Chemistry Reference
In-Depth Information
The metadynamics method was introduced in 2002 by Laio and Parrinello as an
elegant extension of adaptive bias potential methods [
65
]. The authors used a
coarse-grained non-Markovian dynamics in the space defined by a few collective
coordinates
s
i
. With the aid of a history-dependent potential term the minima of
the FES were filled in time, allowing the efficient exploration and accurate deter-
mination of the FES as a function of the collective coordinates. Laio and Parrinello
demonstrated the applicability of this approach in the case of the dissociation of
a sodium chloride molecule in water and in the study of the conformational changes
of a dialanine in solution [
65
].
With the metadynamics approach, the free energy surface
Fð
s
Þ
of a limited set
of collective variables
s
i
can be explored [
68
]. This is done by introducing an
extended Lagrangian with fictitious particles
s
a
for each CV:
X
X
1
2
m
a
_
1
2
k
a
ð
2
s
2
L¼L
CPMD
=
BOMD
þ
a
S
a
ð
r
Þ
s
a
Þ
V
ð
t
;
s
Þ:
(45)
a
a
While the first term on the right-hand side of (45) is the usual AIMD Lagrangian,
the second term is the total kinetic energy of the fictitious particles. For large
enough masses
m
a
they are adiabatically separated from the ionic and electronic
degrees of freedom. Each fictitious particle
s
a
is connected to its actual collective
variable
S
(r) by a harmonic potential [
68
]. The history-dependent biasing potential
[last term in (45)] is introduced in order to enhance sampling. This biasing potential
V
(
t
, s) constitutes a sum of repulsive Gaussian-shaped potential hills:
2
3
"
#
exp
X
2
2
ð
s
s
i
Þ
ðð
s
iþ
1
s
i
Þð
s
s
ÞÞ
4
5
V
ð
t
;
s
Þ¼
H
exp
(46)
2
4
ðD
W
?
Þ
2
W
i
Þ
2
ðD
t
i
<
t
W
i
¼
with s
i
s
iþ
1
s
i
gives the width along the
¼
{
s
a
(
t
i
)} and
H
the height.
D
W
⊥
the size in the orthogonal direction. In the limit of a
long simulation time, the following equation holds [
66
]:
direction of motion and
D
V
ð
t
;
Þ¼
F
ð
Þ:
lim
t
s
s
(47)
!1
Laio and Parrinello stated [
65
] that constructing dynamics on an FES that
depends on a few collective coordinates allowed one to simplify the complexity
of the problem, which depends exponentially on the number of degrees of freedom.
The FES will be smoother than the underlying PES and topologically simpler, with
a greatly reduced number of local minima [
65
]. A history-dependent bias potential
as defined in (46), but applied in a regular MD simulation without applying the
collective variable space, is efficient in finding escapes from the local minima but
will not provide quantitative information about the FES [
65
].
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