Chemistry Reference
In-Depth Information
Figure 11
Schematic representation of the system considered. Black dots indicate
nonlocal representative
atoms (they only interact with other representative atoms), gray
dots indicate
local representative
atoms (they interact with both representative and
constrained atoms), and, finally, small white dots represent
constrained
atoms. The large
circles represent the range of the atomic potential (centered on a local or on a nonlocal
representative atom).
simulations but is obtained by interpolating the positions of the closest represen-
tative atoms using an FE scheme (Figure 11). More precisely, it is their thermally
averaged position that is obtained through interpolation.
The Hamiltonian that is considered for the CG system is
X
N
r
2
2
m
i
þ
ð
p
i
Þ
H
CG
ðf
q
r
g; f
p
r
V
CG
ðf
q
r
g;
b
Þ¼
g;
b
Þ
½
49
i
¼
1
m
i
q
i
are the
momenta of the representative atoms, and
m
i
their effective masses. By impos-
ing the conditions that the total mass of the CG system should be equal to that
of the full-atom system, and that both systems should have the same momen-
tum free energy, the effective masses are determined to be
m
i
¼
a
ðf
q
r
;
p
i
¼
where
V
CG
g;
b
Þ
is the CG potential energy,
b
¼
1
=
k
B
T
n
i
1
m
, where
n
i
is the number of atoms represented by representative atom
i
, and
a
is found
solving the equation
X
n
r
n
i
1
1
a
¼
N
½
50
i
¼
Similar to the quasi-continuum Monte Carlo approach (see above), the CG
potential energy is the PMF for the constrained degrees of freedom
153,206
ln
ð
e
b
V
ðf
q
r
1
b
g;f
q
c
V
CG
ðf
q
r
gÞ
d
f
q
c
g;
b
Þ¼
g
½
51
where
V
is the interatomic potential. This choice for the CG potential energy
guarantees that the ensemble average of any observable
A
that depends only
on the positions of the representative atoms is equal to the ensemble average
