*Article* **Reaching the Maximal Unquenched Orbital Angular Momentum** *L* **= 3 in Mononuclear Transition-Metal Complexes: Where, When and How?**

**Vladimir S. Mironov 1,2**

<sup>1</sup> Shubnikov Institute of Crystallography of Federal Scientific Research Centre "Crystallography and Photonics" of Russian Academy of Sciences, Leninskiy Prospekt 59, Moscow 119333, Russia; mirsa@list.ru

2 Institute of Problems of Chemical Physics, IPCP RAS, Chernogolovka 142432, Russia

**Abstract:** The conditions for achieving the maximal unquenched orbital angular momentum *L* = 3 and the highest magnetic anisotropy in mononuclear 3d complexes with axial coordination symmetry are examined in terms of the ligand field theory. It is shown that, apart from the known linear two-coordinate 3d<sup>7</sup> complex CoII(C(SiMe2ONaph)<sup>3</sup> )<sup>2</sup> characterized by record magnetic anisotropy and single-molecule magnet (SMM) performance (with the largest known spin-reversal barrier *U*eff = 450 cm−<sup>1</sup> ), the maximal orbital angular momentum *L* = 3 can also be obtained in linear two-coordinate 3d<sup>2</sup> complexes (V3+, Cr4+) and in trigonal-prismatic 3d<sup>3</sup> (Cr3+, Mn4+) and 3d<sup>8</sup> (Co<sup>+</sup> , Ni2+) complexes. A comparative assessment of the SMM performance of the 3d<sup>2</sup> , 3d<sup>3</sup> and 3d<sup>8</sup> complexes indicates that they are unlikely to compete with the record linear complex CoII(C(SiMe2ONaph)<sup>3</sup> )2 , whose magnetic anisotropy is close to the physical limit for a 3d metal.

**Keywords:** magnetic anisotropy; single-molecule magnet; unquenched orbital momentum; spinreversal barrier; mononuclear 3d complexes; spin-orbit coupling

### **1. Introduction**

Mononuclear paramagnetic metal complexes with high magnetic anisotropy are of great interest in recent years, largely due to the development of low-dimensional moleculebased magnetically bistable materials featuring slow magnetic relaxation and blocking of magnetization at low temperature—single-molecule magnets (0D, SMMs) [1–9] and single-chain magnets (1D, SCMs) [10–12]. The overall performance of SMMs and SCMs is quantified by two key parameters, the effective energy barrier to magnetization reversal (*U*eff) and the blocking temperature (*T*B), below which the magnetization persists in zero external fields [6–12]. The *U*eff and *T*<sup>B</sup> values depend strongly on the magnetic anisotropy of spin carriers incorporated in the high-spin molecule. The early research was mainly focused on polynuclear transition-metal complexes based on high-spin 3d-ions with enhanced single-ion magnetic anisotropy arising from the zero-field splitting (ZFS) of the ground spin state, such as Mn3+ and Ni2+ [1–3]. Then, after the discovery of slow magnetic relaxation in mononuclear lanthanide complexes (2003) [13] and later in mononuclear transition metal compounds (2010) [14,15], research interest has turned to SMMs involving single paramagnetic ion with high magnetic anisotropy due to the first-order unquenched orbital angular momentum, 4f-complexes [16–24] and special orbitally degenerate 3d-complexes with high axial symmetry [25,26]. These mononuclear 4f- and 3d-SMMs are known as single-ion magnets (SIMs).

At present, it is generally recognized that the unquenched orbital angular momentum of the magnetic metal ions is the most important factor governing large magnetic anisotropy and high SMM characteristics [8]. In this respect, lanthanide ions are especially attractive. In fact, all Ln3+ ions with open 4f*N*-shell exhibit large unquenched orbital angular momentum *L* (ranging from 3 to 6, except Gd3+ with *L* = 0) due to very weak ligand-field (LF) splitting

**Citation:** Mironov, V.S. Reaching the Maximal Unquenched Orbital Angular Momentum *L* = 3 in Mononuclear Transition-Metal Complexes: Where, When and How? *Inorganics* **2022**, *10*, 227. https://doi.org/10.3390/ inorganics10120227

Academic Editor: Peter Segl'a

Received: 28 October 2022 Accepted: 24 November 2022 Published: 27 November 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

energy and strong spin-orbit coupling (SOC) of 4f-electrons; the maximal orbital momentum *L* = 6 occurs in Nd3+ (4f<sup>3</sup> , 4 I9/2), Pm3+ (4f<sup>4</sup> , 5 I4), Ho3+ (4f<sup>10</sup> , 5 I8) and Er3+ (4f<sup>11</sup> , 4 I15/2) ions. Due to the strong SOC of 4f electrons, the orbital momentum *L* and spin *S* are coupled into the total angular momentum *J*; the ground state of free Ln3+(4f*N*) ions corresponds to the <sup>2</sup>*S*+1*L<sup>J</sup>* multiplet with *<sup>J</sup>* <sup>=</sup> *<sup>L</sup>* <sup>−</sup> *<sup>S</sup>* (*<sup>N</sup>* < 7) and *<sup>J</sup>* <sup>=</sup> *<sup>L</sup>* <sup>+</sup> *<sup>S</sup>* (*<sup>N</sup>* > 7). For Ln3+ ions in a ligands coordination environment, the ground *J*-multiplet undergoes LF splitting into 2*J* + 1 sublevels (called Stark sublevels) creating a highly anisotropic ground state [8,20]. In particular, pure Ising-type uniaxial magnetic anisotropy occurs in monometallic lanthanide complexes with distinct axial symmetry, in which axial LF lifts the (2*J* + 1)-fold degeneracy of the ground multiplet into ±*M<sup>J</sup>* microstates with a definite projection of the total angular momentum [16–24]. The energy splitting diagram of ±*M<sup>J</sup>* states (*E* vs. *M<sup>J</sup>* ) depends on both the strength and the intrinsic structure of the axial LF, which is specified by three axial LF parameters *B*20, *B*40, *B*<sup>60</sup> [27]. In the optimal case, the strong field stabilizes the doubly degenerate ground state with maximal *M<sup>J</sup>* projection (*M<sup>J</sup>* = ±*J*) and with large energy separation from excited LF states thereby resulting in a double-well potential with the highenergy barrier *U*eff inherent to high-performance SMMs. In particular, such LF-splitting pattern is observed in square antiprismatic [13], pentagonal-bipyramidal [28,29] and quasilinear metallocene lanthanide complexes [30,31], many of which exhibit exceptionally high *U*eff and *T*<sup>B</sup> values, including a record-setting dysprosium metallocene SMM with *U*eff = 1541 cm−<sup>1</sup> and *T*<sup>B</sup> = 80 K [31]. Recent trends in high-performance 4f SIM are reviewed in ref. [32].

Later on, since the report on the first mononuclear trigonal pyramidal FeII complexes with SMM behavior (2010) [14,15], extensive efforts have been taken to the development of 3d SIMs using diverse approaches to increase magnetic anisotropy and energy barrier; numerous 3d SIMs were reported in the past decade [8,25,26]. However, in contrast to f-block element complexes, in transition metal complexes the orbital angular momentum is commonly quenched (*L* = 0) by a strong LF to produce the spin-only ground state with low second-order magnetic anisotropy [33]. In the majority of high-spin 3d complexes, magnetic anisotropy is typically due to zero-field splitting (ZFS) of the ground spin multiplet 2*S* + 1, which is generally a weak effect resulting from second-order SOC [34]. Hence, the barriers *U*eff of 3d SIMs are mostly low, within a few tens of cm−<sup>1</sup> [8,25,26,33]. The most efficient strategy toward high-performance 3d SIMs takes advantage of unquenched orbital angular momentum (*L* 6= 0) giving rise to the first-order spin-orbit splitting of the ground spin stated with *L* 6= 0, which provides considerably stronger magnetic anisotropy and high energy barriers [16–24]. Unquenched first-order orbital angular momentum can only appear in orbitally degenerate transition metal complexes with high symmetry, such as octahedral CoII and [FeIII(CN)6] 3 complexes (*O<sup>h</sup>* ) [33,34], trigonal-pyramidal (*C*3*v*), [14,15] trigonal-bipyramidal (*D*3*<sup>h</sup>* ), [35] trigonal-prismatic (*D*3*<sup>h</sup>* ) [36,37], pentagonal-bipyramidal (*D*5*<sup>h</sup>* ) [38] and linear (*D*∞*<sup>h</sup>* ) two-coordinate complexes [39–43].

However, apart from unquenched orbital angular momentum, another important condition for the highest SMM performance of mononuclear 3d complexes is the axial limit of the magnetic anisotropy, which has a purely Ising-type nature, with zero transverse components. This case occurs in the LF of axial symmetry, which is produced by tuning the geometric axiality of the ligand environment of the 3d-ion. Depending on the specific type of the coordination polyhedron and electronic configuration of the 3d ion, the axial LF may stabilize the ground orbital doublet with the projection of the orbital angular momentum *M<sup>L</sup>* = 0, ±1, ±2, or ±3. The latter is further split by the first-order SOC into energy levels ±*M<sup>J</sup>* with definite projection *M<sup>J</sup>* of the total angular momentum *J* along the magnetic axis, which ranges from *L* + *S* to |*L* − *S*|. In this regime, spin energy levels are represented by pure |±*MJ*> spin wave functions, in which quantum tunneling of magnetization (QTM) is suppressed both in the ground and exited spin states. In these systems the barrier *U*eff is controlled by the first-order SOC energy ζ3d*LS*, which is proportional to the value of the ground-state orbital angular momentum *ML*; approximately, *U*eff is specified by ζ3d*ML*/2S, which is the energy separation between the ground ±*M<sup>J</sup>* and first exited

±(*M<sup>J</sup>* −1) states [42]. This leads to a pure Ising magnetic anisotropy and double-well potential with two lowest *M<sup>J</sup>* = ±*J* states similar to those in 4f SIMs [16–24]. Accordingly, the highest barriers *U*eff are observed in 3d SIMs with large orbital momentum *ML*, namely, in linear 3d<sup>7</sup> complexes [Fe<sup>I</sup> (C(SiMe3)3)2] <sup>−</sup> (226 cm−<sup>1</sup> ) [40] and [(sIPr)CoIINDmp] (413 cm−<sup>1</sup> ) [41] and also in trigonal-prismatic CoII complex (152 cm−<sup>1</sup> ) [36] featuring *L* = 2, S = 3/2 and *M<sup>J</sup>* = ±7/2 in the ground state. The record barrier belongs to linear two-coordinate cobalt complex, CoII(C(SiMe2ONaph)3)<sup>2</sup> (*U*eff = 450 cm−<sup>1</sup> ) [42] and for the monocoordinated cobalt(II) adatom on a MgO surface (*U*eff = 468 cm−<sup>1</sup> ) [43] which have *L* = 3, S = 3/2 and *M<sup>J</sup>* = ±9/2 in the ground state. ±(*MJ* −1) states [42]. This leads to a pure Ising magnetic anisotropy and double-well potential with two lowest *MJ* = ±*J* states similar to those in 4f SIMs [16–24]. Accordingly, the highest barriers *U*eff are observed in 3d SIMs with large orbital momentum *ML*, namely, in linear 3d7 complexes [FeI (C(SiMe3)3)2]− (226 cm–1) [40] and [(sIPr)CoIINDmp] (413 cm–1) [41] and also in trigonal-prismatic CoII complex (152 cm–1) [36] featuring *L* = 2, S = 3/2 and *MJ* = ±7/2 in the ground state. The record barrier belongs to linear two-coordinate cobalt complex, CoII(C(SiMe2ONaph)3)2 (*U*eff = 450 cm−1) [42] and for the monocoordinated cobalt(II) adatom on a MgO surface (*U*eff = 468 cm–1) [43] which have *L* = 3, S = 3/2 and *MJ* = ±9/2 in the ground state.

the value of the ground-state orbital angular momentum *ML*; approximately, *U*eff is specified by ζ3d*ML*/2S, which is the energy separation between the ground ±*MJ* and first exited

*Inorganics* **2022**, *10*, 227 3 of 12

Considering that a large unquenched orbital angular momentum is extremely important for attaining the highest SMM performance in 3d-SIMs, the relevant question is in which mononuclear 3d complexes, besides the already known record-breaking linear two-coordinate 3d<sup>7</sup> complexes [42,43], the maximum orbital angular momentum *L* = 3 (represented by the ground orbital doublet *M<sup>L</sup>* = ±3) is feasible. In this paper, general conditions for the appearance of the maximum orbital angular momentum *L* = 3 in mononuclear 3d complexes are analyzed in terms of the LF theory. We show that, apart from the linear two-coordinate 3d<sup>7</sup> complex CoII(C(SiMe2ONaph)3)<sup>2</sup> [42], the ground-state orbital doublet *<sup>M</sup><sup>L</sup>* <sup>=</sup> <sup>±</sup>3 can occur in linear 3d<sup>2</sup> complexes and in trigonal-prismatic 3d<sup>3</sup> and 3d<sup>8</sup> complexes. Specific conditions for the LF splitting pattern of 3d orbitals leading to *L* = 3 are established. Considering that a large unquenched orbital angular momentum is extremely important for attaining the highest SMM performance in 3d-SIMs, the relevant question is in which mononuclear 3d complexes, besides the already known record-breaking linear twocoordinate 3d7 complexes [42,43], the maximum orbital angular momentum *L* = 3 (represented by the ground orbital doublet *ML* = ±3) is feasible. In this paper, general conditions for the appearance of the maximum orbital angular momentum *L* = 3 in mononuclear 3d complexes are analyzed in terms of the LF theory. We show that, apart from the linear two-coordinate 3d7 complex CoII(C(SiMe2ONaph)3)2 [42], the ground-state orbital doublet *ML* = ±3 can occur in linear 3d2 complexes and in trigonal-prismatic 3d3 and 3d8 complexes. Specific conditions for the LF splitting pattern of 3d orbitals leading to *L* = 3 are established.

#### **2. Results 2. Results**

Below, the basic conditions for the realization of the maximal orbital angular momentum *M<sup>L</sup>* = ±3 in mononuclear 3d complexes with the axial LF are analyzed. First of all, it is clear that the ground orbital doublet *M<sup>L</sup>* = ±3 can only occur in 3d ions, which in the free-ion state have a 2S+1*F* ground atomic term featuring the maximum orbital angular momentum *L* = 3 for 3d ions, i.e., 3d<sup>2</sup> , 3d<sup>3</sup> , 3d<sup>7</sup> and 3d<sup>8</sup> ions. In transition–metal complexes with the LF of axial symmetry, the lowest <sup>2</sup>*S*+1*F* term splits into four orbital components <sup>2</sup>*S*+1*F*(*ML*) with a projection of orbital momentum *<sup>M</sup><sup>L</sup>* = 0, <sup>±</sup>1, <sup>±</sup>2, <sup>±</sup>3 (Figure 1). The objective of the present study is to establish conditions for the *M<sup>L</sup>* = ±3 orbital doublet to be the ground state. Below, the basic conditions for the realization of the maximal orbital angular momentum *ML* = ±3 in mononuclear 3d complexes with the axial LF are analyzed. First of all, it is clear that the ground orbital doublet *ML* = ±3 can only occur in 3d ions, which in the freeion state have a 2S+1*F* ground atomic term featuring the maximum orbital angular momentum *L* = 3 for 3d ions, i.e., 3d2, 3d3, 3d7 and 3d8 ions. In transition–metal complexes with the LF of axial symmetry, the lowest 2*S*+1*F* term splits into four orbital components 2*S*+1*F*(*ML*) with a projection of orbital momentum *ML* = 0, ±1, ±2, ±3 (Figure 1). The objective of the present study is to establish conditions for the *ML* = ±3 orbital doublet to be the ground state.

**Figure 1.** Energy splitting of the ground-state atomic term 2S+1*F*(*L* = 3) and 3d orbitals in a ligand field of axial symmetry. The orbital composition of the 2*S*+1*F*(*ML* = ±3) states is shown below in the insets to the corresponding figures. **Figure 1.** Energy splitting of the ground-state atomic term 2S+1*F*(*L* = 3) and 3d orbitals in a ligand field of axial symmetry. The orbital composition of the <sup>2</sup>*S*+1*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) states is shown below in the insets to the corresponding figures.

In an axial LF, five 3d orbitals are split into three groups of orbitals with a definite projection of the one-electron orbital angular momentum *ml* on the anisotropy axis *z*, 3*dz*<sup>2</sup> (*ml* = 0), (3*dxz*, 3*dyz*) (*ml* = ±1) and 3*dxy*, 3*dx*<sup>2</sup> *<sup>−</sup> y*2 (*ml* = ±2) (Figure 1). A necessary (but not sufficient) condition for the *ML* = ±3 ground state is the presence of one unpaired electron in each of the two doubly degenerate 3d orbitals with magnetic numbers *ml* = ±1, and *ml* = ±2, i.e., (*xz*, *yz*)1(*xy*, *x*2 *− y*2)1 or (*xz*, *yz*)3(*xy*, *x*2 *− y*2)3. For each of the 3d ions selected above, the LF splitting pattern of 3d-orbitals leading to the 2S+1*F*(*ML* = ±3) ground state is examined In an axial LF, five 3d orbitals are split into three groups of orbitals with a definite projection of the one-electron orbital angular momentum *m<sup>l</sup>* on the anisotropy axis *z*, 3*dz*<sup>2</sup> (*m<sup>l</sup>* = 0), (3*dxz*, 3*dyz*) (*m<sup>l</sup>* = ±1) and 3*dxy*, 3*dx*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> (*m<sup>l</sup>* = ±2) (Figure 1). A necessary (but not sufficient) condition for the *M<sup>L</sup>* = ±3 ground state is the presence of one unpaired electron in each of the two doubly degenerate 3d orbitals with magnetic numbers *m<sup>l</sup>* = ±1, and *m<sup>l</sup>* = ±2, i.e., (*xz*, *yz*) 1 (*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2)<sup>1</sup> or (*xz*, *yz*) 3 (*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2)<sup>3</sup> . For each of the 3d ions selected above, the LF splitting pattern of 3d-orbitals leading to the 2S+1*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state isexamined in terms of the conventional LF theory, which takes into account the interelectron repulsion (quantified in terms of the Racah parameters *B* and *C*) and SOC, ζ3dΣ*<sup>i</sup> lis<sup>i</sup>* . LF

calculations are performed with the full basis set of 3d<sup>N</sup> configurations involving 45 (3d<sup>2</sup> , 3d<sup>8</sup> ) and 120 (3d<sup>3</sup> , 3d<sup>7</sup> ) |*LMLSMS*> microstates. Details of LF calculations are available in the Supplementary Materials. tions are performed with the full basis set of 3dN configurations involving 45 (3d2, 3d8) and 120 (3d3, 3d7) |*LMLSMS*> microstates. Details of LF calculations are available in the Supplementary Materials.

in terms of the conventional LF theory, which takes into account the interelectron repulsion (quantified in terms of the Racah parameters *B* and *C*) and SOC, ζ3dΣ*ilisi*. LF calcula-

### *2.1. 3d<sup>2</sup> Complexes (V3+, Cr4+) 2.1. 3d2 Complexes (V3+, Cr4+)*

*Inorganics* **2022**, *10*, 227 4 of 12

The necessary condition for the <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) orbital doublet to be the ground state is to have the LF spitting pattern *E*(*z* 2 ) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 − *y*2). Such an orbital energy scheme can occur in complexes with linear or quasi-linear two-coordinate complexes [44] similar to the quasi-linear FeII [39] and CoII complexes [42]. For a quantitative evaluation, LF calculations for the energy levels of the 3d<sup>2</sup> configuration were performed using the Racah parameters *B* = 600, *C* = 2900 cm−<sup>1</sup> and SOC constant ζ = 150 cm−<sup>1</sup> , which are typical for V3+ ion [45]. The calculations were performed with fixed orbital energies *E*(*z* 2 ) = 10,000 cm−<sup>1</sup> , *E*(*xy*, *x*2 − *y*2) = 0 and variable orbital energy *E*(*xz*, *yz*) = ∆ (see inset in Figure 2a). The orbital doublet <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) (shown in the solid red line in Figure 2a) is found to be the ground state in the energy range 0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 5400 cm−<sup>1</sup> ; at larger energies, the ground state is the orbital singlet <sup>3</sup>*F*(*M<sup>L</sup>* = 0), whose wave function *Det*||*xy*↑, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2↑|| corresponds to two singly occupied *xy* and *x*2 − *y*2 orbitals with parallel electron spins (Figure 2a). From the point of view of coordination geometry, the only suitable case for the <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state is a linear two-coordination of 3d<sup>2</sup> ion. Numerous 3d complexes of this type were reported in the literature, as outlined in a review article [44]. The necessary condition for the 3*F*(*ML* = ±3) orbital doublet to be the ground state is to have the LF spitting pattern *E*(*z*2) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 *− y*2). Such an orbital energy scheme can occur in complexes with linear or quasi-linear two-coordinate complexes [44] similar to the quasi-linear FeII [39] and CoII complexes [42]. For a quantitative evaluation, LF calculations for the energy levels of the 3d2 configuration were performed using the Racah parameters *B* = 600, *C* = 2900 cm−1 and SOC constant ζ = 150 cm−1, which are typical for V3+ ion [45]. The calculations were performed with fixed orbital energies *E*(*z*2) = 10,000 cm−1, *E*(*xy*, *x*2 *− y*2) = 0 and variable orbital energy *E*(*xz*, *yz*) = Δ (see inset in Figure 2a). The orbital doublet 3*F*(*ML* = ±3) (shown in the solid red line in Figure 2a) is found to be the ground state in the energy range 0 ≤ Δ ≤ 5400 cm−1; at larger energies, the ground state is the orbital singlet 3*F*(*ML* = 0), whose wave function *Det*||*xy*↑, *x*2 *− y*2↑|| corresponds to two singly occupied *xy* and *x*2 *− y*2 orbitals with parallel electron spins (Figure 2a). From the point of view of coordination geometry, the only suitable case for the 3*F*(*ML* = ±3) ground state is a linear two-coordination of 3d2 ion. Numerous 3d complexes of this type were reported in the literature, as outlined in a review article [44].

**Figure 2.** (**a**) Variation of the energies of the 3*F*(*ML*) LF states of V3+(3d2) ion in linear coordination with increasing energy separation Δ = *E*(*xz*, *yz*) − *E*(*xy*, *x*2 *− y*2) between the lowest and first excited sets of 3d orbitals. The range of the stabilization of the 3*F*(*ML* = ±3) ground state (0 ≤ Δ ≤ 5400 cm<sup>−</sup>1) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals in a linear two-coordination, *E*(*xy*, *x*2 *− y*2) = 0, *E*(*z*2) = 10,000 cm−1 (fixed) and *E*(*xz*, *yz*) = Δ (variable); (**b**) *E* vs. *MJ* diagram calculated at Δ = 3000 cm<sup>−</sup>1. The spin-reversal barrier *U*eff ≈ 200 cm<sup>−</sup>1 is approximately estimated by the energy separation between the ground *MJ* = ±2 and the first excited *MJ* = ±3 states. **Figure 2.** (**a**) Variation of the energies of the <sup>3</sup>*F*(*ML*) LF states of V3+(3d<sup>2</sup> ) ion in linear coordination with increasing energy separation ∆ = *E*(*xz*, *yz*) − *E*(*xy*, *x*2 − *y*2) between the lowest and first excited sets of 3d orbitals. The range of the stabilization of the <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 5400 cm−<sup>1</sup> ) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals in a linear two-coordination, *E*(*xy*, *x*2 − *y*2) = 0, *E*(*z* 2 ) = 10,000 cm−<sup>1</sup> (fixed) and *E*(*xz*, *yz*) = ∆ (variable); (**b**) *E* vs. *M<sup>J</sup>* diagram calculated at ∆ = 3000 cm−<sup>1</sup> . The spin-reversal barrier *<sup>U</sup>*eff <sup>≈</sup> 200 cm−<sup>1</sup> is approximately estimated by the energy separation between the ground *M<sup>J</sup>* = ±2 and the first excited *M<sup>J</sup>* = ±3 states.

with a barrier *U*eff of about 200 cm−1, as estimated from the energy gap between ground *MJ* = ±2 and first excited *MJ* = ±3 states (Figure 2b). Remarkably, the *E* vs. *MJ* diagram has an inverse order of ±*MJ* levels compared to the usual double-well spin energy profile (with the maximum *MJ* at the bottom and the minimum *MJ* at the top) due to the antiparallel coupling of the orbital momentum *L* = 3 and spin *S* = 1 of V3+(3d2) ion. However, the actual The calculated *E* vs. *M<sup>J</sup>* diagram indicates that such a 3d<sup>2</sup> complex may be a SMM with a barrier *U*eff of about 200 cm−<sup>1</sup> , as estimated from the energy gap between ground *M<sup>J</sup>* = ±2 and first excited *M<sup>J</sup>* = ±3 states (Figure 2b). Remarkably, the *E* vs. *M<sup>J</sup>* diagram has an inverse order of ±*M<sup>J</sup>* levels compared to the usual double-well spin energy profile (with the maximum *M<sup>J</sup>* at the bottom and the minimum *M<sup>J</sup>* at the top) due to the antiparallel coupling of the orbital momentum *L* = 3 and spin *S* = 1 of V3+(3d<sup>2</sup> ) ion. However, the actual SMM performance of the linear two-coordinate complex of V3+ is expected to be

The calculated *E* vs. *MJ* diagram indicates that such a 3d2 complex may be a SMM

much lower due to the non-Kramers nature of the ground doublet *M<sup>J</sup>* = ±2, which is subject to splitting due to possible deviations from the axiality of the molecular structure, leading to fast under-barrier quantum tunneling of magnetization (QTM) that shortcuts the energy barrier. Hence, in terms of SMM performance, the situation with the linear two-coordinate 3d<sup>2</sup> complex is much less favorable compared to the record cobalt 3d<sup>7</sup> complex CoII(C(SiMe2ONaph)3)<sup>2</sup> [42]. splitting due to possible deviations from the axiality of the molecular structure, leading to fast under-barrier quantum tunneling of magnetization (QTM) that shortcuts the energy barrier. Hence, in terms of SMM performance, the situation with the linear two-coordinate 3d2 complex is much less favorable compared to the record cobalt 3d7 complex CoII(C(SiMe2ONaph)3)2 [42].

SMM performance of the linear two-coordinate complex of V3+ is expected to be much lower due to the non-Kramers nature of the ground doublet *MJ* = ±2, which is subject to

### *2.2. 3d<sup>3</sup> Complexes (Cr3+, Mn4+) 2.2. 3d3 Complexes (Cr3+, Mn4+)*  The ground state of a free 3d3 ion is a 4*F* atomic term, which is split by the axial LF

The ground state of a free 3d<sup>3</sup> ion is a <sup>4</sup>*F* atomic term, which is split by the axial LF into four orbital components, <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3), <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>2), <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>1) and <sup>4</sup>*F*(*M<sup>L</sup>* = 0). The <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) state originates from the (*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2)<sup>1</sup> (*xz*, *yz*) 1 (*z* 2 ) 1 electronic configuration. LF analysis indicates that this configuration can only occur at the *E*(*xy*, *x*2 − *y*2) > *E*(*xz*, *yz*) > *E*(*z* 2 ) LF splitting pattern of 3d orbitals. Comparative LF calculations with atomic parameters *B* = 650, *C* = 3000 cm−<sup>1</sup> and ζ = 250 cm−<sup>1</sup> for Cr3+ ion reveal that the orbital doublet <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) is the ground state when the energy difference ∆ = *E*(*xy*, *x*2 − *y*2) − *E*(*xz*, *yz*) between the energy levels of the (*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2) and (*xz*, *yz*) orbitals is within the range 0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 5850 cm−<sup>1</sup> . Beyond this range, the ground state turns to the <sup>4</sup>*F*(*M<sup>L</sup>* = 0) orbital singlet, which is represented by a mixture of the (*xy*) 1 (*x* <sup>2</sup> <sup>−</sup> *<sup>y</sup>* 2 ) 1 (*z* 2 ) <sup>1</sup> and (*xz*) 1 (*yz*) 1 (*z* 2 ) 1 electronic configurations (Figure 3). into four orbital components, 4*F*(*ML* = ±3), 4*F*(*ML* = ±2), 4*F*(*ML* = ±1) and 4*F*(*ML* = 0). The 4*F*(*ML* = ±3) state originates from the (*xy*, *x*2 *− y*2)1(*xz*, *yz*)1(*z*2)1 electronic configuration. LF analysis indicates that this configuration can only occur at the *E*(*xy*, *x*2 *− y*2) > *E*(*xz*, *yz*) > *E*(*z*2) LF splitting pattern of 3d orbitals. Comparative LF calculations with atomic parameters *B* = 650, *C* = 3000 cm−1 and ζ = 250 cm−1 for Cr3+ ion reveal that the orbital doublet 4*F*(*ML* = ±3) is the ground state when the energy difference Δ = *E*(*xy*, *x*2 *− y*2) − *E*(*xz*, *yz*) between the energy levels of the (*xy*, *x*2 *− y*2) and (*xz*, *yz*) orbitals is within the range 0 ≤ Δ ≤ 5850 cm−1. Beyond this range, the ground state turns to the 4*F*(*ML* = 0) orbital singlet, which is represented by a mixture of the (*xy*)1(*x*<sup>2</sup> − *y*2)1(*z*2)1 and (*xz*)1(*yz*)1(*z*2)1 electronic configurations (Figure 3).

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**Figure 3.** Dependence of the energies of the 4*F*(*ML*) LF states of Cr3+(3d3) ion on the energy separation Δ = *E*(*xy*, *x*2 *− y*2) − *E*(*xz*, *yz*) between the (*xy*, *x*2 *− y*2) and (*xz*, *yz*) sets of 3d orbitals. The range of the stabilization of the 4*F*(*ML* = ±3) ground state (0 ≤ Δ ≤ 5850 cm<sup>−</sup>1) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*z*2) = 0, *E*(*xz*, *yz*) = 5000 cm<sup>−</sup>1 (fixed) and *E*(*xy*, *x*2 *− y*2) = 5000 cm<sup>−</sup><sup>1</sup> + Δ (variable). LF calculations were done with atomic parameters *B* = 650, *C* = 3000 and ζ = 250 cm<sup>−</sup>1. Evidently, the linear two-coordinate geometry of 3d3 complexes cannot provide the **Figure 3.** Dependence of the energies of the <sup>4</sup>*F*(*ML*) LF states of Cr3+(3d<sup>3</sup> ) ion on the energy separation ∆ = *E*(*xy*, *x*2 − *y*2) − *E*(*xz*, *yz*) between the (*xy*, *x*2 − *y*2) and (*xz*, *yz*) sets of 3d orbitals. The range of the stabilization of the <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 5850 cm−<sup>1</sup> ) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*z* 2 ) = 0, *E*(*xz*, *yz*) = 5000 cm−<sup>1</sup> (fixed) and *E*(*xy*, *x*2 − *y*2) = 5000 cm−<sup>1</sup> + ∆ (variable). LF calculations were done with atomic parameters *B* = 650, *C* = 3000 and ζ = 250 cm−<sup>1</sup> .

required orbital pattern *E*(*xy*, *x*2 *− y*2) > *E*(*xz*, *yz*) > *E*(*z*2) since linear coordination is characterized by another order of orbital energies, *E*(*z*2) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 *− y*2), which can result in *ML* = ±1 or *ML* = ±2 ground state. Therefore, unlike the record linear two-coordinate Co2+(3d7) complexes with the *ML* = ±3 ground state, linear 3d3 complexes have a maximum achievable orbital angular momentum of only *ML* = ±2. Among a variety of axial symmetry coordination geometries, the necessary orbital Evidently, the linear two-coordinate geometry of 3d<sup>3</sup> complexes cannot provide the required orbital pattern *E*(*xy*, *x*2 − *y*2) > *E*(*xz*, *yz*) > *E*(*z* 2 ) since linear coordination is characterized by another order of orbital energies, *E*(*z* 2 ) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 − *y*2), which can result in *M<sup>L</sup>* = ±1 or *M<sup>L</sup>* = ±2 ground state. Therefore, unlike the record linear twocoordinate Co2+(3d<sup>7</sup> ) complexes with the *<sup>M</sup><sup>L</sup>* <sup>=</sup> <sup>±</sup>3 ground state, linear 3d<sup>3</sup> complexes have a maximum achievable orbital angular momentum of only *M<sup>L</sup>* = ±2.

splitting pattern *E*(*xy*, *x*2 *− y*2) > *E*(*xz*, *yz*) > *E*(*z*2) can be obtained in the trigonal-prismatic coordination of the 3d3 ion. However, in trigonal-prismatic complexes, the energy order of 3d orbital strongly depends on the polar angle θ of the trigonal prism. In order to evaluate the conditions for stabilization of the *ML* = ±3 ground state, LF calculations are performed in combination with the angular overlap model (AOM) [46–51] using the AOM parameters *e*σ = 9000 cm−1 and *e*σ/*e*π = 4 for Cr3+ ions. These calculations show that the Among a variety of axial symmetry coordination geometries, the necessary orbital splitting pattern *E*(*xy*, *x*2 − *y*2) > *E*(*xz*, *yz*) > *E*(*z* 2 ) can be obtained in the trigonal-prismatic coordination of the 3d<sup>3</sup> ion. However, in trigonal-prismatic complexes, the energy order of 3d orbital strongly depends on the polar angle θ of the trigonal prism. In order to evaluate the conditions for stabilization of the *M<sup>L</sup>* = ±3 ground state, LF calculations are performed in combination with the angular overlap model (AOM) [46–51] using the AOM parameters *e*<sup>σ</sup> = 9000 cm−<sup>1</sup> and *e*σ/*e*<sup>π</sup> = 4 for Cr3+ ions. These calculations show that the ground state

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<sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) with the maximal orbital momentum *<sup>M</sup><sup>L</sup>* <sup>=</sup> <sup>±</sup>3 stabilizes in a compressed trigonal prism with a polar angle θ ranging between 61◦ and 66.5◦ (Figure 4a). ground state 4*F*(*ML* = ±3) with the maximal orbital momentum *ML* = ±3 stabilizes in a com-

pressed trigonal prism with a polar angle θ ranging between 61° and 66.5° (Figure 4a).

**Figure 4.** (**a**) Variation of the energies of the 4*F*(*ML*) LF states of Cr3+(3d3) ion with increasing polar angle θ in a trigonal-prismatic coordination polyhedron. The range of the stabilization of the 4*F*(*ML* = ±3) ground state (61° ≤ θ ≤ 66.5°) is marked with light pink; (**b**) *E* vs. *MJ* diagram calculated at θ = 64°. The spin-reversal barrier *U*eff ≈ 200 = 250 cm<sup>−</sup>1 is approximately estimated by the energy separation between the ground *MJ* = ±3/2 and the first excited *MJ* = ±5/2 states. **Figure 4.** (**a**) Variation of the energies of the <sup>4</sup>*F*(*ML*) LF states of Cr3+(3d<sup>3</sup> ) ion with increasing polar angle θ in a trigonal-prismatic coordination polyhedron. The range of the stabilization of the <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (61◦ <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> 66.5◦ ) is marked with light pink; (**b**) *E* vs. *M<sup>J</sup>* diagram calculated at θ = 64◦ . The spin-reversal barrier *<sup>U</sup>*eff <sup>≈</sup> 200 = 250 cm−<sup>1</sup> is approximately estimated by the energy separation between the ground *M<sup>J</sup>* = ±3/2 and the first excited *M<sup>J</sup>* = ±5/2 states.

SOC splits the ground orbital doublet *ML* = ±3 into four Kramer doublets with the projection of the total angular momentum *MJ* = ±3/2, ±5/2, ±7/2 and ±9/2, listed in order of increasing energy; their energy diagram with the total splitting energy of 762 cm−1 is shown in Figure 4b. These results indicate that such a complex should have a SMM behavior with a barrier of about 250 cm−1, corresponding to the energy separation between the ground *MJ* = ±3/2 and first excited *MJ* = ±5/2 states. However, as opposed to linear twocoordinate CoII complexes [42,43], the trigonal-prismatic 3d3 complexes with doubly degenerate ground state *ML* = ±3 is Jahn–Teller active, subject to distortions that remove orbital degeneracy and thereby reduce the magnetic anisotropy of the complex. Nevertheless, LF/AOM calculations indicate that with moderate departures from the regular *D*3*<sup>h</sup>* geometry, the overall energy pattern of the *JM* states (Figure 4b) does not change much, so the trigonal-prismatic 3d3 complex behaves as an SMM with an estimated barrier of about 150–200 cm−1. SOC splits the ground orbital doublet *M<sup>L</sup>* = ±3 into four Kramer doublets with the projection of the total angular momentum *M<sup>J</sup>* = ±3/2, ±5/2, ±7/2 and ±9/2, listed in order of increasing energy; their energy diagram with the total splitting energy of 762 cm−<sup>1</sup> is shown in Figure 4b. These results indicate that such a complex should have a SMM behavior with a barrier of about 250 cm−<sup>1</sup> , corresponding to the energy separation between the ground *M<sup>J</sup>* = ±3/2 and first excited *M<sup>J</sup>* = ±5/2 states. However, as opposed to linear two-coordinate CoII complexes [42,43], the trigonal-prismatic 3d<sup>3</sup> complexes with doubly degenerate ground state *M<sup>L</sup>* = ±3 is Jahn–Teller active, subject to distortions that remove orbital degeneracy and thereby reduce the magnetic anisotropy of the complex. Nevertheless, LF/AOM calculations indicate that with moderate departures from the regular *D*3*<sup>h</sup>* geometry, the overall energy pattern of the *J<sup>M</sup>* states (Figure 4b) does not change much, so the trigonal-prismatic 3d<sup>3</sup> complex behaves as an SMM with an estimated barrier of about 150–200 cm−<sup>1</sup> .

#### *2.3. 3d7 Complexes (Fe+,Co2+) 2.3. 3d<sup>7</sup> Complexes (Fe<sup>+</sup> ,Co2+)*

Linear two-coordinate 3d7 cobalt complex CoII(C(SiMe2ONaph)3)2 [42] is currently the only known individual mononuclear 3d complex featuring the maximal orbital angular momentum *L* = 3. From the point of view of the LF theory, in the 3d7 configuration the orbital LF splitting pattern *E*(*z*2) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 *− y*2) is a necessary condition for the stabilization of the *ML* = ±3 ground state. Note that the order of the 3d orbitals in the 3d7 configuration is opposite to that in the 3d3 configuration because of the electron–hole symmetry. As in the case of 3d2 and 3d3 complexes, the additional condition is the restriction on the energy separation between the *ml* = ±1 and *ml* = ±2 orbitals, Δ = *E*(*xz*, *yz*) *− E*(*xy*, *x*2 *− y*2), which is 0 ≤ Δ ≤ 6750 cm−1 (Figure 5). Outside this range, the ground state is the orbital singlet 4*F*(*ML* = 0). Formally, an extra condition is a limitation on the total LF splitting energy ΔLF = *E*(*z*2) *− E*(*xy*, *x*2 *− y*2) < 22,000 cm−1, which is, however, too large to occur in a linear two-coordinate 3d7 complex exhibiting much lower total LF splitting energy, around 6000 cm−1 [42]. Linear two-coordinate 3d<sup>7</sup> cobalt complex CoII(C(SiMe2ONaph)3)<sup>2</sup> [42] is currently the only known individual mononuclear 3d complex featuring the maximal orbital angular momentum *L* = 3. From the point of view of the LF theory, in the 3d<sup>7</sup> configuration the orbital LF splitting pattern *E*(*z* 2 ) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 − *y*2) is a necessary condition for the stabilization of the *M<sup>L</sup>* = ±3 ground state. Note that the order of the 3d orbitals in the 3d<sup>7</sup> configuration is opposite to that in the 3d<sup>3</sup> configuration because of the electron– hole symmetry. As in the case of 3d<sup>2</sup> and 3d<sup>3</sup> complexes, the additional condition is the restriction on the energy separation between the *m<sup>l</sup>* = ±1 and *m<sup>l</sup>* = ±2 orbitals, ∆ = *E*(*xz*, *yz*) <sup>−</sup> *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2), which is 0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 6750 cm−<sup>1</sup> (Figure 5). Outside this range, the ground state is the orbital singlet <sup>4</sup>*F*(*M<sup>L</sup>* = 0). Formally, an extra condition is a limitation on the total LF splitting energy ∆LF = *E*(*z* 2 ) <sup>−</sup> *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2) < 22,000 cm−<sup>1</sup> , which is, however,too large to occur in a linear two-coordinate 3d<sup>7</sup> complex exhibiting much lower total LF splitting energy, around 6000 cm−<sup>1</sup> [42].

3200 and 0 cm−1, respectively) in the FeI

the 3d*z*2 orbital in FeI

**Figure 5.** Energies of the 4*F*(*ML*) LF states of Co2+(3d7) ion in a linear coordination as a function of the energy gap Δ = *E*(*xz*, *yz*) *− E*(*xy*, *x*2 *− y*2). The range of the stabilization of the 4*F*(*ML* = ±3) ground state (0 ≤ Δ ≤ 6750 cm<sup>−</sup>1) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*xy*, *x*2 *− y*2) = 0, *E*(*z*2) = 10,000 cm−1 (fixed) and *E*(*xz*, *yz*) = Δ (variable). LF calculations are performed with atomic parameters *B* = 750, *C* = 3800 and ζ = 480 cm<sup>−</sup>1. **Figure 5.** Energies of the <sup>4</sup>*F*(*ML*) LF states of Co2+(3d<sup>7</sup> ) ion in a linear coordination as a function of the energy gap <sup>∆</sup> <sup>=</sup> *<sup>E</sup>*(*xz*, *yz*) <sup>−</sup> *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2). The range of the stabilization of the <sup>4</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 6750 cm−<sup>1</sup> ) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*xy*, *x*2 − *y*2) = 0, *E*(*z* 2 ) = 10,000 cm−<sup>1</sup> (fixed) and *E*(*xz*, *yz*) = ∆ (variable). LF calculations are performed with atomic parameters *B* = 750, *C* = 3800 and ζ = 480 cm−<sup>1</sup> .

In light of these results, it is interesting to consider the difference in the nature of the

(*ML* = ±2) [40] and CoII(C(SiMe2ONaph)3)2 (*ML* = ±3) [42]. The underlying reason is the different order of the actual 3d orbitals, which is *E*(*xz*, *yz*) > *E*(*xy*, *x*2 *− y*2) > *E*(*z*2) (c.a. 5200,

((c.a. 5700, 3000 and 0 cm−1) in CoII complex [42]. A significant decrease in the energy of

bination of the ground orbital state *ML* = ±3, the parallel coupling of the orbital momentum *L* = 3 and spin S = 3/2 and a sufficiently strong SOC leads to the most favorable conditions for obtaining a record spin-reversal barrier *U*eff = 450 cm−1 of CoII(C(SiMe2ONaph)3)2 [42].

(C(SiMe3)3)2]− is caused by a very strong 4s–3d*z*2 mixing [40]. The com-

(C(SiMe3)3)2]<sup>−</sup>

complex [40] and *E*(*z*2) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 *− y*2)

ground state of the isoelectronic linear two-coordinate 3d7 complexes [FeI

*2.4. 3d8 Complexes (Ni2+, Co1+)*  As in the case of 3d2 complexes with *L* = 3, the necessary condition for the occurrence of the *ML* = ±3 ground state in a 3d8 complex is the presence of one unpaired electron on the doubly degenerate 3d orbitals with magnetic quantum numbers *ml* = ±1 and *ml* = ±2, i.e., (*x*2 *− y*2, *xy*)3(*xz*, *yz*)3(*z*2)2. For this, the order of the 3d orbitals should be the same as in 3d3 complexes, i.e., *E*(*xy*, *x*2 *− y*2) > *E*(*xz*, *yz*) > *E*(*z*2). Therefore, this LF splitting pattern can occur in trigonal-prismatic complexes (see Figure 4a), but not in linear two-coordinate complexes (Figure 5). As in the case of other above 3d complexes, there is an additional condition that constrains the energy separation between *ml* = ±1 and *ml* = ±2 orbitals, Δ = *E*(*xy*, *x*2 *− y*2) *− E*(*xz*, *yz*), which is 0 ≤ Δ ≤ 7200 cm−1.(Figure 6). LF/AOM calculations for a In light of these results, it is interesting to consider the difference in the nature of the ground state of the isoelectronic linear two-coordinate 3d<sup>7</sup> complexes [Fe<sup>I</sup> (C(SiMe3)3)2] − (*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>2) [40] and CoII(C(SiMe2ONaph)3)<sup>2</sup> (*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) [42]. The underlying reason is the different order of the actual 3d orbitals, which is *E*(*xz*, *yz*) > *E*(*xy*, *x*2 − *y*2) > *E*(*z* 2 ) (ca. 5200, 3200 and 0 cm−<sup>1</sup> , respectively) in the Fe<sup>I</sup> complex [40] and *E*(*z* 2 ) > *E*(*xz*, *yz*) > *E*(*xy*, *x*2 − *y*2) ((ca. 5700, 3000 and 0 cm−<sup>1</sup> ) in CoII complex [42]. A significant decrease in the energy of the 3d*<sup>z</sup>* <sup>2</sup> orbital in Fe<sup>I</sup> (C(SiMe3)3)2] − is caused by a very strong 4s–3d*<sup>z</sup>* <sup>2</sup> mixing [40]. The combination of the ground orbital state *M<sup>L</sup>* = ±3, the parallel coupling of the orbital momentum *L* = 3 and spin S = 3/2 and a sufficiently strong SOC leads to the most favorable conditions for obtaining a record spin-reversal barrier *U*eff = 450 cm−<sup>1</sup> of CoII(C(SiMe2ONaph)3)<sup>2</sup> [42].

trigonal-prismatic 3d8 complex with variable polar angle θ indicate that the *ML* = ±3

### ground state stabilizes at 61° ≤ θ ≤ 74.2°, corresponding to a compressed trigonal prism *2.4. 3d<sup>8</sup> Complexes (Ni2+, Co1+)*

(Figure 7a). Several trigonal-prismatic cage complexes of NiII were recently reported in [52]. In these complexes, the Ni2+ ion is encapsulated in a frame-type clathrochelate cage, that forces robust trigonal-prismatic coordination. However, in these complexes, the polar angle θ is about 52°, which is well below the critical angle θ > 61° (Figure 7a). Accordingly, the ground state of these complexes was found to be orbital singlet 3*F*(*ML* = 0) with a second-order magnetic anisotropy due to ZFS [48], which agrees well with the results of LF/AOM calculations indicating the *ML* = 0 ground state at θ ≈ 52° (Figure 7a). One more example of a trigonal-prismatic 3d8 complex is given by similar clathrochelate complexes CoI (GmCl2)3(BPh)2 reported in ref. [53]. Unfortunately, these complexes also do not fall As in the case of 3d<sup>2</sup> complexes with *L* = 3, the necessary condition for the occurrence of the *<sup>M</sup><sup>L</sup>* <sup>=</sup> <sup>±</sup>3 ground state in a 3d<sup>8</sup> complex is the presence of one unpaired electron on the doubly degenerate 3d orbitals with magnetic quantum numbers *m<sup>l</sup>* = ±1 and *m<sup>l</sup>* = ±2, i.e., (*x*2 − *y*2, *xy*) 3 (*xz*, *yz*) 3 (*z* 2 ) 2 . For this, the order of the 3d orbitals should be the same as in 3d<sup>3</sup> complexes, i.e., *E*(*xy*, *x*2 − *y*2) > *E*(*xz*, *yz*) > *E*(*z* 2 ). Therefore, this LF splitting pattern can occur in trigonal-prismatic complexes (see Figure 4a), but not in linear two-coordinate complexes (Figure 5). As in the case of other above 3d complexes, there is an additional condition that constrains the energy separation between *m<sup>l</sup>* = ±1 and *m<sup>l</sup>* = ±2 orbitals, <sup>∆</sup> <sup>=</sup> *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2) <sup>−</sup> *<sup>E</sup>*(*xz*, *yz*), which is 0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 7200 cm−<sup>1</sup> .(Figure 6). LF/AOM calculations for a trigonal-prismatic 3d<sup>8</sup> complex with variable polar angle θ indicate that the *M<sup>L</sup>* = ±3 ground state stabilizes at 61◦ ≤ θ ≤ 74.2◦ , corresponding to a compressed trigonal prism (Figure 7a). Several trigonal-prismatic cage complexes of NiII were recently reported in [52]. In these complexes, the Ni2+ ion is encapsulated in a frame-type clathrochelate cage, that forces robust trigonal-prismatic coordination. However, in these complexes, the polar angle θ is about 52◦ , which is well below the critical angle θ > 61◦ (Figure 7a). Accordingly, the ground state of these complexes was found to be orbital singlet <sup>3</sup>*F*(*M<sup>L</sup>* = 0) with a second-order magnetic anisotropy due to ZFS [48], which agrees well with the results of LF/AOM calculations indicating the *M<sup>L</sup>* = 0 ground state at θ ≈ 52◦ (Figure 7a). One more example of a trigonal-prismatic 3d<sup>8</sup> complex is given by similar

clathrochelate complexes Co<sup>I</sup> (GmCl2)3(BPh)<sup>2</sup> reported in ref. [53]. Unfortunately, these complexes also do not fall into the range of existence of the ground state *M<sup>L</sup>* = ±3 due to violation of the <sup>∆</sup> > 0 criterion (which is actually <sup>∆</sup> ≈ −4500 cm−<sup>1</sup> ), resulting in the ground state being the orbital singlet <sup>3</sup>*F*(*M<sup>L</sup>* = 0). into the range of existence of the ground state *ML* = ±3 due to violation of the Δ > 0 criterion (which is actually Δ ≈ −4500 cm−1), resulting in the ground state being the orbital singlet <sup>3</sup>*F*(*ML* = 0). into the range of existence of the ground state *ML* = ±3 due to violation of the Δ > 0 criterion (which is actually Δ ≈ −4500 cm−1), resulting in the ground state being the orbital singlet

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*Inorganics* **2022**, *10*, 227 8 of 12

<sup>3</sup>*F*(*ML* = 0).

**Figure 6.** Variation of the energies of the 3*F*(*ML*) LF states of Ni2+(3d8) ion with increasing energy gap Δ = *E*(*xy*, *x*2 *− y*2) *− E*(*xz*, *yz*). The range of the stabilization of the 3*F*(*ML* = ±3) ground state (0 ≤ Δ ≤ 7200 cm<sup>−</sup>1) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*z*2) = 0, *E*(*xz*, *yz*) = 5000 cm<sup>−</sup>1 (fixed) and *E*(*xy*, *x*2 *− y*2) = 5000 cm<sup>−</sup>1 + Δ (variable). LF calculations are performed with atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm<sup>−</sup>1. **Figure 6.** Variation of the energies of the <sup>3</sup>*F*(*ML*) LF states of Ni2+(3d<sup>8</sup> ) ion with increasing energy gap <sup>∆</sup> <sup>=</sup> *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2) <sup>−</sup> *<sup>E</sup>*(*xz*, *yz*). The range of the stabilization of the <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (0 <sup>≤</sup> <sup>∆</sup> <sup>≤</sup> 7200 cm−<sup>1</sup> ) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*z* 2 ) = 0, *E*(*xz*, *yz*) = 5000 cm−<sup>1</sup> (fixed) and *<sup>E</sup>*(*xy*, *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2) = 5000 cm−<sup>1</sup> <sup>+</sup> <sup>∆</sup> (variable). LF calculations are performed with atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm−<sup>1</sup> . **Figure 6.** Variation of the energies of the 3*F*(*ML*) LF states of Ni2+(3d8) ion with increasing energy gap Δ = *E*(*xy*, *x*2 *− y*2) *− E*(*xz*, *yz*). The range of the stabilization of the 3*F*(*ML* = ±3) ground state (0 ≤ Δ ≤ 7200 cm<sup>−</sup>1) is marked with light pink. Inset: the LF splitting pattern of 3d orbitals, *E*(*z*2) = 0, *E*(*xz*, *yz*) = 5000 cm<sup>−</sup>1 (fixed) and *E*(*xy*, *x*2 *− y*2) = 5000 cm<sup>−</sup>1 + Δ (variable). LF calculations are performed with atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm<sup>−</sup>1.

**Figure 7.** (**a**) Variation of the energies of the 3*F*(*ML*) LF states of Ni2+(3d8) ion with increasing polar angle θ in trigonal-prismatic coordination, as obtained from LF/AOM calculations. The range of the stabilization of the 3*F*(*ML* = ±3) ground state (61° ≤ θ ≤ 74.2°) is marked with light pink; (**b**) *E* vs. *MJ* diagram calculated at θ = 64°. The spin-reversal barrier *U*eff ≈ 850 cm<sup>−</sup>1 is approximately estimated by the energy separation between the ground *MJ* = ±3/2 and the first excited *MJ* = ±5/2 states; the *MJ* = 0 and *MJ* = ±1 states (marked in blue) originate from 2S+1*L* atomic terms other than 3*F*. LF/AOM calculations were performed with *e*σ = 5000 cm<sup>−</sup>1, *e*σ/*e*π = 4 AOM parameters and atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm<sup>−</sup>1. The calculated *E* vs. *MJ* diagram suggests potentially high SMM characteristics of the **Figure 7.** (**a**) Variation of the energies of the 3*F*(*ML*) LF states of Ni2+(3d8) ion with increasing polar angle θ in trigonal-prismatic coordination, as obtained from LF/AOM calculations. The range of the stabilization of the 3*F*(*ML* = ±3) ground state (61° ≤ θ ≤ 74.2°) is marked with light pink; (**b**) *E* vs. *MJ* diagram calculated at θ = 64°. The spin-reversal barrier *U*eff ≈ 850 cm<sup>−</sup>1 is approximately estimated by the energy separation between the ground *MJ* = ±3/2 and the first excited *MJ* = ±5/2 states; the *MJ* = 0 and *MJ* = ±1 states (marked in blue) originate from 2S+1*L* atomic terms other than 3*F*. LF/AOM calculations were performed with *e*σ = 5000 cm<sup>−</sup>1, *e*σ/*e*π = 4 AOM parameters and atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm<sup>−</sup>1. **Figure 7.** (**a**) Variation of the energies of the <sup>3</sup>*F*(*ML*) LF states of Ni2+(3d<sup>8</sup> ) ion with increasing polar angle θ in trigonal-prismatic coordination, as obtained from LF/AOM calculations. The range of the stabilization of the <sup>3</sup>*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) ground state (61◦ <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> 74.2◦ ) is marked with light pink; (**b**) *E* vs. *M<sup>J</sup>* diagram calculated at θ = 64◦ . The spin-reversal barrier *<sup>U</sup>*eff <sup>≈</sup> 850 cm−<sup>1</sup> is approximately estimated by the energy separation between the ground *M<sup>J</sup>* = ±3/2 and the first excited *<sup>M</sup><sup>J</sup>* <sup>=</sup> <sup>±</sup>5/2 states; the *<sup>M</sup><sup>J</sup>* = 0 and *<sup>M</sup><sup>J</sup>* <sup>=</sup> <sup>±</sup>1 states (marked in blue) originate from 2S+1*<sup>L</sup>* atomic terms other than <sup>3</sup>*F*. LF/AOM calculations were performed with *e*<sup>σ</sup> = 5000 cm−<sup>1</sup> , *e*σ/*e*<sup>π</sup> = 4 AOM parameters and atomic parameters *B* = 800, *C* = 3200 and ζ = 600 cm−<sup>1</sup> .

compressed (θ > 61°) trigonal-prismatic Ni2+ complex with a perfect *D*3*h* symmetry. In particular, the spin-reversal barrier *U*eff, assigned to the energy separation between the ground state *JM* = ±4 and the first excited state *JM* = ±3, is as high as 857 cm−1 (Figure 7b), which is significantly larger than in the record cobalt complex CoII(C(SiMe2ONaph)3)2 (450 cm−1) [42]. However, it should be considered that, similarly to the trigonal-prismatic 3d3 The calculated *E* vs. *MJ* diagram suggests potentially high SMM characteristics of the compressed (θ > 61°) trigonal-prismatic Ni2+ complex with a perfect *D*3*h* symmetry. In particular, the spin-reversal barrier *U*eff, assigned to the energy separation between the ground state *JM* = ±4 and the first excited state *JM* = ±3, is as high as 857 cm−1 (Figure 7b), which is significantly larger than in the record cobalt complex CoII(C(SiMe2ONaph)3)2 (450 cm−1) [42]. However, it should be considered that, similarly to the trigonal-prismatic 3d3 The calculated *E* vs. *M<sup>J</sup>* diagram suggests potentially high SMM characteristics of the compressed (θ > 61◦ ) trigonal-prismatic Ni2+ complex with a perfect *D*3*<sup>h</sup>* symmetry. In particular, the spin-reversal barrier *U*eff, assigned to the energy separation between the ground state *<sup>J</sup><sup>M</sup>* <sup>=</sup> <sup>±</sup>4 and the first excited state *<sup>J</sup><sup>M</sup>* <sup>=</sup> <sup>±</sup>3, is as high as 857 cm−<sup>1</sup> (Figure 7b), which is significantly larger than in the record cobalt complex CoII(C(SiMe2ONaph)3)<sup>2</sup>

complex, this complex is Jahn–Teller active and is subject to significant distortions that tend to reduce magnetic anisotropy. In fact, LF/AOM calculations for distorted complexes

complex, this complex is Jahn–Teller active and is subject to significant distortions that tend to reduce magnetic anisotropy. In fact, LF/AOM calculations for distorted complexes (450 cm−<sup>1</sup> ) [42]. However, it should be considered that, similarly to the trigonal-prismatic 3d<sup>3</sup> complex, this complex is Jahn–Teller active and is subject to significant distortions that tend to reduce magnetic anisotropy. In fact, LF/AOM calculations for distorted complexes show that even subtle departures from the regular trigonal-prismatic geometry (*D*3*<sup>h</sup>* ) of the complex cause noticeable splitting of the ground non-Kramers doublet *J<sup>M</sup>* = ±4 (tens of cm−<sup>1</sup> ), which should lead to complete suppression of the SMM behavior due to fast QTM processes in the split ground state *J<sup>M</sup>* = ±4 (Figure 7b).

### **3. Materials and Methods**

### *Ligand Field and Angular Overlap Model Calculations*

Ligand-field calculations for many-electron 3d*<sup>N</sup>* ions (3d<sup>2</sup> , 3d<sup>3</sup> , 3d<sup>7</sup> and 3d<sup>8</sup> ) were performed in terms of the conventional LF theory involving the one-electron LF operator (specified by the set of energies of 3d orbitals in an axial LF, *E*(*xy*, *x*2 − *y*2), *E*(*xz*, *yz*) and *E*(*z* 2 )), interelectron Coulomb repulsion (treated in terms of the Racah parameters *B* and *C*) and the spin-orbit coupling ζ3dΣ*<sup>i</sup> lis<sup>i</sup>* . Angular-overlap model calculations [46–48] for trigonal-prismatic complexes were carried out with the AOM parameters *e*<sup>σ</sup> = 5000 cm−<sup>1</sup> (for Ni2+) and 9000 cm−<sup>1</sup> (Cr3+) at a fixed ratio of *e*σ/*e*<sup>π</sup> = 4. Details of LF and AOM calculations are available in the Supplementary Materials.

### **4. Discussion and Conclusions**

This study has established that the maximum orbital angular momentum *L* = 3 observed in the linear two-coordinate 3d<sup>7</sup> complex CoII(C(SiMe2ONaph)3)<sup>2</sup> [42] (as well as in the monocoordinated Co adatom on a MgO surface [43]) is not a unique single phenomenon, but under specific conditions, it can also be obtained in some complexes of 3d<sup>2</sup> , 3d<sup>3</sup> and 3d<sup>8</sup> transition metal ions. The basic point is that in the free-ion state these ions have the ground-state atomic term 2S+1*F* with maximal orbital angular momentum *L* = 3, which, when split in the ligand field of axial symmetry, can produce the ground orbital doublet <sup>2</sup>*S*+1*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) with a maximum projection *<sup>M</sup><sup>L</sup>* <sup>=</sup> <sup>±</sup>3 of the orbital momentum *L* = 3. For each of these electronic configurations, the LF splitting pattern of 3d orbitals leading to the ground orbital doublet <sup>2</sup>*S*+1*F*(*M<sup>L</sup>* <sup>=</sup> <sup>±</sup>3) has been established. Then, with the help of LF and AOM calculations, it was shown that the maximal orbital angular moment *L* = 3 is realized in the linear two-coordinate geometry for the 3d<sup>2</sup> and 3d<sup>7</sup> ions and in the compressed trigonal-prismatic coordination (with polar angle θ > 61◦ ) for the 3d<sup>3</sup> and 3d<sup>8</sup> ions. In this regard, it is worth noting that the linear two-coordinate geometry, which intuitively appears to be the most preferable for obtaining maximum magnetic anisotropy, in the case of 3d<sup>3</sup> and 3d<sup>8</sup> ions yields only an orbital momentum *L* = 2, not *L* = 3.

These results can be used to assess the prospects for improving the SMM performance of single-ion 3d SMMs. The calculated *E* vs. *M<sup>J</sup>* spin energy diagrams show that the SMM performance of the linear 3d<sup>2</sup> complex and the trigonal-prismatic 3d<sup>3</sup> complex is significantly lower than that of the linear cobalt complex due to weaker spin-orbit coupling and antiparallel coupling of *L* and *S*. On the other hand, although the calculated spin energy diagram for the trigonal-prismatic Ni2+(3d<sup>8</sup> ) complex formally indicates a higher barrier (*U*eff <sup>≈</sup> 850 cm−<sup>1</sup> ) than for the record linear cobalt complex (450 cm−<sup>1</sup> ) [42], in reality, this complex is unlikely to be a good SMM because of the Jahn–Teller distortions resulting in the significant splitting of the ground non-Kramer doublet *M<sup>J</sup>* = ±4 (Figure 7b), that causes fast QTM processes bypassing the barrier. Thus, despite the possible presence of maximum orbital momentum *L* = 3 in the aforementioned 3d<sup>2</sup> , 3d<sup>3</sup> and 3d<sup>8</sup> complexes, they cannot compete as SMMs with linear two-coordinate 3d<sup>7</sup> cobalt complex CoII(C(SiMe2ONaph)3)<sup>2</sup> [42].

In summary, this study has shown that the maximal orbital angular momentum *L* = 3 can occur only in 3d<sup>2</sup> , 3d<sup>3</sup> , 3d<sup>7</sup> and 3d<sup>8</sup> complexes with axial symmetry of the ligand field. However, the SMM performance of 3d<sup>2</sup> , 3d<sup>3</sup> and 3d<sup>8</sup> complexes appeared significantly lower than that of the linear two-coordinate 3d<sup>7</sup> complex CoII(C(SiMe2ONaph)3)<sup>2</sup> with *L* = 3. Thus, these results confirm the earlier conclusion [42] that linear two-coordinate

3d<sup>7</sup> complexes provide the maximal SMM performance of mononuclear 3d complexes (*U*eff = 450 cm−<sup>1</sup> ), which is apparently near the physical limit for a single 3d metal ion.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/inorganics10120227/s1, details of LF and AOM calculations.

**Author Contributions:** Conceptualization, validation, methodology, writing—original draft preparation, writing—review and editing, supervision, V.S.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Russian Science Foundation, project No. 18-13-00264.

**Data Availability Statement:** The data presented in this study are available in this article and in the Supplementary Materials.

**Acknowledgments:** V.S.M. acknowledges support by the Ministry of Science and Higher Education within the State assignment FSRC 'Crystallography and Photonics' RAS in part of the development of computational approaches for analysis of extreme magnetic anisotropy in transition-metal complexes.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

