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U. Fitsch:
"Molecular Dynamics - The Appropriate Combination of Classical Newtonian and Quantum Mechanics in Simulating Many-Body Systems";
Talk: MATHMOD 2012 - 7th Vienna Conference on Mathematical Modelling, Wien; 2012-02-14 - 2012-02-17; in: "Preprints Mathmod 2012 Vienna - Full Paper Volume", F. Breitenecker, I. Troch (ed.); Argesim / Asim, 38 (2012), 411 - 412.

Introduction. A lot of problems in the field of medicine or bio- and nanotechnology deal with many-body
systems. Because of the size of the interacting bodies, called particles, a handling at a molecular and atomic level
is required, which leads to quantum mechanics. An anlaytic or numerical solution of the resulting Schrödinger
equation is only possible in a few simple special cases - even a system consisting of three bodies cannot be
clearly solved. Therefore several approximations and simplifications have to be made when the number of particles
increases. The approach to solve many-body systems makes use of the laws of classical Newtonian mechanics and
the actually required quantum mechanics.
Modelling Approach. The simulation of such systems is based on particle models. In this models the physical
system consists of discrete particles and their interactions, therefore very small as well as very big particles can
be handled. One particle carries some physical properties, like mass, velocity, position, energy or charge, and the
evolution of the system is given by these properties and the interaction of the particles.
Many particle models are based on classical Newtonian mechanics. Newton´s second law leads to a system of
ordinary differential equations of second order. The acceleration of a particle depends on the force acting on it, and
this force depends on the interactions of the particles.
In atomic models one actually needs to make use of the laws of quantum mechanics and solve Schrödinger´s
equation, representing the equation of motion. Its solution doesn´t provide unique trajectories, meaning determined
positions and other physical properties. All statements about this system result from the state function, also called
wave function, y, given as the solution of the Schrödinger equation
i¯h¶y(R;r;t)
¶t = ^H y(R;r; t)
with R representing the positions of the nuclei and r the positions of the electrons.
Under the assumption that the Hamilton operator isn´t explicitely time dependent a seperation approach leads to
the stationary Schrödinger equation
^H
y(R;r) = Ey(R;r)
which is an eigenvalue problem for the Hamilton operator ^H with the energy eigenvalue E.
Simulation Approach. Within the simulation of many-body systems some simplifications have to be made. The
Born-Oppenheimer approximation makes use of the big difference regarding the mass of the atomic nuclei and the
electrons. This allows a separation of the equations of motion of the nuclei and the electrons. Also to be mentioned
is the Hartree-Fock approach to calculate the potential caused by the electrons.
A simplification concerning the potential is the use of the cut-off radius. For one particle only the interactions with
particles in a certain radius are considered. Particles out of this area are treated as if they weren´t there, because
the distance is too wide and therefore the interaction negligible compared to the interactions with closer particles.
For the Lennard-Jones potential U(ri j) with ri j the distance between two particles, the approximation leads to ....

where e is the parameter defining the depth and s the parameter determining the zero-crossing of the potential.
With regard to the boundary conditions it depends on what system is simulated. In case of a closed box one could
work with reflecting boundary conditions. In periodic systems it is common to use periodic boundary conditions.
The approach is based on the assumption that a particle leaving the simulation domain on one side reenters on the
opposite side. Therefore particles located on opposite sides close to the borders interact with each other.