Figure 6. Isosurface plots of the electron density (left) and spin density (right) for complex 5 with initial (A, B) and final (C, D) electronic structure (for complex 8 , see Figure S4 in the Supporting Information). The cutoff radii are set to 0.2 for the electron density and 0.005 for the spin density.
The outliers excluded from the fit presented above are complexes9 and 10 (see SI for the fit including all 20 complexes), which have been noted as outliers in previous studies as well.20,21 The isomer shifts predicted with B3LYP deviate by 0.077 mm s−1 (9 ) and 0.159 mm s−1 (10 ) from experiment, the respective quadrupole splitting values by 1.601 mm s−1(9 ) and 1.848 mm s−1 (10 ). While for9 , δ is thus predicted within the double mean absolute deviation of 0.131 mm s−1, the calculated value for10 is outside this trust region. For both complexes, the error in the quadrupole splitting prediction is far outside the trust region of 0.451 mm s−1 and the maximum deviation of 0.57 mm s−1 for the other members of the correlation line. In both cases, the iron(II) intermediate spin (2S +1 = 3) ion is found in the square-planar coordination sphere of a decorated porphyrin. The difficulties in describing such Fe(II)-porphyrin electronic structures have been known for many years, and the body of computational chemistry literature pertaining to the correct prediction of the relative spin states is vast.65 To briefly review the key points of discussion, in D4h symmetry the two lowest-lying triplet states are 3A2gwith a d-orbital occupation pattern of (xy)2(z2)2(xz)1(yz)1(x2–y2)0and 3Eg with degenerate (xy)2(z2)1(xz)2(yz)1(x2–y2)0or (xy)2(z2)1(xz)1(yz)2(x2–y2)0configurations. Additionally, there is a low-lying5A2g state, and it is difficult to predict the energetic gap between the correct triplet ground state and the higher lying triplet and quintet states. However, for the purpose of estimating the uncertainty of Mössbauer parameter predictions, the key point here is not whether the energetic separation of triplet and quintet states is predicted with quantitative accuracy, but how good the quality of the predicted electron density for the specific triplet states is in comparison with experimental data and in relation to the calibration line.
Given that the use of symmetry can have significant effects,21,132,133 Pápai and Vankó used symmetry constraints to enforce either triplet state in a DFT calculation, leading to quadrupole splittings of 0.40 mm s−1(10 , 3A2g) and 3.08 mm s−1 (10 ,3Eg) with the B3LYP functional. Lowering the symmetry to D2h led mixing of d(xy) and d(z2) orbitals resulting in a computed quadrupole splitting of 1.25 mm s−1, quite close to the experimental value of 1.51 mm s−1. The electronic structures in this manuscript were computed without any symmetry constraints, and virtually no mixing of d(xy) and d(z2) character is found, explaining the pronounced deviation from experiment.
While some of the examples discussed in this section may at first glance look almost trivial, it is important to note that an egregious error in the electronic structure can be quickly spotted with the spin population analysis but may be more difficult to identify based on the spin density contour plots. In most cases, as shown above, such a scenario may be resolved with re-optimising the geometry and/or the electronic structure. An error of this magnitude will most likely result in a drastic difference between the experimental and computed isomer shift. In contrast, it is equally possible that while the spin population is correct, the orbital occupation pattern that leads to this spin population is incorrect. This will influence the isomer shift very little or not at all, as was shown in the discussion of complexes9 and 10 ; however, the prediction of the quadrupole splitting will be much more strongly affected as is expected from the fundamental interactions at play in such a scenario. It is thus clear that treating DFT purely like a black-box method is prone to failure, and we hope to have shown with the above examples that a careful inspection of the electronic structures at least at the level of a spin population analysis, but preferably with an analysis of the MO occupation pattern, is mandatory to achieve reliable results.