# Tools for Predicting the Nature and Magnitude of Magnetic Anisotropy in Transition Metal Complexes: Application to Co(II) Complexes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{S}components of the ground spin state S, such that the components with the largest magnetization +M

_{S}and −M

_{S}become the ground state. As this phenomenon occurs even in the absence of an external magnetic field, it is called zero-field splitting (ZFS). There are two main ZFS parameters: D and E respectively characterize the axial and rhombic anisotropies. Up to now the blocking temperatures under which the property can be observed are only of a few Kelvins. The exploitation of SMMs for practical applications—as classical or quantum bits of information, for instance—thus relies on the possibility of synthesizing molecules with higher blocking temperatures and long coherence times [10,11,12]. In order to achieve high blocking temperatures, several conditions must be fulfilled: (i) The axial parameter D of the ZFS must be negative (by convention) to ensure an easy axis of magnetization. Indeed, a positive value characterizes an easy plane of magnetization and no blocking is observed in the absence of an applied magnetic field [13]; (ii) the absolute value of D must be as large as possible, as it governs the splitting of the Ms components; and (iii) the rhombic parameter E must be strictly zero in order to prevent tunneling. Moreover, E plays also a role on the coherence time as observed experimentally [14,15]. The physical factors governing the sign and magnitude of these parameters are the amplitude of the spin-orbit coupling (SOC) and the symmetry. In symmetry point groups for which the components of the angular momentum (which transform as R

_{x}, R

_{y}and R

_{z}in character tables) appear in the same irreducible representation (such as tetrahedral, octahedral, and icosahedral symmetries), the SOC does not lift the degeneracy of the M

_{S}components, and consequently no axial magnetic anisotropy can occur. A lowering of the symmetry is therefore necessary to observe the property. Molecules with the same symmetry but with different electronic configurations of the magnetic ion (different metal ions or different oxidation states) lead to completely different magnetic anisotropies (easy axis or easy plane and with or without rhombic contribution) so that the ways to improve SMMs are far from being intuitive [16,17,18,19,20].

_{3h}symmetry point group, we have lowered its symmetry to C

_{3v}and changed the ligands to reach the situation that corresponds to a real compound for which experimental data are available. The complexes are studied by means of correlated ab initio calculations including SOC.

## 2. Selection Rules and Computational Information

#### 2.1. Selection Rules for the Spin-Orbit Coupling in Atoms and Molecules

^{th}component of an irreducible tensor of rank k coupling two states labelled by the quantum numbers n, J, M and n′, J′, M′, where J, M and J′, M′ are angular momentum quantum numbers, and n and n′ are additional quantum numbers required to completely specify the states. Since angular momenta are rank 1 tensor operators, the Wigner–Eckart theorem appears to be useful to study SOC interactions. The 3-J symbol of Equation (2) contains all information about angular dependences (and thus selection rules) and the reduced matrix element does not depend on M. The notations are those used in reference [47].

^{2S+1}L terms with themselves), as second-order effects are weaker by orders of magnitude. Nevertheless, as it will be shown in the next section, second-order SOC may be of crucial importance to describe the ZFS in molecules and the selection rules (see Equation (4)) will be helpful to unravel the physics governing the magnetic properties of transition metal complexes.

_{S}are the spin quantum numbers. The SOC Hamiltonian used here is the phenomenological ${\widehat{H}}_{SOC}=A\overrightarrow{L}\xb7\overrightarrow{S}$ as we wish to correlate the SOC in magnetic complexes and in atoms. For symmetry based selection rules, the matrix elements to evaluate are $\u3008\mathsf{\Gamma},S,{M}_{s}|\overrightarrow{L}\xb7\overrightarrow{S}|{\mathsf{\Gamma}}^{\prime},{S}^{\prime},{M}_{S}^{\prime}\u3009$. The application of the Wigner–Eckart theorem on the spin part of the Hamiltonian leads to the expression [49]:

#### 2.2. Computational Information

## 3. Results and Discussion

_{5}]

^{2+}model complex belonging to the D

_{3h}symmetry point group that the main spin-orbit coupling effects arise from electronic states issued from atomic terms, which are coupled though SOC in the Co(II) ion. (ii) How the various selection rules presented in the previous section are selective for the calculation of the ZFS of a real complex. For this second study, geometrical distortions and chemical substitutions were imposed to the model complex to reach the geometry of the real [Co(Me

_{6}tren)(Cl)]

^{+}one [18]. Starting from the initial D

_{3h}geometry (with d

_{Co–N}fixed to 2 Å; note that d

_{Co–N}= 2.15 Å in the real compound, see Figure 1), we applied a progressive angular deformation by varying the angle α between the three equatorial NCH ligands and the xOy plane. The symmetry was thus lowered from D

_{3h}to C

_{3v}. α was increased until 9° and then the NCH ligand located on the z axis was replaced by a Cl atom.

#### 3.1. Reminiscence of the Physics of the Atom in the Molecular Spectrum

_{Co–N}.

^{4}F,

^{4}P, and

^{2}G atomic spectroscopic terms for different reasons. Firstly, the other excited states are higher in energy and their effect on the lowest energy spectrum through SOC is negligible (see the various spectra given in Supporting Information). Secondly, the states coming from

^{4}F and

^{4}P are well separated in energy, and the

^{4}P and

^{2}G terms, despite having different spins, possess similar energies. Finally, as the

^{2}G is coupled to the

^{4}F through SOC while the

^{4}P is not (see selection rules in Equation (4)), the discussion can be limited to these terms for illustrating the idea proposed in this section, i.e., the selection rules for SOC in the isolated metal ion are not (or at least just partially) quenched by the weak applied ligand field. It can be noted that calculations have been performed with all the states of the d

^{7}configuration and the results show that contributions of the other excited states to the low energy spin-orbit states are negligible. Since we are interested in the effect of SOC between

^{4}F and excited states issued from the

^{4}P and

^{2}G atomic terms, we need first to account for the effect of SOC between the states coming from the

^{4}F term. Let us first define what we call the

^{4}F-SO spectrum. The

^{4}F term is split in the four levels

^{4}F

_{9/2},

^{4}F

_{7/2},

^{4}F

_{5/2}, and

^{4}F

_{3/2}by the spin-orbit coupling giving rise to 28 components that can be obtained by the diagonalization of the SOC Hamiltonian matrix when only the 7 spin-orbit free states issued from the

^{4}F term are taken into account. In the same way, we define the (

^{4}F +

^{4}P) SO spectrum (respectively. the (

^{4}F +

^{2}G) SO spectrum), i.e., the lowest 28 components of the spin-orbit states (levels) obtained when the

^{4}F +

^{4}P (respectively

^{4}F +

^{2}G) states coming from

^{4}F and

^{4}P (respectively

^{2}G) terms are included in the spin-orbit state-interaction matrix. The

^{4}F SO spectrum is depicted in Figure 1b. One may notice that the

^{4}F

_{9/2},

^{4}F

_{7/2},

^{4}F

_{5/2}, and

^{4}F

_{3/2}are split by the ligand field in, respectively, 5, 4, 3, and 2 molecular spin-orbit states, each of them being degenerate due to Kramer’s theorem. The three computed spectra (

^{4}F SO, (

^{4}F +

^{4}P) SO and (

^{4}F +

^{2}G) SO) are very similar and their plots only show small differences (the spectra are given in Tables S1–S3 in Supporting Information). We define Δ

_{(4F+}

_{4P)−(4F)}and Δ

_{(4F+2G)−(4F}

_{)}as the sum of the energy shifts (in absolute values, see Tables S4 and S5) experienced by the 14 spin-orbit states between the

^{4}F SO spectrum and the (

^{4}F +

^{4}P) SO and (

^{4}F +

^{2}G) SO spectra, respectively. The values of Δ reflect the SOC effects of the states coming from the

^{4}P or

^{2}G terms on those issued from the

^{4}F term. Δ

_{(4F+4P)−(4F)}and Δ

_{(4F+2G)−(4F)}as a function of d

_{Co–N}are shown in Tables S6 and S7 and Figure S1. The values of Δ when the ligand field is negligible (d

_{Co–N}= 3.5 Å) are in agreement with the selection rules for the isolated atom. Δ

_{(4F+4P)−(4F)}is equal to 0 because the

^{4}F-

^{4}P SOC is not allowed. On the other hand, the

^{4}F-

^{2}G coupling is allowed and Δ

_{(4F+2G)−(4F)}is quite large (≈ 600 cm

^{−1}). When the ligand field increases, Δ

_{(4F+4P)−(4F)}increases to ≈ 40 cm

^{−1}and Δ

_{(4F+2G)−(4F)}slowly decreases to reach ≈ 450 cm

^{−1}at d

_{Co–N}= 2 Å. For a ligand field of this magnitude, the SOC between electronic states coming from the

^{4}F and

^{4}P atomic terms is very small in comparison to the one between states coming from

^{4}F and

^{2}G atomic states. One should keep in mind that the

^{4}P state leads to three spin-free states while the

^{2}G leads to nine spin-free states, however these different degeneracies cannot explain alone such differences in the SOC magnitude. The main result here is that the SOC strength between the molecular ground state (GS) and excited states (ES) can be ranked as GS(

^{4}F) − ES(

^{4}F) $>$ GS(

^{4}F) − ES(

^{2}G) $\gg $ GS(

^{4}F) − ES(

^{4}P). In other words, the main effect of the SOC in the molecular electronic states stem from those related to the atomic

^{4}F and

^{2}G states, while those coming from the

^{4}P state can be neglected.

#### 3.2. Selection Rules for the Calculation of the ZFS in Molecular Complexes

_{6}tren)(Cl)]

^{+}, previously studied by the authors [18], have been theoretically investigated. All these complexes are depicted in Figure 2 where the calculated D values are reported. The low energy spectrum and each excited-state contribution to D are also shown.

#### 3.2.1. Molecular Selection Rules Based on the Symmetry Point Group

_{3}axis, for states with same (respectively different) spin multiplicity, if the coupling occurs through the ${\widehat{L}}_{z}{\widehat{S}}_{z}$ part of ${\widehat{H}}_{SOC}$ the coupling between the ground and excited states leads to a negative (respectively positive) contribution to D, while a coupling caused by ${\widehat{L}}_{y}{\widehat{S}}_{y}+{\widehat{L}}_{x}{\widehat{S}}_{x}$ leads to a positive (respectively negative) contribution.

_{3h}symmetry point group, the model complex exhibits an easy-plane type magnetic anisotropy (D > 0). The main contribution to D comes from the SOC between the ${}^{4}{\mathrm{A}}^{\prime}_{2}$ ground state and the excited ${}^{4}{\mathrm{E}}^{\u2033}$ state which provides a positive contribution to D. Indeed, in the D

_{3h}symmetry point group, ${\widehat{L}}_{x}$ and ${\widehat{L}}_{y}$ span the ${\mathrm{E}}^{\u2033}$ irreducible representation and the tensor product ${{\mathrm{A}}^{\prime}}_{2}\otimes {\mathrm{E}}^{\u2033}\otimes {\mathrm{E}}^{\u2033}={{\mathrm{A}}^{\prime}}_{1}\oplus {{\mathrm{A}}^{\prime}}_{2}\oplus {\mathrm{E}}^{\prime}$ contains the totally symmetric irreducible representation. Another non-negligible contribution comes from the ${}^{2}{\mathrm{A}}^{\prime}_{1}$issued from the

^{2}G term. The interaction is now brought by the ${\widehat{L}}_{z}{\widehat{S}}_{z}$ part of ${\widehat{H}}_{SOC}$ (${\widehat{L}}_{z}$ spans the ${{\mathrm{A}}^{\prime}}_{2}$ irreducible representation and ${{\mathrm{A}}^{\prime}}_{2}\otimes {{\mathrm{A}}^{\prime}}_{2}\otimes {{\mathrm{A}}^{\prime}}_{1}={{\mathrm{A}}^{\prime}}_{1}$). In this symmetry (and for the four depicted excited states), no other coupling is symmetry allowed. As the symmetry is lowered to C

_{3v}, the D value decreases and becomes negative for the last complex, indicating a switch from easy plane to an easy axis type anisotropy. Note that the anisotropy of the [Co(NCH)

_{4}Cl]

^{+}model complex with α = 9° is in quantitative agreement with that of the [Co(Me

_{6}tren)(Cl)]

^{+}complex despite the differences between them. Electron paramagnetic resonance (EPR) spectroscopy studies on [Co(Me

_{6}tren)(Cl)]

^{+}gave D = −8.12 cm

^{−1}. The decrease of D as the distortion increases can easily be explained by noting that, while both couplings between the ${}^{4}\mathrm{A}_{2}$ GS and the excited ${}^{4}\mathrm{E}$ (${}^{4}{\mathrm{E}}^{\u2033}$ in D

_{3h}) and ${}^{2}\mathrm{A}_{1}$ (${}^{2}{\mathrm{A}}^{\prime}_{1}$ in D

_{3h}) states still leads to a positive contribution to D, the coupling between the ${}^{4}\mathrm{A}_{2}$ and the ${}^{4}\mathrm{A}_{1}$ (${}^{4}{\mathrm{A}}^{\u2033}_{2}$ in D

_{3h}) states is now allowed through the ${\widehat{L}}_{z}{\widehat{S}}_{z}$ part of ${\widehat{H}}_{SOC}$ (${\widehat{L}}_{z}$ spans the ${\mathrm{A}}_{2}$ irreducible representation and ${\mathrm{A}}_{2}\otimes {\mathrm{A}}_{2}\otimes {\mathrm{A}}_{1}={\mathrm{A}}_{1}$) and contributes negatively to D.

#### 3.2.2. Molecular Selection Rules Based on the Double Group Theory

_{5}]

^{2+}model complex with D

_{3h}symmetry but the same reasoning could be applied to the C

_{3v}complexes. Figure 3 shows the low energy electronic (O(3)) and spin-orbit (O(3)*) spectra for the isolated ion and electronic (D

_{3h}) and spin-orbit (D

_{3h}*) spectra of the molecular complex. All molecular electronic states can be correlated with an atomic term. The GS ${}^{4}{\mathrm{A}}^{\prime}_{2}$ in the D

_{3h}symmetry point group is split into E

_{1/2}and E

_{3/2}in the D

_{3h}* double group under the effect of the SOC. The ZFS is then described by the energy difference between E

_{1/2}(M

_{S}= ±1/2) and E

_{3/2}(M

_{S}= ±3/2). The ground states E

_{1/2}and E

_{3/2}in D

_{3h}* are coming from the ${}^{4}\mathrm{F}_{9/2}$ in O(3)* and the ground state ${}^{4}{\mathrm{A}}^{\prime}_{2}$ is issued from the atomic ${}^{4}\mathrm{F}$ term in D

_{3h}. As can be seen in the column D

_{3h}*, the excited states which have the same symmetry as the two lowest states in the double group are numerous and selection rules of the double group are therefore less restrictive than those of the symmetry point group.

#### 3.2.3. Atomic Selection Rules

_{3h}symmetry point group and according to the symmetry based selection rule, the ${}^{4}\mathrm{A}_{2}$ ground state can only be coupled with states of ${{\mathrm{A}}^{\prime}}_{1}$ (${\widehat{L}}_{z}{\widehat{S}}_{z}$) and ${\mathrm{E}}^{\u2033}$ (${\widehat{L}}_{x}{\widehat{S}}_{x}+{\widehat{L}}_{y}{\widehat{S}}_{y}$) symmetries. Under a D

_{3h}ligand field, the

^{4}P term is split into ${}^{4}\mathrm{A}_{2}\oplus {}^{4}{\mathrm{E}}^{\u2033}$. Calculations show that only the ${}^{4}{\mathrm{E}}^{\u2033}$ state is coupled to the GS and gives a positive contribution to D of 0.58 cm

^{−1}. The ${}^{2}\mathrm{G}$ state in D

_{3h}symmetry reduces as ${}^{2}{\mathrm{A}}^{\prime}_{1}\oplus 2{}^{2}{\mathrm{E}}^{\prime}\oplus {}^{2}{\mathrm{A}}^{\u2033}_{1}\oplus {}^{2}{\mathrm{A}}^{\u2033}_{2}\oplus {}^{2}{\mathrm{E}}^{\u2033}$. As expected, the ${\mathrm{A}}_{1}^{\prime}$ and ${\mathrm{E}}^{\u2033}$ states are coupled to the GS. Their contributions to D are, respectively, 13.02 and −3.03 cm

^{−1}. The contribution of ${}^{4}{\mathrm{E}}^{\u2033}({}^{4}\mathrm{P})$ is one order of magnitude weaker than the contributions of ${}^{2}{\mathrm{A}}^{\prime}_{1}({}^{2}\mathrm{G})$ and ${}^{2}{\mathrm{E}}^{\u2033}({}^{2}\mathrm{G})$ The SOC strength, and thus the contributions of excited states to D, therefore strongly depends on the atomic origin of the molecular state, and the almost negligible contribution of ${}^{4}{\mathrm{E}}^{\u2033}({}^{4}\mathrm{P})$ to D can be attributed to the atomic selection rules (Equation (4)). Indeed, we expect that the ${}^{4}{\mathrm{E}}^{\u2033}({}^{4}\mathrm{P})$ contribution will decrease if we lower the ligand field as the contribution of the

^{4}P term must reach zero for negligible ligand field. These couplings are summarized in Figure 3.

#### 3.3. Rationalization of the ZFS Nature

- (i)
- If an easy axis of magnetization is needed (negative D value), the symmetry lowering from D
_{3h}to C_{3v}is beneficial because it allows a coupling with the first excited state that brings a negative contribution to D. - (ii)
- In order to obtain a more negative overall D value, one can decrease its positive contribution by destabilizing the
^{4}E(C_{3v}) excited state. As the main determinant of this state coupled to the ground state has a double occupancy in the d_{z}^{2}orbital, ligands with a strong field in axial positions would be appropriate to do so. - (iii)
- Finally, one may question the role of the chemical substitution by chlorine. To answer this query and separate the role of the distortion brought by the change of the value of the angle α from that of the electronic effect of chlorine, we have also substituted an axial NCH ligand in a slightly distorted (α = 3°) complex. In both cases (α = 3° and α = 9°), the main effect comes from the first excited state that for chlorine has a two-fold stronger coupling (see Figure S2) to the ground state, resulting in a much larger negative contribution to D. From this last observation, one may learn that not only the energy differences between the states (diagonal elements of the state interaction matrix) but also the magnitude of the coupling (off diagonal elements which are proportional to the fine structure constant and depend on the chemical nature of the ligand, i.e., electronic effect) can be tuned with the choice of appropriate ligands.

## 4. Summary

_{S}components of the high-spin ground state, the reminiscence of the physics of the atom in the molecular complex is still present. An appropriate a priori selection of these states may lead to both (i) a computational gain as the size of the state-interaction matrix can be reduced, and (ii) a gain in quality of the ab initio results as the number of states used in the average optimization of the orbitals may be reduced to its bare essential. Finally, as theory may guide the synthesis of molecular complexes with targeted magnetic anisotropy, the main object of this work concerns the possible rationalization of the magnitude and nature of the ZFS from analytical derivation using second-order perturbation theory. Regarding this peculiar aspect, the selection of the states that will bring the main contributions to the ZFS is crucial. For this purpose, we have analyzed the selectivity of the various selection rules derived from the Wigner–Eckart theorem. The main conclusions are:

- (i)
- The atomic selection rules are essentially fulfilled. The contribution to D of the single excited state that is coupled to the ground state in the molecular complex but not in the atom is only 0.58 cm
^{−1}and can be safely neglected in any attempt of rationalization of the SOC nature. - (ii)
- The double group selection rule is less restrictive than the symmetry point group ones.
- (iii)
- From the symmetry point group selection rules, one may not only determine the excited states that are coupled to the ground state but also identify which part of the SOC operator, ${\widehat{L}}_{z}{\widehat{S}}_{z}$ or ${\widehat{L}}_{x}{\widehat{S}}_{x}+{\widehat{L}}_{y}{\widehat{S}}_{y}$, is responsible for the coupling and therefore the sign of their contribution to D.

## Supplementary Materials

^{4}F SO spectrum (cm

^{−1}), Table S2: (

^{4}F +

^{4}P) SO spectrum (cm

^{−1}), Table S3: (

^{4}F +

^{2}G) SO spectrum (cm

^{−1}), Table S4: Difference between (

^{4}F) and (

^{4}F +

^{4}P) SO spectra (cm

^{−1}), Table S5: Difference between (

^{4}F) and (

^{4}F +

^{2}G) SO spectra (cm

^{−1}), Table S6: Δ(

^{4}F +

^{4}P) − (

^{4}F) (cm

^{−1}), Table S7: Δ(

^{4}F +

^{2}G) − (

^{4}F) (cm

^{−1}), Figure S1: Δ(

^{4}F +

^{4}P) − (

^{4}F) (blue) and Δ(

^{4}F +

^{2}G) − (

^{4}F) (red) as a function of d

_{Co–N}, Figure S2: Most important contributions to D for the [Co(NCH)

_{4}Cl]

^{+}(α = 3°) complex.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ZFS | Zero-Field Splitting |

CASSCF | Complete Active Space Self-Consistent Field |

SO–SI | Spin-Orbit State Interaction |

SOC | Spin-orbit coupling |

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**Figure 1.**(

**a**) Lowest spin-orbit free states of [Co(NCH)

_{5}]

^{2+}model complex of D

_{3h}symmetry computed at the CASSCF level; (

**b**)

^{4}F spectrum (see text) of the lowest spin-orbit states of [Co(NCH)

_{5}]

^{2+}model complex of D

_{3h}symmetry as a function of d

_{Co–N}computed at the spin-orbit state-interaction (SO–SI) level.

**Figure 2.**Structure of studied complexes and their calculated axial anisotropy parameter D (

**top**). Low energy spectrum of the spin-orbit free states and contributions (in cm

^{−1}) to D of each excited states (

**bottom**). Red arrows indicate couplings between the states through the ${\widehat{L}}_{z}{\widehat{S}}_{z}$ operator while blue arrows indicate couplings through the ${\widehat{L}}_{x}{\widehat{S}}_{x}+{\widehat{L}}_{y}{\widehat{S}}_{y}$ operator.

**Figure 3.**Schematic (not scaled) energy spectrum. As mixing of states with same symmetry in the D

_{3h}* are possible only a few dashed lines are indicated to guide the eyes. Blue plain line arrows indicate allowed couplings between states, the dotted arrow indicates a coupling allowed at the molecular level but forbidden in the atom. Red, green and purple lines respectively indicate E

_{1/2}, E

_{3/2}and E

_{5/2}.

**Figure 4.**All depicted excited states are multireference. Only the main determinants of the ground (bottom) and excited states coupled through SOC are represented. The labels and energetic order of the d orbitals are indicated. Only one of the two determinants equivalent by left-right symmetry is represented. The excitation leading to these excited states and the part either ${\widehat{L}}_{z}{\widehat{S}}_{z}$ (purple arrow) or ${\widehat{L}}_{x}{\widehat{S}}_{x}+{\widehat{L}}_{y}{\widehat{S}}_{y}$ (red arrow) of the spin-orbit coupling (SOC) operator which is involved are also depicted.

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**MDPI and ACS Style**

Cahier, B.; Maurice, R.; Bolvin, H.; Mallah, T.; Guihéry, N.
Tools for Predicting the Nature and Magnitude of Magnetic Anisotropy in Transition Metal Complexes: Application to Co(II) Complexes. *Magnetochemistry* **2016**, *2*, 31.
https://doi.org/10.3390/magnetochemistry2030031

**AMA Style**

Cahier B, Maurice R, Bolvin H, Mallah T, Guihéry N.
Tools for Predicting the Nature and Magnitude of Magnetic Anisotropy in Transition Metal Complexes: Application to Co(II) Complexes. *Magnetochemistry*. 2016; 2(3):31.
https://doi.org/10.3390/magnetochemistry2030031

**Chicago/Turabian Style**

Cahier, Benjamin, Rémi Maurice, Hélène Bolvin, Talal Mallah, and Nathalie Guihéry.
2016. "Tools for Predicting the Nature and Magnitude of Magnetic Anisotropy in Transition Metal Complexes: Application to Co(II) Complexes" *Magnetochemistry* 2, no. 3: 31.
https://doi.org/10.3390/magnetochemistry2030031