Environmental Engineering Reference
In-Depth Information
velocity of each atom. This is a major step in MD simulation,
which involves many technical details such as boundary conditions,
numerical integration algorithm, and heat bath (reservoir) that can
realize the canonical ensemble. Finally, one calculates the ensemble
average of the thermodynamic variables of interest with long
enoughsimulation time.
1.2 Force Field Potential
TheinteratomicforceinMDsimulationisderivedfromthesystem's
potentialenergy.Althoughafullyquantum-mechanicaltreatmentof
the system's Hamiltonian is highly desirable, it can only be applied
to a system with very small size due to the large computational
load. Therefore, tremendous efforts have been put in the past to
develop reliable empirical force field in many material systems that
can accurately describe the interatomic interactions and easily be
applied to a large system size. The simplest model for interatomic
interaction is the van der Waals (vdW) force with pairwise (two-
body)interaction. Forthecovalentbonding,morecomplicated form
of force fields with many-body interaction are used. To illustrate
this in more details, we introduce several widely used force field
potentials as examples.
1.2.1
Pair Potential
The Lennard-Jones (LJ) potential is a widely used pair potential to
model the vdW force. It isdefined as
r
6
,
12
r
V
(
r
)
=
4
ε
−
(1.2)
where
ε
and
σ
arecharacteristicenergyandlengthunit,respectively,
and
r
is the interatomic distance. It is also known as 12-6 potential
and can be expressed as
A
r
12
−
B
r
6
,
V
(
r
)
=
(1.3)
with
A
and
B
denoting the repulsive and attractive constant in
LJ potential. As shown in Fig. 1.1, the equilibrium distance in LJ
