One of the basic purposes of chemical reaction dynamics including state-to-state reaction is to understanding reaction mechanism by analyzing the evolution relationships between quantum states of products and reactants. These relationships show the information of how bonds break and form gradually from reactant to product through the transition state. Within Born-Oppenheimer approximation, the motion of atoms in the molecule relies on the force created by the potential energy surface which is formed by averaging motion of all electrons in the molecule.
A tri-atomic reaction, AB+CA+BC, is the most basic reaction for state-to-state reaction dynamical studies. In this fundamental reaction involving only three normal-mode vibrations plus rotation and translation, both experiments and theoretical research have very much advanced in order to obtain the complete information for diatom-atom reactions. These indicate that vibrational or translational excitation may produce dramatic effects on chemical reactions. One of the most obvious effects is the influence of chemical reactivity. 1 Polanyi has shown that the vibrational excitation is more efficient than the translation excitation for increasing endoergic chemical reactivity for atom-diatom-reaction, namely late barrier. The concept of early barrier/late barrier becomes not so apparent for reactions containing polyatomic reagents since many degree of freedoms involves in polyatomic molecules in which
there are a large number of vibrational modes. The quantum5-7 and quasiclassical trajectory8 methods for atom-diatom reaction have been extended for studying polyatomic reactions. At the same time, several exciting experimental results for state-to-state reaction dynamics of polyatomic reactions have been found. Both these experiments and theoretical research have pointed out that there are essential differences between atom-diatom reactions and polyatomic reactions. Studying these differences open new horizons for scientists.
1-1 Quasi-classical trajectory, quantum and semi-classical methods
Quasi-classical trajectory (QCT), quantum scattering and semi-classical methods are three general methods to study chemical reaction dynamics theoretically. I would like to introduce a brief review of QCT first, the second is quantum method, and the third is semi-classical method. Finally, I will mention problems for each of three methods.1-1.1 Quasi-classical trajectory method
As mentioned above from Born-Oppenheimer approximation, the motion of atoms in molecule is just experiencing force induced by PES formed. If potential energy surface can be constructed as an analytical function of the internal coordinates
of the constituent atoms, it is possible to solve the motion of atoms on the surface for collision trajectory in classical way. The initial conditions can be determined by using coordinates and momentums of reactant molecule semi-classically, and then numerical integration of the Hamilton equation of motion for coordinate Qj and it’s conjugate momentum Pj=mdQj/dt are evaluated. Because of classical mechanical approach, there are several quantum effects that have to be added separately:
1. Particles penetrating into the region are classical forbidden.
2. The zero point energy (ZPE) is absence. Several approaches9-15 have been done to fix the problem of ZPE, but none of them are that reliable.
3. Classical mechanics does not recognize that reactants/products have quantum states, the given integration results may produce the vibrational energy of products different from quantum mechanics. But what we’re interested in is the quantum mechanic features as the vibrational or rotational product distribution, in order to obtain the correspondence, a binning process is needed. How to bin the region for different vibration and rotation quantum number will influence the accuracy of state to state dynamics, but this defect may cancel out for overall distribution.
4. Some of the initial-condition parameters of reactants would not be specified during experiments, but the specification of these parameters is required in doing calculation, such as impact parameter, the orientation of the reactants, the phase of
the reactant vibration with respect to the time of collision, and so on. So, averaging over these uncontrolled parameters of collision in the classical mechanic calculation is required.
The defects of QCT can be easily seen from the mentioned above, the omission of the quantum effects make it experienced difficulties at threshold; when the total energy is slightly above the minimum required to cross the potential barrier, the ZPE energy tends to be converted into translation energy, which is not allowed in quantum mechanics. Besides this general problem for QCT, dealing with large system by QCT is still a challenging problem. The reason for this is that determination of vibrational actions; binning process, for large system is not a standard process. Even if there is standard process for binning, obtaining the PES for large system is still difficult.
1-1.2 Quantum method
Apart from using the Newton mechanics to describe the motion of atoms; their motion is governed by a wave equation in quantum mechanics. The general strategy is to solve the time dependent or time independent Schrodinger equation in the superposition wave function of channels interested and then obtain the quantities needed, e.g. state to state cross section, by giving suitable boundary condition.
For exact quantum calculation, it may consume large amount of computational
time. Since all accessible states; rotational and vibrational states, should be considered in the calculation. Of course, one could calculate all the information, but are these considerable quantities that demanding? Doing that requires huge effort, so a modified method; reduced dimension quantum method is developed for this need. For four, five, or six atoms system, this method is accessible. But for larger system, it’s a big problem.
So there’s reduced dimension quantum method (modified quantum method). However for too large system, the modified quantum method is still not ok.
1-1.3 Semi-classical method
This method would be explained clearly in chapter 2. Same as the two methods mentioned above. There is no general strategy for large system.
1-2 Motivation
As mentioned above, no matter quasi-classical trajectory, modified quantum method or semi-classical method, there still exist difficulties for studying dynamics of large system. So, this motivates us to produce a more general way; which is practical to large system for dealing with state-to-state reaction dynamics. For system more than 5 atoms, it’s not that easy to obtain exact quantum results for reactions. In order to make sure the feasibility of our method, we would like to apply this approach to
OH+H2H2O+H; which has been studied thoroughly from both theory and experiments. Fristly, I would like to give a brief review of the calculations and experiments which have been done for OH+H2H2O+H.
OH+H
2H
2O+H
The thermal rate constant for OH+H2; which is important in combustion chemistry, has been measured over a wide range of temperature(T=250~2000K) for both thermal distribution of reactants16-18 and there is obvious evidence18 that k(T) shows non-Arrhenius behavior with a suggested best fit expression k(T) = 1.66
*10-16*T1.6*exp (-1660K/T) cm3/molecule*s. G.P. Glass19, Spencer et al 20 and R.
Zellner 18,21 investigate the influence of vibrational excitation on reaction rate.
They18,19-20 found that not small amount of enhancement (H2(v=1)/H2(v=0)>100) for H2(v=1), but smaller effect(50% less than the excitation of H2) for OH(v=1); which is reasonable since OH can be seen as a spectator in the reaction. Later, R. Zellner21 thought the large enhancement may be due to the possibilities of the contribution of a translational rate acceleration of OH +H2 in the flow system. After correction, the enhancement is rather small (5~66) compared to before (>100). Many fast-atom experiments22-29 have been done for H+H2O(and it’s isotopomers) in the final vibrational and rotational distribution of H2 and OH were determined.
Isaacson’s30 group reported a reaction path potential, based on Walch and
Dunning’s calculation, and use it in transition state theory calculation for rate constant with complex tunneling effect. Cohen et al31 used Isaacson’s reaction path potential in transition state theory calculation of rate constant with new treatments of anharmoncity of the transition state force field. A potential curve along a tunneling path is also calculated by Truong and Evans32; which is used in calculating the rate constant.
Schatz and Elgersma33 based on the method for saddle point properties developed by Walch and Dunning34 to construct a global potential energy surface. Extensive quasiclassical trajectory calculations33,35,36 based on this PES have been done, they point out interesting mode specificity in the H+H2O reaction in highly vibrational excited states. There are several other QCT studies being reported later37-40
Quantum dynamics studies for the OH+H2 reaction have been reported by Wang
& Bownan41, Clary and coworkers42, Miller and coworkers43 , Zhang44, and Neuhauser45. Those of Miller and coworkers , Zhang, and Neuhauser are full dimension. Wang &
Bownan RD-AB theory. Clary and coworkers use RBA-AB theory.
The cumulative reaction probability for J=0 has been calculated with full dimension; by Miller, RD-AB; by Wang, and RBA-AB, by Clary. The results show that there is a quite good agreement for RD-AB and RBA-AB with full dimension at low E, but a little deviation at higher E.
Miller also calculated the thermal rate constant for OH+H2 by J-shift
approximation. The result for thermal rate constant along RD-AB (J-shift approximation), RBA-AB (centrifugal-sudden approximation), and QCT are also calculated. The RD-AB result is in good agreement with full dimension (RD-AB results are reasonable, since J=0 CPR are almost the same and they both use J shift method), but RBA-AB somewhat higher (The difference comes from using centrifugal-sudden approximation). Finally, QCT’s result show good agreement with full dimension at higher temperatures, but underestimate at lower temperatures (The lack of tunneling effect for QCT method). The comparisons between full dimensions show that taking bending motion as a adiabatic motion still give reliable result and QCT method can’t deal with dynamics at low temperature.