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deduced that the canonical distribution can be sampled using a Langevin-type
equation:
;
M
I
R
I
¼
F
inc
g
R
I
þ X
I
þ X
inc
I
(42)
where
g
is a Langevin friction coefficient,
X
I
a random noise term, and F
BO
¼
inc
F
inc
+
I
. The force from the incomplete SCF optimization is denoted here by F
inc
and the corresponding fully converged force by F
BO
. Sampling of the Boltzmann
distribution requires that the fluctuation dissipation theorem is obeyed:
X
h
X
I
ð
0
ÞX
I
ð
t
Þ
i ¼
6
g
M
I
k
B
T
dð
t
Þ:
(43)
In applications, the friction term is split into two contributions
g ¼ g
D
+
g
L
,
where
g
L
is taken to be an arbitrary constant and
g
D
is determined by requiring
D
is generated.
R
2
1
3
k
B
that the correct temperature
T
¼
M
I
g
D
can be kept small, the Langevin method not only generates the
correct canonical distribution but also dynamical properties are accurately described
[
60
]. In order to achieve this goal, the error of the forces can be consistently reduced
by noting that the last self-consistent cycle can be interpreted as a Harris functional.
The missing force F
BO
If the parameter
F
inc
can then be approximated to a high degree by
Z
dr
@
V
XC
½r
in
@r
in
Dr þ
V
H
½Dr
r
I
r
in
;
(44)
where
r
in
and output density, and
V
XC
and
V
H
are the exchange-correlation and Hartree potential, respectively.
Using the Langevin method, K
Dr
is the difference between input
uhne et al. were able to accelerated their bench-
mark calculations on liquid SiO
2
by one to two orders of magnitude. In 2009,
K
€
€
uhne et al. simulated liquid water with the new method [
60
]. Oxygen-oxygen
radial distribution functions agreed well with other approaches. Because of the
acceleration they were able to estimate for the first time reliably the diffusion
coefficient and shear viscosity of liquid water.
3.5 Enhanced Sampling
3.5.1 Elastic Band and String Methods
Of special importance to AIMD are the enhanced sampling methods termed elastic
band and string methods. Michaelides and coworkers explained in their excellent
free-energy method assessment article that special considerations have to be taken
when rare event simulations are done within AIMD. As the iterative SCF procedure
has an impact on the calculation of the forces in contrast to the exact forces in
empirical simulations, the convergence tolerance of the SCF procedure has to be
chosen carefully, especially for reactive situations [
61
].
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