Occupancy Determination from Resonant X‑ray Diffraction

Abstract

We analyze the effect of integrated resolution, data scaling, structural refinement, and crystal symmetry on the extracted scattering perturbations and their uncertainties for anomalous (resonant) X-ray diffraction on mixed-metal molecular clusters. We probe the metal constituency, substitutional homogeneity, and positional disorder of both biased and unbiased ligand environments. Anomalous X-ray diffraction studies were conducted on (^FtbsL)­Zn₂Ni­(py) (1), [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2), and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3). Analysis of diffraction data collected at energies along the Ni and Zn K-edges on [Zn₂Ni] clusters reveals data resolution, and the accuracy of the structural model greatly impacts the resonant scattering factors (f′, f″) and their uncertainties. Occupancy studies on all three clusters were conducted in three ways and compared: (i) using the f′ values of each metal site refined from diffraction data collected at energies along the Ni and Zn K-edges; (ii) using the f′ value of each metal site refined from diffraction data collected with in-house, Cu Kα radiation; and (iii) refining each metal site as a Zn/Ni disordered pair and fixing the relative Zn-to-Ni occupancies as 2:1 across the core with supporting spectroscopic evidence from scanning electron microscope energy-dispersive spectroscopy. First, we observe highly ordered structural compositions, even when statistical mixing was anticipated; and second, we discovered that in-house X-ray diffraction collection was as precise in composition determination as synchrotron sources for this study.

1. Introduction

Probing the chemical makeup of polynuclear reaction centers is central to understanding how composition imparts functionality. Some of the most important chemical machinery (e.g., biological inorganic cofactors, − catalysts − ) possess multimetallic structural elements that are essential to driving function. Interrogation of these complex structures is traditionally left to synchrotron-enabled anomalous (or resonant) X-ray diffraction − (AXD) or neutron diffraction techniques. − In this study, we seek to address whether in-house X-ray diffraction dual source capabilities enable interrogation of site-specific occupancies in mixed metal inorganic complexes compared to what is acquired from state-of-the-art resonant diffraction studies collected at synchrotron radiation sources. ,,−

The occupational disorder of a series of previously reported heterobimetallic dizinc–nickel complexes (1–3, Scheme ) is examined by collecting diffraction data at energies along the Ni and Zn K-edges. The combination of obtaining spectroscopic (f′ + f″) information with structural precision (f ₀) affords the site-specific signatures for atoms comprising the core. Although the resonant/anomalous effect is maximized at an element’s K-edge, ,,− perturbations to scattering are still present when the incident radiation is significantly off-resonance, thereby allowing for rigorous examination of occupancy using a fixed energy Cu Kα source. Indeed, resonant scattering perturbations refined from Cu Kα data replicate occupancies determined via synchrotron-based AXD.

1. Compounds Surveyed in This Study.

Our study allows us to probe the metal constituency, substitutional homogeneity, and positional disorder of both biased and unbiased ligand environments for a series of previously reported bimetallic Ni–Zn complexes (^FtbsL)­Zn₂Ni­(py) (1), [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2), and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3). Furthermore, we examine which factors (i.e., resolution, scaling methods, uncertainty propagation, radiation sources) impact our ability to refine the anomalous scattering perturbations, which in turn reveal each cluster‘s structural composition. We make two remarkable discoveries: (1) we observe highly ordered structural compositions, even when we expected statistical mixing; and (2) we discovered that anomalous scattering perturbations refined from an in-house X-ray diffraction collection were as precise as synchrotron sources for this study, expanding the capabilities of readily accessible instrumentation.

2. Experimental Section

2.1. X-ray Diffraction Data Collection

All diffraction experiments with incident wavelengths at energies along the Ni and Zn K-edges were performed by using a synchrotron light source. Data collection on a crystal began with full structure determination at 30, 25, or 17.012 keV (depending on the capabilities of the synchrotron). Following the acquisition of the high-energy data set, X-ray absorption scans across the Zn and Ni K-edges were measured. The energies of interest for the AXD studies were selected based on the features of the absorption spectra. Diffraction data at each energy of interest were acquired via a series of ϕ scans, the exact collection strategy for each crystal is included in the Supporting Information (Tables S40–S42). Roughly 1000–4000 reflections were acquired at each energy. The number of diffraction data sets collected was balanced against the lifetime of the crystal under high flux radiation (5–6 h) and the desired resolution of the data. Diffraction experiments utilizing Mo or Cu Kα radiation were performed in-house on a D8 Bruker Venture single-crystal diffractometer equipped with a Photon III–C14 detector and IμS microfocus X-ray source. Immediately following the full structural collections with Mo and Cu Kα, the crystals were removed and loaded into a JEOL 7900F SEM instrument for analysis by energy-dispersive spectroscopy (EDS). The ratio of Ni and Zn in the single crystals was assessed via comparison of the areas under their Kα fluorescence peaks by the EDS software AZtec.

2.2. Data Analysis of Scattering Perturbations

Analysis of anomalous scattering perturbations derived from diffraction data collected at energies along the Ni and Zn K-edges began with compound 1. The low-energy data sets were integrated using SAINT and scaled with a numerical absorption correction using SADABS. The numerical absorption correction was calculated based on crystal size and theoretically calculated absorption coefficient (μ) values from WinGX (Brennan values). The atomic coordinates and displacement parameters from the 30 keV solution were used as the structural basis for each low energy data set. Refinement of the high energy solution required modeling positional disorder for a tert-butyl dimethyl silyl arm of one ligand subunit. The refinement of f′ and f″ was conducted in Jana2020. The positions and displacements of each atom in the asymmetric unit were fixed, minimizing the total number of parameters for each data set to seven, three f′ and f″ terms for each metal, and the scale factor.

3. Background

In the high-energy limit of diffraction, the scattering of X-rays by an atom is described by f ₀ or the normal scattering factor. As the incident radiation approaches the energy of an electron residing in an atomic orbital, a 1s binding energy, for instance, photon absorption and emission processes, perturb the normal scattered wave. These scattering perturbations terms, f′ and f″, are described in eq and Figure .

$$f(\mathit{Q},\omega )=f_0(\mathit{Q})+f^{\prime }(\omega )+i{f}^{″}(\omega )$$ 1

1.

(a) Theoretical anomalous scattering perturbations for a metal atom (Zn⁰) when the incident radiation coincides with a 1s core electron excitation. (b) Argand diagram illustrating the effect of both f′ and f″ on the normal scattering factor f ₀ and the resulting structure factor, F _hkl. (c) Wave depiction of the scattering factors: f ₀, f′, and f″.

The real scattering perturbation term (f′) is a dispersive term. At the absorption edge, f′ will diminish the total scattered amplitude because it is 180° out of phase with f ₀. The imaginary scattering perturbation term (f″) takes on positive values at and above the binding energy due to the photoionized atom recombining with its photoelectron producing a 90° phase shift with respect to f ₀. , In fact, f″ is intimately related to a crystal’s absorption coefficient (μ­(ω)) as described by the Optical Theorem in the quantum description of scattering (eq )

$$f^{″}(\omega )=\frac{mc}{{4\pi e}^2}\omega \mu (\omega )$$ 2

where m is the mass of an electron, e is the charge of an electron, c is the speed of light, and ω is the frequency of the incident radiation.

In an AXD experiment, crystallographic diffraction data are collected at energies along the resonant absorption edge(s) of interest. The scattering perturbation terms, f′ and f″, are refined from the diffracted intensities and plotted as a function of the energy of incident radiation. The changes to f′ and f″ that occur at the resonant edge(s) are then used to assign atomic occupancies in mixed-metal molecules, ,,− ,, materials ,,,,,− ,, and proteins, or determine redox-localization in homometallic inorganic clusters − and biological cofactors. , In the case of heterobimetallic clusters, the perturbation terms can provide information on metal occupancy because |f′| and |f″| are maximized for the metal whose relevant absorption edge is on resonance with the incident radiation. Experimentally refined f′ values at each energy and theoretical f′ values for M₁ and M₂ can be used to solve for the M₁ or M₂ occupancy at each site with eq (eq )­

$$f^{\prime }({\mathrm{M}}_{\mathrm{o}\mathrm{b}\mathrm{s}})=\mathrm{x}{f}^{\prime }({\mathrm{M}}_1)+(1-\mathrm{x})f^{\prime }({\mathrm{M}}_2)$$ 3

where x is the weighting factor of M₁ in each metal site and f′ (M_theoretical) is the theoretical value for either M₁ or M₂.

4. Results

4.1. Extraction of f′ and f″ for Centrosymmetric Crystal: (^FtbsL)­Zn₂Ni­(py) (1)

Refinement of f′ at energies along the Ni K-edge shows a large perturbation in f′ for the pyridine-bound site and a lack thereof for the nonpyridine bound 3- and 4-coordinate sites (Figure S25). At the Zn K-edge, the reverse is true (Figure S27). The large response in the f′ term at one metal’s resonant K-edge and lack thereof at the other K-edge informs us that each site is exclusively occupied by either Ni or Zn. Therefore, the pyridine-bound position was assigned as 100% Ni, and the two other sites were assigned as 100% Zn (Figure c).

2.

Plot of the scattering perturbation terms f′ and f″ for the Ni₁ site (a, red), Zn₁ site (b, black), and Zn₂ site (b, blue) in (^FtbsL)­Zn₂Ni­(py) (1) alongside the Ni (a, purple) and Zn (b, purple) K-edge X-ray absorption spectra of 1 (scaled for visual clarity). The error bars represent 2σ. (c) Solid-state structure of 1 solved from diffraction data collected at 30 keV. Thermal ellipsoids are at 50% probability level. Color scheme: Zn, gray; Ni, purple; N, blue; C, light gray; Si, light pink; F, lime green.

Additionally, f″ was refined for the centrosymmetric crystal system of 1 at energies along both Ni and Zn K-edges (Figure S26 and S28). The accuracy of the refinement is validated by comparison of the f″ curve to the Ni and Zn K-edge X-ray absorption spectra (Figure a,b). As previously mentioned, the scattering perturbation described by the f″ term is related to an absorptive process and therefore should closely align with the measured X-ray absorbance spectrum. The theoretical basis for refinement of f″ in a centrosymmetric crystal system is discussed further below and in the Supporting Information, (Addendum S91). Furthermore, at energies along the Ni K-edge, only the f″ value for the pyridine-bound metal site substantially changes as a function of the incident radiation energy. The reverse is true at the Zn K-edge, further supporting the assignment of each site described in Figure c.

4.2. Effect of Integrated Resolution on f′ and f″ in (^FtbsL)­Zn₂Ni­(py) (1)

To better understand how treatment of diffraction data impacts the extracted values of f′, f″, and their uncertainties, we sought to test the effect of three different aspects of data quality and treatment on their refinement: (1) resolution of the diffracted intensities, (2) data scaling, and (3) the accuracy of the full structural solution used for the atomic coordinates and displacement parameters. The effect of these different factors was tested by using the diffraction data collected for 1 at energies along the Ni K-edge.

Our first comparison focused on the effect of data resolution on the refinement of anomalous scattering perturbations and their associated errors. During integration, the diffraction data collected at each energy were integrated to 0.8, 1.0, 1.2, and 2.0 Å. The absolute values of f′ and f″ for the Ni₁ site at each energy remained consistent within error (Figure a,b). If we focus our analysis on the value of f′ at a single energy (8331 eV) across many resolutions, we can see that as the integrated resolution increases from 0.6 to 2.0 Å, the absolute value of f′ at the Ni₁ site decreases (Figure S41). The results of this analysis for the f″ term are less consistent. Across all energies, the variation in f″ as a function of data resolution is more erratic (Figure S42). The effect of data resolution is more prominent in the refinement of f′ and f″ for the off-resonance Zn₁ and Zn₂ sites (Figures S43–S46). Unlike the Ni₁ site, as the resolution increases from 0.6 to 2.0 Å, the absolute value of f′ at both Zn sites also increases (Figure S43 and S45). Similarly to the Ni₁ site, the f″ terms of the Zn₁ and Zn₂ sites vary erratically as a function of resolution (Figures S44 and S46).

3.

Plot of f′ (a) and f″ (b) for the Ni₁ site of (^FtbsL)­Zn₂Ni­(py) (1) in which the data have been integrated to various resolutions. (c) Plot of the uncertainty in the f′ value of the Ni₁ site in 1 at 8331 eV as a function of data resolution; the line of best fit is y = 0.06x + 0.02 (R ² = 0.96). Plot of f′ (d) and f″ (e) for the Ni₁ site of 1 in which the structural solution has been refined in various ways described in the main text. (f) Plot of the uncertainty in the f′ value of the Ni₁ site in 1 at 8331 eV as a function of the R ₁ value of the high energy structure used as a basis; the line of best fit is y = 2.80x – 0.07 (R ² = 0.96).

For all three metal sites, the uncertainty of the scattering perturbations increases consistently as a function of the resolution (Figure c). The uncertainties in f′ for all three metal sites increase by roughly 0.06 electrons per unit increase in resolution at an energy of 8331 eV (Figures S47–S49). Generally, the uncertainty in f″ is larger across all energies and increases more dramatically as a function of resolution than f′ (0.25 electrons per unit resolution).

To rule out the linearity of this relationship being caused by a smaller data-to-parameter ratio, the uncertainties of f′ and f″ were plotted against this variable for the different resolutions of the data set collected at 8331 eV. In all cases, the data-to-parameter ratio was calculated by dividing the total number of reflections by the number of parameters used in refinement. The results show that the linear relationship does not exist as strongly between the number of reflections and uncertainty in f′ or f″ (indicated by the R ² value of the line of best fit, Figures S50–S52). Therefore, we attribute the increase in uncertainty in both f′ and f″ to the resolution of the data itself and not the decreased number of reflections that results from restricting the integration limits in reciprocal space.

4.3. Effect of Scaling on f′ and f″ in (^FtbsL)­Zn₂Ni­(py) (1)

Our next assessment focused on the effect of data scaling on the refinement of anomalous scattering perturbations and their associated errors. As such, we compared the results of (1) the normal refinement strategy, (2) refinement where the numerical absorption correction was set to zero, and (3) refinement where Friedel’s pairs were treated as equivalent (further details in the Supporting Information). These small changes to the scaling process led to no statistically significant difference between the absolute values or f′, f″, or their associated errors in any of the three metal sites of 1 (Figures S29–S34).

4.4. Effect of the Structural Model on f′ and f″ in (^FtbsL)­Zn₂Ni­(py) (1)

Finally, we turned our attention toward the effect of the refinement of the high energy structural solution on f′, f″, and their associated uncertainties. Although the refinement of the high energy structural solution is separated from the treatment of the diffracted intensities at low energies, the anomalous scattering perturbations are ultimately refined against the predetermined structural model. As such, we selected five different refinement strategies that would systematically vary the associated R ₁ and wR ₂ values of the structural solution. Those strategies are as follows: (1a) positional disorder on the tert-butyl dimethyl silyl arm of one ligand subunit was unmodeled (R ₁ = 0.0495, wR ₂ = 0.1180); (1b) all atoms were isotropically refined (R ₁ = 0.0540, wR ₂ = 0.1313); (1c) hydrogen atoms were excluded from the structural solution (R ₁ = 0.0600, wR ₂ = 0.1806); (1d) hydrogen atoms were excluded from the structural solution and all atoms were refined isotropically (R ₁ = 0.0686, wR ₂ = 0.1999); and (1e) the solvent diethyl ether molecule was deleted and refined as unresolved electron density (R ₁ = 0.0941, wR ₂ = 0.3074). The crystallographic details of each of these solutions are described in the Supporting Information (Table S15–S17).

The foregoing analysis showed that the structural solution has the largest impact on the absolute value and uncertainty of the anomalous scattering perturbations. The effect of the R ₁ value on the absolute value of f′ is consistent for all three metal sites. For Ni₁, Zn₁, and Zn₂, the structural models 1a–1d display an increase in the absolute value of f′ as the R ₁ value increases. However, the solution with the largest R ₁, 1e (where the solvent molecule was deleted and refined as unresolved electron density), decreases the |f′| by as much as 0.25 electrons in the Ni-occupied site and 1.0 electron in the Zn-occupied sites (Figures d and S60, and S62). This trend is more apparent when focusing on the f′ value for each metal site at 8331 eV as a function of R ₁ (Figure ). The effect of the quality of the structural refinement on the f″ curves for all three metal sites is more irregular, similar to that in the resolution analysis. In particular, the pre-edge region of the Ni₁, Zn₁, and Zn₂ sites vary by as many as 3.0 electrons (Figures S59, S61, and S63).

4.

f′ values for the Ni₁ (a), Zn₁ (b, black), and Zn₂ (b, blue) sites of (^FtbsL)­Zn₂Ni­(py) (1) at 8331 eV plotted as a function of R ₁ of the structural model used in refinement.

For all three metal sites, the uncertainties of f′ and f″ increase consistently as a function of the R ₁ of the structural solution. The uncertainties in f′ for all three metal sites increase by roughly 0.03 electrons per 1% increase in R ₁ at an energy of 8331 eV (Figures f and S70–S72). The increase in the uncertainty in the f″ value is less well-defined by a linear fit than that for f′ at 8331 eV; however, when fitted with a linear model, the f″ of each metal site increases by roughly 0.1 electrons per 1% increase in the R ₁ value of the structural solution.

4.5. Extraction of f′ and f″ and Occupancy Studies in Non-Centrosymmetric Crystals: [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2) and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3)

Following the analysis of f′ and f″ in the C ₁-symmetric complex 1, where no metal mixing between core sites is expected, we sought to analyze compounds with C ₃-symmetric ligand architectures, where an even distribution of Ni and Zn in the core was anticipated. As such, the scattering perturbation terms were extracted from X-ray diffraction data collected at energies along the Ni and Zn K-edges for 2 and 3 in an analogous manner to that performed for 1 described in Section . The experimental f′ values for the M₃ site in 2 at energies along the Ni K-edge show a large perturbation as a function of the energy of the incident radiation. The f′ traces of the M₁ and M₂ positions do not feature the same response (Figure S76). Conversely, at energies along the Zn K-edge, the largest perturbations to the f′ term occurred in the M₁ and M₂ positions. There is, however, a small dip in the f′ trace of the M₃ position that indicates minimal Zn-mixing into that site (Figure b). The f″ trace of M₁ and M₂ closely aligns with the Zn K-edge X-ray absorption profile, whereas the f″ trace for the M₃ position remains unperturbed at energies along the Zn K-edge (Figures c and S79). These results indicate that a minimal amount of mixing occurs between the metal sites. The estimated occupancies of each site were calculated using data collected at 8296, 8301, 9631, and 9641 eV and referenced against theoretical f′ and f″ values. Although the use of theoretical references introduces systematic error in occupancy calculations, the synthetic inaccessibility of the all-Zn or Ni equivalent to 2 (or 3) made use of homometallic structurally equivalent clusters impossible. The average occupancies at all four energies yield results consistent with what is seen qualitatively; M₁ site: 0.86(9) Zn, M₂: 0.82(8) Zn, and M₃: 0.89(9) Ni. These freely refined occupancies yield a Zn/Ni ratio of 1.8:1.0, consistent with what is expected for the core (Table S27).

5.

Solid-state structures of the anions of [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2, a) and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3, d), thermal ellipsoids are at the 50% probability level. Color scheme: Zn/Ni site, dark purple; N, blue; C, gray; F, yellow; Si, pink. Plot of f′ (b) and f″ (c) for the three metal sites in 2 at energies along the Zn K-edge overlaid with the Zn K-edge X-ray absorption spectra (scaled for visual clarity). Plot of f′ (e) and f″ (f) for the three metal sites in 3 at energies along the Ni K-edge overlaid with the Ni K-edge X-ray absorption spectra. The error bars for the anomalous scattering components represent 2σ.

The experimental f′ and f″ values of the metal sites in 3 collected at energies along the Ni K-edge showcase a large response at the M₃ position and a small response at the M₁ and M₂ positions as seen in Figure e,f (f″ versus energy for the M₁ and M₂ positions can be found in Figure S83). Unfortunately, corroborating evidence at the Zn K-edge could not be collected due to the crystal’s rapid decomposition on the beamline. Nevertheless, estimated occupancies of each site were calculated using data collected at 8317, 8322, and 8327 eV and referenced against theoretical Ni and Zn f′ values. The average occupancies at all three energies yield results consistent with what is seen qualitatively; M₁ site: 0.82(15) Zn, M₂: 0.58(14) Zn, and M₃: 0.82(13) Ni. The deviation of the Zn/Ni ratio (1.1:1.0) from the expected value of 2:1 is unsurprising given the use of theoretical references (Table S30).

4.6. Extraction of f′ and f″ from Cu Kα Diffraction Data

In addition to conducting anomalous X-ray diffraction studies using synchrotron radiation, we were also interested in testing whether diffraction data collected with an in-house Cu Kα source would be sufficient for refining f′ parameters and solving for metal occupancies. We expect the f′ value of a core site when collected at an energy significantly off resonance (from the K-edge) to better reflect the metal occupancy because there is little effect from the fine absorbance features due to oxidation or ligand environment that typically strongly affect f′ at the K-edge. Diffraction data were collected on a high-quality crystal of 1, 2, and 3 using Cu Kα radiation. The integration, scaling, and f′ refinement of the Cu Kα data were conducted in a similar manner to that of the synchrotron-collected data sets. The only difference in the data treatment is that the full structural solution collected with Cu Kα radiation allows for the refinement of all atomic coordinates, displacement parameters, and anomalous scattering perturbations with one data set. Refinement of the anomalous scattering perturbations in 1, 2, and 3 yielded f′ and f″ values that were used to calculate metal occupancies similarly to what was done with diffraction data collected at energies along the Ni and Zn K-edges. Given that no site-mixing occurs in 1, two references were available to calculate the occupancy of each site in 2 and 3: Ni and Zn f′ values obtained from 1 or theoretically derived values. Occupancies for 2 and 3 were calculated using both references, and ultimately the more appropriate reference was selected based on the closeness of the associated Zn/Ni ratio to the expected value of 2:1 (Tables S34 and S36).

Occupancy calculations for 2 utilizing diffraction data collected with Cu Kα radiation and reference f′ values derived from 1 yielded the following results: M₁ site: 0.862(14) Zn, M₂: 0.975(12) Zn, and M₃: 0.936(14) Ni. These freely refined occupancies yield a Zn/Ni ratio of 2.2:1.0, consistent with what is expected for the core. Identical calculations for 3 utilizing theoretical f′ reference values yielded the following results: M₁ site: 0.92(4) Zn, M₂: 0.97(4) Zn, and M₃: 0.89(4) Ni. These freely refined occupancies yield a Zn/Ni ratio of 2.0:1.0.

4.7. Refinement of Occupancies from Electron Density: 1, 2, and 3

In addition to using the experimentally refined f′ values to calculate occupancy, we also modeled the metal occupancy at each site in all three solutions from both Mo and Cu Kα diffraction data sets as Zn/Ni disordered pairs using the EXYZ and EADP constraints. The occupancies were refined with two variables; the sum of Ni and Zn occupancies at each site is 1, and the total Zn/Ni ratio is 2:1 in the core using the SUMP command. Restraining the relationship between the occupancies of Zn and Ni at each site and across the core is necessary for stable refinement. The use of the SUMP command was justified through additional spectroscopic evidence collected on the crystals via scanning electron microscope energy-dispersive spectroscopy (SEM-EDS). SEM-EDS utilizes the primary electron beam of an SEM instrument to excite core electrons and measure the fluorescence that occurs as a result of relaxation. The ratio of the areas under the Kα curves (resulting from an L-shell to K-shell relaxation) in the fluorescence spectrum for each metal were compared to solid-state Zn and Ni calibration curves to identify the Zn/Ni ratios for each crystal. The Zn/Ni ratios obtained for the crystals of 1, 2, and 3 were 2.05:1.00, 2.00:1.00, and 1.98:1.00 (Figures S5, S12, and S19). The comparison of occupancies calculated via refinement of the diffraction data collected at the Ni and Zn K-edges, refinement of the diffraction data collected with Cu Kα radiation, and the results of the structural refinement utilizing the SUMP command are included in Figure . The results show that the results of occupancy calculations across all methods are consistent for compounds 1, 2, and 3.

8.

Refined occupancies of the predominantly Ni site in complexes 1, 2, and 3 using the various methods indicated.

5. Discussion

5.1. Refinement of f″

Whereas both anomalous scattering perturbations offer spectroscopic insight with structural precision, refinement of the f″ term provides exact absorptive profiles for each unique crystallographic metal site within a complex or material. Indeed, extraction of f″ for both centrosymmetric and noncentrosymmetric crystals revealed valuable insights into the metal-site occupancies for complexes 1, 2, and 3. Unlike the f′ perturbation term, the features of the f″ curve are binary. Due to its dispersive nature, f″ slowly decreases in magnitude as the incident wavelength deviates from the absorptive K-edge. If the incident radiation is above the binding energy of a core electron in the metal atom, then photoelectric absorption occurs. Recombination of the photoelectron with its constituent atom augments the “normal” (f _o ) scattering event. This process is described by the term f″, and as such, the contribution of the imaginary term to scattering is near zero below a resonant absorption edge. It is important to note, however, that at some energetic threshold, the refinement of f″ becomes trivial as scattering becomes increasingly Thompson-like. This transition from resonant scattering conditions at high energies explains the nonsensical values and large errors in the refinement of f″ at the Mo Kα energy.

For complexes 1, 2, and 3, the largest response of the f″ trace to the Ni K-edge position occurred at the predominantly Ni occupied sites (Figure ). This response, or lack thereof, at the predominantly Zn-occupied sites supports the occupancy determined from the f′ values. Given that previously reported anomalous X-ray diffraction studies have noted an inability to refine f″ in centrosymmetric crystal systems, it is worth discussing its refinement in 1. In both cases of crystal symmetry, the scattering perturbation terms are calculated by assessing the change in the structure factor as a function of the incident radiation. The full derivation for the calculation of f′ and f″ for centrosymmetric and noncentrosymmetric crystals is included in the Supporting Information (Addendum S91). Interestingly, f″ carries a larger uncertainty than f′ across all crystals. However, this difference is more pronounced for the centrosymmetric crystal 1 (Tables S38 and 39). This is likely due to the fact that in noncentrosymmetric crystals there is a difference between I _hkl and I _–h–k–l that does not exist for centrosymmetric crystals. This difference provides an additional metric against which to refine, thus improving the precision of the f″ calculation.

6.

f″ values for each metal site in (^FtbsL)­Zn₂Ni­(py) (1, a) in [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2, b) and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3, c), refined from diffraction data collected along the Ni K-edge. In all three plots, the red trace corresponds to the predominantly Ni occupied site in the cluster.

In addition to providing information on the absorption profile with site-specificity, one benefit of f″ refinement is that the accuracy of the f′ and f″ refinement can be compared against a different type of data, X-ray absorption spectroscopy. As seen in Figure , the f″ and XAS profiles for the Ni₁ position at the Ni K-edge and the Zn₁ and Zn₂ positions at the Zn K-edges match. These results, in addition to the small uncertainties in f′ and f″, indicate that data workup strategy described herein accurately refines the experimental anomalous scattering components.

5.2. Impact of Resolution, Scaling, and Structural Modeling on the Refinement of f′ and f″

Analysis of the impact of several factors on the refinement of f′ and f″ revealed that the resolution of the diffraction data collected at each energy does not impact the absolute value of f′ or f″ in a statistically significant way as evidenced by the overlapping error bars in Figure . However, the resolution greatly impacts the uncertainty of each parameter. The increase in the uncertainty of f′ and f″ could be due to two factors. The first of these is that as the resolution decreases, the number of total reflections decreases. The smaller number of reflections may increase the uncertainties across every parameter. The lack of a linear relationship between the error in f′ and f″ and the data-to-parameter ratio of the data set, however, indicate that the cause of the increase in the uncertainties is intrinsic to the resolution itself. This finding makes physical sense because the heaviest atoms in the unit cell contribute most significantly to the higher resolution reflections. As the volume of integrated reciprocal space is decreased, the reflections most impacted by the presence of Ni and Zn are excluded from the refinement, decreasing the certainty in the values of f′ and f″ for each metal site.

The aforementioned changes to the scaling of the data sets had little impact on the absolute value or uncertainties of f′ or f″ for the Ni₁, Zn₁, or Zn₂ sites. The integrated intensities are combined into a set of structure factors and normalized according to symmetry during the data scaling process. We suspect the method of absorption correction (numerical, semiempirical, or refined corrections) may have a larger impact on the refinement of f′ and f″; thus, future studies will analyze this relationship more closely.

The largest impact on the refinement of f′, f″, and their respective uncertainties was the high energy structural solution used to fix the atomic coordinates and displacement parameters. As the R ₁ of the structural solution increased, the absolute value of f′ and f″ deviated further from what was refined using the normal strategy. Additionally, the uncertainties of these quantities increased linearly as a function of R ₁. R ₁ is a statistic used in crystal structure refinement to indicate the accuracy of the structural model with respect to the diffraction data. The equation defining R ₁ is as follows

$$R_1=\frac{\sum ||F_{obs}|-|F_{calc}||}{|F_{obs}|}$$ 4

where F _obs and F _calc are the observed and calculated structure factors. An increase in R ₁ for a given refinement model indicates a less accurate structural model. Despite the fact that only the Zn and Ni atoms experience a significant anomalous scattering effect, every atom across every unit cell contributes to every diffracted intensity in reciprocal space. As such, an accurate structural model is required in order to properly account for the effects of other atoms on each structure factor. The most extreme example, where the solvent diethyl ether molecule was deleted and refined as unresolved electron density, led to anomalous scattering perturbations for all three metal sites that varied dramatically from what was extracted using a structural solution where all atoms are accounted for. These studies indicate the importance of a high-quality, fully refined structural solution prior to refinement of f′ and f″ in Jana2020.

5.3. Results of AXD Occupancy Studies from Synchrotron and Cu Kα Data Sets

AXD studies indicate that each metal site in 1 remains substitutionally homogeneous. The C ₁-symmetric nature of the ligand scaffold in 1 prevents rotation of the cluster across different unit cells. The C ₃ symmetric nature of the ligand in 2 and 3 makes such rotation possible; therefore, in these structures, one would expect no metal occupancy preference between each core site (i.e., each site is 33.3% Ni and 66.6% Zn). Refinement of the diffraction data sets at the Ni and Zn K-edges for 2 and 3, however, indicate that the core is preferentially oriented with respect to crystal packing. The lack of Zn₃ and Ni₃ homometallic standards makes determination of absolute occupancies at each site difficult. Calculation of occupancies via eq referencing theoretical Ni and Zn f′ values indicates that each site is predominantly Ni or Zn, with between 5 and 25% mixing with the other metal. Because the all-Zn and Ni clusters were synthetically inaccessible, we sought to investigate whether theoretical references would be better suited for calculating occupancies further from the absolute K-edge. As such, we turned toward the use of in-house Cu Kα radiation to investigate this hypothesis.

The anomalous signal of crystals containing first-row transition metals when irradiated with Cu Kα X-rays (8042 eV) has been used previously to experimentally phase diffraction data from protein crystals. Given this precedent in macromolecular crystallography, we hypothesized that refinement of f′ components for [Zn₂Ni] clusters would be possible using in-house Cu Kα data. Gratifyingly, f′ was refined with very small uncertainties from Cu Kα diffraction data collected on crystals of 1, 2, and 3 in-house. The f″ term could not be refined accurately because the Cu Kα energy falls far below the absorptive edge for both Ni and Zn. However, because f′ is a dispersive term, its perturbative effect on scattering is present even when the incident radiation is far below resonance. Thus, it can be refined with a very low uncertainty and used to calculate metal occupancies at each site. The results of these occupancy calculations for 2 agree with what is calculated from AXD data. The occupancies calculated for 3 are more reasonable than what is calculated from AXD data, which can be seen from the Zn/Ni ratio summed across the core (Table S36).

We attribute the increase in accuracy of the occupancy calculations from the f′ parameter because Cu Kα radiation is far off resonance from the Ni or Zn K-edge. Although the difference between the f′ value of Zn and Ni is not maximized at this wavelength (Figure ), the issue of convoluting effects of K-edge transitions on the value of f′ is avoided. This was especially important for 2 and 3, where homometallic structural analogues are not synthetically accessible, and thus we relied on theoretical f′ and f″ values to calculate occupancy.

7.

Theoretical f′ and f″ values for Ni and Zn from 4000 to 18,000 eV; energies of Mo Kα and Cu Kα radiation indicated in blue and orange, respectively.

5.4. Determining Occupancy from Structural Refinement via Electron Density Differences

In addition to calculating occupancies from f′ values, we pursued the refinement of metal occupancies from the differences in electron density between each site. Perturbations to atomic scattering factor are measured in units of electrons; thus, we can use theoretical f′ values to calculate the difference in measurable electron density of each metal when irradiated with X-rays of varying energies (Table ). The ∼2.3 and ∼3.6 electron difference between the measurable electron density of Ni and Zn at the Mo and Cu K-edges, respectively, implies that differentiating between the metals at these wavelengths may be possible. To test this claim, each metal site in the structural solutions of 1, 2, and 3 derived from Mo and Cu Kα data sets was modeled as a Zn/Ni disordered pair using the EXYZ and EADP constraints where the occupancies are refined with two free variables via the SUMP command; the sum of Ni and Zn occupancies at each site is 1, and the total Zn/Ni ratio is 2:1 in the core. The use of the SUMP command is justified via confirmation of the Zn/Ni ratio in the crystal through SEM-EDS. Due to the charged nature of electrons, the penetration of the electron beam into the sample during an EDS measurement is small (approximately 1 μm). To account for this shortcoming in assessing the Zn/Ni ratio of the entire crystal, the SEM-EDS spectrum was acquired on 10 × 10 μm² areas of the crystals (or 1 × 10¹⁴ ų) via the Aztec EDS software (Figures S5, S12, and S19). Given that the approximate unit cell sizes of 1, 2 and 3 are ∼20³ ų, this scan area assesses ∼1.25 × 10¹⁰ unit cells. Because the crystallinity, and thus uniformity, of the sample is confirmed via SCXRD studies, we concluded that the calculated Zn/Ni ratio via EDS of a 10 × 10 μm² area approximates the ratio of the entire crystal.

1. Difference in Scattering between Ni and Zn at Various Wavelengths .

incident radiation (eV)	% difference in scattering
8042 (Cu Kα)	13.19
8330 (Ni K-edge)	32.62
9660 (Zn K-edge)	16.95
17,462 (Mo Kα)	7.67
25,000	7.33
30,000	7.26

^aThe normal scattering factor (f _o ) is estimated by the total number of electrons in the atom, and the scattering perturbations are calculated in the XDISP program of WinGX (Brennan values).

The results of refinement using this strategy were consistent between Mo and Cu Kα data sets and furthermore matched the occupancies that were calculated from anomalous scattering components refined from the Cu Kα diffraction data (Figure ). Given the limited accessibility of synchrotrons, we propose alternative strategies to those interested in assessing occupancy in mixed-metal clusters: (1) refinement of f′ for each site from Cu Kα diffraction data, or (2) refinement of each metal site as a disordered pair in conjunction with EDS analysis to gain information on the metals’ ratio within the crystal.

6. Conclusions

Anomalous X-ray diffraction studies are indispensable for assigning metal-site occupancies within heterobimetallic clusters. Herein, we conducted AXD studies at the Ni and Zn K-edges on three previously reported clusters: (^FtbsL)­Zn₂Ni­(py) (1), [K­(C₂₂₂)]­[(^FtbsL)­Zn₂Ni] (2), and [K­(THF)₃]­[(^FtbsL)­Zn₂Ni­(NAd)] (3). The diffraction data from 1 were treated in various ways, revealing the importance of the data resolution and accuracy of the basis structural model on f′, f″, and their uncertainties. These studies revealed the substitutional preferences of Ni and Zn within the trinuclear core. In the C ₁ symmetric complex, 1, Ni solely occupies the pyridine-bound site, and the other two sites are occupied by Zn. In 2 and 3, where the ligand adopts a C ₃ symmetric geometry, Ni mixes into the other two sites as seen in the f′ and f″ refinement curves, although in small amounts. Additionally, refinement of f’ for M₁, M₂, and M₃ in 1, 2, and 3 in using in-house Cu Kα data yields information on metal occupancy similar to what is gathered during anomalous X-ray diffraction studies at a synchrotron radiation source. These in-house studies also suggest that the calculation of metal occupancies using f′ is more reliable further away from the K-edge where the value is less affected by molecular influences such as ligand environment or oxidation state. Future studies will assess the accuracy of AXD occupancy studies as a function of the energetic distance from the metal K-edge. Overall, these results provide multiple strategies for those looking to investigate the metal site occupancies of heterobimetallic clusters.

Glossary

Abbreviations

SCXRD

single crystal X-ray diffraction

AXD

anomalous X-ray diffraction

XAS

X-ray absorption spectroscopy

SEM-EDS

scanning electron microscope-energy-dispersive spectroscopy

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.4c05423.

• Crystallographic data, crystal preparation, energy-dispersive spectra, X-ray diffraction and synchrotron methods, X-ray absorption data, and anomalous scattering refinement data (PDF)

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

The authors declare no competing financial interest.