Introduction

Molecular nanomagnets were reported for the first time at the beginning of the 1990s, when Mn12O12(CH3COO)16(H2O)4 was discovered to display magnetic hysteresis in analogy to classical magnets, but with a quantum tunnelling mechanism for the relaxation of the magnetisation1,2. This polynuclear magnetic complex was the first of a plethora of single-molecule magnets (SMMs). The term was coined for systems behaving as hard bulk magnets below a certain temperature, but where the slow relaxation of the magnetisation is of purely unimolecular origin. Their magnetic behaviour can be approximated to that of an effective anisotropic magnetic moment arising from the exchange interactions between the spins of the metal ions. The reversal of this giant anisotropic spin occurs by populating excited spin states and overcoming an energy barrier. Hence, the thermal dependence of the relaxation rate was described by the Arrhenius equation (Fig. 1b), using this effective energy barrier (Ueff) and a pre-exponential factor (τ0)3. Both parameters were not extracted directly from the hysteresis loop (see Fig. 1c), but rather from the combined frequency- and temperature-dependence of the so-called out-of-phase component of the ac susceptibility (χ”, see Fig. 1a)3,4. Given sufficient experimental information, considering other processes in the fit, such as the Raman process and Quantum Tunnelling of the Magnetisation (QTM), in principle results in more accurate values, which are denoted as Ueff,ff, τ0,ff. It also allows extracting additional parameters C, n to characterise Raman, and τQTM (Fig. 1b).

Fig. 1: Main magnetic concepts employed in this study.
figure 1

Slow relaxation of the magnetisation in SIMs can manifest in different ways. a Spin blocking is often characterised by a temperature- and frequency-dependent out-of-phase ac susceptibility χ”, being TB3H the temperature for the maximum χ” at 103 Hz in the presence of a magnetic field. b These relaxation dynamics have most often been modelled as an Orbach process (black dots), using the Arrhenius equation. Raman (green dots) and quantum tunnelling (red dots) processes can also be relevant. c Magnetic hysteresis can be full (orange) or “pinched”, also known as butterfly (purple), where the latter signals a fast relaxation at zero magnetic field. d Scheme of different relaxation processes: tunnelling involves just the states within the ground doublet, Orbach process takes place via an excited state, and Raman process happens via a virtual state.

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The best metric for slow relaxation is hysteresis temperature (Thyst), the highest temperature at which the system presents magnetic bistability. This translates in open hysteresis as a necessary condition for achieving molecular magnets. The first SMMs exhibited low values of Thyst, which was attributed to their modest effective energy barrier (Ueff ≈ 50 K)5. Initial models based on effective spin Hamiltonians gave rise to the relation Ueff = DSz² and concluded that the best strategy to raise Ueff and, therefore, to improve the maximum Thyst is to maximise the total effective spin (S), rather than the magnetic anisotropy (D)6. Indeed, the latter is a less straightforward target for the synthetic chemist7. Despite great effort toward the synthesis of such systems and an abundance of molecules with increasing values of S, very little progress was made in the first decade in terms of increasing Ueff or Thyst8.

In the 2000s, a novel class of molecular nanomagnets emerged, namely bis-phthalocyaninato (Pc) double deckers9. This second generation of SMMs, commonly known as Single Ion Magnets (SIMs), is based on mononuclear complexes containing a single magnetic ion embedded in a coordination environment. They constitute the smallest molecule-based magnet and their properties arise from strong spin–orbit coupling which, combined with the crystal-field interaction with the surrounding ligands, results in an enhanced magnetic anisotropy when compared to SMMs. Identical data treatment using the Arrhenius equation led to effective energy barriers Ueff up to an order of magnitude higher for SIMs based on rare-earth ions when compared to those of polynuclear metal complexes of the d-block. The characteristic maxima in the out-of-phase component of the ac susceptibility χ” also moved to higher temperatures (Fig. 1a), but Thyst did not increase as significantly.

After the germinal LnPc2, different chemical families such as polyoxometalates10 and metallocenes11 were shown to exhibit slow relaxation of the magnetisation of purely molecular origin (Fig. 2). Initially, Tb3+ and Dy3+ ions were the most commonly studied. These present an equatorially expanded f-electron charge cloud and are known as oblate (see Supplementary Section 1). Success cases were also found for lanthanide ions with axially elongated f-electron charge cloud (prolate ions, e.g., Er3+, Tm3+, Yb3+). The realisation that lanthanide SIMs were not restricted to a single chemical strategy inspired a large community of chemists. As a result, between 2003 and 2019 SIM behaviour was reported in over 600 compounds, and above a third of these compounds actually displayed magnetic hysteresis. No single chemical strategy has dominated in terms of reported examples, although many approaches have been paradigmatic (e.g., the use of radicals12,13 and diketonates14). Recent efforts have been made to offer some perspective15,16,17,18. Nevertheless, anecdotal claims from proven strategies are hard to distinguish, as so many studies pursuing independent approaches have been reported. Modern techniques of data analysis and visualisation can contribute to remedy this knowledge gap. In particular, dashboards are intuitive graphical applications for dynamic data visualisation and information management, of growing popularity in different fields19,20,21.

Fig. 2: Molecular structures of some representative lanthanide-based SIMs from different chemical strategies and some of their chemical descriptors.
figure 2

a Pc double deckers, abbreviated as LnPc2 (Thyst = 2 K)72. b Metallocene complex LnCp*2 (Thyst = 60 K)42, see Supplementary Information 1 for our CN criterion for the metallocene family. c A complex with the introduction of a diamagnetic TM ion near the lanthanide ion, [Zn2DyL2(MeOH)]− (L is a tripodal ligand, 2,2’,2”-(((nitrilotris(ethane-2,1-diyl))tris(azanediyl))tris(methylene))tris-(4-bromophenol)) (Thyst = 11 K)73. d [L2Dy(H2O)5]3+ (L = tBuPO(NHiPr)2), a complex outside the main categories of the present study, which was classified as other families (Thyst = 30 K)46. (colour scheme for atoms: green, Zn; cyan, Tb or Dy; grey, C; blue, N; yellow, P; orange, Br; red, O. Hydrogen atoms are not shown for clarity).

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The present work firstly aims to rationalise the correlations among the different physical variables involved in SIMs. A common hypothesis is that the parameters arising from the ac susceptometry (e.g., Ueff) are well correlated with the experimental values (e.g., Thyst). This, however, has actually been challenged in various ways18,22,23. Over the years, various theoretical approaches have put the focus on the rationalisation of different physical processes and parameters22,23,24,25. Secondly, in order to provide the synthetic efforts with a data-driven chemical design guide, we apply the techniques of third generation computational chemistry26. Here, we started by collecting a high-quality dataset from published data and represent it in an interactive dashboard. Then, we statistically analysed the correlations between molecular descriptors and physical parameters. As a second phase of the work, we expanded this dataset to rationalise the correlations found in the first phase, and to analyse the influence of the shapes of coordination environments on the magnetic dynamics.

Results

A dataset and interactive dashboard for lanthanide SIMs

We built a dataset of the most relevant chemical and physical properties of 1411 lanthanide SIM samples collected from 448 scientific articles (Supplementary file) published between 2003 and 2019 and developed a user-friendly dashboard-style web application named SIMDAVIS (Single Ion Magnet DAta VISualisation) to host it. The dataset contains over 10000 independent pieces of chemical information, as well as over 5000 independent pieces of physical (magnetic) information. Furthermore, the dataset is hierarchically clustered into magnetostructural taxonomies (see Supplementary Sections 4 and 6) in order to pave the way for further analysis, including machine-learning studies, a field that is now on the rise27,28. Indeed, data taxonomies are powerful tools to make sense of data, since they provide ordered representations of the formal structure of knowledge classes or types of objects within a data domain29.

Each chemical family that has been widely explored in this field is claimed to be promising as molecular nanomagnet, usually by citing the best reported case. However, it is crucial to avoid getting distracted by the occasional well-behaving example and instead to evaluate the general behaviour of each chemical strategy. Do members of a family generally present a slow relaxation of the magnetisation, in terms of ac susceptometry and/or hysteresis? To evaluate this against a common reference, our dataset allows comparing the overall performance of samples in each family with the performance of the “Mixed Ligands” and “Others” Families, that act here as a sort of control group. Similarly, since Tb3+ and Dy3+ ions are oblate, as well as the cases where the record results have been obtained, it is also commonly assumed that in general complexes with oblate ions result in better SIM properties compared with prolates. Our dataset should be able to test this.

The question remains of what does one mean by better SIM properties, or, as eloquently put recently, “How do you measure a magnet?”30 Blocking temperature definitions in recent works include the temperature at which the relaxation time is 100 s (TB2), the temperature at which there is a maximum in the zero-field cooled susceptibility (TB1), and the maximum temperature at which hysteresis is observed (Thyst). Unfortunately, only a small part of the articles provided any of these parameters, whereas older bibliography favoured only Thyst. This potentially introduces a severe publication bias that our dataset tackles by including information about ac susceptometry, Ueff and hysteresis (see Supplementary Section 1.1).

Qualitative and quantitative information based on the almost ubiquitous ac susceptibility measurements was invaluable for our analysis. Since there is an ac curve at (or near) 1000 Hz frequency in virtually all works in the field (given that both MPMS and PPMS magnetometers cover this range), we chose the maximum of the out-of-phase ac curve at this frequency as the basis for our most abundant qualitative and quantitative data source. At a given frequency, χ” does not necessarily present a maximum; an external magnetic field facilitates this effect by cancelling QTM. In this study, we register the temperature of this maximum as TB3 (TB3H), the blocking temperature at 103 Hz in absence (in presence) of a magnetic field (Fig. 1a).

Ueff is also very widely employed and assumed to be a good descriptor of magnetic behaviour. In contrast, in Arrhenius-type fits generally little attention is paid to τ0; according to a simplified two-phonon Orbach model, the two variables are supposed to correlate31. At the same time, Ueff is rightfully criticised as an oversimplification that overlooks physically independent mechanisms (notably, Raman) that could be dominating the behaviour (Fig. 1d). Our dataset aimed to answer these questions.

Finally, note that a remnant magnetisation at H = 0 is a defining feature for molecular nanomagnets. If remanence is lacking, it is not feasible to store long-term information on a molecule. One can ask: is a molecule that shows out of phase magnetic susceptibility as a response to alternating current but no hysteresis, really a molecular nanomagnet? There is, however, a link between short- and long-term magnetic memory, so we included the wider definition of SIMs in the present study, as detailed above.

SIMDAVIS allows the chemical community to visualise the key relationships between chemical structures and physical properties in our catalogue of SIMs. Our interactive dashboard can be directly invoked by accessing the internet site where it is located32. It is organised in 6 main tabs: Home, ScatterPlots, BoxPlots, BarCharts, Data (View Data and Download Data) and About SIMDAVIS (Variables, Authors, Feedback&Bugs, Changelog and License) as we can observe in Supplementary Fig. 11.

In the SIMDAVIS dashboard, the most versatile source of graphical information is the “ScatterPlots” tab, where an example plot is explained in Supplementary Fig. 11. The next two tabs display the data in complementary ways. The “BoxPlots” tab allows to examine the distribution of each SIMs quantitative property vs. a categorisation criterion, e.g., we can see the distribution of Ueff values as a function of the coordination elements. The boxplot for each category is shown, including the median and the interquartile range. The “BarCharts” tab allows the exploration of the frequency of different qualitative variables in our dataset. Stacked bar graphs allow the simultaneous analysis of two qualitative variables, e.g., we can display, for each chemical family, the number of samples, which present magnetic hysteresis. The “Data” tab is a powerful interface to browse the dataset, featuring the possibility to choose the data columns to show, ordering in ascending or descending order, and filtering by arbitrary keywords; it also permits downloading all data, including links to the CIF files, when available. Finally, the “About SIMDAVIS” tab contains information about the variables contained in the dataset.

Statistically driven chemical design of SIMs

SIMDAVIS allows the visualisation of the relationships between chemical and physical variables in SIMs, and thereby enables determining the main variables that the synthetic chemist needs to consider to obtain the desired physical properties. We will first analyse this qualitatively employing a series of boxplots, violin plots and bar charts (see Fig. 3 and Supplementary Figs. 11.1–11.6, 12.1–12.5, 13.1–13.2). The full statistical analysis is presented in Supplementary Sections 4, 5 and 6.

Fig. 3: Violin plots and bar charts relating magnetic relaxation behaviour with the main chemical parameters.
figure 3

The width of each violin plot is proportional to the density of data points for this range of values, and the horizontal stripes mark the quartiles. a Values of Ueff for samples in each chemical family. b Values of TB3 for samples in each chemical family. c Values of TB3 for samples containing Tb3+, Dy3+, Er3+. d Values of Thyst for samples containing Tb3+, Dy3+, Er3+.

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First, let us focus on the effective energy barrier Ueff and the blocking temperature TB3 (the temperature for maximum out-of-phase ac susceptibility χ” at 103 Hz, see Fig. 1). From Fig. 3 and Supplementary Figs. 11.1–11.4, we can see that the chemical families with a distinctly good behaviour are LnPc2 and metallocenes, with median values of Ueff > 200 K and TB3 > 30 K. Equivalently, one can see that Dy3+ and Tb3+ present somewhat higher Ueff, TB3 than the rest (Supplementary Figs. 11.3c, 11.4a) and that, in general, oblate ions perform better than prolate ions, for both properties. In addition, non-Kramers ions present higher median TB3 but similar Ueff values compared with Kramers ions.

Now, let us analyse the maximum hysteresis temperature Thyst. The only chemical family with a distinct positive behaviour is the metallocene family. More surprisingly, Er3+ complexes have distinctly high hysteresis temperatures, markedly with a higher median than Dy3+ or Tb3+ complexes. This is in sharp contrast with their relative TB3 values, which are consistently much lower in the case of Er3+ complexes. This not only indicates that searching for equatorial environments, precisely the ones that favour good magnetic properties in Er3+ complexes24, often results in more rigid ligands, but also indicates an underexplored territory. It is certainly possible that chemical modifications of [Er(COT)2]− (or other Er3+ record-bearing complexes) designed to optimise the detrimental effect of molecular vibrations may achieve records that are competitive with DyCp*2. Prolate ions are consistently—and surprisingly—better than the oblate ones, having a higher median value for Thyst. This is again in contrast with the opposite behaviour that is observed for TB3 and Ueff, and possibly again due to the influence of Er3+ complexes with their more rigid equatorial environments. This behaviour of Er3+ is unexpected after a recent theoretical contribution33, which calculated the electronic structure of Er[N(SiMe3)2]3 variants, concluding that no geometrical optimisation can significantly improve Ueff for Er3+. Nevertheless, all the high Ueff cases involving Er3+ in our database are based in the COT ligand, meaning our differing conclusions stem from different chemical strategies.

Finally, the coordination number and the number of ligands do have an influence on the statistically expected hysteresis temperature, with the best ones being 2 and 7 in the case of the coordination number and just 7 for the number of ligands. As we will discuss below, there are chemical insights to be gained from this if one analyses the influence of the coordination environment shape.

To put all these trends into perspective, it is important to numerically analyse the connection between the different variables and the clustering of our data. A lognormal analysis (see Supplementary Section 4.3) shows that the three main chemical variables, namely the chemical family, the lanthanide ion and the coordination elements, are sufficient to reasonably explain the variation of values of the others. This means that there is a limit on the information one can independently extract from the rest of the chemical variables. Multiple correspondence analysis (see Supplementary Sections 4.1, 4.2) suggests a chemical clustering that consists in three small groups, namely Gd3+ complexes, metallocenes and LnPc2 double deckers, and two much larger groups with a large overlap with oblate and prolate ions respectively. A factorial analysis of mixed data considering also all magnetic information available (see Supplementary Section 6) simplifies the clustering to three groups. Again, the two distinct families present a large overlap with metallocenes and LnPc2 double decker chemical families, both of them presenting significantly better properties than the other kinds of samples. Finer clustering categorisations are possible and indeed available in the dataset. These taxonomies can serve to guide future theoretical work. In the current stage, they mainly serve to confirm that, in layman’s terms, all chemical families within our current dataset present basically indistinguishable magnetic behaviours, except metallocenes and double deckers.

Further insight is provided representing the reported behaviour of magnetic hysteresis and ac magnetic susceptibility as a function of different chemical variables, namely (i) chemical family, (ii) metal ion, (iii) coordination number and (iv) coordination elements (Fig. 3, Supplementary Figs. 12.1 to 12.5). Remember that for hysteresis we are limited by the minority of the samples where hysteresis or its absence is reported; in the vast majority of the cases this information is lacking. Nevertheless, it is apparent that certain chemical families such as LnPc2 and metallocenes tend to display hysteresis, with diketonates being in a distant third position. In contrast, complexes based on POMs or on Schiff bases seldomly report hysteresis, and actually tend to not even present out of phase ac signals (Supplementary Fig. 12.2). In terms of metal ions, Dy3+, Tb3+ and Er3+ are clearly the best behaved (Fig. 3 and Supplementary Fig. 12.3).

The suggestion of future directions to guide synthetic efforts requires a more detailed study of the physical mechanisms of relaxation and the predictive power of the different parameters, as well as taking into account the shape of the coordination environment. This information is not immediately available from the literature. We performed these tasks and here we present the results in the following sections.

Orbach mechanism: oversimplified, predictive… a function of vibronic coupling?

A key question is: how much have the analyses in this field been affected by the simplified assumption that SIMs relax via an Orbach mechanism, characterised by τ0 and Ueff? We strived to quantify up to what level the value of Ueff is well correlated with the slow relaxation of the magnetisation. Ultimately, we wanted to verify whether a true correlation with magnetic relaxation is only possible when available data allows the fitting of Ueff,ff. A visual inspection (Fig. 4a), a categorical analysis (Fig. 4b, c), an in-depth statistical analysis of all physical parameters based on the Akaike Information Criterion (Supplementary Section 5.3) and a factorial analysis of mixed data (Supplementary Section 6) conclude that Ueff derived from a simple Arrhenius plot is, by itself, an excellent predictor for magnetic behaviour. Supplementary Section 5.4 presents the full discussion of this question. As an alternate approach to evaluate the validity of the Orbach mechanism, it has been pointed out that frequently, as Ueff increases, τ0 decreases, leaving relaxation times essentially constant5. An approximate relation between τ0 and Ueff can be derived for a two-phonon Orbach process within a Debye phonon model31. Fitting experimental data to it results in a large dispersion but only moderate deviations from the expected parameter range, meaning these approximations can be useful (Supplementary Section 5.5 for details)30.

Fig. 4: Main dependencies between the physical variables.
figure 4

a Dependence between Ueff and Ueff,ff. b Distribution of Ueff for samples depending on their qualitative hysteresis behaviour, c Distribution of Ueff for samples depending on their qualitative ac χ” susceptibility behaviour. For a more complete analysis, see Supplementary Sections 5.1 and 6.

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Having shown that Ueff is very successful—perhaps unreasonably so—at predicting the magnetic behaviour, we turn our attention to the other relaxation pathways, notably Raman and quantum tunnelling of the magnetisation processes. The former is characterised by a prefactor C and an exponent n, whereas the latter by a time τQTM. Since fits including this information are relatively scarce, one needs to note that this phase of the analysis has much less statistical power. We extracted C, n, τQTM from all samples which presented Ueff,ff, and explored possible correlations among the different parameters, in logarithmic form. We found that log(Ueff) seems to correlate quite well with log(C) (see Supplementary Fig. 34, top), and this correlation is perhaps more clear with log(Ueff,ff) (see Fig. 5). In particular, cases with high Ueff,ff (>200 K) present a low C ( < 10−3 s−1) and vice versa. While the number of points is limited, this correlation is statistically sound (see Supplementary Section 9.3). However, these are supposed to be two fully independent mechanisms, so what could be the reason behind this apparent coincidence?

Fig. 5: Relation between Ueff,ff and Raman relaxation parameter C, in logarithmic form.
figure 5

Data points are accompanied by a least square fit including confidence intervals.

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First we need to be aware of the fact that even a fit considering different relaxation mechanisms is a simplification. Indeed it has been shown that the anomalous Raman exponents so often found in these fits come from this fact34. These fits assume that one is studying an experiment with 3 physical processes that can be independently parameterised. Instead, many more processes are simultaneously taking place, including alternate multi-phonon Orbach mechanisms, competing Raman mechanisms dominated by different vibrational frequencies and the direct process. This means it is not surprising that, when fitting an overly complex process with a few parameters, some of them are unphysically correlated.

However, there is also a possible physical reason behind the correlation we found. It is the coupling between spin states and vibrations: a recently recognised key parameter in molecular spin dynamics35,36,37. Spin-vibration coupling is a common factor for both relaxation pathways38, and plays a vital role in both Raman37,39 and Orbach37,40,41 mechanisms. This means that a strong crystal field is not a sufficient condition for a high Ueff. One also needs a low vibronic coupling so that the effective barrier is closer to the total crystal field, rather than the first excited state. This hypothesis is consistent with the interpretation of their own results in the record-setting dysprosoceniums with hysteresis up to 60 K42 and 80 K43. Supporting this interpretation is also the fact that this correlation with Ueff,ff is apparently absent in the case of τQTM (see Supplementary Fig. 36). Furthermore, this vibrationally controlled Ueff would contribute to explain the typically weak correspondence between predicted (or even experimentally determined) energy levels and the Ueff extracted from the spin relaxation experiments, a problem which is often minimised and sometimes justified by a role of QTM. Crucially, according to this idea, LnPc2 and metallocenes would behave as exceptionally good nanomagnets not just because they provide exceptionally strong crystal field, but because they additionally provide exceptionally weak spin-vibrational coupling due to their rigidity, blocking Orbach and Raman processes simultaneously.

We have now obtained a likely rationalisation of why the controversial, oversimplified Ueff is such a good predictor for the magnetic behaviour, and why a parameter that, resulting from a simplified fit, effectively summarises other relaxation mechanisms and correlates so unexpectedly well with the true Ueff,ff. The thermal dependence of the spin relaxation depends on Orbach + Raman, but Ueff is statistically correlated with C and, as can be speculated from the current understanding of spin relaxation34, this can be related to the fact that Ueff and C are affected by the spin-vibrational relaxation. Whether the spin levels are real or virtual, to exchange energy with the thermal bath the spin needs to couple to vibrations. If this hypothesis is proven to be correct, a high Ueff could be understood as acting as a witness for a weak spin-vibrational coupling.

Influence of the coordination environment shape

Since we have established that Ueff is a good predictor for magnetic behaviour and also rationalised how and why it correlates with Raman relaxation, let us now turn our attention to rational chemical design strategies. A key question is: are there coordination polyhedra that are intrinsically well suited to produce high effective barriers? Plotting all Ueff values vs. the closest polyhedron for each complex, this seems to be the case. In contrast to what a cursory review of claims in literature would suggest, preparing lanthanide complexes that present a coordination environment close to D4d is not the best path. A more detailed analysis can be read in the Supporting Information Sections 7–8, including a critical assessment of data scarcity. Let us focus here on a salient case constituted by pentagonal bipyramids (with CN = 7), which present a striking distribution of Ueff, with consistently high values compared with any of the other common polyhedra (see Fig. 6a), as well as a high success rate both in terms of presenting a peak in χ” at 103 Hz and magnetic hysteresis (see Supplementary Fig. 32.2). Indeed, pentagonal bipyramids, much like square antiprisms, present no extradiagonal crystal-field terms therefore minimising spin mixing. Additionally, all of their diagonal terms are in first approximation protected from low-energy vibrations, minimising vibronic coupling (for a longer discussion of this see Supplementary Section 7). Their barriers can be maximised by vertical compression (see Supplementary Fig. 33).

Fig. 6: Relation between Ueff and chemical design.
figure 6

a Dependence between Ueff and coordination polyhedra. b Numbers of ligands that are present in the different coordination polyhedra. c Elements in the coordination sphere that are present in the different coordination polyhedra.

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The natural follow-up question would be how to chemically favour this kind of coordination rather than, for example, capped trigonal prisms, which also present CN = 7. From the dataset, it is obvious that employing non-chelating ligands massively favours the formation of pentagonal bipyramids (see Fig. 6b). The greatest synthetic competitor would seem to be octahedra, that also forms most often from ligands coordinating via a single atom, whereas most other shapes with CN ≥ 8 tend to result from chelating ligands. Similarly, a combination of oxygens and nitrogens is to be avoided, since for CN = 7 this tends to result in capped trigonal prisms; to obtain pentagonal bipyramids, an all-oxygen coordination sphere is often employed instead.

Discussion

We have systematically analysed 448 articles to collect information from over 1400 samples reported over the first 17 years of the field of lanthanide-based SIMs and built a user-friendly dashboard for the visualisation of all the collected data. Moreover, we carried out an in-depth statistical analysis that allowed extracting trends, distinguishing the most relevant variables and grouping the data in clusters based on their chemical and physical properties. From this study, we can highlight two main pieces of information.

In the first place, from the point of view of the parametric characterisation, the simple Arrhenius fit assuming an Orbach process has been proven to be surprisingly meaningful, with the expected approximate relation between τ0 and Ueff. One can therefore perform this oversimplified theoretical fit knowing that the effective energy barrier Ueff has been proven to present a consistently good correlation with SMM behaviour, as well as with Raman parameters C, n. Crucially, we have also shown the different nature of short-term magnetic memory in the form of the blocking temperature TB3 at 103 Hz and its long-term counterpart in the form of maximum hysteresis temperature Thyst. The best strategies that optimise the former are not necessarily the best for the latter.

In the second place, the chemical roadmap for the preparation of lanthanide complexes with higher Thyst becomes now a little clearer. Generally, oblate ions are superior to prolate in terms of ac and Ueff, but not in Thyst. So far, there has been a single chemical strategy to consistently and prolifically produce good magnetic memories, namely sandwiching an oblate ion between two rigid, planar, aromatic ligands; furthermore, the ion should be chosen to result in the most favourable MJ structure, given the electron distribution offered by the ligand. Up to now, only two chemical families are well adapted to this strategy, namely TbPc2 complexes and dysprosium metallocenes. Optimisation is ongoing within these two families, for example TbPc2 complexes featuring a radical Pc display enhanced properties44, and the reduced (divalent) analogues of DyCp*245. We find comparatively little value in further pursuing chemical strategies that have been amply explored and never yielded hysteresis above 10 K, like polyoxometalates, Schiff bases, diamagnetic transition metals placed near the magnetic lanthanide, or radical ligands, except when acting as a bridge or as a part of a TbPc2 complex. On the other hand, we also evidence that there is, of course, value in chemical ingenuity and exploration, in the quest for another successful strategy, which according to our results might well be based on equatorial erbium complexes, since these display consistently high Thyst values. Note that a few complexes included in our data fall into ample families such as mixed ligands or other families, and yet present excellent hysteresis temperatures. It is possible that the next family of record-setters is related to one of the promising candidates in Fig. 7. Two axial phosphine oxide ligands with bulky substituents seem to function in a similar way as metallocenes, despite the five equatorial H2O molecules (see Fig. 7a)46,47. This strategy is not restricted to phosphine oxides and deserves to be explored further: complexes with 7 ligands have median values of Thyst close to 10 K, as high as those with 2 ligands. Indeed, and as pointed out above, axially compressed pentagonal bipyramids are a most promising yet underexplored strategy, and monodentate oxygen-based ligands seem to be a consistent path to achieve them.

Fig. 7: Promising systems for the development of new high-Thyst SIMs, all chemically distinct from each other and from the TbPc2 and metallocene categories47,74,75,76,77,78.
figure 7

a [Dy(Cy3PO)2(H2O)5]3+ (Cy3PO = tricyclohexyl phosphine oxide)47; b [Dy4(bzhdep-2H)4(H2O)4(NO3)4] (bzhdep = pyrazine-2,5-diyl-bis(ethan−1-yl-1-ylidene)-di-(benzohydrazide))74; c [Dy(BIPMTMS)2]− (BIPMTMS = {C(PPh2NSiMe3)2}2−)75; d [Dy(bbpen)Br] (H2bbpen = N,N′-bis(2-hydroxybenzyl)-N,N′-bis(2-methylpyridyl)ethylenediamine)76; e (NNTBS)DyI(THF)2 (NNTBS = fc(NHSitBuMe2)2, fc = 1,1′-ferrocenediyl)77; f [DyLz2(o-vanilin)2]+ (Lz = 6-pyridin-2-yl-[1,3,5]triazine-2,4-diamine)78. (colour scheme for atoms: green, P; cyan, Dy; grey, C; blue, N; yellow, Si; orange, Fe; red, O; magenta, Br; purple, I. Hydrogen atoms are not shown for clarity).

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At the same time, here we provide a catalogue of lanthanide SIMs, together with SIMDAVIS, a dashboard that allows its interactive navigation; this is a type of tool utterly missing in the field of molecular nanomagnets. Perhaps more importantly in the wider perspective of design of new materials48,49,50,51 and new molecules26,52, the dataset curated in this work will serve for machine-learning studies. It can also be employed as an annotated training dataset for the development of new web scraping systems to retrieve chemical data53,54, or even word embeddings55, from the scientific literature. Finally, this work constitutes a step towards the availability of findable, accessible, interoperable, and reusable (FAIR) data in Chemistry56.

Methods

Data gathering

This process started with the collection and organisation of literature data. The following search criterion was applied for the manuscript: articles are searched via Web of Science, employing this code:

TOPIC: TS = ((lanthan* OR 4$f OR “rare$earth”) AND ((single NEAR/1 magnet*) OR “slow relaxation”)) Timespan: 2003–2019.

For an article to be included in the study, it needs to contain data on at least one compound with certain requirements as follows: (a) contain one trivalent lanthanide ion from the set Ln = {Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb} and (b) contain no other paramagnetic entity with the only accepted exception being the presence of a single radical in the coordination sphere and (c) present no strong Ln-Ln interaction, in particular meaning the Ln-Ln distance needs to be larger than 5 Å and more than 3 bridging atoms between neighbouring Ln centres, and there cannot be a radical in the bridge. Additionally, the data needs to include at least one of the following information: (a) whether χ” presents a maximum as a function of T, or a mere frequency-dependence, or neither; (b) χ” vs. T with at least one frequency (f) in the window 0.9 kHz ≤ f ≤ 1.1 kHz and at a field (H) in the window 0 ≤ H ≤ 2 T; (c) Ueff; (d) the presence or absence of hysteresis; (e) Thyst at sweep speeds (v) below 0.3 T/s. The compounds were classified in chemical families: LnPc2, polyoxometalates, Schiff base, metallocenes, diketonates, radicals, TM near Ln, mixed ligands, and other families. Furthermore, we registered for each sample (when available), the lanthanide ion, its concentration, the coordination number and number of ligands coordinated to the lanthanide ions, the coordination elements, the presence of a field-dependent χ” or a maximum, the temperature of said maximum in presence or absence of an external magnetic field, the external magnetic field, the extracted effective energy barrier and relaxation time, either from a simplified Arrhenius fit or from a model considering all relaxation processes, whether these were extracted from the maxima of χ” vs. T at different frequencies or from an Argand fit, the presence of hysteresis in the magnetisation, and the maximum temperature at which it was recorded. Additionally the DOI, the full reference to the original article, and a link to a CIF file were recorded for each sample. The question of publication bias is addressed at the end of Supplementary Section 1.1. Further details including the classification in chemical families and the criteria for data extraction are provided in Supplementary Sections 1 and 2.

SIMDAVIS dashboard

We programmed the dashboard using R language57,58 and shiny59, an open source R package to create the interactive web app. The design aimed to obtain a clean and simple user interface that adapts automatically to any screen size. The R packages readr60, dplyr61, DT62, ggplot263 and rcrossref64 were also employed in the development of the dataset or the app. The dashboard-style web application is available at https://go.uv.es/rosaleny/SIMDAVIS. This interface allows for variables in the analysis, and subsets of the data, to be adjusted and chosen in real time.

Statistical analysis

The statistical analysis was also based on R, a widely used software environment for statistical computing and graphics, and included the Gifi system for Multiple Correspondence Analysis65 (R homals package66, ade4 package67, see details in Supplementary Section 4.1), hierarchical clustering studies (FactoMineR68, see details in Supplementary Section 4.2), lognormal modelling (Poisson’s distribution, see Supplementary Section 4.3), factorial analysis of mixed data (FactoMineR and factoextra69, see details in Supplementary Section 6), as well as Pearson’s product-moment correlation and the Akaike information criterion (AIC)70 (see details in Supplementary Section 5.3). The analysis was repeated and verified an overall excellent qualitative and quantitative consistency in all results between the period 2003-2017 (1044 samples) and 2003–2019 (1405 samples).

SHAPE analysis

We employed a modified version of the pyCrystalField code71 to extract the coordination environments of samples with a cif file in either the COD or the CCDC databases. We employed the SHAPE program to compare these with the reference polyhedra. Elongated and compressed versions of the reference polyhedra were also evaluated. We searched for correlations between the new data and the rest of the dataset.