Single
g crystalydiffraction

Jacob Overgaard
Department off ChemistryhUniversity of Aarhus
DenmarkAcknowledgementsg
Aarhus X‐ray Diffraction Group:
Finn Krebs
Fi
K b Larsen
L
Bo B. Iversen
Henrik F.
F Clausen
Mads R. V. Jørgensen
Helle SvendsenMette S. Schmøkel

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The ambition of this talk...

... to understand the steps involved in this process..

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The ambition of this talk...

... and as a bonus also these steps..

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Overview of the presentation
Part One••••

Understanding of single crystals – and why they are so important in science
Taking crystallography a step further – Extension of the IAM
Experimentalprequirementsq& challengesgSynchrotron radiation – why?

Part Two•Time for some examples

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Single crystals

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What is inside a single-crystal?
Definition: “An arrangement of molecules or atoms, which is periodic in 3D”
Crystal planes are named using Miller indices(reciprocal of the crossing with the axes)

Unit cell denotes a coordinate system

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The building block – the unit cell

Low symmetry

7

2

4

↓
High symmetrys mmetr1.

Triclinic (a≠b≠c, α≠β≠γ≠90)

2.

Monoclinic (a≠b≠c, α≠γ≠90, β=90)3.

Orthorhombic (a≠b≠c, α=β=γ=90)4.

Tetragonal (a=b≠c, α=β=γ=90)5.

Rhombohedral (a=b=c, α=β=γ)6.

Hexagonal (a=b≠c, α=β=90, γ=120)

7.

Cubic (a=b=c, α=β=γ=90)

3

5

6

1
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Atomic parametersTheh atomici coordinatesdi
are referedf d to theh threehbasis
b i vectorsof the unit cell: a, b, c, and are called x,y,z.

ComplicatedSimpleCheiron School 20109

Scattering from a material unit

Incoming field directionof electromagnetic wavea ePhase difference (δ) between
density at 0 and at r.

X = AO − BC
= −r ⋅ s 0 + r ⋅ sInterference patterns arise due to thespatial separation of scattering centers.

δ=

2π

λ

r ⋅ (s - s 0 )= 2πr ⋅ r ∗r = 2 sin θ λ∗Cheiron School 201010

Scattering from a material unitN

The observer sees a wave:

F (r ) = ∑ A j exp(2πir ⋅ r * )*j =1

Scattering from an atom:()

f (r * ) = ∫ ρ (r ) exp 2πir ⋅ r * drVf (r * ) = T [ρ (r )]In other words, the scattering from an atom is thef
off the atomic electron densityy
Fourier transform

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Scattering from a molecule

ASSUMPTION: Chemical bonding does not exist!
The molecular density is a simple sum of non‐interacting atomic densities, ie theIndependent Atom Model (IAM).

ρ mol (r ) = ∑ ρ j (r - rj )
atoms

⎛
⎞
*
Scattering amplitude
l d = ∑⎜⎜ ∫ ρ j (r - rj )exp2πir ⋅ r dr ⎟⎟j =1 ⎝ V⎠(

N

Continuous function

=

N

() (

*
*
f
r
exp
2
π
i
r
⋅
r
∑j
j

)

)

j =1

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Atomic scattering factor

( )

(

)

f atom r * = ∫ ρ (R ) exp 2πiR ⋅ r * dRVf atom (0 ) = ∫ ρ (R )dR = Z
VCheiron School 201013

Now we need single crystals!
N

() (

*
*
f
r
exp
p
2
π
i
r
⋅
r
∑j
j

)

j =1

Scattered radiation amplitude = convolution of unit cell density and Fouriertransform of lattice function
Infinite 3D lattice:+∞L(r ) = ∑ δ (r − nr )n = −∞(1 +∞F (r ) = FM (r )
δ r * − rH*∑
V h ,k ,l = −∞**Cheiron School 2010)14

Reciprocal lattice
1. Only diffracted waves at certain directions where: r* = r*H (Bragg reflections)2. The crystal can be considered an “amplifier” of the signal (Interference)
3. The directions of scattering are given by the reciprocal lattice
Summarized in the Laue equations:

Scattering only takes place in
certain directions:a ⋅ r * = h; b ⋅ r * = k ; c ⋅ r * = lr * = ha* + kb* + lc* = H

This is the reciprocal lattice!
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The structure factor

The crystallographic structure factor is the amplitude of scattering from the unit cell, ie.
the Fourier transform of the unit cell electron density, and it is given by:

F (H ) = ∑ f j exp(2πiH ⋅ r j )∗ T jj!!!! This is the basis of all crystallography !!!!!

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The structure factor

The structure factor can be written in different ways, emphasizing certain
characteristics of it:
F (H ) = ∑ f j exp(2πiH ⋅ r j )N“Normal”jN

Complex noF (H ) = ∑ f j exp(2πiH ⋅ r j ) = AH + iBHj

AH = ∑ f j cos(2πH ⋅ r j )NjN

BH = ∑ f j sin (2πH ⋅ r j )NjFhkl = ∑ f j exp(2πi (hx j + ky j + lz j ))⎛B ⎞F (H ) = F (H ) expp(iϕ H ), ϕ H = arctan⎜⎜ H ⎟⎟⎝ AH ⎠j

Phase

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Historical backgroundnat

Basic structure factor:

F (H ) = ∑ f i (H ) exp(2πiH ⋅ ri )Ti
i =1

f (S ) = ∫ ρ at (r ) exp(2πiS ⋅ r )dr
Born‐Oppenheimer approximation

ρ free (r ) = ∫ ρ (r − u ) p(u )du = ρ static (u ) ⊗ p(u )f atom (r *) = f static (r *)q(r *)In theory, X‐ray scattering gives info about the FULL electron distribution!

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Braggs Lov
X-ray diffraction ”sees” and interacts with the electrons.
To have diffraction in certain directions we requireqa largeg degreegof constructive
interference:

THEREFORE SINGLE-CRYSTALS!
Ewald
sphere2d H sin θ = λaLatticei planeslc(2,2,0)Cheiron School 2010b

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Cold gas

SINGLE CRYSTAL

X‐ray source
Rotation

POWDER

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Normal structure determination (IAM)”Result” of a crystallographic
structure determination is the usual
ORTEP‐drawing which shows the
NUCLEAR PROBABILITYDENSITY FUNCTION
The IAM model is only concernedwith the core electrons/nuclear
density. It works surprisingly well!

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Interpretation of data

ReflectionsR
fl ti
are observedbd overmany frames→Integration:
Reveals the content of the unit cell

Characteristic distance: Gives the dimensionsof the unit cell

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Problem almost solved ☺
I (H ) ∝ F (H )2

IIntensitiest
iti are measureddusing X-ray detectors ☺
MeasuredUnknown!

Crystallographic
(
)
(
)
(
)
F (H) = ∑ f exp 2πiH ⋅ r = F H exp iϕ PHASE PROBLEM!!!
atomerjjj

Solution devised by Karle and Hauptman in 1985 – Nobel prize!
Short version: direct methods use probabilities of sets of structure factors –built in to software and automated – black box

1
ρ uc (r ) = ∑ F (H ) exp(−2πiH ⋅ r )Vc
The electron density reveals ALL about thestructure – we want to know this!
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The IAM model1⎛ Fobs (H )⎞()
Difference Fourier maps: Δρ (r ) =FH−⎟ exp(− 2πiH ⋅ r )⎜∑calc
Vuc H ⎝ k
⎠

1967

2010

Can we quantify this additional information?
YES
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Expanding the atomic model

Stewart (1969): Derives Generalized scattering factors from orbital products.
Suggests that the effects are significant enough to be measured.

Hansen‐Coppens
pp(1978):() Builds on the work byy Stewart,, but uses a modified
version where the angular part of the density functions are based on
spherical harmonic functions :ρ = core + valence + deformation (bonding, lone pair etc)
l max

l

l =0

m =0

ρ (r ) = Pc ρ c + Pv ρ v (κr )κ 3 + ∑ Rl (κ ' r )κ '3 ∑ Plm ± d lm ± (θ , φ )Cheiron School 201025

Characteristics of spherical harmonics
Hansen‐Coppens structure factor:nat

F (H ) = ∑ f i (H ) exp(2πiH ⋅ ri )Ti →
i =1

l max
⎛
⎞
3
F (H ) = ∑ ⎜⎜ Pc f c ,i + Pv f v ,i (H κ )κ + 4π ∑ ∑ Plm ± i l jl (H κ ') d lm ± (β , γ )⎟⎟ exp(2πiH ⋅ ri )Ti
i ⎝l = 0 m = 0 ,l⎠Ex. ff‐orbitalic
Exorbitalic functions(octopoles):Chargehis moved
d fromfredd
to green if the multipole ispopulated in the lsq.
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The radial function

Nodeless, normalized, density
functions:

n +3
ζ
3

Rl = κ 'l(nl + 2)!

(κ ' r )n exp(− κ ' ζ l r )ln=4; ζ=4; κ = 1.0

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Refinement approach

A general entry to charge density modeling
As always, complete an IAM structure determination (SHELX or other)
Study the Fourier difference maps from the above – do they show excess
bonding density or other characteristic trends? If yes, the option is there.
Specialised software is required: XD (WinXD), Valray, Jana2006...
3(xyz)+6(U’s)+1(κ)+25(anharm)+25(P3(xyz)
6(U s) 1(κ) 25(anharm) 25(Plm’s)s) = 60!! parameters available per atom

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Any improvement of the fit?

IAM model: ΔF = (F(obs)‐F(calc))
Lone pair density

0.8
0.6
O1

0.4
0.2
Bonding density
‐0.2
‐0.4
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Improvement

Multipole model: ΔF = (F(obs)‐F(calc))
0 15
0.15
0.10
0.05
0
‐0.05
0 05
‐0.10

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How suitable is your compound?

The suitability factor
The structure factors gives information on the FULL electronic distribution, while
chemical bonding is primarily due to changes in the valence electrons!
The relative core scattering intensity per unit volume determines the suitabilityof the crystal for charge density determination:

S = u .c .
Examples:a p es:Formamide
Tetracyanoethylene
Co‐dimerS8
Th‐complex

V2n∑ core,i

4.7
3.7
0 35
0.35
0.26
0.07

i

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The ideal way to charge density data
The ideal experiment has the following ingredients
Significant diffraction to high‐angle ((sin(θ)/λ)max >1.0 Å‐1)
Spherical or face‐indexed crystal
Very low temperature (liquid He‐flow or closed‐cycle cryostat)High incoming intensity, eg. synchrotron X‐ray source with high brilliance
Neutron data for H‐parametersp(p(position and anisotropic
p adp)
p)
High accuracy as well as high precision (as in low R(int) from Sortav etc.)
low precisionhighaccuracyhighprecision low
accuracy

How can we determine what we have?
Comparison with theoretical calculations
Comparison of data from different sources and using different equipment,ddetectors...
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The actual way to charge density data
Are all CD studies following this route? Some numerical facts:
Maximum sin(θ)/λ > 1.0 Å‐1:
Using He‐cooling:
Using synchrotron or rotating anode:
Combined with neutron data:

93% (recently, this is easier to achieve)
30% (expensive & problematic)
27% (beam time limitations & problematic)~10% (large crystals necessary – SNS??)

73%

16%
Spackman et alal., Acta Cryst
Cryst. 2008,
2008 A64,
A64 465‐475
465 475

11%

Sealed tube Synchrotron Rot anode

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The hydrogen problem…no reasonable estimate of the charge-density parameters
can be obtained without an adequate description of thermal
motion.‘ti‘ (Koritszansky(K it
k & Coppens,C
2001)
X-raysUCHOFe Co NiHydrogen characteristics:
Veryy weak scatterer
No core electrons => asphericity shift
Potential Energy Surface

Neutrons
1
2

54
58

56

62

57
60

WEAK
STRONG
Strong hydrogen bonds have partial covalency
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Hydrogen dynamics•

Accurate multi‐temperature X‐ray studies combined with difference Fourier
methods
th d may ill
illustratet t theth protont movement,
t i.e.
i reveall details
d t il off theth potential
t ti lenergy surface

A. Parkin, K. Wozniak, C. C. Wilson Cryst. Grow. Des. 2007, 7, 1393. Reproduced courtesy of Andrew Parkin, University of Glasgow.

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Combining neutrons with X-rays

•
•
•
•

Grow LARGE crystals (volume > 1 mm3)
Preferably use very low T (~15 K) – matching for
X anddN
From N exp: refine X and Uij values for ALL
atoms.
From X exp: refine an IAM with Uij for heavy
atoms.

IPNS

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Combining neutrons with X-rays•••

A scalingof Uij between X and N is necessary:
Blessing
CompoundTemp# HAprogram UIJXN<ΔU> by Robert<Uii,X/Uii,N>Uij(X) = k*Uij(N) + ΔUij
OxHy
15
0.00076(45)0.97(21)
Apply
thiscomplex
scaling to the H atom Uij(N).109100.00031(23)
0.97 (7)
HowCu(OH2)6(SO4)2(NH4)2
well should X and N compare – somebenchmarknumbers:
Ni(NO2)2(NH3)4
Catalytic triade
KHC2O4

9
28
11

5
50
7

0.00018000018 (27)0.00091 (80)0.00030 (3)0.96
096 (5)
1.01 (2)0.99 (1)

MoO(O2)(HMPA)(dipic)_-Glycine
1-Methyluracill-Alanine
2-Methyl-4-nitroaniline
MBADNP
Acetamide

28
23
21
23
10
20
23

27
5
9
6
11
21
4

0.00134000134 (40)0.00083 (86)0.00021 (24)0.00176 (165)0.00150 (139)0.00223 (189)0.00273 (190)1.32
132 (1)
0.80 (11)1.03 (8)0.62 (5)1.28 (21)1.81 (26)1.62 (15)

25
12
122
10
100
19
114
19
123
7
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School 2010 1

0.00094 (72)0.00113 (103)0.00124 (109)0 00186 (127)0.001860.00049 (39)0.00008 (7)1.17 (28)
1.23 (10)0.83 (4)
1 16 (9)1.16
1.02 (5)1.00 (3)Co(OD2)6SO4
Xylitol
Adenosine
Nit(SMe)Ph
Alloxan
Beryllium metal37

What can we do without neutron data
Utot((H)) = Uint + Uext
atomicvibrations

Rigid body
motionMadsen, A. Ø., Spackman, M. et al:
J. Appl. Crystallogr. 2006, 39, 757‐758 & Acta Cryst. 2008, A64, 465‐475
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Hydrogen multipoles?
Isotropic U(H):( ) only
l b
bond
dd
directedd multipolesllCheiron School 201039

Hydrogen multipoles?
Anisotropic U(H):( ) allll dipoles
d l andd quadropoles
d
l

l=0
l=1
l=2

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Synchrotron
issuesCheiron School 201041

Oblique incidence correction

PathP
th lengthl th (L) throughthhphosphor depends on
incidence angle, αIncoming X‐ray

I ∝L

Phosphor

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Oblique incidence correction
Incompletelabsorptionbin theh phosphor
h h creates an incidencedangle
l dependentddcorrection. The used approach is adopted from work by Wu et al (1) in correctionof APS data using this expression:

I corr=I meas1− T⎛ ln (T ) ⎞1 − exp⎜
⎟
cos
α
⎝
⎠

The absorption was measured as a functionof wavelength on a sample piece ofphosphor. This is used to obtain Tbar values
for the the correction
G. Wu, B. L. Rodrigues, P. Coppens, J. Appl. Cryst. 2002, 35, 356.

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Oblique incidence correction

The effect of the oblique incidence is very significant. Two series of datacollected to highg scatteringg angle.
gRelative correction up to 20%!!

Perpendicularincidence
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Oblique correction now in SADABSThe recurrent use of the oblique program has led the developers at Bruker toincorporatepthis into their software,, despite
p the use of a flood field correctionthat should take care of the problem.

I corr=I meas

I corr
1− Tf⇔=
I meas 1 − (1 − f )1 cosα
⎛ ln (T ) ⎞1 − exp⎜⎟⎝ cos α ⎠f = 1 ‐ T = detector efficiency

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Refinement details
A rough guide to charge density modeling
Initiate with a refinement of the positional and thermal parameters using thehigh‐angle data, typically maintaining a cut‐off of at least 0.8 Å‐1.

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Scattering factor functions

http://xrayweb.chem.ou.edu/notes/crystallography.html#scatter

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Refinement details
A rough guide to charge density modeling
Initiate with a refinement of the positional and thermal parameters using thehigh‐angle data, typically maintaining a cut‐off of at least 0.8 Å‐1.
Fix these parameters. Now start with a κ‐refinement of κ and Pv.
Make clever local coordinate systems & find any possible non‐crystallographic symmetry elements to introduce constraints on the choice ofrefinable Plm’s.

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Local coordinate system

Choose for each atom
two neighbours to define
the
h LCS
LCS.
If, for instance, a mirror
planeliis ⊥ z, thenthselection rules apply:
1
1
2
m
2/m
222
mm2
mmm

any
any
2ԡz
m⊥z
2ԡz, m⊥z
2ԡ 2ԡy2ԡz,2ԡ2ԡz, m⊥ym⊥z, m⊥yyCx

all (l(l,m,±)m ±)(2 l,m, ±)(l,2μ, ±)(l,l‐2μ, ±)(2 l,2μ, ±)(2 ll,2μ,+),2 +) (2l+1,2μ,–)(2l+1 2 )(l,2μ,+)(2l,2μ, +)Cheiron School 201049

Refinement details
A rough guide to charge density modeling
Initiate with a refinement of the positional and thermal parameters using thehigh‐angle data, typically maintaining a cut‐off of at least 0.8 Å‐1.
Fix these parameters. Now start with a κ‐refinement of κ and Pv.
Make clever local coordinate systems & find any possible non‐crystallographic symmetry elements to introduce constraints on the choice ofrefinable Plm’s.
Introduce the multipoles in a sequence of refinement steps and examine theparameters for significance.
Eventually include also κ’, although this may not converge or give large
deviations from unity → KRMM

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Multipole model bias (KRMM)

Multipole model introduces a bias
bias, due to limited flexibility of the radialfunction.
Solution could be more sophisticated model – several sets of multipole
functions but this is not feasible in most cases!
functions,Kappa restricted multipole model (KRMM) method:
Large spread of experimental values of κ’ for similar atoms
B d on tabulatedBasedb l d valuesloff κ’’ fromf
MM off theoreticalhi l structure factorsfDirect import of these leads to improved and more stable fits

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Validating the (final) model
Hirshfeld rigid bond test

‐3
3 Å2)
||uaԡ|‐|u| | bԡ| ∼ 0 (<10(Hirshfeld, F. L. Acta Crystallogr. Sect A. 1976, 32, 239‐244.
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Validating the (final) model
Analysis of the residual density in the unit cell – program jnk2RDA
A number of systematic
errors were introduced andtheir fingerprint in thefractal distribution plot wasclarified.

Course of refinement

κ‐errorMeindl, K.; Henn, J. Acta Crystallogr. Sect A. 2008, 64, 404‐418.
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Validating the (final) model
Analysis of the residual density in the unit cell – program jnk2RDA

w/o extinction

with extinction

Meindl, K.; Henn, J. Acta Crystallogr. Sect A. 2008, 64, 404‐418.
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Validating the (final) model
Shell like behaviour of2
2FF∑ obs ∑ calc

Zhurov, V. V.; Zhurova, E. A.; Pinkerton A. A. J. Appl. Crystallogr. 2008, 41, 340
340‐349.
349.
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Is it worth the effort?

1.
2.
3.
4.
5
5.

An analytical expression for the total electron density distribution! From thiswe can get the following properties:
Outer electronic moments: dipole moments, quadrupole moments etc
d‐orbital populations for metal atoms
Electric field gradients on the nuclear position – correlate with measured
quadrupole splitting energy from Mossbauer spectroscopy
Estimates of the crystal lattice energy
Better U‐values!!!

Q
QTAIM
analysis!!
l i !!
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QTAIM analysis of the electron density
The fundamental improvement is that the experimental electron density is now
described analytically and can thus be analysed using the Atoms in Molecules
theory by Richard Bader:Zero‐flux surface ∇ρ(r)⋅n = 0 separates atomic basin which all contain one
and only one nuclei.
Critical points where ∇ρ=0.
∇ρ=0At the critical points, there are four possibilities for the algebraic sum of λi:‐3, ‐1, +1 or +3 corresponding to a nuclei, a bond, a ring, and a cage.
Atomic properties can be calculated by integration over the atomic basin
basin, forinstance the atomic charge: q(Ω ) = N (Ω ) − Z = ∫ ρ (r )dτ and volume: V (Ω ) = ∫ dτΩΩAll these quantities can be compared to theoretical values!
Difficult to calculate for network structures, heavy atom molecules etc.

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R. F. W. Bader; Atoms In Molecules, Cleardon Press, Oxford 1990
Cheiron School 201058

Atoms in Molecules partioning
In 2D

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Characterization of chemical interactions
Charge depletion∇2ρ(r)( )> 0Charge accumulation∇2ρ(r)( )< 0Ionic bonds

Covalent bonds

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Source function
ρ (r ) = ∫ −
V

= ∑∫−
Ω Ω

39.3%

1
∇ 2 ρ (r')dr'
4π r − r '1∇ 2 ρ (rr')drr' ≡ ∑Ω S (r, Ω )4π r − r '

47.9%

44.7%

S(bcp, 2C) = 78.6%

S(bcp, 2C) = 89.4%

EthaneEthylene

S(bcp, 2C) = 95.8%

Acetylene

Bader, R.F.W., Gatti, C. (1998), Chem. Phys. Lett. 287, 233‐238.
C. Gatti in ”The Quantum Theory of Atoms in Molecules: from Solid State to DNA and Drug Design”, C. Matta
and R. Boyd (Eds.), Wiley‐VCH.
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Energy densities•

Energy densities can be estimated from the topological analysis using
the expression by Abramov:23
31
3π 2 ρ (rbcpp ) 5 3 + ∇ 2 ρ (rbcpp )G (rbcpp ) =
10
6

( )

•

Based on results from neutron diffraction studies and the local virial
theorem, Espinosa proposed a formula to derive the dissociation energyof a HB:h 2
V (rbcp ) =∇ ρ (rbcp ) − 2G (rbcp )4m

E HB = 0.5 ∗ V (rbcp )Abramov, Yu. A. Acta Cryst. 1997, A53, 264.
Espinosa, E. et al., Chem. Phys. Lett. 1998, 285, 170.

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Bond classifications•

A classification based on these properties was suggested:
LL atoms

ρ(rbcp)∇2ρ(rbcp)

G(rbcp)/ρ(rbcp)

H(rbcp)Open shell (C‐C)

Large

Negative<1

Negative

Polar bonds (C‐O) Large?

>1

Negative

Closed shell (HB)

Small

Positi e
Positive>1

Positi e
Positive

HH atoms

ρ(rbcp)∇2ρ(rbcp)

G(rbcp)/ρ(rbcp)

H(rbcp)Shared

Small

∼0

<1

Negative

Donor acceptor

Small

Positive∼1

Negative

Macchi, P. et al JACS, 1998, 120, 13429.

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Conclusions
Take home message
Take‐home• The experimental charge density method can give otherwise unattainable
information on a wide variety of chemically interesting compounds• Synchrotron data can be imperative• A whole new array of properties are available through the use of QTAIM• VeryV llow ttemperaturet
iis stronglytl recommendedd d• Charge density modeling is ”just” advanced structure refinement – but it
requires extreme care in data acquisition and data analysis

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