Introduction 1 2 3 4 5 6 During the development of PM6, efforts were made to minimize the potential for this increase in error. Among these were the construction and use of very large survey and training sets. In contrast to previous methods in which the training set was a subset of the survey set, during the development of PM6 the training set was a superset of the survey set. No solids were used in either the training set or the survey set while PM6 was being developed because inclusion of even one solid in the parameter optimization would have made the whole process extremely slow, which in turn would have precluded optimization of the parameters in any reasonable time. Because of this, solids were excluded from the parameterization, and therefore they form an ideal, clearly defined set of systems for testing the applicability of PM6 to species that were not used in the development of the method. Theory 7 8 9 k 10 11 12 During the development of a procedure to allow PM6 to be applied to solids, various deficiencies and limitations were found in earlier procedures. Some of these, and the resulting modifications that had to be made in order to allow PM6 to be used for modeling solids, will now be described. NDDO error 13 15 16 1 17 γ AB A B R AB 1 \documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{AB} = \frac{1}{{\sqrt {R_{AB}^2 + \frac{1}{4}\left( {\frac{1}{{G_A }} + \frac{1}{{G_B }}} \right)^2 } }}$$\end{document} G A A R Q Q R V 2 2 \documentclass[12pt]{minimal}
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V \propto 4\pi R^2 \left( {\frac{Q}
{{\sqrt {R^2  + \frac{1}
{4}\left( {\frac{1}
{{G_A }} + \frac{1}
{{G_A }}} \right)^2 } }} - \frac{Q}
{{\sqrt {R^2  + \frac{1}
{4}\left( {\frac{1}
{{G_A }} + \frac{1}
{{G_B }}} \right)^2 } }}} \right)
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\begin{document}$$V \propto \int\limits_{R = 10}^\infty {\frac{1}{R}} $$\end{document} The value of this integral is infinity, which means that, if the DSK approximation is used and the integration is done correctly, the potential experienced by an atom of type A arising from the electrostatic contributions of all other atoms would then be either plus infinity or minus infinity, depending on the sign of its partial charge. This is an obviously unphysical result. R 4 4 4 \documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{AB} = \frac{1}{4}\left( {1 - e^{ - 0.05\left( {R - 5} \right)^2 } } \right)  e^{ - 0.05\left( {R - 5} \right)^2 } \left( {R^2  \frac{1}{4}\left( {\frac{1}{{G_A }}  \frac{1}{{G_B }}} \right)^2 } \right)^{{{ - 1} \mathord{\left/{\vphantom {{ - 1} 2}} \right.\kern-\nulldelimiterspace} 2}} $$\end{document} Electrostatic interaction 8 12 18 19 12 4 The potential experienced by each atom in a solid that arises from the partial charges on other atoms falls off rapidly with increasing distance. This is a natural result of the fact that the net charge arising from all atoms in a spherical shell must rapidly converge to zero as the radius increases. An implication of this is that, for large radii, the precise value of the radius used in evaluating the potential is unimportant. Conversely, when the radius is small, and there are relatively few atoms, the potential arising from the associated partial charges is large. In that case, the value of the interatomic separation used is of great importance. This behavior can be used as the basis for modifying the electrostatic sum. At large distances, because the electrostatic effect of the distant atoms is small, the value of the interatomic distance used in calculating the potential can be different from the actual value, and, in fact, can be set to any arbitrary large fixed value. That is, all potentials arising from distant atoms can be treated as if their partial charges were moved in to the surface of a sphere of fixed radius. A result of this is that the gradient or force arising from a charge that was initially outside the sphere would be exactly zero: any potential motion of the central atom in response to the presence of a charge on the surface of the sphere would be accompanied by a simultaneous motion of that charge. A consequence of this is that the gradient of the potential arising from a charge on the surface of the sphere is precisely zero. This modification of the effective interatomic distance (EID) used in evaluating the electrostatic potential completely eliminates all directional effects, in particular all artifacts arising from the use of a finite number of interacting unit cells. If no further modifications to the EID were made, then there would be a discontinuity in the gradient arising from the presence of the sphere. The gradient arising from a partial charge just inside the sphere would be finite, but if that charge were to move just outside the sphere, its gradient would now become zero, and there would be a discontinuity. The presence of such discontinuities would then preclude the gradients being used in subsequent operations such as geometry optimization and calculation of vibrational frequencies. To avoid them, the EID must be further modified to ensure that the gradient arising from an atom near the surface of the sphere drops smoothly to zero as the atom approaches the surface of the sphere. This is most simply accomplished by reducing the EID of an atom as it approaches the surface of the sphere. 1 Fig. 1 Truncation approximation for Madelung integral. C = 30 Å  C C C 4 5 5 \documentclass[12pt]{minimal}
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\begin{document}$$R_{AB}^\prime   = 2R_{AB}  - C/3 - \frac{{3R_{AB}^2 }}{{4C}}$$\end{document} C C 2 Fig. 2 black green  C C C C C C Unlike the Ewald summation, this modified DSK approximation can be used directly in evaluating the electrostatic potential. The new approximation is relatively simple in that the use of error functions and reciprocal space terms are avoided. Solids with unpaired electrons 3 III 2 3 12. d d \documentclass[12pt]{minimal}
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\begin{document}$${{80!} \mathord{\left/ {\vphantom {{80!} {\left( {32!48!} \right) \approx 2.2 \times 10^{22} }}} \right. \kern-\nulldelimiterspace} {\left( {32!48!} \right) \approx 2.2 \times 10^{22} }}$$\end{document} 6 d 2g 4 2g s Applications Organic compounds −1 −1 20 1 1 Densities 1 Table 1 Calculated and X-ray structural parameters for organic compounds   RMS error PM6 X-ray 3 d a b c α β γ a b c α β γ (Z Z)-N N′-Dimethylurea (NIJHUJ) 0.11 9.69 4.36 11.38 90.9 92.3 96.4 10.35 4.57 11.40 90.0 90.0 102.7 1.23 (1.11) 1, 4 Dioxocine 6-carboxylic acid chloride (CUWWIA) 0.29 14.67 12.30 3.90 89.6 91.7 89.2 15.15 12.49 3.81 90.0 87.7 90.0 1.63 (1.59) 1, 3, 5-Triaminobenzene, 1, 3, 5-trinitrobenzene (NIBZAM) 0.05 14.69 6.84 13.82 77.5 90.3 89.9 15.08 6.98 14.06 76.5 90.0 90.0 1.65 (1.55) 1, 3-bis((Pyrid-2-ylamino)carbonyl)adamantane α-ketoglutaric acid (RIZWUS) 0.14 9.97 13.66 10.72 92.0 113.1 67.2 10.53 13.73 10.73 91.8 114.7 67.8 1.42 (1.35) 1,4-diazabicyclo[2.2.2]octane azelaic acid (UNEGEZ) 0.50 30.36 8.33 5.93 91.1 89.1 91.3 28.15 8.60 6.98 90.0 93.0 90.0 1.33 (1.18) 1-diazonia-4-azabicyclo[2.2.2]octane glutarate (UNEFIC) 0.29 10.56 11.67 10.39 89.4 83.1 91.4 10.28 12.46 10.68 90.0 66.1 90.0 1.42 (1.30) 2-(2-(3-Carboxypyridyl))-4-isopropyl-4-methyl-5-oxo-imidazole (JAZCOC01) 0.10 16.07 10.74 7.201 88.7 86.5 85.5 16.31 10.78 7.17 85.6 90.0 90.0 1.40 (1.38) 2, 4, 4, 6, 6-Pentachloro-2-(piperidyl)cyclotriphosphazene (POTKEO) 0.23 8.12 22.50 17.20 89.5 90.1 90.1 8.32 22.01 17.26 90.0 90.0 90.0 1.68 (1.67) 2, 4, 6-Tribromoaniline (BRANIL) 0.09 4.20 13.59 12.99 90.0 90.0 90.0 4.26 14.62 13.44 90.0 90.0 90.0 2.96 (2.62) 2, 4, 6-Trinitro-N-methyl-N-nitroaniline (Tetryl) (MTNANL) 0.30 11.37 7.27 15.96 90.4 72.3 90.4 10.61 7.37 14.13 90.0 84.9 90.0 1.52 (1.73) 2-Pyridone (PYRIDO04) 0.03 5.77 5.93 12.75 90.2 90.4 90.1 5.60 5.79 13.56 90.0 90.0 90.0 1.45 (1.43) 4, 5-bis(Dimethylamino)-1,8-dihydroxynaphthalene (RISBIE) 0.06 10.33 13.75 8.91 107.7 90.3 90.1 10.65 13.77 9.32 107.0 90.0 90.0 1.36 (1.25) 4-Aminobenzoic acid 4-nitroaniline (RILJEB) 0.05 30.90 8.66 4.86 95.0 89.5 91.2 31.25 8.64 4.87 93.7 90.0 90.0 1.41 (1.39) 4-Fluoro-2-(phosphonomethyl)benzenesulfonic acid monohydrate (KIXQIR) 0.17 8.49 9.71 7.60 105.8 108.6 99.4 8.49 9.25 7.92 104.8 110.0 97.4 1.74 (1.75) 4-Hydroxybenzoic acid isonicotinamide (VAKTOR)  5.48 22.29 9.37 90.3 90.0 91.9 6.07 20.67 9.40 90.0 90.0 95.3 1.51 (1.47) 5H-Dibenz(b,f)azepine-5-carboxamide saccharin solvate (UNEZAO) 0.17 12.23 11.46 7.16 69.6 84.6 82.7 12.68 10.45 7.51 75.4 85.7 83.6 1.50 (1.46) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH) 0.23 18.50 15.96 4.24 88.7 89.6 88.9 18.69 16.40 3.95 86.2 90.0 90.0 1.38 (1.43) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH01) 0.11 16.42 8.32 8.57 90.0 89.6 89.7 16.41 8.54 8.50 88.2 90.0 90.0 1.47 (1.47) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH02) 0.11 7.48 11.90 7.91 104.2 117.0 78.6 7.49 11.91 7.79 104.5 116.4 77.80 1.42 (1.44) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH03) 0.17 8.02 11.36 13.66 90.6 90.4 106.5 7.98 11.68 13.32 90.0 90.0 104.8 1.44 (1.44) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH04) 0.20 10.96 12.05 4.37 89.6 90.5 73.0 11.25 12.32 4.59 90.2 91.8 71.2 1.56 (1.43) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH05) 0.12 13.29 22.57 7.91 90.6 90.1 90.5 13.18 22.80 8.02 90.0 90.0 90.0 1.45 (1.43) 5-Methyl-2-((2-nitrophenyl)amino)-3-thiophenecarbonitrile (QAXMEH12) 0.17 7.93 12.16 12.25 90.8 90.9 102.8 8.23 12.31 11.82 90.0 90.0 102.5 1.50 (1.47) 6-(Pyran-4-one-2-yl)-1,4-dioxocine (CUWWEW) 0.19 7.69 10.91 10.96 90.7 74.4 90.6 7.91 10.80 11.58 90.0 70.4 90.0 1.53 (1.46) 9-(2-Hydroxyethyl)adenine (FABFUJ) 0.11 7.18 8.34 13.96 90.2 90.0 89.4 7.34 8.27 13.57 90.0 90.0 90.0 1.42 (1.45) Acetophenone (ACETPH) 0.04 8.71 8.45 9.75 90.9 58.5 90.6 8.56 8.68 10.26 90.0 59.0 90.0 1.31 (1.22) Acetylacetone (LIWPIQ01) 0.04 7.85 4.11 15.48 92.5 90.1 89.8 8.46 4.15 16.03 90.0 90.0 90.0 1.33 (1.18) Acetylcholine chloride (ACHOLC01) 0.17 15.29 5.94 9.62 90.0  89.3 90.2 15.32 6.30 9.89 90.0 90.0 90.0 1.38 (1.26) Adenosine-3″-phosphate dihydrate (ADPOSD) 0.35 5.91 11.62 10.55 88.6 90.0 90.2 6.34 11.90 9.94 87.8 90.0 90.0 1.76 (1.70) Adenosine-5″-diphosphate dihydrate (HMADPH) 0.29 18.50 6.76 8.89 91.7 89.3 93.0 18.44 6.89 9.20 90.0 87.5 90.0 1.75 (1.66) α Glycine 0.10 5.09 5.50 10.77 90.1 90.0 69.6 5.10 5.46 11.97 90.0 90.0 68.3 1.77 (1.61) α Resorcinol 0.05 5.60 9.59 10.38 89.7 89.9 90.1 5.60 9.53 10.53 90.0 90.0 90.0 1.31 (1.30) Anthracene (ANTCEN14) 0.02 10.36 7.32 6.84 89.9 109.4 89.8 11.17 6.02 8.55 90.0 124.6 90.0 1.21 (1.25) Aqua-tris(2-((dimethylamino)methyl)phenyl)borane (QEVYEV) 0.16 19.81 8.41 14.99 89.9 75.1 89.6 20.35 8.57 15.23 90.0 72.3 90.0 1.19 (1.13) Aspartic acid (LASPRT) 0.20 5.13 6.38 7.70 90.0 106.2 90.0 5.14 6.98 7.62 90.0 99.8 90.0 1.83 (1.64) Barium oxalate oxalic acid dihydrate (BAHOXH11)  4.75 10.68 13.09 68.9 90.1 90.1 5.41 12.46 14.45 63.8 90.0 90.0 3.77 (2.67) Benzene (BENZEN) 0.01 9.27 6.79 7.15 90.1 90.2 89.8 9.55 6.92 7.44 90.0 90.0 90.0 1.15 (1.06) Berberine sulfate (CISREB) 0.15 19.91 7.43 27.51 97.6 117.3 90.7 20.37 7.44 27.43 97.7 116.2 85.5 1.61 (1.56) β Glycine 0.08 5.34 6.09 5.14 90.0 65.7 89.8 5.38 6.27 5.08 90.0 66.8 90.0 1.64 (1.59) β Resorcinol 0.04 8.81 4.99 12.34 89.9 88.9 90.0 7.81 5.43 12.62 90.0 90.0 90.0 1.35 (1.37) 26  15.39 20.60 12.81 103.7 89.9 90.0 16.17 21.37 13.23 101.0 90.0 90.0 2.65 (2.33) bis 0.18 12.01 16.11 7.27 75.4 89.7 94.1 12.77 15.77 7.27 73.1 91.1 93.4 1.56 (1.51) bis(Pyridinium) oxalate oxalic acid (DEFCUM)  4.12 11.39 7.77 102.1 99.8 91.0 4.00 11.32 8.44 102.0 97.3 88.6 1.60 (1.52) bis(Urea) oxalic acid (UROXAL01)  12.34 7.00 5.13 84.2 89.8 90.0 12.37 6.88 5.05 83.6 90.0 90.0 1.58 (1.64) Bromo-tris(2-dimethylaminoethyl)amine-manganese(ii) (DAEAMN) 0.27 11.85 11.85 11.83 90.2 90.3 89.7 12.22 12.22 12.22 90.0 90.0 90.0 1.78 (1.62) Calcium acetate chloride pentahydrate (CALCLA) 0.04 13.26 11.20 6.66 119.1 89.7 88.6 13.72 11.51 6.82 116.7 90.0 90.0 1.73 (1.55) Camphor (UGAHUF) 0.06 8.45 26.77 7.32 90.8 91.1 89.9 8.93 27.04 7.38 90.0 90.0 90.0 1.22 (1.14) Cholesteryl acetate (CHOLAD04) 0.12 16.98 9.21 16.28 87.9 71.7 88.4 17.62 9.22 16.52 90.0 72.8 90.0 1.18 (1.11) cis 0.19 5.33 12.60 10.54 89.9 90.1 89.9 5.19 13.55 10.69 90.0 90.0 90.0 2.16 (2.03) Citric acid (CITRAC10) 0.31 11.92 5.38 12.18 89.5 115.2 89.6 11.47 5.62 12.81 90.2 111.5 90.0 1.81 (1.66) Coronene (CORONE) 0.03 16.02 4.79 9.88 90.1 68.8 90.0 16.12 4.70 10.10 90.0 69.1 90.0 1.41 (1.40) Cyclohexane (CYCHEX) 0.02 8.12 6.41 11.23 90.0 107.1 90.0 8.20 6.44 11.23 90.0 108.8 90.0 1.00 (1.00) Cyclotrimethylene-trinitramine (RDX) 0.22 11.24 11.48 14.32 89.8 88.4 89.9 10.71 11.57 13.18 90.0 90.0 90.0 1.60 (1.81) Cystine (LCYSTI10) 0.42 5.34 54.25 5.37 88.4 59.9 89.2 5.42 56.28 5.42 90.0 60.0 90.0 1.78 (1.67) Cytosine (CYTOSM03) 0.04 10.21 7.09 7.79 87.2 90.4 88.4 9.82 7.52 7.73 79.5 90.0 90.0 1.52 (1.53) Dinicotinic acid (DINICA10) 0.18 6.71 11.09 9.68 89.7 111.9 90.1 6.59 11.15 9.70 90.0 107.8 90.0 1.66 (1.64) Dipyridinium bis(hydrogen-oxalate) oxalic acid (DUVLUB)  27.71 8.30 7.35 90.0 83.8 90.0 26.84 8.91 7.45 90.0 87.1 90.0 1.69 (1.60) Disodium adenosine-triphosphate trihydrate (ADENTP) 0.76 6.24 20.58 31.37 90.0 91.4 88.3 7.07 20.88 30.45 90.0 90.0 90.0 2.00 (1.79) DL-2-Amino-4-phosphonobutyric acid monohydrate (CAXDIO01) 0.11 8.11 4.83 19.89 87.6 102.3 90.6 8.40 4.95 20.60 90.0 100.4 90.0 1.76 (1.58) γ Glycine 0.07 5.52 6.93 8.94 114.4 50.9 90.7 5.48 7.04 8.92 113.2 52.1 90.0 1.65 (1.59) Glucose (GLUCSA) 0.19 10.27 4.20 15.12 90.0 90.0 90.0 10.36 4.97 14.84 90.0 90.0 90.0 1.84 (1.57) Guanine 0.06 3.62 16.84 9.88 88.2 85.9 96.2 3.55 16.34 9.69 90.0 90.0 90.0 1.68 (1.79) 8-Azaguanine (AZGUAN01) 0.06 3.67 11.81 16.38 89.6 102.7 89.7 3.56 11.44 16.47 90.0 95.1 90.0 1.46 (1.51) Hydroquinone bis(aniline) (HIBFUT)  5.03 8.21 18.42 76.8 90.0 90.0 5.39 8.22 18.74 77.5 90.0 90.0 1.33 (1.22) Isonicotinamide (EHOWIH) 0.08 9.98 5.80 10.11 90.0 79.3 89.9 10.03 5.73 10.17 90.0 81.8 90.0 1.41 (1.40) Isonicotinamide 3-hydroxybenzoic acid (LUNMEM)  20.70 5.16 22.19 89.9 96.2 90.0 20.87 5.14 22.42 90.0 97.7 90.0 1.47 (1.45) Keggin pentakis(Tetraethylammonium) bis(meso-tetraphenyl-porphyrinato-zinc) tetraconta-oxo-silicon-dodeca-molybdenum bromide(PIJFUJ)  15.14 32.01 14.98 90.1 90.3 90.1 15.22 32.28 15.22 90.0 90.0 90.0 1.88 (1.83) l-Alanine 0.06 5.93 11.67 5.86 90.0 89.9 89.9 5.79 12.26 5.93 90.0 90.0 90.0 1.46 (1.41) Leucine (LEUCIN02) 0.13 13.85 5.24 9.44 89.2 82.4 90.2 14.52 5.30 9.56 90.0 85.8 90.0 1.28 (1.19) 2 0.02 11.23 6.38 7.15 98.3 94.4 88.2 10.88 6.62 6.82 90.0 90.0 90.0 1.34 (1.38) l-Lysine monohydrochloride dihydrate (LYSCLH) 0.11 12.35 5.76 14.38 81.6 90.4 89.8 13.32 5.88 14.98 81.2 90.0 90.0 1.44 (1.25) Malonic acid bis(isonicotinamide) (ULAWEJ)  14.65 14.68 12.25 65.8 81.8 68.8 15.68 14.86 11.94 67.3 85.6 65.7 1.55 (1.49) m-Aminophenol (MAMPOL02) 0.02 8.24 6.12 11.33 90.0 89.5 89.9 8.28 6.10 11.23 90.0 90.0 90.0 1.27 (1.28) m-Aminopyridine (AMIPYR) 0.03 6.30 5.68 15.23 90.1 89.5 111.4 6.19 5.71 15.30 90.0 90.0 110.5 1.23 (1.24) m-Cresol N,N,N′,N′-tetraisopropyloxamide (DUGRIG)  6.62 26.72 11.88 93.4 89.8 89.8 6.86 27.19 12.08 96.1 90.2 89.8 1.15 (1.07) m-Diaminobenzene, 3,5-dinitro-1-cyanobenzene (REDDEJ)  13.07 10.39 9.85 84.0 89.5 90.4 13.26 10.45 9.88 86.7 90.0 90.0 1.51 (1.46) Methionine (LMETON02) 0.11 5.18 16.40 9.31 85.3 93.8 83.3 5.20 14.83 9.49 81.0 90.0 90.0 1.27 (1.37) 2,9 4,8 0.15 6.19 18.35 8.11 96.5 101.6 81.7 6.61 17.89 8.07 102.3 99.8 83.2 1.47 (1.43) m-Hydroxybenzoic acid (BIDLOP) 0.03 23.80 4.97 5.04 90.0 74.3 90.0 23.89 4.94 5.49 90.0 74.3 90.0 1.60 (1.47) 6 6 2 IV 8 4- 0.09 17.96 10.46 9.70 93.3 91.4 89.2 18.51 10.15 9.17 89.8 90.0 90.0 1.46 (1.54) m-Toluidine (FANDOO) 0.08 9.13 5.79 23.12 91.7 80.9 89.4 8.76 5.81 24.87 90.0 79.9 90.0 1.18 (1.14) 2 2 4 0.58 4.23 9.29 5.38 90.1 90.2 86.5 3.45 10.38 5.24 90.0 90.0 90.0 2.11 (2.38) N, N′-(6 6″-dimethylbiphenyl-2 2″-diyl) bis(methylamine) (ENIWIH) 0.07 10.67 11.77 10.15 90.1 90.1 90.0 11.09 12.02 10.57 90.0 90.0 90.0 1.25 (1.13) N, N-2 6-Tetramethyl-4-nitroaniline (FOCVOI) 0.12 7.85 16.17 7.67 90.7 95.6 88.9 7.64 18.08 7.64 90.0 90.0 90.0 1.33 (1.22) N, N-Dimethylaniline (DMAFBZ01) 0.08 7.09 13.20 8.71 82.4 101.2 95.4 6.78 12.33 8.26 80.5 74.0 92.0 1.29 (1.57) Nicotinamide adenine dinucleotide tetrahydrate (CEVYEH11) 0.25 8.89 8.20 11.33 83.2 69.7 103.2 8.84 8.59 11.19 89.4 70.4 103.9 1.66 (1.58) N-Methylurea (MEUREA) 0.09 6.85 6.64 8.08 90.1 91.2 90.3 6.92 6.98 8.48 90.0 90.0 90.0 1.34 (1.20) N, N-Dimethylbenzamide (ODOTOQ) 0.08 6.51 16.19 7.49 90.0 90.6 90.8 6.63 16.30 7.65 90.0 90.0 90.0 1.26 (1.20) N, N-Dimethylurea (WIFKEB) 0.09 9.27 8.49 6.06 90.4 89.6 109.3 9.27 8.60 6.04 90.0 90.0 108.8 1.30 (1.29) o-Aminophenol (AMPHOM02) 0.05 7.50 8.10 19.70 90.0 90.5 90.0 7.85 7.25 19.75 90.0 90.0 90.0 1.21 (1.29) o-Aminopyridine (AMPYRD) 0.04 7.41 5.65 11.62 90.0 85.9 89.9 7.59 5.67 11.71 90.0 84.5 90.0 1.29 (1.25) o-Diaminobenzene (BAGFIY) 0.04 7.72 7.57 9.96 89.9 82.5 90.0 7.72 7.54 10.32 90.0 80.0 90.0 1.25 (1.21) o-Dimethoxybenzene (TUKGEL) 0.06 12.94 9.35 5.58 90.1 89.8 89.9 13.35 9.92 5.53 90.0 90.0 90.0 1.36 (1.25) Oxalic acid (OXALAC06) 0.03 6.91 6.08 8.16 89.9 89.9 90.0 6.56 6.09 7.85 90.0 90.0 90.0 1.74 (1.91) Oxalic acid dihydrate (OXACDH26) 0.04 4.92 11.45 3.44 89.7 90.0 106.9 6.09 11.93 3.47 89.7 90.0 106.9 2.26 (1.73) 23 0.44 22.06 8.89 26.73 90.6 99.1 88.4 23.04 9.04 27.27 90.0 102.2 90.0 1.43 (1.33) p-Aminophenol (AMPHOL01) 0.03 7.95 12.90 4.78 90.1 90.1 90.1 8.18 12.95 5.26 90.0 90.0 90.0 1.48 (1.30) p-Aminopyridine (AMPYRE) 0.04 5.48 11.92 7.19 89.6 90.3 91.0 5.57 12.12 7.32 90.0 90.0 90.0 1.33 (1.26) p-Chloroaniline (CLANIC05) 0.03 8.37 7.30 8.98 90.0 89.7 90.2 8.59 7.24 9.19 90.0 90.0 90.0 1.54 (1.48) p-Diaminobenzene (PENDAM) 0.03 8.29 23.08 5.94 89.9 90.0 93.5 8.37 22.95 5.97 90.0 90.0 93.6 1.27 (1.26) Phenanthrene (PHENAN08) 0.04 9.16 5.73 8.56 90.2 80.9 89.9 9.44 6.14 8.44 90.0 82.0 90.0 1.34 (1.22) Phenolphthalein (NIMDAO) 0.09 11.24 14.58 19.01 90.0 89.8 89.5 11.39 14.82 19.27 90.0 90.0 90.0 1.34 (1.28) Phloroglucinol (PHGLOL) 0.03 12.71 9.35 4.87 89.8 89.6 89.9 12.56 9.37 4.83 90.0 90.0 90.0 1.45 (1.47) p-Hydroxybenzoic acid (JOZZIH) 0.04 6.21 10.05 18.46 89.8 85.7 90.1 6.34 10.46 18.51 90.0 86.8 90.0 1.60 (1.50) Picric acid (PICRAC) 0.15 9.63  19.49 9.38 88.9 90.4 89.8 9.70 19.13 9.25 90.0 90.0 90.0 1.73 (1.77) Potassium hydrogen acetate 0.01 3.91 6.77 25.00 90.3 89.8 82.1 4.01 7.19 23.88 90.0 90.0 81.7 1.60 (1.54) 1-Dimethylamino-8-dimethylammonionaphthalene saccharin dihydrate (AJOHUC) 0.09 8.81 9.54 24.50 90.2 99.6 90.1 9.25 9.17 24.93 90.0 95.8 90.0 1.42 (1.37) Salicylaldoxime (SALOXM) 0.07 12.91 5.45 10.08 90.3 65.4 89.8 13.60 5.08 10.41 90.0 67.1 90.0 1.41 (1.38) 3 2  11.65 11.39 15.40 88.9 78.8 91.7 10.40 10.47 12.35 90.0 68.3 90.0 0.90 (1.45) Sodium hydrogen acetate  16.33 16.33 16.33 89.9 89.9 90.0 15.92 15.92 15.92 90.0 90.0 90.0 1.30 (1.40) Sucrose (SUCROS01) 0.24 10.36 8.59 7.50 90.0 103.0 90.0 10.82 8.68 7.72 90.0 103.0 90.0 1.75 (1.61) Tetramethylammonium dihydrogen phosphate monohydrate (FIJHEL) 0.02 8.01 8.07 12.32  90.1 90.0 98.1 8.52 8.44 12.90 90.0 90.0 99.1 1.59 (1.37) Tetramethylurea (TIDBIR) 0.16 8.69 6.00 11.73 92.1 93.1 91.6 9.97 6.26 10.63 90.0 93.0 90.0 1.27 (1.17) Thymine (THYMIN01) 0.05 6.79 6.96 11.71 78.1 89.8 89.6 6.85 6.78 12.89 75.1 90.0 90.0 1.55 (1.45) trans 0.32 9.53 27.27 5.32 89.0 92.3 88.5 9.44 27.93 5.06 90.0 90.0 90.0 1.85 (1.91) Trinitrotoluene (ZZZMUC01) 0.11 5.43 22.63 14.72 90.1 91.4 90.1 6.08 20.02 14.99 90.0 90.0 90.0 1.67 (1.66) tris(Acetylacetonato) titanium(iv) perchlorate (TIACPC) 0.25 8.97 11.73 9.78 91.0 83.5 92.3 8.68 11.74 9.89 88.5 84.3 92.0 1.45 (1.47) Tryptophan (TRYPTC) 0.13 15.26 5.17 6.90 89.3 78.3 88.9 14.67 5.30 7.45 90.0 81.2 90.0 1.50 (1.40) Tyrosine (LTYROS10)  6.85 20.39 5.90 89.9 90.0 90.0 6.91  21.12 5.83 90.0 90.0 90.0 1.46 (1.41) Urea (UREAXX13) 0.03 4.67 5.58 5.58 90.0 90.0 90.0 4.69 5.57 5.57 90.0 90.0 90.0 1.37 (1.37) Urea nitrate (UREANT02)  7.81 8.20 9.70 89.9 52.8 90.1 7.50 8.20 9.54 90.0 55.8 90.0 1.65 (1.69) 2 4 2  7.23 11.72 11.72 89.7 89.9 89.9 7.36 12.37 12.37 90.0 90.0 90.0 2.20 (1.94) 2 4 2  6.04 9.77 14.57 89.9 90.5 107.7 6.29 10.12 14.58 90.0 90.0 109.5 2.37 (2.22) tris(2,2,6,6-tetramethylheptane-3,5-dionato)-yttrium(iii) (HAHTOZ01) 0.28 9.57 8.89 18.95 89.6 86.4 90.2 10.63 9.98 17.87 90.0 90.0 90.0 1.32 (1.12) d 21 RMS 21 2 4 2- 3 + A related system is barium oxalate oxalic acid dihydrate (CSD entry BAHOXH11) in which there exist polymeric chains of oxalate groups connected by bridging protons. As with oxalic acid, in the optimized PM6 geometry the proton is abstracted by the water molecule to give oxalate groups and hydronium, resulting in an  increase in density of almost 40%. Sodium acetate trihydrate is an ionic solid consisting of sodium ions surrounded by an acetate group and five water molecules, four of which form bridges between pairs of sodium ions. PM6 completely fails to predict the observed structure: the distance between the sodium ions increases considerably, effectively destroying any tendency of the water molecules to form bridges. When these three solids were removed from consideration, the average unsigned error (AUE) in density decreased to 6.1%. The most common intermolecular interaction is hydrogen bonding, which PM6 predicts to be too short by about 0.1 Å, with the result that the average signed error in calculated densities of organic compounds is too high by 3.9%. When a systematic correction to the density was made, the AUE in density decreased still further to 4.8%. Heats of formation 2 Table 2 Comparison of calculated and experimental heats of formation of organic compounds (kcal/mol)  PM6 a Difference (Z Z)-N, N′-Dimethylurea (NIJHUJ) −62.0 −76.3 14.3 2, 4, 6-Tribromoaniline (BRANIL) −30.7 13.8 −44.5 α Glycine −122.9 −126.1 3.2 α Resorcinol −79.3 −88.0 8.7 Anthracene 26.4 30.0 −3.6 β Glycine −121.8 ∼−126.1 ∼4.3 Camphor −66.3 −76.3 10.0 Citric acid −348.7 −369.0 20.3 Cyclotrimethylene-trinitramine (RDX) −5.9 18.9 −24.8 Cystine −237.9 −246.8 8.9 γ Glycine −120.8 ∼−126.1 ∼5.3 l −132.0 −134.1 2.1 Leucine −142.7 −152.3 9.6 m-Aminophenol (MAMPOL0
) −37.6 −47.9 10.3 m-Aminopyridine (AMIPYR) 23.2 14.4 8.8 m-hydroxybenzoic acid −123.7 −142.0 18.4 o-Aminophenol (AMPHOM02) −38.2 −48.1 9.9 o-Aminopyridine (AMPYRD) 21.4 9.4 12.0 o-Diaminobenzene (BAGFIY) 12.9 9.3 3.6 Oxalic acid −175.5 −198.4 22.9 p-Aminophenol (AMPHOL01) −40.7 −46.4 5.7 p-Aminopyridine (AMPYRE) 19.0 10.0 9.0 p-Chloroaniline (CLANIC05) 2.5 −8.0 10.5 p-Diaminobenzene (PENDAM) 5.1 10.1 −5.1 Phenanthrene 24.1 26.2 −2.1 p-Hydroxybenzoic acid −125.9 −145.0 19.2 Picric acid −43.4 −52.1 8.7 Salicylaldoxime −27.0 −43.9 16.9 Sucrose −535.8 −532.0 −3.8 Trinitrotoluene −5.7 −15.1 9.4 Tyrosine −150.7 −163.7 13.0 Urea −65.7 −79.6 13.9 a 32 Heats of sublimation 3 22 Table 3 Comparison of calculated and experimental heats of sublimation   22 PM6 Difference (Z Z)-N, N′-Dimethylurea (NIJHUJ) 22.1 15.9 −6.2 α Glycine 32.6 29.7 −2.9 Anthracene 23.8 33.1 9.3 Aspartic acid 22.9 37.8 14.9 Benzene 10.0 3.2 −6.8 Camphor 12.4 6.5 −5.9 Cyclohexane 9.0 2.1 −7.0 Guanine 44.5 25.2 −19.3 l 31.7 32.6 0.9 Leucine 36.0 28.5 −7.5 m-Aminophenol (MAMPOL02) 23.6 11.8 −11.8 m-Aminopyridine (AMIPYR) 19.3 9.5 −9.8 Methionine 32.0 27.2 −4.8 m-Hydroxybenzoic acid 29.5 15.3 −14.3 N-Methylurea 22.6 17.0 −5.6 N, N-Dimethylbenzamide 21.4 10.0 −11.4 o-Aminophenol (AMPHOM02) 22.3 17.6 −4.7 o-Aminopyridine (AMPYRD) 18.3 9.2 −9.1 o-Diaminobenzene (BAGFIY) 20.4 9.1 −11.3 Oxalic acid 22.3 19.2 −3.1 p-Aminophenol (AMPHOL01) 24.2 19.3 −4.9 p-Aminopyridine (AMPYRE) 20.8 10.4 −10.4 p-Chloroaniline (CLANIC05) 21.7 8.7 −13.0 p-Diaminobenzene (PENDAM) 22.0 16.0 −6.1 Phenanthrene 22.0 30.2 8.2 p-Hydroxybenzoic acid 29.5 14.5 −15.1 Picric acid 25.1 14.6 −10.5 Trinitrotoluene 26.9 14.7 −12.2 Tyrosine 24.1 33.7 9.6 Urea 21.7 17.5 −4.2 Biomolecules The primary objective in developing PM6 was to more accurately model systems of biochemical interest. The applicability of PM6 to the study of crystals of biochemical importance was therefore of interest. Oligopeptides 23 24 2 Because the positions of the hydrogen atoms on the water molecules were not given in the X-ray structure, an estimate of the locations of the 52 hydrogen atoms had to be made before the geometry could be optimized. For this operation, the “ice rules” were used: each oxygen atom in a water molecule was involved in forming two hydrogen bonds and each hydrogen atom formed one hydrogen bond. Of necessity, some of these bonds involved atoms on the peptide. 43 65 11 12 2 4 2 Each initial geometry optimized to give a different final structure. That is, the optimized geometry was very sensitive to the choice of initial locations of the hydrogen atoms assigned to the water molecules. As a result, it was not possible to unambiguously define an optimized PM6 structure; however, all the fully optimized structures were within a few kcal/mol of each other, so one structure was chosen arbitrarily and used in the following analysis. 1 2 2 Acetylcholine 3 2 2 3 3 + Adenosine diphosphate 2 7 4 Adenosine triphosphate Nicotinamide adenine dinucleotide Hydrogen bonding Because of the importance of hydrogen bonding in biochemistry, a range of types of hydrogen bond were examined. Most of the important hydrogen bonds in biochemistry involve a proton positioned between either two oxygen atoms, two nitrogen atoms, or an oxygen and a nitrogen atom, the more exotic bonds, such as those involving halogen ions, while interesting, being of secondary importance. Individual types of hydrogen bonds O–H–O 4 2 3 – 3 2 3 N–H–N 25 2 3 1 9 10 5 3 25 Fig. 3 1,8-bis(hexamethyltriaminophosphazenyl)naphthalene  f f f 4 14 26 4 \documentclass[12pt]{minimal}
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\begin{document}$$Br_3^ -  $$\end{document} − Fig. 4 Detail of bis(1,4-Diazoniabicyclo(2.2.2)octane) bis(1-aza-4-azoniabicyclo(2.2.2)octane) tetrakis(tribromide) dibromide  4 Table 4 Interatomic distances and charges in bis(1,4-Diazoniabicyclo(2.2.2)octane) bis(1-aza-4-azoniabicyclo(2.2.2)octane) tetrakis(tribromide) dibromide Interatomic distances Charges  X-ray PM6  PM6 1 2 2.69 2.70 1 +1.01 2  1.16 2 +1.68 3  1.16 3 +1.01 3 4 2.66 2.70 4 +1.81 5 1 3.17 2.97 1 −0.78 1 2 3.60 3.00 2 −0.54 2 3 2.51 2.48 3 −0.05 3 4 2.59 2.50 4 −0.39 4 5 3.25 2.98 5 −0.45 5 6 2.45 2.47 6 −0.09 6 7 2.69 2.53 7 −0.47 N–H–O 3 + – 2,9 4,8 5 27 Fig. 5 2-(2-(3-Carboxypyridyl))-4-isopropyl-4-methyl-5-oxo-imidazole  π−π stacking π−π stacking occurs in the polycyclic aromatic hydrocarbons where it is the result of Van der Waals (VDW) interactions between the rings. In general, VDW interactions are weaker than hydrogen bonding interactions, and, historically, have been the hardest to model using semiempirical methods. Thus, when the default single determinant wavefunction is used, VDW terms are completely absent, and, in order to mimic the effects of VDW attraction, the normal procedure is for modifications to be made to the core–core interaction. An estimate of how accurately PM6 can reproduce the VDW interaction is provided by anthracene (CSD entry ANTCEN14), benzene (CSD entry BENZEN), and coronene (CSD entry CORONE). In anthracene, the molecules are stacked in a staggered arrangement. While PM6 reproduces the density with good accuracy, the optimized structure predicts the parallel sheets of anthracene molecules to be separated by 5.37 Å rather than the observed 2.83 Å, and pairs of anthracene molecules forming “T” structures rather than the observed “V” configuration. Conversely, both benzene and coronene crystallize with a perfect herringbone packing, and this structure is reproduced with very good accuracy by PM6, the calculated and observed inter-plane distance being essentially identical. Very weak interactions f 2 3 3d Polymorphs 28 1 Co-crystals 6 f f 2 Fig. 6 Crystal structure of the co-crystal of urea and oxalic acid  7 Fig. 7 Co-crystal of isonicotinamide and 3-hydroxybenzoic acid showing PM6 hydrogen-bond lengths (X-ray in parenthesis)  Metal-containing species 2 4 2 2 4 2 2 4 2 2d Solids containing metal complex ions can be regarded as salts: that is, as molecular metal complexes, cations or anions, plus counterions. An example is bromo-tris(2-dimethylaminoethyl)amine-manganese(ii) (CSD entry DAEAMN), where the metal complex consists of the neutral tris(2-dimethylaminoethyl)amine that chelates a manganese dication, the whole complex behaving like a large dication. Two bromide counterions are present for each such complex ion in order to maintain electroneutrality. PM6 predicts the single N–Mn distance to be 2.09 Å versus 2.19 Å in the X-ray structure, and the three N–Mn to be 1.96 Å versus 2.27 Å in the X-ray. Inorganic compounds In general, most inorganic solids differ from crystalline organic compounds in that they do not involve discrete molecules. Instead, they exist as extended covalently or ionically bound infinite systems. A consequence of this is that identification of simple structural units in inorganic solids is often either difficult or impossible. A more subtle consequence is that the band-structure of inorganic solids is usually more complicated than those of organic solids: for the same reciprocal distance, bands arising from inorganic solids generally have a much greater curvature than those for organic solids, this being a consequence of the strong bonds that extend throughout such solids. Conversely, in most organic solids there is an intermolecular gap that effectively confines molecular orbitals to individual molecules. This means that the band-structure of most organic compounds consists of relatively flat bands. In the cluster method, the Γ point represents the entire Brillouin zone so that a much larger cluster must be used when inorganic solids that are not composed of discrete molecules are modeled. In practice this means that the cluster used has to be large enough to contain a sphere of radius 10–12 Å, in contrast to the 7–8 Å used in modeling organic solids. 29 II 2 4 2 4 Some minerals that contain hydrogen atoms did not have the positions of these atoms reported. In those cases a preliminary calculation was carried out in which the hydrogen atoms were positioned in likely sites, and then the positions of those atoms optimized. During this operation, the positions all the other atoms were fixed at the experimental values. 5 6 7 Table 5 Calculated and X-ray structural parameters for inorganic compounds. Values of a, b, c are in Ångstroms; α, β, γ in degrees  PM6 X-ray 3 d a b c α β γ a b c α β γ Elements Diamond (C) 3.59 3.59 3.59 90.0 90.0 90.0 3.57 3.57 3.57 90.0 90.0 90.0 3.44 (3.52) e 2.60 2.60 5.96 90.1 90.1 90.0 2.60 2.60 5.90 90.0 90.0 90.0 2.98 (3.08) Graphite (C) 2.47 2.478 7.17 89.6 90.0 60.0 2.46 2.46 6.70 90.0 90.0 60.0 2.12 (2.28) Silicon (Si) 5.11 5.11 5.11 90.0 90.0 90.0 5.43 5.43 5.43 90.0 90.0 90.0 2.78 (2.33) Sulfur (S) 11.77 14.01 26.81 90.4 90.2 89.8 10.43 12.83 24.36 90.0 90.0 90.0 1.54 (2.09) Group I Halides Cesium bromide (CsBr) 4.35 4.35 4.35 90.0 90.0 90.0 4.30 4.30 4.30 90.0 90.0 90.0 4.29 (4.43) Cesium chloride (CsCl) 4.17 4.18 4.18 90.3 89.8 89.7 4.12 4.12 4.12 90.0 90.0 90.0 3.84 (3.99) Cesium fluoride (CsF) 6.02 6.02 6.02 90.0 90.0 90.0 6.01 6.01 6.01 90.0 90.0 90.0 4.62 (4.64) Cesium iodide (CsI) 4.73 4.73 4.73 90.0 90.0 90.0 4.57 4.57 4.57 90.0 90.0 90.0 4.07 (4.51) Potassium bromide (KBr) 6.79 6.75 6.75 87.6 90.0 90.0 6.61 6.61 6.61 90.0 90.0 90.0 2.55 (2.74) Potassium chloride (KCl) 6.12 6.12 6.12 90.0 90.0 90.0 6.29 6.29 6.29 90.0 90.0 90.0 2.16 (1.99) Potassium fluoride (KF) 5.03 5.03 5.03 90.0 90.0 90.0 5.38 5.38 5.38 90.0 90.0 90.0 3.02 (2.48) Potassium iodide (KI) 7.17 7.17 7.17 90.1 90.0 90.0 7.07 7.07 7.07 90.0 90.0 90.0 2.99 (3.12) Lithium bromide (LiBr) 5.49 5.48 5.48 90.0 90.0 90.0 5.50 5.50 5.50 90.0 90.0 90.0 3.50 (3.46) Lithium chloride (LiCl) 5.13 5.13 5.13 90.0 90.0 90.0 5.14 5.14 5.14 90.0 90.0 90.0 2.09 (2.07) Lithium fluoride (LiF) 4.03 4.04 4.05 90.0 89.9 90.0 4.03 4.03 4.03 90.0 90.0 90.0 2.62 (2.64) Lithium iodide (LiI) 6.04 6.03 6.04 90.0 90.0 90.0 6.03 6.03 6.03 90.0 90.0 90.0 4.04 (4.06) Sodium bromide (NaBr) 6.27 6.20 6.20 85.9 90.0 90.0 6.04 6.04 6.04 90.0 90.0 90.0 2.84 (3.10) Sodium chloride (NaCl) 6.24 6.24 6.24 90.0 90.0 90.0 5.60 5.60 5.60 90.0 90.0 90.0 1.60 (2.21) Sodium fluoride (NaF) 4.79 4.79 4.79 90.2 90.1 89.8 4.61 4.61 4.61 90.0 90.0 90.0 2.54 (2.85) Sodium iodide (NaI) 7.02 7.02 7.02 90.0 90.0 90.0 6.47 6.47 6.47 90.0 90.0 90.0 2.87 (3.67) Rubidium bromide (RbBr) 6.87 6.88 6.88 90.3 90.0 90.0 6.90 6.90 6.90 90.0 90.0 90.0 3.38 (3.35) Rubidium chloride (RbCl) 6.70 6.68 6.68 89.1 90.0 90.0 6.63 6.63 6.63 90.0 90.0 90.0 2.69 (2.76) Rubidium fluoride (RbF) 5.86 5.86 5.86 90.0 90.0 90.0 6.01 6.01 6.01 90.0 90.0 90.0 3.45 (3.20) Rubidium iodide (RbI) 7.84 7.76 7.76 83.3 90.0 90.0 7.35 7.35 7.35 90.0 90.0 90.0 3.01 (3.55) Halides 2 6.21 4.48 4.48 119.9 90.0 90.1 6.96 4.48 4.48 120.0 90.0 90.0 4.51 (4.04) 2 6 3 15.86 22.31 10.24 73.1 92.8 95.9 16.12 22.47 9.55 90.0 90.0 90.0 1.61 (1.60) 2 19.40 3.75 3.75 119.9 89.1 90.9 17.59 3.60 3.60 107.9 90.0 90.0 2.00 (2.41) 3 6 5.67 8.03 5.69 90.1 90.0 89.9 5.46 7.80 5.61 90.0 90.0 90.2 2.69 (2.92) 2 5.49 5.45 5.48 90.0 90.0 90.0 5.46 5.43 5.46 90.0 90.0 90.0 3.17 (3.20) 2 6.22 6.22 6.23 90.0 90.0 90.0 6.20 6.20 6.20 90.0 90.0 90.0 4.83 (4.89) 2 4.28 6.42 6.38 89.9 90.0 90.0 4.20 6.43 6.24 90.0 90.0 90.0 2.10 (2.19) 2 5.83 4.02 4.02 119.1 91.6 88.3 6.26 3.81 3.81 120.0 90.0 90.0 3.71 (3.88) 2 6.28 4.27 4.24 118.2 90.6 89.6 6.88 4.14 4.14 120.0 90.0 90.0 4.61 (4.52) 3 6 10.23 7.35 7.00 90.0 87.1 89.8 9.81 7.09 6.68 90.0 89.7 90.0 2.08 (2.36) 3 8.14 8.13 8.13 90.0 90.0 90.1 7.67 7.67 7.67 91.4 90.0 90.0 2.57 (3.07) 4 4.07 4.07 3.68 90.0 89.9 73.5 3.86 3.86 3.86 90.0 90.0 90.0 1.52 (1.54) 2 3.07 4.62 4.62 90.0 90.0 90.0 3.05 4.63 4.63 90.0 90.0 90.0 3.16 (3.17) Oxides 2 5.36 5.36 7.47 89.9 90.1 89.6 4.98 4.98 6.95 90.0 90.0 90.0 1.86 (2.32) 2 5.19 5.18 5.74 90.0 90.0 120.0 4.91 4.91 5.41 90.0 90.0 120.0 2.24 (2.65) 2 10.29 3.92 3.92 90.1 90.2 89.6 9.52 3.78 3.78 90.0 90.0 90.0 3.36 (3.90) 2 6.09 4.91 5.50 90.0 89.9 90.1 5.13 5.13 5.13 90.0 90.0 90.0 4.97 (6.06) Barium oxide (BaO) 5.64 5.64 5.64 89.8 90.0 90.0 5.63 5.63 5.63 90.0 90.0 90.0 5.69 (5.72) 2 5.19 5.18 5.74 90.0 90.0 60.0 5.00 5.00 5.46 90.0 90.0 60.0 2.24 (2.54) 2 5.34 5.34 8.72 90.0 90.0 119.9 5.05 5.05 8.27 90.0 90.0 120.0 1.86 (2.18) 2 5.42 5.67 9.61 90.0 90.4 90.0 5.14 5.45 9.17 90.0 90.0 90.0 3.59 (4.13) 2 4.94 4.96 3.49 90.0 90.0 89.9 4.74 4.74 3.19 90.0 90.0 90.0 5.86 (7.00) 2 9.55 9.56 9.56 93.9 93.9 93.8 9.40 9.40 9.40 94.3 94.3 94.3 1.38 (1.46) 2 7.43 12.85 7.49 90.0 60.8 90.0 7.17 12.37 7.14 90.0 59.7 90.0 2.56 (2.92) 2 3 14.00 5.43 5.06 89.8 85.8 81.8 12.87 4.54 5.25 86.2 90.0 90.0 3.46 (4.29) 2 3 5.13 5.12 5.12 56.2 56.2 56.2 5.13 5.13 5.13 55.3 55.3 55.3 3.91 (3.99) 2 3 4.83 4.83 12.91 90.0 90.0 120.0 4.76 4.76 12.99 90.0 90.0 120.0 3.91 (3.99) Ice-I (Ice-Ih) 7.45 4.36 7.09 90.5 90.3 89.3 7.83 4.52 7.38 90.0 90.0 90.0 1.04 (0.92) Ice-II 7.22 12.26 5.91 89.9 106.5 91.0 7.78 12.98 6.24 90.0 105.5 90.0 1.43 (1.18) Ice-III 6.69 6.50 6.46 89.9 89.5 90.7 6.73 6.73 6.73 90.0 90.0 90.0 1.28 (1.18) Ice-V 7.02 9.44 8.80 70.1 90.6 92.6 7.53 10.35 9.20 70.7 90.0 89.1 1.53 (1.24) Ice-VI 5.38 5.94 5.90 90.9 89.5 89.0 5.70 6.18 6.18 90.0 90.0 90.0 1.59 (1.37) Ice-VIII 6.44 4.36 4.36 90.0 90.5 90.1 6.41 4.45 4.45 90.0 90.0 90.0 1.96 (1.89) Ice-X (0Gpa) 2.90 2.90 2.90 90.0 90.0 90.0 2.73 2.73 2.73 90.0 90.0 90.0 2.46 (2.96) Ice-X (62Gpa) 2.72 2.72 2.72 90.0 90.0 90.0 2.73 2.73 2.73 90.0 90.0 90.0 2.98 (2.96) Ice-XI 7.62 4.26 7.15 90.1 90.2 89.9 7.83 4.52 7.38 90.0 90.0 90.0 1.03 (0.92) Ice-XIII 9.77 7.05 8.46 90.5 69.5 90.6 10.29 7.47 9.24 90.0 70.3 90.0 1.54 (1.25) Ice-XIV 3.83 8.07 7.84 90.3 90.0 90.3 4.08 8.35 8.14 90.0 90.0 90.0 1.48 (1.29) 3 4 12 36 2 13.01 13.43 13.00 90.4 92.2 89.7 12.51 12.51 12.51 90.0 90.0 90.0 4.37 (5.07) Lime (CaO) 4.95 4.95 4.95 90.0 90.1 90.0 4.82 4.82 4.81 90.0 90.0 90.0 3.07 (3.34) Mordenite (SiO2) 18.68 21.00 7.75 90.2 90.0 90.0 18.13 20.49 7.52 90.0 90.0 90.0 1.58 (1.71) Lead oxide (PbO) 3.83 4.74 3.83 89.9 90.0 89.9 3.98 5.02 3.98 90.0 90.0 90.0 10.68 (9.34) Periclase (MgO) 4.29 4.29 4.29 90.0 90.0 90.0 4.22 4.22 4.22 90.0 90.0 90.0 3.40 (3.57) 3 5.38 5.54 7.84 89.7 89.9 90.1 5.38 5.44 7.64 90.0 90.0 90.0 3.87 (4.04) 2 3.50 4.66 4.73 89.8 90.0 90.0 3.39 4.96 4.96 90.0 90.0 90.0 10.31 (9.55) 2 4.84 4.84 3.12 90.0 90.0 90.0 4.59 4.59 2.96 90.0 90.0 90.0 3.71 (4.25) 2 2.82 4.38 4.38 90.0 90.0 90.0 2.67 4.18 4.18 90.0 90.0 90.0 3.70 (4.28) Zincite (ZnO) 3.33 3.33 5.26 90.0 90.0 60.3 3.25 3.25 5.20 90.0 90.0 60.0 5.33 (5.68) Spinels 2 3 4.58 9.58 5.72 90.3 90.0 90.0 4.40 9.33 5.44 90.0 90.0 90.0 3.36 (3.78) 2 4 8.31 16.63 16.63 90.2 90.0 90.0 8.21 16.41 16.41 90.0 90.0 90.0 3.29 (3.42) Borates 5 2 2 3 7 8.54 8.59 12.11 90.0 90.0 90.0 8.55 8.55 12.09 90.0 90.0 90.0 2.93 (2.95) 2 4 7 2 12.33 11.27 16.20 105.1 93.7 83.0 12.19 10.74 11.89 90.0 73.4 90.0 1.18 (1.70) 2 5 9 2 17.19 6.39 6.18 119.6 99.8 84.6 17.50 6.49 6.31 119.2 100.4 84.0 2.84 (2.69) 5 9 2 6.77 13.62 8.88 97.4 108.2 103.3 6.68 12.87 8.82 110.0 109.1 90.4 1.82 (1.96) Carbonates 3 4.55 7.80 6.08 90.0 90.0 90.1 4.96 7.97 5.74 90.0 90.0 90.0 3.08 (2.93) 3 2 2 2 5.73 5.12 9.50 91.0 90.1 90.0 5.85 5.01 10.34 92.4 90.0 90.0 4.10 (3.78) 3 6.32 6.32 6.32 45.2 45.2 45.2 6.36 6.36 6.36 46.1 46.1 46.1 2.87 (2.73) 3 5.84 6.43  7.09 89.3 91.4 89.3 5.18 6.14 8.49 90.0 90.0 90.0 6.66 (6.57) 2 2 3 2 2 2 2 6.73 5.98 5.75 90.0 90.2 90.0 6.71 5.21 5.58 90.0 90.0 90.0 2.07 (2.45) 3 2 4.74 4.74 15.32 90.0 89.9 120.0 4.81 4.81 16.02 90.0 90.0 120.0 3.08 (2.86) 3 3 4 9.40 9.35 7.67 90.5 90.2 59.8 9.50 9.50 7.82 90.0 90.0 60.0 3.02 (2.88) 2 2 6 8.72 7.85 10.81 91.3 71.3 90.3 8.87 8.23 11.02 90.0 69.8 90.0 1.97 (1.83) 3 19.63 4.16 4.16 90.9 77.8 90.2 15.19 5.63 3.71 90.0 75.5 90.0 2.00 (2.17) 3 14.91 4.57 7.95 90.0 90.2 90.0 15.02 4.63 8.03 90.0 90.0 90.0 3.11 (3.01) 2 3 9.93 6.81 5.66 90.2 90.0 102.0 8.91 6.04 5.24 90.0 90.0 101.3 1.88 (2.55) 2 3 2 5.85 11.81 7.81 90.0 89.3 91.6 5.26 10.72 6.47 90.0 90.0 90.0 1.53 (2.26) 3 5.56 4.71 15.59 90.0 88.9 90.9 4.93 4.27 16.27 90.0 90.0 90.0 4.21 (5.02) 3 4.83 4.83 14.46 89.5 90.0 120.0 4.65 4.65 15.03 90.0 90.0 120.0 4.28 (4.44) 3 6.59 8.65 5.02 90.1 90.0 90.0 6.00 8.36 5.09 90.0 90.0 90.0 3.42 (3.84) 3 4.99 7.26 8.49 89.9 89.9 89.9 5.31 6.43 8.90 90.0 90.0 90.0 4.26 (4.32) 2 3 5.26 8.80 6.23 67.5 90.0 90.0 4.97 8.36 6.20 65.2 90.0 90.0 1.84 (2.10) Group IV Moissanite (SiC) 5.00 5.36 3.09 90.0 90.0 90.0 5.05 5.33 3.08 90.0 90.0 90.0 3.22 (3.22) Silicon carbide (SiC) 4.39 4.39 4.39 90.0 90.0 90.0 4.35 4.35 4.35 90.0 90.0 90.0 3.16 (3.24) Groups III-V Aluminum antimonide (AlSb) 5.94 5.94 5.94 90.0 90.0 90.0 6.14 6.14 6.14 90.0 90.0 90.0 4.71 (4.28) Aluminum arsenide (AlAs) 5.91 5.91 5.91 90.0 90.0 90.0 5.66 5.66 5.66 90.0 90.0 90.0 3.27 (3.73) Aluminum nitride (AlN) 4.40 4.40 4.40 90.0 90.0 90.0 4.37 4.37 4.37 90.0 90.0 90.0 3.20 (3.25) Aluminum phosphide (AlP) 5.49 5.49 5.49 90.0 90.0 90.0 5.42 5.42 5.42 90.1 90.1 90.1 2.33 (2.42) Boron Nitride (BN) 3.64 3.64 3.64 90.0 90.0 90.0 3.61 3.61 3.61 90.0 90.0 90.0 3.42 (3.49) Gallium arsenide (GaAs) 5.66 5.66 5.66 90.0 90.0 90.0 5.65 5.65 5.65 90.0 90.0 90.0 5.31 (5.32) Indium arsenide (InAs) 5.97 5.97 5.97 90.0 90.0 90.0 6.06 6.06 6.06 90.0 90.0 90.0 5.93 (5.67) Group VI Cadmium telluride (CdTe) 5.58 5.58 5.58 90.0 90.0 90.0 6.36 6.36 6.36 90.0 90.0 90.0 9.17 (6.20) Coloradoite (HgTe) 6.61 6.61 6.61 90.0 90.0 90.0 6.32 6.32 6.32 90.0 90.0 90.0 7.56 (8.63) Lead selenide (PbSe) 4.34 4.34 4.35 90.0 90.0 90.0 6.17 6.17 6.17 90.0 90.0 90.0 23.18 (8.10) Lead telluride (PbTe) 7.87 7.87 7.87 90.0 90.0 90.0 7.10 7.10 7.10 90.0 90.0 90.0 4.56 (6.22) 2 3 10.67 4.05 11.40 90.0 87.9 90.0 11.29 3.83 11.21 90.0 90.0 90.0 4.58 (4.66) Stilleite (ZnSe) 5.95 5.95 5.95 90.0 90.0 90.0 5.54 5.54 5.54 90.0 90.0 90.0 4.56 (5.65) Hydrides 3 3 4.82 4.24 5.25 94.4 90.1 90.1 4.99 4.89 5.39 90.0 90.0 90.0 0.96 (0.78) Hydroxides 2 3.20 3.21 4.30 89.8 90.0 120.1 3.14 3.14 4.77 90.0 90.0 120.0 2.53 (2.38) Diaspore (AlO(OH)) 4.17 9.75 3.00 90.0 89.9 89.8 4.40 9.43 2.85 90.0 90.0 90.0 3.26 (3.38) 3 9.04 10.38 4.45 90.1 90.0 90.0 9.74 10.16 4.34 90.0 85.5 90.0 2.48 (2.42) MnO(OH) 7.00 6.23 13.45 90.2 88.4 89.8 5.29 5.80 13.31 89.7 89.6 90.0 2.99 (4.29) 3 10.97 7.26 7.26 120.1 83.4 100.4 6.35 7.04 7.02 119.8 78.5 87.5 0.84 (1.56) Nitrates 4 3 8.82 13.85 10.76 95.8 89.4 72.1 9.49 12.23 10.76 90.0 89.9 91.3 1.71 (1.70) 3 8.96 5.38 6.59 90.6 89.1 89.5 9.16 5.41 6.43 90.0 90.0 90.0 2.11 (2.11) 3 18.66 5.43 4.08 90.1 96.2 90.5 16.83 5.07 4.39 90.0 90.0 90.0 2.06 (2.26) Phosphates 3 4 2 2 8 10.46 26.99 4.58 89.4 104.3 93.0 10.07 27.93 4.67 90.0 105.0 90.0 2.16 (2.13) 3 4 2 5.53 19.97 5.34 88.7 56.9 90.3 5.25 18.67 5.25 90.0 60.0 90.0 3.13 (3.47) 5 4 3 9.28 9.29 6.87 89.8 90.0 59.6 9.37 9.37 6.88 90.0 90.0 60.0 3.28 (3.20) 2 4 2 8.49 11.77 15.90 89.9 90.2 90.1 6.60 10.36 16.87 90.0 90.0 90.0 1.48 (2.05) 3 4 5.49 6.75 11.62 90.0 90.1 90.1 4.93 6.12 10.53 90.0 90.0 90.0 1.79 (2.42) 4 2 9.87 10.06 10.32 90.0 90.0 90.4 10.02 10.68 10.20 90.0 90.0 90.0 2.26 (2.12) 4 2 4 7.75 6.74 10.91 89.9 66.4 90.1 8.03 6.70 11.04 90.0 66.6 90.0 1.68 (1.61) 2 4 7.69 7.57 7.59 88.0 91.0 88.7 6.97 7.45 7.45 90.0 90.0 90.0 2.05 (2.33) 5 7.66 7.71 8.81 90.2 89.8 90.0 7.61 7.31 8.41 90.0 90.0 90.0 2.56 (2.85) 4 8.32 8.33 8.60 89.4 90.5 90.4 9.04 9.04 9.04 90.0 90.0 90.0 2.04 (1.64) Wagnerite (Mg2PO4F) 12.11 12.42 9.65 90.0 74.6 90.0 11.96 12.68 9.64 90.0 71.7 90.0 3.09 (3.11) Sulfates 4 2 2 12.31 12.35 12.27 89.3 89.8 90.5 12.18 12.18 12.19 90.0 90.0 90.0 1.69 (1.74) 2 6 2 6.90 15.33 11.29 90.2 108.0 90.0 7.44 15.58 11.70 90.0 110.2 90.0 2.01 (1.80) 3 4 2 6 7.01 7.00 17.38 89.9 89.7 60.2 6.96 6.96 17.35 90.0 90.0 60.0 2.79 (2.84) 4 4 10.99 5.75 7.29 91.0 90.1 90.2 10.64 5.99 7.78 90.0 90.0 90.0 1.90 (1.77) 4 7.40 6.33 7.00 92.4 91.4 95.1 6.96 8.48 5.40 90.0 90.0 90.0 6.18 (6.32) 4 6.79 8.39 7.09 87.1 90.0 90.2 6.99 6.25 7.00 90.0 90.0 90.0 2.24 (2.96) 2 4 7.55 9.62 6.38 89.9 90.1 90.2 7.48 
0.07 5.76 90.0 90.0 90.0 2.50 (2.67) 4 7.44 5.37 9.34 90.1 89.9 90.0 7.15 5.46 8.88 90.0 90.0 90.0 4.15 (4.47) 4 8.43 5.67 6.93 90.1 89.8 91.1 8.39 5.36 6.89 90.0 90.0 90.0 3.68 (3.94) 4 2 7 11.82 11.81 6.19 89.9 90.0 90.1 11.89 12.01 6.86 90.0 90.0 90.0 1.89 (1.67) 4 8.41 10.59 10.18 50.5 90.1 90.1 8.31 8.53 10.13 76.7 90.0 90.0 2.64 (2.78) 4 2 5.58 14.42 6.71 90.2 106.1 90.2 5.67 15.10 6.49 90.0 118.5 90.0 2.21 (2.34) 2 2 22 3 18 2 4 11.65 11.76 21.87 88.0 90.5 119.2 10.47 10.46 21.19 90.0 90.0 120.0 1.99 (2.59) 4 2 24.30 6.92 9.65 89.8 78.2 90.0 24.41 7.21 10.11 90.0 81.7 90.0 1.91 (1.72) 2 2 4 3 10.70 10.38 10.58 89.3 89.0 90.2 10.43 10.43 10.43 90.0 90.0 90.0 2.52 (2.62) 2 4 2 2 10.73 8.99 13.04 89.9 95.6 90.0 9.85 9.47 11.78 90.0 84.7 90.0 1.95 (2.23) 4 9.85 17.85 8.715 90.2 90.0 90.6 9.80 18.96 8.41 90.0 90.0 90.0 2.36 (2.32) 2 4 6.34 13.28 11.31 90.2 91.1 90.2 5.87 12.30 9.83 90.0 90.0 90.0 1.98 (2.66) 4 4.93 6.36 8.85 90.1 86.6 90.1 4.77 6.75 8.60 90.0 90.0 90.0 3.87 (3.87) Sulfides Cinnabar (HgS) 4.31 4.31 9.87 90.0 90.0 60.0 4.15 4.15 9.51 90.0 90.0 60.0 7.30 (8.17) Galena (PbS) 5.87 5.87 5.87 90.0 90.0 90.0 5.94 5.94 5.94 90.0 90.0 90.0 7.86 (7.60) Greenockite (CdS) 6.70 7.15 4.13 90.0 89.9 89.9 6.75 7.16 4.13 90.0 90.0 90.0 4.85 (4.80) Hawleyite (CdS) 3.84 3.84 3.84 90.0 90.0 90.0 5.83 5.83 5.83 90.0 90.0 90.0 5.02 (4.84) Metacinnabar (HgS) 6.29 6.29 6.29 90.0 90.0 90.0 5.85 5.85 5.85 90.0 90.0 90.0 6.21 (7.71) Niningerite (MgS) 5.20 5.20 5.20 90.0 90.0 90.0 5.20 5.20 5.20 90.0 90.0 90.0 2.66 (2.66) Sphalerite (ZnS) 5.41 5.41 5.41 90.0 90.0 90.0 5.43 5.43 5.43 90.0 90.0 90.0 4.08 (4.04) Wurtzite (ZnS) 3.84 3.84 6.30 90.0 90.0 90.0 3.81 3.81 6.23 90.0 90.0 60.0 4.03 (4.12) Vanadates, tungstates, chromates, molybdates 4 8.36 6.30 9.35 94.7 88.9 90.8 7.60 6.07 9.44 90.0 90.0 90.0 5.48 (6.17) 4 5.00 5.35 4.64 89.9 95.4 89.9 4.95 5.70 4.73 90.0 90.0 90.0 8.17 (7.55) 2 5 5.67 7.74 15.67 42.1 91.8 88.2 5.67 7.14 14.00 64.8 90.0 90.0 7.90 (7.07) 4 5.45 5.73 14.43 91.0 87.8 92.4 5.24 5.24 11.38 90.0 90.0 90.0 4.25 (6.12) 2 4 10.57 6.05 7.70 89.9 89.8 90.3 10.39 5.92 7.66 90.0 90.0 90.0 2.62 (2.74) 5 4 3 7.78 11.97 16.74 87.4 92.3 91.6 7.34 10.33 17.89 90.0 90.0 90.0 6.04 (6.93) 4 5.62 12.49 5.59 90.3 90.0 89.2 5.47 12.18 5.47 90.0 90.0 90.0 6.22 (6.69) 2 4 8.40 8.40 8.40 90.0 90.0 90.0 8.33 8.33 8.33 90.0 90.0 90.0 5.24 (5.36) 2 8 9.43 9.49 9.50 90.2 89.7 89.8 9.18 9.18 9.18 90.0 90.0 90.0 4.59 (5.04) d e 31 Table 6 Calculated and X-ray structural parameters for silicates. Values of a, b, c are in Ångstroms; α, β, γ in degrees  PM6 X-ray d a b c α β γ a b c α β γ Cyclosilicate 3 9 6.76 11.60 9.82 90.0 90.3 90.0 6.60 11.43 9.71 90.0 90.0 90.0 3.57 (3.75) 3 2 6 18 19.09 19.06 9.02 90.0 90.0 120.0 18.43 18.43 9.24 90.0 90.0 90.0 2.51 (2.63) 2 6 3 3 6 18 4 16.14 16.13 7.39 89.9 89.9 60.2 15.95 15.95 7.21 90.0 90.0 60.0 2.99 (3.14) Inosilicate 7 8 23 2 9.06 18.38 5.67 90.0 100.0 90.0 9.51 18.19 5.33 90.0 101.9 90.0 2.79 (2.87) 2 6 5.36 9.11 9.22 90.0 76.3 90.0 5.25 8.90 9.75 90.0 74.4 90.0 3.29 (3.28) 3 5.38 8.96 17.69 90.1 89.9 90.0 5.18 8.82 18.23 90.0 90.0 90.0 3.13 (3.20) 2 6 8.68 9.08 5.72 90.1 106.2 90.1 9.47 8.61 5.24 90.0 107.6 90.0 3.10 (3.30) 2 6 9.79 5.26 9.79 75.6 90.2 90.1 9.16 5.29 9.98 74.5 90.0 90.0 3.36 (3.52) 2 6 5.69 9.39 10.07 90.0 76.2 90.2 5.27 8.72 9.58 90.0 72.6 90.0 2.89 (3.59) 2 6 9.52 6.11 8.04 64.0 86.2 87.4 8.39 5.22 9.46 69.9 90.0 90.0 2.95 (3.18) 2 5 8 22 2 9.30 18.83 5.58 90.1 77.4 90.0 9.84 18.05 5.27 90.0 75.3 90.0 2.83 (2.98) 3 9.16 11.12 7.63 100.0 102.1 82.7 10.12 11.07 7.31 99.5 100.5 83.4 3.10 (2.93) Nesosilicate 2 2 7 7.91 7.73 4.77 90.0 87.2 90.1 7.84 7.84 5.01 90.0 90.0 90.0 3.11 (2.94) 2 5 6.13 7.96 5.90 97.1 85.9 87.9 5.56 7.90 7.80 90.0 90.0 90.0 3.78 (3.14) 2 4 12.52 11.38 5.77 89.8 90.2 89.0 9.59 11.17 5.70 90.0 90.0 90.0 3.17 (4.26) 2 2 3 2 2 2 4.83 15.12 4.41 90.0 99.9 90.0 4.78 14.32 4.63 90.0 100.3 90.0 3.03 (3.09) 2 4 4.77 6.10 10.23 90.1 89.9 90.1 4.76 6.00 10.22 90.0 90.0 90.0 3.14 (3.20) 3 3 2 3 11.75 11.75 11.75 90.0 90.0 90.0 11.86 11.85 11.85 90.0 90.0 90.0 3.69 (3.59) 2 5 7.96 5.77 5.65 82.1 84.9 78.3 7.85 7.13 5.57 78.9 90.0 74.0 4.29 (3.67) 2 3 3 2 3 11.36 11.37 11.36 89.8 89.7 89.8 11.55 11.55 11.55 90.0 90.0 90.0 3.65 (3.48) 2 5 12.49 7.36 5.73 89.8 90.0 90.0 11.55 7.68 7.49 90.0 90.0 90.0 4.09 (3.24) 5 7.44 7.00 8.43 89.9 89.6 113.9 7.07 6.57 8.72 90.0 90.0 113.9 3.24 (3.52) 2 4 2 7.11 4.18 8.95 89.7 89.1 89.6 9.33 4.42 8.39 90.0 90.0 90.0 4.60 (3.54) 3 2 3 12 12.07 12.06 12.06 89.9 90.1 90.0 12.00 12.00 12.00 90.0 90.0 90.0 3.79 (3.85) 2 4 8.94 8.93 8.95 108.8 108.7 108.7 8.62 8.62 8.63 107.9 107.9 107.9 3.93 (4.26) 4 6.09 6.08 8.53 90.6 90.1 90.3 6.61 6.61 5.98 90.0 90.0 90.0 3.85 (4.66) Phyllosilicate 2 8 2 16 2 15 9.55 9.81 14.91 92.1 85.9 89.0 8.97 8.97 15.77 90.0 90.0 90.0 2.08 (2.28) 10 2 3 3 2 4 2 9 5.45 9.39 14.16 89.3 97.2 89.2 5.24 9.07 14.29 90.0 97.0 90.0 2.57 (2.64) 2 2 4 2 7 5.14 20.70 7.60 90.4 90.0 91.1 4.95 20.50 7.28 90.0 90.0 90.0 1.76 (1.93) 2 2 5 4 6.59 9.65 5.34 87.5 77.8 88.6 7.40 8.94 5.16 89.9 75.1 88.3 2.59 (2.60) 2 2 5 5.37 14.78 6.46 90.0 90.0 90.0 4.78 14.65 5.68 90.0 90.0 90.0 1.95 (2.50) 2 2 3 10 2 4.36 5.88 18.43 90.4 88.8 89.8 4.65 5.48 18.49 90.0 90.0 90.0 2.90 (2.91) 2 5 9.43 5.63 20.70 99.6 90.1 90.0 8.90 5.16 18.64 100.8 90.0 90.0 2.21 (2.85) 3 4 10 2 5.57 9.61 9.79 91.1 81.2 90.2 5.29 9.17 9.46 90.4 81.3 89.9 2.43 (2.78) Sorosilicate 4 2 2 2 16.26 4.92 8.75 90.1 90.0 90.0 15.27 4.57 8.71 90.0 90.0 90.0 2.26 (2.60) 2 2 8 8.70 8.19 8.24 90.0 90.0 90.0 8.04 8.75 7.73 90.0 90.0 90.0 2.78 (3.00) 2 2 7 8.04 8.04 4.82 86.4 86.9 89.8 7.83 7.83 5.01 90.0 90.0 90.0 3.36 (3.39) 4 2 7 2 2 5.40 10.61 8.32 90.1 89.9 90.0 5.12 10.73 8.37 90.0 90.0 90.0 3.36 (3.48) 2 2 7 2 2 7.56 6.33 13.34 90.1 90.0 90.0 8.80 5.85 13.14 90.0 90.0 90.0 3.27 (3.09) 2 3 3 10 9.37 6.70 6.82 88.4 79.2 105.0 9.48 6.81 6.96 82.9 84.1 108.6 3.01 (2.90) 2 3 4 2 7 15.61 5.52 9.95 90.0 90.0 90.1 16.22 5.54 10.03 90.0 90.0 90.0 3.52 (3.35) 2 2 7 7.44 4.46 8.98 89.9 89.7 73.4 6.65 4.69 8.62 90.0 90.0 77.8 3.00 (3.27) Tectosilicate 3 8 8.76 12.69 7.04 89.6 65.0 90.2 8.15 12.87 7.11 93.1 63.5 89.7 2.46 (2.62) 11 24 18.67 21.00 7.75 90.2 90.0 90.0 18.13 20.49 7.52 90.0 90.0 90.0 1.58 (1.71) 3 4 4 16 10.15 10.29 8.26 90.0 90.0 61.1 10.05 10.05 8.38 90.0 90.0 60.0 2.57 (2.64) 2 2 8 7.05 9.37 9.38 90.1 90.2 90.1 8.40 9.48 9.00 90.0 89.3 90.0 4.40 (3.81) 4 3 3 12 8.91 8.90 8.90 90.1 90.1 90.0 8.88 8.88 8.88 90.0 90.0 90.0 2.28 (2.30) 2 5 5 20 2 13.69 13.57 12.74 80.7 75.8 84.2 13.25 13.06 13.10 90.0 90.0 90.0 2.37 (2.36) d Table 7 Comparison of calculated and experimental heats of formation of inorganic compounds (kcal/mol)   PM6 a Difference 2 −189.4 −217.7 28.3 2 −193.2   2 −193.2   2 −189.4   2 −190.7   2 −187.2   2 −191.2   2 −160.7   Aluminum nitride (AlN) −42.7 −76.0 33.3 Aluminum phosphide (AlP) −26.5 −39.8 13.3 4 −70.0 −75.1 5.1 4 3 −88.0 −87.4 −0.6 4 2 4 −251.5 −282.2 30.7 2 −229.3 −224.4 −4.9 2 5 −665.2 −619.5 −45.7 4 −221.9 −219.9 −2.0 4 −275.3 −342.9 67.6 3 −269.0 −288.6 19.6 2 4 −377.5 −343.6 −33.9 2 −286.4 −263.0 −23.4 Barium oxide (BaO) −182.2 −131.0 −51.2 4 −383.8 −352.1 −31.7 Boron (B) −8.2 0.0 −8.2 Boron Nitride (BN) −75.5 −60.8 −14.7 2 −229.8   2 −176.6 −221.0 44.4 3 −271.7 −288.6 16.9 2 −108.8 −127.5 18.7 2 −59.8 −138.1 78.3 4 −297.4 −291.6 −5.8 2 −153.9 −153.3 −0.6 2 3 −490.6 −549.9 59.3 Cinnabar (HgS) −53.3 −13.9 −39.4 Coloradoite (HgTe) −20.4 −10.0 −10.4 2 3 −370.9 −400.5 29.6 3 6 −871.0 −792.8 −78.2 3 −324.2 −370.2 46.0 2 −207.1 −293.0 85.9 2 4 −452.9 −520.3 67.4 2 −305.3 −288.5 −16.8 Galena (PbS) −24.8 −24.0 −0.8 Gallium arsenide (GaAs) −35.5 −17.0 −18.5 Graphite (C) 1.3 0.0 1.3 Greenockite (CdS) −85.1 −38.7 −46.4 Hawleyite (CdS) −85.0 −38.7 −46.3 2 −151.4 −190.1 38.7 Indium arsenide (InAs) −17.6 −14.0 −3.6 Lead selenide (PbSe) −124.7 −24.6 −100.1 Lead telluride (PbTe) 6.2 −16.9 23.1 Lime (CaO) −116.2 −151.8 35.7 3 −224.8 −265.7 40.9 2 −116.9 −125.3 8.4 2 −67.2 −87.0 19.8 2 4 −372.5 −331.5 −41.0 3 −238.3 −261.7 23.4 Lead oxide (PbO) −85.9 −52.3 −33.6 2 −21.6 −66.3 44.7 Periclase (MgO) −96.9 −143.7 46.8 2 −230.1 −225.6 −4.5 2 −234.9 −268.7 33.8 Silicon (Si) −13.5 0.0 −13.5 Silicon carbide (SiC) −37.6 −15.6 −22.0 3 −175.7 −194.3 18.6 3 −159.6 −111.8 −47.8 Sphalerite (ZnS) −40.4 −49.2 8.8 2 4 −493.7 −549.5 55.8 Sulfur (S) 1.5 0.0 1.5 2 4 −375.9 −391.2 15.3 Wurtzite (ZnS) −39.4 −46.0 6.6 Zincite (ZnO) −84.6 −83.8 −0.8 4 −211.0 −234.9 24.0 4 −461.0 −529.9 68.9 a 32 Elements sp 3 sp 2 30 31 f f 32 f f 33 f Sulfur forms eight-membered rings, with 16 rings per unit cell. Because the unit cell is so large, and because there is a distinct insulating gap between each ring, the approximation that Γ represents the entire Brillouin zone is valid even when only a single unit cell is used. Within each ring, the sulfur–sulfur distance is 2.04 Å, in perfect agreement with the 2.04 Å observed, but the inter-ring distance is badly predicted, resulting in a calculated density of 1.54 g/cc, considerably less than the observed 2.06 g/cc. This lack of inter-ring interaction is the likely cause of the calculated heat of formation being 1.45 kcal/mole-atom, rather than being nearer to the reference 0.0 kcal/mole-atom. Halides 8 Table 8 Comparison of calculated and experimental heats of formation of alkali metal halides (kcal/mol) Salt Molecule Crystal PM6 1 PM6 (NaCl) PM6 (CsCl) 32 Lithium fluoride −81.5 −86.0 −138.2 −139.7 −147.4 Lithium chloride −46.8 −53.9 −107.4 −108.2 −97.6 Lithium bromide −36.8 −38.1 −89.6 −87.3 −83.9 Lithium iodide −19.4 −16.1 −72.0 −73.2 −64.6 Sodium fluoride −69.6 −64.0 −123.2 −125.7 −137.5 Sodium chloride −43.4 −49.4 −87.7 −86.8 −98.3 Sodium bromide −34.2 −37.6 −99.4 −101.7 −86.4 Sodium iodide −19.0 −19.8 −75.2 −75.4 −68.8 Potassium fluoride −78.1 −74.9 −185.7 −183.5 −135.9 Potassium chloride −51.2 −53.4 −112.0 −118.3 −104.4 Potassium bromide −43.0 −44.3 −99.8 −113.2 −94.1 Potassium iodide −30.0 −30.0 −82.5 −93.9 −78.4 Rubidium fluoride −79.2 −101.3 −152.8 −111.3 −133.3 Rubidium chloride −54.7 −61.3 −117.9 −108.2 −104.1 Rubidium bromide −43.7 −50.8 −107.7 −103.5 −94.3 Rubidium iodide −32.1 −24.0 −66.4 −69.6 −79.8 Cesium fluoride −85.2 −81.9 −155.9 −115.3 −132.6 Cesium chloride −57.4 −62.9 −128.4 −130.2 −105.8 Cesium bromide −50.0 −55.0 −106.0 −117.6 −97.0 Cesium iodide −36.3 −43.5 −86.0 −98.4 −82.8 2 2 2 2 2 3 2 2 6 2+ 34 6 3- 3 6 Oxides 2 35 2 2 7 36 8 9 Fig. 8 2 Left right  Fig. 9 2 Left right  2 37 2 3 12 d 9 Table 9 f  Experimental a a a PM6 a (Å) 4.76 4.85 4.69 5.28 4.83 c (Å) 12.99 13.12 12.33 15.47 12.91 f −400.5 −266.5 −241.9 −236.4 −370.9 a 12 2 10 10 Fig. 10 2  Table 10 Interatomic distances in rutile Distance PM6 X-ray Ti-O 2.022 1.981 Ti-O’ 2.080 1.948 Ti-Ti 3.116 2.959 3 3 38 5 2 + 39 40 5 2 + 2h 2h 41 42 2 s 2h 2h 6 6 2 + IV 8 4- Other AB-type solids A number of solids of the type AB are formed from elements of Group III and V, while others involve elements of Group IV, and still more involve elements of Group VI and heavy elements. Some of these occur naturally, such as wurtzite and sphalerite (zinc sulfide), and coloradoite (mercury telluride), while others are formed synthetically, often by chemical vapor deposition methods. In most of these materials, each atom of type A is tetrahedrally coordinated to four atoms of type B. This results in two types of packing, best exemplified by the two polymorphs of zinc sulfide. For convenience all compounds of this type will be grouped together. Many of these compounds are semiconductors, and have small band-gaps between the occupied and virtual orbitals. A consequence of this is that the cluster used must be very large in order to minimize errors arising from the cluster approximation. In addition, it was anticipated that the narrow band-gap would give rise to difficulties in solving the self-consistent field equations—such difficulties had frequently occurred when molecules that had small HOMO–LUMO gaps were being studied. Surprisingly, the SCF equations were solved using default options, albeit more iterations than normal were needed. Symmetry was used to accelerate the geometry optimization of those solids that had the sphalerite structure; such solids have only one adjustable parameter. In most cases, the calculated density for the optimized structure was close to that expected, the exceptions being cadmium telluride, which PM6 predicts to be too dense, and coloradoite, where the density was predicted to be too low. Carbonates, nitrates, and borates 11 3 3 3 3 11 3 3 d 2g g II 6 1g d- II Fig. 11 Unit cell of calcite. Crossed-eyes stereo view  Table 11 Interatomic distances in calcite Distance PM6 X-ray Ca-O 2.297 2.389 C-O 1.289 1.210 Ca-Ca 3.989 4.036 3 3 3 3 2 3 2 3 Nitrates PM6 correctly reproduces the structures of ammonium nitrate and potassium nitrate (niter), but the predicted structure of sodium nitrate, nitratine, is completely incorrect; as with other sodium compounds, the predicted Na–Na distances are unrealistically small. Borates 2 5 9 2 12 Fig. 12 Unit cell of parahilgardite. Crossed-eyes stereo view  Molybdates, tungstates chromates vanadates, sulfates, and phosphates 4 2- 4 3- 4 2- 4 2- 4 2- 4 + 2+ 2+ 4 13 12 Fig. 13 Unit cell of fluorapatite. Crossed-eyes stereo view  Table 12 Ca 1 Ca 2 Ca 3 Distance PM6 X-ray 1 1 5.415 5.408 1 2 3.450 3.456 1 2.305 2.338 3 2.386 2.375 3 2.330 2.361 P-O 1.580 1.585 4 2 4 2 4 14 13 15 14 Fig. 14 4 Top bottom  Fig. 15 2 4 3 4 2 4 Left right  Table 13 4 Distance X-ray PM6 O(H) ⋯O 2.63 2.63 O–H 0.73 1.04 S–S 4.40 4.60 Table 14 2 4 Distance X-ray PM6 O(H)⋯O 2.49 2.53 O–H 1.07 1.20 O–H 1.43 1.33 P–P 4.11 4.26 2 4 PM6 reproduces the structures of most of the phosphates, but, as expected, in the case of sodium phosphate the predicted structure was qualitatively incorrect. Silicates 4 2 6 4 4 2 5 4 4- 2- 5 Double and triple tetrahedra: sorosilicates 2 7 6- 2 2 7 2 2 7 36 2 7 6- 2h Chains: inosilicates 7 8 23 2 Cyclosilicates 3 9 3 2 6 18 2 6 3 3 6 18 4 15 16 + 2+ 3+ 4+ 2- 3 3- 6 18 12- Fig. 16 Unit cell of beryl. Crossed-eyes stereo view  Table 15 Interatomic distances and angles in beryl Geometric quantity X-ray PM6 Al-Be 2.754 2.656 Al-O 1.928 1.906 Si-O(Al) 1.655 1.610 Si-O(Si) 1.611 1.607 Si-O-Si 168.1 165.3 Al-O-Si 128.6 137.1 Sheets: phyllosilicates 17 16 Fig. 17 Unit cell of talc. Crossed-eyes stereo view  Table 16 Interatomic distances in talc Distance PM6 X-ray Mg-O(H) 2.060 2.062 Mg-O(Si) 2.084 2.080 Si-O(Mg) 1.629 1.624 Si-O(Si) 1.674 1.622 a 3.762 3.095 a 6 4 10 8 II III 6 4 10 8 2 8 10 8 Frameworks: tectosilicates 2 5 5 20 2 Discussion With a few exceptions, the geometries of individual organic molecules and ions and their packing arrangement in the crystal lattice were reproduced with good accuracy. In the original PM6 article it was shown that the average error in predicted bond-lengths in organic compounds was about 2–3%. The effect of intermolecular forces on the geometries of the component molecules is likely to be small, so by implication the accuracy of prediction of the geometries of those molecules would be similar to that reported in the earlier article. By far the largest structural error in organic solids involves intermolecular separations. The values of these are determined by several forces, from the weak VDW and π-stacking attractions, in cyclohexane and coronene, through simple hydrogen bonding of the type found in sucrose, to very strong intermolecular interactions, usually ionic, best typified by the Zwitterions, such as the simple amino acids. Errors in intermolecular separations could be determined by a direct comparison of calculated and reference structures. However, because errors in intermolecular separation have a direct impact on the unit cell dimensions, and consequently on the density, a convenient and reliable estimate of the accuracy of prediction of intermolecular separations of molecules in the unit cell can be obtained from a comparison of predicted and X-ray densities. 2 – The structures of most of the inorganic solids were predicted with useful accuracy, with the highest accuracy being exhibited by minerals in which metal atoms interact with oxygen, and the oxygen then interacts with a main-group element. Most of the silicates, phosphates, and sulfates fall into this group. One reason for this can be attributed to the procedur
 used in developing PM6, in that a large quantity of reference data for systems that had metal-oxygen or main-group oxygen bonds was used during the parameterization. This naturally resulted in increased emphasis being placed on the structure and thermochemistry of oxygen-containing systems. On the other hand, inorganic systems exhibit a much wider range of types of interaction than those found in organic chemistry, and this makes the general application of PM6 to inorganic solids more difficult. An implication of the fact that PM6 uses the Voityuk diatomic core-core term is that parameters must be present for each pair of elements in a system, unless the pair of elements is separated by a distance sufficiently large that there would be no significant core-core interaction. Only a limited number of types of interactions were surveyed in this work; even within that number, several instances were found in which the PM6 values of the Voityuk parameters were severely in error and gave rise to results that were nonsense. Because of this, great care should be exercised in determining the suitability of PM6 for modeling solids that include diatomic interactions of types not found in any of the species reported here. 18 2 PM6 X-ray Fig. 18 Comparison of calculated and X-ray densities  Systems that are badly predicted Of the systems whose properties were badly predicted, three types could be identified. In the first group, illustrated by lead selenide and 2, 4, 6-tribromoaniline (TBA), the origin of the error could be traced to the values of individual diatomic parameters used in PM6. These were either incorrect, as in lead selenide, where the PM6 value for Voityuk’s lead–selenium core–core repulsion was much too small, and in TBA where the bromine–nitrogen parameters were either physically unrealistic or absent. Errors of this type could be easily corrected by carrying out a small parameterization operation involving only the faulty parameters and using a training set consisting of examples of the two atoms in close proximity. The second type of error was found in some solids for which no equivalent error was found in the isolated molecule. This is best illustrated by the Group I halides, where PM6 predicted the lowest energy structure to be either rock salt or cesium chloride, almost at random. During the development of PM6, only small representatives of I-VII species were used. Apparently, these systems were too small to allow the lattice properties to be accurately characterized. Errors of this type were therefore not immediately obvious, and only became apparent when full solid-state calculations were done. The implication is that future parameterizations should include solids in the training set. The best solids for use in parameterization would be those that were badly predicted by PM6. Addition of solids to the parameterization would be unlikely to cause a significant increase in error in the prediction of molecular properties. Although the use of solids in parameterization is impractical at present, it is likely that increases in computer power will make such calculations possible in the not-too-distant future. The third type of error is specific to organic compounds, where PM6 predicts some ions to be too stable. Thus the aminophenols are predicted to exist in the solid state as the Zwitterions rather than as the neutral species. The hydrate of oxalic acid was predicted to exist as oxalate and hydronium ions, a prediction completely in variance with the X-ray structure. No instances were found where PM6 underestimated the stability of ions relative to their neutral counterparts. That is, the error was completely systematic and unequivocally indicated a fault in the parameterization. The origin of this error most likely lies in the set of atomic electronic parameters, rather than in the diatomic core–core parameters, and as such could only be corrected by a re-parameterization. There was no evidence that any faults were due to an underlying defect in the set of approximations. Accuracy of geometry vs hardness In general, the structures of solids that are mechanically extremely hard, seven or more on Moh’s scale, are predicted with good accuracy, whereas the geometries of many softer solids, such as most organic species, the transition metal carbonates, and various layer silicates, particularly the micas, are predicted with significantly less accuracy. This inverse relationship of mechanical hardness and computational accuracy can be rationalized by consideration of the interatomic forces involved. In hard solids, all atoms are connected to adjacent atoms by strong covalent bonds, and, of their nature, those bonds have large force constants. In general, semiempirical methods predict such geometries with good accuracy. Any tendency to deviation from the expected geometry would result in a large energy penalty, therefore distortions in predicted geometries are likely to be small. Conversely, in mechanically weak solids some atoms interact only weakly with adjacent atoms. Thus the layers in the micas and in boric acid are held together by low energy electrostatic and VDW forces. Much effort was expended during the development of PM6 in attempting to accurately model hydrogen bonds and other weak interactions because of their importance in biochemical systems.