Chemistry Reference
In-Depth Information
P
"Real" System
Q
Damping Region
(a)
(b)
Figure 12
Schematic representation of: (a) damping region BC (the region of interest is
surrounded by a border region where damping of the outgoing waves takes place), and
(b) coupled system for exact nonreflective BC (P is the region of interest and may contain
inhomogeneities, while Q is the defect-free medium that surrounds it).
In general, damping methods are very convenient to use because they do not
require precalculations and are very easily implemented, but their effectiveness
is completely dependent on the size of the damping region and, often, a signif-
icant number of atoms must be added to the system for this method to work
properly. Moreover, there is no a priori way to estimate the size of the damp-
ing region. In summary, all three methods can be very effective in suppressing
phonon reflection, and only the specific problem under examination dictates
which one is the most convenient to use.
We conclude this section with an example of exact nonreflective BCs, the
method developed by Cai et al.
214
This method is designed to minimize the
wave reflection between a region of interest P, and a defect-free medium Q
around it [Figure 12(b)]. The region P may contain inhomogeneities and
must be large enough that, for moderate temperatures, the interactions
between P and Q (P-Q), and within P (P-P), can be considered as harmonic.
The idea behind this approach is to represent the effects of Q on P in terms of a
set of response functions that are estimated numerically during test simulations
prior to the real calculations. These response functions provide a numerical set
of boundary conditions for P that preserves the correct dynamic because it con-
tains the information related to the degrees of freedom in region Q. More
in detail, the generalized Langevin equation (GLE) (the equation of motion
for the degrees of freedom
x
i
N
in P when the explicit degrees of free-
dom associated with medium Q have been replaced by an implicit formulation)
takes the form
185-188,223
¼
1
; ...;
ð
t
X
N
x
i
¼
q
V
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x
i
þ
d
t
1
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ð
t
Þ
x
j
ð
t
t
Þ
q
0
j
¼
½
56
X
N
þ
1
b
ij
ð
t
Þ
x
j
ð
0
Þþ
R
i
ð
t
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j
¼
