Atomic Motion in Crystals,positional disorder below the

resolution limit andtheir effects on structure

Jürg Hauser and Hans-Beat BürgiUniversity of Berne, Switzerland

Results of a ‘crystal structure‘ analysis

Atomic coordinates x, y, z

Interatomic distancesand angles

Static structure

Atomic displacement Parameters (ADPs)U11, U22, U33, U12, U13, U23

???

Structural dynamics and disorder?

?

- does not provide ´crystal structures´, but a unit cell showing the distribution of atoms averaged over the time of the experiment and the space occupied by the crystal.

- does not measure the chemically interesting bond lengths and angles, but mean atomic positions, dynamic excursions and static displacements.

Crystal structure analysis

A cautionary remark at the outset

Questions: Effect of motion and disorder on bondlengths and angles?Dynamic processes in crystals?

Effects absorbed by ADPs

Average over entire crystal (= space average):Differences in atomic positions smaller than the resolution limit (ca. 0.5 Å) due to positional and orientational disorder

Average over time of experiment (= time average):Atomic displacements arising from dynamic processes faster than hours, e.g. molecular vibrations, conformational equilibria, etc.

rms-displacement or PEANUT representation

Equi-probability ellipsoid

Models of motion

Atomic Einstein or mean-field model

Generalized Einstein or molecular mean-field model

Lattice-dynamical model

1-D harmonic oscillator (atomic Einstein model)

- Harmonic potential function V = k (Δx)2 /2

- Quantized energy states Ev = hν (v + ½) = ħ (k/m)1/2 (v + ½)

- Wavefunctions Ψv(Δx)

-Probability density of state v Pv (Δx) = |Ψv(Δx)|2

- Total probability density is the Boltzmann- weighted sum P(Δx,T) = Σv |Ψv(Δx)|2 exp{hν (v + ½)/(2kBT)}/ Σvexp{hν (v + ½)/(2kBT)}

Mean-square displacement amplitude

- Total probability density is Gaussian

P(Δx,T) = Σv |Ψv(Δx)|2 exp{hν (v + ½)/(2kBT)}/

Σvexp{hν (v + ½)/(2kBT)}

= (2π < Δx2>)–1 exp{- Δx2/(2< Δx2>)}

(with 68% probability interval, ±1σ)

- mean square displacement amplitude

is temperature-dependent

< Δx2> = Δx2 P(Δx,T) dΔx

= h/(8π2mν) coth(hν/2kBT)

T(K)

<Δx2>

P(Δx)

Δx

2-D anisotropic harmonic oscillator

b

a

b

a

v = xa+yb = ζ(a*a)+η(b*b) P(v) = (2π)-1(detU-1)1/2 exp(-vTU-1v/2) U = U11 U12 = <ζζ> <ζη> U12 U22 <ζη> <ηη>

P(ζa*a+ηb*b) = P(ζa*a-ηb*b) U12 = <ζη> = 0

P(ζa*a+ηb*b) > P(ζa*a-ηb*b) U12 = <ζη> ≠ 0 (>0)

K.N. Trueblood, et al., Acta Cryst. A52 (1996) 770-781

P(v)

ab

3-D anisotropic harmonic oscillator

v = xa+yb+zc = ζ(a*a)+η(b*b)+θ(c*c)P(v) = (2π)-3/2(detU-1)1/2 exp(-vTU-1v/2)

U11 U12 U13 <ζζ> <ζη> <ζθ> U = U12 U22 U23 = <ζη> <ηη> <ηθ> U13 U23 U33 <ζθ> <ηθ> <θθ>

Equiprobability surface (ellipsoid)vT U-1 v = const = -ln{P(v) (2π)3/2(detU-1)-1/2}

Mean-square amplitude (difference-)surface<u2(n)> = nT U n , <Δu2(n)> = nT ΔU n

Rms amplitude (difference-)surface (PEANUT)<u2(n)>1/2 = (nT U n)1/2 etc.

n

Influence of P(v) on scattering

Fk(h) = { ∫ [ ∫ ρk(t-v) P(v)d3v] exp(2πih.t)d3t} exp(2πih.<rk>)

= fk(h) Tk(h) exp(2πih.<rk>)

Atomic form factor . Temperature factor . Phase factor with the well known expression

Tk(h) = exp [-2π2<(h.v)2>k] = exp [-2π2hTU(k)h>] = exp [-2π2 (Uk,11h2a*2 + Uk,22k2b*2 + Uk,33l2c*2

+ 2Uk,12hka*b* + 2Uk,13hla*c* + 2Uk,23klb*c*)]

The damping effect of atomic displacements on scattering intensity was first noticed by P. Debye

Verh. Dtsch. Phys. Ges. 15 (1913) 738

Effects of motion

on structure?

Something

is missing!

O-H distance Observed by neutron diffraction:

T 10 K 100 K 295 K 500 K 700 K 900 K

O-H (Å) 0.939(7) 0.947(7) 0.945(7) 0.945(6) 0.942(6) 0.929(8)

Expected 0.97(1) Å

translation

libration

O -- H .......... F

Foreshortening of interatomic distances by riding motion

<ΔXB2(φ)> = do

2<φ2> [Å2]

<ΔZB(φ)> = do<φ2>/2 [Å] = <ΔXB

2(φ)>/2do

(from ΔZB(φ)/do= cosφ–1 = – φ2)

Correction for librational displacements in the X- and Y-directions:

do = do‘ + {<ΔXB

2(φ)> + <ΔYB2(φ‘)>}/ 2do

‘ [Å]

W.R. Busing, H.A. Levy, Acta Cryst 17 (1964) 142

<φ2>1/2

A B

ΔXB

ΔZB

do

do‘

General situation with two atoms moving

d = {[do-(ΔZA-ΔZB)]2+(ΔXA-ΔXB)2 +(ΔYA-ΔYB)2}1/2 [Å]

<d> - do = {(<ΔXA2>+<ΔXB

2>-2<ΔXAΔXB>) +(<ΔYA

2>+<ΔYB2>-2<ΔYAΔYB>)}/2do

<d> - do = {<ΔX2> + <ΔY2>}/2do

ΔXB

ΔZBΔYB

ΔXAΔZA

ΔYAd

A B

do e2 e1

e3

<ΔXA> = <ΔYA > =

<ΔZA> = 0, etc

Bond Length Corrections

Δd = <ΔX2> /(2dobs)

W.R. Busing, H.A. Levy, Acta Cryst 17 (1964) 142

<ΔX2> = <ΔX2H> + <ΔX2

O> – 2 <ΔXOΔXH> [Å2]

Upper Limit <ΔX2H> + <ΔX2

O> + 2{<ΔX2O><ΔX2

H>}1/2

Independent Motion <ΔX2H> + <ΔX2

O>

H riding on O <ΔX2H> – <ΔX2

O>

Lower Limit <ΔX2H> + <ΔX2

O> – 2{<ΔX2O><ΔX2

H>}1/2

<ΔX2O> <ΔX2

H>

translation

libration

Indirect retrieval of correlation terms from temperature dependence of ADPs

ADPs Generalized Einstein Model

ADPs, determined at several T’s [Å2]

δi = (ħ/2ωi) coth(ħωi/2kT); H.B. Bürgi, S.C. Capelli, Acta Cryst., A56 (2000) 403

libration and translation (ω, V)

disorder (ε), (~temperature independent)

+

// mV'T),/(Vm x

HO )T(XX

HO )T(XX 2121

H )T(X 2

O )T(X 2

1

Correlation ADPs <ΔXO ΔXH (T) > from model [Å2]

// mV'T),/(Vm x

HO )T(XX

HO )T(XX 2121

H )T(X 2

O )T(X 2

1

Comparison of Corrections<ΔX2> = <ΔX2

O> + <ΔX2H> – cross term [Å2]

M. Kunz, G. A. Lager, H.-B. Bürgi, M. T. Fernandez-Diaz, Phys. Chem. Minerals 33 (2006) 17

Average O-H distance 0.976 Å Vibration frequencies ┴ to O–H bond 888, 338 cm-1

Vibration frequencies ║ to O–H bond 3514, 263 cm-1

O-H

dis

tan

ce

0 200 400 600 800 1000T(K)

1.08

1.06

1.04

1.02

1.00

0.98

0.96

0.94

0.92

Upper limit

Lower LimitObserved

Indep. Motion

T-DEPENDENCE.Riding motion

Conclusions from discussion of diatomic molecular fragment

1) The lack of interatomic or correlation amplitudes between atoms is always a problem in interpreting atomic displacement parameters (ADPs) and the influence of motion on structure.

2) The best opportunity for interpreting ADPs arises when as many correlation amplitudes as possible are known or can be estimated with reliability.

3) This is the case for a rigid molecule. The instantaneous displacements of its atoms can be represented in terms of librational and translational displacements of the molecule as a whole.

Instantaneousatomic displacement in a rigid body

ΔXAe1 +ΔYAe2 + ΔZAe3 = l1xrA + l2xrA + l3xrA + t1 + t2 + t3 libration translation

0 rA3 –rA2

[ΔXA ΔYA ΔZA] = [l1 l2 l3] -rA3 0 rA1 + [t1 t2 t3] rA2 –rA1 0

Libration and translation axes parallel to e1, e2, e3, a cartesian coordinate system, e.g. the system of inertia

A

e2

e1

e3

rA1rA2

rA3

rA

Mean square displacements of atom k in terms of mean square libration, translation and screw

coupling motion

0

0

0

100

010

001

0100

0010

0001

)(

11

13

23

332313332313

232212322212

131211312111

333231332313

232221232212

131211131211

12

13

23

kk

kk

kkkk

kk

kk

calc

rr

rr

rr

LLLSSS

LLLSSS

LLLSSS

SSSTTT

SSSTTT

SSSTTT

rr

rr

rr

kU

Tk

Tkkkcalc k rSSrrLrT

R

I

LS

STRIU T

kTk

)(

to be represented in terms of <titj> = Tij <lilj> = Lij <litj> = Sij

<ΔXkΔXk> <ΔXkΔYk> <ΔXkΔZk> Uk,11 Uk,12 Uk,13

U(k) = <ΔXkΔYk> <ΔYkΔYk> <ΔYkΔZk> = Uk,12 Uk,22 Uk,23

<ΔXkΔZk> <ΔYkΔZk> <ΔZkΔZk> Uk,13 Uk,23 Uk,33

Determination of the elements Tij, Lij, Sij

The elements of T, L and S are determined by a least–squares procedure from

δΣk,l,m (Uk,lm,obs- Uk,lm,calc)2/δTij = 0

δΣk,l,m (Uk,lm,obs- Uk,lm,calc)2/δLij = 0

δΣk,l,m (Uk,lm,obs- Uk,lm,calc)2/δSij = 0

- Note that the problem is linear in the 21 unknowns Tij, Lij, Sij

- The trace of S, tr(S) = S11 + S22 + S33, is found to be indeterminate! It is arbitrarily set equal to zero. This fact is a remaining consequence of the lack of information on interatomic correlation amplitudes as discussed above.

Symmetry restrictions on Tij, Lij, Sij

An instantaneous displacement coordinate of a rigid body is called s or symmetric if it maintains symmetry, a or asymmetric if it destroy symmetry. Expectation values <titj> = Tij, <lilj> = Lij, <litj> = Sij between two symmetric or two asymmetric coordinates may differ from zero; expectation values between a symmetric and an asymmetric coordinate are identically equal to zero

<ss> or <aa> 0<sa> or <as> ≡ 0

Example 1: tris(bicyclo[2.1.1]hexeno)benzene(symmetry: 2)

- The twofold axis coincides with l2

- instantaeous diplacements l2, t2 maintain twofold symmetry- instantanous displacements l1, l3, t1, t3 destroy it- thus L12 = L23 = T12 = T23 = S12 = S21 = S23 = S32 = 0

- check this result in the practice session

l2, t2

l1, t1

Example 2: 2,2‘-dimethylstilbene(symmetry: 1bar)

C7

C8

C1

C3C2

C5

C4

C6

- The inversion centre sits in the middle of the molecule - instantaeous diplacements l1, l2 l3 maintain inversion symmetry- instantanous displacements t1, t2, t3 destroy it- thus all elements of S are zero: Sij = 0

- check this result in the practice session

How do

we know

whether a

molecule

behaves

as a rigid

body?

Rigid-bond and rigid-body tests

- calculate the mean-square displacement of atom A in the direction of atom B and of atom B in the direction of atom A (|nAB| = 1) ΔUAB = nT

AB (UA - UB) nAB

- If atoms A and B are connected through a covalent bond, ΔUAB is expected to be small (< 0.001 A2, for atoms at least as heavy as carbon, << 0.001 A2 for atoms heavier than F, so called ‘Hirshfeld test’)

- If the ΔUIJ–values for an entire group of atoms I,J = A, B, C, …, Z fulfill the Hirshfeld test, the group of atoms {A, B, C, …, Z} may be considered to form a rigid body.

Example 1: tris(bicyclo[2.1.1]hexeno)benzene(symmetry: 2)

Rigid-body analysis and discussion thereof in the practice session

Example 2: 2,2‘-dimethylstilbene(symmetry: 1bar)

C7

C8

C1

C3C2

C5

C4

C6

Rigid-body analysis and discussion thereof in the practice session

Effects of rigid body motion on structure

- Translation corresponds to linear motion: no effect on structure.

- Libration corresponds to curvilinear motion analogous to that discussed for the diatomic fragment: bonds are foreshortened, there are (usually small) changes in bond angles. The ominous correlation element, <ΔX2> = <ΔX2

A> + <ΔX2B> – 2 <ΔXAΔXB>,

in the expression for distance correction, Δd = <ΔX2>/(2dobs), is obtained from T, L and S.

- The indeterminacy in Tr(S) does not affect these corrections. Note that corrections of intermolecular distances cannot be made.

- It is always better to avoid corrections by doing a better experiment. If molecular geometry is important, measure at the lowest available temperature.

Some caveats

Cases in which any interpretation of ADPs has to be taken with a grain of salt (or better: two!)

- insufficient resolution: U‘s from standard structure determination absorb effects arising from nonspherical atomic valence densitites (Mo-radiation, 2θ ~ 50 deg, corresponding to a resolution of ~0.85 Å, Hirshfeld recommends a resolution of 0.5 Å, if this problem is to be avoided). The effect may amount to as much as 0.002 Å2, especially in aromatics.

- U‘s, even of ‘rigid‘ molecules, represent librations, translations as well as intramolecular deformations. The contribution of the latter is indiscriminately absorbed into T, L and S.

Pathological U‘s and ΔU‘s

More cases in which any interpretation of ADPs has to be taken with a grain of salt (or better: two!)

- Molecules with low-energy vibrations, e.g. torsions and angle-bends (i.e. nonrigid molecules!!!)

- Disorder with a good chemical explanation, e.g.High spin/low spin mixtures in spin crossover compoundsMolecules with dynamic Jahn-Teller effectsFluxional molecules in general

- Anharmonic motion: potentials are no longer quadratic, ADPs are Gaussian fits to non-Gaussian probability density functions.

- Absorption and (pseudo-)extinction, incomplete data

K. Chandrasekhar and H. B. Bürgi, Acta Cryst. (1984). B40, 387-397

Pathological ΔU‘s from 33 spin crossover compounds

Some conclusions

- Our standard anisotropic displacement parameters cannot account for curvilinear motion

- This deficiency implies systematic errors in <rk>

- Distances between mean atomic positions are not the same as mean interatomic distances

- Corrections for curvilinear motion require interatomic or correlation displacement amplitudes which are not available from a single-temperature diffraction measurement

- If a reasonable estimate of these correlation amplitudes can be obtained, a zeroth-order correction of molecular geometry is possible (libration, independent and riding motions)

Some literatureElementary aspects:K.N. Trueblood, H.-B. Bürgi, H. Burzlaff, J.D. Dunitz, C.M. Grammacioli, H.H. Schulz, U. Shmueli, and S.C. Abrahams, Acta Cryst A52 (1996) 770-781 (Definitions and coordinate transformations)V. Schomaker and K.N. Trueblood, Acta Cryst. B24 (1968) 63-76 (On the rigid body motions of molecules in crystals)W.R. Busing, H.A. Levy, Acta Cryst 17 (1964) 142 (On distance corrections) M. Kunz, G. A. Lager, H.-B. Bürgi, M. T. Fernandez-Diaz, Phys. Chem. Minerals 33 (2006) 17 (On the problem of interatomic correlation of motion)

Advanced topics:V. Schomaker, K.N. Trueblood, Acta Cryst B54 (1998) 507-514 H.B. Bürgi, Acta Cryst. B45 (1989) 383-390 (both articles on segmented i.e. semi-rigid bodies)H.B. Bürgi, S.C. Capelli, Acta Cryst. A56 (2000) 403-412S.C. Capelli, M. Förtsch, H.B. Bürgi. Acta Cryst. A56 (2000) 413-424H.B. Bürgi, S.C. Capelli, H. Birkedal, Acta Cryst. A56 (2000) 425-435 (All 3 articles on dynamics of molecules in crystals from multi-temperature anisotropic displacement parameters)

Reviews:J.D. Dunitz, V. Schomaker, K.N. Trueblood, J. Phys. Chem. 92 (1988) 856-867J.D. Dunitz, E.F. Maverick, K.N. Trueblood, Angew. Chem. Int. Ed. Engl. 27 (1988) 880-895

Ueq

Ueq = 1/3{ U11 + U22 + U33

+ 2U12 a*b*ab cos γ + 2U13 a*c*ac cos β + 2U23 b*c*bc cos α}

Ueq = 1/3{ U11 + U22 + U33 + 2U12a*b*a.b + 2U13a*c*a.c + 2U23b*c*b.c}