Simple Alternative to Neural Networks for Predicting Sublimation

Jan 6, 2012 - Simple Alternative to Neural Networks for Predicting Sublimation Enthalpies .... Journal of Molecular Graphics and Modelling 2017 72, 22...
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Simple Alternative to Neural Networks for Predicting Sublimation Enthalpies from Fragment Contributions Didier Mathieu* CEA, DAM, Le Ripault, F-37260 Monts, France S Supporting Information *

ABSTRACT: Taking advantage of an extended data set of sublimation enthalpies recently used to develop an artificial neural network for the prediction of this property, an alternative model based on 35 atom and ring contributions is presently reported. The values predicted using both approaches are remarkably similar, although the present one is much simpler and less empirical.

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INTRODUCTION As it determines the stability of crystal structures, the sublimation enthalpy ΔH sub is a property of practical significance. For instance, in the fields of pharmacology or nonlinear optics, some criteria to assess the stability of molecular crystals before their actual synthesis are desirable. Since ΔHsub represents the contribution of intermolecular interactions to the energy content of materials, it should also be taken into account in order to estimate the performance of energetic materials or reactivity hazards of chemical substances. Finally, ΔHsub arises as an intermediary quantity in the evaluation of other thermochemical properties, including solid vapor pressure. Since sublimation enthalpies have been measured for only a small fraction of all molecular compounds of practical interest, predictive methods are frequently needed. Nevertheless, only rather few models for ΔHsub have been published so far. They have been very recently reviewed in this journal by Gharagheizi et al.1 As pointed out by these authors, early models based on additivity schemes are mostly restricted to specific chemical series or simple compounds.2−5 In fact, because of a lack of experimental data, it is not possible to develop a model for ΔHsub based on well-established additivity schemes such as the Benson6 or UNIFAC7 group contribution methods. Another popular approach involves an empirical expression for ΔHsub in term of molecular electrostatic potentials (MEPs).8−13 While MEPs provide useful descriptors to characterize intermolecular interactions, present implementations of this approach exhibit a significant limitation. As a result of a spurious quadratic dependence on the molecular surface area, they cannot be applied to extended molecules such as large n-alkanes.14 In this context, the artificial neural network (ANN) model described in ref 1 yields significant progress. Taking advantage of 1348 sublimation enthalpies from the DIPPR 801 database,15 it appears to exhibit an unprecedented predictive power, with all calculated values within 30 kJ/mol from experimental counterparts and typical deviations characterized by values of 0.9854, 3.54%, and 4.21 kJ/mol for the determination coefficient (R2), average percent error (APE), and root-meansquare error (RMSE), respectively. This ANN model appears to be applicable to virtually any pure compound, including inorganic species, salts, and © 2012 American Chemical Society

organometallic compounds. In fact, a close examination of the data set indicates that most compounds in the database are molecular crystals devoid of ionic species or metals. For the other compounds, the ANN might be unreliable for the following reasons: (1) some elements are not statistically well represented in the database; for instance, there is only one entry involving each of the elements Ga, Ge, and As; (2) the data set exhibits some errors for unusual elements, presumably owing to ambiguities in the assignment of implicit hydrogens; for instance, arsine is described as AsH5 instead of AsH3; (3) bond orders are ill-defined for organic ions owing to the fact that a qualitatively correct description of their electronic structure may require a superposition of canonical structures; as a result, their decomposition into functional groups is ambiguous. In fact, these difficult cases are systematically included in the training set of the ANN model. As a result, the predictive power of the method for these compounds remains to be demonstrated using a more comprehensive data set. In contrast, the ANN proves reliable for molecular organic crystals. However, it exhibits a number of drawbacks. Like any model based on artificial neural networks, it does not provide immediate insight into the factors contributing to the property of interest, in contrast to additivity schemes. In addition, it is relatively tedious to develop, extend, and apply, since ANNs involve many empirical nonlinear parameters to be optimized and their use requires specific software. Finally, ANN models do not lend themselves to advanced computer-aided design strategies, as emphasized recently in the context of flash point prediction.16 Taking advantage of the data reported in ref 1, the present paper reports on a much simpler model based only on 35 additive contributions to ΔHsub. In order to avoid the abovementioned issues with compounds containing unusual elements and salts, they are removed from the original database. Finally, the present data set is made of 1300 molecular compounds provided as Supporting Information. Additivity schemes are clearly not well-suited to estimate ΔHsub for salts since their cohesion is dominated by Coulomb interaction and thus Received: Revised: Accepted: Published: 2814

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an atom X in a molecule, its contribution to ΔHsub should decrease either as its coordination number nc or the size of its neighbors increases. This expectation relies on the fact it gets somewhat buried into the molecule, with a reduced surface available to interact with surrounding species. This simple picture suggests that a fragmentation scheme similar to the one recently introduced to predict the densities of molecular crystals24 might be valuable for ΔHsub as well. Indeed, this procedure precisely introduces transferable atomic contributions depending on the coordination number of the atoms and on the sizes of their neighbors. In order to avoid too many parameters to be fitted, all neighbors are assumed to have the same size except for hydrogen atoms. The contribution of any atom is then assigned a label Xnc-nH where X and nH stand, respectively, for the atomic symbol and number of hydrogen neighbors of the atom.24 For instance, terminal −COOH groups in carboxylic acids are described in term of three atomic fragments labeled respectively C3-0, O1-0 and O2-1. This nomenclature was first introduced in the original implementation of this fragmentation procedure.24 In this approach, the contribution of any hydrogen atom is dumped into that of its neighbor. Therefore, an explicit H1-0 parameter would be redundant and introduces linear dependencies in the regression equations. Moreover, in contrast to the previous density model based on similar fragments,24 no explicit hydrogen bond contribution is used in the present work, as the associated parameter proves to be statistically illdefined and does not yield any significant improvement. Instead, the contribution of hydrogen bonding to ΔHsub is implicitly dumped into the values of the parameters associated with proton donors, such as atoms labeled O2-1, N3-1, or N3-2 according to present notations. On the other hand, explicit ring contributions are retained. For a given ring, they depend on its size and aromatic character. More specifically, four contributions labeled R3 to R6 are introduced for ring sizes r = 3 to r = 6, and a single contribution R>6 is introduced for larger rings. Two specific contributions R5a and R6a are used for aromatic rings. Thus, a total of seven parameters is used to account for the role of rings. Three additional parameters are introduced to account for crowded atoms surrounded by elements beyond the first row of the periodic table, i.e., heavier than neon. These atoms are thereafter referred to as big atoms. A given atom X gets more crowded as either hydrogen or first-row atoms among its neighbors are substituted with big atoms. Accordingly, its contribution to the crystal cohesion is expected to decrease, depending on the number n of big atoms introduced among its neighbors. As a matter of fact, it turns out that introducing three Bn parameters (n = 2,3,4) proves very fruitful to account for these negative contributions to ΔHsub. No explicit B1 descriptor is required as the screening influence of an atom by a single big atom is already included in the contribution of the latter. Accordingly, such a descriptor would only introduce linear dependencies. Finally, the model thus obtained involves only 35 adjustable parameters. Moreover, their physical significance is straightforward, as they simply represent additive contributions to ΔHsub. The values of these parameters, hereafter denoted h(i) for the 35 constitutive descriptors i, are obtained in this work through a linear regression based on a singular values decomposition. The associated standard deviations δ(i) are used to assess their statistical significance. Despite its simplicity, the present model

decreases as bigger ions are considered. Alternative procedures, based either on systematic packing of ions17 or simple expressions based on ionic volumes and charges are available for such compounds.18−21

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PRESENT MODEL In sharp contrast with ref 1, the present work assumes that a linear expression is suitable to express ΔHsub in term of molecular fragments. This assumption relies on both theoretical and empirical grounds. In theory, the harmonic approximation for the vibrations of a molecule in a crystal, combined with the virial and equipartition theorems, yield an approximate expression for ΔHsub in terms of the lattice energy Elatt and temperature T:22 ΔHsub = − Elatt − 2RT −1

(1)

−1

where R ≃ 8.31 J mol K is the ideal gas constant. Because the thermal contribution −2RT decreases only by about 3 kJ/ mol as T increases by 200 K, while ΔHsub values range from 30 to 250 kJ/mol, the first contribution −Elatt is clearly predominant. A linear expression of this term into additive fragment contributions naturally arises from a mean field approximation for the crystalline environment of each fragment. An examination of experimental ΔHsub values of the homologous series further supports this assumption of linearity as illustrated in Figure 1 for n-alkanes.23 Notwithstanding the

Figure 1. Sublimation enthalpies ΔHsub of n-alkanes versus number NC of carbon atoms in the chain. Experimental values (○) and ANN values from ref 1 (□).

significant uncertainties affecting the ΔHsub data for chain lengths NC > 20, it is clear from this plot that the slight nonlinearity afforded by the ANN is not justified by present experimental values and that a simple additivity scheme would yield similar performance. The procedure employed for decomposing molecules into additive fragments and associated constitutive descriptors is a critical aspect of additivity models. Because the interactions of a given atom with neighboring molecules depend on the surroundings of this atom within the molecule under focus, the definition of additive fragments must take this influence into account, while avoiding the introduction of too many parameters needing to be fitted. For this purpose, simple considerations are usually very useful. In particular, focusing on 2815

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yields ΔHsub values very similar to the predictions of the ANN model, as detailed below.

observed as the ANN is applied to compounds in its own training set, as indicated in Table 1. Since the errors observed on applying the present scheme on the external test set are not significantly worse than the residual deviations obtained after a fit of the data in the training set, the parameters are eventually fitted against the whole data set of 1300 compounds. The overall performance obtained is again remarkably close to that reported for the ANN, as summarized in the two bottom rows of Table 1. Therefore, in spite of being much more flexible, with a 5-fold increase of the number of input variables and many more adjustable parameters involved, the ANN model yields no improvement to the fit. For both models, typical errors are close to 4 kJ/mol, with the most significant ones close to 30 kJ/mol. In fact, the errors from both models are significantly correlated, as shown in Figure 3 where present deviations

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RESULTS In order to assess the predictive value of the present method, the data set is first split into a training set of 814 compounds and an external test set of 486 compounds. The model is then fitted against the training set and subsequently applied to the test set. Figure 2 shows fitted and predicted ΔHsub values versus

Figure 2. Calculated versus observed ΔHsub values for the model fitted against the reduced training set: fitted values (black circles) and predicted values for the test set (red squares).

Table 1. Comparison of the Performance of the Present Model with the Earlier ANN Modela data set

N

R

2

APE

RMSE

MIN

Quality of Fit on Training Sets present model 814 0.983 3.44 3.97 −23.5 ANN model 1080 0.985 3.53 4.34 −28.0 Prediction Performance on External Test Set present model 486 0.988 2.54 4.07 −26.4 ANN model 134 0.988 3.31 3.54 −14.0 present modelb 134 0.992 2.61 3.04 −12.7 Performance for the Entire Data Set present modelc 1300 0.986 3.07 4.00 −25.7 ANN model 1300 0.986 3.39 4.14 −25.8

Figure 3. Present versus ANN deviations (residuals) from experiment (kJ/mol).

MAX

from experimental ΔHsub values are plotted against corresponding errors obtained using the ANN. A significant determination coefficient R2 = 0.52 is observed between both quantities. There are only three compounds for which present predictions are significantly worse than ANN predictions, namely, benzamide and tert-dodecylmercaptan for which ΔHsub is underestimated and 2,6-di-tert-butyl-p-cresol for which it is overestimated. The latter error may be easily explained by the fact that the hydroxyl group is crowded and thus not readily available for intermolecular hydrogen bonding. The three most severe overestimations from present model are for 1,2,3benzenetriol (+29.7 kJ/mol), salicylic acid (+28.7 kJ/mol), and pentaerythritol (+21.6 kJ/mol), three compounds characterized by significant intramolecular hydrogen bonding. The next most severe overestimations are for relatively large compounds, especially tricosane (+21 kJ/mol) and dibutyl phthalate (+18.2 kJ/mol). The most significant underestimations are for large compounds such as 1-triacontene (−25.7 kJ/mol) or tertdodecylthiol (−23.1 kJ/mol). For the latter, the ANN model performs significantly better with a predicted ΔHsub value only 1.7 kJ/mol below experiment. Notwithstanding this compound, the main errors observed using the present model are often close to those observed in ref 1, where the following deviations from experiment were obtained for the above-mentioned molecules: 1,2,3-benzenetriol, +10.7 kJ/mol; salicylic acid,

+28.8 +27.0 +29.2 +14.3 +13.4 +29.7 +27.0

a Determination coefficient R2, average percent error (APE), and root mean square error (RMSE) in kJ/mol, calculated using fitted (resp. predicted) ΔHsub values for training (resp. test) sets of N compounds. b Assessment of the present model using the same test set as in ref 1 for comparison. cResults obtained using the present model fitted against the entire data set.

experiment. As detailed in Table 1, the predictions on the 486 compounds in the external test set are quite satisfactory, with R2 = 0.988, RMSE ≃ 4 kJ/mol, and all values within 30 kJ/mol from experiment. In fact, the predictive power of this model appears to be quite similar to that of the ANN model. The narrower range of errors reported for the ANN predictions presumably stems from the small size of the corresponding test set which did not include any of the most significant outliers. Indeed, deviations from experiment close to ±30 kJ/mol are 2816

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Table 2. Present Fragments i, Associated Standard Contributions h(i), Corresponding Standard Deviations δ(i) (kJ/mol), and Corresponding Numbers of Occurrences Nocc(i)

+18.3 kJ/mol; pentaerythritol, +23.1 kJ/mol; tricosane, +23.3 kJ/mol; 1-triacontene, −25.8 kJ/mol. On the other hand, the ANN exhibits significant errors for compounds for which the present model performs well. Deviations from experiment for the ANN and the present method are, respectively, +27.7 and +1.4 kJ/mol for 9,10anthracenedione, +17.8 kJ/mol and +3.7 kJ/mol for 3pyridinecarbonitrile, −23.1 and −2.1 kJ/mol for RDX, and −17.2 and +8.0 kJ/mol for HMX. For the prediction of ΔHsub, the ANN clearly exhibits no advantage over the simple additive model reported here.

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PARAMETER VALUES The parameters of the present model are compiled in Table 2. It is gratifying to observe that their values are statistically welldefined and consistent with a priori expectations. The only exception is B4, for which the standard deviation δ(B4) is as much as 16% of the actual value h(B4). This significant uncertainty stems from a lack of data, carbon tetrachloride being the only compound in the data set for which this parameter occurs. Since present atomic contributions are designed on the basis of simple geometrical criteria, ignoring complexities associated with hydrogen bonding, it is of interest to focus on a first stage on atoms not bound to any hydrogen in order to identify general trends. The contributions of these atoms to ΔHsub are plotted in Figure 4 as a function of their coordination numbers nc. Striking regularities emerge from this graph. First, it is clear that the contribution of an atom to ΔHsub decreases roughly linearly with nc, in line with the anticipated screening effect of its neighbors. Moreover, notwithstanding halogens, the contributions of different elements from the same row are very similar, provided they are not bound to any hydrogen atom and exhibit the same value of nc. More generally, whatever the element considered, adding a new non-hydrogen neighbor decreases the corresponding atom contribution to ΔHsub by 12−13 kJ/mol. The fact that crowded atoms exhibit negative contributions to ΔHsub might appear puzzling at first glance as such atoms cannot be too close to another molecule and therefore can only contribute positively to the cohesion of the crystal. However, such negative contributions are easy to understand, keeping in mind the fact that non-hydrogen neighbors also partially screen each other. In the lack of explicit parameters to explicitly account for this mutual screening of geminal atoms, the latter can only be taken into account within the present scheme by an effective decrease of the contribution of the central atom to the cohesion of the crystal, hence the negative values listed in Table 2. For atoms bonded to hydrogen, the above-mentioned regularities are not observed owing to additional complexities associated with hydrogen bonding. In particular, while in the lack of a hydrogen neighbor, C, N, and O atoms exhibit similar contributions to ΔHsub as long as they share the same value of nc, the situation is very different as hydrogen neighbors are considered. For instance, h(N3-1) is about 3 times larger than h(C3-1) owing to the fact that the former implicitly includes hydrogen bonding effects. The dependence of the h parameters on the number nH of hydrogen neighbors (for a fixed value of nc) is illustrated in Figue 5. It is gratifying to observe that these parameters decrease with nc and increase with nH, as expected from the fact that they include the contribution of the neighboring hydrogens. This observation may also be explained

i

h(i)

δ(i)

Nocc(i)

C2-0 C2-1 C3-0 C3-1 C3-2 C4-0 C4-1 C4-2 C4-3 N1-0 N2-0 N3-0 N3-1 N3-2 O1-0 O2-0 O2-1 F1-0 Si4-0 Si4-1 S2-0 S2-1 S4-0 Cl1-0 Br1-0 R3 R4 R5 R5a R6 R6a R>6 B2 B3 B4

9.066 7.174 −2.293 5.522 10.800 −15.498 −2.434 6.856 11.893 23.002 9.792 −2.524 14.194 24.151 19.430 8.674 31.680 9.559 −11.622 −2.073 15.302 22.736 −9.134 14.650 17.671 9.304 6.874 8.258 9.266 6.877 14.111 6.345 −3.316 −3.961 −6.603

0.046 0.188 0.004 0.001 0.007 0.012 0.004 0.000 0.001 0.060 0.027 0.011 0.012 0.008 0.005 0.002 0.002 0.006 0.036 0.260 0.019 0.039 0.230 0.005 0.045 0.070 0.131 0.019 0.067 0.016 0.045 0.127 0.033 0.131 1.076

40 6 640 582 138 139 360 885 935 29 19 59 54 74 395 322 265 33 18 4 43 26 5 123 18 15 8 58 23 89 347 9 33 12 1

by atoms becoming more accessible to neighboring molecules as some of their neighbors are substituted with hydrogen atoms. The only exception to this general trend concern carbon with nc = 2. Indeed, h(C2-1) happens to be smaller than h(C20). With only six ΔHsub values determining the exact value of h(C2-1), this anomaly might be a spurious consequence of experimental uncertainties. However, in view of the small values obtained for the standard deviations reported in Table 2, alternative explanations might be preferred. In particular, the value of h(C2-0) is specially large because of the very polar character of the cyano group to which this parameter is most often associated.

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CONCLUSION

The simplicity of the present model is remarkable, as it allows a fairly accurate estimation of ΔHsub for most organic compounds using only pencil and paper or a hand calculator, i.e., without dedicated software. On the other hand, using freely available molecular toolkits such as Pybel,25 it can be easily applied to extended molecular databases. Being one of the most reliable 2817

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This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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Figure 4. Atom contributions to ΔHsub for atoms not bound to any hydrogen plotted as a function of the coordination number nc of the atom. Circles, squares, and diamonds refer to first, second, and third row atoms, respectively.

Figure 5. Dependence of the atom contributions to ΔHsub on the number nH of neighboring hydrogen atoms, for diverse values of the coordination numbers nc of the atoms.

procedure presently available, it is a method of choice for any chemical engineer in need of ΔHsub estimates. Another interesting conclusion from the present results is the fact that in some cases, simple additivity schemes may be competitive with recent state-of-the-art modeling methodologies, provided the corresponding fragments are carefully defined. While popular additivity models rely on standardized fragmentation schemes, such as UNIFAC or Benson groups, this work illustrates the interest of custom fragmentation schemes specifically designed for the target property to be predicted.

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REFERENCES

(1) Gharagheizi, F.; Sattari, M.; Tirandazi, B. Prediction of Crystal Lattice Energy Using Enthalpy of Sublimation: A Group ContributionBased Model. Ind. Eng. Chem. Res. 2011, 50, 2482−2486. (2) Horvath, A. L. Molecular Design; Elsevier: Amsterdam, The Netherlands, 1992. (3) Charlton, M. H.; Docherty, R.; Hutchings, M. G. Quantitative structure-sublimation enthalpy relatioship studied by neural networks, theoretical crystal packing calculations and multilinear regression analysis. J. Chem. Soc. Perkin Trans. 2 1995, 2023−2030. (4) Ouvrard, C.; Mitchell, J. B. Can we predict lattice energy from molecular structure? Acta Crystallogr., Sect. B 2003, 59, 676−685. (5) Keshavarz, M. H. Improved prediction of heats of sublimation of energetic compounds using their molecular structure. J. Hazard. Mater. 2010, 177, 648−659. (6) Benson, S. Current status of group additivity. Abs. Papers Am. Chem. Soc. 1996, 212, 56. (7) Gmehling, J. Present status and potential of group contribution methods for process development. J. Chem. Thermodyn. 2009, 41, 731−747. (8) Politzer, P.; Murray, J. S.; Grice, M. E.; Desalvo, M.; Miller, E. Calculation of heats of sublimation and solid phase heats of formation. Mol. Phys. 1997, 91, 923−928. (9) Rice, B. M.; Pai, S. V.; Hare, J. Predicting heats of formation of energetic materials using quantum mechanical calculations. Combust. Flame 1999, 118, 445−458. (10) Kim, C. K.; Lee, K. A.; Hyun, K. H.; Park, H. J.; Kwack, I. Y.; Kim, C. K.; Lee, H. W.; Lee, B. S. Prediction of physicochemical properties of organic molecules using van der Waals surface electrostatic potentials. J. Comput. Chem. 2004, 25, 2073−2079. (11) Politzer, P.; Ma, Y.; Lane, P.; Concha, M. C. Computational prediction of standard gas, liquid, and solid-phase heats of formation and heats of vaporization and sublimation. Int. J. Quantum Chem. 2005, 105, 341−347. (12) Byrd, E. F. C.; Rice, B. M. Improved Prediction of Heats of Formation of Energetic Materials Using Quantum Mechanical Calculations. J. Phys. Chem. A 2006, 110, 1005−1013. (13) Hu, A.; Larade, B.; Dudiy, S.; Abou-Rachid, H.; Lussier, L.-S.; Guo, H. Theoretical Prediction of Heats of Sublimation of Energetic Materials Using Pseudo-Atomic Orbital Density Functional Theory Calculations. Propel. Explos. Pyrotech. 2007, 32, 331−337. (14) Mathieu, D.; Bougrat, P. Model equations for estimating sublimation enthalpies of organic compounds. Chem. Phys. Lett. 1999, 303, 369−375. (15) Project 801, Evaluated Process Design Data, Public Release Documentation, 2006. (16) Patel, S. J.; Ng, D.; Mannan, M. S. QSPR Flash point prediction of solvents using topological indices for application in computer aided molecular design. Ind. Eng. Chem. Res. 2009, 48, 7378−7387. (17) Beaucamp, S.; Bernand-Mantel, A.; Mathieu, D.; Agafonov, V. Ab initio solid-state heats of formation of molecular salts from ion packing and crystal modelling: application to ammonium crystals. Mol. Phys. 2004, 102, 253−258. (18) Beaucamp, S.; Mathieu, D.; Agafonov, V. Parametrization of Semiempirical Models against ab Initio Crystal Data: Evaluation of Lattice Energies of Nitrate Salts. J. Phys. Chem. B 2005, 109, 16469− 16473. (19) Jenkins, H. D. B.; Glasser, L. Inorg. Chem. 2006, 45, 1754. (20) Gutowski, K. E.; Rogers, R. D.; Dixon, D. A. Accurate Thermochemical Properties for Energetic Materials Applications. II. Heats of Formation of Imidazolium-, 1,2,4-Triazolium-, and

ASSOCIATED CONTENT

S Supporting Information *

Table with sublimation enthalpies in kJ/mol, derived from experiment and using the present model and the ANN network described in ref 1 and also defines the training and test sets employed to assess the predictive value of the present model. 2818

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Tetrazolium-Based Energetic Salts from Isodesmic and Lattice Energy Calculations. J. Phys. Chem. B 2007, 111, 4788−4800. (21) Byrd, E. F. C.; Rice, B. M. A comparison of methods to predict solid phase heats of formation of molecular energetic salts. J. Phys. Chem. A 2009, 113, 345−352. (22) Gavezzotti, A.; Filippini, G. Theoretical Aspects and Computer Modeling; Gavezzotti, A., Ed.; Wiley and Sons: Chichester, U.K., 1997; pp 61−97. (23) The well-known odd−even variation in the solid-state properties of alkanes26 extends in principle to enthalpies of sublimation. However, while this is clearly the case for standard enthalpies of sublimation obtained after a correction for temperature effects,14 this subtle odd−even variation does not significantly affect experimental values considered in this work, as clear from Figure 1 (24) Beaucamp, S.; Mathieu, D.; Agafonov, V. Optimal partitioning of molecular properties into additive contributions: the case of crystal volumes. Acta Crystallogr., Sect. B 2007, 63, 277−284. (25) O’Boyle, N. M.; Morley, C.; Hutchinson, G. R. Pybel: a Python wrapper for the OpenBabel cheminformatics toolkit. Chem. Cent. J. 2008, 2, 5. (26) Boese, R.; Weiss, H.-C.; Bläser, D. The melting point alternation in the short-chain n-alkanes: single-crystal X-ray analyses of propane at 30K and of n-butane to n-nonane at 90K. Angew. Chem., Int. Ed. 1999, 38, 988−991.

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