Absolute Structure Determination and Light-Atom Structures...iso freely. • Data collected by Trixie Wagner (Novatis), Olly Presly (Agilent) or me Parsons, Flack & Wagner Acta Cryst

Absolute Structure Determination

and Light-Atom Structures

Simon Parsons

The University of Edinburgh

What This is About • X-ray crystallography is

so popular because it

directly produces

images of molecules.

No other technique

does this.

• One problem with the

technique is that

characterisation of the

absolute configuration

of a molecule is difficult.

• This information is often

vitally important though.

Absolute Structure & The Flack

Parameter

_2 2

model single single( ) (1 ) | ( ) | | ( ) |I x F x F h h h

Precision of x

Suppose I refine a structure and get x = 0.00(8).

Precision of x

Suppose I refine a structure and get x(u) = 0.00(8).

Notice that x may be –ve.

This is unphysical but may happen statistically.

0.24 = 3u u

σ

esd

sd

su

x = 0 x = 1

Physical range of x

Statistical range of x

x = 0 x = 1

How wide should the

distributions be?

x = 0 x = 1

x = 0 x = 1

Model right Model wrong Don’t know

Flack & Bernardinelli J. Appl. Cryst. (2000), 33, 1143

x = 0 x = 1

3u 6u 3u

12u = 1

u = 1/12 = 0.08 ~ 0.1

Refinement of x in SHELXL

TWIN -1 0 0 0 -1 0 0 0 -1

BASF 0.1

N value esd shift/esd parameter

1 4.86497 0.02065 0.000 OSF

2 -0.03716 0.26951 0.000 BASF 1

3 0.00881 0.00260 0.000 EXTI

The Problem

• The requirement that u is less than 0.1 is

actually quite difficult to attain for light atom

structures (C,H,N,O compounds).

• A precise determination of x requires large

values of f ” relative to f0+f ’.

• But even with Cu-radiation f ”(O) etc. are

small.

• Alanine C3H7NO2 x = -0.04(27) [100 K data]

The Problem

• If Friedel’s law held

exactly absolute

structure determination

by X-ray crystallography

would be impossible.

• But anomalous

scattering introduces

deviations which carry

absolute structure

information.

• Magnitudes depend on

– Elements present

– Wavelength of X-

rays used.

f” values Mo Cu

C 0.002 0.009

N 0.003 0.018

O 0.006 0.032

S 0.124 0.558

Friedif (Friedifstat)

Flack & Shmueli. Acta Cryst. (2007) A63, 257

Friedif = 104 χ

Friedif

Flack & Bernardinelli. Acta Cryst. (2008) A64, 484

Test Data Sets • Good crystals – 23 data sets

• Mostly 100 K data collection. High redundancy.

• Complete or nearly complete Friedel data

• Cu-Kα radiation. CCD Instruments

• Multiscan absorption correction.

• Merged in SORTAV (defaults)

• Refine against F2 with all data. Shelx weights.

• Spherical scattering factors – Dittrich et al. Acta Cryst (2006) A62, 217

• Refine H positions and Uiso freely.

• Data collected by Trixie Wagner (Novatis), Olly Presly (Agilent) or me

Parsons, Flack & Wagner Acta Cryst (2013) B69, 249

Test Data Sets

Formula Space Group Redundancy R1(F>4u(F))

C3H7NO2 P212121 25 2.19

C25H31NO5 P212121 15 2.48

C19H26N6O* P212121 14 4.25

C16H18N2 P32 13 2.83

C27H48 P21 18 2.88

*Some disorder

Friedif and refined x

Code Formula Friedif x (Normal Ref’t)

L-Alanine C3H7NO2 34 -0.04(27)

GKO02 C25H31NO5 32 0.01(15)

R-CYCLO C19H26N6O 21 -0.02(27)

TWA16A C16H18N2 13 0.00(69)

Cholestane C27H48 9 -0.01(77)

Expected distribution of 23

values of x/u

Refinement of x

Code x (Normal Ref’t)

L-Alanine -0.04(27)

GKO02 0.01(15)

R-CYCLO -0.02(27)

TWA16A 0.00(69)

Cholestane -0.01(77)

Distribution of x/u compared

to a unit Gaussian for 23

Structures

Chi2 = 0.03

Precise Absolute Structure

Determination

• Is there a way to get lower (more

realistic) standard uncertainties?

• Post refinement methods

– Probability the model hand is correct

– Estimate of x (2 methods)

• Obtaining x during refinement

– Restraints

Right or Wrong?

Bayesian Methods

(PLATON/BIJVOET)

y = Hooft parameter = x calculated by Bayesian methods

Hooft, Straver & Spek. J. Appl. Cryst. (2008), 41, 96.

Bayesian Methods

Calculate structure factors F2single = F2

c with x = 0.

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fz

u F

h h

h

21( ) exp

22

hh

zp z

(observations | x 0) ( )hp p z

Bayesian Methods

Calculate structure factors with x = 0.

Fc2 Fo2 Sigma(Fo2)

-6 2 1 68.55 65.70 0.71

6 2 1 67.61 64.50 0.71

2 2

(67.61 68.55) (64.50 65.70)

0.71 0.71z

Assumes measurement errors

are distributed with a Gaussian

pdf.

21( ) exp

22

hh

zp z

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fz

u F

h h

h

Bayesian Methods

Calculate structure factors F2single = F2

c with x = 0.

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fz

u F

h h

h

21( ) exp

22

hh

zp z


Bayesian Methods

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fq

u F

h h

h

21(q ) exp

22

hh

qp

(observations | x 1) (q )hp p

Bayesian Methods:

Right or Wrong Structure?

(obs | 0)

(obs | 0) (

(x 0)( 0 | obs)

(x 0) (obs | 1) x 1)

pp x

p

p x

p x p px

Bayesian Methods

Code Friedif x (Normal Ref’t) P2(true)

L-Alanine 34 -0.04(27) 1.000

GKO02 32 0.01(15) 1.000

R-CYCLO 21 -0.02(27) 1.000

TWA16A 13 0.00(69) 1.000

Cholestane 9 -0.01(77) 1.000

Calculation of x(u) by

Bayesian Methods • This analysis can be

extended to obtain a value of x (aka y).

• Instead of calculating probabilities at y = 0 and 1, calculate for the range 0-1, and build up a distribution.

• Equations expressed in terms of

γ (gamma) = 1-2y

Hooft, Straver & Spek. J. Appl. Cryst. (2008), 41, 96.

Hooft Parameter

Code Friedif x (Normal Ref’t) y

L-Alanine 34 -0.04(27) 0.01(4)

GKO02 32 0.01(15) 0.03(3)

R-CYCLO 21 -0.02(27) -0.02(4)

TWA16A 13 0.00(69) 0.02(7)

Cholestane 9 -0.01(77) -0.04(9)

Chi2 for 23 structures = 0.83 (~1)

When Errors are Non-

Gaussian (‘Poor Data’)

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fz

u F

h h

h

21( ) exp

22

hh

zp z


Hooft, Straver & Spek. J. Appl. Cryst. (2010), 43, 665

When Errors are Non-

Gaussian (‘Poor Data’)

2 2

single

2

( ) ( )

( ( ))

o

h

o

F Fz

u F

h h

h


Student-t

ν = 5

Hooft, Straver & Spek. J. Appl. Cryst. (2010), 43, 665

12 2

1

2( , ) 1

2

zp z

Example

Riebenspies & Bhuvanesh Acta Cryst. (2013), B69, 288

‘Quotient’ Methods

• Systematic errors like absorption may drown-out

anomalous differences.

• Measure Friedel opposites in such as way that

absorption errors are the same for both.

• Stoe - measure

I(h) at 2, , and

I(-h) at -2, -, and

• The quotient I(h)/I(-h) is free from absorption and

extinction errors. Also scale-free.

Le Page, Gabe & Gainsford. J. Appl. Cryst. (1990), 23, 406

Quotients


2 2 2 22 2single single

2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )( ) ( )(1 2 )

( ) ( ) ( ) ( ) | ( ) | | ( ) |o o

o o

F FF FI Ix

I I F F F F

h hh hh h

h h h h h h

Quotients


2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )(1 2 )

( ) ( ) | ( ) | | ( ) |o o

o o

F FF Fx

F F F F

h hh h

h h h h

This can be

calculated from

your data set.

This can be calculated

(Fc2 for a model refined

with x = 0).


2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )(1 2 )

( ) ( ) | ( ) | | ( ) |o o

o o

F FF Fx

F F F F

h hh h

h h h h

single( ) (1 2x)Q ( )oQ h h


2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )(1 2 )

( ) ( ) | ( ) | | ( ) |o o

o o

F FF Fx

F F F F

h hh h

h h h h


y = mx

Fit graph to y = mx

Equate gradient m to (1-2x)

Solve for Flack:

Gradient = 0.893(50) = 1 – 2x

Flack = 0.05(3)

Implemented in XPREP and Shelxl-2013.

No TWIN/BASF instructions!


Code Friedif x (Normal Ref’t) x(QUOT)

L-Alanine 34 -0.04(27) 0.01(4)

GKO02 32 0.01(15) 0.02(3)

R-CYCLO 21 -0.02(27) 0.00(4)

TWA16A 13 0.00(69) 0.18(8)

Cholestane 9 -0.01(77) -0.01(13)

Differences

2 2 2 2 2 2

single single

single

( ) ( ) (1 2 )(| ( ) | | ( ) | )

(1 2 ) ( )

o o

o

F F x F F

D x D

h h h h

h h


The Post-Refinement Problem

A potential problem with post-refinement methods is

that any correlations involving x are lost.

But…

These quantities can also be applied as restraints.

Need to code F2(h) etc. in terms of x’s, U’s, occs and

so on.


2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )(1 2 )

( ) ( ) | ( ) | | ( ) |o o

o o

F FF Fx

F F F F

h hh h

h h h h

Facilities in Topas Academic 5

• A symbolic equation can be written for the quotient.

• Like a function or subroutine in Fortran

fn FPC(h, k, l)

{

return

2.31000*Exp( -20.84390*s2(h,k,l)) +

1.02000*Exp( -10.20750*s2(h,k,l)) +

1.58860*Exp( -0.56870*s2(h,k,l)) +

0.86500*Exp( -51.65120*s2(h,k,l)) +

( 0.23370);

}

prm !FPPC 0.00910

fn X1O1(h, k, l) = tpi*(h*xO1 + k*yO1 + l*zO1);

...

fn QUOT(h, k, l) = (1-2*ENANTIO)*( (

U1O1(h, k ,l)*(FPO(h,k,l)*Cos(X1O1(h, k, l)) - FPPO*Sin(X1O1(h, k, l)))

+

U2O1(h, k ,l)*(FPO(h,k,l)*Cos(X2O1(h, k, l)) - FPPO*Sin(X2O1(h, k, l)))

+

...

restraint = ( 0.01642 - QUOT( 5, 2, 3 ))/ 0.00978;

restraint = ( 0.02984 - QUOT( 4, 3, 2 ))/ 0.00968;

restraint = ( -0.02283 - QUOT( 3, 1, 9 ))/ 0.00924;

restraint = ( 0.04175 - QUOT( 1, 2, 8 ))/ 0.02252;

Refine with intensity data merged in the centrosymmetric Laue

group and Q (or D) applied as restraints.

Code Friedif x(QUOT)

Post Refine

x(QUOT)

Refine

L-Alanine 34 0.01(4) 0.01(3)

GKO02 32 0.02(3) 0.03(3)

R-CYCLO 21 0.00(4) -0.02(4)

TWA16A 13 0.18(8) 0.14(8)

Cholestane 9 -0.01(13) 0.00(11)

Summary Code x(TWIN) y(HOOFT) x(QUOT)

Post Refine

x(QUOT)

Refine

L-Alanine -0.04(27) 0.01(4) 0.01(4) 0.01(3)

GKO02 0.01(15) 0.03(3) 0.02(3) 0.03(3)

R-CYCLO -0.02(27) -0.02(4) 0.00(4) -0.02(4)

TWA16A 0.00(69) 0.02(7) 0.18(8) 0.14(8)

Cholestane -0.01(77) -0.04(9) -0.01(13) 0.00(11)

Validation – Alanine (34)

Cholestane (9)

TWA16A

Outlier omission 0.18(8) 0.08(8)

Outlier Detection

• Remove Bijvoet pairs if Do(h) > 2Dc, max

• For quotient calculations remove data

where Fo2(h)/u(Fo

2(h)) and

Fc2(h)/u(Fo

2(h)) are < 3.


2 2 2 2 2 2

single single

| ( ) | | ( ) |( ) ( )(1 2 )

( ) ( ) | ( ) | | ( ) |o o

o o

F FF Fx

F F F F

h hh h

h h h h

Normal Probability Plots

Validation

Mandelic acid

Why might this work?

Conclusions • Several methods for obtaining precise absolute

structure determination for light-atom structures.

• Post refinement calculations are OK. But still work to

do on the effects of completeness.

• Validation is important.

• Still work to do on automatic detection of outliers.

• Transformation into sensitive and insensitive

components is important. Still work to do on the

reasons the methods work - or why L.S. doesn’t.

Acknowledgements

• Howard Flack (Geneva)

• Trixie Wagner (Novartis)

• Alan Coelho (Topas)

• Richard Cooper & David Watkin (Oxford)

• George Sheldrick (Göttingen)

Documents

Absolute Structure Determination and Light-Atom Structures...iso freely. • Data collected by Trixie Wagner (Novatis), Olly Presly (Agilent) or me Parsons, Flack & Wagner Acta Cryst