Real-World Predictions from Ab Initio Molecular Dynamics Simulations - Multiscale Molecular Methods in Applied Chemistry - page 113

Using Newton's second law (3) the acceleration ( R I ) of the particles can be
obtained from the forces F I acting on the particles.

2.2 Obtaining the Forces and Integrating
the Equations of Motion

Traditional molecular dynamics simulations use pre-defined analytical potentials.
The potentials most commonly work with the pairwise additivity approximation
( U ( R IJ ) ). This means that an analytical expression of the pair potential, a potential
between each set of atoms I and J , is parameterized such that good structural and/or
thermodynamics quantities can be expected [ 12 ]. Alternatively, electronic structure
calculations are carried out for a pair of particles as a function of distance and the
analytical expression has to be fitted to these energy points on the potential energy
surface [ 12 ]. It also means that the Born-Oppenheimer approximations have to be
valid, i.e., a separation of nuclear and electronic variables is possible and coupling
terms (non-diagonal and diagonal) can be neglected [ 13 , 14 ]. From the analytical
potentials the forces are then obtained by taking the derivatives with respect to the
positions

F I ¼ @

U

@ R I ¼ @

ð R I Þ U ð R IJ Þ

R IJ :

(4)

@

t is introduced and a
numerical step-by-step integration of the equations of motion is carried out. Taking
the Taylor series expansion in

In order to propagate the atoms, a small discrete time step D

D t gives

1
2 D

1
6 D

R I ð R I ð

t 2 t 3 B I ð

R I ð t þ D t Þ¼ R I ð t ÞþD t t Þþ t Þþ t Þþ (5)

and

1
2 D

R I ð

Þ¼ R I ð t R I ð

t 2 B I ð

t þ D t t ÞþD t Þþ t Þþ (6)

The time evolution of the system is followed by applying integration algorithms
(the so-called integrator) in an MD computer program. One can obtain these
integrators from the Taylor expansion around t +

t and by combining
the resulting equations. The following form is the velocity Verlet (Stromer-Verlet)
integrator:

D t and t D

F I ð Þ

2 M I D

t

Þþ R I ð

t 2

R I ð t þ D t Þ¼ R I ð t t ÞD t þ ; (7)