Figure 1. (A) Schematic environment of an FeNC site with the iron ion shown as an orange circle. The extent of the graphene-like environment (black lines) is unknown as indicated by grey dashed lines; note that conjugation is not shown. The nitrogen (blue circle) donation may occur from six- or five-membered rings; the latter will result in local distortions and defects (green lines). Axial ligands may be present, but their number and chemical character is unknown (half blue/half red circles). (B) Generic Mössbauer spectrum showing the definition of isomer shift δ and quadrupole splitting ΔE Q.
A Mössbauer spectrum of a typical FeN4 environment shows a doublet with two defining features: the isomer shift δ and the quadrupole splitting ΔE Q, both sketched in Figure 1B.1 These arise from the interaction between the charge densities of the iron nucleus and the surrounding electrons. Specifically, the isomer shift reflects the electron density at the iron nucleus, also known as the contact density, and the quadrupole splitting is indicative of an electric field gradient, i.e. the degree of asymmetry within the electron density. While the isomer shift provides information about the iron oxidation state and spin state, the quadrupole splitting can help to differentiate between different electronic states with the same multiplicity. We note that the simultaneous evaluation of both parameters is important to distinguish signals of sites for which the isomer shift or quadrupole splitting by themselves cannot provide an unambiguous assignment (see below).
The numerical values for isomer shift and quadrupole splitting expected for iron in various oxidation and spin states are illustrated in Figure 2. While generally, the isomer shift is higher for lower oxidation states, it can be seen that the observed regions overlap for different oxidation states and different spin states. Fe(II) as one of the relevant oxidation state for the resting state of FeNC-catalysts shows good differentiability between its high spin ( = 2,δ  =  0.59–1.45 mm s−1) and low spin ( =  0, δ  =  –0.16–0.50 mms) states, although intermediate spin ( =  1,δ  =  0.26–0.49 mm s−1) centers are found at the high end of the range expected for low spin complexes. Iron in oxidation state +III is found between –0.17–0.67 mm s−1 with some overlap in the observed regions for all three spin states =  1/2, =  3/2 and =  5/2. Since Fe(II) and Fe(III) may both be present in FeNC catalysts, it is important to note that the isomer shift regions of all ferric spin states overlap with those of ferrous low spin and intermediate spin to some extent. This implies that additional information, such as the quadrupole splitting values shown in Figure 2B, will be needed to assign oxidation and spin states. Lower and higher oxidation states are shown for completeness; while Fe(I) is less relevant to the FeNC intermediates during ORR, higher oxidation states are likely important for the later stages of catalysis.
Computationally, the isomer shift and quadrupole splitting can be predicted with good accuracy using hybrid functionals and suitable basis sets.14,19-22 Details on the computational approach and the expected error margins are given below. A recurring problem in computational iron chemistry is the prediction of the correct spin state energies,61 the correct spin state being obviously very important to achieve a reliable Mössbauer prediction. When using density functional theory, the selection of an appropriate density functional and basis set in combination with a good knowledge of ligand field theory and MO theory appears to be sufficient to identify shortcomings in many cases, e.g. when the electronic structure obtained by the self-consistent field (SCF) procedure is not the lowest-lying electronic state.61-63 In other cases of course, the electronic structure will have non-negligible multireference character or substantial mixing of low-lying excited states, and in such scenarios DFT is prone to failure. Wavefunction approaches such as the complete active space SCF or density matrix renormalisation group methods for large active spaces in combination with extensive basis sets can lead to accurate predictions of relative spin state energies.64,65 The more recently introduced multiconfiguration pair-density functional theory also shows promising results.66,67 While one might thus think that post-Hartree–Fock-methods are the ideal approach to obtain both more accurate spin state energetics and more reliable Mössbauer parameter predictions, these types of calculation remain far from routine for large molecules with complicated electronic structures. For the problem targeted —a screening of iron sites embedded in extended π-systems, paralleling the strategy that has contributed greatly to resolving many questions in bioinorganic chemistry13,15,20,23,68,69— a density functional theory based approach is clearly better suited. Given the limitations and challenges set out above, it is however equally clear that the reliability and pitfalls of Mössbauer parameter predictions must be carefully calibrated specifically for FeN4 sites.