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Continuous Symmetry and Chirality Measures

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## Continuous Symmetry and Chirality Measures

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**Continuous Symmetry and Chirality Measures**David Avnir Institute of Chemistry The Hebrew University of Jerusalem Harvard, Boston, January 28, 2013**“Near” C2 symmetry: HIV Protease mutant V82A complexed**with A77 inhibitor What, quantitatively, is the C2 symmetry content of that protein?**Gradual changing chirality and C2-ness in aggregates**Is it possible to quantify these changes?**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?”**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:**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**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**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.**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**E**C3 C32 Measuring the degree of C3-ness (S(C3)) of a triangle Ch. Dryzun**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**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)**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**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**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**C3v**Td D4h Cv 0 33.33 72.22 100 65.73 S(Td) 0 1 The full scale of the CSM**The symmetry map of 13,000 transition metal ML4 complexes**S. Alvarez, P. Alemany, JACS 2004**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**110o**70o Several possible pathways for this transformation Spread Compression Twist**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)**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)**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°**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**Stone-Wales Enantiomerizations in Fullerenes**Y. Pinto, P. Fowler (Exeter)**The sensitivity of energy/chirality dependence on the size**of the fullerene**Temperature and pressure effects**on symmetry and chirality**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**Temperature and pressure effects on the chirality and**symmetry of extended materials: Quartz Low Quartz SiO2, P3221**The building blocks of quartz**SiO4 Si(OSi)4 SiSi4 -O(SiO3)4-**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**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**The building blocks of quartz:**All are chiral! SiO4 Si(OSi)4 SiSi4 -O(SiO3)4-**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)**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.**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**Correlations between continuous symmetry and spectral**properties**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**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