# Methods and Models of Theoretical Calculation for Single-Molecule Magnets

^{*}

## Abstract

**:**

## 1. Introduction

_{12}} was discovered in the early 1990s with typical slow relaxation of magnetization [7,8]. After almost one decade, the first lanthanide-based SMM [Pc

_{2}Tb][N(C

_{4}H

_{9})

_{4}] was reported [9], which exhibits slower magnetization relaxation owing to the fact that some rare-earth ions can provide stronger magnetic anisotropy and be more sensitive to the crystal field [10]. After years of efforts to improve the effective energy barrier (U

_{eff}) and magnetic blocking temperature (T

_{B}), as two figure-of-merits, multitudinous SMMs have been synthesized, which can be classified as mono-, bi- and multi-nuclear molecules due to the diversity of synthesis methods [11,12,13,14,15,16,17]. Among them, two families of dysprosium(III) SMMs, namely the pentagonal–bipyramidal (PB) family with local D

_{5h}symmetry and the cyclopentadienyl (Cp) family with a sandwich structure, are broadly paid attention to, especially for the representative complexes of [Dy(py)

_{5}(O

^{t}Bu)

_{2}][BPh

_{4}] (U

_{eff}= 1815 K, T

_{B}= 14 K) and [(η

^{5}-Cp*)Dy(η

^{5}-Cp

^{iPr5})][B(C

_{6}F

_{5})

_{4}] (U

_{eff}= 2220 K, T

_{B}= 77 K) [18,19]. Evidently, the material is far from large-scale practical application, as its blocking temperature is still far below the room temperature. How to build higher performance SMMs is undoubtedly an urgent problem for synthetic chemists and theoretical researchers.

## 2. Mononuclear Single-Molecule Magnets

_{T}, specifically: ${U}_{eff}=\left|D{S}_{\mathrm{T}}^{2}\right|$ (S

_{T}is an integer) or ${U}_{eff}=\left|D\left({S}_{\mathrm{T}}^{2}-\frac{1}{4}\right)\right|$ (S

_{T}is a half-integer). Some studies have pointed out that D and S

_{T}cannot increase simultaneously [30], making the improvement of the energy barrier a challenge. For rare-earth SMMs, the situation is much more complicated, and their spin Hamiltonian cannot be directly expressed as an aforementioned equation. In total, four sections need to be considered: spin–orbit (SO) coupling, crystal field effect, exchange coupling and Zeeman splitting (Equation (2)). Similarly, here, the last item should be ignored. Although the magnitude of the SO effect is larger than the crystal field effect in the 4f system, based on chemical synthesis modification, the contribution of the crystal field to the anisotropic energy barrier is also undeniably significant. The operator on the crystal field can be written as Equation (3). In the formula, ${\widehat{O}}_{k}^{q}$ is the Wybourne operator (the real combinations of the spherical harmonics), and ${\widehat{B}}_{k}^{q}$ is the crystal field parameter of rank k and projection q, reflecting the symmetry of the crystal field: when q is 0, it represents the axial crystal field parameter; when q is not 0, it stands for the equatorial one. When the latter value is large, the overall axial anisotropy of the entire molecule is weakened, leading to quantum tunnelling behavior for Kramers ions with half-integer spin, which is not conducive to enhancing the high energy barrier of SMMs. Therefore, some studies have pointed out that [31,32,33] regulating ligands to synthesize complexes with high geometric symmetry, such as C

_{∞v}, D

_{∞h}, D

_{4d}, D

_{5h}and D

_{6d}, can realize the equatorial crystal field parameters close to zero, helping to suppress quantum tunnelling and improve the blocking temperature and energy barrier. Nevertheless, compared with 3d systems, the existence of other thermal relaxation processes and quantum tunnelling effects makes the above energy barrier relationship unsuitable for rare-earth SMMs [34]. There are four types of relaxation processes existing: the direct relaxation process, Orbach process, Raman process and quantum tunnelling process (QTM). The direct relaxation process should be negligible without an external magnetic field. Hence, the relationship between relaxation time (τ) and temperature (T) can be described using Equation (4), where ${\tau}_{0}^{-1}$ is a pre-exponential factor of the Arrhenius term, C is the coefficient in the Raman process term and ${\tau}_{QTM}^{-1}$ expresses the QTM process. The Orbach process is a two-phonon process through a successive transition between microstates m

_{J}split from ground state J. The Raman process is a two-phonon process leading to a relatively low temperature zone. QTM normally takes place in the lowest temperature zone, independent of temperature and existing between ground doublets m

_{J}, as well as thermally assisted quantum tunnelling of magnetization (TA-QTM). All of the possible magnetic relaxation paths in rare-earth SMMs are shown in Figure 1 [35]. Among them, the existence of Raman and QTM processes, especially the latter, significantly descends the anisotropic energy barrier, so it needs to be suppressed. Furthermore, while constructing mononuclear SMMs with high performance, researchers have also been trying to explore and reveal the specific mechanism of the slow relaxation process for further guiding chemical design and synthesis.

_{3})

_{3})

_{2}]

^{−}with a 3d

^{7}electron configuration and 3/2 as the ground-state spin, exhibiting high performance that U

_{eff}arrives at 226 cm

^{−1}, and the hysteresis loop can be observed below 4.5 K [43]. In this compound, the existence of D

_{∞}

_{h}local symmetry and a weak ligand field environment induces specific crystal field splitting: ${d}_{xz},{d}_{yz}{d}_{xy},{d}_{{x}^{2}-{y}^{2}}{d}_{{z}^{2}}$ from CASSCF and NEVPT2 calculations considering state averaging over the 10 quartets and 40 doublet states (Figure 3). The calculated energy gap between the ground doublet and the first excited doublet is 210 cm

^{−1}, which is close to the experimental fitting value, suggesting Orbach relaxation passes through the latter doublet. In light of the results, they point out that a low coordination number and a low oxidation state matching with a weak ligand field create a large axial magnetic anisotropy and set out the vision for the future that the energy of ${d}_{{z}^{2}}$ orbital needs a lower and minimum quenching of orbital angular momentum, which can be realized by modifying ligands.

_{eff}as a result of the less orbital degeneracy of the ground state. Moreover, the Fe(III) complexes without intermediate spin S = 3/2 show positive D values, while the existence of intervening spin S = 3/2 makes it possible to generate a large negative D value. Our group carried out a detailed study on Fe[N(SiMe

_{3})

_{2}]

_{3}with perfect local D

_{3h}symmetry to explore its magnetic anisotropy (Figure 4) [44]. Three classes of methods were performed to acquire D: least-square fitting toward data from high-frequency EPR (HF-EPR) and ab initio calculation and data fitting based on spin Hamiltonian. In addition to the experimental value obtained by the first way, the theoretical values were generated from the other two means. As expected, the D values from distinctive aspects are basically in accordance with each other (−1.15, −1.48 and −1.62, respectively), and the Orbach energy barriers are 6.91 cm

^{−1}by magnetic dynamic analysis, as well as 6.90 cm

^{−1}by calculation. CASSCF with NEVPT2 calculation was performed to understand the origin of easy-axis magnetic anisotropy and acquire the information on the energy level of the 3d orbitals. In this case, the active space was determined as CAS(5,5), and 1 sextet state, 24 quartet states and 75 doublet states were chosen for the state average. From the d-orbital energy diagram, a weak z-axis crystal field reduces the energy of ${d}_{{z}^{2}}$ orbital, while a strong crystal field in the xy plane improves the energy of ${d}_{xy}$ and ${d}_{{x}^{2}-{y}^{2}}$ orbitals, making the ground state electron configuration (${d}_{{z}^{2}}$)

^{1}(${d}_{xz}$)

^{1}(${d}_{yz}$)

^{1}(${d}_{{x}^{2}-{y}^{2}}$)

^{1}(${d}_{xy}$)

^{1}. This result is suggestive of the fact that the energy difference between the

^{6}A

_{1}ground state and the quartet excited states produced by

^{4}G furnishes axial magnetic anisotropy.

^{Dipp}NH) to synthesize one case of mononuclear SMM (

**1**), whose experimental energy barrier is nearly 803 K (Figure 6) [48]. In sharp contrast, the energy barrier of similar analogous alkoxide ligand SMM (

**2**) is merely 53 K. Ab initio calculations at the SA CASSCF/RASSI level were performed to understand the magnetic properties from the microscopic electronic structure using Molcas software. In the RASSCF module, 21 sextets, 224 quartets and 490 doublets optimized are considered, and, in the RASSI module, 21 sextets, 128 quartets and 130 doublets are constructed and diagonalized in spin–orbit (SO) coupling Hamiltonian. Then, these SO states are transmitted toward SINGLE_ANISO, which is a magnetism of complexes program and calculates zero-field splitting and the pseudospin Hamiltonians for Zeeman interaction, as well as temperature- and field-dependent magnetic properties. Ultimately, the parameters and information we are concerned with are printed, comprising crystal field parameters, g values, susceptibility curves, magnetization curves and transition magnetic moment matrix. The calculated U

_{eff}of

**1**and

**2**are 759 K and 585 K; however, the probability of QTM between the first excited doublets in

**2**is 0.5 ${\mu}_{B}^{2}$, which is non-ignorable, leading to the existence of a smaller experimental fitting value. DFT calculations reveal that the Mayer bond order of Dy-X (X = N or O) in

**1**and

**2**are 2.23 (σ + π) and 1.03 (σ), indicating that the formation of multiple bonds enhances the rigidity of axial ligands, weakens the intramolecular vibration, suppresses the Raman process and improves the magnetic anisotropy. Moreover, the theoretical prediction shows that the U

_{eff}of a linear molecule with Cl ions and THF eliminated arrives at almost 4000 K, along with attenuation of the QTM process. Therefore, one of the strategies to improve the performance of SMM is to introduce multiple bonds in the axial coordination sites.

_{2}(py)

_{5}][BPh

_{4}] (HL=1-phenylethanol) (

**3**and

**4**, Figure 7), which has an energy barrier of 1130 (20) cm

^{−1}[49], and it also exhibits hysteresis at a blocking temperature of 22 K, which is much higher than [Dy(O

^{t}Bu)

_{2}(py)

_{5}]

^{+}. The crystal structure and intermolecular packing diagram were investigated, and the appearance of C-H…π and π…π interactions would increase the rigidity of the whole molecule. A DFT calculation confirms that there are strong intramolecular C-H…π interactions (~20.4 KJ/mol), intermolecular π…π (~5.7 KJ/mol) and C-H…π (~19.1 KJ/mol) interactions. An ab initio spin dynamic study demonstrates that the existence of such forces is beneficial to improve molecular rigidity, leading to the increase of blocking temperature.

_{5}(O

^{t}Bu)

_{2}]

^{+}, [Er{N(SiMe

_{3})

_{2}}

_{3}Cl]

^{−}and [Dy(CpMe

_{3})

_{2}Cl] and acquired relevant principles through combining DFT structural optimization with the field keyword available using Gaussian 09, and, then, the electronic structure and magnetic properties were studied by means of the CASSCF method in Molcas software. For Ln(III) with oblate electron density, such as Dy(III), exerting EEF along the equatorial direction is beneficial to heighten the energy barrier, while for Ln(III) with elongated electron density, such as Er(III), the adaptable direction must be near the axial Ln-L bond. Despite the physical method being represented for the first time, the strength of the electric field, which effectively enhances the energy barrier of SMMs, is about ca. 10

^{9}V/m, which is almost impossible to reproduce in an experimental method.

## 3. Bi- and Multi-Nuclear Single-Molecule Magnets

_{ij}is the exchange coupling constant between i and j. For compounds with relatively weak SO coupling, particularly those containing transition metal ions, density functional theory combined with the broken symmetry (DFT-BS) approach proposed by Noodleman is a universal method to calculate the J value [52,53]. In the process of derivation, to produce a mixed spin state wave function, the single configuration model, which consists of nonorthogonal magnetic orbitals, was combined with unrestricted Hartree–Fock (HF) theory or density functional theory (DFT), putting forward that it is in the direct ratio between the J value and the energy difference within the high spin state, E(S

_{max}), and the mixed spin state, E

_{B}, indicating that this exchange constant can be calculated by Equation (6) when both energies are acquired, where S and A

_{1}(S) mean spin quantum number and square of Clebsh–Gordan coefficient. Ultimately, due to $\sum}_{S=0}^{{S}_{max}}{A}_{1}\left(S\right)\xb7S\left(S+1\right)$ being equal to S

_{max}, J can be obtained from Equation (7). One of the superiorities of this means is handling larger compounds comprising hundreds of atoms, while for circumstances with stronger SO coupling, for instance, when there are existing lanthanide ions, it cannot meet the requirements toward the accuracy of the parameters.

_{total}) within both magnetic centers primarily consists of magnetic exchange (J

_{exch}) and dipole–dipole exchange (J

_{dip}), namely J

_{total}= J

_{exch}+ J

_{dip}. This relation is in a position to be expressed by the Hamiltonian as Equation (8) when the ground state of magnetic centers can be regarded as the Ising limit state [54]. Where J

_{dip}, as a long-range interaction, frequently plays a dominant role in Ln(III) dimer complexes and can be calculated directly via Equation (9) under Ising approximation. Where g

_{1}, g

_{2}and g

_{1z}, g

_{2z}are g tensors of two magnetic centers and their components in the z-direction; r is the distance within both lanthanide ions; θ is the angle between the anisotropic axes of two magnetic sites; and μ

_{B}

^{2}is a constant, almost 0.43297 cm

^{−1}/T. The formula illustrates that the dipole–dipole interaction depends on the distance between the two dipoles and the angle between the magnetic moments of the magnetic centers. The positive or negative sign of J

_{tota}

_{l}intimates that the interaction between two magnetic sites is ferro- or antiferro-magnetic, respectively.

_{T}-T data. This fitting method based on the Lines model has been successful in numerous molecules [55,56,57,58]. Meanwhile, this model is particularly suitable for the following circumstances: interactions within isotropic spins, between isotropic spin and Ising spin and both Ising spins. In the Molcas software package, the specific calculation processes are as follows: primarily, the magnetic property and electronic structure information of a single magnetic ion need computing through the SINGLE_ANISO module, and, then, a file named “ANISOINPUT” should be invoked to the POLY_ANISO module to output exchange the coupling constants cooperated with molecular susceptibility and magnetization data. To simplify the calculation, when dealing with multinuclear systems, the whole molecule is capable of being segregated into several independent binuclear sections, but it will bring about large errors. Consequently, one more reliable approach is to retain the magnetic centers that we calculate while replacing the other ones to diamagnetic ions of the same charge, for example, replacing Co(II) with Zn(II), or Dy(III) with La(III), Lu(III), Sc(III), etc. Figure 8 describes a simple flow chart of the whole calculation process.

^{III}Mn

^{IV}(μ-O)

_{2}(μ-OAc)DTNE]

^{2+}for the first time (Figure 9) [59]. After geometry optimization toward the structures of high spin (hs) and broken symmetry states (bs), utilizing different density functionals and basis sets, g tensors, hyperfine parameters, exchange coupling constant J and nuclear quadrupole coupling constants were determined from the DFT-BS method via ORCA and ADF software, which are consistent with the experimental results. Compared with single crystal data, the optimized coordinates from pure functional BP are more pinpoint than those from hybrid functional B3LYP. B3LYP overestimates the spin densities of Mn ions, which is traceable in the hybrid functional with reduced self-interaction error, and, for the calculation of J constant, B3LYP is superior to BP, which produces three or four times larger values. Moreover, molecular orbitals (MOs) for the bs state in this system are conducive to understanding the exchange interaction at the level of electronic structure, and MOs 123–126 indicate that exchange interaction passes through μ-oxo atoms.

_{2}CCF

_{3})

_{3}(C

_{2}H

_{5}OH)

_{2}](L=N,N-bis(3-ethoxy-salicylidene)-1,2-diamino-2-methylpropanato) with DFT calculations to comprehend magnetic coupling and magneto-structural relations (Figure 10) [60]. After testing and comparing several functional methods, they recommend using a B3LYP hybrid functional and effective core potential (ECP) basis set to obtain the calculated value of magnetic exchange constant (−5.8 cm

^{−1}), which is closely consistent with the experimental value (−4.42 cm

^{−1}). When amply considering the relativistic effect of the rare-earth ion system, zeroth-order regular approximation (ZORA) or Douglas–Kroll–Hess (DKH) means should be performed. After analyzing MOs and spin density distribution, they found that 5d orbitals of Gd(III) acquire charge densities from 3d orbitals of Cu(II) and 4f ones of Gd(III) through charge transfer, showing a direct interaction between both sites. Furthermore, an authentic exponential relation within J and O-Cu-O-Gd dihedral angles was established.

_{3}tacn)

_{2}(cyclam)NiMo

_{2}(CN)

_{6}]

^{2+}, [−(Me

_{3}tacn)

_{2}(cyclam)Ni-Cr

_{2}(CN)

_{6}]

^{2+}, [(Me

_{3}tacn)

_{6}MnMo

_{6}(CN)

_{18}]

^{2+}and [(Me

_{3}tacn)

_{6}MnCr

_{6}(CN)

_{18}]

^{2+}(Me

_{3}tacn = N,N’,N’’’-trimethyl-1,4,7-triazacyclononane), were investigated using diverse functionals in ADF and Gaussian software to evaluate the magnetic couplings within Ni(II) and Mo(III), Ni(II) and Cr(III), Mn(II) and Mo(III) and Mn(II) and Cr(III), respectively (Figure 11). For models A and B, the ferromagnetic exchange interaction was promoted after replacing Mo(III) with Cr(III) using Operdew, OPBE, O3LYP and B3LYP functionals, while the antiferromagnetic exchange interaction was enhanced after the same operation for C and D under the usage of XCs and B3LYP functionals. In brief, this type of substitution indeed strengthens the magnetic exchange interaction in this system.

_{3}Si)

_{2}N]

_{2}(THF)Ln}

_{2}(μ-η

^{2}:η

^{2}-N

_{2})

^{−}(Ln = Gd, Tb, Dy, Ho, Er) (Figure 12), whose Dy(III) and Tb(III) homologues possess evident slow magnetic relaxation properties attributed to the high magnetic anisotropy of central ions and the strong magnetic coupling between them: their U

_{eff}are 178 K and 327 K, their blocking temperature approaches 8.3 K and 14 K (the temperature in which hysteresis loop opens, T

_{B}) and the coercive field of hysteresis loop of the latter is up to 5 T at 11 K, making itself the hardest SMM at that time. Gao and his co-workers studied the magnetic anisotropy and coupling effect for this series using DFT and CASSCF calculations [64]. The exchange coupling constants (J) of them were computed via the spin-projected approach, the results indicating that coupling effects in Ln-N

_{2}

^{3−}are stronger than in Ln-Ln, and the types of coupling for them are antiferromagnetic except for Er(III) due to its orthogonal magnetic orbitals between Er(III) and the radical. The comparison between Tb

_{2}N

_{2}

^{3−}and Er

_{2}N

_{2}

^{3−}intimates large magnetic anisotropy for the mononuclear fragment combined with strong coupling within Ln-N

_{2}

^{3−}, which leads to a high energy barrier in this dinuclear system. Thus, the introduction of free radical bridging becomes a novel synthesis idea for high-energy barrier lanthanide dinuclear SMMs.

^{δ}

^{−}…M

^{n+}, considered as inverse hydrogen bonds (IHBs) [65]. DFT calculations confirm that IHBs in this system are relatively strong and larger than 24 kJ/mol (B-H

^{δ}

^{−}…Dy

^{3+}) in order of magnitude by establishing model complexes extracted from original geometry and considering Grimme’s D3 dispersion correction. The electrostatic potential (ESP) plots of Dy

^{3+}and B-H

^{δ}

^{−}surface explain the formation of this type of interaction, showing that the surface of carboranyl with negative electrostatic potential has a strong interaction with Dy

^{3+}. The distribution of IHBs is reflected more intuitively via fingerprint plots, color-mapped isosurface graphs of Hirshfeld surface and variable sections based on the independent gradient model (IGM) (Figure 13), which are visual analyses means of intra- and inter-molecular interactions. Among these complexes, three dimer ones were studied which are named

**2Dy**,

**4Dy**and

**6Dy**. The combination of imidazolin-iminato ligands and Dy(III) can promote strong magnetic axiality, and the introduction of IHBs leads to the evident exchange biasing effect. In the aspect of magnetism, the Zeeman splitting diagrams explain this phenomenon with the help of CASSCF calculations and magnetic exchange constant fittings, and the magnetic exchange constant of a single crystal sample reaches −2.0 cm

^{−1}, which is equivalent to that of a single atom-bridged binuclear complex system. Moreover, the bonding strengths of B-H

^{δ}

^{−}…M

^{n+}IHBs are cation dependent and decreased to 14 kJ/mol for B-H

^{δ}

^{−}…Na

^{+}. The total exchange constant (J

_{total}) decreases, and the exchange biasing effect disappears with the more distant path of magnetic exchange. This work is conducive to the understanding of the influence of IHBs on the magnetism of SMMs.

_{8}Dy

_{8}(mdea)

_{16}(CH

_{3}COO)

_{16}]∙CH

_{3}CN∙18H

_{2}O ({Fe

_{8}Dy

_{8}}), which contains 16 alternating magnetic centers (Figure 14) [17]. From the dc magnetic measurement, a local “S” shape curve was observed at 0.23 Tesla and 0.5 K. Combining with ab initio calculations and high-frequency/high-field electron paramagnetic resonance spectroscopy experiments (HF-EPR), the characteristic of a net toroidal moment was determined. Meanwhile, its isomorphic compounds of {Fe

_{8}Y

_{8}} and {Al

_{8}Dy

_{8}} were synthesized by replacing Fe(III) and Dy(III) with diamagnetic Al(III) and Y(III) to obtain magnetic interactions and understand the origin of the annular magnetic moment. The calculated Zeeman spectrum reveals that the strong ferromagnetic exchange interaction between Fe(III) and Dy(III) ensures a large energy gap between the first excited state and the ground state, which belongs to non-magnetic quadruple degenerate states. From the perspective of potential applications, the insertion of Fe, to some extent, can make compounds avoid interference of external magnetic field and realize information storage more steadily.

_{2h}and a large size of cage, matrix-diagonalization techniques cannot be utilized to fit its susceptibility curve. They first used CMC methods to study the discrete cluster, and the high-spin ground state was corroborated. Considering its actual crystal structure, three types of exchange parameters were determined as J

_{1}, J

_{2}and J

_{3}. The constant J

_{1}accounts for all superexchange interactions bridged by two atoms in this cage, while J

_{2}and J

_{3}describe couplings mediated through single hydroxide and oxide bridges, respectively (Figure 15). The best fitting gives J

_{1}= −44 cm

^{−1}, J

_{2}= −13 cm

^{−1}and J

_{3}= −10 cm

^{−1}, and these constants perfectly reproduced the susceptibility behavior of this cage in the temperature range of 300–40 K.

_{11}complex with S = $\frac{11}{2}$ to calculate J values and understand its magnetic behavior [67]. Ten coupling constants were introduced due to its low symmetry. After DFT calculations using the PBE functional, CMC simulations were performed to output the susceptibility curve by using the set of calculated exchange parameters, and this curve is close to the experimental data. Then, they took the calculated values as the starting point and continued modifying the shape of the susceptibility curve through CMC simulations, until outputting the magnetic susceptibility curve in near agreement with the experimental data (Figure 16). These results indicate that the trend of all J values is the same as other polynuclear Fe(III) complexes, and μ

_{3}-O bridging ligands are beneficial to construct stronger antiferromagnetic coupling. Accordingly, the combination of DFT calculations and CMC simulations can produce consistent results with the experimental susceptibility curve.

_{21}and Cu

_{16}, and their molecular configurations are tricorne and saddlelike cyclic, respectively [68]. Their temperature-dependent susceptibility curves hint at overall antiferromagnetic interactions. Then, QMC simulations were performed to study exchange interactions between copper ions. Considering different bridging modes, four exchange constants were defined for Cu

_{21}, while two coupling parameters were defined for Cu

_{16}in view of the nearest Cu…Cu distances (Figure 17). For the former complex, to avoid overparameterization, they supposed J

_{3}= J

_{4}in light of both interactions accounting for Cu(II) ions bridged by the pyrazolato and phenoxo groups. The magnetic susceptibility curve through fitting is consistent with that of the dc magnetic measurement. The data show ferro- and antiferro-magnetic interactions in Cu

_{21}and strong antiferromagnetic interactions in Cu

_{16}.

## 4. Conclusions and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**(

**a**) Crystal structure of the [Fe(C(SiMe

_{3})

_{3})

_{2}]

^{−}anion. Fe, purple; C, gray; Si, cyan. For clarity, all hydrogen atoms are omitted. (

**b**) Calculated 3d orbital energy for [Fe(C(SiMe

_{3})

_{3})

_{2}]

^{−}.

**Figure 4.**(

**a**) Crystal structure of the Fe[N(SiMe

_{3})

_{2}]

_{3}. Fe, purple; C, gray; N, blue. For clarity, all hydrogen atoms are omitted. (

**b**) Calculated 3d orbital energy for Fe[N(SiMe

_{3})

_{2}]

_{3}.

**Figure 5.**The shape of electron density in 4f trivalent free ion (reproduced with permission from [47]).

**Figure 6.**(

**a**,

**b**) Crystal structure of

**1**and

**2**. (

**c**) Geometric structure of the model complex with linear Im

^{Dipp}NH ligands coordinated. (

**d**,

**e**) Isosurfaces of HOMO orbitals for

**1**and

**2**. Dy, dark cyan; C, gray; N, blue; Cl, green; O, red. All hydrogen atoms are omitted for clarity (adapted with permission from [48]).

**Figure 7.**(

**a**) Crystal structure of

**3**(left) and

**4**(right) and intramolecular interactions. (

**b**) Intermolecular π…π (left) and C-H…π (right) interactions between the phenyl groups of neighboring cations. Dy, magenta; C, gray; N, blue; O, red. All hydrogen atoms are omitted for clarity (adapted with permission from [49]).

**Figure 8.**Simple flow chart on magnetic exchange calculation steps. A trinuclear dysprosium compound is attached to explain that specifically. The antimagnetic ions Lu(III) are applied to replace other Dy(III) ions in the calculation for the mononuclear magnetic properties.

**Figure 9.**(

**a**) Structure and atomic label of [Mn

^{III}Mn

^{IV}(μ-O)

_{2}(μ-OAc)DTNE]

^{2+}. (

**b**) MOs 123–126 of this compound at bs state calculated utilizing B3LYP functional. Four α unpaired electrons and three β ones present three potential superexchange pathways, respectively, indicating interaction passes through µ-oxo atoms: MO 123 represents a crossed σ/π pathway containing in-plane oxo p-orbitals, and MOs 124 and 125 can be characterized as symmetric π/π pathways involving metal d-orbitals and out-of-plane oxo p-orbitals (adapted with permission from [59]).

**Figure 10.**(

**a**) Crystal structure of [LCuGd(O

_{2}CCF

_{3})

_{3}(C

_{2}H

_{5}OH)

_{2}]. Cu, orange; Gd, dark cyan; C, gray; N, blue; O, red; F, light cyan. All hydrogen atoms are omitted for clarity. (

**b**) Mechanism diagram of magnetic exchange at Cu(II) and Gd(III) sites. The 5d orbitals of Gd(III) accept charge densities from its 4f orbitals and 3d orbitals of Cu(II) (adapted with permission from [60]).

**Figure 11.**Structures of four model complexes marked A (

**a**), B (

**b**), C (

**c**) and D (

**d**). All hydrogen atoms are omitted for clarity (adapted with permission from [61]).

**Figure 12.**(

**a**) Crystal structures of {[(Me

_{3}Si)

_{2}N]

_{2}(THF)Ln}

_{2}(μ-η

^{2}:η

^{2}-N

_{2})

^{−}(Ln = Gd, Tb, Dy, Ho, Er). (

**b**) Model structure of single fragment compound for this system in CASSCF calculations. Ln, dark cyan; La, turquoise; C, gray; N, blue; O, red; Si, blackish green. All hydrogen atoms are omitted for clarity.

**Figure 13.**Cross-section coloring maps based on electron density (ED) for lanthanacarborane dimers. The red circles represent the distribution of IHBs. (

**a**–

**c**) B-H

^{δ−}…Dy

^{3+}interactions in

**2Dy**and

**4Dy**; (

**d**) B-H

^{δ−}…Na

^{+}interactions in

**6Dy**.

**Figure 14.**Crystal structures of [Fe

_{8}Dy

_{8}(mdea)

_{16}(CH

_{3}COO)

_{16}]∙CH

_{3}CN∙18H

_{2}O. Dy, dark cyan; Fe, purple; C, gray; N, blue; O, red. All hydrogen atoms are omitted for clarity.

**Figure 15.**The molecular skeleton of {${\mathrm{Fe}}_{10}^{\mathrm{III}}$} cage and exchange coupling scheme. Fe-O bonds shown as full lines; J

_{1}shown as open lines; J

_{2}shown as dashed full lines; J

_{3}shown as dashed open lines (adapted with permission from [66]).

**Figure 16.**The molecular structure of Fe

_{11}complex (

**a**) and magnetic susceptibility curves (

**b**). The black circles and the white circles and squares correspond to experimental data, fitting results from DFT J values and CMC simulations with the calculated values as the starting point, respectively. All hydrogen atoms are omitted for clarity (adapted with permission from [67]).

**Figure 17.**The coupling models of two high-nuclearity copper cage compounds: Cu

_{21}and Cu

_{16}. (

**a**) Four J exchange parameters were defined considering different bridging modes. (

**b**) Two J exchange parameters were defined in view of the nearest Cu…Cu distances (adapted with permission from [68]).

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Luo, Q.-C.; Zheng, Y.-Z.
Methods and Models of Theoretical Calculation for Single-Molecule Magnets. *Magnetochemistry* **2021**, *7*, 107.
https://doi.org/10.3390/magnetochemistry7080107

**AMA Style**

Luo Q-C, Zheng Y-Z.
Methods and Models of Theoretical Calculation for Single-Molecule Magnets. *Magnetochemistry*. 2021; 7(8):107.
https://doi.org/10.3390/magnetochemistry7080107

**Chicago/Turabian Style**

Luo, Qian-Cheng, and Yan-Zhen Zheng.
2021. "Methods and Models of Theoretical Calculation for Single-Molecule Magnets" *Magnetochemistry* 7, no. 8: 107.
https://doi.org/10.3390/magnetochemistry7080107