4.3 RESULTS AND DISCUSSION
4.3.1 Initial state selected dynamics
4.3.1.1 Total reaction probabilities
0 0.1 0.2 0.3
Probability
J = 0
0 0.05 0.1 0.15 0.2
J = 10
0 0.05 0.1 0.15 0.2
Probability
J = 20
0 0.05 0.1 0.15
J = 40
0 0.05 0.1 0.15
0.25 0.3 0.35 0.4 0.45 0.5
Probability
J = 50
Ec (eV)
0 0.05 0.1 0.15
0.25 0.3 0.35 0.4 0.45 0.5 J = 60
Ec (eV)
Figure 4.4: Same as Figure 4.3 for He + NeH+(v= 0, j= 1)→HeH+ + Ne.
excited rotational states, respectively. Excellent agreements observed between TDQM- CC and TIQM results suggest that both sets of the results are well converged and accurate. As can be seen in Figures 4.3 and 4.4, both the CC and CS probabilities ex- hibit dense oscillations for lowJ values, which may be an indication of an intermediate complex getting formed in the deep potential well during the reaction. Such dense os- cillatory total reaction probabilities have been seen for many other bimolecular complex forming reactions.8,10,13,14,21,25,27,31–38 For larger J values, the peaks in the probability curves become broader. The other significant observation is that the amplitudes of the oscillations are larger for the CS probabilities than those in the CC results. CC prob- ability for a particular value of J is calculated by summing over the probabilities of all the individual K states included in the CC calculation. As a result, the resonance features of the individual K-dependent probabilities are diminished in the resultant CC probabilities. As it is seen in Figure 4.3, the reaction has a threshold of ∼ 0.29 eV for ground ro-vibrational state of the reactants. The threshold increases for higherJ values due to the increase of centrifugal potential. For v = 0 reactions, the CC probabilities
−0.3
−0.2
−0.1 0 0.1 0.2
4 6 8 10 12
R (a.u.) (v,j)=(0,0), J=40
Effective Potential Energy (eV)
NK=1 NK=2 NK=3 NK=4 NK=5 NK=6 NK=7 NK=8
Figure 4.5: Effective potential energy profiles for the He + NeH+reaction atJ = 40.
Horizontal axis corresponds to the collinear minimum energy path. NK is the number ofK states used.
for larger J values (e.g., J = 50, 60) have smaller threshold energies than the corre- sponding thresholds in CS probabilities. The NK-dependent effective potential energy profiles, plotted in Figure 4.5, address this difference. Here, NK is the number of K states included in QM-CC calculations. The effective potential energies (E(R, r)) for a particular J andNK are calculated as
E(R, r) = F
2µRR2 +V(R, r), (4.1)
whereF is the smallest eigenvalue obtained by diagonalizing the centrifugal tridiagonal matrix of Eq. 2.24. V(R, r) is the potential energy of the system along the collinear minimum energy path. As the angular kinetic energy decreases with the increase in K (see Eq. 2.24), the term F, described in Eq. 4.1 also decreases. This results in a lower effective potential energy. Thus, more the number ofK states included in the CC calculations, less is the value of the effective potential energy and this feature is reflected in Figure 4.5. On the other hand, in the CS calculations,NK is always equal to one and the centrifugal barrier is independent of NK.
In Figures 4.3 and 4.4 , the similarities between CC and CS reaction probabilities for low J values (J = 10) are quite clear but differences increase with the increase of
0 0.05 0.1 0.15 0.2
0.2 0.3 0.4 0.5
(v,j)=(0,0), J=10
Probability
0 0.03 0.06 0.09 0.12
0.2 0.3 0.4 0.5
(v,j)=(0,1), J=10
Probability
Ec (eV) K=0
K=1 K=2 K=3 K=4 K=5 K=6 K=7
0 0.02 0.04 0.06 0.08
0.2 0.3 0.4 0.5
(v,j)=(0,0), J=40
0 0.02 0.04 0.06 0.08
0.2 0.3 0.4 0.5
Ec (eV) (v,j)=(0,1), J=40
Figure 4.6: K-dependent total reaction probabilities for two different ro-vibrational andJ states obtained from TDQM-CC calculations.
total angular momentum. These findings are in accordance with the results for other atom-molecular ion systems8,21,32 for which it has been observed that Coriolis coupling becomes more important as the value of J increases.
The contributions of eachKstate to the total CC reaction probabilities for (v, j) = (0,0) and (v, j) = (0,1) andJ = 10, 40 are presented in Figure 4.6. It is obvious from this figure that for (v, j) = (0,0) and (0, 1) and J = 10, the major part of the reaction probabilities comes from first threeK states and the contributions from largerK states are negligibly small. However, forJ = 40, the initial WP is channeled to largerK states to overcome the centrifugal barrier and more numberK states contribute significantly to the reaction probabilities. In the CS calculations, the out of plane rotations of molecule are restricted and only oneKstate (hereK= 0) is included during quantum calculations.
As a result, it can be seen in Figure 4.3 that there are no significant differences between the CS and CC reaction probabilities for (v, j) = (0,0) and J = 10, where major share of the CC reaction probabilities comes from K = 0 state. However, it is seen in Figure 4.4 that the differences between CS and CC reaction probabilities increase a bit for (v, j) = (0,1) and J = 10, where K = 0 and 1 states have almost similar contribution
0 0.1 0.2 0.3
Probability
J = 0
0 0.05 0.1 0.15 0.2
J = 10
0 0.05 0.1 0.15 0.2
Probability
J = 20
0 0.05 0.1 0.15 0.2
J = 40
0 0.05 0.1 0.15
0.25 0.3 0.35 0.4 0.45 0.5
Probability
Ec (eV) J = 50
0 0.05 0.1 0.15
0.25 0.3 0.35 0.4 0.45 0.5 J = 60
Ec (eV)
Figure 4.7: Comparison between (v, j) = (0, 0) and (0, 1) TDQM-CC total reaction probabilities.
to the total CC reaction probability.
Effect of rotational excitation of the reactants on the title reaction is presented in Figure 4.7. The overall behavior of the reaction probabilities is similar for both the initial states, which is in accordance with many other endothermic reactions.8,10,13,14,32,34,35
The thresholds for the probabilities for (v, j) = (0,1) are smaller than corresponding (v, j) = (0,0) probabilities due to increment in internal energy for the rotationally excited reactants. A comparison in magnitudes of the probabilities at different collision energies for both the initial states does not show any special trend. It is seen that the reaction probabilities for J = 0 and (v, j) = (0,1) are very much comparable to the corresponding (v, j) = (0,0) probabilities near the threshold regions, but at higher collision energies (v, j) = (0,1) probabilities are slightly larger than the (v, j) = (0,0) probabilities. Probabilities for (v, j) = (0, 1) are larger and smaller for J = 40 and J
= 10, 20, respectively, than the results for (v, j) = (0, 0) case. But the probabilities for J = 50 and 60 for both the initial states are comparable to each other. It seems that
0 0.1 0.2 0.3 0.4 0.5
Probability
J = 0
0 0.1 0.2 0.3 0.4
0.5 J = 10
0 0.1 0.2 0.3
Probability
J = 20
0 0.1 0.2 0.3 0.4
0.5 J = 30
0 0.1 0.2
0.3 J = 50
Probability
0 0.05 0.1 0.15 0.2
0.25 J = 60
0 0.05 0.1 0.15 0.2
0 0.1 0.2 0.3 0.4 0.5
J = 70
Ec (eV)
Probability
0 0.02 0.04 0.06 0.08
0 0.1 0.2 0.3 0.4 0.5
J = 80
Ec (eV)
Figure 4.8: Same as Figure 4.3 for He + NeH+(v= 1, j= 0)→HeH+ + Ne.
the rotational excitation of the reactants to its first excited state has less effect on the reaction probabilities.
In Figure 4.8, total reaction probabilities computed via TDQM-CC, TDQM-CS and TIQM approaches are presented for a few selected Js for He + NeH+(v = 1, j = 0) → HeH+ + Ne reaction. It is clear in Figure 4.8 that TIQM results agree quite well with the TDQM-CC results in almost entire collision energy range. Vibrational excitation of the reactants is found to enhance the title reaction to a large extent. As is seen in Figure 4.1, vibrational excitation makes the reaction thermodynamically exothermic, which results in a strong enhancement in the reaction probabilities. For (v, j) = (1,0) the reaction probabilities for lowerJ values start without any threshold, which is typical for barrierless exothermic reactions and processes with reactants in vibrationally excited states.21,33,36–38 Differences between the CC ans CS results for larger Js are clearly noticeable in Figure 4.8.
0 2 4 6 8 10
(v,j) = (0,0) (a)
TDQM−CC TDQM−CS TIQM
0 2 4 6 8
0.25 0.3 0.35 0.4 0.45 0.5
Integral Cross Section (a02 )
(v,j) = (0,1) (b)
0 40 80 120 160 200
0 0.1 0.2 0.3 0.4 0.5
(v,j) = (1,0)
Ec (eV) (c)
Integral Cross Section (a02 )
Figure 4.9: Total integral reaction cross sections for He + NeH+ → HeH+ + Ne reaction for different initial reactant states computed via TDQM-CC, TDQM-CS and
TIQM approaches.