Molecular dynamics thesis

2 Molecular dynamics
2.1 General principles
The principles of a classical molecular dynamics simulation are quite
straightforward. It is simply a matter of solving Newtons equations of motion for a number
of atoms. All that is needed for this is an initial list of the positions and velocities of the
atoms in the system and a description of particle interactions (see section 2.2). These
interactions determine the forces acting on the atoms and since the masses of the atoms are
known, their accelerations can be calculated. Positions, velocities and accelerations are
enough to determine the time evolution of the system.
Once the state of the system (determined by just a list of positions and velocities,
called a configuration) is known at a particular time, all other information about the system
at that time can be calculated. For instance, the temperature can be calculated simply by
determining the average kinetic energy from the velocities.
The equations governing a molecular dynamics simulation are
F
i
= −
U
r
i
,
(1)
a
i
=
F
i
m
i
,
(2)
where F
i
is the force acting on atom i , U is the total potential energy of the system, r
i 
is
the position of atom i, a
i  
is the acceleration of atom i , and m
i 
is the mass of atom i.
When a particular form has been chosen to describe the interactions (see section 2.2), eqns.
(1) and (2) can be rewritten in a more explicit form.
2.2 Interaction potentials
The time evolution of a system starting from an initial configuration is determined by
the interactions between the atoms. Therefore it is very important that the interactions are an
accurate model of how real atoms interact.
In this thesis low energy Mo-Mo interactions are modelled by the Embedded Atom
Method (EAM). This method was chosen because EAM provides a more accurate
description than pair potentials for systems containing pronounced defects, such as
vacancies, vacancy clusters, and free surfaces [1]. EAM is based on the idea that one part of
the potential energy of an atom is determined by a contribution from a pair potential and that
the other part is determined by the electron density in which the atom is embedded. The
equation describing the EAM is
U
=
F
i
i
=
1
N
∑
(
i
(r
i
))
+
1
2
ij
(r
ij
)
j
=
1,j
≠
i
N
∑
i
=
1
N
∑
,
(3)
in which F
i
  is the embedding function, i.e. the function that describes how the electron
density part of the potential energy of atom i  at position r
i
depends on the electron density
i 
at that position, and 
ij 
is the pair potential contribution of two atoms i  and j  separated
by a distance r
ij
.
The electron density is derived from atomic wave functions, with some strong
simplifications. The angular dependence of the wave function is averaged out.
Consequently the electron density that one atom experiences only depends on the distances

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to other atoms. The total density at a certain point is calculated by simply adding densities
contributed by different atoms,