Quantum Trajectory calculations

In the fourth part, we study the effect of Cl on IC. It was applied to study the TtTt ->nTT transition of the pyrazine molecule. In this nonadiabatic process, the Cl of the TCK and nir PESs is believed to play a major role in the nonadiabatic fs transition. In fact, the Cl has been widely proposed to play the key factor in an IC, and quantum trajectory calculations have been used to calculate the IC rates [45]. However, this method cannot properly take into account of the initial conditions of the population and coherence of the system created by the fs pumping laser. In this chapter, we propose to develop a method to calculate the IC with conical intersections. It should be known that for the IC between Si and So in most molecules (in these cases, the energy gap between and So is of several eV), the surface crossings do not take place due to the anharmonic effect in the two PESs. Thus, the Cl should not play any role in these cases. We have proposed one method to calculate the IC rate of mt of the pyrazine molecule. The... 

For other hydrogen-transfer reactions, satisfactory ab initio potential energy surfaces [115, 116] and approximate quantum trajectory calculations of isotope effects [136] are also available in a few instances, but for the most part experimental measurements have been compared with values calculated on the basis of empirical or semi-empirical potential surfaces, making use of transition-state theory, and discussion has centred on the pros and cons of the surface used and the type of tunnelling correction made [3, 19]. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Quantum mechanical calculation of molecular dynamics trajectories can sim ulate bon d breakin g and frtrm ation.. Although you dt) n ot see th e appearance or disappearan ce ofhonds, you can plot the distan ce between two bonded atom s.. A distan ce excccdi n g a theoretical bond length suggests bond breaking. 

Quasiclassical calculations are similar to classical trajectory calculations with the addition of terms to account for quantum effects. The inclusion of tunneling and quantized energy levels improves the accuracy of results for light atoms, such as hydrogen transfer, and lower-temperature reactions. 

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

How to Find Instanton Trajectory and How to Incorporate Accurate ab initio Quantum Chemical Calculations. 

Now, the general formulation of the problem is finished and ready to be applied to real systems without relying on any local coordinates. The next problems to be solved for practical applications are (1) how to find the instanton trajectory qo( t) efficiently in multidimensional space and (2) how to incorporate high level of accurate ab initio quantum chemical calculations that are very time consuming. These problems are discussed in the following Section III. A. 2. 

We can calculate the thermal rate constants at low temperatures with the cross-sections for the HD and OH rotationally excited states, using Eqs. (34) and (35), and with the assumption that simultaneous OH and HD rotational excitation does not have a strong correlated effect on the dynamics as found in the previous quantum and classical trajectory calculations for the OH + H2 reaction on the WDSE PES.69,78 In Fig. 13, we compare the theoretical thermal rate coefficient with the experimental values from 248 to 418 K of Ravishankara et al.7A On average, the theoretical result... 

The analytic potential energy surfaces, used for the Cl + CH3Clb and Cl + CHjBr trajectory studies described here, should be viewed as initial models. Future classical and quantum dynamical calculations of SN2 nucleophilic substitution should be performed on quantitative potential energy functions, derived from high-level ab initio calculations. By necessity, the quantum dynamical calculations will require reduced dimensionality models. However, by comparing the results of these reduced dimensionality classical and quantum dynamical calculations, the accuracy of the classical dynamics can be appraised. It will also be important to compare the classical and quantum reduced dimensionality and classical complete dimensionality dynamical calculations with experiment. 

The utility of spline functions to molecular dynamic studies has been tested by Sathyamurthy and Raff by carrying out quassiclassical trajectory and quantum mechanical calculations for various surfaces. However, the accuracy of spline interpolation deteriorated with an increase in dimension from 1 to 2 to 3. Various other numerical interpolation methods, such as Akima s interpolation in filling ab initio PES for reactive systems, have been used. 

The classical description is quite different from the quantum. In classical dynamics we describe the coordinates and momenta simultaneously as a function of time and can follow the path of the system as it goes from reactants to products during the collision. These paths, called trajectories, provide a fnotion picture of collision process. The results of any real collision can be represented by computing a large number of trajectories to obtain distribution of post-collisions properties of interest (e.g. energy or angular distribution). In fact, the trajectory calculation means the transformation of one distribution function (reagent distribution, pre-collision) into another (product distribution, post-collision), which is determined by PE function. 

Another major, future advance in the quantum chemical computation of potential energy surfaces for reaction dynamics will be the ability to routinely compute the energies of molecular systems on the fly . The tedious and time-consuming process of fitting computed quantum chemical values to functional forms could be avoided if it were possible to compute the PES as needed during a classical trajectory or quantum dynamics calculation. For many chemical reactions, it should be practical in the near future to prudently select a sufficiently rapid and accurate electronic structure method to facilitate dynamics computations on the fly. 

As a first example, let us consider the time-dependent mean position of a normal mode xj of the system. In a mixed quantum-classical calculation, this observable is directly given by the quasiclassical average over the nuclear trajectories x t), that is. 

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