2.1. Implementation

The DiSCaMB library, capable of calculating scattering factors for the Hansen–Coppens multipole model (Chodkiewicz et al., 2018), was extended to support TAAM by adding the capability to recognize atom types and handle TAAM parameters from the MATTS2021 pseudoatom databank. The details of the atom-typing algorithm implementation and new databank format MATTS2021 have been published elsewhere (Jha et al., 2022). The DiSCaMB library was further integrated into NoSpherA2.

For this study, we have used NoSpherA2 in Olex2-1.5 (Dolomanov et al., 2009) for the refinements. It incorporates the TAAM scattering factors available in the DiSCaMB library into the olex2.refine module and permits the use of these aspherical atomic form factors via a text file in the .tsc file format (Kleemiss et al., 2021). The aspherical atomic form factors in the .tsc file are used during the least-squares refinement against experimental intensities in an iterative cycle until convergence is achieved. The DiSCaMB component necessary to run TAAM refinements under NoSpherA2 can be downloaded from https://4xeden.uw.edu.pl/.

2.2. Crystal structures used for the TAAM refinement

A diverse set of already published crystal structures was chosen for this study, which included small organic molecules, co-crystals, multi-component systems, disordered structures, COFs and twinned structures (Fig. 1). Structure 1 [molecular formula (MF) = C₈H₈N₂] is a simple organic system and contains half a molecule in the asymmetric unit (Michaels et al., 2017). Structure 2 (MF = C₁₂H₁₅N₇O₂) is a co-crystal (Jarzembska et al., 2013). In cases 1 and 2 the reported data set contained only merged reflections. Structures 3–6 are two-component systems (Bhowal et al., 2021) having MFs C₃₁H₂₆N₆O₅ (3), C₄₄H₂₅N₁₁ (4), C₂₃H₂₃N₅O₄ (5) and C₂₆H₂₂N₆ (6). In these cases, data collected at two different temperatures (100 and 300 K) were also considered to highlight the effect of temperature on the refinement statistics. Structure 6 has half a molecule of both components in the asymmetric unit. Structure 7 (MF = C₁₉H₂₃N₃O₃) shows disorder in a terminal alkyl chain; the disorder was not modelled in the reported structure (Yi-Hua et al., 2019). Structures 8 and 9 are COF systems with masked solvent disorder (Ma et al., 2018); in the case of 9 there is also disorder in the ring. Structure 10 (MF = C₇₇H₉₃N₃O₆) is a hydrogen-bonded liquid crystal based on resveratrol having two different components in a 3:1 ratio in the asymmetric unit (Blanke et al., 2020). Structure 11 (MF = C₂₀H₂₆N₂O₈S₂) is twinned and contains two molecules in the asymmetric unit (Shafiq et al., 2009). Structure 12 (MF = C₆H₁₀NNaO₃) is a network alkali metal coordination complex with organic molecules (Clegg & Tooke, 2013). The coordinated metal–organic complex structure 12 represents an ideal case for hybrid IAM/HAR/TAAM refinement.

Figure 1 Visualization of structures used for spherical and aspherical refinement in this study. Atom labels are shown on the structures; the same atom labels indicate the symmetry-generated atoms. In COF structures 8 and 9, only the atoms in the asymmetric unit are labelled for clarity. In structure 10, out of the three identical molecules in the asymmetric unit, atoms of only one molecule are shown labelled.

2.3. Refinement process using IAM

The initial geometries taken from the published structures were re-refined using the default IAM framework (Wilson & Geist, 1993) with olex2.refine in Olex2-1.5 (Dolomanov et al., 2009) without making any changes in resolution limit or diffraction data. Hydrogen atoms were refined with isotropic displacement parameters freely without any restraints or constraints. The most commonly used spherical density model for bonded hydrogen in the IAM approach recommended by Stewart et al. (1965) was used for hydrogen-atom refinement. The published results from SHELXL (Sheldrick, 2015) IAM refinement were used for comparison.

2.4. Refinement process using TAAM in NoSpherA2

The structures obtained from IAM were further refined with olex2.refine using the TAAM approach and MATTS databank (Jha et al., 2022). The DiSCaMB library (Chodkiewicz et al., 2018) integrated into NoSpherA2 (Kleemiss et al., 2021) was used for the transfer of aspherical atomic form factors into the .tsc file format (Kleemiss et al., 2021). Hydrogen atoms were refined with isotropic displacement parameters freely without any restraints or constraints. The maximum number of iterative cycles was set to ten in NoSpherA2 for the TAAM refinement; however, convergence was usually achieved after four to five cycles.

2.5. Refinement process using HAR in NoSpherA2

The structures obtained from IAM were also refined with olex2.refine using HAR in NoSpherA2. ORCA version 4.2.1 was used for wavefunction calculation (Neese, 2012). HAR in NoSpherA2 was performed using the B3LYP functional (Lee et al., 1988) and various basis sets. Hydrogen atoms were refined with isotropic displacement parameters freely without any restraints or constraints. The integration accuracy, self-consistent field (SCF) threshold and SCF strategy (convergence) were set to normal. Ten iterative cycles were set in the NoSpherA2 setting. The calculations were performed on a laptop computer with a single-node 64-bit Intel i7 processor with four cores running at 2.60 GHz and 16 GB RAM.

2.6. Refinement process using hybrid-mode IAM–TAAM and HAR–TAAM in NoSpherA2

Hybrid-mode refinements were performed in two ways. In the first case, IAM–TAAM hybrid refinement was used for the coordinated metal–organic complex, where the metal atoms were refined spherically using IAM and organic parts were treated aspherically using TAAM. In the second case of hybrid HAR–TAAM refinement, the metal atoms were treated using HAR and organic molecules were treated using TAAM. The metal and organic parts were parametrized either with formal charge assigned to them or as neutral moieties. A level of theory of B3LYP/6-31G(d,p) was used for HAR in hybrid refinement. The hybrid refinements were performed like the disorder case, where separate .tsc files were generated for each part and combined before the final refinement. Other refinement settings for HAR are described in the previous section. More details can be found in the supporting information.

3.1. Comparison of IAM refinements obtained from SHELXL as originally reported and from olex2.refine used in this study

The reported structures (Fig. 1) were originally refined with SHELXL (Sheldrick, 2015). In this paper, we have used olex2.refine (Dolomanov et al., 2009) for the IAM refinement. The initial models obtained from IAM using olex2.refine were used as a basis for further refinements using TAAM or HAR. Although both SHELXL and olex2.refine are based on least-squares refinements and the same IAM scattering model, there are subtle differences in how the reliability factors (R1) and residual densities are calculated by these two programs. While these differences were not so apparent for good-quality data when refined with SHELXL or olex.2refine, the differences became more prominent when the data were of poor quality or the data were collected at high temperature. Here we have compared the refinement statistics of structures 1–11 which are purely organic systems refined at atomic resolutions (d_min in the range of 0.65–0.89 Å). Structure 12, a network structure consisting of an aqueous sodium salt of methyl pyridone, will be discussed separately.

In structures 1 and 2 the data sets were merged and are of good quality (Fig. S1); there were no differences in reliability factors and residual densities obtained from SHELXL and olex2.refine (Table 1). For structures 3–6, the data quality was poor with low I/σ, especially in the high-resolution region (Table 1 and Fig. S1), and the model did not fit well to the data, as indicated by a very high R1 for the high-resolution region [2θ > 35° (sin θ/λ > 0.42 Å⁻¹); θ is the incident angle, I/σ the signal-to-noise ratio and λ the incident beam wavelength]. The differences between refinements in SHELXL and olex2.refine were visible in the refinement statistics, especially in residual densities (Table 1). For some of the structures 3–6, data collected at low temperature (4_100 K and 6_100 K) were of better quality than those collected at ambient temperatures (4_300 K and 6_300 K) (Fig. S1). The differences in refinement statistics resulting from the usage of different software were also minimal in structures 4_100 K and 6_100 K. The reported structure 7 showed disorder in the terminal alkenyl chain which was not modelled, as can be seen from the large displacement ellipsoids and residual densities [Fig. 1 and Table 1 (7)]. The R1 and residuals were reduced after modelling the disorder in olex2.refine. The major and minor conformers have occupancies of 0.65 and 0.35, respectively. In the case of COFs 8 and 9, again the data quality was poor due to trapped solvent which could not be modelled properly and was masked in the IAM refinement (Fig. S1). Differences in R1 and the residual densities in the case of 8 and 9 obtained from SHELXL and olex2.refine were apparent (Table 1). For structure 10, the data quality was good and there was not much difference between SHELXL and olex2.refine results. However, a high residual density was observed in structure 10 (Table 1). In the case of the twinned structure 11, the data quality was poor at high resolution [2θ > 35° (sin θ/λ > 0.42 Å⁻¹)] and slightly higher residuals were observed in olex2.refine compared with the reported values (Table 1 and Fig. S1).

3.2. Comparison of IAM and TAAM refinements

The improvement in refinement statistics from TAAM refinement compared with IAM has been discussed on numerous occasions (Bąk et al., 2011; Sanjuan-Szklarz et al., 2016; Jha et al., 2020). TAAM refinement has been generalized on a set of small organic molecules using the MATTS2021 databank to achieve a better description of atom positions, hydrogen-bond distances and anisotropic displacement parameters (Jha et al., 2020). However, it was not possible to perform such a comparison on structures involving disorder, twinned structures, MOFs etc. Integration of TAAM into NoSpherA2 now enables refinement and comparison of the improvements in these classes of crystal structures.

In the following we will compare the results obtained from IAM and TAAM refinements for structures 1–11. Structures 1 and 2 show an improvement of ∼1 percentage point in R1 and >50% of the electron-density residuals after TAAM refinements, compared with IAM. This is in good agreement with our previous study (Jha et al., 2020). For the multi-component structures 3–6, only structures 4_100 K and 6_100 K showed considerable improvement in R1 and residual densities upon the description of the aspherical density. As mentioned earlier, the data quality of structures 4_100 K and 6_100 K was much better than that for the other multi-component structures 3–6 (Fig. S1). For 3, 4_300 K, 5 and 6_300 K the TAAM refinement showed a small improvement in R1; however, the residual densities in these structures were found to be even higher than for the IAM refinement. We also compared the residual electron-density map and fractal dimension plot for structure 3_100 K as one of the representatives of structures 3–6 (Figs. S2 and S3). A parabolic shape of a fractal dimension plot is an indicator of Gaussian noise distribution on a residual map and data are considered devoid of any systematic error (Meindl & Henn, 2008). The residual electron-density map from TAAM was found to be more populated and less featureless compared with that from IAM; also the fractal dimension plot showed behaviour that deviated more from ideal Gaussian noise in TAAM than in IAM. These observations again highlight the poor quality of the data and the sensitivity of TAAM refinement in detecting problematic data (Jha et al., 2020).

In the data set of structure 7, the data quality was good (Fig. S1), but the initial TAAM refinement led to higher residual densities than the corresponding IAM refinement, clearly supporting the observation that there is unmodelled disorder in the structure. After appropriate modelling of the disorder, TAAM refinement showed further improvements in R1, residual densities and fractal plots [Table 1 and Fig. S7(a)]. The fractal dimension plot of the reported IAM structure of 7 with disorder unmodelled deviates greatly from the parabolic shape and indicates the biases in the model [Fig. S7(a)]. The fractal dimension plots after proper disorder treatment are parabolic for IAM and TAAM refinement, but the plot is much steeper for TAAM refinement [Fig. S7(a)]. COFs 8 and 9 both had disordered solvents, which could not be modelled and were masked. These refinements of structures with masked solvent disorder using TAAM are only possible in NoSpherA2. Structures 8 and 9 also showed issues with data quality, which is reflected in the TAAM refinements by a high R1 and residual densities (Table 1). In structure 8, the data at high resolution were found to be mostly contributing to noise (Fig. S1). The residual electron-density map and fractal dimension plot indicated a small improvement in TAAM compared with IAM for structure 8 (Figs. S4 and S5). Structure 10 shows higher residual density after TAAM refinements than after IAM, although the data quality for structure 10 appears to be without problems (Fig. S1). TAAM refinement results in an almost featureless residual density map except around the C=C bond and ring of one component in the structure [Fig. 2(b)], which clearly shows the presence of disorder that was not visible in the IAM results [Fig. 2(a)]. The TAAM refinement, which included proper modelling of the disorder, further improved the structure model (R1_gt = 3.31% compared with 3.71% for TAAM with disorder not modelled) and gives a featureless residual density map [peak/hole = 0.36/−0.35 (e Å⁻³) compared with 0.72/−0.46 (e Å⁻³) for TAAM with disorder not modelled] [Table 1 and Fig. 2(c)]. The minor conformer refined to an occupancy of 0.0430 (16) for TAAM and to 0.048 (2) for IAM. Structure 11 is twinned; the data quality was poor and we observed a higher residual density in TAAM compared with IAM. The comparison of the C—C bond-length precision obtained from IAM and TAAM on structures 1–11 showed significant changes; the TAAM refinement showed better precision compared with IAM in all cases except in 8 and 9 (Table S1 and Fig. S6). In the case of 8 and 9 there was unmodelled solvent disorder which was left out of the refinement and, since TAAM is sensitive to such unmodelled atoms, this may lead to higher values of fitting statistics and lower precision compared with IAM.

Figure 2 Comparison of residual density maps obtained for structure 10 from IAM (a) and TAAM [(b), (c)] refinements. The presence of disorder is easily visible in the residual density obtained from the TAAM refinement (b), but was not distinguishable from the unmodelled bonding density in IAM (a). The disorder-treated TAAM (c) clearly shows a featureless residual density map. The contour level was set to 0.35 e Å−3.

We can conclude that, when employing TAAM refinements, any unmodelled features, like disorder or data quality issues, are more clearly exposed. Similarly to our earlier observations (Jha et al., 2020) and despite the issues of data quality, it was always possible to observe significant improvements in the geometry, displacement parameters and hydrogen-bond lengths in TAAM compared with IAM. The detailed comparisons of these individual parameters will be exemplified on a selected model structure later in the text.

3.3. Comparison of IAM and hybrid refinement on selected model structures

Hybrid refinements were performed for structure 12 using IAM–TAAM and HAR–TAAM and compared with the IAM results. The hybrid refinements were performed with and without assignment of formal charges to the sodium atom and hy­droxymethyl­pyridone, respectively.

The improvements in R1 and electron-density residuals from hybrid refinement compared with the IAM results were found to be similar to those of pure HAR (Woińska et al., 2016) and TAAM refinements (Jha et al., 2020) in structures with good data quality. For the hybrid IAM–TAAM refinement of structure 12, without assigning any formal charges, an improvement of ∼0.7 percentage points in R1 and ∼0.13 e Å⁻³ in residual electron density was found (Table 2). We observed no significant changes in refinement statistics between the IAM–TAAM neutral and IAM–TAAM and IAM–HAR with formal charge hybrid models (Table 2). The comparison of displacement parameters U_eq (non-hydrogen atoms) and U_iso (hydrogen atoms) between IAM and hybrid refinements showed lower values for the hybrid refinements (Fig. 3). However, there were no significant differences found in U_eq (non-hydrogen atoms) and U_iso (hydrogen atoms) among the different types of hybrid refinements (Fig. 3). The U_iso for hydrogen atoms in the hybrid refinements showed the opposite trend to those for pure TAAM refinement, where the U_iso values for hydrogen atoms were found to be higher than those from IAM (Jha et al., 2020).

Figure 3 Comparison of (a) Ueq (Å2) for non-hydrogen atoms and (b) Uiso (Å2) for hydrogen atoms obtained for structure 12 from IAM and hybrid refinements.

There were three types of bonds involving hydrogen atoms in structure 12, namely the hydrogen atom attached to (a) oxygen in water (H₂O), (b) carbon in the aromatic ring [C(ar)—H] and (c) carbon in a terminal methyl group (C—Csp³—H₃). A comparison of the X—H bond lengths shows a significant improvement in hybrid refinements compared with the IAM refinement in all three cases, with the data approaching corresponding neutron bond lengths (Allen & Bruno, 2010). There is no significant difference in the X—H bond lengths between different hybrid-mode refinements. The improvement in bond lengths compared with IAM was approximately 4, 12 and 7% for H₂O, C(ar)—H and C—Csp³—H₃ bond lengths, respectively (Fig. 4).

Figure 4 Comparison of the X—H average bond lengths of various bond types [H2O, C(ar)—H, C—Csp3—H3] for structure 12 with neutron bond lengths as defined previously (Allen & Bruno, 2010). The O—H bonds in water molecules are compared with the corresponding neutron bond lengths taken from Woińska et al. (2016). The numbers in parentheses in the bond type labels indicate the number of occurrences.

3.4. Comparison of accuracy and computational costs in IAM, HAR and TAAM refinements on selected model structures

We used structures 7 and 3_100 K for the comparison of IAM, HAR and TAAM refinements. Structure 7 has strong and high-resolution data. Additionally, it shows disorder. Structure 3_100 K has comparatively weak data. The IAM refinement (SHELXL or olex2.refine) on both structures converges in less than 10 s (Tables 2 and S2). However, R1 and the residual densities were higher in the IAM results compared with the aspherical refinements in the case of structure 7, while for structure 3_100 K the residuals were lower.

We used B3LYP along with different basis sets for the wavefunction calculation for HAR for structures 7 and 3_100 K. R1 and the residual densities improved with higher basis sets; however, the time taken for the refinement increased exponentially with the complexity of the basis sets in HAR. It took around 7–8 min for a small basis set (3-21G) to run one cycle of HAR, which consists of wavefunction calculation, scattering factor calculation from the wavefunction and finally the least-squares refinement. To achieve convergence the whole process was repeated iteratively, and it took 24 min for 3_100 K (Table S2) and 39 min for structure 7 (Table 3). Moderate basis sets such as 6-31G(d,p) took around 1–1.5 h to achieve convergence in HAR. It took around 5–6.5 h for a higher basis set such as Def2-TZVP in HAR to achieve convergence (Tables 3 and S2). The improvement in R1_gt and the residuals going from 3-21G to Def2-TZVP was around 0.13 percentage points and 0.05/−0.06 e Å⁻³, respectively, for 3_100 K (Table S2). For structure 7 R1 improves by ∼0.15 percentage points and there was no significant difference in residuals (Table 3), while there was a tenfold increase in time in both cases.

The refinement statistics for TAAM were found to be comparable to those obtained by HAR. A close resemblance was found with HAR using 6-31G(d,p) (Tables 3 and S2). The B3LYP functional and 6-31G(d,p) basis set were also used for the calculation of wavefunctions for the creation of the MATTS2021 databank (Jha et al., 2022) used in this study. With a larger basis set (Def2-TZVP) in HAR R1 decreases, but the highest and lowest residuals remain almost the same. One cycle of TAAM refinement, consisting of databank transfer, scattering factor calculation and finally the least-squares refinement, took around 20 s, which is on the same order as that of the IAM refinement. To achieve convergence iteratively the TAAM refinement took 2 and 5 min for structure 3_100 K (Table S2) and 7 (Table 3), respectively.

We also compared the fractal dimension plots from the different models for structure 7. The comparison of the fractal dimension plots between TAAM and HAR with different basis sets for disorder-treated structure 7 [Fig. S7(b)] shows a parabolic shape with minimal variations among the different models [Fig. S7(b)].

Additionally, a comparison of the atomic displacement parameters between IAM, HAR and TAAM for non-hydrogen (anisotropic, U^ij) and hydrogen atoms (isotropic, U_iso) was performed. The U^ij values of non-hydrogen atoms were compared by focusing on their equivalent isotropic displacement parameters (U_eq) (Fischer & Tillmanns, 1988). Aspherical refinements – both HAR and TAAM – showed a 10–11% decrease in all atomic displacement parameters compared with the corresponding atomic displacement parameter from IAM [Figs. 5(a) and S8(a)] for non-hydrogen atoms. There was no significant difference between the average U_eq of non-hydrogen atoms obtained from HAR using different basis sets and TAAM. Significant differences appeared in U_iso values for hydrogen atoms when using HAR or TAAM. In the case of structure 7, the average U_iso from HAR was ∼9% smaller compared with the IAM results, while the average U_iso using TAAM was higher (∼28%) compared with the IAM results [Fig. 5(a)]. The overall difference between HAR and TAAM average U_iso values for hydrogen atoms in 7 was ∼34%. In structure 7, the nitro­gen- and carbon-bound hydrogen atoms showed higher U_iso values and the oxygen-bound hydrogen atoms lower U_iso values in the TAAM refinement compared with IAM and HAR [Fig. 5(b)]. In 3_100 K, the U_iso values for C—H atoms from HAR and TAAM showed a different trend, with higher values compared with those from IAM [Fig. S8(b)]. The U_iso values for O—H and N—H hydrogen atoms from HAR and TAAM were slightly smaller than those from IAM. The U_iso of hydrogen atoms obtained from HAR using different basis sets showed very similar values. The average TAAM U_iso for C—H hydrogen atoms increased by approximately 34%, while after HAR the corresponding difference was only around 16% compared with the IAM results [Fig. S8(b)].

Figure 5 Comparison of displacement parameters of structure 7 after IAM, HAR and TAAM refinements: (a) Ueq (Å2) for non-hydrogen atoms, (b) Uiso (Å2) for hydrogen atoms.

A comparison of the X—H bond lengths reveals that the IAM bond lengths were the shortest in both structures 7 (Fig. 6) and 3_100 K (Fig. S9) compared with HAR, TAAM and neutron bond lengths, which is consistent with earlier findings (Woińska et al., 2016; Jha et al., 2020). The bond lengths obtained from TAAM refinement were comparable to those obtained by HAR in both structures. The TAAM bond lengths were moderately longer compared with those from HAR employing the 3-21G basis sets and slightly shorter in the case of the higher basis sets, 6-31G(d,p) and Def2-TZVP (Figs. 6 and S9). The biggest differences appeared in the case of water molecules in both structures 7 and 3_100 K. In structure 7 with good data quality, the O—H bond lengths of the water molecule from both HAR and TAAM were comparable to neutron bond lengths and much longer than those from IAM (Fig. 6). In the case of structure 3_100 K, the O—H bond lengths were overestimated in all aspherical refinements and were the longest for TAAM (Fig. S9). This confirms that, irrespective of the data quality, there is a significant improvement in structure models for both bond lengths and displacement parameters from any aspherical refinement compared with the spherical IAM refinement.

Figure 6 Comparison of X—H average bond lengths of various bond types [Z2—N—H, H2O, C(ar)—G, C=Csp2—H, C2—Csp3—H2, Z3—Csp3—H] for structure 7 with neutron bond lengths as defined previously (Allen & Bruno, 2010). The O—H bonds in water molecules are compared with the corresponding neutron bond lengths taken from Woińska et al. (2016). The numbers in parentheses in the bond type labels indicate the number of occurrences.