BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 140

1.3.3 SOME IMPORTANT SIMULATION ALGORITHMS THAT APPLIED
IN PSD CALCULATION

1.3.3.1 THE MD ALGORITHM
Solving Newton's equations of motion does not immediately suggest activity at
the cutting edge of research. Continuing to discuss, for simplicity, a system com-
posed of atoms with coordinates and potential energy we introduce the atomic
moment ain terms of which the kinetic energy may be written Then the energy,
or Hamiltonian, may be written as a sum of kinetic and potential terms Write the
classical equations of motion as. This is a system of coupled ordinary differential
equations. Many methods exist to perform step-by-step numerical integration of
them. Characteristics of these equations are: (a) they are 'stiff,'that is, there may
be short and long timescales, and the algorithm must cope with both; (b) calculat-
ing the forces is expensive, typically involving a sum over pairs of atoms, and
should be performed as infrequently as possible [157-160].
Also we must bear in mind that the advancement of the coordinates fulls two
functions: (i) accurate calculation of dynamical properties, especially over times
as long as typical correlation times of properties a of interest (we shall dene this
later); (ii) accurately staying on the constant-energy hypersurface, for much lon-
ger times , in order to sample the correct ensemble. To ensure rapid sampling of
phase space, we wish to make the time step as large as possible consistent with
these requirements. For these reasons, simulation algorithms have tended to be of
low order (i.e., they do not involve storing high derivatives of positions, veloci-
ties, etc.). This allows the time step to be increased as much as possible without
jeopardizing energy conservation. It is unrealistic to expect the numerical method
to accurately follow the true trajectory for very long time The 'ergodic' and 'mix-
ing' properties of classical trajectories, that is, the fact that nearby trajectories
diverge from each other exponentially quickly, make this impossible to achieve.
All these observations tend to favor the Verlet algorithm in one form or another,
and we look closely at this in the following section. For historical reasons only, we
mention the more general class of predictor-corrector methods, which have been
optimized for classical mechanical equations:

p

r  =

i

and p  = f

(114)

i i i

m

i

1.3.3.2 THE VERLET ALGORITHM
There are various, essentially equivalent, versions of the Verlet algorithm, includ-
ing the original method and a 'leapfrog' form. Here we concentrate on the 'veloc-
ity Verlet' algorithm, whichmay be written:

1 1

� �

+d= +d

(115)

( ) ( )

p t t p t tf t

€ ‚ i

ƒ „

i i

2 2