What is the nature of the Fe–O2 binding in oxymyoglobin and oxyhemoglobin?
TL;DR: It is not really "one or the other" - it is more of "somewhere in between", like a resonance hybrid. If I had to choose, I'd personally lean towards Weiss's $\ce{Fe(III)-O2-}$ model. However, there will always be people who disagree, and there is still no general consensus amongst the scientific community.

There is an excellent - and recent - perspective article by Bren et al.,1 describing the research carried out on the structure of hemoglobin ever since Pauling and Coryell's initial report2 on the diamagnetism of oxyhemoglobin.



Model systems

A common technique in bioinorganic chemistry is to synthesise "inorganic versions" of enzyme active sites; for example, much work has been done on dinitrogen complexes in the hope of understanding the enzyme nitrogenase. Myoglobin and hemoglobin are not enzymes in the strict sense of the word,3 but the same technique applies: if stable $\ce{Fe(II)-O2}$ complexes could be synthesised, they might prove to be more amenable to experimental investigation.

The main problem was the fact that dioxygen readily oxidised simple Fe(II) complexes to Fe(III). Initially, Co(II) complexes (which only have one unpaired electron) were used: EPR studies showed that the unpaired electron was localised on oxygen,4,5 and the length of the O–O bond resembled that of superoxide:6

$$\begin{array}{ccccc}
\text{Species} &amp; \ce{O2^2-} &amp; \ce{O2^-} &amp; \ce{O2} &amp; \ce{Co(II)-O2}\text{ complex} \\ \hline
\text{Bond order} &amp; 1 &amp; 1.5 &amp; 2 &amp; ? \\
\text{Bond length / Å} &amp; 1.49 &amp; 1.28 &amp; 1.21 &amp; 1.26
\end{array}$$

Of course, Co(II) complexes do not necessarily have the same electronic structure as their Fe(II) counterparts, so such data can only be of limited use. The key to overcoming the early difficulties (of Fe(II) oxidation) was the fact that such oxidations tend to proceed via dimerisation; therefore, if a sterically hindered Fe(II) complex could be made, the oxidation could be prevented. Based on this, Collman and coworkers synthesised so-called "picket fence" complexes,7 which reversibly bound dioxygen, the concept of which is shown below:



Access of other iron centres, as well as the imidazole ligand, to the top face of the metal ion was blocked by the bulky substituents on the porphyrin ring (in one example, $\ce{-C6H4NHC(O)^tBu}$ was chosen). The reported data from different "picket fences" strongly favoured the iron(III)–superoxide model: in particular, the O–O stretching frequencies ranged from $1163\text{–}\pu{1150 cm−1}$ (cf. $1200\text{–}\pu{1050 cm−1}$ in superoxide complexes, $\pu{1556 cm-1}$ in $\ce{O2}(^3\Sigma_\mathrm{g}^-)$, and $\pu{1484 cm-1}$ in $\ce{O2}(^1\Delta_\mathrm{g})$).8



A controversy over magnetic data

It was realised that a key difference between the two schemes was in their predicted magnetic susceptibilities. For the $\ce{Fe(II)-O2}$ binding scheme, one would expect the compound to be diamagnetic at all accessible temperatures (to obtain a magnetic state would require electronic promotion). On the other hand, the $\ce{Fe(III)-O2-}$ binding scheme would be expected to have a temperature-dependent magnetic susceptibility, as antiferromagnetic coupling could be overcome with higher temperatures. In other words, there is a thermally accessible triplet state.

In 1977, Cerdonio et al. published a paper claiming to have observed a temperature-dependent magnetic susceptibility for $\ce{HbO2}$.9 However, their findings were refuted in a later 1984 paper by Savicki et al., which found that $\ce{HbO2}$ was in fact diamagnetic at all temperatures.10 The incorrect magnetic data was attributed to residual deoxyhemoglobin, which is paramagnetic, and in a private communication with Savicki, Cerdonio agreed that his earlier findings were in error.



Spectroscopic data on the proteins

Using infrared spectroscopy, the O–O stretching frequencies in both oxymyoglobin11 and oxyhemoglobin12 were identified as $\pu{1103 cm−1}$ and $\pu{1107 cm-1}$ respectively, again consistent with superoxide.

Furthermore, $\ce{^57Fe}$ Mössbauer spectra of $\ce{HbO2}$ showed large quadrupole splittings, which is consistent with iron being in the +3 oxidation state.13



Computational studies

Early quantum mechanical calculations by Karplus14 showed that the properties of $\ce{HbO2}$ could be explained by a mixture of two $\ce{Fe(II)-O2}$ resonance forms: one with both bonding partners in the singlet state, and one with both in a triplet state. The latter was actually put forth as a separate - third - model, most notably by Goddard in 1975.15

Of course, these are fairly primitive by today's standards. In 2008, Shaik et al. revisited the topic,16 and provided an overview of the numerous computational studies performed up until then, which had failed to reach a conclusion. Instead of trying to discern the appropriate bonding model from indirect parameters, such as the net charge on oxygen or the form of the occupied MOs, the authors opted to transform a CASSCF wave function into a generalised valence bond wave function. They concluded that the Weiss $\ce{Fe(III)-O2-}$ model was dominant for a gas-phase oxygen–heme complex, and that the interaction with the protein further reinforces the $\ce{Fe(III)-O2-}$ character of the wave function.



More recent experimental work

In an entirely separate endeavour, Gray and coworkers found that polyhalogenated iron porphyrins, which possessed unusually high Fe(III)/Fe(II) reduction potentials, did not bind to oxygen.17 This has been taken as evidence for the necessity of electron transfer from Fe(II) to dioxygen in $\ce{Hb-O2}$ binding, and hence the identification of the bonding as being $\ce{Fe(III)-O2-}$ charge-transfer in nature.

X-ray absorption spectra by Wilson et al. further complicate the matter, though. Firstly, their investigation of picket fence porphyrins led to the conclusion that these generally showed characteristics of both models, but were more in line with an Fe(II) description. The effective nuclear charge on iron was found to be closer to that of Fe(II) than to that of Fe(III), and the spectra lacked a low-energy transition characteristic of Fe(III) complexes (arising from the hole in the $\mathrm{t_{2g}}$ orbitals on iron).18

A further study on the protein itself revealed that crystalline $\ce{HbO2}$ mainly has $\ce{Fe(II)-O2}$ character, whereas solution $\ce{HbO2}$ has $\ce{Fe(III)-O2-}$ character.19 They therefore concluded that the bonding was multiconfigurational in nature, and that "secondary changes in the protein environment, crystal packing, and so forth, can tip the electronic structure in either direction".



Notes and references

1. Bren, K. L.; Eisenberg, R.; Gray, H. B. Discovery of the magnetic behavior of hemoglobin: A beginning of bioinorganic chemistry. Proc. Natl. Acad. Sci. 2015, 112 (43), 13123–13127. DOI: 10.1073/pnas.1515704112.

2. Pauling, L.; Coryell, C. D. The Magnetic Properties and Structure of Hemoglobin, Oxyhemoglobin and Carbonmonoxyhemoglobin. Proc. Natl. Acad. Sci. 1936, 22 (4), 210–216. DOI: 10.1073/pnas.22.4.210.

3. Hemoglobin is famously known as an "honorary enzyme"; for an overview of its history, see e.g. Brunori, M. Hemoglobin is an honorary enzyme. Trends Biochem. Sci. 1999, 24 (4), 158–161. DOI: 10.1016/S0968-0004(99)01380-8.

4. Hoffman, B. M.; Diemente, D. L.; Basolo, F. Electron paramagnetic resonance studies of some cobalt(II) Schiff base compounds and their monomeric oxygen adducts. J. Am. Chem. Soc. 1970, 92 (1), 61–65. DOI: 10.1021/ja00704a010.

5. Getz, D.; Melamud, E.; Silver, B. L.; Dori, Z. Electronic structure of dioxygen in cobalt(II) oxygen carriers, singlet oxygen of dioxygen(1-). J. Am. Chem. Soc. 1975, 97 (13), 3846–3847. DOI: 10.1021/ja00846a064.

6. Gall, R. S.; Rogers, J. F.; Schaefer, W. P.; Christoph, G. G. The structure of a monomeric oxygen carrying cobalt complex: dioxygen-N,N'-(1,1,2,2-tetramethyl)ethylenebis(3-tert-butylsalicylideniminato)(1-benzylimidazole)cobalt(II). J. Am. Chem. Soc. 1976, 98 (17), 5135–5144. DOI: 10.1021/ja00433a015.

7. Collman, J. P.; Gagne, R. R.; Reed, C. A.; Halbert, T. R.; Lang, G.; Robinson, W. T. Picket fence porphyrins. Synthetic models for oxygen binding hemoproteins. J. Am. Chem. Soc. 1975, 97 (6), 1427–1439. DOI: 10.1021/ja00839a026.

8. Collman, J. P.; Brauman, J. I.; Halbert, T. R.; Suslick, K. S. Nature of O2 and CO binding to metalloporphyrins and heme proteins. Proc. Natl. Acad. Sci. 1976, 73 (10), 3333–3337. DOI: 10.1073/pnas.73.10.3333.

9. Cerdonio, M.; Congiu-Castellano, A.; Mogno, F.; Pispisa, B.; Romani, G. L.; Vitale, S. Magnetic properties of oxyhemoglobin. Proc. Natl. Acad. Sci. 1977, 74 (2), 398–400. DOI: 10.1073/pnas.74.2.398.

10. Savicki, J. P.; Lang, G.; Ikeda-Saito, M. Magnetic susceptibility of oxy- and carbonmonoxyhemoglobins. Proc. Natl. Acad. Sci. 1984, 81 (17), 5417–5419. DOI: 10.1073/pnas.81.17.5417.

11. Maxwell, J. C.; Volpe, J. A.; Barlow, C. H.; Caughey, W. S. Infrared evidence for the mode of binding of oxygen to iron of myoglobin from heart muscle. Biochem. Biophys. Res. Commun. 1974, 58 (1), 166–171. DOI: 10.1016/0006-291X(74)90906-1.

12. Barlow, C. H.; Maxwell, J. C.; Wallace, W. J.; Caughey, W. S. Elucidation of the mode of binding of oxygen to iron in oxyhemoglobin by infrared spectroscopy. Biochem. Biophys. Res. Commun. 1973, 55 (1), 91–95. DOI: 10.1016/S0006-291X(73)80063-4.

13. Lang, G.; Marshall, W. Mössbauer effect in some haemoglobin compounds. Proc. Phys. Soc. 1966, 87 (1), 3–34. DOI: 10.1088/0370-1328/87/1/302.

14. Case, D. A.; Huynh, B. H.; Karplus, M. Binding of oxygen and carbon monoxide to hemoglobin. An analysis of the ground and excited states. J. Am. Chem. Soc. 1979, 101 (16), 4433–4453. DOI: 10.1021/ja00510a001.

15. Goddard, W. A., III; Olafson, B. D. Ozone Model for Bonding of an O2 to Heme in Oxyhemoglobin. Proc. Natl. Acad. Sci. 1975, 72 (6), 2335–2339. DOI: http://10.1073/pnas.72.6.2335.

16. Chen, H.; Ikeda-Saito, M.; Shaik, S. Nature of the Fe−O2 Bonding in Oxy-Myoglobin: Effect of the Protein. J. Am. Chem. Soc. 2008, 130 (44), 14778–14790. DOI: 10.1021/ja805434m.

17. Grinstaff, M. W.; Hill, M. G.; Labinger, J. A.; Gray, H. B. Mechanism of catalytic oxygenation of alkanes by halogenated iron porphyrins. Science 1994, 264 (5163), 1311–1313. DOI: 10.1126/science.8191283.

18. Wilson, S. A.; Kroll, T.; Decreau, R. A.; Hocking, R. K.; Lundberg, M.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. Iron L-Edge X-ray Absorption Spectroscopy of Oxy-Picket Fence Porphyrin: Experimental Insight into Fe–O2 Bonding. J. Am. Chem. Soc. 2013, 135 (3), 1124–1136. DOI: 10.1021/ja3103583

19. Wilson, S. A.; Green, E.; Mathews, I. I.; Benfatto, M.; Hodgson, K. O.; Hedman, B.; Sarangi, R. X-ray absorption spectroscopic investigation of the electronic structure differences in solution and crystalline oxyhemoglobin. Proc. Natl. Acad. Sci. 2013, 110 (41), 16333–16338. DOI: 10.1073/pnas.1315734110.