11/6/2013. Refinement. Fourier Methods. Fourier Methods. Difference Map. Difference Map Find H s. Difference Map No C 1

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1 Refinement Fourier Methods find heavy atom or some F s phases using direct methods locate new atoms, improve phases continue until all atoms found in more or less correct position starting point of refinement solvent or unexpected atoms next harder to find in a Fourier map do not know where to look or what to look for; symmetry related atoms clutter map use a difference map 1 (x, y, z) = ( F o F c ) cos (hx + ky + lz c ) V h k l Fourier Methods Difference Map _atom_sites_solution_primary _atom_sites_solution_secondary _atom_sites_solution_hydrogens direct difmap geom should just contain random noise difmap vecmap heavy direct geom disper isomor difference Fourier map realspace vector search heavy atom method structure invariant direct methods inferred from neighboring sites anomalous dispersion techniques isomorphous structure methods atoms not included in phase will stand out H s often located this way when structure is nearly complete Difference Map No C 1 Difference Map Find H s H H 1

2 Final Difference Map when structure complete (all atoms and H) final difference map largest, most negative and average e density given in cif (e /Å 3 ) _refine_diff_density_max _refine_diff_density_min _refine_diff_density_rms should be < 1e /Å 3 and random max/min often near heavy atom; ripples of Fourier summation Au Atom Type Assignment 1. peak heights. magnitude of displacement parameters (temperature factors) 3. refinement with alternative atoms 4. chemistry (and Cambridge Structural Database) knowledge usually final refinement when all non-h atoms located involves adjusting the 9N + 1 (Number of atoms in asymmetric unit) (or more) parameters (p) to minimize the function: R = w hkl ( F o kf c ) sum of the squares of the difference k = 1/K minimized by taking derivative wrt each parameter, setting equal to zero and approximating results with truncated Taylor series; gives n equations (number of parameters) with n unknowns (p j parameter shifts) m m m m c,r c,r c,r c,r c,r c,r kf kf kf kf kf kf wr Δ p1+ wr Δ p wr Δ pn = wrδfr p1 p1 p p1 pn p1 kf c,r kfc,r kfc,r kfc,r kfc,r kfc,r wr Δ p + wr Δ p wr Δ pn = wrδfr p p1 p p pn p r=1 r=1 r=1 r=1 m m m m r=1 r=1 r=1 r=1 m m m m c,r c,r c,r c,r c,r c,r kf kf kf kf kf kf wr Δ p1 + wr Δ p wr Δ pn = wrδfr p n p1 pn p pn pn r=1 r=1 r=1 r=1 a ij = kf a 11 x 1 + a 1 x a 1n x n = v 1 a 1 x 1 + a x a n x n = v m c,r c,r wr r=1 pi pj a n1 x 1 + a n x a nn x n = v n kf x j = p j m kfc,r vi = wrδfr p r=1 i SHELX a11 a... a1n x1 v1 a1 a... an x v = an1 an... ann xn vn Ax = v x = A 1 v elements of A 1 usually given as b i j minimizes: R = w hkl ( F o kf c ) F r equations are similar: replaced by F o kf c kf c,r kf c,r replaced by kf c p i p i _refine_ls_matrix_type full atomblock _refine_ls_structure_factor_coef Fsqd

3 Estimate of Standard Deviations (esd s) least-squares is nonlinear run cycles until convergence usually want at least 10 reflections/parameter: _refine_ls_number_reflns 3753 _refine_ls_number_parameters 63 shifting parameters allows an estimate of the standard deviation in p s ( p ); calculate p j / p (shift/error) of each parameter structure converges when p j / p 0 b ij w ( F o F c ) 1/ pi = Nreflections N parameters 1/N 1/ ref ; more reflections = better esd s F (w) ; good reflections = better esd s _refine_ls_shift/su_max _refine_ls_shift/su_mean Correlation Coefficients Goodness of Fit (GOF, GooF or S) off diagonal elements of A 1, b i j, correlation of parameters i and j ij = b i j / ((b i i ) 1/ (b jj ) 1/ ) S = w ( F o kf c ) N reflections N parameters 1/ 0 to ±1 no correlation completely correlated indicates refinement problems: disorder or missed symmetry Largest correlation matrix elements U13 F1 / U11 F U13 F3 / U33 F U Cu / OSF U13 F1 / U33 F U1 F1 / U3 F U33 Cu / OSF z F1 / x F U13 F3 / U11 F3 fit of distribution of F with distribution expected from w if w are correct and model good, S = 1.0 _refine_ls_goodness_of_fit_ref 1.04 _refine_ls_restrained_s_all 1.04 used to adjust w: forced 1 _refine_ls_weighting_scheme calc _refine_ls_weighting_details 'calc w=1/[\s^^(fo^^)+(0.0345p)^^ p] where P=(Fo^^+Fc^^)/3' Parameters Temperature Factors too large: 1. atom position incorrect: smeared out to eliminate 1. scale factor: K 9N + 1 parameters are:. atom type wrong (more e than should have): spreads e over a larger volume to lower e density. x, y, z of each non-h atom of asymmetric unit O (8 e ) but should be N (7 e ) 3. temperature factor (thermal parameters) a. isotropic: one parameter first, refine all atoms isotropic to identify any problems: atoms with U iso > 0.10 or < 0.01 may be in error 3. Disorder too small: atom type wrong (fewer e than should have): concentrates e in smaller volume to raise e density N (7 e ) but should be O (8 e ) 3

4 Disorder 1. static disorder atom in two (or more) position with about same energy; see units cells with atom in different positions Parameters 9N + 1 parameters are: 1. scale factor: K. x, y, z of each non-h atom of asymmetric unit no disorder static disorder. dynamic disorder atom moving relatively large distances from rest position 3. temperature factor (thermal parameters) b. anisotropic: six parameters atoms refined anisotropically before locating H s if: 1. there is sufficient data; otherwise just heavy atoms. the atom is not disordered Temperature Factors Constraints exact mathematical condition: one or more variables expressed exactly in terms of other variables, and eliminated for example, coordinates and anisotropic parameters for atoms on special positions: [ π (U 11 h a* + U k b* + U 33 l c* + U 1 hka*b* + U 13 hla*c* + U 3 klb*c*)] f = f o e cross terms related to orientation if very elliptical, maybe split into two atoms: partial occupancy Special position constraints for Cu x = z = U3 = 0 U1 = 0 sof = 0.5 Constraints geometrical constraints: rigid-group refinements (benzene ring) and riding models for hydrogen atom refinement calculated H: 1.0 Å from C at 10 o as attached C shifts (least-squares), H moves with it Restraints additional information, not exact but subject to probability; inserted into least-squares as additional parameters 1. two or more atoms sharing the same site: sof s expressed as variables so sum is fixed as a constant (e.g. 1.0); disorder here ~½ the time sof = () here ~½ the time sof = 0.466() make chemically, but not crystallographically, equivalent distances equal (within an esd) 3. anisotropic parameters: isolated or disordered atoms made approximately isotropic to prevent them blowing-up 4

5 Solving Structure 1. heavy atom or direct methods to get approximate phases. Fourier synthesis with ~phases and F o to find atoms 3. calculate better phases and repeat step ; watch R value 4. when all atoms found, check difference map 5. least-squares refine scale factor, x, y, z and isotropic temperature factors until R converged 6. refine anisotropically until convergence, if you have data; look for small shift/error 7. difference map to find H s or calculate positions 8. refine all non-h parameters to 0 shift/error Finally Is structure chemically reasonable!?! Structure Validation - An Emotive Issue. D.J. Watkin & R.A. Cooper, Chemical Crystallography Laboratory, Oxford, UK In the days of photographic data collection, validation was implicit because of the pains-taking labour needed to resolve a structure, and external validation was more-or-less limited to checking for typing errors in the long lists of numbers. Serial diffractometers and computers changed this. Improvements in software made it possible for non-specialists to undertake analyses. However, it became evident that do-it-yourself analyses were not all of a high quality. It fell to the referees not only to assess the scientific merit of a paper, but also to verify that the work was technically sound. Structure Validation: Automated Detection of Poor and Incorrect Single Crystal Structures. A.L.Spek, Bijvoet Center, Utrecht University, Padualaan 8, 3584 CH, Utrecht, Netherlands. Validation of a reported single crystal structure determination is a long-standing issue requiring an independent structure evaluation system. Dick Marsh (Space group corrections) and Dick Harlow (ORTEP-of-the-Year) clearly demonstrated the need for this already for many years. It is becoming even more urgent in view of the current practice where non-professionals carry out structure determinations using user-friendly-firmware and referees with limited expertise or given insufficient access to the supporting data for independent evaluation. Example of Bad Crystallography paper submitted for publication in Acta Cryst. C Cu + L, confirmed with routine X-ray. R < 7% Br L + An ORTEP-of-the year Award Winner Cu + Br (HBr used in synthesis) H-atoms needed/found on sp 3 N and CO Pca 1 should have been Pcam 5

6 described as dichloride actually disordered CO and Cl unusual coordination number of 1 true dichloride structure correct structure wrong space group P1 Z = correct space group P1 Z = missing H s NO 3 actually CO 3 note: relatively large N temperature factor 6

7 Wrong Structures usually exhibit one or more of three symptoms: unusual bond distances unreasonable thermal ellipsoids impossible intra- and interatomic nonbonded contacts 7