Molecular Dynamics Simulation
While fully quantum mechanical simulations provide the exact description of collisional energy transfer, such calculations are not practical for the treatment of large polyatomic systems. However, it is believed that classical trajectory methods can often provide useful accuracy in describing many aspects of the dynamics of molecular collisions. This is particularly true when the molecular mass is high, and the internal quantum energy level spacing is small. Both these conditions hold for the Kr+Az system. A typical classical trajectory will start with a large Kr-Az separation and an incoming velocity commensurate with the collision energy. The Az molecule is initially excited with a fixed internal energy that is distributed randomly. The trajectory is then propagated in time and the collision takes place.
The propagation is continued until the Kr-Az separation becomes
sufficiently large to insure that the collision is over. To simulate the
dynamics of the system, a large ensemble of trajectories is used that
randomly sample the impact parameter and the internal vibrational
and rotational states of the target molecule. The final observables are
then constructed using the distributions of final conditions obtained
from the ensemble. Thus, the simulation can be regarded as
consisting of three separate steps: (1) creating the correct ensemble
of initial conditions, (2) numerically propagating the trajectories
through the collision, and (3) extracting the observables from the final
conditions of the trajectories.
A. The Preparing of Initial Conditions
The initial conditions for the classical trajectories provide the boundary conditions necessary to solve the differential equations of motion.
These initials conditions are chosen to model the experimental
conditions being considered. Here we seek to describe the molecular
beam collision of Kr and Az at fixed collision energy. The initial
conditions of Az (i.e., initial Cartesian coordinates and momentum for
each atom in Az) are chosen by microcanonical normal mode
sampling which is an exact microcanonical sampling technique
21for
harmonic oscillators and a complementary approximate method to
orthant sampling for anharmonic oscillators. The initial vibrational
energy of Az consists of the excitation energy, 4.66 eV, and the zero
point energy of vibration, 4.1eV. Thus, the initial vibrational energy of
Az is 8.76 eV or 197.69 kcal-mol
-1. The microcanonical sampling
method produces a trajectory of fixed total internal energy, but is
randomly distributed over all internal modes of vibration. The
rotational temperature of Az produced in the experiment is 2 K. Thus,
the rotational coordinates of the Az molecule used in the simulation
are randomly selected according to canonical Monte Carlo sampling
at this temperature. The initial translational energy of the Az-Kr
system is held fixed at the experimental collision energy. The initial
separation distance between Kr and azulene is 10 A
0, which is
sufficiently large to insure that the intermolecular interaction is
negligible.
The impact parameter, b, for the Kr-Azulene collision is chosen randomly by
2 1 max R b b =
where R is a random number from 0 to 1. The cutoff value, b
max,is
6.175 A
0, which is sufficiently large to include all hard collisions. At
each collision energy, 250,000 trajectories are employed to model the
dynamics.
B. Running the Trajectory
One computes the time-development of the coordinates of each atom from a set of initial conditions by solving the classical equations of motion. The classical equations of motion used here are Hamilton’s equations of motion,
p H k q . k = ∂ ∂
q H k p k = ∂ ∂
− .
where q
kand p
kare the canonical coordinate and momentum of the k
thdegree of freedom. Thus, there are a total of 108 coupled differential equations that are solved simultaneously for each trajectory. The Hamilton’s equations of motion are numerically solved at a discrete grid times to obtain the complete trajectory versus time. The integration is performed using a sixth-order Adams-Moulton integrator and the equations of motion are integrated by using a computer program, Venus.
20To converge the integration, one has to use a sufficiently small
timestep. If the timestep is too big, the solutions solved may have a
significant error. On the other hand, if the timestep is too small, it
takes a long time to get the information needed. One has to compare
how the size of timestep affect the results and find the optimum
timestep. The conservation of total energy to six significant figures
provides a criterion for the convergence of the integration. The
C. Analyzing the Data
While each classical trajectory is solved at thousands of timesteps throughout the collision process, only a relatively small number of dynamical variables need to be saved per trajectory to describe collision observables. For each trajectory, the following initial variables are stored:
1. Initial internal energy of azulene 2. Initial rotational energy
3. Impact parameter
During the collision process, the number of turning points occurring between Kr and the Az center of mass is monitored and stored. This quantity is useful in identifying long-lived collision complexes. At the end of the collision, the following quantities are stored:
1. The total time taken by a collision event 2. Final relative translational energy of Kr-Az 3. The scattering angle
4. The collisional energy transfer
In addition to this information collected for all 250,000 trajectories in the ensemble, a more detailed analysis is conducted on selected trajectories. There, the full trajectory (q,p) versus time is retained.
This information is used to construct an animated representation of
the collision that is useful to analysis the mechanism of energy transfer.
To compute the energy transfer cross section, dσ/d∆E(≡dσ/dE’), we use the histogram binning technique
22. A set of energy bins is defined,
∆E i , with a spacing of 50 cm
-1. Then, the number of trajectories with energy transfer within each bin is computed, N i . The cross section is given by
π ε σ
N b N E
d E
d
i 2 imax
)
( =
∆
∆
where N=250,000 is the total number of trajectories and ε=50 cm -1 is the bin size. The differential cross section is obtained using histogram binning for both the final energy transfer and scattering angle
22. A set of angular bins is defined, θ
k. The θ
kis given by
max
1 max
2
cos k k k
k
=
−− θ
where k=0,…,k
maxand k
maxis the total number of angular bins. Then, the number of trajectories with energy transfer within each bin and each angular bin is computed, N k . The differential cross section is given by
22N N b k Ed
d E
d
i k4 )
(
2maxmax 2