CHARGE DENSITY STUDY
how & whyyFinn Krebs Larsen
Aarhus University,
Denmark

Cheiron School 20091

Cheiron School 20092

Aarhus University,

Denmark
Department
DepartmentofofChemistry
Chemistry
9 9Profs.
Profs.
2222Associate
AssociateProfs.
Profs.
4 4AssistantfAssistantProfs.
Profs.
3838techs/Adm
techs/Adm
6060Post
Postdocs
docs~100
StStudents
d t~100Phd
PhdStudents~80~80Masters
Mastersstudents
students>200>200undergrad.
undergrad.majors
majorsCheiron School 20093

More than just pretty pictures
In 2D

Cheiron School 20094

Outline
Part 1••••
Part 2Modelling in crystallographya) Atomic Thermal Motionb) Electron Density Distribution
E
Experimentalil requirementsi& challenges
h ll•

Selected examples
p from our libraryy of structures•

1) Transition metal complex. Jahn‐Teller effect.
Comparison with theory. Topological analysis••

2) Thermoelectric materials. CoSb3•

3) Hydrogen bonding•4)) Coordinationdpolymersl•5) Chrome wheel complexes•

Conclusions
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5

F (H ) = ∫ ρ (r ) exp(2πiH ⋅ r )dr• The structure factor is the Fourier transform of
periodic electron density.
ythe p• Descrete values for H = h∙a* + k∙b* + l∙c* forinteger values of h,h k,k and l in reciprocal space

Cheiron School 20096

Experimental Structure Factor|Fobs| can be deduced by applying corrections to
the integrated reflectivity, R(H), which can be measured in adiffraction experiment.
R(H)( ) = k ∙ L ∙ P ∙ Tabsorb∙Eextinc∙|Fobs|||2
K : scale factor
L P : Lorentz‐Polarization
L∙P
L
P l i i correctioniTabsorb : absorption correction
Eextinc : extinction correction
CRYSTA OGRAPHYCRYSTALLOGRAPHY
Solve the structure and find the phase Æ Fobs
Cheiron School 20097

Total density :1⎛ Fobs (H ) ⎞ρ (r ) =⎟ exp(− 2πiH ⋅ r )⎜∑Vuc H ⎝ k ⎠Pyrene,y, C16H10

Hazell, Larsen and Lehmann
Acta Cryst. (1972). B28, 2977‐2984
Cheiron School 2009

8

MEM charge density
Nishibori, Sunaoshi, Yoshida, Aoyagi, Kato,
Takata and Sakata
Acta Cryst. (2007). A63, 43‐52

Si at 100 K
measured at beamline BL02B2 at Spring‐8

Diamond at 100 K
measured at beamline BL02B2 at Spring‐8
p gCheiron School 20099

Modelling
IAM ( independent atom model )nat

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

f (S ) = ∫ ρ at (r ) exp(2πiS ⋅ r )dr

AtomicScattering factor : fi
Coordinates : ri = xi, yi, zi
Temperature factor : Ti
Cheiron School 200910

Atomic Thermal Motion• Harmonic , isotropic• Debye approximation•

Harmonic, anisotropic2<u 2>)(sinθ/λ)2Tharm,iso=exp(‐8πpjCheiron School 200911

Atomic Temperature factor
Tj(H) = Tj,harmonic(H)∙T(H) Tj,anharmonic(H)Tj,harmonic = Tj,aniso = exp(‐2π2(U11h2a*2 + U22k2b*2 + U33l2c*2+ 2U23klb*c* + 2U13hla*c* + 2U12hka*b*))
Tj,anharmonic = exp(‐2π2 (1‐ i4/3π3Cjklhjhkhl +2/3π4Djklmhjhkhlhm))
cubic and quartic Gram‐Charlier expansion
Cheiron School 200912

Thermal parameters
Beryllium hcp structure

Nuclear density distribution

Anharmonic motion

•
•
•
•
•

Atomic Temperature factor T(H)harmonic, isotropic :harmonic anisotropic :harmonic,
Anharmonic, cubic
Anharmonic,, quarticqCheiron School 2009

Number of thermal parameters
1
6
6 + 106 + 10 + 15
13

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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Structure Refinement• Least‐squares minimisation of• Σ w(Fobs2 – Fcalc2)2= minimum• Refine scale factor,
factor positional,
positional and thermal atomic
parameters.
• Then check the difference Fourier !

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The IAM model1⎛ Fobs (H )⎞()
Difference Fourier maps: Δρ (r ) =−FH⎟ exp(− 2πiH ⋅ r )⎜∑calc
Vuc H ⎝ k
⎠

1967

2008

Cheiron School 200916

Expanding the atomic model
Stewart (1969): Develops generalized scattering factors from orbital products
and envisions their use in modeling of experimentally measured X‐ray
sstructure
uc u e factors
ac o s as eelectron
ec o dedensitys y sstudies.
ud es Sugges
Suggestss thata thee eeffects
ec s aaree
significant enough to be measured.
Hansen‐Coppens (1978): Builds on the work by 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

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

3

l =0

Cheiron School 2009

m =0

17

Sources
Part 1

Part 2

Webpages on CDDB
Philip Coppens:”X‐ray charge densities
and Chemical Bonding
Bonding”

htt //
http://www.chem.au.dk/~uorg/cddb
h
dk/~
/ ddb

http://xd.chem.buffalo.edu/
Cheiron School 200918

Characteristics of spherical harmonics
Hansen Coppens structure factor:
Hansen‐Coppensnat

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
Cheiron School 200919

Density Parameters fors, p, d, and f‐orbital functions
Number of charge density parameters in multipole model
Monopole :Dipolesp:
Quadrupoles :Octopoles :Hexadecapoles :

1
3
5
7
9

In total pr atom :9 ‐ 25

Expansion or contraction of the atomic electron density aredescribed with Kappa
pp parametersp
via the radial function
Cheiron School 200920

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

Cheiron School 200921

Experimental implications
Besides a scale factor electron density modelling requires refinement ofmany parameters
t per atomt
Positional parameters
Thermal parameters
Charge density parameters3
6 Æ 31
9 Æ 25

Atomic site symmmetry may reduce the number of parameters, but often
approximately 25 parameters per atom is needed in accurate studies.
In least‐squares
least squares refinement 10 reflections per parameter is advocated,
i.e. 250 reflections per atom. Extensive data sets are necessary for charge
density studies.
10,000 independent accurate reflections for 40 atom structure!

Solution : Low‐temperaturepandshort‐wavelength synchrotron radiation
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Tharm,iso=exp(‐8π=exp( 8π2<uj2>)(sinθ/λ)2
The Debye approximation for atomic thermal motion as function of temperature gives a hint.
T Æ 0 K results in temperature factor Æ 1.
<u2>
2

Approximate intensity gain factors
by lowering temperature

Temperature

Lowering temperatureSinθ/lθ/lI(100(K)/I(290
)/ (
K))

I(10
( K)/I(100
)/ (
K))

1.0

100

5

1.3

2 500

10

1) gives more significant reflections and2) increases accessible sint/l range
3) makes anharmonic contributions insignificant
insignificant,Cheiron School 200923

Thermal Diffuse Scattering (TDS)

R(H)meas(T) = R(H)(T) (1+α)α is correction factor for first order TDS.
Can only be calculated
if elastic constants are known

0 0
0.10

TDS is at minimum at low temperatures Æ
betterbee accuaccuracy
acy in measurements
easu e e s
Cheiron School 200924

Advantages of cryocrystallography
1) Reduced atomic thermal motiona) better resolutionb) less TDSc) less anharmonicity

more data
better data
simpler model

2) Reduced radiation damage
Th crystall lastsThelllongeri theinh beamb

Th colderThe
ld theth better!
b tt !
Cheiron School 200925

And then of course preferably use

SYNCHROTRON RADIATIONHigh brilliance, tunability and generally short wave legths allows :use of :tiny crystals for less absorption and extinction generally, andchoice of :short wavelength below absorption edges for heavy atoms

Cheiron School 200926

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...
Cheiron School 200927

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
Cheiron School 200929

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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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.

Cheiron School 200931

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.

Cheiron School 200932

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 200933

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

Cheiron School 200934

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.
Cheiron School 200935

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.
Cheiron School 200936

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.
Cheiron School 200937

3d-electron
3dpopulation analysis

Copper Tuttons’ Salt

Cheiron School 200938

Copperpp Tuttons’ Salt
(ND4)2Cu(SO4)2∙6D2O

Deuterated ammonium and sulphate groups are hydrogenbonded to Cu2+ ion, which is octahedrally hydrated bydeuterated water molecules.
Geometry changes with temperature and pressure.
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2+
2
Cu (D2O)6 chromophore

2 ‐ (D
Cu2+( 2O))6 chromophore
h
h
shows
h
Jahn‐Tellerhll distortion.
dTwo Cu‐O distances are significantly shorter than the third :Cu‐O7=2.081(6) Å, Cu‐O8=2.242(7) Å, and Cu‐O(9)=1.927(6) ÅCheiron School 200940

d‐orbitals are energy split in an octahedral field
Cu2+ has 9 3d
3d‐electrons
electrons Æ Expectation that 3dxy, 3dxz, 3dyz and 3dz2 orbitals arefilled wih electron pairs, while 3dx2‐y2 is only partly populated
The long distance Cu‐O8 must correspond to a fully occupied
3 dz2 pushing O8 away from Cu leaving 3 dx2‐y2 partly occupied .
This explains the shorter distances between Cu and O7 and O9.
The z direction is thus along Cu‐O8, x and y directions along Cu‐O7 and Cu‐O9.
Cheiron School 200941

Model static deformation densities

z‐axis vertical

x‐y plane

Figgis, Iversen, Larsen and Reynolds, Acta Cryst. 1993, B49, 794‐806
Cheiron School 200942

3d electron population analysis
3d‐electron
The static deformation maps of the deuterated Copper Tuttons’
Tuttons salt show:a) Water oxygen lone pairs directed towards Cu2+b) Flow of charge from SO42‐ to Cu2+c) Deficiency of charge density in 3dx2‐y2 orbitald) The z axis is directed from Cu2+ towards O8
Valence function populations are in good accord with theoretical calculations
Chandler et al. J. Chem. Soc. Faraday Trans. 1992 2, 88, 1961‐1969.
3dxy
3dxz
3dyz
3dx2‐y2
3dz2

9K
2.14(2)2.02(2)2 01(2)2.01(2)
1.26(2)2.07(2)

Theory
1.95
1.96
1.96
1 96
1.23
1.98

Total

9.50

9.08

Cheiron School 200943

Water molecules

Static deformation

Ab initio theoretical deformation

Almost quantitative comparison
Cheiron School 200944

Is it worth the effort?
What else? Topological analysis!
 When the experimental electron density is described analytically, its
topology can be analysed in the same way as theoretically calculated
densities are analysed using the Atoms in Molecules QTAIM (quantum
theoretical atoms in molecules) theory byRichard BaderTHE INTERNATIONAL SERIES OF MONOGRAPHS ON CHEMISTRY:ATOMS IN MOLECULES – A Quantum Theory
Clarendon Press Oxford – 1990
Topological analysis opens possibilities for comparisons with a wide variety ofexperimental measurements
Cheiron School 200945

Topology
p gyof the C2H4, ethylene molecule

Gradient vector field∇ρ(r)Attractor basins, critical points

Contour map of the charge density
Cheiron School 200946

Topological analysis of the electron density
9 Zero‐flux surface ∇ρ(r)⋅n = 0 separates atomic basinsbasins,which each contains one and only one nucleus.
9 Critical p
points where ∇ρ=0.
ρ
9 At the critical points, there are four possibilities for the
algebraic sum of the eigenvalues , λi of ∇2ρ:‐3, ‐1, +1 or +3 corresponding to a nucleus, a bond‐, aring‐, or a cage‐critical point.
9 Atomic properties can be calculated by integration over
the atomic basin, for instance :gandthe volume
the atomic chargeq (Ω ) = N (Ω ) − Z = ∫ ρ ( r ) d τΩCheiron School 2009

V (Ω ) = ∫ dτΩ47

R. F. W. Bader; Atoms In Molecules, Cleardon Press, Oxford 1990
Cheiron School 200948

Atoms in Molecules partioning
C6N7(NH2)3 – melem ‐ in 2D

Cheiron School 200949

Physical and chemical characterizations
 Basic crystallographic analysis gives information on chemical composition
composition,
structural geometry and atomic thermal motion.
 MultipolarM l i l analysis
l i can determinedi d‐orbitald bi l populations
l iffor metall atoms.
 Topological QTAIM analysis furthermore allows calculation of
1.
2.
3.
4.
5.

Atomic volumes and charges
Characterization of chemical interactions – chemical bonds
Outer electronic moments: dipole moments, quadrupole moments etc
Electric field gradients on the nuclear position – to be correlated withmeasured quadrupole splitting energy from Mossbauer spectroscopy
Estimates of the crystal lattice energy
Cheiron School 200950

Characterization of chemical interactions
the negative Laplacian,Laplacian ‐∇∇2ρ(r)
Charge depletion−∇∇2ρ(r)( )< 0Charge accumulation−∇∇2ρ(r)( )> 0Ionic bonds

Covalent bonds

Cheiron School 200951

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.
Cheiron School 200952

Other descriptors
The source function, f. Inst. for characterization of hydrogen bondsρ (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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MORE EXAMPLES
first CoSb3

All made with XD – could be done withother programs like JANA,
JANA Valray,
ValrayMollyy etc.

Cheiron School 20095454

THERMOELECTRIC
MATERIALS
CoSb3

Thermoelectrics

Peltierl i effect
ff observedbdb
byJ. Peltier in 1834

Thermoelectrics
Thermoelectric Figure of Merit
Electrical conductivity
Seebeck coefficientTh
Thermall conductivity
d ti itTemperature

I summary, we are looking
Inl ki forf a;
Phonon Glass,Electron Crystal

State of the Art Thermoelectric Materials
State‐of‐the‐Art

Ba8Ga16Ge30

Doped CoSb3•

Doping atoms – f.f inst. La, Ba, Sn andd Ni – improve theh thermoelectrichlFigure of Merit primarily by decreasing the thermal conductivity.

•

Theory: Bertini and Gatti J.Chem. Phys. (2004) 121, 8983‐8989•

DopingDi atoms can replacelat atomic
i positionsii
or go iinto iinterstitial
ii lposition (0,0,0).

•

Doping atoms can rattle in voids in the structure decreasing the latticecontribution to the thermal conductivity.

•

Structural implications tested by neutron and X‐ray synchrotron powder
diffraction.

Co(1‐x) NixSb3 Neutron diffraction
Neutron powder diffraction:
Investigate the Ni doping in the Co(1‐x) NixSb3 system.
Ni and Co are almost indistinguishable by X‐ray diffraction,
they differ a factor of ~16 in neutron diffraction.
X-raysUCHOFe Co Ni

Neutrons
1
2

54
58

56

62

57
60

ElementxyzGroup VG
Transition metalInterstitial

0.00
0
00
0.25
0.50

0.33
0
33
0.25
0.50

0.16
0
16
0.25
0.50

Co(1‐x) NixSb3 Neutron diffraction
Disagreement on stoichiometry between
neutron powder diffraction and atomic absorption.
Interstitial sites 0.5 0.5 0.5 site appears to beoccupied by Ni giving a formal stoichiometry of
Ni0.4(Co7.8Ni2.2)Sb30

The table shows the Ni content in %,and the mass density
density.
SampleFramework Framework + interstial Atomic absorption Neutron density Measured density
0.22(1)0.25(2)0.30
7.643(5)7.62(7)
Co7Ni3Sb30
Co8Ni2Sb30

0.13(2)0.15(3)0.19

7.643(4)7.9(1)

Co9Ni1Sb30

0.04(2)0.05(3)0.08

7.633(3)7.74(7)

Co10Sb30

0007.646(4)-Additionally 1%‐2% Ni can be added due to impurities

CoSb3 Impurities
The high resolution and high intensity of the synchrotron radiation
makes it possible to refine content of impurity.

Identified Phases:
CoSb3 a=9.02Å
fraction in weight % 96.4(3)CoSb2 a=6.49Å b=6.37Å c=6.52Å fraction in weight % 3.6(1)CoN
a=3 57Å
a=3.57Å
fraction in weight % < 0.1
01

Conclusion – substituted Skutterudites
Neutron powder diffraction revealed:‐ A deviation between intended Ni content and actual Ni content.
‐ Ni on theh interstitiall site.
Synchrotron powder diffraction:‐ Impurity phases could be quantified.
Follow‐up
p work on purep
CoSb3:‐ MEM density study from powder synchrotron X‐ray data.
‐ Benchmark single crystal X‐ray diffraction study including multipole
modelling, and topological analysis.

Spring 8 MEM results
Spring‐8
Co : blue
Sb : yellow

Ohno, Sasaki, Nishibori, Aoyagi, Sakata and Iversen,
Physical Review (2007) B76, 064119.

Spring 8 MEM densities
Spring‐8

(I)

[010]

[010]

(II)

[100]

[100]

Sb‐Sb (I) 2.86 Å 0.50 eÅ‐3
Sb Sb (II) 2.98
Sb‐Sb
2 98 Å 00.35
35 eÅ‐33

Co

Co‐Sb

2.53 Å 0.52 eÅ‐3

CoSb3 iso‐surface
iso surface plot and schematical model

Iso‐surface at 0.40 eÅ‐3

Co8Sb12
Pentagonal Dodecahedron

Grey ball lies in centre of the void potentially occupied by rattling atoms

Pure CoSb3 Skutterudite

Combined theoretical and experimental diffraction study.
study

CoSb
C
Sb3, Mr=424.2,
M 424 2
cubic symmetry, space group Im‐3
a = 9.0165(1) Å at T = 10 K
Th
Theoryb GattibyG tti ett al.,l Mil
MilanoSingleg crystaly
X‐rayy diffraction studyy includingg multipole
p modelling,g andtopological analysis. Based on data collected at
Aarhus University (Larsen et al.),al )University of Toledo (Pinkerton et al.) and at synchrotrons
APS (Yu‐Sheng Chen) andSPring‐8 (Kunihisa Sugimoto)Conventional X‐ray diffractometers
HUBER, scintillation detector – Bruker APEX2 CCD
Radiation: Sealed tube AgKa ‐ Rotating anode MoKa

Aarhus HUBER

Toledo APEX2

Advanced Photon Source Synchrotron
Argonne National Laboratory – Chem Mat Cars beamline

Radiation: λ = 0.4201 ‐ Detector: Bruker AXS CCD

Superp Photon ringg 8GeV
SPring-8

High Brilliance X-ray Beam
By Synchrotron Radiation

Data comparisons
λ

No
N
meas

U i
Unique<N>

ExtincEti(1 3 0)

Rint

a Å

i θ/
sinθ/
λ

140

9.0082(7)1.250.5608

33077

1071

30.9

50.5%

0.032

20

-

9.0204(1)1.320.7107

59113

1305

45.3

37.6%

0.034

APS (08)16

15

9 0268(1)9.0268(1)

1 19
1.19

0 44280.4428

13553

2294

59
5.9

Spring-810-9.0206(3)1.67

0.4117

56969

2559

22.3

1.5%

0.037

Spring-8
MEM

10

powd
er

1.8

0.4207

9256

1957

4.2

0

0.018

D t
Data

T/
K

Xtall
Xt
Ø μm

C ll
Cell

Huber

11

Toledo

0 043
0.043

CoSb3 – 49 model parameters
cubic symmetry, space group Im‐3 (# 204)

Sb: 24g, m-symmetry
IAM : 0 y z, u11 u22 u33 u12; 15 anh.
CD : Pv P11± P20 P22± P31± P33± P40 P42± P44± κ’ κ’’
Co: 8c, -3 symmetry,yy x = y = z = 0.25;
u11= u22= u33 ; u12= u23= u13
IAM : u11 u12
CD : Pv P20 P40 P43± κ’ κ’’
Besides scale factor and extinction parameter
In total: 38 (Sb) + 9 (Co) + 2 = 49 parameters.

Parameter comparisonsAarhusToledoAPS (08)

Spring-8 (09)Spring-8(MEM)x, y, z(Co)¼, ¼, ¼¼, ¼, ¼¼, ¼, ¼¼, ¼, ¼¼, ¼, ¼

X(Sb)

0

0

0

0

0

Y(Sb)

0.33542(2)0.33544(2)0.335222(4)0.335225(5)0.33510(1)Z(Sb)

0 15780(2)0.15780(2)0 15778(2)0.15778(2)0 157864(4)0.157864(4)0 157858(5)0.157858(5)0 15805(1)0.15805(1)0 00218(9)0.00218(9)0 00229(9)0.00229(9)0 00235(2)0.00235(2)0 00228(1)0.00228(1)0 00289(3)0.00289(3)0.00005(2)0.00005(1)0.00097(3)Coordinates

Thermalparameters
U11(C )
U11(Co)U12(Co)U11(Sb)0.00181(6)0.00192(7)0.00200(1)0.00200(1)0.00205(1)U22(Sb)0.00233(6)0.00249(7)0.00240(1)0.00246(1)0.00393(2)U33(Sb)0.00202(6)0.00216(6)0.00217(1)0.00220(1)0.00246(2)0.00009(1)0.00011(1)0.00000(1)U23(Sb)

Topological analysis of the multipole
modelsSpring‐8
dist

∇ 2ρ

ρ
eÅ
Å-33

Å

theory
ρ

eÅ
Å-55

∇ 2ρ
eÅ
Å-33

Co‐Sb

2.53

0.41

1.82

0.39

1.28

Sb‐Sb
Sb
Sb (I)

2.86

0.33

0.56

0.50

‐1.61
1.61

Sb‐Sb (II)

2.97

0.33

0.51

0.47

‐ 0.86eÅ
Å-55

Density profiles for Sb‐Sb(long) interaction

Figureg
2a. ρ pprofile of Sb‐Sb ((long)g) interaction.

3

Lap(rho) / e*Ang.^-5

2

1

Figure 2b. ∇2ρ profile of Sb-Sb (long) interaction.
0

-1

-2

-3
0.988

1.088

1.188

1.288

1.388

1.488
r / Ang.

1.588

1.688

1.788

1.888

1.988

Maps of the negative Laplacianin the Sb4 plane
Theory (Gatti,Milano)
Spring‐8 (2009)

State of affairs
Single crystal results from Aarhus,
Aarhus Toledo
Toledo, APS and Spring‐8 – positional andthermal parameters agree perfectly, density parameters almost.
Most thermalMhl parameters off SSpring‐8i 8 powder
d MEM study
d deviated ivery
significantly from single crystal results.
Electron Diffraction measurements of low‐order structure factors would bedesirable.
Comparison of topological measures for experiment and theory indicate
B3LYP/3‐21G as best theoretical model.
Calculating Bader atomic charges, energies and topological classification ofbonds, etc. is next.

Benchmark CoSb3 study.
Some Conclusions
Conventional data: Compare well – although rotating anode
gives relatively more significant weak intensities
intensities.
Synchrotron data. VASTLY increased number of uniquereflections . Strongly decreased extinction.
extinction
Spring‐8 powder diffraction (BL02B2 beamline).
beamline) MEM; very
good qualitative electron density description, but
Spring‐8
Spring8 single crystal study (BL02B1 beamline). Most extensive
and superior for multipolar and topological analysis.

Conclusions
l i• The experimental charge density method can giveotherwise unattainable information on a wide variety ofchemically interesting compounds• Synchrotron data can be imperative• A whole new array of properties are available
through the use of QTAIM• Very low temperature is strongly recommended• Charge density modeling is ”just” advanced structurerefinement – but it requires extreme care in dataacquisition and data analysis
Cheiron School 200979

Acknowledgements
Aarhus Charge Density People:
B Brummerstedt
BoBt dt Iversen
I
Jacob Overgaard
Henrik F. Clausen
Mads R. V. Jørgensen
Rasmus D. Poulsen
Mette S. Schmøkel

Cheiron School 200980