Rietveld Refinement with GSAS

Recent Quote seen in Rietveld e-mail: “Rietveld refinement is one of those few fields of intellectual endeavor wherein the more one does it, the less one understands.” (Sue Kesson) Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned”

Demonstration – refinement of fluroapatite

R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory

Rietveld refinement is multiparameter curve fitting

(lab CuK a B-B data) I obs I calc I o -I c + | | Refl. positions  Result from fluoroapatite refinement – powder profile is curve with counting noise & fit is smooth curve 

NB: big plot is sqrt(I) 2

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 c 2 ) False minimum c 2 Least-squares cycles True minimum – “global” minimum parameter c 2 surface shape depends on parameter suite

3

Fluoroapatite start – add model (1 st choose lattice/sp. grp.)

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

4

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

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

5

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 above NB

2

– trend with 2

Q

(or TOF) important

peak shift too sharp wrong intensity a – too small LX - too small Ca2(x) – too small

6

Difference curve – what to do next?

Characteristic “up-down-up”  profile error NB – can be “down-up down” for too “fat” profile    Dominant error – peak shapes? Too sharp?

Refine profile parameters next (maybe include lattice parameters)

NB - EACH CASE IS DIFFERENT 7

Result – much improved!

 maybe intensity differences left – refine coordinates & thermal parms.

8

Result – essentially unchanged

Ca F PO 4  Thus, major error in this initial model – peak shapes

9

So how does Rietveld refinement work?

Rietveld Minimize

M R

 

w

(

I o



I c

) 2 Exact overlaps - symmetry I c I o

S

I c Incomplete overlaps Residuals:

R wp

 

w(I



o



wI o

2

I c )

2 Extract structure factors: Apportion I o to

S

i c by ratio of I & apply corrections c

F o

2



1

Lp

 

I c I c

10

Rietveld refinement - Least Squares Theory

Given a set of observations G obs and a function G calc



g ( p 1 , p 2 , p 3 ..., p n ) then the best estimate of the values p i minimizing M

 

w ( G o



G c ) 2 is found by This is done by setting the derivative to zero



w ( G o



G c )



G c



p j



0 Results in n “normal” equations (one for each variable) - solve for p i 11

Least Squares Theory - continued

Problem - g(p i ) is nonlinear & transcendental (sin, cos, etc.) so can’t solve directly Expand g(p i ) as Taylor series & toss high order terms G c ( p i )



G c ( a i )

 

i



G



p i c



p i a



i p - initial values of p i = p i - a i (shift) i Substitute above



w

  

G

 

i



G c



p i



p i

  

G c



p j



0



G



G o



G c ( a i ) Normal equations - one for each



p i ; outer sum over observations Solve for



p i - shifts of parameters, NOT values 12

Least Squares Theory - continued

Rearrange



w



G



p 1 c

 

n



i



1



G c



p i



p i

 

.

.

.



w



G c



p n

 

i n

 

1



G c



p i



p i

   

G



G c



p 1 w G



G c



p n Matrix form: Ax=v a i , j

 

w



G c



p i



G c



p j x j

 

p j v i

 

w (



G )



G c



p i 13

Least Squares Theory - continued

Matrix equation Ax=v Solve x = A -1 v = Bv; B = A -1 This gives set of



p i to apply to “old” set of a i repeat until all x i ~0 (i.e. no more shifts) Quality of fit – “

c

2 ” = M/(N-P)



1 if weights “correct” & model without systematic errors (very rarely achieved) B ii =

s

2 i – “standard uncertainty” (“variance”) in



p i (usually scaled by

c

2 ) B ij /(B ii *B jj ) – “covariance” between



p i &



p j Rietveld refinement - this process applied to powder profiles G calc - model function for the powder profile (Y elsewhere) 14

Rietveld Model: Y c = I o {

S

k h F 2 h m h L h P(



h ) + I b }

Least-squares: minimize M=

S

w(Y o -Y c ) 2 I o - incident intensity - variable for fixed 2

Q

k h - scale factor for particular phase F 2 h - structure factor for particular reflection m h - reflection multiplicity L h - correction factors on intensity - texture, etc.

P(



h ) - peak shape function - strain & microstrain, etc.

I b - background contribution 15

Peak shape functions – can get exotic!

Convolution of contributing functions Instrumental effects Source Geometric aberrations Sample effects Particle size - crystallite size Microstrain - nonidentical unit cell sizes

CW Peak Shape Functions – basically 2 parts: Gaussian – usual instrument contribution is “mostly” Gaussian

P(



k ) = H k

p 

2 k 2 / H k ] = G

Lorentzian – usual sample broadening contribution

P(



k ) =

p

2 H k 1 1 + 4



k 2 2 /H k = L H - full width at half maximum - expression from soller slit sizes and monochromator angle



- displacement from peak position

Convolution – Voigt; linear combination - pseudoVoigt

CW Profile Function in GSAS

Thompson, Cox & Hastings (with modifications) Pseudo-Voigt

P (  T )   L (  T ,  )  ( 1   ) G (  T ,  )

Mixing coefficient

  j 3   1 k j (   ) j

FWHM parameter

  5 i 5   1 c i  g 5  i  i

18

CW Axial Broadening Function

Finger, Cox & Jephcoat based on van Laar & Yelon Debye-Scherrer cone 2

Q

Scan H Slit 2

Q

min 2

Q

i 2

Q

Bragg Depend on slit & sample “heights” wrt diffr. radius H/L & S/L - parameters in function (typically 0.002 - 0.020)



Pseudo-Voigt (TCH) = profile function 19

How good is this function?

Protein Rietveld refinement - Very low angle fit 1.0-4.0

°

peaks - strong asymmetry “perfect” fit to shape 20

Bragg-Brentano Diffractometer – “parafocusing”

Focusing circle Diffractometer circle X-ray source Receiving slit Incident beam slit Sample displaced

Divergent beam optics

Beam footprint Sample transparency

21

CW Function Coefficients - GSAS

Shifted difference

 T '   T  S s cos Q  T s sin 2 Q

Sample shift s

  p

RS s 36000 Sample transparency



eff

 

9000

p

RT s Gaussian profile Lorentzian profile



2 g

 

U tan 2

Q 

V tan

Q 

W



P cos 2

Q 

X cos

Q 

Y tan

Q

(plus anisotropic broadening terms) Intrepretation?

22

Crystallite Size Broadening

b* a*

Lorentzian term - usual K - Scherrer const. Gaussian term - rare particles same size?



d*=constant



d *



2

Q    

d



d 2



2

Q Q

cot cot d

Q Q

sin

Q  

d



d 2 cos

Q

p



180 K

 p

" LX " p

 p

180 K



" GP "

Microstrain Broadening

b* a*

Lorentzian term - usual effect Gaussian term - theory?

Remove instrumental part



d



cons tan t d



d



d



d * d *

 Q

cot

Q 

2

Q 

2



d tan

Q

d S

 p

100 % 180 " LY " S

 p

100 % 180



" GU "

Microstrain broadening – physical model

Model – elastic deformation of crystallites Stephens, P.W. (1999). J. Appl. Cryst. 32, 281-289.

Also see Popa, N. (1998). J. Appl. Cryst. 31, 176-180.

d-spacing expression

1 2

d hkl



M hkl

 a 1

h

2  a 2

k

2  a 3

l

2  a 4

kl

 a 5

hl

 a 6

hk

Broadening – variance in M hkl

s 2 

M hkl

 

i ,



j S ij



M

 a

i



M

 a

j

25

Microstrain broadening - continued

Terms in variance



M

 a 1 

h

2 , 

M

 a 2 

k

2 , 

M

 a 3 

l

2 , 

M

 a 4 

kl

, 

M

 a 5 

hl

, 

M

 a 6 

hk

Substitute – note similar terms in matrix – collect terms



M

 a

i



M

 a

j

         

h h h h h 2 h 2 2 3 l 4 k kl 3 l k 2 2 h 2 k 2 k 4 k 2 l 2 k 3 l hk 2 l hk 3 h 2 l 2 k 2 l 2 l 4 kl 3 hl 3 hkl 2 h 2 kl k 3 l kl 3 k 2 l 2 hkl 2 hk 2 l h 3 l hk 2 l hl 3 hkl 2 h 2 l 2 h 2 kl h 3 hk hkl hk h 2 kl h 2 k 3 2 k 2 l 2

        

26

Microstrain broadening - continued

Broadening – as variance

s

2



M hkl

  

HKL S HKL h H k K l L , H



K



L



4

3 collected terms

General expression – triclinic – 15 terms

s

2



M hkl

 

2 4

 

S 400 S 310 S 211 h h h 4 2 3



k kl S 040

 

k S 103 4 S 121



hl hk 3 S 004 2 l

 

l 4 S 031



k S 112 3 3



l S 220 hkl h



2



S 130 2 k hk 2 3

 

S 202 S 301 h h 2 3 l l 2

 

S 022 S 013

  

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

s 2 

M hkl

  2 

S

400

h S

301

h

4 3

l



S

040

k

 4

S

103

hk

 3 

S

 004

l

4  3

S

4

S

121

hk

2 202

l h

2

l

2  3 (

S

220

h

2

k

2 

S

022

k

2

l

2 ) 

Cubic – m3m – 2 terms

s

2



M hkl



S 400



h 4 k 4 l 4

 

3 S 220



h 2 k 2



h 2 l 2



k 2 l 2



27

Example - unusual line broadening effects in Na parahydroxybenzoate

Broad lines Sharp lines Directional dependence Lattice defects?

Seeming inconsistency in line broadening - hkl dependent 28

H-atom location in Na parahydroxybenzoate Good



F map allowed by better fit to pattern



F contour map H-atom location from x-ray powder data 29

Macroscopic Strain

Part of peak shape function #5 – TOF & CW d-spacing expression;

a

ij

from recip. metric tensor

1 2

d hkl



M hkl

 a 1

h

2  a 2

k

2  a 3

l

2  a 4

kl

 a 5

hl

 a 6

hk

Elastic strain – symmetry restricted lattice distortion TOF:

ΔT = (

d

11 h 2 +

d

22 k 2 +

d

33 l 2 +

d

12 hk+

d

13 hl+

d

23 kl)d 3

CW:

ΔT = (

d

11 h 2 +

d

22 k 2 +

d

33 l 2 +

d

12 hk+

d

13 hl+

d

23 kl)d 2 tan

Q

Why? Multiple data sets under different conditions (T,P, x, etc.) 30

Symmetry & macrostrain

d

ij

– restricted by symmetry e.g. for cubic



T =

d

11 h 2 d 3

for TOF Result: change in lattice parameters via change in metric coeff.

a

ij ’ =

a

ij -2

d

ij /C for TOF

a

ij ’ =

a

ij -(

p

/9000)

d

ij for CW Use new

a

ij ’ to get lattice parameters e.g. for cubic

a



1

a '

ij

31

Nonstructural Features

Affect the integrated peak intensity and not peak shape Bragg Intensity Corrections: Extinction Preferred Orientation Absorption & Surface Roughness Other Geometric Factors L h

Extinctio n

Sabine model - Darwin, Zachariasen & Hamilton Bragg component - reflection E b = 1 1+x Laue component - transmission E l = 1 E l =

p

2 x

  

x 2 4 5x 3 48 . . . x < 1 1 3 128x 2 . . .

  

x > 1 Combination of two parts E h = E b sin 2

Q

+ E l cos 2

Q

Sabine Extinction Coefficient

x  E x F h V 2

Crystallite grain size =

80% 60%

Increasing wavelength (1-5 Å) E h

40%

E x

20% 0%

0.0

25.0

50.0

75.0

2

Q

100.0

125.0

150.0

What is texture? Nonrandom crystallite grain orientations

Random powder - all crystallite orientations equally probable - flat pole figure Pole figure - stereographic projection of a crystal axis down some sample direction Loose powder Metal wire (100) random texture (100) wire texture Crystallites oriented along wire axis - pole figure peaked in center and at the rim (100’s are 90

º

apart) Orientation Distribution Function - probability function for texture 35

Texture - measurement by diffraction

Non-random crystallite orientations in sample Incident beam x-rays or neutrons Sample (220) (200) (111) Debye-Scherrer cones

•

uneven intensity due to texture

•

also different pattern of unevenness for different hkl’s

•

Intensity pattern changes as sample is turned 36

Preferred Orientation - March/Dollase Model

Uniaxial packing Ellipsoidal Distribution assumed cylindrical Ellipsoidal particles Spherical Distribution R o - ratio of ellipsoid axes = 1.0 for no preferred orientation

O h  1 M j n   1   R 2 o cos 2   sin 2 R o     3 2

Integral about distribution - modify multiplicity

Texture - Orientation Distribution Function - GSAS

Probability distribution of crystallite orientations - f(g)

f(g) = f(

F 1 ,Y,F 2

) f(g) =



l=0 l



m=-l l



n=-l C l mn T l mn (g)

F 1 Y F 2 F 1 ,Y,F 2

Euler angles T l mn = Associated Legendre functions or generalized spherical harmonics

Texture effect on reflection intensity - Rietveld model

A

(

h

,

y

) 

l

   0 2

l

4 p  1

m l

   

l n



l



l C l mn K l m

(

h

)

K l n

(

y

) • • • • •

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, C

l mn

, and 3 orientation angles - sample alignment 39

Absorption

X-rays - independent of 2

Q

- flat sample – surface roughness effect - microabsorption effects - but can change peak shape and shift their positions if small (thick sample) Neutrons - depend on 2

Q

smaller effect and



but much - includes multiple scattering much bigger effect - assume cylindrical sample Debye-Scherrer geometry

Model - A.W. Hewat

A h  exp(  T 1 A B   T 2 A 2 B  2 )

For cylinders and weak absorption only i.e. neutrons - most needed for TOF data not for CW data – fails for



R>1 GSAS – New more elaborate model by Lobanov & alte de Viega – works to



R>10 Other corrections - simple transmission & flat plate

Surface Roughness – Bragg-Brentano only

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

Nonuniform sample density with depth from surface Most prevalent with strong sample absorption If uncorrected - atom temperature factors too small Suortti model Pitschke, et al. model

S R  p  p  1    q  1 

exp

 q  

exp

q 

sin

Q S R  1  p  1

sin

Q 1  p  q

sin

2  pq Q (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 L p = 1 + M cos 2 2

Q

2sin 2

Q

cos

Q

L p = 1 2sin 2

Q

cos

Q

L p = d 4 sin

Q

Solvent scattering – proteins & zeolites?

Contrast effect between structure & “disordered” solvent region Babinet’s Principle: Atoms not in vacuum – change form factors f = f o -Aexp(-8

p

Bsin 2

Q

/



2 ) Carbon scattering factor uncorrected 6 f C 4 2 Solvent corrected 0 0 5 10 15 20 2

Q

44

Background scattering 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 - empirical Chebyschev power series - ditto Exponential expansions - air scatter & TDS Fixed interval points - brute force Debye equation - amorphous background (separate diffuse scattering in GSAS)

Debye Equation - Amorphous Scattering

real space correlation function especially good for TOF terms with amplitude A i sin( QR i ) exp(



QR i 1 B i Q 2 2 ) vibration distance

Neutron TOF fused silica “quartz”

47

Rietveld Refinement with Debye Function

1.60Å Si O 4.13Å 3.12Å 2.63Å 5.11Å 6.1Å

a -quartz distances

7 terms R i –interatomic distances in SiO 2 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 48

Non-Structural Features in Powder Patterns

Summary 1. Large crystallite size - extinction 2. Preferred orientation 3. Small crystallite size - peak shape 4. Microstrain (defect concentration) 5. Amorphous scattering - background

Time 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” 50

Complex structures (even proteins)

Too many parameters – “free” refinement fails Known stereochemistry: Bond distances Bond angles Torsion angles (less definite) Group planarity (e.g. phenyl groups) Chiral centers – handedness Etc.

Choice: rigid body description – fixed geometry/fewer parameters stereochemical restraints – more data 51

Constraints vs restraints

Constraints – reduce no. of parameters Derivative vector After constraints (shorter)

  v F i  R il U lk S kj  F  p j

Rigid body User Symmetry Rectangular matrices Derivative vector Before constraints (longer) Restraints – additional information (data) that model must fit Ex. Bond lengths, angles, etc.

52

Space group symmetry constraints

Special positions – on symmetry elements Axes, mirrors & inversion centers (not glides & screws) Restrictions on refineable parameters Simple example: atom on inversion center – fixed x,y,z What about U ij ’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 & U 11 ,U 22 ,U 33 , & U 13 on m: y fixed & U 11 ,U 22 , U 33 , & U 13 variable variable Rietveld programs – GSAS automatic, others not 53

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 uses fraction & multiplicity derived from sp. gp.

Others use occupancy If two atoms in site – Ex. Fe/Mg in olivine Then (if site full) F Mg = 1-F Fe 54

Multi-atom site fractions - continued

If 3 atoms A,B,C on site – problem Diffraction experiment – relative scattering power of site “1-equation & 2-unknowns” unsolvable problem Need extra information to solve problem – 2 nd diffraction experiment – different scattering power “2-equations & 2-unknowns” problem Constraint: solution of J.-M. Joubert Add an atom – site has 4 atoms A, B, C, C’ so that F A +F B +F C +F C’ =1 Then constrain so



F A = -



F C and



F B = -



F C’ 55

Multi-phase mixtures & multiple data sets

Neutron TOF – multiple detectors Multi- wavelength synchrotron X-ray/neutron experiments How constrain scales, etc.?

I c  I b  I d  S h  p S ph Y ph

Histogram scale Phase scale Ex. 2 phases & 2 histograms – 2 S h & 4 S ph Only 4 refinable – remove 2 by constraints Ex.



S 11 = -



S 21 &



S 12 = -



S 22 – 6 scales 56

Rigid body problem – 88 atoms – [FeCl 2 {OP(C 6 H 5 ) 3 } 4 ][FeCl 4 ]

P2

b

1 /c a=14.00Å b=27.71Å c=18.31Å =104.53

V=6879Å 3 264 parameters – no constraints Just one x-ray pattern – not enough data!

Use rigid bodies – reduce parameters

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

57

Rigid body description – 3 rigid bodies

FeCl 4 – tetrahedron, origin at Fe Cl 4 Cl 1

x z

Cl 2 Fe - origin

y

1 translation, 5 vectors Fe [ 0, 0, 0 ] Cl 1 Cl 2 [ sin(54.75), 0, cos(54.75)] [ -sin(54,75), 0, cos(54.75)] Cl 3 [ 0, sin(54.75), -cos(54.75)] Cl 4 [ 0, -sin(54.75), -cos(54.75)] D=2.1Å; Fe-Cl bond Cl 3 58

Rigid body description – continued

C 6 C 4 D 2 C 2 C 1 D 1 P x D PO – linear, origin at P C 6 – ring, origin at P(!) (ties them together) O z C 5 P [ 0, 0, 0 ] O [ 0, 0 1 ] D=1.4Å C 3 C 1 -C 6 [ 0, 0, -1 ] D 1 =1.6Å; P-C bond C1 [ 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 ] D 2 =1.38Å; C-C aromatic bond 59

Rigid body description – continued

Rigid body rotations – about P atom origin For PO group – R 1 (x) & R 2 (y) – 4 sets For C 6 group – R 1 (x), R 2 (y),R 3 (z),R 4 (x),R 5 (z) 3 for each PO; R 3 (z)=+0, +120, & +240; R 4 (x)=70.55

Transform: X’=R 1 (x)R 2 (y)R 3 (z)R 4 (x)R 5 (z)X C C x C C C R 5 (z) C 47 structural variables y P R 2 (y) R R 1 4 (x) (x) O R 3 (z) z Fe 60

Refinement - results

R wp =4.49% R p =3.29% R F 2 =9.98% N rb =47 N tot =69 61

Refinement – RB distances & angles

OP(C 6 ) 3 R 1 (x) R 2 (y) R 3 (z) a 1 122.5(13) -71.7(3) 27.5(12) 68.7(2) 2 -76.6(4) -15.4(3) 51.7(3) 68.7(2) 3 69.3(3) 12.8(3) -10.4(3) 68.7(2) 4 -158.8(9)

}

69.2(4) -53.8(9) 147.5(12) 267.5(12) 171.7(3) 291.7(3) 109.6(3) 229.6(3) 66.2(9) 186.2(9) 68.7(2) PO orientation

}

C 3 PO torsion (+0,+120,+240)

p

− C-P-O angle Fe-Cl = 2.209(9)Å R 5 (z) R 2 (y- PO) x R 4 (x) R 1 (x - PO) z R 3 (z) Fe 62

Packing diagram – see fit of C 6 groups

63

Stereochemical restraints – additional “data”

M

        

w i

f f f f f f f f a d v h x f Y R

  

t p

 

w i w i

   

w w i w i w i w i w i w i i

 

ci

4 

v oi

= 1/

s

2

(  

a oi



d



Y oi

  

h oi x oi



oi

  

R ci



p ci

  )

a ci

 2

Y ci d v ci h ci x ci

4

ci

 4  2  2  2  2  2

Powder profile (Rietveld)* Bond angles* Bond distances* Torsion angle pseudopotentials Plane RMS displacements* van der Waals distances (if v

oi

<v

ci

) Hydrogen bonds Chiral volumes** “

/y

” pseudopotential weighting factor f x - weight multipliers (typically 0.1-3) 64

For [FeCl 2 {OP(C 6 H 5 ) 3 } 4 ][FeCl 4 ] - 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)Å Number = 4 + 4 + 12 + 72 = 92 Bond angles: O-P-C, C-P-C & Cl-Fe-Cl = 109.5(10) – assume tetrahedral C-C-C & P-C-C = 120(1) – assume hexagon Number = 12 + 12 + 6 + 72 + 24 = 126 Planes: C 6 to 0.01 – flat phenyl Number = 72 Total = 92 + 126 + 72 = 290 restraints

A lot easier to setup than RB!!

65

Refinement - results

R wp =3.94% R p =2.89% R F 2 =7.70% N tot =277 66

Stereochemical restraints – superimpose on RB results

Nearly identical with RB refinement Different assumptions – different results 67

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 68

Residue rigid body model for phenylalanine

Q ijk t xyz

y 3t xyz +3Q ijk + y + c 1 + c 2 = 9 variables vs 33 unconstrained xyz coordinates c 1 c 2

69

Q ijk – Quaternion to represent rotations

In GSAS defined as: Q ijk = r+ai+bj+ck – 4D complex number – 1 real + 3 imaginary components Normalization: r 2 +a 2 +b 2 +c 2 = 1 Rotation vector: v = a x +b y +c z ; u = (a x +b y +c z )/sin(

a

/2) Rotation angle: r 2 = cos 2 (

a

/2); a 2 +b 2 +c 2 = sin 2 (

a

/2) Quaternion product: Q ab = Q a * Q b

≠ Q b * Q a

Quaternion vector transformation: v’ = QvQ -1 70

How effective? PDB 194L – HEWL from a space crystallization; single xtal data,1.40

Å resolution

21542 observations; 1148 atoms (1001 HEWL) X-Plor 3.1 – R F ~4600 variables = 25.8% GSAS RB refinement – R F =20.9% ~2700 variables RMS difference 0.10Å main chain & 0.21Å all protein atoms

RB refinement reduces effect of “over refinement” 71

194L & rigid body model – essentially identical

72

Conclusions – constraints vs. restraints

Constraints required space group restrictions multiatom site occupancy Rigid body constraints reduce number of parameters molecular geometry assumptions Restraints add data molecular geometry assumptions (again) 73

GSAS - A bit of history

GSAS – conceived in 1982-1983 (A.C. Larson & R.B. Von Dreele) 1 st version released in Dec. 1985

•

Only TOF neutrons (& buggy)

•

Only for VAX

•

Designed for multiple data (histograms) & multiple phases from the start

•

Did single crystal from start Later – add CW neutrons & CW x-rays (powder data) SGI unix version & then PC (MS-DOS) version also Linux version (briefly HP unix version) 2001 – EXPGUI developed by B.H. Toby Recent – spherical harmonics texture & proteins New Windows, MacOSX, Fedora & RedHat linux versions All identical code – g77 Fortran; 50 pgms. & 800 subroutines GrWin & X graphics via pgplot EXPGUI – all Tcl/Tk – user additions welcome Basic structure is essentially unchanged 74

Structure of GSAS

1. Multiple programs - each with specific purpose editing, powder preparation, least squares, etc.

2. User interface - EXPEDT edit control data & problem parameters for calculations - multilevel menus & help listings text interface (no mouse!) visualize “tree” structure for menus 3. Common file structure – all named as “experiment.ext” experiment name used throughout, extension differs by type of file 4. Graphics - both screen & hardcopy 5. EXPGUI – graphical interface (windows, buttons, edit boxes, etc.); incomplete overlap with EXPEDT but with useful extra features – by B. H. Toby 75

PC-GSAS – GUI only for access to GSAS programs

pull down menus for GSAS programs (not linux)

76

GSAS & EXPGUI interfaces

GSAS – EXPEDT (and everything else): 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 On console screen Keyboard input – text & numbers 1 letter commands Layers of menus – menu help – tree structure Type ahead thru layers of menus Macros (@M, @R & @X commands) Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e-5’ all allowed Drag & drop for e.g. file names 77

GSAS & EXPGUI interfaces

EXPGUI:

Access to GSAS Typical GUI – edit boxes, buttons, pull downs etc.

Liveplot – powder pattern

78

Unique EXPGUI features (not in GSAS)

    CIF input – read CIF files (not mmCIF) widplt/absplt coordinate export – various formats instrument parameter file creation/edit widplt Sum Lorentz FWHM (sample) Gauss FWHM (instrument)

79

Powder pattern display - liveplot

Zoom (new plot) updates at end of genles run – check if OK!

cum. c 2 on

80

Powder pattern display - powplot

I o I c Refl. pos.

I o -I c

“publication style” plot – works OK for many journals; save as “emf” can be “dressed up”; also ascii output of x,y table 81

Powplot options – x & y axes – “improved” plot?

Sqrt(I) Refl. pos.

I o



I c

rescale y by 4x Q-scale (from Q= p /sin q )

82

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).

83