Strategies to control the geometry and symmetry around lanthanide centres for tailored luminescence and magnetism

Domain: chemistry
Published: 2026-02-17
Editorial outcome: minor_revision

Abstract
The luminescent and magnetic properties of trivalent lanthanides (Ln3+) are indispensable for many emerging technologies, but exacting fine control over these properties requires an understanding of how to purposefully engineer the coordination geometry and site symmetry of Ln3+ centres. Here, we use the Cambridge Structural Database to extract the structures of 12,670 eight-coordinate Ln3+ centres and use Continuous Shape Measures, Continuous Symmetry Operation Measures, and a new structural similarity-based network analysis to survey geometry and structure trends. This survey is then leveraged to deliver concrete strategies for controlling the coordination geometry of eight-coordinate Ln3+ centres using familiar concepts like Ln3+ metal size and ligand denticity, bite angle, flexibility, shape, symmetry, and size. Ultimately, we present a roadmap for targeting each of the six common eight-coordinate geometries – hexagonal bipyramidal, cubic, square antiprismatic, dodecahedral, bicapped trigonal prismatic, and snub disphenoid – which are each demonstrated to have unique use-cases in diverse research areas. The effects of crystal packing and non-covalent interactions are also illustrated, allowing fine-grained control over the geometry and symmetry of Ln3+ centres. This work ultimately serves to inform the deliberate design of Ln3+ coordination complexes and materials for applications including data-storage, quantum information processing, lighting, thermometry, and medical bioimaging.

Introduction
Due to their unique luminescent and magnetic properties, the lanthanide elements are integral to the development of advanced electronic technology including lighting and displays, bioimaging, anti-counterfeiting, thermometry, refrigeration, data-storage, and spintronics1–4. Crucially, while the functionality of trivalent lanthanide (Ln3+) centres depends on both the geometry and the electronic structures of the surrounding ligands or host matrix, the extreme sensitivity of luminescence and magnetism to the symmetry around the Ln3+ centre has demanded increasingly fine control over coordination geometry in the design of new Ln3+ coordination complexes for these applications. This fact has been demonstrated in the design of highly-emissive compounds, phosphors, and lasers5–14, upconverting nanoparticles (UCNPs)15–17, 4f single-molecule magnets (SMMs)18–24, NMR shift reagents and MRI contrast agents25–28, magnetocaloric refrigerants29, and spin qubits30–32. For example, the strong quantum tunnelling of magnetization (QTM) observed with lanthanide SMMs remains a longstanding challenge for achieving slow magnetic relaxation and realizing Angstrom-scale data-storage; avoiding QTM is primarily an issue of site symmetry18,19. In the design of Ln3+ coordination complexes as radiopharmaceuticals, MRI contrast agents, and bioimaging agents geometry is also an important consideration; macrocyclic chelates that uniformly encapsulate the Ln3+ centre show the best stability and kinetic inertness under physiological conditions33–36. Finally, the synergy between structure and lanthanide geometry in supramolecular assemblies like metal-organic frameworks (MOFs), helicates, and metal-organic polyhedra suggests a versatile strategy for tuning properties37–40.

Ultimately, all of the above examples demonstrate the intimate relationship between physical properties and the coordination geometry of lanthanide coordination complexes. However, it is well established that because of their contracted f-orbitals Ln3+ centres exhibit diverse and unpredictable geometries that are predominantly the result of steric hindrance40–42. How can we then learn to predictably control the geometry in lanthanide coordination complexes? We here in turn to the Cambridge Structural Database (CSD) – an invaluable resource containing over one million crystal structures, thus facilitating comprehensive structural analyses1,34,43–46. We analysed structures containing eight-coordinate Ln3+ centres – since they are the most common, followed closely by nine-coordinate centres – and we herein provide an overview of all such lanthanide complexes in the CSD catalogued by structure, coordination geometry, and Ln3+ site symmetry, and then leverage the findings of this survey to prescribe useful strategies for understanding and controlling the coordination geometry of Ln3+ centres (Fig. 1).

Results
Overview of survey methodology
The CSD Python API was used to extract eight-coordinate Ln3+ fragments from the CSD, allowing the ligand atoms to be any combination of the elements N, O, F, P, S, Cl, or Br (LnX8)47. The coordination number of the fragments was validated using the ChemEnv Voronoï method from the Materials Project pymatgen Python library48–51. Eight coordinate Ln3+ centres can adopt the following relevant geometries (Fig. 1): square antiprismatic (SAP), dodecahedral (DD), bicapped trigonal prismatic (BTP), cubic (CU), hexagonal bipyramidal (HBP), and snub disphenoid (SD) – where SD is topologically equivalent to DD but with regular faces and non-equivalent bond lengths; the other five geometries tested all have equivalent bond lengths52. Many algorithms and programs have been developed to assign coordination geometries to crystal structure fragments; pertinently, the program Shape8 was developed to analyse eight-coordinate lanthanide model complexes in the study of plutonium-sequestering systems53–56. Here, we use the Continuous Shape Measure (CShM) method that is widely implemented via the SHAPE v2.1 and ChemEnv tools48,57–59. The CShM method calculates the percent deviation between the atomic coordinates and a reference polyhedron that has been oriented and scaled to minimize the deviation, allowing geometries to be assigned by determining the reference polyhedron with the smallest deviation (CShM value); the CShM value also gives some measure of the distortion of the coordination sphere, allowing the geometry to be treated continuously instead of discretely57,60. Note that a CShMs study of eight-coordinate centres and octanuclear clusters was previously conducted by Alvarez and coworkers61, with the analysis focusing on transition metals (TMs) rather than lanthanide centres.

To categorize the Ln3+ centres by site symmetry we employ the Continuous Symmetry Operation Measure (CSoM) method recently introduced by Sørensen and coworkers62,63, which can directly determine the best point symmetry description – a preferred method over simply inferring the point symmetry from the coordination geometry (Fig. 1). The CSoM method autonomously locates the highest order principal axis of a coordination sphere and then calculates the percent deviation from each symmetry operation in a point group, allowing each fragment to be assigned an appropriate point group. The CSoM value (the percent deviation from the ideal symmetry of the point group) also provides a parameter to describe the symmetry continuously, in an analogous fashion to the CShM parameter.

Lastly, to categorize the Ln3+ centres by structure we developed a new method that uses the similarity algorithm in the CSD Python API to build a similarity matrix of all fragments, which via network analysis allows for the partitioning of the dataset into structurally similar subsets. Combining information of geometry and structure allows for a rigorous evaluation of the trends of eight-coordinate Ln3+ centres in the CSD. Further details of the dataset construction and the CShM, CSoM, and network analyses are included in the Supplementary Information, which includes additional citation to references64–66.

Part 1: Overview of eight-coordinate lanthanide centres in the CSD
Searching the CSD (version 5.45, June 2024) for eight-coordinate Ln3+ sites yielded 12,670 LnX8 fragments from 9,132 crystal structures. Of the lanthanide elements (La–Lu), the radioactive element Pm is absent, the elements Dy, Gd, Tb, and Eu are over-represented, and the element Tm has the fewest examples (Figure S1). These metrics reflect the extensive research targeting magnetic properties with Dy3+, Gd3+, and Tb3+ compounds and photoluminescence research with Tb3+ and Eu3+ compounds. As for the ligand X atoms, 80% are O, 18% are N, and 1% are Cl, reflecting the hard Lewis acidic nature of the Ln3+ centres – the remaining X atoms are S, Br, F, or P atoms, with <250 instances of each. Further, 47% of hits are LnO8 structures; another 47% of the dataset contain the LnO8-xNx (x=1–8) permutations; and the remaining 6% of hits contain 70 different LnX8 donor atom combinations – with 13 combinations being unique. The donor set statistics are also included in Figure S1. The average R-factor of the crystal structures used for the dataset was 5.0 ± 1.9%, and a histogram of R-factor values is included as Figure S2.

Continuous shape measure analysis and geometry trends
The CShM analysis results are shown in Fig. 2. Each fragment was categorized by geometry based on the closest reference polyhedron, and those not within 3 % of any ideal geometry were classified as distorted (DIST). Note that the closest geometries SAP, DD, BTP, and SD are all ~3% deviated with respect to each other, providing the basis for this DIST threshold (the full 0-100% scale for CShM values is not relevant in the context of coordination geometries, as 100% corresponds to a non-physical case of all nine LnX8 atoms occupying the same position). A histogram of the CShM values (Fig. 2a) illustrates that Ln3+ centres are in general only slightly distorted from some ideal geometry – with an average CShM of 1.2 ± 0.8% – in contrast to the common literature description of Ln3+ complexes as ‘distorted’. The inset shows the average CShM values for each coordination geometry, revealing that Ln3+ centres with SAP, CU, and DD geometry are on average the least distorted from ideal, while those with HBP, BTP, and SD are most distorted besides the DIST structures themselves. Overall, the SAP geometry was found to be most common (46% of structures), followed by DD (36%) and BTP (13%) – as shown in Fig. 2b. These three geometries are the low-energy configurations for eight-coordinate structures that best minimize inter-ligand repulsion52. Comparatively, there are very few structures with the geometries HBP, CU, and SD, and only 3% of structures are classified as DIST.

Figure 2c–f are shape maps, where the structures are placed in scatterplots according to their percent deviation from two reference polyhedra67. We chose the SAP–DD and SAP–CU shape maps to visualize the data, as DD and SAP are the two most common coordination geometries while SAP and CU are unique in that they have an obvious stereochemical relationship (interconverting via a twist of the top face) and are also very distorted (11%) with respect to each other61. The data in Fig. 2c, d are colour-coded by geometry; the positions of the six ideal polyhedra are also indicated, and interconversion paths between the polyhedra are shown as black lines. Figure 2e and f are included as complementary heatmap plots. Together, these four plots reveal the distribution of geometries for Ln3+ complexes: most structures lie within the SAP–DD–BTP region – as these three geometries best minimize inter-ligand repulsion – some structures also extend towards SD, a small number of structures are localized around CU and HBP, and the interspace is sparsely populated with primarily DIST structures (light grey). Furthermore, instead of local maxima near each ideal polyhedron, the heatmaps in Fig. 2e, f demonstrate that there is only one maximum near SAP. This illustrates the continuous nature of coordination geometry for eight-coordinate Ln3+ centres, and the importance of the SAP shape as an optimal arrangement.

Minimal distortion path analysis was next conducted, to characterize geometries distorted between the ideal cases60. Consider that to transform a cube into a square antiprism the top face can be twisted by 45°. Stopping at any point along this transformation provides a geometry that is neither CU or SAP but is distorted between the two – quantifiable via the twist angle or the path position, the latter being the percent conversion between CU (0%) and SAP (100%). Note that path position can be used for any pair of end-members, even though transformations between other polyhedra do not have obvious geometric parameters akin to the twist angle.

From the minimal distortion path analysis, some LnX8 complexes were found to have geometries upon the CU→SAP and SAP→DD interconversion paths and all other possible paths showed low statistical significance (see the Supplementary Information for details). 558 structures were located on the CU→SAP path, having an average path position of 75 ± 14% distorted towards SAP, which corresponds to a 34 ± 6° twist of the top face. This distortion is likely common considering that for four bidentate ligands to generate perfect SAP geometry around a Ln3+ centre (with the chelates spanning the two square faces of the polyhedron) then the ligands must have a bite angle matched to the 74.86 ° vertex-centre-vertex angle of the ideal SAP polyhedron. However, bite angles for ligands that form six-membered chelate rings, such as the ubiquitous β-diketonates, are typically ca. 72 °, falling short of perfect SAP geometry and instead coinciding with the vertex-centre-vertex angle of the 75% CU→SAP distorted polyhedron. That is, limitations of chelating ligands force distortions away from ideal geometry. As for the SAP→DD interconversion, 487 structures were found to lie upon the distortion pathway, with an average path position of 63 ± 14%. Nonetheless, the majority of LnX8 complexes do not have geometries that can be described as minimally distorted between any two ideal polyhedra, indicating that distortions of the coordination geometry generally occur in an unsystematic fashion (where case-by-case ligand requirements cause pinching of angles or elongation of bonds).

Continuous symmetry operation measure analysis and point group trends
Each of the LnX8 fragments (excluding DIST structures) were assigned point groups, with structures where no symmetry elements were present assigned the trivial point group C1. A deviation threshold of 5% was used to decide whether a symmetry element was present or not, based on reasoning outlined in the Supplementary Information, although note that the actual symmetry may be lower than the point symmetry derived from the CSoM analysis – for example, Sørensen et al.62,68 propose that a 0.1-1% CSoM value may be enough to cause symmetry breaking in lanthanide luminescence. The distribution of point groups in the dataset is shown in Fig. 3a (see Table S1 for a full tabulation). Cs is the most common point group, followed by D4d, D2d, and C2v symmetry – reflecting the prevalence of SAP, DD, and BTP structures, respectively. Next are C2 and C1 symmetry, and then 15 other point groups occur within the remaining 5% of structures.

Partitioning the structures into those with homogenous and heterogenous donor atom sets shows that 87.1 % of the homogenous structures have ideal symmetry (the highest symmetry allowed by the geometry) in contrast to only 4.4 % of the heterogenous structures. Alternatively, plotting CSoM versus CShM values for all homogenous structures shows that CSoM values can at most increase by ca. 2% for each 1% increase in CShM value (Figure S3). Thus, while heterogenous donor atom sets can be used to lower the symmetry, distortions of the geometry can also have an impact. This is important as while heterogenous donor atom sets can be easily built into coordination complexes, the donor atoms in host materials like fluorides, oxides, or nitrides cannot be easily substituted to lower the site symmetry. Assuming a ca. 1 % CSoM upper threshold, the CSoM versus CShM relationship indicates that just a ca. 0.5% deviation from ideal geometry (as measured via the CShM method) may be enough for symmetry-lowering to be observed in emergent physical properties, although this threshold must be benchmarked against more experimental results before it is put into use62.

Analysing by geometry (Fig. 3b), most (ca. 40–50%) SAP, DD, and BTP structures have ideal symmetry while 25–30% have Cs symmetry and the remainder are assigned to some other subgroup. In the less common HBP, CU, and SD configurations, the ideal symmetry occurs almost all the time and C1 symmetry is never encountered, although this may be due to the low number and small variety of structures reported for those geometries.

Inspecting Fig. 3c, the maximum of the CSoM histogram plot occurs at 0% deviation (Fig. 3c), indicating that it is more common to have near-perfect point symmetry than it is to have near-perfect geometry (cf. Figure 2a). Overall, the average CSoM of the dataset is 1.3 ± 1.0% (ignoring C1-symmetric structures) and by comparing the insets of Fig. 2a and Fig. 3c it can be seen that the average CShM and CSoM values appear to be correlated for the HBP through BTP subsets, although the SD structures show unusually small CSoM values (average CSoM of only 0.53%), in contrast to being the most deviated from ideal geometry.

Geometry Trends of Lanthanide Centres with Monodentate Ligands
Next, it is worthwhile to consider the subset of LnX8 fragments with only monodentate ligands. Filtering the dataset for LnX8 fragments where the Ln centre is acyclic (not involved in any chelate rings) yielded 949 monodentate LnX8 fragments from 872 crystal structures, and their geometry trends are displayed in Fig. 4.

As expected, the monodentate subset displays far less polyhedral distortion than the parent dataset (Fig. 4a vs. Fig. 2a), with an average CShM of 0.6 ± 0.3% and a max CShM of only 2.3% (thus containing no fragments in the DIST category). There are also no HBP or SD monodentate structures; the majority of structures are SAP (57%), followed by DD (35%), BTP (6%), and CU (1%; 11 structures), as shown in Fig. 4b. This indicates that LnX8 centres will not naturally adopt HBP or SD geometry or have DIST geometry unless ligands are used that enforce the geometry. Even the cases with CU geometry are enforced by hydrogen bonding; they are all Ln(OH2)8 molecules enclosed in highly-symmetrical molecular or supramolecular cages69–71. Hence, when unfettered from the influences of chelating ligands or hydrogen-bonded pseudo-chelate rings, Ln3+ centres appear to adopt SAP, DD, and to a small extent, BTP geometry.

Structural trends of lanthanide coordination complexes
The structural trends of the Ln3+ fragments were analysed using a network graph: the LnX8 fragments were expanded to include all atoms within three bonds of the Ln3+ centres, chemically-identical structures were combined into proportionally-sized nodes, and then edges were drawn between nodes with > 80 % similarity, allowing structurally-similar subsets to be identified using the Louvain community detection algorithm72. Of the 12,670 total fragments, 1449 were unique (subset size of 1), 7357 belonged to 1479 subsets of size less than 40 – representing the majority of the dataset (58%) – and 3864 (or 30%) belonged to 26 subsets of size 40 or greater; these 26 are shown in Fig. 5, colour-coded by subset. The structure descriptions of subsets 1–53 (of size 25 or greater) are included in Table S2. Structure descriptions were determined by inspecting the highest-connected node in each subset.

A wealth of information can be gleaned from the network analysis and Fig. 5 provides a compelling vista of the broader trends in lanthanide coordination chemistry. For example, the prevalence of β-diketonate (β-DK-) ligands is demonstrated by their appearance in the three largest subsets; 2,2’-bipyridine (bipy) derivatives are also ubiquitous and the largest subset contains Ln(β-DK)3(bipy)-type compounds that are well-known model complexes for both luminescence and magnetism research73–76. Subsets also feature monodentate acetates, nitronyl nitroxides, phosphine oxides, and common O-donor solvent molecules like water, N,N-dimethylformamide (DMF), dimethylsulfoxide (DMSO), and 1,2-dimethoxyethane (DME). Bis-tetradentate ligand arrangements are also common, such as with polyoxotungstates (POTs), phthalocyanine (HPc) macrocycles, 2,2’-bipyridine-6,6’-dicarboxylate (dcbipy) and N,N’-ethylenebis(salicylimine) (Hsalen) chelates, and Zn2+-based compartmental ligands (subset 26). Further diverse ligand classes are encountered in the smaller subsets, as described in Table S2.

Most striking in Fig. 5 is the central network feature composed of subsets 4, 5, 9, 13, 17, and 20 and accounting for 795 structures. These subsets originate from polynuclear SMM research and feature f-for-d-f assemblies supported by polydentate ligands; due to their complexity and variety, more in-depth analysis of the elements and ligand-types in each subset was conducted (see Tables S3 and S4 for details)4. Briefly, subset 4 contains Ln-TM clusters, subset 5 shows less presence of TMs and a greater proportion of Dy3+ centres, and subsets 9, 13, 17, and 20 are exclusively Dy3+-based with ca. 1% Dy-TM heteronuclear clusters. The clusters generally feature μ-hydroxo or μ-oxo bridges and supports from phenolate- or hydrazonate-derived polydentate ligands. The intricate metal-ligand arrangements in these subsets prevented a straight-forward analysis of structure-geometry relationships, so they will not be discussed in detail in Part 2. In general, the cluster-based subsets showed an uptick in DIST geometries, indicating that distorted geometries generally arise in structurally restricted environments.

Part 2: Strategies to control the geometry of eight-coordinate lanthanide centres
The geometry, point symmetry, and structure information described in Part 1 are next harnessed to propose strategies to influence the coordination spheres of Ln3+ centres. Many different geometry and symmetry combinations are of interest to chemistry and materials science researchers: in general, low symmetry compounds are preferred for luminescence applications, sites with high symmetry are often required to unlock magnetic behaviour, and in either case small distortions of the coordination sphere can have a great impact on properties. In addition, multifunctional lanthanide materials are desirable targets – for instance, SMMs with luminescence thermometry functionality for built-in temperature calibration – and these systems may have even more nuanced symmetry requirements77,78. Lastly, each application requires that ligands have certain electronic features. For example, the ligand triplet state energy must be tuned when designing highly-emissive compounds or luminescence thermometers; low-lying ligand-metal charge transfer transitions are detrimental to emissive Eu3+ complexes but beneficial for Yb3+ centres; and highly covalent Ln3+-ligand bonds can push Ce3+ luminescence from the ultraviolet into the deep-red but can also reduce magnetic axiality for SMMs79–83. Therefore, it is often difficult to design ligands that exert certain geometries around Ln3+ centres while also accommodating for important electronic aspects.

In view of these delicate design criteria for lanthanide coordination complexes, we present a variety of perspectives on coordination geometry. First, the interplay of how different sized Ln centres interact with different sized chelate rings are discussed, as this forms the basis for understanding the behaviour of more complicated systems. Next, we look at the effect of temperature on geometry. Then, specific strategies to target each of the six geometries (SAP, DD/SD, BTP, CU, and HBP) are put forth, and we illustrate the advantages of each geometry and point symmetry within the context of luminescence-based (phosphors, UCNPs, lasers, bioimaging, and quantum communication) and magnetism-based (SMMs and spin qubits) applications. Lastly, we conclude with how supramolecular interactions and crystal packing can be used to further control and tune geometry and site symmetry.

Geometry trends across the lanthanide series
The lanthanide series is often treated as a monolith, to the detriment of a more nuanced understanding of these elements. For example, while it is well-known that the lanthanide contraction leads to a preference for nine-coordination for the larger early-series lanthanides but eight-coordination for the smaller late-series elements, to the best of our knowledge a corresponding trend in geometry preferences within each coordination number has not been explicitly explored except in specific systems41. To isolate from other structural factors that affect geometry, the structurally-similar subsets built from the network analysis can be utilized, allowing for an analysis of how geometry distributions change as a function of lanthanide size.

Subsets 1, 3, and 6 of the Ln(β-DK)3(bipy), Ln(β-DK)4, and Ln(DMF)8 structures were used for the analysis, as they contain instances of most of the lanthanide elements and they represent simple lanthanide coordination complexes that used different bidentate or monodentate ligands. The Ln elements were split into four groups based on size (trivalent eight-coordinate ionic radii): the early lanthanides La–Nd (1.16–1.11 Å); the early-mid lanthanides Sm–Gd (1.08–1.05 Å); the mid-late lanthanide Tb–Ho (1.04–1.02 Å); and the late lanthanide Er–Lu (1.00–0.98 Å); the geometry preferences and average CShM and CSoM values of these four lanthanide groups are shown in Fig. 6 for the three subsets84. Note that DIST or C1-symmetric structures were omitted from the calculations of the average CShM and CSoM values, respectively.

In all three cases, the geometry distributions (Fig. 6a–c) show that the proportion of SAP structures increases as the Ln size decreases. We can speculate that with smaller Ln3+ centres the inter-ligand repulsion increases, putting a greater pressure to adopt SAP geometry and maximize the inter-ligand distance. Further, the three subsets illustrate that this pressure can occur to different degrees, as the Ln(β-DK)4 subset shows much more drastic behaviour than the other two subsets.

It is also interesting to note that the only Ln(β-DK)4 fragments with CU geometry were found in the early lanthanide group, indicating that CU geometry most often occurs when inter-ligand repulsion is reduced; i.e., larger Ln3+ centres have less of a penalty to adopt CU geometry. Inspection of further subsets reveals a consistent trend of CU centres occurring only in the early-series; specifically in the Ln-TM clusters, Ln(OH2)8, Ln(POT)2, Ln(Pc)2, and Ln(bipyO2)4 (bipyO2 = 2,2’-bipyridine-N,N’-dioxide) subsets (subsets 4, 9, 12, 15, and 25, respectively). In fact, the perchlorate salts of the Ln(bipyO2)4 cation show an interesting phenomenon where the Ln3+ centre geometries can be accurately placed along the CU→SAP path – with the Lu, Nd, and La structures having path positions of 68.6, 50.4, and 12.3 %, respectively – indicating a step-wise distortion towards CU geometry as the size of the Ln3+ centre increases (Figure S4a)85,86.

Looking next at Fig. 6d–i, as the Ln size decreases the average CShM and CSoM values decrease for the Ln(β-DK)3(bipy) and Ln(β-DK)4 subsets, indicating a shift to less distorted geometries. While this may in part be driven by increasing inter-ligand repulsion, the absence of a trend in the CShM and CSoM values when monodentate DMF ligands are used points to a different mechanism. As will be discussed in the next section, when chelating ligands are used the geometry becomes contingent on how well the chelates can fit around the Ln3+ centre. The trends with the Ln(β-DK)3(bipy) and Ln(β-DK)4 subsets suggest that these bidentate ligands struggle to enclose the larger early-series lanthanide elements, leading to distorted coordination spheres, while the monodentate DMF ligands cause no such distortions. Swapping one β-DK ligand for a bipy ligand (subset 3→1) even exacerbates the distortions, as bipy ligands form smaller five-membered chelate rings.

Thus, the identity (and size) of a Ln3+ centre does affect the coordination geometry, implicating the lanthanide contraction as another handle to control Ln3+ geometry. This phenomenon is also relevant to phosphor or UCNP design, where doping Ln3+ centres into a host matrix may cause local geometry distortions and symmetry lowering and modulate the material’s luminescence properties16,87.

The effect of ligand chelate ring size
On the other side of the coin, a careful inspection of how chelate ring size effects Ln3+ geometry completes the picture of how Ln3+ centres interact with simple chelate motifs. This can be achieved via an analysis of the coordination geometry trends of Ln3+ centres coordinated by five- or six-membered chelate ring (5- or 6CR) bidentate ligands, keeping all other ligands monodentate (see Supplementary Information for details). The Supplementary Information and Table S5 also contain data for the types of ligands encountered in the analysis, which are primarily bipy- or oxalate-type for the 5CR ligands and β-DK-type for the 6CR ligands.

As shown in Fig. 7a, as 5CR ligands are added to the coordination sphere a shift to favour DD occurs in the geometry distribution. On the other hand, although there were statistically too few mono-6CR fragments to be included, an opposite shift from DD to SAP occurs when four 6CR ligands are added to the coordination sphere compared to two or three such ligands (Fig. 7b). This suggests that 5- and 6CR ligands impose preferences for DD and SAP geometry, respectively.

Extracting the average X-Ln-X ligand bite angles (Fig. 7c and d) from the two datasets provided 66 ± 2 and 72.1 ± 1.8° for 5- and 6CR ligands, respectively. Comparing these values to the vertex-centre-vertex angles of the ideal SAP (74.86°) and DD (73.69°) polyhedra, the DD geometry has a slightly smaller polyhedral angle and thus can better match to the smaller bite angles of the 5CR ligands. Indeed, Hoffmann et al.52 stipulate that DD geometry is preferred when using ligands with smaller bite angles, which is verified on a large scale by this analysis.

However, as noted in the Ln3+ size analysis, since the bite angles imposed by 5CR ligands are too small to perfectly match the DD polyhedron, 5CR ligands cause distortions of the geometry. This is demonstrated by Fig. 7e, where the average CShM value increases as 5CR ligands are added to the coordination sphere. On the other hand, adding 6CR ligands to the coordination sphere does not affect the average CShM (Fig. 7f), as the bite angles are better matched to ideal geometries. Thus, while small-bite angle 5CR ligands may be useful to bias towards DD geometry, a concomitant distortion of the geometry appears unavoidable.

Four-membered chelate ring (4CR) motifs can occur in polynuclear systems when two donor atoms are bridged by a second Ln3+ centre or a TM centre. However, the same geometry distribution analysis could not be conducted with these chelates, as a statistically relevant number of chelate rings could not be isolated in the same manner (i.e., with all other ligands monodentate). Therefore, the average X-Ln-X bite angles in Ln3+-bridged and TM-bridged 4CR motifs were extracted from the parent dataset, providing 70 ± 4 and 68 ± 4°, respectively. As with the 5CR ligands, the smaller and more rigid bite angles from the TM bridges impose a bias towards DD geometry (vida infra).

Lastly, we highlight the ligand bipyO2 that forms seven-membered chelate rings (7CR) and can uniquely accommodate different geometries like SAP, CU, and DD without distorting the coordination sphere. This is due to its flexibility: the planarity of the bipy backbone is broken by the repulsion between the N-oxide groups, allowing the ligand to twist and assume the necessary bite angle to match the geometry (Figure S4b). When the adaptability is removed by using the rigid ligand 1,10-phenanthroline-N,N’-dioxide (phenO2), only SAP geometry is observed for the analogous Ln(phenO2)4 compounds (Ln=Gd, Er, Yb)88,89. The flexibility of 7CR ligands like bipyO2 may pose a unique opportunity for the design of lanthanide complexes, given the bite angle limitations of 5- and 6CR ligands.

The effect of temperature on coordination geometry
Next, we examine if temperature has a noticeable effect on geometry, given that different temperature regimes are often used to collect crystal structural, magnetic, and luminescence data and correlating these data is crucial to establishing structure-property relationships. Thus, the structures in the LnX8 dataset were checked in the CSD for additional depositions measured at different temperatures, which had initially been filtered out of the dataset as duplicates. Seven such LnX8 structures were reported at three or more temperature points across a broad temperature range, either as part of variable-temperature experiments or from independently deposited measurements. The geometries, CShM values, point groups, and CSoM values of these structures were extracted as a function of temperature, to elucidate the effect of thermal motion on the coordination environment (Figure S5 and Table S6).

First, the isostructural compounds Ln(DMF)3(OH2)3Fe(CN)6·H2O (Ln=Ce, Sm) can be analysed, which represents simple structures with just monodentate ligands90–93. The Ln3+ centres have SAP geometry that upon heating shift closer to ideal, with CShM values decreasing by -0.1 % per 100K. This occurs because the molecules are subject to packing forces; the balance of intra- and intermolecular forces distorts the coordination spheres away from ideal geometry but as the temperature increases the ligand atoms move to distribute more equally to accommodate for increased thermal motion94. Thus, the average structure seen by diffraction techniques resembles a more ideal geometry as temperature increases.

This behaviour is also reflected in datasets where chelating ligands are involved. First, in a protonated Tb(TPP)2 (TPP = tetraphenylporphyrin) structure the Tb3+ centre has a geometry that is distorted 74.8% along the CU→SAP path, and the deviation from SAP decreases by -0.07% per 100K as the TPP ligands rotate towards an ideal 45 ° twist angle95,96. Next are two Dy(β-DK)3(bipy) compounds with SAP geometry, which have CShM values of 1.383% and 0.657% at their lowest temperatures. The former more distorted centre shifts towards ideal by -0.03% per 100K upon heating, while the latter is much closer to ideal and shows essentially no change in CShM value with temperature97–100. Indeed, this latter example was used to confirm the validity of using structural data to interpret magnetic anisotropy measurements made at much lower temperatures100. On the other hand, the HBP compound [Dy(OtBu)Cl(18-crown-6)]- shows a drastic structural evolution with temperature; the axial t-butoxide and chloride ligands are bent at 30-80K but straighten out upon heating, causing a decrease in CShM of -1.10 % per 100K from 100-273K101. Further, analysis of the CSoM values shows that the Dy3+ point symmetry switches from C2v to C6v at 150K. From ab initio calculations, these changes predict a ca. 10 % increase in the energy barrier to magnetization reversal upon heating from 30 to 273K101. In this case, the HBP polyhedron is delicate and the exposed axial ligands are prone to distortive packing effects. Lastly, there is one example where the geometry distorts with heating: two Dy3+ centres in a rhomboid [Zn2Dy2] cluster show an increase in CShM value of 0.06 and 0.01% per 100K caused by twisting of a metastable cis-N-N bond within the ligand scaffold102. Overall, each dataset illustrates that temperature is usually a minor factor for Ln3+ coordination geometry and CShM values change by at most 0.1% per 100K, except in the case of the unique HBP structure.

Strategies to target each eight-coordinate lanthanide geometry
Looking beyond general factors, we can next use the network subsets to establish specific structure-geometry relationships. This is accomplished by extracting the geometry and point group distributions for each subset (Table S2) and identifying ligands that strongly bias toward certain geometries. Fifteen such subsets are shown in Fig. 8, which form the basis of the following strategies for targeting SAP, DD, SD, BTP, CU, and HBP geometries.

Targeting Ln3+ Centres with Square-Antiprismatic Geometry
SAP geometry is the most common configuration adopted by eight-coordinate Ln3+ centres and so is nominally easy to target; however, enacting subtle control over this geometry and causing small distortions has proved vital for unlocking the luminescent and magnetic properties of many systems. As an example, in the luminescence of Eu3+ centres with near-SAP geometry there is an interplay between the 5D0→7F2 and 5D0→7F4 transitions; the former becomes forbidden under D4d symmetry while the latter begins to dominate103. The 5D0→7F2 transition is well-known for its intense red ca. 614 nm emission that is useful for displays and bioimaging, while the 5D0→7F4 transition occurs at ca. 700 nm where the human eye is insensitive to light and is thus detrimental to display applications. However, Eu3+ complexes have been of interest in plant cultivation and the 5D0→7F4 transition in particular has good overlay with the absorption band of photochrome PFR104,105. D4d symmetry is also beneficial for SMMs as the high symmetry causes transverse crystal field parameters to vanish, suppressing QTM19. Thus, sandwiching Ln3+ centres between either tetradentate POTs or Pc2- rings confers near-ideal D4d symmetry and excellent SMM properties: double-decker Ln(Pc)2 complexes were the first mononuclear lanthanide complexes discovered to exhibit SMM behaviour and the [Er(WO5)18]9- was reported as the first fully-inorganic analogue to the mononuclear Ln(Pc)2 SMMs23,24. Due to structural and steric differences between the Pc2- and POT ligands, the Ln(Pc)2 complexes generally show axial elongation of the SAP polyhedron while the [Ln(WO5)18]9- compounds show slight axial compression, thus matching the anisotropic electron densities of oblate and prolate Ln3+ centres, respectively23. Small distortions of the D4d symmetry in the [Ln(WO5)18]9- compounds also causes a spin clock transition in the Ln=Ho3+ member; this is behaviour that can protect against quantum decoherance from long-range dipolar interactions in spin qubit systems30.

The Ln(POT)2 and Ln(Pc)2 examples show that SAP geometry can be specifically targeted using rigid, C4v-symmetric tetradentate ligands where the donor atoms are arranged as a square (subsets 12 and 15; 97.39 and 93.48 % SAP geometry, respectively). This has also been demonstrated in hybrid Ln(Pc)(POT) complexes and with [Ln{M(SAc)4}2]- (M=Pd, Pt; HSAc=thioacetic acid) complexes, where square-planar Pd2+ or Pt2+ centres play the role of directing SAP geometry106–111. The symmetry can also be broken by rotating the tetradentate ligands; in triple-decker Dy2(Pc)3 complexes dual magnetic relaxation behaviour is observed unless the twist angle is less than 45°, and the twist angle can be controlled by influencing the steric interactions between the rings from different peripheral substituents20. Similar phenomena have been shown in heteroleptic Dy2(Pc/Por)3 (H2Por = porphyrin) compounds (subset 27) and quadruple-decker Dy2Cd2(Pc)4 SMMs112,113.

Another important class of compounds showing SAP geometry is the Ln(DOTA)4- (H4DOTA=2,2’,2”,2”’-(1,4,7,10-tetraazacyclododecane-1,4,7,10-tetrayl)tetraacetic acid) family, of which 20 examples were manually identified from the dataset and are summarized in Table S7. Such Ln(DOTA) complexes are benchmark compounds for molecular magnetism, are well-known as MRI contrast agents, and have been used as radiopharmaceuticals25,114–117. They are well know to exhibit a twisted SAP (TSAP) geometry, with the cyclen base making one square face of the polyhedron and the carboxylate arms wrapping around the Ln3+ centre to form the twisted top face. As discussed in the CU→SAP distortion path analysis, ideal SAP geometry can only be reached with a 74.86° bite angle; however, the carboxylate arms of the DOTA4- macrocycle form small-bite angle 5CRs. Because of this, all 20 of the Ln(DOTA) structures lie upon the CU→SAP distortion path with an average path position of 58 ± 3% (corresponding to a 26.1 ± 1.4° twist angle and a 71.08 ° vertex-centre-vertex angle) and are all C4-symmetric with low CSoM values (<0.047%). Thus, DOTA-type ligands are a reliable method to access a C4-symmetric environment.

Targeting Ln3+ centres with dodecahedral and snub disphenoid geometry
Both DD and SD geometry have D2d point symmetry, thus providing a lower-symmetry environment than SAP. This is not optimal for SMM design, as transverse crystal-field terms are present, leading to relaxation via QTM19. However, an asymmetric DD geometry has been implicated as the source of luminescence intensity enhancement in Eu3+ and Sm3+ complexes and Nd3+ lasers; moving to lower symmetry generally affords a stronger emission, as the selection rules for the formally forbidden f-f transitions become relaxed and the radiative lifetime shortens9,11,14. In Er3+ UCNPs it has been demonstrated that lowering the symmetry too far from D2d to C1 can produce a detrimental effect; the lifetimes of the reservoir levels become too short, preventing efficient upconversion15. Further, the increase in crystal-field splitting upon symmetry-lowering can broaden the typically sharp Ln3+ f-f transitions, reducing emission colour purity. This is demonstrated by the classic red phosphors Y2O3:Eu3+ and YVO4:Eu3+ that have C2 and D2d site symmetry, respectively; the latter shows a sharper 5D0→7F2 transition that provides a more intensely-red emission colour118,119. Isostructural GdVO4:Eu3+ nanoparticles also show intense red luminescence and have been used as multifunctional bioimaging and MRI contrast agents120.

As was discussed in the chelate ring size analysis, DD geometry can be targeted using small chelate rings. The network analysis further corroborates this trend; Ln(en)4 structures (en=1,2-diaminoethane) are 96.30% DD with an average en bite angle of 67.5 ± 0.7° (subset 42) and the family of toroidal {Ln3Fe6} clusters121,122 (subset 47) are 80.77% DD due to the small bite angles imposed from neighbouring Fe centres (average 67.5±0.9 °). Another interesting case is the Ln(OOCR)8 structures (subset 23). Here there are no chelating ligands, yet 95.35% of the structures have DD geometry. Inspection of the crystal structures reveals hydrogen-bonded pseudo-chelate rings between adjacent monodentate ligands, involving the dangling OH groups of the carboxylic acids. This gives pseudo 5CRs with bite angles of ca. 71–73°, thus likely causing the DD preference. A similar phenomenon occurs in the Ln(O=CNHR)8 structures (subset 53; 76.00% DD geometry), due to hydrogen bonds involving the amide NH groups.

Bis-nitronyl nitroxide and bis-phosphine oxide adducts of Ln(β-DK)3 provide another useful comparison (subsets 2 and 14, respectively). The monodentate nitronyl nitroxide ligands generally prefer to bind from opposite sides to give a C2-symmetric environment, which predisposes towards DD geometry (71.43% DD and only 11.98% SAP); on the other hand, subset 14 is primarily SAP. This difference is likely due to the steric profiles of the ligands; the nitronyl nitroxides are small and planar, while the phosphine oxides are often tri-aryl substituted with large cone angles. The steric bulk demands larger inter-ligand angles with the surrounding β-DK- ligands, causing a switch from DD to SAP. Therefore, another handle to control between SAP and DD geometry may be the steric bulk of monodentate ligands.

SD geometry is far less common than DD geometry (occurring only 0.7% of the time in the parent dataset) and yet also offers access to D2d symmetry. To that end, the Ln(dcbipy)2 (Hdcbipy=2,2’-bipyridine-6,6’-dicarboxylic acid) and Ln(salen)2 subsets particularly stand out, exhibiting 48.15% and 13.95% SD, respectively (subsets 19 and 24). This large bias towards SD geometry occurs because both dcbipy and salen ligands are planar tetradentate ligands, which leads to encapsulation of the Ln3+ centre in a near-perfect D2d-symmetric geometry (which is likely why the SD structures have the lowest average CSoM compared to the other geometry subsets). Note that whether this bis(ligand) encapsulation leads to DD or SD geometry depends on the bond lengths, as DD geometry is spherical with equivalent vertex-to-centre lengths while SD geometry is non-spherical (with four short and four long vertex-to-centre lengths; but with regular faces)61. Here, the ligands are predisposed towards SD geometry because they form two long bonds with neutral N-donor atoms and two short bonds with anionic O-donor atoms. In fact, 88.1 % of the SD hits in the parent dataset have N4O4 donor atom sets – with average Ln-N and Ln-O distances of 2.30 ± 0.06 and 2.54 ± 0.07 Å, respectively – whereas only ca. 0-6% of structures with other geometries are N4O4-type. Note that the anisotropic bond lengths arranged at axial and equatorial positions along the S4 axis may increase magnetic anisotropy and thus benefit SMM behaviour123,124.

Targeting Ln3+ centres with bicapped trigonal prismatic geometry
After DD/SD geometry, BTP has the lowest point symmetry (C2v) and thus has limited use in SMM design. Yet as previously mentioned, there can be advantages to targeting low-symmetry sites for luminescence applications, and in a few cases BTP geometry has been integral to accessing certain properties. For example, the 5D0→7F0 transition of Eu3+ centres can have an extremely narrow linewidth and a long lifetime and so is applicable for quantum computing, memory, or communication applications125,126. However, this transition is forbidden except at low symmetry sites with Cn, Cs, or Cnv symmetry103. Leveraging the ultra-narrow linewidth and uniquely large linear Stark effect imposed on the 5D0←7F0 transition by C2v symmetry, a two-qubit quantum gate based on electric dipole-dipole interactions has recently been demonstrated in a Eu(β-DK)4-based molecular crystal127,128. Lower-symmetry sites that cause linewidth broadening – like the distorted BTP sites in the host material GdScO3 – have also been critical in designing Yb3+ and Tm3+-based sub-100 fs pulsed lasers, as the minimum pulse duration is inversely dependent on the spectral width129,130. On the other hand, BTP often occurs unintentionally in cases when researchers are targeting SAP geometry for SMM applications. For example, the tetradentate calixarene ligand TBC[4](OMe)2 (H2TBC[4](OMe)2 = p-tert-butylcalix[4](OMe)2(OH)2 arene) has been utilized to target SAP geometry, operating in a similar way as a Pc2- or POT ligand to provide one of the square faces131,132. However, when combined opposite a POT ligand or two β-DK ligands, a distorted BTP geometry is obtained due to the C2v symmetry of the TBC[4](OMe)2 ligand. In view of this, it is useful to understand the structural features that lead to BTP to both target the geometry and avoid it as a low-symmetry pitfall.

Inspecting the family of Ln(POT)(salo)2 structures (saloH=salicylaldehyde) with 100% BTP geometry and C2v symmetry (subset 30) shows that while SAP geometry is provided when two POT groups ligate a Ln3+ centre, replacing one rigid tetradentate POT with two more sterically flexible bidentate ligands can cause a switch to BTP. This is because both BTP and SAP geometry share a common square face, but the Ln3+ centre sits deeper into the pocket in the BTP arrangement (bite angle of 81.79° at the square face) which matches better with the POT ligand (viz. the axial compression of the Ln(POT)2 structures). Similarly, while the {Ln[M(SAc)4]2}- (M = Pd, Pt) family exhibits SAP geometry, complexes with only one [M(SAc)4]- metalloligand and four monodentate ligands adopt BTP geometry, as the compressed Ln-[M(SAc)4]- distance facilitates an attractive M···Ln interaction133. Thus, BTP can be targeted by using one wide-base C4v-symmetric tetradentate ligand and combining it with either four monodentate, two bidentate, or one C2v symmetric tetradentate ligand like TBC[4](OMe)2.

One other subset with 100% BTP geometry is subset 37 of fragments from toroidal {Ln3Fe} clusters utilizing the tetradentate tripodal ligand TEA3- (H3TEA=triethanolamine). The BTP geometry likely occurs because the tripod shape is preorganized to cap one face of the trigonal prism, and the other four ligands arrange accordingly. Thus, tripodal ligands may be advantageous for targeting capped or bicapped coordination geometries, although in this case the neighbouring Fe3+ centres constrain and distort the geometry leading to an avg. CShM of 2.0 ± 0.3%. Additionally, replacing the four other ligands with an additional tripodal TEA3- ligand as in subset 51 does not further enforce the BTP geometry but instead leads to 61.29% of the structures having DIST geometry.

Targeting Ln3+ centres with cubic geometry
In contrast to BTP is CU, which is the geometry with the highest symmetry (Oh). However, this leads to many features that are undesirable for targeting highly luminescent compounds or accessing SMM behaviour: the existence of a centre of symmetry causes the f-f transitions to be formally forbidden and anisotropic ligand field precludes an energy barrier to magnetization reversal31,103. However, such high symmetry Oh systems do exhibit complex energy level manifolds, and CU [LnPd12(AsPh)8O32]5- polyoxopalladates show potential as spin qubits upon moderate axial compression and application of a small magnetic field to help quench decoherence from hyperfine interactions31. Also note that two-photon absorption (TPA) is not parity-forbidden and so cubic hosts (CaF2, Cs2NaGdCl6, etc.) are used to isolate TPA lines from single-photon processes when studying Ln3+ electronic transitions134,135.

To understand how to target CU geometry using ligands, the subset of Ln(triNTB)2 (triNTB=tris(2-benzimidazolylmethyl)amine) structures is revelatory (subset 34). Here, 100 % of the structures have CU geometry (compared to only 1% in the parent dataset), due to the two tetradentate tripodal ligands capping the Ln3+ centre in an S6-symmetric fashion. The first example of a Ln(triNTB)2 structure was reported in 1999, and although the coordination geometry was erroneously described as distorted BTP, the authors point out that the geometry may be enforced by pi-pi stacking between the benzimidazole arms of the interlaced tripods136. Clearly, the bis-triNTB arrangement provides a reliable route to CU geometry, although no other subsets in the network manifested a similar bias towards CU geometry.

Targeting Ln3+ centres with hexagonal bicapped prismatic geometry
Last is HBP geometry, which due to its rarity is not encountered in any of the 52 subsets of size 25 or larger. Thus, the 32 fragments with HBP geometry were manually checked and found to all feature axial monodentate ligands and equatorial hexa-dentate macrocyclic or metallocrown ligands; HBP geometry is intrinsically tied to such six-fold symmetric ligand motifs and can thus be rationally targeted. Importantly, the macrocyclic and metallocrown ligands are built from all-5CR or alternating 5- and 4CR chelate motifs, respectively, whose bite angles in this case match well with the ideal 60 ° angle of the equatorial hexagon. Additionally, bulky axial phosphine oxide or siloxide ligands are often used to stabilize the geometry and prevent additional ligands from entering the coordination sphere137.

As with CU geometry, HBP geometry is less applicable for luminescent applications as there exists a centre of symmetry (D6h point group). On the other hand, HBP complexes are of interest as SMMs, as the high symmetry reduces QTM and the combination of weakly interacting equatorial ligands with anionic electron-donating axial ligands can enhance the magnetic anisotropy of oblate-shaped Ln centres138. For example, Dy3+ complexes with hexadentate Schiff-base macrocycles and electron-rich axial siloxide ligands exhibit very high energy barriers to magnetization reversal (1124-1800 K) – thus rivalling the state-of-the-art lanthanide metallocenes but having the advantage of being air-stable139–141. Many researchers have attempted to broaden SMM design by utilizing more uncommon Ln3+ elements; D6h symmetry is particularly useful for Ce3+ and Sm3+ centres as all transverse crystal-field terms vanish19,137. Overall, the scarcity of Ln3+ complexes with HBP geometry represents a compelling opportunity for further SMM design.

Influencing coordination geometry and site symmetry using crystal packing
Lastly, one under-appreciated variable that can influence solid-state Ln3+ coordination geometry is supramolecular assembly, which includes effects from crystal packing and intermolecular non-covalent interactions37. The effects of non-covalent interactions have already been seen in the earlier Ln(OH2)8, Ln(OOCR)8, and Ln(O=CNHR)8 complexes, where hydrogen-bonding motifs direct the adoption of CU and DD geometry, respectively; and the interplay between crystal packing and thermal motion has already been illustrated. Understanding how supramolecular effects can guide the geometry and symmetry around Ln3+ centres is useful for further tailoring the properties of lanthanide complexes.

One case where crystal packing forces have been utilized is in the study of Ln(DOTA) compounds (Fig. 9a and b). Generally, nine-coordinate water-capped Ln(DOTA)(OH2) complexes are more common, with the eight-coordinate form only occurring as a minor species in solution for lanthanides heavier than Ho3+117,142. However, it has been shown that eight-coordinate Ln(DOTA) complexes can be specifically targeted by considering packing effects and judiciously selecting an appropriate counter-ion143. Crystallizing with Me4N+ cations complements the tetragonal symmetry of the Ln(DOTA) complexes and provides a packed structure where strict C4 symmetry is applied to the Ln3+ centre. This allowed an investigation of the slow magnetisation relaxation behaviour of the Ln=Er and Yb compounds compared to their lower-symmetry water-capped analogues.

A similar example is with the [Ln{Pt(SAc)4}2]- complexes (Fig. 9c, d)111. Packing with symmetry-compatible PPh4+ cations brings the geometry to nearly ideal SAP (twist angle of the square faces = 44.52-44.61°), while using Et4N+ cations causes the geometry to distort along the CU→SAP path, lowering the symmetry from D4d to C4111. This structural control over the point symmetry enabled a direct investigation of the QTM behaviour of these complexes, illustrating a threefold increase in the tunnel splitting for the Dy3+ case upon switching from the D4d-symmetric to C4-symmetric environment110.

Finally, we draw from our own research with Ln(bipyO2)4 complexes (Fig. 9e, f). As noted previously, the perchlorate salts have geometries that lie on the CU→SAP path, with the position depending on the size of the Ln3+ centre. However, if combined with larger, tetragonally-shaped [Au(CN)2]- anions to give the [RE(bipyO2)4][Au(CN)2]3·H2O (RE=La, Ce, Eu, Tb, Lu, Y, Sc) series, then the ions form a square close-packed arrangement based on an S4-symmetric configuration of the Ln(bipyO2)4 cation that has DD geometry5,144. In this system, the geometry does not change as a function of Ln3+ size (Figure S6a and S6b), as now the geometry is being enforced by the surrounding efficient crystal packing. As a result of the lower-symmetry DD geometry accessed in this series, the Eu3+ complex shows a higher quantum yield than its perchlorate analogue.

These three examples reveal how counterions can regulate the shape of Ln3+ complexes and indirectly influence the configuration at the Ln3+ site, providing additional control over the geometry. The counterions must have a shape and symmetry that pack well with the desired conformer, although it is difficult to predict these relationships a priori. In simple systems, it may be sufficient to choose a counterion whose symmetry matches that of the geometry; for instance, a rare CU Nd(β-DK)4 complex can be accessed by crystallizing it with an S6-symmetric bistris(pyrazolyl)borate Ln3+ complex145.

A related concept is to use MOF or coordination polymer architecture to impose a strict geometry, either by incorporating the Ln3+ centre into the framework or encapsulating a Ln3+ complex within the pores18. For example, cyanometallate linkers have diverse shapes and symmetries and D4d-symmetric [W(CN)8]4- linkers have been used to assemble Ln3+ coordination polymers with SAP geometry71. MOFs as a vehicle to control the Ln3+ geometry for SMM development has also been previously reviewed39.

Methods
Technical details of the CSD search and dataset construction, ChemEnv coordination number validation, CShM analysis, CSoM analysis, similarity-based network analysis, minimum distortion path analysis, and chelate ring size analysis are included in the Supplementary Methods section of the Supplementary Information.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Authors report the strategies to control the geometry and symmetry around lanthanide centres for development of functional trivalent lanthanide complexes. The coordination geometry and symmetry affected lanthanide electronic structures, which are related to luminescent and magnetic properties. Authors use the Cambridge Structural Database to extract the structures of 12,697 eight-coordinate Ln3+ centres and use CShM analysis, CSMO analysis, and structural similarity-based network analysis to survey geometry and structure trends. The present strategy is new and meaningful for development of functional lanthanide complexes. Thus, the present paper can be attractive for the various readers in the field of material chemistry, photo-chemistry based on metal complexes, and so on. I would recommend it for acceptance after addressing the following issues.\n1)Limitation of the prediction\nThe coordination structures affected the electronic structure of lanthanide complexes, which are related to their functionality. However, the function from lanthanide complexes is determined by synergetic effect between coordination structures and π-electronic structures of ligands (small orbital mixing between π and 4f orbital, π-π excited state level, and so on). For example, most luminescent thermometers based on lanthanide complexes are based on thermally-assisted back energy transfer from lanthanide to ligands. In this case, the π-π* T excited level can become a significant factor. There is a lack of discussion from the perspective in the present paper.\n2) Coordination structures depending on the temperatures\nMagnetic properties are often measured by cryotemperature (< 50 K), and luminescence properties are measured under various temperatures (4K-500K). Author should include the explanation of the structural deviations determined by SCXRD analyses, which depend on temperature.\n3) Minor comment\n“intense 5D →7F transitions of interest in plant cultivation”\n0 4\nBut, intense 5D →7F transitions have also been interested in plant cultivation.\n0 2\n(For ex. Scientific Reports, 12, Article number: 17155 (2022)).\nThese sentences might cause confusion.", "reviewer_2": "This article reports on the factors that are at play in determining the geometries of eight-coordinate lanthanide complexes by data mining the CSD database and performing geometry minimization calculations. This is useful article for the field, and it will likely be heavily referenced. My issue with this article is the lack of novelty. This is more of a review and does not present significantly new concepts. The geometry minimization programs have been around since around 2000 with K. N. Raymond's Shape8 and the concept has been studied on and off since the 1960's. The primary conclusion that the article reaches in that repulsion primarily dictates the geometry, but that's well established. Nevertheless, that was a lot of work and is definitely useful. I think it would be welcomed in Coordination Chemistry Reviews. In my mind, it's not the kind of article that you expect to see publish in Nat. Commun.", "reviewer_3": "In this work, Lenzoff and Karpiuk present a detailed analysis of the geometry of octacoordinate complexes with lanthanides and how this determines some interesting properties of these compounds, such as luminescence and magnetic behavior.\nThe manuscript is divided into two parts: 1) a detailed shape analysis of all the compounds found in the CSD; 2) different strategies for controlling the geometry of these complexes through the use of ligands, etc. I believe that the inclusion of a third part that systematically specifies the relationship between form and functon(e.g., magnetism) would complete the work, but I will understand that the authors consider that out of the scope of the present work.\nMore importantly, from my point of view, the information in the second part (pages 12 to 19) could be organized in a clearer way, especially regarding the requirements for achieving a given geometry. Perhaps a larger table or some kind of graph would help to provide a simpler and more comprehensive overview. Furthermore, I think that there is an overuse of specific examples of compound families, while clear guidelines for achieving specific geometries are lacking. On the other hand, the nature of the lanthanide center is only addressed at the end (p. 16) when it seems essential in determining the geometry. I understand that it is not easy to combine all the information into a \"roadmap\" for a given geometry, but I think the authors should make an effort to facilitate the use of their results as a useful tool.\nThat said, the work done is significant in both volume and quality. I think the choice of continuous shape measures to quantify the degree of distortion in the lanthanide compounds is very successful. This tool is especially powerful for extracting information from high-coordination compounds where unusual geometries may appear. Access to the CSD has been systematically provided, which is essential when working with such large data sets. I think the potential of this work is enormous and that it will be useful to a wide range of chemists. Therefore, I recommend its publication in Nature Communication if the authors address the comments made in the previous paragraph regarding the second part of the manuscript on the presentation of results, as well as the following minor points:\n- The coordination bonds in the CSD are defined arbitrarily. Has a control mechanism been used to include or discard spheres with ill-defined coordination numbers?\n- In the context of the work, is the information obtained from the shape maps in Figure 2 really relevant?\n- On page 6, a brief summary of the trends regarding the symmetry point groups is provided, and only a cursory reference is made to a relationship between shape distortions and symmetry (2% increase in CSoM value for each 1% increase in CShM value…). I find this a very interesting point. In my opinion, the authors could have further explored the relationship between shape distortion and symmetry and how this affects achieving the desired properties of luminescence, magnetism, etc.\n- It is especially interesting the analysis of chelate ligands on pages 11 and 12. I wonder if, given the 60/40 ratios for SAP and DD in both 5CR and 6CR rings, the nature of the X and Y atoms has been evaluated in order to result in one geometry or the other.\n- It's also intriguing that for 5CR rings, where 60% of the structures are SAP, there isn't a single angle close to the 73.69 degree angle expected by the geometry. Does the fact that they are all smaller reflect a specific type of distortion? Which ligands would favor larger angles?"}, "author_response": "Response to the Reviewers\nReviewer 1\nGeneral Comment: Authors report the strategies to control the geometry and symmetry around\nlanthanide centres for development of functional trivalent lanthanide complexes. The coordination\ngeometry and symmetry affected lanthanide electronic structures, which are related to luminescent\nand magnetic properties. Authors use the Cambridge Structural Database to extract the structures\nof 12,697 eight-coordinate Ln3+ centres and use CShM analysis, CSMO analysis, and structural\nsimilarity-based network analysis to survey geometry and structure trends. The present strategy is\nnew and meaningful for development of functional lanthanide complexes. Thus, the present paper\ncan be attractive for the various readers in the field of material chemistry, photo-chemistry based\non metal complexes, and so on. I would recommend it for acceptance after addressing the following\nissues.\nResponse: We thank the reviewer for their positive reception to this work and the recommendation\nto publish after addressing the comments below.\nComment 1: Limitations of the prediction: the coordination structures affected the electronic\nstructure of lanthanide complexes, which are related to their functionality. However, the function\nfrom lanthanide complexes is determined by synergetic effect between coordination structures and\nπ-electronic structures of ligands (small orbital mixing between π and 4f orbital, π-π excited state\nlevel, and so on). For example, most luminescent thermometers based on lanthanide complexes are\nbased on thermally-assisted back energy transfer from lanthanide to ligands. In this case, the π-π*\nT1 excited level can become a significant factor. There is a lack of discussion from the perspective\nin the present paper.\nResponse: The reviewer is correct that while this work focuses on the symmetry aspects of\nlanthanide coordination complex design, there are also electronic effects involving the ligand\nscaffold that play a large role in determining magnetic, luminescent, and other properties. Although\nwe intend for this work to inform design from a symmetry perspective, we had not originally\nmentioned or discussed other important electronic effects. Accordingly, we have made the\nfollowing change in the introduction:\nOld: Crucially, increasingly fine control over the coordination geometry of trivalent lanthanide\n(Ln3+) centres is being demanded in the design of new Ln3+ coordination complexes for such\napplications, as both luminescent and magnetic properties are extremely sensitive to the symmetry\naround the Ln3+ centre.\nNew: Crucially, while the functionality of trivalent lanthanide (Ln)3+ centres depends on both the\ngeometry and the electronic structures of the surrounding ligands or host matrix, the extreme\nsensitivity of luminescence and magnetism to the symmetry around the Ln3+ centre has demanded\nincreasingly fine control over coordination geometry in the design of new Ln3+ coordination\ncomplexes for these applications.\nWe have also added the following sentences to the beginning of Part 2:\n[…] Many different geometry and symmetry combinations are of interest to chemistry and materials\nscience researchers […] New: Lastly, each application requires that ligands have certain electronic\nfeatures. For example, the ligand triplet state energy must be tuned when designing highly-emissive\ncompounds or luminescence thermometers; low-lying ligand-metal charge transfer transitions are\ndetrimental to emissive Eu3+ complexes but beneficial for Yb3+ centres; and highly covalent Ln3+-\n2\nligand bonds can push Ce3+ luminescence from the ultraviolet into the deep-red but can also reduce\nmagnetic axiality for SMMs.(five new references added) Therefore, it is often difficult to design\nligands that exert certain geometries around Ln3+ centres while also accommodating for important\nelectronic aspects.\nComment 2: Coordination structures depending on the temperatures: magnetic properties are\noften measured by cryotemperature (< 50 K), and luminescence properties are measured under\nvarious temperatures (4K-500K). Author should include the explanation of the structural\ndeviations determined by SCXRD analyses, which depend on temperature.\nResponse: We agree that temperature is an important variable when evaluating the structures and\ngeometries of Ln3+ complexes, especially considering that crystal structural data is often collected\nabove liquid nitrogen temperature while spectroscopic or magnetic properties are collected at much\nlower temperatures and correlating these data is important for establishing structure-property\nrelationships. Therefore, we have added a new section called The Effect of Temperature on\nCoordination Geometry to Part 2, analysing seven structures from the LnX dataset deposited at\n8\nthree or more temperatures in the CSD. A new table and figure have also been generated from the\nanalysis and added to the Supporting Information. The section concludes that temperature has a\nminor influence on coordination geometry, generally getting slightly closer to ideal with heating\ndue to increased thermal motion demanding a more spherical ligand field. This is in line with the\nwork of Qian and Baldoví et al. “Does the thermal evolution of molecular structures critically affect\nthe magnetic anisotropy?” Chem. Sci. 6, 4587–4593 (2015), although we also highlight recent work\nby Ding et al. “Studies of the Temperature Dependence of the Structure and Magnetism of a\nHexagonal-Bipyramidal Dysprosium(III) Single-Molecule Magnet.” Inorg. Chem. 61, 227–235\n(2022) that describes a very temperature-sensitive hexagonal bicapped pyramidal complex\n(references added to text).\nComment 3: “Intense 5D →7F transitions of interest in plant cultivation” But,\n0 4\nintense 5D →7F transitions have also been interested in plant cultivation. (For ex. Scientific\n0 2\nReports, 12, Article number: 17155 (2022)). These sentences might cause confusion.\nResponse: According to the reviewer’s comment, we have amended the mentioned sentence to\navoid any confusing statements. The following change has been made, as also highlighted in the\nmanuscript, and a citation to the reference mentioned by the reviewer (Hasegawa, 2022) has also\nbeen added.\nOld: However, Eu3+ complexes with intense 5D →7F transitions are of interest in plant cultivation,\n0 4\nas the emission overlaps well with the absorption band of photochrome PFR. (Cui, 2024)\nNew: However, Eu3+ complexes have been of interest in plant cultivation and the 5D →7F\n0 4\ntransition in particular has good overlay with the absorption band of photochrome P .(Cui, 2024;\nFR\nHasegawa, 2022)\nReviewer 2:\nGeneral Comment: This article reports on the factors that are at play in determining the\ngeometries of eight-coordinate lanthanide complexes by data mining the CSD database and\nperforming geometry minimization calculations. This is useful article for the field, and it will likely\nbe heavily referenced. My issue with this article is the lack of novelty. This is more of a review and\ndoes not present significantly new concepts. The geometry minimization programs have been\naround since around 2000 with K. N. Raymond's Shape8 and the concept has been studied on and\n3\noff since the 1960's. The primary conclusion that the article reaches in that repulsion primarily\ndictates the geometry, but that's well established. Nevertheless, that was a lot of work and is\ndefinitely useful. I think it would be welcomed in Coordination Chemistry Reviews. In my mind, it's\nnot the kind of article that you expect to see publish in Nat. Commun.\nResponse: We thank the reviewer for their careful consideration of our paper and for their\nconstructive feedback; we also appreciate their kind words, and we likewise agree that this paper\nwill be of great use to many researchers. We acknowledge their suggestion to consider Coord.\nChem. Rev. for this manuscript but we still maintain that Nature Communications is an appropriate\nvenue for this work, and we hope to make the case to this reviewer in the responses below.\nComment 1: The geometry minimization programs have been around since around 2000 with K.\nN. Raymond's Shape8 and the concept has been studied on and off since the 1960's.\nResponse: While the study of eight-coordinate metal complexes and the development of geometry\nminimization algorithms has been ongoing since the 1960s, we point out that we developed a novel\nsimilarity-based network analysis method for this project and that we also used the new Continuous\nSymmetry Operation Measures method reported just recently in 2023 by Sørensen et al. The\ninsights we have gleaned from crystallographic data in the CSD is significant and sheds light on\ncutting-edge lanthanide research, which we feel makes it appropriate for Nature Communications.\nThis work would not be suitable for Coord. Chem. Rev. as it is not a review of the field but contains\nsubstantial new results gained from data mining, network analysis, and crystal structural analysis.\nWe do thank the reviewer for bringing the program Shape8 to our attention, which like the CShM\nmethod finds the minimum distortion between a coordination polyhedron and eight-coordinate\nreference polyhedra (formalized through differences in dihedral angles between the two polyhedra,\ninstead of distances between vertices). To recognize this prior art focused on the study of eight-\ncoordinate actinide and lanthanide complexes, as well as earlier foundational work by Porai-\nKoshits and Aslanov (1972) and Muetterties and Guggenberger (1974), we have made the\nfollowing additions to the introduction of the manuscript:\nOld: To match the LnX fragments to these geometries, we use the Continuous Shape Measure\n8\n(CShM) method developed by Avnir and coworkers, which calculates the percent deviation\nbetween the atomic coordinates and a reference polyhedron that has been oriented and scaled to\nminimize the deviation.\nNew: Many algorithms and programs have been developed to assign coordination geometries to\ncrystal structure fragments; pertinently, the program Shape8 was developed to analyse eight-\ncoordinate lanthanide model complexes in the study of plutonium-sequestering systems.(four new\ncitations added) Here, we use the Continuous Shape Measure (CShM) method that is widely\nimplemented via the SHAPE v2.1 and ChemEnv tools.\nComment 2: The primary conclusion that the article reaches in that repulsion primarily dictates\nthe geometry, but that's well established.\nResponse: In the paper introduction, we recognize that lanthanide geometry is primarily dictated\nby inter-ligand repulsion (sterics) and indicate that the goal of the work is to thus learn how to\npurposefully influence the geometry: “However, it is well-established that because of their\ncontracted f-orbitals Ln3+ centres exhibit diverse and unpredictable geometries that are\npredominantly the result of steric hindrance. How can we then learn to predictably control the\ngeometry in lanthanide coordination complexes?” (pg. 2, ln 19). We believe that the article\n4\nultimately illustrates that ligands can be chosen to gain a degree of control of the geometry, which\nprogresses beyond the crutch of ‘repulsion’ and demonstrates that lanthanide geometry is not an\nuncontrollable variable.\nReviewer 3:\nGeneral Comment: In this work, Lenzoff and Karpiuk present a detailed analysis of the geometry\nof octacoordinate complexes with lanthanides and how this determines some interesting properties\nof these compounds, such as luminescence and magnetic behavior. The manuscript is divided into\ntwo parts: 1) a detailed shape analysis of all the compounds found in the CSD; 2) different\nstrategies for controlling the geometry of these complexes through the use of ligands, etc. I believe\nthat the inclusion of a third part that systematically specifies the relationship between form and\nfunction (e.g., magnetism) would complete the work, but I will understand that the authors consider\nthat out of the scope of the present work.\nResponse: The reviewer recommends a third part to the article that describes exactly how each\ngeometry can affect/tune properties, building on Part 2 that describes how to target each geometry.\nIn our approach, we aimed to briefly discuss such relationships throughout Part 2 as we covered\nhow to target each geometry, although we see how collating all of these examples into a third part\nmay benefit the reader. As mentioned by the reviewer, such a part would be justifiable if it provided\na systematic examination of the relationship between symmetry and properties, which we do\nbelieve would be beyond the scope of the work. In any case, other articles have already\nsystematically described the relationship between symmetry/coordination geometry and SMM\nproperties (Liu et al. Symmetry strategies for high performance lanthanide-based single-molecule\nmagnets, Chem. Soc. Rev. 2018) and Eu3+ luminescence (Binnemans, Interpretation of\neuropium(III) spectra. Coord. Chem. Rev. 2015) and we have cited these references throughout the\ntext.\nGeneral Comment: More importantly, from my point of view, the information in the second part\n(pages 12 to 19) could be organized in a clearer way, especially regarding the requirements for\nachieving a given geometry. Perhaps a larger table or some kind of graph would help to provide a\nsimpler and more comprehensive overview. Furthermore, I think that there is an overuse of specific\nexamples of compound families, while clear guidelines for achieving specific geometries are lacking.\nOn the other hand, the nature of the lanthanide center is only addressed at the end (p. 16) when it\nseems essential in determining the geometry. I understand that it is not easy to combine all the\ninformation into a \"roadmap\" for a given geometry, but I think the authors should make an\neffort to facilitate the use of their results as a useful tool.\nResponse: In light of the reviewer’s comment on organization, clarity (providing general\nguidelines), and depth of analysis, we have spent time and effort improving Part 2, as follows:\n1) We have re-ordered the sections to first discuss Geometry Trends Across the Lanthanide Series\n(as recommended by the reviewer), followed by The Effect of Ligand Chelate Ring Size. We have\nalso sought to extract more information from our results in these sections (see the plots of avg.\nCShM/CSoM versus Ln size or number of chelate ligands), which has further clarified the\nrelationships. We believe the two sections now provide an improved general overview of the\ninterplay between Ln size and chelate ring size, which are essentially the ‘building blocks’ of\nlanthanide coordination complexes. Specifically, plots d-i have been added to Fig. 6 and the\nfollowing paragraph has been updated:\nmid-late (Tb-Ho), and late (Er-Lu) lanthanides. For subset 1 of 717 Ln(β-DK)3(bipy) structures,\nsubset 3 of 338 Ln(β-DK)4 structures, and subset 6 of 172 Ln(DMF)8 structures, respectively: (a-c)\nshows the geometry distributions versus lanthanide size with the key at the bottom of (c); (d-f)\nshows the average CShM values versus lanthanide size; and (g-i) shows the average CSoM values\nversus lanthanide size.\nOld: In a similar vein, the DIST structures in the Ln(DMF) subset only occur in the early-series\n8\nlanthanide group. Again, larger Ln centres have a lower penalty to adopting distorted coordination\nspheres where inter-ligand distances are not optimized. This behaviour also holds true for subset\n10, which are structures derived primarily from Ln3+-substituted POTs. In the cluster-based subsets\n4 and 5 and in subset 17 of the Ln(dcbipy) hits, DIST structures were encountered in all groups\n2\nexcept of the smallest lanthanides in the late-series group, which corroborates the idea that smaller\nLn3+ centres prefer to adopt geometries that are closer to ideal.\nNew: Looking next at Figs. 6d-6i, as the Ln size decreases the average CShM and CSoM values\ndecrease for the Ln(β-DK) (bipy) and Ln(β-DK) subsets, indicating a shift to less distorted\n3 4\ngeometries. While this may in part be driven by increasing inter-ligand repulsion, the absence of a\ntrend in the CShM and CSoM values when monodentate DMF ligands are used points to a different\nmechanism. As will be discussed in the next section, when chelating ligands are used the geometry\nbecomes contingent on how well the chelates can fit around the Ln3+ centre. The trends with the\nLn(β-DK) (bipy) and Ln(β-DK) subsets suggest that bidentate ligands struggle to enclose the\n3 4\nlarger early-series lanthanide elements, leading to distorted coordination spheres, while the\nmonodentate DMF ligands cause no such distortions. Swapping one β-DK ligand for a bipy ligand\n(subset 3 → 1) even exasperates the distortions, as bipy ligands form smaller five-membered\nchelate rings.\n6\n2) Next, we have added a short section on The Effect of Temperature on Coordination Geometry,\nbased on the suggestion of the first reviewer.\n3) Next follows the section Strategies to Target each Eight-Coordinate Lanthanide Geometry.\nHere, we have added the following paragraph to introduce the section:\nNew: Looking beyond general factors, we can next use the network subsets to establish specific\nstructure-geometry relationships. This is accomplished by extracting the geometry and point group\ndistributions for each subset (Table S2) and identifying ligands that strongly bias towards certain\ngeometries. Fifteen such subsets are shown in Fig. 8, which form the basis of the following\nstrategies for targeting SAP, DD, SD, BTP, CU, and HBP geometries.\nFig. 8 (which previously appeared earlier in the manuscript) has been simplified and also re-\norganized to easily indicate which geometries are targeted by which subsets (the subsets were\npreviously just ordered by descending subset size).\nFigure 8 Subsets from the network analysis showing a strong geometry preference, labelled with the subset\nnumber, the description, and the proportion of the dominant coordination geometry and point group.\nSubsets are organized into those that target (a) SAP, (b) DD, (c) SD, (d) BTP, (e) CU, and (f) HBP. *The\nLn(DOTA), Ln[Pt(SAc) ] , Ln(LN6)(OR) , and Ln[Cd (HA) ](OR) structures were not identified within\n4 2 2 3 3 2\nthe network subsets of size 25 or larger but were manually identified (see Table S7 for the Ln(DOTA)\nstructures). Abbreviations: POT = polyoxotungstate, Hβ-DK = β-diketone, H Pc = phthalocyanine, HSAc =\n2\nthioacetic acid, H DOTA = 2,2’,2”,2”’-(1,4,7,10-tetraazacyclododecane-1,4,7,10-tetrayl)tetraacetic acid, en\n4\n7\n= 1,2-diaminoethane, H dcbipy = 2,2’-bipyridine-6,6’-dicarboxylic acid, and H salen = 2,2’-\n2 2\nethylenebis(nitromethylidene)diphenol, saloH = salicaldehyde, H TEA = triethanolamine, triNTB = tris(2-\n3\nbenzimidazolylmethyl)amine, HA = hydroxamic acid.\n4) Next follows the sections to target SAP, DD/SD, BTP, CU, and HBP geometries. Here, we have\nmade a concerted effort to make the sections more focused and provide general guidelines, with\nthe philosophy of using specific examples from the network analysis to illustrate more general\nprinciples. We have also removed several examples that are too specific or information that is\nsuperfluous. The alterations are detailed below:\nIn Targeting Ln3+ Centres with Square-Antiprismatic Geometry:\nRemoved: However, it is evident that distortions from perfect symmetry can also be advantageous.\nThe Ln(β-DK) (bipy) subset (72.00 % SAP; 89.93 % C symmetry) shows that D symmetry can\n3 s 4d\nbe removed using asymmetric ligand sets, which contributes to enhanced luminescence intensity\nfor these compounds. Likewise, Ln3+ centres chelated by two tridentate tripodal Tp- ligands and\none β-DK- are primarily C symmetric with SAP geometry (subset 38), such that Tb3+ members of\n2\nthis subset show high luminescence quantum yields above 70 % while Dy3+ members show zero-\nfield SMM behaviour\nIn Targeting Ln3+ Centres with Dodecahedral and Snub Disphenoid Geometry:\nRemoved: To understand why dcbipy selects better for the SD geometry than salen, consider that\nthe former ligand has all 5CRs, while salen has one 5CR and two 6CRs; thus, the former case better\nsatisfies the criterion for SD geometry of having regular faces (equal distances between the donor\natoms). Accordingly, the dcbipy ligand is identified as a reliable route to SD geometry and D\n2d\nsymmetry. Note that a bis(ligand) arrangement is required, as when one of the ligands is replaced\nby four monodentate ligands (subset 40) the geometry distribution switches to 100 % DD.\nInterestingly, as the strict ligand requirements for SD geometry coincide with the necessities for an\nideal D environment, this is likely why the SD hits have the lowest average CSoM compared to\n2d\nthe other geometry subsets.\nIn Targeting Ln3+ Centres with Bicapped Trigonal Prismatic Geometry:\nNew (general guideline): Thus, BTP can be targeted by using one “wide-base” C -symmetric\n4v\ntetradentate ligand and combining it with either four monodentate, two bidentate, or one C\n2v\nsymmetric tetradentate ligand like TBC[4](OMe) .\nx y\ncluster structures utilizing the tetradentate tripodal ligand TEA3- (H TEA = triethanolamine). As\n3\nwill be demonstrated below, two tetradentate tripods generally chelate in a staggered fashion along\nthe S axis to give CU geometry; however, subset 37 contains trinuclear Ln(TEA) Fe fragments\n6 2 2\nwhere the bridging Fe3+ centres enforce an eclipsed configuration of the TEA ligands, leading to a\nhighly distorted geometry around the Ln3+ centres. Yet, replacement of one of the TEA chelates\nwith four monodentate ligands gives a subset with 100 % BTP Ln3+ centres (subset 51),\ndemonstrating that easing away from a highly sterically demanding arrangement allows relaxation\nto the nearest non-distorted configuration (BTP in this case).\nNew (more general): One other subset with 100 % BTP geometry is subset 37 of fragments from\ntoroidal {Ln Fe } clusters utilizing the tetradentate tripodal ligand TEA3- (H TEA =\nx y 3\ntriethanolamine). The BTP geometry likely occurs because the tripod shape is preorganized to cap\n8\none face of the trigonal prism, and the other four ligands arrange accordingly. Thus, tripodal ligands\nmay be advantageous for targeting capped or bicapped coordination geometries, although in this\ncase the neighbouring Fe3+ centres constrain and distort the geometry leading to an avg. CShM of\n2.0±0.3 %. Additionally, replacing the four other ligands with an additional tripodal TEA3- ligands\nas in subset 51 does not further enforce the BTP geometry but instead leads to 61.29 % of the\nstructures having DIST geometry.\nIn Targeting Ln3+ Centres with Cubic Geometry:\nRemoved: The magnetoluminescent properties of Ln(triNTB) (Ln = Dy, Ho, Tb) compounds have\n2\nalso been reported.\nRemoved: Nonetheless, further methods of accessing CU geometry will be demonstrated below\nwhen discussing the lanthanide contraction and supramolecular effects.\nIn Targeting Ln3+ Centres with Hexagonal Bicapped Prismatic Geometry:\nNew (more general guidelines): Importantly, the macrocyclic and metallocrown ligands are built\nfrom all-5CR or alternating 5- and 4CR chelate motifs, respectively, whose bite angles in this case\nmatch well with the ideal 60 ° angle of the equatorial hexagon. Additionally, bulky axial phosphine\noxide or siloxide ligands are often used to stabilize the geometry and prevent additional ligands\nfrom entering the coordination sphere.\nThe general guidelines to target each geometry are also now summarized in a new ‘roadmap’\nfigure, which is included in the conclusion:\ncoordination geometries and their point groups, their example applications or limitations, and relevant\nstrategies to target these geometries using representative monodentate, bidentate, or polydentate ligands.\nAbbreviations: UCNP = upconverting nanoparticle, SMM = single-molecule magnet, bipy = 2,2’-\nbipyridine, en = 1,2-diaminoethane, Hβ-DK = beta-diketone, and bipyO = 2,2’-bipyridine-N,N’-dioxide.\n2\n5) Lastly, for the organization of Part 2, the Influencing Coordination Geometry and Site Symmetry\nusing Crystal Packing remains as the last section, although we have added a figure to illustrate the\nthree examples discussed:\nfilling style and hydrogen atoms and solvent water molecules omitted for clarity. (a) water-capped Ln3+\ncentres in Na[Ln(DOTA)(OH )]·4H O and (b) non-water capped Ln3+ centres in\n2 2\n[Me N] [Ln(DOTA)]Cl ·5H O. (c) CU→SAP intermediate geometry in [Et N][Ln{Pt(SAc) } ] and (d)\n4 3 2 2 4 4 2\nSAP geometry in [PPh ][Ln{Pt(SAc) } ]. (e) CU→SAP intermediate geometry in\n4 4 2\n[Ln(bipyO ) ](ClO ) ·2H O and (f) DD geometry in [Ln(bipyO ) ][Au(CN) ] ·H O. Element colours: C,\n2 4 4 3 2 2 4 4 3 2\ngrey; N, blue; Au, gold; Cl, green; Ln, light-blue; Na, purple; O, red; P, orange; Pt, dark-cyan; S, yellow.\nWe hope you agree that the manuscript has become more focused, organized, and insightful, to the\nbenefit of the reader.\nGeneral Comment: That said, the work done is significant in both volume and quality. I think the\nchoice of continuous shape measures to quantify the degree of distortion in the lanthanide\ncompounds is very successful. This tool is especially powerful for extracting information from high-\ncoordination compounds where unusual geometries may appear. Access to the CSD has been\nsystematically provided, which is essential when working with such large data sets. I think the\npotential of this work is enormous and that it will be useful to a wide range of chemists. Therefore,\nI recommend its publication in Nature Communication if the authors address the comments made\nin the previous paragraph regarding the second part of the manuscript on the presentation of\nresults, as well as the following minor points.\nResponse: Thank you very much for your support of this research, and we hope we have addressed\nthe points made both above and below.\nComment 1: The coordination bonds in the CSD are defined arbitrarily. Has a control mechanism\nbeen used to include or discard spheres with ill-defined coordination numbers?\nResponse: Indeed, to the best of our knowledge the CSD uses preset bonding radii to determine if\natoms are bonded are not, providing the number of neighbours reported when using the CSD Python\nbe ill-defined, especially with large CNs and distorted spheres. We had not initially accounted for\nthis when extracting the eight-coordinate fragments from the CSD and had used the fragments\nfound in the substructure search as-is (except for 3 structures listed in Table S6 “Details of Excluded\nEntries”, which were manually identified and removed as nine-coordinate structures).\nTo account for this oversight, we have validated the CN of each fragment using the ChemEnv tool\nfound in the Materials Project pymatgen Python library (Hautier et al. Acta Cryst. B 2020).\nChemEnv is a program used to implement CShM analysis and assign coordination environments to\nstructures in the Materials Project database, and the method does not presuppose a CN but\ndetermines the CN using a Voronoï analysis of the surrounding atoms. Overall, all but 27 (0.2 %)\nof the fragments were found to be assigned correctly (see details below); these erroneous\nfragments were removed from the dataset and analyses, tables, and plots were re-made as\nnecessary to account for the change, even though none of the findings or conclusions of the\narticle have changed.\nDetails of the ChemEnv analysis have been added to the manuscript and ESI as appropriate. Below\nis a brief outline of the analysis. For the CN analysis, the ChemEnv recommended maximum\ndistance factor of 1.41 was used, which only considers atoms within 1.41× the distance to the\nclosest neighbour as potential neighbours (e.g., if the closest Ln-O bond is 2.30 Å then this will\nconsider all atoms within 3.24 Å). The “SimplestChemEnvStrategy” was implemented for the\nCShM analysis, using the recommended distance cutoff of 1.4 and angle cutoff of 0.3. Out of\n12,769 fragments, 12,553 were correctly assigned as eight-coordinate by ChemEnv, while 8, 102,\n30, and 4 fragments were assigned CNs of 7, 9, 10, and 12, respectively. It must be noted that these\nlatter fragments have truly ill-defined coordination environments, and there is some difficulty on\ndeciding how to partition these structures; for example, manual inspection of many of the 144\nstructures that ChemEnv assigned as CN≠8 indicated that eight-coordinate would still in our eyes\nbe a ‘reasonable’ description. Also, for most of these 144 structures the CShM measured for the\nnon-eight coordinate spheres were higher (more distorted) than the original CShMs (CN=8). For\nonly 27 fragments was the newly assigned non-eight coordinate geometry a good fit based on the\nCShM value. Inspection of these 27 fragments indeed indicated that CN≠8 gave better descriptions\nof the coordination spheres, and we decided that the criteria of CShM (CN≠8) < CShM (CN=8)\nwas suitable for deciding when to reject lanthanide fragments with ill-defined coordination spheres.\nThe 27 fragments re-determined to have non-eight coordinate spheres were from 21 crystal\nstructures. 20 of the fragments had been previously categorized as distorted (DIST) while seven\nhad been categorized as bicapped trigonal prismatic (BTP). In most cases the ill-defined nature of\nthe coordination sphere was due to longer Ln-O bond lengths (~3 Å) from peripheral ligand arms\noften with ether donor atoms, weakly-interacting solvent molecules, or O atoms constrained within\npolyoxometalate clusters.\nWe thank the reviewer for their diligence in considering our methodology and think that these\nchanges have not only improved the analysis but provide useful knowledge for future CSD\nanalyses.\nComment 2: In the context of the work, is the information obtained from the shape maps in Figure\n2 really relevant?\npossible coordination geometries in the database and the interconversion between different\ngeometries, and this justifies the short description of the shape maps in Part 1.\nComment 3: On page 6, a brief summary of the trends regarding the symmetry point groups is\nprovided, and only a cursory reference is made to a relationship between shape distortions and\nsymmetry (2% increase in CSoM value for each 1% increase in CShM value…). I find this a very\ninteresting point. In my opinion, the authors could have further explored the relationship between\nshape distortion and symmetry and how this affects achieving the desired properties of\nluminescence, magnetism, etc.\nResponse: Indeed, we also found this relationship between CSoM and CShM values interesting,\nbut we think expanding upon it is beyond the scope of this article. Essentially, we discussed that\nsymmetry lowering can be caused by donor atom difference or by distortions of the coordination\ngeometry, and when the CShM increases by 1% the CSoM shows a ca. 2 % likewise increase.\nHowever, to our understanding the CShM and CSoM values are on relative scales, so this trend is\nonly meaningful if the CShM/CSoM values are put within the broader context of many structures\n– as done in this work – or if the CShM/CSoM values were benchmarked against experimentally\nmeasured properties to derive the relationship where for example a certain increase in CShM/CSoM\ncauses symmetry lowering to be observed in the physical properties. Indeed, an excellent report by\nVillads Nielsen and Thomas Sørensen has just now appeared in Nature Communications expanding\nupon their CSoM method and discussing potential threshold values (“yardsticks”) for symmetry-\nbreaking to occur. We agree with them that adaptation by the broader community of their new\nCSoM method will help refine such threshold values, and in view of that and the reviewer’s\ncomment we have revised the section in question, as shown below.\nOld: […] note that the actual symmetry may be lower than the point symmetry derived from the\nCSoM analysis […] for example, Sorensen et al. have shown that a 1.15 % deviation is enough to\ncause symmetry breaking for the luminescence of a ten-coordinate Eu3+ centre.\nNew: […] for example, Sørensen et al.(reference added) propose that a 0.1-1 % CSoM value may\nbe enough to cause symmetry breaking in lanthanide luminescence.\nOld: To understand the causes of symmetry-lowering in Ln3+ complexes, the structures were\npartitioned into those with homogenous and heterogenous donor atom sets. 87.0 % of the 6,329\nhomogenous structures were found to have ideal symmetry (the highest symmetry allowed by the\ngeometry), in contrast to only 4.4 % for the 5,918 heterogenous structures. Also, C -symmetric\n1\nstructures only occurred in the latter set. This implies that symmetry lowering in LnX complexes\n8\nis mainly driven by donor atom differences, as opposed to distortions of the coordination geometry\n– where there is only a ca. 2% increase in CSoM value for each 1% increase in CShM value (Fig.\nS3).\nNew: Partitioning the structures into those with homogenous and heterogenous donor atom sets\nshows that 87.1 % of the homogenous structures have ideal symmetry (the highest symmetry\nallowed by the geometry) in contrast to only 4.4 % of the heterogenous structures. Alternatively,\nplotting CSoM versus CShM values for all homogenous structures (Fig. S3) shows that CSoM\nvalues can at most increase by ca. 2 % for each 1% increase in CShM value (Fig. S3). Thus, while\nheterogenous donor atom sets can be used to lower the symmetry, distortions of the geometry can\nalso have an impact. This is important as while heterogenous donor atom sets can be easily built\ninto coordination complexes, the donor atoms in host materials like fluorides, oxides, or nitrides\ncannot be easily substituted to lower the site symmetry. Assuming a ca. 1 % CSoM upper threshold,\n13\nthe CSoM versus CShM relationship indicates that just a ca. 0.5 % deviation from ideal geometry\n(as measured via the CShM method) may be enough for symmetry-lowering to be observed in\nemergent physical properties, although this threshold must be benchmarked against more\nexperimental results before it is put into use.\nComment 4: It is especially interesting the analysis of chelate ligands on pages 11 and 12. I\nwonder if, given the 60/40 ratios for SAP and DD in both 5CR and 6CR rings, the nature of the X\nand Y atoms has been evaluated in order to result in one geometry or the other.\nResponse: We have extended the analysis of bidentate ligands to consider the nature of the X and\nY atoms. First, the Y atoms are almost always C atoms. As for the X donor atoms, the 5CR ligands\ndo show a variety of donor atom combinations, which causes different bite angles. However, we\nfound that the preference for DD geometry was present no matter the X donor atoms or other ligand\ncharacteristics such as ligand planarity. Considering the 6CR ligands, they are almost entirely beta-\ndiketonate-type ligands. We have added the following sentence to the manuscript\n[…] Thus, it is critical to understand the effect that chelate ring size imposes on the geometry of\nLn3+ centres, which can be achieved via an analysis of the coordination geometry trends of Ln3+\ncentres coordinated by 5- or 6CR bidentate ligands (with all other ligands monodentate; see\nSupporting Information for details). New: The Supporting Information and Table S5 also contain\ndata for the types of ligands encountered in the analysis, which are primarily bipy- or oxalate-type\nfor the 5CR ligands and β-DK-type for the 6CR ligands.\nNext, the following paragraphs have been added to the “Chelate Ring Size Analysis” section of the\nSupporting Information. A table containing the ligand analysis results has also been added to the\nSupporting Information, as reproduced below.\nNew: The ligand types were also analysed, indicating that the 5CR ligands have 96.38 % C atom\nbackbones (Y atoms), are 87.8 % planar, and have NN (bipy-type), OO (catecholate/oxalate-type),\nNO (picolinate-type), or SX (X= O or F) donor atoms, in decreasing order of occurrence (Table\nS5). The average ligand bite angle increases smoothly from NN (64±2 °) to SX (67.5±1.2 °) as the\ndonor atom size increases, and while it increases to 67.6±0.8 ° for non-planar NN (en-type) ligands\nit decreases from 67±2 ° to 65±2 ° when going from planar OO to non-planar OO (DME-type)\nligands. Regardless, a preference for DD geometry persists throughout; tetrakis-5CR fragments\nwith all planar NN or planar OO ligands are 67.86 and 60.53 % DD, respectively, and Ln(en)\n4\nstructures (cf. subset 43) are 96.3 % DD. Analysis of the 6CR ligands indicated that they are 91.78\n% planar OO ligands with 92.91 % C atom backbones, which corresponds to common β-DK-type\nligands.\nBite angles were determined from the X-Ln-X angle, while ligand planarity of the 5CR ligands was\ndetermined from the X-Y-Y-X torsion angle (θ < 20 ° = planar and θ ≥ 20 ° = non-planar; the non-\nplanar ligands typically had θ ≈ 60, corresponding to the gauche configuration observed with sp3\nhybridized backbones). Ligand planarity of the 6CR ligands was determined from the average of\nthe X-Y-Y-Y and Y-Y-Y-X torsion angles, using the same 20 ° threshold.\nTable S5. Analysis of Bidentate Ligands\nAtoms Common Ligand (%) Angle (°)\nFive-Membered Chelate Ring Ligands\nNN Planar 2,2’-bipyridine 462 34.50 64±2\nOO Planar catecholate/oxalate 434 32.41 67±2\nNO Planar picolinate 251 18.75 65.2±1.8\nNN Non- ethane-1,2-diamine 127 9.48 67.6±0.8\nplanar\nOO Non- 1,2-dimethoxyethane 33 2.46 65±2\nplanar\nSX (X = O Planar 28 2.09 67.5±1.2\nor F)\nNO Non- 4 0.30 64.3±0.3\nplanar\nSix-Membered Chelate Ring Ligands\nOO Planar beta-diketonate 3628 91.78 72.0±1.7\nOO Non- 219 5.54 74±3\nplanar\nNO Planar 83 2.10 70.0±1.6\nNO Non- 13 0.33 69.4±1.5\nplanar\nNN Planar 8 0.20 68.3±1.0\nNN Non- 2 0.05 71±2\nplanar\nComment 5: It's also intriguing that for 5CR rings, where 60% of the structures are SAP, there\nisn't a single angle close to the 73.69 degree angle expected by the geometry. Does the fact that\nthey are all smaller reflect a specific type of distortion? Which ligands would favor larger angles?\nResponse: The reviewer refers to the 5CR ligand analysis, where adding 5CR ligands to the\ncoordination sphere causes an ultimate shift to 60% DD, although the bite angles all fall short of\nthe corresponding ideal angle of 73.69 degrees. The reviewer correctly points out that this should\ninduce some sort of distortion. Indeed, we have supplemented the 5CR and 6CR ligand analysis by\nplotting the average CShM value as a function of the number of bidentate ligands in the\ncoordination sphere (Fig. 7e and 7f below). The new results show that the average CShM value\nincreases smoothly as 5CR ligands are added (r2 = 0.95), while the CShM value is not affected by\nadding 6CR ligands to the coordination sphere.\n(b) ligands in the coordination sphere (n). Histograms of 5CR (e) and 6CR (d) ligand bite angles (°), as displayed in the\ninsets. Plots of average CShM (%) versus number of 5CR (e) or 6CR (f) ligands in the coordination sphere (n), where\nthe n = 0 data point corresponds to the subset of LnX8 fragments with only monodentate ligands.\nThis demonstrates that 5CR ligands do cause a distortion of the geometry, because the bite angles\nare not matched to the angles of the ideal polyhedra. The specific distortion would be ‘pinching’ of\ndifferent vertex pairs in the polyhedron to accommodate the small bite angles. From the new ligand\ntype analysis discussed in the previous Response, we can see that non-planar 5CR ligands like\nethylene diamine (en) can offer larger bite angles, but they still cannot reach the ideal angle for a\nDD polyhedron and the subset of Ln(en) structures (subset 43) still exhibits a high average CShM\n4\nof 1.13±0.14 % (Table S2). Of course, the 6CR ligands favour larger and better-matched bite angles\nand so they do not cause any distortion of the geometry. To account for the new results, the\nfollowing paragraph has been added to the analysis:\n[…] Indeed, Hoffmann et al. stipulate that DD geometry is preferred when using ligands with\nsmaller bite angles, which is verified on a large scale by this analysis. […] New: However, as\nnoted in the Ln3+ size analysis, since the bite angles imposed by 5CR ligands are too small to\nperfectly match the DD polyhedron, 5CR ligands cause distortions of the geometry. This is\ndemonstrated by Fig. 7e, where the average CShM value increases as 5CR ligands are added to the\ncoordination sphere (r2 = 0.95). On the other hand, adding 6CR ligands to the coordination sphere\ndoes not affect the average CShM (Fig. 7f), as the bite angles are better matched to ideal geometries.\nThus, while small-bite angle 5CR ligands may be useful to bias towards DD geometry, a\nconcomitant distortion of the geometry appears unavoidable.\n16"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have adequately addressed all of my concerns, and I consider the manuscript acceptable for publication.", "reviewer_3": "The authors have made a great effort to address all the reviewers' comments, resulting in an excellent article that certainly deserves to be published in Nature Commun."}, "author_response": ""}]

Decision letter