the reaction is endothermic (by 0.293 eV) with reactants in the ground ro-vibrational states, but vibrational excitation of the reactant makes the process exothermic. As shown in Figure 4.1 there exists a potential energy well (of maximum depth 0.521 eV with respect to the reactant asymptote) along the minimum energy path. The minimum of the potential energy well corresponds to a collinear He-H-Ne geometry with NeH and HeH bond distances of 2.102 and 1.804 bohr, respectively.
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
Reaction Coordinate
Energy (eV)
He+NeH+
(v,j)=(0,0-2) (v,j)=(1,0)
(v,j)=(0,0-2) (v,j)=(1,0)
Ne+HeH+
[HeHNe]+
Figure 4.1: Schematic potential energy profile of reactants and products for the He + NeH+→HeH++ Ne reaction. Energy of He+NeH+ asymptote is set as zero.
Table 4.1: Numerical parameters used in the TDQM simulations (All parameters are given in atomic units).
Number ofRgrid points 220
Number ofrgrid points 150
Number of angular grid points 120
Rmin 0.2
rmin 0.5
δR 0.1
δr 0.12
Centre of initial wave packet 14.0
Damping coefficients alongRandr 0.001, 0.001 Starting points of damping alongR andr 15.5, 13.22
Analysis point alongr 12.98
Number of Chebyshev iterations 30000
Eq. 2.46. Different initial state specific total reaction probabilities are computed within a collision energy range 0.0005-0.5 eV by calculating the total flux of the energy de- pendent WP going through a fixed surface. The TDQM calculations were performed using a Fortran code developed by us and parallelized using MPI as well as OPENMP algorithm. Many test runs were carried out to check the convergence of each parame- ter used in the TDWP studies. The final parameters used in the TDQM calculations are given in Table 4.1. For v = 0, the values of Jmax are 67 and 74 for TDQM-CS and TDQM-CC calculations, respectively, while for v = 1, Jmax = 91 was enough for both the TDQM-CS and TDQM-CC calculations to converge the integral cross sections within the above mentioned energy range.
The calculations of reaction probabilities including all K states for high J values are highly computationally demanding. However, for the title reaction, converged total reaction probabilities are obtained using truncated helicity basis. In Figure 4.2, the convergence of the TDQM-CC reaction probabilities with respect to Kmax (maximum value ofK) is shown for different initial reactant states at largeJ values. Forv= 0 and 1, J = 50 and 70 respectively, are chosen to investigate the Kmax dependence on total reaction probabilities. As can be seen in Figures 4.2a and 4.2b, very small differences appear between the reaction probabilities obtained usingKmax= 7 and 11. Considering the computational cost and time factor as well as accuracy of the calculations,Kmax = min(J, 7) were used for all v = 0 calculations. However, for v = 1, a large number of
J’s were needed to converge the cross sections within the investigated energy range. As it is observed in Figure 4.2c, Kmax = 11 results slightly overestimate the probabilities obtained using Kmax = 7. But reaction probabilities obtained by using Kmax = 11 and 13 are literally having the same values. Hence, Kmax = 11 was used for J > 50 calculations andKmax = min(J, 7) were used in rest of the calculations.
0 0.05 0.1 0.15 0.2 0.25
(v,j)=(0,0), J=50 (a)
Probability
Kmax=6 Kmax=7 Kmax=11
0 0.05 0.1 0.15 0.2 0.25
0.2 0.25 0.3 0.35 0.4 0.45 0.5
(v,j)=(0,1), J=50 (b)
Probability
Kmax=6 Kmax=7 Kmax=11
0 0.05 0.1 0.15
0 0.1 0.2 0.3 0.4 0.5
(v,j)=(1,0), J=70 (c)
Probability
Ec (eV) Kmax=7
Kmax=11 Kmax=13
Figure 4.2: Dependence of total reaction probabilities on the Kmax values. (a,b,c) Results are shown for three different initial states and for two differentJ values.
As mentioned already, all the TIQM calculations are performed using the ABC code
Table 4.2: Numerical parameters employed in the TIQM calculations.
Maximum hyperradius (ρmax/a0) 30 Ec≤0.01 eV
25 0.01 eV< Ec≤0.5 eV 22 Ec>0.5 eV
Number of log derivative propagation sectors (nsec) 290 Ec≤0.01 eV
234 0.01 eV< Ec≤0.5 eV 200 Ec>0.5 eV
Maximum rotational quantum number (jmax) 30 Maximum internal energy (emax/eV) 2.0
Helicity truncation parameter (Kmax) min(11, J) Ec≤0.6 eV min(15, J) Ec>0.6 eV
developed by Manolopoulos and co-workers.29Many test runs are performed to converge the state-to-state reaction probabilities with respect to different initial parameters. The optimal values of the parameters used to obtain the TIQM results reported here are listed in Table 4.2. Optimal values of the parameters emax and jmax as mentioned in Table 4.2 define the coupled-channel basis set for J = 0 with 261 basis functions. The number of basis functions in the truncated helicity basis set forJ ≥11 withKmax = 11 is 2224 and forJ ≥15 withKmax = 15 is 2327.
QCT studies are performed only for the He + NeH+(v = 0, j = 0) → Ne + HeH+(v0, j0) reaction. As mentioned already, these QCT calculations are carried out using a newly developed code by us which uses a fourth-order Runge-Kutta method for numerical integration. A fixed time step of 2 a.u. was used which guaranteed conser- vation of the total energy and the angular momentum up to eighth and ninth decimal places, respectively. For all the trajectories, the He atom was separated initially from the center of mass of NeH+ molecule by a distance of 30 a.u. Trajectory calculations were stopped when the distance between He and H atoms exceeded 35 a.u. or distance between Ne and H atoms exceeded 30 a.u. Product internal states have been assigned using both HB and GB formulations. J-sampling technique is used to calculate ICSs and DCSs at particular collision energies by running batches of 2000000 - 5000000 tra- jectories. In addition, total and vibrational state resolved reaction probabilities have been computed for J = 0 and 50, and total ICSs have been computed as a function of collision energy. The method of moments expansion in Legendre polynomials approach is used to calculate probabilities and ICSs as a function of collision energy.30