An initial velocity is assigned to each atom, and Newton's laws are applied at the atomic level to propagate the system's motion through time (see `Classical and Quantum Mechanics - in a Nutshell' above). Thus, dynamical properties such as time correlation functions and transport coefficients (e.g., diffusion constants, bulk viscosities) can be calculated from a sufficiently long MD trajectory.
Once again, Newton's second law is: , where is the sum of all forces acting on atom i that results in its acceleration . The acceleration is the second derivative of the position with respect to time: . In words, it is the rate of change of the velocity , which in turn, is the rate of change of the position .
The `Leap Frog' algorithm is one method commonly used to numerically integrate Newton's second law.
We obtain all atomic positions at all times and all atomic velocities at intermediate
times . This method gets its name from the way in which positions and velocities are calculated in an alternating
sequence, `leaping' past each other in time:
The energy of an isolated system (as opposed to, for example, one in contact with a thermal bath) is conserved in nature, but it may not be in simulations. Energy conservation can be violated in simulations because of an insufficiently short integration time step , an inadequate cutoff method applied to long-range (electrostatic and Lennard-Jones) forces, or even bugs in the program. Of course, energy conservation alone is not sufficient to ensure a realistic simulation. The realism of the dynamics trajectory depends on the empirical potential energy function , the treatment of long-range forces, the value of , etc.