Chemistry Reference
In-Depth Information
functions. A similar relation applies to the momenta,
p
m
j
. The CG energy is
then defined as the average energy of
the canonical ensemble on this
constrained (Eq. [45]) phase space:
E
ð
u
k
;
u
k
Þ¼h
H
MD
i
u
k
;
u
k
ð
dx
m
dp
m
H
MD
e
b
H
MD
¼
=
Z
½
46
ð
dx
m
dp
m
e
b
H
MD
ð
u
k
;
u
k
Þ¼
Z
½
47
!
Y
j
d
ð
u
j
X
X
p
m
f
j
m
m
m
u
m
f
j
m
Þ
d
u
j
¼
½
48
m
m
where
is a
three-dimensional delta function. The delta functions enforce the mean-field
constraint (Eq. [45]). The explicit form of the full CG energy (Eq. [46]) for
a monoatomic harmonic or anharmonic solid is derived in Ref. 204.
b
¼
1
=ð
Þ
(
T
¼
temperature),
Z
is the partition function, and
d
ð
u
Þ
kT
Applications
Test applications of the method include calculation of the
phonon spectra for solid argon and tantalum in three dimensions.
Quasi-continuum Coarse-Grain Alternative to Molecular Dynamics
In order to extend the quasi-continuum method (discussed above) to
include not only finite-temperature effects but also a dynamical description of
the system, Dupoy et al. introduced the ''molecular dynamics without all the
atoms'' method in 2005.
205
This procedure, which essentially is a coarse-grain
(CG) alternative to molecular dynamics, is based on the
potential of mean force
(PMF) concept, which was first introduced by Kirkwood in 1935.
206
Here, a
variety of equilibrium and nonequilibrium properties of large systems are calcu-
lated using only a limited number of degrees of freedom. However, the tricky
part in applying the PMF method is making it computationally efficient. Dupoy
et al. proposed to expedite the calculation of the PMF by coupling it to the
QC method, i.e., making use of finite-element interpolation to determine the
position of the
constrained atoms
and introducing the local harmonic approxi-
mation
147-149
(see above) and the Cauchy-Born rule
88,100,101
(see above) when
determining the state of the system under strain.
The basic structure of this methodology is the following. Following the
QC procedure, the atoms of an
N
-atom system are separated into
representative
and
constrained
ones. The representative atoms are the only atoms actively con-
sidered in the simulations. Their positions are indicated as
f
q
r
, their number as
N
r
, and they can be either
nonlocal
, if only interacting with other representative
atoms, or local, if interacting with constrained atoms as well. The
constrained
atoms are atoms whose position
g
f
q
c
g
is not directly determined in the
