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Exchange Coupling and Magnetic Anisotropy in a Family of Bipyrimidyl Radical-Bridged Dilanthanide Complexes: Density Functional Theory and Ab Initio Calculations Yi-Quan Zhang,* Cheng-Lin Luo, and Qiang Zhang The origin of the magnetic anisotropy energy barriers in a series of bpym2 (bpym 5 2,20 -bipyrimidine) radical-bridged dilanthanide complexes [(Cp*2Ln)2(l-bpym)]1 [Cp* 5 pentamethylcyclopentadienyl; Ln 5 GdIII (1), TbIII (2), DyIII (3), HoIII (4), ErIII (5)] has been explored using density functional theory (DFT) and ab initio methods. DFT calculations show that the exchange coupling between the two lanthanide ions for each complex is very weak, but the antiferromagnetic Ln-bpym2 couplings are strong. Ab initio calculations show that the effective energy barrier of 2 or 3 mainly
comes from the contribution of a single TbIII or DyIII fragment, which is only about one third of a single Ln energy barrier. For 4 or 5, however, both of the two HoIII or ErIII fragments contribute to the total energy barrier. Thus, it is insufficient to only increase the magnetic anisotropy energy barrier of a single Ln ion, while enhancing the Ln-bpym2 C 2014 Wiley Periodicals, couplings is also very important. V Inc.
Introduction
the optimized structures for 4 and 5 can be found in Supporting Information.
In the past decade, much work has been performed to pursue the single-molecule magnets (SMMs) with high energy barrier.[1–4] Recently, Long and coworkers reported a series of N32 2 radical-bridged dilanthanide complexes [{[(Me3Si)2N]2(THF)Ln}2 (l-g2:g2-N2)]2 (Me 5 methyl; THF 5 tetrahydrofuran; Ln 5 GdIII, TbIII, DyIII, HoIII, ErIII).[5,6] Among them, the blocking temperature of the terbium complex is up to 13.9 K, which is the highest one discovered to date, although its relaxation barrier (227 cm21) is not the largest one.[6] A series of calculations on them have shown that both of the intramolecular coupling interactions and the single Ln anisotropies have important contributions to the total relaxation barrier.[7] Compared to the N32 radical-bridged dilanthanide com2 plexes,[5,6] the bpym2 (bpym 5 2,20 -bipyrimidine) radical bridging the two Ln ions in a series of complexes [(Cp*2Ln)2(lbpym)]1 [Cp* 5 pentamethylcyclopentadienyl; Ln 5 GdIII (1), TbIII (2), DyIII (3)] reported recently by Long and coworkers[8] is more complex, and the Ln-bpym2 couplings are so much different from those of Ln-N32 2 . Moreover, the effective energy barrier of Tb2bpym2 is far smaller than that of Dy2bpym2, while complex TbN32 2 has the largest energy barrier. To understand it, density functional theory (DFT) and ab initio calculations on these Ln2bpym2 systems have been performed by us. The general structure of complexes 1–3 is shown in Figure 1. A detailed description of the structures can be found in Ref. [8]. To extend our research, we first substituted two DyIII of complex [(Cp*2DyIII)2(l-bpym)]1 using HoIII and ErIII, respectively, to obtain models [(Cp*2HoIII)2(l-bpym)]1 (4) and [(Cp*2ErIII)2(l-bpym)]1 (5), and then, optimized the structures of 4 and 5 using DFT (see the following computational details). The optimized structures of 4 and 5 are very similar to that of [(Cp*2DyIII)2(l-bpym)]1. The Cartesian coordinates of
DOI: 10.1002/jcc.23565
Computational Details Similar to the Ln2N32 compounds, the Ising exchange cou2 0 pling constants J~ex and J~0 ex , and the isotropic Jex and J ex of complexes 2–5 also have the relationships (assuming the orientations of the local anisotropy axes on Ln1 and Ln2 are strictly parallel)[9–11]: J~ex 54SLn Srad Jex J~0 ex 54SLn1 SLn2 J0ex
(1)
where SLn1 , SLn2 , and Srad are the spin operator of the two Ln ions and of the bpym2 radical, respectively, (Srad 5 1/2). Based on eq. (1), Chibotaru and coworkers[9–13] have successfully obtained the Ising exchange couplings of Ln–Ln and Ln-transition metal systems using ab initio method combined with Lines model.[14] For Ln-radical systems, ab initio method combined with Lines model is difficult to give Ln-radical coupling as radical has more than one magnetic center ions. However, DFT methods are very effective to calculate the isotropic exchange coupling constant in the absence of spin-orbit coupling.[15,16] Thus, we can first use DFT to calculate the isotropic exchange coupling constant, and then, use eq. (1) to obtain Y.-Q. Zhang, C.-L. Luo, Q. Zhang Physical Department, Jiangsu Key Laboratory for NSLSCS, School of Physical Science and Technology, Nanjing Normal University, Nanjing 210023, People’s Republic of China E-mail: [email protected] Contract/grant sponsor: Natural Science Foundation of Jiangsu Province of China; contract/grant number: BK2011778; Contract/grant sponsor: China Postdoctoral Science Foundation funded project; contract/grant number: 2012M520104. C 2014 Wiley Periodicals, Inc. V
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Figure 1. General structure of complexes [(Cp*2Ln)2(l-bpym)]1 [Ln 5 GdIII (1), TbIII (2), DyIII (3)]; H atoms have been omitted for clarity. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
the Ising exchange one. In a recent work,[17] Chibotaru and coworkers succeeded in using the above approach to obtain the Ln–Ln Ising exchange coupling constants. Our previous work has also shown that DFT combined with eq. (1) in the calculation of the Ln-radical couplings is reasonable and effec[7] tive for the Ln2N32 2 compounds. To optimize the structures of complexes 4 and 5 and calcu0 late the isotropic exchange coupling constants Jex and J ex of [18] complexes 1–5, Orca 2.9.1 calculations were performed with the popular hybrid functional B3LYP proposed by Becke[19,20] and Lee et al.[21] Triple-f with one polarization function TZVP[22,23] basis set was used for all atoms, and the scalar relativistic treatment (ZORA) was used in all calculations. The large integration grid (grid 5 6) was applied to Ln for ZORA calculations. Tight convergence criteria were selected to ensure that the results are well converged with respect to technical parameters. The spin-unrestricted procedure was used in all of the above calculations. Through calculate the energies of three spin states: the highspin state (SHS 5SLn1 1Srad 1SLn2 ), the first low-spin state (flip the spin on bpym2 radical; SLS1 5SLn1 2Srad 1SLn2 ), and the second low-spin state (flip the spin on one Ln ion: SLS2 5SLn1 1Srad 2SLn2 ) (the obtained low-spin state is not the pure spin state, which is a broken-symmetry one mixed by the high- and low-spin states), we used the spin-projected approach[24–26] to obtain the Ln0 bpym2 and Ln–Ln coupling constants Jex and J ex (Srad 51=2; SLn1 5 SLn1 5SLn2 ). The equations are as follows: ELS1 2EHS 4ðSLn 11=2Þ
(2)
ðELS2 2ELS1 Þ12Jex 4SLn 2 12SLn
(3)
Jex 5 0
J ex 5
Complete-active-space self-consistent field (CASSCF) calculations on individual lanthanide fragments have been carried out with MOLCAS 7.8 program package.[27] For each complex, we have used two structural approximations: A, a reduced fragment of the molecule; B, the complete entire molecule (see Fig. 2). For A, the only removed atoms are those connected to the Ln from the opposite side of the molecule, and we used the closed-shell La31 ab initio embedding model potentials (La.ECP.deGraaf.0s.0s.0e-La(LaMnO3.)[28] to take into account the influence of the neighboring Ln ion. For B, one of the Ln ions is replaced by the diamagnetic LuIII ion. In all calculations 2
Journal of Computational Chemistry 2014, DOI: 10.1002/jcc.23565
on individual lanthanide fragment, each bpym2 radical was treated as bpym which has a zero spin. For CASSCF calculations, the basis sets for all atoms are atomic natural orbitals from the MOLCAS ANO-RCC library: ANO-RCC-VTZP for Ln ions; VTZ for close C and N; VDZ for distant atoms. The calculations used the second-order Douglas–Kroll–Hess Hamiltonian, where scalar relativistic contractions were taken into account in the basis set. The active electrons in seven active orbitals include all f electrons in all calculations. First, we calculated all possible roots available for a given active space for each spin multiplet using state-averaged CASSCF method, and then mixed the maximum number of spin-free state which was possible with our hardware in the following spin-orbit coupling calculation by the RASSI module. For the Tb fragment, we mixed the roots coming from the following spin-free states: all from seven septets; all from 140 quintets; 68 from 500 triplets. Similarly, 21 sextet spin states, 128 quadruplet spin states, and 130 doublet spin states were mixed for the Dy fragment. For the Ho fragment, 35 quintet spin states, 150 triplet spin states, and 120 singlet spin states were mixed. For the Er fragment, 35 quartet spin states and 112 doublet spin states were mixed.
Results and Discussions Evaluation and comparison of the exchange interactions The Cartesian coordinates of the optimized structures of Ho2bpym2 and Er2bpym2 are in the Supporting Information. During each optimization, the structure of the bpym2 has been constrained. The optimized molecular geometries of 4–5 using B3LYP are very similar to those of 1–3. Only the Ln-N
Figure 2. Structures of the calculated Ln fragments A (left) and B (right) for each complex (Ln 5 TbIII, DyIII, HoIII, and ErIII)
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’ values (cm21) of five complexes [(Cp* Ln) (l-bpym)]1. Table 1. Ground-state spin multiplets, and the calculated and experimental Jex and Jex 2 2
2S11 Jex J’ ex
Gd2bpym2 (1)
Tb2bpym2 (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
14 218.3 (210.0[a]) 21.2
12 218.9 21.6
10 221.9 21.1
8 227.1 21.5
6 227.3 21.4
[a] The result was obtained from Ref. [8].
distances have a little difference within 0.05 A˚. The calculated 0 Jex and J ex of five complexes [(Cp*2Ln)2(l-bpym)]1 using B3LYP functional are shown in Table 1. 0 The calculated Jex and J ex values in Table 1 indicate that the Ln-bpym2 and Ln–Ln couplings are all antiferromagnetic and the Ln-bpym2 couplings are so much stronger than those of Ln–Ln. The experimental fitting also shows that the Ln-bpym2 and Ln–Ln couplings of complexes 1–3 are all antiferromagnetic.[8] To investigate the Ln-bpym2 couplings, we first gave the calculated spin values on each Ln ion and bpym2 radical of five complexes in their low-spin states (see Table 2). From the spin populations in Table 2, we only know the spins on Ln and bpym2 are almost localized. Thus, we further gave the spin density distribution map of complex Tb2bpym2 in the first low-spin state, which is shown in Figure 3 (the other complexes have the similar spin density distributions which are shown in Supporting Information Figure S1), where the spins on TbIII are almost localized, and those on bpym2 are a little more diffused although the total spin population on bpym2 is near to 21.0, but not delocalize onto the TbIII ions. Moreover, the spin densities on bpym2 have a larger space distribution around the central carbon atoms which have longer distances from Ln compared to those between Ln and N32 2 . The strength of the calculated Ln-bpym2 exchange interactions shown in Table 1 follows the experiment trend: GdIII [5,6] TbIII < DyIII,[8] which is opposite to that of Ln2N32 2 systems. As usual, the Ln–Ln couplings mainly caused by spin polarization are usually very weak for the contracted nature of the Ln 4f orbitals. From Figure 3, however, the spin densities on bpym2 have a much larger space distribution around the central carbon atoms which are far from the Ln ions, which results in the small overlap between the magnetic orbitals on Ln and bpym2, so as to the weaker Ln-bpym2 antiferromagnetic couplings compared to the strong Ln–N32 interactions according 2 to Kahn’s theory.[29] Thus, contrary to the Ln2N32 2 systems, the spin polarization of the Ln ions may dominate the Ln-bpym2 exchange couplings due to the weak magnetic orbital interac-
tions. However, B3LYP has no way to accurately evaluate the contributions of the weak overlap between Ln and bpym2, and the spin polarization of Ln to the Ln-bpym2 exchange coupling, so as to the larger calculated Ln-bpym2 exchange coupling constants than the experimental ones. As usual, DyIII would have more contracted valence orbitals than GdIII or TbIII due to its slightly smaller ionic radius, and so the overlap between the magnetic orbitals on DyIII and bpym2 is smaller. Based on the above analysis, however, the spin polarization of Ln ions dominates the Ln-bpym2 exchange couplings. From Table 1, the ratio of the spin population on the 4f orbitals to the total one is the largest for DyIII among complexes 1–3, which indicates that the spin polarization involving the empty 5d orbitals of DyIII is the strongest.[8] Thus, the DyIII-bpym2 antiferromagnetic exchange interactions are stronger than those of GdIII-bpym2 or TbIII-bpym2. For the same reason, the HoIII-bpym2 antiferromagnetic exchange interactions are the largest among complexes 1–4. It is unexpected that the ErIII-bpym2 antiferromagnetic coupling is a little larger than that of HoIII-bpym2. The previous article[7] has shown that the ErIII-N32 couplings are ferromag2 netic due to the very small overlap between the magnetic orbitals on ErIII and N32 for the smallest ionic radius of ErIII. 2 Similarly, the ferromagnetic contribution of the magnetic orbital exchange interactions to the ErIII-bpym2 coupling are larger than the antiferromagnetic one arising from the overlap. Thus, if not considering the spin polarization of ErIII, the ErIIIbpym2 coupling should be ferromagnetic. However, the magnetic orbital exchange interactions are much weak in Er2N32 2 system as the distances between ErIII and the central carbon atoms of bpym2 are so much long. Thus, the spin polarization of ErIII favoring the antiferromagnetic contribution to the ErIIIbpym2 coupling overwhelms the competing magnetic orbital exchange interactions, which leads to the ErIII-bpym2 antiferromagnetic coupling. Besides, a little larger ErIII-bpym2 antiferromagnetic coupling compared to that of HoIII-bpym2 is attributed to the opposite contributions of the spin
Table 2. Total spin populations on Ln and bpym2, and the individual spin populations for each of the 4f and 5d orbital levels of Ln1 and Ln2 of five complexes in the first low-spin state.
Ln1
Ln2
Bpym2
5d 4f total 5d 4f total
Gd2bpym2 (1)
Tb2bpym2 (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
0.134 6.934 7.104 0.134 6.934 7.104 20.992
0.103 5.942 6.065 0.104 5.942 6.068 20.986
0.081 4.945 5.040 0.081 4.941 5.039 20.974
0.061 3.947 4.020 0.061 3.947 4.020 20.965
0.043 2.950 3.000 0.041 2.954 2.999 20.960
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Figure 3. Spin density distribution map of complex Tb2bpym2 in the first low-spin state (blue and red regions indicate positive and negative spin populations, respectively; the isodensity surface represented corresponds to a value of 0.002 e2 bohr23). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
polarization of ErIII and the exchange interactions between the magnetic orbitals on ErIII and bpym2.
Ab initio calculations on mononuclear Ln fragments The orientations of the local magnetic axes, the g tensors, and the lowest spin-orbit multiplets of mononuclear Ln fragments (see Fig. 2) of complexes 2–5 were calculated using CASSCF. Considering the high-symmetry environment around two Ln ions for each complex, the local anisotropy axes of Ln fragments using model B in Figure 2b for each of complexes 2–5 are almost parallel. For the similar orientations of the local magnetic axes on the Tb, Dy, and Ho fragments, we only gave one of them in Figure 4 (a for Tb; b for Er). Figure 4 shows that the local magnetic axes on the Er fragments of complex Er2bpym2 have the different orientations compared to those of the other complexes. Moreover, the local magnetic moments in Figure 4 should be parallel because the Ln-bpym2 couplings are so much stronger than the Ln–Ln coupling for each complex. The calculated g tensors of the lowest effective S~ on the Tb, Dy, Ho, and Er fragments using the program SINGLE_ANISO[30] are given in Table 3. From Table 3, the calculated g values of models A and B for each of 2–4 are almost the same, while they have a little difference for 5. The much larger values of gz on TbIII, DyIII, HoIII, and ErIII ions than gx and gy indicate that the Ln-bpym2 exchange interactions will be of the Ising type. The CASSCF energies of the lowest multiplets corresponding to each spin-orbit coupling ground state of four Ln fragments are listed in Table 4. In the above B3LYP calculations of the Ln-bpym-exchange couplings, the strong 4f–5d interactions may have an important
influence on the couplings. In the CASSCF calculations, however, we did not consider the influence of the strong 4f–5d interactions on the spin-orbit coupling, although we do not know whether such influence is important. If we extend the active space to a larger one to include the 5d ions, the possible roots available for each spin multiplet (in the state-averaged CASSCF calculation, we should include all the available roots for each multiplet) will exceed the capacity of CASSCF and our hardware. As each Kramers doublet are degenerate for the Dy or Er fragment, we only gave one energy level for each doublet in Table 4. But, we gave all the non-Kramers states for the Tb or Ho fragment. As the thermal energy is much larger than the small splitting between the lowest two energy levels of the Tb or Ho fragment, the two levels can be supposed as the doubly degenerated ground states. If the Orbach relaxation process is dominant, the energy barrier is related to the separation DEi between the lowest two doublets of a single Ln fragment. Therefore, the energy separation of the lowest two non-Kramers states of the Tb fragment is 144.4 cm21 for model A and 154.3 cm21 for model B, and that of the Ho fragment is 68.4 cm21 for A and 54.3 cm21 for B. For each of the Dy and Er fragments, however, as the lowest and the second lowest Kramers doublets are both degenerate, the energy separations of the lowest two Kramers doublets of the Dy and Er fragments are 348.0 cm21 for A, 315.5 cm21 for B and 64.4 cm21 for A, 58.7 cm21 for B, respectively. Although, the energy separation DEi (144.4 cm21) between the lowest two Kramers doublets of the Tb fragment is a little [7] larger than that of the Tb fragment (133.0 cm21) of Tb2N32 2 , why is the effective energy barrier Ueff of complex Tb2bpym2 (44.0 cm21) far smaller than that of Tb2N32 (227.0 cm21)? 2 2 Moreover, why is the Ueff of Dy2bpym only 87.8 cm21 although the single Dy energy barrier is up to 348.0 cm21 far larger than that of the Dy fragment in Dy2N32 2 ? Next section will present the analysis of the above results combined single Ln ion anisotropies with Ln-bpym2 exchange couplings. Origin of the energy barrier For transition-metal systems, we can obtain the total D value through project the single-site anisotropies (Di) onto the spin ground state S.[31] X 2Si 21 D5 di D i di 5 (4) Nð2NSi 21Þ i where i numbers the N metal centers and the di values are projection coefficients. Equation (4) is derived in the strongexchange limit for a spin cluster. The anisotropic interaction tensor Dij is ignored as dipolar and anisotropic interactions
Table 3. Main values of the g tensors of the lowest effective S~ on the Tb, Dy, Ho, and Er fragments. Tb (2)
gx gy gz
4
Dy (3)
Ho (4)
Er (5)
A
B
A
B
A
B
A
B
0.000 0.000 17.799
0.000 0.000 17.792
0.002 0.003 19.940
0.002 0.003 19.867
0.000 0.000 17.574
0.000 0.000 17.952
1.103 1.571 14.281
0.107 0.457 15.137
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Table 4. CASSCF energies of the lowest Kramers doublets of 3 and 5 and non-Kramers states or singlet (mJ 5 0) of 2 and 4 with two models of A and B (cm21). Tb(2)
Dy (3)
A
B
0.000 0.090 144.432 145.449 319.311 322.548 462.161 509.295 575.734 741.474 755.648 876.330 878.406
0.000 0.044 154.311 154.537 324.399 334.222 464.596 538.245 591.711 805.265 815.243 999.758 1001.208
Ho (4)
Er (5)
A
B
A
B
A
B
0.000 347.994 497.794 568.741 640.769 717.273 806.296 1097.127
0.000 315.534 478.236 545.702 611.025 680.983 789.798 1100.177
0.000 0.362 68.393 72.163 154.636 169.219 201.906 231.269 238.763 284.618 329.325 347.987 388.215 414.957 418.099 487.581 490.515
0.000 0.668 54.292 56.261 115.790 124.382 135.270 183.943 192.189 261.248 276.162 305.660 321.882 375.734 376.198 444.239 446.363
0.000 64.415 118.940 202.748 255.332 310.317 417.914 485.764
0.000 58.704 138.865 186.109 265.528 313.912 422.299 438.049
yield only minor contributions. For most of lanthanide-metal SMMs, the energy barrier comes from a single Ln ion due to the very weak Ln–Ln exchange coupling. But, for radicalbridged dilanthanide complexes, the strong interactions of the two Ln ions transmitted by the radical may make both Ln ions contribute to the total energy barrier. A recent work[7] has successfully given the relationship between the total energy barrier DE with separated single Ln barriers DEi (the energy separation between the lowest two doublets). Similarly, if assuming that all complexes are in the strong-exchange limit, the approximate relationships between the total energy barrier DE and the single Ln barrier DEi for each complex are as follows.
Figure 4. Orientations of the local magnetic axes on the Tb (a) and Er (b) fragments.
DE51:87DEi for Tb2 N32 2
(5)
DE51:94DEi for Dy2 N32 2
(6)
DE51:90DEi for Ho2 N32 2
(7)
Er2 N32 2
(8)
DE51:94DEi for
To assess the relative strength between magnetic anisotropy and exchange interaction for the investigated bpym2 radicalbridged dilanthanide complexes, we also use the ratio Ecorr/DEi where Ecorr represents the exchange correlation energy (Ecorr 5 4jJexjSLnSrad) which is often used in single-chain magnets to describe the creation energy of one domain wall.[32] The calculated single Ln barrier DEi of model B, the total energy barrier DE according to eqs. (5–8), the Ln-bpym2 exchange correlation energy Ecorr, the experimental Ueff, and the ratios Ecorr/DEi and DEi/Ueff for each of 2–5 are shown in Table 5. Table 5 shows that the exchange correlation energies Ecorr of 2 and 3 are both far smaller than the corresponding single Ln anisotropy energy barrier DEi, and thus, the effective energy barriers Ueff mainly come from the single Cp*2Ln(l-bpym)La fragments. Our previous work[7] has shown that the total energy barrier of the Dy2N22 or Er2N32 system is about one 2 2 third of a single Dy or Er ion energy barrier DEi as the Ecorr is far smaller than the DEi for each of them. It is expected that the Ueff values of 2 and 3 are also close to one third of the calculated single Tb and Dy fragment energy barriers, respectively. From observing the DEi/Ueff values in Table 5, it is obvious that eqs. (5) and (6) are invalid for 2 and 3 as they are not in the strong-exchange limit. For 4 or 5, however, as the correlation energy Ecorr is much larger than the DEi, we deduced that the Ueff values of 4 and 5 will be close to the DE obtained using eqs. (7) and (8), respectively. The above results indicate that the relatively weaker Tbbpym2 exchange interactions lead to the Ueff value of Tb2bpym2 being far smaller than that of Tb2N32 2 . Although, the DEi of a single Dy fragment for 2 is up to 315.5 cm21, the Journal of Computational Chemistry 2014, DOI: 10.1002/jcc.23565
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Table 5. Calculated single Ln barrier DEi, the total energy barrier DE, the Ln-bpym2 exchange correlation energy Ecorr, the ratios Ecorr/DEi and DEi/ Ueff for each complex, and the experimental effective energy barrier Ueff of 2–5 (cm21).
Tb2bpym (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
154.3 288.5 113.4 44.0 0.73 6.6
315.5 612.1 109.5 87.8 0.35 7.0
54.3 103.2 108.4
58.7 113.9 81.9
2.0
1.40
2
DEi DE Ecorr Ueff Ecorr/DEi DE/Ueff
Ueff value is much smaller than that of Dy2N32 2 , which is attributed to its far smaller Ecorr/DEi value. Due to the larger Ecorr/ DEi values of 4 and 5, the two HoIII or ErIII ions both contribute to the total energy barrier. From the above analysis, we conclude that enhancing Ln-radical exchange interactions will be especially important to increase the total energy barrier, rather than only increase the single Ln-ion energy barrier. As the current molecular magnetism theory is far from perfect, and the calculation methods are not accurate enough, the relationships between the intramolecular exchange interactions, the single-ion anisotropies, and the total effective energy barriers for lanthanide-metal SMMs have no way to be clearly interpreted by us. Some problems, what’s the accurate value of Ecorr/DEi to make both Ln ions contribute to the total energy barrier, whether the above conclusions can also be applied to other dilanthanide systems and so forth are waiting to be clarified. We hope this work will be an exploration on the origin of the slow relaxation of the magnetization which is the most important one in present-day molecular magnetism.
Summary B3LYP and CASSCF methods were used to calculate the Lnbpym2 exchange couplings and the single Ln energy barrier to explore the origin of the total magnetic anisotropies for five Ln2bpym2 SMMs. B3LYP calculations show that the Ln–Ln couplings are very weak, but the Ln-bpym2 couplings are relatively stronger. Moreover, due to the predominant strongest spin polarization involving the empty 5d orbitals of ErIII, the ErIII-bpym2 antiferromagnetic exchange interactions are the strongest among five Ln2bpym2 systems. CASSCF calculations on four Ln fragments indicate that the single Ln anisotropy energy barriers DEi of 2 and 3 are far larger than those of 4 and 5, but their effective energy barriers are much smaller. Moreover, the Ueff values of 2 and 3 are also close to one third of the calculated single Tb and Dy fragment energy barriers, respectively, and those of 4 and 5 are close to the DE obtained using eqs. (7) and (8). Thus, to enhance the total energy barrier, it needs to enhance the Ln-bpym2 couplings when improves the single Ln fragment barrier at the same time.
6
Journal of Computational Chemistry 2014, DOI: 10.1002/jcc.23565
Keywords: single-molecule magnet energy barrier exchange coupling complete-active-space self-consistent field B3LYP
How to cite this article: Y. Q. Zhang, C. L. Luo, Q. Zhang. 2014. J. Comput. Chem., DOI: 10.1002/jcc.23565
]
Additional Supporting Information may be found in the online version of this article.
[1] J. D. Rinehart, J. R. Long, Chem. Sci. 2011, 2, 2078. [2] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. Sorace, P. Sainctavit, M.-A. Arrio, E. Otero, L. Joly, J. C. Cezar, A. Cornia, R. Sessoli, Nature 2010, 468, 417. [3] R. Sessoli, A. K. Powell, Coord. Chem. Rev. 2009, 253, 2328. [4] D. Gatteschi, R. Sessoli, J. Villain, Molecular Nanomagnets; Oxford University Press: Oxford, 2006. [5] J. D. Rinehart, M. Fang, W. J. Evans, J. R. Long, Nat. Chem. 2011, 3, 538. [6] J. D. Rinehart, M. Fang, W. J. Evans, J. R. Long, J. Am. Chem. Soc. 2011, 133, 14236. [7] Y. Q. Zhang, C. L. Luo, B. W. Wang, S. Gao, J. Phys. Chem. A 2013, 117, 10873. [8] S. Demir, J. M. Zadrozny, M. Nippe, J. R. Long, J. Am. Chem. Soc. 2012, 134, 18546. [9] L. Chibotaru, L. Ungur, A. Soncini, Angew. Chem. Int. Ed. 2008, 47, 4126. [10] G. Novitchi, G. Pilet, L. Ungur, V. V. Moshchalko, W. Wernsdorfer, L. F. Chibotaru, D. Luneau, A. K. Powell, Chem. Sci. 2012, 3, 1169. [11] J. Long, F. Habib, P.-H. Lin, L. Korobkov, G. Enright, L. Ungur, W. Wernsdorfer, L. F. Chibotaru, M. Murugesu, J. Am. Chem. Soc. 2011, 133, 5319. [12] K. Bernot, J. Luzon, L. Bogani, M. Etienne, C. Sangregorio, M. Shanmugam, A. Caneschi, R. Sessoli, D. Gatteschi, J. Am. Chem. Soc. 2009, 131, 5573. [13] M. L. Kahn, R. Ballou, P. Porcher, O. Kahn, J.-P. Sutter, Chem. Eur. J. 2002, 8, 525. [14] M. E. Lines, J. Chem. Phys. 1971, 55, 2977. [15] E. Ruiz, J. Comput. Chem. 2011, 32, 1998. [16] E. Ruiz, J. Cano, S. Alvarez, P. Alemany, J. Comput. Chem. 1999, 20, 1391. [17] V. Vieru, L. Ungur, L. F. Chibotaru, J. Phys. Chem. Lett. 2013, 4, 3565. [18] F. Neese, ORCA–an ab initio, density functional and semiempirical program package, Version 2.9.1; Max-Planck institute for bioinorganic chemistry: M€ ulheim an der Ruhr, Germany, 2012. [19] A. D. Becke, J. Chem. Phys. 1993, 98, 5648. [20] A. D. Becke, Phys. Rev. A 1988, 38, 3098. [21] C. Lee, W. Yang, R. G. Parr, Phys. Rev. B 1988, 37, 785. [22] A. Schafer, H. Horn, R. Ahlrichs, J. Chem. Phys. 1992, 97, 2571. [23] A. Schafer, C. Huber, R. Ahlrichs, J. Chem. Phys. 1994, 100, 5829. [24] L. Noodleman, J. Chem. Phys. 1981, 74, 5737. [25] L. Noodleman, E. J. Baerends, J. Am. Chem. Soc. 1984, 106, 2316. [26] L. Noodleman, D. A. Case, Adv. Inorg. Chem. 1992, 38, 423. [27] G. Karlstr€ om, R. Lindh, P.-A˚. Malmqvist, B. O. Roos, U. Ryde, V. Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Comput. Mater. Sci. 2003, 28, 222. [28] L. Seijo, Z. Barandiaran, Computational Chemistry: Reviews of Current Trends, Vol. 4; World Scientific, Inc.: Singapore, 1999; pp. 55-152. [29] O. Kahn, Molecular Magnetism; VCH: New York, 1993. [30] L. F. Chibotaru, L. Ungur, The Computer Program SINGLE_ANISO; University of Leuven: Leuven, Belgium, 2006. [31] A. Bencini, D. Gatteschi, Electron Paramagnetic Resonance of Exchange Coupled Systems; Springer: Berlin, 1990. [32] C. Coulon, H. Miyasaka, R. Cl erac, Struct. Bonding 2006, 122, 163.
Received: 14 November 2013 Revised: 30 January 2014 Accepted: 31 January 2014 Published online on 00 Month 2014
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Exchange Coupling and Magnetic Anisotropy in a Family of Bipyrimidyl Radical-Bridged Dilanthanide Complexes: Density Functional Theory and Ab Initio Calculations Yi-Quan Zhang,* Cheng-Lin Luo, and Qiang Zhang The origin of the magnetic anisotropy energy barriers in a series of bpym2 (bpym 5 2,20 -bipyrimidine) radical-bridged dilanthanide complexes [(Cp*2Ln)2(l-bpym)]1 [Cp* 5 pentamethylcyclopentadienyl; Ln 5 GdIII (1), TbIII (2), DyIII (3), HoIII (4), ErIII (5)] has been explored using density functional theory (DFT) and ab initio methods. DFT calculations show that the exchange coupling between the two lanthanide ions for each complex is very weak, but the antiferromagnetic Ln-bpym2 couplings are strong. Ab initio calculations show that the effective energy barrier of 2 or 3 mainly
comes from the contribution of a single TbIII or DyIII fragment, which is only about one third of a single Ln energy barrier. For 4 or 5, however, both of the two HoIII or ErIII fragments contribute to the total energy barrier. Thus, it is insufficient to only increase the magnetic anisotropy energy barrier of a single Ln ion, while enhancing the Ln-bpym2 C 2014 Wiley Periodicals, couplings is also very important. V Inc.
Introduction
the optimized structures for 4 and 5 can be found in Supporting Information.
In the past decade, much work has been performed to pursue the single-molecule magnets (SMMs) with high energy barrier.[1–4] Recently, Long and coworkers reported a series of N32 2 radical-bridged dilanthanide complexes [{[(Me3Si)2N]2(THF)Ln}2 (l-g2:g2-N2)]2 (Me 5 methyl; THF 5 tetrahydrofuran; Ln 5 GdIII, TbIII, DyIII, HoIII, ErIII).[5,6] Among them, the blocking temperature of the terbium complex is up to 13.9 K, which is the highest one discovered to date, although its relaxation barrier (227 cm21) is not the largest one.[6] A series of calculations on them have shown that both of the intramolecular coupling interactions and the single Ln anisotropies have important contributions to the total relaxation barrier.[7] Compared to the N32 radical-bridged dilanthanide com2 plexes,[5,6] the bpym2 (bpym 5 2,20 -bipyrimidine) radical bridging the two Ln ions in a series of complexes [(Cp*2Ln)2(lbpym)]1 [Cp* 5 pentamethylcyclopentadienyl; Ln 5 GdIII (1), TbIII (2), DyIII (3)] reported recently by Long and coworkers[8] is more complex, and the Ln-bpym2 couplings are so much different from those of Ln-N32 2 . Moreover, the effective energy barrier of Tb2bpym2 is far smaller than that of Dy2bpym2, while complex TbN32 2 has the largest energy barrier. To understand it, density functional theory (DFT) and ab initio calculations on these Ln2bpym2 systems have been performed by us. The general structure of complexes 1–3 is shown in Figure 1. A detailed description of the structures can be found in Ref. [8]. To extend our research, we first substituted two DyIII of complex [(Cp*2DyIII)2(l-bpym)]1 using HoIII and ErIII, respectively, to obtain models [(Cp*2HoIII)2(l-bpym)]1 (4) and [(Cp*2ErIII)2(l-bpym)]1 (5), and then, optimized the structures of 4 and 5 using DFT (see the following computational details). The optimized structures of 4 and 5 are very similar to that of [(Cp*2DyIII)2(l-bpym)]1. The Cartesian coordinates of
DOI: 10.1002/jcc.23565
Computational Details Similar to the Ln2N32 compounds, the Ising exchange cou2 0 pling constants J~ex and J~0 ex , and the isotropic Jex and J ex of complexes 2–5 also have the relationships (assuming the orientations of the local anisotropy axes on Ln1 and Ln2 are strictly parallel)[9–11]: J~ex 54SLn Srad Jex J~0 ex 54SLn1 SLn2 J0ex
(1)
where SLn1 , SLn2 , and Srad are the spin operator of the two Ln ions and of the bpym2 radical, respectively, (Srad 5 1/2). Based on eq. (1), Chibotaru and coworkers[9–13] have successfully obtained the Ising exchange couplings of Ln–Ln and Ln-transition metal systems using ab initio method combined with Lines model.[14] For Ln-radical systems, ab initio method combined with Lines model is difficult to give Ln-radical coupling as radical has more than one magnetic center ions. However, DFT methods are very effective to calculate the isotropic exchange coupling constant in the absence of spin-orbit coupling.[15,16] Thus, we can first use DFT to calculate the isotropic exchange coupling constant, and then, use eq. (1) to obtain Y.-Q. Zhang, C.-L. Luo, Q. Zhang Physical Department, Jiangsu Key Laboratory for NSLSCS, School of Physical Science and Technology, Nanjing Normal University, Nanjing 210023, People’s Republic of China E-mail: [email protected] Contract/grant sponsor: Natural Science Foundation of Jiangsu Province of China; contract/grant number: BK2011778; Contract/grant sponsor: China Postdoctoral Science Foundation funded project; contract/grant number: 2012M520104. C 2014 Wiley Periodicals, Inc. V
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Figure 1. General structure of complexes [(Cp*2Ln)2(l-bpym)]1 [Ln 5 GdIII (1), TbIII (2), DyIII (3)]; H atoms have been omitted for clarity. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
the Ising exchange one. In a recent work,[17] Chibotaru and coworkers succeeded in using the above approach to obtain the Ln–Ln Ising exchange coupling constants. Our previous work has also shown that DFT combined with eq. (1) in the calculation of the Ln-radical couplings is reasonable and effec[7] tive for the Ln2N32 2 compounds. To optimize the structures of complexes 4 and 5 and calcu0 late the isotropic exchange coupling constants Jex and J ex of [18] complexes 1–5, Orca 2.9.1 calculations were performed with the popular hybrid functional B3LYP proposed by Becke[19,20] and Lee et al.[21] Triple-f with one polarization function TZVP[22,23] basis set was used for all atoms, and the scalar relativistic treatment (ZORA) was used in all calculations. The large integration grid (grid 5 6) was applied to Ln for ZORA calculations. Tight convergence criteria were selected to ensure that the results are well converged with respect to technical parameters. The spin-unrestricted procedure was used in all of the above calculations. Through calculate the energies of three spin states: the highspin state (SHS 5SLn1 1Srad 1SLn2 ), the first low-spin state (flip the spin on bpym2 radical; SLS1 5SLn1 2Srad 1SLn2 ), and the second low-spin state (flip the spin on one Ln ion: SLS2 5SLn1 1Srad 2SLn2 ) (the obtained low-spin state is not the pure spin state, which is a broken-symmetry one mixed by the high- and low-spin states), we used the spin-projected approach[24–26] to obtain the Ln0 bpym2 and Ln–Ln coupling constants Jex and J ex (Srad 51=2; SLn1 5 SLn1 5SLn2 ). The equations are as follows: ELS1 2EHS 4ðSLn 11=2Þ
(2)
ðELS2 2ELS1 Þ12Jex 4SLn 2 12SLn
(3)
Jex 5 0
J ex 5
Complete-active-space self-consistent field (CASSCF) calculations on individual lanthanide fragments have been carried out with MOLCAS 7.8 program package.[27] For each complex, we have used two structural approximations: A, a reduced fragment of the molecule; B, the complete entire molecule (see Fig. 2). For A, the only removed atoms are those connected to the Ln from the opposite side of the molecule, and we used the closed-shell La31 ab initio embedding model potentials (La.ECP.deGraaf.0s.0s.0e-La(LaMnO3.)[28] to take into account the influence of the neighboring Ln ion. For B, one of the Ln ions is replaced by the diamagnetic LuIII ion. In all calculations 2
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on individual lanthanide fragment, each bpym2 radical was treated as bpym which has a zero spin. For CASSCF calculations, the basis sets for all atoms are atomic natural orbitals from the MOLCAS ANO-RCC library: ANO-RCC-VTZP for Ln ions; VTZ for close C and N; VDZ for distant atoms. The calculations used the second-order Douglas–Kroll–Hess Hamiltonian, where scalar relativistic contractions were taken into account in the basis set. The active electrons in seven active orbitals include all f electrons in all calculations. First, we calculated all possible roots available for a given active space for each spin multiplet using state-averaged CASSCF method, and then mixed the maximum number of spin-free state which was possible with our hardware in the following spin-orbit coupling calculation by the RASSI module. For the Tb fragment, we mixed the roots coming from the following spin-free states: all from seven septets; all from 140 quintets; 68 from 500 triplets. Similarly, 21 sextet spin states, 128 quadruplet spin states, and 130 doublet spin states were mixed for the Dy fragment. For the Ho fragment, 35 quintet spin states, 150 triplet spin states, and 120 singlet spin states were mixed. For the Er fragment, 35 quartet spin states and 112 doublet spin states were mixed.
Results and Discussions Evaluation and comparison of the exchange interactions The Cartesian coordinates of the optimized structures of Ho2bpym2 and Er2bpym2 are in the Supporting Information. During each optimization, the structure of the bpym2 has been constrained. The optimized molecular geometries of 4–5 using B3LYP are very similar to those of 1–3. Only the Ln-N
Figure 2. Structures of the calculated Ln fragments A (left) and B (right) for each complex (Ln 5 TbIII, DyIII, HoIII, and ErIII)
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’ values (cm21) of five complexes [(Cp* Ln) (l-bpym)]1. Table 1. Ground-state spin multiplets, and the calculated and experimental Jex and Jex 2 2
2S11 Jex J’ ex
Gd2bpym2 (1)
Tb2bpym2 (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
14 218.3 (210.0[a]) 21.2
12 218.9 21.6
10 221.9 21.1
8 227.1 21.5
6 227.3 21.4
[a] The result was obtained from Ref. [8].
distances have a little difference within 0.05 A˚. The calculated 0 Jex and J ex of five complexes [(Cp*2Ln)2(l-bpym)]1 using B3LYP functional are shown in Table 1. 0 The calculated Jex and J ex values in Table 1 indicate that the Ln-bpym2 and Ln–Ln couplings are all antiferromagnetic and the Ln-bpym2 couplings are so much stronger than those of Ln–Ln. The experimental fitting also shows that the Ln-bpym2 and Ln–Ln couplings of complexes 1–3 are all antiferromagnetic.[8] To investigate the Ln-bpym2 couplings, we first gave the calculated spin values on each Ln ion and bpym2 radical of five complexes in their low-spin states (see Table 2). From the spin populations in Table 2, we only know the spins on Ln and bpym2 are almost localized. Thus, we further gave the spin density distribution map of complex Tb2bpym2 in the first low-spin state, which is shown in Figure 3 (the other complexes have the similar spin density distributions which are shown in Supporting Information Figure S1), where the spins on TbIII are almost localized, and those on bpym2 are a little more diffused although the total spin population on bpym2 is near to 21.0, but not delocalize onto the TbIII ions. Moreover, the spin densities on bpym2 have a larger space distribution around the central carbon atoms which have longer distances from Ln compared to those between Ln and N32 2 . The strength of the calculated Ln-bpym2 exchange interactions shown in Table 1 follows the experiment trend: GdIII [5,6] TbIII < DyIII,[8] which is opposite to that of Ln2N32 2 systems. As usual, the Ln–Ln couplings mainly caused by spin polarization are usually very weak for the contracted nature of the Ln 4f orbitals. From Figure 3, however, the spin densities on bpym2 have a much larger space distribution around the central carbon atoms which are far from the Ln ions, which results in the small overlap between the magnetic orbitals on Ln and bpym2, so as to the weaker Ln-bpym2 antiferromagnetic couplings compared to the strong Ln–N32 interactions according 2 to Kahn’s theory.[29] Thus, contrary to the Ln2N32 2 systems, the spin polarization of the Ln ions may dominate the Ln-bpym2 exchange couplings due to the weak magnetic orbital interac-
tions. However, B3LYP has no way to accurately evaluate the contributions of the weak overlap between Ln and bpym2, and the spin polarization of Ln to the Ln-bpym2 exchange coupling, so as to the larger calculated Ln-bpym2 exchange coupling constants than the experimental ones. As usual, DyIII would have more contracted valence orbitals than GdIII or TbIII due to its slightly smaller ionic radius, and so the overlap between the magnetic orbitals on DyIII and bpym2 is smaller. Based on the above analysis, however, the spin polarization of Ln ions dominates the Ln-bpym2 exchange couplings. From Table 1, the ratio of the spin population on the 4f orbitals to the total one is the largest for DyIII among complexes 1–3, which indicates that the spin polarization involving the empty 5d orbitals of DyIII is the strongest.[8] Thus, the DyIII-bpym2 antiferromagnetic exchange interactions are stronger than those of GdIII-bpym2 or TbIII-bpym2. For the same reason, the HoIII-bpym2 antiferromagnetic exchange interactions are the largest among complexes 1–4. It is unexpected that the ErIII-bpym2 antiferromagnetic coupling is a little larger than that of HoIII-bpym2. The previous article[7] has shown that the ErIII-N32 couplings are ferromag2 netic due to the very small overlap between the magnetic orbitals on ErIII and N32 for the smallest ionic radius of ErIII. 2 Similarly, the ferromagnetic contribution of the magnetic orbital exchange interactions to the ErIII-bpym2 coupling are larger than the antiferromagnetic one arising from the overlap. Thus, if not considering the spin polarization of ErIII, the ErIIIbpym2 coupling should be ferromagnetic. However, the magnetic orbital exchange interactions are much weak in Er2N32 2 system as the distances between ErIII and the central carbon atoms of bpym2 are so much long. Thus, the spin polarization of ErIII favoring the antiferromagnetic contribution to the ErIIIbpym2 coupling overwhelms the competing magnetic orbital exchange interactions, which leads to the ErIII-bpym2 antiferromagnetic coupling. Besides, a little larger ErIII-bpym2 antiferromagnetic coupling compared to that of HoIII-bpym2 is attributed to the opposite contributions of the spin
Table 2. Total spin populations on Ln and bpym2, and the individual spin populations for each of the 4f and 5d orbital levels of Ln1 and Ln2 of five complexes in the first low-spin state.
Ln1
Ln2
Bpym2
5d 4f total 5d 4f total
Gd2bpym2 (1)
Tb2bpym2 (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
0.134 6.934 7.104 0.134 6.934 7.104 20.992
0.103 5.942 6.065 0.104 5.942 6.068 20.986
0.081 4.945 5.040 0.081 4.941 5.039 20.974
0.061 3.947 4.020 0.061 3.947 4.020 20.965
0.043 2.950 3.000 0.041 2.954 2.999 20.960
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Figure 3. Spin density distribution map of complex Tb2bpym2 in the first low-spin state (blue and red regions indicate positive and negative spin populations, respectively; the isodensity surface represented corresponds to a value of 0.002 e2 bohr23). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
polarization of ErIII and the exchange interactions between the magnetic orbitals on ErIII and bpym2.
Ab initio calculations on mononuclear Ln fragments The orientations of the local magnetic axes, the g tensors, and the lowest spin-orbit multiplets of mononuclear Ln fragments (see Fig. 2) of complexes 2–5 were calculated using CASSCF. Considering the high-symmetry environment around two Ln ions for each complex, the local anisotropy axes of Ln fragments using model B in Figure 2b for each of complexes 2–5 are almost parallel. For the similar orientations of the local magnetic axes on the Tb, Dy, and Ho fragments, we only gave one of them in Figure 4 (a for Tb; b for Er). Figure 4 shows that the local magnetic axes on the Er fragments of complex Er2bpym2 have the different orientations compared to those of the other complexes. Moreover, the local magnetic moments in Figure 4 should be parallel because the Ln-bpym2 couplings are so much stronger than the Ln–Ln coupling for each complex. The calculated g tensors of the lowest effective S~ on the Tb, Dy, Ho, and Er fragments using the program SINGLE_ANISO[30] are given in Table 3. From Table 3, the calculated g values of models A and B for each of 2–4 are almost the same, while they have a little difference for 5. The much larger values of gz on TbIII, DyIII, HoIII, and ErIII ions than gx and gy indicate that the Ln-bpym2 exchange interactions will be of the Ising type. The CASSCF energies of the lowest multiplets corresponding to each spin-orbit coupling ground state of four Ln fragments are listed in Table 4. In the above B3LYP calculations of the Ln-bpym-exchange couplings, the strong 4f–5d interactions may have an important
influence on the couplings. In the CASSCF calculations, however, we did not consider the influence of the strong 4f–5d interactions on the spin-orbit coupling, although we do not know whether such influence is important. If we extend the active space to a larger one to include the 5d ions, the possible roots available for each spin multiplet (in the state-averaged CASSCF calculation, we should include all the available roots for each multiplet) will exceed the capacity of CASSCF and our hardware. As each Kramers doublet are degenerate for the Dy or Er fragment, we only gave one energy level for each doublet in Table 4. But, we gave all the non-Kramers states for the Tb or Ho fragment. As the thermal energy is much larger than the small splitting between the lowest two energy levels of the Tb or Ho fragment, the two levels can be supposed as the doubly degenerated ground states. If the Orbach relaxation process is dominant, the energy barrier is related to the separation DEi between the lowest two doublets of a single Ln fragment. Therefore, the energy separation of the lowest two non-Kramers states of the Tb fragment is 144.4 cm21 for model A and 154.3 cm21 for model B, and that of the Ho fragment is 68.4 cm21 for A and 54.3 cm21 for B. For each of the Dy and Er fragments, however, as the lowest and the second lowest Kramers doublets are both degenerate, the energy separations of the lowest two Kramers doublets of the Dy and Er fragments are 348.0 cm21 for A, 315.5 cm21 for B and 64.4 cm21 for A, 58.7 cm21 for B, respectively. Although, the energy separation DEi (144.4 cm21) between the lowest two Kramers doublets of the Tb fragment is a little [7] larger than that of the Tb fragment (133.0 cm21) of Tb2N32 2 , why is the effective energy barrier Ueff of complex Tb2bpym2 (44.0 cm21) far smaller than that of Tb2N32 (227.0 cm21)? 2 2 Moreover, why is the Ueff of Dy2bpym only 87.8 cm21 although the single Dy energy barrier is up to 348.0 cm21 far larger than that of the Dy fragment in Dy2N32 2 ? Next section will present the analysis of the above results combined single Ln ion anisotropies with Ln-bpym2 exchange couplings. Origin of the energy barrier For transition-metal systems, we can obtain the total D value through project the single-site anisotropies (Di) onto the spin ground state S.[31] X 2Si 21 D5 di D i di 5 (4) Nð2NSi 21Þ i where i numbers the N metal centers and the di values are projection coefficients. Equation (4) is derived in the strongexchange limit for a spin cluster. The anisotropic interaction tensor Dij is ignored as dipolar and anisotropic interactions
Table 3. Main values of the g tensors of the lowest effective S~ on the Tb, Dy, Ho, and Er fragments. Tb (2)
gx gy gz
4
Dy (3)
Ho (4)
Er (5)
A
B
A
B
A
B
A
B
0.000 0.000 17.799
0.000 0.000 17.792
0.002 0.003 19.940
0.002 0.003 19.867
0.000 0.000 17.574
0.000 0.000 17.952
1.103 1.571 14.281
0.107 0.457 15.137
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Table 4. CASSCF energies of the lowest Kramers doublets of 3 and 5 and non-Kramers states or singlet (mJ 5 0) of 2 and 4 with two models of A and B (cm21). Tb(2)
Dy (3)
A
B
0.000 0.090 144.432 145.449 319.311 322.548 462.161 509.295 575.734 741.474 755.648 876.330 878.406
0.000 0.044 154.311 154.537 324.399 334.222 464.596 538.245 591.711 805.265 815.243 999.758 1001.208
Ho (4)
Er (5)
A
B
A
B
A
B
0.000 347.994 497.794 568.741 640.769 717.273 806.296 1097.127
0.000 315.534 478.236 545.702 611.025 680.983 789.798 1100.177
0.000 0.362 68.393 72.163 154.636 169.219 201.906 231.269 238.763 284.618 329.325 347.987 388.215 414.957 418.099 487.581 490.515
0.000 0.668 54.292 56.261 115.790 124.382 135.270 183.943 192.189 261.248 276.162 305.660 321.882 375.734 376.198 444.239 446.363
0.000 64.415 118.940 202.748 255.332 310.317 417.914 485.764
0.000 58.704 138.865 186.109 265.528 313.912 422.299 438.049
yield only minor contributions. For most of lanthanide-metal SMMs, the energy barrier comes from a single Ln ion due to the very weak Ln–Ln exchange coupling. But, for radicalbridged dilanthanide complexes, the strong interactions of the two Ln ions transmitted by the radical may make both Ln ions contribute to the total energy barrier. A recent work[7] has successfully given the relationship between the total energy barrier DE with separated single Ln barriers DEi (the energy separation between the lowest two doublets). Similarly, if assuming that all complexes are in the strong-exchange limit, the approximate relationships between the total energy barrier DE and the single Ln barrier DEi for each complex are as follows.
Figure 4. Orientations of the local magnetic axes on the Tb (a) and Er (b) fragments.
DE51:87DEi for Tb2 N32 2
(5)
DE51:94DEi for Dy2 N32 2
(6)
DE51:90DEi for Ho2 N32 2
(7)
Er2 N32 2
(8)
DE51:94DEi for
To assess the relative strength between magnetic anisotropy and exchange interaction for the investigated bpym2 radicalbridged dilanthanide complexes, we also use the ratio Ecorr/DEi where Ecorr represents the exchange correlation energy (Ecorr 5 4jJexjSLnSrad) which is often used in single-chain magnets to describe the creation energy of one domain wall.[32] The calculated single Ln barrier DEi of model B, the total energy barrier DE according to eqs. (5–8), the Ln-bpym2 exchange correlation energy Ecorr, the experimental Ueff, and the ratios Ecorr/DEi and DEi/Ueff for each of 2–5 are shown in Table 5. Table 5 shows that the exchange correlation energies Ecorr of 2 and 3 are both far smaller than the corresponding single Ln anisotropy energy barrier DEi, and thus, the effective energy barriers Ueff mainly come from the single Cp*2Ln(l-bpym)La fragments. Our previous work[7] has shown that the total energy barrier of the Dy2N22 or Er2N32 system is about one 2 2 third of a single Dy or Er ion energy barrier DEi as the Ecorr is far smaller than the DEi for each of them. It is expected that the Ueff values of 2 and 3 are also close to one third of the calculated single Tb and Dy fragment energy barriers, respectively. From observing the DEi/Ueff values in Table 5, it is obvious that eqs. (5) and (6) are invalid for 2 and 3 as they are not in the strong-exchange limit. For 4 or 5, however, as the correlation energy Ecorr is much larger than the DEi, we deduced that the Ueff values of 4 and 5 will be close to the DE obtained using eqs. (7) and (8), respectively. The above results indicate that the relatively weaker Tbbpym2 exchange interactions lead to the Ueff value of Tb2bpym2 being far smaller than that of Tb2N32 2 . Although, the DEi of a single Dy fragment for 2 is up to 315.5 cm21, the Journal of Computational Chemistry 2014, DOI: 10.1002/jcc.23565
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Table 5. Calculated single Ln barrier DEi, the total energy barrier DE, the Ln-bpym2 exchange correlation energy Ecorr, the ratios Ecorr/DEi and DEi/ Ueff for each complex, and the experimental effective energy barrier Ueff of 2–5 (cm21).
Tb2bpym (2)
Dy2bpym2 (3)
Ho2bpym2 (4)
Er2bpym2 (5)
154.3 288.5 113.4 44.0 0.73 6.6
315.5 612.1 109.5 87.8 0.35 7.0
54.3 103.2 108.4
58.7 113.9 81.9
2.0
1.40
2
DEi DE Ecorr Ueff Ecorr/DEi DE/Ueff
Ueff value is much smaller than that of Dy2N32 2 , which is attributed to its far smaller Ecorr/DEi value. Due to the larger Ecorr/ DEi values of 4 and 5, the two HoIII or ErIII ions both contribute to the total energy barrier. From the above analysis, we conclude that enhancing Ln-radical exchange interactions will be especially important to increase the total energy barrier, rather than only increase the single Ln-ion energy barrier. As the current molecular magnetism theory is far from perfect, and the calculation methods are not accurate enough, the relationships between the intramolecular exchange interactions, the single-ion anisotropies, and the total effective energy barriers for lanthanide-metal SMMs have no way to be clearly interpreted by us. Some problems, what’s the accurate value of Ecorr/DEi to make both Ln ions contribute to the total energy barrier, whether the above conclusions can also be applied to other dilanthanide systems and so forth are waiting to be clarified. We hope this work will be an exploration on the origin of the slow relaxation of the magnetization which is the most important one in present-day molecular magnetism.
Summary B3LYP and CASSCF methods were used to calculate the Lnbpym2 exchange couplings and the single Ln energy barrier to explore the origin of the total magnetic anisotropies for five Ln2bpym2 SMMs. B3LYP calculations show that the Ln–Ln couplings are very weak, but the Ln-bpym2 couplings are relatively stronger. Moreover, due to the predominant strongest spin polarization involving the empty 5d orbitals of ErIII, the ErIII-bpym2 antiferromagnetic exchange interactions are the strongest among five Ln2bpym2 systems. CASSCF calculations on four Ln fragments indicate that the single Ln anisotropy energy barriers DEi of 2 and 3 are far larger than those of 4 and 5, but their effective energy barriers are much smaller. Moreover, the Ueff values of 2 and 3 are also close to one third of the calculated single Tb and Dy fragment energy barriers, respectively, and those of 4 and 5 are close to the DE obtained using eqs. (7) and (8). Thus, to enhance the total energy barrier, it needs to enhance the Ln-bpym2 couplings when improves the single Ln fragment barrier at the same time.
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Journal of Computational Chemistry 2014, DOI: 10.1002/jcc.23565
Keywords: single-molecule magnet energy barrier exchange coupling complete-active-space self-consistent field B3LYP
How to cite this article: Y. Q. Zhang, C. L. Luo, Q. Zhang. 2014. J. Comput. Chem., DOI: 10.1002/jcc.23565
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Additional Supporting Information may be found in the online version of this article.
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Received: 14 November 2013 Revised: 30 January 2014 Accepted: 31 January 2014 Published online on 00 Month 2014
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