R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory Rietveld Refinement with GSAS & GSAS-II Talk will mix both together.

R.B. Von Dreele, Advanced Photon SourceArgonne National Laboratory

Rietveld Refinement with GSAS & GSAS-II

Talk will mix both together

What does GSAS do in powder pattern analysis?

Thanks to Lynn McCusker for maze

Includes:- Rietveld refinement- Results- Powder pattern plots- For publication

- Bond lengths & angles- Other geometry- CIF (& PDB) files of result- Fourier maps & (some)

display- Texture (polefigures)- UtilitiesMissing:- Indexing- Structure solutionMust go elsewhere for these.

3

Form of GSAS

genles

disagl

fourier

expedt

forplot

powplot

PC-GSAS – thin wrapper GUI

GSAS programs – each is a Fortran exe (common library of routines)

.EXP file, etc.

Keyboard interface only

4

genles

disagl

fourier

forplot

powplot

Form of GSAS & EXPGUI

widplt

liveplot

GUI

expedt

expgui

EXPGUI – incomplete GUI access to GSAS but with extras

Keyboard & mouse

GSAS & EXPGUI interfaces

5

EXPEDT data setup option (<?>,D,F,K,L,P,R,S,X) >EXPEDT data setup options: <?> - Type this help listing D - Distance/angle calculation set up F - Fourier calculation set up K n - Delete all but the last n history records L - Least squares refinement set up P - Powder data preparation R - Review data in the experiment file S - Single crystal data preparation X - Exit from EXPEDT

GSAS – EXPEDT (and everything else) – text based menus with help, macro building, etc.(1980’s user interface!)

EXPGUI: access to GSASTypical GUI – edit boxes,buttons, pull downs etc.Liveplot – powder pattern display (1990’s user interface)

GSAS-II: A fresh start

GSASII – fresh start

Fill in what’s missing from GSAS:- Indexing- Structure solution

Base code – pythonMixed in old GSAS FortranGraphics – matplotlib,OpenGLModern GUI – wxPythonMath – numpy,scipyCurrent: python 2.7All platforms: Windows, Max OSX & Linux

7

GSAS-II – python code model

Fast core processing codes(a few fortran routines)

Slow GUI code – wxPython

& common project filename.gpx

Fast code – numpy array routinesPython – ideal for this

GSAS-II: A screen shot – 3 frame layout + console

Data tree

Data window

Graphics window

Data tabs

Main menu

Submenu

Drawing tabs

NB: Dialog box windows will appear wanting a response

9

Rietveld results - visualization

Easy zoom

I/s(I)

Normal Probability

10

Complex peak broadening models

m-strain surface

NB: mm size & mstrain units

11

Variance-covariance matrix displayUseful diagnostic! High V-covV?

Forgot a “hold”

Highly coupled parmsNote “tool tip”

Structure drawing

PolyhedraVan der Waals atoms Balls & sticksThermal ellipsoidsAll selectable by atom

13

Result from fluoroapatite refinement – powder profile is curve with counting noise & fit is smooth curve

NB: big plot is sqrt(I) Old GSAS example!

Rietveld refinement is multiparameter curve fitting

Iobs +Icalc |Io-Ic |

)

Refl. positions

(lab CuKa B-B data)

14

So how do we get there? Beginning – model errors misfits to pattern Can’t just let go all parameters – too far from best model (minimum c2)

c2

parameter

False minimum

True minimum – “global” minimum

Least-squares cycles

c2 surface shape depends on parameter suite

15

Fluoroapatite start – add model (1st choose lattice & space group)

important – reflection marks match peaks Bad start otherwise – adjust lattice parameters (wrong space group?)

16

2nd add atoms & do default initial refinement – scale & background

Notice shape of difference curve – position/shape/intensity errors

17

Errors & parameters? position – lattice parameters, zero point (not common)

- other systematic effects – sample shift/offset shape – profile coefficients (GU, GV, GW, LX, LY, etc. in GSAS) intensity – crystal structure (atom positions & thermal parameters)

- other systematic effects (absorption/extinction/preferred orientation)

NB – get linear combination of all the aboveNB2 – trend with 2 Q (or TOF) important

a – too small LX - too small Ca2(x) – too small

too sharppeak shift wrong intensity

18

Difference curve – what to do next?

Dominant error – peak shapes? Too sharp? Refine profile parameters next (maybe include lattice parameters) NB - EACH CASE IS DIFFERENT

Characteristic “up-down-up”profile errorNB – can be “down-up-down” for too “fat” profile

19

Result – much improved!

maybe intensity differences remain– – refine coordinates & thermal parms.

20

Result – essentially unchanged

Thus, major error in this initial model – peak shapes

CaFPO4

Pawley/Rietveld refinement

21

Exact overlaps - symmetryIncomplete overlaps

Io

SIc

2o

2co

wp wI

)Iw(IR

Residual:

2coR )II(wM

Ic

Minimize

Processing:GSAS – point by pointGSAS-II – reflection by reflection

Least Squares Theory

i

ii

cicic p

p

I)a(I)p(I

This is done by setting the derivative of MR to zero

ai - initial values of pi

Dpi = pi - ai (shift)

)a(III0p

Ip

p

IIw ico

j

c

ii

i

c

Normal equations - one for each Dpi; outer sum over observationsSolve for Dpi - shifts of parameters, NOT values

Matrix form: Ax=v & B = A-1 so x = Bv = Dp

i

cijj

j

c

i

cj,i p

I)I(wvpx

p

I

p

Iwa

2coR )II(wMMinimize

23

Least Squares Theory - continued

Matrix equation Ax=v

Solve x = A-1v = Bv; B = A-1

This gives set of Dpi to apply to “old” set of ai

repeat until all xi~0 (i.e. no more shifts)

Quality of fit – “c2” = M/(N-P) 1 if weights “correct” & model without

systematic errors (very rarely achieved)

Bii = s2i – “standard uncertainty” (“variance”) in Dpi

(usually scaled by c2)

Bij/(Bii*Bjj) – “covariance” between Dpi & Dpj

Rietveld refinement - this process applied to powder profiles

Gcalc - model function for the powder profile (Y elsewhere)

24

Rietveld Model: Yc = Io{SkhF2hmhLhP(Dh) + Ib}

Io - incident intensity - variable for fixed 2Q

kh - scale factor for particular phase

F2h - structure factor for particular reflection

mh - reflection multiplicity

Lh - correction factors on intensity - texture, etc.

P(Dh) - peak shape function - strain & microstrain, etc.

Ib - background contribution

Least-squares: minimize M=Sw(Yo-Yc)2

Convolution of contributing functions

Instrumental effects

Source

Geometric aberrations

Sample effects

Particle size - crystallite size

Microstrain - nonidentical unit cell sizes

Peak shape functions – can get exotic!

Gaussian – usual instrument contribution is “mostly” Gaussian

G - full width at half maximum – expression from soller slit sizes and monochromator angle & sample broadeningD- displacement from peak position

CW Peak Shape Functions – basically 2 parts:

Lorentzian – usual sample broadening contribution

Convolution – Voigt; linear combination - pseudoVoigt

𝑮 (𝜟 ,𝜞 )=√𝟒 𝒍𝒏𝟐𝜞 √𝝅

𝒆𝒙𝒑 (−𝟒 𝒍𝒏𝟐 𝜟𝟐

𝜞𝟐 )𝑳 (𝜟 ,𝜞 )= 𝟐

𝝅 𝜞𝟏

(𝟏+𝟒𝜟𝟐

𝜞𝟐 )

27

CW Profile Function in GSAS & GSAS-II

Thompson, Cox & Hastings (with modifications)

Pseudo-Voigt

Mixing coefficient

FWHM parameter

𝑷 (𝜟 ,𝜸 ,𝚪 )=𝜼𝑳 (𝜟 ,𝑯 )+(𝟏−𝜼 )𝑮 (𝜟 ,𝑯 )

𝜼=∑𝒋=𝟏

𝟑

𝒌 𝒋( 𝜸𝜞 )𝟐

𝑯=𝟓√∑𝒊=𝟏

𝟓

𝒄 𝒊 𝜞𝟓− 𝒊𝜸 𝒊

Where Lorentzian FWHM = g and Gaussian FWHM = G

28

CW Axial Broadening FunctionFinger, Cox & Jephcoat based on van Laar & Yelon

2QBragg2Qi2Qmin

Ä Pseudo-Voigt (TCH)= profile function

Depend on slit & sample “heights” wrt diffr. radiusH/L & S/L - parameters in function (combined as S/L+H/L; S = H)(typically 0.002 - 0.020)

Debye-Scherrer cone

2Q Scan

Slit

H

29

How good is this function?

Protein Rietveld refinement - Very low angle fit1.0-4.0° peaks - strong asymmetry “perfect” fit to shape

30

Bragg-Brentano Diffractometer – “parafocusing”

Diffractometercircle

Sampledisplaced

Receiving slit

X-ray sourceFocusing circle

Divergent beam optics

Incident beamslit

Beam footprintSample transparency

31

CW Function Coefficients – GSAS & GSAS-II

Sample shift

Sample transparency

Gaussian profile

Lorentzian profile

(plus anisotropic broadening terms) Intrepretation?

Shifted difference 𝜟′=𝜟+𝑺𝒔𝒄𝒐𝒔 𝜣+𝑻 𝒔 𝒔𝒊𝒏𝟐𝜣

𝒔=−𝝅 𝑹𝑺𝒔

𝟑𝟔𝟎𝟎𝟎

𝝁𝒆𝒇𝒇=−𝟗𝟎𝟎𝟎𝝅𝑹𝑻 𝒔

𝚪𝟐=𝐔𝐭𝐚𝐧 𝚯+𝐕 𝐭𝐚𝐧𝚯+𝐖+ 𝐏𝐜𝐨𝐬 𝚯❑

𝟐❑

𝟐

𝛄=𝐗

𝐜𝐨𝐬𝚯+𝐘𝐭𝐚𝐧𝚯

NB: P term not in GSAS-II; sample shift, meff refined directly as parameters

Crystallite Size Broadening

a*

b*

Dd*=constant

dcot

d

d*d 2

sincot2

cosd

d2 2

Lorentzian term - usualK - Scherrer const. "LX"

K180p

Gaussian term - rareparticles same size?

"GP"K180

p

NB: In GSAS-II size is refined directly in mm

Microstrain Broadening

a*

b*

ttanconsdd

cot

*d*d

dd

tand

d22

Lorentzian term - usual effect

"LY"180

%100S

Gaussian term - theory? (No, only a misreading)

Remove instrumental part

"GU"180

%100S

NB: In GSAS-II mstrain refined directly; no conversion needed)

34

Microstrain broadening – physical model

Stephens, P.W. (1999). J. Appl. Cryst. 32, 281-289.Also see Popa, N. (1998). J. Appl. Cryst. 31, 176-180.

Model – elastic deformation of crystallites

hkhlkllkhMd hklhkl

6542

32

22

12

1

d-spacing expression

j,i ji

ijhkl

MMSM2

Broadening – variance in Mhkl

35

hkM

hlM

klM

lM

kM

hM

654

2

3

2

2

2

1

,,,,,

2222233

2222323

2222332

23342222

32322422

33222224

khklhlhkhklhkkh

klhlhhklhllhklh

lhkhkllkkllkklh

hklhlklllklh

hklhklklkkkh

khlhklhlhkhh

MM

ji

Microstrain broadening - continued

Terms in variance

Substitute – note similar terms in matrix – collect terms

36

42 LKH,lkhSMHKL

LKHHKLhkl

2112

2121

2211

3013

3301

3130

3031

3103

3310

22022

22202

22220

4004

4040

4400

2

4

2

3

hklSlhkSklhS

klSlhShkSlkShlSkhS

lkSlhSkhSlSkShSM hkl

Microstrain broadening - continuedBroadening – as variance

General expression – triclinic – 15 terms

Symmetry effects – e.g. monoclinic (b unique) – 9 terms

lhkShkSlhS

lkSkhSlhSlSkShSM hkl

2121

3103

3301

22022

22220

22202

4004

4040

4400

2

42

)(33

3 collected terms

Cubic – m3m – 2 terms

222222220

444400

2 3 lklhkhSlkhSM hkl

37

Example - unusual line broadening effects in Na parahydroxybenzoate

Sharp lines

Broad lines

Seeming inconsistency in line broadening- hkl dependent

Directional dependence - Lattice defects?

38

H-atom location in Na parahydroxybenzoateGood DF map allowed by better fit to pattern

DF contour mapH-atom locationfrom x-ray powder data

39

Macroscopic Strain

hkhlkllkhMd hklhkl

6542

32

22

12

1

Part of peak shape function #5 – TOF & CWd-spacing expression; aij from recip. metric tensor

Elastic strain – symmetry restricted lattice distortion

TOF:

ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d3

CW:

ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d2tanQ

Why? Multiple data sets under different conditions (T,P, x, etc.)

NB: In GSAS-II generally available (CW only at present)

40

Symmetry & macrostrain

dij – restricted by symmetry

e.g. for cubicDT = d11h2d3 for TOF (in GSAS)

'ij

a

1

Result: change in lattice parameters via change in metric coeff.aij’ = aij-2dij/C for TOFaij’ = aij-(p/9000)dij for CWUse new aij’ to get lattice parameterse.g. for cubic

Bragg Intensity Corrections: Lh

Extinction

Absorption & Surface Roughness

Preferred Orientation/Texture

Other Geometric Factors

Affect the integrated peak intensity and not peak shape

Nonstructural Features

} diagnostic: Uiso too small!

Sabine model - Darwin, Zachariasen & Hamilton

Bragg component - reflection

Laue component - transmission

Extinction – only GSAS for now

Eh

= Eb

sin2Q + El c o s2Q

Eb

= 1 + x

1

Combination of two parts

El = 1 -

2x +

4x2

- 4 8

5 x3 . . . x < 1

El = px

2 éêë

1 - 8 x1 -

1 2 8 x23 . . . ùú

û

x > 1

Sabine Extinction Coefficient

Crystallite grain size =

2Q

0%

20%

40%

60%

80%

0.0 25.0 50.0 75.0 100.0 125.0 150.0

Eh

Increasingwavelength(1-5 Å)

2

hx V

FEx

√𝐸𝑥

44

Random powder - all crystallite orientations equally probable - flat pole figure

Crystallites oriented along wire axis - pole figure peaked in center and at the rim (100’s are 90 apart)

Orientation Distribution Function - probability function for texture

(100) wire texture(100) random texture

What is texture? Nonrandom crystallite grain orientations

Pole figure - stereographic projection of a crystal axis down some sample direction

Loose powder

Metal wire

45

Texture - measurement by diffraction

Debye-Scherrer cones • uneven intensity due to texture • also different pattern of unevenness for different hkl’s• Intensity pattern changes as sample is turned

Non-random crystallite orientations in sample

Incident beamx-rays or neutrons

Sample(111)

(200)

(220)

Spherical Distribution

Ellipsoidal Distribution -assumed cylindrical

Ellipsoidal particles

Uniaxial packing

Preferred Orientation - March/Dollase Model

Integral about distribution- modify multiplicity

Ro - ratio of ellipsoid axes = 1.0 for nopreferred orientation

2

3n

1j o

222

oh R

sincosR

M

1A

47

• Projection of orientation distribution function for chosen reflection (h) and sample direction (y)

• K - symmetrized spherical harmonics - account for sample & crystal symmetry

• “Pole figure” - variation of single reflection intensity as fxn. of sample orientation - fixed h

• “Inverse pole figure” - modification of all reflection intensities by sample texture - fixed y - Ideally suited for neutron TOF diffraction

• Rietveld refinement of coefficients, Clmn, and 3

orientation angles - sample alignment NB: In GSAS-II as correction & texture analysis

Texture effect on reflection intensity – Sph. Harm. model

)()(12

4),(

0

yKhKCl

yhA nl

ml

l

lm

l

ln

mnl

l

Absorption

X-rays - independent of 2Q - flat sample – surface roughness effect - microabsorption effects - but can change peak shape and shift their positions if small (thick sample)

Neutrons - depend on 2Q and l but much smaller effect - includes multiple scattering much bigger effect - assume cylindrical sample Debye-Scherrer geometry

Diagnostic: thermal parms. too small!

Model - A.W. Hewat

For cylinders and weak absorption onlyi.e. neutrons - most needed for TOF datanot for CW data – fails for mR>1

GSAS & GSAS-II – New more elaborate model by Lobanov & alte de Viega – works to mR>10

Other corrections - simple transmission & flat plate (GSAS only for now)

)ATATexp(A 22B2B1h

Nonuniform sample density with depth from surfaceMost prevalent with strong sample absorptionIf uncorrected - atom temperature factors too smallSuortti model Pitschke, et al. model

Surface Roughness – Bragg-Brentano & GSAS only

High angle – more penetration (go thru surface roughness) - more dense material; more intensity

Low angle – less penetration (scatter in less dense material) - less intensity

pqp1

q1p1S

2

R

sinsin qq1p

qq1pSR

exp

sinexp

(a bit more stable)

Other Geometric Corrections

Lorentz correction - both X-rays and neutrons

Polarization correction - only X-rays

X-rays

Neutrons - CW

Neutrons - TOF

Lp

= 2 sin2Q c o sQ

1 + M c o s22Q

Lp

= 2 sin2Q c o sQ

1

Lp

= d4

sinQ

52

Solvent scattering – proteins & zeolites?

Contrast effect between structure & “disordered” solvent region

Babinet’s Principle:Atoms not in vacuum – change form factors

(GSAS only)

f = fo-Aexp(-8pBsin2Q/l2)

0

2

4

6

0 5 10 15 20

2Q

fC

uncorrected

Solvent corrected

Carbon scattering factor

Manual subtraction – not recommended - distorts the weighting scheme for the observations& puts a bias in the observations

Fit to a function - many possibilities:

Fourier series - empiricalChebyschev power series - dittoExponential expansions - air scatter & TDS (only GSAS) Fixed interval points - brute forceDebye equation - amorphous background(separate diffuse scattering in GSAS; part of bkg. in GSAS-II)

Background scattering

real space correlation functionespecially good for TOFterms with

Debye Equation - Amorphous Scattering

)QB21

exp(QR

)QRsin(A 2

ii

ii

amplitudedistance

vibration

55

Neutron TOF - fused silica “quartz”

56

Rietveld Refinement with Debye Function

7 terms Ri –interatomic distances in SiO2 glass 1.587(1), 2.648(1), 4.133(3), 4.998(2), 6.201(7), 7.411(7) & 8.837(21)Same as found in a-quartz

1.60Å

Si

O

4.13Å

2.63Å3.12Å

5.11Å 6.1Å

a-quartz distances

Summary

Non-Structural Features in Powder Patterns

1. Large crystallite size - extinction

2. Preferred orientation

3. Small crystallite size - peak shape

4. Microstrain (defect concentration)

5. Amorphous scattering - background

58

When to quit?

Stephens’ Law – “A Rietveld refinement is never perfected,

merely abandoned”Also – “stop when you’ve run out of things to vary”

What if problem is more complex?Apply constraints & restraints

“What to do when you have

too many parameters

& not enough data”

59

Complex structures (even proteins)Too many parameters – “free” refinement failsKnown stereochemistry:Bond distancesBond anglesTorsion angles (less definite)Group planarity (e.g. phenyl groups)Chiral centers – handednessEtc.

Choice: (NB: not GSAS-II yet!)rigid body description – fixed geometry/fewer parametersstereochemical restraints – more data

60

Constraints vs restraints

Constraints – reduce no. of parameters

jkjlkil

i p

FSUR

v

F

Rigid body User Symmetry

Derivative vectorBefore constraints(longer)

Derivative vectorAfter constraints(shorter)

Rectangular matrices

Restraints – additional information (data) that model must fitEx. Bond lengths, angles, etc.

61

Space group symmetry constraints

Special positions – on symmetry elementsAxes, mirrors & inversion centers (not glides & screws)Restrictions on refineable parametersSimple example: atom on inversion center – fixed x,y,zWhat about Uij’s?

– no restriction – ellipsoid has inversion center

Mirrors & axes ? – depends on orientation

Example: P 2/m – 2 || b-axis, m ^ 2-fold

on 2-fold: x,z – fixed & U11,U22,U33, & U13 variableon m: y fixed & U11,U22, U33, & U13 variableRietveld programs – GSAS, GSAS-II automatic, others not

62

Multi-atom site fractions

“site fraction” – fraction of site occupied by atom“site multiplicity”- no. times site occurs in cell“occupancy” – site fraction * site multiplicity

may be normalized by max multiplicity

GSAS & GSAS-II uses fraction & multiplicity derived from sp. gp.Others use occupancy

If two atoms in site – Ex. Fe/Mg in olivineThen (if site full) FMg = 1-FFe

63

If 3 atoms A,B,C on site – problemDiffraction experiment – relative scattering power of site“1-equation & 2-unknowns” unsolvable problemNeed extra information to solve problem –2nd diffraction experiment – different scattering power“2-equations & 2-unknowns” problem

Constraint: solution of J.-M. JoubertAdd an atom – site has 4 atoms A, B, C, C’ so that FA+FB+FC+FC’=1Then constrain so DFA = -DFC and DFB = - D FC’ NB: More direct in GSAS-II as constraints are on values!

Multi-atom site fractions - continued

64

Multi-phase mixtures & multiple data sets

Neutron TOF – multiple detectorsMulti- wavelength synchrotronX-ray/neutron experimentsHow constrain scales, etc.?

p

phphhdbc YSSIII

Histogram scale Phase scale

Ex. 2 phases & 2 histograms – 2 Sh & 4 Sph – 6 scalesOnly 4 refinable – remove 2 by constraintsEx. DS11 = -DS21 & DS12 = -DS22

65

Rigid body problem – 88 atoms – [FeCl2{OP(C6H5)3}4][FeCl4]

264 parameters – no constraintsJust one x-ray pattern – not enough data!Use rigid bodies – reduce parameters

P21/ca=14.00Åb=27.71Åc=18.31Åb=104.53V=6879Å3

V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, 174-181 (2003)

66

Rigid body description – 3 rigid bodies

FeCl4 – tetrahedron, origin at Fe

z

x

y

Fe - origin

Cl1

Cl2

Cl3Cl4

1 translation, 5 vectorsFe [ 0, 0, 0 ]Cl1 [ sin(54.75), 0, cos(54.75)]Cl2 [ -sin(54,75), 0, cos(54.75)]Cl3 [ 0, sin(54.75), -cos(54.75)]Cl4 [ 0, -sin(54.75), -cos(54.75)]D=2.1Å; Fe-Cl bond

67

PO – linear, origin at PC6 – ring, origin at P(!)

Rigid body description – continued

P OC1

C5 C3

C4 C2

C6z

x

P [ 0, 0, 0 ]O [ 0, 0 1 ]D=1.4Å

C1-C6 [ 0, 0, -1 ]D1=1.6Å; P-C bondC1 [ 0, 0, 0 ]C2 [ sin(60), 0, -1/2 ]C3 [-sin(60), 0, -1/2 ]C4 [ sin(60), 0, -3/2 ]C5 [-sin(60), 0, -3/2 ]C6 [ 0, 0, -2 ]D2=1.38Å; C-C aromatic bond

DD1D2

(ties them together)

68

Rigid body description – continuedRigid body rotations – about P atom originFor PO group – R1(x) & R2(y) – 4 setsFor C6 group – R1(x), R2(y),R3(z),R4(x),R5(z)

3 for each PO; R3(z)=+0, +120, & +240; R4(x)=70.55Transform: X’=R1(x)R2(y)R3(z)R4(x)R5(z)X

47 structural variables

P

O

C

C C

C C

C

z

x

y

R1(x)

R2(y)R3(z)

R5(z) R4(x)

Fe

69

Refinement - results

Rwp=4.49%Rp =3.29%RF

2 =9.98%Nrb =47Ntot =69

70

Refinement – RB distances & angles OP(C6)3 1 2 3 4R1(x) 122.5(13) -76.6(4) 69.3(3) -158.8(9) R2(y) -71.7(3) -15.4(3) 12.8(3) 69.2(4)R3(z)a 27.5(12) 51.7(3) -10.4(3) -53.8(9)R3(z)b 147.5(12) 171.7(3) 109.6(3) 66.2(9)R3(z)c 267.5(12) 291.7(3) 229.6(3) 186.2(9)R4(x) 68.7(2) 68.7(2) 68.7(2) 68.7(2)R5(z)a 99.8(15) 193.0(14) 139.2(16) 64.6(14)R5(z)b 81.7(14) 88.3(17) 135.7(17) -133.3(16)R5(z)c 155.3(16) 63.8(16) 156.2(15) 224.0(16)P-O = 1.482(19)Å, P-C = 1.747(7)Å, C-C = 1.357(4)Å, Fe-Cl = 2.209(9)Å

z

x

R1(x - PO)

R2(y- PO)R3(z)

R5(z) R4(x)

Fe

} Phenyl twist

p − C-P-O angle

C3PO torsion(+0,+120,+240)

} PO orientation

}

71

Packing diagram – see fit of C6 groups

72

Stereochemical restraints – additional “data”

4

2

2

4

2

4

2

2

2

)( ciiR

cioiix

cioiih

cioiiv

ciip

ciit

cioiid

cioiia

cioiiY

Rwf

xxwf

hhwf

vvwf

pwf

twf

ddwf

aawf

YYwfM

Powder profile (Rietveld)*

Bond angles*

Bond distances*

Torsion angle pseudopotentials

Plane RMS displacements*

van der Waals distances (if voi<vci)

Hydrogen bonds

Chiral volumes**

“ /f y” pseudopotentialwi = 1/s2 weighting factorfx - weight multipliers (typically 0.1-3)

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For [FeCl2{OP(C6H5)3}4][FeCl4] - restraints

Bond distances: Fe-Cl = 2.21(1)Å, P-O = 1.48(2)Å, P-C = 1.75(1)Å, C-C = 1.36(1)ÅNumber = 4 + 4 + 12 + 72 = 92Bond angles:O-P-C, C-P-C & Cl-Fe-Cl = 109.5(10) – assume tetrahedralC-C-C & P-C-C = 120(1) – assume hexagonNumber = 12 + 12 + 6 + 72 + 24 = 126Planes: C6 to 0.01 – flat phenylNumber = 72Total = 92 + 126 + 72 = 290 restraints

A lot easier to setup than RB!!

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Refinement - results

Rwp=3.94%Rp =2.89%RF

2 =7.70%Ntot =277

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Stereochemical restraints – superimpose on RB results

Nearly identical with RB refinementDifferent assumptions – different results

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New rigid bodies for proteins (actually more general)

Proteins have too many parameters Poor data/parameter ratio - especially for powder data Very well known amino acid bonding –

e.g. Engh & Huber Reduce “free” variables – fixed bond lengths & angles Define new objects for protein structure –

flexible rigid bodies for amino acid residues Focus on the “real” variables –

location/orientation & torsion angles of each residue Parameter reduction ~1/3 of original protein xyz set

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txyz

Qijk

y

c1

c2

Residue rigid body model for phenylalanine

3txyz+3Qijk+y+c1+c2 = 9 variables vs 33 unconstrained xyz coordinates

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Qijk – Quaternion to represent rotations

In GSAS defined as: Qijk = r+ai+bj+ck – 4D complex number – 1 real + 3 imaginary components

Normalization: r2+a2+b2+c2 = 1

Rotation vector: v = ax+by+cz; u = (ax+by+cz)/sin(a/2)

Rotation angle: r2 = cos2(a/2); a2+b2+c2 = sin2(a/2)

Quaternion product: Qab = Qa * Qb ≠ Qb * Qa

Quaternion vector transformation: v’ = QvQ-1

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Conclusions – constraints vs. restraints

Constraints required space group restrictionsmultiatom site occupancy

Rigid body constraintsreduce number of parametersmolecular geometry assumptions

Restraintsadd datamolecular geometry assumptions (again)

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Citations: GSAS:A.C. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los Alamos National Laboratory Report LAUR 86-748 (2004). EXPGUI:B. H. Toby, EXPGUI, a graphical user interface for GSAS, J. Appl. Cryst. 34, 210-213 (2001). GSAS-II:None yet except the web site https://subversion.xor.aps.anl.gov/pyGSASWe’ll have a paper soon.

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Thank you - Questions from future Crystallographers?