Trajectory computations

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. 

Equations (4) are called quasi-Hamiltonian because, even though they employ generalized velocities, they describe the motion in the space of canonical variables. Accordingly, numerical trajectories computed with appropriate integrators will conserve the symplectic structure. Eor example, an implicit leapfrog integrator can be expressed as... 

A full description of the outline and the capabilities of the NEWTON-X package is given elsewhere [38], In brief, the nuclear motion is represented by classical trajectories computed by numerical integration of Newton s equations using the Velocity-Verlet algorithm [39], Temperature influence can be added by means of the Andersen thermostat [40], The molecule is considered to be in some specific... 

The first step in the dynamics, the S Sj deactivation, is completed in only 52 1 fs, presenting the expected monoexponential decay profile, as can be seen in Figure 8-1 la. This figure shows the fraction of trajectories in each state between 0 and 200 fs for the 35 trajectories computed. Between 20 and 30 fs and again at 37 fs it is possible to observe a revival of the S2 state occupation. The fraction of trajectories in Sj is not shown in the figure for sake of clarity. It is just complementary to the fraction of trajectories in S2. Thus, a revival in S2 is companied by a decrease in Sj occupation. The revivals in the S2 occupation occur when the total number of... 

This chapter concentrates on the experimental determination of product energy distributions and their interpretation in terms of the dynamics of collisions. In a volume concerned with excited states, it seemed appropriate to bias the article in favor of the spectroscopic methods and results, at the expense of the crossed-beam studies. Electronically adiabatic reactions, which pass smoothly from reactants to products in their electronic ground states, are emphasized, since the results of such processes may be compared with trajectories computed with the equations of classical, rather than wave, mechanics, and the effects of kinematic factors on the sharing of energy can be explored. 

A BRIEF INTRODUCTION TO MOLECULAR DYNAMICS TRAJECTORY COMPUTATIONS... 

While the outcome of an individual trajectory depends in a sensitive way on the details at the surface, the ensemble average is robust. In other words, if an ensemble of initial conditions of the cluster is reused, the yield of dissociation is essentially unchanged even though the new run samples a somewhat different set of conditions at the surface. Similarly, the ensemble-averaged yield is unchanged if a new set of cluster initial conditions is used. We will return to this point later but it is an important practical consideration since it means that a finite number of trajectory computations suffices. 

The GCE model is more detailed than the TCE and VFD chemistry models. However, only through the availability of detailed information about the dependence of reaction cross sections on the precollision states of the colliding particles can the parameters g, P, and 7 be determined. In the results section, such information for the exchange reaction involving formation of nitric oxide is available from detailed trajectory computations and is used to determine suitable parameters for the GCE model for this reaction. 

Wadsworth and Wysong made a detailed assessment of the threshold line model (and other dissociation models) for hydrogen dissociation by making comparison with quasiclassical trajectory computations. They found that the original forms of the threshold line models proposed by Macheret and Rich have to be extended to more complete forms in order to avoid singularities at specific values of the collision energies. Some of their findings are shown in the results section. 

The subscript E on the right-hand side denotes fixed E (undamped trajectory). Computing x(t)x(0) without damping is consistent with the low-friction limit where damping is assumed to be small on the time scale associated with Z t) [see Eq. (5.29)]. Comparing the two results for (dE/dt)j.=o using Eq. (5.49) we get the result Eq. (5.57). Equation (5.57) provides a convenient numerical way to compute e(E) all one needs is to run a trajectory over the undisturbed molecular motion at the given E for a time of several t. ... 

Figure 7. Two laser measurements of the translational energy dependence of the reaction cross section for H + D2 - HD + D. Rectangles experimental results [K. Tsukiyama, B. Katz, and R. Bersohn, J. Chem. Phys. 84, 1934 (1986) and private communication]. Points Classical trajectory computations on the best available ab initio potential energy surface [by N. C. Blais and D. G. Truhlar (triangles) and by I. Schechter (circles)].

Bill Hase received his Ph.D. in chemistry in 1970, working in the research area of experimental physical chemistry under the direction of John W. Simons at New Mexico State University. His research included studies of the methylene singlet-triplet energy gap and of the unim-olecular decomposition of vibrationally excited alkane and alkylsilane molecules prepared by chemical activation. His career as a computational chemist began during his postdoctoral work with Don Bunker at the University of California, Irvine. In 1973 he joined the Chemistry Department at Wayne State University, where he remained until 2004, when he assumed the Robert A. Welch Chair in Chemistry at Texas Tech University. He remembers that his hrst computational chemistry classical trajectory computer program was written in assembly language and run on a PDP-10. 

Figure 2.25 Variation in the three in-plane CO-Cr-CO angles along the entire CrlCOlg and Cr(CO)5 trajectory computed at the CASSCF/6-31G quasi-classical/TSH level. In the first phase a symmetric bend is excited and the angles a and y increase. The second phase corresponds to vibrational energy transfer from symmetric to antisymmetric bending coordinates. In the final phase, the molecule oscillates in a square planar minimum energy well with a frequency of 98 cm 1. (Adapted from Paterson, M.J., Hunt, P.A. and Robb, M.A., J. Phys. Chem. A, 106, 10494-10504, 2002.)...

The results obtained by averaging over the stochastic trajectories computed according to (3.25) are equivalent to those obtained via the solution of the corresponding Fokker-Planck equation. In many cases it is more convenient to work with the Langevin equation. We discuss a specific example in detail in Section XII. 

The time autocorrelation function can be written as a transition dipole correlation function, a form that is equally useful for an inhomogeneously broadened spectrum. This is the form that is extensively used to discuss the spectral effects of the environment (32-34). The dipole correlation function also provides for the novice an intuitively clear prescription as to how to compute a spectrum using classical dynamics. For the expert it points out limitations of this, otherwise very useful, approximation. The required transformation is to rewrite the spectrum so that the time evolution is carried by the dipole operator rather than by the bright state wave packet. The conceptual advantage is that it is easier to imagine what the classical limit will be because what is readily provided by classical mechanics trajectory computations is the time dependence of the coordinates and momenta and hence, of functions thereof. In other words, in our mind it is easier to... 

M. Ben-Nun and R. D. Levine, Conservation of zero-point energy in classical trajectory computations by a simple semiclassical correspondence, J. Chem. Phys. 101 8768 (1994). 

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

Therefore, the difference between the trajectory computed with the method of undetermined parameters (i.e., using the ["y]) and the trajectory computed with the analytical method (i.e., using the [) (to))) is of 0(8f ). This is the same order of error assumed in the integration algorithm and present in the analytical method before this modification. Hence the modification in the analytical method leading to the method of undetermined parameters does not introduce an error of lower order in 8r than the error already present. The errors in the trajectory computed with the analytical method and the trajectory computed with the method of undetermined parameters are both of 0(8t ), whereas in the latter method the constraints are perfectly satisfied at every time step (compare Eq. [13] with Eq. [36]). 

The harmonic oscillator is of course an integrable system. When the underlying system is a chaotic Hamiltonian system, the trajectories from any collection of initial conditions become disordered (even while preserving the energetic constraints defined by their initial states). The hope (in some cases, naive) is that typical trajectories computed using a symplectic method applied to such a chaotic system provide space filling curves on the = E perturbed energy surface. 

Figure 18 shows the VADW rotational distribution for 0( P) reacting with the tertiary hydrocarbon HC(CH3)3 s Isobutane Into v =l at Etr 22.2 kJ mor (Clary et al. (271). The VADW result is compared with experimental measurements (Andresen and Luntz (21) and with quasiclassical trajectory computations (Luntz and Andresen (561). The... 

Aoiz FJ, Banares L, Herrero VJ (1998) Recent results from quasiclassical trajectory computations of elementary chemical reactions. J Chem Soc-Faraday Trans 94 2483-2500... 

W.L. Hase. Mercury a general Monte-Carlo classical trajectory computer program, QCPE 453. 

Trajectories computed for later times were directed northeast, east, and southeast, into the area of the former Soviet Union. These trajectories were computed for the lowest atmospheric layer, with a height less than about 200 to 300 m. Because some pollutants may have risen to higher altitudes, we also computed the trajectories of air masses in other atmospheric layers. The transport directions at higher altitudes turn in a clockwise direction. The contribution of these pollutants to observational data in Finland is therefore small. 

The measured ammonia and ammonium concentrations at the Finnish EMEP measurement stations of Virolahti and Ahtari show a maximum on March 21, 1989 this time corresponds to computed arrival times. However, such maximum values are not uncommon in springtime. A detailed analysis of backward trajectories, computed for these measurement stations, and the measured concentrations do not show conclusively that the measured maximum values would have been caused by the accident (Kukkonen et al., 1993). Most of the NH due to the accident may have escaped the available measurement stations. 

We also received a number of phone calls from individual citizens in Finland after the accident, mainly reporting eye irritation. These observations were made during the evening of March 21 in a limited area, on or near the southern coastline of Finland, about 10 to 50 km west of Helsinki. The location and time of these observations are consistent with the results of the trajectory computations in the lowest atmospheric layer (Fig. 40.2). We received the phone calls during March 22 to 25, before any information about the accident had been disseminated to the public in Finland. 

Hase WL. MERCURY a general Monte-Carlo elassieal trajectory computer program, QCPE 3, 453, 1983. An updated version of this eode is VENUS96. Hase WL, Duehovic RJ, Hu X, Komornik A, Lim KF, Lu D-H, Peslherbe GH, Swamy KN, van de Linde SR, Varandas AJC, Wang H, Wolf RJ. QCPE Bull. 1996 16 43. 

Compute (IQ ms). Each vehicle computes the trajectory for all the level of services that the vehicle supports in each test case using the acquired information since the last round. The time costs of all the advanced driver assistance systems is 0 n) with preprocessing time of 0 n log(n)), where n is the number of vehicles. During our three vehicle experiments, we observed a sub-millisecond trajectory computation cost but for redundancy reasons we assume 10 ms. 

---