Systematic Theoretical Study of the Zero-Field Splitting in

Sep 9, 2010 - Density Functional Theory versus Multireference Wave Function ... For the estimation of the spin−orbit coupling (SOC) part of the zfs,...
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J. Phys. Chem. A 2010, 114, 10750–10758

Systematic Theoretical Study of the Zero-Field Splitting in Coordination Complexes of Mn(III). Density Functional Theory versus Multireference Wave Function Approaches Carole Duboc,*,† Dmitry Ganyushin,‡ Kantharuban Sivalingam,‡ Marie-Noe¨lle Collomb,† and Frank Neese*,‡,§ De´partement de Chimie Mole´culaire, UMR-5250, Laboratoire de Chimie Inorganique Redox, Institut de Chimie Mole´culaire de Grenoble FR- CNRS-2607, UniVersite´ Joseph Fourier Grenoble 1/CNRS, BP-53, 38041 Grenoble Cedex 9, France, Institut fu¨r Physikalische und Theoretische Chemie, UniVersity of Bonn, Wegelerstrasse 12, 53115 Bonn, Germany, and Max-Planck-Institut fu¨r Bioanorganische Chemie, Stiftstrasse 34-36, D-45470 Mu¨lheim an der Ruhr, Germany ReceiVed: August 18, 2010

This paper presents a detailed evaluation of the performance of density functional theory (DFT) as well as complete active space self-consistent field (CASSCF)-based methods (CASSCF and second-order N-electron valence state perturbation theory, NEVPT2) to predict the zero-field splitting (zfs) parameters for a series of coordination complexes containing the Mn(III) ion. The physical origin of the experimentally determined zfs’s was investigated by studying the different contributions to these parameters. To this end, a series of mononuclear Mn(III) complexes was chosen for which the structures have been resolved by X-ray diffraction and the zfs parameters have been accurately determined by high-field EPR spectroscopy. In a second step, small models have been constructed to allow for a systematic assessment of the factors that dominate the variations in the observed zfs parameters and to establish magnetostructural correlations. Among the tested functionals, the best predictions have been obtained with B3LYP, followed by the nonhybrid BP86 functional, which in turn is more successful than the meta-hybrid GGA functional TPSSh. For the estimation of the spin-orbit coupling (SOC) part of the zfs, it was found that the coupled perturbed SOC approach CP is more successful than the Pederson-Khanna method. Concerning the spin-spin interaction (SS), the restricted openshell Kohn-Sham (ROKS) approach led to a slightly better agreement with the experiment than the unrestricted KS (UKS) approach. The ab initio state-averaged CASSCF (SA-CASSCF) method with a minimal active space and the most recent implementation that treats the SOC and SS contributions on an equal footing provides the best predictions for the zfs. The analysis demonstrates that the major contribution to the axial zfs parameter (D) originates from the SOC interaction but that the SS part is far from being negligible (between 10 and 20% of D). Importantly, the various excited triplet ligand field states account for roughly half of the value of D, contrary to popular ligand field models. Despite covering dynamic correlation contributions to the transition energies, NEVPT2 does not lead to large improvements in the results as the excitation energies of the Mn(III) d-d transitions are already fairly accurate at the SA-CASSCF level. For a given type of coordination sphere (e.g., elongated or compressed octahedron), the magnetic anisotropy of the Mn(III) ion, D, does not appear to be highly sensitive to the nature of the ligands, while the E/D ratio is notably affected by all octahedral distortions. Furthermore, the introduction of different halides into the coordination sphere of Mn(III) only leads to small effects on D. Nevertheless, it appears that oxygen-based ligands afford larger D values than nitrogen-based ligands. Introduction A large number of experimental and theoretical techniques are employed to characterize the physical properties of transition-metal ion complexes and correlate the results to their electronic structures and their reactivity.1,2 The magnetic anisotropy, also called zero-field splitting (zfs, D and E being the axial and rhombic parameters of the zfs, respectively), is among the most important properties that characterize the geometric and electronic environment of a given transition-metal ion with a spin greater than 1/2.3 In particular, in the field of molecular magnetism, the exceptional properties of the well-known Mn12 * To whom correspondence should be addressed. E-mail: carole.duboc@ ujf-grenoble.fr (C.D.); [email protected] (F.N.). † Universite´ Joseph Fourier Grenoble 1/CNRS. ‡ University of Bonn. § Max-Planck-Institut fu¨r Bioanorganische Chemie.

complex4 have opened an entirely new area of research.5 In fact, such nanomagnets could constitute the elementary constituent for data storage in quantum computers.6,7 The efficiency of these molecules as elementary quantum bits relies mostly on the existence of a large magnetic anisotropy, more precisely, a large and negative D value. In order to put the design of singlemolecule magnets on a rational basis, it is of great importance to develop theoretical tools that can reliably predict the magnetic anisotropy in arbitrary compounds. A precise experimental determination of the zfs is generally achieved using EPR spectroscopy.8-11 Depending on the magnitude of D, the frequency of the EPR experiments must be properly chosen. In order to reach the high-field limit that allows for a precise determination of D, high-field EPR (HFEPR) is required whenever |D| > ∼0.5 cm-1, which is usually

10.1021/jp107823s  2010 American Chemical Society Published on Web 09/09/2010

Zero-Field Splitting in Coordination Complexes of Mn(III) the case for Mn(III). HF-EPR is also necessary in order to unambiguously determine the sign of D.7,12 Since the information contained in this single parameter is complex, sophisticated theoretical calculations of the zfs are required for extracting molecular electronic structure level information. Indeed, the zfs is known to arise from the direct electron-electron magnetic dipole spin-spin (SS) interaction between unpaired electrons (to first-order in perturbation theory) together with the spin-orbit coupling (SOC) of electronically excited states into the ground state (to second-order in perturbation theory).3,13-15 To approach this problem, methods based on DFT and ab initio wave function based approaches have been used recently.16-27 Among the first-row transition-metal ions, Mn(III) is the most promising candidate for producing large negative and axial zfs values that are required for single-molecule magnet behavior.28 It is well-known that the large e × E Jahn-Teller effect that is characteristic of octahedral Mn(III) complexes usually leads to axial elongation associated with a negative D value.29 The case of Mn(III) is special in this respect because the high-spin d4 configuration together with the relatively high oxidation state provides particularly large distortions and hence also D values. Second- and third-row transition-metal ions would feature larger spin-orbit couplings, but the stronger metal-ligand bonds lead to low-spin complexes. The same is usually true for higher oxidation states than +III in first-row transition-metal complexes. They either have low-spin ground states or are too reactive for the construction of single-molecule magnets. Unfortunately, the zfs of only a few mononuclear Mn(III) complexes has been predicted by quantum chemistry so far.13,28,30,31 The zfs’s of three relatively simple complexes have been calculated with the ab initio method (CASSCF) and a minimal active space consisting of only the metal d-based molecular orbitals. Despite this simplicity, this appears to be a suitable and readily affordable method for the calculation of the zfs.13,30,31 Consensus has not yet been reached how to best calculate the zfs in a DFT framework. Three alternatives appear to be available, (a) the method of Pederson and Khanna based on perturbation arguments,21 (b) the two-component DFT method advocated by Reviakine et al.19 and (c) the linear response approach proposed by us.17 The relationship between the relative merits of these methods has been discussed before.17,23,30-35 However, none of these approaches has yet given highly accurate predictions for the zfs. From the limited experience gained so far, it seems that the calculated D values from DFT for Mn(III) systems appear to be consistently underestimated by a factor of at least 1/3. The best results are obtained with the linear response approach and inclusion of the direct spin-spin coupling.13,30,31,35 The results are reasonably consistent for different functionals and basis sets. However, hybrid GGA functionals appear to produce slightly better predictions than “pure” functionals. Since very few numerical approximations are made in the evaluation of the SOC and SS parts, it was argued that the underestimation of D is an intrinsic feature of the present-day functionals that treat, in particular, spin-flip contributions incorrectly.13,17 Interestingly these investigations have revealed that the DSS part is far from being negligible since it represents up to 1/3 of the total D value.13,35,36 While the ab initio approach gives more satisfactory predictions of D with respect to DFT, previous CASSCF studies have concentrated on DSOC.13,30,31 Concerning the rhombicity parameter E/D, no tendency for over- or underestimation can yet be deduced from the results, and frequently, a significant scatter has been observed

J. Phys. Chem. A, Vol. 114, No. 39, 2010 10751 in these values.13,30,31 A more systematic study performed on several complexes with various coordination sphere is now necessary to confirm these observations. The main aim of the present paper is to determine how accurately the zfs can be predicted for Mn(III) complexes, especially in a wave function based ab initio framework, and to investigate the magnetostructural variations of the zfs in Mn(III) complexes more systematically. To this end, we have studied a series of mononuclear Mn(III) complexes for which the zfs has been accurately determined by HF-EPR and their structures resolved by X-ray diffraction.37-43 The calculations have been carried out with the ORCA program, which allows for a consistent calculation of the SOC and SS parts of the zfs at both the DFT and CASSCF levels.44 In the DFT studies, we have compared the hybrid meta-GGA functional TPSSh, the hybrid B3LYP, and nonhybrid BP86 functionals. The reason for investigating TPSSh is that it has recently been shown that TPSSh gives superior results for hyperfine couplings, exchange couplings, and spin-state energies compared to other hybrid or GGA functionals.45,46 Concerning the prediction of DSS, two DFT approaches are used that are based on restricted open-shell Kohn-Sham (ROKS) and unrestricted KS (UKS) reference determinants.18,23,47-49 Although the ROKS method has led to distinctly better agreement with experiment in the case of organic radicals,45 no noticeable difference has been found in the case of Mn(II) complexes.23 This can be explained by the fact that, in the case of Mn(II), the major contribution to D usually originates from the SOC contribution,35,36,50 which,surprisingly,isnottrueforMn(III).13,28,30,31 In the ab initio CASSCF framework, we have focused our attention on the importance of the excited states for correctly predicting the DSOC part of the zfs. In order to investigate the impact of dynamic correlation on the predictions, we have resorted to the recently developed second-order N-electron valence perturbation theory (NEVPT2)51-53 that was recently implemented in an efficient form into a development version of the ORCA program. In the NEVPT2 calculations, the wave functions are left at the SA-CASSCF level, but the state energies that enter the quasi-degenerate perturbation theory are perturbed to second order using the original NEV contraction scheme (the strongly contracted variant) together with the Dyall partitioning for H0.54 Theory The zfs, expressed in terms of the D tensor, is the leading spin Hamiltonian (SH) parameter for systems with a ground state spin of S > 1/2.3,15 It describes the lifting of the degeneracy of the 2S + 1 magnetic sublevels MS ) S, S-1, ..., -S (which are exactly degenerate at the level of the Born-Oppenheimer Hamiltonian) in the absence of an external magnetic field. To first order in perturbation theory, the zfs arises from the direct electron-electron magnetic dipole spin-spin interaction between unpaired electrons. To second order, contributions arise from spin-orbit coupling of electronically excited states into the ground state. These effects can be phenomenologically collected in a Hamiltonian of the form

ˆ zfs ) SˆDSˆ H

(1)

where Sˆ is the fictitious spin of the ground state. In a coordinate system that diagonalizes the D tensor, eq 1 can be written as shown in eq 2, in which D and E represent the axial and rhombic terms, respectively.

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ˆ zfs ) D Sˆz2 - 1 S(S + 1) + E(Sˆ2x - Sˆ2y ) H 3

(

)

Duboc et al.

(2)

The choice of the axes is based on the convention 0 e E/D e 1/3.8 Density Functional Formalism. The SS contribution of the D tensor can be calculated on the basis of a single ground-state Slater determinant (e.g., the Kohn-Sham determinant), as in eq 3

Dkl(SS)

g2e R2 ) 4 S(2S - 1)

R-β R-β - Pµκ Pντ } × ∑ ∑ {PµνR-βPR-β κτ µν

κτ

2 〈µν|r-5 12 {3r12,kr12,l - δklr12}|κτ〉

(3)

Here, S ()2) is the total spin of the electronic ground state, ge ()2.002319...) is the free-electron g value, R (∼1/137) is the fine structure constant, PR-β is the spin density matrix in the atomic orbital basis, and µ, ν, κ, and τ are the basis functions. -5 2 {3r12,kr12,l - δklr12 } represents the SS coupling The operator r12 13,18 Spin density matrices derived between a pair of electrons. from restricted ROKS and UKS approaches will be considered in this paper for comparison. The results of ROKS calculations and of those that are obtained on the basis of the spinunrestricted natural orbital (UNO) determinant are virtually indistinguishable, and hence, the latter is used for simplicity. In this paper, the SOC operator is represented by an effective one-electron with the spin-orbit mean field (SOMF) method55 in the implementation described previously.56 An uncoupled perturbation theory has been described by Pederson and Khanna (PK) for the SOC contribution based on the UKS formalism.21 Although this approach is, perhaps, the most widely used, it generally underestimates DSOC, especially in the case of transition-metal monomers. A recently derived linear response method has been proposed based on a coupled perturbed SOC approach (CP).17,20 Both approaches have been compared in this work for evaluating the performance of the CP versus PK methods to the zfs. Four types of excitations have been considered to calculate the DSOC part.13 Qualitatively, they take the form of (i) excitation of a spin-down (β) electron from a doubly occupied MO (DOMO) to a SOMO leading to states of the same spin S as that od ground state (β f β), (ii) excitation of a spin-up (R) electron from a SOMO to a virtual MO (VMO), also giving rise to states of total spin S (R f R), (iii) excitations between two SOMOs that are accompanied by a spin-flip and giving rise to states of S′ ) S - 1 (R f β), and (iv) “shell-opening” transitions from a DOMO to a VMO, leading to states of S′ ) S + 1 (β f R). Multiconfigurational Ab Initio Formalism. The ab initio calculations were based on a complete active space selfconsistent field (CASSCF) treatment. The active space was composed of the five metal d-based orbitals, which are occupied by four electrons. The SOC matrix was diagonalized in the basis of the 5 lowest quintet and 35 lowest triplet states. Singlet states were excluded from the calculations because their contribution to D is negligible.13,30 The spin-spin contribution to the zfs has been evaluated precisely as an expectation value of the twoelectron spin-spin coupling operator over the many-electron ground-state wave function in a way described earlier.57,58 The SOC contribution to the zfs is evaluated in two ways, using quasi-degenerate perturbation theory (QDPT)14 together

TABLE 1: Experimental zfs Parameters of the Synthetic Mononuclear Mn(III) Complexes and Nature of the Coordination Sphere of the Mn(III) Ion

[Mn(terpy)Cl3]a (1) [Mn(pterpy)Cl3]b (2) [Mn(bpea)F3]c (3) [Mn(terpy)F3] (4) [Mn(bpea)(N3)3] (5) [Mn(terpy)(N3)3] (6) [Mn(tpp)Cl]d (7) [Mn(tpfc)(OPPh3)]e (8) [Mn(dbm)2(py)2]f (9) [Mn(OD2)6]3+ (10)

Dexp (cm-1)

E/Dexp

coordination sphere

ref

-3.46 -3.53 -3.67 -3.83 +3.50 -3.29 -2.29 -2.69 -4.50 -4.49

0.12 0.09 0.19 0.11 0.23 0.15 0 0.01 0.09 0.06

N3Cl3 N3Cl3 N3F3 N3F3 N6 N6 N4Cl N4O N2O4 O6

37 37 38 38 38 39 40 41 42 43

a terpy ) 2,2′:6′,2′′-terpyridine. b pterpy ) 4′-phenyl-,2′:6′,2′′terpyridine. c bpea ) N,N-bis(2-pyridylmethyl)-ethylamine. d tpp ) tetraphenylporphyrin. e tpfc ) 5,10,15-tris-(pentafluorophenyl) corrole; OPPh3 ) triphenylphosphine oxide. f dbm ) 1,3-diphenyl-1,3-propanodionate; py ) pyridine.

with a matching procedure.13 The results are a reasonably consistent and direct application of the original second-order perturbation equations22 using the SA-CASSCF wave functions. In the NEVPT2 calculations, only the perturbed energies enter the calculation while the wave functions remain at the SACASSCF level. This approach is comparable to the calculation of SOC effects in the framework of the CASPT2 method in conjunction with the restricted active space state interaction (RASSI) procedure;59-61 see the related discussions and references in refs 62 and 63. Computational Details All calculations have been performed with the ORCA program package.44 The zfs parameters have been calculated on the basis of the X-ray structures. For a discussion of this point, see refs 23, 30, 34, and 64. The DFT-based zfs calculations were performed with the hybrid B3LYP,65 hybrid meta-GGA TPSSh,66 and nonhybrid BP8667,68 functionals using the TZVP basis set69 and taking advantage of the RI approximation with the auxiliary TZV/J Coulomb fitting basis sets.70 Increased integration grids (Grid4 in ORCA convention) and tight SCF convergence criteria were used. CASSCF calculations were performed as described previously.13 We have taken the simplest possible approach in which only the five metal d-based orbitals are in the active space. In this respect, these calculations resemble an ab initio version of a complete ligand field theory. The energies of the d-d multiplets are often already wellpredicted by this simple approach.71 The structures of the hypothetical small models were fully optimized using the BP86 functional67,68 and the TZVP basis set.69 Scalar relativistic effects were taken into account for the halide derivates using our implementation of van Wu¨llen’s model potential approximation72 to the ZORA equations73 in conjunction with the one-center approximation.74 Experimental Data Calculations have been performed on 10 complexes for which the structures have been determined by X-ray diffraction and the zfs’s were measured by HF-EPR spectroscopy. This set of compounds covers a large range of D values ranging from +3.50 to -4.50 cm-1. Table 1 summarizes the experimental data, and the X-ray structures are shown in Figure 1. As mentioned above, the calculations were performed on the experimental molecular

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Figure 1. Representation of the 10 Mn(III) complexes studied in this work. Hydrogen atoms are omitted for clarity. Color code: Mn (pink); C (gray); O (red); N (blue); F (yellow); Cl (green); P (orange).

geometries. Optimization of the geometries generally leads to a significant deterioration of the theoretical predictions (less than 6% for the Mn(III) ion).23,30,34,64 The zfs parameters in Table 1 have been measured on powder samples, as described in refs 37-43. Results Synthetic Complexes. DFT Calculations. We have tested different methodological choices including various density functionals and methods for calculating DSOC and DSS in an effort to define the best DFT-based methodology for predicting the zfs parameters for Mn(III) complexes. This complements earlier work along the same lines.13,28,30,31 The quality of a given treatment is best judged from the standard deviation (SD) of the linear regression analysis since it directly relates to the predictive power of the theoretical method (in the ideal case, both the slope and the correlation coefficient are unity of course). The results of the analysis are presented in Table 2. TPSSh Versus B3LYP and BP86. From the data in Table 2, it emerges that TPSSh is an unexpectedly poor choice for the calculation of D tensors. The standard deviation and the correlation coefficient are significantly inferior to those from the B3LYP and BP86 results. In some cases, the calculated TPSSh D values match the experimental values slightly better than those from B3LYP (D9 ) -4.50 (exp), -3.34 (B3LYP), and -4.56 cm-1 (TPSSh)). However, the sign of D is frequently in error in the TPSSh calculations, and its predicted magnitude is either under- or overestimated. The B3LYP- and BP86derived D values consistently underestimate the experimental

TABLE 2: Calculated versus Experimental D Parameters for the Test Set of the Synthetic Mn(III) Complexesa

BP86-CP (UNO) B3LYP-CP (UNO) TPSSh-CP (UNO)b BP86-PK(UNO) B3LYP-PK (UNO) TPSSh-PK (UNO)b B3LYP-CP (UKS) CASSCF NEVPT2

correlation coefficient

slope

standard error (SD)

0.951 0.987 0.888 0.947 0.968 0.897 0.953 0.993 0.994

0.618 0.712 0.964 0.535 0.597 0.895 0.640 0.957 0.995

0.499 0.290 1.317 0.451 0.438 1.243 0.578 0.277 0.264

a Linear regression results (the SS contribution is included in all calculations). b For the calculations with the TPSSh functional, the sign of D has not been considered in the analysis.

D values, but the sign of D is correct in all cases. This result was somewhat unexpected since the TPSSh functional has been shown to improve results over other hybrid or GGA functionals for hyperfine couplings, exchange couplings, and spin-state energies.45,46 On the other hand, TPSSh has shown rather poor performance in a recent benchmark study for the g tensors of small radicals.75 A closer analysis shows that the deviations observed in this work are almost entirely due to the SOC part and not to the SS part that is rather insensitive to the choice of functional. Hence, it appears that TPSSh is rather grossly inaccurate for magnetic response property calculations. Even if B3LYP shows a slightly better performance than BP86, considering the significantly higher computational cost of the

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TABLE 3: Experimental and Calculated zfs Parameters (D in cm-1 and E/D) and D Contributions (in cm-1) of the Synthetic Mononuclear Mn(III) Complexes with the B3LYP Functional Using the CP and the UNO Approaches for the SOC and SS Contributions, Respectively 1 2 3 4 5 6 7 8 9 10 a

Dexp

DDFT

DSOC

R f Ra

β f βa

R f βa

β f Ra

DSS

E/Dexp

E/DDFT

-3.46 -3.53 -3.67 -3.83 +3.50 -3.29 -2.29 -2.69 -4.50 -4.49

-2.25 -2.30 -2.43 -2.17 2.56 -2.79 -1.26 -1.82 -3.34 -3.16

-1.68 -1.87 -1.81 -1.85 1.76 -2.05 -0.78 -1.37 -2.14 -1.95

-0.55 -0.46 -0.51 -0.50 0.44 -0.59 -0.29 -0.34 -0.56 -0.50

-0.29 -0.30 -0.11 -0.09 0.11 -0.26 0.24 -0.08 -0.14 -0.03

-1.06 -1.31 -1.20 -1.28 1.19 -1.10 -0.73 -0.98 -1.55 1.44

0.22 0.20 0.01 0.02 0.02 -0.10 -0.01 0.03 0.11 0.02

-0.57 -0.43 -0.63 -0.32 0.81 -0.74 -0.47 -0.45 -1.20 -1.22

0.12 0.09 0.19 0.11 0.23 0.15 0 0.01 0.09 0.06

0.09 0.10 0.25 0.21 0.13 0.20 0.18 0.03 0.11 0.05

The excitations contributing to the total DSOC value.

calculations, BP86 remains the most efficient DFT method for the prediction of zfs’s of large, mononuclear Mn(III) complexes. CP Versus PK. The SOC part of D has been calculated based on the two competing approaches proposed by PK21 and ourselves (CP17). From the results in Table 2, it emerges that the CP method yields larger D values that are closer to those from experiment in all cases. This is consistent with previously published benchmark calculations performed on mononuclear Mn(II) and Mn(III) complexes30,31,34 and is mainly due to the revised prefactors for the spin-flip terms that have been more rigorously derived in the CP theory.17 Indeed, the parts of DSOC that is most noticeably method-dependent are the contributions of the R f β spin-flip excitations. Furthermore from the SDs coefficients, the CP method is more successful than the PK approach. UNO Versus UKS. The SS contribution to D is far from being negligible for the Mn(III) ion. Previous studies have revealed that it accounts for as much as 30% of the total D value.13,30,31 It is therefore important to define the best approach to calculate DSS. In Table 2, we have compared the results obtained with both the UKS and UNO methods. A trend to larger DSS values is observed with the UNO approach relative to the UKS calculations that amount to 0-20%. This difference must be attributed to spin-polarization that for two-electron spindependent observables is unphysical in spin-unrestricted approaches. The increased D values from the UNO estimation, again, lead to an improvement of the predicted D values compared to experiment. However, the improvements are not as significant as those for organic radicals where the SS part strongly dominates D.18 From this comparison, the combination of the B3LYP functional, the CP approach for the SOC part of D, and the UNO variant for the calculation of the SS part of D emerges as our favorite DFT-based approach with BP86 together with CP, and UNO is the method of choice if efficiency of the calculations is a point of concern. This former combination has been used for the remainder of this work. Sign of D. The prediction of the sign of D is not straightforward because it becomes ambiguous when E/D approaches the rhombic limit (E/D ≈ 1/3). For all Mn(III) complexes tested in this study, the sign of D is correctly predicted in the B3LYP calculations, even for those complexes that are characterized by large E/D values (Table 3). This is different from the case met in Mn(II) complexes, where the calculations were shown to be unreliable once E/D became larger than about 0.2.23 This difference between Mn(II) and Mn(III) may originate from the Jahn-Teller character of Mn(III). This effect is intimately linked to the structure of the complex, and the sign of D can, in the

overwhelming majority of cases, be successfully predicted from classical ligand field theory.8,76 SS and SOC Contributions. Table 3 shows that the SS contribution is significant since its magnitude varies from 15 to 30% of the final D value. Interestingly, the sign of DSS is always the same as that of the final D value. This result confirms the importance of this interaction for calculating the zfs of Mn(III) complexes and puts the conclusions of ref 13 onto more firm ground. Comparable results were obtained for other Mn(II) and Mn(III) complexes.13,31,35,77 Concerning the DSOC part, the major contribution arises from the R f β spin-flip excitations (around 70% of DSOC), while the β f R contributions are essentially negligible. In a qualitative picture, the R f β excitations are dominated by ligand field quintet-triplet excited states, while the β f R contributions correspond to ligand-tometal charge-transfer transitions. Since SOC between the ground state and d-d excited states is much larger than the corresponding SOC with LMCT states, this result is sensible. It is noteworthy that in the DFT calculations, the spin-lowering excitations dominate over the spin-conserving excitations that are the only ones being typically considered in ligand field treatments. E/D Ratio. The experimental E/D values of the test set are in the range of 0-0.23. Previous studies reported that the DFTcalculated E/D ratios are typically overestimated.13,30 Our results show no clear trend in this respect. E/D is overestimated for complexes 3, 4, 6, and 7, underestimated for 5, and wellpredicted for 1, 2, 9, and 10. Once more, this confirms the considerable problems that DFT has in the prediction of E/D. A similar conclusion has been reached in systematic studies on the Mn(II) ion.23,36 Apparently, the five contributions that enter the calculation of the D tensor are difficult to obtain with sufficient precision and balance in order to pick up the relatively small differences between the axial tensor components. Ab Initio Calculations. The ligand field treatment of the highspin d4 configuration attributes the major part of the zfs to the SOC between the ground-quintet configuration and the lowlying spin quintet states that arise from excitations within the d shell.76 However, consistent with our earlier discussion13 and the DFT results above, the correctly spin-coupled triplet ligand field excited states should also be included in the treatment. Properly spin-coupled multiplet states are readily obtained from multiconfiguration wave function based ab initio calculations, of which the state-averaged CASSCF method is the simplest and computationally most affordable one. In this case, the SOC effects are obtained from QDPT,14,78 which amounts to diagonalizing the Born-Oppenheimer, SOC, and SS operators in a basis of quintet and triplet roots of the Born-Oppenheimer

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TABLE 4: Experimental and Calculated zfs Parameters (D in cm-1 and E/D) and D Contributions (in cm-1) of the Synthetic Mononuclear Mn(III) Complexes with the CASSCF Ab Initio Approach 1 2 3 4 5 6 7 8 9 10

Dexp

DCASSCF

DSOCSA-CASSCF

DSOCNEVPT2

DSS

DS)2

E/Dexp

E/DCASSCF

-3.46 -3.53 -3.67 -3.83 +3.50 -3.29 -2.29 -2.69 -4.50 -4.49

-3.66 -3.69 -3.54 -3.66 +3.39 -3.45 -1.98 -2.43 -3.85 -4.13

-3.19 -3.23 -3.08 -3.17 +2.97 -2.98 -1.57 -1.94 -3.36 -3.63

-3.23 -3.30 -3.11 -3.19 +3.18 -3.09 -1.71 -2.47 -3.63 -3.84

-0.47 -0.47 -0.46 -0.48 +0.41 -0.47 -0.41 -0.49 -0.49 -0.49

-1.79 -1.60 -1.82 -1.89 +1.57 -1.72 -0.32 -0.49 -1.77 -2.00

0.12 0.09 0.19 0.11 0.23 0.15 0 0.01 0.09 0.06

0.10 0.10 0.25 0.10 0.26 0.16 0.26 0.01 0.02 0.04

Hamiltonian (extended to treat all M ) S, S-1, ..., -S components of a given state ΨSM I ). Since the results are obtained in form of the (real) eigenvalues and (complex) eigenvectors of the SOC+SS extended Hamiltonian, a matching procedure must be applied in order to determine the SH parameters.13 Here, one has to fit the lowest-energy levels (the five lowest roots of the relativistic Hamiltonian that are dominated by the nonrelativistic quintet ground-state configuration) to the SH. The drawback of this approach is that higher-order terms in the SH are combined into the biquadratic SH parameters. However, since the quartic zfs parameters are at least 1-2 orders of magnitude smaller than the biquadratic ones (unless high symmetry forces the latter to be 0), this is a very small price to pay. Alternatively (and slightly less accurately), the entire D tensor can be calculated form perturbation theory using the general expression developed earlier.22 Unless the ground state is nearly orbitally degenerate, the two approaches yield very similar D and E/D values, and the perturbation approach has the benefit of yielding the entire D tensor and its orientation. In the calculations reported below, all the five possible quintet states in the d shell as well as 35 triplet states were included in the calculations. None of the 100 singlet states have been considered since they have been found to provide a negligible contribution to the zfs.13 This is sensible because these states are only expected to contribute to fourth- and higher-order the D values. In Table 4, all of the zfs values calculated for the presently investigated synthetic mononuclear Mn(III) complexes are reported. The D values are considerably closer to the experimental values than the DFT results, as confirmed by the SD, the slope of the regression line, and the linear regression coefficient (Figure 2 and Table 2).

Figure 2. Comparison of correlations between calculated and experimental D parameters from DFT (B3LYP functional with the CP and UNO approaches to calculate the SOC and SS contributions, respectively) and ab initio (CASSCF) approaches.

As established by previous studies,13,30,31 the SOC contribution calculated by the SA-CASSCF method is noticeably larger than that by DFT, leading to values that agree better with experiment (Figure 2). It is pleasing to observe that by including dynamic correlation into the calculation by the NEVPT2 scheme, the results further improve, though only slightly. Dynamic correlation usually works to increase the absolute value of D (by 0-20%) by slightly lowering the excitation energies that enter prominently into either the perturbation22 or QDPT14 procedures (Tables 2 and 4). Since with calculation of the NEVPT2 correction takes much less time than the CASSCF calculation itself, the NEVPT2 approach can be generally recommended for such calculations. This conclusion likely also holds for other dN configurations in which the application of DFT-based methods is less successful. The integration of the SS part into the wave function based ab initio calculations considerably improves the prediction of D relative to the experimental numbers. If both effects are fully treated, the SD is only 0.277 (SA-CASSCF) or 0.264 cm-1 (NEVPT2) (Table 1). Furthermore, a slope of essentially unity is obtained with the NEVPT2 approach () 0.995), and the correlation coefficient demonstrates the excellent quality of the results (R ) 0.994; Table 2). In the wave function based ab initio calculations, the DSOC part contributes between 80 and 88% of the total D value. The DSS part contributes to the zfs with an almost constant value (0.46-0.49 cm-1) in all complexes. This might be understood from the fact that the 10 systems all feature similar ligands, and hence, there is not a large variation in the metal-ligand covalency that would markedly influence an expectation value property. Thus, the DSS parameter is not sensitive to the geometry and/or the nature of the coordination sphere of the Mn(III) ion. The results are nevertheless still unexpected since with comparable ab initio methods, the calculated SS interaction is sensitive to the chemical environment of the Co(II) ion.79 On the other hand, the DSS values calculated by DFT cover a larger range of magnitudes (0.31-1.32 cm-1) (Table 3). The importance of the triplet states for the zfs has been estimated from separate calculations that either only involve the quintet states or involve both the quintet and triplet states (Table 4). In each case, the triplet states contribute about half of the total D value, thus confirming and emphasizing again the crucial role played by these states. Except for complex 7, the E/D ratios obtained by the SACASSCF (or NEVPT2) method are predicted with exceptionally high quality, as demonstrated by the standard deviation of only 0.032. Hence, these methods are much more successful than DFT in this respect. Hypothetical Models. We have studied a series of hypothetical models which are computationally straightforward to handle, namely, trans- and cis-[Mn(NH3)4(X)2]+, with X

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) I, Br, Cl, F (labeled trans,cis-MX), and [MnL6]3+, with L ) NH3, OH2 (labeled MN6 and MO6). The main aim is to investigate the effect of the nature and the arrangement of the halide ligands and the influence of nitrogen- versus oxygen-based ligands on D. Geometric Structure. Geometry optimizations at the BP86/ TZVP level (with the introduction of scalar relativistic effects at the ZORA level72,73 for the halide derivatives) were performed starting from pseudo-octahedral coordination geometries (Table S2 and Figure S1, Supporting Information). The optimized structures correspond to octahedra with a clear axial geometry for the cis-MX, trans-MF, MN6, and MO6 models. Except for trans-MF, which displays a compressed geometry, in the other systems, the octahedra are elongated. On the other hand, for the trans-MX models (X ) I, Br, Cl), the manganese-ligand (M-L) bond lengths are noticeably different along the three principal axes and avoid an unambiguous assignment of the nature of the tetragonal distortion from the relative comparison of the M-L distances (as frequently proposed80). Nevertheless, in these three models, it can be established that an elongation occurs when comparing their metrical data to those of MN6. Electronic Structure. We have investigated the electronic structure of the models using the set of quasi-restricted molecular orbitals (QRMOs)13,81 since this approach was successfully employed in previous works (Figure S2, Supporting Information).20,35,36,82 Except for trans-MF, the empty d-based MO is the Mn-dx2-y2 based molecular orbital (for simplicity, henceforth referred to simply as the dx2-y2 orbital) for all models, in agreement with an elongated octahedral geometry. The most surprising result is certainly the fact that the nature of the halide has no significant effect on the energy level diagram in both configurations (cis versus trans). A noticeable difference is observed between the electronic structures of MN6 and MO6, with a significant splitting of the dxz and dyz orbitals for MO6. Despite the slight deviations of the MnO6 core in [Mn(OD2)6]3+ present in CsMn(SO4)2 · 12D2O from idealized D4h symmetry, a significant magnitude for the rhombic zfs parameter (E) has been experimentally measured by different techniques (elastic83 and inelastic neutron scattering84 as well as HF-EPR43 experiments). It has been proposed that this rhombicity originates from the π-anisotropic nature of the Mn(III)-water interaction. In the proposed experimental structure (as in our MO6 model), the coordination mode of the water molecules is trigonal planar, which enhances the π-anisotropy with respect to a trigonal pyramidal coordination mode. Therefore, our result confirms that even if the coordination mode of the water ligands on the Mn(III) ion has a negligible effect on the geometric structure, it plays a significant role in the electronic structure of the complexes and leads to rhombic magnetic anisotropy (see below). EPR Parameters. The zfs parameters of the models have been calculated using the SA-CASSCF ab initio method presented above (Table 5). The total D magnitudes all fall in the range of 2.94-4.25 cm-1. The calculated D values for cis-MCl, cis-MF, and MO6 agree very well with the experimental data determined on the chloro (1,2), fluoro (3,4), and aqua (10) complexes, respectively, thus demonstrating that one can indeed learn something meaningful from studying such small-model complexes. The major contribution to D arises from DSOC (85-90%), while the DSS part, again, shows a quasi-constant magnitude between 0.45 and 0.51 cm-1. The three lowest excited triplet states (originating from 3T1g) contribute between 50 and 65% to D (DS)2 contributes between 35 and 50%).

Duboc et al. TABLE 5: Calculated zfs Parameters and the D Contributions (in cm-1) for the Theoretical Models Using the CASSCF Ab Initio Approach I

cis-M cis-MBr cis-MCl cis-MF trans-MI trans-MBr trans-MCl trans-MF MN6 MO6

D

DSOC

DSS

DS)2

E/D

-2.94 -3.32 -3.68 -3.82 -3.04 -3.35 -3.55 4.10 -3.85 -4.25

-2.47 -2.84 -3.20 -3.32 -2.67 -2.89 -3.09 3.63 -3.37 -3.76

-0.47 -0.48 -0.48 -0.51 -0.46 -0.46 -0.45 0.47 -0.48 -0.50

-1.07 -1.44 -1.81 -1.93 -1.07 -1.40 -1.68 1.77 -1.82 -2.26

0.03 0.02 0.01 0.00 0.32 0.26 0.25 0.00 0.00 0.10

The magnitude of D increases with increasing electronegativity of the halide. This tendency is unexpected since contradictory results have been found with several other transition-metal ions3 such as high-spin Ni(II)85 and Mn(II)86,87 complexes. This explains our previous experimental results that showed that D increases from the chloro (4) to the fluoro (1) complexes. The most important difference between the cis and trans configurations arises from the E/D ratio, which is in perfect agreement with the electronic structure of the models; the cis systems are purely axial, whereas the trans systems are rhombic (except trans-MF). For trans-MF, the degeneracy of the dxz and dyz orbitals is reflected in the vanishing E/D ratio, and the positive sign of D agrees with an empty dz2 MO. Interestingly, the calculated E/D for MO6 is close to that determined experimentally (E/Dcalc ) 0.10l E/Dexp ) 0.06). The absence of a comparable rhombicity for MN6 confirms the hypothesis that the trigonal planar coordination mode of the water molecules is certainly at the origin of π-anisotropy, leading to a significant E value. Discussion and Conclusion The investigations presented in this paper have been focused on the assessment of theoretical methods for estimating the zfs parameters in Mn(III) complexes and analyzing the different contributions to D with the aim of understanding the physical origin of the magnetic anisotropy. Although it has been demonstrated in a previous work that the use of different hybrid GGA functionals, such as B3LYP, does not notably influence the calculated D values,28 we have tested here the hybrid meta-GGA functional TPSSh. This functional has previously been shown to significantly improve the prediction of the nuclear hyperfine coupling in radicals and transition-metal ion complexes (Mn, Ni and Cu compounds) and lead to better exchange couplings and also to better spinstate energies than other functionals.45,46 However, in the case of the Mn(III) ion, the results obtained with TPSSh are disappointing because the sign of D is not reliably reproduced and the magnitude of D is not predicted. Hence, TPSSh appears to have a problem with magnetic response properties. Similarly disappointing results were found for the g tensors of small radicals.75 On the other hand, B3LYP, as was already demonstrated,28,30 leads to fairly reasonable predictions of the zfs in Mn(III) complexes. We have investigated different DFT approaches to estimate the SS and SOC contributions to D. For the spin-spin interaction, the UNO and UKS approaches lead to good predictions, but the UNO results are preferred. For the estimation of DSOC, it was confirmed here that the CP approach is slightly superior to the PK method.

Zero-Field Splitting in Coordination Complexes of Mn(III) A significant result of this present study is the surprisingly high quality of the prediction of D with the ab initio SACASSCF method, as demonstrated by the excellent correlation found between experimental and calculated D parameters. This theoretical framework leads to the lowest standard deviation, the best correlation coefficient, as well as a slope of the correlation line close to 1. Furthermore, this method even affords reasonably accurate E/D ratios, whereas DFT appears to fail in this respect. Even better results are obtained if dynamic correlation contributions to the excitation energies are considered in the framework of the NEVPT2 second-order many-body perturbation theory. Since the computational cost of the NEVPT2 is smaller than that of the necessarily preceding CASSCF calculation, the NEVPT2 scheme can be generally recommended for D tensor calculations. At a second level of analysis, the different contributions to D have been quantified. The largest contribution arises from the SOC interaction and especially from the R f β spin-flip excitations that account for about half of the total D value throughout the test set. More surprisingly, the SS part, even if it is far from being negligible, has an almost constant value for all Mn(III) complexes and models. Unexpectedly, from the investigation of the simplified theoretical models, it has been shown that the electronic structure is particularly sensitive to neither the nature of the halides present in the coordination sphere of the Mn(III) ion nor their configuration. This confirms the experimental results obtained on synthetic models for which no significant differences in D havebeenobservedbetweenfluoro,chloro,andazidocomplexes.37-39 Furthermore, the fact that comparable D values are found for the synthetic and theoretical Mn(III) complexes reveals that the constraints brought about by the polydentate nature of the ligands in the synthetic complexes investigated here do not noticeably contribute to D. Our study also confirms the importance of the coordination mode of water molecules in Mn(III) complexes. A water molecule bound to Mn(III) via a trigonal planar coordination mode induces a splitting of the dxz and dyz based molecular orbitals. The resulting rhombicity manifests itself, both experimentally and theoretically, in elevated E/D values. In the course of designing new nanomagnets, the definition of magnetostructural correlations for the transition-metal ions that serve as elementary building blocks is of particular interest. For Mn(III), the results presented here are, to some extent, disappointing since they show that the zfs is not highly sensitive to the coordination sphere of this ion. Thus, very sterically enforcing ligands may be necessary in order to strongly influence the inherently large and negative zfs of Mn(III). The incorporation of a halide ligand in the coordination sphere decreases the D value compared to nitrogen- or oxygen-based ligands. The only advice that can be voiced at this point is to prefer oxygenbased ligands over nitrogen-based ligands in order to achieve large D values. Other types of ligands, such as thiolates, should be tested as well. Work along these lines is in progress in our laboratories. It is important to explore these chemical variations in the context of mononuclear complexes as it was recently argued that large clusters may not be necessary in order to achieve single-molecule magnet behavior.28,88-90 Acknowledgment. C.D. and M.N.C. thank the Agence Nationale pour la Recherche (Grant No. ANR_JC09_435677) for financial support. F.N., D.G., and K.S. thank the Special Research Unit SFB 813 (Chemistry at Spin Centers) for financial support.

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