Attaining record-high magnetic exchange, magnetic anisotropy and blocking barriers in dilanthanofullerenes

Abstract

While the blocking barrier (U_eff) and blocking temperature (T_B) for “Dysprocenium” SIMs have been increased beyond liquid N₂ temperature, device fabrication of these molecules remains a challenge as low-coordinate Ln³⁺ complexes are very unstable. Encapsulating the lanthanide ion inside a cage such as a fullerene (called endohedral metallofullerene or EMF) opens up a new avenue leading to several Ln@EMF SMMs. The ab initio CASSCF calculations play a pivotal role in identifying target metal ions and suitable cages in this area. Encouraged by our earlier prediction on Ln₂@C₇₉N, which was verified by experiments, here we have undertaken a search to enhance the exchange coupling in this class of molecules beyond the highest reported value. Using DFT and ab initio calculations, we have studied a series of Gd₂@C_2n (30 ≤ 2n ≤ 80), where an antiferromagnetic J_Gd⋯Gd of −43 cm⁻¹ was found for a stable Gd₂@C₃₈-D_3h cage. This extremely large and exceptionally rare 4f⋯4f interaction results from a direct overlap of 4f orbitals due to the confinement effect. In larger cages such as Gd₂@C₆₀ and Gd₂@C₈₀, the formation of two centre-one-electron (2c-1e⁻) Gd–Gd bonds is perceived. This results in a radical formation in the fullerene cage leading to its instability. To avoid this, we have studied heterofullerenes where one of the carbon atoms is replaced by a nitrogen atom. Specifically, we have studied Ln₂@C₅₉N and Ln₂@C₇₉N, where strong delocalisation of the electron yields a mixed valence-like behaviour. This suggests a double-exchange (B) is operational, and CASSCF calculations yield a B value of 434.8 cm⁻¹ and resultant J_Gd–rad of 869.5 cm⁻¹ for the Gd₂@C₅₉N complex. These parameters are found to be two times larger than the world-record J reported for Gd₂@C₇₉N. Further ab initio calculations reveal an unprecedented U_cal of 1183 and 1501 cm⁻¹ for Dy₂@C₅₉N and Tb₂@C₅₉N, respectively. Thus, this study offers strong exchange coupling as criteria for new generation SMMs as the existing idea of enhancing the blocking barrier via crystal field modulation has reached its saturation point.

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Introduction

Single molecule magnets (SMMs) are of prime interest in molecular magnetism due to their potential application in memory storage devices, qubits, etc.1,2 The figure of merit of an SMM is determined by the blocking barrier for magnetisation reversal (U_eff) and blocking temperature (T_B), the temperature below which opening of magnetic hysteresis is observed. These U_eff and T_B values are generally very high for lanthanides, thanks to their strong spin–orbit coupling.^3–10 The enhancement of T_B as high as 80 K in “Dysprocenium” complexes was an important breakthrough, replenishing the hope for potential applications in information storage devices.^11–15 Among others, important bottlenecks that are likely to hamper the futuristic application of these SMMs are (i) enhancing the blocking temperature beyond 80 K (ii) obtaining molecules that are stable under ambient conditions so that fabrication can be attempted (iii) retaining their intriguing magnetic properties upon fabrication – many of the best transition metal SMMs failed these criteria.^16–21

To address the first challenge, among other strategies that could help enhance the barrier height/blocking temperature is the quenching of quantum tunnelling of magnetisation (QTM), which is prevalent at low temperatures. If a robust magnetic exchange between two Ln³⁺ ions is induced, it can act as a perturbation to reduce the degeneracy of Kramers doublets (KDs). This quenches the QTM and gives rise to large U_eff and T_B values.^22–24 However, obtaining a large exchange coupling between two Ln³⁺ metal ions is a formidable task as 4f orbitals are deeply buried, leading to a weak/no interaction in dinuclear or polynuclear Ln³⁺ complexes.^8,25–30

In this regard, lanthanide encapsulated fullerenes (called endohedral metallofullerenes or EMFs) are gaining tremendous attention for various reasons: (a) they offer stability to guest molecules which are otherwise unstable;³¹ (b) thanks to their strong π cloud, fabrication of such molecules on graphene/HOPG/CNTs and other surfaces is straightforward;^31–39 (c) during this process guest molecules stay intact, and hence they are unlikely to lose their characteristics upon fabrication;³¹ (d) as fullerenes are made of pure carbon, and the source of nuclear spin of the guest molecules can be controlled, they offer a nuclear spin free system – a key criterion for some qubit applications.^31,40 These key advantages mentioned here directly address the aforementioned goals (ii) and (iii), making them superior to traditional coordination chemistry/organometallic SMMs/SIMs.

One way to attain strong exchange coupling in lanthanide SMMs is to employ radical–Ln exchange which is substantially larger due to the direct exchange between 4f–2p orbitals.^41–43 In the search for a stronger exchange in Ln–radical systems, using a combination of DFT and ab initio methods, we have predicted a record high magnetic exchange coupling for a Gd₂@C₇₉N radical fullerene complex and also suggested a very large blocking barrier for the Dy analogue.^44,45 Both these predictions were proved in a span of few years independently by two groups,^46–49 and Gd₂@C₇₉N is found to have a very large spin relaxation time opening up a new avenue in spin-based qubits.^40,46 While a Ln–radical exchange could solve this problem,⁴¹ the majority of the conventional lanthanide–radical systems are highly reactive and could pose a challenge in accomplishing the aforementioned goals (ii) and (iii).^{41,43,50–55}

In this connection, if a robust exchange is induced between two Ln³⁺ ions, this will be very rewarding. One strategy to enhance the exchange coupling is to induce a weak Ln⋯Ln bond, which is possible if two ions are brought very close to each other directly. The metal–metal bonds in transition metal complexes are common but are scarce for lanthanides.^56–58 Inspired from the report that even noble gas elements such as He form He⋯He bonds under confinement, we devise such models for lanthanides that can offer very large 4f–4f exchange interactions.^59–63 In line with this idea, we have explored various Gd₂@C_2n (2n = 30–52, 60, 80) complexes in search of a stronger exchange and found Gd³⁺⋯Gd³⁺ exchange as high as −43 cm⁻¹. In the second approach, we have extended our study to air-stable azafullerene radical analogues such as Ln₂@C_59/79N (Ln = Gd, Tb, Dy). Using ab initio calculations, we have computed the double-exchange parameter B in these azafullerene cages. We have exploited the presence of double exchange to design SMMs based on Dy and Tb and unveil a new line of prediction with models exhibiting a U_eff value exceeding 1500 cm⁻¹.

Results and discussion

Achieving large exchange coupling in lanthanides is challenging as the 4f orbitals of lanthanides are deeply buried and interact weakly with ligand orbitals. The highest magnetic exchange between two Ln³⁺ ions is estimated in a {Gd₂Cr₂} complex where J_Gd–Gd is +1.4 cm⁻¹ (Ĥ = −JŜ₁Ŝ₂).²⁸ As the Ln⋯Ln distance plays a crucial role in controlling the 4f–4f exchange interaction, a large J is expected if two Ln³⁺ ions are confined in a fullerene cage. With this goal, we begin our study with Gd₂ endohedral fullerenes by varying the cage size from C₃₀ to C₈₀. We have analysed the structure, binding energy, and magnetic properties within the DFT framework for two low energy conformers of the fullerene cages among various close-lying isomers.

Structure and bonding in Gd₂@C_2n (2n = 30–48, 52, 40, 80)

The C₃₀ fullerene is the smallest cage where encapsulation leads to a stable geometry, as steric strain dominates over the metal-cage stabilisation in C₂₈ and lower cages (Fig. 1, Table S1 and Appendix S1–S25†). For Gd₂@C₃₀, a C_2v isomer is found to be stable by 52.3 kJ mol⁻¹ compared to the D_5h isomer due to stronger Gd–C interactions in the former as affirmed by the AIM analysis (see Table 1 for larger cages and Tables S1–S4 in the ESI†). In larger cages, the stability can be rationalised using (i) the number of APRs (Table S2†) and (ii) the nature of Gd–C interaction as obtained from the AIM analysis (see Fig. 1, S1–S26 and Tables S3–S27 in the ESI†).

Fig. 1 The optimized structures of (and Gd–C bond length range) (a) Gd₂@C₃₀-D_5h (2.140–2.350 Å), (b) Gd₂@C₆₀-I_h (2.400–2.407 Å), and (c) Gd₂@C₅₉N-C_s (2.400–2.407 Å). The corresponding spin density plots for the high spin state are given in figures (d–f) with an isosurface value of 0.006 e⁻ bohr⁻³. Colour code: Gd-pink, C-grey, N-blue.

Table 1 The estimated J and the binding energy (kJ mol⁻¹) of chosen conformers in Gd₂@C_2n. Next to the symmetry label, the Gd⋯Gd distance is given in parentheses (Å). The value of spin density of each metal centre in the HS configuration of all Gd₂@C_2n has been given below the exchange values in parentheses. All the J and B values are shown in cm⁻¹

Gd2@C2n	J Gd⋯Gd (cm^−1)		Binding energy (kJ mol^−1)		ΔE (kJ mol^−1)
a The binding energy has been calculated with respect to electronic energy. In all other isomers, the binding energy has been calculated with respect to electronic and thermal free energies. b Here ‘n’ represents the total number of atoms, including the one nitrogen atom.
2n = 30	C 2v (2.185)	D 5h (2.224)	C 2v	D 5h	C 2v	D 5h
	J Gd–Gd = −49.8 (6.96; 6.95)	J Gd–Gd = −62.7 (6.94; 6.94)	1160.9	1149.8	0.0	52.3
2n = 32	C 2 (2.207)	D 3 (2.272)	C 2	D 3	C 2	D 3
	J Gd–Gd = −12.0 (6.93; 6.95)	J Gd–Gd = −15.5 (6.93; 6.93)	907.3	620.2	166.5	0.0
2n = 34	C s (2.283)	C 2 (2.266)	C s	C 2	C s	C 2
	J Gd–Gd = −11.6 (6.94; 6.94)	J Gd–Gd = −13.7 (6.94; 6.94)	279.1	392.1	73.0	0.0
2n = 36	C s (2.400)	D 2d (2.269)	C s	D 2d	C s	D 2d
	J Gd–Gd = −8.3 (6.92; 6.97)	J Gd–Gd = −26.3 (6.96; 6.95)	61.0	330.0	0.0	37.2
2n = 38	C 1 (2.443)	D 3h (2.734)	C 1	D 3h	C 1	D 3h
	J Gd–Gd = −7.0 (6.97; 6.93)	J Gd–Gd = −43.4 (6.99; 6.99)	−108.1	−63.0	0.0	478.1
2n = 40	C 2v (2.376)	D 2 (2.400)	C 2v	D 2	C 2v	D 2
	J Gd–Gd = −4.9 (7.00; 7.00)	J Gd–Gd = −10.3 (6.99; 6.99)	0.7	−173.2	320.6	0.0
2n = 42	C 1 (2.495)	D 3 (2.430)	C 1	D 3	C 1	D 3
	J Gd–Gd = −3.0 (6.99; 6.99)	J Gd–Gd = −7.6 (7.00; 7.00)	−337.3	−193.1	0.0	60.8
2n = 44	C s (2.608)	D 2 (2.549)	C s	D 2	C s	D 2
	J Gd–Gd = 0.2 (6.98; 7.02)	J Gd–Gd = −8.4 (7.00; 7.00)	−386.9	−308.0	157.1	0.0
2n = 46	C 1 (2.728)	C s (2.796)	C 1	C s	C 1	C s
	J Gd–Gd = −1.4 (6.99; 7.00)	J Gd–Gd = −0.3 (6.98; 6.98)	−494.8	−537.0	42.1	0.0
2n = 48	C 1 (2.836)	C 2v (3.002)	C 1	C 2v	C 1	C 2v
	J Gd–Gd = −0.4 (7.00; 7.00)	J Gd–Gd = 0.7 (6.99; 6.99)	−560.2	−534.3	0.0	410.6
2n = 52	C s (3.282)	D 2d (2.324)	C s	D 2d	C s	D 2d
	J Gd–Gd = −1.3 (7.02; 6.99)	J Gd–Gd = 2.7 (7.04; 7.04)	−595.2	−991.4	224.3	0.0
2n = 60^b	Gd2@C60-Ih (3.056)	Gd2@C59N-Cs (3.056)	Gd2@C60-Ih	Gd2@C59N-Cs
	J 1 = 869.8, J2 = 0.08, J3 = 40.2, B = 434.8 (7.53; 7.53)	J 1 = 869.8, J2 = 0.08, B = 434.8 (7.54; 7.54)	−369.4^a	−389.1^a
2n = 80^b	D 5h (3.818)	C 2v (4.074)	D 5h	C 2v	D 5h	C 2v
	J 1 = 404.6, J2 = 0.03, B = 202.1, J3 = −41.3 (7.54; 7.54)	J 1 = 351.3, J2 = 0.03, B = 175.6, J3 = −95.5 (7.52; 7.53)	−987.6^a	−840.5^a	0.0	96.7
2n = 80^b	Gd2@C79N-Cs-1 (3.816)	Gd2@C79N-Cs-2 (4.107)	Gd2@C79N-Cs-1	Gd2@C79N-Cs-2	Gd2@C79N-Cs-1	Gd2@C79N-Cs-2
	J 1 = 404.6, J2 = 0.03, B = 202.1 (7.55; 7.55)	J 1 = 351.3, J2 = 0.03, B = 175.6 (7.51; 7.53)	−958.0^a	−733.7^a	0.0	157.7

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Considering the Gd³⁺ ionic radius,⁶⁴ a Gd⋯Gd distance less than 2.5 Å (van der Walls radii) is likely to suggest a weak interaction or even a metal–metal bond. Such interactions are expected to reflect on J_Gd–Gd values with smaller values indicate weaker Gd⋯Gd interactions and not a metal–metal bond. Therefore, to compare the metal–metal interaction in Gd₂@C_2n with 30 ≤ 2n ≤ 52, the magnetic exchange J_Gd–Gd between two Gd³⁺ ions has been estimated using DFT calculations (B3LYP/TZV, Ĥ = −JŜ_Gd1Ŝ_Gd2, see computational details and Table 1). The J_Gd–Gd is found to be antiferromagnetic in all Gd₂@C_2n (2n ≤ 52) EMFs with the exception of Gd₂@C₄₄-C_s, Gd₂@C₄₈-C_2v, and Gd₂@C₅₂-D_2d EMFs having a ferromagnetic coupling (Table 1). The value in Table 1 suggests the decrease in antiferromagnetic interaction with the decrease in cage size. Within the same cage, the J_Gd–Gd value increases for a higher symmetry isomer. The largest antiferromagnetic J_Gd–Gd was estimated for Gd₂@C₃₀-D_5h (−62.7 cm⁻¹). This is several orders of magnitude larger than the experimentally known largest 4f–4f interaction. For the Gd₂@C₃₀-C_2v isomer, the J_Gd–Gd decreases to −49.6 cm⁻¹ despite a shorter Gd⋯Gd distance compared to the D_5h isomer. This is due to stronger 4f–4f overlaps (Tables S28–S29†). Although the Gd⋯Gd distances are very similar for C₃₀ and C₃₂, the J_Gd–Gd value is significantly smaller in Gd₂@C₃₂ (see Table 1) due to symmetry constraints and the associated 4f–4f overlaps (see Tables S30–S49 in ESI†). Further increase in the cage size only nominally decreases the J_Gd–Gd values with several exceptions, though lower symmetry models follow the trend (see Fig. S27†). A net ferromagnetic interaction is observed in Gd₂@C₄₄-C_s, Gd₂@C₄₈-C_2v and Gd₂@C₅₂-D_2d cages due to a meagre contribution to the antiferromagnetic part of J (see Tables S42, S46 and S48†). Orbital orthogonality of 4f-orbitals and dipolar contributions due to shorter Gd⋯Gd distance leads to a net ferromagnetic coupling in these examples. A very large 4f–4f overlap suggests a possibility of direct 4f–4f interactions between two lanthanide ions, which are hard to observe in classical coordination chemistry. The binding energy becomes positive for Gd₂@C_2n with 2n ≤ 36 and negative for Gd₂@C_2n with 2n > 36 (see Table 1) except for the Gd₂@C₄₀-C_2v isomer, where it is thermoneutral (0.7 kJ mol⁻¹). Thus, it suggests that the large antiferromagnetic interaction is feasible for the isomers of Gd₂@C_2n with 2n > 36.

The magnitude of the spin density of the two Gd³⁺ ions increases with an increase in ring size, supported by the contour plots of the electron density map obtained from AIM analysis (Fig. S28–S54 and Tables S50, S51†). Particularly a sudden jump in the magnitude of spin density is noted for Gd₂@C₆₀, with nearly one electron found between the two Gd ions (see Fig. 1, S50–S52†). Our NBO analysis reveals that this electron is delocalised in the formally empty orbitals, which are hybridised among 6s, 6p, and 5d orbitals (6s6p^0.115d^0.33, see Fig. S55†). Thus, it suggests a strong valence delocalisation where one unpaired electron is delocalised to vacant 5d/6s/6p orbitals of each Gd³⁺ ion leading to a type-III class of mixed valence systems (Gd^2.5+⋯Gd^2.5+, see later).⁶⁵

Mechanism of the formation of Gd₂@C_2n

To further investigate the unusual behaviour wherein the cage size decides the magnitude of the spin density present between the Gd³⁺ ion, we have analysed the formation of Gd₂@C_2n from the HOMO–LUMO gap perspective. In the formation of dimetallofullerene Gd₂@C_2n, we can presume that two Gd atoms donate three electrons each from their frontier orbitals (5d and 6s orbitals) to the three lowest unoccupied molecular orbitals (LUMOs) of the C_2n cage resulting in Gd₂⁶⁺@C_2n⁶⁻.⁶⁶ Thus, the formation of Gd₂@C_2n depends on the energy gap between the frontier orbitals of Gd₂ and the LUMOs of the C_2n cage. If the LUMOs of the C_2n cage are found to be lower in energy than the frontier orbitals of the Gd₂ fragment, a large stabilisation occurs after the electron transfer. Quite interestingly, this is the case for the C_2n cage with 2n ≤ 52, which favours the transfer of six electrons from Gd₂ (with the Gd–Gd distance <3.0 Å, ignoring 4f orbitals, the valence electron configuration is σ²_gσ¹_uπ¹_gπ²_u)⁶⁷ to the C_2n cage (Fig. 2a for Gd₂@C₃₀ and Fig. S56 and S57† for Gd₂@C₅₂-D_2d and Gd₂@C₄₈-C_2v). As the ring size increases, the LUMOs of the C_2n cage destabilised. In Gd₂@C₆₀ with the Gd⋯Gd distance of 3.056 Å, the bonding in the Gd₂ fragment before encapsulation is found to be σ²_gπ³_uσ¹_u (ignoring the 4f orbitals, Fig. 2b). After encapsulation, the five electrons are fully transferred to the cage except one σ¹_g electron (here the β electron in Fig. 2b for Gd₂@C₆₀-I_h) resulting in a 2c-1e⁻ bond between two Gd atoms. This is due to the comparable energy of the beta (6s/5d) σ¹_g orbital with the LUMO of the C_2n cage.

Fig. 2 The MO diagrams corresponding to the formation of (a) Gd₂@C₃₀-C_2v and (b) Gd₂@C₆₀-I_h isomers. The three black and red horizontal lines correspond to the energy of the occupied and empty orbitals of the C_2n fullerene ring, respectively. The blue and pink horizontal lines correspond to the energy of α and β orbitals of the Gd₂ fragment. We have shown the three lowest unoccupied α orbitals of the C_2n fullerene cage with an isosurface value of 0.055 e⁻ bohr⁻³. The three highest occupied α orbitals for Gd₂ in the C₃₀ and C₆₀ fullerene cage are also shown (isosurface 0.06 e⁻ bohr⁻³). Colour code: Gd-pink, C-grey.

Estimation of magnetic exchange in Gd₂@C₅₉N-C_s, Gd₂@C₇₉N-C_s-1, and Gd₂@C₇₉N-C_s-2

The most sensitive parameter that yields insight into the spin density distribution discussed in the last section is the corresponding exchange coupling J_Gd–Gd. Here we intend to compute this parameter and analyse this with respect to the cage size. The mechanism of formation of Gd₂@C_2n suggests the presence of one unpaired electron between two Gd³⁺ ions and another conjugate electron in the fullerene cage for Gd₂@C₆₀-I_h, Gd₂@C₈₀-D_5h, and Gd₂@C₈₀-C_2v isomers (see Table S50†). For these molecules, a complex set of magnetic coupling emerges: (i) the coupling between Gd³⁺ and the radical that reside inside the cage (J₁), (ii) the second one describes the coupling between two Gd³⁺ ions (J₂), (iii) the third one describes the coupling between two radicals (J₃) and (iv) in addition to these isotropic exchange coupling values, a strong electron delocalisation of the radical between two Gd³⁺ ions suggests a double-exchange (parameter B) being operative between two Gd ions (in a fully delocalised case, Gd₂^2.5+). All these exchanges have been illustrated in Scheme S2.† This is similar to a type-III mixed-valence system^68,69 where the spin Hamiltonian parameters are estimated using the following Hamiltonian⁷⁰

H = −J(S_A·S_B·O_A + S_A·S_B·O_B) + BT_AB, (1)

where J and B denote the exchange interaction and delocalisation parameter, respectively, S_A and S_B are the total spin multiplicity of centres A and B, respectively, O_A and O_B are the localisation operator, and T_AB is the electron transfer operator (see computational details for more information).

The presence of one unpaired electron in the fullerene cage of Gd₂@C₆₀-I_h, Gd₂@C₈₀-D_5h, and Gd₂@C₈₀-C_2v leads to polymerisation or aggregation, and often, these complexes are not isolable.^71,72 There are two strategies available to demonstrate their existence (i) by transforming them into a chemically stable form with one-electron reduction/substitution at the ring position. This has been adapted to stabilise the Dy₂@C₈₀-I_h molecule by chemically transforming it to Dy₂@C₈₀(CH₂Ph).^72–75 (ii) By substituting one of the carbon with the nitrogen atom yielding azafullerenes such as Ln₂@C₇₉N and other analogues.^{49,70,76–79} Here, we have adopted the second approach where one carbon atom is substituted by nitrogen in Gd₂@C_60/80 isomers yielding Gd₂@C_59/79N molecules (see Appendix S26–S28† for optimised coordinates) possessing C_s symmetry (here Gd₂@C₇₉N-C_s-1 is derived from Gd₂@C₈₀-D_5h and Gd₂@C₇₉N-C_s-2 is derived from Gd₂@C₈₀-C_2v, see Table 1). Upon substitution, as expected, the spin density of the cage in Gd₂@C_59/79N was seized (see Fig. 1e and f for the 665 isomer, see ref. 44). While Gd₂@C₇₉N is a well-characterised and thoroughly studied molecule, Gd₂@C₅₉N is not known. However, the X-ray structure of C₅₉N and encapsulation of some metal ions are experimentally studied, and their existence has been proved beyond ambiguity.^35,80–86 Particularly, K₆C₅₉N has been isolated and characterised thoroughly. This suggests that the C₅₉N⁶⁻ cage is a stable molecular fragment and can encapsulate Ln³⁺ cations similar to those hypothesised here.^87–89

This type-III mix valence moiety of Gd₂@C₅₉N-C_s, Gd₂@C₇₉N-C_s-1, and Gd₂@C₇₉N-C_s-2 isomer represents a multireference wave function as the unpaired electron is not localised on a particular centre. Therefore, a multireference method such as the state-average CAS(15,15)SCF set up was employed to estimate the double exchange parameter (B) (Fig. 3, see computational details).⁹⁰ As per the CASSCF calculations, the additional radical electron resides in a hybrid orbital containing coefficients from 6s, 5p_z, 6p_z, and 5d_z² orbitals of Gd₁ and Gd₂ centres (see Table S55† for composition). The set of spin Hamiltonian parameters obtained from the CASSCF calculations are as follows, Gd₂@C₅₉N-C_s (Gd₂@C₇₉N-C_s-1) [Gd₂@C₇₉N-C_s-2]: J₁ = +869.8 (+404.6) [+351.3] cm⁻¹, J₂ = 0.08 (0.03) [0.03] cm⁻¹ and B = +434.8 (+202.1) [+175.6] cm⁻¹ (Table 1). For all three complexes, the J₁ interaction is found to be extremely large, and this is due to the involvement of the diffuse virtual 6s and 6p_z and 5d_z² orbitals of Gd ions, while the J₂ coupling between two Gd³⁺ ions is found to be very small as the 4f orbitals are only weakly interacting here. It is worth mentioning that we have previously reported a very large J₁ value of +400 cm⁻¹ (Ĥ = −JŜ₁Ŝ₂) in Gd₂@C₇₉N using the UB3LYP/TZV setup.⁴⁴ The estimated J₁ value by our ab initio approach for Gd₂@C₇₉N lies in the range of 350–405 cm⁻¹, and this is in line with the DFT calculations and experimental reports (−350 ± 20 cm⁻¹ using Ĥ = J₁(Ŝ_Gd1Ŝ_rad + Ŝ_Gd2Ŝ_rad) + J₂(Ŝ_Gd1Ŝ_Gd2) in ref. 40 and a J₁ value of 170 ± 10 cm⁻¹ using the Hamiltonian Ĥ = −2J₁(Ŝ_Gd1Ŝ_rad + Ŝ_Gd2Ŝ_rad) − 2J₂(Ŝ_Gd1Ŝ_Gd2) in ref. 76; see ESI† for the discussion of J₃). These large exchange values have potential application in qubits as they enhance the quantum coherence required for qubit applications.⁴⁰

Fig. 3 The fifteen active orbitals of Gd₂@C₅₉N with CAS (15,15) active space for the S = 15/2 state. Colour code: Gd-pink, C-grey, N-blue.

A case study of magnetic anisotropy in Dy₂@C₅₉N and Tb₂@C₅₉N

As heterofullerenes yield larger J_s and homofullerene yields relatively smaller antiferromagnetic J_s, the former is the best suited to design SMMs. The antiferromagnetic J_s in homofullerene yields diamagnetic ground states, and smaller ferromagnetic J_s observed in larger cage sizes such as C₅₂ did not yield any appealing SMM characteristics. To harness SMMs in this class, heterodinuclear lanthanides with unequal m_J states were modelled. Models such as PrEr@C₃₈-D_3h yield a reasonable U_cal value with robust QTM quenching (ca. 109 cm⁻¹, see ESI†) but are not substantial to serve as a synthetic target.

Therefore, we aim to estimate the magnetic anisotropy in the Dy₂@C₅₉N and Tb₂@C₅₉N. It is noteworthy to mention that the record-breaking magnetic anisotropy is previously achieved in Dy₂@C₇₉N and Tb₂@C₇₉N molecules.^44,77 The metal centre in Dy₂@C₅₉N is found to interact in an η⁶ fashion with the C₅₉N cage, which creates a strong uniaxial anisotropy (see Fig. 1c) as a long Dy⋯Dy bond (3.056 Å) induces a weak ligand field in the opposite site of cage binding. Thus, the coordination can be compared with Dy³⁺–O, which perfectly suits the oblate ground state.^91,92 The easy axis of magnetisation is found to be nearly collinear with the Dy–Dy axis with a very small angle (2.0 (1.1°) for Dy1(Dy2), Fig. S72†). The calculated g_z values of KD1 (∼19.97) imply an Ising ground state for both the Dy centres (Tables S70, S71 and Fig. S71†), with the relaxation predicted to proceed via the first excited state for Dy2 (U_cal = 250.5 cm⁻¹) and third excited state for the Dy1 centre (U_cal = 483.7 cm⁻¹, Fig. S71 and Tables S70, S71†). The very large axial crystal field parameter⁹³B_kq (k = 0, q = 0) compared to the non-axial crystal field parameter B_kq (k = 0, q ≠ 0) suggests significant axiality for both the Dy centres in Dy₂@C₅₉N (see Table S72†). Furthermore, the axial CF parameters are found to be slightly larger in Dy₂@C₅₉N compared to Dy₂@C₇₉N, suggesting a larger axiality of the former compared to the latter.⁴⁴

The ab initio calculations on Tb₂@C₅₉N reveal a negligible tunnel splitting in the ground pKDs (0.025(0.013) cm⁻¹ for Tb1(Tb2), see Tables S73, S74 and Fig. S73†). Further, the ground state g_z value (g_z = 17.921(17.919) for Tb1(Tb2) centre^94,95) suggests the Ising nature of the ground state. The ground anisotropy axis of the Tb1(Tb2) centre is oriented along the pseudo C₆ axis of the hexagonal ring and nearly collinear with the Tb–Tb axis (the tilting angle becomes 0.45 and 1.50° for Tb1 and Tb2 centres, respectively, Fig. 4). However, the significant tunnel splitting (0.115 and 0.102 cm⁻¹ for Tb1(Tb2), Fig. S73†) in the first excited pKDs reinforces the magnetisation relaxation via this state. This leads to the U_cal value of 227.6 and 233.2 cm⁻¹ for Tb1 and Tb2 centres, respectively (see Tables S73, S74 and S70†).

Fig. 4 The POLY_ANISO computed relaxation mechanism of Tb₂@C₅₉N-C_s. The anisotropy axis of the metal (represented by yellow) and radical (represented by red) centre are shown on the right. The thick black line represents the magnetic moment of KDs. The red arrows imply the QTM for ground KD and TA-QTM for higher excited KDs. The blue dotted arrows indicate a possible Orbach process. The green arrows represent the mechanism of magnetic relaxation. Colour code: Tb-blue violet, C-grey, N-blue.

To explore the mechanism of magnetisation relaxation in the exchange-coupled Dy₂@C₇₉N and Tb₂@C₇₉N systems, we have simulated the exchange-coupled energy spectrum using the POLY_ANISO module (see Table 1 and computational details). For the computed ground state, a large magnetic moment of ca. 21 and 19 μ_B for Dy₂@C₅₉N and Tb₂@C₅₉N respectively, was obtained (see Fig. 4 and S72†) with negligible tunnel splitting or QTM effects. The first, second, and third excited states are found to possess negligible tunnel splitting/TA-QTM, which is reflected in negligible g_x/g_y and very large g_z values (see Tables S75, S76, Fig. 4 and S72†). The magnetic moment in the fourth excited state is very small, and it results in sizeable QTM for Dy₂@C₅₉N and Tb₂@C₅₉N. Therefore, the magnetisation relaxation for this exchange-coupled system is expected via the fourth excited state yielding a record-high U_cal value of 1183.3 and 1501.8 cm⁻¹ for Dy₂@C₅₉N and Tb₂@C₅₉N, respectively (see Fig. 4, 5, and S72†). These gigantic U_cal values are two times larger than Dy₂@C₇₉N/Tb₂@C₇₉N estimates, thanks to a very large ferromagnetic exchange.^44,77 The other relaxation process due to intermolecular interactions is expected to be minimal due to confinement, which is likely to yield large T_B values. As our predictions on Gd₂@C₇₉N and Dy₂@C₇₉N are proved by experiments lately, with Dy₂@C₇₉N yielding an attractive blocking temperature (24 K), these smaller cages, if made, could enhance T_B values even further.^40,44,47,76

Fig. 5 Diagrammatic representation of the estimated single ion and exchange-coupled U_cal for Dy₂@C₅₉N-C_s and Tb₂@C₅₉N-C_s.

Conclusions

To this end, we have employed an array of theoretical tools in search of finding lanthanide encapsulated fullerenes with very large blocking barriers and blocking temperatures. Various ideas, such as enhancing the coupling between two lanthanide ions by bringing them close to each other in the confined space, have been tested, and the main conclusions drawn from this works are summarised below.

(i) Sourcing the large J_Gd–Gd exchange via confinement: in search of increasing the magnetic exchange (J_Gd–Gd) between two lanthanide ions via confinement, we have varied the cage size from C₃₀ to C₈₀ where the Gd⋯Gd distance ranging from 2.185 Å to 4.107 Å is observed. Here smaller cages (C_2n, 2n ≤ 52) yield a weaker Gd⋯Gd interactions with a stable Gd₂@C₃₈-D_3h complex having a record-high exchange for any 4f–4f interaction (J_Gd⋯Gd = −43.4 cm⁻¹). A strong 4f–4f orbital overlap between two Gd³⁺ ions suggests the Gd–Gd bond formation under confinement. As the exchange is antiferromagnetic, these are not ideal for SMMs, however among hetero dilanthanide EMFs, some promising SMMs are identified.

(ii) Ab initio estimation of double exchange in endohedral azafullerenes: the larger cages (Gd₂@C₆₀ and Gd₂@C₈₀) lead to the formation of a two-centre-one-electron Gd–Gd bond due to the comparable energy of the highest occupied orbitals of Gd₂ and lowest unoccupied orbitals of the fullerene cage. Here we have studied Gd₂@C_59/79N complexes where the delocalisation of the electron between two Gd centres is treated via a double-exchange parameter. A protocol to compute the double-exchange using ab initio CASSCF calculations is proposed, and this methodology yields spin Hamiltonian parameters that are in excellent agreement with experiments for Gd₂@C₇₉N. The application of this method in Gd₂@C₅₉N unveils a massive J_Gd–rad exchange (J_Gd–rad = +869 cm⁻¹) which is two times larger than the record-high J reported for Gd₂@C₇₉N.

(iii) Record-high blocking barrier for Dy₂@C₅₉N and Tb₂@C₅₉N: the huge ferromagnetic J_Gd–rad exchange found in the C₅₉N cage quenches the QTM significantly and yields a very high U_cal value of 1502 cm⁻¹ for Tb₂@C₅₉N – the largest reported for any lanthanide EMF. This opens up the possibility of generating large magnetic anisotropy without relying on a stronger ligand field.

Computational details

All the DFT calculations have been performed using the Gaussian09 suite with the B3LYP functional.^96,97 There are several isomers with different symmetries possible for a chosen fullerene cage. To estimate the effect of symmetry of the fullerene cage in the associated magnetic properties, we have chosen two low energy conformers of the C_2n (n = 15–24, 26, 30, 40) fullerene from http://www.nanotube.msu.edu/ and encapsulated two Gd atoms.⁹⁸ The optimisation of the resulting Gd₂@C_2n has been performed using the UB3LYP/CSDZ(Gd), SVP (rest) methodology.^99,100 Further single point energy calculations were performed with an Ahlrichs' triple-ξ valence (TZV) basis set to obtain an excellent numerical estimate of energy/magnetic coupling.¹⁰¹ A quadratic convergence SCF method was used throughout all the calculations.¹⁰² One high spin (HS, the spin on both Gd³⁺ centres is “up”) and one broken symmetry (BS, the spin on one Gd³⁺ centre is “up” and another Gd³⁺ centre is “down”) configuration was used to estimate the magnetic exchange. The magnetic exchange has been calculated using the Hamiltonian Ĥ = −JŜ₁Ŝ₂, where .^103,104 Additionally, we have performed AIM (atoms in molecules) analysis with the AIM2000 programme package to determine the coordination number of the Gd³⁺ ion inside the fullerene cage along with the nature of Gd–C and Gd–Gd bonds.¹⁰⁵

To estimate the double exchange in Gd₂@C₇₉N and Gd₂@C₅₉N molecules, ab initio CASSCF calculations have been performed using the MOLCAS 8.4 programme package.¹⁰⁶ We have employed [Ln·ANO-RCC⋯8s7p5d3f2g1h], [C·ANO-RCC⋯3s2p1d] and [N·ANO-RCC⋯3s2p1d] contraction schemes in the basis set for Gd, C and N, respectively.^106,107 The DKH Hamiltonian was used to take into account the scaler relativistic effect.¹⁰⁸ The Cholesky decomposition technique was used to reduce the size of the disk space.¹⁰⁹ The CASSCF calculations have been performed with the CAS (15,15) active space.⁹⁰ The active space includes seven 4f orbitals from each Gd atom and one orbital for the unpaired electron. Within the active space, we have computed the energy of the S = 13/2, 11/2, 9/2, 7/2, 5/2 and 3/2 states with two roots while the energy of S = 15/2 and 1/2 states has been estimated using only one root. Further details on the computational methods are elaborated in the ESI.†

The magnetic anisotropy in the Gd₂@C₃₈-D_3h and Gd₂@C₅₂-D_2d isomer has been estimated by replacing the isotropic Gd metal centres with Dy, Er and Pr. For Gd₂@C₅₉N model, anisotropic calculations were perforemd using Dy and Tb ions. The CASSCF calculations have been performed with minimal CAS(n,7) active space (n = number of 4f electrons) for Pr, Dy, Tb and Er using the MOLCAS 8.4 programme package.¹⁰⁶ We have computed the energies of the 21 triplets and 28 singlets of Pr³⁺, 7 septets, 140 quintets and 195 triplets for Tb³⁺, 21 sextets for Dy³⁺, and 35 quartets and 112 doublets for Er³⁺ within the size of the active space. Thereafter, the computed spin-free states (7 septets, 105 quintets and 112 triplets for Tb³⁺) have been mixed in RASSI-SO to obtain the spin–orbit coupled energies. Finally, the g tensors, QTM/TA-QTM, etc. of the metal centre have been computed by SINGLE_ANISO, which interfaces with the RASSI-SO. After calculating the magnetic anisotropy of the individual metal centres, they have been coupled by POLY_ANISO using the Lines model to compute the energy of the exchange-coupled system.¹¹⁰ The magnetic exchange computed with the DFT and ab initio approach has been scaled with 5/7, 6/7, and 3/7 for Dy, Tb and Er centres, respectively, to estimate exchange coupled energy levels.

Author contributions

GR conceived the idea and SD performed all the calculations and interpret and analyse the data. Both wrote the manuscript together.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

SD would like to thank UGC for the SRF fellowship and Dr Hélène Bolvin, UPS, Toulouse for her helpful suggestions. GR would like to thank DST/SERB (CRG/2018/000430, DST/SJF/CSA03/2018-10; SB/SJF/2019–20/12; SPR/2019/001145) for funding.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1sc03925c

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