Fields in the middle and bottom panels show magnified versions of the DCSs. 136 5.3 Total and vibrationally resolved cross sections for the reaction Ne + HeH+(v = . 0, j = 0) → NeH+(v0) + He for different values of the collision energy calculated by TIQM, TDQM-CC and QCT Methods.
STUDIES ON BIMOLECULAR REACTIONS
For the no-barrier reaction occurring mainly in the ISM, 68 excitation functions exhibit large value at very low energies due to the long-range attractive interactions. Numerous sharp oscillations originating from the quasi-bound states of the intermediate complex are manifested in the reaction probability curves, which are the signatures of indirect mechanisms associated with these reactions.
PROTON-RARE GAS SYSTEMS
RgH + systems
Emission spectra for ArH+ 92 and KrH+ 93 have been observed in hole cathode discharge experiments in rare gases and hydrogen mixtures. A number of studies have been carried out on the stability of RgH+ molecules using various theoretical methods.94–101 The binding energies of RgH+ increase with the increase in the atomic number of rare gases.
RgH + 2 systems
A potential well with a depth of ~0.54 eV corresponding to the equilibrium structure is present on the minimum energy path for the Ne+H+2 process. Dense resonances originating from the potential well were seen in the QM probability curves for the proton transfer reaction in the low energy region.
PROBLEM FORMULATION FOR THIS THESIS
THESIS OVERVIEW
Molecular energy calculations
- The Born-Oppenheimer approximation
- Solving the TISE
The coefficientsχi(R) in Eq. 2.2 are functions of the nuclear coordinates only. where E is the total energy of the system. However, when the electronic states are very close, the terms likehψj|∇α|ψii are large and cannot be neglected.
Construction of analytical PES
- Interpolation
- Fitting
VABC(3) (RAB, RAC, RBC) are the one-, two-, and three-body (3B) interaction energies at the corresponding geometries. The two-body interaction energies Vi(2) used in Eq. 2.15 can be calculated at different geometries by interpolation or by fitting the functional forms of Eq.
TIME DEPENDENT QUANTUM DYNAMICS
- The Hamiltonian operator
- Chebyshev real wave packet propagation
- Action of Hamiltonian operator on the wave function
- Damping function
- Preparation of the initial wave packet
- Total reaction probability
- Probabilities via interpolation
- Integral cross section and rate constant
Use of absorbing potential or damping function frees the procedure from unnecessary reflection of the wave function at the grid edges. In this work, a Legendre DVR is used as the associated Legendre polynomials (Plm(cosθ)), which are the eigenfunctions of the angular momentum operator. To evaluate the action of the rotational kinetic energy operators on the wave function, a transformation matrix Fn,j is used.
Here, xd is the starting point of the damping function and Ax is the power of the damping function.
TIME INDEPENDENT QUANTUM DYNAMICS
The initial state-specific thermal rate constants as a function of temperature (kvj(T)) are calculated from the initial state selected total ICSs as. The coupled-channel hyperradial equations are solved using a constant reference potential log derivative method between ρmax to ρmin in nsec evenly spaced sectors. The S-matrix for all the channels is then calculated from the final log-derived matrix by comparing with the asymptotic shapes.
To converge the reaction probabilities obtained through ABC calculations, three important convergence parameters semax, jmax and Kmax are first determined, which define the coupled channel basis set for the calculations.
QUASICLASSICAL TRAJECTORY CALCULATION
- The Hamiltonian
- Hamilton’s equations of motion
- Initial conditions
- Sampling of initial conditions for j > 0
- Integration
- Conservation check
- Product analysis
- Reaction probability calculation
- Integral cross section
- Differential cross section
The rotational quantum number of the product diatom can be calculated as a continuous number by equating p. The vibrational quantum number (v0) of the product diatom is determined following the procedure described in Ref. The Dunham expansion coefficients, Yl, m, are calculated by fitting the QM ro-vibrational energies of the product diatom to the expansion form, i.e., Eq.
Finally, the ro-vibrational states of the diatom product are assigned by rounding the integers to appropriate integers.
BOUND STATE CALCULATION FOR ABC
- Hamiltonian
- Initial wave packet
- Propagator
- Damping function
- Autocorrelation function
- Energy Spectrum
The initial wavepacket for the time propagation is chosen arbitrarily with a suitable overlap with the eigenstate of the systems as73. Here, x is the starting point of the damping function and ∆x is the width of the damping function. The intensity of the energy spectrum depends on the choice of the initial wave packet due to different overlaps with the eigenstates of the systems.
The WFs corresponding to eigenstates with eigenenergy En can be calculated by FT of the time-evolved WF (Ψ(t)) axis.
THE [HeHNe] + SYSTEM
- Electronic structure calculations
- The global minimum
- The local minima
- Generation of the global PES
- Global analytical PES
- Bound states of [HeHNe] + complex
The bond lengths (Req) of the most stable structures obtained via different theoretical approaches are shown in Table 3.1. Finally, over 20,000 triatomic ab initio energies were calculated, covering all major regions of the triatomic system. The linear and nonlinear parameters of the functional forms obtained by the fitting are shown in Tables 3.3 and 3.4 for the diatom and triatomic, respectively.
The contour plots of the analytical PES for six different He-H-Ne angles are shown in Figure 3.2.
THE [NeHNe] + SYSTEM
- Electronic structure calculations
- The global minimum
- The local minimum and the transition state
- Generation of the global PES
- Global analytical PES
- Bound states of [NeHNe] + complex
The contour maps of the analytical PES for four different internal bond angles are shown in Figure 3.10. Figure 3.11(b) presents contour plots of the interaction potential for different ∠ NeHNe values with equal NeH+ distances. In Figures 3.12 and 3.13, the contour maps of the analytical PES are presented in reactant Jacobi coordinates for Ne+NeH+ reactive systems.
For most geometries, there are very small differences between the analytic and ab initio energies.
CONCLUSIONS
Excellent agreements between analytical and ab initio energies are found in the strong interaction region and asymptotic regions for both PESs. The calculated global analytical PESs for these two systems are used in the next three sections to study the dynamics of reactive scattering. TDQM calculations were performed within CC and CS frameworks, and the importance of including Coriolis coupling in QM calculations is also discussed.
The rest vibrational states of some selective reactants and products are also depicted in the same image.
METHODS
Different initial state-specific total reaction probabilities are calculated within a collision energy range eV by computing the total flux of the energy-dependent WP passing through a solid surface. 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 of K) is shown for different initial reactant conditions at large J values.
Forv= 0 and 1, J = 50 and 70, respectively, are chosen to investigate the dependence of Kmax on the total reaction probability.
RESULTS AND DISCUSSION
Initial state selected dynamics
- Total reaction probabilities
- Total integral cross sections and opacity functions
As shown in Figure 4.3, the reaction has a threshold of ~0.29 eV for the ground-level vibrational state of the reactants. As a result, Figure 4.3 shows that there are no significant differences between the CS and CC response probabilities for (v, j) = (0,0) and J = 10, with the majority of the CC response probabilities coming from is from K = 0 state. As shown in Figure 4.1, vibrational excitation makes the reaction thermodynamically exothermic, resulting in a strong increase in reaction probabilities.
The opacity functions, the reaction probabilities as a function of specific collision energies, are plotted as (2J+ 1)PJ(Ec) in Figure 4.11.
State-to-state dynamics
- Rotational distributions
- Differential cross sections
QCT-HB approach fails to describe the threshold region, but reproduces the average behavior of the QM ICSs forEc≥0.5 eV. The overall shapes of the QCT distributions agree reasonably well with the QM distributions at all Ec's. However, the forward peaks of the TIQM DCSs at Ec = 0.5 and 0.65 eV are ~ 2 times larger than corresponding QCT DCSs.
Both QCT GB and HB approaches successfully describe the general behavior of TIQM DCSs.
Understanding the reaction mechanisms
The contributions of the reactive trajectories with collision timeτ to the ICS (στ) are plotted as solid blue lines in Figure 4.19 at Ec = 0.35 eV and 0.8 eV. Normalized cumulative distributions (σc) of the ICSs with respect to collision time are shown as dashed red lines in the. As can be seen in the cumulative distribution function plotted in Figure 4.19 for Ec = 0.8 eV, ~ 95% of the reaction proceeds via direct mechanisms.
Although τ is not exactly the lifetime of a colliding complex, for a tightly bound complex (as is the case for the trajectories presented in Fig. 4.21 (a) and (b)), τ can be approximated as the lifetime of the complex.
CONCLUSIONS
The smallest rotation period τrot of the intermediate complex formed in a complexing trajectory is estimated from the equilibrium geometry of the [HeHNe]+ complex, i.e. the collinear [HeHNe]+, asτrot = 2πI/L~. The Alchemy of Heaven: Searching for Meaning in the Milky Way; New York: Anchor Books/Doubleday, 1995. A number of collision energies have been chosen to investigate the state-to-state dynamics for the title reaction for ground ro-vibrational reactant state.
Quantum, statistical and quasiclassical trajectory studies of the Ne + HeH+→NeH++ He reaction on the ground electronic state.
INTRODUCTION
METHODS
In Figure 5.2, total reaction probabilities calculated, including different numbers of helicity terms, are plotted for different initial conditions and J = 60. All the TIQM calculations are performed using the ABC code of Manolopoulos and co-workers.37 Initial state selected total reaction probabilities for min selected Js and total ICSs at few selected energies were calculated to compare with the TDQM results. Convergence between three and five decimal places for the state-to-state response probabilities was achieved using this set of parameters.
In the QCT approach, the set of Hamilton's first-order differential equations of motion is numerically integrated with standard Monte Carlo sampling of the initial conditions.
RESULTS AND DISCUSSION
Initial state selected dynamics
- Total reaction probabilities
- Total integral cross sections
- Rate constants
Thus, the vibrational excitation of the reactant molecules reduces the reactivity of the title collision to a large extent in the low energy region. Here, it is worth noting that τ is not exactly the lifetime of the complex, but close. Contrary to the TDQM results, no decreasing behavior of the ICS TIQM is seen in the ultra-low energy regime with decreasing collision energy.
However, the long-range part of the PES used in the present set of calculations may not be accurate enough to account for the cross sections in the low-energy region.
State-to-state dynamics
- State-to-state cross sections
- Differential cross sections
A much finer detail on the dynamics of the title reaction is obtained with rotational distributions. The state-state cross sections for the reaction Ne + HeH+(v = 0, j = 0) → NeH+(v0, j0) + He were calculated at different values of the collision energy by TIQM, QCT-GB and QCT - Methods HB. The angular distribution at Ec = 0.01 eV obtained by the TIQM and QCT approaches are compared in Figure 5.12.
As observed for the rotational distributions at Ec= 0.3 eV, TIQM and QCT methods seem to describe the same overall dynamical features (see Figure 5.13 right panels).
Understanding the reaction mechanisms
However, as observed in Figure 5.16, for the trajectories with very large J, dispersion of the products occurs in a wider range of θ(θ≈0◦−120◦). The contributions of the reactive trajectories to the total ICSs with respect to the collision timeτ are presented as solid blue lines in Figure 5.17 atEc= 0.005 eV and 0.5 eV. The cumulative distribution of the ICS with respect to the collision time (as described in Chapter 4, Equation 4.3) is also plotted as dashed red lines in the insets of Figure 5.17.
For most of the indirect trajectories, τrot> τ is found, making the response nonstatistical.
CONCLUSIONS
Numerous resonances seen in the QM probability curves for small J values indicate the presence of metastable states of the intermediate collisional complex formed during the reaction. Rotational or vibrational excitation of the reactants significantly decreases the reactivity of the reaction in the low energy regime. In this chapter, the initial state-selected dynamics of the Ne + NeH+→NeH++ Ne reaction are reported by performing quantum mechanical studies of the electronic ground state.
In this chapter, the distribution dynamics of the reaction Ne + NeH+ → NeH+ + Ne is investigated by means of QM calculations.
METHODS
RESULTS AND DISCUSSION
Total reaction probabilities
Integral cross sections
Analysis of resonance peaks in the ICSs
Rate constants
CONCLUSIONS
