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Article

What Is More Important When Calculating the Thermodynamic Properties of Organic Crystals, Density Functional, Supercell, or Energy Second-Order Derivative Method Choice?

by
Aleksandr S. Dubok
1,2 and
Denis A. Rychkov
1,3,*
1
Institute of Solid State Chemistry and Mechanochemistry SB RAS, 630090 Novosibirsk, Russia
2
Laboratory of Physicochemical Fundamentals of Pharmaceutical Materials, Novosibirsk State University, 2 Pirogova Str., 630090 Novosibirsk, Russia
3
SRF “SKIF”, Boreskov Institute of Catalysis, 630559 Koltsovo, Russia
*
Author to whom correspondence should be addressed.
Crystals 2025, 15(3), 274; https://doi.org/10.3390/cryst15030274
Submission received: 28 January 2025 / Revised: 13 March 2025 / Accepted: 13 March 2025 / Published: 16 March 2025
(This article belongs to the Special Issue Computational Research on Crystals)
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Abstract

:
Calculation of second-order derivatives of energy using the DFT method is a valuable approach for the estimation of both the thermodynamical and mechanical properties of organic crystals from the first principles. This type of calculation requires specification of several computational parameters, including the functional, supercell, and method of phonon calculations. Nevertheless, the importance of these parameters is presented in the literature very modestly. In this work, we demonstrate the influence of these computational parameters on the accuracy of calculated second-order derivatives using the practical example of pyrazinamide polymorphs, including the plastically bending α form and the β, γ, and brittle δ form. The effects of the settings used on the resulting enthalpies of the polymorphic modifications of pyrazinamide are compared: supercell setting (primitive cell vs. appropriate supercell) has a much stronger impact than functional (PBE-D3BJ vs. Hamada rev-vdW-DF2) which in turn affects results significantly more than the method for second-order derivative computation (FD vs. DFPT approach). Finally, we propose some suggestions for choosing the right settings for calculating second-order derivatives for molecular crystals.

1. Introduction

Polymorphism is a widespread phenomenon that occurs in both organic and inorganic crystals [1,2,3]. The polymorphism of organic crystals has received considerable attention, mainly in the pharmaceutical field, due to its practical role [4,5,6,7,8]. Depending on the structure of the crystal lattice, one and the same substance may have not only different bioavailability [9,10,11] but also different solubility rates [12,13], shelf-life stability [14,15,16,17,18,19], and manufacturing parameters [20,21,22]. In addition, different polymorphic modifications can have different mechanical properties, which may directly affect the production of drug forms, including tableting and phase transitions under pressure [23,24,25,26].
While the prediction of the thermodynamic stability and solubility of organic crystals [27,28,29,30] from first principles [31,32,33,34,35,36,37] is an important unsolved problem in materials science, with it being performed thus far at the first stage using mainly force field methods [38,39,40], additional examination of their computation using non-empirical DFT methods is required. Second-order derivative calculations using the DFT method have proven to be a valuable approach for the estimation of both the thermodynamical stability and the mechanical properties from the first principles [41,42,43]. This can be achieved using phonon modes and elastic tensor analysis performed readily using the second-order derivative calculation results. However, second-order derivative calculations are accompanied by several practical challenges. Firstly, the functional used in the computation process should be optimal in terms of accuracy and computational cost ratio and suitable for the chosen method of phonon calculation [44,45]. Secondly, second-order derivative calculations usually require the construction of a supercell to account for fluctuations disproportionate to the primitive cell [46,47]. The rule of thumb states that supercell vectors should be of the order of 1 to 1.5 nanometers. Nevertheless, the use of large supercells greatly increases the demand for computational resources. Finally, there are two different approaches to calculating second-order derivatives: a finite differences approach and the density functional perturbation theory method [48]. In a finite differences approach (FD), the second-order force constants are computed using finite differences of the forces when each ion is displaced in each independent direction. The number and step size of every displacement should be selected carefully. This may increase the amount of displacement required to calculate the second derivatives and also result in a sufficient increase in the computational cost. Density-functional perturbation theory (DFPT) provides a way to compute the second-order linear response to ionic displacement analytically. This approach requires the usage of non-local dispersion correction functionals but does not require the construction of supercells if not only zone-center (Γ-point) frequencies are calculated. However, not all computational software implements the calculations of phonon dispersion at different q points, including the so-called “gold standard” Vienna ab initio simulation package (VASP 5.4.4). Calculation of vibrations at q ≠ 0 usually requires increased computational cost compared to the q = 0 computations for the primitive cell but should save computational resources compared to supercell usage [49]. A brief overview of DFPT method implementations in various solid-state calculation software is given in Table 1.
Fundamental computational parameters such as energy cutoff, k-points sampling, and pseudopotential quality are explicitly discussed in the classical literature [50,51,52]. Nevertheless, the question of the importance of the functional, supercell, and method of phonon calculations is presented in the literature very modestly and mainly in the manuals of the abovementioned software, several books, and professional community discussions. In this work, we demonstrate the influence of these computational parameters on the accuracy of calculated second-order derivatives using the practical example of pyrazinamide polymorphs (Figure 1) [53,54,55,56].
Recently, the relative stability of pyrazinamide polymorphs over a wide temperature range has been studied using experimental and computational methods [57,58,59,60,61,62,63,64,65]. Differential scanning calorimetry and vapor pressure measurements were used to describe the polymorphs’ stability in the provided experiments [57,58,64]. Electronic structure and phonon calculations using a finite differences approach were performed to determine the temperature dependence of the thermodynamic properties of α, β, γ, and δ pyrazinamide [65]. It was confirmed that the α form was a stable polymorph at room temperature, whereas the β form appeared to be a low-temperature form instead of the previously suggested δ form according to the latest experiments and our previous calculations (Figure 2) [64,65].
Due to the significant amount of obtained data on pyrazinamide polymorph stability in a wide temperature range, it is a convenient example to estimate the influence of various computational parameters on the accuracy/computational cost ratio for practical calculations.

2. Materials and Methods

2.1. Crystallographic Structures

The crystal structures of pyrazinamide polymorphs were chosen from the CCDC database with the lowest R-factors (<4%) solved and refined at 100K temperature: PYRZIN22 (α) [60], PYRZIN18 (β) [57], PYRZIN19 (γ) [57], and PYRZIN16 (δ) [66]. The disordered γ form was treated as two independent structures with “A” and “B” positions only, and energies were calculated further by summing energies in the ratio of 0.866 and 0.134, respectively. These structures were used as starting guesses for all subsequent calculations. All pyrazinamide structures were converted to VASP format using the CIF2Cell code [67] for periodic DFT calculations.

2.2. Periodic DFT Calculations

All periodic DFT calculations were performed using the Vienna ab initio simulation package (VASP 5.4.4) [68,69,70,71] using either the PBE [72] or rev-vdw-DF2 [73] functionals for all cases after functional and dispersion correction benchmarking. A plane-wave basis set with a kinetic energy cutoff of 550 eV and projector augmented wave (PAW) atomic pseudopotentials [74,75] were employed during the DFT calculations. Dense k-point Monkhorst–Pack meshes [76] of 5 × 2 × 1, 1 × 6 × 2, 3 × 7 × 2, and 4 × 4 × 2 were used for the α, β, γ (both “A” and “B”), and δ forms, respectively. A 4 × 2 × 1 mesh was also used for functional and dispersion correction scheme benchmarking to speed up calculations. Typical energy convergence criteria of 1 meV/atom and additional energy criteria of total 0.1 kJ/mol were used to achieve tighter convergence for structural optimization. Gaussian smearing was chosen (ISMEAR = 0) with 0.1 eV width smearing (SIGMA = 0.1). Dispersion correction was used for electronic structure calculations: Grimme D3 with the Becke–Johnson damping function (D3BJ) [77] for the PBE functional and nonlocal correlation energy EcNL for the rev-vdW-DF2 functional of Hamada [73], also known as vdW-DF2-B86R. Armiento–Mattsson (AM05) [78,79,80], Perdew–Burke-Ernzerhof (PBE) [72], revised PBE for solids (PBEsol) [81], Perdew–Wang (PW91) [82], revised PBE from Zhang and Yang (revPBE) [83], and the revised PBE from Hammer et al. (RPBE) [84] density functionals were also employed for studying the unit cell volume deviation of pyrazinamide polymorphic forms. Likewise, additional dispersion correction schemes such as the dDsC dispersion correction method (dDsC) [85,86], the DFT-D2 method of Grimme (DFT-D2) [87], the DFT-D3 method of Grimme with the zero-damping function (DFT-D3) [88], the DFT-D3 method with the Becke–Johnson damping function (DFT-D3(BJ)) [77], the many-body dispersion energy method (MBD@rsSCS) [89,90], no dispersion correction, the Tkatchenko–Scheffler method (TS) [91], and the Tkatchenko–Scheffler method with iterative Hirshfeld partitioning (TS/HI) [92,93] were also employed.
All structures were fully optimized, both the unit cell and ion positions (ISIF = 3). The temperature dependence of Gibbs free energy was treated by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite differences approach (IBRION = 6) for the PBE functional with D3BJ dispersion correction, and both the finite differences approach (IBRION = 6) and the density functional perturbation theory approach (IBRION = 8) for the rev-vdW-DF2 functional. Symmetry was used to reduce the number of displacements for a finite differences approach. A full list of the used symmetry parameters for this and previous [65] work is provided in Table S1.
The second-order derivatives of the total energy with respect to the position of the ions were computed using the default primitive cells as well as supercells of 3 × 2 × 1 (α), 1 × 3 × 1 (β), 2 × 3 × 1 (γ), and 2 × 2 × 2 (δ) constructed to increase the accuracy of the performed calculations (Figure S1). The k-point meshes were reduced for calculations using the supercells in order to preserve the chosen k-point mesh density (2 × 1 × 1, 1 × 2 × 2, 2 × 2 × 2, and 2 × 2 × 1 for the α, β, γ, and δ form supercells, respectively).

2.3. Phonon Calculations

The dynamical matrix (constructed and diagonalized), and the phonon modes and frequencies of each system reported in the OUTCAR file were further extracted to proceed with the Phonopy 2.17.1 package [94,95] to calculate the zero-point energy correction term (ZPE), temperature-dependent vibrational energy, entropy, and free energy. Mesh samplings of the reciprocal space used for the calculation of thermal properties in the Phonopy 2.17.1 package were equal to the meshes used for electronic calculations in VASP.

3. Results

3.1. Electronic Structure Energies and Unit Cell Parameters

Depending on the chosen calculation method, the absolute values of the energies and therefore the relative values of the obtained energies of polymorphic modifications are different [96,97,98]. Therefore, it is important to assess the effect of the different calculation routes and parameters to assess their importance and determine which are the best ones to provide a balanced description of both states so that the relative energies and structural characteristics of the different polymorphs are as close as possible to the experimental results. Figure 3 shows the results of various internal functional benchmarking in terms of the accuracy of unit cell parameter modeling for all four polymorphs (five structures due to γ form crystallographic disorder) of pyrazinamide. The D3BJ dispersion correction scheme was used for all tested functionals, which could affect the poor PBEsol results for these molecular crystals [99,100].
In this particular case, classical PBE outperformed all other tested functionals, showing not only reasonable volumes for all pyrazinamide polymorphs but also the lowest dispersion over various crystalline forms and moderate calculation times (Table S2). Dispersion correction schemes were further tested for the PBE functional to evaluate the best combination (Figure 4, Table S2). Summing up the relative deviation from the experimental unit cell parameters, energy difference (Table S3), and computational time (Table S4), PBE-D3BJ proved to be an optimal choice, which coincides well with the lattice constants and energy benchmarking in work [101]. Thus, all further calculations and comparisons were performed over the PBE-D3BJ level of theory.
The electronic energies for the α, β, and δ forms and γ (A) and γ (B) substructures of the γ form were calculated after optimization. From the obtained electronic energies (the electronic energy of the γ form was obtained by averaging over the γ (A) and γ (B) substructures with their disorder weights), relative energies were calculated, where the β form was chosen as a relative zero energy (Table 2). The series of polymorphs’ relative energies were the same for the PBE-D3BJ and rev-vdW-DF2 methods: β < δ < α < γ. It should also be noted that the values of relative polymorphic energies obtained with the rev-vdW-DF2 method are generally lower than those obtained with PBE-D3BJ. Both functionals were chosen to provide close geometries and energies of solids, whereas the rev-vdw-DF2 functional can be used in conjunction with the DFPT method to obtain energy second-order derivatives.
All polymorphs’ relative energies are less than 4.5 kJ/mol. This coincides well with the upper limit of polymorph energy of 7 kJ/mol obtained by J. Nyman et al. in a study on 1061 experimentally determined ordered crystal structures of 508 polymorphic organic substances [102] and confirmed computationally by Aurora J. Cruz-Cabeza et al. [2]. However, the γ (B) value of the substructure of the disordered γ form was found to be higher than 7 kJ/mol for both methods. This coincides well with the energy difference found for other disordered structures, e.g., tolazamide II disordered structures [103], and confirms possible outliers for the abovementioned empirical rule. Since disordered crystals are often stabilized by a configurational entropy term, their electronic energies are generally found to be higher than the values typical of ordered structures. Thus, it can be claimed that both methods give reasonable values in qualitative agreement with each other.

3.2. Gibbs Energy Calculations at 0 K

The primary way to correct the energies of polymorphic modifications using second-order derivative calculations is to calculate the zero-point vibrational energies (ZPEs) to obtain the thermodynamic potential—the internal energy U, equal to the Helmholtz energy F at 0 K [104,105]. Since the polymorphic modifications of pyrazinamide are modeled at 1 bar pressure, the PV term that distinguishes the Helmholtz energy F from the Gibbs energy G, as well as the internal energy U from enthalpy H, can be neglected.
The ZPE values were obtained from second-order derivative calculations for all polymorphic modifications (Table S5) and added to the electronic energies (Table 3). When ZPE was taken into account, the series of relative energies of the polymorphs did not significantly change but provided varying stabilities of δ and α forms for both the PBE-D3BJ and rev-vdW-DF2 levels of theory (β < δ/α < γ). The lack of supercell usage leads to a slight underestimation of δ form stability.

3.3. Thermodynamic Potential Calculations in the 0 K–470 K Temperature Range

The difference between the supercell and primitive cell usage becomes significant when calculating thermodynamic parameters, such as enthalpies and Gibbs energies in a wide range of temperatures (Table 4).
FD calculations using the PBE-D3BJ method with supercells provide enthalpies very close to the experimental values, while DFPT/rev-vdW-DF2 with supercells results in slightly worse agreement with the DSC results [64]. All calculations without supercells mainly show enthalpy overestimation for carefully measured phase transitions (Table 4).
Putting exact values aside, it is valuable to find correlations between methods, which may be a consequence of the usage or absence of supercells, functional choice, or second-order derivative calculation algorithms. A very good correlation between calculations with the use of supercells (as well as the use of primitive cells) was found, which shows the crucial role of supercell construction (Table 5).
Since computational results are mainly affected by supercell use or its absence, we provide a brief comparison of FD and DFPT methods with the same functional and dispersion correction scheme-FD/rev-vdW-DF2 (primitive cell) and DFPT/rev-vdW-DF2 (primitive cell) levels of theory. The calculated enthalpy results have an excellent correlation (r > 0.98), showing almost no difference between the FD and DFPT methods. FD/rev-vdW-DF2 also shows a reasonable correlation of 0.90 with FD/PBE-D3BJ (both without supercells). From this analysis, it is possible to compare the effects of the settings used on the resulting enthalpies of the polymorphic modifications of pyrazinamide: the supercell setting (primitive cell vs. appropriate supercell) has a stronger impact than the functional (PBE-D3BJ vs. rev-vdW-DF2), which in turn affects the results more than the method for second-order derivative computation (FD vs. DFPT).
To better understand how the chosen level of theory (PBE-D3BJ vs. rev-vdW-DF2) as well as the supercell settings (primitive cell vs. supercell with at least 1 nm in each direction) affect the estimation of the relative thermodynamic stability of polymorphs, an attempt was made to analyze the temperature dependences of the Gibbs energy and its enthalpic H and entropic TS terms (Figure 5).
In this figure, the energies are taken with respect to the β form for each temperature point. Therefore, the thermodynamic parameter plots for the β form become horizontal and equal to 0 at this representation. The γ form is treated separately as γ (A), γ (B), and γ and γ (mix), where γ is calculated as a superposition of disordered structures proportional to their ratio in the final structure γ = x1 × γ (A) + x2 × γ (B) and γ (mix) 287 is γ, corrected by T × Smix, where Smix is calculated as the entropy of mixing of ideal solution Smix = R 288 × [x1 × ln(x1) + x2 × ln(x2)], where x1 = 0.866 and x2 = 0.134 for the A disorder and B disorder, correspondingly. The original data (uncorrected to the β form for each temperature) are provided in Figure S2. Calculations performed without supercells predicted that the β form was stable over the entire temperature range, which contradicts the experimental data. The use of supercells reduces the differences in Gibbs energies between polymorphic modifications and leads to the prediction of the thermodynamical possibility of β→ α transition at temperature points between 260 and 270 K and 470 and 480 K (the data have been extrapolated above 470 K for FD/PBE-D3BJ and DFPT/rev-vdW-DF2, respectively. The experimental data suggest that the β form is stable at low temperatures while the α form is stable under ambient conditions. Therefore, the calculated dependences reproduce qualitatively (although not always quantitatively) the α and β forms’ relative stability when supercells are used. Supercell absence drastically affects both enthalpy and entropy terms, while functional choice mainly differs in TS terms, keeping enthalpy trends close; the latter coincides well with previous experimental work (Tables S9–S11) [106].
However, both levels of theory (FD/PBE-D3BJ and DFPT/rev-vdW-DF2) poorly predict the thermodynamics of the disordered high-temperature γ form even with the appropriate supercell setting employed and the mixing entropy correction being introduced to take into account entropic stabilization of disordered structures [61]. The Gibbs energy curve for the γ form calculated using FD/PBE-D3BJ with mixing entropy correction crosses the β form curve and slowly approaches the α form curve and goes almost parallel to the δ form curve. The Gibbs energy curve for the γ form calculated using DFPT/rev-vdW-DF2 moves away from the other polymorph forms (and seems to overestimate the stability of low-temperature phases). Since the FD/PBE-D3BJ level of theory correctly predicted the δ → γ and α → γ transition enthalpy, we can conclude that this deviation from the experiment can be explained by insufficient accuracy in accounting for the entropic stabilization of disordered structures both from theory or XRD experiments, which has already been discussed by N. Wahlberg et. al. [61].

4. Conclusions and Discussion

The effects of the settings used to calculate the thermodynamic parameters of pyrazinamide polymorphs were compared: supercell setting (primitive cell vs. appropriate supercell) has a much stronger impact than the functional choice (PBE-D3BJ vs. rev-vdW-DF2), which in turn affects the results more than the method for second-order derivative computation (finite differences vs. DFPT). This is consistent with the rule of thumb that a suitable supercell has to be used as well as an assumption that finite differences and DFPT should give, in principle, the same results.
A supercell of at least 1 nm in each direction is necessary to obtain reliable thermodynamical potentials of pyrazinamide polymorphs not only for the FD but also for the DFPT approach if the implementation of this method involves the calculation of only gamma point displacements. Thus, given that calculations at q ≠ 0 should also break translational symmetry in the same way as supercell construction, the DFPT method should not be considered as a more accurate alternative to a finite-difference approach (but may be faster).
Functionals and dispersion correction scheme should be tested to verify minor deviation from experimental data prior to thermodynamic calculations of molecular crystals. Both lattice constants and relative energies should be examined, taking into account the accuracy/computational costs ratio. Not only mean but also dispersion values should be taken into account to guarantee an optimal choice for all examined polymorphs.
Although DFPT and FD methods should, in principle, give similar results, it should be borne in mind that the usage of the DFPT method implies the usage of the semi-local functionals, which have not been as widely used as PBE-D3BJ, for example. It follows that the use of DFPT together with the semi-local functional may require additional validation on known experimental data and may not result in the same compromise of accuracy and speed of computation that the widely used PBE-D3BJ provides.
As a result of our comparison of different levels of theory for calculations on pyrazinamide polymorphs systems, we can make some suggestions for choosing the right settings for calculating second derivatives. Firstly, we strongly recommend using a suitable supercell (the length of basis vectors is at least 1 nm) for the calculation of second-order derivatives. Secondly, reasonable functional and dispersion correction choice is substantial for the correct modeling of polymorphic systems. Finally, we recommend choosing the method of calculating second derivatives (DFPT or FD) based on the availability of well-established functionals for the target problem; if there are no semi-local or non-local functionals among these functionals, then it is preferable to use FD, for which a wider range of functionals is available. In the case of polymorphic modifications of molecular organic crystals, we advise to verify all obtained results using available experimental data.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/cryst15030274/s1. Table S1: VASP symmetry options for various calculations in this and a previous study; Figure S1: Constructed supercells of pyrazinamide polymorphs: (a) α form viewed along the a axis; (b) β form viewed along the b axis; (c) γ form viewed along the b axis, where disordered atom positions are marked as single atoms; and (d) δ form viewed along the a axis; Table S2: Summary of deviation of cell volumes of pyrazinamide polymorph crystalline structures, obtained through optimization using different functionals and dispersion corrections, with respect to the experimental data (PYRZIN22, PYRZIN18, PYRZIN19, and PYRZIN16 for the α, β, γ (both “A” and “B”), and δ forms, respectively; Table S3: Electronic energies of pyrazinamide polymorphs (relative to the β form), obtained using different functionals and dispersion corrections; Table S4: CPU time (in seconds) required for the structural optimization of pyrazinamide polymorph structures using different functionals and dispersion corrections. All of the calculations of CPU time are provided for (2 x) 6-core Intel Xeon X5670, 2.93 GHz (Westmere), with 24 GB RAM; Table S5: Relative ZPE energies for pyrazinamide polymorphs calculated using different methods; Figure S2: Calculated relative thermodynamic parameters for pyrazinamide polymorphs as a function of temperature: (a–c) FD/PBE-D3BJ supercell; (d–f) FD/PBE-D3BJ primitive cell; (g–i) DFPT/rev-vdW-DF2 supercell; (j–l) DFPT/rev-vdW-DF2 primitive cell. From left to right: Gibbs free energy; enthalpy of crystal structures; entropy as the −T × S term. The energies taken with respect to the β form at 0 K. The γ form is treated separately as γ (A), γ (B), and γ and γ (mix) where γ is calculated as a superposition of disordered structures proportional to their ratio in the final structure γ = x1 × γ (A) + x2 × γ (B); γ (mix) is γ, corrected by T × Smix., where Smix is calculated as the standard entropy of mixing Smix = R × [x1 × ln(x1) + x2 × ln(x2)], where x1 = 0.866, and x2 = 0.134 for the A disorder and B disorder, correspondingly; Table S6: Experimental and calculated enthalpies for pyrazinamide polymorph phase transitions at experimental temperatures from the work [64]. This table copies Table 4 from the main text to provide readers with a more convenient comparison of the data obtained from various sources (Tables S7 and S8); Table S7: Experimental and calculated enthalpies for pyrazinamide polymorph phase transitions at experimental temperatures from work [57]; Table S8: Experimental and calculated enthalpies for pyrazinamide polymorph phase transitions at experimental temperatures from work [58]; Table S9: Experimental and calculated entropies for pyrazinamide polymorph phase transitions at experimental temperatures from work; Table S10: Heat capacity temperature dependance (cv(T) in J/mol/K) for pyrazinamide polymorphs calculated using the FD/PBE-D3BJ supercell setting; Table S11: Heat capacity temperature dependance (cv(T) in J/mol/K) for pyrazinamide polymorphs calculated using the DFPT/rev-vdW-DF2 supercell, kJ/mol setting; Table S12: Experimental and optimized unit cell parameters of the α-pyrazinamide polymorph (PYRZIN22) using different functionals and dispersion correction schemes; Table S13: Experimental and optimized unit cell parameters of the β-pyrazinamide polymorph (PYRZIN18) obtained using different functionals and dispersion corrections schemes; Table S14: Experimental and optimized unit cell parameters of the γ(A)-pyrazinamide polymorph (PYRZIN22) obtained using different functionals and dispersion corrections schemes. Table S15: Experimental and optimized unit cell parameters of the γ(B)-pyrazinamide polymorph (PYRZIN22) obtained using different functionals and dispersion corrections schemes. Table S16: Experimental and optimized cell parameters of the δ-pyrazinamide polymorph (PYRZIN22) obtained using different functionals and dispersion corrections schemes.

Author Contributions

Conceptualization, D.A.R.; methodology, D.A.R. and A.S.D.; software, D.A.R. and A.S.D.; validation, D.A.R. and A.S.D.; formal analysis, A.S.D.; investigation, D.A.R. and A.S.D.; resources, D.A.R.; data curation, D.A.R. and A.S.D.; writing—original draft preparation, D.A.R. and A.S.D.; writing—review and editing, D.A.R. and A.S.D.; visualization, A.S.D.; supervision, D.A.R.; project administration, D.A.R.; funding acquisition, D.A.R. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the RSF (Russian Science Foundation) project 23-73-10142 (https://rscf.ru/en/project/23-73-10142/) (accessed on 10 March 2025).

Data Availability Statement

Data are available in the Supplementary Material section and upon reasonable request.

Acknowledgments

The Siberian Branch of the Russian Academy of Sciences (SB RAS) Siberian Supercomputer Center is gratefully acknowledged for providing the supercomputer facilities (http://www.sscc.icmmg.nsc.ru) (accessed on 10 March 2025). The authors also acknowledge the Supercomputing Center of the Novosibirsk State University (http://nusc.nsu.ru) (accessed on 10 March 2025) for providing computational resources.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:
FDFinite difference
DFPTDensity functional perturbation theory
DFTDensity functional theory
PAWProjector augmented wave (atomic pseudopotentials)
D3BJGrimme D3 with Becke–Johnson damping (function)
PBEPerdew–Burke–Ernzerhof (functional)
rev-vdW-DF2Revised van der Waals density (functional)
AM05Armiento–Mattsson (functional)
PBEsolRevised PBE for solids (functional)
PW91Perdew–Wang (functional)
revPBERevised PBE from Zhang and Yang (functional)
RPBERevised PBE from Hammer et al. (functional)
dDsCdDsC dispersion correction method (dDsC)
DFT-D2DFT-D2 method of Grimme (dispersion correction)
DFT-D3DFT-D3 method of Grimme with zero-damping function (dispersion correction)
DFT-D3BJDFT-D2 method of Grimme with Becke–Johnson damping (dispersion correction)
TSTkatchenko–Scheffler method (dispersion correction)
TS/HITkatchenko–Scheffler method with iterative Hirshfeld partitioning (dispersion correction)
MBD@rsSCSMany-body dispersion energy method (dispersion correction)

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Figure 1. Primitive cells of pyrazinamide polymorphs: (a) α form viewed along the a axis; (b) β form viewed along the b axis; (c) γ form viewed along the b axis, where disordered atom positions are marked as single atoms; and (d) δ form viewed along the a axis. Colors are: oxygen—red, nitrogen—blue, carbon—grey, hydrogen—white.
Figure 1. Primitive cells of pyrazinamide polymorphs: (a) α form viewed along the a axis; (b) β form viewed along the b axis; (c) γ form viewed along the b axis, where disordered atom positions are marked as single atoms; and (d) δ form viewed along the a axis. Colors are: oxygen—red, nitrogen—blue, carbon—grey, hydrogen—white.
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Figure 2. An energy–temperature schematic diagram for the four solid phases based on experimental (a) and computational (b) results.
Figure 2. An energy–temperature schematic diagram for the four solid phases based on experimental (a) and computational (b) results.
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Figure 3. Unit cell volume deviation of pyrazinamide polymorphic structures optimized using different functionals with respect to experimental data (PYRZIN22, PYRZIN18, PYRZIN19, and PYRZIN16) for the α, β, γ (both “A” and “B”), and δ forms, respectively.
Figure 3. Unit cell volume deviation of pyrazinamide polymorphic structures optimized using different functionals with respect to experimental data (PYRZIN22, PYRZIN18, PYRZIN19, and PYRZIN16) for the α, β, γ (both “A” and “B”), and δ forms, respectively.
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Figure 4. Relative unit cell volume deviation of pyrazinamide polymorphic structures optimized using the PBE functional and different dispersion correction schemes with respect to the experimental data (PYRZIN22, PYRZIN18, PYRZIN19, and PYRZIN16) for the α, β, γ (both “A” and “B”), and δ forms, respectively.
Figure 4. Relative unit cell volume deviation of pyrazinamide polymorphic structures optimized using the PBE functional and different dispersion correction schemes with respect to the experimental data (PYRZIN22, PYRZIN18, PYRZIN19, and PYRZIN16) for the α, β, γ (both “A” and “B”), and δ forms, respectively.
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Figure 5. Calculated relative thermodynamic parameters for pyrazinamide polymorphs as a function of temperature (a–c) FD/PBE-D3BJ supercell; (d–f) FD/PBE-D3BJ primitive cell; (g–i) DFPT/rev-vdW-DF2 supercell; (j–l) DFPT/rev-vdW-DF2 primitive cell. From left to right: Gibbs free energy; enthalpy of crystal structures; and entropy as the −T × S term.
Figure 5. Calculated relative thermodynamic parameters for pyrazinamide polymorphs as a function of temperature (a–c) FD/PBE-D3BJ supercell; (d–f) FD/PBE-D3BJ primitive cell; (g–i) DFPT/rev-vdW-DF2 supercell; (j–l) DFPT/rev-vdW-DF2 primitive cell. From left to right: Gibbs free energy; enthalpy of crystal structures; and entropy as the −T × S term.
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Table 1. DFPT implementation in the most popular plane-wave DFT software. q = 0 means that only Gamma point calculations (at the center of the Brillouin zone) are supported whereas q ≠ 0 means that generic q-points (also known as q-vectors) are available.
Table 1. DFPT implementation in the most popular plane-wave DFT software. q = 0 means that only Gamma point calculations (at the center of the Brillouin zone) are supported whereas q ≠ 0 means that generic q-points (also known as q-vectors) are available.
SoftwareVASP
(5.4.4/6.4.3)		CASTEP
(23.1.1)		Quantum Espresso
(7.3)		ABINIT
(10.2.3)
DFPT implementationq = 0q = 0 and q ≠ 0q = 0 and q ≠ 0q = 0 and q ≠ 0
Table 2. Relative electronic structure energies (kJ/mol) of pyrazinamide polymorphs obtained from plane-wave periodic DFT calculations using different methods. The energies were accidentally swapped for these two methods in our previous study [65] and are corrected herein by switching off the symmetry option.
Table 2. Relative electronic structure energies (kJ/mol) of pyrazinamide polymorphs obtained from plane-wave periodic DFT calculations using different methods. The energies were accidentally swapped for these two methods in our previous study [65] and are corrected herein by switching off the symmetry option.
Calculation Methodαβγ(A)γ(B)γδ
PBE-D3BJ, kJ/mol *2.7 (III)0 (I)3.110.04.1 (IV)2.1 (II)
rev-vdW-DF2, kJ/mol *2.1 (III)0 (I)2.07.02.6 (IV)1.1 (II)
* The relative stability rank is given in brackets.
Table 3. Relative Gibbs energy * of pyrazinamide polymorphs calculated as a sum of full electronic energies and ZPE vibrations using the finite difference approach (PBE-D3BJ) and DFPT (rev-vdW-DF2) with and without the supercell.
Table 3. Relative Gibbs energy * of pyrazinamide polymorphs calculated as a sum of full electronic energies and ZPE vibrations using the finite difference approach (PBE-D3BJ) and DFPT (rev-vdW-DF2) with and without the supercell.
Calculation MethodAΒγ(A)γ(B)γδ
FD/PBE-D3BJ supercell, kJ/mol2.0 (II–III)0 (I)3.19.64.0 (IV)2.0 (II–III)
FD/PBE-D3BJ primitive cell, kJ/mol1.8 (II)0 (I)2.99.93.8 (IV)2.2 (III)
DFPT/rev-vdW-DF2 supercell, kJ/mol1.5 (III)0 (I)1.96.82.6 (IV)1.0 (II)
DFPT/rev-vdW-DF2 primitive cell, kJ/mol1.3 (II–III)0 (I)1.87.12.5 (IV)1.3 (II–III)
FD/rev-vdW-DF2 primitive cell, kJ/mol1.3(II)0 (I)1.97.22.6 (IV)1.5 (III)
* The relative stability rank is given in brackets.
Table 4. Experimental and calculated enthalpies for pyrazinamide polymorph phase transitions at different experimental temperatures [64].
Table 4. Experimental and calculated enthalpies for pyrazinamide polymorph phase transitions at different experimental temperatures [64].
MethodΔ → αΒ → γΔ → γA → γ
1 Ttrans, K299314326399
1 Exp. ΔH, kJ/mol0.42.22.11.6
FD/PBE-D3BJ supercell, kJ/mol0.2–4.51.71.6
FD/PBE-D3BJ primitive cell, kJ/mol2.03.04.92.9
DFPT/rev-vdW-DF2 supercell, kJ/mol0.72.11.10.3
DFPT/rev-vdW-DF2 primitive cell, kJ/mol2.11.84.72.8
FD/rev-vdW-DF2 primitive cell, kJ/mol1.81.94.63.1
1 Data according to ref. [64]. A full list of experimental data on the temperatures and enthalpies of phase transitions in comparison with the calculated values is provided in the Supplementary Information section (Tables S6–S8).
Table 5. Correlation matrix constructed from the calculated enthalpies at different levels of theory for the phase transitions of pyrazinamide polymorphs at experimental temperatures from ref. [64]. The values in the table correspond to the Pearson correlation coefficients for each pair of different levels of theory. This coefficient has a value from 1 to −1, which corresponds to a perfect direct to inverse linear relationship, respectively; a value of 0 means that the two sets of enthalpies are linearly independent. Colors are provided for clarity.
Table 5. Correlation matrix constructed from the calculated enthalpies at different levels of theory for the phase transitions of pyrazinamide polymorphs at experimental temperatures from ref. [64]. The values in the table correspond to the Pearson correlation coefficients for each pair of different levels of theory. This coefficient has a value from 1 to −1, which corresponds to a perfect direct to inverse linear relationship, respectively; a value of 0 means that the two sets of enthalpies are linearly independent. Colors are provided for clarity.
FD/PBE-D3BJ SupercellFD/PBE-D3BJ Primitive CellDFPT/Rev-Vdw-DF2 SupercellDFPT/Rev-vdW-DF2 Primitive CellFD/Rev-Vdw-DF2 Primitive Cell
FD/PBE-D3BJ supercell10.190.85−0.26−0.16
FD/PBE-D3BJ primitive cell0.1910.180.900.91
DFPT/rev-vdW-DF2 supercell0.850.181−0.24−0.24
DFPT/rev-vdW-DF2 primitive cell−0.260.90−0.2410.98
FD/rev-vdW-DF2 primitive cell−0.160.91−0.240.981
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Dubok, A.S.; Rychkov, D.A. What Is More Important When Calculating the Thermodynamic Properties of Organic Crystals, Density Functional, Supercell, or Energy Second-Order Derivative Method Choice? Crystals 2025, 15, 274. https://doi.org/10.3390/cryst15030274

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Dubok AS, Rychkov DA. What Is More Important When Calculating the Thermodynamic Properties of Organic Crystals, Density Functional, Supercell, or Energy Second-Order Derivative Method Choice? Crystals. 2025; 15(3):274. https://doi.org/10.3390/cryst15030274

Chicago/Turabian Style

Dubok, Aleksandr S., and Denis A. Rychkov. 2025. "What Is More Important When Calculating the Thermodynamic Properties of Organic Crystals, Density Functional, Supercell, or Energy Second-Order Derivative Method Choice?" Crystals 15, no. 3: 274. https://doi.org/10.3390/cryst15030274

APA Style

Dubok, A. S., & Rychkov, D. A. (2025). What Is More Important When Calculating the Thermodynamic Properties of Organic Crystals, Density Functional, Supercell, or Energy Second-Order Derivative Method Choice? Crystals, 15(3), 274. https://doi.org/10.3390/cryst15030274

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