Atomic trajectories, computation

Classical ion trajectory computer simulations based on the BCA are a series of evaluations of two-body collisions. The parameters involved in each collision are tire type of atoms of the projectile and the target atom, the kinetic energy of the projectile and the impact parameter. The general procedure for implementation of such computer simulations is as follows. All of the parameters involved in tlie calculation are defined the surface structure in tenns of the types of the constituent atoms, their positions in the surface and their themial vibration amplitude the projectile in tenns of the type of ion to be used, the incident beam direction and the initial kinetic energy the detector in tenns of the position, size and detection efficiency the type of potential fiinctions for possible collision pairs. 

Having updated the position, we again calculate the force on each atom for the new configuration of the system and complete the next iteration by obtaining the new acceleration and integrating. In this way we compute the atomic trajectories as a function of time. 

The experimental and computational study of bacterial thioredoxin, an E. coli protein, at THz frequencies is presented. The absorption spectrum of the entire protein in water was studied numerically in the terahertz range (0.1 - 2 THz). In our work, the initial X-ray molecular structure of thioredoxin was optimized using the molecular dynamical (MD) simulations at room temperature and atmospheric pressure. The effect of a liquid content of a bacterial cell was taken into account explicitly via the simulation of water molecules using the TIP3P water model. Using atomic trajectories from the room-temperature MD simulations, thioredoxin s THz vibrational spectrum and the absorption coefficient were calculated in a quasi harmonic approximation. 

Latterly, increasing use has also been made of Quantum Molecular Dynamics (QMD), based on the pioneering work of Car and Parrinello (1985) (see Chapter 8). The Car-Parrinello method makes use of Density Functional Theory to calculate explicitly the energy of a system and hence the interatomic forces, which are then used to determine the atomic trajectories and related dynamic properties, in the manner of classical MD. As an ab initio technique, QMD has the advantage over classical simulation methods that it is not reliant on interatomic potentials, and should in principle lead to far more accurate results. The disadvantage is that it demands far greater computing resources, and its application has thus far been limited to relatively simple systems. 

Although precise motions of individual atoms are computed in the MD simulation, the complex nature of their trajectories makes their interpretation difficult without statistical methods, which employ functions that express the probability of a given event or condition. Some of the commonest of these are now... 

Initially, counterions were distributed randomly around the nanocapsule, which was filled and surrounded by water. In the internal cavity of the POM 172 water molecules were placed 72 waters fulfilling the molybdenum coordination sphere of the pentagonal Mo(Mo)s units and a structureless 100 H2O cluster. Following a standard protocol [24], a large number of configurations were collected through the MD trajectory and analyzed. Then, the radial (RDF) and spatial (SDF) distribution functions of the centers of the capsule-oxygen water atoms were computed. 

Classical mechanics provides a direct route from the potential energy surface to the dynamics of the collision, namely, the (numerical) solution of the classical equations of motion for the atoms. The solution uses Newton s law of motion to determine the position of each atom as a function of time. This output is known as a trajectory. It allows us to visualize how each atom moves as the reaction is taking place. Trajectory computations are carried out for two purposes. First, as a diagnostic of trends, i.e., features of the dynamics arising from different featnres of the surface or from changes in reactants energies, masses, and so... 

J. Phys. Chem. 83,1000 (1979)]. Trajectory computations for D + FCI DF + Cl, not shown in the figure, indicate that the DF vibrational distribution is essentially the same as that of HF when the results are plotted vs. fj, the fraction of the available energy in the vibration of the product. As discussed in Section 5.1.5.1, the reaction is expected to proceed by two mechanisms, a direct abstraction and a migratory route where the H atom approaches from the direction of Cl. The migratory route is favored by higher-impact-parameter coiiisions and so leads to more forward scattering [for earlier results for the H -r F2 HF -r F reaction see N. Jonathan,... 

Figure 5.17 CO vibrational state distribution following the highly exoergic 0( P) + CS CO + S( P) reaction. Dots experimental [adapted from G. Hancock, B. A. Ridley, and I. W. M. Smith, J. Chem. Soc. Faraday Trans 68, 2117 (1972)]. Open symbols trajectory computations for thermal (A) and translationally hot ( ) atoms [adapted from D. Summerfield at al., J. Chem. Phys. 108,1391 (1997)]. Dashed line a fit of the distribution by a linear vibrational surprisal as discussed in Section 6.4 [adapted from H. Kaplan, R. D. Levine, and J. Manz, Chem. Phys. 12, 447 (1976)].

Figure 5.19 Bond distances (in atomic units) vs. time from a classical trajectory computation for the KCi + NaBr KBr + NaCI reaction [P. Brumer, Ph.D. thesis (1972)]. The most stabie structure of the tetratomic ionic intermediate is shown as an insert.

It is quite straightforward to perform quasiclassical trajectory computations (QCT) on the reactions of polyatomic molecules providing a smooth global potential energy surface is available from which derivatives can be obtained with respect to the atomic coordinates. This method is described in detail in Classical Trajectory Simulations Final Conditions. Hamilton s equations are solved to follow the motion of the individual atoms as a function of time and the reactant and product vibrational and rotational states can be set or boxed to quantum mechanical energies. The method does not treat purely quantum mechanical effects such as tunneling, resonances. or interference but it can treat the full state-to-state, eneigy-resolved dynamics of a reaction and also produces rate constants. Numerous applications to polyatomic reactions have been reported. ... 

If A transforms to B by an antara-type process (a Mdbius four electron reaction), the phase would be preserved in the reaction and in the complete loop (An i p loop), and no conical intersection is possible for this case. In that case, the only way to equalize the energies of the ground and excited states, is along a trajectory that increases the separation between atoms in the molecule. Indeed, the two are computed to meet only at infinite interatomic distances, that is, upon dissociation [89]. 

In numerous cases an atomically detailed picture is required to understand function of biological molecules. The wealth of atomic information that is provided by the Molecular Dynamics (MD) method is the prime reason for its popularity and numerous successes. The MD method offers (a) qualitative understanding of atomic processes by detailed analysis of individual trajectories, and (b) comparison of computations to experimental data by averaging over a representative set of sampled trajectories. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

Figure 5 Time dependence of RMSD of atomic coordinates from canonical A- and B-DNA forms m two trajectories of a partially hydrated dodecamer duplex. The A and B (A and B coiTespond to A and B forms) trajectories started from the same state and were computed with internal and Cartesian coordinates as independent variables, respectively. (From Ref. 54.)...

---