4.3 RESULTS AND DISCUSSION
4.3.2 State-to-state dynamics
0 2 4 6 8 10
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Cross Section (a02 )
Ec (eV)
TDQM TIQM QCT-GB QCT-HB
Figure 4.13: Total integral cross sections plotted as a function of collision energies calculated by means of different theoretical approaches.
corresponding QM ones in the threshold region, similar to this kind of observations no- ticed for many other systems.39–41 The resonance features of the QM probabilities can not be reproduced via QCT calculation because of inherent limitations associated with QCT methodology. In QM approach, all the regions of the phase space are described by the WF, and thus probabilities with resonances can be obtained. But in QCT, a Monte Carlo sampling is done over the phase space and the results obtained from QCT calculations are computed as the averaged values. However, resonance peaks have been observed in QCT probabilities for F + OH system due to appearance of a large number of trappedtrajectories.42
A comparison between the QCT and QM total ICSs is presented in Figure 4.13.
QCT-GB approach successfully reproduces the QM ICSs near the threshold region but differences can be seen atEc= 0.45 - 0.65 eV range. QCT-HB approach fails to describe the threshold region but reproduces the average behavior of the QM ICSs forEc≥0.5 eV. Similar observations were made for many other endothermic reactions.41,43–45
4.3.2.1 Rotational distributions
TIQM and QCT product rotational distributions at Ec = 0.35, 0.5, 0.65 and 0.8 eV are presented in Figure 4.14. The overall shapes of the QCT distributions fairly agree with the QM distributions at all Ecs. However, neither of the QCT binning methods
0 0.5 1 1.5 2
0 1 2 3 4 5
Ec = 0.35 eV
v' = 0 Cross Section (a02 )
0 2 4 6 8
Ec = 0.5 eV
v' = 0 TIQM QCT-GB QCT-HB
0 0.5 1 1.5 2
0 2 4 6 8 10
v' = 0
v' = 1
Ec = 0.65 eV
Cross Section (a02 )
HeH+ Rotational State 0 2 4 6 8 10 12 Ec = 0.8 eV
v' = 0
v' = 1
HeH+ Rotational State Figure 4.14: TIQM and QCT product rotational state distributions for the He + NeH+(v = 0, j = 0) → HeH+(v0, j0) + Ne reaction calculated at different collision energies. QCT results obtained following both HB and GB approaches are plotted.
is able to reproduce the correct distribution around the maxima of the QM results for v0= 0. QCT-HB results at Ec= 0.35 eV are not plotted in Figure 4.14 because at this energy QCT-HB approach fails to calculate the correct ICSs (See Figure 4.13, QCT-HB results differ to a large extent from the exact QM results). At Ec = 0.35 eV, QCT- GB method successfully reproduces the average behavior of the TIQM distribution, which is related to the excellent agreement between QCT-GB and QM ICSs at that energy (See Figure 4.13). Rotational distribution for v0 = 0 at Ec = 0.5 eV shows a better agreement of QCT-HB results than that of QCT-GB with TIQM distributions for low product rotational states and around the maximum. Although the QCT-HB and QM ICSs are nearly indistinguishable atEc= 0.5 eV, QCT-HB distribution is little hotter than its QM counterpart. As can be seen in Figure 4.14, HeH+ with v0 = 1
is formed only via QCT-HB methods at Ec = 0.65 eV, though this state is quantum mechanically closed. Rotational state resolved cross sections for v0 = 1 calculated via different methods atEc= 0.8 eV agree quite well. It is quite obvious in Figure 4.14 that the rotational distributions obtained from QCT-GB method have excellent agreements with QM distributions for higher j0 values at all Ecs reported here. Formation of the products with rotational states beyond allowed quantum states can be seen in QCT-HB results, whereas those states are suppressed by following the GB approach.
4.3.2.2 Differential cross sections
0 2 4 6
8 Ec = 0.35 eV
0 20 40 60 80
Differential Cross Section (a02 )
Ec = 0.5 eV
TIQM QCT-GB QCT-HB
0 4 8 12 16
0 30 60 90 120 150 180
0 15 30 45 60
0 30 60 90 120 150 180
CM Scattering Angle (˚) Ec = 0.65 eV
0 4 8 12 16
0 30 60 90 120 150 180
Figure 4.15: Total differential cross sections calculated for He + NeH+(v= 0, j= 0)
→HeH+ + Ne reaction at Ec = 0.35 (top), 0.5 (middle) and 0.65 eV (bottom). The insets in the middle and bottom panels show the zoomed versions of the DCSs.
0 10 20 30 40
50 Ec = 0.8 eV, Total
0 4 8 12 16
0 30 60 90 120 150 180
0 10 20 30 40
Differential Cross Section (a02 )
v' = 0
TIQM QCT-GB QCT-HB
0 0.5 1 1.5 2 2.5
0 30 60 90 120 150 180
CM Scattering Angle (˚) v' = 1
Figure 4.16: Total and vibrational state resolved differential cross sections for the He + NeH+(v= 0, j= 0)→HeH+(v0) + Ne reaction atEc= 0.8 eV. The inset in the top
panel shows the zoomed versions of the DCSs.
Total differential cross sections obtained by means of TIQM, QCT-GB and QCT-HB at three different values of collision energy with reactants in ground ro-vibrational state are presented in Figure 4.15. It is observed in Figure 4.15 (top) that atEc= 0.35 TIQM DCS shows a forward peak, which is almost double in magnitude of the backward peak.
However, QCT-GB DCS is totally asymmetric and the products clearly favor a forward scattering. The intensity of the forward peak of the QCT-GB DCS is greater than the TIQM one, whereas reverse is true for the backward peak. As can be seen in Figure 4.15 (middle and bottom panels), the overall shapes of the DCS curves obtained from all the methods atEc= 0.5 and 0.65 eV are similar in nature with a marked preference for the forward direction. At these energies, both the QCT approaches produce similar
results and the backward peaks of the QCT DCSs are in an excellent agreement with the TIQM one. However, the forward peaks of the TIQM DCSs at Ec = 0.5 and 0.65 eV are∼ 2 times larger than corresponding QCT DCSs.
In Figure 4.16, total and vibrational state resolved TIQM and QCT DCSs calculated at 0.8 eV are plotted. All the DCS curves are asymmetric in nature with intense peaks at
∼0◦. Both the QCT GB and HB approaches successfully describe the overall behavior of the TIQM DCSs. However, the forward and backward peaks of the total DCSs computed via QCT approaches slightly underestimate the TIQM ones, and, as products with v0
= 0 state contributes more to the total DCSs, similar observations are noticed for v0
= 0 results. For v0 = 1, peaks observed at ∼ 0◦ and ∼ 180◦ in the QCT-HB DCSs are in good agreement with the exact results. The clear preference of the products to get forward scattered with a predominant peak at ∼0◦, as observed in the DCS plots, indicates that the overall dynamics is not governed by intermediate complex formation with longer lifetimes.