Predictions of the Spin Configuration in Mn12 Molecular Magnets

Aug 22, 2014 - University of Central Florida, 12424 Research Parkway, Suite 400, Orlando ... centers i and j with the spin states Si and Sj. The posit...
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Predictions of the Spin Configuration in Mn12 Molecular Magnets Made Accurate with the Help of Hubbard U on the Ligand Atoms Shruba Gangopadhyay,†,‡ Artem ̈ E. Masunov,*,†,‡,§ and Svetlana Kilina∥ †

NanoScience Technology Center, ‡Department of Chemistry, and §Department of Physics and Florida Solar Energy Center, University of Central Florida, 12424 Research Parkway, Suite 400, Orlando, Florida 32826, United States ∥ Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota 58108, United States ABSTRACT: We predict Heisenberg exchange coupling J values between all metal centers for two different Mn12-based molecular complexes. Both wheels are reported to have the ground spin state of multiplicity 15 and weak antiferromagnetic couplings between two identical halves. The correct sign and order of the exchange couplings for all six magnetic interactions are predicted for the first time and are in agreement with experimental observations. Empirical tuning of the Hubbard repulsion term U for both metal and ligand atoms and geometrical optimization of the ground spin state were found to be crucial for accurate prediction of J values in the magnetic wheels.

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INTRODUCTION Single-molecule magnets (SMMs) are a class of polynuclear transition metal complexes, characterized by a large spin ground state and considerable negative anisotropy. These properties result in slower magnetization relaxation and slower decoherence rates. For these reasons SMMs are expected to be promising materials for molecular spintronics,1−8 which exploits both the spin and charge of an electron in memory and logical devices and quantum computers9 of the future. Instead of the classical bits, which can take only one value (1 or 0), quantum computing operates with quantum bits (qubits), prepared as a quantum superposition state of 1 and 0. Such a superposition state can simultaneously store multiple values in the same register. Therefore, any operation on this register becomes massively parallelized, which speeds up calculations by orders of magnitude. The electron spin is a preferred candidate for a qubit. However, spin−spin interactions, as well as interactions with the environment, lead to decoherence of a superposition state and limit the time available for quantum operations. Therefore, such interactions have to be well understood to be controlled and reduced. In order to design SMMs for quantum computation, one needs to accurately predict their spin−spin interactions, reflected in magnetic properties. The most important theoretical characteristic related to magnetic properties is the Heisenberg exchange coupling parameter, J. This parameter appears in the model Heisenberg Hamiltonian that can be written in a general form as H = −∑ Jij SiSj

SMMs, and accurate predictions on the J value for different SMMs still present an open problem. Here, we focus on investigations of magnetic exchange couplings in SMMs with a wheel-shaped topology and, specifically, single-stranded Mn12 wheels.10 We consider two different molecules, Mn12(ada) and Mn12(mda). These SMM systems have the general chemical formula [Mn12Rda8(CH3COO)14]·x(CH3CN), where the Rda2− are dianions of N-R diethanolamine, R = a (allyl), abbreviated as Mn12(ada), and R = m (methyl), abbreviated as Mn12(mda), and x = 7 for allyl and x = 1 for methyl. Both systems were reported to have high ground spin state with the total spin S = 7.11,12 However, S = 8 also was suggested by another experiment,44 thus still leaving ambiguities. In addition, there are long-standing debates2,12−15 on whether the magnetic dimer model is applicable to these molecules. Thus, Ramsey and coauthors12 suggested that the Mn12(ada) wheel acts as a magnet dimer with a weak antiferromagnetic coupling between two identical halves, each including six strongly coupled Mn centers. An alternative magnetization model beyond a dimer was suggested14 for the Mn12(mda) wheel. This discussion has been followed by two more papers,13,15 which argue that Mn12 wheels have the weakest coupling center along the first and sixth Mn ions and, thus, do not support an appropriateness of the magnetic dimer model. Firstprinciple atomistic calculations can help to get more insights into this question and better interpret experimental data. The density functional theory (DFT) approach was shown to make accurate predictions on the structure and properties of molecules and solids, such as aggregation and crystal formation,16−19 reaction rates and mechanisms,20−23 and linear and nonlinear optical properties.24−27 To date, DFT in plane wave basis is the only first-principle method capable of describing

(1)

where Jij represents the coupling between the two magnetic centers i and j with the spin states Si and Sj. The positive J values indicate the ferromagnetic ground state, and the negative ones indicate the antiferromagnetic ground state. For the system of two equivalent magnetic centers, the J value can be calculated from the first-principles using total energies of the high and low spin states. However, such a “dimer” approach is questionable for © 2014 American Chemical Society

Received: May 14, 2014 Revised: July 24, 2014 Published: August 22, 2014 20605

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the electronic structure of the large molecules, such as Mn12−, Fe8−, and V15− SMMs.28−34 Recent DFT calculations utilizing the commonly used functionals, such as the hybrid functional B3LYP11,35 and the generalized gradient approximation (GGA) type functional PBE,35 have shown the lowest magnitude of the Heisenberg exchange coupling in Mn1−Mn6, thus confirming the magnetic dimer approach. However, the GGA functional35 was not able to reproduce the correct nature of magnetic coupling observed in experiments,11 whereas the B3LYP functional provides the order of lowest magnitude of the ground spin with some ambiguity and, hence, allows multiple interpretations of the calculated results. Motivated by all these contradictions,10,11,23−30,34−36 we aim to establish a computational protocol that unambiguously predicts magnetic behavior and accurate magnetic exchange couplings for both Mn12(ada) and Mn12(mda) molecules. Computations of magnetic couplings represented by J values in molecular systems have posed a long-standing problem in quantum chemistry, since calculations of this kind require an accurate description of electron correlation, both static and dynamic, as well as a reliable mapping between computed observables and parameters featured in the model Heisenberg Hamiltonian. For example, self-interaction error in pure DFT (e.g., GGA-type functionals) is known to artificially delocalize electron density and bring inaccuracies in the prediction of the small energy effects associated with magnetic interactions. Two modifications of DFT are commonly used to partly alleviate this artificial delocalization: hybrid functionals (hybrid DFT) and DFT+U. In hybrid functionals, a fraction of GGA exchange is replaced with the Hartree−Fock (HF) exchange to reduce selfinteraction error. In the DFT+U approach, the onsite Coulomb repulsion term, the Hubbard term U, is added to reduce overdelocalization of an electronic orbital. Both modifications had been used for predictions of J values in SMMs. Hybrid functionals are commonly utilized in broken symmetry DFT (BSDFT), as suggested by Noodleman37−41 and Yamaguchi.42−45 The variations of BSDFT formalism were also proposed by Ruiz46 and Nishino47 with a more sophisticated scheme taking into account the overlap between magnetic orbitals.48−50 Despite a wide use of hybrid functionals in predicting magnetic properties of SMMs,51 several limitations of this approach have been reported for systems with (i) bridging acetate ligands,52 (ii) mixed valence metal atoms,53 and (iii) artifactual instability of ferromagnetic coupling.53 Therefore, the hybrid DFT might not be the most accurate approach in modeling spin−spin interactions in Mn12(ada) and Mn12(mda) wheels. The DFT+U approach was also used for J value calculations for different molecular magnets1,3,5,6,54−56 and organometallic compounds.57−61 The general strategy in DFT+U is to add the Hubbard U-term for d orbitals on metal atoms. Recently, the use of the Hubbard U-parameter for p orbitals on ligand atoms was found necessary to predict correct spin states in different bimolecular manganese complexes,62,63 including both ferro- and antiferromagnetic couplings, mixed valence compounds, and acetate-bridged complexes. The presence of π-dative bonding in the latter was identified as the reason for delocalization of magnetic orbitals and breakdown of the simple superexchange picture.63 Here we refer to this methodology as DFT+Umetal+ligand and adopt it for studies of Mn12(ada) and Mn12(mda) SMMs as the most accurate approach in predictions of J values. Earlier we investigated in detail the values of U-terms for both d and p electrons in the DFT+Umetal+ligand approach to correctly predict antiferromagnetic exchange parameters J between Mn1 and

Mn6′ centers in the Mn12(mda) wheel.63 Here, using our DFT +Umetal+ligand method, we report predictions for all six nearestneighbor Heisenberg exchange parameters J in Mn12(ada) and Mn12(mda)SMM systems having similar coordination environments and the same ground spin states.11,36 Our calculations help to confirm the validity of the DFT+Umetal+ligand approach, get insights into the mechanisms of spin−spin interactions, and put to rest the questions on applicability of a dimer model for Mn12 wheels.

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METHODS AND COMPUTATIONAL DETAILS All the reported calculations were performed using the older and later versions64 of the Quantum-ESPRESSO computational Scheme 1. Pairs of Spin States Used for Prediction of Six Exchange Parameters in Mn12 Wheelsa

a Here (S=0)nm refers to the spin alignment used for calculating corresponding Jnm. The symbols 1, 3, 5 and 1′, 3′, 5′ are Mn3+; 2, 4, 6 and 2′, 4′, 6′ are Mn2+ with four and five unpaired electrons, respectively; i represents spin-up and i ̅ represents spin-down orientation (i = 1−6, 1′−6′).

software. The PBE exchange−correlation functional, Vanderbilt ultrasoft pseudo-potentials,65 and a plane-wave basis set were utilized for all our calculations. The energy cutoffs for the wave functions and charge densities were set at 25 and 250 Ry, respectively. To ensure total energy convergence, all calculations used the spin-polarized approach with the Marzari−Vanderbilt66 cold smearing (smearing factor 0.0008) and local Thomas− Fermi mixing mode. This combination allows for improving the SCF convergence. To calculate the Heisenberg exchange parameter, we first optimized both systems in their ground high spin state (S = 7) using pure DFT. To obtain desirable spin states, we used different starting magnetization of transition metal atoms and verified with Löwdin spin densities after SCF convergence. The geometry was optimized in the high spin state starting from atomic coordinates, taken from X-ray diffraction data11,36 with added hydrogen atoms. The optimization of initial geometries was found to be critically important for the accuracy of the final results. 20606

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In application of the DFT+U method, we followed the protocol used in our previous work.63 We employed the simplified rotationally invariant DFT+U formulation, implemented by Cococcioni in the Quantum-ESPRESSO-4.0.1 package.67 The values of the U parameter for both the metal atom and the ligand atoms (O and N) were empirically adjusted to the optimum values 2.10 eV for Mn, 1.00 eV for O, and 0.20 eV for N that fit the experimental spin splitting energies for the benchmark set of five small dinuclear manganese complexes in a variety of oxidation states (+2, +3, and +4).63 Next, we constructed six hypothetical low-spin states, by partitioning the wheel into two equal sets of six atoms each and inverting the spins for one of these sets. Assuming the secondneighbor magnetic interactions to be negligible, the energy for each low spin state (S = 0)nm allows us to calculate the magnetic coupling parameters as Jnm =

E(S = 0)nm − E(S = 7) 2 × Sn × Sm

(2)

E(S=0)nm is the energy of (S = 0)nm alignments shown in the second column of Scheme 1, E(S=7) is the energy of the spin configuration shown in the third column of Scheme 1. E(S=7) is the energy of the experimental spin ground state of the wheel. Sn = 5/2 and Sm = 2 (n = 2, 4, 6 and m = 1, 3, 5 according to Scheme 1). Because molecules are symmetric Jnm = Jmn. Note that our approach is different from the one used by Cano et al.35 In their work the energies for 32 different high, low, and intermediate total spin states were calculated, and quantum Monte Carlo was used to simulate the magnetic behavior based on presumed Jvalues for both the first- and second-neighbor interactions. They found that the second-neighbor J-values necessary to reproduce the state ladder turned to be rather small. This finding confirms our assumption that only nearest neighbors dominate in magnetic interactions.

Figure 1. (a) Schematic diagram of magnetic coupling in Mn12 wheels. The green ball represents Mn3+, and yellow represents Mn2+. Symmetrically identical atoms are labeled with a prime. The spin states of the atomic centers are also shown. All Mn3+ (labeled 1, 1′, 3, 3′, 5, 5′) are spin-up, Mn2+ (2, 2′, 6, 6′) are spin-down, and two pentacoordinated Mn2+ (4, 4′) are spin-up. (b) Schematic for representation of antiferromagnetic Mn12 wheels.

which is consistent with experimental data.11,12 From Figure 1a, one can see that out of six magnetic interactions present in the wheel two are ferromagnetic and four antiferromagnetic. The antiferromagnetic interactions are confirmed to be Mn(1)− Mn(2), Mn(2)−Mn(3), Mn(3)−Mn(4), and Mn(5)−Mn(6), and ferromagnetic interactions are Mn(3)−Mn(4) and Mn(4)− Mn(5). To date, there are no quantitative measurements of the exchange coupling parameters reported for these molecules. The qualitative predictions suggest the weakest magnetic coupling between Mn(1) and Mn(6′).11,12 Several DFT studies predicting J values for the Mn12(mda) wheel11,35 have been published. We compare the previous and our current DFT results in Table 1. The first BSDFT calculation on the [Mn12] wheel was performed by Foguet-Albiol11 et al. (presented in sixth column of Table 1). To reduce the computational expense, the wheel was fragmented into several tri- and binuclear clusters. This study predicted Mn(1)−Mn(6′) centers to couple ferromagnetically, resulting in the S = 0 ground state, which contradicts the experimental data.11 Cano et al.35 calculated the Heisenberg exchange parameters for the whole magnetic wheel using a variety of spin states and fitting the J values using their Monte Carlo technique. Their results for the pure and hybrid DFT functionals are compared in columns 4 and 5 of Table 1. The weakest magnetic interaction Mn(1)− Mn(6′) corresponds to the ferromagnetic spin coupling when the pure functional PBE is used, which is switched to an antiferromagnetic one when the hybrid functional B3LYP is used. The last one provides a correct state. However, the B3LYP calculation predicts all the J values to be roughly of the same order of magnitude. This contradicts the conclusion that according to experimental observables, the remaining magnetic centers should have much stronger coupling.11 Our own results obtained by the pure PBE functional (column 7) are comparable

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RESULTS AND DISCUSSION The motivation behind the calculations of Heisenberg exchange constants for two Mn12 wheels (Mn12(mda) and Mn12(ada)) arise from a long-standing debate.2,12−15 The original magnetization studies on the Mn12(ada) wheel12 in 2008 revealed that the wheel behaves as a nanomagnet dimer. Another paper was published the same year14 that appears to contradict the Ramsays conclusions.12 The discussion has followed,13,15 where the latest comment15 suggests that a detailed theoretical investigation would help. No DFT calculations on these two molecules have been published since then. Earlier B3LYP11,35 and PBE35 DFT studies reported the lowest magnitude of the Heisenberg exchange constant between Mn1 and Mn6 centers, but the correct magnitude of the magnetic coupling could not be reproduced. Although the correct ground spin state was predicted in the latest B3LYP calculation,35 the J magnitude is questionable. Here we establish a computational protocol which can give an accurate description of the magnetic behavior for both large Mn12 molecular wheels. As illustrated in Figure 1, both wheel molecules include alternating Mn2+ and Mn3+ ions arranged in a single stranded wheel, symmetrical under inversion. The ion Mn3+ has d4 highspin electron configuration and the Jahn−Teller distorted octahedral coordination. The Mn2+ ions are found in two different coordination environments: one is octahedral and another one is pentagonal bipyramidal. The calculated spin alignments provide the ground spin state S = 7 for both wheels, 20607

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Table 1. Calculated Heisenberg Exchange Parameters of Mn12(mda) Obtained from our DFT+U and Other DFT Calculations Reported in the Literaturea Jij (cm−1) Mn···Mn length (Å) J16 J12 J23 J34 J45 J56

Mn12(mda)

Mn12(ada)

X-ray

opt

PBEb

B3LYPb

B3LYPc (fragment)

PBE (X-ray)

PBE (opt)

PBE+U (opt)

PBE+U (opt)

3.46 3.21 3.15 3.17 3.18 3.20

3.44 3.21 3.18 3.17 3.15 3.21

+1.2 −6.0 −14.9 +10.9 +9.2 −5.4

−3.5 −5.6 −2.5 +6.3 +5.4 −5.9

+0.04 −2.8 −9.2 +7.0 +8.0 −5.0

+4.6 −20.8 −26.8 50.5 56.9 −13.6

−7.4 −8.6 −31.3 8.1 5.3 −5.4

−0.8 −3.7 −23.5 44.0 54.1 −14.2

−2.4 −23.9 −31.0 57.6 45.9 −35.5

a The Jij values are labeled according to Figure 1 (first column). The second and third columns compare the Mni−Mnj distances from X-ray data and our optimized structure, respectively. The fourth and fifth columns show the results obtained by PBE and B3LYP functionals, as reported by Cano et al.35 The sixth column is the BSDFT value obtained by fragmenting the Mn12(mda) wheel.11 The seventh and eighth columns are J values calculated by our DFT approach using a pure PBE functional for X-ray and optimized geometries, respectively. The last two columns represent J values obtained by our DFT+Umetal+ligand for the optimized geometries of Mn12(mda) and Mn12(ada), respectively. bReference 35. cReference 11.

to the previously published35 coupling parameters, despite the difference in the formalism used for J extraction and the basis set (plane waves vs Gaussian). Importantly, the geometry optimization changes the coupling of the Mn(1)−Mn(6′) centers from incorrect ferromagnetic to correct antiferromagnetic, although its absolute value appears to be too large, comparable to four other magnetic interactions in the ring. Geometry optimization of the molecular structure slightly changes the interatomic distances by ca. 0.01 Å (compare the columns 2 and 3 in Table 1). Due to this small change in the geometry, the Mn(1)−Mn(6′) coupling becomes antiferromagnetic, in agreement with experiment. We emphasize the importance of geometry optimization for Mn12 wheels. The comparison between experimental (X-ray) and optimized Mn− Mn lengths for the Mn12(mda) wheel is reported in Table 1. The optimized structures of both Mn12(mda) and Mn12(ada) wheels are illustrated in Figure 2a. Experimentally, slightly different crystal structures of Mn12(mda) wheels have been reported.11,36 Also it was shown35 that variations in the crystal structure obtained from two different experimental sources11,36 have resulted in different ground magnetic states. Thus, the B3LYP calculation35 based on the crystal structure reported in ref 11 has shown the lowest Heisenberg exchange constant for Mn(1)− Mn(2), whereas those calculations obtained from geometries reported in ref 36 have reproduced the smallest J between the Mn(5) and Mn(6) magnetic centers. These findings point to the sensitivity of the magnetic state of Mn12 wheels to slight changes in their geometries, similar to what our calculations have shown. One can hypothesize that small fluctuations in experimental atomic coordinates are due to static disorder brought by a flexibility of the solvent molecules in the lattice. Such fluctuations in solvent positions might result in different molecular conformations with very small energy differences. There is another experimental aspect that might be related to this discrepancy: the X-ray analysis is often carried out at a temperature higher than that at which the magnetic experiments are performed, which is 4 K liquid helium temperature. Therefore, the reported geometries might not exactly correlate with those corresponding to the magnetic measurements. The use of the Hubbard parameter U (column 9) in the DFT procedure reduces antiferromagnetic coupling Mn(1)−Mn(6′) by an order of magnitude and increases the ferromagnetic couplings Mn(3)−Mn(4) and Mn(4)−Mn(5) by 5−10 times. Thus, our DFT+Umetal+ligand approach improves the values of J, making them in qualitative agreement with the experiment.12 In

fact, the magnetic study on a similar wheel by Ramsey et al.12 suggests that it is a dimer of two strongly coupled six-atom sets, connected with a very weak antiferromagnetic interaction. Ramsey et al. also correctly inferred that the weakest coupling is observed for Mn(1)−Mn(6′) centers. Next we applied a similar computational protocol (optimization, followed by DFT+Umetal+ligand) to another magnetic wheel, Mn12(ada).36 This wheel is reported to have the same ground spin state (S = 7) as the Mn12(mda) molecule12 and a similar spin alignment. The calculated exchange parameters are reported in the last column in Table 1. For Mn12(ada), our calculation predicts that the weakest J is obtained for Mn(1)−Mn(6), which is an order of magnitude smaller than the next weak coupling for Mn(1)−Mn(2). Despite the quantitative difference in values of J, both wheels are qualitatively similar in terms of their magnetic exchange. This proves a validity of our DFT+Umetal+ligand approach and confirms that U-parameters are not sensitive to variation in the ligand environment, when the main wheel structure is preserved. Another interesting feature in J couplings obtained by our DFT+Umetal+ligand approach is the relatively high magnitude of the ferromagnetic coupling for Mn(3)−Mn(4) and Mn(4)−Mn(5) centers, which was not predicted by other DFT studies.11,35 Recent inelastic neutron scattering measurements68 also point toward a significantly higher magnitude of ferromagnetic coupling in the Mn(3)−Mn(4)−Mn(5) fragment of Mn12(mda). According to Meier and Loss,69 a strongly coupled multicenter fragment can be described as a single large spin center. Considering that model, both wheels, especially Mn12(ada), fit in the model for perfect antiferromagnetic wheel alternating spin-up and spin-down magnetic moments, as illustrated in Figure 1a. To investigate the details of electronic structure and estimate the ability of these molecules to act as qubit in molecular spintronics, we performed the Löwdin population analysis, results shown in Table 2. These results confirm different spin states for different manganese atoms: spin-up for octahedral Mn 3+ , spin-down for octahedral Mn2+, and pentagonal bipyramidal (PBP) Mn3+. From spin polarization of ligand atoms (N, μ-O, and acetate O), one may conclude that the nitrogen atoms connected with bipyramidal and octahedral coordinated Mn ions have different spin densities. The acetate oxygen and oxo-bridged oxygen (μ-O), responsible for superexchange have similar spin polarization values, whereas diagrams 20608

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Table 2. Löwdin Population Analysis for Mn12(mda) and Mn12(ada) Mn(1), Mn(3), Mn(5) Mn(2), Mn(6) Mn(4) NOh NPBP OAc μ-O

mda

ada

3.89 −4.75 4.77 0.06 0.02 0.02 0.02

3.92 −4.76 4.77 0.05 0.02 0.02 0.02

to the splitting of the highest-energy PDOS peak into two. The contributions to the spin density from each type of atoms in the ground states (S = 7) are presented in Figures 3 and 4 for mda and ada wheels, respectively. One can see from Figure 3 that nitrogen atoms involved in octahedral coordination of Mn ions contribute to the highest occupied molecular orbital (HOMO), along with Mn3+ (Mn(3)), whereas acetate O and Mn2+ (Mn(4)) contribute to the lowest unoccupied molecular orbital (LUMO). The JT distorted d4 orbitals of Mn3+ demonstrate the splitting of eg levels, whereas the octahedral d5 of Mn2+ spin-down possesses degenerate eg levels. In the case of the ada wheel the oxygen atom involved in superexchange is taking part as an acceptor, whereas in mda there is no such interaction. Another difference is evident from the PDOS diagram in mda: the Mn(3) (Figures 3 and 4) is involved in HOMO where in the ada wheel the HOMO level is nearly degenerate between Mn(3) and Mn(1). From Figures 3 and 4, one can see that Mn3+ has the smaller highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gap compared to Mn2+ (both octahedral and pentagonal bipyramidal coordination). If we compare the DOS of Mn(1) and Mn(3) in Figures 3 and 4, both the cases Mn(3) show a little wider HOMO−LUMO energy gap than Mn(1). According to Hoffmann and Hay70 the antiferromagnetic coupling is stronger when the HOMO− LUMO energy difference is higher. On this basis, we can conclude that the presence of Mn1 reduces the antiferromagnetic coupling. A similar observation can be made from Figures 3 and 4 for Mn2+ octahedral and PBP Mn2+, and the PBP manganese demonstrates a wider HOMO−LUMO energy gap than octahedral bivalent manganese. This electronic structure analysis can justify why the Mn(3)−Mn(4)−Mn(5) shows stronger magnetic coupling than other Mn−Mn couplings. Though these energy differences are fairly small, they are significant in comparison to the other exchange parameters in this wheel, which are on the order of 1 cm−1.

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CONCLUSIONS We applied the DFT+Umetal+ligand method to calculate all nearestneighbor Heisenberg exchange parameters in molecular magnets Mn12(mda) and Mn12(ada). Our calculations successfully confirm the S = 7 ground spin states. The weakest coupling parameter is found for Mn(1)−Mn(6′), thus supporting an accuracy of a dimer model. The stronger ferromagnetic coupling for the Mn(3)−Mn(4) and Mn(4)−Mn(5) pairs is also predicted, in agreement with recent inelastic neutron scattering observations.68 The details of electronic structure of the molecular magnetic wheels was analyzed in detail. Löwdin population and PDOS plots help to rationalize the stronger ferromagnetic coupling in the Mn(3)−Mn(4)−Mn(5) zone by quantitative comparison of the HOMO−LUMO energy gaps in different magnetic centers.

Figure 2. (a) Two Mn12 magnetic wheels: top, Mn12(mda); bottom, Mn12(ada). (b) Jahn−Teller distortion in Mn3+ sites in the Mn12 wheel.

of the electronic structure (below) reveal the substantial differences. The detailed analysis of atomic contribution to spin density can be carried out with the assistance of projected density of states (PDOS), presented in Figures 2−4. One can compare the PDOS in ideal octahedral coordination vs Jahn−Teller (JT) distorted octahedral coordination of d4 orbitals of Mn3+ (Figure 2b). The JT distortion is expected to break degeneracy in eg orbitals, as illustrated in the inset of Figure 2b. This corresponds 20609

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Figure 3. Density of states plotted for Mn12(mda) in the S = 7 ground state. Positive values present spin-up and negative values present spindown densities.

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Figure 4. Density of states plotted for Mn12(ada) in the S = 7 ground state. Positive values present spin-up and negative values present spindown densities.

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AUTHOR INFORMATION

ACKNOWLEDGMENTS This work was supported in part by the U.S. National Science Foundation (CHE-0832622). S.K. acknowledges financial support of the Alfred P. Sloan Research Fellowship BR2014073. The authors gratefully acknowledge the use of computational resources at The Stokes Advanced Research Computing Center, University of Central Florida (UCF), and the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

Corresponding Author

*A. E. Masunov. Tel.: +1(407)374-3783. E-mail: amasunov@ ucf.edu. Present Address

S.G.: Department of Physics, University of California, Davis, CA 95616, USA and IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA. Notes

The authors declare no competing financial interest. 20610

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