Transition path sampling

In this section, new reactive pathways for the O-O bond breaking upon coordination of an hydrogen peroxide molecule to an iron(II) ion in aqueous solution will be generated, using the transition path sampling technique. To this end, a point on an existing reaction pathway has to be chosen as the starting configuration of a new pathway. In the present sampling procedure, a configuration file with the atomic configurations and wavefunction coefficients was saved every 1000 time steps (145 femtoseconds). One of the saved configurations was randomly chosen as the starting point for a new pathway. The first new successful reaction path initiated from a point on our initial pathway, will be referred to as the first generation path; a successful reaction path initiated from a point on this first generation path makes than a second generation path, etcetera. A sequence of reactive pathways is a series of subsequent generations. Figure 7.4 illustrates the procedure for four pathways; the circles denote the chosen starting points where on each $n^{\rm th}$ generation path a new $(n+1)^{\rm th}$ generation path branches off. On a certain pathway, the time between its own starting point and the starting point of the next pathway, varied between 0.29 and 0.58 picoseconds, which is in a sense the time that the system is allowed to relax on a pathway before a next generation one branches off.

**Figure 7.4:** R $_{FeO}$ during the last four reaction pathways from sequence B, with their starting points denoted by the circles. That is, pathway no. 7 was generated by simulating backwards and forward in time starting from the configuration at t=3.303 ps of pathway no. 6. The configuration at pathway 7 at t=3.594 ps, was the starting configuration of pathway 8, etcetera. Note the shorter Fe $^{IV}$ =O distance in pathway no. 7 (at about t=4 ps) compared to the Fe $^{IV}$ -OH distance in the later pathways.)

We calculated two sequences (sequence A and sequence B), each with a length of 10 generations. For the first generation of each sequence we took one of the 20 trajectories calculated earlier to verify the transition state location on the initial reaction pathway, shown in the right-hand-side plots of figure 7.3. That is, the first generation pathways for both sequences A and B branch off at the transition state point of our initial pathway, with random atomic momenta drawn from a

K Boltzmann distribution. Of course, figure 7.3 shows only halves of reactive pathways (namely from the TS point to either the product state or the reactant state), so the other half was calculated, for our two first generations, by integrating the equations of motion backwards in time from the starting configuration of the reactive pathway (i.e. the TS point).

In table 7.2, which shows some characteristics of the computed pathways for sequence A (first 11 rows) and sequence B (last 9 rows), we see that the first generation pathways of sequences A and B are quite different from each other. For sequence A, the pathway does not show the direct mechanism of our initial pathway in which the ferryl ion (Fe $^{\rm {IV}}$ O $^{2+}$ ) is formed via a rebound of the leaving OH. radical, which abstracts the hydrogen of the intermediate Fe $^{\rm {III}}$ -OH. Instead, the leaving OH. radical jumps after a lifetime of $\tau_{\rm {OH\mbox{\.{}}}}=149$ fs via two solvent water molecules and terminates at a water ligand of a periodic image of the iron complex in a neighboring copy of the unit cell to form the dihydroxo iron(IV) moiety. This is indeed the first step of the two-step mechanism that we have seen before in simulations starting from hydrogen peroxide already coordinated to pentaaquairon(II).[171,146] In a second step (see last column in table 7.2), the ferryl ion was formed by donation of a proton by one of the hydroxo ligands to the solvent. For sequence B, the first generation pathway shows the rebound mechanism, although also an incipient OH. radical shunt via two solvent water molecules is observed, but this OH.-shunt is not completed, the motion of the solvent hydrogens is reversed when the Fe $^{\rm {III}}$ -OH intermediate donates its hydrogen to the leaving OH. radical.

From these two first generation pathways, we successively generated new pathways by taking a configuration from a path and changing the atomic momenta. The momenta were changed by randomly drawing new momenta from a gaussian distribution of

K or

K and adding these to the old momenta (and correcting for any total momentum of the system). The success rate of obtaining a new pathway connecting reactants and products was about 50%. In both sequences, accidentally, the fifth generation pathway was started from an unsuccessful fourth generation pathway that started in the product well and recrossed back to the product well, although the iron-oxygen distances reached a separation of more than 4 Å (which is our order parameter defining the reactant well) in both fourth generation pathways.

**Table:** Compilation of the characteristics of the reaction paths generated in sequences A and B. The O $^\beta$ H. radicals formed in the paths of the first generations have relatively long lifetimes $\tau_{\rm{OH\mbox{\.{}}}}$ . In sequence A already from the first generation no direct (*i. e.* rebound) mechanism is observed, but in the first generations, untill generation 7 (except for generation 3) the H abstraction takes place along a long wire to the next box, to a water ligand of the periodic ''copy'' complex to form dihydroxo iron(IV) (and ferryl ion in the second step). The O $^\beta$ H. radicals in later generation paths of sequence A live shorter and terminate via a short wire at an aqua ligand of the initial iron complex (``same''). In the second step the acidic dihydroxo iron(IV) species can transform into a trihydroxo species or a ferryl ion (last column). In the generations 1-7 of sequence B the O $^\beta$ H. radical abstracts the O $^\alpha$ H hydrogen to form the ferryl ion in one step (direct mechanism). Only in the later generations the transition paths relaxes to the abstraction through a H-bond wire, either a long wire (generations 8 and 9) or finally (generation 10) a short one.

generation	mechanism	$\tau_{\rm {OH\mbox{\.{}}}}$ / fs	# HO in	terminating	final obs.
			H-bond wire	Fe complex	species


Relaxation sequence A
1	long wire	149	2	copy	ferryl
2	long wire	380	2	copy	ferryl
3	short wire	322	0	same	ferryl
5	long wire	289	2	copy	Fe $^{\rm {IV}}$ (OH)
6	long wire	265	2	copy	Fe $^{\rm {IV}}$ (OH)
7	long wire	150	3	copy	ferryl
7	short wire	70	1	same	ferryl
8	short wire	73	1	same	Fe $^{\rm {IV}}$ (OH)
9	short wire	70	1	same	Fe $^{\rm {IV}}$ (OH)
9	short wire	65	1	same	Fe $^{\rm {IV}}$ (OH)
10	short wire	66	1	same	Fe $^{\rm {IV}}$ (OH)

table 7.2 continues on next page...

continuing table 7.2...

generation	mechanism	$\tau_{\rm {OH\mbox{\.{}}}}$ / fs	# HO in	terminating	final obs.
			H-bond wire	Fe complex	species


Relaxation sequence B
1	both	101	2	both	ferryl
2	direct	121	0	same	ferryl
3	direct	133	0	same	ferryl
5	direct	251	0	same	ferryl
6	direct	248	0	same	ferryl
7	direct	141	0	same	ferryl
8	long wire	43	2	copy	Fe $^{\rm {IV}}$ (OH)
9	long wire	42	2	copy	Fe $^{\rm {IV}}$ (OH)
10	short wire	93	1	same	Fe $^{\rm {IV}}$ (OH)

Accidentally, for both sequences the 4 $^{th}$ generation path was rejected because it recrossed from products to products, but was still used to generate a successful 5 $^{th}$ generation reaction path.

The O $^\beta$ H. radical grabs simultaneously H $^\alpha$ and initiates an (unsuccessful) shunt to a periodic copy of the iron complex. See also text.

Taking a closer look at table 7.2, we see trends along the two sequences which could indicate that indeed our initial pathway is an atypical one and relaxation towards more representative pathways does take place. For example, the time that the leaving OH. radical remains intact before abstracting a hydrogen from the complex or a solvent water (the lifetime $\tau_{\rm {OH\mbox{\.{}}}}$ , which is measured from the moment of O-O lysis, defined as $R_{\rm OO}>2.0$ Å, until the first H-abstraction by OH.), is seen to decrease in both sequences. Secondly, in both sequences the followed mechanisms change via or from the ``long-wire two-step'' mechanism (in which the OH. radical in the first step terminates via a wire of two or three solvent waters at a water ligand of the periodic image of the complex) to the ``short-wire two step'' mechanism. In the latter case, the leaving OH. radical terminates in the first step at an adjacent water ligand (thus stays in the same unit cell) via one bridging solvent molecule, forming the dihydroxo iron(IV) complex. Figure 7.5 shows in four snapshots the H

coordination to iron(II), the O-O lysis, and the OH. radical shunt and termination, of such a short-wire step. In fact, this new ``short-wire'' reaction pathway was already predicted in previous work where we discussed the possibilities for a radical shunt in a very large box containing a single pentaaqua iron(II) hydrogen peroxide complex.[171] In our present pathway relaxation procedure it indeed appears spontaneously.

**Figure 7.5:** Four snapshots of the 10 $^{th}$ generation reaction path of sequence A, which had a total length of 2.68 ps, showing the formation of the dihydroxo iron(IV) complex by H-abstraction from an HO ligand by the leaving OH. radical via one bridging solvent water molecule. The bridging solvent molecule which is hydrogen bonded to HO from t=0 ps is also shown, but for simplicity, the other solvent water molecules are left out.

The last column in table 7.2 displays the last observed iron complex. It shows that not always the ferryl ion is formed, but instead in many cases an [Fe $^{\rm {IV}}$ (H

(OH)

] $^{+}$ complex. This is due to the dynamic equilibrium between the acidic dihydroxo species and its conjugate base, the hydrolyzed trihydroxo species, by proton donation to the solvent: