Experimental determination of partial charges with electron diffraction

Abstract

Atomic partial charges, integral to understanding molecular structure, interactions and reactivity, remain an ambiguous concept lacking a precise quantum-mechanical definition^1,2. The accurate determination of atomic partial charges has far-reaching implications in fields such as chemical synthesis, applied materials science and theoretical chemistry, to name a few³. They play essential parts in molecular dynamics simulations, which can act as a computational microscope for chemical processes⁴. Until now, no general experimental method has quantified the partial charges of individual atoms in a chemical compound. Here we introduce an experimental method that assigns partial charges based on crystal structure determination through electron diffraction, applicable to any crystalline compound. Seamlessly integrated into standard electron crystallography workflows, this approach requires no specialized software or advanced expertise. Furthermore, it is not limited to specific classes of compounds. The versatility of this method is demonstrated by its application to a wide array of compounds, including the antibiotic ciprofloxacin, the amino acids histidine and tyrosine, and the inorganic zeolite ZSM-5. We refer to this new concept as ionic scattering factors modelling. It fosters a more comprehensive and precise understanding of molecular structures, providing opportunities for applications across numerous fields in the chemical and materials sciences.

Fig. 1. With iSFAC, the chemical environment in the crystal structure matters.

The symbols ‘−’ and ‘+’ refer to negative and positive partial charge values of the respective atoms, respectively.

Fig. 2. Plot of atomic scattering factors f(s) compared with data resolution s (Å⁻¹).

a, iSFAC modelling refines individual scattering factors f(s) for each atom. This enables the extraction of partial charges for each atom. b, Conventional refinement assigns one scattering factor per element type.

Fig. 3. Experimental partial charges for ciprofloxacin hydrochloride.

The numbers in brackets denote the standard uncertainties of the last significant digit. For example, −0.50(3) for Cl1 means −0.50e ± 0.03e.

Fig. 4. Experimental partial charges for amino acids.

a,b, Experimental partial charges for tyrosine (a) and histidine (b). Element types are grouped by colour boxes: oxygen (light red), nitrogen (blue), hydrogen (pale yellow) and carbon (white).

Fig. 5. Comparison of computed energies for histidine conformations.

Energy for four different possible configurations of the N1–H1⋯O2 hydrogen bond calculated with DFT. DS1 and DS5 are observed in the crystal structure of histidine. DS1_HFIX has H1 fixed at N1 in the riding atom position, and an optimized configuration of the dimer (opt).

Fig. 6. Experimental partial charges for ZSM-5 across three different datasets (DS1, DS2 and DS3).

The red bars represent negative charges, and the blue bars denote positive charges.

Extended Data Fig. 1. Comparison between iSFAC modelling and computational methods.

Most of the partial charges are either determined via a) electron density-based, b) electrostatic potential-based, or c) electronegativity-based approaches. (d) Pearson correlation coefficients of the individual experimental partial charge distributions in histidine with quantum-mechanical calculations using B3LYP or ωB97XD with the basis sets 6-31G, 6-311G, and def2-tzvp. Plot grouped into electron-density-based, electrostatic-potential-based, and electronegativity-based approaches, respectively. Dark red boxes ignore the contribution of protons, only modelling partial charges for non-hydrogen atoms. Green boxes represent complete iSFAC modelling of partial charges for both hydrogen and non-hydrogen atoms based on Eq (3). Orange boxes are based on the values as for the dark red boxes, amended by one average value for the partial charge of each hydrogen atoms. See Sec. 1.7 for an in-depth discussion.

Extended Data Fig. 2. Juxtaposition of partial charges for L-histidine from two polymorphs.

Both polymorphs have similar crystal packing, but one is in spacegroup P2₁2₁2₁, the other one in spacegroup P2₁ with β = 98°.

Extended Data Fig. 3. Robustness of iSFAC modelling.

(a)-(f): with respect to data resolution, (g)/(h): with respect to data completeness. (a) Resolution dependence of partial charges for ZSM-5. The high-resolution limit was systematically reduced from 0.7Å to 0.95 Å in steps of 0.05 Å, and to 1.00 Å and 1.20 Å. Results remain consistent up to 0.95 Å; deviations become noticeable only at 1.00 Å and above. (b) Resolution dependence of partial charges for tyrosine. The high-resolution limit was systematically reduced from 0.75Å to 1.20 Å in steps of 0.05 Å. (c) For ciprofloxacin, at resolutions worse than 0.85Å, deviations in the positions of O3 and H3 become apparent. These deviations arise from the instability of the H3 position when the resolution is worse than 0.85 Å. (d) Restraining the O3-H3 bond distance recovers the partial charge values close to the high-resolution data. (e) For histidine, at resolutions worse than 1.00Å, deviations in the positions of C4 and H4B become evident, affecting their partial charges. (f) Restraining the H4B-C4 distance recovers the partial charge values close to the high-resolution data. (g) Spread of partial charges with leave-one-out refinement. Crosses mark the values using all reflections (red: anionic, blue: cationic). (h) Spread of U_ij values with leave-one-out refinement without iSFAC modelling, i.e. with conventional refinement. The comparison shows that the maxima and minima outliers are within expectations for crystallographic refinement.

Extended Data Fig. 4. Partial charges for zeolite ZSM-5.

(a) Comparison of partial charges for zeolite ZSM-5 with data collected at room temperature (RT) and at –110 ^°C. Inset shows the estimated standard uncertainties for the partial charges, and the isotropic ADP values U_iso. As expected, the errors are reduced for data collected at lower temperature. Otherwise, we do not observe any systematic trends. The Pearson correlation coefficient of 0.91, inset at the top left, compares the values for the low temperature and room temperature measurements. The correlation was computed with the moduli of the partial charges for the oxygens, becaue otherwise, the correlation would be overestimated (0.98). (b) Visual representation of the asymmetric unit of ZSM-5 zeolite, highlighting its unique framework structure characterized by 12 T-sites (tetrahedrally coordinated atoms) and 26 oxygen atoms.

Extended Data Fig. 5. Variation of partial charges with crystal thicknes for L-histidine.

(a) partial charges δq for four data sets of L-histidine of for different crystals. (b) picture of the crystal for DS2 (c) picture of the crystal for DS4 (d) picture of the crystal for DS5 (e) picture of the crystal for DS6 (c)-DS 4 and (e)-DS 6 are thickest, which the underlying lacey carbon hardly visible, (d)-DS 5 is thinnest.

Extended Data Fig. 6. iSFAC modelling against X-ray data.

High quality data sets of (a) Ca tartrate and (b) L-histidine, CCDC ID KAGGEI, were refined with iSFAC modelling. In neither case refinement converges, even after thousands of cycles of least square refinement with SHELXL, there are still non-zero shifts. The resulting partial charges are outside physical plausibility ≫ 1 and ≪ − 1, and protons with negative partial charges. (c) For comparison, iSFAC partial charges from the ED data.

Extended Data Fig. 7. Comparison of electron scattering factors.

Electron scattering curves for some ions, illustrating the difference between Eq. (2), as published in, and the classical Mott-Bethe formula.

Extended Data Fig. 8. Comparison of the R₁ factor between conventional and iSFAC modeling.

(a) Tyrosine: the upper section compares the R₁ factor between conventional and iSFAC models, while the table below presents the percentage change (Δ) in R₁-iSFAC compared to conventional R₁. (b) Histidine: the top panel shows the R₁ factor comparison between conventional and iSFAC models, with the table below indicating the percentage change (Δ) between the two. (c) ZSM5: similarly, the top panel compares the R₁ factor between conventional and iSFAC models, with the table underneath displaying the percentage change (Δ) in R₁-iSFAC compared to R₁.

Extended Data Fig. 9. iSFAC modelling enhances hydrogen signal.

Hydrogen omit maps for tyrosine ((b), with iSFAC modelling; (c), without) and histidine ((e), with iSFAC modelling; (f), without) illustrate the enhanced signal of hydrogen with iSFAC modelling ((b) for tyrosine, (e) for histidine) compared with conventional modelling. OMIT maps are shown as isosurface maps at 3σ. Note: All protons in (a) and (d) were refined freely and unconstrained. (g) Hydrogen networking in tyrosine crystal packing.

Extended Data Fig. 10. ESP maps.

Tyrosine (a-c) and Histidine (d-f): Comparison of the experimental electrostatic potential map (E-ESP, a/d respectively) with the quantum-mechanical electrostatic potential map (QM-ESP, b/e respectively). (c) and (f) show the orientation of the molecules, the next-neighbours composition for the QM maps, and highlight the protons (labelled yellow with “H”) responsible for the positive patches of the carboxyl groups and (for tyrosine, c) the hydroxyl group. QM-ESP computed with ORCA Version 6.0.1 and MULTIwfn^, with the B3LYP density functional and the 6-311G(d) basis set.