Transcript Atomic Motion in Crystals, positional disorder below the

Atomic Motion in Crystals,
positional disorder below the
resolution limit and
their effects on structure
Jürg Hauser and Hans-Beat Bürgi
University of Berne, Switzerland
Results of a ‘crystal structure‘ analysis
Atomic displacement
Parameters (ADPs)
U11, U22, U33, U12, U13, U23
Atomic coordinates
x, y, z
Interatomic distances
and angles
Static structure
?
???
Structural dynamics
and disorder?
A cautionary remark at the outset
Crystal structure analysis
- does not measure the chemically interesting bond lengths
and angles, but mean atomic positions, dynamic excursions
and static displacements.
- 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.
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
Equiprobability
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 Boltzmannweighted sum
P(Δx,T) = Σv |Ψv(Δx)|2 exp{hν (v + ½)/(2kBT)}/
Σvexp{hν (v + ½)/(2kBT)}
Mean-square displacement amplitude
P(Δx)
Δx
- 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)
<Δx2>
T(K)
- mean square displacement amplitude
is temperature-dependent
< Δx2> =  Δx2 P(Δx,T) dΔx
= h/(8π2mν) coth(hν/2kBT)
2-D anisotropic harmonic oscillator
P(v)
v = xa+yb = ζ(a*a)+η(b*b)
P(v) = (2π)-1(detU-1)1/2 exp(-vTU-1v/2)
a
b
U = U11 U12
U12 U22
=
<ζζ> <ζη>
<ζη> <ηη>
b
a
P(ζa*a+ηb*b) = P(ζa*a-ηb*b)
U12 = <ζη> = 0
b
a
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
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>
n
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.
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
libration
Effects of motion
on structure?
O
--
H
..........
Something
F
is missing!
translation
O-H distance
T
O-H (Å)
10 K
100 K
Observed by neutron diffraction:
295 K
500 K
700 K
900 K
0.939(7) 0.947(7) 0.945(7) 0.945(6) 0.942(6) 0.929(8)
Expected
0.97(1) Å
Foreshortening of interatomic distances by
riding motion
<ΔXB2(φ)> = do2<φ2> [Å2]
do
do
ΔXB
‘
ΔZB
<φ2>1/2
A
<ΔZB(φ)> = do<φ2>/2 [Å]
= <ΔXB2(φ)>/2do
(from ΔZB(φ)/do= cosφ–1 = – φ2)
B
Correction for librational
displacements in the X- and
Y-directions:
do = do‘ + {<ΔXB2(φ)> + <ΔYB2(φ‘)>}/ 2do‘
[Å]
W.R. Busing, H.A. Levy, Acta Cryst 17 (1964) 142
General situation with two atoms moving
e2
do
d
ΔYA
ΔZA
A
ΔXA
e1
e3
ΔYB
ΔZB
<ΔXA> = <ΔYA > =
ΔXB
B
<ΔZA> = 0, etc
d = {[do-(ΔZA-ΔZB)]2+(ΔXA-ΔXB)2 +(ΔYA-ΔYB)2}1/2
<d> - do = {(<ΔXA2>+<ΔXB2>-2<ΔXAΔXB>)
+(<ΔYA2>+<ΔYB2>-2<ΔYAΔYB>)}/2do
<d> - do = {<ΔX2> + <ΔY2>}/2do
[Å]
libration
Bond Length Corrections
Δd = <ΔX2> /(2dobs)
<ΔX2O>
<ΔX2H>
W.R. Busing, H.A. Levy,
Acta Cryst 17 (1964) 142
translation
<ΔX2>
= <ΔX2H> + <ΔX2O> – 2 <ΔXOΔXH> [Å2]
Upper Limit
<ΔX2H> + <ΔX2O> + 2{<ΔX2O><ΔX2H>}1/2
Independent Motion
<ΔX2H> + <ΔX2O>
H riding on O
<ΔX2H> – <ΔX2O>
Lower Limit
<ΔX2H> + <ΔX2O> – 2{<ΔX2O><ΔX2H>}1/2
Indirect retrieval of correlation terms from
temperature dependence of ADPs
< D XO D XH (T) > 
 < D XO2 (T) >
-1 / 2
-1 / 2
x
=
+
d
w
e
m
V
(
1
/
,
T)
V'
m
<
< D X2H (T) > 
 D XO D XH (T) >
ADPs
Generalized Einstein Model
ADPs, determined
at several T’s [Å2]
libration and
translation (ω, V)
+
Correlation ADPs
<ΔXO ΔXH (T) >
from model [Å2]
δi = (ħ/2ωi) coth(ħωi/2kT);
disorder (ε),
(~temperature
independent)
H.B. Bürgi, S.C. Capelli, Acta Cryst., A56 (2000) 403
Comparison of Corrections
O-H distance
<ΔX2> = <ΔX2O> + <ΔX2H> – cross term [Å2]
1.08
1.06
1.04
1.02
1.00
0.98
0.96
0.94
0.92
Upper limit
Indep. Motion
T-DEPENDENCE.
Riding motion
Lower Limit
Observed
0
200
400
600
T(K)
Average O-H distance
Vibration frequencies ┴ to O–H bond
Vibration frequencies ║ to O–H bond
800
1000
0.976 Å
888, 338 cm-1
3514, 263 cm-1
M. Kunz, G. A. Lager, H.-B. Bürgi, M. T. Fernandez-Diaz, Phys. Chem. Minerals 33 (2006) 17
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.
Instantaneous
atomic displacement
in a rigid body
e3
A
rA3
rA2
rA
rA1
e2
e1
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
Mean square displacements of atom k in terms
of mean square libration, translation and screw
coupling motion
U(k) =
<ΔXkΔXk> <ΔXkΔYk> <ΔXkΔZk>
<ΔXkΔYk> <ΔYkΔYk> <ΔYkΔZk> =
<ΔXkΔZk> <ΔYkΔZk> <ΔZkΔZk>
to be represented in terms of <titj> = Tij
1
U calc (k ) = 0
0
U calc (k ) =
0
0
0
rk 3
1
0
- rk 3
0
0
1
rk 2
- rk1
I
Rk 
T
S T

 T11
T
- rk 2   12
T
rk1   13
S
0   11
 S12

 S13
S
L

Uk,11 Uk,12 Uk,13
Uk,12 Uk,22 Uk,23
Uk,13 Uk,23 Uk,33
<lilj> = Lij
T12
T13
S11
S12
T22
T23
S 21
S 22
T23
S 21
T33
S31
S31
L11
S32
L12
S 22
S32
L12
L22
S 23
S33
L13
L23
<litj> = Sij
S13   1
S 23   0
S33   0

L13   0
L23   rk 3

L33  - rk1
0
1
0
- rk 3
0
rk1
0 
0 
1 

rk 2 
- rk1 

0 
 I 
T
T
=
T
+
r

L

r
+
r

S
+
S

r
k
k
k
k
R T 
 k
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)
l1, t1
l2, t2
- 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
Example 2:
2,2‘-dimethylstilbene
(symmetry: 1bar)
C7 C1
C6
C5
C4
C2 C3
C8
- 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 = nTAB (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
C6
Example 2:
2,2‘-dimethylstilbene
(symmetry: 1bar)
C7 C1
C5
C4
C2 C3
C8
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> = <ΔX2A> + <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 anglebends (i.e. nonrigid molecules!!!)
- Disorder with a good chemical explanation, e.g.
High spin/low spin mixtures in spin crossover compounds
Molecules with dynamic Jahn-Teller effects
Fluxional 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
Pathological ΔU‘s from 33
spin crossover compounds
K. Chandrasekhar and H. B. Bürgi, Acta Cryst. (1984). B40, 387-397
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 literature
Elementary 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) 403-412
S.C. Capelli, M. Förtsch, H.B. Bürgi. Acta Cryst. A56 (2000) 413-424
H.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) 856-867
J.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}