Structural parameters obtained from calculations on a model compound bearing sterically small bridging carboxylates, [Mn₂(μ-O)₂(μ-O₂CH)₂(bpy)₂]⁺, are in good agreement with the experimentally determined single crystal X-ray structure.

There is calculated to be minimal Mn⋯Mn bonding despite contraction of the Mn⋯Mn distance relative to related complexes.

Although our calculations support assignment of [Mn₂(μ-O)₂(μ-O₂CH)₂(bpy)₂]⁺ as a valence-trapped Mn^IIIMn^IV configuration involving high-spin Mn^III, a delocalized configuration arising from low-spin Mn^III is calculated to lie very close in energy.

The Mn^III and Mn^IV centers are found to be strongly antiferromagnetically coupled in all other dinuclear complexes.^13,35–40

Density functional theory calculations on the model compound [Mn₂(μ-O)₂(μ-O₂CH)₂(bpy)₂]⁺ reproduce the crystallographic structure very satisfactorily.

Our geometry optimization on the S = 1/2 ‘broken symmetry’ (BS) state of the model compound, in C_2v symmetry, yields a structure for which all of the key bond lengths and bond angles, summarized in Table 1 and Fig. 2, agree with the crystallographic data to within ±2.5%.

In particular, the BS calculation demonstrates that the asymmetry in bond lengths between Mn atoms and the carboxylate O atoms arises through electronic effects rather than through steric considerations, since the bulky carboxylate substituent of the crystallographically isolated species is replaced with H in our model compound.

The important structural features of the BS geometry are also evident in the geometry optimized for the S = 7/2 state in C_2v symmetry, although the Mn–Mn bond length found in this ferromagnetically coupled structure is over 0.1 Å longer than the experimental value.

These optimizations commenced in C_s symmetry but spontaneously converged towards the C_2v optimized geometries for both the S = 1/2 and S = 7/2 states.

In contrast, calculations on the S = 7/2 state in D_2h symmetry led to a structure with enforced delocalization of all of the Mn-based electrons.

As well as (understandably) failing to reproduce the observed intermetallic asymmetry, the S = 7/2 D_2h calculations yielded a total energy markedly higher than either the S = 1/2 or S = 7/2 C_2v optimized geometries (see Table 2), thereby underlining the energetic preference for a ‘valence-trapped’ d³d⁴ (Mn^IVMn^III) configuration rather than a ‘delocalized’ d^3.5d^3.5 configuration.

The latter calculation spontaneously converges to a structure of D_2h symmetry, indicating that in this mode of coupling a delocalized configuration is preferred.

The total energy for this configuration is only 14 kJ mol⁻¹ above that of the S = 1/2 C_2v (BS) optimized geometry.

While the latter species does not describe a tenable solution to the Mn^IIIMn^IV complex’s electronic configuration, its relative energy (only 19 kJ mol⁻¹ above that of the S = 1/2 C_2v (BS) optimized geometry) emphasizes that the energetic preference for the C_2v geometry, with distinctly different coordination environments around the two metal atoms, is really rather slight.

Total bond energies and relative energies, for the various electronic configurations of [Mn₂(μ-O)₂(μ-O₂CH)₂(bpy)₂]ⁿ⁺ (n = 0, 1, 2) are given in Table 2.

For the d⁴d⁴ (Mn^IIIMn^III) dinuclear complex, valid single-determinant descriptions are possible for BS (M_S = 0) and for S = 1, 2, 3, and 4.

The S = 4 configuration explored corresponds to a ferromagnetically coupled, high-spin Mn^III dimer, while the S = 2 configuration consists of two low-spin Mn^III centers which are ferromagnetically coupled.

Our calculations show that there is a considerable energetic preference for low-spin Mn^III, which contrasts with the results of our previous calculations on di-μ-oxo-bridged (and mixed oxo-/carboxylato-bridged) Mn^III dimers for which high-spin Mn^III is preferred.^30,31

Again, the antiferromagnetic broken-symmetry solution is the preferred (lower-energy) description of this dinuclear complex.

It is also worthwhile to compare the geometries of the homonuclear Mn^IIIMn^III and Mn^IVMn^IV complexes with that of the broken-symmetry Mn^IIIMn^IV minimum; the central bond lengths and bond angles for all of these structures are included in Table 1.

There is very good agreement between the calculated Mn^IIIMn^IV geometry and the optimized Mn^IVMn^IV structures, insofar as the coordination around the Mn^IV center (Mn^B) of the mixed-valence complex is concerned.

However, the agreement between geometric parameters featuring Mn^A (the identified Mn^III center of the mixed-valence complex) and the Mn^III centers in the broken-symmetry Mn^IIIMn^III geometry is much poorer.

Better general agreement with the geometry surrounding Mn^A is seen for the ferromagnetically coupled, high-spin (S = 4) Mn^IIIMn^III structure also summarized in Table 1.

Note, however, that all of the ferromagnetically-coupled structures detailed in Table 1 have metal–metal separations which significantly exceed those of the antiferromagnetically-coupled (broken-symmetry) minima, and of the crystallographically-isolated complex.

A detailed perusal of the composition of the metal-based molecular orbitals (MOs) in the BS geometry of the Mn^IIIMn^IV complex (see Tables 3–5) reveals that none of the magnetic electrons is strongly delocalized between both metal centers.

Instead, the 39a₁ (↑), 17b₂ (↑), and 10a₂ (↑) MOs show strong mixing between Mn^A and the oxo bridges, with the corresponding spin-down MOs similarly mixed between Mn^B and the oxo bridges.

The remaining occupied valence metal-based orbital, 40a₁ (↑), shows a modest degree (14%) of Mn^B character but is principally identifiable as Mn^A with mixing from the immediately attached oxygens of the carboxylate ligands.

The intermetallic interaction here is an example of ‘crossed exchange’ involving the J_{x²−y²/z²} pathway, resulting in transfer of charge from a majority-spin, e_g-derived orbital on Mn^III (here d_x²−y²) to a minority-spin, t_2g-derived orbital on Mn^IV (here d_z²).

Composition of the occupied 39a₁, 10a₂, and 17b₂ orbitals, as detailed in Table 3, clearly shows that the magnetic exchange between metal atoms is dominated by superexchange via the oxo bridges, in keeping with the results of our earlier investigations of di-³⁰ and tri-μ-oxo-⁴⁹ and mixed oxo- and carboxylato-bridged³¹ Mn^IIIMn^IV dinuclear complexes.

In fact, the di-μ-oxo-bridged species [(NH₃)₄Mn^III(μ-O)₂Mn^IV(NH₃)₄]³⁺, which we have previously studied,³⁰ is seen to bear a very close resemblance to the species under investigation in the present work.

This resemblance is reflected both structurally, in the elongation of bonds from the Mn^III atom to the μ-oxo bridges, and the more pronounced elongation of bonds from Mn^III to the axially-coordinated atoms (respectively, the N of NH₃, or the O of the carboxylate bridge), and in the electronic configuration determined for the S = 1/2 broken symmetry state.

Consistent trends are also seen when we compare the present results with those of our study on Mn^III(μ-O)₂(μ-O₂CH)Mn^IV and Mn^III(μ-O)(μ-O₂CH)₂Mn^IV complexes:³¹ in each case, the BS (S = 1/2) complex can be described as a largely valence-trapped structure consisting of high-spin Mn^III antiferromagnetically coupled to Mn^IV, although the Mn^III(μ-O)(μ-O₂CH)₂Mn^IV core is rather more electronically complex and displays configurational mixing from low-spin Mn^III.

The mixed μ-oxo/μ-carboxylato bridged complexes also invariably display markedly longer bonds for Mn^III–O than for Mn^IV–O (whether to oxo or carboxylato O atoms), but the elongation of the Mn^III–O (carboxylato) bonds within the present S = 1/2 complex (bond lengths of 2.27 Å are 0.3 Å longer than the corresponding Mn^IV–O distances) is rather greater than is seen in the earlier study,³¹ presumably because in the present work these carboxylato O atoms lie on the Jahn–Teller axis.

Nevertheless, for all of the mixed μ-oxo/μ-carboxylato bridged complexes which we have studied,³¹ the disparity between Mn^III–O and Mn^IV–O is more extreme in the ferromagnetically coupled S = 7/2 configuration than for S = 1/2.

Another consequence of the ‘crossed exchange’ interaction is that mixing of the occupied Mn^III e_g-derived orbital with a t_2g-derived Mn^IV orbital leads to a lessening in the Jahn–Teller distortion of the Mn^III center.

The carboxylic oxygens lie on the nominal Jahn–Teller axis in this instance, and the shorter Mn^III-O_carb distance of 2.27 Å seen for the S = 1/2 Mn^IIIMn^IV dimer than that found for the S = 7/2 structure (2.32 Å) is consistent with the structural implications of the crossed-exchange mechanism.

Further structural analysis—for example, of the variation in Mn–O–Mn angles as a function of the bridging combination within different modelled complexes—can be given: not surprisingly, the Mn–O–Mn angles found here, of 87.4° (BS) and 91.7° (S = 7/2), are uniformly less than those which we have previously reported for the complexes containing Mn^III(μ-O)₂Mn^IV (97.7° (BS)), Mn^III(μ-O)₂(μ-O₂CH)Mn^IV (94.2° (BS); 93.4° (S = 7/2)), and Mn^III(μ-O)(μ-O₂CH)₂Mn^IV (130.5° (BS); 124.7° (S = 7/2)) cores , reflecting the more compact Mn–Mn distance in the present case.

The Mn₂O₂ core geometry may well not hold a significant direct influence over the crossed-exchange pathway, since this appears to operate through mixing of near-degenerate orbitals rather than through overlap, but the antiferromagnetic pathways J_xy/xy and J_yz/yz (which are, respectively, analogous to the J_xz/xz and J_yz/yz pathways explored in our study of the di-μ-oxo-bridged complexes)³⁰ are dependent on the overlap of Mn-centered and O-centered orbitals and thus should be affected by the Mn–Mn and Mn–O distances and the Mn–O–Mn angles.

Using a notation scheme similar to that employed in Table 3, our earlier study reported percentage compositions of the 17a₁ orbital of 31% N_axial (attached to Mn^A), 48% Mn^A (principally d_z² character), 2% O_br, 12% Mn^B (principally d_x²−y² character), and 0% N_axial (attached to Mn^B).

As with the HOMO 40a₁ in the S = 1/2 (BS) [Mn₂(μ-O)₂(μ-O₂CH)₂bpy₂]⁺ complex which forms the focus of the present study, the 17a₁ HOMO in the di-μ-oxo-bridged complex studied previously³⁰ is therefore a crossed exchange pathway.

Note that, due to differences in the axis assignments of the two studies (see Fig. 2 for the axis assignment used here), the occupied majority-spin e_g-derived orbital on Mn^III is identified, in the earlier study, as d_z² and the vacant minority-spin t_2g-derived orbital on Mn^IV is d_x²−y².

If we assess the ratio of ‘Mn^IV’ to ‘Mn^III’ character in the highest-occupied a₁-symmetry orbital of di-μ-oxo-bridged complexes featuring, respectively, 0,³⁰ 2 (this work), and 1³¹ carboxylato bridge(s), we find that this ratio consistently increases from 0.25 to 0.3 to 0.55 as the energy gap between the highest occupied and the lowest unoccupied α-spin a₁-symmetry orbitals is reduced from 1.35 to 1.29 to 0.85 eV.

The extent of configurational mixing evident between high-spin Mn^III (contaminated by Mn^IV character) and Mn^IV (contaminated, in turn, by low-spin Mn^III character) is greater than that seen in our previous calculations on complexes containing the Mn(μ-O)₂Mn core, but less than in complexes featuring a Mn(μ-O)₂(μ-O₂CH)Mn centre.