A Comparison of Electron Density from Hirshfeld-Atom Refinement , X-Ray Wavefunction Refinement and Multipole Refinement on Three Urea Derivatives

MM HAR (+/-) [a] HAR [a] HAR (+/-) [b] HAR [b] O1-C1 1.2559(5) 1.2541(4) 1.2538(4) 1.2537(4) 1.2534(4) C1-N1 1.3500(5) 1.3497(4) 1.3498(4) 1.3498(4) 1.3499(4) C1-N2 1.3513(5) 1.3520(4) 1.3522(4) 1.3520(4) 1.3523(4) N1-C11 1.4488(6) 1.4498(4) 1.4498(4) 1.4498(5) 1.4498(5) N1-H1 1.009 1.016(9) 0.998(9) 1.01(1) 0.988(9) N2-H2A 1.009 0.99(1) 0.97(1) 0.99(1) 0.97(1) N2-H2B 1.009 1.008(9) 1.000(9) 1.011(9) 1.002(9) C11-H11A 1.067 1.101(9) 1.100(9) 1.093(9) 1.094(9) C11-H11B 1.066 1.09(1) 1.10(1) 1.08(1) 1.09(1) C11-H11C 1.066 1.095(9) 1.090(9) 1.10(1) 1.09(1) [a] HAR and HAR(+/-) were performed using the blyp/cc-pVTZ level of theory. [b] HAR and HAR(+/-) were performed using the blyp/cc-pVDZ level of theory. The bond distances N-H and C-H were set to average distances from neutron diffraction in the multipole refinement, but freely refined in both variations of HAR.


Introduction
Our interest in urea derivatives was initially fuelled by their hydrogen-bonding patterns.The molecules analysed here mostly form N-H … O type hydrogen-bonded networks, where each oxygen atom is involved in two or three hydrogen bonds.These interactions are responsible for the arrangements of molecules in the solid state.Moreover, these interactions have an influence on molecular geometry: urea derivatives are a good example where molecular solid-state and gas-phase structures differ considerably, as was already confirmed for urea. 1 The substitution pattern of the urea derivatives named below leads to another aspect, which is the competition of weak intermolecular C-H … p and classical N-H … O hydrogen bonding.
In order to study these effects on the level of their electron density distribution, the single-crystal X-ray structures of the three urea derivatives N-methylurea, N-phenylurea and N,N9-diphenylurea (named here murea, phurea and dphurea) were re-determined.Conventional crystal structure determinations were known from the literature. 2 There are some earlier experimental and theoretical studies on adducts of phenyland diphenylurea with nitro-substituted acids, 3 halides and oxyanions, 4 as well as reports of new co-crystals of methylurea, 5 but charge density studies have not been reported to date.
Last but not least, urea compounds provide ideal examples and test cases.The small size of the molecules investigated enabled us to compare in a reasonable time three different approaches to model the non-spherical electron density distribution, which is not taken into account in conventional X-ray single crystal structure analysis: the Hansen-Coppens multipole model (MM), 6 Hirshfeld-atom refinement (HAR) 7 with and without the presence of a cluster of surrounding point charges and dipoles and ultimately X-ray wavefunction refinement (XWR). 8The latter two approaches are computationally rather demanding.The performance of these electron density models is reported.

Refinement procedures
(Sigma-Aldrich).High-resolution synchrotron data were measured at beamline F1 (at a temperature of T = 100 K) and D3 (at T = 8 K) of the storage ring DORIS III at the HASYLAB/DESY, Hamburg (Germany).Wavelengths of 0.6000(2) Å and 0.5166(2) Å were used for phurea/dphurea and murea, respectively.The XDS program 9 was used for integration of frames and for preliminary data reduction of all data sets.Final sorting, merging and application of the oblique incidence correction 10 for the datasets were performed with the program SADABS. 11Further experimental details and crystallographic data are presented in Table 1 and in the ESI. 3 Multipole refinement X-ray structures were solved with SHELXS and initially refined using SHELXL97. 12The resulting Independent Atom Models (IAM) for all three investigated compounds were used as input for multipole refinement with the XD2006 program package, 13 in which the electron density model of Hansen and Coppens is implemented. 6Multipole refinements were carried out on F 2 , using all data except the unobserved intensities with negative values, which were excluded from calculations.The multipole model can be summarized as follows: local mm2-symmetry was introduced for carbon atom C1 in all structures (for the atomic numbering scheme see Fig. 1); local m-symmetry was applied in all planar fragments such as phenyl rings for phurea and dphurea as well as for nitrogen atoms N1 and N2 in all structures; threefold symmetry was imposed on the C11 methyl atom for murea.To allow a detailed description of density rearrangements due to hydrogen-bonding, symmetry restrictions were absent for the oxygen atoms in the urea moiety, since atom O1 is involved as a multi-centered acceptor.Furthermore, multipole populations of chemically equivalent atoms were constrained to be the same: C15, C16, C25, C26 were constrained to C12, C13, C22, C23; H13...H16 as well as H23...H26 were constrained to H12 and H22 for phurea and dphurea, respectively; H11B, H11C were constrained to H11A for murea.All non-hydrogen atoms were treated up to the hexadecapolar level, whereas bond-directed dipoles were the highest multipoles applied for all H-atoms.Moreover, k parameters were refined independently, except one common k for H atoms, that was kept constant at a value of 1.13.In the case of murea, one k parameter for the N2 atom was kept fixed in the final stages of refinement because of high correlation.For all multipole models, bond lengths to H atoms were set to standard neutron distances. 14Prior to the final cycles of refinements, anisotropic displacement parameters (ADPs) for hydrogen atoms were obtained by combining molecular rigidbody contributions with average values for internal vibrational contributions from neutron diffraction using the SHADE2 server. 15An ORTEP 16 plot of each molecular structure with the atom numbering scheme is shown in Fig. 1, using the results of the multipole refinement.

Hirshfeld-atom refinement
Using the starting geometry from the multipole models of all urea derivatives, the Hirshfeld-atom refinement technique 7 was applied using the Tonto program. 17In this method a single-point energy calculation is performed on the starting geometry.The molecular electron density is then partitioned into atomic electron density fragments by Hirshfeld's stockholder scheme 18 using spherically averaged atomic electron densities calculated for the respective atoms in their spin ground states for partitioning.Thus obtained non-spherical atoms are subsequently used as scattering factors in a leastsquares refinement of positions and ADPs.The energy calculation and least-squares refinement are repeated to convergence.For the three urea derivatives all positions and ADPs were refined for non-hydrogen atoms, while for hydrogen atoms only positions were refined.This was because not all three datasets allowed one to obtain physically reasonable values for the atomic displacement parameters of H atoms. Instead, ADPs from the SHADE2 server were used. 15This further had the advantage of allowing a fair comparison of topological parameters resulting from XD2006 and Tonto refinements, since SHADE ADPs were mandatory to describe anisotropic thermal motion of H atoms in XD refinements.In Hirshfeld-atom refinement point charges and point dipoles can easily be obtained following Hirshfeld's definition.The numerical integration used a Lebedev grid 19 and 35 (40) radial points for H (non-H atoms) according to the formula of Mura and Knowles. 20Dipoles were simulated by charges of the opposite sign separated by a distance of 0.001 a.u.In order to simulate the effects of the crystal field, such point charges and dipoles were then placed on atomic sites around the molecule; all sites on whole molecules with at least one atom within a radius of 8 Å were included.The above refinement procedure was then applied in a second round of refinements.
To summarize: two types of Hirshfeld-atom refinements were performed for each urea derivative: one with charges and dipoles (denoted by the symbols +/2) and one without using a cluster of point charges around the central molecule.The former refinements take into account the change in the electron density distribution during the refinement process.

X-ray wavefunction refinement
The structural models from the Hirshfeld-atom refinements without a surrounding cluster of point charges and dipoles were subsequently used as input geometry for the wavefunction fitting procedure.Note that this procedure is slightly different from those described in Grabowsky et al. in ref. 8c, who did use a cluster of surrounding charges, and have called it X-ray wavefunction refinement (XWR).Since constraint models provide any substantial difference between the calculated and experimental structure factors, we do not expect that the results will be significantly different.Density functional theory wavefunctions were constrained to fit the experimental data by introducing a variable mixing parameter l.Contributions of the experimental data to the model were increased in steps of 0.02 until the method ceased to converge.The final l values of 0.2, 0.1 and 0.08 were reached for murea, phurea and dphurea, respectively.The size of these values depend on the sigmas of a given set of reflections and also on how significant an energy penalty is presented by a particular reflection.Hence they are highly dependent on the particular system, the Hamiltonian used for modeling, and the conditions of the experiment.
All basis-set calculations (HAR and XWR) were carried out on F, and negative intensities were omitted from the data like for multipole refinement so that a comparable number of reflections were used in all refinements.All density functional calculations were performed using the BLYP functional 21 in combination with the Dunning cc-pVDZ and cc-pVTZ basis sets 22 using the Tonto software package. 17

Topological analysis and integrated atomic properties
After wavefunctions or a set of multipole parameters were obtained, Bader's Quantum Theory of Atoms In Molecules (QTAIM) 23 was applied.It allows one to analyse the electron density distribution from multipole refinement as well as that obtained from unconstrained and constrained wavefunctions.Topological analysis and integration of atomic properties were performed with the XDPROP and TOPXD modules implemented in XD2006 for the multipole model, while AIM2000 24 and AIMALL 25 were used for the obtained wavefunctions.

Comparison of figures of merit
Table 2 gives the various figures of merit obtained for the three different methods.The R factor and goodness-of-fit (S) are usually used as a measure of agreement between the calculated and experimental structure factors in X-ray structure determinations, however in ref. 7 it was shown that the x 2 agreement statistic is a better measure of the fitting accuracy in HAR, especially in regards to the ADP changes.The value x 2 = 1 indicates that, on average, the calculation result lies within the acceptable experimental error bounds.For all three compounds the x 2 value is better for the HAR using surrounding charges rather than not.This is not unexpected, since all of the molecules are involved in medium strength intermolecular interactions.However, differences are small.In more detail, murea gives the best value of x 2 .This may be due to the very low measurement temperature of 8 K.An important observation is that for murea and dphurea, the x 2 from the HAR is better than the corresponding value from the multipole refinement.This is significant because the multipole model has the ability to fit the electron density, whereas HAR does not.Hence, the HAR procedure can fit the experimental data better than a multipole refinement despite using fewer parameters.
Of course, the best figures of merit were obtained after performing the constrained wavefunction procedure.However, in agreement with previous reports, 26 the changes in x 2 after constraint refinement are much more variable than the R factors.

Comparison of residual and deformation electron density maps
The subsequent step in our analysis of the performance of the three methods is a comparison of residual and deformation electron density maps obtained in each case.The first observation is that the residual density plots (see Fig. S1-S3, ESI 3 ) obtained after HAR, and then after the constraint (the XWR procedure) are essentially identical and featureless in comparison with those from multipole refinements, where some characteristics, e.g. a minimum at most positions of atomic nuclei is noticeable.
Fig. S4-S6 (in the ESI3) show that the deformation density maps from triple-zeta HAR+/2, HAR and XWR models are essentially the same.This indicates that the effect of the crystal environment is small, visually confirming the earlier results from the x 2 agreement statistics in Table 2.
If we compare the deformation densities between all three molecules coming from the multipole models, we observe significant changes on the O1 atom in murea and phurea.On the other hand, the deformation density is a little better on the O1 atom in dphurea.Since in dphurea there are fewer hydrogen bond contacts to O1 than in the other molecules, we suggest that the deformation in the multipole model may be associated with the intermolecular hydrogen bonds.
A comparison of multipole and wavefunction based deformation density plots shows significant differences for the same atoms discussed above.Generally the results from wavefunction fitting are less distorted and more in line with chemical intuition.

Comparison of molecular geometries
From the inspection of Table 3 (and Tables S1-S3, ESI 3 ) one can see that there is a good agreement between the three sets of experimental geometries of all urea derivatives.A comparison shows that the non-hydrogen bond lengths mostly agree within 3s.There are however three exceptions: O1-C1 for murea, C13-C14 for phurea and C14-C15 for dphurea, where  a HAR and HAR(+/2) were performed using the blyp/cc-pVTZ level of theory.
the differences are much bigger, but do not exceed 0.003 Å.
The disagreement with the O1-C1 bond length may be correlated with the previously mentioned distortions in the deformation map for murea; however the other deviations occur in the phenyl rings.
It is also seen that the standard deviations of non-hydrogen bond lengths after Hirshfeld-atom refinement are systematically improved in most cases when compared with the multipole-model results.It is worth noting that the HAR calculations reproduce rather well the hydrogen bond lengths from neutron diffraction, which were elongated and fixed prior to multipole refinement but were freely refined in HAR.Both variations of Hirshfeld-atom refinement (with and without a cluster of charges) lead to C(N)-H bond lengths close to neutron diffraction, see Tables S1-S3 in the ESI 3 for details.
The differences do not exceed 0.04 Å with a maximum standard deviation of 0.01 Å in the worst cases.These findings are very encouraging for a broader application of HAR.They suggest that this method can give accurate bond distances for X-H bonds from X-ray diffraction, which match the results from neutron diffraction quite well.Such results can also be expected for normal-resolution data (Mo Ka with h ¢ 25u).Hirshfeld-atom refinement can hence give us the most accurate geometries available from X-ray diffraction.

Comparison of topological properties
Since all three compounds are very similar, we were interested to find out whether or not a comparison of their chemical bonds in terms of bond-topological properties as defined by Bader's Quantum Theory of Atoms In Molecules 23 can be indicative of subtle differences caused by different packing environments in each crystal structure.For murea and phurea the electron density values (see Table 4 and additionally Tables S4-S6, ESI3) at the bond critical point agree within three standard uncertainties, whereas the values for the negative Laplacian can deviate considerably, as it becomes especially obvious for the polar C-O and C-N bonds.
The limited flexibility of the multipole model was discussed before 27 and our results further substantiate those findings.The results from HAR and XWR, which are based on a basis-set electron density model show a high consistency and a lower spread than the multipole-model results.This highlights the limitations of the Hansen-Coppens multipole model and the strength of the more sophisticated basis-set model.For dphurea the agreement between the electron densities at the bond critical point is a little worse than for the other two derivatives.Several values from multipole refinement deviate by more than 3s from the basis-set methods.Still the trends seen in the strength of chemical bonds are consistent in all three compounds.While the highest spread is indeed seen for the polar bonds, an obvious trend in the different intermolecular forces does not seem to be extractable from the topological analysis of covalent chemical bonds in the three structures and earlier findings about the small influence of the crystal field on the strength of covalent bonds are confirmed.

Comparison of integrated atomic properties
As concerns the atomic volumes (see Table 5 and more detailed Tables S7-S9, ESI 3 ) the same internal consistency of atomic volumes and charges as defined by Bader's theory 28 can be seen.The highest disagreement in atomic volumes is again observed for such atoms that are involved in polar bonds, e.g.O1 and N1-N2, whereas atomic charges of these atoms agree rather well.This overall consistency is observed in all three urea derivatives.It remains open whether the discrepancies between the polar bonds are due to chemical factors or if they can be explained by a perturbation from different electron density models used.All in all, trends like the increase of charge with increasing electronegativity are very well confirmed.
For all three compounds a systematic shift in volume is observed in the multipole model for the H atoms to the C,N,O atoms they are attached to.This points to a limitation of the multipole model when only bond directed dipoles are refined,

Conclusions
High quality data of three urea derivatives were collected.Diffraction data were evaluated by three methods: the Hansen-Coppens' multipole model (MM), Hirshfeld-atom refinement (HAR) and X-ray wavefunction refinement (XWR).
All three approaches show a significant improvement over the refinements carried out with the commonly used independent atom model.As concerns the geometry of urea derivatives, multipole refinement and Hirshfeld-atom refinement give very similar results.This is an interesting result because it shows that the more demanding embedded cluster-of-charges wavefunction calculations may not be necessary.Hirshfeld-atom refinement has an additional benefit, namely that positional parameters of hydrogen atoms can be freely refined -and the bond distances to H atoms then agree favourably with the results from neutron diffraction.
When including point charges and dipoles to model the effect of the surrounding molecules on the asymmetric unit the fit in Hirshfeld-atom refinement consistently improves, although the change remains small.That is why we consider the method with charges as the best current choice for small molecule structure refinement, especially since high-resolution data are not required.
As concerns figures of merit one can routinely get better results with a cc-pVTZ basis-set by Hirshfeld-atom refinement than with a multipole refinement, although only xyz and U ij are adjusted to the X-ray data.The number of parameters is hence much smaller than in a multipole refinement.Although triple-zeta bases gave the best results, a double-zeta basis set leads to comparable results.This is interesting because for larger systems it may only be possible to use the smaller double-zeta bases.
From the three current methods investigated X-ray wavefunction refinement based on the geometry from Hirshfeldatom refinement gives the best possible results, although sometimes a triple zeta basis (here: BLYP/cc-pVTZ) is required for this statement to hold.Hence we recommend that all users benefit more frequently from combining Hirshfeld-atom refinement and X-ray constrained wavefunction fitting, the procedure known as X-ray wavefunction refinement (XWR).

Fig. 1
Fig.1ORTEP16 of molecular structures from multipole refinements of murea (a) phurea (b) and dphurea (c) including the general atomic numbering schemes.H atom labels are the same as for the heavy atoms they are attached to.Anisotropic displacement parameters (ADPs) are drawn at 50% probability.

Table 2
Figures of merit (R factors in %) based on the comparison of observed structure factors with those obtained from the different calculations for urea derivatives

Table 3
Selected bond lengths/Å for urea derivatives obtained from the various models and methods a

Table 4
Selected topological parameters for urea derivatives obtained from the various models and methods a

Table 5
Selected integrated atomic properties for urea derivatives obtained from the various models and methods a Basis-set calculations were performed using the blyp/cc-pVTZ level of theory.b V 001 /Å 3 is the basin volume cut at r = 0.001 a.u.c N 001 /e is the corresponding electron population.d Atomic charge Q/e.