the ensemble [4]. This work evaluates the newly developed periodic extension of the molecular GFN-FF force field [5] for molecular crystals named mcGFN-FF on lattice energy and unit cell volume benchmarks.
Early FFs were limited to mostly organic molecules or specific applications. Examples are DREIDING [6] for organic and a few inorganic main group elements, and AMBER [7] or CHARMM [8] for proteins. The first universal force field, parameterized for almost the entire periodic table, was introduced with UFF [9]. Being applicable to systems including elements up to radon, the GFN-FF [5] found a good balance between generality and accuracy in addition to its ease of use. These features motivated the implementation of periodic boundary conditions for the GFN-FF published by Gale et al. leading to the pGFN-FF [10]. In their publication, the authors present various adjustments needed to accurately describe various forms of solids. To stay with the original focus of GFN-FF on non-covalent interactions, the mcGFN-FF is specifically designed for molecular crystals.
The next chapter explains the theory of the changes applied in the periodic implementation. Subsequently, different benchmark studies are presented and discussed to demonstrate the performance of mcGFN-FF. The method’s versatility is showcased through optimizations of three systems with intricate electronic structures. Finally, limitations are discussed for the energetic screening of polymorphs and off-target calculations on covalently bound metal-organic frameworks.
2 Theory
To apply the GFN-FF method to molecular crystals, periodic boundary conditions are introduced using the supercell method. In this approach, copies of the unit cell are generated around it until all interactions within given cutoffs are considered.
The mcGFN-FF energy E mcGFN-FF can be split into covalent (cov) and non-covalent interactions (NCI) with a total of ten additive contributions according to
The contributions are the bonded repulsion energy
Moving from single molecules to periodic boundary conditions mainly changes the effect of non-covalent interactions. Therefore, changes to the electrostatic energy, London dispersion, hydrogen-bond energy, and non-bonded repulsion have been applied to optimize the performance of the method for molecular crystals.
The electrostatic energy
is calculated from the partial charges
where the first term converges in real space and the second in reciprocal space. This introduces the Ewald splitting parameter ξ that determines the ratio of the two terms. The golden section search (GSS) [12] is applied to obtain an optimal Ewald splitting parameter. This algorithm finds a local minimum of a strictly unimodal function within a given interval. Here, the optimal parameter is the argument of the minimum of
where Δf is the absolute value of the difference between estimates of the largest contribution in reciprocal space f rec and the largest contribution in real space f real to the electrostatic energy. The norms of the smallest reciprocal r rec and real space r real lattice vector are used to obtain the largest contributions. The local minimum is searched in the interval from 10−8 to 2.
The dispersion energy
is calculated from the dispersion coefficients
The periodic gradient and stress tensor are implemented to obtain equilibrium geometries using the L-BFGS algorithm [14].
3 Computational details
The optimizations for the mcVOL22 benchmark set were performed with the Vienna Ab Initio Simulation Package (VASP) version 6.3.2 [15–18] utilizing the r2SCAN meta-GGA density functional [19] with added D4 dispersion correction [20, 21]. The projector-augmented-wave (PAW) method [22] is applied with POTCAR files taken from the VASP library as listed in Table S6 in the Supporting Information. The calculations include plane waves up to 900 eV and use an automatic mesh of k-points as listed in Table S7. Gaussian smearing with a width of 0.01 eV is applied to aid the convergence of the electronic self-consistent field equations. Structures are considered as converged with an energy difference smaller than 1.2 × 10−4 eV, gradient norm smaller than 1.3 × 10−2 eV/Å, and stress tensor norm smaller than 5.7 × 10−2 eV/Å3. Converged values are also listed in Table S7. Final reference volumes were obtained by fitting the Birch-Murnaghan equation of state [23] to energy-volume data points. The data points were calculated with single-point calculations on scaled unit cells with scaling factors ranging from 0.98 to 1.04 with a step size of 0.02.
All calculations for the GFN-FF and mcGFN-FF were performed with a development version of xtb [24]. The final implementation will be freely available on GitHub.
Calculations with the pGFN-FF were performed with the General Utility Lattice Program (GULP) version 6.1.0 [25, 26]. In their publication, the authors present two extensions to the force field. First, the use of the Wolf sum (W) to handle Coulomb interactions and calculate robust charges, and second the damping of the three-body bond correction (dATM). Results for pGFN-FF W + dATM are taken from the original publication, while results for pGFN-FF W were calculated with the GULP program. Calculations with GFN1-xTB [27] and UFF were performed with the Amsterdam Modeling Suite (AMS) [28]. To ensure accurate results for GFN1-xTB, the k-space sampling quality was set to “very good” as implemented in the utilized DFTB engine [29].
All energies and unit cell volumes presented in the results section are calculated through free optimizations. In the boxplots discussed below, we exclude outliers defined as deviations that are either larger than the third quartile plus 1.5 times the interquartile range or smaller than the first quartile minus 1.5 times the interquartile range.
4 Results and discussion
4.1 Studied benchmarks
Herein, the various test sets for evaluating the mcGFN-FF are introduced.
The X23 benchmark [30] comprises a diverse set of 23 small organic molecules with different binding motives. In the revision of the benchmark [31], lattice energies are calculated from experimental sublimation enthalpies by correcting for harmonic vibrational contributions obtained as an average over four density-functional approximations. Reference unit cell volumes were calculated by considering thermal and zero-point energy effects.
The G60 benchmark [32] primarily consists of rigid molecular crystals and includes multiple compounds containing chlorine or nitro groups. The original publication mentions partially low accuracy of the experimental sublimation enthalpies, with an estimated standard deviation of 10 %. A revision of the benchmark [33] provides a constant correction of 2RT for the experimental values. Here, the revised values are taken as reference values. Since theoretical lattice energies allow a direct comparison of methods, results with sHF-3c [34] reference lattice energies and unit cell volumes as calculated in the revision are presented in the Supporting Information.
To gauge the performance of the force field on larger biological systems, lattice energies and unit cell volumes are evaluated for a set of eight peptide structures (PEP8) each consisting of six or seven amino acids. The initial geometries were taken from Schmitz et al. [35] and optimized with r2SCAN-3c [36].
The ICE10 [37] benchmark provides reference lattice energies and unit cell volumes for ten ice polymorphs.
Ionic liquids (ILs) have become popular in the chemical industry, among other things, due to being good reaction solvents with a low vapor pressure [38]. To test the accuracy of the mcGFN-FF on these difficult systems, back-corrected unit cell volumes of five IL crystals (IL5) were extracted from C̆ervinka [39]. The reference values were calculated from the experimental unit cell volumes by approximating the thermal expansion with the quasi-harmonic approximation using B3LYP-D3 [40, 41]/pob-TZVP-rev2 [42].
4.2 Results for the fit set
To optimize the performance of mcGFN-FF for molecular crystals, the s
8 dispersion scaling factor for interactions between atoms belonging to different molecules (intermolecular) is adjusted as well as the newly introduced scaling factors c
i
for the electrostatic energy, the hydrogen-bond energy, and the non-bonded repulsion energy. These four scaling factors were fitted simultaneously on the revised energies of the X23 benchmark [31]. In the fitting procedure, the lattice energies are obtained from single-point calculations on structures optimized with the original parameterized periodic implementation of the GFN-FF. The final optimum scaling factors are c
ES = 0.800,
Deviation from back-corrected experimental sublimation enthalpies for compounds of the revised X23 benchmark. The key includes mean absolute deviations and mean deviations for the tested methods in kcal/mol.
As a first validation of the fit, the deviations from back-corrected experimental unit cell volumes are depicted in Figure 2 as percentage values. Even though the volumes were not used in the reparameterization of the periodic GFN-FF, the MARD is substantially reduced from 10.2 % to 4.7 %. The underestimation of unit cell volumes is reduced from −9.7 % to −2.1 %. With an MARD of 12.0 % pGFN-FF W + dATM provides slightly worse volumes than GFN-FF. Similar to the lattice energies, GFN1-xTB performs slightly worse than mcGFN-FF with an MARD of 5.5%.
Deviation from back-corrected experimental unit cell volumes of the revised X23 benchmark. The deviations are given as percentage values relative to the reference.
4.3 Molecular crystal volumes benchmark set
In order to cover a wider variety of atom types and structural properties we present the mcVOL22 benchmark set. The set provides r2SCAN-D4 reference volumes for 22 molecular crystals that contain phosphorus, sulfur, and chlorine atoms. Illustrations of representative systems are shown in Figure 3 and the whole set is depicted in the Supporting Information. The deviations from the reference volumes are depicted in Figure 4 for the tested methods. The slight overbinding of the crystals with a mean relative deviation (MRD) of −4.4 % for GFN1-xTB is consistent with the results on the X23 benchmark. The higher-level method only shows one larger deviation for mcv14. With an MRD of −7.7 % too small cell volumes, mcGFN-FF is the best-performing force field on this benchmark, followed by UFF with 8.0 %. All GFN-FF type methods show the largest deviation for mcv02, a small rigid system including phosphorus. The universal force field has the largest deviation of 22 % for mcv01 which contains Cl− counter ions.
Selected structures from the mcVOL22 benchmark for molecular crystal volumes.
Deviation from r2SCAN-D4 unit cell volumes. The key includes mean absolute relative deviations and mean relative deviations for the tested methods.
4.4 Statistical evaluation
The results for the presented benchmarks are depicted in Figure 5. The MARD for the energies does not include the X23 benchmark since it was used to fit the mcGFN-FF. Due to small values for lattice energies and unit cell volumes, the ICE10 benchmark has a high impact on the MARD and therefore values without this benchmark are provided as indicated. The boxplots show that the adjustments in the mcGFN-FF consistently reduce the systematic overbinding present in an unmodified periodic implementation of GFN-FF. Overall, our reparameterization improves the results significantly, with an exception for the ICE10 benchmark. In this case, the original method already provides accurate results, leading to larger errors in the adjusted variant due to the reduced stabilization of the crystals. As indicated by the MARD, mcGFN-FF performs comparable to GFN1-xTB, while GFN-FF shows similar results as pGFN-FF. Though the UFF provides accurate cell volumes, it shows the largest standard deviation overall.
Boxplots for deviations in lattice energies and relative deviations in unit cell volumes for various test sets. The provided MARDs are calculated as averages over the given benchmarks. Since the X23 energies are used to fit the mcGFN-FF, they are excluded from the corresponding MARD for all methods. Maximum and minimum deviations do not include outliers.
4.5 Showcases
In order to demonstrate the versatility of the mcGFN-FF we present three systems with intricate electronic structures, including a system with a molybdenum-silicon triple bond [43], an osmium system with B
that scales with the optimization step i.
Illustrations of systems with intricate electronic structures that were optimized with mcGFN-FF under periodic boundary conditions. The experimental structure is depicted as a blue overlay. To make the structures easier to recognize, hydrogen atoms that are bound to carbon are hidden and not the entire unit cell is shown. The RMSD and volume deviation are calculated between the entire unit cell of the experiment and the optimized structure from mcGFN-FF.
The structures were downloaded from the Cambridge Structural Database (CSD) with identifiers LIDWED, DIXRIN, and AXOBAT respectively. The metallasylidyne system (a) shows the smallest structural deviation with a root mean squared deviation (RMSD) of 0.53 Å from the experimental crystal structure and a −8.2 % smaller unit cell. After optimization, the silicon-molybdenum triple bond is 0.18 Å longer with a length of 2.47 Å and the silicon-molybdenum single bond becomes 0.15 Å longer with a length of 2.6 Å. The osmium system (b) has slightly larger deviations with an RMSD of 0.55 Å and a −12.2 % smaller unit cell. Finally, the siladodecahedrane system (c) has an RMSD of 0.91 Å and a −20.3 % smaller unit cell. For systems b and c the inclusion of chloroform molecules in the crystal structures is an additional challenge. System a was optimized in 43.7 min and 608 optimizations steps, system b was optimized in 12.5 min and 313 steps, and system c was optimized in 23.7 min and 318 steps. Even though there are noticeable changes in geometry, it is remarkable that the force field can optimize these systems at all without major structural distortions compared to experiment.
4.6 Polymorph ranking on ROY
Six polymorphs of the well-studied 5-methyl-2-[(2-nitrophenyl)amino]-3-thiophenecarbonitrile crystal (dubbed ROY) [47], [48], [49], [50], [51], [52], [53] are ranked energetically with FF methods and GFN1-xTB. The experimental reference is taken from Ref [54]. The polymorphs are named yellow prisms (Y), red prisms (R), orange plates (OP), orange needles (ON), yellow needles (YN), and orange-red plates (ORP). As all polymorphs lie within an energy range of 1.0 kcal/mol, the energetic ranking becomes a tough challenge for all tested methods. The results for this “real-life” application case are depicted in Figure 7. The enthalpic contributions are negligible for comparison to FF methods, as shown by high-level calculations from Beran et al [47]. All force fields assign a rank of two to the most stable form (Y), while GFN1-xTB assigns a rank of four. The widening of the energy range ΔE range = E max − E min is similar for the GFN-FF related methods with approximately 10 kcal/mol. For GFN1-xTB the widening is smaller with 5.7 kcal/mol and significantly higher for UFF with 19.1 kcal/mol. With a nitro group and a cyano group, the molecule itself is a challenging system let alone energetically ranking its polymorphs with low-cost methods. Therefore, the results should only be considered relative to each other.
Calculated energy difference between the most stable form (Y) and other ROY polymorphs plotted against experimental enthalpy difference. The molecular structure of ROY is illustrated in the figure.
4.7 POLY59
The POLY59 benchmark [55] comprises five sets of eleven to fifteen polymorph structures, numbered from 22 through 26 and each set has one (or five for set 23) reference structure that should yield the lowest energy. To evaluate this benchmark, all sets of structures have to be sorted energetically and the rank of the reference structure is summed up over the five sets. The ranking is performed via single-point calculations on given TPSS-D3 structures. Since set 23 has five references the best score is 19 and the worst score is 109. In order to provide an additional reference point, assigning the rank of the reference structure with an equal distribution yields a summed score of 64. Results for the tested FFs as well as results from the benchmark publication are listed in Table 1. With a score (∑) of 42 and 43 respectively, pGFN-FF W and GFN-FF are the best-performing FFs. Though mcGFN-FF is just slightly worse than HF-3c, with a score of 67 compared to 65, the ranking is worse than in the unmodified version although we note a slightly improved energy range. The UFF is not able to work reliably on DFT structures, as reflected in the score of 84 and the immense average energy range between the polymorph with the lowest and the highest energy
Rank of the energetically lowest polymorph for the five sets according to different methods. The score (∑) for each method is calculated as the sum over the different set ranks. For set 23 there are 5 “lowest” polymorphs α − ϵ. The average energy range from the energetically lowest to the highest polymorph
| Rank | 22 | 23α | 23β | 23γ | 23δ | 23ϵ | 24 | 25 | 26 | Score ∑ |
|
||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| mcGFN-FF | 10 | 10 | 1 | 14 | 8 | 13 | 4 | 2 | 5 | 67 | (57) | 21.8 | (20.4) |
| GFN-FF | 5 | 5 | 2 | 7 | 4 | 8 | 3 | 3 | 6 | 43 | (44) | 22.4 | (27.9) |
| pGFN-FF W | 4 | 5 | 1 | 7 | 4 | 8 | 4 | 3 | 6 | 42 | (43) | 24.6 | (28.3) |
| UFF | 6 | 6 | 11 | 12 | 10 | 15 | 9 | 4 | 11 | 84 | (48) | 549.6 | (24.6) |
| HF-3c | 5 | 49 | – | – | – | – | 1 | 5 | 8 | 65 | – | 4.9 | – |
| DFTB3-D3 | 7 | 52 | – | – | – | – | 1 | 6 | 8 | 74 | – | 7.0 | – |
| M06L | 1 | 41 | – | – | – | – | 1 | 1 | 7 | 51 | – | 3.5 | – |
| PBE-D3 | 1 | 21 | – | – | – | – | 1 | 1 | 1 | 25 | – | 2.5 | – |
Though the accurate energetic ranking of polymorphs with FF methods is not possible yet, testing the methods for the generation of crystal structure ensembles could be worthwhile. In addition, mcGFN-FF may be useful for initial screening steps in multi-level computational workflows.
4.8 Off-target evaluation on 12 metal-organic frameworks (MOF12)
Though mcGFN-FF was specifically designed for molecular crystals it is evaluated for 12 covalently bound metal-organic frameworks (MOFs) using PBE-D3 reference cell volumes [56]. Note that some optimizations with GFN-FF had to be restarted after deleting the topology file or needed the construction of supercells to ensure convergence. The frameworks each include one of the following elements: silver, cadmium, cobalt, copper, dysprosium, iron, lithium, zinc, or samarium. Figure 8 shows boxplots of the relative deviations from PBE-D3 unit cell volumes for the tested methods. Even though mcGFN-FF is not developed for these systems, it performs comparable to GFN1-xTB.
Boxplots of the relative deviations from PBE-D3 unit cell volumes for 12 MOFs.
5 Conclusions
A periodic extension of the GFN-FF for molecular crystals, named mcGFN-FF has been presented. To introduce periodic boundary conditions into the existing implementation, the Ewald summation is applied to converge the electrostatic energy. Herein, the Ewald splitting parameter is determined via the golden section search algorithm. In mcGFN-FF the intermolecular non-covalent energy contributions are adjusted, to reduce the systematic overbinding of molecular crystals observed in the original method.
While the original periodic implementation already shows good and robust results across the diversified benchmarks, mcGFN-FF provides significantly more accurate lattice energies and unit cell volumes in almost all cases. The only exception is the ICE10 benchmark. Here, the unmodified implementation already yields accurate results and the reduced non-covalent binding in mcGFN-FF leads to larger errors.
In principle, the mcGFN-FF can be applied to any periodic system, but the focus on molecular crystals mandates treating other systems with caution. Showing comparable results as low-cost QM methods for polymorph ranking, we suggest the use of mcGFN-FF in the first steps of crystal structure prediction workflows. With promising results on the transition metal showcases, the method poses a useful tool for gathering first insights on intriguing chemical problems.
To this end, precompiled binaries as well as the source code will be freely available on the corresponding Github website (https://github.com/grimme-lab/xtb).
6 Supporting Information
Detailed Benchmark results, illustrations of the mcVOL22 systems, and used POTCAR files for VASP calculations are given as supplementary material and are available online (https://doi.org/10.1515/znb-2023-0088). Furthermore, geometry files, used fit data, and reference values for the mcVOL22 and PEP8 benchmarks are available as a ZIP file.
Dedicated to Professor Thomas Bredow of the University of Bonn on the occasion of his 60th birthday.
Acknowledgment
The authors thank S. Ehlert for providing the code for the L-BFGS algorithm, U. Huniar (3DS) and J. M. Mewes for the development version of TURBOMOLE including the periodic r2SCAN-3c implementation, and T. Bredow for assistance with the VASP calculations. Furthermore, the authors thank S. Spicher, S. Ehlert, T. Gasevic, and M. Bursch for many helpful discussions.
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Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: The authors state no conflict of interest.
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Research funding: None declared.
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Data availability: The raw data can be obtained on request from the corresponding author.
References
1. Medina-Franco, J. L., Chávez-Hernández, A. L., López-López, E., Saldívar-González, F. I. Chemical multiverse: an expanded view of chemical space. Mol. Inf. 2022, 41, 12, 2200116; https://doi.org/10.1002/minf.202200116.Search in Google Scholar PubMed PubMed Central
2. Campos, K. R., Coleman, P. J., Alvarez, J. C., Dreher, S. D., Garbaccio, R. M., Terrett, N. K., Tillyer, R. D., Truppo, M. D., Parmee, E. R. The importance of synthetic chemistry in the pharmaceutical industry. Science 2019, 363, 9, eaat0805; https://doi.org/10.1126/science.aat0805.Search in Google Scholar PubMed
3. Theerthagiri, J., Karuppasamy, K., Lee, S. J., Shwetharani, R., Kim, H.-S., Pasha, S. K. K., Ashokkumar, M., Choi, M. Y. Fundamentals and comprehensive insights on pulsed laser synthesis of advanced materials for diverse photo- and electrocatalytic applications. Light: Sci. Appl. 2022, 11, 47, 250; https://doi.org/10.1038/s41377-022-00904-7.Search in Google Scholar PubMed PubMed Central
4. Bowskill, D. H., Sugden, I. J., Konstantinopoulos, S., Adjiman, C. S., Pantelides, C. C. Crystal structure prediction methods for organic molecules: state of the art. Annu. Rev. Chem. Biomol. Eng. 2021, 12, 593–623; https://doi.org/10.1146/annurev-chembioeng-060718-030256.Search in Google Scholar PubMed
5. Spicher, S., Grimme, S. Robust atomistic modeling of materials, organometallic, and biochemical systems. Angew. Chem. Int. Ed. 2020, 59, 15665–15673; https://doi.org/10.1002/anie.202004239.Search in Google Scholar PubMed PubMed Central
6. Mayo, S. L., Goddard, W. A., Olafson, B. D. DREIDING: a generic force field for molecular simulations. J. Phys. Chem. 1990, 94, 8897–8909; https://doi.org/10.1021/j100389a010.Search in Google Scholar
7. Weiner, S. J., Kollman, P. A., Nguyen, D. T., Case, D. A. An all atom force field for simulations of proteins and nucleic acids. J. Comput. Chem. 1986, 7, 230–252; https://doi.org/10.1002/jcc.540070216.Search in Google Scholar PubMed
8. Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J., Swaminathan, S., Karplus, M. CHARMM: a program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 1983, 4, 187–217; https://doi.org/10.1002/jcc.540040211.Search in Google Scholar
9. Rappe, A. K., Casewit, C. J., Colwell, K. S., Goddard, W. A. I., Skiff, W. M. UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J. Am. Chem. Soc. 1992, 114, 10024–10035; https://doi.org/10.1021/ja00051a040.Search in Google Scholar
10. Gale, J. D., LeBlanc, L. M., Spackman, P. R., Silvestri, A., Raiteri, P. A universal force field for materials, periodic GFN-FF: implementation and examination. J. Chem. Theory Comput. 2021, 17, 7827–7849; https://doi.org/10.1021/acs.jctc.1c00832.Search in Google Scholar PubMed
11. Ewald, P. P. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 1921, 369, 253–287; https://doi.org/10.1002/andp.19213690304.Search in Google Scholar
12. Kiefer, J. Sequential minimax search for a maximum. Proc. Am. Math. Soc. 1953, 4, 502–506; https://doi.org/10.1090/s0002-9939-1953-0055639-3.Search in Google Scholar
13. Becke, A. D., Johnson, E. R. A density-functional model of the dispersion interaction. J. Chem. Phys. 2005, 123, 9, 154101; https://doi.org/10.1063/1.2065267.Search in Google Scholar PubMed
14. Liu, D. C., Nocedal, J. On the limited memory BFGS method for large scale optimization. Math. Program. 1989, 45, 503–528; https://doi.org/10.1007/bf01589116.Search in Google Scholar
15. Kresse, G., Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558–561; https://doi.org/10.1103/physrevb.47.558.Search in Google Scholar PubMed
16. Kresse, G., Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6, 15–50; https://doi.org/10.1016/0927-0256(96)00008-0.Search in Google Scholar
17. Kresse, G., Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 1996, 54, 11169–11186; https://doi.org/10.1103/physrevb.54.11169.Search in Google Scholar PubMed
18. Kresse, G., Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 1758–1775; https://doi.org/10.1103/physrevb.59.1758.Search in Google Scholar
19. Furness, J. W., Kaplan, A. D., Ning, J., Perdew, J. P., Sun, J. Accurate and numerically efficient r2SCAN meta-generalized gradient approximation. J. Phys. Chem. Lett. 2020, 11, 8208–8215; https://doi.org/10.1021/acs.jpclett.0c02405.Search in Google Scholar PubMed
20. Caldeweyher, E., Ehlert, S., Hansen, A., Neugebauer, H., Spicher, S., Bannwarth, C., Grimme, S. A generally applicable atomic-charge dependent London dispersion correction. J. Chem. Phys. 2019, 150, 19, 154122; https://doi.org/10.1063/1.5090222.Search in Google Scholar PubMed
21. Caldeweyher, E., Mewes, J.-M., Ehlert, S., Grimme, S. Extension and evaluation of the D4 London-dispersion model for periodic systems. Phys. Chem. Chem. Phys. 2020, 22, 8499–8512; https://doi.org/10.1039/d0cp00502a.Search in Google Scholar PubMed
22. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953–17979; https://doi.org/10.1103/physrevb.50.17953.Search in Google Scholar PubMed
23. Birch, F. Finite elastic strain of cubic crystals. Phys. Rev. 1947, 71, 809–824; https://doi.org/10.1103/physrev.71.809.Search in Google Scholar
24. Bannwarth, C., Caldeweyher, E., Ehlert, S., Hansen, A., Pracht, P., Seibert, J., Spicher, S., Grimme, S. Extended tight-binding quantum chemistry methods. WIREs Comput. Mol. Sci. 2021, 11, 49, e1493; https://doi.org/10.1002/wcms.1493.Search in Google Scholar
25. Gale, J. D. GULP: a computer program for the symmetry-adapted simulation of solids. J. Chem. Soc., Faraday Trans. 1997, 93, 629–637; https://doi.org/10.1039/a606455h.Search in Google Scholar
26. Gale, J. D., Rohl, A. L. The general utility lattice program (GULP). Mol. Simul. 2003, 29, 291–341; https://doi.org/10.1080/0892702031000104887.Search in Google Scholar
27. Grimme, S., Bannwarth, C., Shushkov, P. A robust and accurate tight-binding quantum chemical method for structures, vibrational frequencies, and noncovalent interactions of large molecular systems parametrized for all spd-block elements (Z = 1–86). J. Chem. Theory Comput. 2017, 13, 1989–2009; https://doi.org/10.1021/acs.jctc.7b00118.Search in Google Scholar PubMed
28. Rüger, R., Franchini, M., Trnka, T., Yakovlev, A., van Lenthe, E., van Vuren, T., Klumpers, B., Soini, T. Theoretical Chemistry; AMS 2022; Vrije Universiteit: Amsterdam, The Netherlands. http://www.scm.com.Search in Google Scholar
29. Rüger, R., Yakovlev, A., Philipsen, P., Borini, S., Melix, P., Oliveira, A., Franchini, M., van Vuren, T., Soini, T., de Reus, M., Asl, M. G., Teodoro, T. Q., McCormack, D., Patchkovskii, S., Heine, T. Theoretical Chemistry; AMS DFTB 2023; SCM; Vrije Universiteit: Amsterdam, The Netherlands. http://www.scm.com.Search in Google Scholar
30. Reilly, A. M., Tkatchenko, A. Understanding the role of vibrations, exact exchange, and many-body van der Waals interactions in the cohesive properties of molecular crystals. J. Chem. Phys. 2013, 139, 12, 024705; https://doi.org/10.1063/1.4812819.Search in Google Scholar PubMed
31. Dolgonos, G. A., Hoja, J., Boese, A. D. Revised values for the X23 benchmark set of molecular crystals. Phys. Chem. Chem. Phys. 2019, 21, 24333–24344; https://doi.org/10.1039/c9cp04488d.Search in Google Scholar PubMed
32. Maschio, L., Civalleri, B., Ugliengo, P., Gavezzotti, A. Intermolecular interaction energies in molecular crystals: comparison and agreement of localized møller–Plesset 2, dispersion-corrected density functional, and classical empirical two-body calculations. J. Phys. Chem. A 2011, 115, 11179–11186; https://doi.org/10.1021/jp203132k.Search in Google Scholar PubMed
33. Cutini, M., Civalleri, B., Corno, M., Orlando, R., Brandenburg, J. G., Maschio, L., Ugliengo, P. Assessment of different quantum mechanical methods for the prediction of structure and cohesive energy of molecular crystals. J. Chem. Theory Comput. 2016, 12, 3340–3352; https://doi.org/10.1021/acs.jctc.6b00304.Search in Google Scholar PubMed
34. Caldeweyher, E., Brandenburg, J. G. Simplified DFT methods for consistent structures and energies of large systems. J. Phys.: Condens. Matter 2018, 30, 40, 213001; https://doi.org/10.1088/1361-648x/aabcfb.Search in Google Scholar
35. Schmitz, S., Seibert, J., Ostermeir, K., Hansen, A., Göller, A. H., Grimme, S. Quantum chemical calculation of molecular and periodic peptide and protein structures. J. Phys. Chem. B 2020, 124, 3636–3646; https://doi.org/10.1021/acs.jpcb.0c00549.Search in Google Scholar PubMed
36. Grimme, S., Hansen, A., Ehlert, S., Mewes, J.-M. r2SCAN-3c: a “Swiss army knife” composite electronic-structure method. J. Chem. Phys. 2021, 154, 18, 064103; https://doi.org/10.1063/5.0040021.Search in Google Scholar PubMed
37. Brandenburg, J. G., Maas, T., Grimme, S. Benchmarking DFT and semiempirical methods on structures and lattice energies for ten ice polymorphs. J. Chem. Phys. 2015, 142, 11, 124104; https://doi.org/10.1063/1.4916070.Search in Google Scholar PubMed
38. Singh, S. K., Savoy, A. W. Ionic liquids synthesis and applications: an overview. J. Mol. Liq. 2020, 297, 23, 112038; https://doi.org/10.1016/j.molliq.2019.112038.Search in Google Scholar
39. Červinka, C. Tuning the quasi-harmonic treatment of crystalline ionic liquids within the density functional theory. J. Comput. Chem. 2022, 43, 448–456; https://doi.org/10.1002/jcc.26804.Search in Google Scholar PubMed
40. Lee, C., Yang, W., Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785–789; https://doi.org/10.1103/physrevb.37.785.Search in Google Scholar PubMed
41. Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652; https://doi.org/10.1063/1.464913.Search in Google Scholar
42. Laun, J., Bredow, T. BSSE-corrected consistent Gaussian basis sets of triple-zeta valence with polarization quality of the fifth period for solid-state calculations. J. Comput. Chem. 2022, 43, 839–846; https://doi.org/10.1002/jcc.26839.Search in Google Scholar PubMed
43. Ghana, P., Arz, M. I., Chakraborty, U., Schnakenburg, G., Filippou, A. C. Linearly two-coordinated silicon: transition metal complexes with the functional groups M ≡Si-M and M=Si=M. J. Am. Chem. Soc. 2018, 140, 7187–7198; https://doi.org/10.1021/jacs.8b02902.Search in Google Scholar PubMed
44. Zhu, Q., Zhu, C., Deng, Z., He, G., Chen, J., Zhu, J., Xia, H. Synthesis and characterization of osmium polycyclic aromatic complexes via nucleophilic reactions of osmapentalyne. Chin. J. Chem. 2017, 35, 628–634; https://doi.org/10.1002/cjoc.201600478.Search in Google Scholar
45. Zhu, C., Zhu, Q., Fan, J., Zhu, J., He, X., Cao, X.-Y., Xia, H. A metal-bridged tricyclic aromatic system: synthesis of osmium polycyclic aromatic complexes. Angew. Chem. Int. Ed. 2014, 53, 6232–6236; https://doi.org/10.1002/anie.201403245.Search in Google Scholar PubMed
46. Bamberg, M., Bursch, M., Hansen, A., Brandl, M., Sentis, G., Kunze, L., Bolte, M., Lerner, H.-W., Grimme, S., Wagner, M. [Cl@Si20H20]-: parent siladodecahedrane with endohedral chloride ion. J. Am. Chem. Soc. 2021, 143, 10865–10871; https://doi.org/10.1021/jacs.1c05598.Search in Google Scholar PubMed
47. Beran, G. J. O., Sugden, I. J., Greenwell, C., Bowskill, D. H., Pantelides, C. C., Adjiman, C. S. How many more polymorphs of ROY remain undiscovered. Chem. Sci. 2022, 13, 1288–1297; https://doi.org/10.1039/d1sc06074k.Search in Google Scholar PubMed PubMed Central
48. Yu, L., Stephenson, G. A., Mitchell, C. A., Bunnell, C. A., Snorek, S. V., Bowyer, J. J., Borchardt, T. B., Stowell, J. G., Byrn, S. R. Thermochemistry and conformational polymorphism of a hexamorphic crystal system. J. Am. Chem. Soc. 2000, 122, 585–591; https://doi.org/10.1021/ja9930622.Search in Google Scholar
49. Lévesque, A., Maris, T., Wuest, J. D. ROY reclaims its crown: new ways to increase polymorphic diversity. J. Am. Chem. Soc. 2020, 142, 11873–11883; https://doi.org/10.1021/jacs.0c04434.Search in Google Scholar PubMed
50. Chen, S., Xi, H., Yu, L. Cross-nucleation between ROY polymorphs. J. Am. Chem. Soc. 2005, 127, 17439–17444; https://doi.org/10.1021/ja056072d.Search in Google Scholar PubMed
51. Tan, M., Shtukenberg, A. G., Zhu, S., Xu, W., Dooryhee, E., Nichols, S., Ward, M. D., Kahr, B., Zhu, Q. ROY revisited, again: the eighth solved structure. Faraday Discuss. 2018, 211, 477–491; https://doi.org/10.1039/c8fd00039e.Search in Google Scholar PubMed
52. Gushurst, K. S., Nyman, J., Boerrigter, S. X. M. The PO13 crystal structure of ROY. CrystEngComm 2019, 21, 1363–1368; https://doi.org/10.1039/c8ce01930d.Search in Google Scholar
53. Chen, S., Guzei, I. A., Yu, L. New polymorphs of ROY and new record for coexisting polymorphs of solved structures. J. Am. Chem. Soc. 2005, 127, 9881–9885; https://doi.org/10.1021/ja052098t.Search in Google Scholar PubMed
54. Yu, L. Polymorphism in molecular solids: an extraordinary system of red, orange, and yellow crystals. Acc. Chem. Res. 2010, 43, 1257–1266; https://doi.org/10.1021/ar100040r.Search in Google Scholar PubMed
55. Brandenburg, J. G., Grimme, S. Organic crystal polymorphism: a benchmark for dispersion-corrected mean-field electronic structure methods. Acta Crystallogr. 2016, B72, 502–513; https://doi.org/10.1107/s2052520616007885.Search in Google Scholar
56. Nazarian, D., Ganesh, P., Sholl, D. S. Benchmarking density functional theory predictions of framework structures and properties in a chemically diverse test set of metal–organic frameworks. J. Mater. Chem. A 2015, 3, 22432–22440; https://doi.org/10.1039/c5ta03864b.Search in Google Scholar
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