Schematic representation of the coordinates used for the H + HCY (Y = H3, CH5, C2H7) system. The two bond distances H–H a and H a –C are denoted r 1 and r 2, respectively. R denotes the Jacobi coordinate given by the distance between the center of mass (COM) of the H2 fragment (⊙) and the COM of the CY moiety (•). The second Jacobi coordinate, r, is identical to the H–H a bond distance r 1.
Schematic representation of the coordinates used to calculate the 1D MEP of the hydrogen exchange reaction H + H a CY → HH a + CY. The abstracted hydrogen, H a , is described in Cartesian coordinates; all other coordinates are represented in internal (Z-matrix) coordinates. In the Z-matrix, the first atom is a dummy atom (denoted as •), which therefore defines the origin of the (Cartesian) coordinate system. Furthermore, the distance between the H- and C-atom, r HC, is given by sum of the distances and , i.e., r HC = r 1 + r 2. The choice of the origin of the coordinate system implies that the position z of the transferred H-atom is z = (1/2)(r 2 − r 1). Note that (large) negative z values correspond to reactant geometries (H + H a CY).
Upper panel: Potential, V 0, at CCSD(T)-F12a/VTZ-F12//MP2-cc-pVTZ level (solid, red line) and ZPE at MP2/cc-pVTZ level postprojection of the two active DOFs for H + CH4. Black circles represent the actually calculated ab initio PES points. Lower panel: Equilibrium values of the HC bond distance, and Hessian matrix element, H HC, corresponding to this DOF.
Contour plot of the 2D PES for H + CH4 including ZPE corrections of the 3N − 8 spectator modes. Upper panel: CCSD(T)-F12a/VTZ-F12 ab initio data (568 points) fitted to the analytical double Morse function [Eq. (11)]. Lower panel: (1 + 1)D PES calculated by harmonic extension perpendicular to the MEP [Eq. (13)].
1D cuts at fixed values of ρ of the 2D PES fit (dashed-dotted, red lines) and (1 + 1)D PES (solid, green lines). The two 1D cuts correspond to the transition state (ρ* = 4 a 0, lower panel) and asymptotic region (ρ* = 20 a 0, upper panel). Dots (blue) represent the CCSD(T)-F12a/VTZ-F12 ab initio data.
Cumulative reaction probability, , for H + CH4 → H2 + CH3 vs. total energy, E, relative to the lowest reactant state energy, E 0. The solid (red) line shows the result obtained by using the (1 + 1)D PES [Eq. (13)], while the dashed (green) line shows the result employing the 2D PES fit [Eq. (11)].
Arrhenius plot of the thermal rate constant for the reaction H + CH4 →H2 + CH3. Solid (black) and dashed-dotted (purple) lines refer to 2D quantum results, using the (1 + 1)D and 2D fitted PESs, respectively. The dashed (green) line is the TST result (with kJ/mol). Triangular (red) data points show the previous 2D quantum results by Banks et al. (Ref. 14). Experimental data by Sutherland et al. (Ref. 31) and Baulch et al. (Ref. 30) are depicted as (blue) circles and squares, respectively. In the latter case, the error bars signify the estimated uncertainty factor of 1.58.
Potential, V 0, at CCSD(T)-F12a/VTZ-F12//MP2-cc-pVTZ level and ZPE at MP2/cc-pVTZ level postprojection along the MEP for H + C2H6 → H2 + C2H5.
Arrhenius plot of the thermal rate constant for H + C2H6. Solid (black) and dashed (green) lines correspond to quantum and TST results, respectively. Experimental data of Bryukov et al. (Ref. 37) are shown as (blue) circles [Arrhenius parameters for T = 357–826 K: A = (1.04 ± 0.4) × 10−10, B = 0, α = 4357 ± 191]. Experimental data recommended by Tsang and Hampson (Ref. 32) are shown as (blue) squares with error bars signifying the estimated uncertainty factor of 3.
Schematic representation of the propane (C3H8) molecule. In the text, abstraction of H1-atoms is denoted primary or n-C3H7 channel. Likewise, abstraction of H2–atoms is referred to as secondary or i-C3H7 channel.
Potential, V 0, at CCSD(T)-F12a/VTZ-F12//MP2/cc-pVTZ level along the MEP for the reaction H + C3H8 → H2 + C3H7. Solid (red) and dashed (green) lines show the channels n-C3H7 and i-C3H7, respectively.
Arrhenius plot of the thermal rate constant for H + C3H8 → H2 + n-C3H7 (left panel) and H + C3H8 → H2 + i-C3H7 (right panel). Solid (black) and dashed (green) lines refer to quantum and TST results of this paper, respectively. Triangles (red) denote the previous quantum calculations of Kerkeni and Clary (Ref. 53). Experimental results, as recommended by Tsang (Ref. 73), are shown as (blue) circles. The experimental data are fitted to an Arrhenius expression [Eq. (10)] with parameters for n-C3H7: A = 2.2 × 1018, B = 2.54, α = 3400 and for i-C3H7: A = 2.16 × 1018, B = 2.4, α = 2250. The uncertainty factor of the experimental data is estimated to be 1.3 in the temperature range T = 300–900 K increasing to 3 at T = 2500 K.
Branching ratio for the production of n-C3H7 and i-C3H7 as a function of the temperature (in K). Solid (black) line represents the 2D quantum result employing the (1 + 1)D PES. Experimental results of Tsang (Ref. 73), are shown as (blue) circles. It should be noted that the experimental results are subject to a significant error, which is not shown for clarity. The (red) triangles denote previous 2D quantum calculations by Kerkeni and Clary (Ref. 53).
Relative error of the thermal rate constant with respect to decreasing number of ab initio grid points. The reference k(T) was calculated with 56 points corresponding to a grid spacing of Δz = 0.1 Å (see Fig. 12). Upper and lower panels show the primary and secondary channel, respectively.
H + CH4 coupled cluster ab initio energies (in hartree), and reaction energetics (in kJ/mol) relative to the H + CH4 reactant state. The depth of the product vdW well, ΔV vdW, is given relative to the H2 + CH3 product state (ΔE = 0). The notation VnZ refers to a correlation consistent polarized valence n-ζ (n = double (D), triple (T), quadruple (Q)) Dunning basis set,58 i.e., cc-pVnZ. VnZ-F12 denotes the corresponding explicitly correlated MOLPRO basis set.63 All geometries are optimized at MP2/cc-pVTZ level of theory.
H + CH4 harmonic frequencies (in cm−1) calculated at MP2/cc-pVTZ level. ZPEs (in kJ/mol) are given in the bottom row. Quantities postprojection of the two active DOFs are denoted in parentheses.
H + CH4 reaction energies (in kJ/mol) at CCSD(T)(-F12a) level using different basis sets. ZPE corrections and geometries at MP2/cc-pVTZ level.
Numerical parameters of the scattering calculations.
Reaction energetics (in kJ/mol) relative to the reactant state at CCSD(T)-F12a level for H + C2H6 → H2 + C2H5. ZPE corrections and geometries at MP2/cc-pVTZ level.
H + C2H6 harmonic frequencies (in cm−1) at MP2/cc-pVTZ level. Quantities postprojection of the two active DOFs are denoted in parentheses. ZPEs (in kJ/mol) are given in the bottom row.
Reaction energetics (in kJ/mol) relative to the reactant state at CCSD(T)-F12a level for H + C3H8 → H2 + C3H7. ZPE corrections and geometries at MP2/cc-pVTZ level.
H + C3H8 harmonic frequencies (in cm−1) at MP2/cc-pVTZ level. Quantities postprojection of the two active DOFs are denoted in parentheses. ZPEs (in kJ/mol) are given in the bottom row.
Thermal rate constants for reactions (R1)–(R3). Rates given in cm3 molecule−1 s−1 with powers of ten in parentheses.
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