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

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 3d⁷ complex Co^II(C(SiMe₂ONaph)₃)₂ characterized by record magnetic anisotropy and single-molecule magnet (SMM) performance (with the largest known spin-reversal barrier U_eff = 450 cm⁻¹), the maximal orbital angular momentum L = 3 can also be obtained in linear two-coordinate 3d² complexes (V³⁺, Cr⁴⁺) and in trigonal-prismatic 3d³ (Cr³⁺, Mn⁴⁺) and 3d⁸ (Co⁺, Ni²⁺) complexes. A comparative assessment of the SMM performance of the 3d², 3d³ and 3d⁸ complexes indicates that they are unlikely to compete with the record linear complex Co^II(C(SiMe₂ONaph)₃)₂, whose magnetic anisotropy is close to the physical limit for a 3d metal.

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 molecule-based magnetically bistable materials featuring slow magnetic relaxation and blocking of magnetization at low temperature—single-molecule magnets (0D, SMMs) [1,2,3,4,5,6,7,8,9] and single-chain magnets (1D, SCMs) [10,11,12]. 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 [6,7,8,9,10,11,12]. The U_eff and T_B 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⁺ [1,2,3]. 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 [16,17,18,19,20,21,22,23,24] 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 energy and strong spin-orbit coupling (SOC) of 4f-electrons; the maximal orbital momentum L = 6 occurs in Nd³⁺ (4f³, ⁴I_9/2), Pm³⁺ (4f⁴, ⁵I₄), Ho³⁺ (4f¹⁰, ⁵I₈) and Er³⁺ (4f¹¹, ⁴I_15/2) 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 ^2S+1L_J multiplet with J = L − S (N < 7) and J = L + S (N > 7). For Ln³⁺ ions in a ligands coordination environment, the ground J-multiplet undergoes LF splitting into 2J + 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 (2J + 1)-fold degeneracy of the ground multiplet into ±M_J microstates with a definite projection of the total angular momentum [16,17,18,19,20,21,22,23,24]. The energy splitting diagram of ±M_J states (E vs. M_J) depends on both the strength and the intrinsic structure of the axial LF, which is specified by three axial LF parameters B₂₀, B₄₀, B₆₀ [27]. In the optimal case, the strong field stabilizes the doubly degenerate ground state with maximal M_J projection (M_J = ±J) and with large energy separation from excited LF states thereby resulting in a double-well potential with the high-energy 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 quasi-linear metallocene lanthanide complexes [30,31], many of which exhibit exceptionally high U_eff and T_B values, including a record-setting dysprosium metallocene SMM with U_eff = 1541 cm⁻¹ and T_B = 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 Fe^II 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 2S + 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⁻¹ [8,25,26,33]. The most efficient strategy toward high-performance 3d SIMs takes advantage of unquenched orbital angular momentum (L ≠ 0) giving rise to the first-order spin-orbit splitting of the ground spin stated with L ≠ 0, which provides considerably stronger magnetic anisotropy and high energy barriers [16,17,18,19,20,21,22,23,24]. Unquenched first-order orbital angular momentum can only appear in orbitally degenerate transition metal complexes with high symmetry, such as octahedral Co^II and [Fe^III(CN)₆]³ complexes (O_h) [33,34], trigonal-pyramidal (C_3v), [14,15] trigonal-bipyramidal (D_3h), [35] trigonal-prismatic (D_3h) [36,37], pentagonal-bipyramidal (D_5h) [38] and linear (D_∞h) two-coordinate complexes [39,40,41,42,43].

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_L = 0, ±1, ±2, or ±3. The latter is further split by the first-order SOC into energy levels ±M_J with definite projection M_J 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 |±M_J> 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 ζ_3dLS, which is proportional to the value of the ground-state orbital angular momentum M_L; approximately, U_eff is specified by ζ_3dM_L/2S, which is the energy separation between the ground ±M_J and first exited ±(M_J −1) states [42]. This leads to a pure Ising magnetic anisotropy and double-well potential with two lowest M_J = ±J states similar to those in 4f SIMs [16,17,18,19,20,21,22,23,24]. Accordingly, the highest barriers U_eff are observed in 3d SIMs with large orbital momentum M_L, namely, in linear 3d⁷ complexes [Fe^I(C(SiMe₃)₃)₂]⁻ (226 cm⁻¹) [40] and [(sIPr)Co^IINDmp] (413 cm⁻¹) [41] and also in trigonal-prismatic Co^II complex (152 cm⁻¹) [36] featuring L = 2, S = 3/2 and M_J = ±7/2 in the ground state. The record barrier belongs to linear two-coordinate cobalt complex, Co^II(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 M_J = ±9/2 in the ground state.

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 3d⁷ complexes [42,43], the maximum orbital angular momentum L = 3 (represented by the ground orbital doublet M_L = ±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 Co^II(C(SiMe₂ONaph)₃)₂ [42], the ground-state orbital doublet M_L = ±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

Below, the basic conditions for the realization of the maximal orbital angular momentum M_L = ±3 in mononuclear 3d complexes with the axial LF are analyzed. First of all, it is clear that the ground orbital doublet M_L = ±3 can only occur in 3d ions, which in the free-ion state have a ^2S+1F 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 ^2S+1F term splits into four orbital components ^2S+1F(M_L) with a projection of orbital momentum M_L = 0, ±1, ±2, ±3 (Figure 1). The objective of the present study is to establish conditions for the M_L = ±3 orbital doublet to be the ground state.

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_l on the anisotropy axis z, 3d_z2 (m_l = 0), (3d_xz, 3d_yz) (m_l = ±1) and 3d_xy, 3d_{x2 − y2} (m_l = ±2) (Figure 1). A necessary (but not sufficient) condition for the M_L = ±3 ground state is the presence of one unpaired electron in each of the two doubly degenerate 3d orbitals with magnetic numbers m_l = ±1, and m_l = ±2, i.e., (xz, yz)¹(xy, x2 − y2)¹ or (xz, yz)³(xy, x2 − y2)³. For each of the 3d ions selected above, the LF splitting pattern of 3d-orbitals leading to the ^2S+1F(M_L = ±3) ground state is examined 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, ζ_3dΣ_il_is_i. LF calculations are performed with the full basis set of 3d^N configurations involving 45 (3d², 3d⁸) and 120 (3d³, 3d⁷) |LM_LSM_S> microstates. Details of LF calculations are available in the Supplementary Materials.

2.1. 3d² Complexes (V³⁺, Cr⁴⁺)

The necessary condition for the ³F(M_L = ±3) orbital doublet to be the ground state is to have the LF spitting pattern E(z²) > E(xz, yz) > E(xy, x2 − y2). Such an orbital energy scheme can occur in complexes with linear or quasi-linear two-coordinate complexes [44] similar to the quasi-linear Fe^II [39] and Co^II 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(z²) = 10,000 cm⁻¹, E(xy, x2 − y2) = 0 and variable orbital energy E(xz, yz) = Δ (see inset in Figure 2a). The orbital doublet ³F(M_L = ±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 ³F(M_L = 0), whose wave function Det||xy↑, x2 − y2↑|| corresponds to two singly occupied xy and x2 − y2 orbitals with parallel electron spins (Figure 2a). From the point of view of coordination geometry, the only suitable case for the ³F(M_L = ±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].

The calculated E vs. M_J diagram indicates that such a 3d² complex may be a SMM with a barrier U_eff of about 200 cm⁻¹, as estimated from the energy gap between ground M_J = ±2 and first excited M_J = ±3 states (Figure 2b). Remarkably, the E vs. M_J diagram has an inverse order of ±M_J levels compared to the usual double-well spin energy profile (with the maximum M_J at the bottom and the minimum M_J 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 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 M_J = ±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 3d² complex is much less favorable compared to the record cobalt 3d⁷ complex Co^II(C(SiMe₂ONaph)₃)₂ [42].

2.2. 3d³ Complexes (Cr³⁺, Mn⁴⁺)

The ground state of a free 3d³ ion is a ⁴F atomic term, which is split by the axial LF into four orbital components, ⁴F(M_L = ±3), ⁴F(M_L = ±2), ⁴F(M_L = ±1) and ⁴F(M_L = 0). The ⁴F(M_L = ±3) state originates from the (xy, x2 − y2)¹(xz, yz)¹(z²)¹ electronic configuration. LF analysis indicates that this configuration can only occur at the E(xy, x2 − y2) > E(xz, yz) > E(z²) 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 ⁴F(M_L = ±3) is the ground state when the energy difference Δ = E(xy, x2 − y2) − E(xz, yz) between the energy levels of the (xy, x2 − y2) and (xz, yz) orbitals is within the range 0 ≤ Δ ≤ 5850 cm⁻¹. Beyond this range, the ground state turns to the ⁴F(M_L = 0) orbital singlet, which is represented by a mixture of the (xy)¹(x² − y²)¹(z²)¹ and (xz)¹(yz)¹(z²)¹ electronic configurations (Figure 3).

Evidently, the linear two-coordinate geometry of 3d³ complexes cannot provide the required orbital pattern E(xy, x2 − y2) > E(xz, yz) > E(z²) since linear coordination is characterized by another order of orbital energies, E(z²) > E(xz, yz) > E(xy, x2 − y2), which can result in M_L = ±1 or M_L = ±2 ground state. Therefore, unlike the record linear two-coordinate Co²⁺(3d⁷) complexes with the M_L = ±3 ground state, linear 3d³ complexes have a maximum achievable orbital angular momentum of only M_L = ±2.

Among a variety of axial symmetry coordination geometries, the necessary orbital splitting pattern E(xy, x2 − y2) > E(xz, yz) > E(z²) 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 M_L = ±3 ground state, LF calculations are performed in combination with the angular overlap model (AOM) [46,47,48,49,50,51] using the AOM parameters e_σ = 9000 cm⁻¹ and e_σ/e_π = 4 for Cr³⁺ ions. These calculations show that the ground state ⁴F(M_L = ±3) with the maximal orbital momentum M_L = ±3 stabilizes in a compressed trigonal prism with a polar angle θ ranging between 61° and 66.5° (Figure 4a).

SOC splits the ground orbital doublet M_L = ±3 into four Kramer doublets with the projection of the total angular momentum M_J = ±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 M_J = ±3/2 and first excited M_J = ±5/2 states. However, as opposed to linear two-coordinate Co^II complexes [42,43], the trigonal-prismatic 3d³ complexes with doubly degenerate ground state M_L = ±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_3h geometry, the overall energy pattern of the J_M 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.

2.3. 3d⁷ Complexes (Fe⁺,Co²⁺)

Linear two-coordinate 3d⁷ cobalt complex Co^II(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(z²) > E(xz, yz) > E(xy, x2 − y2) is a necessary condition for the stabilization of the M_L = ±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 m_l = ±1 and m_l = ±2 orbitals, Δ = E(xz, yz) − E(xy, x2 − y2), which is 0 ≤ Δ ≤ 6750 cm⁻¹ (Figure 5). Outside this range, the ground state is the orbital singlet ⁴F(M_L = 0). Formally, an extra condition is a limitation on the total LF splitting energy Δ_LF = E(z²) − E(xy, x2 − y2) < 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].

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 3d⁷ complexes [Fe^I(C(SiMe₃)₃)₂]⁻ (M_L = ±2) [40] and Co^II(C(SiMe₂ONaph)₃)₂ (M_L = ±3) [42]. The underlying reason is the different order of the actual 3d orbitals, which is E(xz, yz) > E(xy, x2 − y2) > E(z²) (ca. 5200, 3200 and 0 cm⁻¹, respectively) in the Fe^I complex [40] and E(z²) > E(xz, yz) > E(xy, x2 − y2) ((ca. 5700, 3000 and 0 cm⁻¹) in Co^II complex [42]. A significant decrease in the energy of the 3d_z² orbital in Fe^I(C(SiMe₃)₃)₂]⁻ is caused by a very strong 4s–3d_z² mixing [40]. The combination of the ground orbital state M_L = ±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 Co^II(C(SiMe₂ONaph)₃)₂ [42].

2.4. 3d⁸ Complexes (Ni²⁺, Co¹⁺)

As in the case of 3d² complexes with L = 3, the necessary condition for the occurrence of the M_L = ±3 ground state in a 3d⁸ complex is the presence of one unpaired electron on the doubly degenerate 3d orbitals with magnetic quantum numbers m_l = ±1 and m_l = ±2, i.e., (x2 − y2, xy)³(xz, yz)³(z²)². For this, the order of the 3d orbitals should be the same as in 3d³ complexes, i.e., E(xy, x2 − y2) > E(xz, yz) > E(z²). 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_l = ±1 and m_l = ±2 orbitals, Δ = E(xy, x2 − y2) − E(xz, yz), which is 0 ≤ Δ ≤ 7200 cm⁻¹.(Figure 6). LF/AOM calculations for a trigonal-prismatic 3d⁸ complex with variable polar angle θ indicate that the M_L = ±3 ground state stabilizes at 61° ≤ θ ≤ 74.2°, corresponding to a compressed trigonal prism (Figure 7a). Several trigonal-prismatic cage complexes of Ni^II 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 ³F(M_L = 0) with a second-order magnetic anisotropy due to ZFS [48], which agrees well with the results of LF/AOM calculations indicating the M_L = 0 ground state at θ ≈ 52° (Figure 7a). One more example of a trigonal-prismatic 3d⁸ complex is given by similar clathrochelate complexes Co^I(GmCl₂)₃(BPh)₂ reported in ref. [53]. Unfortunately, these complexes also do not fall into the range of existence of the ground state M_L = ±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 ³F(M_L = 0).

The calculated E vs. M_J diagram suggests potentially high SMM characteristics of the compressed (θ > 61°) trigonal-prismatic Ni²⁺ complex with a perfect D_3h symmetry. In particular, the spin-reversal barrier U_eff, assigned to the energy separation between the ground state J_M = ±4 and the first excited state J_M = ±3, is as high as 857 cm⁻¹ (Figure 7b), which is significantly larger than in the record cobalt complex Co^II(C(SiMe₂ONaph)₃)₂ (450 cm⁻¹) [42]. However, it should be considered that, similarly to the trigonal-prismatic 3d³ 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_3h) of the complex cause noticeable splitting of the ground non-Kramers doublet J_M = ±4 (tens of cm⁻¹), which should lead to complete suppression of the SMM behavior due to fast QTM processes in the split ground state J_M = ±4 (Figure 7b).

3. Materials and Methods

Ligand Field and Angular Overlap Model Calculations

Ligand-field calculations for many-electron 3d^N ions (3d², 3d³, 3d⁷ and 3d⁸) 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, x2 − y2), E(xz, yz) and E(z²)), interelectron Coulomb repulsion (treated in terms of the Racah parameters B and C) and the spin-orbit coupling ζ_3dΣ_il_is_i. Angular-overlap model calculations [46,47,48] for trigonal-prismatic complexes were carried out with the AOM parameters e_σ = 5000 cm⁻¹ (for Ni²⁺) and 9000 cm⁻¹ (Cr³⁺) at a fixed ratio of e_σ/e_π = 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 3d⁷ complex Co^II(C(SiMe₂ONaph)₃)₂ [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 3d², 3d³ and 3d⁸ transition metal ions. The basic point is that in the free-ion state these ions have the ground-state atomic term ^2S+1F with maximal orbital angular momentum L = 3, which, when split in the ligand field of axial symmetry, can produce the ground orbital doublet ^2S+1F(M_L = ±3) with a maximum projection M_L = ±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 ^2S+1F(M_L = ±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 3d² and 3d⁷ ions and in the compressed trigonal-prismatic coordination (with polar angle θ > 61°) for the 3d³ and 3d⁸ 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 3d³ and 3d⁸ 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_J spin energy diagrams show that the SMM performance of the linear 3d² complex and the trigonal-prismatic 3d³ 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⁺(3d⁸) complex formally indicates a higher barrier (U_eff ≈ 850 cm⁻¹) than for the record linear cobalt complex (450 cm⁻¹) [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_J = ±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 3d², 3d³ and 3d⁸ complexes, they cannot compete as SMMs with linear two-coordinate 3d⁷ cobalt complex Co^II(C(SiMe₂ONaph)₃)₂ [42].

In summary, this study has shown that the maximal orbital angular momentum L = 3 can occur only in 3d², 3d³, 3d⁷ and 3d⁸ complexes with axial symmetry of the ligand field. However, the SMM performance of 3d², 3d³ and 3d⁸ complexes appeared significantly lower than that of the linear two-coordinate 3d⁷ complex Co^II(C(SiMe₂ONaph)₃)₂ with L = 3. Thus, these results confirm the earlier conclusion [42] that linear two-coordinate 3d⁷ complexes provide the maximal SMM performance of mononuclear 3d complexes (U_eff = 450 cm⁻¹), which is apparently near the physical limit for a single 3d metal ion.