Programming heterometallic 4f–4f′ helicates under thermodynamic control: the circle is complete

Abstract

Three non-symmetrical segmental ligand strands L4 can be wrapped around a linear sequence of one Zn²⁺ and two trivalent lanthanide cations Ln³⁺ to give quantitatively directional [ZnLn₂(L4)₃]⁸⁺ triple-stranded helicates in the solid state and in solution. NMR speciations in CD₃CN show negligible decomplexation at a millimolar concentration and the latter helicate can be thus safely considered as a preorganized C₃-symmetrical HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ platform in which the thermodynamic properties of (i) lanthanide permutation between the central N₉ and the terminal N₆O₃ binding sites and (ii) exchange processes between homo- and heterolanthanide helicates are easy to access (Ln = La, Eu, Lu). Deviations from statistical distributions could be programmed by exploiting specific site recognition and intermetallic pair interactions. Considering the challenging La³⁺: Eu³⁺ ionic pair, for which the sizes of the two cations differ by only 8%, a remarkable excess (70%) of the heterolanthanide is produced, together with a preference for the formation of the isomer where the largest lanthanum cation lies in the central N₉ site ([(La)(Eu)] : [(Eu)(La)] = 9 : 1). This rare design and its rational programming pave the way for the preparation of directional light-converters and/or molecular Q-bits at the (supra)molecular level.

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Introduction

The lack of radial node characterizing the atomic orbitals having n − l = 1 (n and l are the principal and azimuthal quantum numbers, respectively), often referred to as the primogenic effect,¹ significantly contributes to the inner-shell pseudo-atomic character of the valence 4f orbitals in the trivalent lanthanides Ln³⁺ ([Xe]4fⁿ, n = 0–14).^2–4 The main consequence for chemistry results in a (very) similar reactivity of Ln³⁺ along the complete lanthanide series, which is only smoothly modulated by the stepwise 1% contraction of the ionic radii between adjacent elements.^5–8 The molecular recognition of specific Ln³⁺, beyond the standard electrostatic trend,^5–8 is therefore mainly lacking for coordination complexes in solution.⁹ This prevents the planned design of heterometallic polynuclear f–f′ assemblies in solution under thermodynamic control, except for some rare reports of deviations from statistical distributions in solution.^10–14 Consequently, the planned implementation of pure heterometallic f–f′ molecular complexes in solution mainly relies on multistep strategies which exploit the kinetic inertness provided by the complexation of Ln³⁺ within rigid, highly preorganized and often anionic receptors (Fig. A1-1 in Appendix 1, see ESI†).^15–26 By broadening the perspective, one realizes that energy barriers, responsible for kinetic inertness, thermodynamic stability and selectivity, may greatly benefit from long-range stacking interactions accompanying crystallization processes,^27–29 and serendipitous pure f–f′ assemblies are therefore reported in crystalline materials (Fig. A1-2 in Appendix 1, see ESI†).^30,31–40 The use of statistical doping has made it possible to temporarily circumvent these limits and myriads of doped ionic solids,⁴¹ nanoparticles,^42,43 metal–organic frameworks^30,44,45 or solid-state molecular aggregates and clusters^46–49 have been prepared and explored for improving lighting and optical signaling in materials. However, the recent recognition³⁵ that the (very) minor magnetic coupling operating between two different lanthanide Kramer's ions in non-statistical molecular heterometallic f–f′ entities represents a keystone for the design of basic information units in quantum computers, that is Q-bits,⁵⁰ reactivates the efforts aiming at the preparation of pure (i.e. non-statistical) heterometallic lanthanide molecular complexes under thermodynamic control. With this in mind, Aromi and coworkers have developed some remarkable and versatile scaffolds consisting of fused didentate β-diketonate and tridentate 2,6-dipicolinate units for the formation of different binding pockets, which display size discriminating effects along the lanthanide series in the solid state, when three ligands are wrapped around two (Scheme 1)^35–37 or more trivalent cations (Fig. A1–2d in Appendix 1, see ESI†).^38–40

Scheme 1 a) Permutation (eqn 1-L1) and exchange (eqn 2-L1) equilibria proposed for the heterometallic HHT-[(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)] complexes and (b) associated gas-phase DFT-computed energy changes.³⁶ corresponds to the difference of nine-coordinate lanthanide ionic radii in the considered metallic pair.⁵¹ The linear trendlines are only a guide to the eye. (c) Crystal structure of HHT-[(L1)₃LaEr(NO₃)(H₂O)(pyridine)].³⁵

Focusing on HHT-[(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)],³⁶ impressive deviations (1 ≤ E^A,B_perm ≤ 11 kJ mol⁻¹ and −120 ≤ ΔH^A,B_exch ≤ −14 kJ mol⁻¹, Scheme 1b) from the expected statistical distributions (ΔG^{A,B, stat}_perm = 0 kJ mol⁻¹ for eqn 1-L1 and ΔG^{A,B, stat}_exch = −RT ln(4) = −3.4 kJ mol⁻¹ for eqn 2-L1) could be predicted by gas-phase DFT calculations assuming that the molecular structures observed in the crystalline state (Scheme 1c) are pertinent to initiate gas-phase modelling.³⁶

To the best of our knowledge, related experimental thermodynamic data are available only for homometallic/heterometallic exchange eqn (2-L2) operating in the triple-stranded [(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ helicates (n = 0, 1, 2) in acetonitrile (Scheme 2a).¹⁰ One can note that the measured free energy changes −6 ≤ ΔG^A,B_exch ≤ −14 kJ mol⁻¹ in solution (Scheme 2b) drastically differ from the optimistic gas-phase DFT-predictions reported for HHT-[(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)] (Scheme 1b).³⁶

Scheme 2 a) Thermodynamic exchange equilibria (eqn 2-L2) and (b) associated free energy changes of the triple-stranded [(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ helicates (x = 0, 1, 2) in CD₃CN at room temperature.¹⁰ Only the HHH isomer is shown, but mixtures of HHH and HHT isomers exist in solution. corresponds to the difference of nine-coordinate lanthanide ionic radii in the metallic pair.⁵¹ The dashed linear trendline is only a guide to the eye. (c) Crystal structure of HHH-[(L2)₃LaEu](ClO₄)₆.¹⁰

The unavoidable ligand permutation, which interconverts C₃-symmetrical HHH-[(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ with its C₁-symmetrical HHT-[(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ counterpart, severely limits further thermodynamic analysis and the target (trivial) lanthanide exchange process involving two well-defined and different binding sites proposed by Aromi in eqn (1-L1) for HHT-[(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)] (Scheme 1) escaped quantification in solution with L2.

Connecting the three strands to a non-labile tripod seems to be the obvious choice for avoiding ligand scrambling and permutation, but structural constraints imposed by the helical wrapping of the strands required considerable synthetic efforts and delicate chemical design, which have been only approached once for the preorganized heterometallic dinuclear C₃-symmetrical [L3LaLu]⁶⁺ podate (Scheme 3a).^52–55 The free energy change estimated for the searched La : Lu permutation summarized in eqn (1-L3) amounts to ΔG^{La, Lu}_perm = 12.1(1) kJ mol⁻¹ (CD₃CN at room temperature), but it entirely relies on the assumption that the mixing rule ΔE^mix_1–2 = ΔE^{La, Lu}_1–2 − ½(ΔE^{La, La}_1–2 + ΔE^{Lu, Lu}_1–2) = 0 is obeyed.⁵⁶

Scheme 3 a) Thermodynamic permutation equilibrium (eqn 1-L3) estimated for [(L3)LaLu]⁶⁺ in CD₃CN at room temperature.⁵² (b) Removal of HHH/HHT isomerism in the dinuclear-lanthanide triple-stranded helicates HHH-[(L4₃Zn)(Ln^A)(Ln^B)]⁸⁺ by using [Zn(pyridine-benzimidazole)₃] as a preorganized non-covalent tripod.⁵⁷

Rejuvenated by the challenge of preparing pure f–f′ complexes under thermodynamic control, which are required for the preparation of molecular magnetic Q-bits,³⁶ we have re-engaged the fight with the use of a terminal non-covalent HHH-[Zn(pyridine-benzimidazole)₃] tripod which preorganizes the three strands for their concomitant efficient binding around two successive Ln³⁺ guests in two well-defined and different coordination sites. We therefore propose in this work to close the loop with a novel and efficient preparation of the segmental ligand L4 so that one can access the thermodynamically self-assembled HHH-[(L4₃Zn)Ln^ALn^B]⁸⁺ helicates for which both lanthanide permutation eqn (1-L4) and lanthanide exchange eqn (2-L4) can be deciphered in solution (Scheme 3b).⁵⁷

Results and discussion

Preparation and structures of L4 and its homolanthanide triple-helical complexes HHH-[(L4₃Zn)Ln₂](CF₃SO₃)₈ (Ln = La, Eu, Lu)

Taking advantage of previous synthetic efforts,⁵⁷ the strategy for preparing the segmental didentate–tridentate–tridentate ligand L4 has been optimized (Scheme A2-1 in Appendix 2). L4 could be thus efficiently prepared in seven steps with a global yield of 5.6% from commercially available 2,5-lutidine (1a), together with previously synthesized 4,4′-methylenebis(N-methyl-2-nitroaniline) (2c),⁵⁸ 6-(diethylcarbamoyl)picolinic acid (3d)⁵⁹ and N²,N²-diethyl-N⁶-methyl-N⁶-(4-(4-(methylamino)-3-nitrobenzyl)-2-nitrophenyl)pyridine-2,6-dicarboxamide (8).⁵⁹ The ¹H-NMR of the free ligand L4 shows a total of 32 signals accounting for the 55 protons, which confirms an average C_s-symmetry on the NMR timescale (Fig. S1†). The absence of NOE correlations between the pyridine meta-protons and the benzimidazole methyl groups implies anti conformations of the donor N-atoms of the α,α′-diimine units, which are typical of unbound polyaromatic benzimidazole-pyridine segments in solution,^59–61 a trend further confirmed in the crystal structure of L4·C₃H₈O (Fig. S2, S3 and Tables S1–S3†). The subsequent reaction of the segmental ligand L4 (3.0 eq., 15 mM) with stoichiometric amounts of Zn(CF₃SO₃)₂ (1.0 eq., 5 mM) and Ln(CF₃SO₃)₃ (2.0 eq., 10 mM, Ln = La(III), Eu(III), Lu(III)) in CDCl₃/CD₃CN (1 : 2) quantitatively affords the target homolanthanide HHH-[(L4₃Zn)Ln₂]⁸⁺ triple-stranded helicates within a few hours according to global equilibrium 3 (Fig. S4–S6†).

The final ¹H-NMR spectra of HHH-[(L4₃Zn)Ln₂]⁸⁺ show the exclusive (>98%) formation of a single C₃-symmetrical helicate in solution for each assembly process (Fig. 1 and S7–S15†). A consequence of the close vicinity of the three wrapped strands is highlighted in the diamagnetic complexes HHH-[(L4₃Zn)Ln₂]⁸⁺ (Ln = La, Lu) by the unusually low chemical shifts (4.95 ≤ δ ≤ 5.85 ppm) recorded for the aromatic protons 8, 12, 20 and 24 which are located in the shielding region of a benzimidazole ring of an adjacent strand (Fig. 1 and S7†).

Fig. 1 ¹H-NMR spectrum of the HHH-[(L4₃Zn)La₂]⁸⁺ complex (2 : 1 CD₃CN/CDCl₃, 400 MHz, 298 K). The aromatic region was expanded for clarity.

One further notes that the small ionic radius of Lu(III) leads to a tighter wrapping of the ligand strands, which shifts the ¹H-NMR signals of the benzimidazole protons 8, 12, 20 and 24 from 5.07 ≤ δ ≤ 5.85 ppm in HHH-[(L4₃Zn)La₂]⁸⁺ toward 4.95 ≤ δ ≤ 5.34 ppm in HHH-[(L4₃Zn)Lu₂]⁸⁺ (Fig. S16 and Table S4†). The replacement of the diamagnetic La³⁺ or Lu³⁺ cations with fast-relaxing paramagnetic Eu³⁺ in HHH-[(L4₃Zn)Eu₂]⁸⁺ results in the expected^11,59 downfield shifts of the singlet signals of the benzimidazole protons 12, 20 and 24 (12.00 ≤ δ ≤ 14.62 ppm) which are located close to the Eu³⁺ paramagnetic centers in the final complex (Table S5 and Fig. S8†). Additional proof for the formation of the desired complex is provided by the high-resolution ESI-TOF spectra recorded in acetonitrile, which display peaks corresponding to the series of triflate adducts HHH-{[(L4₃Zn)Ln₂](CF₃SO₃)_n}⁽⁸⁻ⁿ⁾⁺ (n = 2, 4, 5; Ln = La, Eu), although only at low relative intensities (Fig. S17 and S19†). The theoretical isotopic patterns nicely match the experimental peaks (Fig. S18 and S20†). The rest of the peaks, which have been assigned by their isotopic patterns, correspond to partial dissociation of one or more ligand strands and/or of one or more metal ions (Tables S6 and S7†). Due to the complete lack of ¹H-NMR evidence supporting significant decomplexation of HHH-[(L4₃Zn)Ln₂]⁸⁺ complexes (Ln = La, Eu, Lu) at millimolar concentrations in solution, the dissociation observed in the HR-MS spectra are assigned to the gas-phase reaction accompanying the ESI process.

Considering the labile character of both Zn(II) and Ln(III) (Ln = La, Eu, Lu) in solution, the formation of the desired trinuclear homolanthanide helicates within a few hours contrasts sharply with the fast (within seconds) self-assembly of the analogous dinuclear dimetallic [ZnLa(6)₃]⁵⁺ complex, where the shorter segmental ligand 6 (Fig. A2–1 in Appendix 2, see ESI†) corresponds to L4 after the removal of the central tridentate 2,6-bis(benzimidazole)pyridine unit.⁵⁹ The elongation of the ligand strand in L4 increases the complexity of the supramolecular system, which in turn increases the number of possible intermediates and reversible steps required before converging toward the thermodynamic products.^62,63 An ultra-simplistic consideration of the whole self-assembly process as being modeled with equilibrium (3) predicts a negligible dissociation rate constant since (i) k₁ can be estimated around 1000 M⁻⁶ h⁻¹ when one takes into account a characteristic time constant of 2 hours for the formation of 50% of the final HHH-[(L4₃Zn)Ln₂]⁸⁺ helicate in solution at a millimolar concentration and (ii) for HHH-[(L4₃Zn)Lu₂]⁸⁺ in acetonitrile.⁵⁷

Slow diffusion of isopropanol and diethyl ether, respectively, into solutions of HHH-[(L4₃Zn)Eu₂]⁸⁺ and HHH-[(L4₃Zn)La₂]⁸⁺ in acetonitrile yielded single crystals of [(L4₃Zn)Eu₂](CF₃SO₃)₈·12(C₃H₈O) and [(L4₃Zn)La₂](CF₃SO₃)₈·7.25(CH₃CN) of suitable quality for X-ray diffraction analysis (Fig. 2 and S21–S28, Tables S8–S25†). The X-ray structure of analogous [(L4₃Zn)Lu₂](CF₃SO₃)₈ was reported previously,⁵⁷ but the limited quality of the datasets collected at this time (despite using a synchrotron radiation source) did not allow a detailed analysis of the structure.

Fig. 2 Molecular structures of (a) HHH-[(L4₃Zn)Eu₂]⁸⁺ as observed in the crystal structure of [(L4₃Zn)Eu₂](CF₃SO₃)₈·12(C₃H₈O) with highlighted intermetallic distances (color code: C = grey, N = blue, O = red) and (b) HHH-[(L4₃Zn)La₂]⁸⁺ as found in the crystal structure of [(L4₃Zn)La₂](CF₃SO₃)₈·7.25(CH₃CN) with the three wrapped strands shown in different colors.

The crystal structures unambiguously confirm the formation of the desired homolanthanide d–f–f triple-stranded helicates, showing the three metal ions almost linearly aligned along the pseudo-C₃ axis (average Zn–Ln_c–Ln_t angle 176(2)°, Fig. 2a and Table S26†) and the three ligand strands helically wrapped around them in a head-to-head-to-head fashion (Fig. 2b). The intermetallic distances between adjacent cations average to 8.7(2) Å (Table S26†) lie within the range of distances previously reported in a number of polynuclear lanthanide helicates based on similar oligo-pyridyl-benzimidazole scaffolds (Scheme 2c and Fig. 3).^{10,59,64–68} The tighter wrapping of the ligand strands around the smallest lanthanide ions, previously mentioned when discussing the large upfield shift of the ¹H-NMR signals of the benzimidazole protons in HHH-[(L4₃Zn)Lu₂]⁸⁺ (Fig. S16†), leads to increasingly longer intermetallic distances as the size of the coordinated lanthanide ions reduces (entries 1 and 2 in Table S26†). While the average helical pitches (14.1–14.4 Å) do not vary significantly between the three complexes (entry 8 in Table S26†), and closely mirror those measured for previous homolanthanide helicates (Fig. S29†), a detailed analysis of each helical portion defined by eight almost parallel facial planes F1–F8 (Fig. S22, S23 and S25–S28†) showed significant local variations of the wrapping mode (Table S25†). In all three helicates, the tight rotations observed around each binding site alternate with severely relaxed helical twists associated with the diphenylmethane linkers. The helicity within the terminal [Ln_tN₆O₃] binding site is the most irregular out of the three coordinating units due to the difference in angular rotation caused by the carboxamide-pyridine moiety on one side, and by the pyridine-benzimidazole motif on the other side.⁵⁹

Fig. 3 Variation of the average bond valences ν_Ln-donor calculated with eqn (4) for (a) Ln(III)_c in the central N₉ binding site and (b) Ln(III)_t in the terminal N₆O₃ site as a function of the inverse of the nine-coordinate lanthanide ionic radii⁵¹ in HHH-[(L4₃Zn)Ln₂]⁸⁺ (Ln = La(III), Eu(III), and Lu(III)). Standard deviations of the averages are shown with vertical error bars. The dashed traces are only a guide to the eye.

The geometries of the coordination spheres around the three cations were analyzed with the software SHAPE,^69–71 the final scores of which point toward a pseudo-octahedral arrangement for the [ZnN₆] units (Table S26, entries 11 and 12†). Due to the poor stereochemical preferences of the lanthanide ions,^72,73 comparable SHAPE scores are obtained for various geometries of the nine-coordinate Ln³⁺ sites. At the more flexible terminal [Ln_tN₆O₃] sites, the lowest scores for all three lanthanides point to the tricapped trigonal prism geometry (Table S26, entries 20 and 21†). In the central [Ln_cN₉] sites, a tricapped trigonal prism geometry is adopted by the largest La(III) cation, while a spherical capped square antiprism geometry is observed around the smaller Eu(III) and Lu(III) cations (Table S26, entries 15 and 16†). One finally notices that the Zn–N bond distances do not vary significantly in the different HHH-[(L4₃Zn)Ln₂]⁸⁺ helicates (Ln = La, Eu, Lu; Fig. S30†). This implies sufficient flexibility within the wrapped strands for overcoming significant structural constraints accompanying the lanthanide contraction along the 4f-series.

As expected for the flexible scaffold found in HHH-[(L4₃Zn)Ln₂]⁸⁺, the Ln–N and Ln–O bond distances mirror the lanthanide contraction along the series (Fig. S30†),^51,74,75 but a more detailed analysis of the Ln–N and Ln–O bond strengths, corrected for the lanthanide contraction, can be assessed by calculating the bond valences ν_ij with eqn (4), where R_ij is the bond valence parameter associated with a given set of metal ion i and donor atom j,^76–78d_ij is the distance between the i–j pair, and b = 0.37 is a universal scaling constant (Fig. 3 and Table S27†).⁷⁹

ν_ij = e^{[(R_ij−d_ij)/b]} (4)

The largest bond valences, diagnostic of the strongest metal–ligand affinities, were found between the lanthanides and the O-donors in the terminal N₆O₃ site (Fig. 3b). Interestingly, a sharp decrease of the Ln_t–O bond valence observed when going from La(III) to Eu(III) is compensated by an increase of the Ln_t–N_bz and Ln_t–N_py interactions. The resulting concave trend detected for both Ln_t–N_bz and Ln_t–N_py bond valences in the terminal N₆O₃ site is not reproduced in the central N₉ site (Fig. 3a), where the average Ln_c–N_bz interaction decreases regularly throughout the series while the Ln_c–N_py interaction remains roughly constant. Altogether, the terminal N₆O₃ site exhibits a weak preference for binding mid-range Ln³⁺ while the central N₉ site penalizes the binding of the smaller lanthanides in HHH-[(L4₃Zn)Ln₂]⁸⁺, a tendency previously established for the related [LnN₉] and [LnN₆O₃] sites found in the homolanthanide D₃-symmetrical [(L5)₃Ln₂]⁶⁺ and [(L6)₃Ln₂]⁶⁺ helicates (Fig. S29†).⁸⁰

Formation and speciation of heterolanthanide triple-helical complexes HHH-[(L4₃Zn)Ln^ALn^B]⁸⁺ in solution (Ln^A, Ln^B = La, Eu, Lu)

The reaction of the segmental ligand L4 (3.0 eq.) with a 1 : 1 : 1 mixture of Zn(CF₃SO₃)₂ (1.0 eq.), La(CF₃SO₃)₃ (1.0 eq.) and Eu(CF₃SO₃)₃ (1.0 eq.) in a 1 : 2 mixture of CDCl₃/CD₃CN was followed by ¹H-NMR until the equilibrium was reached (Fig. S31†). The comparison of the ¹H-NMR spectrum of the final mixture with those of the free ligand L4 and the previously characterized homolanthanide complexes HHH-[(L4₃Zn)La₂]⁸⁺ and HHH-[(L4₃Zn)Eu₂]⁸⁺ demonstrates the formation of a single major new species displaying the characteristic features of a C₃-symmetric triple-stranded helicate (Fig. S32†). The rest of the (minor) signals correspond to the homolanthanide complexes, with no trace of the free ligand (Fig. 4). One of the two possible heterolanthanide isomers strongly dominates the speciation (Table 1, second column), thus confirming that the two different lanthanide binding sites exhibit size-discriminating effects. With the help of correlation and NOE spectroscopy (Fig. S33†), the ¹H-NMR spectrum of the main product could be fully assigned to HHH-[(L4₃Zn)LaEu]⁸⁺, where specific paramagnetic-induced chemical shifts (Fig. S34†) ascertain that the Eu(III) cation occupies the terminal N₆O₃ binding site in the major heterolanthanide isomer (Fig. 4). As expected, the high-resolution ESI-TOF spectrum of the mixture confirmed the co-existence of both the homo- and heterolanthanide complexes in the gas-phase (Fig. S35†).

Fig. 4 ¹H-NMR spectrum of a 1 : 1 : 1 : 3 mixture of Zn(CF₃SO₃)₂, La(CF₃SO₃)₃, Eu(CF₃SO₃)₃ and L4 at equilibrium (2 : 1 CD₃CN/CDCl₃, 400 MHz, 298 K). The aromatic region was expanded for clarity. The signals highlighted in green represent HHH-[(L4₃Zn)Eu₂]⁸⁺ and those in pink represent HHH-[(L4₃Zn)La₂]⁸⁺.

Table 1 Speciation (mole fraction) at equilibrium following the reaction of L4 (3.0 eq., 15 mM) with a mixture of Zn(CF₃SO₃)₂ (1.0 eq.), Ln^A(CF₃SO₃)₃ (1.0 eq.) and Ln^B(CF₃SO₃)₃ (1.0 eq.). Thermodynamic descriptors and related free energies associated with the permutation (eqn (5)) and exchange (eqn (6)) equilibria (1 : 2 mixture of CDCl₃/CD₃CN, 298 K)

Ln^A–Ln^B	La–Eu	Eu–Lu	La–Lu
ΔRCN=9^LnA^, LnB /Å	0.096	0.088	0.184
x(HHH-[(L43Zn)Ln^A2]^8+)	0.14(1)	0.28(2)	0.25(2)
x(HHH-[(L43Zn)Ln^B2]^8+)	0.16(2)	0.37(4)	0.24(2)
x(HHH-[(L43Zn)Ln^ALn^B]^8+)	0.63(5)	0.27(3)	0.46(3)
x(HHH-[(L43Zn)Ln^BLn^A]^8+)	0.07(1)	0.08(1)	0.05(1)
	0.11(2)	0.30(5)	0.11(2)
	5.4(4)	3.0(4)	5.4(5)
	22(4)	1.2(2)	4.3(6)
	−7.5(4)	−0.4(4)	−3.6(4)
u ^mix1–2	1.4(2)	0.46(8)	0.6(1)
ΔE^mix1–2 /kJ mol^−1	−0.8(4)	1.9(4)	1.2(5)

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The formation of the HHH-[(L4₃Zn)LaEu]⁸⁺ complex as the main product of the self-assembly of a 1 : 1 : 1 mixture of Zn(CF₃SO₃)₂, La(CF₃SO₃)₃, and Eu(CF₃SO₃)₃ with 3.0 eq. of L4 is consistent with the stereochemical preference of the central N₉ site for larger lanthanide ions and that of the terminal N₆O₃ site for smaller ones.^10,11,80 In this context, replacing La(III) with Lu(III) in the mixture should make the coordination of the Eu(III) cation now more favorable in the central N₉ site made of three wrapped bis(benzimidazole)pyridine segments while the terminal N₆O₃ site should preferentially accommodate the smaller Lu(III), hence yielding HHH-[(L4₃Zn)EuLu]⁸⁺ as the major heterolanthanide isomer in solution. The latter prediction was confirmed by following the reaction of L4 (3.0 eq., 15 mM) with a 1eq : 1eq : 1eq mixture of Zn(CF₃SO₃)₂, Eu(CF₃SO₃)₃ and Lu(CF₃SO₃)₃ in a 1 : 2 mixture of CDCl₃/CD₃CN with the help of ¹H-NMR techniques. The ¹H-NMR spectrum recorded at equilibrium (after 24 hours, Fig. S36†) showed non-negligible amounts of the homolanthanide helicates HHH-[(L4₃Zn)Eu₂]⁸⁺ and HHH-[(L4₃Zn)Lu₂]⁸⁺ together with one major set of unidentified peaks that corresponded to the heterolanthanide HHH-[(L4₃Zn)EuLu]⁸⁺ isomer (Fig. S37–S39,† column 3 in Table 1).

Finally, the reaction of L4 (3.0 eq., 15 mM) with a 1 eq : 1 eq : 1 eq mixture of Zn(CF₃SO₃)₂, La(CF₃SO₃)₃ and Lu(CF₃SO₃)₃ was the fastest to reach the equilibrium, showing little to no evolution in the ¹H-NMR spectrum after only a few hours at room temperature (Fig. S40†). Similarly to the previous mixtures, two homolanthanide helicates HHH-[(L4₃Zn)La₂]⁸⁺ and HHH-[(L4₃Zn)Lu₂]⁸⁺ are formed, along with only one of the two possible heterolanthanide isomers (Fig. S41–S42,† column 4 in Table 1). The absence of the open-shell Eu(III) probe in the mixture makes the assignment of the ¹H-NMR spectrum of the heterolanthanide HHH-[(L4₃Zn)LaLu]⁸⁺ complex harder since the chemical shifts of the protons surrounding the central and the terminal sites are not as different as with the paramagnetic helicates. However, the size difference between La(III) and Lu(III) has been shown to affect the tightness of the wrapping of the ligand strands (Fig. S16†), which results in ¹H-NMR signals for the central isolated benzimidazole singlets which are diagnostic for the binding of the largest La(III) cation in the central N₉ site, while Lu(III) lies in the terminal site in HHH-[(L4₃Zn)LaLu]⁸⁺ (Fig. S42†). The permutated HHH-[(L4₃Zn)LuLa]⁸⁺ isomer could not be detected in the final mixture and its mole fraction was thus set at the limit of accuracy (x ≤ 0.05) estimated for our ¹H-NMR experimental setup (Table 1).

Thermodynamic rationalization of the formation of heterolanthanide triple-helical complexes HHH-[(L4₃Zn)Ln^ALn^B]⁸⁺ in solution (Ln^A, Ln^B = La, Eu, Lu)

In the absence of significant complex dissociation at millimolar concentrations, as demonstrated for the stoichiometric mixing of L4 (3.0 eq.) with Zn(CF₃SO₃)₂ (1.0 eq.), Ln^A(CF₃SO₃)₃ and Ln^B(CF₃SO₃)₃ (1.0 eq.) in solution, the four interconverting helicates HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ (n = 0, 1, 2) are related by the generic thermodynamic permutation equilibrium (1) (eqn 1-Lk in Schemes 1–3, further generalized below as eqn (5)) and exchange equilibrium (2) (eqn 2-Lk in Schemes 1–3, further generalized below as eqn (6)), where HHH-[(L4₃Zn)]²⁺ is considered as a rigid platform for the complexation of Ln^A and Ln^B in the two appended and preorganized N₉ and N₆O₃ binding sites (Scheme 4). The equilibrium concentrations are written between double vertical bars | | in eqn (5) and (6).

Scheme 4 Microscopic thermodynamic formation constants for HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ (n = 0, 1, and 2) and their modeling with the site binding model.^81,82 See main text for the definitions of f^Lnj_i and u^Lni_1–2, Ln^j.

Focusing on HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺, (eqn (5)) and (eqn (6)) can be modeled with the help of microscopic formation constants to give eqn (7) and (8) within the frame of the site binding model, where f_i^Lnj is the intermolecular microscopic affinity of the nine-coordinate site i for the entering lanthanide Ln^j in the preorganized HHH-[(L4₃Zn)]²⁺ receptor and is the Boltzmann factor measuring the intermetallic pair interactions operating between adjacent Lnⁱ and Ln^j cations in [(Lnⁱ)(Ln^j)] (Scheme 4).^81,82

The last term of eqn (8) corresponds to the square of , which is related to the mixing energy ΔE^mix_1–2 in eqn (9).⁵⁶

When ΔE^mix_1–2 = 0, the pair interaction energies obey the mixing rule , which corresponds to a non-cooperative behavior and results in a random distribution of the two different metal ions among the coordination sites.⁵⁶ Deviations from the mixing rule can be assigned to either cooperative processes (ΔE^mix_1–2 > 0), which are characterized by the clustering of identical metals along the strands, or anti-cooperative processes (ΔE^mix_1–2 < 0), which correspond to an alternation of the different metals.^56,81,82

The experimental permutation energies (orange markers) and exchange energies (blue markers) obtained for HHH-[(L4₃Zn)(Ln^A)(Ln^B)]⁸⁺ (entries 7 and 9 in Table 1) are summarized in Fig. 5. One immediately notices the systematic positive permutation energies (top of Fig. 5), which reflect the preferred formation of the heterolanthanide isomer where the larger cation lies in the central N₉ binding site and the smaller cation occupies the terminal N₆O₃ binding site (, eqn (7)). The combination of eqn (7) and (8), pertinent to and , provides an elegant experimental access to the balance of the intermetallic pair interactions as measured by u^mix_1–2 in eqn (10), and consequently to the associated mixing energies −0.8 ≤ ΔE^mix_1–2 ≤ 1.9 kJ mol⁻¹ operating in HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ (n = 0, 1, 2; Table 1, entry 11).

Fig. 5 Free energies for permutation ( in eqn (5), orange markers) and for exchange ( in eqn (6), blue markers) observed in solution at room temperature for HHH-[(L4₃Zn)(Ln^A)(Ln^B)]⁸⁺ (Table 2) as a function of the difference of the nine-coordinate lanthanide ionic radii (ΔR^{A, B}_CN=9).⁵¹ The full traces correspond to statistical behaviours.

When the two different lanthanide cations are larger than Gd³⁺ (= belong to the first half of the lanthanide series), as illustrated in HHH-[(L4₃Zn)(La)_(2−n)(Eu)_n]⁸⁺ (n = 0, 1, 2), then ΔE^{La, Eu}_1–2 < ½(ΔE^{La, La}_1–2 + ΔE^{Eu, Eu}_1–2) and the associated value of u^mix_1–2 = 1.4(2) boosts the formation of the heterolanthanide HHH-[(L4₃Zn)(La)(Eu)]⁸⁺ complexes to reach , which lies much beyond the statistical value of ΔG^stat_exch = −RT ln(4) = −3.4 kJ mol⁻¹ (bottom of Fig. 5, blue trace). As soon as one lanthanide cation of the pair belongs to the second part of the lanthanide series, as exemplified in heterolanthanide [(Eu)(Lu)] and in [(La)(Lu)] helicates, the reverse situation occurs with and the balance of pair interactions tend to discard the formation of heterolanthanide HHH-[(L4₃Zn)(Ln^A)(Ln^B)]⁸⁺ complexes.

The origin of the latter driving force is far from being obvious, but it can be traced back to related trends observed for the thermodynamic self-assemblies of symmetrical dinuclear [(L5₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (n = 0, 1, 2)⁶⁵ and [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (n = 0, 1, and 2),^66,83 trinuclear [(L7₃)(Ln^A)_(3−n)(Ln^B)_n]⁹⁺ (n = 0, 1, 2, 3)¹¹ and tetranuclear [(L8₃) (Ln^A)_(3−n)(Ln^B)_n]¹²⁺ (n = 0, 1, 2, 3, and 4)¹² helicates (Fig. S29†) based on the segmental ligands L5–L8 (Scheme 5, see Appendices 3–4 for a comprehensive thermodynamic analysis, see ESI).

Scheme 5 Chemical structures of segmental ligands used for the self-assemblies of heterometallic dinuclear [(L5₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (n = 0, 1, 2)⁶⁵ and [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (n = 0, 1, 2),^66,83 trinuclear [(L7₃)(Ln^A)_(3−n)(Ln^B)_n]⁹⁺ (n = 0, 1, 2, 3)¹¹ and tetranuclear [(L8₃) (Ln^A)_(4−n)(Ln^B)_n]¹²⁺ (n = 0, 1, 2, 3, and 4)¹² helicates.

The two adjacent N₆O₃ binding units found in [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ are not able to induce deviations from statistical mixtures in solution (Fig. A3-1a in Appendix 3, see ESI†) and one systematically obtains u^mix_1–2 = 1 for any lanthanide pairs (ΔE^mix_1–2 = 0 in Table S29†). In contrast, the two connected N₉ sites implemented in [(L5₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ induce positive mixing energies (Fig. A3-1b and ΔE^mix_1–2 > 0 in Table S28†), which favor homometallic matching beyond statistical distributions, as long as at least one lanthanide of the metallic pairs belongs to the second half of the lanthanide series. Moving from two adjacent identical nine-coordinated binding sites, as found in [(L5₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (N₉–N₉) or in [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (N₆O₃–N₆O₃), toward two different connected N₉ and N₆O₃ sites in HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ brings a novel dimension to the size discriminating process. Firstly, due to the presence of the central constrained N₉ site,⁸⁴ the mixing rule ΔE_1–2^mix > 0 discards the formation of the heterolanthanide HHH-[(L4₃Zn)(Ln^A)(Ln^B)]⁸⁺ isomers as soon as the small Lu³⁺ cation is considered as a member of the lanthanide pair in [(La)(Lu)] and [(Eu)(Lu)]. Secondly, the non-zero permutation energies provided by the existence of the two different binding sites (N₉–N₆O₃) may partially compensate for the latter detrimental effect, and the formation of the heterolanthanide HHH-[(L4₃Zn)(Ln^A)Lu]⁸⁺ isomers still represents 40–50% of the speciation in solution (Table 1, entries 4 and 5). Finally, when two large lanthanides are bound in HHH-[(L4₃Zn)LaEu]⁸⁺, both the balance of the intermetallic interactions (ΔE^mix_1–2 < 0) and site selectivity contribute favorably and synergistically to a large deviation of statistics with the formation of up to 70% of the heterolanthanide HHH-[(L4₃Zn)LaEu]⁸⁺ and HHH-[(L4₃Zn)EuLa]⁸⁺ in solution, which exist moreover in a |[(La)(Eu)]|/|[(Eu)(La)]| = 9 : 1 ratio (Table 1 and Fig. 5).

Conclusions

Puzzled by preliminary, partial and explorative studies reported for [(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)] (Scheme 1),³⁶ [(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (Scheme 2)¹⁰ and [(L3)LaLu]⁶⁺ (Scheme 3),⁵² which claimed for some selective lanthanide recognition to form f–f′ complexes under thermodynamic control in solution, one realizes that any pertinent discriminating effects, if they exist, should be quantitatively addressed with the help of two simple free energy descriptors measuring (i) the intermolecular affinity of a given preorganized binding site i for the entering lanthanide Ln^j (ΔG^Lnj_{aff, i} = −RT ln(f^Lnj_i)) and (ii) the balance of the intermetallic interactions operating within adjacent pairs of lanthanides .⁵⁶ Although a direct access to these two crucial thermodynamic descriptors proved to be (very) difficult,⁸⁰ an indirect approach appears to be possible since the experimentally accessible permutation equilibrium (eqn (7)), which accompanies the distribution of the various heterolanthanide isomers, and the exchange equilibrium (eqn (8)), which measures the amounts of homo- versus heterolanthanide complexes formed, reflect these thermodynamic parameters in solution. With this in mind, the lack of reliable and complete speciations addressed for the non-symmetrical [(L1)₃Ln^ALn^B(NO₃)(H₂O)(pyridine)],³⁶ [(L2)₃(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ (Table 2, entry 1)¹⁰ and [(L3)LaLu]⁶⁺ (Table 2, entry 2)⁵² complexes limits further rational thermodynamic analysis. In contrast, the detailed solution studies reported for the symmetrical dinuclear triple-stranded [(L5₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ and [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ helicates in solution can be used to initiate the thermodynamic exploration.⁸⁰ The strict statistical behavior observed for the loading of pairs of lanthanide cations in [(L6₃)(Ln^A)_(2−n)(Ln^B)_n]⁶⁺ indicates that the sequence of two adjacent semi-flexible N₆O₃ binding sites is not able to induce any significant recognition along the lanthanide series (Table 2, entry 3). The same behaviour characterizes the binding of the two large La³⁺ and Eu³⁺ cations in [(L5₃)(La)_(2−n)(Eu)_n]⁶⁺ (Table 2, entry 4). However, when at least one lanthanide of the pairs is smaller than Gd³⁺, the two adjacent N₉ binding sites in the latter complexes [(L5₃)(Ln^A)_(2−n)(Ln^A)_n]⁶⁺ show a global preference for the formation of homometallic complexes due to cooperative intermetallic mixing energies ΔE^mix_1–2 ≈ 2 kJ mol⁻¹ (Table 2, entry 5). The symmetrical trinuclear [(L7₃)(Ln^A)_(3−n)(Ln^B)_n]⁹⁺ (N₆O₃–N₉–N₆O₃, Table 2 entry 6) and tetranuclear [(L8₃)(Ln^A)_(4−n)(Ln^B)_n]¹²⁺ (N₆O₃–N₉–N₉–N₆O₃, Table 2 entry 7) helicates confirm these trends with the appearance of sizeable unbalanced intermetallic interactions (i.e. ΔE^mix_1–2 ≠ 0) only when a sequence of two adjacent N₉ site binding sites exists.

Table 2 Summary of the thermodynamic free energy changes relevant to address the difference in intermolecular affinity and in intermetallic interactions which control the speciation of f–f′ helicates under thermodynamic control in solution beyond statistical distributions mentioned in columns 4 and 7

Helicate	Binding sites		ΔG^statperm (kJ mol^−1)	ΔE^mix1–2 (kJ mol^−1)		ΔG^statexch (kJ mol^−1)	Condition	Favoured species^a	Ref.
a Homo resp. hetero = preference for homometallic, resp. heterometallic lanthanide complexes; statistical = no preference. b The triple-stranded helicates exist as variable mixtures of HHH (N9–N6O3) and HHT (N8–N7O2) isomers. c R represents an organic tripod. d The mixing energy is arbitrarily fixed to ΔE^mix1–2 = 0.
[(L2)3(Ln^A)(2−n)(Ln^B)n]^6+	N9–N6O3/N8–N7O2 ^b	—	0	—	−14.1 to −6.4	−3.4	—	Hetero	10
[(L3)3(La)(2−n)(Lu)n]^6+	R–N9–N6O3 ^c	12.1	0	0^d	—	−3.4	—	—	52
[(L6)3(Ln^A)(2−n)(Ln^B)n]^6+	N6O3–N6O3	0	0	−0.6 to 0.4	−3.5 to −3.3	−3.4	—	Statistical	83
[(L5)3(Ln^A)(2−n)(Ln^B)n]^6+	N9–N9	0	0	0	−3.4	−3.4	R ^LnA ≥ R^Gd and R^LnB ≥ R^Gd	Statistical	65
[(L5)3(Ln^A)(2−n)(Ln^B)n]^6+	N9–N9	0	0	1.6 to 2.3	−0.2 to 1.2	−3.4	R ^LnA < R^Gd or R^LnB < R^Gd	Homo	65
[(L7)3(Ln^A)(3−n)(Ln^B)n]^9+	N6O3–N9–N6O3	2.2 to 4.2	3.34	−0.6 to 0.2	−10.8 to −5.4	−5.4	—	Hetero	67
[(L8)3(La)(4−n)(Lu)n]^12+	N6O3–N9–N9–N6O3	—	—	−2	−47.5	−22.2	—	Hetero	12
[(L43Zn)(Ln^A)(2−n)(Ln^B)n]^8+	[ZnN6]–N9–N6O3	3.0 to 5.4	0	1.2 to 1.9	−3.6 to −0.4	−3.4	R ^LnA < R^Gd or R^LnB < R^Gd	Homo	This work
[(L43Zn)(Ln^A)(2−n)(Ln^B)n]^8+	[ZnN6]–N9–N6O3	5.4	0	−0.8	−7.5	−3.4	R ^LnA ≥ R^Gd and R^LnB ≥ R^Gd	Hetero	This work

---

When a constrained sequence of two adjacent N₆O₃ and N₉ binding sites is ensured via the connection of the ligand strands to a covalent sulfur tripod in [(L3)LaLu]⁶⁺ (Table 2, entry 2) or to a non-covalent [ZnN₆] podand in the HHH-[(L4₃Zn)(Ln^A)_(2−n)(Ln^B)_n]⁸⁺ helicates (Table 2, entries 8 and 9), both the specific binding site affinities (via, eqn (7)) and the intermetallic mixing energies (via, eqn (8)) can be exploited for boosting the formation of one targeted heterolanthanide isomer in solution. The systematic preference of the central N₉ site for binding the largest lanthanide of the Ln^A: Ln^B pair favors the formation of the heterolanthanide HHH-[(L4₃Zn)Ln^ALn^B]⁸⁺ isomer when R^LnA > R^LnB. However, the unfavorable mixing energy accompanying the distribution of the two lanthanides within the two sites as soon as one is smaller than Gd³⁺ limits this drift with the formation of only 51% of the heterolanthanide complexes for the La : Lu pair and 35% for the Eu : Lu pair (Table 2, entry 8). The latter restriction is removed when the two lanthanides belong to the first half of the series as demonstrated for the challenging La³⁺: Eu³⁺ ionic pair in HHH-[(L4₃Zn)LaEu]⁸⁺ (Table 2, entry 9; the sizes of the two cations differ by only 8%), where the latter isomer accounts for 63% of the speciation in solution under stoichiometric conditions (|L4₃Zn|_tot = |La|_tot = |Eu|_tot). This largely exceeds the 25% predicted by the statistical distribution. The road to the selective formation of f–f′ heterometallic complexes under thermodynamic control is still a long one, but the use of non-covalent tripods for programming specific sequences of binding sites, as demonstrated here for HHH-[(L4₃Zn)Ln^ALn^B]⁸⁺ helicates, corresponds to a major step forward in the rational design of heterolanthanide complexes obtained under thermodynamic control.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Financial support from the Swiss National Science Foundation is gratefully acknowledged (grant 200020_207313).

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Footnote

† Electronic supplementary information (ESI) available. CCDC 2320733–2320735. For ESI and crystallographic data in CIF or other electronic format see DOI: https://doi.org/10.1039/d4dt00610k

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