Chemistry Reference
In-Depth Information
this tutorial, we rewrite the Verlet method in the equivalent ''velocity Verlet''
form. The inside of the integration loop is given by
h
2
M
1
F
i
v
n
þ1=2
i
¼
v
i
þ
½
6a
hv
n
þ1=2
i
r
n
þ1
i
¼
r
i
þ
½
6b
h
2
M
1
F
n
þ1
¼
v
n
þ1=2
i
v
n
þ1
i
þ
½
6c
i
In contrast to the constant energy regime described above, it is sometimes
desirable to perform simulations at a fixed temperature. This can be accom-
plished by the Langevin dynamics model:
10
Mr
¼
F
ð
r
;
v
;
t
Þ :¼r
r
V
ð
r
Þ
g
v
þ
R
ð
t
Þ
½
7
where
is a vector of normally distributed
random variables with zero mean and covariance
g
>
0 is a friction coefficient and
R
ð
t
Þ
T
t
0
Þ
h
R
ð
t
Þ
R
ð
i¼
2
g
k
B
TM
d
t
0
Þ
ð
t
, where
k
B
is Boltzmann's constant,
T
is the simulation temperature,
and
is the Dirac delta function. A natural extension of discretization [6] gives
the following time discretization scheme:
11
d
h
2
M
1
F
i
v
n
þ1=2
i
¼
v
i
þ
ð
r
n
;
v
n
þ1=2
;
Þ
½
8a
t
n
hv
n
þ1=2
i
r
n
þ1
i
¼
r
i
þ
½
8b
h
2
M
1
F
i
¼
v
n
þ1=2
i
v
n
þ1
i
ð
r
n
þ1
;
v
n
þ1=2
þ
;
Þ
½
8c
t
n
þ1
Molecular Dynamics Potential
The interactions of polyatomic molecules are frequently modeled by pair
potentials, both Lennard-Jones and electrostatic, between all constituent
atoms. The model potential used must also account for intramolecular geome-
tries by including the ''bonded'' terms: bond lengths, bond angles, and dihe-
dral angles. The result is the molecular modeling potential function that
generally is of the form
V
b
V
a
V
d
V
i
V
LJ
V
C
V
ð
r
Þ¼
þ
þ
þ
þ
þ
½
9
where
V
b
,
V
a
,
V
d
, and
V
i
are sums over various pairs, triples, and quadruples
of spatially localized bonded groups of atoms representing bonds, angles, dihe-
dral angles, and improper dihedral angles, respectively:
X
X
X
V
b
V
ij
V
a
V
ijk
V
b
V
ijkl
; ...
¼
¼
¼
½
10
bonds
angles
dihed
