Conformational isomerization in Co(acac)₂via spin-state switch: a computational study

Abstract

Conformational dynamics of ligands in transition metal complexes can give rise to interesting physical, chemical, spectroscopic, and magnetic properties of the complexes. The changing ligand environment often affects the d-orbital splitting pattern that allows multiple possible ways of electron arrangement in the frontier molecular orbitals, resulting in several closely spaced electronic states with different orbital and spin symmetries. The system can explore these states with either thermal or photophysical means. In the present work, we demonstrate the possibility of a spin transition in Co(acac)₂ assisted by a conformational rearrangement of the ligand. Electronic structure calculations show that the complex adopts a square-planar and tetrahedral geometry with low-spin and high-spin electronic configurations, respectively. A spin-conserved conformational change involves a larger energy barrier in both high- and low-spin states. On the other hand, a low-lying minimum-energy-crossing point exists between the two spin-states that provides a low-energy pathway for conformational isomerization between the two isomers. While the spin-assisted isomerization from a tetrahedral to square planar form requires crossing a 10 kcal mol⁻¹ barrier, the reverse barrier is only 2 kcal mol⁻¹. The calculation of the magnetic properties of the complex reveals a large magnetic anisotropy barrier of 57.6 cm⁻¹ for this complex in the high-spin state.

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1. Introduction

The spin-crossover (SCO) describes a phenomenon where a molecule alters its ground spin state triggered by external stimuli, such as temperature, pressure, light irradiation, or external electric and magnetic fields.^1–4 Since Cambi's pioneering discovery of the SCO effect in the tris(dithiocarbamato)Fe(III) complex,⁵ it has become a well-known characteristic of the first-row transition metal ions with d⁴ to d⁷ electron configurations. These metal ions are often paired with ligands characterized by intermediate field strength, primarily N- and O-donors, which provide labile electronic configurations for the metal-complexes.^2,6–9

While a considerable amount of research exists for iron-based SCO complexes,^4,8,10–13 exploration of Co(II) based SCO systems is rather limited.^2,14–17 The most commonly studied hexa-coordinated Co(II) complexes involve either an N₆ or an N₄O₂ coordination sphere, achieved through a combination of monodentate and chelating ligands. SCO in Co(II) complexes was initially observed in complexes with a [CoN₆]²⁺ core where an intermediate magnetic moment at room temperature was reported.^18,19 Since then, N₆ based ligands, e.g., the tridentate 2,2′;6′,2′′-terpyridine (terpy) ligand and its numerous derivatives have proven highly effective in constructing Co(II) based SCO complexes.^20–22 Complexes with N₄O₂ coordination spheres include Co(II)-semiquinone compounds that demonstrate valence tautomerism,^14,23,24 and the well-studied quadridentate salen-type Schiff base Co(II) complexes have also been reported for their SCO behaviour.^19,25,26

The spin-state switch is typically associated with a change in metal–ligand bond distances, arising from the changes in the electronic population of the metal–ligand bonding and anti-bonding orbitals.^27,28 However, there are instances where spin-state transition accompanies conformational isomerization in metal complexes.^19,29–31 The four-coordinated metal β-ketomine based complexes are a prime example, where the spin-state switch involves square-planar to tetrahedral isomerization, characterized by spectral, magnetic, and NMR methods.²⁹ This conformational isomerism has been observed in several bis-chelate complexes, especially those with α,β-unsaturated β-ketoamines.^32–35 Similarly, the [Co(PEt₃)₂(NCS)₂] complex exhibits a transition from a tetrahedral to a dimeric-pyramidal conformation as it switches from high-spin (HS) to low-spin (LS) states.³⁶ Monomeric bis-chelate Co(II) complexes, such as [Co(SDPM)₂] (with SDPM: 2,2′,6,6′-tetramethyl-5-mercaptohept-4-en-2-one), also show the transition between planar and tetrahedral conformations.³⁰ Additionally, Co(II) triazene 1-oxide bis(chelates) complexes, such as [Co(OMeN₃C₆H₄Me-4)₂], demonstrate a planar to tetrahedral transition in solvents that do not coordinate with the metal ion.³¹

Among the metal bis-acetylacetonate (acac) complexes, the five-coordinated [Ni(acac)₂(pyridine)] complex is known to exist in three polytopal forms (a square-pyramid and two distinct trigonal bipyramids), separated by low-energy barriers.³⁷ Bis-chelate bis-(propane-1,3-dionato)Ni(II) and bis-(N-methyl-oxyvinylaldiminato)Ni(II) complexes have been studied computationally to elucidate the isomerization between the square-planar to tetrahedral forms and the role of spin states in governing the transition.³⁸ [Mg(acac)₂] is also known to exhibit square-planar and tetrahedral forms, although the involvement of the spin-state in its isomerism is unlikely.³⁹

On the other hand, the structural and electronic characterization of Co(acac)₂ has been contentious and remains a topic of ongoing debate. Although the initial study of Cotton and Holm suggested a square-planar geometry,⁴⁰ the later studies show overwhelming evidence for a tetrahedral structure of Co(acac)₂ at room temperature.^41–44 From molecular weight measurements and visible- and near-infrared spectroscopy, it was found that tetrahedral Co(acac)₂ units oligomerise in non-coordinating solvents.⁴¹ In dilute solutions, the compound predominantly adopts a mononuclear tetrahedral configuration, but at higher concentrations, it shows octahedral coordination due to additional interactions with diketonate oxygen atoms from adjacent units.⁴¹ UV-visible-NIR spectroscopy lends further support to the tetrahedral Co(acac)₂ monomer, as reflected in the comparable spectra of Co(DPM)₂ (DPM: dipivaloylmethanido).⁴¹ Later on, the X-ray crystal structure confirmed a tetrahedral geometry of the Co(acac)₂ monomer in a tetrameric assembly.^42,43 In 2000, Burgess et al. reported a square planar crystal structure for Co(acac)₂.⁴⁵ However, a later study in 2010 suggested this complex to be of copper instead of cobalt.^44,46 A square-planar arrangement of Co(acac)₂ is reported in coordinating solvents, where two solvent molecules occupy the two axial sites of the metal ion.⁴⁴ Temperature-dependent EPR study shows a HS ground state for Co(acac)₂L₂ (L = water and ethanol) below 40 K,⁴⁷ which is also supported by theoretical studies.⁴⁸ The computational modelling of the L_2,3-edge X-ray absorption spectra of the Co(acac)₂ complex confirms the contribution from both the HS and LS states.⁴⁹

While the previous experimental and theoretical studies have focused on characterizing the ground spin-states and the relative stabilities of the square-planar and tetrahedral geometries of the Co(acac)₂ complex, the mechanism underlying their interconversion has remained unexplored. In this study, we elucidate the potential switching mechanism between these two geometries and examine the influence of spin-states on this process. Additionally, we investigate the magnetic anisotropy related to the two ground spin-states and explore the mechanisms governing the relaxation of magnetization.

2. Results and discussion

2.1. Molecular geometry analysis

Molecular geometry optimizations with density functional theory (DFT) methods show that the Co(acac)₂ complex adopts a tetrahedral-like geometry in the HS state (⁴TD) and a square-planar like geometry in the LS state (²SQP), Fig. 1. The 〈S²〉 values were used to examine the spin-contamination associated with the DFT calculations. For the ⁴TD systems, all the functionals produced 〈S²〉 values between 3.76 and 3.77 (theoretical value 3.75). For the ²SQP system, the 〈S²〉 values computed by TPSSh-D3, B3PW91-D3, and M05-D3 (0.77, 0.81, and 0.88, respectively) deviated marginally from the ideal value (0.76), while the rest of the functionals showed no spin-contamination (〈S²〉 = 0.76) (see Table S1 in the ESI†). To understand the effect of dispersion correction on the optimized geometry, the molecular geometries and spin-splitting energies of both the ⁴TD and ²SQP complexes were also examined with and without considering the dispersion interaction. Using B3LYP and CAM-B3LYP functionals (with def2-TZVP basis), we find that the dispersion correction with either the B3LYP or CAM-B3LYP functional has a negligible effect on the geometry, as evident from the near-zero RMSD values (Table S2 in the ESI†).

Fig. 1 The optimized geometries of Co(acac)₂ with (a) a tetrahedral structure in the HS state (⁴TD) and (b) a square-planar structure in the LS state (²SQP). Atom colour: Co in blue, O in red, C in grey, and H in white. The four coordinating oxygen atoms are numbered.

The Co–O distances are found to vary between 1.93 and 1.95 Å in the ⁴TD state optimized with different functionals employed together with the def2-TZVP basis set. This is consistent with the Co–O bond distances observed in the experimental geometry.^42–44 All bond parameters of the complexes are provided in Table S3 in the ESI.† In the optimized geometries of ⁴TD, the two O–Co–O bite angles are around 94.9° and the two exterior O–Co–O angles are around 117.2°, deviating from the ideal tetrahedral value of 109.5°. On the other hand, in the ²SQP geometry, the O–Co–O angles deviate by ±5° from their ideal value of 90°. Such deviations from the ideal bond angles are also consistent with the shape analysis.⁵⁰ Since the deviations in the angles are more pronounced in the ⁴TD geometry, a shape deviation parameter of 1.6 is obtained for ⁴TD, as compared to 0.09 for the ²SQP geometry (see Table S3 in the ESI†). In the ²SQP geometry, comparatively shorter Co–O bond distances (1.85–1.89 Å) are observed. Therefore, the spin-transition from ⁴TD to the ²SQP state not only influences the mode of coordination but is also associated with the reduction in the Co–O distances. The 3d-based MOs for the HS geometry, involve e-like (3d_x²−y² and 3d_z²) and t₂-like (3d_xy, 3d_yz, and 3d_xz) orbitals (Fig. 2). In the TD geometry, ligands occupy alternating corners of a cube relative to the metal atom at the centre, and the d_x²−y² and d_z² orbitals (referred to as the e-orbitals) are lower in energy, as their lobes point between the ligand directions. The d_xy, d_yz, and d_xz (t₂ orbitals) are higher in energy due to their lobes being closer to the ligand axes. For the LS state in a square-planar geometry, the orbitals are identified as e_g (3d_xz, 3d_yz), a_1g (3d_z²), b_2g (3d_xy), and b_1g (3d_x²−y²). In the SQP geometry, the ligands lie in the xy plane, with the z-axis perpendicular to the ligand plane. The d_x²−y² has the highest energy due to the strong interaction with the four planar ligands. The d-orbitals in the ⁴TD and ²SQP geometries are designated according to their idealized symmetry. The loss of orbital degeneracy in the HS state geometry indicates the deviation from an ideal tetrahedral geometry. In the LS state, the antibonding d_x²−y² orbital carries no electrons (Fig. 2), thus shortening the metal–ligand bond length in the ²SQP state as compared to the ⁴TD state, where all d-orbitals are singly or doubly occupied.

Fig. 2 The d-orbital energy levels of Co(acac)₂ at the optimized geometry in the HS state (⁴TD), in the LS state (²SQP), and at the minimum-energy cross point between the HS and LS states in the LS configuration (²MECP) and HS configuration (⁴MECP). The orbital energy levels are shown with respect to the corresponding barycentre.

2.2. Adiabatic energy differences

We employed four basis sets (def2-SVP, 6-311G(d,p), def2-TZVP, and def2-TZVPP) with the B3LYP-D3 functional and compared the geometry and adiabatic energy. The size of the basis sets was observed to have a minimal influence on the geometries of the complex (see Table S4†), although it affected the adiabatic energies of the HS and LS states. With the def2-SVP basis set (consisting of 297 basis functions), the computed energy difference is 11.9 kcal mol⁻¹. This value decreases to 7.6 kcal mol⁻¹ with the 6-311G(d,p) basis set (382 basis functions). Conversely, computation with def2-TZVPP (with 694 basis functions) yielded an energy difference of 8.9 kcal mol⁻¹, very close to the value calculated with the def2-TZVP (with 563 basis functions) basis set (8.6 kcal mol⁻¹). Based on the observed convergence (Table S5 in the ESI†) of the adiabatic energy difference with the def2-TZVP basis set, we carried out all other DFT calculations using this basis set.

While different DFT functionals resulted in very similar optimized geometries, the spin-splitting energies are found to vary substantially with different functionals, as observed in several other systems.^51–53 Pure DFT functionals BP86-D3 and BLYP-D3 suggest a more stable ²SQP geometry compared to the ⁴TD complex (Fig. 3 and Table S1 in the ESI†). Conversely, the hybrid DFT functionals estimate a more stable ⁴TD state than the ²SQP state. For instance, B3LYP*-D3 (having 15% Hartree–Fock (HF) exchange), B3LYP-D3 and B3PW91-D3 (both incorporating 20% HF exchange) suggest adiabatic spin-state energy differences of 10.3, 8.6 and 11.1 kcal mol⁻¹, respectively, between the LS and HS states. The range-separated CAM-B3LYP-D3 indicates an adiabatic energy difference of 11.9 kcal mol⁻¹, whereas the meta-hybrid-GGA functionals TPSSh-D3 (10% HF exchange), M05-D3 (28% HF exchange), and M06-D3 (27% HF exchange) predict energy differences of 7.0, 13.3 and 4.7 kcal mol⁻¹, respectively. The B2PLYP optimization was found to be very expensive. Instead, we performed single-point energy calculation with B2PLYP/def2-TZVP (using the B3LYP-D3/def2-TZVP optimized geometry). This is justified since the optimized geometry shows negligible dependence on functional, basis set, and dispersion correction. The double-hybrid B2PLYP estimates a significantly larger spin-splitting energy (31.5–33.9 kcal mol⁻¹) as compared to the other functionals (see Table S6 in the ESI†). Employing the complete active space second-order perturbation theory (CASPT2) method, Radoń et al. have estimated this energy difference to be 13 kcal mol⁻¹.⁴⁸ Although hybrid functionals with 10–15% Hartree–Fock (HF) exchange have been shown to provide the best estimates of spin-splitting energies, their accuracy varies from system to system.^54,55 Such deviation in the spin-splitting energies across various functionals is previously noted.⁵² The dispersion correction lowers the spin-splitting energy by ∼2 kcal mol⁻¹ with the B3LYP functional, while with the CAM-B3LYP functional, it shows a small change of 0.4 kcal mol⁻¹. The range-separated functional CAM-B3LYP includes the correction terms for the long-range interactions. Hence, the dispersion correction has a marginal contribution to the geometry and energy with CAM-B3LYP (see Table S2 in the ESI†).

Fig. 3 The energy of the square-planar Co(acac)₂ in the LS (²SQP) state, as compared to that of the tetrahedral geometry in the HS state (⁴TD), obtained from different DFT functionals, and using DLPNO-CCSD(T) and NEVPT2 methods.

The significant variation in the adiabatic energy differences (−5.3 to 13.3 kcal mol⁻¹) calculated with different DFT functionals prompted us to conduct a further comparison with the domain-based local pair natural orbital coupled cluster method (DLPNO-CCSD(T)). While CCSD(T) is widely acknowledged as the “gold standard” for electron correlation recovery in single-reference methods, its application is limited to relatively small systems.^56–61 The recent development of the domain-based local pair natural orbital coupled cluster method (DLPNO-CCSD(T)) allows larger systems to be studied with near DFT speed and near CCSD(T) accuracy.^{56,57,62–64} Therefore, we compared the DFT-computed adiabatic energy difference between the ⁴TD and ²SQP complexes with those obtained from the DLPNO-CCSD(T) method. The DLPNO-CCSD(T) computed energy difference is 16.9 kcal mol⁻¹, higher than those generated by pure, hybrid DFT functionals, reparametrized hybrid functionals, and hybrid meta-GGA functionals (Table S1†). However, the DLPNO-CCSD(T) method preserves the trend seen with the hybrid DFT functionals, i.e., the ⁴TD state is more stable than the ²SQP state. We opted to utilize the B3LYP-D3/def2-TZVP method for the geometry calculations and subject the molecular geometries to high-level wavefunction theory methods to evaluate the energetics and other observables. The B3LYP functional has also been used in other computational work on Co(II) complexes.^47,49,65

The impact of non-coordinating solvents was investigated by optimizing the complex with B3LYP-D3/def2-TZVP in benzene and toluene solvents with a PCM model. An energy difference of ±1 kcal mol⁻¹ was observed between the gas phase and solvent energetics (see Table S7†). The relativistic effect is also analysed using PBE0, TPSSh, and B3LYP functionals (Table S8 in the ESI†). A negligible change in the energy is found with the DKH/ZORA scalar relativistic effect. The energy difference with and without relativistic corrections is within 1 kcal mol⁻¹, in agreement with the observation in other 3d-metal complexes.⁶⁶

The energy difference between the two spin-states is also determined by the multireference methods by using (7,5) and (7,10) active spaces. In the first-row transition-metal complexes, efficient treatment of electron correlation in a multi-configuration framework requires a second set of nd-orbitals to model chemical bonding⁶⁷ and energetics,⁶⁸ known as the “double d-shell effect”.⁶⁹ The lowest spin-free energies of the HS and LS geometries (with one quartet and three doublet roots) using (7,5) and (7,10) active spaces for the Co(acac)₂ complex are provided in Table S9 of the ESI.† There is not much difference in energetics with the two active spaces when computed with the complete active space self-consistent field (CASSCF) method. However, when dynamical electron correlation is included by employing the N-electron valence state perturbation theory (NEVPT2), a significant change in energy is observed in (7,5) and (7,10) active spaces for both the ⁴TD and ²SQP conformers, justifying the use of double d-shells in d⁷ systems. Furthermore, we considered two additional active spaces that include ligand orbitals in addition to the five 3d-orbitals and the five double d-shell orbitals. By including one and two ligand orbitals, we get (9,11) and (11,12) active spaces, respectively (Fig. S9 to S12 in the ESI†). The spin-state energy gap from the (9,11) active space shows a significant difference from that of the (7,10) active space at CASSCF, NEVPT2, and quasi-degenerate (QD)-NEVPT2 level of theories. Including the second ligand orbital (i.e., (11,12)) changes the spin-state gap marginally (Table S9 in the ESI† and Fig. 3). For all active spaces, NEVPT2 and QD-NEVPT2 yield very similar results (Table S9 in the ESI†).

2.3. Mechanism of spin-state switching

To assess how the mode of ligand coordination impacts the relative stability of the spin-states, we conducted a one-dimensional potential energy scan along the dihedral plane of the four oxygen atoms coordinated to the metal ion. This dihedral angle connects the ⁴TD geometry to the ²SQP geometry (Fig. 4). The doublet state lies 32 kcal mol⁻¹ above the quartet at the tetrahedral geometry. As we progress towards the square planar geometry by altering the dihedral angle, the energy of the quartet state gradually rises while the doublet state gets stabilized (Fig. 4). Notably, at a dihedral angle of approximately −10° (nearing a square planar geometry), the two spin-states intersect (see Fig. 4). At the square-planar geometry, the ²SQP state appears <1 kcal mol⁻¹ below the ⁴TD state. The one-dimensional dihedral scan shows how the doublet and quartet states change their energy and provides insight into the transition structure along the quartet and doublet potential energy curves. In a non-coordinating solvent medium (benzene), the potential-energy profile along this dihedral angle follows a similar pattern (Fig. S1 in the ESI†). The presence of the solvent reduces the spin-splitting energy, resulting in a decrease of 2 kcal mol⁻¹ in the crossing energy compared to the gas-phase crossing point (see Fig. S1 in the ESI†).

Fig. 4 The one-dimensional potential energy scan of the ⁴TD and ²SQP states along the rotation of the dihedral angle (1, 2, 3 and 4 highlighted in the inset) using the B3LYP-D3/def2-TZVP method.

A transition state (TS) search was carried out (with the B3LYP-D3/def2-TZVP method) in the HS electronic configuration that connects two equivalent forms of the tetrahedral geometry (⁴TD and ⁴TD′, which can be generated by a 180° rotation along the dihedral angle formed by the four oxygen centres coordinated to Co). The transition state (TS1) exhibits a four-coordinate square-planar geometry, with all the O–Co–O angles approximating 90° and all Co–O bond lengths elongated (1.99 Å) relative to the ²SQP structure (Table S12†). TS1 has an activation energy barrier of 11 kcal mol⁻¹ (see Fig. 5 and Table S13†). It is characterized by a single imaginary vibrational mode with a frequency of −21 cm⁻¹ (Table S14 and Fig. S2†). Similarly, within the doublet spin multiplicity, the transition state search yielded a four-coordinated distorted tetrahedral structure (TS2) that connects two equivalent forms of the square-planar geometry (²SQP and ²SQP′). The TS2 has Co–O bond lengths ranging from 1.91 Å to 1.96 Å and O–Co–O angles varying from 93° to 124° (see Table S12†). TS2 has an activation energy barrier of 13 kcal mol⁻¹ and is associated with a single imaginary vibrational mode with a frequency of −123 cm⁻¹ (see Fig. 5, and Table S13†). The transition states are also confirmed by IRC calculations (Fig. S4†).

Fig. 5 Schematic energy diagram of spin-state switching in Co(acac)₂. ⁴TD and ⁴TD′ correspond to minima in the quartet PE curve, and ²SQP and ²SQP′ are the minima in the doublet PE curve. TS1 and TS2 represent the transition states along the quartet and doublet PE curves, respectively. MCEP indicates the minimum energy crossing point between the HS and LS curves.

Investigation of the potential energy (PE) surfaces in both spin manifolds suggests that the molecular orientation in the ⁴TD and ²SQP geometries is entirely governed by the spin state. As a result, we did not observe any square-planar minimum in the quartet PE profile or the tetrahedral minimum in the doublet PE profile. Instead of finding a pathway for conformational transition within a particular spin- state, we tried to find a pathway that involves a change in the spin-state. To that end, the minimum energy crossing point (MECP) between the HS and LS state was optimized using the B3LYP-D3/def2-TZVP method.

The computed MECP represents the minimum energy geometry where the ⁴TD and ²SQP states are degenerate, and the spin transition from ⁴TD to ²SQP can occur through this critical point. The optimized MECP results in a structure reminiscent of the square-planar geometry with elongated Co–O bonds. The Co–O bond length and the angles O₂–Co–O₃, O₄–Co–O₁ in the MECP structure acquire values in between the corresponding values in the ⁴TD and ²SQP geometries. The d-orbital energy levels at the MECP structure (Fig. 2, Table S10†) show a similar d-orbital energy ordering in both HS and LS electronic configurations. In the MECP geometry, the ligands lie in the xy-plane, with the z-axis perpendicular to the ligand plane. The reduced gap between the antibonding d_x²−y² and the rest of the d-orbitals in the MECP geometry indicates a reduced energy gap between the HS and LS states. The energy (electronic energy with zero-point energy correction) and bond parameters of all the structures are given in Tables S12 and S13,† respectively. The calculated MECP is 10.3 kcal mol⁻¹ above the ⁴TD state and 1.7 kcal mol⁻¹ higher than the ²SQP geometry. In Fig. 5, a schematic diagram shows different pathways for conformational transition. Since the ²SQP state lies about 8.6 kcal mol⁻¹ above the ⁴TD state, its equilibrium population is expected to be low from a thermodynamic point of view. It is clear that the spin-assisted conformational transition involves a comparatively smaller barrier. In particular, a very small barrier between the ²SQP state and the MECP (<2 kcal mol⁻¹) indicates the possibility of a spontaneous conformational transition from a square-planar to a tetrahedral structure. However, the tetrahedral to square-planar conformational transition involves a barrier inaccessible through thermal means. This explains the lack of structural evidence of square-planar Co(acac)₂, despite several attempts.^40,44–46

2.4. Magnetic properties of the Co(acac)₂ complexes

Pietrzyk and co-workers investigated the magnetic properties of the Co(acac)₂ complexes by EPR measurements.⁴⁷ From the unresolved EPR spectra, the zero-field splitting (ZFS) parameters could not be experimentally measured. Consequently, the spin-Hamiltonian parameters were computed using the DFT method. The magnetic properties involve excited electronic states of a molecule, where the multi-configuration interactions dominate. DFT is well known to fall short in describing such interactions. It further suffers from strong functional dependence in describing the sign and magnitude of the ZFS parameters.⁵² Although the ground state energy can be reliably estimated from single-reference methods, such as DFT or DLPNO-CCSD(T), both ground and excited states, coupled via the spin–orbit operators, contribute to the magnetic properties of the complex. Hence, we have evaluated the magnetic parameters using the multi-configurational second-order perturbation theory methods. The spin–orbit coupling was treated a posteriori.

To explore the magnetic anisotropy of the Co(acac)₂ complex, we conducted state-averaged (SA)-CASSCF calculations on the molecular geometries optimized with the B3LYP-D3/def2-TZVP method. Our approach involved utilizing an active space of (7,5), representing seven metal d-electrons distributed across five 3d-orbitals. Additionally, the influence of the second d-shell was taken into account by incorporating the five 4d-orbitals into an active space of (7,10). Since the CASSCF method takes into account only the static electron correlation, the dynamical electron correlation was evaluated by subjecting the SA-CASSCF wavefunctions to NEVPT2 calculations.

In the ⁴TD geometry, the (7,5)-SA-CASSCF computed ground (Q₀) and the first excited quartet states (Q₁) are separated by 1898 cm⁻¹, which increases to 1985 cm⁻¹, when the (7,10) active space is used. The Q₂ and Q₃ states form a quasi-degenerate pair in both the active spaces. The energy differences between Q₀ and Q₂ states were estimated to be 5238 cm⁻¹ with (7,5) and 5525 cm⁻¹ with (7,10) active spaces. The SA-CASSCF energetics computed by the two active spaces are found to be consistent with each other. Including dynamical electron correlation increases the energy separation between the Q₀ and Q₁ states from 1985 cm⁻¹ in (7,10) CASSCF to 2256.4 cm⁻¹. Similarly, the gap between the Q₀ and Q₂ states also increased to 6447.2 cm⁻¹ (compared to 5237.8 cm⁻¹ in CASSCF), while the near degeneracy of the Q₂ and Q₃ states is maintained. Conversely, in the ²SQP geometry, the scalar-relativistic energy levels demonstrate significant sensitivity to the incorporation of the second d-shell and dynamical electron correlation (see Table 1). The gap between the lowest energy LS state and the first excited LS state (D₀ and D₁, respectively) decreases from 1503 cm⁻¹ to 939 cm⁻¹ when the double-d-shell effect is included in the (7,5)-SA-CASSCF. It decreases to 317 cm⁻¹ upon accounting for the dynamic electron correlation through the NEVPT2 level of theory with the (7,10) active space.

Table 1 The energy (in cm⁻¹) of the lowest four spin-free and spin–orbit states of HS and LS conformers of the Co(acac)₂ complex with SA-CASSCF and NEVPT2 methods by using 10 quartet roots in the ⁴TD geometry and 40 doublet roots in ²SQP in different active spaces

State (active-space)	Spin-free states		Spin–orbit states
	CASSCF	NEVPT2	CASSCF	NEVPT2
^4TD	0 (Q0)	0 (Q0)	0	0
CAS(7,5)	1898.2 (Q1)	2256.4 (Q1)	68.9	61.2
	5237.8 (Q2)	6447.2 (Q2)	1988.8	2337.2
	5240.2 (Q3)	6450.1 (Q3)	2145.2	2470.3
^4TD	0 (Q0)	0 (Q0)	0	0
CAS(7,10)	1984.7 (Q1)	2170.2 (Q1)	62.3	57.6
	5524.7 (Q2)	5986.5 (Q2)	2066.4	2245.8
	5527.2(Q3)	5988.7 (Q3)	2206.4	2375.4
^2SQP	0 (D0)	0 (D0)	0	0
CAS(7,5)	1503.4 (D1)	170.5 (D1)	1569.3	703.2
	2021.6 (D2)	1420.2 (D2)	2463.8	2031.0
	8595.1 (D3)	9722.7 (D3)	8761.0	10 104.1
^2SQP	0 (D0)	0 (D0)	0	0
CAS(7,10)	938.5 (D1)	317.3 (D1)	1104.9	734.4
	1625.6 (D2)	1374.9 (D2)	2125.9	1966.1
	8378.9 (D3)	8592.5 (D3)	8601.9	8931.9

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The scalar-relativistic states computed by the SA-CASSCF/NEVPT2 methods are subjected to spin–orbit mixing, which results in Kramers doublets for the odd electron systems with S > 1/2. When a (7,5) active space is used for the SA-CASSCF/NEVPT2 calculations, the ground Kramers doublets are separated from the first excited Kramers pairs by 61.2 cm⁻¹, which shifts to 57.6 cm⁻¹, with the (7,10) active space. This separation gets marginally affected (within 2–4 cm⁻¹) in the presence of the energetically closed doublet states that are known to contribute to anisotropy by spin–flip excitations.⁷⁰ Analysis of the spin–orbit states suggests that the low-lying Kramers doublets are primarily composed of the non-degenerate Q₀ states, with marginal contribution from the Q₁ states, and this remains the same, irrespective of the choice of the active space for the SA-CASSCF/NEVPT2 calculations (see Table S16†).

The spin–orbit states are subjected to the SINGLE_ANISO module to evaluate the magnetic parameters, such as zero-field splitting (ZFS) parameters and g-tensors. For systems with unsaturated coordination numbers and unquenched angular momentum, spin–orbit coupling plays a significant role in governing the axial (D) and rhombic (E) ZFS parameters. The (7,5)-SA-CASSCF/NEVPT2/RASSI/SINGLE_ANISO calculation results D = −30.6 cm⁻¹ and E = −0.003 cm⁻¹. A similar calculation with the (7,10) active space slightly lowers the magnitude of D to −28.8 cm⁻¹, although the E-value remains unchanged (Table S17†). The negative values of D and a small E/D ratio indicate easy-axis type anisotropy in the ⁴TD of Co(acac)₂ and suggest the complex is a potential single-molecule magnet (SMM). The ZFS parameters are known to vary with the dihedral angle.^71,72 In Co(acac)₂, the HS state of the complex converts to the LS state with a dihedral change. Given that the ZFS parameters are not defined for the LS states (S = 1/2), the dihedral dependence of ZFS parameters is not explored in this work.

SMMs exhibit a preferential direction of magnetization, also known as “easy axis” and can retain magnetization for a considerable period of time. The g-tensors associated with the Kramers pairs indicate the preferential direction of magnetization in a particular spin–orbit state. The g-tensors were computed for pseudospin S = 1/2. According to the Kramers theorem, pseudospin S = 1/2 corresponds to Kramers doublets that remain degenerate in the absence of any applied magnetic field and, hence, exhibit no ZFS.⁷³ While our computed axial ZFS parameter is consistent with that observed from DFT,⁴⁷ the magnitude of the g-tensors is different. While the DFT computations show isotropic g-tensors, our calculation shows for the ground spin–orbit pairs, the g-tensors are g_x = g_y = 0, and g_z = 7.6 (with both the active spaces), indicating the Z-axis as the preferential direction (“easy axis”) of magnetization. On the other hand, in the first excited spin–orbit states, the magnitude of the g-tensors changes significantly. The increase in the magnitude of g_x and g_y to 4.3 and a simultaneous reduction of g_z to 2.6 suggest a decrease in the uniaxial magnetic anisotropy. Such changes in g-values are also accompanied by deviation in the magnetisation direction (Table S18†). Such non-coincidence of the excited-state anisotropy axis with respect to that of the ground Kramers doublet leads to fast relaxation of the magnetization.⁷⁴ To probe the mechanism of the relaxation of magnetization, we computed the transition-magnetic-moment matrix elements between the connecting pairs of spin–orbit states of the opposite magnetic moments (Fig. 6). The calculated transition-magnetic-moment matrix elements between the ground Kramers doublets are very small (0.0005μ_B), suggesting a negligible quantum tunnelling of magnetization from the ground spin–orbit pairs. It is consistent with the uniaxial magnetic anisotropy predicted by the magnitudes of the g-tensors. Conversely, the higher transition magnetic moment matrix element between the first excited spin–orbit states (1.45μ_B) suggests significant relaxation of magnetization. This is also consistent with the increase in the transverse components of the g-tensors (g_x = g_y > g_z) than the axial component in the first excited spin–orbit states. Such relaxation of magnetization from the first-excited spin–orbit pairs is known as thermally-assisted quantum tunnelling of magnetization (TA-QTM), which results in an effective magnetic anisotropy barrier (U_eff) of 61.2 cm⁻¹, with the (7,5) active-space. The inclusion of the second d-shell marginally reduces the U_eff to 57.6 cm⁻¹. A large U_eff indicates the potential of Co(acac)₂ as a single-molecule magnet.

Fig. 6 The mechanism of relaxation of magnetization using CAS(7,10) in the ⁴TD complex.

The anisotropic g-tensors and the negative axial ZFS parameter fit the criteria for a molecule to behave as a single-molecule magnet. Furthermore, the anisotropic g-tensors of the ground spin–orbit states are also consistent with the small relaxation of magnetization from the ground spin–orbit pairs. Reduction in the anisotropy in the first-excited spin–orbit pairs results in the relaxation of magnetization from the first-excited spin–orbit pairs by the TA-QTM process.

3. Computational details

The geometry optimization of the Co(acac)₂ complex at the HS and LS states was carried out using a number of DFT functionals, together with the def2-TZVP basis sets on all the elements.^75,76 Two pure DFT functionals BP86^77,78 and BLYP;⁷⁹ four hybrid DFT functionals B3PW91,^80,81 B3LYP,^79,82,83 PBE0,⁸⁴ and CAM-B3LYP;⁸⁵ one reparametrized hybrid functional B3LYP*;⁸⁶ three hybrid meta-GGA functionals TPSSh,⁸⁷ M05,⁸⁸ and M06;⁸⁹ and one double-hybrid functional B2PLYP⁹⁰ were employed. Grimme's D3 dispersion correction with the original damping function was used in all the geometry optimizations.^91,92 No symmetry restrictions were used. All the geometries were confirmed as minima using harmonic vibrational analysis. The absence of imaginary frequency confirms the stability of the computed molecular geometries in the respective spin states.

The potential energy scan was carried out in both HS and LS multiplicities along the dihedral angle that connects the tetrahedral structure with the square-planar structure. The transition state between the tetrahedral and square-planar structures was determined in the quartet and doublet potential energy surfaces. These transition state structures were confirmed through the imaginary frequencies and further validated through the intrinsic reaction coordinate (IRC) calculations that connect two correct minima. The crossing between the LS and HS potential-energy surfaces provides the first estimation of the crossing point between the two geometries. To evaluate the mechanism of spin-state switching and associated structural changes between the HS and LS states, a search for the minimum energy crossing point (MECP) was carried out by using a program developed by Harvey et al.,⁹³ where the location of the MECP is determined by employing a constrained optimization in both spin states.^94,95 All the optimization and frequency calculations described in this work were done using Gaussian 16 software.⁹⁶

The adiabatic energy differences calculated using a series of DFT functionals were compared with those determined from DLPNO-CCSD(T) computations.^97–100 In the DLPNO-CCSD(T) calculations, B3LYP-computed unrestricted Kohn–Sham determinants served as the reference. The elements of the first coordination sphere, Co and O atoms, were treated with the aug-cc-pwcvtz basis set,¹⁰¹ and the remaining elements were treated with the cc-pvdz basis set.^101,102 The auxiliary basis sets (AuxJ, AuxC, and AuxJK) were generated using the “Autoaux” generation procedure.¹⁰³ The DLPNO-CCSD(T) method can fall short when the states are closely spaced and multireference interactions dominate.^104,105 However, the ability of the method to capture the dynamic correlation can be assessed from the so-called T₁ diagnostic.¹⁰⁶

To evaluate the magnetic properties of the systems, the DFT-optimized geometries were subjected to the multi-configurational SA-CASSCF^107,108 followed by NEVPT2 calculations.^109–111 Dynamical correlation effects, which are not captured by the CASSCF method, are incorporated through the NEVPT2 and QD-NEVPT2 approaches.¹¹² The scalar-relativistic effects were treated using the Douglas–Kroll–Hess (DKH) Hamiltonian. For multireference calculations the DKH-def2-TZVP basis set was used for Co and O, and DKH-def2-SVP was used for C and H atoms. The auxiliary basis sets were generated by using the “Autoaux” procedure, as mentioned previously. The convergence criteria were set to be tighter than the default by choosing the “tightscf” option. For SA-CASSCF calculations, an active space of seven-d electrons in five 3d-orbitals (7,5) was used. The effect of the double d-shell was taken into account by accommodating another set of five d-orbitals, with (7,10) as the active space. Both the active spaces were used for state-averaged calculations of 10 quartet states and 40 doublet states. A comparison of energetics and magnetic parameters obtained from these two active spaces enables us to elucidate the role of the double d-shell in governing these properties. We also include the doubly occupied ligand orbitals into the active space, expanding it by accommodating one or two ligand-based orbitals along with five 3d- and 4d-atomic orbitals of the metal center, leading to (9,11) and (11,12) active spaces. The adiabatic separation energy between the ground quartet state of the ⁴TD complex and the three-fold degenerate doublet ground state of the ²SQP complex was obtained from these active spaces and was compared with those obtained from active spaces (7,5) and (7,10), different DFT functionals and the DLPNO-CCSD(T) method (Fig. 3). The energy levels of the d-orbitals were computed using ab initio ligand field theory (AILFT).^113,114 The scalar-relativistic states (10 quartets and 40 doublets) obtained by (7,5) or (7,10)-SA-CASSCF/NEVPT2 calculations were subjected to the RASSI module to evaluate the spin–orbit mixing of states.¹¹⁵ The zero-field splitting (ZFS) parameters, g-tensors, and effective barrier of magnetic anisotropy (U_eff) were determined using the SINGLE_ANISO¹¹⁶ module. All DLPNO-CCSD(T), CASSCF, NEVPT2, and QD-NEVPT2 calculations were performed using Orca 5.0.4 software.^117–119

4. Conclusions

The geometry of Co(acac)₂ has been a contentious topic for several decades. In this study, we have examined the relative stability of the Co(acac)₂ complex in tetrahedral and square-planar geometries. The relative energy differences of the two geometries are computed with different DFT functionals and are further benchmarked against the DLPNO-CCSD(T) method. The hybrid functionals and DLPNO-CCSD(T) methods predict a stable tetrahedral geometry in a HS electronic configuration and a stable square-planar geometry in a LS configuration. Once the relative stabilities are characterized, the inter-conversion mechanism between the two conformers is explored. Our investigation shows that the conformational arrangement of the molecule in the tetrahedral and square-planar geometries is governed by a spin-state switch. The conformation change within a given spin-multiplicity (either quartet or doublet) involves a large barrier. On the other hand, when a spin-state switch is allowed, the complex finds a relatively low-lying minimum-energy-cross point between ⁴TD and ²SQP states. This crossing point lies 10 kcal mol⁻¹ above the ⁴TD state, while it is only 2 kcal mol⁻¹ above the ²SQP state, explaining why it has been difficult to isolate and characterize ²SQP Co(acac)₂.^40,44–46 With multi-reference CASSCF/NEVPT2 methods together with spin–orbit coupling, the magnetic properties of the ⁴TD state of Co(acac)₂ are examined. The effects of the double d-shell and dynamic electron correlation on the spin-state energetics, g-tensors, ZFS parameters, and magnetic anisotropy barrier are evaluated. Our investigation shows that in the ⁴TD geometry, the ground spin–orbit state is magnetically anisotropic, and the molecule exhibits an anisotropy barrier of 57.6 cm⁻¹.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgements

SJ acknowledges the UGC, India, New Delhi, for the research fellowship. SRC acknowledges the support from IIT Kharagpur and the University of South Dakota. SM acknowledges the support from the SERB, DST, Government of India (CRG/2022/004088 and SR/FST/CSII-026/2013), and the CSIR, New Delhi, India (01(2987)/19/EMR-II). This work used the resources of the Paramshakti supercomputing facility of IIT Kharagpur and Paramporul supercomputing facility of NIT Trichy established under the National Supercomputing Mission of the Government of India and supported by CDAC, Pune.

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Footnote

† Electronic supplementary information (ESI) available: Additional tables and figures. The coordinates of the optimized geometries. See DOI: https://doi.org/10.1039/d5dt00052a

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