R.E. Dinnebier &: S.].L. Billinge, "Powder
`
`Dtflraction - Theory and Practice”, RSC
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`Publishing, 2008, pp. 266-281
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`ISBN: 978-0-8540-1-231-9
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`A caiulogue record far this book is available from the British Library
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`\'r_" The Royal Sociciy 0FChcmist1'y 2008
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`page.
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`Published by The: Royal Society of Chemistry.
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`Chapter 9
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`the scattering process that gives rise to features in the observed pattern to avoid
`significant systematic errors.
`The result of a Taylor series approximation is that the computed shifts. Ari,-.
`is not accurate yielding a fully minimized solution to the problem. but are a
`(hopefully) better approximation. Consequently, the new parameter values are
`used for a subsequent refinement cycle;
`this process is
`repeated until
`the
`parameter shifts are less
`than some fraction of their estimated errors as
`obtained from the diagonal elements of the B matrix.
`Implicit in this approach is that an initial estimate of all the parameters [a,-_.
`Equation (4)] must be provided beforehand. These estimates must place the
`value of the minimization function within one {preferably the “best") well in
`the M—surface.
`
`The quality of the least squares refinement
`functions:
`
`is indicated by some residual
`
`'|y0_'
`
`&=z%fi
`
`)!{i
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`'
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`I,
`1'
`
`_
`
`ill
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`Rt.-In «'3
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`on
`
`U2)
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`the "weighted“ residual. R
`Notice that
`that is
`...,,. contains the factor, M,
`minimized by the least squares; thus it is th
`e only statistically relevant residual,
`The reduced 13 or “goodness of fit“
`as:
`is defined from the minimization function
`
`£=Mamu—Mp
`and the “expected RM," from:
`
`R»--ptcxpi = R»--M
`
`rm
`
`(143
`
`If the weights for the observations are chosen
`"properly" (tie. as reciprocal
`variances) then the value of reduced X: will be somewhat greater than unity for
`an optimal refinement. Misscaled variances will drive this value away from
`unity without affecting the value of R,,._,,. As recently noted by Mercier at (tf..M'l5
`inversion of the A matrix developed in a Rietveld refinement is frequently
`plagued by poor numerical conditioning due to the extreme range of the
`individual partial derivatives. This had been noted for some time by many
`developers of the Rietveld refinement codes (tag. GSAS by Larson and Von
`Dreelcm) and one solution is to employ at normalization process to the matrix
`before inversion. The simplest is to apply the following:
`
`V/.4 A__,-,-
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`9.3
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`9.3.1
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`A crj
`arrar.
`occur
`nates
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`scale
`This
`relini
`desci
`mete
`thest
`Tlics
`symi
`rclat
`the 6
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`Cizctpter 9
`
`P [ o.o,01
`010.0, 1]
`D=i.4A
`
`-1]
`
`ti,
`e,—c6|o,
`o]=i.eA
`CJ{ 0,
`0. O]
`C2]sin(60)‘ 0, 4:21
`C3I—:.in(60}, 0, 4:2]
`C4[:;in(60), 0, -3:2]
`cs [—:.in(60). 0. —3r2]
`Cam,
`0,
`.21
`D2=1t38A
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`Figure 9.7 Rigid body descriptions of the PO and PC6 groups‘
`
`Rigid body rotations — about P atom origin
`For PO group — R,(x) 8; R31’;-J — 4 sets
`For Ce SFOUP — R10}. R20). Rgiiil. R40‘). R5{2J
`3 For each P0; R3(z)=O, 120, éit 240; R_1(.'f.}=70.55
`Transform: X'=R1{it)R2{yJR3(z)R,[>t)R5(i:)X
`
`R|(9-)
`
`Rim
`———Fe
`4? struetu ra] variables
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`Figure 9.8 Assembly of PO and Cg, rigid bodies with applied rotations.
`
`Each triphenylphosphine oxide ligand is then assembled from one PC and
`three Cg, rigid bodies (Figure 9.8). The first
`two rotations, R[(x) and R3(y)
`position the OP group about the P atom position such that the O atom forms a
`reasonable bond with Fe; there are four sets of these for the four OP(C5)3
`ligands giving eight parameters. Each P atom has an x,_;r,z crystallographic
`coordinate for [2 more parameters. The three additional rotations R3(z), R4(_x)
`and R5(z} are applied to each of the three C6 groups to position them about
`each P atom. The P50.) rotations describe the “twist" ofeacli Cg group about its
`P43 bond; these are 12 more parameters. The initial P4{x) rotation is 70.55° and
`P3(z) rotations are 0°, 120° and 240° apart to give tetrahedral stereochernistry
`for the POC3 bonds. These are refined so that the P400 rotation is identical for
`all Four OP(C.=,)3 ligands and the three P312) rotations on each ligand retain their
`relative 120" differences; this yields five more parameters. Figure 9.8 shows the
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`Rierveid Rcfflrtemem
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`‘i
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`5 l 33H
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`:2
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`.._.i'____
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`Figure 9.1]
`
`Superposition of the rigid body and stereocheinical restrained refine-
`ments of Fe{0P(C¢,H5)_i]4Cl2[FeCl,,].
`
`body model and the stereochemical restraint model. These constrztittirestraiii
`the refinement in slightly diflerent ways.
`
`9.3.5 Protein Powder Refinements
`
`The crystal structures of proteins represent an extreme in the number of atom
`positions needed to describe them compared to tltosc structures more com-
`monly studied by powder diffraction. For example, the well—known tctragonal
`crystal structure ofhert egg white lysozyme has 1001 nonhydrogen atoms within
`the protein molecule; another 160 or so water molecules and salt ions are also
`present. This gives over 3000 atomic _1',y,: coordinates. Nonetheless, a Rietveld
`refinement of these structures front powder diffraction data can be performed
`by extending the suite of restraints to include all stereochemieal features that
`show characteristic values. ml" The suite of restraints given in Equation (24) is
`then:
`
`M : E l1‘-T‘i(Yni — Y{'J')2 E Wrularjf — aft)? E ll’-Ifildof _ dd}:
`i mt’-‘il_p:'rl2 + f\ E W.\'r(x!.\'i' _ xvi): + E wt-t(1'r:i' _ "rild
`7»-fl, Z it-,,,(ii,,, — ii,,)3 +_;, Z !t',,[—i'(-J1 +_/',i Z lt'R,(—R“)2
`
`with additional terms for chiral volume, .\.‘, van der Waals “bump", 5'. hydrogen
`bond. ii. torsion angle. 1. and torsion angle pair. R. restraints. The characteristic
`values for these restraints are obtained from the results of high—resolution
`single—erystal protein structures“ and those of small peptides.” The torsion
`angle and paired torsion angle restraints present it particular difficulty as these
`
`'r'tupIei' 9
`
`ictoy-“
`5 im_
`ted iron‘!
`
`"still in at
`ture. but
`suite ol‘
`ll Feline-
`
`length
`l
`id 73 C--
`P—-C, I2
`‘. C C--C
`
`g can be
`aositions.
`lll1ll‘tll7,‘il-
`ot‘ the R8
`:ltovi't‘l in
`
`11 Figure
`:re, most
`:sults are
`
`the rigid
`
`‘ complex
`-r )_ Calcu-
`ittons are
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`Chapter 9
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`have no single preferred value (unlike, for example, a bond angle)” These
`restraints can be formulated as pseudopotentials'9'2°‘24 determined from the
`observed distribution ofthese values in high-resolution protein structures. This
`allows the restraint to have multiple preferred values and can accommodate the
`coupling of the two torsion angles, qb and if/, as present in, for example,
`the
`protein backbone.”
`
`ACKNOWLEDGEMENT
`
`This work was supported by the U. S. DOE,!OS,lBES under Contract No.
`DE—AC-O2-06CHl 1357.
`
`REFERENCES
`
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`. K. Shankland. L. McBride, W. I. F. David. N. Shanklund and G. Steele.
`J. Appl. C.ij’_\'m{logi'.. 2002, 35. 443- 454.
`. V. Jorik. I. Ondrejkovicovail, R. B. You Dreele and H. Ehrenberg, (‘i-_y.s*1.
`Res. Tcrrlmol, 2003. 38, l74—l8l.
`. R. B. Von Dreelc. J. Appl. (Tr_i-‘.rta!fogi'., I999, 32.
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`Ric: veid Rogfirremerrr
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`. R. B. Van Dreeic. J. Appf. Cry-srw‘!ogr., 2007, 40, 133443.
`. A. L Morris. M. W. MacArthur, E. C}. Hutchinson and I. M. Thornion.
`Prmeins, 1992, 12, 345-364.
`. R. A. Engh and R. Huber. /{cm C'ry.s'raHogr., Sect. A, 1991, 47, 392-400.
`. E. Dodson, Arm Cry:-mIfogr., Sect. D, 1998, 54, 1109- I118.
`. R. B. Von Dreeie. Acm Cr_v.sraHogr._ Sect. D, 2005. 61, 22-32.
`. G. N. Ramachandran. C. Ramakrishnan and V. Sasisekharan,
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`.1. Mol.
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