Lecture 11 : Non-covalent interactions 2 : Structures of Liquids
11.5
Classical Molecular Dynamics.
In this method, the time evolution of the system of N particles is studied using the classical (Newtonian) equations of motion ( = m ). Since it is not possible to find the explicit formulae for all the positions
i (t) and velocities
i(t) as a function of time, the equations of motion are integrated using algorithms whose accuracy depends on the discretized time step t used in the dynamics. In this method, the forces on each particle are calculated at each time step using
i =
-
uij(rij)
(11.8)
One of the simplest algorithms is the Verlet algorithm.
In this algorithm, the displacement of a particle
i at time t +
t is obtained in terms of its displacements at times t and t -
t as follows
i (t+ t) = 2i(t) - i (t- t) + i /m t 2
(11.9)
The velocity of the particle does not play a dynamical role in this algorithm, but its value can be easily obtained from
i(t) = [ i(t+ t) - i(t- t)] / 2t
(11.10)
In this method, the position and velocities of all the particles are obtained during each step of the simulation and are written to the hard disc. Typically 105 to 106 configurations are generated in the simulation. In these configurations, N, V and E (energy) of each member is the same.
From the distances between all the pairs of particles in all the members (rij = [(xj-xi)2 + (yj-yi)2 + (zj-zi)2] 1/ 2 ) g(r) can be obtained.
To determine i (t +t), we need the value of i at two time steps t and t- t. This is fine for all intermediate steps except the first step. In the first step only
i(0) is known. To get i(0+t) we may use the well known Newton's formula s = ut +1/2 at2, i.e.,
= m 
). Since it is not possible to find the explicit formulae for all the positions
i (t) and velocities
i(t) as a function of time, the equations of motion are integrated using algorithms whose accuracy depends on the discretized time step 
t used in the dynamics. In this method, the forces on each particle are calculated at each time step using 
uij(rij) 
i at time t +
t is obtained in terms of its displacements at times t and t -
