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A theoretical study of the dynamics of the Al + H2O reaction in the gas-phase
Chemical Physics 382 (2011) 92–97
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A theoretical study of the dynamics of the Al + H2O reaction in the gas-phase Sonia Álvarez-Barcia, Jesús R. Flores ⇑ Departamento de Química Física, Universidad de Vigo, Facultad de Química, E-36310 Vigo (Pontevedra), Spain
a r t i c l e
i n f o
Article history: Received 28 September 2010 In final form 7 March 2011 Available online 14 March 2011 Keywords: Al hydroxides H2 generation Al reactivity AlOH2 HAlOH
a b s t r a c t The dynamics of the Al + H2O reaction in the gas phase has been studied by means of TST and RRKM theories, including the treatment of tunneling. The main reaction path involves isomerization into a HAlOH species, which dissociates into AlOH + H. There is a second mechanism which involves hydrogen atom elimination from a Al-OH2 complex, but it only becomes relevant at very high temperatures. Our theoretical results for the variation of the rate coefficient with temperature are coherent with the experimental findings [R.E. McClean, H.H. Nelson, M.L. Campbell, J. Phys. Chem. 97 (1993) 9673] in the sense that they can be fitted to a double exponential form. Our explanation for that behavior is the variation of the transmission coefficient with temperature. We find formation of AlO + H2 to be insignificant and the lifetime of Al-OH2 much larger than that of HAlOH. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The reaction of Al with water is of practical importance in areas as rocket propulsion or explosives [1–3]. The Al atom is found to interact with water in the gas phase to produce chemiluminiscence [1,2,4]; such an outcome is attributed by Oblath and Gole [1] to the formation of a HAlOH radical or to a more complex species. In their studies [1,4], the emission resulting from the bombardment of water vapor by Al atoms in their electronic ground state, under single collision conditions, was recorded. Jones and Brewster have employed the exploding wire technique, in connection with spectroscopic measurements, to study the Al + H2O combustion process; under the high energy conditions of this experiment the reaction seems to proceed to the point of generating Al2O3 particles [2]. McClean et al. have studied the kinetics of the title reaction in the gas phase over an ample temperature range (298–1174 K) by means of a laser-induced fluorescence (LIF) technique, which monitors the decay of Al atoms (H2O being in excess) at approximately 21 Torr of total pressure (which includes Ar as the carrier gas). At room temperature, they have not found any pressure dependence in the interval 10–100 Torr. Finally they have fitted their results to the following biexponential form k(T)(cm3 s1) = (1.9 ± 1.5) 1 0 1 2 e x p [ ( 0 . 8 8 ± 0 . 4 4 kcal/mol)/RT] + (1.6 ± 0.7)1010 exp[(5.7 ± 0.9 kcal/mol)/RT]. Based on the existing information about the possible intermediates and products, they have concluded that there should be two molecular mechanisms, one
⇑ Corresponding author. Tel.: +34 986812288; fax: +34 986812556. E-mail address: fl[email protected] (J.R. Flores). 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.03.004
Al + H2O ? HAlOH ? H + AlOH and the other would generate somehow AlO (probably Al + H2O ? HAlOH ? AlO + H2). It must be noted that the conclusion that interaction of Al and water generates HAlOH and AlOH has also been obtained from the co-condensation and matrix isolation studies of Hauge et al. [5]. Those experiments have involved co-condensation of Al atoms with water in an excess of Ar at 15 K (the infrared spectra were obtained immediately after deposition). The interpretation that a HAlOH adduct is formed has been confirmed by the condensation and EPR spectroscopic experiments of Joly et al. [6]. Further condensation experiments of Douglas et al. [7] have also suggested that HAlOH is formed and allowed for its electronic spectrum to be recorded. In a theoretical study of the electronic structure and spectra of the AlOH2 system [8], which has completed previous theoretical work [9–14], we have concluded that the Al atom and a water molecule form a weakly bound complex Al-OH2 (as had been pointed out much earlier by Kurtz and Jordan [9]) and also that elimination of H either directly or after isomerization into HAlOH or of H2 from HAlOH is hindered by very significant barriers (DE0 = 7.4, 3.8 and 7.4 kcal/mol vs. Al(2P1/2) + H2O(1A1), respectively). It must be noted that the Al–H2O complex has been studied by the ZEKE-PFI technique (zero electron kinetic energy plus pulsed field ionization), which would suggest that the title reaction should be extremely slow [15]. On the other hand, we have also showed through further theoretical studies that additional water molecules can have a catalytic effect on the title reaction, to the point that it becomes barrier-less with just one hydration water [16]. We therefore think that a theoretical dynamical study of the title reaction is very much needed in order to interpret the many experimental results and gain some additional insights.
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2. Theoretical approach: the basic aspects In previous studies of Al–water systems [8,16], we have employed reasonably accurate procedures for the computation of the potential energy surfaces. The procedure employed in Ref. [8] employs QCISD/6-311++G(2df,p) geometry optimizations, B3LYP/ 6-311++G(3df,2p) ZPE determinations, and obtains the electronic energies by adding core-correlation corrections, the effect of triple excitations and scalar relativistic corrections to RHF-UCCSD(1) energies (1 indicates that an extrapolation to the basis set limit of both the restricted Hartree–Fock and the UCCSD correlation energy has been performed). Spin–orbit corrections have also been added. We will name the results from Ref. [8] ‘‘benchmark’’ results. We have employed those data to construct an energy profile, which is shown in Fig. 1. It is readily seen that we have three major processes (a) Al + H2O ? Al-OH2 [TS1H] ? AlOH + H (b) Al + H2O ? Al-OH2 [TS12] ? HAlOH ? AlOH + H (c) Al + H2O ? Al-OH2 [TS12] ? HAlOH [TS2_H2] ? AlO + H2 It should be pointed out that we have found no saddle point linking Al-OH2 to AlO + H2. Obviously energetic considerations should favor process (b); it must also be noted that, although we could optimize a saddle point for the HAlOH ? AlOH + H process, it turns out it is a very loose one [8] (very low imaginary frequency of 33i cm1 at the B3LYP/6-311++G(2d,2p) level), while TS2_H2 (as well as TS12 and TS1H) is a tight one. In other words the HAlOH ? AlOH + H process should also be dynamically favored with respect to HAlOH [TS2_H2] ? AlOH + H. Moreover, when AlOH + H are generated they can react to generate Al-OH2 (via TS1H), HAlOH or AlO + H2 through TS_AlOH_H, i.e. (a) AlOH + H [TS1H] ? Al-OH2 (b) AlOH + H ? HAlOH (c) AlOH + H [TS_AlOH_H] ? AlO + H2
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Note that in all processes a van der Waals complex AlOHH could be an intermediate to AlOH + H generation. We have employed the following strategy. First we have used the Gaussrate program developed by Truhlar’s group [17], which performs direct dynamics computations using transition state theory (TST) employing the methods programmed in the Polyrate code [18], in order to study processes (a) and (b). We have anticipated that reactions (a) and (b) should be by far the most important and their behavior could explain the LIF experiments of McClean et al. [3] (which monitor the decrease of the Al concentration). One of the reasons for using Gaussrate/Polyrate (instead of semi-classical trajectory methods for instance) is that tunneling should be a crucial effect in this case so one needs a highly developed code in such respect. On the other hand, the potential energy surface shown schematically in Fig. 1 is already quite complex, so one would want to examine all possible processes together. For such purpose we have employed a simpler TST in which we determine energy-dependent rate coefficients for each process ki ðEÞ ¼ N – i ðE E0;i Þ=hqi ðEÞ and the (E-dependent) kinetic equations are integrated over E, using the energy distribution f(E) which results of the Al + H2O ? Al-OH2 process [19]. In other words, we examine the evolution of Al-OH2 complex, which has gained its activation energy in the Al–H2O collision, in the absence of further collisions. For such purpose we have employed the methods employed in the Multiwell program [20,21] and its utilities.
3. The study of processes (a) and (b) through TST As we have mentioned before, such processes should be the most relevant for the explanation of the LIF kinetics [3]. For process (a) we have defined a reaction scheme with a well in the reactant’s region, namely (Al-OH2); the saddle point (TS1H) connects directly to the products. For process (b) two wells are defined, one is Al-OH2, and the other HAlOH, which connects to AlOH + H. We have computed a minimum energy path (MEP) at the B3LYP/6311++G(2d,2p) level, that represents a reasonable compromise between accuracy and computational simplicity. For some compu-
Fig. 1. Schematic energy profile. The relative electronic energies are the values computed by the methods of Ref. [8] and are given in kcal/mol.
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tations, the energies of the stationary points are corrected through the ISPE approach (interpolated single-point energies) [22,18]; this is a dual-dynamics approach, where the higher-level of electronic structure method has been described briefly in the former section and completely in Ref. [8]. Given the complicated nature of that higher-level electronic structure method, and the fact that the B3LYP/6-311++G(2d,2p) geometries and vibrational frequencies are quite similar to those computed with the QCISD/6311++G(2df,p) or B3LYP/6-311++G(3df,2p) methods, we have decided to use the simple ISPE technique which improves only the energies. We have employed three types of TST, namely conventional TST, where the transition state is located at the saddle point, canonical variational TST (CVT), (where its position is varied for each temperature, being common for the whole canonical ensemble), and microcanonical variational TST (lVT), where its position depends on the energy [23]. Given that processes (a) and (b) depend on tight saddle points, one should not expect dramatic differences between the three methods. In order to deal with tunneling we have employed the zero-curvature tunneling method, mostly for comparison purposes, as well as the small-curvature (SCT) and large curvature (LCT) tunneling methods [24–26]. We have also employed optimized tunneling methods, namely the canonical optimized multidimensional tunneling method (COMT) [27] where the transmission coefficient is assigned the largest of the SCT and LCT (thermally averaged) values for each temperature, and the lOMT method [26], which is a microcanonical version, in which the transmission factor is computed by taking as tunneling probability the largest between the corresponding LCT and SCT values (the closest of the two to the value one would get by minimizing the action integral). In this particular case however, SCT appears to deliver the largest values so the OMT results are in practice coincident with SCT results. Some important details of the computation are the following. The Al atom has six electronic states Al(2P1/2, 2P3/2). Without spin–orbit coupling we have three terms at short Al–O distances, which for (non-planar) Cs geometries are named X2A00 , 12A0 and 22A0 [8]. The third term is repulsive, while the other two are attractive (see Ref. [8]), 2A00 is actually the electronic ground state of AlOH2. Although 12A0 is attractive, the lowest excited electronic state at the geometries of TS12 and TS1H lies at 27.2 kcal/mol and 14.6 kcal/mol, respectively, at the CASSCF/6-311++G(d,p) level [28,29]. The six electronic states of Al are included in the electronic factor of the partition function of the reactants (while only the ground (doublet) term is considered in TS1H and TS12 due to the
sizeable energy gap with respect to the first excited term). Vibrational frequencies are scaled by a factor of 0.96. The results are presented in Tables 1 and 2 (detailed information including that of the reverse processes is given in Tables S1 and S2 of the Supplementary material; at the end of it we also present ISPE data). In the last columns of each table we also include the results computed with the uncorrected B3LYP/6-311++G(2d,2p) MEP (i.e. without ISPE correction) for the LCT and lOMT methods. It is readily seen that the variational determination of the transition state (models lVT and CVT vs. TST) has a rather modest impact in the reaction rates; the lVT and CVT versions deliver virtually coincident results, except for the highest temperatures. As we have said, it turns out that the large curvature results, CVT/LCT, are always lower than their SCT counterparts, so the CVT/COMT results are equal to the CVT/SCT values; moreover lOMT and COMT results are also coincident. However, as one would expect, tunneling has a big difference at the lowest temperatures. For process (b), the transmission coefficient is about 7 at 300 K, although it goes to just about 1.1 at 1200 K (lCVT/SCT). Process (b) (i.e. isomerization of Al-OH2 to HAlOH plus fragmentation to AlOH + H) is much faster than process (a) (i.e. fragmentation of Al-OH2 to AlOH + H), even at 1200 K there is a ratio of about 8 (lCVT/SCT), which is six times larger at 300 K. In the case of process (b), there is a remarkable difference between the dual-level and the B3LYP/6-311++G(2d,2p) results, especially at the lower temperatures. The main reason is that the barrier height (computed as DVMEP with respect to the reactants [18], MEP indicates the minimum energy path) is 1.43 kcal/mol lower at the B3LYP/6-311++G(2d,2p) level than at the ‘‘benchmark’’ level; such a difference has a very big impact at the lowest temperatures. The ‘‘benchmark’’ procedure of Ref. [8] is estimated to be in error by less than 0.5 kcal/mol. In order to further test the sensitivity of the values with respect to the barrier height, we have elevated the B3LYP/6-311++G(2d,2p) value by 0.8 kcal/mol and 1.8 kcal/mol, instead of 1.3 kcal/mol as we do in the ISPE procedure. The rate (kb) at 298 K (in cm3 s1) goes from 2.9 1013 (1.3 kcal/mol) to about twice, 5.8 1013 (0.8 kcal/mol) (note that the experimental result at 298 K (4.6 ± 0.8 1013 cm3 s1) [3] is somehow halfway) or half 1.4 1013 (1.8 kcal/mol) (all are lCVT/SCT values). The temperature dependence is the most interesting aspect. As we have pointed out, McClean et al. [3] find that the rate constants measured between 298 K and 1174 K can be described by an expression with two exponentials, namely k(T) = (1.9 ± 1.5) 1012 exp((0.88 ± 0.44)/RT) + (1.6 ± 0.7) 1010 exp((5.7 ± 0.9)/
Table 1 Rate coefficients (cm3 s1) for the Al + H2O [TS12] ? AlOH + H process (b), computed with several tunneling models with a B3LYP/6-311++G(2d,2p) reaction coordinate corrected with the interpolated single-point energies (ISPE) scheme. The last two columns are for pure non-ISPE values.a T(K)
200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 a
TST
1.98E15 1.22E14 4.21E14 1.06E13 2.17E13 3.90E13 6.38E13 1.41E12 2.62E12 4.35E12 6.68E12 9.66E12 1.34E11 1.78E11
CVT
1.85E15 1.15E14 4.01E14 1.01E13 2.08E13 3.74E13 6.13E13 1.35E12 2.51E12 4.16E12 6.37E12 9.18E12 1.27E11 1.68E11
CVT/ZCT
9.70E14 1.47E13 2.31E13 3.60E13 5.44E13 7.93E13 1.12E12 2.04E12 3.38E12 5.20E12 7.56E12 1.05E11 1.41E11 1.82E11
The complete set of results is provided in the Supplementary material.
CVT/SCT
1.57E13 2.09E13 2.98E13 4.32E13 6.24E13 8.82E13 1.22E12 2.16E12 3.51E12 5.35E12 7.73E12 1.07E11 1.43E11 1.85E11
lCVT/SCT 1.57E13 2.09E13 2.97E13 4.32E13 6.24E13 8.81E13 1.22E12 2.15E12 3.52E12 5.37E12 7.75E12 1.07E11 1.43E11 1.85E11
Non-ISPE LCG4-lCVT/LCT
lCVT/lOMT
6.041E13 8.263E13 1.119E12 1.134E12 1.537E12 2.045E12 2.655E12 3.394E12 5.235E12 7.655E12 1.065E11 1.420E11 1.828E11 2.296E11
8.35E13 1.04E12 1.34E12 1.75E12 2.26E12 2.87E12 3.62E12 5.47E12 7.91E12 1.09E11 1.45E11 1.86E11 2.33E11 2.83E11
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Table 2 Rate coefficients (cm3 s1) for the Al + H2O [TS1H] ? AlOH + H process (a), computed with several tunneling models with a B3LYP/6-311++G(2d,2p) reaction coordinate corrected with the interpolated single-point energies (ISPE) scheme. The last two columns are for pure non-ISPE values.a T(K)
200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 a
TST
2.22E20 1.18E18 1.74E17 1.24E16 5.55E16 1.84E15 4.92E15 2.28E14 7.21E14 1.79E13 3.75E13 6.98E13 1.19E12 1.88E12
CVT
1.96E20 1.06E18 1.58E17 1.13E16 5.10E16 1.69E15 4.53E15 2.10E14 6.62E14 1.64E13 3.43E13 6.34E13 1.07E12 1.69E12
CVT/ZCT
2.85E15 3.53E15 4.89E15 7.36E15 1.17E14 1.89E14 3.05E14 7.45E14 1.63E13 3.20E13 5.72E13 9.49E13 1.48E12 2.19E12
CVT/SCT
3.82E15 4.59E15 6.11E15 8.82E15 1.34E14 2.10E14 3.31E14 7.83E14 1.68E13 3.27E13 5.82E13 9.62E13 1.50E12 2.21E12
lCVT/SCT 3.88E15 4.64E15 6.16E15 8.88E15 1.35E14 2.11E14 3.33E14 7.89E14 1.70E13 3.31E13 5.90E13 9.80E13 1.53E12 2.27E12
Non-ISPE LCG4-lCVT/LCT
lCVT/lOMT
4.02E14 5.59E14 8.23E14 8.38E14 1.30E13 1.98E13 2.97E13 4.33E13 8.41E13 1.48E12 2.40E12 3.64E12 5.24E12 7.19E12
4.72E14 6.35E14 9.22E14 1.39E13 2.09E13 3.10E13 4.48E13 8.59E13 1.51E12 2.43E12 3.68E12 5.28E12 7.23E12 9.57E12
The complete set of results is provided in the Supplementary material.
RT); where k(T) is given in cm3 s1 and the energies are in kcal/ mol. The presence of these two exponentials (with two limiting activation energies) is, of course, suggestive of the existence two molecular mechanisms. McClean et al. [3] used the limited knowledge of the PES available at that time to suggest that those two channels could be process (b), which would account for the low activation energy, and a process producing AlO, probably through HAlOH (they could not possibly detect the AlO produced by the reaction because of the experimental conditions). In Fig. 2 we have represented Ln(ka + kb) as computed by the lCVT and lCVT/SCT methods (i.e. without and with tunneling, respectively). One finds that the lCVT/SCT curve displays what could be considered a biexponential behavior, but one immediately sees as well that the lCVT curve is almost linear. Moreover, we have isolated kb (recall kb ka except for the higher temperatures) in Fig. 3 and represented its values computed by the same methods; it is clear that without tunneling we almost have a straight line. Therefore the bi-exponential form is not due to molecular mechanisms (a) and (b) (i.e. saddle points TS1H and TS12) but mostly to the variation of the transmission coefficient with temperature. Our ka + kb values (lCVT/SCT) could be represented by k(T)(cm3 s1) = 5.3
1012 exp(1.74/RT) + 3.2 1010 exp(7.0/RT), within the 300– 1200 K (and by k(T) = 1.4 1012 exp(0.90/RT) + 1.8 1010 exp(5.6/RT) in the 200–1200 K range) where the exponential parameters are quite close to the upper bound of the experimental ones (5.7 ± 0.9 kcal/mol and 0.88 ± 0.44 kcal/mol) for the first fit and remarkably close for the second. Finally, we have studied the AlOH + H [TS_AlOH_H] ? AlO + H2 reaction at thermal equilibrium. Its rate could be meaningful to the LIF experiments if the products of processes (a) and (b), i.e. the reactants of that reaction, reach thermal equilibrium by collisions. The results are presented in Table S3 of the Supplementary material. It must be noted that the 298 K equilibrium constant is very low 1.0 107. The thermal rate constant at 298 K is very low as well 2.6 1025 cm3 s1 (lCVT/SCT).
Fig. 2. Representation of Ln(ka + kb) with respect to T1. ka + kb are measured in cm3 s1 and computed by the lCVT method (squares) or the lCVT/SCT method (circles) (the latter includes tunneling).
Fig. 3. Representation of Ln(kb) with respect to T1. kb are measured in cm3s1 and computed by the lCVT method (squares) or the lCVT/SCT method (circles) (the latter includes tunneling).
4. More complex molecular mechanisms The fact that we obtain a bi-exponential rate, which is very close to the experimental results without considering production of AlOð2 Rþ Þ þ H2 ð1 Rþ g Þ does not imply, of course, that it should not happen, so we have investigated the formation of
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AlOð2 Rþ Þ þ H2 ð1 Rþ g Þ more deeply. As we have already mentioned, we have found in our study of the PES no direct route linking AlOH2 to those products; HAlOH has to be formed and then eliminate molecular hydrogen through TS2_H2 (process (c) of Section 2). We also wanted to study other processes (d)–(f). For such purpose we have used the microcanonical expression – ki ðEÞ ¼ N – i ðE E0;i Þ=hqi ðEÞ, where N i ðE E0;i Þ is the number of quantum states above the barrier (E0,i) of process i and qi(E) is the density of states of the reactants, and integrated the E-dependent rate equations corresponding to two reaction networks shown below. We have used the Multiwell program and its utilities [20,21] to compute the number and densities of states and programs of ours to perform the integrations. It should be recalled that McClean et al. have not found any dependence of rate constant with pressures up to 110 Torr (at room temperature) [3]; which implies that collision energy transfer does not play an important role under the experimental conditions. In other words, the energy of the collision complex is normally conserved till the system breaks into the products or back into the reactants. We have employed two reaction networks. The first (Network I) is still quite simple; isomerization of HAlOH into the cis form and further molecular hydrogen elimination are taken into account.
In the second Network (II) the van der Waals complex AlOHH (noted as AlOH–H) is taken into account as well as some reverse processes:
The transition states of processes (1) and (8), which do not have saddle points, have been determined by CVTST at T = 300 K. For process (5) we have taken as the TS location the loose saddle point which appeared in the QCISD/6-311++G(2df,p) computations of Ref. [8]. For process (8) we have employed a QCISD(T)/aug-cc-pVTZ reaction coordinate corrected for BSSE (Basis Set Superposition Error) by the counterpoise method [30]. We have employed the relative energies computed by the methods of Ref. [8] in combination with B3LYP/6-311++G(2d,2p) vibrational frequencies and rotational constants for coherence with Section 3. Only one rotation has been considered as active in all intermediates and transition states (the ‘‘K’’ rotor), except for linear structures; for the reactants or products all rotations are considered active. Vibrations are treated in a quantum harmonic way, except for the Al–O stretching of Al-OH2 and the H–H stretching of AlOH–H; given that the dissociation energy is so small and that such stretching modes are closely associated to the dissociation reaction coordinate we have employed Multiwell’s anharmonicity model [20]. Tunneling is treated by a one-dimensional asymmetrical Eckart barrier [20]. The rate coefficients of processes (1) (direct and reverse) are related by microscopic reversibility; in other words by using k1 ðEÞqr ðEÞ ¼ k1 ðEÞqAl-OH2 ðEÞ we can compute k1(E). In both networks we have chosen not to include isomerization into a H2AlO
isomer, for preliminary computations have indicated it is not an important process and, in any case, it would only reduce the AlO + H2 formation rate, where what we want to find is an upper bound. The results are shown in the first two rows of Table 3. In the case of Network I we see that the rate of production of AlO + H2 is very small as compared to that of AlOH + H. The reason is that process (5) is so much faster than process (4). The cis–trans isomerization is extremely fast as compared to processes (5) or (4) and (2) (i.e. the reverse of process (2)). Given that both isomers are very close in energy and that HAlOH has a very large energy excess their proportions are roughly 50%. As we have already pointed out TS2_H2 is not only much higher in energy than AlOH + H but it is also a very tight transition state. As one would expect from the results of the former section, production of AlOH + H through process (5) is much faster than through process (6). This simple network ignores of course the production of AlO + H2 by reaction of the main products AlOH + H. The average lifetimes of Al-OH2 and HAlOH (cis and trans) are 5.9 and 0.07 ps. Note that, despite of its weak Al–O bond, the lifetime of Al-OH2 is in fact longer than that of HAlOH. The latter species is a deep minimum in the PES (see Fig. 1); the reasons for such a low lifetime are its very large excitation energy and the large rate of process (5). Note that this difference in the lifetimes of Al-OH2 and HAlOH, as well as the decay of the rate coefficient with diminishing temperature may explain why Duncan and coworkers obtain the spectrum of Al-OH2 in their ZEKE-PFI spectroscopic experiments [15]. In the second Network (II) we take into account explicitly the AlOHH van der Waals complex, which can be considered an intermediate to AlOH + H formation, but can also form HAlOH, Al-OH2 and finally AlO + H2, which is a mechanism additional to process (4) that may result in an increased rate of formation of those products. By using AlOHH as an intermediate, which can originate AlO + H2, we intend to compute an upper bound to AlO + H2 formation in the absence of collisions. As shown in Table 3 the overall rate is almost equal as in the first network, but the AlO + H2/ AlOH + H branching ratio increases to 3.1 105, which is still a small value. As a tool to assess the relative importance of the different process in a simple way we have computed an individual thermal average of the corresponding ki(E) by using the energy distribution p(E) obtained in the formation of Al-OH2 (the ‘‘chemical activation’’ distribution [20]); note that those values have nothing to do with the high-pressure limits, which assume a Boltzmann distribution. The results are in Table 3. It is readily seen that the rates of pro-
Table 3 Rate coefficients of Networks I and II. The unimolecular rates are averaged over the energy distribution obtained in process (1) and given in s1. The rate coefficients of the common processes of both networks are equal.
k(AlO + H2)a k(AlOH + H)a k(1) k(2) k(2) k(5) k(6) k(3) k(3) k(4) k(5) k(6) k(8) k(7) a
In cm3 s1.
Network II
Network I
1.1E17 3.5E13 5.7E+12 2.4E+09 2.1E+07 1.3E+14 2.8E+07 1.6E+13 1.4E+13 1.3E+05 1.2E+14 3.4E+05 2.2E+13 9.3E+07
9.0E21 3.5E13
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cesses (8) and (5) are similar in magnitude and much larger than that of process (7), which has a tight saddle point. The increase of the branching ratio comes from process (7) but also from flux to HAlOH. Perhaps we should stress again that, in the case of both networks, it is assumed that no collisions other than those between the two reactants or the two products take place. Ultimately, in a system at thermal equilibrium the AlO + H2/AlOH + H ratio is given by the equilibrium constant, which is 1.1 107 at 300 K, and still low (2.7 103) at 1200 K. Those results and the analysis of the reaction networks lead us to conclude that the products AlO + H2 are not likely to have a very significant branching ratio under any pressure conditions in an Al + H2O collision.
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Ministry of Education (Spain). The services provided by the ‘‘Centro de Supercomputación de Galicia’’ (CESGA) are also acknowledged. Appendix A. Supplementary data Tables S1–S3, which contain detailed results for the rates of the Al + H2O TS12 ? AlOH + H, Al + H2O TS1H ? AlOH + H and AlOH + H TS_AlOH_H ? AlO + H2 processes. Tables S4–S6 contain the ISPE corrections and S7 contains ZPE energies and details of Fig. 1. Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemphys. 2011.03.004. References
5. Conclusions We have performed a theoretical study of the dynamics of the Al + H2O reaction by means of TST and RRKM theories including tunneling, with sophisticated models in the first case. We have found that the main reaction path involves isomerization into a HAlOH species, which dissociates into AlOH + H. A second mechanism involving hydrogen atom elimination from a Al-OH2 adduct only becomes important at very high (>1000 K) temperatures. We have found no direct route in the PES to produce AlO + H2. The results are coherent with the experiment in the sense that the variation of the total rate with temperature can be fitted to a bi-exponential form and that the values are close (for instance at 298 K the experimental value is 4.6(±0.8) 1013 cm3 s1 and the theoretical one is 2.9 1013 cm3 s1). Our interpretation of the apparent bi-exponential behavior is not the existence of two molecular mechanisms, but the change of the transmission coefficient with temperature. The moderate discrepancy between the experimental and the theoretical values can be attributed to the models employed in our calculations. We have studied the possible generation of AlO + H2 by a secondary process AlOH + H ? AlO + H2 by RRKM theory. We have assumed that the energy is maintained in every collision up to the generation of the final product and proposed two reaction networks; we find that the AlO + H2/AlOH + H has to be very small, un upper bound could be 3.1 105 (at 300 K). On the other hand, the equilibrium constant of the AlOH + H ? AlO + H2 reaction is about 1.1 107 at 300 K (2.7 103 at 1200 K), which suggests that even if collisions with other particles in the system take place, formation of AlO + H2 would not be significant. Acknowledgements We acknowledge the financial support of the Xunta de Galicia through the project INCITE09314252PR and the program INCITE08ENA314104ES. S.A.B acknowledges a FPU grant from the
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A theoretical study of the dynamics of the Al + H2O reaction in the gas-phase Sonia Álvarez-Barcia, Jesús R. Flores ⇑ Departamento de Química Física, Universidad de Vigo, Facultad de Química, E-36310 Vigo (Pontevedra), Spain
a r t i c l e
i n f o
Article history: Received 28 September 2010 In final form 7 March 2011 Available online 14 March 2011 Keywords: Al hydroxides H2 generation Al reactivity AlOH2 HAlOH
a b s t r a c t The dynamics of the Al + H2O reaction in the gas phase has been studied by means of TST and RRKM theories, including the treatment of tunneling. The main reaction path involves isomerization into a HAlOH species, which dissociates into AlOH + H. There is a second mechanism which involves hydrogen atom elimination from a Al-OH2 complex, but it only becomes relevant at very high temperatures. Our theoretical results for the variation of the rate coefficient with temperature are coherent with the experimental findings [R.E. McClean, H.H. Nelson, M.L. Campbell, J. Phys. Chem. 97 (1993) 9673] in the sense that they can be fitted to a double exponential form. Our explanation for that behavior is the variation of the transmission coefficient with temperature. We find formation of AlO + H2 to be insignificant and the lifetime of Al-OH2 much larger than that of HAlOH. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The reaction of Al with water is of practical importance in areas as rocket propulsion or explosives [1–3]. The Al atom is found to interact with water in the gas phase to produce chemiluminiscence [1,2,4]; such an outcome is attributed by Oblath and Gole [1] to the formation of a HAlOH radical or to a more complex species. In their studies [1,4], the emission resulting from the bombardment of water vapor by Al atoms in their electronic ground state, under single collision conditions, was recorded. Jones and Brewster have employed the exploding wire technique, in connection with spectroscopic measurements, to study the Al + H2O combustion process; under the high energy conditions of this experiment the reaction seems to proceed to the point of generating Al2O3 particles [2]. McClean et al. have studied the kinetics of the title reaction in the gas phase over an ample temperature range (298–1174 K) by means of a laser-induced fluorescence (LIF) technique, which monitors the decay of Al atoms (H2O being in excess) at approximately 21 Torr of total pressure (which includes Ar as the carrier gas). At room temperature, they have not found any pressure dependence in the interval 10–100 Torr. Finally they have fitted their results to the following biexponential form k(T)(cm3 s1) = (1.9 ± 1.5) 1 0 1 2 e x p [ ( 0 . 8 8 ± 0 . 4 4 kcal/mol)/RT] + (1.6 ± 0.7)1010 exp[(5.7 ± 0.9 kcal/mol)/RT]. Based on the existing information about the possible intermediates and products, they have concluded that there should be two molecular mechanisms, one
⇑ Corresponding author. Tel.: +34 986812288; fax: +34 986812556. E-mail address: fl[email protected] (J.R. Flores). 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.03.004
Al + H2O ? HAlOH ? H + AlOH and the other would generate somehow AlO (probably Al + H2O ? HAlOH ? AlO + H2). It must be noted that the conclusion that interaction of Al and water generates HAlOH and AlOH has also been obtained from the co-condensation and matrix isolation studies of Hauge et al. [5]. Those experiments have involved co-condensation of Al atoms with water in an excess of Ar at 15 K (the infrared spectra were obtained immediately after deposition). The interpretation that a HAlOH adduct is formed has been confirmed by the condensation and EPR spectroscopic experiments of Joly et al. [6]. Further condensation experiments of Douglas et al. [7] have also suggested that HAlOH is formed and allowed for its electronic spectrum to be recorded. In a theoretical study of the electronic structure and spectra of the AlOH2 system [8], which has completed previous theoretical work [9–14], we have concluded that the Al atom and a water molecule form a weakly bound complex Al-OH2 (as had been pointed out much earlier by Kurtz and Jordan [9]) and also that elimination of H either directly or after isomerization into HAlOH or of H2 from HAlOH is hindered by very significant barriers (DE0 = 7.4, 3.8 and 7.4 kcal/mol vs. Al(2P1/2) + H2O(1A1), respectively). It must be noted that the Al–H2O complex has been studied by the ZEKE-PFI technique (zero electron kinetic energy plus pulsed field ionization), which would suggest that the title reaction should be extremely slow [15]. On the other hand, we have also showed through further theoretical studies that additional water molecules can have a catalytic effect on the title reaction, to the point that it becomes barrier-less with just one hydration water [16]. We therefore think that a theoretical dynamical study of the title reaction is very much needed in order to interpret the many experimental results and gain some additional insights.
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2. Theoretical approach: the basic aspects In previous studies of Al–water systems [8,16], we have employed reasonably accurate procedures for the computation of the potential energy surfaces. The procedure employed in Ref. [8] employs QCISD/6-311++G(2df,p) geometry optimizations, B3LYP/ 6-311++G(3df,2p) ZPE determinations, and obtains the electronic energies by adding core-correlation corrections, the effect of triple excitations and scalar relativistic corrections to RHF-UCCSD(1) energies (1 indicates that an extrapolation to the basis set limit of both the restricted Hartree–Fock and the UCCSD correlation energy has been performed). Spin–orbit corrections have also been added. We will name the results from Ref. [8] ‘‘benchmark’’ results. We have employed those data to construct an energy profile, which is shown in Fig. 1. It is readily seen that we have three major processes (a) Al + H2O ? Al-OH2 [TS1H] ? AlOH + H (b) Al + H2O ? Al-OH2 [TS12] ? HAlOH ? AlOH + H (c) Al + H2O ? Al-OH2 [TS12] ? HAlOH [TS2_H2] ? AlO + H2 It should be pointed out that we have found no saddle point linking Al-OH2 to AlO + H2. Obviously energetic considerations should favor process (b); it must also be noted that, although we could optimize a saddle point for the HAlOH ? AlOH + H process, it turns out it is a very loose one [8] (very low imaginary frequency of 33i cm1 at the B3LYP/6-311++G(2d,2p) level), while TS2_H2 (as well as TS12 and TS1H) is a tight one. In other words the HAlOH ? AlOH + H process should also be dynamically favored with respect to HAlOH [TS2_H2] ? AlOH + H. Moreover, when AlOH + H are generated they can react to generate Al-OH2 (via TS1H), HAlOH or AlO + H2 through TS_AlOH_H, i.e. (a) AlOH + H [TS1H] ? Al-OH2 (b) AlOH + H ? HAlOH (c) AlOH + H [TS_AlOH_H] ? AlO + H2
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Note that in all processes a van der Waals complex AlOHH could be an intermediate to AlOH + H generation. We have employed the following strategy. First we have used the Gaussrate program developed by Truhlar’s group [17], which performs direct dynamics computations using transition state theory (TST) employing the methods programmed in the Polyrate code [18], in order to study processes (a) and (b). We have anticipated that reactions (a) and (b) should be by far the most important and their behavior could explain the LIF experiments of McClean et al. [3] (which monitor the decrease of the Al concentration). One of the reasons for using Gaussrate/Polyrate (instead of semi-classical trajectory methods for instance) is that tunneling should be a crucial effect in this case so one needs a highly developed code in such respect. On the other hand, the potential energy surface shown schematically in Fig. 1 is already quite complex, so one would want to examine all possible processes together. For such purpose we have employed a simpler TST in which we determine energy-dependent rate coefficients for each process ki ðEÞ ¼ N – i ðE E0;i Þ=hqi ðEÞ and the (E-dependent) kinetic equations are integrated over E, using the energy distribution f(E) which results of the Al + H2O ? Al-OH2 process [19]. In other words, we examine the evolution of Al-OH2 complex, which has gained its activation energy in the Al–H2O collision, in the absence of further collisions. For such purpose we have employed the methods employed in the Multiwell program [20,21] and its utilities.
3. The study of processes (a) and (b) through TST As we have mentioned before, such processes should be the most relevant for the explanation of the LIF kinetics [3]. For process (a) we have defined a reaction scheme with a well in the reactant’s region, namely (Al-OH2); the saddle point (TS1H) connects directly to the products. For process (b) two wells are defined, one is Al-OH2, and the other HAlOH, which connects to AlOH + H. We have computed a minimum energy path (MEP) at the B3LYP/6311++G(2d,2p) level, that represents a reasonable compromise between accuracy and computational simplicity. For some compu-
Fig. 1. Schematic energy profile. The relative electronic energies are the values computed by the methods of Ref. [8] and are given in kcal/mol.
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tations, the energies of the stationary points are corrected through the ISPE approach (interpolated single-point energies) [22,18]; this is a dual-dynamics approach, where the higher-level of electronic structure method has been described briefly in the former section and completely in Ref. [8]. Given the complicated nature of that higher-level electronic structure method, and the fact that the B3LYP/6-311++G(2d,2p) geometries and vibrational frequencies are quite similar to those computed with the QCISD/6311++G(2df,p) or B3LYP/6-311++G(3df,2p) methods, we have decided to use the simple ISPE technique which improves only the energies. We have employed three types of TST, namely conventional TST, where the transition state is located at the saddle point, canonical variational TST (CVT), (where its position is varied for each temperature, being common for the whole canonical ensemble), and microcanonical variational TST (lVT), where its position depends on the energy [23]. Given that processes (a) and (b) depend on tight saddle points, one should not expect dramatic differences between the three methods. In order to deal with tunneling we have employed the zero-curvature tunneling method, mostly for comparison purposes, as well as the small-curvature (SCT) and large curvature (LCT) tunneling methods [24–26]. We have also employed optimized tunneling methods, namely the canonical optimized multidimensional tunneling method (COMT) [27] where the transmission coefficient is assigned the largest of the SCT and LCT (thermally averaged) values for each temperature, and the lOMT method [26], which is a microcanonical version, in which the transmission factor is computed by taking as tunneling probability the largest between the corresponding LCT and SCT values (the closest of the two to the value one would get by minimizing the action integral). In this particular case however, SCT appears to deliver the largest values so the OMT results are in practice coincident with SCT results. Some important details of the computation are the following. The Al atom has six electronic states Al(2P1/2, 2P3/2). Without spin–orbit coupling we have three terms at short Al–O distances, which for (non-planar) Cs geometries are named X2A00 , 12A0 and 22A0 [8]. The third term is repulsive, while the other two are attractive (see Ref. [8]), 2A00 is actually the electronic ground state of AlOH2. Although 12A0 is attractive, the lowest excited electronic state at the geometries of TS12 and TS1H lies at 27.2 kcal/mol and 14.6 kcal/mol, respectively, at the CASSCF/6-311++G(d,p) level [28,29]. The six electronic states of Al are included in the electronic factor of the partition function of the reactants (while only the ground (doublet) term is considered in TS1H and TS12 due to the
sizeable energy gap with respect to the first excited term). Vibrational frequencies are scaled by a factor of 0.96. The results are presented in Tables 1 and 2 (detailed information including that of the reverse processes is given in Tables S1 and S2 of the Supplementary material; at the end of it we also present ISPE data). In the last columns of each table we also include the results computed with the uncorrected B3LYP/6-311++G(2d,2p) MEP (i.e. without ISPE correction) for the LCT and lOMT methods. It is readily seen that the variational determination of the transition state (models lVT and CVT vs. TST) has a rather modest impact in the reaction rates; the lVT and CVT versions deliver virtually coincident results, except for the highest temperatures. As we have said, it turns out that the large curvature results, CVT/LCT, are always lower than their SCT counterparts, so the CVT/COMT results are equal to the CVT/SCT values; moreover lOMT and COMT results are also coincident. However, as one would expect, tunneling has a big difference at the lowest temperatures. For process (b), the transmission coefficient is about 7 at 300 K, although it goes to just about 1.1 at 1200 K (lCVT/SCT). Process (b) (i.e. isomerization of Al-OH2 to HAlOH plus fragmentation to AlOH + H) is much faster than process (a) (i.e. fragmentation of Al-OH2 to AlOH + H), even at 1200 K there is a ratio of about 8 (lCVT/SCT), which is six times larger at 300 K. In the case of process (b), there is a remarkable difference between the dual-level and the B3LYP/6-311++G(2d,2p) results, especially at the lower temperatures. The main reason is that the barrier height (computed as DVMEP with respect to the reactants [18], MEP indicates the minimum energy path) is 1.43 kcal/mol lower at the B3LYP/6-311++G(2d,2p) level than at the ‘‘benchmark’’ level; such a difference has a very big impact at the lowest temperatures. The ‘‘benchmark’’ procedure of Ref. [8] is estimated to be in error by less than 0.5 kcal/mol. In order to further test the sensitivity of the values with respect to the barrier height, we have elevated the B3LYP/6-311++G(2d,2p) value by 0.8 kcal/mol and 1.8 kcal/mol, instead of 1.3 kcal/mol as we do in the ISPE procedure. The rate (kb) at 298 K (in cm3 s1) goes from 2.9 1013 (1.3 kcal/mol) to about twice, 5.8 1013 (0.8 kcal/mol) (note that the experimental result at 298 K (4.6 ± 0.8 1013 cm3 s1) [3] is somehow halfway) or half 1.4 1013 (1.8 kcal/mol) (all are lCVT/SCT values). The temperature dependence is the most interesting aspect. As we have pointed out, McClean et al. [3] find that the rate constants measured between 298 K and 1174 K can be described by an expression with two exponentials, namely k(T) = (1.9 ± 1.5) 1012 exp((0.88 ± 0.44)/RT) + (1.6 ± 0.7) 1010 exp((5.7 ± 0.9)/
Table 1 Rate coefficients (cm3 s1) for the Al + H2O [TS12] ? AlOH + H process (b), computed with several tunneling models with a B3LYP/6-311++G(2d,2p) reaction coordinate corrected with the interpolated single-point energies (ISPE) scheme. The last two columns are for pure non-ISPE values.a T(K)
200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 a
TST
1.98E15 1.22E14 4.21E14 1.06E13 2.17E13 3.90E13 6.38E13 1.41E12 2.62E12 4.35E12 6.68E12 9.66E12 1.34E11 1.78E11
CVT
1.85E15 1.15E14 4.01E14 1.01E13 2.08E13 3.74E13 6.13E13 1.35E12 2.51E12 4.16E12 6.37E12 9.18E12 1.27E11 1.68E11
CVT/ZCT
9.70E14 1.47E13 2.31E13 3.60E13 5.44E13 7.93E13 1.12E12 2.04E12 3.38E12 5.20E12 7.56E12 1.05E11 1.41E11 1.82E11
The complete set of results is provided in the Supplementary material.
CVT/SCT
1.57E13 2.09E13 2.98E13 4.32E13 6.24E13 8.82E13 1.22E12 2.16E12 3.51E12 5.35E12 7.73E12 1.07E11 1.43E11 1.85E11
lCVT/SCT 1.57E13 2.09E13 2.97E13 4.32E13 6.24E13 8.81E13 1.22E12 2.15E12 3.52E12 5.37E12 7.75E12 1.07E11 1.43E11 1.85E11
Non-ISPE LCG4-lCVT/LCT
lCVT/lOMT
6.041E13 8.263E13 1.119E12 1.134E12 1.537E12 2.045E12 2.655E12 3.394E12 5.235E12 7.655E12 1.065E11 1.420E11 1.828E11 2.296E11
8.35E13 1.04E12 1.34E12 1.75E12 2.26E12 2.87E12 3.62E12 5.47E12 7.91E12 1.09E11 1.45E11 1.86E11 2.33E11 2.83E11
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Table 2 Rate coefficients (cm3 s1) for the Al + H2O [TS1H] ? AlOH + H process (a), computed with several tunneling models with a B3LYP/6-311++G(2d,2p) reaction coordinate corrected with the interpolated single-point energies (ISPE) scheme. The last two columns are for pure non-ISPE values.a T(K)
200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 a
TST
2.22E20 1.18E18 1.74E17 1.24E16 5.55E16 1.84E15 4.92E15 2.28E14 7.21E14 1.79E13 3.75E13 6.98E13 1.19E12 1.88E12
CVT
1.96E20 1.06E18 1.58E17 1.13E16 5.10E16 1.69E15 4.53E15 2.10E14 6.62E14 1.64E13 3.43E13 6.34E13 1.07E12 1.69E12
CVT/ZCT
2.85E15 3.53E15 4.89E15 7.36E15 1.17E14 1.89E14 3.05E14 7.45E14 1.63E13 3.20E13 5.72E13 9.49E13 1.48E12 2.19E12
CVT/SCT
3.82E15 4.59E15 6.11E15 8.82E15 1.34E14 2.10E14 3.31E14 7.83E14 1.68E13 3.27E13 5.82E13 9.62E13 1.50E12 2.21E12
lCVT/SCT 3.88E15 4.64E15 6.16E15 8.88E15 1.35E14 2.11E14 3.33E14 7.89E14 1.70E13 3.31E13 5.90E13 9.80E13 1.53E12 2.27E12
Non-ISPE LCG4-lCVT/LCT
lCVT/lOMT
4.02E14 5.59E14 8.23E14 8.38E14 1.30E13 1.98E13 2.97E13 4.33E13 8.41E13 1.48E12 2.40E12 3.64E12 5.24E12 7.19E12
4.72E14 6.35E14 9.22E14 1.39E13 2.09E13 3.10E13 4.48E13 8.59E13 1.51E12 2.43E12 3.68E12 5.28E12 7.23E12 9.57E12
The complete set of results is provided in the Supplementary material.
RT); where k(T) is given in cm3 s1 and the energies are in kcal/ mol. The presence of these two exponentials (with two limiting activation energies) is, of course, suggestive of the existence two molecular mechanisms. McClean et al. [3] used the limited knowledge of the PES available at that time to suggest that those two channels could be process (b), which would account for the low activation energy, and a process producing AlO, probably through HAlOH (they could not possibly detect the AlO produced by the reaction because of the experimental conditions). In Fig. 2 we have represented Ln(ka + kb) as computed by the lCVT and lCVT/SCT methods (i.e. without and with tunneling, respectively). One finds that the lCVT/SCT curve displays what could be considered a biexponential behavior, but one immediately sees as well that the lCVT curve is almost linear. Moreover, we have isolated kb (recall kb ka except for the higher temperatures) in Fig. 3 and represented its values computed by the same methods; it is clear that without tunneling we almost have a straight line. Therefore the bi-exponential form is not due to molecular mechanisms (a) and (b) (i.e. saddle points TS1H and TS12) but mostly to the variation of the transmission coefficient with temperature. Our ka + kb values (lCVT/SCT) could be represented by k(T)(cm3 s1) = 5.3
1012 exp(1.74/RT) + 3.2 1010 exp(7.0/RT), within the 300– 1200 K (and by k(T) = 1.4 1012 exp(0.90/RT) + 1.8 1010 exp(5.6/RT) in the 200–1200 K range) where the exponential parameters are quite close to the upper bound of the experimental ones (5.7 ± 0.9 kcal/mol and 0.88 ± 0.44 kcal/mol) for the first fit and remarkably close for the second. Finally, we have studied the AlOH + H [TS_AlOH_H] ? AlO + H2 reaction at thermal equilibrium. Its rate could be meaningful to the LIF experiments if the products of processes (a) and (b), i.e. the reactants of that reaction, reach thermal equilibrium by collisions. The results are presented in Table S3 of the Supplementary material. It must be noted that the 298 K equilibrium constant is very low 1.0 107. The thermal rate constant at 298 K is very low as well 2.6 1025 cm3 s1 (lCVT/SCT).
Fig. 2. Representation of Ln(ka + kb) with respect to T1. ka + kb are measured in cm3 s1 and computed by the lCVT method (squares) or the lCVT/SCT method (circles) (the latter includes tunneling).
Fig. 3. Representation of Ln(kb) with respect to T1. kb are measured in cm3s1 and computed by the lCVT method (squares) or the lCVT/SCT method (circles) (the latter includes tunneling).
4. More complex molecular mechanisms The fact that we obtain a bi-exponential rate, which is very close to the experimental results without considering production of AlOð2 Rþ Þ þ H2 ð1 Rþ g Þ does not imply, of course, that it should not happen, so we have investigated the formation of
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AlOð2 Rþ Þ þ H2 ð1 Rþ g Þ more deeply. As we have already mentioned, we have found in our study of the PES no direct route linking AlOH2 to those products; HAlOH has to be formed and then eliminate molecular hydrogen through TS2_H2 (process (c) of Section 2). We also wanted to study other processes (d)–(f). For such purpose we have used the microcanonical expression – ki ðEÞ ¼ N – i ðE E0;i Þ=hqi ðEÞ, where N i ðE E0;i Þ is the number of quantum states above the barrier (E0,i) of process i and qi(E) is the density of states of the reactants, and integrated the E-dependent rate equations corresponding to two reaction networks shown below. We have used the Multiwell program and its utilities [20,21] to compute the number and densities of states and programs of ours to perform the integrations. It should be recalled that McClean et al. have not found any dependence of rate constant with pressures up to 110 Torr (at room temperature) [3]; which implies that collision energy transfer does not play an important role under the experimental conditions. In other words, the energy of the collision complex is normally conserved till the system breaks into the products or back into the reactants. We have employed two reaction networks. The first (Network I) is still quite simple; isomerization of HAlOH into the cis form and further molecular hydrogen elimination are taken into account.
In the second Network (II) the van der Waals complex AlOHH (noted as AlOH–H) is taken into account as well as some reverse processes:
The transition states of processes (1) and (8), which do not have saddle points, have been determined by CVTST at T = 300 K. For process (5) we have taken as the TS location the loose saddle point which appeared in the QCISD/6-311++G(2df,p) computations of Ref. [8]. For process (8) we have employed a QCISD(T)/aug-cc-pVTZ reaction coordinate corrected for BSSE (Basis Set Superposition Error) by the counterpoise method [30]. We have employed the relative energies computed by the methods of Ref. [8] in combination with B3LYP/6-311++G(2d,2p) vibrational frequencies and rotational constants for coherence with Section 3. Only one rotation has been considered as active in all intermediates and transition states (the ‘‘K’’ rotor), except for linear structures; for the reactants or products all rotations are considered active. Vibrations are treated in a quantum harmonic way, except for the Al–O stretching of Al-OH2 and the H–H stretching of AlOH–H; given that the dissociation energy is so small and that such stretching modes are closely associated to the dissociation reaction coordinate we have employed Multiwell’s anharmonicity model [20]. Tunneling is treated by a one-dimensional asymmetrical Eckart barrier [20]. The rate coefficients of processes (1) (direct and reverse) are related by microscopic reversibility; in other words by using k1 ðEÞqr ðEÞ ¼ k1 ðEÞqAl-OH2 ðEÞ we can compute k1(E). In both networks we have chosen not to include isomerization into a H2AlO
isomer, for preliminary computations have indicated it is not an important process and, in any case, it would only reduce the AlO + H2 formation rate, where what we want to find is an upper bound. The results are shown in the first two rows of Table 3. In the case of Network I we see that the rate of production of AlO + H2 is very small as compared to that of AlOH + H. The reason is that process (5) is so much faster than process (4). The cis–trans isomerization is extremely fast as compared to processes (5) or (4) and (2) (i.e. the reverse of process (2)). Given that both isomers are very close in energy and that HAlOH has a very large energy excess their proportions are roughly 50%. As we have already pointed out TS2_H2 is not only much higher in energy than AlOH + H but it is also a very tight transition state. As one would expect from the results of the former section, production of AlOH + H through process (5) is much faster than through process (6). This simple network ignores of course the production of AlO + H2 by reaction of the main products AlOH + H. The average lifetimes of Al-OH2 and HAlOH (cis and trans) are 5.9 and 0.07 ps. Note that, despite of its weak Al–O bond, the lifetime of Al-OH2 is in fact longer than that of HAlOH. The latter species is a deep minimum in the PES (see Fig. 1); the reasons for such a low lifetime are its very large excitation energy and the large rate of process (5). Note that this difference in the lifetimes of Al-OH2 and HAlOH, as well as the decay of the rate coefficient with diminishing temperature may explain why Duncan and coworkers obtain the spectrum of Al-OH2 in their ZEKE-PFI spectroscopic experiments [15]. In the second Network (II) we take into account explicitly the AlOHH van der Waals complex, which can be considered an intermediate to AlOH + H formation, but can also form HAlOH, Al-OH2 and finally AlO + H2, which is a mechanism additional to process (4) that may result in an increased rate of formation of those products. By using AlOHH as an intermediate, which can originate AlO + H2, we intend to compute an upper bound to AlO + H2 formation in the absence of collisions. As shown in Table 3 the overall rate is almost equal as in the first network, but the AlO + H2/ AlOH + H branching ratio increases to 3.1 105, which is still a small value. As a tool to assess the relative importance of the different process in a simple way we have computed an individual thermal average of the corresponding ki(E) by using the energy distribution p(E) obtained in the formation of Al-OH2 (the ‘‘chemical activation’’ distribution [20]); note that those values have nothing to do with the high-pressure limits, which assume a Boltzmann distribution. The results are in Table 3. It is readily seen that the rates of pro-
Table 3 Rate coefficients of Networks I and II. The unimolecular rates are averaged over the energy distribution obtained in process (1) and given in s1. The rate coefficients of the common processes of both networks are equal.
k(AlO + H2)a k(AlOH + H)a k(1) k(2) k(2) k(5) k(6) k(3) k(3) k(4) k(5) k(6) k(8) k(7) a
In cm3 s1.
Network II
Network I
1.1E17 3.5E13 5.7E+12 2.4E+09 2.1E+07 1.3E+14 2.8E+07 1.6E+13 1.4E+13 1.3E+05 1.2E+14 3.4E+05 2.2E+13 9.3E+07
9.0E21 3.5E13
S. Álvarez-Barcia, J.R. Flores / Chemical Physics 382 (2011) 92–97
cesses (8) and (5) are similar in magnitude and much larger than that of process (7), which has a tight saddle point. The increase of the branching ratio comes from process (7) but also from flux to HAlOH. Perhaps we should stress again that, in the case of both networks, it is assumed that no collisions other than those between the two reactants or the two products take place. Ultimately, in a system at thermal equilibrium the AlO + H2/AlOH + H ratio is given by the equilibrium constant, which is 1.1 107 at 300 K, and still low (2.7 103) at 1200 K. Those results and the analysis of the reaction networks lead us to conclude that the products AlO + H2 are not likely to have a very significant branching ratio under any pressure conditions in an Al + H2O collision.
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Ministry of Education (Spain). The services provided by the ‘‘Centro de Supercomputación de Galicia’’ (CESGA) are also acknowledged. Appendix A. Supplementary data Tables S1–S3, which contain detailed results for the rates of the Al + H2O TS12 ? AlOH + H, Al + H2O TS1H ? AlOH + H and AlOH + H TS_AlOH_H ? AlO + H2 processes. Tables S4–S6 contain the ISPE corrections and S7 contains ZPE energies and details of Fig. 1. Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemphys. 2011.03.004. References
5. Conclusions We have performed a theoretical study of the dynamics of the Al + H2O reaction by means of TST and RRKM theories including tunneling, with sophisticated models in the first case. We have found that the main reaction path involves isomerization into a HAlOH species, which dissociates into AlOH + H. A second mechanism involving hydrogen atom elimination from a Al-OH2 adduct only becomes important at very high (>1000 K) temperatures. We have found no direct route in the PES to produce AlO + H2. The results are coherent with the experiment in the sense that the variation of the total rate with temperature can be fitted to a bi-exponential form and that the values are close (for instance at 298 K the experimental value is 4.6(±0.8) 1013 cm3 s1 and the theoretical one is 2.9 1013 cm3 s1). Our interpretation of the apparent bi-exponential behavior is not the existence of two molecular mechanisms, but the change of the transmission coefficient with temperature. The moderate discrepancy between the experimental and the theoretical values can be attributed to the models employed in our calculations. We have studied the possible generation of AlO + H2 by a secondary process AlOH + H ? AlO + H2 by RRKM theory. We have assumed that the energy is maintained in every collision up to the generation of the final product and proposed two reaction networks; we find that the AlO + H2/AlOH + H has to be very small, un upper bound could be 3.1 105 (at 300 K). On the other hand, the equilibrium constant of the AlOH + H ? AlO + H2 reaction is about 1.1 107 at 300 K (2.7 103 at 1200 K), which suggests that even if collisions with other particles in the system take place, formation of AlO + H2 would not be significant. Acknowledgements We acknowledge the financial support of the Xunta de Galicia through the project INCITE09314252PR and the program INCITE08ENA314104ES. S.A.B acknowledges a FPU grant from the
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