Anharmonic Motion in Experimental Charge Density Investigations

Article pubs.acs.org/JPCA

Anharmonic Motion in Experimental Charge Density Investigations Regine Herbst-Irmer,*,† Julian Henn,‡ Julian J. Holstein,†,§ Christian B. Hübschle,† Birger Dittrich,† Daniel Stern,† Daniel Kratzert,† and Dietmar Stalke† †

Institut für Anorganische Chemie, Universität Göttingen, Tammannstraße 4, 37077 Göttingen, Germany Laboratory of Crystallography, Universität Bayreuth, 95440 Bayreuth, Germany

‡

S Supporting Information *

ABSTRACT: In the charge density study of 9-diphenylthiophosphinoylanthracene the thermal motion of several atoms needed an anharmonic description via Gram−Charlier coefficients even for data collected at 15 K. As several data sets at different temperatures were measured, this anharmonic model could be proved to be superior to a disorder model. Refinements against theoretical data showed the resemblance of an anharmonic model and a disorder model with two positions very close to each other (∼0.2 Å), whereas these two models could be clearly distinguished if the second position is 0.5 Å apart. The refined multipole parameters were distorted when the anharmonic motion was not properly refined. Therefore, this study reveals the importance of detecting and properly handling anharmonic motion. Unrefined anharmonic motion leads to typical shashlik-like residual density patterns. Therefore, careful analysis of the residual density and the derived probability density function after the refinement of the Gram− Charlier coefficients proved to be the most useful tools to indicate the presence of anharmonic motion.

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INTRODUCTION Structure and the Various Data Sets. 9-Diphenylthiophosphinoylanthracene molecules {(Ph2PS)C14H9} absorb UV and emit visible light and are therefore of interest as photoluminescent materials.1−3 To investigate the reason for this behavior, a charge density study was performed on this compound. Four data sets of 9-diphenylthiophosphinoylanthracene were collected at different temperatures and on different diffractometers: data sets 1, 2, and 4 were measured on a Bruker TXSMo rotating anode equipped with INCOATEC Helios mirror optics and an APEX II detector. Data set 1 was measured at 15 K, data set 2 at 100 K, and data set 4 at room temperature. Data set 3 was measured on an INCOATEC IμS with QUAZAR mirror optics and an APEX II detector at 100 K. Details of all data sets are given in Table 1. Data sets 1−3 were collected to a resolution suitable for a charge density study, whereas data set 4 at room temperature was measured for comparison purposes only. Data sets 1 and 2 were collected on the same diffractometer to investigate the temperature dependence, whereas data set 3 was intended for evaluation of the influence of the X-ray source in comparison to data set 2. The structure crystallizes in space group P1̅ with two molecules in the asymmetric unit (Figure 1 and additional figures in the Supporting Information). The two molecules differ only marginally in geometry, whereas their vibrational behavior appears to be markedly different. Refinements. For all data sets an independent atom model refinement with SHELXL4 was performed initially (Table 2). For the high resolution data sets 1−3 at 15 and 100 K, respectively, the highest residual density peaks were, as © 2012 American Chemical Society

expected, located in the bonds of the aromatic ring systems, whereas the highest peak for data set 4 at room temperature was found close to one of the sulfur atoms. Asphericity of the electron density distribution can be taken into account in a multipole refinement via the Hansen− Coppens formalism:5 lmax

ρat (r ) = Pcρc (r ) + Pνκ 3ρv (κr ) +

∑ κ′3Rl(κ′r) l=0

l

∑ Plm±dlm±(θ ,ϕ) m=0

where the density is divided into a core density, a spherical valence density, and an aspherical valence density. The parameters κ and κ′ are used to describe the expansion and contraction of the density. This approach requires up to 25 + 2 additional parameters per atom for the multipole populations and the κ and κ′ parameters. To derive a good starting model for the multipole refinement, an invariom refinement was applied using the programs XD.6 Here the multipole parameters are theoretically calculated and therefore no additional parameters are introduced. 7 The invariom database was extended by calculating new model compounds to derive invarioms for the fragments containing phosphorus and sulfur (for more details see Supporting Information.) Because of the insufficient Received: October 9, 2012 Revised: November 16, 2012 Published: December 17, 2012 633

dx.doi.org/10.1021/jp309985e | J. Phys. Chem. A 2013, 117, 633−641

The Journal of Physical Chemistry A

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Table 1. Crystallographic Data for Data Sets 1−4, λ = 0.710 73 Å data set temp [K] source res [Å]/[Å−1] data unique dataa Rint multiplicity Rpim completeness ⟨I⟩ ⟨I/σ⟩ max. res with 99% completeness [Å] a

1 15 Bruker TXS 0.45/1.11 108638 42657 (18745) 0.025 (0.042) 2.34 (1.42) 0.0151 (0.040) 91.7 (85.9) 4.7 (1.2) 24.31 (12.59) 0.65

2 100 Bruker TXS 0.44/1.14 299913 48281 (22298) 0.022 (0.046) 6.00 (2.95) 0.008 (0.031) 96.7 (94.0) 3.2 (0.4) 42.18 (17.35) 0.54

3 100 INCOATEC IμS 0.45/1.11 276805 45135 (20261) 0.022 (0.067) 5.99 (2.87) 0.008 (0.046) 97.7 (95.3) 3.8 (0.5) 37.27 (11.69) 0.48

4 293 Bruker TXS 0.78/0.64 55339 8932 (2702) 0.028 (0.151) 6.19 (4.45) 0.012 (0.081) 99.8 (99.6) 6.8 (1.0) 27.25 (6.33) 0.78

Data in brackets refer to the highest resolution shell, e.g., 0.55−0.45 Å for 1 and 3, 0.54−0.44 Å for data set 2, and 0.88−0.78 Å for data set 4.

programs were used and the number of reflections used for the refinement differed. To avoid convergence problems in this first stage of refinement, only reflections with I > 3σ(I) were used in the invariom refinement. For the high-resolution data sets 1−3 at 15 and 100 K, respectively, the invariom model was a good starting point for the multipole refinement. This is essential in a multipole refinement, where parameter correlation and convergence problems can occur. In a second step, chemical constraints were employed as much as possible. The multipole parameters of chemically equivalent atoms in both molecules were thus fixed to be identical. In addition, the highest possible local atomic site symmetry was applied. All these constraints were generated by the program MolecoolQT,8 which uses the functionality of the program InvariomTool.9 Residual Density Analysis. To decide on the quality of the model, the residual electron density should be checked carefully. After an appropriate refinement, it should be “flat and featureless”. This can be verified by graphical inspection with the program MoleCoolQt or by residual density analysis (RDA).10 When the residual density is plotted against the fractal dimension, a small parabolic curve with a maximum

Figure 1. Molecule 2 of {(Ph2PS)C14H9} at 100 K.

resolution for the room-temperature data set 4 the invariom approach was the only way to take the aspherity of the electron density into account via the multipole model. As can be seen in Table 2, the residual electron density improved greatly after invariom refinement. Note that the R values given in Table 2 are not directly comparable, because different refinement Table 2. Various Stages of Refinement data set SHELX R1 (I > 2σ(I)) wR2 (all data) res density [e Å−3]

1 at 15 K

2 at 100 K

3 at 100 K

4 at RT

0.0285 0.0856 −0.33/0.65

0.0290 0.1003 −0.85/1.41

0.0326 0.1101 −0.79/1.33

0.0375 0.1255 −0.39/0.24

XD Invariom R(F2) (I > 3σ(I)) res density [e Å−3] data/parameter

0.0204 −0.27/0.52 52.2

0.0239 −0.83/1.23 61.9

0.0252 −0.77/1.13 55.4

0.0261 −0.30/0.30 12.4

XD Multipole Harm. R(F2) (I > 3σ(I)) res density [e Å−3] data/parameter

0.0191 −0.28/0.45 41.3

0.0198 −0.83/1.05 50.0

0.0197 −0.79/0.95 43.9

XD Anharmonic R(F2) (I > 3σ(I)) res density [e Å−3] data/parameter

0.0189 −0.20/0.27 40.1

0.0178 −0.24/0.25 44.0

0.0179 −0.19/0.32 39.4

634

0.0247 −0.13/0.14 11.9



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close to 3 is expected for a flat and featureless distribution of the residual density.

Fourier truncation errors was minute for the low resolution data set but showed the strongest peaks for the 100 K data sets, indicating this effect to be of only marginal importance. In addition, Fourier truncation error peaks cause a different pattern of residual electron density than do the shashlik-like features that we observed. • Problems with the local coordinate system could be excluded by refining all multipole parameters up to the hexadecapole level for sulfur and phosphorus and by testing different coordinate systems. • A refinement of sulfur/oxygen disorder was tested but did not improve the residual density. Positional disorder of only the sulfur atom did not appear to be a chemically reasonable explanation. • Core polarization was recently described as the cause of residual density peaks close to atom positions.11,12 However, these peaks were mainly observed for theoretical data or for experimental data at extremely high resolution. In any case, they were closer to the atomic position than in the present case. • We therefore considered anharmonic motion as a possible explanation for the observed phenomena. Anharmonic Motion. In the program XD the Gram− Charlier expansion13 is implemented:

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DISCUSSION The Problem. The multipole refinement led to a residual density distribution that was by no means flat and featureless, especially for the two data sets at 100 K, the most frequently employed temperature for experimental charge density data collection (Figure 2).

T (H ) = (1 − 4 /3 π 3iCjklhjhk hl + 2/3 π 4 iDjklmhjhk hlhm + ...) T0(H )

Refinement up to the third order involves 10 additional parameters per atoms, whereas for the fourth order 15 further parameters are required. We tried this approach and applied Gram−Charlier coefficient refinement up to the third order for atom S2. Indeed, the residual density improved considerably. For data sets 1−3 at 15 and 100 K, respectively, a further improvement was achieved by applying fourth-order coefficients. The resulting residual density for data set 1 at 15 K displayed no particular extremum close to any atom. For the other data sets the highest extrema were now located close to the phosphorus atom P2. Again, refinement of the Gram−Charlier coefficients up to third order improved the residual map. However, the residual density distribution in the 100 K data sets still exhibited pronounced shoulders, and a similar shashlik-like density pattern occurred close to the carbon atoms of the neighboring phenyl ring. Subsequent application of Gram− Charlier coefficients up to third order for these atoms improved the residual density likewise (Figure 3 and residual density plots for both molecules in the Supporting Information). Impact on Multipole Populations. The typical alternating residual density caused by neglect of anharmonic motion was found mainly at atom S2 but was also visible at S1 for the 100 K data sets 2 and 3. To our surprise, this residual density decreased significantly after introducing Gram−Charlier coefficients only for atom S2. This could be explained by the constrained refinement of the multipole populations. The dipole populations in particular varied in the course of the refinements (Table 3). The invariom refinement assumed cylindrical symmetry, so only P10 was populated. After relaxing the symmetry constraints, the population of P11+ refined to significant values whereas the population of P10 vanished. After refining the Gram−Charlier coefficients, they reappeared close to the invariom values. This effect can also be seen in a harmonic refinement without constraining the multipole parameters of atom S2 to atom S1. The multipole populations

Figure 2. Residual density analysis (left column) and residual density isosurfaces (right column) close to the sulfur atom S(2) after multipole refinement. The green density is positive and red negative (data set 1, ±0.1 e Å−3; data set 2, ±0.085 e Å−3; data set 3, ±0.094 e Å−3; data set, 4 ±0.074 e Å−3).8 (See Supporting Information for residual density for both molecules.)

The highest residual density peaks and holes were close to the sulfur atom of the more strongly vibrating molecule for all data sets. Even for the 15 K structure, a strange shashlik-like pattern of alternating maxima and minima (red−green−red− green) was observed. Relaxing the symmetry constraints for the sulfur and phosphorus atoms and refining all multipole parameters for these atoms up to the hexadecapole level did not cure the problem. To explain this phenomenon, we reasoned as follows: • Problems with the data sets (low-quality data, absorption, twinning, etc.) could be excluded. We collected data on four different crystals, but the effects were similar for all data sets; therefore, bias due to absorption could be excluded. In space group P1̅ with no higher metric symmetry, only nonmerohedral twinning would be possible, which was excluded by careful inspection of the diffraction frames. The effect of 635



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of atom S1 remained close to the theoretical values, whereas they differed significantly for atom S2. Hence, in anharmonic refinement the multipole populations of both sulfur atoms were precisely adjusted. Similar, but smaller, effects were also observed for the phosphorus atoms. The obvious lessons to learn from these observations are the following: • It is important to introduce Gram−Charlier coefficients gradually, because typical “anharmonic” residual density patterns at one atom can be caused by anharmonic motion of a second atom constrained to the first. • The neglect of modeling anharmonic motion causes bias in the multipole populations. Consequently, properties derived from the electron density model are also affected. This clearly emphasizes the importance of detecting and modeling anharmonic motion appropriately. Anharmonic Motion versus Disorder. Anharmonic refinement of the sulfur and phosphorus atoms as well as of one entire phenyl ring obviously raises the suspicion that the whole moiety might be disordered. The green residual density peaks suggest at first sight a second position of the phenyl ring with a reduced site occupation factor (sof) (see Table 6 experimental data with harmonic model). Disorder models in XD represent a significant challenge, because it is impossible to refine occupancies directly. Hence, SHELXL refinements were employed to judge the occupancies of the second virtual site. We defined a second position at the residual density peaks close to the atoms and refined it with restraints to interatomic distances and to the anisotropic displacement parameters. Table 4 shows the results of the various SHELXL refinements. Disorder refinement seemed to improve the model. Clearly, with only the independent atom model the residual density was not so easy to evaluate, because with data at high resolution the deficiencies in the IAM approach were clearly apparent in the residual density (e.g. density in bonds and at positions of lone pairs), but the high peaks at sulfur and phosphorus atoms disappeared. Upon refinement, the second position approached the main position much closer than in the starting model (from ca. 0.5 Å to ca. 0.2 Å). Furthermore, there seemed to be a dependence of the site occupation factor on temperature: namely, the higher the temperature, the higher the site occupation factor of the minor fraction. To exclude the effect of different resolution and to diminish the residual density in covalent bonds, we also tested refinements with a resolution cut to 0.84 Å. Astonishingly, the occupancy of the second position seemed to decrease when the resolution was lowered to 0.84 Å. Moreover, all site occupation factors were nearly identical (0.01) for the high resolution data sets (1−3) at 15 and 100 K, respectively, whereas now the refinement of the second position converged to the position of the residual density on the bonds for data sets 1−3. For the room temperature data set 4, however, the second position stayed similar to the previous refinement. In quintessence, for

Figure 3. Residual density analysis (left column) and residual density isosurfaces (right column) close to the sulfur atom S(2) after anharmonic refinement: data set 1, S2 4th order; data set 2, S2 4th order; P2, C47 to C52 3rd order; data set 3, S2 4th order, P2, C47 to C52 3rd order; data set 4, S2 and P2 3rd order. The green density is positive and red negative (data set 1, ±0.1 e Å−3; data set, 2 ±0.08 e Å−3; data set 3, ±0.088 e Å−3; data set 4, ±0.071 e Å−3)8.

Table 3. Dipole Populations at Different Refinement Steps for Data Set 2 at 100 K P11+ invariom harmonic refinement no symmetry constraints S1 and S2 constrained anharmonic refinement no symmetry constraints S1 and S2 constrained harmonic refinement no symmetry constraints

P11-

P10

0 −0.105(5)

0 0.004(5)

−0.081 −0.005(6)

0.001(5)

0.010(5)

−0.050(6)

S1: 0.004(7)

0.028(7)

−0.054(8)

S2: −0.240(7)

−0.025(7)

0.032(8)

Table 4. Disorder Refinement with SHELXL

res [Å] R1 (I > 2σ(I)) wR2 (all data) sof max. res dens within 1 Å to S2 [e Å−3]

1 at 15 K

1at 15 K

2 at 100 K

2 at 100 K

3 at 100 K

3 at 100 K

4 at RT

4 at RT

0.45 0.0284 0.0853 0.08(1) 0.26

0.84 0.0272 0.0699 0.0094(7) 0.25

0.44 0.0276 0.0970 0.121(5) 0.27

0.84 0.0267 0.0709 0.0101(6) 0.21

0.45 0.0313 0.1078 0.101(6) 0.27

0.84 0.0271 0.0721 0.0107(6) 0.19

0.78 0.0364 0.1228 0.23(2) 0.15

0.84 0.0331 0.1124 0.16(2) 0.15

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Table 5. Disorder Refinements with Various Occupancies of the Minor Position Compared to Anharmonic Refinement with XD Using Data Set 2 at 100 K refinement 2

R(F ) (I > 3σ(I)) res dens within 1 Å to S2 [e Å−3] C−C range main pos [Å] C−C range minor pos [Å] parameter data/param

harm.

sof 0.08

sof 0.1

sof 0.12

sof 0.14

sof 0.16

3rd order SPPH

4th order

0.0198

0.0183

0.0182

0.0182

0.0182

0.0182

0.0178

0.0178

−0.83/1.05

−0.16/0.31

−0.14/0.27

−0.14/0.23

−0.14/0.22

−0.15/0.21

−0.11/0.21

−0.11/0.15

1.392−1.400

1.390−1.400

1.390−1.400

1.389−1.401

1.389−1.401

1.389−1.402

1.391−1.400

1.391−1.400

1.360−1.417

1.361−1.415

1.363−1.414

1.365−1.413

1.366−1.412

854 47.6

854 47.6

854 47.6

854 47.6

854 47.6

911 44.7

926 44.0

831 49.0

To confirm the improvements seen by modeling anharmonic motion, further tests using theoretical data were accomplished. We generated three sets of theoretical data by calculating Fcalc values from the refined XD parameters: For data set A we generated data using the starting position of the disorder model with a sof of 0.02 for the minor site. For data set B the final model of the disorder refinement with a sof of 0.12 was employed. Data set C was created from the anharmonic model up to the fourth order for sulfur and up to the third order for phosphorus and the phenyl carbon atoms. For all three data sets noise was added to mimic experimental data.10 Against these data three models were tested: In model 1 only harmonic motion but no disorder was considered. In model 2 anharmonic motion up to the third order for the phenyl carbon atoms and phosphorus and up to the fourth order for sulfur was taken into account. For data sets A and B even fourth-order anharmonicity for phosphorus was tried. In model 3 a second position with occupancy of 0.12 was included. Table 6 summarizes the results of the various refinements. Refining only harmonic motion led to positive and negative density close to the atomic positions for all data sets. Using the anharmonic model for data set A, high residual density remained. It is improved for data set B, but, especially close to one carbon atom, there were clear extrema in the residual density. Data set C showed only noise. The experimental data did not show residual density at the carbon atoms, whereas there was small positive density left at P and S. The disorder model of course exhibited only noise for data sets A and B, whereas for data set C, as well as for the experimental data, there were positive and negative residual peaks close to one carbon, the phosphorus and the sulfur atom. R values for the disorder and the anharmonic model differed only little, so again they remained indecisive about the two models. As here theoretical data were employed, true values for the monopole and multipole populations were known. So a possible deviation from the reference values could be evaluated for the different models (details in the Supporting Information). It became apparent that the largest deviation from the true values was obtained using the harmonic model, whereas the “wrong” model (anharmonic for data A and B and disorder for data C) led to only small deviations. These findings revealed that it is better to refine Gram−Charlier coefficients than to use the harmonic approximation in cases where the residual density indicates that there is either anharmonic motion or minor disorder. Temperature Dependence. We demonstrated that disorder can clearly be distinguished from anharmonic motion if both positions are far apart, as seen in data set A. If the two

the low temperature data sets high resolution seemed to be a prerequisite for resolving the disorder. Although a multipole refinement of a disordered structure certainly diminishes the quality of a charge density model, a refinement with XD was performed for data set 2 at 100 K. As sof and the monopole occupation are strongly correlated, a combined refinement of these parameters would not be reasonable. Therefore, several different occupancies from 0.08 to 0.16 were assumed and tested. Starting models for the second position were provided by using the residual density peaks fitted to an optimal geometry (defined by the first position). Refinement with fixed coordinates of this second position gave implausible results. Free refinement of the coordinates was therefore performed (unfortunately the use of restraints is not possible in XD) (Table 5). Anisotropic displacement parameters, monopole and multipole populations of the second position, were constrained to the first position. Hydrogen atoms of the major position were refined freely, whereas for the minor position they were constrained to ideal bond lengths and to U values 1.2 times the Ueq of the attached carbon atom. Again, the minor site approached the position of the major site. The residual density became much flatter, but the geometry of the minor position was distorted (Table 5). The R values and also the residual density seemed to be relatively independent of the sofs assumed. Probably the other parameters compensated for this influence. We next switched to the program MoPro14 to have the opportunity to refine the coordinates using restraints. Assisted by the restraints, the geometry improved, but the residual density became worse. In MoPro there is also the possibility of refining occupancies, but the refinement was extremely unstable probably because of the above-mentioned correlation; the sof of the minor component became smaller and the R values increased. Comparing the results of the two different descriptions (anharmonic motion or disorder) the anharmonic model provided a slightly better model for the following reasons. • The R values differed insignificantly with respect to the two models. • The geometry of the second position in the disorder refinement was chemically implausible. • To show only the effects in this region, we calculated the residual density descriptors for a cuboid containing this SPPh moiety and no further atoms. These were much better for the anharmonic approach (Table 6). • There was relatively high residual density for the disorder refinement, especially close to carbon atom C50. • In the disorder refinement the two positions were so close to each other that this seemed to model anharmonic motion. 637



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ellipsoid describes the density of two positions. Therefore, decreasing the temperature should facilitate the discrimination of the two different positions. Here we worked with experimental data at different temperatures. As mentioned above, the lower temperature did not help to pinpoint a tentative second position for a disorder model, whereas in the anharmonic model the Gram−Charlier coefficients became smaller with lower temperature. The Gram−Charlier coefficients have no direct physical meaning15 but may be tested for the expected temperature dependence of the probability density function. Above (half) Debeye temperature linear dependence of the second-order Uij values and a quadratic dependence of the third-order coefficients are expected.16−18 Figure 4a depicts the graph of the Ueq of atom

Table 6. Refinement against Theoretical and Experimental Dataa

Figure 4. Ueq (a) and significant Cjkl (b) of atom S2 against T and T2, respectively.

S2 against the temperature and Figure 4b a plot of the two third-order coefficients Cjkl that showed significant values (>3σ) for all temperatures. Both graphs show the expected temperature dependence (see, for example, Figure 3.2 in ref 19). We may summarize by stating that for this structure, every result favors a refinement utilizing an anharmonic model. Parameter Correlations. The observed effects on the dipole populations might lead to the assumption that multipole populations and Gram−Charlier coefficients are correlated and should therefore not be refined together. Although some multipole refinements using an anharmonic description of several atoms have recently been described (for example, see refs 20−31), there were some reservations concerning this procedure.32,17 Even in the XD manual there is the warning referring to Mallinson et al.: “There are likely to be high correlations between the Gram−Charlier coefficients and the

a

The residual density analysis plot is calculated for a cuboid containing the SPPh group of molecule 2.

positions are in close proximity, the differences between the two approaches are much smaller and in unfavorable cases the boundaries might be blurred totally. In such cases data sets collected at different temperatures might assist an informed decision. The difference between anharmonic motion and a second disordered position is similar to dynamic and static disorder in an IAM refinement. Dynamic disorder leads to elongated anisotropic displacement parameters that get smaller with decreasing temperature. With static disorder an elongated 638



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performed with XD in two separate steps by refining with one data file with the working set reflections and then calculating an R value with the received model without any refinement using the test reflections. Recently, some investigations were performed whereby Rfree was used in MoPro refinements of small molecules to determine optimum values for weights for restraints and the use of constraints.20,31,36 To test the benefit of calculating an Rfree value, we switched again to the program MoPro. Details of the procedure are described in the Supporting Information. A decrease in Rfree was detectable, but the differences were minute, smaller than those between the individual Rfree values of the various test sets. Furthermore, there was also a decline for refinements where unreasonable parameters, such as anharmonic motion up to the fourth order for S and P of the first molecule, were added. We saw only trends similar to those for the R values; hence Rfree is not a meaningful tool in the current investigation. • Residual density We noticed a typical residual density pattern if anharmonic motion was not taken into account. For third-order anharmonic motion, one would expect the shashlik-like pattern of alternating peaks and holes close to the atomic position. Fourth-order anharmonic motion produces residual density of the same sign (Figure S6 in the Supporting Information). This typical pattern should disappear if Gram−Charlier coefficients are included in the refinement. • Significance of the refined values The coefficients Cjkl and Djklm have (in general) no direct physical meaning, but one can examine the resulting probability density distribution (pdf):15,25 In theory there should be no negative regions, but because of the limited accuracy of the refined coefficients, the pdfs may have some negative features far from the center, with small absolute values. If negative regions are found close to the center of the pdf, especially if they are surrounded by positive areas, it is likely that the refined anharmonic coefficients are meaningless.18 In the program MoleCoolQt8 an option to calculate and graphically represent pdfs of anharmonically refined atoms has recently been implemented. These should carefully be examined if they show holes or strange shapes (see Figure 5 for the pdfs of the sulfur and phosphorus atoms, whereas pdfs of carbon atoms are listed in the Supporting Information). The values itself should be significant,19 which means at least one Cjkl should refine to a value larger than 3σ. • Resolution dependence Kuhs18,37 introduced a rule for estimating the minimum data resolution for meaningful refinement of anharmonic thermal parameters (Gram−Charlier coefficients) for each anisotropic atom: Qn = 2n1/2(2π)−1/2(2 ln 2)1/2⟨u2⟩−1/2. These criteria were then tested on all data sets. The details are described in the Supporting Information. Because of the low temperature of data set 1 at 15 K, only the sulfur and phosphorus atoms were considered for refinement with Gram−Charlier coefficients. Only anharmonic refinement of atom S2 improved the model, although Kuhs’ rule was not fulfilled. As the effect of anharmonic motion increases with elevated temperature, more atoms have to be checked for the 100 K data sets 2 and 3. As the anharmonic motion should only be temperature dependent, both data sets should show residual density patterns for anharmonic motion for the same atoms. Only the slightly better resolution of data set 2 could be an

multipole populations for the same atom in the least-squares refinement, and great care should be taken to ensure that a true minimum is reached.”32 Therefore, we checked the correlations of the refined parameters and noticed the following: Third-order coefficients Cjjj are highly correlated with the positional parameters x, y, z, and this also holds for Uij values with fourth-order coefficients Djklm (for correlation coefficient ccij: 0.8 < ccij < 0.9). This can easily be explained by generating a theoretical Gaussian probability density function and adding a third- and fourthorder curve (Supporting Information, Figure S6). A third-order curve shifts the maximum, whereas a fourth-order curve narrows or widens the Gaussian curve. If we compare the Uij values after anharmonic, third- and fourth-order refinement, we again observe practically no difference between harmonic and third-order refinement, whereas the fourth-order refinement led to slightly higher values in accordance with theory. Within the Cjkl and the Djklm values a correlation similar to the values for the correlation between monopoles and κ (0.6 < ccij < 0.8) can be observed. However, correlations coefficients between multipoles and Gram−Charlier coefficients were below 0.3. This can be explained by the predominantly highorder reflections being affected by anharmonic motion, whereas low-order data describe the multipole populations.33 However, this could be different for heavier elements like transition metals, where d orbitals are involved, which could explain the different findings of Mallison et al..32 This also explains why we obtained smaller occupancies of the minor position in the IAM model after pruning the data. Furthermore, it explains why anharmonic motion refinement has recently become more popular. The experimental data are becoming better and better, the resolution is increasing, and therefore effects such as anharmonic motion are progressively more detectable.34 Avoiding Overfitting. There is still the concern about potential overfitting. After Gram−Charlier coefficients were introduced for the atoms of the SPPh moiety, the remaining residual density peaks were now close to further carbon atoms. Of course the Gram−Charlier coefficients could also be used here, but the question occurs whether this really improves the model or if this would just lead to overfitting of the data. What criteria can be used to justify adding 10 (or even 25) additional parameters per affected atom? The following criteria could be considered to validate the use of anharmonic refinement: • R-values The R value is of course not the most reliable figure, because with overfitting, the R value might decrease without improving the model. As anharmonic motion mainly influences high resolution data, the effect on the R value of adding Gram− Charlier coefficients can be fairly small (see, for example, Table 2, especially for data set 1 at 15 K). In protein crystallography the Rfree value was introduced35 to account for the problem of overfitting. This is an R value calculated for a certain amount (e.g. 5−10%) of reflections not used in the refinement. This would normally not make sense (and is also not necessary) for a routine structure determination of a small molecule, because this value is only reliable if there are at least 500−1000 reflections in the Rfree test set. A complete high resolution data set of sufficient multiplicity should contain sufficient reflections, so the use of Rfree should in principle be feasible. This is implemented in the multipole refinement program MoPro and could in principle be 639



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temperature parameters. The probability distribution must remain physically reasonable. Rfree calculations are indecisive, because the differences are insignificant and physically unreasonable refinements may give slightly smaller values. Kuhs’ rule is fulfilled for the lighter carbon atoms but seems to be too strict for heavier atoms such as sulfur and phosphorus. Resolution Dependence. Having found such large effects of anharmonic motion, we wondered why these have rarely been discussed in the literature38−45 and are recently becoming more popular (for example, see refs 20−31). Therefore, we checked whether a resolution-dependent bias in the measured intensities could also cause similar features: For the 15 K data set the residual density analysis plots improved when the data were cut to (sin Θ /λ)max = 1 Å−1 (Supporting Information). Residual density plots are often shown up to this resolution, and then the typical residual density distribution for anharmonic motion becomes invisible. For our 100 K data sets the anharmonic motion is so extreme that even at 0.6 Å−1 a shoulder is visible in the residual density plots.

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CONCLUSIONS • Anharmonic motion does not seem to be modeled by multipole refinement but requires an additional treatment. • The deconvolution of anharmonic motion and disorder of two atoms in close proximity is difficult, the more so if there is only one data set at one single temperature. Although the anharmonic treatment requires more parameters, it could be the superior solution because the handling of disorder is still commonly not accepted in experimental charge density investigations and is challenging with the refinement program XD. • Kuhs’ rule for minimum resolutions seems to be too strict, at least for heavier elements such as sulfur and phosphorus. • A combination of several criteria should be used to decide whether to refine anharmonic motion. The residual density, in particular, should show the typical pattern and an improvement after modeling anharmonic motion should be apparent. The pdf should show a reasonable shape without any strange holes. • If anharmonic motion is real, refinement of Gram−Charlier coefficients in the presence of multipole parameters is feasible without high correlations, although this could be different for heavier elements like transition metals, where d orbitals are involved. • Neglecting anharmonic motion can lead to bias in the multipole parameters and to wrongly derived properties. • As refinement of anharmonic motion seems to be important, especially for high resolution data sets, the residual density should be examined for the whole resolution range. As modern data are constantly improving, proper treatment of anharmonic motion will become increasingly important.

Figure 5. Graphical representation of the probability function at the 50% probability level.

argument for anharmonic refinement for an atom against data set 2 compared to 3. For both data sets, anharmonic refinement up to the fourth order for atom S2 and up to the third order for atoms P2 and the carbon atoms of the connected phenyl ring C47 to C52 should be taken into account, although again Kuhs’ rule is not fulfilled for the phosphorus atom and for the sulfur atom only for third-order refinement. Further carbon atoms showed the typical residual density pattern, which disappeared after refinement of Gram−Charlier coefficients, but as the refinement process is not at a final stage, further refinement options (relaxing chemical constraints, symmetry constraints, improvement of the hydrogen parameters, including all data, weighting, ...) should be tested first. If then the residual density peaks have still not disappeared, an anharmonic description should be used, as unresolved residual density seems to cause more harm than an anharmonic model used to describe disorder (see the comparison of disorder versus anharmonicity). For data set 4 at room temperature anharmonic refinement up to the third order for atoms S2 and P2 was regarded as the best model, although Kuhs’ rule was only fulfilled for S2 but not for P2. This decision was mainly based on the improvement in the residual density and on pdfs. To summarize the benefit of the above-mentioned criteria for anharmonic refinement, we can state the following: The most important indicators are the residual density map and the calculated pdfs. Refinement of Gram−Charlier coefficients is only reasonable if a typical residual density pattern is observed before applying the anharmonic approximation, and this vanishes upon refinement of higher order

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ASSOCIATED CONTENT

S Supporting Information *

Further details about the crystallographic data of the different refinement steps, figures of the molecular structures of both molecules at all measured temperatures, plots of the residual density before and after anharmonic refinements, tables about the invariom values and the used local symmetry, tables of the refined monopole and multipole populations after refinement against the theoretical data sets, further information of the Rfree refinements and the required minimum resolution for a meaningful anharmonic refinement following Kuhs’ rule, tables 640



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of all refined Gram−Charlier coefficients, plots of all pdfs, RDA plots with different resolution cut-offs, and DMSDA tables. This material is available free of charge via the Internet at http://pubs.acs.org

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 0049-551-393007. Present Address §

Global Phasing Ltd., Sheraton House, Castle Park, Cambridge CB3 0AX, United Kingdom. Author Contributions

The manuscript was written using contributions from all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Deutsche Forschungsgemeinschaft (DFG, charge density priority program 1178, DI 921/3-2), the Danish National Research Foundation (DNRF) funded Center for Materials Crystallography (CMC) for support, and the Land Niedersachsen for providing a fellowship in the Catalysis for Sustainable Synthesis (CaSuS) Ph.D. program. We thank Prof. G. M. Sheldrick and Dr. D. Leusser for useful discussions. Julian Henn thanks the DFG for financial support (Grant HE 4573/3-1).

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REFERENCES

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Anharmonic Motion in Experimental Charge Density Investigations

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