Carbon dioxide (CO 2 ) emissions into the atmosphere result in great part from combustion of fossil fuel, and this greenhouse gas is believed to be a potential source of global warming and a driver of climate change. In the context of clean combustion, hydrogen (H 2 ) fuel is considered as a nearly ideal and renewable energy source. [1][2][3] In contrast to fossil fuels, its combustion in air produces water vapor and small amounts of nitrogen oxides; it produces no carbon dioxide (CO 2 ) nor any compounds that will impact the ozone layer. [1][2][3] However, a mitigating consideration is that hydrogen fuel has a low-energy density, and therefore expensive to store and transport. [1][2][3] Another simple and widely available compound that admits to clean combustion is ammonia (NH 3 ). In the thermal DeNO x process, [4][5][6] NH 3 is often supplied to transform NO x species to N 2 and H 2 O. Combustion of ammonia liberates water vapor, molecular nitrogen, and nitrogen oxides (NO x ) [7][8][9][10] , all of which are environmentally innocuous. Given these favorable properties, it is not surprising that ammonia (NH 3 ) has been predicted to become a viable carbon-free energy source, especially in the transportation sector. [7][8][9][10] We are currently interested in studying the mechanism, thermochemistry, and kinetics of fundamental reactions relevant to ammonia combustion. In a previous work, 11 we have reported thermal rate constants for the OH-oxidation of NH 3 . In this work, we calculate the thermochemistry and the thermal rate constants for the similar H + NH 3 ⇌ H 2 + NH 2 reaction using the same theoretical approach as that taken previously. 11 The H-abstraction reaction H + NH 3 ⇌ H 2 + NH 2 has been fairly well studied both experimentally [12][13][14][15][16][17][18] and theoretically. [19][20][21][22][23][24][25][26][27][28][29][30] In addition, a nine-dimensional global potential energy surface of NH 4 and quantum dynamic calculations have been reported. [19][20][21][22][23][24][25][26][27][28][29][30] Therefore, a large amount of data is available to compare with our high-accuracy extrapolated ab initio thermochemistry (HEAT)/semiclassical transition state theory (SCTST) results.
The same approaches that have been used successfully for the OH + NH 3 reaction 11 are reemployed here. Briefly, the high-level HEAT-456QP method-which can yield an accuracy of 1 kJ/mol-is used to compute thermochemistry and provide rovibrational parameters as well as anharmonic constants for the purpose of chemical kinetics. Except the CCSDTQP energy calculations that use MRCC program, 31 the other calculations are done using CFOUR quantum chemistry program. 32 The forward and reverse rate coefficients for the title reaction as displayed in Figure 1 computed at the high-pressure limit using Equations 1 and 2, respectively:
where
where v and r stand for vibrational and rotational degrees of freedom, respectively. The SCTST code 33,34 of the MULTI-WELL software package 35,36 was used to compute the sum of fully coupled vibrational states (𝐺 ≠ 𝑣 (𝐸 𝑣 )) using a bin size of 1 cm -1 and a ceiling energy of 50,000 cm -1 . Assume that the TS can be approximated as a rigid-rotor symmetric top, 37 and the density of external rotational states (𝜌 ≠ 𝑟 ) was counted directly using the following equation:
Recently, we have studied fourth-order vibrational perturbation theory (VPT4) applied to the one-dimensional symmetric Eckart potential, 38 and we have found that some effects of higher order perturbation theory within the SCTST framework can heuristically be included by a rescaling of the barrier frequency used in the VPT2 treatment (Equation 5). 38 This approach, designated as "SCTST/VPT2+", seems to work well for the OH + NH 3 → H 2 O + NH 2 reaction, 39 especially in the deep-tunneling regime. Note that this approach is available in the SCTST code of the MULTIWELL software package (keyword "VPT4A").
In this work, we used both SCTST/VPT2 and SCTST/VPT2+ approaches to calculate thermal rate coefficients. However, the empirical adjustment defined by SCTST/VPT2+ lowers the barrier frequency 𝜔 F by about 1.2%, resulting in a slight widening of the barrier. This small change reduces the calculated rate constants marginally for the temperature range of 300-2500 K (see below). Therefore, unless mentioned otherwise, SCTST/VPT2+ approach will be used for the following discussion.
Table 1 shows a comparison of theory and experiment for spectroscopic parameters for NH 3 , NH 2 , and H 2 , which are well established experimentally. As can be seen in Table 1, the CCSD(T) calculations are close to the experimental results apart from the (large amplitude) umbrella inversion motion of NH 3 , but this agreement is still adequate for the purposes of determining zero point energies and densities of states needed for kinetic calculations. Table 2 shows the forward reaction pathway being endothermic by 11.8 ± 1.0 kJ/mol, which is in almost perfect agreement with the ATcT value of 11.5 ± 0.2 kJ/mol. [40][41][42][43] In addition, Table 2 also reveals individual contributions of various terms to a total activation energy. Inspection of Table 2 shows that the SCF method significantly overestimates the total barrier by nearly a factor of two. The CCSD(T) correction is the most important: it recovers most of the electron correlation, together with the SCF (188.91 ± 0.12 kJ/mol), which are taken from ATcT. [40][41][42][43] contribution, and brings the barrier close to the final value. Another important contribution is zero-point energy (ZPE), which cuts the barrier by 6.7 kJ/mol. The full triple, quadru-ple, and pentuple excitations that go beyond CCSD(T)) lower the barrier by an additional 1.1 kJ/mol. Contributions of the remaining terms are smaller, but essential for an accuracy of ca. 1 kJ/mol. Taking all individual contributions into account, the forward barrier height is obtained to be 57.9 ± 1 kJ/mol. To the best of our knowledge, HEAT-456QP is the highest level of theory that has been applied to the title reaction. Previous studies using various levels of theory reported the vibrationally adiabatic barrier height in the range of 55-65 kJ/mol. [19][20][21][22][23][24][25][26]
NH 3 has a well-known inversion ("umbrella") vibrational mode, for which the fundamental frequency is overestimated (with VPT2) by 29 cm -1 as compared to experiment (see Table 1). The following approximate procedure was used to treat this umbrella motion as an independent, one-dimensional hindered internal inversion (1DHI). First, we used the AE-CCSD(T)/aug-cc-pCVQZ level of theory to construct an inversion potential energy curve and an effective moment of inertia as a function of the inversion angle. Second, we solved a one-dimensional Schrödinger equation to obtain a set of eigenvalues of the 1DHI. Then, we directly counted the Finally, we convolved the 1DHI with the remaining vibrations to gain the quantum vibrational states for NH 3 . It should be mentioned that the NH 3 inversion calculation used here gives a tunneling splitting of 1.3 cm -1 , in acceptable, and arguably good agreement with the experimental value of 0.79 cm -1 . 44 In addition, it yields an inversion ZPE of 507 cm -1 , which is 18 cm -1 lower than the VPT2 value of 525 cm -1 . This will impact the reaction enthalpy and the forward rate constants, but not the reverse ones, which are independent of the partition function of NH 3 . A comparison of 1DHI with VPT2 approach is displayed in Figure 2. Inspection of Figure 2 shows a ratio of 1DHI partition function (including the change of 18 cm -1 in ZPE) over that of VPT2 as a function of temperature. As can be seen, the ratio decreases with temperature: it falls from ca. 1.09 at 300 K to ca. 0.92 at 2500 K. Overall, the 1DHI treatment changes the NH 3 vibrational partition function, but by no more than 10%.
For the reverse reaction (H 2 + NH 2 → H + NH 3 ), Figure 3 shows the calculated rate constants increasing significantly with temperature. k(T) rises by more than six orders of magnitudes between 300 and 2500 K: it is 1.5 × 10 -18 cm 3 molecule -1 s -1 at 300 K to 5.3 × 10 -12 cm 3 molecule -1 s -1 at 2500 K. The ab initio k(T) values agree essentially quantitatively with all experimental results, which were measured by four different groups. To the best of our knowledge, this is the first study to achieve such accurate results for this reaction over an extensive temperature range.
To have a careful comparison of theory with experiment for the forward reaction (H + NH 3 → H 2 + NH 2 ), we divide temperature into two regimes, namely, low to moderate T (450-1000 K) and moderate to high T (1000-2000 K). For T = 450-1000 K (see Figure 4A), our calculated rate constants agree well with the results of Ko et al., 18 but marginally over- estimate those of Hack et al. 13 and Marshall and Fontijn. 17 (According to Ko et al., 18 the former results are more reliable than the latter two.) Note that both data sets were measured by the same group. For T = 1000-2000 K (see Figure 4B), again we slightly overestimate experiments by about 20%: our values are at the upper end of experimental results.
The ab initio thermal rate constants from 300 to 2500K for both forward and reverse reactions can then be fit to a three parameter Arrhenius formula: (𝐵∕𝑇 ) , cm 3 molecule -1 s -1 with A = exp(-44.8359), n = 2.58163, and B = -3290 (K).
𝑘(𝑇 ) H+NH 3 = 𝐴𝑇 𝑛 𝑒 (𝐵∕𝑇 ) , cm 3 molecule -1 s -1 with A = exp(-39.8793), n = 2.23036, and B = -5237 (K). Quantum mechanical tunneling effects play an important role for the title reaction, especially at low temperatures. Note that experimental results at T ≤ 298 K are not available because the high barrier prevents reaction to proceed at an experimentally detectable rate at these conditions. Figure 5 shows the tunneling effects declining sharply with temperature: the enhancement is greater than 100 at 200 K, and then drops to ca. 7 at room temperature, and about 3 at 400 K. When temperature is more than 1400 K, the tunneling effects become unimportant.
It is of importance to compare the HEAT/SCTST rate constants with other theoretical calculations. 24,26 Tables 3 and4 (also see Figures 3 and4) display comparisons for the forward and reverse reactions of H + NH 3 ⇌ H 2 + NH 2 , respectively. We chose to compare the SCTST results with two recent sets of theoretical k(T) data calculated with the CVT/𝜇OMT 24 and CVT/LAT 26 techniques of POLYRATE program, 45 based on the same PES-2009, 24 which was constructed using CCSD(T,fc)/cc-pVTZ. 20,24 Both of these sets of k(T) are in close agreement with available experimental data. There are other CVT/LAT results reported earlier, 19,20 which are not included here for comparison because they were based on older and less accurate potential energy surfaces (PESs). 19,20 Inspection of Table 3 (also see Figure 4) shows that the SCTST results are in better agreement with those of Corchado et al. 26 than those of Garcia et al. 24 All three sets of theoretical results slightly overestimate experiment at the high-temperatures characteristic of combustion. 16 Overall, the agreement between two different techniques (HEAT/SCTST vs. CVT/𝜇OMT) is excellent in the whole temperature range of 200-2000 K. The main differences largely arise from the choice of potential energy surface. Both kinetics approaches are nearly identical for this reaction so long as the same energetics are used. 24 However, there are significant differences between the calculated SCTST rate constants and CVT/𝜇OMT results for the reverse reaction, H 2 + NH 2 → H + NH 3 . Table 4 displays the difference defined as a ratio of SCTST over CVT/𝜇OMT rate constants. As seen in Table 4 (also see Figure 3), the difference declines with increasing temperature: it is a factor of ca. 8.8 at 200 K, 2.0 at 600 K, and reduces to ca. 1.3 at 2000 K. This problem is likely due to a possible energy error of about 3 kJ/mol for NH 2 in PES-2009 24 (e.g., ΔH f,ok (NH 2 ) = 185.8kJ/mol from PES-2009 as compared to ATcT's value [40][41][42][43] of 188.9 ± 0.1 kJ/mol). It should be noted here that the SCTST rate constants for this reverse reaction agree very well with all available experimental date (see Figure 3). This result implies that the equilibrium constants from the two PESs are different.
The thermochemistry for the title reaction has been studied using the high-accuracy HEAT-456QP method, followed by chemical kinetic calculations using Miller's SCTST theory to obtain thermal rate coefficients at the high-pressure limit. The forward reaction pathway is endothermic by 11.8 ± 1 kJ/mol, which agrees very well with the ATcT value of 11.5 ± 0.2 kJ/mol. [40][41][42][43] The forward and reverse barrier heights (including ZPE corrections) were estimated to be 57.9 ± 1 and 46.1 ± 1 kJ/mol, respectively. The highly accurate thermochemistry in this work leads to a significant improvement of ab initio thermal rate constants, which now agree well with experimental results for the reverse reaction, for which all previously theoretical calculations were not successful. This certainly demonstrates the significant importance of accurate thermochemistry, even in kinetics calculations. Furthermore, thermal rate constants above 2500 K where experimental results are not available have been provided in Table S1 (see the Supporting Information Material) for kinetic modeling.