Computational studies of polymorphs: principles Angelo Gavezzotti

1. Computational studies of polymorphs: principles Angelo Gavezzotti Dipartimento di Chimica Strutturale Università di Milano

2. molecule intramolecular intermolecular computational box

3. computational box for succinic anhydride: ideal crystal

4. succinic anhydride: a real dynamic snapshot

5. succinic anhydride: liquid

6. The key to molecular simulation is the potential intermolecular model potential: way of representing intermolecular interactions and of calculating quantitatively the energies involved

7. chemical cohesion: electrons and nuclei electr(on)ic interactions atoms stick together but solid matter is impenetrable chemical bond(ing)

8. Richard P.Feynman

9. The Hellmann-Feynman Electrostatic Theorem: in the Born-Oppenheimer approximation, nuclei see a static, smeared out electron cloud and the forces acting at nuclei are just coulombic forces exerted by other nuclei and by the electron cloud if the exact wavefunction is known, any chemical bond or bonding is: electrons between nuclei, more +/- attraction than +/+ and -/- repulsion

10. Empirical methods do not rely on a wavefunction, but on parameter fitting. in setting up the potential model, one needs to define the interaction centers i.e. , e.g., repulsion between what?

11. organic crystals

12. The whole heat of sublimation of a 15-atom molecule is about 25% of a single C-H bond energy Average bonding effect in a crystal: 100/15  6 kJ/atom Strong hydrogen bond: up to 150 kJ/mol Weak hydrogen bond: 30-60 kJ/mol (e.g. in amides or carboxylic acids) How can these weak potentials be described and calculated?

14. close-packing... molecules are held at equilibrium by the balance between attraction and repulsion between their electron distributions the arrangement of nuclei of outer atoms is the result, not the cause of this equilibrium

15. In 1970, computers were very slow. Hence the atom-atom idea: • nuclear positions are reference locations for potentials • E(crystal) = ij E(atom i, atom j) • glycine: a 10-site object fast and incredibly reliable for many applications

16. kJ/mol units

17. Slowly, into chemical thinking crept the atom-atom 'prejudice': • condensed phases can be understood in terms of localized atom-atom bonds, just like molecules • that is: whenever the distance between two nuclei is short, they are joined by a chemical bond, and the sum of these bonds is what determines the crystal structure

18. an alternative view more physical? (more convenient?)

19. "We shall say that there is a chemical bond between two atoms or two groups of atoms in case that the forces acting between them are such as to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent chemical species.” L.Pauling, The Nature of the Chemical Bond

21. The PIXEL idea: Evaluate electron density by quantum chemistry for the isolated molecule Each e-pixel is a site Glycine: a 7300-site object

22. Since our basis set is not complete, and our wavefunction is not dynamically adjusted to polarization, and……………. we cannot calculate Hellman-Feynman forces exactly. We adopt a different point of view: the intermolecular interaction is divided into a coulombic, a polarization,a dispersion and a repulsion part Rigid electron densities, no covalent or charge transfer parts

23. coulombic energy: a sum of coulombic terms over pixels and nuclei (parameter-less) polarization energy: a sum of linear polarization terms over pixels with local polarizability (one parameter) dispersion energy: a sum over pixel-pixel London-type terms (one parameter) repulsion energy: proportional to the overlap integral between electron densities (one parameter, determined by some meta-parameters)

24. E(PIXEL,total) = E(coul)+E(pol)+E(disp)+E(rep) Are these 'the' coulombic, polarization, etc...energies? No: each method defines its own energy partitioning But comparisons of the different kinds of energetic contribution over polymorphs or similar compounds can be very revealing.

25. The PIXEL approach shifts the focus of the analysis from the nuclei to the electron density From atom-atom chemical bonds to molecule-molecule chemical bonding move from qualitative geometrical criteria to quantitative energy criteria

26. computer modeling of crystal polymorphism: - analysis of existing crystal structures what are the relevant features? - generation of new crystal structures a more technical problem - comparing calculated properties for different polymorphs energy, entropy, density, stress tensors, morphology need accurate potential

27. ORDERS OF MAGNITUDE distance : 1 angstrom species against result Point P q = 1 electron field = 1.44 1011 V m-1 Point P q = 1 electron EPOL = -695 kJ mol-1 1 Å3 polarizability q = 1 electron q = 1 electron ECOUL = +1389 kJ mol-1 Intermolecular interactions are weak only because large energies balance out……!!

28. kJ/mol units

29. The OPiX computer program package* • ZipOpec module: packing analysis (atom-atom energies) • Prom/Sorter module: polymorph generator (atom-atom energies) • Pixel module: Pixel calculations for clusters and crystals *ask me, or write to angelo.gavezzotti@unimi.it

30. Explain the crystal structure of naphthalene, naphthoquinone and naphthoic acid •draw packing diagrams (usually a mess) • look at C...H distances, C-H...p, p…p, etc • look at dipoles, quadrupoles, simple electrostatic arguments • look at C-H...O distances • look at O-H...O distances • ………

31. Replace atom-atom analysis by molecule-molecule analysis each pair of molecules in the crystal is characterized by • a distance between centers of mass • coulombic, dispersion and repulsion energy Generate many computational crystal structures for a given molecule • analyze the crystal energy landscape

32. B naphthalene PIXEL energies A C Ecoul+pol Edisp Erep Etot (kJ/mol) A -8 -24 13 -18 B -6 -14 6 -13 C -3 -8 5 -6 PS -6 -51 25 -32 (Parallel Stack)

33. naphthalene Ecoul Epol Edisp Erep Etot (kJ/mol) -23 -11 -96 58 -72 -23 -11 -91 56 -68 -7 -7 -88 42 -61

34. naphthoquinone distance Ecoul+pol Edisp Erep Etot (kJ/mol) 4.07 -5 -46 14 -37 4.38 -3 -37 10 -30

35. naphthoquinone E,F C O…H distance Ecoul+pol Edisp Erep Etot (kJ/mol) C 2.52 -18 -9 12 -15 E,F 2.47,2.49 -17 -10 15 -12

36. exptl

37. 2-naphthoic acid E F A Ecoul+pol Edisp Erep Etot (kJ/mol) A -224 -14 184 -54 E -11 -5 6 -10 F -4 -8 7 -4

38. B 2-naphthoic acid C Ecoul+pol Edisp Erep Etot (kJ/mol) B -9 -38 20 -27 C -6 -17 5 -18

39. Naphthoic acid, ab projection Corrugation to avoid CH…CH electrostatic repulsion

40. some conventional wisdom: hydrogen bonds always form therefore, they are the strongest interaction in a crystal neighbor molecules are attracted to one another the closer, the more attracted

41. 2,6-dinitro-3-acetaminotoluene Ecoul Epol Edisp Erep Etot stack -26 -7 -50 21 -62 H-bond -41 -15 -19 44 -31 H-bond is not most stabilizing interaction

42. ref A B things nearby are not always things stabilizing glycine zwitterion: Ecoul Epol Edisp Erep Etot Ref-A -123 -27 -11 83 -78 Ref-B +51 -10 -9 12 +44

43. computational polymorphs of parabanic acid Ecoul Epol Edisp Erep Etot, Pixel Etot, DMA exptl -107.9 -42.9 -72.0 134.1 -88.7 8 -111.4 1 fc5 -106.9 -45.0 -81.4 138.4 -94.8 1 -109.0 2 af28 -86.1 -32.1 -87.3 113.7 -91.9 2 -106.8 5 Same total energy for large differences in partial contributions

45. at the other extreme: large energy differences for small geometry differences acetic acid hydrogen-bonded dimer

46. crystal polymorphism: a matter of energy landscapes instead of energy points

47. caffeine* 1,7-dimethylxanthine *No ordered anhydrous crystal structure is known Ullrich Griesser knows everything

48. kJ/mol units

49. 1,7-dimethylxanthine 100 crystal structures kJ/mol units

50. THE THERMODYNAMIC RULE free energy: enthalpy and entropy differences ENTHALPY Intermolecular bonding Electric interactions between molecular electron densities ENTROPY if no bonds are broken or formed: mainly from different collective vibration modes