Continuous Symmetry and Chirality Measures

1. Continuous Symmetry and Chirality Measures David Avnir Institute of Chemistry The Hebrew University of Jerusalem Harvard, Boston, January 28, 2013

7. “Near” C2 symmetry: HIV Protease mutant V82A complexed with A77 inhibitor What, quantitatively, is the C2 symmetry content of that protein?

8. Gradual changing chirality and C2-ness in aggregates Is it possible to quantify these changes?

10. Since achirality relates to symmetry, similar questions pop up also in the context of chirality: “By how much is one molecule more chiral than the other?”

11. In fact, asymmetry and chirality are very common: Given a sufficiently high resolution in space or time it is quite difficult to find a fully symmetric, achiral molecule. Consider watching methane on a vibrational time-scale: Only one in zillion frames will show the following:

12. Given a sufficiently high resolution in space or time it is quite difficult to find a fully symmetric, achiral molecule Spatial resolutions: Often, symmetry is lost at the condensed phase: # An adsorbed molecule # A matrix-entrapped molecule # A molecule packed in the crystal # A molecule in the glassy state # A molecule within a cluster

13. A methodology is needed in order to quantify the degree of symmetry and the degree of chirality: # Comparing different molecules # Following changes within a single molecule

14. The proposed methodology for a symmetry-measure design: Find the minimal distance between the original structure, and the one obtained after the G point-group symmetry is operated on it.

15. The continuous symmetry measure : The original structure : The symmetry-operated structure N : Number of vertices d : Size normalization factor * The scale is 0 - 1 (0 - 100): The larger S(G) is, the higher is the deviation from G-symmetry H. Zabrodsky

16. E C3 C32 Measuring the degree of C3-ness (S(C3)) of a triangle Ch. Dryzun

17. The measure is the collection of distances between the blue and the (original) red All three triangles are superimposed. The set of 9 points is C3-symmetric. Its blues average is a C3-symmetric triangle

18. S(G) as a continuous chirality measure G: The achiral symmetry point group which minimizes S(G) Achiral molecule: S(G) = 0 The more chiral the molecule is, the higher is S(G)

19. The Continuous Shape Measure * The CSM estimates the distance to an a-priori unknown shape with the desired symmetry * The Shape Measure estimates the minimal distance to a specific pre-selected shape (any shape) * For ML6: # Shape: What is the degree of ML6-octahedricity (S(L6-Oh))? # Symmetry: What is the degree of Oh-ness (S(Oh))?D4h-ness (S(D4h)? And of S(D2h)? S. Alvarez, P. Alemany

20. Some properties of the symmetry measure * The measure is a global structural parameter: It takes into account all bond angles and bond lengths * A full profile of symmetry and chirality values is obtained * All values are comparable either within the same molecule or between different ones * The computational tools are efficient * Analytical solutions have been obtained for many types of symmetry * The shape of the nearest symmetric object is an outcome * The measure is well behaved, and its correlations with physical/chemical parameters agree with intuition

21. The CSM values of an AB4 species with respect to tetrahedricity and planar-squareness Perfect tetrahedron- Td Distortedtetrahedron S(Td) = 33.3 S(D4h) = 0 S(Td) = 10.6 S(D4h) = 7.84 S(Td) = 0 S(D4h) = 33.3 Planar square – D4h

22. C3v Td D4h Cv 0 33.33 72.22 100 65.73 S(Td) 0 1 The full scale of the CSM

24. The most chiral monodentate complex

26. Trends within families and classificationsSymmetry maps

27. The symmetry map of 13,000 transition metal ML4 complexes S. Alvarez, P. Alemany, JACS 2004

28. 30 25 20 15 10 5 0 0 5 10 15 20 25 30 CuCl42-:The tetrahedral to planar-square symmetry map and pathway S(D4h) S(Td) S. Keinan

29. 110o 70o Several possible pathways for this transformation Spread Compression Twist

30. 30 25 20 15 Spread Twist 10 Compression 5 0 0 5 10 15 20 25 30 The tetrahedral to planar-square transformation  CuCl42- S(D4h) S(Td)

31. Energy in Hartree (relative energy in kcal/mol) -2032.95 (136.8 kcal/mol) -2033.00 (105.4 kcal/mol) 30 (74.1 kcal/mol) -2033.05 (42.67 kcal/mol) -2033.15 -2033.10 -2033.10 (11.29 kcal/mol) 25 -2033.15 S(D ) -2033.05 -2033.20 -2033.00 J J -2033.168 (0 Kcal/mol) 20 Spread simulation 15 10 5 0 0 5 10 15 20 25 30 35 S(T ) d Minimal energy and minimal symmetry values coincide • S(D4h)

32. Tetracoordinated Bis-Chelate Metal Complexes M(L-L')2: The [M(bipy)2] family 110o 70o Twist L-M-L bond angles: # Spread From 90° to 109.4° #Two Twist pathways: The bidentate nature is introduced by keeping the two opposite L-M-L bond angles constant at typical 82 and 73°

33. We (mainly S. Alvarez) analyzed similarly all MLn families with n from 4 to 10 4 Chem. Eur. J., 10, 190-207 (2004). 5J. Chem. Soc., Dalton Trans., 3288-3303 (2000). 6New J. Chem., 26, 996-1009 (2002). 7Chem. Eur. J., 9, 1281-1295 (2003). 8 Chem. Eur. J., 11, 1479 (2005). 9Inorg. Chem., 44, 6939-6948 (2005). 10 Work in progress

34. Symmetry or chirality as reaction coordinates

35. Stone-Wales Enantiomerizations in Fullerenes Y. Pinto, P. Fowler (Exeter)

36. Hückel energy changes along the enantiomerization

37. The sensitivity of energy/chirality dependence on the size of the fullerene

38. Temperature and pressure effects on symmetry and chirality

39. Changes in the degree of octahedricity with temperature CuCl64- Temp (oK) S(Oh) Data: Wei, M. & Willett, R.D. Inorg. Chem. (1995) 34, 3780. Analysis: S. Keinan

40. Temperature and pressure effects on the chirality and symmetry of extended materials: Quartz Low Quartz SiO2, P3221

41. The building blocks of quartz SiO4 Si(OSi)4 SiSi4 -O(SiO3)4-

42. Combining temperature and pressure effects through symmetry analysis b S(C2) of a four tetrahedra unit: A measure of helicity A correlation between global and specific geometric parameters

43. GeO 4 4 4 SiO 4 4 4 Predicting the high pressure symmetry behavior of quartz based on the isostrucutral GeO2 D. Yogev-Einot , D. Avnir; Acta Cryst. (2004) B60 163-173

44. The building blocks of quartz: All are chiral! SiO4 Si(OSi)4 SiSi4 -O(SiO3)4-

45. How small can the measure be and still indicate chirality? The error bar # Typical limit: In quartz, S(Chir) of SiO4 = 0.0007 # For S values near zero, the error bar is not symmetric: The + and - are different. # If the lower bound of S touches 0.00000, then the molecule is achiral. M. Pinsky et al, “Statistical analysis of the estimation of distance measures” J. Comput. Chem., 24, 786–796 (2003)

46. 1.17 1.12 Le Chatelier a t/a 1.07 1.02 0.97 98 298 498 698 898 1098 Temperature (°K) The optical rotation of quartz Le Chatelier, H. Com. Rend de I'Acad Sciences1889, 109, 264.

47. Chirality, SiSi4 Le Chatelier a t/a Chirality a t/a 0 Temperature (°K) 115 years later: Interpretation and exact match with quantitative chirality changes SiSi4 Crystallography: Kihara, 1990. Analysis: D. Yogev-Einot

48. Correlations between continuous symmetry and spectral properties

49. 15000 14000 13000 12000 11000 10000 9000 8000 7000 0 5 10 15 20 25 30 35 Jahn-Teller effects and symmetry: The d-d splitting in Cu complexes max d-d (cm-1) S(Td) Data: Halvorson, 1990. Analysis: S. Keinan

50. 250 a=b=c=(CH2)3 200 150 a=b=c=(CH2)2 a=c=(CH2)3; b=(CH2)2 100 a=c=(CH2)2; b=(CH2)3 50 1 2 3 4 5 6 7 Changes in transition probability as a function of octahedricity CuN4O2 Chromophores: +2H2O  [cm-1M-1] S(Oh) Data: P. Comba, 1999