Classical trajectory Monte Carlo

Fig. 13.6 Classical trajectory Monte Carlo (CTMC) ionization and charge transfer cross sections, with statistical standard deviations, for specific initial f plotted against reduced impact speed v/ve. The cross sections are given in units of a 2, where a = rraQ. Circles are for = 2, squares for i = 14. Also included for comparison are points from approximate CTMC calculations with the Na target core held fixed for i = 14 (triangles) and ( = 2...

Fig. 4.26. Plot of the ratio aPS/aH for positron and proton collisions with helium, versus the velocity of the projectiles (following Schultz and Olson, 1988). The curve is the result of their classical trajectory Monte Carlo calculation. The triangles are based on the positron data of Fromme et al. (1986) whilst the circles are from Diana et al. (1986b).

All these contributions add up to a total antihydrogen formation cross section of approximately 2 x 10-15 cm2 in the antiproton energy range 2-10 keV where the charge-exchange production mechanism is likely to be most effective. This value is consistent with results obtained by Ermolaev, Bransden and Mandal (1987), who used the classical trajectory Monte Carlo method, and also with the results of a recent experiment (Merrison et al., 1997) which measured the hydrogen atom formation cross section via reaction (8.22). 

Classical-trajectory Monte Carlo calculation for collision processes of e++(e-p) and pA + p). Phys. Rev. A 32 2640-2644. 

The fits of Janev et al. [12] stem from a compilation of the results obtained with different theoretical approaches (i) semi-classical close-coupling methods with a development of the wave function on atomic orbitals (Fritsch and Lin [16]), molecular orbitals (Green et al [17]), or both (Kimura and Lin [18], (ii) pure classical model - i.e. the Classical Trajectory Monte Carlo method (Olson and Schultz [19]) - and (iii) perturbative quantum approach (Belkic et al. [20]). In order to get precise fits, theoretical results accuracy was estimated according to many criteria, most important being the domain of validity of each technique. 

An important difference between the wave treatment and the SCA is that energy conservation is retained exactly and when the energy of the projectile is less than the energy required to excite the state under consideration the cross section is zero. This is called a threshold. In fig. 5.8 is plotted the measured ratio for ionization produced by equal-velocity positron and proton projectiles incident on helium. Just above the threshold, 24eV, Uie electron cross section falls below that due to the heavier proton. The ratio is compared to a CTMC (classical trajectory Monte Carlo) prediction and also to the ratio of the PWBA to the SCA cross sections showing the importance of the mass of the projectile to the result. The CTMC method will be discussed in more detail shortly. 

Classical Trajectory Monte Carlo Calculations (CTMC) have been reviewed recently by Schulz et al. [5.24]. Classical treatments of collisions retain with complete kinematic accuracy many of the features which we think of as the physics of a reaction. Their advantage is they are very flexible and can be quickly changed to describe many varied experimental systems. They are however not inexpensive in terms of computing time essentially statistical in nature, it is quite time consuming to produce meaningful cross sections for unlikely events. 

Kuntz, P. J. and Schmidt, W. E, A classical trajectory Monte Carlo model for the injection of electrons into gaseous argon, /. Chem. Phys., 76,1136,1982. 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

The MD/QM methodology [18] is likely the simplest approach for explicit consideration of quantum effects, and is related to the combination of classical Monte Carlo sampling with quantum mechanics used previously by Coutinho et al. [27] for the treatment of solvent effects in electronic spectra, but with the variation that the MD/QM method applies QM calculations to frames extracted from a classical MD trajectory according to their relative weights. 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

While the two methods are, at face value, quite different in the ways in which full quantum dynamics is reduced to quantum-classical dynamics, there are common elements in the manner in which they are simulated. The Trotter-based scheme for QCL dynamics makes use of the adiabatic basis and is based on surface-hopping trajectories where transitions are sampled by a Monte Carlo scheme that requires reweighting. Similarly, ILDM calculations make use of the mapping hamiltonian basis and also involve a similar Monte Carlo sampling with reweighting of trajectories in the ensemble used to obtain the expectation values of quantum operators. 

Goursaud, S., Sizun, M., and Fiquet-Fayard, F. (1976). Energy partitioning and isotope effects in the fragmentation of triatomic negative ions Monte Carlo Scheme for a classical trajectory study, J. Chem. Phys. 65, 5453-5461. 

The Monte Carlo method includes both a random sampling scheme and an importance sampling scheme. Both sampling schemes have been used in Section 4.1 on classical trajectory calculations. 

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