Origin of the relaxation barriers in a family of MRe^IV(CN)₂ single-chain magnets (M = Mn^II, Ni^II, and Co^II): a theoretical investigation

Abstract

Through investigation of the origin of the relaxation barriers and the magneto-structural correlations in three (DMF)₄MRe^IVCl₄(CN)₂ (DMF = dimethylformamide; M = Mn^II (1), Ni^II (2), and Co^II (3)) single-chain magnets (SCMs) in the intermediate regime between the Ising and the Heisenberg limits using hybrid functional theory (B3LYP) and complete active space second-order perturbation theory (CASPT2) methods, we succeeded in obtaining the magnetic anisotropy and correlation energy barriers Δ_A and Δ_ξ in terms of D (magnetic anisotropy parameter), J (magnetic exchange coupling constant), and S (spin value). B3LYP calculations show that the antiferromagnetic Re^IV–Mn^II coupling interactions decrease with the increase of the Mn–N–C angle, while the ferromagnetic Re^IV–Ni^II coupling interactions increase with the increase of the Ni–N–C angle. But, the Re^IV–Co^II exchange interactions with the variation of the Co–N–C angle are almost the same. The total energy barriers Δ_τ of 1 and 2 mainly come from the contribution of the transverse anisotropy of Re^IV, while the Mn^II and Ni^II ions only transmit the exchange coupling between Re^IV and almost have no contributions. For 3, however, both of the Re^IV and Co^II ions have contributions to the total energy barrier.

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Introduction

For single-chain magnets (SCMs) when the anisotropy energy is not sufficiently higher than the exchange energy, they do not fall within the Ising limit with sharp domain walls and instead possess broad domain walls. In this intermediate regime between the Ising and the Heisenberg limits, the expressions of the magnetic anisotropy and correlation energy barriers Δ_A and Δ_ξ in terms of D (magnetic anisotropy parameter), J (magnetic exchange coupling constant), and S (spin value) were still not clear.^1–3 Thus, exploring the expressions is vitally important, which will help us to understand the origin of the relaxation barriers so as to know how to increase them. Recently, Long and co-workers¹ reported a series of (DMF)₄MRe^IVCl₄(CN)₂ (DMF = dimethylformamide; M = Mn^II, Ni^II, Co^II, and Fe^II) SCMs, and demonstrated that the correlation energy Δ_ξ is far smaller than 4|JS₁S₂| since they do not fall within the Ising limit with sharp domain walls. Then, they used experimental methods to study the origin of the energy barrier for one of them, (DMF)₄Mn^IIRe^IVCl₄(CN)₂, and demonstrated that the magnetic anisotropy energy barrier Δ_A mainly comes from the contribution of the transverse anisotropy of Re^IV.² However, they did not give the expression of the correlation energy barrier Δ_ξ, and also they did not investigate the other three complexes.² In a recent work,³ we have done some research on one of them, R₄Fe^IIRe^IVCl₄(CN)₂,^1,4 and gave some interesting results. To probe the general rule, we will select the other three (DMF)₄MRe^IVCl₄(CN)₂ (M = Mn^II (1), Ni^II (2), Co^II (3)) SCMs which have much different magnetic properties from R₄Fe^IIRe^IVCl₄(CN)₂ to explore the origin of the relaxation barriers and the expressions of Δ_A and Δ_ξ in terms of D, J, and S.

Compared to the energy barriers and the intramolecular exchange interactions of R₄Fe^IIRe^IVCl₄(CN)₂,^1,4 those of the other three complexes MRe^IVCl₄(CN)₂ (M = Mn^II (1), Ni^II (2), Co^II (3)) are so much different. The magnetic anisotropies of Mn^II and Ni^II in 1 and 2 are far smaller than those of Fe^II, and the Re^IV–Ni^II and Re^IV–Co^II ferromagnetic interactions are weaker than those of Re^IV–Fe^II. Moreover, the total energy barrier of 1 is far smaller than those of Fe^IIRe^IV although the Re^IV–Mn^II exchange interactions are close to those of Re^IV–Fe^II. Besides, the total energy barriers of 2 and 3 are very close although the magnetic anisotropies of Ni^II and Co^II are so much different. To investigate the above problems, hybrid functional theory (B3LYP) and complete active space second-order perturbation theory (CASPT2) methods were performed to explore the origin of the energy barriers. Considering the similar structures of 1–3, we only gave the general structure in Fig. 1. They added a solution of M(ClO₄)₂ (M = Mn^II (1), Ni^II (2), Co^II (3)) in 1.5 mL of DMF to a solution of (Bu₄N)₂[trans-ReC_l4(CN)₂] in 1.5 mL of DMF, which was allowed to stand for 8 hours to 12 days, to obtain the three complexes. A detailed description of the experimental details and the molecular structures can be found in ref. 1.

Fig. 1 General structure of (DMF)₄MRe^IVCl₄(CN)₂; H atoms are neglected for clarity.

Computational details

The spin Hamiltonian of three MRe^IVCl₄(CN)₂ SCMs in the absence of spin–orbit coupling (SOC) is expressed as:where J represents the exchange coupling constant for the interaction between neighboring Re^IV and M centers, and S_Re and S_M are the local spins of Re^IV (S = 3/2) and M (S = 5/2 for Mn^II, 1 for Ni^II, 3/2 for Co^II), respectively. Long and co-workers showed that using an isotropic Heisenberg model to describe the exchange coupling of R₄MRe^IVCl₄(CN)₂ is appropriate irrespective of the highly anisotropic [ReCl₄(CN)₂]²⁻ unit inside the chain.^1,4

Orca 2.9.1 calculations⁵ using the popular hybrid functional B3LYP proposed by Becke^6,7 and Lee et al.⁸ were performed to obtain the isotropic exchange coupling constant J. Triple-ζ with one polarization function TZVP⁹ basis set was used for all atoms, and the scalar relativistic treatment (ZORA) was used in all calculations. The large integration grid (grid = 6) was applied to Re^IV and M (M = Mn^II, Co^II, Ni^II) for ZORA calculations. Tight convergence criteria were selected to ensure that the results are well-converged with respect to technical parameters. We employed two model structures of A and B (see Fig. 2(a) and (b)) for each complex, and calculated two high- and one low-spin state energy for each model: S_HS = S_M1 + S_Re + S_M2 for model A; S_HS = S_Re1 + S_M + S_Re2 for model B, S_LS = S_M1 − S_Re + S_M2 with the flipped spins of Re^IV for model A; S_LS = S_Re1 − S_M + S_Re2 with the flipped spins of M for model B. The HS state is a “real” state corresponding to the pure high spin state. But, the LS state is a spin contaminated state not a pure low spin state. The Re^IV–M coupling constants J were then obtained as eqn (2)–(4) according to the spin Hamiltonian H = −2JS_Re(S_M1 + S_M2) for model A and H = −2JS_M(S_Re1 + S_Re2) for model B using the spin-projected approach.¹⁰

Fig. 2 Structures of models A (DMF)₄M[Re^IVCl₄(CN)₂]₂ (a) and B [(DMF)₄MCN]₂Re^IVCl₄(CN)₂ (b); H atoms are neglected for clarity.

Since density functional theory (DFT) methods in the calculation of D and E are not accurate enough to describe the magnetic anisotropies for our investigated systems,^3,11 MOLCAS 7.8 program package¹² with CASPT2 was used to calculate the D and E values of the Re^IV (Re^IVCl₄(CN)₂Zn₂) and M [M(DMF)₄(CN)₂Ba₂] fragments extracted from each complex. To calculate the Re^IV fragment (see Fig. 3(a)), the influence of the neighboring M ions have been simulated by the closed-shell Zn^IIab initio embedding model potentials (AIMP; Zn.ECP.Lopez-Moraza.0s.0s.0e-AIMP-KZnF₃).¹³ The only removed atoms are those connected to the Zn^II AIMP from the opposite side of the molecule. Similarly, the neighboring Re^IV ions have been simulated by the closed-shell Ba^II AIMP (Ba.ECP.Pascual.0s.0s.0e-AIMP-BaF₂)¹³ for each M fragment (Fig. 3(b)).

Fig. 3 Calculated Re^IV (Re^IVCl₄(CN)₂Zn₂) (a) and M [M(DMF)₄(CN)₂Ba₂] (b) fragments; H atoms are neglected for clarity.

The basis sets for all atoms are atomic natural orbitals from the MOLCAS ANO-RCC library: ANO-RCC-VTZP for magnetic center ions Re^IV and M; VTZ for close C and N; VDZ for distant atoms in the first complete-active-space self-consistent field (CASSCF) calculations. The scalar relativistic contractions were taken into account in the basis set. After that, the effect of the dynamical electronic correlation was applied using CASPT2. And then, the SOC was handled separately in the restricted active space state interaction (RASSI) procedure. The active space is (3,5), (5,5), (8,10), and (7,10) for the Re^IV, Mn^II, Ni^II and Co^II fragments, respectively. The mixed spin-free states are 40, 100, 25, and 50 for the Re^IV, Mn^II, Ni^II, and Co^II fragments, respectively.

Results and discussion

Exchange coupling interactions

The calculated and experimental J values of three (DMF)₄MRe^IVCl₄(CN)₂ are shown in Table 1, where the Re^IV–Mn^II couplings of 1 are antiferromagnetic, but those of 2–4 are ferromagnetic.

Table 1 Calculated and experimental J values (cm⁻¹) of models A and B for four SCMs, (DMF)₄MRe^IVCl₄(CN)₂ (M = Mn^II (1), Ni^II (2), Co^II (3), Fe^II (4))

		1		2		3		4
		A	B	A	B	A	B	A	B
a Data from ref. 3.
J	Cal.	−7.5	−6.9	5.7	5.5	4.2	3.9	5.8^a	5.4^a
	Exp.^1	−5.4(4)		3.7(3)		2.4(3)		4.8(4)

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For 1–4, both of the calculated J values of models A and B for each complex are close to the experimental one. Moreover, the obtained J values using model B may be better in the calculations of J for all complexes.

To explore the mechanism of the Re^IV–M exchange couplings, we gave three magnetic orbitals localized on Re^IV(t_2g³) and five on Mn^II(t_2g³e_g²) of 1 in Fig. 4 in the low-spin state (the magnetic orbitals on Re^IV, Ni^II and Co^II of 2–3 can be found in Fig. S1 and S2 in the ESI†).

Fig. 4 Magnetic orbitals on Re^IV (left) and Mn^II (right) of 1 in the low-spin state.

For 1, the antiferromagnetic interactions dominate over the competing ferromagnetic interactions between Re^IV(t_2g³) and Mn^II(t_2g³e_g²) centers due to one more t_2g unpaired electron of Mn^II results in the antiferromagnetic Re^IV–Mn^II coupling. It is evident that the Re^IV–Ni^II coupling in 2 is ferromagnetic for the orthogonal magnetic orbitals on Re^IV(t_2g³) and Ni^II(e_g²) metals (see Fig. S1 in the ESI†). For 3, the Co^II(t_2g¹e_g²) center houses one less unpaired electron in a π-type orbital, which leads to the Re^IV–Co^II ferromagnetic interactions (see Fig. S2 in the ESI†).

To thoroughly investigate the relationship between J and the M–N–C angle for the investigated three SCMs, we calculated J with the M–N–C angle changing from 140° to 180° using model B (see Fig. 2(b)) for 1–3, respectively. The obtained J values of 1–3 with the M–N–C angle ranging from 140° to 180° are shown in Table 2.

Table 2 Dependence of the J values (cm⁻¹) on the M–N–C angle ranging from 140° to 180° for 1–3

	140°	145°	150°	155°	160°	165°	170°	175°	180°
1	−9.9	−8.3	−7.4	−6.7	−6.1	−5.5	−5.8	−5.7	−4.9
2	2.3	2.9	3.2	4.7	5.9	6.7	6.9	7.2	8.6
3	−1.9	0.5	2.2	3.7	4.6	5.0	5.1	5.0	4.9

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According to Kahn's theory,¹⁴ the square of the mean overlap integral S_ij between magnetic orbitals on the paramagnetic centers i and j, which favors the antiferromagnetic contribution J_AF, can be correlated to the Δ_ij for polynuclear complexes proposed by Ruiz and co-workers.¹⁵

where ρⁱ_HS, ρⁱ_LS, ρ^j_HS, and ρ^j_LS are the spin populations of the i and j involved in the exchange interaction in the high (HS) or low (LS) spin configurations, respectively.

The variation of J can be correlated with the changes in the J_AF term, which usually control the magneto-structural correlations.¹⁶ The spin populations on Re^IV and M were obtained using Mulliken Population Analysis¹⁷ calculated using the B3LYP functional. The strength of the J_AF is linearly dependent on Δ_ij.¹⁴

|J_AF| ∝ S_ij² ∝ Δ_ij (6)

Fig. 5 gives the relationship between Δ_ij and the M–N–C angle ranging from 140° to 180° for 1–3, respectively.

Fig. 5 Dependence of the Δ_ij on the M–N–C angle ranging from 140° to 180° for 1–3.

For 1, the decrease of the Δ_ij with the Mn–N–C angle ranging from 140° to 180° leads to the decrease of the absolute values of S_ij, and to the decrease of the J_AF term. Thus, the antiferromagnetic Re^IV–Mn^II coupling constant J (J = J_F + J_AF) decrease with the increase of the Mn–N–C angle. The decrease of the Δ_ij with the increase of the Ni–N–C angle leads to the decrease of the absolute values of S_ij, and to the decrease of the J_AF term, which will strengthen the ferromagnetic Re^IV–Ni^II coupling in 2. For 3, however, the variation of the Δ_ij with the Co–N–C angle ranging from 140° to 180° is small, which results in the small differences in the Re^IV–Co^II coupling constants.

Expressions of the anisotropy and correlation energy barriers

CASPT2 could give the most accurate D values for complexes [Re^IVCl₄(CN)₂]²⁻,^1,2 (DMF)₄ZnReCl₄(CN)₂,² [TPA^2C(O)NHtBuFe^II(CF₃SO₃)]⁺ (TPA^2C(O)NHtBu = 6,6′-(pyridin-2-ylmethylazanediyl)bis(methylene)-bis(N-tert-butylpicolinamide))¹⁸ and (TPA^2C(O)NHtBu)Fe^IIRe^IVCl₄(CN)₂¹⁹ compared to those calculated by DFT and CASSCF methods.³ For other transition metal systems containing Mn^II, Ni^II, and Co^II ions, CASPT2 also performs well in the calculations of D.^11,20 Thus, in the following calculations, CASPT2 will be used.

For one-dimensional systems falling within the Ising limit, Δ_ξ can be expressed as 4|JS₁S₂|. In the cases of 1–3, however, the experimental Δ_ξ values are all far smaller than 4JS_ReS_M.¹ It is evident that 1–3 do not fall within the Ising limit. To explore the constitution of the domain walls and the expressions of Δ_ξ and Δ_A in terms of D, J and S, we firstly calculated the D_i and E_i values of the Re^IV (Re^IVCl₄(CN)₂Zn₂) and M ((DMF)₄M(CN)₂Ba₂) fragments extracted from three SCMs (see Fig. 3), respectively, which are shown in Table 3.

Table 3 Calculated D_i, E_i and (D_i − E_i)/J values (cm⁻¹) of the Re^IV (Re^IVCl₄(CN)₂Zn₂) and M ((DMF)₄M(CN)₂Ba₂) fragments where J values are the experimental Re^IV–M coupling constants (cm⁻¹) for 1–3

	1		2		3
	Re^IV	Mn^II	Re^IV	Ni^II	Re^IV	Co^II
D i	11.3	−0.29	16.7	−2.9	19.5	−5.6
E i	−1.91	−0.11	−2.30	−0.73	−2.03	−0.86
(Di − Ei)/J		0.15		0.59		1.97
2|Ei|/J	0.71		1.24		1.69
J	−5.4		3.7		2.4

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As we know, when all D_i/J values are larger than 4/3 in one SCM for homo-spin chains with collinear anisotropy axes, sharp domain walls are present with a creation energy given by Δ_ξ = 4|J|S₁S₂.²¹ For our investigated hetero-spin MRe^IV(CN)₂ SCMs, assuming that the anisotropy axes on Re^IV and M are parallel, we deduced that when D_i/J is larger than 2.5 for Re^IV and 5/6 for Mn^II of 1, 1 for Re^IV and 2 for Ni^II of 2, 1.5 for both of Re^IV and Co^II of 3, sharp domain walls are present according to Barbara and co-workers' theory (Δ_A > 1/3Δ_ξ).²² Since the rhombic anisotropy of M will shortcut the energy barrier, we simply use D_i − E_i to replace D_i to define the ratio (D_i − E_i)/J. For Re^IV in three SCMs, however, their D values are all positive. Thus, we use 2|E_i|/J to define the ratio for the Re^IV fragments (for an easy-plane system, the anisotropy energy associated with the reversal of a single Re^IV spin within a chain is given by Δ_A = 2|E|S² (E = (D_x − D_y)/2)).²

From the 2|E_i|/J and (D_i − E_i)/J values of Re^IV and M in Table 3, three SCMs have different conditions. For 1, the 2|E_i|/J and (D_i − E_i)/J values of Re^IV and Mn^II are smaller than 2.5 and 5/6, respectively. For 2, the 2|E_i|/J of Re^IV is larger than 1, while the (D_i − E_i)/J of Ni^II is smaller than 2. For 3, however, both 2|E_i|/J and (D_i − E_i)/J values of Re^IV and Co^II are larger than 1.5. To explore the origin of the magnetic anisotropies, we firstly gave the general easy plane and axis on the Re^IV and M fragments of 1–3 in Fig. 6.

Fig. 6 Alignment of the local easy plane and hard axis on the Re^IV (Re^IVCl₄(CN)₂Zn₂), and the local easy axis on the M ((DMF)₄M(CN)₂Ba₂) fragments of 1–3.

In three MRe^IV(CN)₂ SCMs, the easy planes on Re^IV or the easy axes on M are strictly parallel. Fig. 6 shows that the easy axes on M do not lie in the easy planes on Re^IV for three SCMs. The included angles between the easy axes on M and the hard axes on Re^IV for 1–3 are 62.2°, 65.6° and 67.1°, respectively.

For 1, due to the very small magnetic anisotropy energy barrier of Mn^II, it can be approximately regarded as an isotropic ion, and so the spins on Mn^II will be orientated to the direction of the spins on Re^IV under the strong Re^IV–Mn^II antiferromagnetic coupling interactions. When the spins on one Re^IV begin to flip, the spins on the neighboring Mn^II flip simultaneously, and those of Re^IV on the other side also flip following them due to its small (D_i − E_i)/J value. When the spins of Re^IV on the other side start to flip, the exchange interaction on the left Re^IV from the central Mn^II is the largest (see Fig. 7).

Fig. 7 Alignment of the spins on one Re^IV (left or front), the central Mn^II, and the other Re^IV (right or behind) when the spins on the other Re^IV begin to flip.

At this moment, the spins on the central Mn^II, the other neighboring Re^IV and the spins on the left Re^IV have the directions shown in Fig. 7 (the weak next neighboring exchange interactions are omitted). When the Δ_A/Δ_ξ of the right Re^IV is smaller than 1/3, the spins on it begin to flip. Thus, we deduced that when the included angle between the spins on the right Re^IV and the central Mn^II is close to 64.1°, the spins on the right Re^IV start to flip (due to the mixing of the magnetic anisotropy combined with the strong exchange coupling for Re^IV, it has much more quantum states than the normal spin states). In conclusion, the total energy barrier Δ_τ of 1 is closely equal to the creation energy of flipping the spins on one Re^IV, which needs to overcome the anisotropic energy barrier Δ_A, 2|E|S², and the correlation energy barrier Δ_ξ coming from the exchange interaction on the left Re^IV from the central Mn^II, which is about 2JS_ReS_Mn(cos 0 − cos 64.1) (64.1° is the included angle between the spins on the right Re^IV and the central Mn^II when the spins on the right Re^IV start to flip). The calculated and experimental Δ_A, Δ_ξ and Δ_τ values are shown in Table 4.

Table 4 Calculated and experimental Δ_A, Δ_ξ, and Δ_τ values of 1 (cm⁻¹)

	Δ A		Δ ξ		Δ τ
	Cal.	Exp.^1	Cal.	Exp.^1	Cal.	Exp.^1
1	7.6	12.0	22.8	19.0	30.4	31.0

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Since one domain wall contains more than one unit cell of Re^IVCl₄(CN)₂(DMF)₄ due to its 2|E_i|/J being much smaller than 2.5, the Δ_A obtained only from the fragment of Re^IV is inappropriate. Thus, the calculated Δ_A of 1 is smaller than the experimental one in Table 4. The calculated Δ_ξ is a little larger than the experimental one since the weak exchange interaction from the next neighboring Mn^II on the central Mn^II is omitted by us. The apparently better Δ_τ value results from the error cancellation in the calculations of the Δ_A and Δ_ξ. Considering the complex of the constitution of the domain wall, further work should be done to explore it thoroughly.

For 2, one unit cell of Re^IVCl₄(CN)₂(DMF)₄ can be regarded as a domain wall since its 2|E_i|/J value is larger than 1, while the (D_i − E_i)/J value of Ni^II is far smaller than 2. When the spins on one Re^IV were flipped by 180°, the ones on the neighboring Ni^II will not be of the same direction to those of Re^IV due to the exchange interaction from the other side Re^IV. Thus, the spins on the central Ni^II will be orientated to the direction perpendicular to its original one under the competing interactions of the two neighboring Re^IV around Ni^II when the spins on one Re^IV were completely flipped to the opposite direction (see Fig. 8).

Fig. 8 Alignment of the spins on one Re^IV (left or front), the central Ni^II, and the other Re^IV (right or behind) when the spins on the left or front Re^IV completely flip to the opposite direction.

Different from 1, the spins on the right Re^IV of 2 will remain still when the spins on the left Re^IV were completely flipped to the opposite direction since its 2|E_i|/J value is larger than 1. Thus, the total energy barrier Δ_τ of 2 is closely equal to the sum of the magnetic anisotropy energy barrier of one Re^IV, 2|E|S², and the correlation energy barrier Δ_ξ, which is closely equal to half of 4JS_ReS_Ni (see Fig. 8). The calculated and experimental Δ_A, Δ_ξ and Δ_τ of 2 are shown in Table 5.

Table 5 Calculated and experimental Δ_A, Δ_ξ and Δ_τ values (cm⁻¹) of 2

	Δ A		Δ ξ		Δ τ
	Cal.	Exp.^1	Cal.	Exp.^1	Cal.	Exp.^1
2	9.2	11.2	11.1	8.8	20.3	20

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From Table 5, the calculated Δ_A is a little smaller than that of the experiment since the weak contribution of the Ni^II is omitted, while the calculated Δ_ξ is a little larger because the weak next neighboring Ni^II–Ni^II exchange coupling is not considered. Thus, the obtained Δ_τ is fortunately close to the experimental one.

For 3, since the 2|E_i|/J and (D_i − E_i)/J values of Re^IV and Co^II in Table 3 are all larger than 1.5, it can be regarded as a spin-canted system. Considering the nonparallel magnetic axes on Re^IV and M for 3, the criterion ratio Δ_A/Δ_ξ should be smaller than 1.5. Thus, it is further confirmed that 3 can be closely regarded as a spin-canted system. The alignment of the local ground magnetic axes on the Re^IV (Re^IVCl₄(CN)₂Zn₂) and Co^II (Co^IIR(CN)₂Ba₂) fragments was shown in Fig. 9.

Fig. 9 Alignment of the local ground magnetic axes on the Re^IV (Re^IVCl₄(CN)₂Zn₂) and Co^II (Co^IIR(CN)₂Ba₂) fragments.

For the Re^IV fragment, the ground spins were orientated to the direction of the g_x axis since its D is positive and E is negative. Thus, the included angle θ of the ground spins on the Re^IV and Co^II of 3 is 63.1°, and so the calculated Δ_ξ = 4JS_ReS_Co cos 63.1 (using the experimental J value) is 9.8 cm⁻¹, which is close to the experimental value (see Table 6).

Table 6 Calculated and experimental Δ_A, Δ_ξ and Δ_τ of 3

Δ A		Δ ξ		Δ τ
Cal.	Exp.^1	Cal.	Exp.^1	Cal.	Exp.^1
8.1, 9.5	8.5	9.8	8.5	17.9, 19.3	17.0

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From Table 6, the magnetic anisotropy energy barriers Δ_A of Re^IV and Co^II are 8.1 cm⁻¹ and 9.5 cm⁻¹, respectively, both of which are close to the experimental one. A little larger Δ_A of Co^II is due to the overestimated magnetic anisotropy energy barrier of Co^II using 2|D|, which is usually larger than the effective energy barrier for the mixing of the ground and excited levels.²³ Moreover, the calculated Δ_ξ is a little larger since 3 could not be completely regarded as a spin-canted system for the D of Co^II and the E of Re^IV being not far larger than J (D ≫ J and E ≫ J).

Contrary to the Fe^IIRe^IV(CN)₂ SCM,^1,3 the total energy barriers Δ_τ of 1 and 2 mainly come from the contribution of the transverse anisotropy of Re^IV, while the Mn^II and Ni^II ions only transmit the exchange couplings between Re^IV and almost have no contributions. For 3, however, both of Re^IV and Co^II ions have contributions to the total energy barrier.

Conclusions

Hybrid functional and complete active space second-order perturbation theory were used to explore the origin of the relaxation barriers and the Δ_ξ and Δ_A in terms of D, J, and S in a series of MRe^IV(CN)₂ single-chain magnets. Calculations show that the antiferromagnetic Re^IV–Mn^II coupling interactions decrease with the increase of the Mn–N–C angle, while the ferromagnetic Re^IV–Ni^II coupling interactions increase with the increase of the Ni–N–C angle. However, the variation of the Re^IV–Co^II exchange interactions with the Co–N–C angle ranging from 140° to 180° is very small.

The 2|E_i|/J and (D_i − E_i)/J values of Re^IV and M of three complexes in Table 3 are so much different. For 1, the total energy barrier Δ_τ is closely equal to the creation energy of flipping the spins on one Re^IV, which needs to overcome the anisotropic energy barrier Δ_A, 2|E|S², and the correlation energy barrier Δ_ξ coming from the exchange interaction on the left Re^IV from the central Mn^II, which is about 2JS_ReS_Mn(cos 0 − cos 64.1). For 2, the total energy barrier Δ_τ is closely equal to the sum of the magnetic anisotropy energy barrier of one Re^IV, 2|E|S², and the correlation energy barrier Δ_ξ, 2JS_ReS_Ni. For 3, it can be approximately regarded as a spin-canted system, and thus the Δ_ξ is equal to 4JS_ReS_Cocos 63.1 and the Δ_A is equal to the magnetic anisotropy energy barrier of one Re^IV or Co^II.

The above results show that simply enhancing the Re^IV–M coupling is not always effective to increase the total energy barriers of MRe^IV(CN)₂. To obtain the high energy barrier for single-chain magnets, we must enhance the magnetic anisotropy energy barriers of single magnetic ions and the intramolecular coupling at the same time.

Acknowledgements

This project is funded by the Natural Science Foundation of Jiangsu Province of China (BK2011778), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4nj01491j

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