Rebound mechanism

In this section, we present the results for the methane-to-methanol oxidation by the aqua iron(IV)oxo species, following the oxygen-rebound mechanism. This mechanism consists of two steps: first the hydrogen abstraction from methane by the iron(IV)oxo species, producing iron(III)hydroxo and a .CH$_3$ radical, and second, the collapse of the .CH$_3$ radical onto the iron(III)hydroxo oxygen, forming Fe(II)CH$_3$OH. Figure 8.2 shows the energy profile for these reaction steps along with the intermediate complex geometries. For this mechanism, we have also computed the zero-point energy corrections, the temperature dependent enthalpy corrections for a temperature of $T=300$ K and the entropy term $-T\Delta S$ (see table 8.3), the sum of which is added to the internal energy to give the free energy, indicated in figure 8.2 between parentheses.

Figure 8.2: Geometries and energy profile (in kcal/mol) of the intermediate steps along the oxygen-rebound mechanism of the methane-to-methanol oxidation by the pentaaqua iron(IV)oxo species. The energy of the separated reactants is set to zero. The free energies are indicated by the dotted levels and the numbers between parenthesis.

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Table 8.3: DFT energies (kcal/mol) for the intermediates in the oxygen-rebound mechanism of the methane-to-methanol oxidation by the pentaaqua iron(IV)oxo species (the proposed active species in the Fenton's reagent), compared to biochemical methane oxidations by MMO and P450. The lowest three rows show the zero-point energy correction, the temperature dependent corrections and the entropy term.

	Reactant	Trans.	Inter-	Free	Trans.	Product	Free
	Complex	State 1	mediate	Radic.	State 2	Complex	Products
Fenton$^a$	-2.1	1.3	-0.1	14.4	3.0	-46.6	-20.9
MMO$^b$	2.3	13.8	2.8		7.6	-47.7
MMO$^c$	-1.5	23.2	11.3	23.7	20.6	-41.8	-34.3
P450$^d$	-1.0	26.7	23.6		29.1	-36.9
$\Delta$ZPE$^a$	-0.8	-4.9	-3.3	-3.9	-3.5	1.8	0.1
$\Delta E^\mathrm{300K}_\mathrm{int}$$^a$	0.4	-4.4	-1.7	-1.8	-2.1	4.2	-1.3
$-T\Delta S$ $^a$	7.2	9.7	6.6	-3.4	7.8	4.6	1.7

$^a$ Our work. $^b$ P.E.M. Siegbahn, DFT-B3LYP + PCM, MMO compound Q model (HCOO)$_2$-(C$_3$N$_2$H$_4$)Fe($\mu$-O)$_2$($\eta^2$-HCOO)Fe(C$_3$N$_2$H$_4$)(HCOO)(H$_2$O), ref. Si01. $^c$ Basch et al, DFT-B3LYP, MMO compound Q model: cis-(H$_2$O)(NH$_2$)Fe($\mu$-O)$_2$($\eta^2$-HCOO)$_2$Fe(NH$_2$)(H$_2$O), ref. BaMuMoMo01. $^d$ Ogliaro et al, DFT-B3LYP, high-spin P450 compound I model: FeO(C$_{20}$N$_4$H$_{12}$)(SCH$_3$), ref. OgHaCoFiViSh.

Table 8.4: Relevant bond lengths (in Å) and angles as well as Mulliken charges and spin populations, for the reaction intermediates in the oxygen-rebound mechanism of the methane-to-methanol oxidation by the tetraaqua iron(IV)oxo species.

	Free	React.	Trans.	Inter-	Free	Trans.	Prod.	Free
	React.	Comp.	State 1	mediate	Radic.	State 2	Comp.	Prod.
$R$FeO	1.61	1.63	1.72	1.77	1.77	1.79	2.09
$R$OC		2.91	2.55	2.78		2.50	1.47
$R$OH		1.76	1.20	1.03	0.98	1.01	0.97	1.43
$R$CH	1.10	1.15	1.34	1.76		1.70	1.99	0.97
$\angle$HOC		1.	0.	1.		30.	108.	108.
$\angle$FeOH		179.	176.	155.	179.	151.	120.
$q$Fe	1.35	1.36	1.46	1.43	1.49	1.43	1.30	1.33
$q$O	-0.20	-0.30	-0.48	-0.36	-0.25	-0.37	-0.08	-0.09
$q$C	0.30	0.34	0.34	0.30	0.00	0.33	0.30	0.22
$q$CH$_3$	0.08	0.13	0.32	0.32	0.00	0.34	0.21	0.02
$q$H	-0.08	-0.11	0.04	-0.03	0.07	-0.02	0.05	0.07
$s$Fe	3.14	3.40	3.97	4.11	4.20	4.08	3.83	3.85
$s$O	0.69	0.59	0.38	0.40	0.52	0.35	0.04	0.00
$s$C	0.00	-0.10	-0.50	-0.75	1.08	-0.65	0.00	0.00
$s$H	0.00	-0.04	-0.03	-0.01	0.03	-0.01	0.00	0.00

Figure 8.3: Contour diagram of the potential energy surface of the bound .CH$_3$ radical in the neighborhood of the hydroxo ligand, which were both produced in the first step of the oxygen-rebound mechanism. Each mark (``+'') indicates the carbon position of the methyl radical at which the energy was obtained from a geometry optimization of the complex, constraining only the C-O distance and the H-O-C angle. Oxygen is located at the point (0,0), and R2=0 axis is chosen to lie on the O-H bond, as indicated. With .CH$_3$ initially positioned close to (0,3), it can follow a bent channel with a small barrier of 3.1 kcal/mol, towards the steep well at (1.4,-0.5) forming methanol bound to iron(II).

The interaction between the methane substrate and the iron(IV)oxo complex is again very weak, equal to $-$2 kcal/mol. The first reaction step, the transfer of a CH$_4$ hydrogen from methane to the oxo ligand, forming an Fe(III)OH complex with a bound .CH$_3$ radical, is surprisingly easy. The overall reaction energy is $+2.0$ kcal/mol, and the barrier is only 3.4 kcal/mol (1.3 kcal/mol with respect to the free reactants). Release of the .CH$_3$ group, producing a free radical, was calculated from the energy difference of the separate geometry optimized structures to cost 14.4 kcal/mol. In contrast to H-abstraction from an isolated CH$_4$ molecule which costs 103 kcal/mol (calculated at the DFT-BP+ZPE level of theory), the energy surface is very flat due to the simultaneous formation of the OH bond with the breaking of the CH bond. Also compared to the formation of the free .CH$_3$ radical from the reactant complex, which costs 16.5 kcal/mol, the reaction energy needed to go from Fe(IV)O$\cdots$HCH$_3$ to Fe(III)OH$\cdots$CH$_3$ is very small. This is obviously due to fairly strong (14.5 kcal/mol) bond between the .CH$_3$ radical and the Fe(III)OH complex. We have not analyzed this bond in detail, but we note that there is a significant charge transfer contribution from CH$_3$ to the iron complex concomitant with the H-abstraction (see also the substantial absolute increase of $q$O and $q$CH$_3$ as $R$OH decreases and $R$CH increases in table 8.4, whereas the charge on CH$_3$ becomes zero for the free radical formation).

For the second step in the oxygen-rebound mechanism, we find the formation of methanol bound to pentaaqua iron(II) to be very exothermic (46.6 kcal/mol), with again a very small barrier, equal to 3.1 kcal/mol. We did not succeed in finding this second transition state from geometry optimization along the unstable mode of the Hessian, due to convergence problems (which is a notorious problem of weakly bound radicals to transition metal complexes, see e.g. ref. Si01). However, from the potential energy surface shown in figure 8.3, we could determine the transition state structure to be the one shown in figure 8.2. This potential energy surface was computed from a geometry optimization for each point (indicated by the crosses) constraining only the C-O bond distance and the H-O-C angle. The oxygen of the hydroxo complex is situated at the coordinates (0,0) in the plot and the $R2=0$ axis is depicted to be along O-H. The energy of the Fe$^{\rm {III}}$-OH$\cdots$CH$_3$ intermediate, which is a local minimum on the energy surface, was set to zero in figure 8.2 The potential energy surface shows that there exists a narrow channel, through which the bound methyl radical can rotate around the hydroxo hydrogen to bind with the oxygen. The channel is very flat up to an H-O-C angle of about 60 degrees, with a small maximum at an angle of 30 degrees (transition state no.2), but then enters a very steep well associated with the irreversible formation of methanol. From the zero-Kelvin energy, this second step seems thus much more likely than formation of the free methyl radical. However, if we include zero-point energy effects, temperature corrections and entropy effects, we see that the free energy barriers are similar, namely 5.2 and 5.3 kcal/mol, respectively (see figure 8.2).

This rebound mechanism has also been theoretically predicted to be the reaction mechanism for the methane-to-methanol oxidation by the enzymes P450 and MMO (as aforementioned in the introduction). In these complexes, a Fe$^{\rm IV}$O species is ligated by either a porphyrin ring via four nitrogens (P450) or octahedrally by none-heme ligands via connecting oxygens (MMO), which makes especially the latter interesting to compare our results with. Literature values for the energetics are compiled along with our results in table 8.3. We see that also in these biological complexes the methane is initially very weakly bound in the reactant complex and that the overall formation of the product complex is very exothermic. The reaction barriers in between, however, are significantly higher for the oxidation by the enzymes than by the aqua ligated iron oxo species. For methane oxidation by P450, Ogliaro et al. found a barrier of 26.7 kcal/mol for the hydroxylation, and 29.1 kcal/mol for the rebound step, although the latter barrier vanishes if the system is allowed to cross to the low-spin surface[160]. Cytochrome P450 is known to be incapable of oxidizing methane, in contrast to MMO. Still, Basch et al find a barrier for H-abstraction by MMO which is almost as high as the one for P450, equal to 23.2 kcal/mol[186]. Siegbahn found a much lower barrier with MMO, namely 13.8 kcal/mol, which is still 11 kcal/mol higher than the barrier we find for the oxidation with aqua iron oxo[185]. The cited work on P450 and MMO was performed using the hybrid B3LYP functional, which is known to sometimes give better transition state energies in cases where these barriers are underestimated by GGA functionals, see for example ref. bernd2. However, a comparison of these two functionals for the hydroxylation of H$_2$ by a bare iron oxo species showed that there is good agreement on the reaction barriers among the B3LYP and BP86 functionals, although the overall reaction exothermicy was found to be 10 kcal/mol smaller by the BP86 functional compared to the B3LYP functional[168]. Since discrepancies between the functionals are typically amplified in such small bare atom reactions, we expect much smaller discrepancies between energetics calculated with B3LYP and BP86 for the present complexes.

The O-H and C-H distances in the first transition state (i.e. the one for the hydroxylation step) (see table 8.4) are very similar to those found for the biochemical hydroxylations. For MMO, Siegbahn found $R_{{\rm OH}}$=1.24 Å and $R_{{\rm CH}}$=1.30 Å, and Basch et al found $R_{{\rm OH}}$=1.21 Å and $R_{{\rm CH}}$=1.33 Å, which compares well with the ones in our transition state ($R_{{\rm OH}}$=1.20 Å and $R_{{\rm CH}}$=1.34 Å). For the P450 transition state, Ogliaro et al found $R_{{\rm OH}}$=1.09 Å and $R_{{\rm CH}}$=1.50 Å, which lies relatively closer to the Fe$^{\rm {III}}$-OH$\cdots$CH$_3$ intermediate state. Summarizing, we find that, apart from the higher barrier for hydroxylation in the MMO case, the rebound mechanism for methane-to-methanol oxidation by the pentaaqua iron(IV)oxo species is quite similar to the mechanism for the methane oxidation by MMO, calculated by Siegbahn. The hydrated ferryl ion in vacuo is found to be a highly reactive species, very well capable of hydroxylation and oxidation reactions with an organic substrate.