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Cite this:DOI: 10.1039/c5cs00898k

Determination and prediction of the magneticanisotropy of Mn ions†

Carole Duboc

This tutorial is dedicated to the investigation of magnetic anisotropy using both electron paramagnetic

resonance (EPR) spectroscopy for its experimental determination and quantum chemistry for its

theoretical prediction. Such an approach could lead to the definition of magneto-structural correlation

essential for the rational design of complexes with targeted magnetic properties or for the identification

of unknown reactive metallic species involved in catalysis. To illustrate this combined approach the high

spin MnII, MnIII and MnIV ions have been taken as specific examples. The first part deals with the analysis

of the EPR experiments as a function of the ions under investigation and the conditions of the

measurements, specifically: (i) EPR spectra recorded under high vs. low frequency conditions with

respect to magnetic anisotropy, (ii) EPR spectra of non-integer (Kramers) vs. integer (non-Kramers) spin

states and (iii) mono- vs. multi-frequency EPR spectra. In the second part, two main quantum chemical

approaches, which have proven their capability to predict magnetic anisotropy, are described. More

importantly, these calculations give access to the different contributions of zero field splitting, key

information for the full understanding of magnetic anisotropy. The last part demonstrates that such a combined

experimental and theoretical approach allows for the definition of magneto-structural correlations.

Key learning points(1) Why magnetic anisotropy is a key physical parameter to understand the magnetic behaviour and the reactivity of transition metal complexes.(2) How magnetic anisotropy can be accurately determined by multi-frequency EPR spectroscopy including high field and high frequency EPR.(3) How quantum chemistry can help in the understanding of the origin of magnetic anisotropy and in the rationalization of magneto-structural correlation.(4) Why systematic investigations are required to define the appropriate theoretical methods to predict the magnetic anisotropy of a specific metal.

1. Introduction

Magnetic anisotropy reflects how the magnetization of a transitionmetal complex will tend to orient preferentially along one orseveral directions. In the domain of magnetic materials, thepeculiar properties of single-molecule magnets (SMMs) will bein part controlled by the magnetic anisotropy. This field ofresearch is particularly developed because SMMs have potentialapplications in many domains including spintronics, quantumcomputing and data storage. On the other hand, magneticanisotropy is also used as a probe for the identification andcharacterization (oxidation and spin states, coordinationsphere) of transition metal-based species involved in biologicalor chemical processes because of its high sensitivity to the

structural and electronic environment of the metal. For transitionmetal complexes with S 4 1/2, the magnetic anisotropy comesfrom both the Zeeman effect (via the effect of an externalmagnetic field) and zero field splitting (ZFS) (in the absenceof an external magnetic field), defined by the g and D tensors,respectively. ZFS removes the degeneracy of the 2S + 1 spinmicrostates (called Zeeman levels, MS) associated with a givenS, as a consequence of molecular electronic structure and/orspin density distribution. Two interactions contribute to ZFS: (i)the direct electron–electron magnetic dipole spin–spin coupling(SSC) between unpaired electrons (to first order in perturbationtheory) and (ii) the spin–orbit coupling (SOC) of electronicallyexcited states into the ground state (to second order in perturbationtheory). From the spin-Hamiltonian formalism (eqn (1)), the Dtensor is described by the D and E parameters, which representits axial and rhombic contributions, respectively. Note thatD a 0 if the symmetry is lower than cubic and E a 0 if thesymmetry is lower than axial. The consequence of the axialcontribution to the ZFS on the energy of the MS levels is the

University Grenoble-Alpes, CNRS, UMR 5250, Departement de Chimie Moleculaire,

301 rue de la Chimie, 38041 Grenoble cedex 9, France.

E-mail: [email protected]

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5cs00898k

Received 7th December 2015

DOI: 10.1039/c5cs00898k

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generation of Kramers (odd spin) or non-Kramers (even spin)doublets. Its rhombic contribution leads to a further splitting ofthe MS levels. The spin Hamiltonian formalism dealing withmononuclear and polynuclear metal complexes has been extensivelyinvestigated and is well documented in the review of Boca,1 theZFS Hamiltonian being that given in eqn (1)

HZFS = D[Sz2 � S(S + 1)/3] + (Sx

2 � Sy2) (1)

with S the spin operator describing the spin projection along agiven axis.

The experimental determination of ZFS parameters is farfrom being straightforward and several techniques can be appliedincluding indirect methods such as magnetometry and variable-temperature variable-field magnetic circular dichroism, or directmethods such as inelastic neutron scattering and frequencydomain magnetic resonance spectroscopy. However, the mostprevalent and powerful technique to determine ZFS parametersremains the electronic paramagnetic resonance (EPR) spectro-scopy for a large range of ZFS values. Meanwhile, significanttheoretical efforts have allowed the development of appropriatequantum chemical methodologies to predict ZFS parameters togive insight into the physical origin of the magnetic anisotropyand/or to help in the analysis of experimental observations.

This tutorial review will focus on the experimental determinationof ZFS parameters by means of multi-frequency EPR spectroscopy(Section 2) and on their prediction by quantum chemicalmethods including DFT and ab initio approaches (Section 3).More specifically these two points will be illustrated through theexamples of high-spin Mn-based complexes in different oxidationstates from +II to +IV. Manganese undoubtedly represents one ofthe best candidates for such a tutorial for several reasons. (i) Thehigh spin state is preferred for MnII, MnIII and MnIV leading tointeger (S = 2 for MnIII) and non-integer (S = 5/2 and 3/2 for MnII

and MnIV, respectively) spin systems. (ii) The D parameter hasbeen accurately determined for these ions by multi-frequency

EPR spectroscopy, especially through the investigation of seriesof MnII and MnIII complexes, and predicted in different quantumchemical frameworks through systematic studies. (iii) The MnII,MnIII and MnIV complexes offer a broad range of D-magnitudes(values of 0.01 cm�1 up to 5 cm�1) in a large variety ofcoordination spheres. (iv) In the context of SMMs, manganesewas a special case since the single molecule magnet behaviour hasbeen first discovered in the famous [Mn12O12(CH3CO2)16(H2O)4]complex composed of eight MnIII and four MnIV ions,2 andremains widely used for the development and investigation ofsuch molecules, including single ion SMMs.3,4 (v) Because asignificant number of Mn-based complexes are implicated infundamental biological reactions or important synthetic catalyticprocesses, the definition of appropriate fingerprints to identifyintermediate species is of great interest.5

The last section will discuss the magnetostructural correlationsthat can be established from combined experimental andtheoretical investigations and how they can help to predicteither a D value for a specific environment or a structure from aD value. Such predictions can be very helpful for the rationaldesign of SMMs or the elucidation of catalytic mechanisms.

The scope of this review will mainly target mononuclearMn complexes whose structures have been resolved by X-raydiffraction in most of the cases.

2. Multi-frequency EPR spectroscopy:the most powerful tool for determiningZFS parameters

EPR spectroscopy allows the investigation of the spin andoxidation states of all paramagnetic transition metal complexesand provides information on the geometric and electronicstructures of these systems. The basic concepts of EPR will bedescribed using the simplest case of an S = 1/2 spin statesystem. When a magnetic field (B0) is applied to an unpairedelectron, the magnetic moment associated with its spin inter-acts with B0. As a result, the so-called electronic Zeemaninteraction lifts the degeneracy of the MS = �1/2 energy levels.When the energy of the microwave (GHz) radiation (hn =radiation frequency) provided by the EPR spectrometer exactlymatches the energy splitting between these two levels (DE), anabsorption occurs corresponding to the transition of an electronfrom the lower to the higher level (eqn (2)).

DE = hn = gmBB0 (2)

with g being the g-factor (or the Lande g-factor), a proportionalityconstant, whose value depends on the nature of the matterunder study.

To improve the signal to noise ratio, an oscillating magneticfield is applied parallel to B0, to produce the derivative shape ofthe absorption spectrum. In cw-EPR experiments, the frequencyof the radiation is kept constant and the applied magnetic fieldis swept until the resonance condition is fulfilled. From theresonance field, a g-factor can be determined that deviatesfrom that of the free electron (ge = 2.0023192778) because of

Carole Duboc

Carole Duboc received her PhDfrom the University of Grenoblein 1998 under the supervision ofProfessor Marc Fontecave andDoctor Stephane Menage.Following postdoctoral positionat the University of Minnesota,with Professor William Tolman,she joined the High MagneticField Laboratory at Grenoble in2000 and the Department ofMolecular Chemistry at Grenoblein 2007, where she is now CNRSsenior researcher. Her current

research interests are on the elucidation of the electronic structure oftransition metal ion complexes using an original approach combiningexperimental and theoretical data and on bio-inspired complexescontaining metal–sulfur bond(s) to develop structural and/orfunctional models of metalloenzymes.

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spin–orbit coupling. The generated small local field adds to B0

producing Dg shifts, whose magnitude depends on the environ-ment of the unpaired electron. Additional features can appear inthe EPR spectrum due to the interaction between the unpairedelectron and the nuclear spin (I) of the metal ion or of the boundatoms, called the hyperfine or superhyperfine interaction,respectively. The nuclear spin I splits each MS level into 2I + 1nuclear levels, called MI and transitions occur between two MI

levels so that DMI = 0. When I a 0, the EPR signal is thereforesplit into several equidistant lines (2nI + 1, where n = thenumber of equivalent nuclei).

In this ‘isotropic’ description (fluid solution), a single transitionassociated with giso (for the isotropic g-factor) is observed in theresulting EPR spectrum arising from the average of all molecularorientations with respect to B0. In such conditions, the isotropichyperfine coupling constant, Aiso, can be measured from thedistance in field between two of these lines. Interestingly,because Aiso is directly related to the extent of delocalizationof the unpaired electron(s) over the molecule, it can provideuseful information regarding the distribution of the spin densityon the different atoms.

However, when EPR experiments are conducted on powderor frozen solution, the Zeeman interaction becomes ‘anisotropic’.In this case, the magnetic axes related to specific orientationsof the molecule can be defined, associated with three differentg-factors (often referred to gx,y,z). Accordingly, the shape of theEPR spectrum will depend on the symmetry of the transitionmetal complex: one, two or three lines are observed for anisotropic (gx = gy = gz = giso), axial (gx = gy = g> a gz = gJ)or rhombic (gx a gy a gz) system, respectively. This descriptionalso applies for the anisotropic hyperfine interaction with threeA-values (Ax, Ay, Az) related to three orientations that are usuallycollinear to the magnetic axes because the orientation of theg and A tensors is generally consistent with the molecularsymmetry axes of the paramagnetic centre.

For paramagnetic systems with S 4 1/2, there is an additionalterm in the Hamiltonian corresponding to ZFS leading to eqn (3)

HS = SgB + SAI + SDS (3)

From these three terms, corresponding to the Zeeman effect,the hyperfine interaction and ZFS, respectively, the followingparameters can be extracted: the g matrix elements (gx, gy and gz),the A matrix elements (Ax, Ay and Az) and the D and E parametersfrom the symmetric traceless D tensor (Dx + Dy + Dz = 0).Conventionally D = 3Dz/2 and E = (Dx � Dy)/2 with|Dz| Z |Dy| Z |Dx|, leading to 0 r E/D r 1/3.

In the case of a neat powder, the intermolecular dipole–dipole interactions together with the D-strain (distribution ofthe principal D-value due to small structural variations amongthe paramagnetic centres in the sample) lead to a broadeningof the lines, which generally preclude observation of thehyperfine structure in the spectrum. The anisotropy of theZeeman interaction is very small for high spin MnII (S = 5/2),MnIII (S = 2) and MnIV (S = 3/2). Therefore, the correspondingcomplexes are characterized by g-values close to ge (with Dg o0.05). Consequently, the shape of the EPR spectra of these Mn

ions only depends on the ZFS terms. Fig. S2 (ESI†) is dedicatedto give a more general view of the expected shape of EPR spectrarecorded under experimental conditions that can resolveg-anisotropy and ZFS.

Two experimental conditions can be distinguished whichlead to different spectral shapes (see below): (i) high-frequencylimit conditions when the Zeeman effect is the dominantinteraction. Here, hn c |D|, so that the quantum energyprovided by the radiation of the EPR spectrometer is muchlarger than the magnitude of D, (ii) low-frequency conditionswhen ZFS is the dominant interaction. Here, |D| c hn. Notethat the accurate determination of ZFS parameters usually requireshigh frequency limit conditions.

EPR experiments can be carried out at different frequencies,typically at X-(9.4 GHz, 0.3 cm�1, g = 2 at 0.330 T) and, lesscommonly, Q-(34 GHz, 1.2 cm�1, g = 2 at 1.250 T) bandfrequencies combined with magnetic fields generally in therange 0–1.7 T, but also by means of high-frequency and high-field EPR (HF-EPR) spectrometers (from 3 cm�1/95 GHz up to33 cm�1/1 THz). Therefore, depending on the ZFS, appropriatefrequency(ies) should be used.

EPR data for transition metal ions are usually recorded atlow temperature (below 30 K) due to the relaxation properties ofthe ions. Different physical phenomena related to phonon-induced transitions contribute to the spin lattice relaxation,T1. At low temperature, the direct spin-phonon processes aredominant, leading to long T1 values and thus to intense signals.When the temperature increases, the two-phonon processes(Raman and Orbach processes) became efficient, leading to adecrease of T1 and thereby broadening of the EPR features.Consequently, a strong decrease of the intensity of the EPRfeatures is observed until its possible complete disappearance.

Additional single-crystal EPR experiments can also allow thedefinition of the orientation of the anisotropy axes (D tensororientation), data that are essential to fully understand themagnetic behavior of single ion SMMs for instance. However,such intricate experiments remain rare because they requirecrystals of decent size, and a detailed study of the angulardependence of the EPR spectra. Besides, the presence of morethan one type of molecule or orientations of the molecule in thecrystal drastically complicates the analysis.

2.1 High-spin MnII

Only the high-spin MnII ion (S = 5/2) will be discussed here: lowspin MnII complexes are rare and not relevant in the context ofthis tutorial, as they have an S = 1/2 spin state. The magneticanisotropy of MnII is usually characterized by small D-magnitudes(o1 cm�1, Table 1), except for a polyoxometalate complex, whichdisplays the largest magnetic anisotropy for MnII (D = +1.460 cm�1).6

When |D| c hn (where ZFS is the dominant interaction), as inthe case of X-band EPR experiments on MnII with |D| 4 0.30 cm�1,the three Kramers doublets (|�1/2i,|�3/2i,|�5/2i) will be separatedfrom each other by larger energies than o0.3 cm�1 (X-bandfrequency) corresponding to low frequency conditions measure-ments (Fig. 1). Consequently, transitions can only occur insideeach Kramers doublet, also called intra-doublet transitions.

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Transitions along each direction (x, y and z) are expected foreach doublet and the corresponding spectra can be described interms of ‘‘effective’’ S = 1/2 spectra with the three g-values foreach Kramers doublet also called effective g values (geff). The E/Dratio can be estimated from the resonance field (in terms of geff)

of the observed transitions, using calculated two-dimensionalgraphs called rhombograms (Fig. 1). The investigation of therelative intensity of the different intra doublet transitions as afunction of temperature can also provide an estimation of |D|and its sign (Fig. 1). At low temperature, only the |�1/2i (|�5/2i)

Table 1 Experimental and calculated ZFS parameters for mononuclear MnII complexes

Complexa Coord. number Coord. sphere Dexpb (cm�1) Eb (cm�1) E/D Dcalc

c (cm�1) Ref.

Iodide complexes[Mn(tppo)2(I)2] 4 O2I2 0.906 0.223 0.246 +1.264d 36 and 37[Mn(terpy)(I)2] 5 N3I2 +1.000 +0.190 0.190 +1.863 7 and 37trans-[Fe(Mn)(pyr)4(I)2] 6 N4I2 0.932 0.020 0.021 38cis-[Mn(tpa)(I)2] 6 N4I2 �0.600 �0.095 0.158 �0.868 27

Bromide complexes[Mn(tppo)2(Br)2] 4 O2Br2 0.507 0.134 0.263 +0.715d 36 and 37[Mn(terpy)(Br)2] 5 N3Br2 +6050 +0.159 0.26 +0.983 7 and 37trans-[Ru(Mn)(pyr)4(Br)2] 6 N4Br2 0.665 0.001 0.002 38cis-[Mn(tpa)(Br)2] 6 N4Br2 �0.360 �0.073 0.203 �0.510 27

Chloride complexes[Mn(tppo)2(Cl)2] 4 O2Cl2 0.165 0.045 0.273 +0.288d 36 and 37[Mn(terpy)(Cl)2] 5 N3Cl2 �0.260 �0.075 0.29 �0.459 7 and 37[Mn(tmc)(Cl)]+ 5 N4Cl 0.250 0.25 33trans-[Ru(Mn)(pyr)4(Cl)2] 6 N4Cl2 0.220 0 0 38cis-[Mn(tpa)(Cl)2] 6 N4Cl2 +0.115 +0.020 0.174 +0.155 27[Mn(L5

2)(Cl)](PF6) 6 N5Cl 0.180 0.060 0.33 +0.254 39[Mn(L4

3)(Cl)(OH2)](ClO4) 6 N4OCl +0.110 +0.016 0.14 +0.150 39

N/O-based ligands complexes[Mn(terpy)(NCS)2] 5 N5 �0.300 �0.050 0.170 �0.504 7 and 28K11Na[As2W20Mn(OH2)3O66] 5 O5 +1.460 +0.033 0.230 36 and 37trans-[Fe(Mn)(pyr)4(NCS)2] 6 N6 0.010 0.003 0.300 38cis-[Mn(tpa)(NCS)2] 6 N6 �0.085 �0.015 0.176 �0.075 28[Mn(L5

2)(OH2)](BPh4)2 6 N5O �0.137 0 0 �0.089 39[Mn(pb)(CF3CO2)2] 6 N4O2 �0.071 �0.005 0.070 �0.029 40[Mn(terpy)(CF3CO2)2(OH2)] 6 N3O3 0.068 0.021 0.308 �0.041 40[Mn(mal)(OH2)3] 6 O6 +0.023 +0.007 0.324 41[Mn(L6

2)(OH2)](ClO4)2 7 N6O �0.127 0 0 �0.030 39[Mn(LOH

2)(OH2)](ClO4)2 7 N5O2 0.137 0.013 0.093 42[Mn(bpea)(NO3)2] 7 N3O4 +0.086 +0.008 0.093 +0.075 40

a A schematic representation of all ligands is given in Fig. S1 of the ESI. b The sign is reported when it has been determined. c Calculated values bythe CP-DFT approach. d Calculations based on optimized structures.

Fig. 1 (left) Energy levels of an S = 5/2 spin system of axial symmetry (for D 4 0) in the absence and in the presence of a magnetic field. The expectedEPR transitions between each intra Kramers doublets are represented as black arrows. (middle) Energy levels of an S = 5/2 spin system with D 4 0 andE/D = 0.1 in the absence of a magnetic field.34 (right) Rhombograms, calculated diagrams representing the expected three g effective values(corresponding to x, y and z axes) for each of the three Kramers doublet for an S = 5/2 system as a function of the E/D value.


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doublet is populated for D 4 0 (D o 0) and leads to thecorresponding transitions (see the rhombograms in Fig. 1).On increasing the temperature, the |�3/2i doublet becomespopulated and new transitions are observed, whose intensity asa function of the temperature is directly correlated to the energygap of 2D (4D) between the two doublets.

When |D| { hn (where the Zeeman effect as the dominantinteraction), in the case of HF-EPR experiments on MnII with|D| 4 0.30 cm�1, the MS levels vary linearly with B corres-ponding to high-frequency limit conditions measurements(Fig. 2, example of the energies of the Zeeman levels as afunction of the magnetic field when applied along the z-axis).Five transitions are then expected for each orientation of themagnetic field along any of the principal directions of D(Fig. 3). The central line corresponds to the contribution ofthe |�1/2i- |+1/2i transitions along the three axes, centred atg C ge. With respect to the position of this central line, the fieldposition of the other transitions linearly depends on the ZFSparameters (Fig. 3). For an axial system (E/D = 0 with Dx = Dy a Dz),

seven transitions shifted by �4D/gmB, �2D/gmB, �D/gmB and 0are expected at high temperature when all Zeeman levels arepopulated. In the opposite situation of a completely rhombicsystem (E/D = 1/3 with Dx = 0 and Dy = Dz), only five transitions at�4D/gmB, �2D/gmB and 0 are expected.

Low-temperature HF-EPR spectra may directly provide thesign of D. Under these conditions, only the lowest MS levels ofthe multiplet are populated (Boltzmann population) and thetransitions associated with the z-axis are located on the low-field side of the spectrum (compared to g = 2) for D o 0, whilethe reverse is true for D 4 0. An unambiguous confirmation ofthe D-sign is obtained by comparing the low-temperatureexperimental data with simulations calculated with both signs.

The mononuclear MnII complexes investigated by EPRspectroscopy can be into two classes: the complexes with N-,O- and S-based ligands with 0.01 o |D| o 0.30 cm�1 and thecomplexes with halide ligands with 0.10 o |D| o 1.00 cm�1.Consequently, Q-band EPR experiments (sometimes combinedwith X-band EPR experiments at different temperatures) usuallyallows the precise determination of the ZFS parameters for thefirst class, while for MnII halide complexes, HF-EPR is requiredto meet high frequency limit conditions.

As an example, 230 GHz powder spectra of a MnII complexrecorded at different temperatures is shown in Fig. 2, displayingfeatures between 5 and 12 T.7 At 5 K, the furthest transitionfrom the centre of the spectrum (g = 2 at 8.21 T), which can beassociated with the |�5/2i - |�3/2i transition along z islocated on the high field part consistent with D 4 0 (Fig. 3).From its field location at 10.95 T, D can be estimated as+0.64 cm�1 (the field difference of 2.74 T between this lineand g = 2 is equal to 4|D|, see Fig. 2). The E value can beestimated as 0.17 cm�1 from the |�5/2i - |�3/2i transitionalong y at 5.82 T (the field difference of 2.39 T between this lineand g = 2 is equal to 2|3E + D|, see Fig. 2). The E/D ratio can thusbe estimated as 0.27. The shape of the spectrum is consistentwith such rhombicity with the close proximity of the |�5/2i-|�3/2i transitions along y and z observed below 6 T at 30 K(Fig. 3). From these estimates, the accurate values of the ZFSparameters (D = +0.605 cm�1, E = +0.159 cm�1, E/D = 0.26) havebeen achieved by simulating the spectra through full diagonalizationof the Hamiltonian eqn (2).

It has been shown that the ZFS parameters can be alsodetermined from the central |�1/2i - |+1/2i transitions (ataround g = 2) under specific conditions, i.e. in solutions or indiluted diamagnetic matrixes.8 The EPR spectrum resulting ofthe addition of the |�1/2i- |+1/2i transitions along x, y and zis complex because it arises not only from the nuclear hyperfineinteraction but also from ZFS. Typically, a sextet (small spectralwidth of about 300 mT) originating from transitions betweenthe six nuclear sublevels (IMn = 5/2 with DMI = 0, see above)of the two Zeeman levels (MS = �1/2) is expected (Fig. 4).However, when ZFS becomes the dominant interaction withrespect to the hyperfine interaction, additional transitionsinvolving the simultaneous change in the nuclear and electronicspin state are observed. Because their intensity depends onthe ratio between the zero field and the Zeeman interactions,

Fig. 2 (top) Energy diagram of the six MS levels deriving from S = 5/2along the z-axis under an applied magnetic field. The reported transitions(red lines) correspond to an energy of 230 GHz. (bottom) Powder experimental230 GHz-EPR spectra recorded on [Mn(terpy)(Br)2] at different temperatures(5, 12.5 and 30 K). EPR parameters extracted from these spectra:D = +0.605 cm�1, E = 0.159 cm�1, E/D = 0.26, gx = 1.985, gy = 1.975,gz = 1.965. This figure has been reproduced with permission of Springer.35


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observations of such spectra will change as a function of therecording conditions. As an example, Fig. 4 displays HF-EPR spectrarecorded on [Zn0.97(Mn)0.03(terpy)2]2+ at 30 K (|D| = 0.058 cm�1).9

While in high frequency limit conditions (where the Zeemaneffect is the dominant interaction), i.e. 285 GHz, only a sextet isobserved from which only the g- and A-values can be measured.At lower frequencies (i.e. 95 GHz, where ZFS becomes thedominant interaction), a complex multiplet appears, fromwhich the ZFS parameters can be also determined using afull-matrix diagonalization procedure. As a result, dependingon the magnetic anisotropy of the complex and the parametersto be determined, the choice of EPR frequency is crucial fordiluted systems.

2.2 MnIV

D-values in MnIV complexes (S = 3/2) range from 0.2 cm�1 up to2.3 cm�1 (Table 2). Consequently, high frequency limit conditionscan be reached only using HF-EPR. Under low frequencyconditions, i.e. at X- and Q-band frequencies, only intra Kramersdoublet (|�1/2i and |�3/2i) transitions can be observed and theE/D value can be extracted by using rhombograms (Fig. 5).Practically, only a strong low-field signal is usually observedand its position ranges from geff = 4.1 for a pure axial system(E/D = 0) to geff = 5.5 for a totally rhombic one (E/D = 1/3). This isdue to the fact that MnIV preferentially presents a positive Dvalue (see Table 2) leading to |�1/2i as the ground doublet.Therefore, at low temperature (below 10 K) only the |�1/2idoublet is populated, while no signal is observed when thetemperature increases due to the relaxation properties of suchparamagnetic species (see above).

When high frequency limit conditions are met, the analysisof the resulting S = 3/2 EPR spectra is comparable to those ofMnII, the only difference is that fewer features are expected(absence of the |�5/2i- |�3/2i and |+3/2i- |+5/2i transitions,see Fig. 3). More recently, it has been shown that D can beprecisely determined by combining X- and Q-band EPR experi-ments in the case of moderate values (D = +0.88 and +0.67 cm�1),i.e. intermediate conditions between high and low field ones.However, such investigation necessitates a cautious analysis oftemperature dependent multi-frequency EPR spectra.

On the other hand, a recent HF-EPR study performed on aMnIV complex, for which a D-value was extracted from a Q-band

Fig. 3 Schematic representation of the first order predictions for the EPR transitions for an S = 5/2 spin system, such as MnII, as a function of E/D in highfrequency limit conditions. Magnetic field positions of the transitions are shown with the respect to the g = 2 field value and in the case of D 4 0. Notethat the central line centred at g = 2 corresponds to the addition of the transitions |�1/2i- |+1/2i along the three axis (x, y and z). This representation isalso consistent with an S = 3/2 spin system, such as MnIV. In this case the diagram is simplified with six lines less (the |�5/2i- |�3/2i and |+3/2i- |+5/2itransitions along the three axis as dotted lines).

Fig. 4 Experimental (exp) and simulated (sim) 25 K EPR spectra recordedon [Zn(Mn)(terpy)2]2+ at 95 (A) and 285 (B) GHz. Parameters used for thesimulation: |D| = 0.058 cm�1, |E| = 0.006 cm�1, E/D = 0.104, giso = 2.00,Aiso = 71 � 10�4 cm�1. This figure has been reproduced with permission ofSpringer.35


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EPR spectrum (D = +0.47 cm�1)10 has revealed a much largerD-value (D = +2.289 cm�1),11 demonstrating that in the absenceof high frequency limit conditions, a precise determination ofD cannot be achieved based on a single EPR frequency.

Multi-frequency HF-EPR spectra of a mononuclear MnIV

complex are displayed in Fig. 6.11 This is an interesting examplebecause a near zero-field transition can be observed at 60 GHzfrom which a direct estimation of the D-magnitude can beafforded (|D| = 1.00 cm�1). Indeed such a transition corre-sponds to the energy gap (2|D|, see Fig. 5) between the twoKramers doublets of the quartet state (MS = �1/2 and �3/2).This estimation is confirmed by the 151 GHz EPR spectrum.From the EPR line at 3.35 T assigned to the |�3/2i - |�1/2itransition with B0 along the z-axis, |D| can be estimated as0.96 cm�1 (field difference of 2.05 T between this line and g = 2(5.40 T) is equal to 2|D|). The low-field position of this line isconsistent with a negative D. The resonances at 6.30 and 6.68 Tare assigned to |�3/2i- |�1/2i transitions, with B0 along the xand y-axis respectively, and their splitting (6|E|) is coherentwith a moderate rhombicity (|E| = 0.059 cm�1 leading to an E/Destimation of 0.06). From these estimates, the accurate valuesof the ZFS parameters (D = �0.997(6) cm�1, E = �0.054(3) cm�1,E/D = 0.054) were obtained from a two-dimensional field/frequency

map (Fig. 6). From this map, it can also be deduced thatfrequencies larger than 200 GHz (about 6 cm�1, thus six timelarger than the magnitude of D) and a field above 3.5 T are requiredto observe the nine-line pattern corresponding to ‘‘pure’’ |�3/2i-|�1/2i, |�1/2i - |+1/2i and |+1/2i - |+3/2i transitions expectedfor a moderate rhombic S = 3/2 system (Fig. 3).

Note that EPR spectra of MnIV-containing inorganic materialscan be also rec multiplicity order. As an example, the localstructure and oxidation state of the MnIV ion in LiCo1�2xNixMnxO2

have been investigated by HF-EPR. The ZFS parameters have beendetermined through measurements carried out on powder sampleunder high frequency limit conditions. Such investigation canprovide key information to help in the rational design of thesepotential electrode materials.12

2.3 High-spin MnIII

The electronic properties of high-spin MnIII are of interest inthe context of this tutorial because they lead to an integer spin,S = 2, characterized by larger magnetic anisotropy than thoseof MnII and MnIV, (|D| generally lies in the small range from2.5–4.5 cm�1, Table 3). These so-called non-Kramers systemsare used to be qualified as ‘‘EPR silent’’ species, but they candisplay EPR signatures either at X-band under certain conditions

Table 2 Experimental and calculated ZFS parameters for mononuclear MnIV complexes

Complexa Coord. sphere Dexpb (cm�1) Eb (cm�1) E/D Metho. Dcalc (cm�1) Ref.

[Mn(Me3tacn)(OMe)3]+ N3O3 +0.245c 0.00 0.00 CP-DFT +0.163 11 and 43NEVPT2 �0.13 25

[Mn(bigH)2]4+ N2O4 �0.997(6)c �0.054(3) 0.054 CP-DFT �0.695 11[MnLN2O4] N2O4 +1.650c 0.00 0.00 CP-DFT +1.341 44

NEVPT2 +0.96 25[Mn(LSO2)2] O4S2 +2.289(5)c +0.323(4) 0.141 CP-DFT +1.377 11Mn4+ doped into LiAlO2 O6 �0.586c �0.026 0.044 CP-DFT �0.711 12[MnH3buea(OH)] N4O +0.88(5)d +0.150 0.31(1) CP-DFTe +1.38 13 and 25

+0.67(5)d +0.114 0.17(1) NEVPT2e �3.02 25[Mn(OH)2(Me2EBC)] N4O2 0.75(25)d 0.11 0.15(2) CP-DFTe +0.56 25

NEVPT2e +0.25 25[Mn(OH)(O)(Me2EBC)] N4O2 +1.2(4)d 0.26 0.22(1) CP-DFTe +1.08 25

NEVPT2e �2.70 25

a A schematic representation of all ligands is given in Fig. S1 of the ESI. b The sign is reported when it has been determined. c Determined byHF-EPR. d Determined by X-band EPR. e Calculations based on optimized structures.

Fig. 5 (left) Energy levels of an S = 3/2 spin system of axial symmetry (for D 4 0) in the absence and in the presence of a magnetic field. The expectedEPR transitions between each intra Kramers doublets are represented as black arrows. (middle) Energy levels of an S = 3/2 spin system with D 4 0 andE/D = 0.1 in the absence of a magnetic field.34 (right) Rhombograms, calculated diagrams representing the expected three g effective values(corresponding to x, y and z axes) for each of the two Kramers doublet for an S = 3/2 system as a function of the E/D value.


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or at high frequencies. For high spin MnIII, the MS levels are splitinto two doublets (|�1i, |�2i) and one singlet (|0i) (Fig. 7).

In low frequency conditions, only intra-doublet transitions canoccur inside each non-Kramers doublet. However, these transitions

are difficult to observe with a regular set-up, in which B1 isperpendicular to B0 (applied magnetic field), but with a specificset-up, mainly developed at X-band frequency, in which B1 isparallel to B0. In such conditions, the selection rules are different,

Table 3 Experimental and calculated ZFS parameters for mononuclear high spin and intermediate spin MnIII complexes

Complexa Coord. number Coord. sphere Dexp (cm�1) E (cm�1) E/D Metho. Dcalc (cm�1) Ref.

S = 2N/O/C complexes[Mn(TBHP8Cz)] 4 N4 �2.60(2) 0.015 0.006 45[Mn(tpfc)(OPPh3)] 5 N4O �2.69(2) �0.030(3) 0.011 CP-DFT �1.82 26 and 46

CASSCF �2.43[Mn(NCTPP)(py)2] 6 N5C �3.084(3) �0.608(3) 0.197 47[Mn(bpea)(N3)3] 6 N6 +3.50(1) �0.82(1) +0.234 CP-DFT +2.56 26 and 48

CASSCF +3.39[Mn(terpy)(N3)3] 6 N6 �3.29(1) �0.48(1) �0.146 CP-DFT �2.79 16

�3.45[Mn(opbaCl2)(py)2] 6 N4O2 �3.246(2) �0.115(1) 0.035 CP-DFT �2.53 49

CASSCF �3.47[Mn(dbm)2(py)2](ClO4) 6 N2O4 �4.504(2) �0.425(1) 0.094 CP-DFT �3.34 26 and 50

CASSCF �3.85[Mn(acac)3] 6 O6 �4.52(2) 0.25(2) 0.055 CP-DFT �3.12 32 and 51

CASSCF �3.67[Mn(OH2)6]2+ 6 O6 �4.491(7) 0.248(5) 0.055 CP-DFT �3.16 26 and 52

CASSCF �4.13

Halide complexes[Mn(tpp)Cl] 5 N4Cl �2.290(5) 0.00(1) 0 CP-DFT �1.26 15, 26 and 53

CASSCF �1.98[Mn(cyclam)I2]I 6 N4I2 +0.604 +0.034 0.056 31[Mn(cyclam)Br2]Br 6 N4Br2 �1.1677(7) �0.0135(6) 0.116 54[Mn(terpy)Cl3] 6 N3Cl3 �3.53 0.30 0.085 CP-DFT �2.25 26 and 55

CASSCF �3.66[Mn(terpy)F3] 6 N3F3 �3.82(2) 0.75(2) �0.196 CP-DFT �2.17 26 and 48

CASSCF �3.66

S = 1[Tp2Mn](SbF6) 6 N6 +17.97(1) +0.42(2) 0.023 DFT +10.78 18[Tp*2Mn](SbF6) 6 N6 +15.89(2) +0.04(1) 0.003 DFT +9.94 18

a A schematic representation of all ligands is given in Fig. S1 of the ESI.

Fig. 6 (left) Experimental powder HF-EPR spectra of a MnIV complex with a N2O4 coordination sphere, recorded at 60, 101, and 151 GHz and at 4.5–7 K.The spectra are approximately normalized to the amplitude of the strongest line. (right) Field vs. frequency map of resonances observed in this MnIV

complex at 4.5–7 K. The squares are experimental HF-EPR resonances, while the curves were simulated using the following best-fit spin Hamiltonianparameters: S = 3/2, D = �0.997(6) cm�1, E = �0.054(3) cm�1, gx = 1.994(4), gy = 1.988(4), gz = 1.954(13). Color caption: red for B0||x, blue for B0||y; blackfor B0||z, and green for off-axis turning points. The dashed vertical bars represent the three frequencies at which spectra shown on the left wererecorded. Reprinted with permission from ref. 11. Copyright 2016 American Chemical Society.


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allowing the observation of intra non Kramers doublet transi-tions by notably increasing their intensity compared to that of theregular transitions.13 However, a spectrum can be observed onlyif 3E2/D o 0.3 cm�1 at X-band frequency. It is thus possible todetect transitions from the two doublets at around g = 8 and g = 4for the |�2i - |+2i and |�1i - |+1i transitions, respectively.From these transitions, the ZFS parameters can be determined bya simulation approach. Note that a silent parallel mode EPRspectrum is not proof of the absence of a MnIII system, it onlymeans that either 3E2/D 4 0.3 cm�1 or E/D = 0.

The ZFS parameters of a series of MnIII complexes have beenprecisely investigated by HF-EPR, as MnIII represented a ‘‘casestudy’’ at the end of the 1990s for the ZFS determination ofX-band EPR-silent species with an integer spin state.14,15 Because|D| is usually large enough to prevent high frequency conditionsfrom occurring in a reasonable frequencies range (95–500 GHz),multifrequency HF-EPR experiments are required to analyze thedata and allow an estimate of the ZFS parameters. As an example,Fig. 8 shows an experimental HF-EPR spectrum recorded on[Mn(terpy)(N3)3] at 285 GHz.16 In the low field part of thespectrum, the high frequency limit conditions are not met andthe relation between the energy of the MS levels and the field is nolonger linear, while in the high field part, the linear relation isagain observed along the three principal molecular axes (seeFig. 8 the evolution of the energy of the Zeeman levels as afunction of the magnetic field along the x and y axes). If E/D = 0(Dx = Dy a Dz), eight transitions shifted from the centre of thespectrum (with g C ge) by �0.5D, �D, �0.5D and �3D areexpected, while when E/D = 1/3 (Dx = 0 and Dy = Dz) only fivetransitions are expected at �D, �3D and g = 2. In the presentcase, a feature at 6.50 T at 475 GHz, associated with the |�2i-|�1i transition along the z-axis allows the estimation of |D|(3.27 cm�1) since the field difference between its position andthe centre of the spectrum (g = 2, 17.00 T at 475 GHz)corresponds to |D|. Its low field position is consistent with anegative D.

Interestingly, the HF-EPR spectrum of [Mn(terpy)(N3)3] alsoreveals unexpected features: some of the x- and y-transitions appearas a doublet instead of a singlet (for instance the |�2i - |�1itransition along y at 13.5 T at 285 GHz). This has been rationalizedby the fact that because of the low symmetry of the complex,

the D and g tensors are no longer collinear as is usuallyhypothesized to a first approximation. This example illustrateswell that caution has to be taken in this step of the analysis andthat one should be aware of all approximations made duringthe simulation process.

Recently it has been shown that HF-EPR can also be adaptedto investigate the properties of materials, especially oxide-basedmaterials based on the hexagonal phase of YInO3 doped withMnIII. Interestingly, HF-EPR experiments have been carried outat room temperature and the ZFS parameters have been determinedwith precision for the isolated MnIII ions to give a completedescription of the electronic structure of the material. In addition,analysis of the line width of the EPR transitions demonstrated thatEPR can be used to probe the dopant level of the materials.17

Intermediate S = 1 spin state is extremely rare for MnIII.However a recent work reports on the precise determination ofthe ZFS through a multifrequency EPR investigation and on itsprediction by DFT and ligand field theory calculations.18 Therefore,even if this point is not further described in this tutorial, it isinteresting to point out that such complexes lead to the largestZFS values reported for a manganese ion (Table 3).

3. Quantum chemistry for predictingand rationalizing magnetic anisotropy

As mentioned in the introduction, two interactions contributeto ZFS: spin–spin coupling (SSC) and spin–orbit coupling (SOC).While SSC represents the main contribution when organicmolecules are involved, for metal-based systems the SOC partbecomes predominant. However, to obtain a reasonable predictionof the ZFS parameters, both contributions have to be considered.Quantum chemistry is the ideal tool for connecting the spin-Hamiltonian parameters to microscopic interactions occurringin the full molecular Hamiltonian. Over the last decade, thedevelopment of various frameworks to predict the ZFS para-meters has been impressive. Among them, two approaches thatwill be the heart of the following section have attracted particularattention: density functional theory (DFT) techniques and wave-function methods based on multiconfigurational self-consistentfield (SCF) techniques (such as Complete Active Space SCF

Fig. 7 (left) Energy levels of an S = 2 spin system of axial symmetry (for D 4 0) in the absence and in the presence of magnetic field. The expected EPRtransitions between the two intra non-Kramers doublets are represented as black arrows. (middle) Energy levels of an S = 2 spin system with D 4 0 andE/D = 0.1 in the absence of magnetic field.34 (right) Schematic representation of the first order predictions for the EPR transitions for an S = 2 spin system,such as MnIII, as a function of E/D in high frequency limit conditions. Magnetic field positions of the transitions are shown with the respect to the g = 2field value and in the case of D 4 0.


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(CASSCF) and related extensions). It should be noticed here thatLigand Field Theory (LFT), which has remained for half a centurythe standard tool to predict the ZFS parameters and to rationalizetheir origin, is at the origin of a renewed interest since LFT-basedmethodologies are currently under development includingLF-DFT and ab initio DFT.11,19–21

3.1 DFT

As far as metal-based systems are concerned, the main focushas been on the treatment of the SOC contribution to ZFS. Thisis a particular challenge for open shell systems because the totalspin does not remain a good quantum number in unrestrictedKohn–Sham framework (other excited states are taken into account).In 2007, Neese et al. implemented the SOC formalism in a practicalway based on a DFT framework using a quasi-restricted DFTapproach that produced mathematical expressions revised fromthose of the Pederson–Khanna (PK) model, with additionalprefactors for a better description of the spin-flip contributions.22

A recently derived linear response method has been proposedbased on a Coupled Perturbed SOC approach (CP-DFT), onwhich the rest of this section will be focused.23

Interestingly, the different SOC contributions can be extractedfrom these calculations giving important insight into the magneticanisotropy of the systems. Four types of excitations (Fig. 9)have been considered in order to calculate DSOC: those of

spin-conserving excitations (i) from a doubly occupied molecularorbital (DOMO) to a singly occupied MO (SOMO) (b- b) and (ii)from a SOMO to a virtual MO (VMO) (a - a) and those of spin-flip excitations (iii) between two SOMOs with a resultingS � 1 spin state (a - b) and (iv) from a DOMO to a VMO witha resulting S + 1 spin state (ba).

The SSC contribution in a DFT framework was first achievedby Petrenko et al. and then implemented for large-scale applicationin different electronic structure variants by several authors.24

In line with this approach, DFT calculations have beenapplied to predict both the sign and magnitude of D for manyMn-based systems (Fig. 10). It should be noticed that in contrastto MnIII and MnII, investigations related to the prediction ofthe ZFS for MnIV are scarce and have been reported only veryrecently.11,25 Even if the magnitude is not always correctlyreproduced when compared to the experimental data, suchDFT calculations are very useful to help in the rationalization ofexperimental trends by properly defining the different contributionsto the total D-value (DSSC and those of DSOC). In a general manner, ithas been shown for high spin MnII, MnIII and MnIV, that the bestpredictions are obtained when calculations have been carried outbased on the experimental X-ray structures rather than fully on theoptimized structures despite the later being of good quality. This isespecially true for MnII, for which D is very sensitive to any structuralmodifications.

Fig. 8 (Top) Plots of energy vs. field for the five energy levels arising from an S = 2 spin state using the following parameters: D =�3.29 cm�1, E = 0.50 cm�1,giso = 2.00. The field is parallel to the molecular axis indicated in each diagram. The observed resonances at 285 GHz are indicated by vertical bars along.(Bottom) Experimental and simulated powder HF-EPR spectra of polycrystalline [Mn(terpy)(N3)3] measured at 285 GHz. The parameters used for thesimulation are D = �3.29 cm�1, E = 0.51 cm�1, y = 121, gx = 2.00, gy = 1.98, gz = 2.01 with y the angle the angle between the g and D tensors along x and y.


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The prediction of the sign of D is not straightforward andthe quality of the prediction will depend on the E/D ratio.Indeed, the sign becomes ambiguous when E/D approachesthe rhombic limit (E/D = 1/3). However, for Jahn–Teller systems,such as MnIII, the sign of D is reproduced well because it isintimately related to the Jahn–Teller distortion as predictedfrom the classical ligand field theory: a positive (negative) signis associated with a compressed (elongated) tetragonal axis(Fig. 10). On the other hand for MnII, and to a lesser extentfor MnIV, it has been shown that the calculated sign of D isunreliable once E/D becomes larger than 0.2.

DFT calculations lead to a systematic underestimation of Dfor MnIV and MnIII with a low impact on the quality of theprediction from the choice of the functional (the GGA BP86 vs.the hydrid B3LYP).25,26 On the other hand, as far as MnII isconcerned, the choice of the functional becomes critical. WhileB3LYP produces a better prediction of D in the case of halide-based MnII complexes, BP86 has proven to be the best functionalwhen only N-/O-based ligands are present.27,28 It should benoticed that all systematic studies, regardless of the oxidationstate of the Mn, have revealed that the CP approach is moresuccessful than the PK method, mainly due to the revisedprefactors for the spin-flip terms that have been more rigorouslyderived in the CP theory.

As mentioned above, DFT has proven to be a powerfulmethod for the understanding of the origin of magnetic anisotropy.For MnIV and MnIII, DSSC represents about 20 to 30% of the totalD-value (Dtot), a much smaller contribution than that of DSOC

but significant enough not to be neglected in the calculations.On the other hand, for MnII, the relative SSC contribution versusSOC depends on the structural properties of the complexes,including the nature of the ligands and the coordination number:(i) for halide MnII complexes, DSSC contributes from 2% to 20% ofDtot, (ii) for five-coordinate MnII complexes about 30% and (iii) forsix-coordinate MnII complexes from 2% up to 90%. These resultsunderline that the SSC cannot be neglected when predicting D,especially for high-spin d5 systems.

Generally, the major contribution to Dtot for these high-spinMn ions is DSOC but with different origins that depend on theiroxidation state. In the case of MnIII (S = 2) the major contributioncomes from a - b spin flip excitations (50 to 70% of DSOC),which can be seen as S = 1 excited states, while for MnIV from the

a- b and to a lesser extent the a- a excitations. In the complexcase of MnII, the origin of DSOC also depends on the coordinationsphere of the complexes i.e. the nature of the ligands and thecoordination number. In the halide complexes, all four excitationsequivalently contribute to DSOC with different signs, with eachcontribution being about four times larger in magnitude than Dtot.The quality of the prediction of Dtot thus depends on a precisecalculation of small differences between large contributions ofvarying signs and of different physical origins. For MnII complexeswith only N- and/or O-based ligands, DSOC mainly arises fromthe a - b and to a lesser extent the a - a excitations and allexcitations display contributions with the same sign of Dtot

when the Mn is five-coordinated. For the six-coordinate systems,as mentioned above, Dsoc usually represents the minor contributionof Dtot and is characterized by a very small magnitude whose originvaries in a non-rational manner. At this point, it clearly appears thatboth the range and the origin of D differ for each of these threeMn ions, demonstrating the interest in such calculations as amethod of rationalizing the observed experimental data.

DFT has considerable problems in giving good predictionsof the E/D ratio, particularly for compounds with small ZFS,MnII is a good example of this because of the apparent difficultyin determining the relatively small differences between the DX

and DY components of the D tensor.An additional interest of DFT calculations is to provide

insight into the orientation of the principal tensor axes versusthe molecular frame of the investigated system. Although suchkey information could help in understanding the origin of theobserved magnetic behavior of a system, experimental data arerarely available because this requires single-crystal EPR investigation.

3.2 Multireference wave function approaches

Although multireference wave function approaches generallylead to better results that DFT for the prediction of the ZFSparameters, it is much more expensive in computational costand less accessible. Indeed a preliminary analysis of theelectronic structure of the system under study should be carriedout to properly define the active space considered for thecalculations. Description of these different methods has beenthe subject of many reviews29,30 and will not be developed in thecontext of this tutorial. In general, the calculations requirethe determination of the electronic states from a state-averaged

Fig. 9 Schematic representation of the four excitations contributing to the DSOC part (example of a d5 MnII), two of them leading to the same spin statewhile the two others to different spin states (S + 1 and S � 1). VOMO for virtual occupied molecular orbital, SOMO for single occupied molecular orbitaland DOMO for double occupied molecular orbital, a for a spin down electron and b for a spin up electron.


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multi-configurational SCF calculation using the CASSCF schemefor selecting relevant electronic configurations. The CASSCFcalculations give a good representation of the static electroncorrelation. In addition, dynamical correlation can be includedin a perturbative treatment using the n-electron valence state

perturbation theory (NEVPT2) method. The SOC contribution iscommonly calculated within a quasi-degenerate perturbation theory(QDPT) formalism, in which the SOC operator is diagonalized ina basis of multiconfigurational wavefunctions from a full CIcalculation in a limited set of active electrons and orbitals. TheSSC contribution is calculated on the basis of a single root CASSCFwave function, which amounts to a mean-field approximation.

As mentioned above, the delicate step for CASSCF calculationsis to define the right active space. Indeed in a CASSCF wave-function, the occupied orbital space is divided into a set of‘‘inactive’’ or closed-shell orbitals (doubly occupied) and a set of‘‘active’’ orbitals (varying occupations). Most commonly theactive space contains the five 3d orbitals. However, when theelectron density is partly delocalized onto the ligand(s), additionalorbitals, more specifically the lowest virtual orbitals, should beadded into the active space. The definition of the active spacealso requires setting the number of electrons involved in thecalculations. Finally, the number of electronic configurationsrelated to different spin state multiplicities taken into accounthas to be determined but this number, like the choice of orbitalsin the active space, impacts the calculation time and musttherefore be chosen carefully.

The number of investigations related to CASSCF (NEVPT2)-predicted ZFS parameters for MnII and MnIV are small. For MnII,the quality of the prediction is comparable to the DFT-calculatedvalues, while for MnIV the unique study reveals worse predictedvalues (Fig. 10). On the other hand, for MnIII, this approach hasproven to be remarkably efficient in predicting both the sign andthe magnitude of D, especially with NEVPT2 correction (a linearregression coefficient of 0.994 and a standard deviation of 0.264,systematic study on 10 complexes) (Fig. 10).26 In the case of MnIII,the active space can be limited to the five metal d-based orbitalsand all excited 5 quintet (S0 = S = 2) and 35 triplet (S0 = S � 1 = 1)states. The prediction of DSOC in the CASSCF framework leads toa better general agreement between experiment and theory withrespect to the DFT that underestimates DSOC. The tripletsprovide about 50% of DSOC, confirming the key contributionplayed by these S � 1 states. The DSSC part is not negligible withcontribution up to 20% (about 0.50 cm�1) for MnIII complexeswith various chemical environments, demonstrating that DSSC

is not sensitive to the nature of the coordination sphereand/or to the geometry of the MnIII ion. Interestingly, the multi-configurational wavefunction approach also leads to an excellentprediction of the E/D ratio.

4. Magneto-structural correlation

Magneto-structural correlation is a challenging objective but itis also an essential requirement if ZFS is to be used as a probefor the identification of species (structure and geometry) duringa reaction, or for the rational design of metal-based systemswith targeted magnetic properties. While systematic investigationshave been reported on MnII and MnIII, this is not the case for MnIV.From Table 2, the conclusions that can be drawn for MnIV are thatthe preferred sign of D is positive and the magnitude of D is larger

Fig. 10 Comparison of CP-DFT (black squares) or CASSCF/NEVPT2 (redcircles) calculated D values with experimental data for MnIV (top), MnIII

(middle) and MnII (bottom). A black line shows as an eye-guide representingthe exact match with experimental results. The black circle on the top panelhighlights a wrong prediction of the D sign. (top) Reprinted with permissionfrom ref. 11. Copyright 2016 American Chemical Society.


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than 2 cm�1. Consequently the case of MnIV will not be furtherdiscussed in this section.

From Tables 1 and 3, it clearly appears that D is much moresensitive in the case of MnII than MnIII to the environment ofthe ions. For high-spin MnIII, it can be seen that |D| lies in asmall range from 2.29 to 4.52 cm�1 with the exception of[Mn(cyclam)I2]I.31 In all other complexes, the sign of D isconsistent with the nature of the distortion, i.e. negative(positive) for D o 0 (D 4 0), while the cyclam complex displaysan unexpected small and positive D value for an elongatedoctahedral MnIII ion. In this particular complex, it has beenshown that valence bond configuration interaction can contributeto D with a large positive contribution leading to a positive value(the system is characterized by a low-energy ligand-to-metal chargetransfer transition). Regarding the evolution of the magnitude ofD, it can be concluded that (i) low-coordinate complexes (4 and 5)leads to smaller D with respect to six-coordinate complexes, (ii)oxygen-based ligands give larger D than do N-based ligands (thelargest magnitude of D has been measured for [Mn(acac)3]),32 (iii)ligands with a large SOC contribution, such as halide, give asmaller D with |DBr| o |DCl| o |DF|.

MnII is unambiguously the most investigated Mn ion,because D can be determined by both X- and Q-band EPR inmost cases. MnII is very sensitive to the nature of the ligandsand the coordination number, with magnitudes of D rangingfrom 0.01 to 1.00 cm�1. Considering complexes with onlyN- and O-based ligands, D is very sensitive to the coordinationnumber of MnII: larger magnitudes of D are found for five-coordinate complexes (|D| 4 0.250 cm�1) compared to bothsix- (0.010 o |D| o 0.137 cm�1) and to a lesser extent seven-coordinate ones (0.068 o |D| o 0.137 cm�1). However, the ratiobetween the N- and O-based ligands cannot be discriminatedbased on the D-values of the corresponding MnII complexessince no trend can be observed.

On the other hand, for the MnII complexes that contain atleast one halide, the D-values are mainly dictated by the natureof the halide ligand with |DCl| o |DBr| o |DI|: in particular,0.110 o |DCl| o 0.260 cm�1, 0.359 o |DBr| o 0.665 cm�1 and0.590 o |DI| o 1.000 cm�1. It is interesting here to notice thatthis behavior is opposite to that of MnIII. A theoretical investigationhas pointed out that the observed trend for MnII is correlated tothe SOC constant of the halide (zX; X = Cl�, Br� and I�) with themajor contribution originating from the interference betweenmetal- and halide-SOC contributions, proportional to zMnzX (zMn,SOC constant for Mn).27 Although the majority of the investigatedMnII complexes are di-halides, it appears that the number ofhalides has no influence of the final D-value (similar D-valuesare observed for [Mn(terpy)(Cl)2] and [Mn(tmc)(Cl)]+).7,33 Similarly,D is sensitive neither to the coordination number nor the nature ofthe other ligands. However, it should be noted that in the specificcase of octahedral complexes, the relative position of two halideligands drastically affects D: its magnitude is two times largerwhen the halide are in a trans-rather than a cis-configuration.

Interestingly, a surprisingly large D of +1.460 cm�1 has beenreported for a MnII complex, in which two polyoxometalatesplays the role of coordinating ligands with a resulting O6

coordination sphere around the MnII ion. This unexpectedresult has still not been rationalized.6

5. Conclusion

EPR spectroscopy combined with theoretical calculations is oneof the most powerful approaches for the investigation of magneticanisotropy of paramagnetic transition metal complexes, especiallyfor those with magnitudes of D not larger than a dozen cm�1 (aprecise number is difficult to provide because it will depend onthe D-sign, the spin state of the metal ion and the E/D ratio). Inthis context, we have focused this tutorial on manganese,because its three oxidation states +II, +III and +IV give rise tothree spin states with different behaviors and D-ranges. Thistutorial emphasizes that the origin of the ZFS is surprisinglycomplex and depends not only on the nature of the metal ion(spin state and oxidation state) but also on the nature of theligand(s). The main conclusions that can be drawn are that (i) inmost cases, a multi-frequency EPR approach is required to preciselydetermine the ZFS parameters, (ii) each case should be treatedindependently and no general rule can be applied to define themost appropriate theoretical methodology to predict D and (iii)calculations are essential for the rationalization of the experimentalobservations, allowing an understanding of the physical origin ofthe electronic parameters. Therefore, for each metal ion, at differentoxidation and spin states, systematic studies should be performedto define helpful magneto-structural correlations.

Acknowledgements

I gratefully thank my students and co-workers, whose namesappear in the references cited, and especially Maylis Orio,Marcello Gennari and Allan Blackman for their helpful commentsand remarks. This work was supported by the French NationalAgency for Research no. ANR-09-JCJC-0087, Labex arcane (ANR-11-LABX-003) and COST Action CM1305 ECOSTBio (Explicit ControlOver Spin-States in Technology and Biochemistry).

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