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20
DRUG SOLUBILITY, REACTION THERMODYNAMICS, AND CO‐CRYSTAL SCREENING

Karin Wichmann and Christoph Loschen

COSMO logic GmbH & Co. KG, Leverkusen, Germany

Andreas Klamt

COSMO logic GmbH & Co. KG, Leverkusen, Germany and Institute of Physical and Theoretical Chemistry, University of Regensburg, Regensburg, Germany

20.1 INTRODUCTION

20.1.1 Methods for Compounds in Solution

There is a variety of computational methods for the treatment of compounds in solution. The scope of this chapter is not to give an overview of them, but to concentrate on applications of COSMO‐RS, a very efficient method for the a priori prediction of thermophysical data of liquids. COSMO‐RS combines unimolecular quantum chemical calculations that provide the necessary information for the evaluation of molecular interaction in the fluid phase, with a very fast and accurate statistical thermodynamic procedure. It has established itself as an alternative to structure‐based group contribution methods (GCMs) on the one hand and to force field‐based simulation methods on the other hand. Because of its special approach, COSMO‐RS is a generally applicable method for compounds in solution. It has been applied successfully in such diverse areas as solvent screening, partitioning behavior, liquid–liquid and vapor–liquid equilibria, and ADME property prediction and for such diverse compound types as drugs, pesticides, common organic compounds, halocarbons, and ionic liquids. COSMO‐RS is used in chemical, pharmaceutical, agrochemical, and petrochemical industry.

In this contribution, three application fields important in drug development and drug production will be considered: solubility prediction and prediction of free energy of reaction in solution and co‐crystal screening. Solubility prediction methods are important during the drug design and development process, because in the early drug design phase compounds are often only virtually considered by computational drug design methods or the synthesized amount of substance is insufficient for experiments. In both cases the only tools for the selection of promising drug candidates with adequate solubility are computational methods that predict the solubility with sufficient accuracy just from the chemical structure of the compound. A method requiring experimental data for solubility prediction is unfeasible in this situation, since such data will not be available.

Prediction of thermodynamic properties of compounds in solution is also important in industrial process development. Here, specifically reaction energies and equilibrium constants of reaction in solution are of particular interest when a new process is developed or alternative pathways for existing processes are explored. The Gibbs free energy of reaction varies with the choice of the solvent or solvent mixture, and hence the chosen solvent system can strongly influence the process in solution. Generally, experimental data for the equilibrium constant or the free energy of reaction in solution are rare but are available relatively straightforward from a computational approach. Prediction of thermochemical data like heat of reaction and heat of vaporization furthermore helps in designing a chemical process such that process hazards can be prevented.

Finally, the prediction of compounds that form co‐crystals together with drugs is of high interest for pharmaceutical research and development, because co‐crystalline materials often show modified properties such as improved solubility or dissolution behavior.

In order to identify new co‐crystals, usually large sets of excipients are screened experimentally, whereas selection of compounds is done by trial and error or according to some empirical rules. Computational methods such as COSMO‐RS allow for a rational design and a more focused search for new co‐crystal candidates, and hence valuable resources can be saved.

20.1.2 COSMO

In conventional quantum chemistry, molecules are treated as isolated particles at a temperature of 0 K. In physical reality however, the major part of reactions takes place in solution and at higher temperatures. Since direct treatment of a large number of molecules is computationally very demanding, solvent effects are often treated indirectly by continuum solvation models, where the solute is embedded in a dielectric continuum and the solvent is represented by a mean interaction with a surrounding dielectric medium. The interaction of the solute with such a dielectric solvent is taken into account in the quantum chemical calculation by polarization charges that arise from the dielectric boundary condition.

The “COnductor‐like Screening MOdel” (COSMO) is an efficient variant of dielectric continuum solvation methods [1]. In quantum chemical COSMO calculations, the solute molecules are calculated in a virtual scaled conductor environment, i.e. the scaled boundary condition of a conducting medium is used, where the molecule is ideally screened and not the exact dielectric boundary condition. In such a conducting environment, the solute molecule induces a polarization charge density σ on the interface between the molecule and the conductor, i.e. on the molecular surface. These charges act back on the solute and generate a more polarized electron density than in vacuum. During the quantum chemical self‐consistency algorithm, the solute molecule is thus converged to its energetically optimal state in a conductor with respect to electron density. Due to the analytic gradients available for the COSMO energy contributions, the molecular geometry can be optimized using conventional methods for calculations in vacuum. The quantum chemical calculation has to be performed once for each molecule of interest.

20.1.3 COSMO‐RS

As discussed in more detail elsewhere, the simple dielectric continuum models suffer from a number of insufficiencies [2, 3]. The polarization charge density σ resulting from unscaled COSMO calculations (also called screening charge density σ), which is a good local descriptor of the molecular surface polarity, is used to extent the model toward “real solvents” (COSMO‐RS).

In COSMO‐RS, a liquid is considered to be an ensemble of closely packed ideally screened molecules, as shown in Figure 20.1. In this picture, each piece of surface has one direct contact partner but is still separated from its partner by a thin film of conductor. Since the conducting medium that was assumed to surround the molecules in the COSMO calculation is not existent in reality, the energy difference between the pairwise contacts and the ideally screened situation has to be defined as a local electrostatic interaction energy that results from the removal of the conductor film between the molecules. Considering a contact on a region of molecular surface of area a_eff (effective contact area), and considering that the two contacting pieces of molecular surface have average ideal screening charge densities σ and σ′ in the conductor, it is possible to calculate this interaction energy as the energy that is necessary to remove the residual screening charge density σ + σ′ from the contact. In the special case of σ = −σ′, the contact is an “ideal electrostatic contact,” and the interaction energy is zero, because the two molecules screen each other as well as the conductor. In the general case however, σ + σ′ does not vanish, and the arising electrostatic interaction energy is

(20.1)

where

e_misfit(σ,σ′) is the misfit energy density on the contact surface.
α′ is a general constant that can be calculated approximately but in COSMO‐RS is fitted to experimental data as fine‐tuning.

2 Schematics of contacting molecular cavities and contact interactions, illustrated by 2 boxes containing condensed formulas involving O–C–O and H–O–H. — **FIGURE 20.1** Schematic illustration of contacting molecular cavities and contact interactions.

The misfit term (Eq. 20.1) subsumes the polarization response of the molecules to the electrostatic misfit quite well [2, 4, 5].

Hydrogen bonding interactions are to some extent already covered by the description of electrostatic interactions, but we still have to parameterize the additional hydrogen bonding energy resulting from interpenetration of the atomic electron densities in some reasonable way. This energy should only be relevant if two sufficiently polar pieces of surface of opposite polarity are in contact, and its strength increases with increasing polarity of both surface pieces. Taking the screening charge density σ as a local measure of polarity, the following function realizes such behavior:

(20.2)

with σ_don = min(σ, σ^′) and σ_acc = max(σ, σ^′). General parameters are σ_hb, the threshold for hydrogen bonding, and c_hb, the coefficient for the hydrogen bond strength. Both parameters have to be adjusted to experimental data. With Eq. (20.2), the hydrogen bond interaction energy is zero, unless the more negative of the two screening charge densities is less than the threshold −σ_hb and the more positive exceeds σ_hb. Because positive molecular regions have negative screening charge, the negative σ now is the donor part of the hydrogen bond and the positive is the acceptor. In this case, the hydrogen bonding energy is proportional to the product of the excess screening charge densities (σ_don + σ_hb)(σ_acc − σ_hb).

Van der Waals (vdW) interactions are described by element‐specific parameters τ in COSMO‐RS. The τ parameters have to be fitted to experimental data. Then, the vdW energy gain of a molecule X during the transfer from the gas phase to any solvent is given by

(20.3)

The vdW term is spatially nonspecific. Because E_vdW is independent of any neighboring relations, it is not really an interaction energy, but may be considered as an additional contribution to the energy of the reference state in solution. Currently nine of the vdW parameters (for elements H, C, N, O, F, S, Cl, Br, and I) have been optimized. For the majority of the remaining elements, reasonable estimates are available. Nonadditive vdW corrections are used for a few element pairs, but they are of minor importance for the topics of this contribution.

The transition from microscopic surface interaction energies to macroscopic thermodynamic properties of a liquid is possible via a statistical thermodynamic procedure. The exact solution of the thermodynamic problem would require sampling of all different arrangements of all molecules of the systems, weighting the contribution of each arrangement by its Boltzmann factor. This direct approach, which is used in the molecular dynamics and Monte Carlo‐type methods, is very time consuming and requires compromises regarding sampling and regarding the accuracy of the energy evaluations. COSMO‐RS follows a different concept. The basic approximation is that the ensemble of interacting molecules may be replaced by the corresponding ensemble of independent pairwise interacting surface pieces. This approximation implies the neglect of any neighborhood information of surface pieces on the molecular surface and the loss of steric information. The advantage of this approximation is the extreme reduction of the complexity of the problem, which allows for a fast and exact solution. It should be noted that GCMs as UNIFAC are also based on the assumption of independent pairwise interacting surfaces.

Since the screening charge density σ is the only descriptor determining the interaction energy terms in Eqs. (20.1) and (20.2), the ensemble of surface pieces characterizing an ensemble S is sufficiently described by its composition with respect to σ. For this purpose we introduce the molecular σ‐profile p^X(σ), which is a histogram of the screening charge densities σ on the surface of a molecule X (Figure 20.2). The σ‐profile can easily be derived from the COSMO files produced as output of the quantum chemical COSMO calculation for molecule X, applying a local averaging algorithm in order to take into account that only screening charge densities σ averaged over an effective contact area are of physical meaning in COSMO‐RS [2].

Graph illustrating σ-profiles of common solvents with 4 discrete waves for water, hexane, acetone, and chloroform. — **FIGURE 20.2** σ‐Profiles of common solvents.

The σ‐profile for the entire solvent of interest S, which might be a mixture of several compounds, p_S(σ), is given by the weighted surface area normalized sum of the σ‐profiles of the components X_i:

(20.4)

where is the COSMO surface of a compound X_i in the system.

Under the condition that there is no free surface in the bulk of the liquid, i.e. each piece of molecular surface has a direct contact partner, the statistical thermodynamics of the system can be solved using the exact equation:

(20.5)

In this equation, μ_S(σ) is the chemical potential of an average molecular contact segment of size a_eff in the ensemble S at temperature T, and e(σ,σ′) is the interaction energy functional e(σ,σ′) = e_misfit(σ,σ′) + e_hb(σ,σ′).

Since μ_S(σ) appears on both sides of Eq. (20.5), it must be solved by iteration, starting with μ_S(σ′) = 0 on the right‐hand side. Fortunately, the solution converges rapidly, and μ_S(σ) can be computed up to numerical precision within milliseconds on a personal computer. For a formal derivation of Eq. (20.5), we refer to Ref. [4].

Now it is straightforward to define the chemical potential of a solute X in the ensemble S by

(20.6)

where the residual part, i.e. the part resulting from the interactions of the surfaces in the liquid, is given by the surface integral of function μ_S(σ) over the solute surface, which is expressed using the σ‐profile of the solute in Eq. (20.6). The second part is the combinatorial contribution, which arises from the different shapes and sizes of the solute and solvent molecules. Expressions based on the surface areas and volume ratios of solvents and solutes, similar to standard chemical engineering expressions as Staverman–Guggenheim, are used in the context of COSMO‐RS [6]. COSMO surface areas and volumes are used for the evaluation of the combinatorial term. Hence Eq. (20.6) can be completely evaluated based on the information resulting from the COSMO calculations of the individual compounds.

The chemical potential of Eq. (20.6) is a pseudo‐chemical potential [7], i.e. the standard chemical potential without the concentration term RT ln x_i. We will shortly use the term chemical potential for the pseudo‐chemical potential from Eq. (20.6) throughout this contribution. Providing the chemical potential of an arbitrary compound X in almost arbitrary solvents and mixtures as a function of temperature and concentration, Eq. (20.6) allows for the prediction of almost all thermodynamic properties of compounds or mixtures, such as activity coefficients, partition coefficients, or solubility, as shown in the flowchart for a COSMO‐RS property prediction in Figure 20.3.

Flowchart of a property prediction procedure with COSMO and COSMO-RS from molecular structure to mixture equilibrium data, with a graph with 3 areas for liquid, liquid+vapor, and vapor. — **FIGURE 20.3** Flowchart of a property prediction procedure with COSMO and COSMO‐RS.

As mentioned above, the COSMO‐RS method depends on a small number of adjustable parameters. Some of the parameters are predetermined from physics, while others are determined from selected properties of mixtures. The parameters are not specific to functional groups or types of molecule. As a result, COSMO‐RS is the least parameterized of all quantitative methods for the prediction of chemical properties in the liquid phase [5].

20.1.4 Treatment of Conformers in COSMO‐RS

Many molecules can adopt more than one conformation, and different conformers of one molecule can have different σ‐profiles. The chemical potentials of the individual conformers and hence the conformer distribution as well as the chemical potential of the compound represented by an ensemble of conformers depend on the composition of the system and the temperature. Thus, it is essential for property prediction with COSMO‐RS to take conformers with different σ‐profiles into account, each described by individual quantum chemical COSMO calculations. The relative contributions of the conformers are determined by an iterative procedure using the Boltzmann weight of the free energies of the conformers in the liquid phase. This results in a thermodynamically fully consistent treatment of multiple molecular conformations.

20.2 SOLUBILITY PREDICTION WITH COSMO‐RS

For the calculation of the solubility of a liquid compound X in a solvent S, we require the chemical potentials of X in S and in its pure liquid state, and . If is sufficiently small, so that the solvent behavior of the X‐saturated solvent S is not significantly influenced by the solute X, then the decadic logarithm of the solubility is given by

(20.7)

with the molecular weight MW, the solvent density ρ, and . In the case of high solubility (typically for solubility >10 wt. %), Eq. (20.7) becomes approximate, and the true solubility would have to be derived from a detailed search for a thermodynamic equilibrium of a solvent‐rich and a solute‐rich phase. But, in general, at least for the purpose of estimating drug solubility, Eq. (20.7) is sufficiently accurate.

If the zeroth‐order as initially provided by Eq. (20.7), using infinite dilution of X in S, is resubstituted into the solubility calculation via , a better approximation for is achieved. In other words, the solute chemical potential is computed for the solvent–solute mixture with the finite mole fraction of X in S that was predicted by the zeroth‐order . Then, using Eq. (20.7) with the new and the resulting values, an improved solubility is computed. Iterating this process to convergence, an iterative solubility can be achieved, which is also implemented in our COSMO‐RS program COSMOtherm and allows for the accurate prediction of solubility values even for cases of high solubility (solubility up to 50 wt. %). Thus, except for rare cases of very high solubility, a complicated search for a multiphase thermodynamic equilibrium of a solvent‐rich and a solute‐rich phase can be avoided, but instead Eq. (20.7) and its iterative refinement can be used.

Drugs are mostly solid at room temperature. Because the solid state of a compound X is related to its liquid state by the free energy difference , which is positive in the case of solids, a more general expression for solubility reads

(20.8)

For liquid compounds, is negative and Eq. (20.8) reduces to Eq. (20.7).

If melting point temperature T_melt and heat of fusion ΔH_fus or entropy of fusion ΔS_fus are known experimentally for a solid compound, the free energy of fusion ΔG_fus in Eq. (20.8) can be estimated from

(20.9)

(20.10)

Equations (20.9) and (20.10) can be complemented by an additional temperature‐dependent term using the heat capacity of fusion ΔCp_fus in order to obtain good absolute predictions, but data for ΔCp_fus are rarely available from experiment.

The free energy of fusion of new compounds is often not known, because experimental measurements can be cumbersome and substance may be scarce. Computational prediction of requires evaluation of the free energy of a molecule of compound X in its crystal, i.e. the crystal structure needs to be known or predicted. In general however, crystal structure prediction in particular for larger drugs is not yet feasible from a practical viewpoint due to the involved complexity. As an alternative, a QSPR approximation for ΔG_fus can be used in COSMOtherm, which is based on a few rather obvious factors that should influence crystallization. Larger molecules should have larger ΔG_fus than smaller ones, compounds with more polarity and hydrogen bonding ability should have larger ΔG_fus than less polar ones, and also rigidity should give rise to larger ΔG_fus. We found that a good regression equation for can be achieved by a combination of the descriptors V^X (the cavity volume from the COSMO calculation as size descriptor), (the number of ring atoms in X as a descriptor of the compounds’ rigidity), and (the compounds’ chemical potential in water as a combined measure of polarity and hydrogen bonding) [8]:

(20.11)

20.2.1 Relative Solubility and Solubility Screening

The computational prediction of the relative solubility of a drug candidate in a variety of solvents with COSMO‐RS is straightforward and can be done without wasting any of the substance in this step. The required DFT/COSMO calculations can be done even before the compound comes to the development laboratory, and a COSMO‐RS solubility screening can be already completed when the work in the development department starts.

Experimental data for melting point and free energy of fusion can often be obtained through differential scanning calorimetry. If melting point temperature and heat of fusion or entropy of fusion are known for the compound in question, the free energy of fusion ΔG_fus can be calculated according to Eq. (20.9) or (20.10), and a solubility screening for absolute solubilities can be done. If data for ΔG_fus are not known, an estimated ΔG_fus may be used, either from the QSPR model implemented in COSMOtherm or from an external model.

EXAMPLE PROBLEM 20.1 Relative and Absolute Solubility of Acetaminophen in Pure Solvents at 30 °C

Experimental data for the solubility of acetaminophen in pure solvents were reported by Granberg and Rasmuson [9]. We use this data set to validate the calculated acetaminophen solubilities. Furthermore, a melting point temperature of T_melt = 441.2 K, a heat of fusion of ΔH_fus = 26.0 kJ/mol, and an entropy of fusion of ΔS_fus = 59.0 J/mol have been reported for acetaminophen [10]. These data will be used to compute absolute solubility predictions for acetaminophen in the solvent data set.

With the case study of acetaminophen solubility, we want to show first the prediction of relative solubility. For relative solubility calculations, we do not make use of any experimental data like melting point temperature or enthalpy of fusion, as that kind of data is usually not available in the early drug design phase.

Solute and solvents for this example were calculated on the BP86/TZVP level of theory, which corresponds to the higher quantum chemical level for which the COSMOtherm program is parameterized. Different conformations of the compounds were taken into account where the conformers showed different σ‐profiles and COSMO energies. All compounds of the data set, including conformers, are available from the COSMObase, a collection of validated COSMO files for common compounds and solvents (Figure 20.4).

For the calculation of relative solubility in pure solvents, we do not use the iterative refinement procedure, since the assumed value of ΔG_fus = 0 kcal/mol will influence computed zeroth‐order solubility and hence the iterative refinement. Therefore, the relative solubilities are calculated in infinite dilution in the respective solvent at 30 °C (Figure 20.5).

Calculation results can be read from the output and table files (Figure 20.6). The relative solubility of a compound can be calculated from the chemical potentials of the compound in the solvent and in its pure state as

(20.12)

Since a logarithmic solubility value larger than 0 indicates only that the two compounds are miscible, there is a cutoff at 0 in the COSMOtherm results for the logarithmic solubility in Figure 20.6. However, in order to provide insight in the whole range of the solvent data set independent of potential misestimates of ΔG_fus, positive values for log S_rel(x) are allowed here for both relative and absolute solubility predictions. Relative solubility data in Table 20.1 were calculated directly from the chemical potential differences as described in Eq. (20.12).

The relative solubility predictions correlate well with the experimental data, revealing an overall shift of 1.8 log units, which arises mainly from the neglect of ΔG_fus and is therefore irrelevant for real solubility considerations. The predicted relative solubility data apparently fall into two groups, as can be seen from Figure 20.7: Solubility data in alcoholic solvents are grouped together on a rather straight line with a slope of 0.7 and a relative shift of 1.4 log units compared with the experimental data, while the second, more scattered group has a larger shift (≥2 log units), but a similar slope. There is one severe outlier in the data set, carbon tetrachloride, where the acetaminophen solubility is, in contrast to the trend observed in the other solvents, underestimated by 1.5 log units. Since the experimental value for the solubility in carbon tetrachloride comes from a single measurement, and the other solvents appear to be described reasonably by the model prediction apart from a general overestimation due to the missing free energy of fusion term, we tend to consider this experimental value as questionable.

Indeed, in a personal communication, the original authors of Ref. [9] stated that a more recently determined value for the solubility of acetaminophen in CCl₄ of log S (mg/g)= −2.5 may be more accurate [12]. This would also perfectly correspond with the respective COSMO‐RS prediction.

Using the experimental data for T_melt and ΔH_fus, absolute solubilities of acetaminophen in the solvent data set are also computed. The absolute predictions are in good quantitative agreement with experimental data, as shown in Figure 20.8. Of the 26 solvents, 4 are predicted with a positive log S(x): tetrahydrofuran, diethylamine, N,N‐diethylformamide, and dimethyl sulfoxide. The rmse for all solvents is 0.77 log units. There are, however, four significant outliers in the data set; one of them, carbon tetrachloride, deviates by almost 3 log units from the flawed experimental solubility as has already been discussed above. The other outliers are overpredicted solubilities. For diethylamine, the predicted solubility is 1.64 log units too high. We attribute this error to the known systematic error of COSMO and COSMO‐RS for secondary and tertiary aliphatic amines [5]. Another outlier is the predicted solubility of acetaminophen in 1,4‐dioxane (317.14 mg/g), which deviates by 1.27 log units from the value reported by Granberg and Rasmuson (17.1 mg/g). It is noteworthy that the deviation is much less (0.56 log units) if compared with the experimental solubility reported by Romero et al. (86.9 mg/g) [13]. The fourth outlier in the data set is in tetrahydrofuran, where the predicted solubility deviates by 1.13 log units from the experimental data. Here, we do not have an explanation for the error of the model prediction. However, since tetrahydrofuran and 1,4‐dioxane are rather similar solvents, the question arises whether the uncertainty of the experimental data for tetrahydrofuran might be comparable with the case of 1,4‐dioxane, where there is a deviation between the published experimental data from the different sources.

With the four outliers removed, the overall rmse reduces to 0.46 log units. While the rmse is very small for alcohols (0.10 log units), the calculated solubilities in polar aprotic solvents like acetone or hydrophobic solvents like toluene are systematically overpredicted. The rmse for the remainder of the data set without the alcoholic solvents is 0.86 log units. It should be noted that the estimate of ΔG_fus based on ΔH_fus and T_melt itself may have an error of the order of 0.5 log units, making these absolute deviations uncertain.

EXAMPLE PROBLEM 20.2 Solubility of Acetaminophen in Binary Mixtures of Water–Acetone and Toluene–Acetone at 25 °C

Experimental data for acetaminophen solubility in water–acetone and acetone–toluene binary solvents were also reported by Granberg and Rasmuson [14] and are used here for comparison with the model prediction.

As in the previous example, the COSMO files of the compounds were calculated on the BP86/TZVP level of theory. Absolute solubility predictions are calculated using the experimental data for ΔG_fus of acetaminophen and employing the iterative refinement procedure for the solubility. The calculations are done for the compositions that were measured by Granberg and Rasmuson (Figure 20.9).

The predicted solubilities of acetaminophen in the binary solvent system can be extracted from the COSMOtherm table file shown in Figure 20.10. Table 20.2 lists the predicted solubilities together with the experimental data.

Figures 20.11 and 20.12 show the prediction results for the acetaminophen solubility in water–acetone and acetone–toluene binary solvent mixtures. The solubility of acetaminophen in the water–acetone binary mixture is nonideal, with a solubility peak at approximately 70% mass fraction of acetone. The nonideal solubility behavior is also captured by the model prediction, with the solubility peak slightly shifted to higher acetone content of the binary solvent. Since the solubility in pure acetone is overpredicted by 0.59 log units, the model prediction for the binary mixture does not show the strong decrease in solubility for very high acetone content of the solvent mixture that is found in the experiment.

The prediction results for the acetone–toluene system (Figure 20.12) are consistent with the trends exhibited by the experimental data. Again, we see effects of the overpredicted solubility in pure acetone and the lower predicted solubility in pure toluene, but the relatively ideal solubility behavior of acetaminophen in the acetone–toluene binary mixture is found also by the model prediction.

Screenshot of Select compounds with parameterization TZVP for tempjob window displaying a data entry field for search list labeled paracetamol. Under select DBs to search is a checked box for Database-BP-TZVP-COSMO. — **FIGURE 20.4** Database view in COSMOthermX. Databases can be searched and columns are sortable.

Screenshot of COSMOthermX-tempjob window with edit tab selected. On the left pane is compounds tab selected displaying 2 folders for 4–hydroxyacetanilide and ethanol. On the right pane is properties tab selected. — **FIGURE 20.5** Overview of COSMOthermX with compound list, solubility panel, and input section. When the solute state is set to liquid, ΔG_fus = 0 is used, while with solid solute state, given or estimated values for ΔG_fus are used. The iterative refinement can also be set in the solubility panel. In the solvent frame, the solvent composition is set to pure for the respective solvent. Pictured here are settings for absolute solubility using the iterative refinement procedure.

No alt text required. — **FIGURE 20.6** Excerpt from the COSMOtherm table file for the solubility calculation of acetaminophen in pure solvents. Log10(x_solub) indicates the logarithmic solubility in mole fractions.

TABLE 20.1 Experimental, Predicted Relative, and Predicted Absolute Solubilities of Acetaminophen in Pure Solvents

Source: Data from Granberg and Rasmuson [9].

Solvent	Experimental		Predicted Relative			Predicted Absolute			Error
Solvent	c_S (g/kg)	Log S(mg/g)	Log S(x)	S (mg/g)	Log S (mg/g)	Log S(x)	S (mg/g)	Log S (mg/g)	Error
Water	17.39	1.24	−1.6819	174.53	2.24	−3.0745	7.07	0.85	−0.39
Methanol	371.61	2.57	0.6155	19 465.33	4.29	−1.0119	459.02	2.66	0.09
Ethanol	232.75	2.37	0.4809	9929.62	4.00	−1.0687	280.11	2.45	0.08
1,2‐Ethanediol	144.3	2.16	0.1829	3711.29	3.57	−1.2535	135.86	2.13	−0.03
1‐Propanol	132.77	2.12	0.2182	4157.36	3.62	−1.2441	143.37	2.16	0.03
2‐Propanol	135.01	2.13	0.3858	6115.50	3.79	−1.1268	187.86	2.27	0.14
1‐Butanol	93.64	1.97	0.0690	2390.35	3.38	−1.3675	87.50	1.94	−0.03
1‐Pentanol	67.82	1.83	−0.0639	1480.10	3.17	−1.4866	55.92	1.75	−0.08
1‐Hexanol	49.71	1.70	−0.1756	987.32	2.99	−1.5932	37.75	1.58	−0.12
1‐Heptanol	37.43	1.57	−0.2630	709.92	2.85	−1.6779	27.31	1.44	−0.14
1‐Octanol	27.47	1.44	−0.3589	507.95	2.71	−1.7716	19.64	1.29	−0.15
Acetone	111.65	2.05	0.9328	22 297.15	4.35	−0.8016	411.01	2.61	0.57
2‐Butanone	69.99	1.85	0.6508	9380.65	3.97	−0.9153	254.79	2.41	0.56
4‐Methyl‐2‐pentanone	17.81	1.25	0.1126	1956.04	3.29	−1.2844	78.41	1.89	0.64
Tetrahydrofuran	155.37	2.19	1.6842	101 306.53	5.01	0.0527	2366.72	3.37	1.18
1,4‐Dioxane	17.08	1.23	1.0644	19 898.20	4.30	−0.7332	317.14	2.50	1.27
Ethyl acetate	10.73	1.03	0.0872	2097.21	3.32	−1.2415	98.38	1.99	0.96
Acetonitrile	32.83	1.52	0.1079	4720.92	3.67	−1.2689	198.25	2.30	0.78
Diethylamine	1316.9	3.12	3.5201	6 845 852.30	6.84	1.4457	57 683.45	4.76	1.64
N,N‐Dimethylformamide	1012.02	3.01	2.2018	276 105.51	5.44	0.4567	4965.64	3.70	0.69
Dimethyl sulfoxide	1132.56	3.05	3.3062	3 915 389.90	6.59	0.2699	3601.53	3.56	0.50
Acetic acid	82.72	1.92	0.3232	5298.62	3.72	−1.1632	172.87	2.24	0.32
Dichloromethane	0.32	−0.49	−1.8354	26.00	1.41	−3.1820	1.17	0.07	0.56
Chloroform	1.54	0.19	−1.8884	16.37	1.21	−3.2647	0.69	−0.16	−0.35
Carbon tetrachloride	0.89	−0.05	−4.5149	0.03	−1.52	−5.9289	0.00	−2.94	−2.89
Toluene	0.34	−0.47	−3.0598	1.43	0.16	−4.4573	0.06	−1.24	−0.77

Log S(x) indicates the logarithmic solubility in mole fractions.

Image described by caption. — **FIGURE 20.7** Predicted relative solubility of acetaminophen versus experimental data in pure solvents at 303.15 K. Triangles represent relative solubility data, with empty triangles (Δ) representing alcoholic solvents and water and solid triangles (▴) representing the remainder of the solvent data set. One outlier (carbon tetrachloride) is represented by a solid diamond (♦).

Image described by caption and surrounding text. — **FIGURE 20.8** Predicted absolute solubility of acetaminophen versus experimental data in pure solvents at 303.15 K. Empty triangles (Δ) represent absolute solubility data of alcoholic solvents and water; solid triangles (▴) represent the remainder of the solvent data set. Four outliers (carbon tetrachloride, diethylamine, 1,4‐dioxane, tetrahydrofuran) represented by solid diamonds (♦).

Screenshot of COSMOthermX-tempjob window displaying tempjob dialog box. On the left pane is Compounds tab, while on the right is Properties tab with Multiple solutes tab selected. Solid is marked under State of solute. — **FIGURE 20.9** Input section of COSMOthermX with entries for a list of solvent compositions.

TABLE 20.2 Experimental and Predicted Solubilities of Acetaminophen in Water–Acetone and Acetone–Toluene Binary Solvent Mixtures

Source: Data from Granberg and Rasmuson [14].

% Mass Fraction			Experimental		Predicted			Error
Water	Acetone	Toluene	c_S (g/kg)	Log S (mg/g)	Log S(x)	S (mg/g)	Log S (mg/g)	Error
100	0	0	14.90	1.17	−3.1508	5.93	0.77	−0.40
93	7	0	28.18	1.45	−2.7658	13.69	1.14	−0.31
85	15	0	53.0	1.72	−2.3994	29.99	1.48	−0.25
80	20	0			−2.1978	45.87	1.66	1.66
75	25	0			−2.0125	67.47	1.83	1.83
70	30	0	150.0	2.18	−1.8424	95.66	1.98	−0.20
65	35	0			−1.6879	130.58	2.12	2.12
60	40	0			−1.5500	171.24	2.23	2.23
55	45	0			−1.4290	215.47	2.33	2.33
50	50	0	327.0	2.51	−1.3242	260.58	2.42	−0.10
30	70	0	454.6	2.66	−1.0260	408.68	2.61	−0.05
15	85	0	420.3	2.62	−0.8851	452.18	2.66	0.03
7	93	0	302.2	2.48	−0.8364	438.41	2.64	0.16
3	97	0	197.1	2.29	−0.8247	415.72	2.62	0.32
0	100	0	99.8	2.00	−0.8285	386.32	2.59	0.59
0	95	5	91.7	1.96	−0.8517	359.46	2.56	0.59
0	90	10	82.4	1.92	−0.8774	332.39	2.52	0.61
0	85	15	75.7	1.88	−0.9062	305.14	2.48	0.61
0	80	20	66.4	1.82	−0.9386	277.63	2.44	0.62
0	70	30	52.8	1.72	−1.0165	222.79	2.35	0.63
0	60	40	37.08	1.57	−1.1186	168.79	2.23	0.66
0	50	50	26.56	1.42	−1.2563	117.58	2.07	0.65
0	30	70	8.55	0.93	−1.7149	37.19	1.57	0.64
0	20	80	3.39	0.53	−2.1018	14.50	1.16	0.63
0	15	85	2.12	0.33	−2.3663	7.68	0.89	0.56
0	7	93	0.78	−0.11	−2.9938	1.73	0.24	0.35
0	0	100	0.37	−0.43	−4.6734	0.03	−1.46	−1.03

Log S(x) indicates the logarithmic solubility in mole fractions.

20.3 CHEMICAL REACTIONS IN SOLUTION

Calculation of reaction energies in the gas phase is a standard application in quantum chemistry. The computational prediction of free energies of reaction in solution is more involved, but still a rather straightforward procedure. Generally, the free energy of reaction is the difference of the total free energies of the reactants and the products of the reaction. For a reaction

where A and B are the reactants with stoichiometric coefficients a and b and C and D are the reaction products with stoichiometric coefficients c and d, the free energy of reaction can be calculated from the difference of the sums of free energies on both sides of the reaction:

(20.13)

The free energies of the reactants and products, and thus the free energy of reaction, depend on the conditions under which the reaction takes place. The free energy of reaction in the gas phase differs from the free energy of reaction in solution, and it is different in each specific solvent.

In solution, the Gibbs free energy of a species is

(20.14)

Here E_gas(i) is the gas‐phase energy of the compound, and for computational predictions of the reaction free energy, it should be taken from an adequate quantum chemical (DFT or post‐Hartree–Fock) level. ΔG_solv(i), the free energy of solvation, describes the change of the free energy that occurs when the compound is dissolved from the gas phase into the liquid phase. This contribution to the total free energy of a compound can be computed using COSMO‐RS.

Using the gas‐phase energies of the compounds and the free energies of solvation of the compounds, the free energy of reaction in solution can be calculated according to a thermodynamic cycle as depicted in Figure 20.13.

Schematic cycle for computation of a free energy change in solution, with A(gas)+B(gas), C(gas)+D(gas), A(sol)+B(sol), and C(sol)+D(sol) linked by arrows labeled ΔGr(gas), ΔGr(sol), ΔGsolv(C), ΔGsolv(D), etc. — **FIGURE 20.13** Cycle for computation of a free energy change in solution.

In order to compute the lower horizontal leg of the cycle, corresponding to the reaction in solution, we have to take the appropriate sums and differences of the upper horizontal leg, i.e. the gas‐phase reaction, and the vertical legs, i.e. the solvation energies of the compounds:

(20.15)

ΔG_r(gas) can be calculated from the chemical potential of the compounds in the gas phase. As already mentioned, the quantum chemical gas‐phase energy E_gas of a compound is computed in vacuum at absolute zero. Furthermore, E_gas does not account for vibrational motion that is present even at T = 0 K. The so‐called zero‐point energy or zero‐point vibrational energy (ZPE) can be computed quantum chemically from the vibrational frequencies of the compound and is a standard correction to E_gas. Using the ZPE, the free energy of a compound can be calculated as

(20.16)

As a further refinement for the gas‐phase energies of the compounds and the resulting reaction energy, the temperature‐dependent thermal contributions to the free energy μ_vib of the molecule can be calculated. From vibrational frequencies, the molecular translational, rotational, and vibrational partition functions, q_trans, q_rot, and q_vib, can be calculated, thus enabling prediction of thermodynamic functions at temperatures other than 0 K and finite pressure:

(20.17)

The other terms required for Eq. (20.15), the free energies of solvation ΔG_solv, can be obtained from a COSMOtherm prediction of the reverse process, i.e. from a vapor pressure prediction. The partial vapor pressure P(i) that is calculated by COSMOtherm corresponds to the pure compound vapor pressure times the activity coefficient and is related to ΔG_solv by

(20.18)

with P being the reference pressure at which the reaction takes place.

Note that if the reactant or product compounds are present in the mixture at a finite concentration with a mole fraction x(i) (e.g. if the reaction takes place in bulk reactant liquid), an entropic contribution RT ln(x(i)) of the compound has to be added to the compounds’ free energy G(i).

20.3.1 Heat of Reaction

The heat of reaction or reaction enthalpy in solution can be calculated by a procedure similar to the free energy of reaction. Instead of the free energy of solvation of the compounds, we make use of the heat of vaporization ΔH_vap. Since ΔH_vap is the enthalpy that is needed to transfer the compound from the liquid phase to the gas phase, it has to be subtracted from the gas‐phase energy to obtain the enthalpy of the compound in solution:

(20.19)

ZPE corrections or thermal correction terms for the enthalpy in the gas phase can also be used as corrections to the gas‐phase energies of the compounds for the calculation of the heat of reaction.

Similarly to Eq. (20.13) for the free energy of reaction ΔG_r, the heat of reaction ΔH_r can be calculated from the enthalpy of the compounds:

(20.20)

20.3.2 Equilibrium Constants

The equilibrium constant K of a reaction is related to the free energy of reaction by

(20.21)

For a reaction in an ideal solution, i.e. in infinite dilution, the equilibrium constant can be calculated using the reaction free energy in solution according to Eq. (20.15). The free energies of the individual compounds can be computed using different quantum chemical correction terms as described above.

20.3.3 Accuracy

In the described procedure the free energies of the compounds are calculated from two main contributions, the quantum chemical gas‐phase energy and the free energy of solvation. The accuracy of the resulting reaction energy is determined mainly by the accuracy of the underlying quantum chemical method. With DFT methods like the BP86 functional, errors of the absolute reaction energy can be in the range of 10 kcal/mol or more [15, 16]. However, for relative reaction energies of one reaction in different solvents in a solvent screening application, i.e. if we are looking at the variation of the solvation energy only, calculated as the COSMOtherm contribution in the liquid phase, the accuracy is much higher. For such relative reaction energy predictions considering ΔG_solv or ΔH_vap from the COSMOtherm vapor pressure prediction only, an accuracy of 0.5 kcal/mol can be expected.

For a higher accuracy of absolute predictions of the reaction energy or enthalpy, it follows that a more accurate quantum chemical method should be applied for the calculation of the gas‐phase energy, e.g. the MP2 or coupled‐cluster methods combined with adequate basis sets. Quantitative improvement of the total free energy of a compound G(i) can also be achieved by the ZPE and thermal corrections to the gas‐phase energy of the compound.

20.3.4 Calculation of the Free Energy of Reaction and Heat of Reaction

A procedure for the computational prediction of the free energy of reaction ΔG_r or heat of reaction ΔH_r using quantum chemical gas‐phase energies and free energies of solvation or heats of vaporization from COSMO‐RS is described in the following:

Compute the reactant and product molecules using the DFT methods for which COSMOtherm is parameterized. Both COSMO and gas‐phase quantum chemical calculations are required. Then use COSMOtherm to obtain the ΔG_solv and/or ΔH_vap values.
Compute gas‐phase energies E_gas of the reactant and product molecules with a high‐level ab initio method, e.g. the coupled‐cluster method.
Compute vibrational frequencies for reactant and product molecules to obtain the ZPE correction or the thermal corrections to the gas‐phase energies of the compounds. Vibrational frequency calculations at a DFT level of theory are usually sufficiently accurate.
Combine E_gas, ZPE or thermal corrections, and ΔG_solv to compute the compounds G(i) according to Eqs. (20.16) and (20.17) or E_gas, ZPE or thermal corrections, and ΔH_vap to compute the compounds H(i).
Calculate the free energy of reaction ΔG_r or the heat of reaction ΔH_r from the difference of the sums of free energies or enthalpies on both sides of the reaction as described by Eqs. (20.13) and (20.20).

EXAMPLE PROBLEM 20.3 Estimate the Heat of Reaction ΔH_r for the Reduction of Nitrobenzene to Aniline in the Liquid Phase in THF at 25 °C:

Following the procedure described above, we calculate the gas‐phase energies and the heats of vaporization. In this example, we calculate the gas‐phase energies of the compounds on the MP2/TZVPP quantum chemical level. Furthermore, we employ the zero‐point vibrational energy (ZPE) and thermal corrections to the enthalpy in the gas phase to refine the calculated heat of reaction.

For the calculation of the heat of vaporization with COSMOtherm, we have to provide COSMO files for all compounds involved in the reaction, including the solvent, and the gas‐phase energies for the reactants and products. For the compounds involved here, the COSMO files and gas‐phase energies are available from the database included in the COSMOtherm package. However, to exemplify the procedure, we will give a short overview of the required quantum chemical calculations.

Since we require the gas‐phase energy as well as the screening charge surface of the compounds for the calculation of the heat of vaporization, we need to do geometry optimizations both in the gas phase and in the conductor using the COSMO model with ideal screening. COSMOtherm is parameterized for the BP86/TZVP and the BP86/SVP//AM1 quantum chemical levels. Here, we will use the higher one of the two levels, i.e. the BP86 functional and the TZVP basis set. The QC calculations are performed using the TURBOMOLE [17] quantum chemical program suite.

For the QC calculation we have to provide 3D structures of the compounds, which can be generated by an external tool or built with the molecular builder of the TURBOMOLE graphical user interface. With the starting geometries, the following QC calculations are performed for each compound:

A gas‐phase geometry optimization on the BP86/TZVP quantum chemical level.
A geometry optimization on the BP86/TZVP/COSMO quantum chemical level.
A gas‐phase geometry optimization on the MP2/TZVPP quantum chemical level.
A vibrational frequency calculation with the optimized BP86/TZVP molecular gas‐phase structure to obtain the ZPE.
Using the results from the vibrational frequency calculations, we also compute the thermal contributions with the corresponding interactive tool of the TURBOMOLE suite.

These steps are described in more detail in the Appendix. Further information about how the quantum chemical calculations are carried out can also be found in the documentation of the TURBOMOLE program suite and the TmoleX documentation.

The heats of vaporization H_vap are calculated with COSMOtherm at a temperature of 25 °C and a solvent composition of pure tetrahydrofuran (THF). For the COSMO‐RS vapor pressure prediction, quantum chemical gas‐phase energies from the BP/TZVP level are used to calculate the chemical potential of the compound in the gas phase.

The calculated data for the heats of vaporization of the compounds can be extracted from the COSMOtherm output file (Figure 20.14). In Table 20.3 the results for the individual energy contributions of the compounds from both the QC calculations and the COSMOtherm calculation are tabulated together with the resulting data for the heat of reaction. The data for the heat of reaction were calculated from ΔH_r = [H(3) + 2 ⋅ H(4)] − [H(1) + 3 ⋅ H(2)], taking into account the stoichiometry of the reaction (Eq. 20.20). For the enthalpy values of the compounds H(i), different correction terms for the quantum chemical gas‐phase energies were employed. For comparison, heats of reaction using the quantum chemical gas‐phase energies from the MP2/TZVPP level and the BP86/TZVP level are tabulated.

The catalytic reduction of aromatic nitro compounds in the gas phase is known to be a highly exothermic process. For the gas‐phase reduction of nitrobenzene, a heat of reaction of ΔH_r = −131 ± 3 kcal/mol was published [18]. Absolute values for the heat of reaction of the reduction of nitrobenzene to aniline in THF solution could not be found in the literature, but heats of reaction for the reduction of other R‐NO₂ compounds in solution have been found independent of R or solvent to be in the range of −125 to −130 kcal/mol.¹

Using MP2/TZVPP gas‐phase energies for the compounds, thermal corrections, and the heat of vaporization from the COSMO‐RS prediction, the calculated heat of reaction in solution is ΔH_r = −127 kcal/mol. With ZPE correction only, the calculated heat of reaction in solution is ΔH_r = −125 kcal/mol. Both predicted values are well inside the range of the experimental data for comparable reactions. In contrast, when ZPE and thermal corrections are ignored, the heat of reaction is overestimated by several kcal/mol with a value of ΔH_r = −141 kcal/mol. Table 20.3 also shows heats of reaction calculated from the BP86/TZVP gas‐phase energies. The heat of reaction without ZPE or thermal corrections is ΔH_r = −125 kcal/mol, which agrees well with experimental data, but when the correction terms, which should in general lead to a better prediction, are included, the heat of reaction in solution is significantly underestimated with predicted values of ΔH_r = −109 kcal/mol and ΔH_r = −111 kcal/mol, respectively. However, it should be noted that absolute errors in the range of 10–20 kcal/mol are not unusual for pure DFT functionals like BP86 [14, 15].

EXAMPLE PROBLEM 20.4 Estimate Reaction Free Energy, Equilibrium Constant, and Equilibrium Composition for the Reaction of 1‐Methoxy‐2‐Propanone 1 and Isopropylamine 2

First, the gas‐phase energies of the reactants and products of the transamination reaction (Figure 20.15) are calculated quantum chemically. In this example, we also employ the MP2 level of theory and the TZVPP basis set. As starting structures for the gas‐phase geometry optimizations, the 3D structures from the BP86/TZVP level are used, which are available from the COSMObase, a database of validated COSMO files and gas‐phase structures. With these structures we perform gas‐phase geometry optimizations using the MP2/TZVPP method and basis set combination.

As a further refinement for the gas‐phase energies, vibrational frequency calculations for the compounds are carried out, and the thermal contributions from the vibrational frequencies to the gas‐phase energy are computed using the corresponding tool of the TURBOMOLE suite. These calculations are carried out on the BP86/TZVP level.

In our next step for the calculation of the reaction free energy, the free energy of solvation of each compound is calculated from Eq. (20.18) using the partial pressures from the vapor pressure prediction of the COSMOtherm program. Reactants, products, and solvent for the reaction are taken from the COSMObase. The conditions for the vapor pressure calculation are set to a temperature of 50 °C, and the solvent composition is set to pure water. Quantum chemical gas‐phase energies of the compounds are used to calculate the chemical potentials of the compounds in the gas phase. The calculated partial pressures can be extracted from the COSMOtherm output file (Figure 20.16).

Energy terms and corrections for the reactants and products from this procedure are tabulated in Table 20.4. Calculated data for the free energy of reaction ΔG_r are also tabulated using different QC correction terms. ΔG_r is calculated according to the stoichiometry of the reaction as ΔG_r = [G(3) + G(4)] − [G(1) + G(2)].

With the MP2/TZVPP gas‐phase energies and the solvation free energies of the compound, the free energy of reaction in solution is ΔG_r = −3.27 kcal/mol, corresponding to an equilibrium constant of ln K = 5.10. The zero‐point vibrational energy corrections for the reactants and products have very little influence on the overall reaction energy (ΔG_r = −3.29 kcal/mol, ln K = 5.12 kcal/mol), but when thermal corrections are included in the free energies G(i) of the compounds, the free energy of reaction decreases to ΔG_r = −2.20 kcal/mol, and the equilibrium constant decreases to ln K = 3.43. The equilibrium constant indicates that the equilibrium position of the reaction lies on the right‐hand side of the reaction equation. The relative amount of reactants at equilibrium is 0.08 each, and the relative amount of products at equilibrium is 0.42 each. The experimental equilibrium constant for the reaction of 1‐methoxy‐2‐propanone with isopropylamine is K = 7.8 (ln K = 2.05) [11]. Thus, the calculated free energy of reaction and equilibrium constant are in excellent agreement with the experimental data.

TABLE 20.3 Gas‐Phase Energies, Heat of Vaporization, Zero‐Point Vibrational Energies, Thermal Corrections, Total Enthalpies of Compounds, and Heat of Reaction for the Hydrogenation of Nitrobenzene

	1	2	3	4	ΔH_r
	1	2	3	4	MP2/TZVPP	(BP86/TZVP)
H_vap	12.02	1.35	14.54	10.24
E_gas (BP86/TZVP) (Hartree)	−436.944 122	−1.177 446	−287.715 215	−76.465 165
E_gas (MP2/TZVPP) (Hartree)	−435.958 485	−1.164 647	−287.000 283	−76.323 461
ZPE (BP86/TZVP) (Hartree)	0.099 295	0.009 850	0.113 236	0.020 640
ΔH_T (thermal corrections)	67.36	8.26	75.45	15.32
H = E_gas − H_vap	−273 580.11	−732.17	−180 109.95	−47 903.93	−141.19	(−125.06)
H = E_gas + ZPE − H_vap	−273 517.80	−725.99	−180 038.89	−47 890.98	−125.08	(−108.95)
H = E_gas + ΔH_T − H_vap	−273 512.75	−723.92	−180 034.50	−47 888.61	−127.22	(−111.09)

Enthalpy terms are in kcal/mol; quantum chemical gas‐phase energies and ZPE are in Hartree.

Reaction schematic of compound 1 added with compound 2 in the presence of enzyme and H2O, 50°C producing compound 3 added with compound 4. — **FIGURE 20.15** Catalytic transamination of 1‐methoxy‐2‐propanone and isopropylamine.

TABLE 20.4 Energy Terms and Energy Corrections for the Reactants and Products for Reaction 2 and Free Energies of Reaction with the Various Correction Terms

	1	2	3	4	ΔG_r	ln K
Log₁₀(P(i) [mbar])	2.720 32	4.112 02	3.123 39	3.755 18
G_solv	−0.41	1.64	0.18	1.12
E_gas (MP2/TZVPP) (Hartree)	−307.100 753	−174.113 883	−288.440 644	−192.779 317
ZPE (BP86/TZVP) (Hartree)	0.112 968	0.117 222	0.149 297	0.080 872
μ_vib (thermal corrections)	54.44	227.770	71.54	32.03
G = E_gas (MP2) + G_solv	−192 709.05	−109 256.47	−180 999.06	−120 969.74	−3.27	5.10
G = E_gas (MP2) + G_solv + ZPE	−192 638.16	−109 182.91	−180 905.38	−120 918.99	−3.29	5.12
G = E_gas (MP2) + G_solv + μ_vib	−192 660.99	−109 202.03	−180 927.52	−120 937.70	−2.20	3.43
Relative amount of compound in equilibrium	0.08	0.08	0.42	0.42

The relative amount of the compounds in equilibrium has been calculated from the equilibrium constant ln K = 3.43. Free energy terms and chemical potential are in kcal/mol; quantum chemical gas‐phase energies and ZPE are in Hartree.

20.4 SCREENING OF CO‐CRYSTALS

Co‐crystalline materials consisting of an active pharmaceutical ingredient (API) and an excipient, usually termed coformer (co‐crystal former), have attracted strong interest in recent years by pharmaceutical research and development. This is mostly due to the fact that co‐crystals provide a leverage to modify the physicochemical properties of a drug substance, in particular its solubility and dissolution behavior and hence its bioavailability. In addition, they open up new interesting possibilities from an intellectual property and patent perspective. This has generated the need for the rational design of such systems and for approaches that go beyond a simple trial‐and‐error process.

The application of COSMO‐RS theory to this field emerged rather recently when it was demonstrated that the excess enthalpy (or mixing enthalpy) of the supercooled co‐crystal melt as computed by COSMOtherm gives rather good correlation with the materials’ tendency to co‐crystallize [19].

The excess enthalpy H_ex of a mixture is defined as the enthalpy difference between the mixture (H_mix) and the pure state (H_pure) for each component i with mole fraction x_i:

(20.22)

The excess enthalpy H_ex corresponds to the mixing enthalpy ΔH_mix of the system, whereas the latter usually refers to integer moles of the reactants. The more negative the excess enthalpy, the more the compounds prefer the mixture over their pure states. In this respect H_ex of a binary system may also be understood as a quantifier of complementarity of two structures. Figure 20.17 shows a histogram of the computed H_ex for about 500 molecule pairs including experimentally confirmed co‐crystals (blue) and failed co‐crystallization attempts (red). It is quite apparent that there is a significant correlation with the tendency to co‐crystallize, in particular at regions with strongly negative H_ex. For positive H_ex co‐crystal formation becomes significantly less probable, whereas in the area slightly below 0 kcal/mol, the discrimination between co‐crystals and non‐co‐crystals is more difficult. A similar picture as in 4.1 would be obtained using the Gibbs free energy of mixing ΔG_mix, which is related to H_ex:

(20.23)

Here, G_ex is the excess free energy and S_ex the excess entropy in the mixture. In retrospective, the relation of liquid‐phase properties with co‐crystal formation can be better understood by the examination of the thermodynamic cycle for co‐crystallization (Figure 20.18) that links the free energy of co‐crystal formation Δ_rG_cocrystal to the free energy of mixing ΔG_mix.

In this thought experiment the drug A and the coformer B are transferred into their respective supercooled liquid state by expending the free energy of fusion ΔG_fus. Mixing of A and B yields the co‐crystal A_mB_n in its supercooled liquid state (ΔG_mix), which then upon solidification gives the final co‐crystal. The correlation of the mixing or excess energies (H_ex,ΔG_mix) with co‐crystal formation can be explained by the assumption that the remaining terms, the sum of ΔG_fus(A) and ΔG_fus(B) of reactants A and B, and the −ΔG_fus(AB) of the co‐crystal are of similar magnitude and hence cancel each other partially when going along the thermodynamic cycle:

(20.24)

Hence, the term ΔΔG_fus (or at least its variance) needs to be small compared with ΔG_mix in order to have a good prediction (ranking). Furthermore, the overall entropic changes related to the conversion of solids A and B to solid A_mB_n will be comparably small; thus the enthalpy, i.e. H_ex, and not the free energy should be the prevailing quantity in this process. Although the drug–coformer interaction in co‐crystals is in many cases dominated by intermediate to strong hydrogen bonding, electrostatics and vdW interaction also play an important role in the condensed state, all of which are properly taken into account by COSMO‐RS theory.

So far only thermodynamic effects have been considered; kinetic effects however may also play a role for crystallization. Therefore, for rather large molecules, it is recommended to introduce an additional term, i.e. a penalty, that takes into account the molecular flexibility that poses a major barrier for crystallization. It was found that using this number of rotatable bonds as a simple additional linear term gives an improvement in predictivity for flexible systems. This can be rationalized by the assumption that the probability of crystallization is an exponentially decreasing function of the molecular flexibility that becomes an essentially linear term when being converted to the energy domain.

Table 20.5 shows the result of the excess enthalpy‐based screening for the anti‐malaria drug artemisinin against 67 potential coformers. Experimentally only two co‐crystals, namely, resorcinol and orcinol, have been found. The data from the experimental screening has been taken from the study of Karki et al. [20].

TABLE 20.5 Results of the Co‐crystal Screening for the Anti‐malaria Drug Artemisinin and 67 Coformers, Where for the Sake of Conciseness Only the Top and the Bottom of the Whole Coformer Set Are Presented

Source: Experimental data taken from Ref. [19].

Rank	Coformer	Co‐crystal	H_ex (kcal/mol)
1	Ethanedisulfonic acid	No	−3.86
2	1,3‐Dihydroxynaphthalene	No	−2.93
3	Phloroglucinol	No	−2.53
4	Benzenesulfonic acid	No	−2.01
5	*Resorcinol*	*Yes*	*−1.98*
6	Olivetol	No	−1.94
7	*Orcinol*	*Yes*	*−1.84*
8	Gentisic acid	No	−1.80
9	Cyanuric acid	No	−1.78
10–63	…	No	…
64	Lysine	No	0.78
65	Arginine	No	0.82
66	L‐Serine	No	0.82
67	Alanine	No	0.92

Coformers are sorted due to increasing excess enthalpy.

Both experimentally confirmed co‐crystals are placed at the fifth and seventh positions on the whole list of 67 tested coformers. This highlights the basic idea of a COSMO‐RS‐based prescreening that is not to obtain absolute predictions but to improve the chances to successfully find a coformer for a follow‐up experimental search. As expected it shows some disagreement with experiment, which has mainly two reasons. Firstly, several approximations have been made from a model point of view, the severest one certainly the total neglect of crystalline long‐range order. Secondly, from an experimental point of view, it can never be ruled out that some of the low excess enthalpy pairs indeed form a co‐crystal that just has not been found in the experiment. Hence, it is possible that some of the false‐positive low excess pairs may still form a co‐crystal at the right experimental conditions. In this context a study by Corner et al. has to be mentioned where COSMOtherm was used to prescreen coformers for the highly polymorphic drug precursor ROY [21]. Here, one of the top predicted coformers (pyrogallol) did not form a co‐crystal but instead an amorphous complex with ROY. In other words, ROY could be stabilized in its amorphous form due to a COSMOtherm‐predicted strong intermolecular interaction with pyrogallol. This is an important finding considering that usually the amorphous form shows a higher solubility/bioavailability than its crystalline counterparts.

So far, the excess enthalpy is merely a quantity that ranks the potential coformers relatively and does not give absolute prediction or 1/0 classification of co‐crystal formation. However, according to Figure 20.17, the probability to form a co‐crystal decreases strongly close above 0 kcal/mol, and pairs having a positive excess enthalpy can be ruled out as co‐crystal formers with high confidence. The introduction of a fixed optimal threshold for a co‐crystal vs. non‐co‐crystal classification is possible, but the exact size of the threshold may vary with the specific targeted drug under consideration.

A thorough statistical analysis of a whole array of drug and coformer sets has been carried out in Reference [19] and in [23] using the area under the curve (AUC) metric from a receiver operating characteristic (ROC) curve. Using an up‐to‐date COSMOtherm parameterization, the overall performance including the penalty term amounts to an AUC of about 0.83 (AUC = 0.81 without the penalty term) on a set of 20 drugs. In other words, the approach ranks any drug–excipient pair that forms a co‐crystal before any non‐co‐crystal pair by a probability of about 83%.

An interesting aspect is related to the original objective of improving solubility via transferring the drug into a co‐crystal: apart from kinetic effects during dissolution, the thermodynamic solubility can only be improved as compared with the pure drug if there is significant interaction between drug and coformer in solution, thereby reducing the activity of the drug as compared with its state without the coformer. This will however strongly correlate with the strength of the excess enthalpy. Hence screening for low excess enthalpy mixtures will simultaneously improve the chance of finding coformers that improve the overall solubility.

Solvates are strongly related to co‐crystal as they differ only by the fact that one of the components is liquid at room temperature. Consequently, some attempts have been undertaken to correlate the excess enthalpy with the tendency of a drug to form a solvate [19, 24]. Again, the correlation is significant, however in many cases less pronounced than for co‐crystals. Usually interactions between solvents and drug are smaller than for coformer and drug, and packing effects are more important. In particular solvents having the right shape to fit into voids or channels of the crystalline may form inclusion compounds in the crystalline state that will show a H_ex close to zero and hence are out of the scope of the current approach. However, it was shown recently that overall predictions can be improved by considering the shape of the solvents and the topology of the drug [25].

A special subset of solvates are hydrates where the solvent that crystallizes together with the drug is water. Abramov tested the excess enthalpy approach for a set of 61 diverse hydrate‐ and non‐hydrate‐forming drugs [22]. The aim of this study was to predict not only the propensity of the pharmaceutical compounds to form a hydrate but also the resistance of co‐crystals to hydration at high humidity conditions. Besides G_ex and H_ex, also other descriptors were taken into account such as the octanol–water partition coefficient (clog P), topological surface area (TPSA), and different donor–acceptor counts.

As a result it was demonstrated that the excess energy approach based on COSMO‐RS offers the most efficient way to probe the tendency of a drug to form a hydrate. Moreover, for the investigated caffeine and theophylline co‐crystals, the size of the excess enthalpy correlated well with the improved resistance toward hydration at 98% RH conditions.

EXAMPLE PROBLEM 20.5 Compute the Excess Enthalpy for Bicalutamide and a Set of Potential Coformers

The experimental data for this data set is taken from the work of Zaworotko and coworkers [26] where the drug bicalutamide and several coformers where experimentally screened for their ability to form co‐crystals. The target molecule bicalutamide is a nonsteroidal antiandrogen and mainly used in the treatment of prostate cancer. The coformers used in this example are shown in Table 20.6. For a larger list of potential coformers, it is recommended to provide COSMOthermX with a list of compound names, for example, in the form of a text file. A subsequent database request is then carried out, and the compounds that are available are automatically added to the compound list. Once all the compounds are selected, the further procedure just consists in selecting the co‐crystal module, choosing the right target API, and then adding the potential coformers. After submitting the job the excess enthalpies and mixing free energies are computed according to Eqs. (20.22) and (20.23).

The results of the screening are shown in Table 20.6. As all coformers are rather small and stiff, it was not necessary to take the molecular flexibility into account here, and only the excess enthalpy is shown. Table entries that have formed a co‐crystal/non‐co‐crystal in the laboratory are labeled with a 1 or 0, respectively. The table is sorted according to increasing excess enthalpy, and except for 1,2‐bis(4‐pyridyl)ethane, the top entries with the lowest excess enthalpy form indeed a co‐crystal.

Although co‐crystals are most often dominated by hydrogen bonds, this case demonstrates that it is important to consider all intermolecular interactions in a screening. For example, taking only hydrogen bonding into account may lead to the false high ranking of essentially hydrophobic compounds such as 1‐hexadecanol that are barely miscible with bicalutamide [27].

TABLE 20.6 COSMO‐RS‐Based Co‐crystal Screening for Bicalutamide (Example Problem 20.5)

Coformer	H_ex (kcal/mol)	Co‐crystal
1,2‐Bis(4‐pyridyl)ethane	−1.57	0
trans‐1,2‐Bis(4‐dipyridyl)ethylene	−1.21	1
4,4′‐Bipyridine	−1.13	1
1‐Naphthol	−0.99	0
1,3,5‐Trihydroxybenzene	−0.89	0
4,4‐Biphenol	−0.71	0
4‐Phenylpyridine	−0.68	0
1,3‐Dihydroxybenzene	−0.62	0
4‐Cyanopyridine	−0.24	0
3‐Cyanopyridine	−0.19	0
3‐Cyanophenol	0.01	0
1‐Naphtalenecarbonitrile	0.01	0
4‐Cyanophenol	0.10	0
1,3‐Benzenedicarbonitrile	0.13	0
1,4‐Benzenedicarbonitrile	0.13	0
3‐Hydroxypyridine	0.30	0
5‐Hydroxyisoquinoline	0.31	0
1‐Hexadecanol	1.28	0

Entries are ordered by increasing excess enthalpy H_ex.

20.5 CONCLUSION AND OUTLOOK

In this chapter we presented an overview of three different applications of COSMO‐RS in the drug development process. The power and main benefit of the COSMO‐RS model is that properties in solution can be obtained from ab initio calculation without any experimental input. It does not require external data for modeling and can also be used when empirical models are not parameterized. Complex multifunctional molecules and new chemical functionalities are treated on the same footing as simple organic molecules.

Prediction of relative drug solubility with COSMO‐RS is based on a consistent thermodynamic modeling of interactions in the solvent and the supercooled state of the drug. Solvent mixtures can be treated in the same way as pure solvents and with similar accuracy. Absolute solubility prediction is limited by the availability of free energy of fusion data.

Although COSMO‐RS in its present state cannot be proven to be more accurate than more empirical models with many adjusted parameters, its strength is the essential independency from experimental data. This allows for independent modeling and avoids errors resulting from erroneous experimental data on which empirical models rely. Potential improvement to the current COSMO‐RS solubility prediction model includes a more accurate fusion term for absolute solubility prediction, and improvement of the COSMO‐RS interaction terms themselves, and requires more reliable experimental data as are available at present.

With COSMO‐RS, solubility prediction is also possible for salts. This is important as many drugs are formulated as salts. Solvent systems involving ionic liquids can also be treated with very good accuracy [28]. Furthermore, different conformational forms of molecules can be used for solubility screening, and the relative weight of conformers in different solvents can be determined. This feature basically allows for examination of conditions influencing the crystallization process and solvent screening for conformational polymorphism and pseudopolymorphism.²

Another application of COSMO‐RS frequently used in pharmaceutical and agrochemical industry deals with reaction modeling. In principle, reaction equilibrium, mechanism, rate, and by‐product formation may be solvent dependent. Here we investigated the influence of solvation on the free energy and heat of reaction and the reaction equilibrium only. A straightforward procedure for the computational prediction of the free energy of reaction and heat of reaction has been shown, and the effect of the employed quantum chemical level on the absolute heat of reaction has been demonstrated. Elsewhere, it has been shown that although, depending on the quantum chemical level, absolute values for the free energy of reaction may differ substantially from experimental data, general trends are predicted correctly [29].

Finally, the prediction of co‐crystal formation can be carried out using COSMO‐RS. Although not being a mere solution process, it is however strongly related to it. In fact, the problem of co‐crystal formation that cannot be treated yet rigorously in all its complexity is approximated by assuming the co‐crystal to be a supercooled mixture of API and coformer. This is of course a strong simplification, with a larger error bar than the usual COSMO‐RS applications. Nevertheless, as it was demonstrated in this chapter, it allows to guide experimental work quite efficiently.

All applications of COSMO‐RS require quantum chemical calculations of compounds, taking into account the various molecular conformations of the compounds. This constitutes the computationally most demanding part of the procedure but has to be done only once per compound. The COSMO files of the involved compounds can be reused for other projects and all types of properties. Thus, if combined with a database of precalculated COSMO files for common compounds, thermodynamic property calculations with COSMOtherm can be carried out quite fast and efficiently.

20.A DETAILS OF COSMO AND GAS‐ PHASE CALCULATIONS

For later use of the COSMO files in COSMOtherm, the details of the quantum chemical COSMO calculation should be consistent with one of the parameterizations of COSMOtherm.

There are three levels of different qualities mainly used in COSMOtherm: a lower, computationally less expensive level; an intermediate level, which is sufficiently accurate for most problems; and a higher level, which is computationally more time consuming and also resolves some particular problem cases. As the intermediate level is a good compromise between accuracy and efficiency, it has been used throughout in this contribution and requires the following:

BP86 DFT geometry optimization with a TZVP quality basis set and the RI approximation applied in the gas phase and in the conductor.
For COSMO calculations only: COSMO applied in the conductor limit (ε = ∞) using optimized element‐specific COSMO radii for the cavity construction.
If more than one conformation is considered to be potentially relevant for a compound, QC calculations have to be done for all conformations.

20.B STEPS FOR CALCULATING THE FREE ENERGY OR ENTHALPY OF A COMPOUND

Apart from the heat of vaporization or free energy of solvation that is calculated by COSMOtherm, the free energies and enthalpies of compounds involved in the reaction examples are composed from several quantum chemical contributions.

20.B.1 Gas‐Phase Energy

The BP86/TZVP gas‐phase energy is obtained from a gas‐phase geometry optimization of the compound using the settings described in Appendix 20.A. The resulting energy value can be found in output files of the TURBOMOLE program suite. The protocol file job.last comprises the output of the last complete cycle and information about the settings that were applied in the program run, e.g. convergence criteria. The gas‐phase energy can be read from “total energy” line in the output section displayed in Figure 20.B.1. If the graphical user interface TmoleX is used, the gas‐phase energy can also be taken from the Energy block of the Job Results panel. This panel also displays information about the run of the job (Figure 20.B.2).

Screenshot of TmoleX 4.2-TmoleXProject_3 window displaying arrows linking boxes labeled geometry, method, etc. A dialog box below has a graph of energy vs. geometry cycle with a descending line with square markers. — **FIGURE 20.B.2** Job Results panel of the graphical user interface TmoleX of the TURBOMOLE program suite.

The MP2/TZVPP gas‐phase energy is obtained from an MP2 geometry optimization employing the TZVPP basis set (also called def‐TZVPP) and the RI approximation. The module used for this type of calculation is ricc2 (Figure 20.B.3). In TmoleX, the method has to be set to RI‐MP2 in the “Level” section of the “Level of Theory” panel. In the MP2 calculations for this contribution, the 1s orbitals of elements C, N, and O were kept frozen. Details about settings for frozen core orbitals can be found in the TURBOMOLE documentation.

20.B.2 Correction Terms from Vibrational Frequency Calculations

For the ZPE contribution, a vibrational frequency calculation is required. This type of calculation can be done with the aoforce module of TURBOMOLE. The ZPE can be taken either directly from the aoforce.out file or from the interactive module freeh, which also allows for calculation of the molecular partition functions at temperatures other than 0 K and finite pressure. Results of the freeh module are printed to standard I/O (Figure 20.B.4). Note that vibrational frequency calculations are based on the assumption of a harmonic oscillator and partition functions are computed within the assumption of an ideal gas and no coupling between degrees of freedom. Vibrational frequency calculations can also be started from TmoleX. Start the job as a single‐point calculation with the “Frequency Analysis” radio button ticked in the “Job Selection” section of the single‐point calculation panel.

The thermal enthalpy contribution ΔH_T can be calculated from the thermal energy contribution printed in the freeh output. This is done via ΔH_T = ΔG_T + RT, where ΔG_T is the value in the “energy” column of the freeh output, T is the temperature, and R is the gas constant R = 8.314 472 J/K∙mol.

For the calculation of the thermal contribution to the free energy of a compound, μ_vib, the freeh value for “chem. pot.” should be used and not the value from the “energy” column.

After converting the contributions from the individual steps to consistent units, the total enthalpy H of a compound can be calculated as the sum of the individual contributions. For compound nitrobenzene on the MP2/TZVPP level of theory, this involves the sum H = E_gas(MP2) + ΔH_T − H_vapconsisting of the following contributions listed in Table 20.3:

E_gas(MP2) = −435.958 485 Hartree = −273 568.09 kcal/mol from the MP2 gas‐phase geometry optimization.
ΔH_T = ΔG_T + RT = 279.36 kJ/mol + 8.314 472 J/(K∙mol) · 298.15 K = 281.84 kJ/mol = 67.36 kcal/mol from the vibrational frequency calculation and subsequent computation of partition functions with the freeh module.
H_vap = 12.02 kcal/mol from the COSMO‐RS vapor pressure prediction.

Note that quantum chemical energies are usually expressed in Hartree, which is the atomic unit of energy. The conversion factor to the kcal/mol unit system is 627.5095 kcal/mol.

The free energy G of a compound can similarly be calculated from

For compound isopropylamine, this requires the following energy terms, listed in Table 20.4:

E_gas (MP2) = −174.113 883 Hartree = −109 258.16 kcal/mol from the MP2 gas‐phase geometry optimization.
μ_vib = 227.77 kJ/mol, the chemical potential value (“chem. pot.”) from a vibrational frequency calculation and subsequent computation of partition functions with the freeh module.
G_solv = 1.64 kcal/mol, calculated from ΔG_solv(i) = RT ln(10)[log₁₀(P(i)) − log₁₀(P)] (Eq. 20.18). The partial pressure of isopropylamine, log₁₀(P(i)) = 4.112 02 mbar, was to be taken from a COSMOtherm vapor pressure prediction (Figure 20.16), and the reference pressure P was assumed to be 1 bar.

ABBREVIATIONS

BP86	approximate DFT functional, consisting of Becke’s exchange functional [30] and Perdew’s correlation functional [31]
COSMO	COnductor‐like Screening MOdel
COSMO‐RS	COnductor‐like Screening MOdel for Real Solvents
DFT	density functional theory
GCM	group contribution method
MC	Monte Carlo method
MD	molecular dynamics method
MP2	second‐order Møller–Plesset perturbation theory
QC	quantum chemistry/quantum chemical
rmse	root mean square error
TZVP, TZVPP	Ahlrichs’ triple‐zeta valence polarization basis sets [32]
vdW	van der Waals interaction

SYMBOLS

σ	screening charge density
ΔG_fus	free energy of fusion
ΔG_solv	free energy of solvation
ΔH_vap	heat of vaporization
E_gas	gas‐phase energy
μ	chemical potential

REFERENCES

1. Klamt, A. and Schüürmann, G. (1993). COSMO: a new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient. J. Chem. Soc. Perkin Trans. 2 799–805.
2. Klamt, A. (1995). Conductor‐like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena. J. Phys. Chem. 99: 2224–2235.
3. Klamt, A. (2005). COSMO‐RS: From Quantum Chemistry to Fluid Phase Thermodynamics and Drug Design. Amsterdam: Elsevier.
4. Gmehling, J. (1998). Present status of group‐contribution methods for the synthesis and design of chemical processes. Fluid Phase Equilibr. 144: 37–47.
5. Klamt, A., Jonas, V., Bürger, T., and Lohrenz, J.C.W. (1998). Refinement and parameterization of COSMO‐RS. J. Phys. Chem. A 102: 5074–5085.
6. Klamt, A. and Eckert, F. (2000). COSMO‐RS: a novel and efficient method for the a priori prediction of thermophysical data of liquids. Fluid Phase Equilibr. 172: 43–72.
7. Ben‐Naim, A. (1987). Solvation Thermodynamics. New York and London: Plenum Press.
8. Klamt, A., Eckert, F., Hornig, M. et al. (2002). Prediction of aqueous solubility of drugs and pesticides with COSMO‐RS. J. Comput. Chem. 23: 275–281.
9. Granberg, R.A. and Rasmuson, A.C. (1999). Solubility of paracetamol in pure solvents. J. Chem. Eng. Data 44: 1391–1395.
10. Manzo, R.H. and Ahumada, A.A. (1990). Effects of solvent medium on solubility. V: Enthalpic and entropic contributions to the free energy changes of Di‐substituted benzene derivatives in ethanol: water and ethanol: cyclohexane mixtures. J. Pharm. Sci. 79: 1109–1115.
11. Matcham, G., Bhatia, M., Lang, W. et al. (1999). Enzyme and reaction engineering in biocatalysis: synthesis of (S)‐methoxyisopropylamine (= (S)‐1‐methoxypropan‐2‐amine). Chimia 53: 584–589.
12. Mota, F.L., Carneiro, A.P., Queimada, A.J. et al. (2009). Temperature and solvent effects in the solubility of some pharmaceutical compounds: measurements and modeling. Eur. J. Pharm. Sci. 37: 499.
13. Romero, S., Reillo, A., Escalera, B., and Bustamante, P. (1996). The behavior of paracetamol in mixtures of amphiprotic and amphiprotic‐aprotic solvents. Relationship of solubility curves to specific and nonspecific interactions. Chem. Pharm. Bull. 44: 1061–1064.
14. Granberg, R.A. and Rasmuson, A.C. (2000). Solubility of paracetamol in binary and ternary mixtures of water + acetone + toluene. J. Chem. Eng. Data 45: 478–483.
15. Koch, W. and Holthausen, M.C. (2001). A Chemist's Guide to Density Functional Theory. Weinheim: Wiley‐VCH.
16. Cramer, C.J. (2002). Essentials of Computational Chemistry. Chichester: Wiley.
17. TURBOMOLE V7.2 2017 (2007). A development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 1989–2007. TURBOMOLE GmbH, since 2007.
18. Macnab, J.I. (1981). The role of thermochemistry in chemical process hazards: catalytic nitro reduction processes. I. Chem. E. Symp. Ser. 68 (3/S): 1–15.
19. Abramov, Y.A., Loschen, C., and Klamt, A. (2012). Rational coformer or solvent selection for pharmaceutical cocrystallization or desolvation. J. Pharm. Sci. 101: 3687.
20. Karki, S., Friscic, T., Fabian, L., and Jones, W. (2010). New solid forms of artemisinin obtained through cocrystallisation. CrystEngComm 12: 4038–4041.
21. Corner, P.A., Harburn, J.J., Steed, J.W. et al. (2016). Stabilisation of an amorphous form of ROY through a predicted co‐former interaction. Chem. Commun. 52: 6537–6540.
22. Abramov, Y.A. (2015). Virtual hydrate screening and coformer selection for improved relative humidity stability. CrystEngComm 17: 5216–5224.
23. Loschen, C. and Klamt, A. (2015). Solubility prediction, solvate and cocrystal screening as tools for rational crystal engineering. J. Pharm. Pharmacol. 67: 803–811.
24. am Ende, D., Klamt, A., Loschen, C. et al. (2014). Prediction of Energetic‐Energetic Cocrystal Forms and Their Competition with Solvate Formation [Internet]. Atlanta, GA, USA. https://aiche.confex.com/aiche/2014/webprogram/Paper383003.html (accessed 12 October 2018).
25. Loschen, C. and Klamt, A. (2016). Computational screening of drug solvates. Pharm. Res. 33: 2794.
26. Bis, J.A., Vishweshwar, P., Weyna, D., and Zaworotko, M.J. (2007). Hierarchy of supramolecular synthons: persistent hydroxylpyridine hydrogen bonds in cocrystals that contain a cyano acceptor. Mol. Pharm. 4: 401–416.
27. Musumeci, D., Hunter, C.A., Prohens, R. et al. (2011). Virtual cocrystal screening. Chem. Sci. 2: 883–890.
28. Diedenhofen, M., Eckert, F., and Klamt, A. (2003). Prediction of infinite dilution activity coefficients in ionic liquids using COSMO‐RS. J. Chem. Eng. Data 48: 475–479.
29. Peters, M., Greiner, L., and Leonhard, K. (2008). Illustrating computational solvent screening: prediction of standard Gibbs energies of reaction in solution. AIChE J. 54: 2729–2734.
30. Becke, A.D. (1988). Density‐functional exchange‐energy approximation with correct asymptotic behaviour. Phys. Rev. A 38: 3098–3100.
31. Perdew, J.P. (1986). Density‐functional approximation for the correlation‐energy of the inhomogenous electron gas. Phys. Rev. B 33: 8822–8824.
32. Schäfer, A., Huber, C., and Ahlrichs, R. (1994). Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr. J. Chem. Phys. 100: 5829–5835.

NOTES

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