2 CHARGE DENSITY STUDY ON A LITHIUM SULFUR YLIDE
2.5 The IAM Model
2.5.1 Basics of X-ray Scattering and IAM Modeling
Due to the proportions of atoms, X-ray beams instead of visible light have to be used to gain a picture of the contents of crystals. This is quite problematic because no lens-systems appropriate for the construction of an X-ray microscope are available.
Thus, a direct view at atoms and electron distributions is impossible. Nevertheless, the diffraction patterns discovered 1912 by Friedrich, Knipping, and Laue[173] permit another way to gain an insight into electron distributions in crystals. The positions and intensities of the reflections from elastic scattering of the X-rays are connected to the electron density (atomic positions and their interactions) by Fourier trans-formation. The kinematic theory of scattering by Born[174] provides a connection between the X-ray intensities I and the scattering amplitudes F(H) (Eq. 2-2).
Eq. 2-2: I ~F
( )
H 2Although the kinematic theory is strictly valid only for crystals thinner than 1 µm, the scattering of typical crystals (d ≥ 0.1 mm) is still well described due to their mosaicity.
Thus, application of the dynamic theory,[175,176] which takes all energy exchanges into account, is not needed.
The scattering amplitudes are the Fourier transforms of the static electron density in the crystal (Eq. 2-3).
Eq. 2-3: F
( )
H =∫
Vρ( ) (
r exp2πiHr)
drThe expression of the structure factor F(H) can be approximated by describing the electron density of the crystal as a summation over the nuclei-centered atomic densities (Eq. 2-4).
This implicates that the ED (electron density) or CD (electronic charge density) ρ(r) can be derived directly from the experiment. In a real experiment, this is subject to some restrictions. The observed structure factors are affected by experimental errors.
Only a finite number of reflections can be collected, thus Fourier truncation errors occur. Last but not least, the phase information is lost in the course of the experiment due to the measurement of intensities, i.e. squared structure factors. These limitations make it necessary to model the ED. The calculated structure factors Fcalc
of the parameterized models are compared to the experimental ones Fobs and refined by least-squares and gradient-descent methods.
Eq. 2-5 gives an expression for the structure factor, where fq(H) is the scattering factor and tq(H) the temperature factor.
Eq. 2-5:
( )
=∑ ( ) ( ) (
π)
The duration of the diffraction experiment largely exceeds the period of thermal vibrations of atoms. Therefore, the electron density ρ(r) is the time-average of the atomic electron densities, which can be described as pure vibrational states[177] within the convolution approximation.[178] In most cases a harmonic approximation with nuclear displacement vectors u and the MSDA (mean square displacement amplitude) U is sufficient (Eq. 2-6).
Eq. 2-6:
( ) ( ) ( )
⎟Sometimes anharmonic effects are not negligible. In those cases the Gram-Charlier-expansion can be used to model small anharmonicities.[179]
Within the IAM (independent atom model) approach the crystal electron density is described as superposition of spherical atomic densities. This is commonly referred to as promolecule density ρpro(r) (Eq. 2-7) and leads to a spherical scattering factor (Eq. 2-8).
Eq. 2-7:
( )
=∑ ( )
′The IAM parameters for modeling the crystal electron density are the three fractional coordinates (x, y, z) and the six anisotropic displacement parameters Uij for non-hydrogen atoms or one isotropic displacement parameter Uiso for hydrogen atoms, respectively.
The IAM approach neglects all density deformations, either from atomic interactions or from lone-pair densities. For heavy atoms with predominant core electron scattering this approximation is valid. For lighter atoms, however, the determined parameters, especially the temperature factors, are biased.[180] This is particularly true for hydrogen atoms. Their whole electron density is shifted towards the bonded atom and hence the interatomic distances are determined too small. This has to be compensated by shifting the hydrogen atoms to distances derived from neutron diffraction experiments.[181]
Figure 2-14: Resolution dependence of X-ray scattering amplitudes by (a) K- and (b) L-shell of carbon.
The bias introduced for all other elements can be reduced significantly if only high-order data are used. At high Bragg angles the influence of polarization effects from bonding is minimized, because predominantly contracted densities (core densities) contribute to these reflections. The decrease of scattering power at high resolution affects valence densities more than core densities (cf. Figure 2-14).
The application of a high-order refinement with sinθ/λ ≥ 1.00 Å results in reliable structural parameters for all non-hydrogen atoms. The results obtained are in good agreement with those from other methods, e.g. neutron diffraction.[163,182-184]
The nucleus-centered finite multipole formalism by Hansen and Coppens,[185] applied later on during this charge density study, is, as the name suggests, based on an atomic description of the crystal as starting model. The independent atoms of this description, the so-called promolecule, do not and should not account for the bond densities, which will be treated by the multipole model. Thus, bias-free results of the independent atom modeling are of great importance.
In all refinements the function M
(
pi,k)
(Eq. 2-9) is minimized using statistical weights wH (Eq. 2-10).Eq. 2-9:
(
,)
=∑ [ obs( )
2− calc( )
2]
2 =min
H
HkF H F H
w k
p M i
Eq. 2-10: wH =1σH2
The weighting scheme is not refined but kept fixed at statistical weights, because the program for multipole modeling comes with no feature for adjustment of the weighting scheme during refinement. Hence, the weights from IAM, which are optimized for the spherical atom refinement, would introduce bias into the multipolar modeling.
The results of the refinements can be verified by comparison of the calculated to the observed structure factors. Commonly used criteria are the residuals R1 (Eq. 2-11) and wR2 (Eq. 2-12). If the model is refined against F2, the wR2 is more significant.
Eq. 2-11:
Additionally, the GoF (goodness of fit), a figure of merit showing the relation between deviation of Fcalc from Fobs and the over-determination of refined parameters, is calculated (Eq. 2-13).
The structure was solved with direct methods using SHELXS.[186] All IAM refinements were performed on F2 with SHELXL.[187]
In order to introduce as little bias as possible in the IAM from bond densities, the positional and anisotropic displacement parameters of all non-hydrogen atoms were refined with high-order data (sinθ/λ ≥ 1.00 Å-1), exclusively. After this step all parameters of the non-hydrogen atoms were kept fixed by means of the AFIX 1 instruction in SHELXL.[187]
The hydrogen atoms were localized in a second step via difference Fourier syntheses from low-order data (sinθ/λ ≤ 0.50 Å-1). Refinement of the hydrogen atom parameters was achieved with the same cut-off without distance restraints. An iso-tropic riding model was applied for the thermal motion parameters, with Uiso
constrained to 1.5 Ueq of the pivot atom for the methyl groups and 1.2 Ueq of the pivot atom for all others. After converged refinement the hydrogen atoms were shifted along their bonding vectors to neutron diffraction distances of 1.085 Å.[181]
Figure 2-15 shows the IAM model obtained after convergent refinement.
Figure 2-15: ADP representation of the asymmetric unit of [(thf)Li2{H2CS(NtBu)2}]2 (4); all except the methylene hydrogen atoms are omitted for clarity, ADP's are depicted at the 50 % probability level.
Only statistical weights wH were used to avoid the introduction of bias into the multipole refinement. This should be kept in mind when comparing the quality criteria shown in Table 2-7 to those of standard X-ray structures.
Table 2-7: Quality criteria after the IAM refinement of [(thf)Li2{H2CS(NtBu)2}]2 (4) with high-order data only.
no. of reflections 41445 R1 (I > 4σ(I)) 0.0784 no. of reflections
(I > 4σ(I))
28200 wR2 (all data) 0.1177
no. of parameters 511 GoF 7.638
The high GoF value is due to a systematic underestimation of the uncertainties of the reflections at higher Bragg angles. Hence, the model quality seems to be too bad compared to the data quality and the GoF is far from its optimal value of 1. Therefore, for the discussion of all geometrical features a threshold of three esd's (experimental standard deviations) is used.