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Charge Density from X-ray Diffraction. Methodology
Ignasi [email protected]
Master on Crystallography and Crystallization, 2012
Outline
I. Charge density in crystals
II. The multipolar refinement
2
III. Methodology
IV. Example
I. Charge density in crystals
II. The multipolar refinement
3
III. Methodology
IV. Example
Charge density from X-ray diffraction
In X-ray diffraction we “see” the electrons.
From the electrons, we find where the nuclei are.
There is a lot of information on the electron
4
There is a lot of information on the electron distribution about the interaction of the atoms in the crystal (chemical bonds, atomic and molecular charges, intermolecular interactions…).
Charge density: - We are interested in ALL the charge distribution in the crystal, not just the nuclei.- We are going to use X-ray diffraction for mapping the electron distribution inside the crystal.
( ) ( ) ( )∑ ⋅⋅⋅=a
aa iTf rHHHHF π2exp)(
Atomic position in the crystal lattice
The intensity I of the reflection H depends on the structure factor F
( ) ( )2HFH ∝I
The structure factor depends on the crystal structure
5
Atomic form factorScattering from the atomType of atom
Atomic coordinatesThermal vibration
Electron density: probability of finding an electron at r
( ) ( )∫⋅⋅= rrH rH def i
aaπρ 2
In X-ray diffraction, the form factor depends on the electron shell
Cusp at the nuclear position
6
Fast decay from the nuclear position Spherical symmetry
(for an isolated atom)
The independent atom approximation• ρa(r) in the crystal = ρa(r) for the independent (isolated) atom• The environment of the atom has no effect on its ρa(r).
100
1000
10
1515
( ) ( )∫⋅⋅= rrH derHf iIA
aIA
aπρ 2
7
0,0 0,5 1,0 1,5 2,00,01
0,1
1
10
0 0.5 1 1.5 20
5
10
0
Pt
20 sth
With the IAM approximation:• faIA(H) has spherical symmetry.• faIA(H) depends only on the chemical element.
Ex. Phosphorus
The interaction of an atom with its environment perturbs its ρa(r).In a crystal:• ρa(r) does not present spherical symmetry.• ρa(r) is different for each atom in the asymmetric unit.
In the IA approximation, we suppose this
In the crystal, we have this
8
Ex. P-atom in H3PO4
Isodensity surfaceρ(r) = constant
Bonding effects: Deviations of the IA approximation that introduce systematic errors in the crystal structure.
Ex. Systematic underestimation of X-H bonding distances
Correct ρa(r)
Correct ra
IA ρa(r)
IA ra(atomic position from
9
X H
(atomic position from X-ray diffration)
Bonding effects introduce small errors in ra and Ua in crystal structures from X-ray diffraction.
A crystal structure from X-ray diffraction presents bonding effects.
Total electron density: Superposition of ρa(r)
)()( aarrr −=∑ρρ )()( a
IA
a
IA rrr −=∑ ρρ
Independent Atom Model (IAM) of the electron density: Superposition of ρa
IA(r)
10
)()()( rrr ρρρ ∆+= IA
Deformation density: All the information about the effect of any kind of interatomic interactions (chemical bonding, intermolecular interactions…) is here.
2D representation of the electron density
Isodensity contour: Constant electron density
Electron density is always positive
11
Contours are truncated at a ρmax just to make the interpretation easier
positive
)(rρ )(rIAρ )(rρ∆= +
12
)(rIAρ
)(rρ∆
)(rρ It is the target of a charge density determination
It is determined from the crystal structure. It is the main contribution to the X-ray structure factors.
It cannot be determined from the crystal structure.Its contribution is ~3% of the X-ray structure factors.
ρ>0 ρ<0
Accurate structure factors
Lower experimental error
Deformation density can be observed
13
Experimental ρ Experimental deformation ρ
I. Charge density in crystals
II. The multipolar refinement
14
III. Methodology
IV. Example
Generalized scattering factor:• Specific for each atom in the asymmetric unit.• Spherical symmetry can be relaxed.
( ) ( ) ( )∑ ⋅⋅⋅⋅−⋅=a
aaT
aa ipf rHHUHHHF ππ 2exp2exp),( 2
Atomic parameters:• Specific for each atom in the asymmetric unit.• Determined from a least squares fit against the experimental structure
15
( ) ( )∫⋅= rrH rH deppf i
aaaaπρ 2,,
( ) ( ) ( )aIAaaa prp ,, rr ρρρ ∆+=
against the experimental structure factors.
( )rρ ( )rρ
• Atomic electron densities superpose in the bonding regions.• There are many partitions of the electron density in atomic contributions.•Assignment of portion of electron density to an specific atom does not mind that electrons “belong” to the atom.• Instead of atoms : pseudoatoms
16
( )rαρ ( )rβρ ( )rαρ ( )rβρ
Two possible partitions of the total electron density in atomic electron densities
( )rρ ( )rcoreρ ( )rvalρ=
= +
+
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• Very large peaks at the atomic positions.• Not perturbed by the atomic environment• It does not depend on the atomic parameters
( )rcoreρ
( )rvalρ • Diffuse in the space.• Perturbed the atomic environment.• It depends on the atomic parameters.
10
100
1000
10
15
fa(H) is decomposed in core and valence contributions
( ) ( ) ( )aaaaa prp ,, val,core, rr ρρρ += ( ) ( ) ( )aaaaa pfHfpf ,, val,core, HH +=
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0,0 0,5 1,0 1,5 2,00,01
0,1
1
0 0.5 1 1.5 20
5
20 sth
Only the valence fa(H) is perturbed by the atomic environment• Core contribution: Bonding effects are negligible• Valence contribution: Bonding effects can be significant
Ex. P-atom
As the atom is larger, the contribution of the valence electron density to the atomic scattering factor• takes place at lower angles.• is a smaller fraction of the total scattering.
This puts a limit to the size of the atoms whose electron density can be determined.
19
TotalCoreValence
C S Ti Zr
The suitability factor: estimate of the suitability of a given crystal for X-ray charge density analysis.
∑= 2
core
cellunit
N
VS As S is smaller, the
determination of the experimental electron density is more challenging.
20
S falls as the size of the heaviest atom in the crystal increases.
Schiøtt, Int J Quant Chem, 96 (2004) 23
The Hansen-Coppens model of the electron density
( )γβκ
πκ
,'
4)()(max
0val core lm
l
l
l
lmllm
lva Y
HjPi
HfPHff ∑∑
= −=
+
+=H
( ) ( ) ( ) ( ) ( )∑∑++=max
33 ,''l l
YrRPrPr φθκκκρκρρ r
Kappa termWith spherical symmetry
Multipolar termWithout spherical symmetry
21
( ) ( ) ( ) ( ) ( )∑∑= −=
++=max
0
3val
3 ,''l
l
lmlmllmv YrRPrPr φθκκκρκρρ corer
( ) ( ) ( ) ( ) ( )rNYrRPrP v
l
l
l
lmlmllmv val
0
3val
3max
,'' ρφθκκκρκρ −+=∆ ∑∑= −=
r
=
lm
v
a
P
P
p ',κκNumber of electrons
Expansion/contraction
Non-spherical deformation
Electrons in the valence shell of the neutral atom
( ) ( )∑∑= −=
max
0
3 ,''l
l
l
lmlmllm YrRP φθκκ
( )γβκ
π ,'
4max
0lm
l
l
l
lmllm
l YH
jPi∑∑= −=
Multipolar term
Expansion in spherical harmonics.
22
Spherical harmonics are similar to atomic orbitals.
Each deformation consists in transferring electrons from the negative to the positive regions of the spherical harmonic.
Coppens, X-Ray Charge Densities and Chemical Bonding, 1997
or( )2)()(∑ − HH calobs kFF ( )222 )()(∑ − HH calobs kFF
)()( HH IAMobs kFF −
The multipolar refinement is the minimization of
In the multipolar refinement, differences between experiment and IA model are taken as the contribution of ∆ρ to the X-ray scattering.
23
( ))()(∑ −H
HH calobs kFF ( ))()(∑ −H
HH calobs kFF
with
( )γβκ
πκ
,'
4)(),(max
0val core lm
l
l
l
lmllm
lvi Y
HjPi
HfPHfpf ∑∑
= −=
+
+=H
( ) ( ) ( ) ( )∑ ⋅⋅⋅=i
iiiical iTpf rHuUHHF π2exp,,
and
( )rρ Two kind of parameters
Structural parameters
Atomic parameters
lmv PP ,',, κκ
αα Ur ,
)(rρ∆
24
• Bonding effects: Contribution of ∆ρ is already in the structural parameters.• Structural parameters have more weight in the least squares refinement.• Contribution of ∆ρ must be removed from the model before starting the multipolar refinement.
lmv PP ,',, κκ
Before the multipolar parameter, bonding effects must be removed.
∑ ⋅−
−=H
rHHHH iicalobsres eeFF
kVcal πϕρ 2)()()(
11
Residual maps
25
Structure with bonding effects. ∆ρ is in the structural parameters.
Bonding effects corrected. ∆ρ appears in the residual maps.
After multipolarrefinement. ∆ρ is in the atomic parameters
1-
lim
Å0.17.0sin −
~λ
θ
High order reflections only present core contribution.
C
High order refinement: Reflections above a threshold angle only
Method 1 for removing bonding effects
26
TotalCoreValence
At the end of the refinement:IA model without bonding effects.
The choice of the threshold value is a compromise between • Minimization of valence shell contribution.• Data set large enough.
H
Hydrogen atoms have no core
Hydrogen atoms do not contribute to the high order reflections.
Reliable structural parameters for H-atoms cannot be obtained from X-ray diffraction data.
27
TotalCoreValence
ρ(r) around H-atoms is inaccurate.
Choices:a) Stay with the approximate ρ(r) for the H-atoms.b) Use additional information for a correct estimation of structural parameters of H-atoms
Method 2 for removing bonding effects
( ) ( ) ( )∑ ⋅⋅⋅=i
iii iTb rHuUHF π2exp,
Neutron scattering length. Depends on the nucleus
Neutron diffraction: Elastic scattering of neutrons by the atomic nuclei
X-ray diffraction F’s Neutron diffraction F’s
28
F.T.
F.T.Electron distribution
Nuclear distribution (structure)
Nuclear distribution (structure)
Independent atom approximation
Bonding effects are related to deviations from this approximation
No bonding effects in the neutron diffraction structure
( ) ( ) ( )∑ ⋅⋅⋅=i
iii iTb rHuUHF π2exp,
Scattering power does not depend on the atom size.
No special treatment is required to H-atoms.
29
atoms.
Anisotropic thermal parameters for H-atoms
Coppens, X-Ray Charge Densities and Chemical Bonding, 1997
I. Charge density in crystals
II. The multipolar refinement
30
III. Methodology
IV. Example
)(rρ∆ It is ~3% of the X-ray structure factors.
We need very high quality structure factors
• High quality crystals• Accurate experiment
• High data redundancy• High order data• Longer time per frame• Use of synchrotron radiation
1.- The experimental structure factors
31
• Use of synchrotron radiation• Complex data reduction
• Integration of measured intensities• Accurate correction of absorption• Accurate equivalent merging
DREADD package for data reduction oriented to charge density studies
http://classes.uleth.ca/200903/chem4000a
http://www.synchrotron-soleil.fr/
Low temperature• In some cases, helium cooling (~ 20 K) is needed.
( ) ( ) ( )∑ ⋅⋅⋅⋅−⋅=a
aaT
aa ipf rHHUHHHF ππ 2exp2exp),( 2
• Ua decrease with T, F(H) increases with T
32
ρres for an IAM at 120 K ρres for an IAM at RT
• T increases, ∆ρ becomes more diffuse
Use of short wavelength / high energy radiation
θλ sin2H=
Total
Zr
• The size of the Ewald sphere increases.
• Reflections shift to lower angles
More reflections to be measured
33
TotalCoreValenceMore reflections with
significant contribution of ρval
Energies up to 100 KeV (0.015 Å) have been used in charge density studies
Aslanov, Crystallographic Instrumentation, 1998
2.- The independent atom model
If using neutrons, X-ray and neutron thermal parameters should match.
• Both X-ray and neutron diffraction data must be collected at the same temperature (same cell parameters)• Experimental errors such as absorption and extinction are absorbed into the Ua.
34
Anisotropic differences
Good agreement
Temperature differences
Coppens, Acta Cryst A84 (1984) 184
Approximate thermal vibration.
The treatment of the hydrogens
Approximate ρ(r) around H-atoms.
35
Anisotropic thermal parameters for H-atoms allow a detailed model of ρ(r) around these atoms.
Munshi Acta Cryst. A64 (2008) 465
Accurate ρ(r) around H-atoms.
2.- The multipolar refinement X-X methodUiso for H-atoms
)(HobsF
High order refinement
Multipolar refinement
Step 1:• Low angle reflections• Pv, κ, κ’, Plm
Step 2:
36
ra and Ua non-H atomsH-atoms:• Bonding distance fixed to the average neutron bonding distance• Uiso estimated from the bonded atom.
Step 2:• All reflections• ra, Ua, Pv, κ, κ’, Plm• H-atoms:
• Bonding distance as in the high order refinement.• Uiso refined.
( )rρ
X-N method
)(, HobsXF )(, HobsNF
Crystal structure determinationMultipolar refinement
2.- The multipolar refinement
37
ra and Ua all atoms
Step 1:• Low angle reflections• Pv, κ, κ’, Plm
Step 2 (optional):• All reflections• ra, Ua, Pv, κ, κ’, Plm• H-atoms fixed
( )rρ
X-(X+N) method
)(, HobsXF )(, HobsNF
Crystal structure
Multipolar refinement
Step 1:• Low angle reflectionsHigh order refinement
2.- The multipolar refinement
38
Crystal structure determination
ra and Ua H-atoms
• Low angle reflections• Pv, κ, κ’, Plm
Step 2:• All reflections• ra, Ua, Pv, κ, κ’, Plm• H-atoms fixed
( )rρ
High order refinement
ra and Ua non-H atoms
Thermal ellipsoid scaling
1.- Get transformation by comparing thermal ellipsoids of non-hydrogen atoms. Minimization of
Neutron
( )2
∑ ∆−− UUU NX q
39
High order X-ray 2.- Apply transformation to hydrogen-atom thermal ellipsoids
UUU ∆+= NX q
( ) ( ) ( ) ( ) ( )∑∑= −=
++=max
0
3val
3 ,''l
l
l
lmlmllmv YrRPrPr φθκκκρκρρ corer
• A non-spherical atom has spatial orientation.• A local coordinate axis must be defined for each atom.• In most cases, axis are oriented along chemical bonds.
40
( ) ( ) ( ) ( ) ( )∑∑= −=
++=max
0
3val
3 ,''l
l
l
lmlmllmv YrRPrPr φθκκκρκρρ corer
Ex. Parameters
O-atom 9 3 15 = 27
P-atom 9 3 24 = 36
ra, Ua Pv, κ, κ’ Plm
Restrictions are often introduced in order to reduce the number of parameters in
41
Restrictions are often introduced in order to reduce the number of parameters in the multipolar refinement:• Same k and k’ for atoms with similar chemical environment.• Same population parameters for atom with the same chemical environment.• Fix total charge of ions and molecules.• Local symmetry conditions.
Besides this:• Neutrality: not a constraint but a condition that must fulfill the total electron density.
( ) ( ) ( ) ( ) ( )∑∑= −=
++=max
0
3val
3 ,''l
l
l
lmlmllmv YrRPrPr φθκκκρκρρ corer
Some Plm can be set to zero by local symmetry conditions.Ex. P in the H3PO4
+11P 20P +21P
42
3-fold symmetry along axis z
0≠lmPKK ,1,0,13 −== iim
+22P30P
31P −32P +33P8 from 24 Plm parameters nonzero
( ) ( ) ( ) ( ) ( )∑∑= −=
++=max
0
3val
3 ,''l
l
l
lmlmllmv YrRPrPr φθκκκρκρρ corer
Electron configuration of the valence shell
Shape of the deformation terms
( ) ''','
4)(),(max
0val core iffY
HjPi
HfPHfpf lm
l
l
l
lmllm
lvi ++
+
+= ∑∑= −=
γβκ
πκ
H
Anomalous scatteringAnharmonic thermal vibration
43
( )( )( )2222 )(,)(∑ −H
HHH calcalobs FgFkyF
( ) ( ) ( ) ( )∑ ⋅⋅⋅=i
iiiical icTpf rHuUHHF π2exp,,,
Extinction model
vibration
Angle dependence of the scale factor
A lot to things to check, correct, optimize…
Phases are very important• Completely different models can present almost identical |Fcal|• In the case of acentric crystals, restrictions must be imposed for avoiding unphysical models
44
Ex. ADP
Residuals in the P-O-P plane.
Q = Molecular charge of NH4.
111 222P)2(1.0+=Q 3.0+=Q 5.0+=Q
7.0+=Q 9.0+=QPérès Acta Cryst A55 (1999) 1038
3.- Validating the model
• Good statistics (R factors, goodness of fit…)• Residual maps• Experimental deformation maps
( )∑ ⋅−−=∆H
rHHH HH iiIAM
ical eeFeF
VIAMcal πϕϕρ 2)()(
exp )()(1
ρ∆
45
resρ∆
resρ∆ expρ∆
3.- Validating the model
• Uiso(H-atom) > Uiso(bonded atom)• Rigid Bond Test
221 Å001.0<∆−∆
46
2∆1∆Bonding distance between non-hydrogen covalently bonded atoms remains constant through thermal vibration.
Vibration amplitudes along the bond should be equal for the bonded atoms.
Comparison with theoretical calculations
In most cases, agreement is qualitative because of
• Measurement errors in the experimental electron density.
• Method-inherent errors in the calculations.
3.- Validating the model
47
Best results observed for • Periodic ab initio calculations• Purely organic molecular crystals
Ex. P-nitroaniline• Experimental• Theoretical
Volkov Acta Cryst A56 (2000) 332
The electron density is the starting point for further analysis
Deformation density
Topological analysis
48
analysis
Laplacian
Electrostatic potential
I. Charge density in crystals
II. The multipolar refinement
49
III. Methodology
IV. Example
2-methyl-4-nitroanilineHoward et al. J Chem. Phys. (1992) 97 5616-5630
C7H8N2O2Ia (Monoclinic)
V 696.9 Å3
T 125 KRefl. measured 7348Refl. independent 3743Refl. observed 2045
50
Refl. observed 2045 (sin(θ)/λ)max 1.08 Å-1
Rint(F2) 0.032
Local axis
C,N,O l = 1, 2, 3H l = 1
H-atoms- d(X-H) fixed- Uiso fixed- κ=1.16
51
I Crystal structureII High orderIII Kappa without structural parametersIV Kappa with structural parametersV Multipole without local symmetry restrictionsVI Multipole with local symmetry restrictionsVII VALRAY multipolar model
Final results
Residuals Deformation Dipole moment
( ) rrrµ dT∫= ρ
Kappa 48 D
µ
52
Rigid bond test∆max = 0.0012 Å2
Qualitative agreement with theoretical calculations
Multipolar 25 D
Theory 9 D
Large increase attributed to crystal field effects
2-methyl-4-nitroaniline revisitedWhitten et al. J. Phys. Chem. (2006) 110 8763-8776
53
X-rays NeutronsV 698.89(8) Å3 689.1(4) Å3
T 100 K 100 KRefl. measured 26425 1482Refl. independent 5683 848Refl. observed 5055 847(sin(θ)/λ)max 1.27 Å-1 0.66 Å-1
Rint(F2) 0.025 0.013
Final results
Residuals DeformationDipole moment
Experimental 11.3 D
Theory
( ) rrrµ dT∫= ρ
54
Excellent agreement with theoretical calculations
Theory
Periodic 11.7 DCrystal geom. 9.0 DIsolated 7.1 D
Increase in dipole moment due to crystal field effects.
R=0.016 for 5055 obs. refl. and 383 pars.
Final remarks- The methodology for the multipolar refinement is well established for
crystals of small organic molecules in centric space groups.
- Multipolar refinement can be performed in crystals with transition metals. However there is some complexity involved in the treatment of these atoms.
- If hydrogen atoms are relevant, anisotropic vibration parameters for these atoms are needed.
55
these atoms are needed.
- In the case of acentric space group, there is a risk of model indeterminacy. Restrictions should be imposed with care.
- Neutron diffraction is very helpful for the treatment of the thermal vibration (H-atoms, anharmonicity).
- Theoretical calculations are useful for validating the experimental electron density. However, theoretical electron densities are not necessarily better than the experimental ones.
