Further Insight into the Reaction FeO+ + H2 â Fe+ + H2O

Article pubs.acs.org/JPCA

Further Insight into the Reaction FeO+ + H2 → Fe+ + H2O: Temperature Dependent Kinetics, Isotope Effects, and Statistical Modeling Shaun G. Ard,† Joshua J. Melko,†,# Oscar Martinez Jr.,† Vladimir G. Ushakov,‡,§ Anyang Li,∥ Ryan S. Johnson,∥ Nicholas S. Shuman,† Hua Guo,∥ Jürgen Troe,§,⊥ and Albert A. Viggiano*,† †

Air Force Research Laboratory, Space Vehicles Directorate, Kirtland AFB, Albuquerque, New Mexico 87117-5776, United States Institute of Problems of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia § Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg 11, D-37077 Göttingen, Germany ∥ Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States ⊥ Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany # Department of Chemistry, University of North Florida, Jacksonville, Florida 32224, United States ‡

S Supporting Information *

ABSTRACT: The reactions of FeO+ with H2, D2, and HD were studied in detail from 170 to 670 K by employing a variable temperature selected ion flow tube apparatus. High level electronic structure calculations were performed and compared to previous theoretical treatments. Statistical modeling of the temperature and isotope dependent rate constants was found to reproduce all data, suggesting the reaction could be well explained by efficient crossing from the sextet to quartet surface, with a rigid near thermoneutral barrier accounting for both the inefficiency and strong negative temperature dependence of the reactions over the measured range of thermal energies. The modeling equally well reproduced earlier guided ion beam results up to translational temperatures of about 4000 K.

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INTRODUCTION The reaction between iron oxide cation and molecular hydrogen1,2 6

small value of k is primarily due to a small spin-conversion probability, or to the barrier on the quartet surface. Likewise, the origin of the negative temperature coefficient has to be understood. It might arise from the complex-forming character of the reaction, where such a phenomenon is typical,11 from the decreasing spin-conversion probability with increasing energy,12 or from other effects. Since its early days,13 experimental studies have provided a wealth of data on this reaction.3,13−16 At the same time, considerable quantum-chemical effort has been invested to characterize the energy profile of the reaction and its intermediate structures.9,17−21 Though this has led to convincing insight into the pathway of this reaction and of many other bond activation reactions induced by transition metal oxide cations,3,8,16 the quantitative understanding of the rate constant is still in a rudimentary state. The most advanced approach by ref 19 applied nonadiabatic transition state theory in an effort to determine whether the barrier crossing or the spin conversion was truly rate-determining. In spite of being still in a simple form, this work was able to achieve reasonable

1

FeO+( Σ+) + H 2( Σ+) → Fe+(6D) + H 2O(1 A1) ΔHr0(0K) = − 151 ± 6 kJ mol−1 → Fe+(4 F) + H 2O(1 A1) ΔHr0(0K) = − 127 ± 6 kJ mol−1 → FeOH+ + H

ΔHr0(0K) = − 2 ± 16 kJ mol−1 (1.1)

is a simple example of bond activation by a transition metal oxide cation3−5 (for energetics, see below). It is highly exothermic and spin-allowed. Nevertheless, near room temperature, it has a small rate constant (k ∼ 10−11 cm3 s−1) with a negative temperature dependence (k ∝T−1). These intriguing properties attracted great attention6 and made it a prototype for the understanding of “two-state reactivity” (TSR).7−9 Although the sextet potential energy surface of reaction 1.1 was shown to have too high a barrier to explain the observed reaction,10 a spin conversion to a quartet surface with a much smaller barrier allows the system to circumvent the high sextet barrier and a second spin conversion may lead back to the sextet product shown in eq 1.1. The basic question then arises whether the © 2014 American Chemical Society

Received: June 5, 2014 Revised: July 31, 2014 Published: July 31, 2014 6789

dx.doi.org/10.1021/jp5055815 | J. Phys. Chem. A 2014, 118, 6789−6797

The Journal of Physical Chemistry A

Article

Ions are extracted and injected into a quadrupole mass filter where FeO+ is mass selected. The ions are then focused before introduction to a laminar flow tube via a Venturi inlet, where ∼104 to 105 collisions with a He buffer gas act to thermalize the ions and carry them downstream. Previous work in our laboratory has shown that some excited electronic states of Fe+ are inefficiently quenched27 and may be present in the flow tube. FeO+, however, is expected to be fully quenched due to its extra degrees of freedom. In either case, linear decays in parent ion signal on a semilogarithmic scale were observed over several orders of magnitude, indicative of a single rate constant. The temperature of the flow tube is variable over a large range either by resistive heating devices (300−700 K), pulsed liquid nitrogen (120−220 K), or recirculating methanol chillers (220−300 K). Operating pressures of 0.4 Torr of He are maintained in the flow tube throughout all experiments. The neutral reagent (H2, HD, or D2) is added 59 cm upstream of the end of the flow tube, with typical reaction times on the order of 4 ms, dependent on helium buffer flow (varied from 10 to 13 std. L min−1), and temperature. After traveling the length of the flow tube, the core of the flow is sampled through a truncated nosecone with a 2 mm aperture. The remainder of the flow is pumped away by a roots pump through a throttled gate valve that acts to maintain the desired pressure within the flow tube. After the nose cone, the primary ions and product ions are guided by a lens stack to a quadrupole mass filter for analysis and are subsequently detected using an electron multiplier operated in counting mode. Rate constants are derived by monitoring the decay of the primary ion as a function of the neutral reagent flow. Measurements were made from approximately 170 to 670 K and temperature dependences of the rate constants determined. Errors in the rate constants are estimated to be ±25% absolute and ±15% relative to each other.26

agreement with the room temperature rate constant, concluding the barrier crossing to be rate-determining. The aim of the present work is to expand upon this, employing a combination of advanced statistical modeling, electronic structure calculations, as well as a significantly broader set of experimental data. We are motivated to analyze reaction 1.1 after our successful modeling of the bond activation of methane22 by FeO+, FeO+ + CH4 → FeOH+ + CH3 → Fe+ + CH3OH

(1.2)

in which we applied the statistical adiabatic channel model (SACM)23 to the primary complex formation24 and general statistical rate theory25 to the subsequent intrinsic dynamics. It was important to account for the multidimensional character of the potential energy surface and, in particular, the conservation of total angular momentum during the reaction (i.e., to account for rotational effects). On the other hand, the details of spin conversion did not have a large effect because such processes were not rate-determining in the sequence of intrinsic steps. These features may not be universal such that it is of interest to test whether reaction 1.1 can also be described by carefully formulated statistical theory and to determine if crossing of the quartet barrier rather than spin conversion is also ratedetermining for this reaction. Although many details of the potential energy surface of the reaction are not yet known to chemical accuracy, it appears worthwhile to investigate these questions on a level similar to our work on reaction 1.2. On the experimental side, the present work extends the database by studying reaction 1.1 with H2, HD, and D2 over a wide temperature range, employing a variable temperature selected ion flow tube (VT-SIFT)26 apparatus. The comparison with guided ion beam (GIB)16 data over a large range of translational energies then provides a unique opportunity for testing the kinetic modeling. On the theoretical side, an extension of calculations of molecular parameters along the reaction path became necessary. Even with advanced quantumchemical methods barrier heights are notoriously difficult to ascertain (estimated uncertainties19 up to 8 kJ mol−1), which is particularly frustrating in this case because that range of uncertainty changes the sign of the barrier. Therefore, we treated the dominant barrier height as a fit parameter in the statistical calculations. In addition, as many further fine details of the potential energy surface are not yet available but play some role, we had to introduce a second effective fit parameter of the potential. The barrier height is most sensitive to the current data and the second parameter to the GIB data reported previously. The fact that we can reproduce the available experimental observables, obtained over wide ranges of temperature and energy, by using only the two fit parameters of the potential appears encouraging. After the successful kinetic analysis of reactions 1.1 and 1.2, one may hope that more complex-forming bond activation processes can be analyzed in an analogous way.

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THEORETICAL METHODS Calculation Techniques. Previous attempts10,14,17,19,20 to model the potential energy surfaces of this reaction have mostly been based on density functional theory (DFT) methods and, in particular, the B3LYP functional. This functional has been shown to provide a reasonably good picture for this reaction and other transition-metal containing systems. Our initial approach followed these early efforts, and in fact, the geometrical parameters used in the following statistical model were produced with the B3LYP functional as defined in the CBS-QB3 optimization procedure.28 In addition, a number of higher level methods were explored to account for electron correlation effects and multireference character. These methods include the complete active space self-consistent field (CASSCF),29,30 coupled cluster singles doubles and perturbative triples (CCSD(T)),31 complete active space second-order perturbation theory (CASPT2),32,33 and multireference configuration interaction (MRCI)34,35 approaches. The CASSCF, CASPT2, and MRCI calculations used the MOLPRO suite of electronic structure programs,36 and the other calculations used the Gaussian 09 package.37 Statistical Modeling. We base our kinetic modeling on the schematic diagram of the minimum energy paths (MEP) of the reaction in its sextet and quartet states such as shown in Figure 1. At the same time we employ the rotational constants and vibrational frequencies of the stationary points of the MEPs from the DFT calculations given in the previous section and summarized in the Supporting Information. We model only the

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EXPERIMENTAL METHODS All measurements were performed on the Air Force Research Laboratory’s variable temperature selected ion flow tube instrument, which has been described in detail elsewhere.26 Briefly, Fe+ ions are created using an electron impact source in the presence of a 10% mixture of Fe(CO)5 in He, with N2O additionally supplied in the source for the formation of FeO+. 6790



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WA and WB are numbers of rotational states of reactants and products convoluted with vibrations. For channel A (in this case a linear rotor + linear rotor), WA,rot(E,J) is calculated by phase space theory (PST), leading to WA,rot(E ,J ) =

∑

P(j1 ,j2 ,j ,L ,E ,J ) (1.6)

j1 , j2 , j , L

where j1 and j2 are the quantum numbers of rotational angular momenta of the reactant rotors, j corresponds to their combined angular momentum, L to the orbital angular momentum, J to the total angular momentum (we note that, because of the small reduced mass of the FeO+ + H2 system, J ≈ L would be a poor approximation; in that respect, the present reaction differs from FeO+ + CH4). P is equal to unity provided that j1 + j2 ≥ j, j2 + j ≥ j1, j + j1 ≥ j2, j + L ≥ J, L + J ≥ j, J + j ≥ L, and B1j1(j1+1) + B2j2(j2 + 1) + Ecent,A(L) < E (the Bi are the rotational constants of the reactants in energy units). Otherwise, P is equal to zero. The centrifugal energy Ecent,A(L) for the assumed charge-dipole + induced dipole potential is given by38

Figure 1. Qualitative potential surface for the reaction of FeO+ + H2 → Fe+ + H2O. The Fe atom is denoted orange (large), the O atom is red (medium), and the H atoms are white (small).

absolute rate constants ktot for product formation, leaving branching between 6Fe+ and 4Fe+ and between Fe+ and FeOH+ for later work, although no FeOH+ is observed in the present thermal measurements. Following the kinetic modeling of ref 22, the thermal rate constant ktot then is expressed by k tot =

1 hQ

∑ (2J + 1) ∫ W̃A J

Ecent(L) =

(1.3)

WB,rot(E ,J ) =

×

n

k tot ≈

kT hQ

∑ (2J + 1) ∫ W̃A e−E /kT J

dE kT

(1.8)

kT hQ

∑ (2J + 1) ∫ WTSe−E /kT J

dE kT

(1.9)

which is obtained from eq 1.1 for WA ≫ WTS, WB ≫ WTS, and WA ≈ W̃ A, corresponding to a bottleneck at TS. For the conditions of the present experiments, the differences were insignificant. We note, however, that eq 1.9 becomes invalid for T → 0 and, thus, does not approach the Langevin rate constant. Instead, for T → 0, eq 1.3 has to be used. We also note that differences between eqs 1.3 and 1.9 become significant at T > 1000 K where eq 1.3 has to be employed. The simplified eq 1.9 holds down to temperatures of about 50−100 K. Only at temperatures around 1 K the transition from a marked negative temperature coefficient of k tot (T) to the temperature independent Langevin value, kLangevin = 1.5 × 10−9 cm3s−1, takes place. The range of contributing J needs special considerations. The MEP of Figure 1 generally implies J = 0. With increasing J,

(1.4)

where υ = κ/μ, μ is the reduced mass of the reactant pair, and κ = (2μEtr)1/2, Qrovib is the product of the rovibrational partition functions of the reactants and n are their rovibrational states. With a thermal distribution of translational energies, Etr, eq 1.4 leads to eq 1.3. When T approaches zero, in the case of a negative energy TS (submerged barrier), one has WA ≪ WB, and WA ≪ WTS such that eq 1.1 approaches k tot(T → 0) =

∼ j,k ,l

0

J

WBWTS WTSWA + (WTS + WA )WB

∑ P(j ,k ̃,L ,E ,J )

with P equal to unity for j + L ≥ J, L + J ≥ j, J + j ≥ L, |k|̃ ≤ j, and Bl j(j+1) + (Al − Bl)k2̃ + Ecent,B(L) ≤ E (k̃ and j denote the rotational quantum numbers and Al and Bl, in energy units, denote the rotational constants of the symmetric top). We note that calculations for o- and p-H2 (or D2) were done separately but at 300 K differed only insignificantly. WTS(E,J) is the usual counted number of rovibrational states at the transition state TS (which was treated as a symmetric top; further refinements proved to be unnecessary). It differs from zero as long as E > E0,TS(J) where E0,TS(J) = E0,TS(J=0) + ETS,rot(J). E0,TS(J=0) is the energy of the submerged barrier of TS at 0 K and ETS,rot(J) is the rotational energy of TS. We calculated ktot(T) alternatively with eq 1.1 and with its simplified form of

∑ ∑ (2J + 1)e−E /kT n

(1.7)

for L(L + 1)ℏ ≥ 2μqμD and zero otherwise (μD denotes the dipole moment of the neutral). For channel B (in this case an atom + symmetric top), WB,rot(E,J) is approximated by

WTSWB e−E / kT dE WTSWA + (WTS + WA )WB

κ π ℏ2 μ κ 2Q rovib

8αμ2 q2 2

where Q stands for the product of the relevant electronic, translational, rotational, and vibrational partition functions of the reactants, the Wi are the energy (E) and total angular momentum (quantum number J) dependent “numbers of open channels” in SACM language23 (or “numbers of activated complex states”) at the positions i; A denotes the entrance to the complex from the primary reactants, B the exit toward the products, and TS the intermediate bottleneck. W̃ A stands for the primary formation of the [FeOH2]+ complex from FeO+ and H2 and WA for its back-dissociation, which may differ according to the experimental distributions of reactant states; however, the difference in the present case turned out not to be relevant. For completeness, we also present the rate constant ktot(Etr) as a function of the translational energy Etr relevant for GIB experiments (with rovibrational temperatures T0), k tot(Etr) = υ⟨σ ⟩ =

[L(L + 1)ℏ2 − 2μqμD ]2

(1.5)

1/2

which here is the Langevin rate constant 2πq(α/μ) ; see, e.g., ref 38 (q denotes the ionic charge and α is the polarizability of the neutral reactant). 6791



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and HD are shown in Table 1, along with the observed temperature dependences. The reactions are found to proceed

however, the energy of compact structures like TS increases much more strongly than the energy of the separated reactants (and products). In addition, the potential well of the intermediate complex [FeOH2]+ may become increasingly shallow and finally disappear. We have simulated these effects by limiting J by an upper limit Jmax(E). In practice, we have chosen Jmax(E) = min(Jmax,A,Jmax,TS). Here, Jmax,A corresponds to that J where WA(E,J) ≈ W̃ A becomes zero, and Jmax,TS is defined by BTSJmax,TS (Jmax,TS +1) = E − E0,TS(J =0)

Table 1. Room Temperature Rate Constant (k), Reaction Efficiency (k/kLangevin), and Temperature Dependence for the Reactions of FeO+ with H2, D2, and HDa reaction FeO+ Fe+ FeO+ Fe+

(1.10)

At small energy, Jmax,A corresponds to Ecent,A and is determined by PST as expressed by Jmax,PST(E). With increasing energy, the limiting centrifugal maximum moves into the range of [FeOH2]+ whose well finally disappears. Without further knowledge of the potential (i.e., the depth and shape of the intermediate well) the latter effect is difficult to quantify. In addition, WA(E,J) will be decreased by rigidity effects from the anisotropy of the potential; see, e.g., ref 39. We mimic these combined effects by employing Jmax, A = min[Jmax,PST (E),Jmax ]

+ + + +

H2 → H2O D2 → D2O

FeO+ + HD → Fe+ + HDO

k (300 K) (10−12 cm3 s−1) 10 5.1 8.7

k (lit. values) (10−12 cm3 s−1) 10 ± 4,b 8.8 ± 2.6c 6.9 ± 2.8,b 4.2 ± 1.3,c 2.5 ± 1d 8.1 ± 3.2,b 7.7 ± 2.3c

efficiency (k/ kLangevin)

T dep

0.0066

T−1±0.1

0.007

T−1±0.1

0.0047

T−1±0.1

a

Errors in the rate constants are estimated at 10% relative and 25% absolute. bICR derived rate from reference 16. cSIFT derived rate from reference 16. dGIB derived rate from reference 16.

very inefficiently, agreeing well with previously published room temperature rate constants obtained by ICR and SIFT apparatuses. The reaction of FeO+ with D2 was observed to react with a rate constant twice as high as that determined by the GIB experiment, just outside of the combined error limits of the two experiments.16 The authors of the GIB experiment, however, admit the lowest energy data points from which the room temperature rate constant is derived may have larger experimental error due to transmission inefficiencies at very low energies. The rate constants are found to vary strongly with temperature for each of the three reactions, scaling as ∼T−1. As was observed in the GIB experiment, the negative energy dependence of the rate constant signifies no barrier in excess of the reactant energies. The results from the present work are shown in Figure 2. Computational Results. In Figure 1, the schematic potential energy surfaces (PESs) for both the sextet and quartet states of the system are shown. To assess the large number of PESs found in the literature and produced in this work, a few benchmark values are chosen: the splitting between

(1.11)

where Jmax is a constant fit parameter. The sensitivity of the modeled ktot vs Jmax is an indication for the importance of the described effects. Two further aspects require additional discussion: the role of tunneling and the participation of electronic states. In our modeling as described below, we obtained fitted energies of the TS, which for the reactions of FeO+ with H2, HD, and D2 within the modeling uncertainties just differ by the differences in the TS zero-point energies. As the TS barrier along the MEP is relatively broad (imaginary frequency corresponds to about 1000 cm−1), tunneling could only play a role at relatively low temperatures. At the same time, there is, however, also reflection above the barrier, which partly compensates for tunneling. The remainder would form a minor contribution to the fitted TS energy. However, we found no evidence for such a contribution. We have done our final modeling under the assumption that the approaching reactants in the [FeOH2]+ adduct have sufficient time to switch from the sextet to the quartet potential such that the overall spin-conversion probability P is close to unity and that TS is quartet. When the potential well of [FeOH2]+ becomes shallow with increasing J, such that the lifetime of this complex becomes too short to allow for efficient spin conversion, P may fall below unity. We did not find evidence for such a behavior. However, such an effect may also be included in the fitted Jmax parameter. The conclusion that probabilities for spin conversion do not enter overall rates of sequences of processes is not new. In relation to reactions like 1.1 and 1.2 it was considered in refs 19 and 40−42. One may, in addition, also remember that spinforbidden thermal dissociation reactions, proceeding by sequences of pressure-proportional collisional activation and pressure-independent spin conversion, only at high pressures are governed by spin conversion; see, e.g., refs 12, 43, and 44. We finally note that the reaction 1.1 with HD instead of H2 has two transition states TS whose energy difference here was fixed at the difference of the TS zero-point energies.

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Figure 2. Rate constants (circles) and statistical modeling (full lines) for the reactions of FeO+ with H2, HD, and D2 (black, red, and green, respectively). The modeled values shown are for ETS = −0.33 kJ mol−1 and Jmax = 40 for H2, ETS = −0.48 and −2.03 kJ mol−1 and Jmax = 49 for HD, and ETS = −1.81 kJ mol−1 and Jmax = 40 for D2.

RESULTS AND DISCUSSION Experimental Results. The room temperature rate constants measured for the reactions of FeO+ with H2, D2, 6792



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Table 2. Various Calculated and Previously Reported Energies of Key Features of the PESs in kJ/mol method modeled values (this work) expt B3LYP CBS-QB3 CCSD(T)

CASSCF CASPT2 MRCI a b c a

Fe+ (6−4)

basis

FeO+ (6−4)

6

TS1 − 6INT1

ΔErxn

4

TS1

−0.33 TZVP na cc-PVDZ/SDD cc-PVTZ/SDD at above geometry cc-PVTZ aug-cc-pVTZ/cc-pVTZ for H aug-cc-pVTZ/cc-pVTZ for H AVTZ B3LYP (BP86) DMC CCSD(T)// B3LYP

24.1d 28.0 −2.9 88.8 88.8 49.2 105.2 95.5 99.4 −10.6 (−18.3) 30.9 28.0

47.3e 61.8 24.1 −41.5 52.1 47.3

41.5 33.8 (53.1) 52.1 53.1

102.3 103.2 110.0 104.2

14.5 5.8 34.7 22.2

−154f −159.2 −144.7 −194.9 −217.1

99.4 (117.7) 107.1 96.5

44.4 56.0 12.5 0.0 (0.0) 5.8 −5.8

−258.6 −158.2 −167.9 −161.1 −208.4 −159.2

Reference 18. bReference 20. cReference 19. dReference 47. eReference 48. fDerived from refs 1 and 2.

the ground state sextet and the first quartet of Fe+ and FeO+, the barrier height of 4TS1 relative to the sextet reactant asymptote, and the barrier height of 6TS1 relative to 6INT, which provides information about the performance of various electronic structure methods in predicting the barrier height in a single electronic state and the overall exothermicity of the reaction (ΔErxn) from sextet reactants to sextet products. These values are summarized in Table 2 for B3LYP, CBS-QB3, CCSD(T), CASSCF, CASPT2, and MRCI, as well as previously reported studies using B3LYP, diffusion Monte Carlo (DMC), and CCSD(T) single-point results at B3LYP geometries. CASSCF was performed with 13 active electrons in 11 orbitals; CASPT2 and MRCI were carried out using the CAS-configurations as the reference configuration state functions. This active space is similar to previous CASSCF/ MRCI studies on FeO+.45,46 No spin−orbit coupling was considered. Experimental splittings between the sextet and quartet states are available for Fe+ and FeO+, being 24.1 and 47.3 kJ/mol, respectively.47,48 B3LYP does a fair job of capturing these values and is comparable to CCSD(T)/cc-PVTZ and DMC. CBS-QB3 fails badly with the Fe+ atom, reversing the order of the sextet and quartet states. As far as overall exothermicity of the reaction is concerned, B3LYP, CBS-QB3, and CASPT2 all perform well, falling within 9.6 kJ/mol of the experimental value of 154 ± 6 kJ/mol.1,2 CASSCF and CCSD(T) fare the worst, overestimating the energy by at least 48 and as much as 96 kJ/mol. The success of the CASPT2 is not surprising because it maintains the property of size constancy necessary to evaluate the ΔErxn. The failure of CCSD(T) is likely due to the use of the SDD effective core potential of ref 49 on Fe and an unconverged basis set. However, full electron calculations were not attempted other than for Fe+ and FeO+, due to formidable computational costs. The key quantity investigated in this work is the relative barrier height of 4TS1, which shows considerable variation among methods. Importantly, due to the fact that the barrier is so close to the energy of the reactants, slight variation will cause the sign of the barrier energy to change, giving qualitatively different behavior at this rate-determining step. Of the methods surveyed here only one predicted a slightly negative barrier, as might be expected from the negative temperature dependence of the reaction rate. The CCSD(T) single-point calculation on B3LYP geometries of ref 19 finds this barrier to be −5.8 kJ/mol. The second lowest barrier, 0.0 kJ/mol, was found at the CASPT2D//NLSD level of theory.

DMC and CBS-QB3 gave the next lowest barriers at +5.8 kJ/ mol, followed closely by MRCI and B3LYP values at +12.5 and +14.5 kJ/mol, respectively. The highest barriers are found in CCDS(T), CASSCF, CASPT2, and MRCI calculations. The failure of CCSD(T) is again likely due to the use of the SDD potential for Fe, and there is evidence that the failure of CASPT2 and MRCI are due to a choice of an active space too small to accurately capture the correlation and the multireference character of the wave functions. Unfortunately, a larger active space is prohibitively costly, but we believe that this is the most appropriate methodology for an accurate description of the reaction path. Because a possible source of error in the height of 4TS1 is due to errors in the calculated splitting of the two spin states of FeO+, the absolute barrier height along the sextet potential energy surface can provide insight into the accuracy of the method independent of the splitting. In general, most methods find the height of 6TS1 to be around 96.5 kJ/mol above 6INT1, and the largest values correspond to methods that overestimate the 4TS1 barrier. The lowest value, of 96.5 kJ/ mol at CCSD(T)//B3LYP, also corresponds to the lowest barrier found for 4TS1. This methodology was also very accurate for the sextet-quartet splitting of Fe+ and FeO+ species. In Table 3 we have compared the geometry of 4TS1 obtained using CBS-QB3, CCSD(T), and CASSCF. In general, the Table 3. Bond Lengths (Å) and Angles (deg) of 4TS1 Calculated at Various Methods 4

a

TS1

RFe−O

RH−H′

RFe−H

RFe−H′

AO−Fe−H

AOFeH′

CBS-QB3 CCSD(T) CASSCF DFTa B3LYPb B3LYPc

1.603 1.656 1.759 1.63 1.62 1.612

0.916 0.922 0.944 1.01 0.93 0.916

1.703 1.760 1.925

1.665 1.704 1.825 1.65 1.689 1.687

52.2 48.6 40.9

83.7 79.4 69.9

Reference 10. bReference 18. cReference 21.

transition-state structure becomes progressively looser in that all bond lengths increase in this order. The CBS-QB3 and CCSD(T) results are more similar to each other than CASSCF, in which bond lengths can be as much as 0.2 Å longer. The CBS-QB3 values are closest to previously published geometries,18,19,21 albeit this is not to be unexpected considering these previous studies are also DFT based. 6793



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cause of the discrepancy between the GIB data and that of the SIFT and ICR measurements is purely speculative, we note that the fitted values of both the TS energy and the value of Jmax are essentially independent of this discrepancy. Furthermore, we note that the TS energy is primarily a function of the current thermal data, whereas the Jmax parameter is primarily determined by the higher energy range data (GIB), highlighting the complementary nature of the two experiments; the fitted curves of Figure 2 depend weakly on Jmax up to about 700 K. Figure 4 illustrates the sensitivity of the fit on the two

Despite a known problem in calculating the splitting of nearly degenerate spin states in Fe(II) compounds with sulfur ligands,50 B3LYP is known to perform quite well in determining bond dissociation energies of MO+ complexes with respect to other DFT functionals.51 It has also been found to provide reasonable geometries52 and is commonly used to provide starting geometries for more accurate single-point evaluations, as is the case in ref 19. In this spirit we have chosen to utilize the geometries, rotational constants and vibrational frequencies as found in the CBS-QB3 method, which is B3LYP with the reasonably large 6-311G basis with Fe, O, and H receiving 2d, d, and p polarization functions, respectively. The molecular parameters obtained and used in the statistical modeling are given in the Supporting Information. Statistical Modeling. The results of our modeling of the present thermal rate constants ktot(T) over the range 173−673 K, as well as the experimental points from the present work, are illustrated in Figure 2. The curve for H2 is fitted with E0,TS(J=0) = −0.33 kJ mol−1 and Jmax = 40, whereas TS energies of −0.48 and −2.03 kJ mol−1 and Jmax = 49 were obtained for HD, and −1.81 kJ mol−1 and Jmax = 40 for D2. Plots of ktot(Ttrans) for the FeO+ reaction with D2 are shown in Figure 3, including GIB

Figure 4. Model dependence on Jmax for the reactions of FeO+ with H2, HD, and D2 (experimental points and full lines for Jmax = 40, 49, 40 as in Figure 2; dashed lines for Jmax= ∞ and ETS= −0.07 kJ mol−1 for H2, −0.36 and −1.91 kJ mol−1 for HD, and −1.38 kJ mol−1 for D2).

parameters. Like for the FeO+ + CH4 reaction in ref 22, the restriction of the range of contributing J increasingly matters only with increasing Ttrans such that the GIB data could not be reproduced with Jmax = ∞ at all. Increasing Jmax for D2 beyond the fitted value of 40, analogous to the GIB data for FeO+ + CH4 from ref 22, would result in a more pronounced upturn of the modeled ktot(Ttrans) with increasing temperature. The increasing sensitivity of ktot(Ttrans) on Jmax at large Ttrans appears understandable because rotations then increasingly modify the effective potential in the [FeOH2]+ adduct range until the well of this complex ceases to exist. The fitted TS energies from Figures 2−4 are all near zero, but slightly negative. Within the estimated uncertainty of 8 kJ mol−1 of present quantum chemical calculations, they agree with the quantum chemical values given above and particularly well with the result from ref 19. As ktot(T) sensitively depends on the TS energies and the dependence on Jmax for the temperature range of the present work is only weak (Figure 4), the fitted experimental energies of the present work would appear more precise than the quantum chemical values. We note in addition that the isotope dependence of the fitted experimental energy values reproduces the results from the present quantum chemical calculations. This is concluded from the following comparison. Our modeling leads to fitted TS energies of −0.33 kJ mol−1 (for FeO+ + H2), −0.48 and −2.03 kJ mol−1 (for the two transition states of FeO+ + HD), and −1.81 kJ mol−1 (for FeO+ + D2). Employing the reactant and TS vibrational frequencies as given in the Supporting Information (and fixing the fitted value for FeO+ + H2), the vibrational zero-point energy-corrected TS energies from the

Figure 3. Experimental and modeled rate constants as a function of translational temperature (for both SIFT and GIB data, from 163 to 4000 K) for the reaction of FeO+ + D2.

data at translational temperature up to 4000 K.16 The present data (as well as the SIFT and ICR data from ref 16) differ by more than the combined error limits from the GIB data in the overlapping range. At energies above 300 K, the SIFT experiment has more rotational and vibrational energy available than the GIB experiment where reactant internal energies remain at 300 K, though this is unlikely to be the entire cause of the discrepancy, which may be due to a systematic error in one or both of the experiments. To test the robustness of the modeling, we have tried scaling both sets of data. Scaling the GIB data by a factor of 1.5, we see excellent agreement with predictions from the above model. Conversely, if we instead scale the present data down by a factor of 1.5, the model would fit the entire energy range well with a TS energy of −0.71 kJ/ mol and a Jmax of 34. Finally, if we instead fit to the entire data set ignoring the GIB data below 1000 K we find a reasonable fit to the data with a TS energy of −2.25 kJ/mol and a Jmax of 33. These fits are shown in Figure 3. As any reasoning as to the 6794



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calculations would be −0.74 and −2.30 for FeO+ + HD and −2.24 for FeO+ + D2. As the zero-point energy-corrected TS energies from the experiments and the quantum chemical calculations (anharmonicity and other uncertainties neglected) differ by no more than 0.43 kJ mol−1, we find our modeling as well reproduces the expected isotope effects as one would hope. We also observe that the temperature dependence of the experimental isotope effects is very well reproduced by the present modeling, Figures 2 and 4. We return our attention to the primary question still being debated about this reaction: is the reactivity governed by the transition state crossing or by the spin conversion? It has long been assumed that the low efficiency and negative temperature dependence of this reaction suggested the spin conversion was rate-determining. Reference 19 recently concluded the TS state to be the bottleneck, employing a semiclassical calculation of the spin conversion. As discussed above, the models shown here use the assumption that the TS crossing is rate limiting, and the spin conversion takes place faster than the TS crossing. Though the current data do not allow for enough constraint to vary the effective probability, P, of spin conversion as an additional fitting parameter, a limiting case of a spinconversion-limited reaction is quite illustrative. In this case P would equal 0.1, as the reaction approaches 10% of the collision rate at low temperatures. The best fit of the data employing a rate limiting 10% curve crossing probability is shown in Figure 5. The TS barrier then would have to be much lower (−14.8 kJ

dynamics. We also do not see conditions where spin-conversion probabilities enter the rate constant (although the spin conversion may play a role in high J reaction events and be hidden in the fitted Jmax values, see above). One has to emphasize the crucial importance of rotational effects in the kinetic modeling. In this respect the present work goes far beyond the statistical modeling given in ref 19 where rotational effects were not considered. Meanwhile, it was estimated in ref 19 that spin conversion within the [FeOH2]+ complex is about 2 orders of magnitude faster than the TS barrier crossing. This conclusion is in agreement with the present results. Likewise, the order of the isotope effect at 300 K calculated in ref 19 corresponds to that modeled here. However, this cannot be represented by conventional transition state theory, dealing with a single transition state only, which was not designed for as complicated reactions as the present system. One may finally ask what in the present case determines the temperature coefficient of ktot(T), which experimentally below 400 K was found to be about T−1. Although the Langevin equation based on eq 1.3 leads to a temperature independent rate constant, the situation looks different for the now relevant eq 1.7. Neglecting vibrational partition functions, the factor kT/hQ is proportional to T−5/2. Again neglecting vibration of TS (and neglecting the small energy difference between the entrance channel A and TS), the sum ∑J(2J + 1)WTS(E,J) is proportional to E3/2, which leads to ktot ∝ T−1. Other factors apparently only contribute to a smaller extent. Unlike other complex-forming bimolecular reactions based on eq 1.1, where the ratio WTS/(WA + WB) combined with “rotational channel switching” dominates the temperature dependence, here the energy dependence of WTS suffices to rationalize the temperature coefficient of ktot.

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CONCLUSIONS We have measured the rate constants for reactions of FeO+ with H2, HD, and D2 from 173 to 673 K for the first time. Statistical modeling of these data has allowed for several key insights into this fundamental reaction. Contrary to many earlier interpretations of this reaction, the crossing from the sextet potential surface to the quartet surface has enough time to occur in the 6FeOH2+ adduct, such that the quartet transition state is rate limiting. Nevertheless, it should be recognized that our work supports the general concept of the TSR mechanism that the reaction occurs because the transition to the quartet state allows the system to circumvent the high barrier on the initial sextet potential. We also determine quartet TS energies of −0.33 kJ mol−1 for the reaction with H2, −0.48 and −2.03 kJ mol−1 for the two HD transition states, and −1.81 kJ mol−1 for the D2 reaction. These independently modeled values are consistent with expectations based on zero-point energy differences, lending added credibility to the model. Attempts to model the reaction as limited by the spin conversion resulted in a much lower transition state barrier, but a temperature dependence that poorly represented the data. The calculations illustrate the difficulties in modeling these reactions. The combination of multiple near-degenerate electronic states, the multireference nature of the system (particularly at the transition states where bond breaking and formation are occurring) and electron correlation all present in this system prove to be challenging for current quantum chemical models. By building on the strength of DFT to provide reasonable geometries for a statistical model, we are able to probe these elusive energetic details. The integration of statistical modeling, quantum chemical calculations, and kinetic measurements over

Figure 5. Modeling of unadjusted SIFT (circles) and GIB (triangles) data for FeO+ + D2 (full line as in Figure 3 assuming the transition state crossing (TS) is rate-determining, dashed line assuming spin conversion with 10% efficiency (SC) is rate-determining, see text).

mol−1) to account for the magnitude of the rate constant, but this would also result in a steeper temperature dependence, which less well reflects the data. (For even lower P, the TS barrier would have to be even lower.) From this example, we would conclude (in agreement with ref 19) that the quartet TS crossing is indeed rate limiting for this reaction. However, we cannot exclude a minor contribution from a mechanism with rate-determining spin conversion. In summary, the present kinetic modeling based on detailed statistical rate theory is able to reproduce the available experiments surprisingly well. This allows one also to extrapolate the existing data over very wide ranges of temperatures. We find no evidence for nonstatistical intrinsic 6795



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a wide energy range, with significant emphasis on the near thermal range, has provided unique insight into the reaction mechanism for this important reaction, as well as provided theoretical benchmarks for the rate limiting transition state, otherwise inaccessible experimentally.

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ASSOCIATED CONTENT

S Supporting Information *

Parameters employed in the modeling including stationary point vibrational and rotational constants. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*A. A. Viggiano. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Discussions of this work with P. Armentrout, J. Harvey, and S. Shaik are gratefully acknowledged. R.S.J., H.G., and A.L. are supported by the Department of Energy (DE-FG0205ER15694). J.T. acknowledges support from the EOARD Grant Award FA8655-11-1-3077. N.S.S. and A.A.V. are grateful for the support of the Air Force Office of Scientific Research for this work under Project AFOSR-2303EP. O.M. acknowledges the support of the National Research Council. J.J.M. and S.G.A. acknowledge Boston College Institute of Scientific Research.

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NOTE ADDED IN PROOF Most recently another high level quantum chemical calculation has been reported53 leading to an energy of −1.6 kJ mol−1 for 4 TS1. This value is in very good agreement with the value derived from statistical modeling of the present experiments of −0.33 kJ mol−1.

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Further Insight into the Reaction FeO+ + H2 â Fe+ + H2O

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Further Insight into the Reaction FeO+ + H2 â Fe+ + H2O