Chemistry Reference
In-Depth Information
methodology, is more accurate than the previous two because it does not make
use of the local harmonic approximation, i.e., it preserves the coupling of the
vibrations of different atoms. The key idea is to keep the amount of computa-
tion manageable by calculating the Helmholtz free energy in the reciprocal
space while using Bloch's theorem
156
with the Born-von Karman boundary
conditions.
157
The authors applied it to the investigation of the effect of tem-
perature and strain on phonon and elastic properties in silicon.
Because of the reciprocal representation (details on the calculation can be
found in Refs. 110,154, and 155), the dynamical matrix
D
for a representative
atom
a
is reduced to a 6
6 matrix of the form
2
3
e
ik
R
0
e
ik
ð
R
0
ba
F
1
x
Þ
N
b
¼
11
j
N
b
¼
12
j
ð
a
;
b
Þ
ð
a
;
b
Þ
1
M
ba
1
;
k
1
;
k
4
5
D
ð
k
Þ¼
e
ik
ð
R
0
ba
þ
F
1
e
ik
R
0
½
24
21
j
x
Þ
22
j
N
b
¼1
ð
a
;
b
Þ
N
b
¼1
ð
a
;
b
Þ
ba
;
k
;
k
a
¼
1
j
;
k
¼
1
;
2
;
3
pq
j
where
k
is the wave vector,
ð
a
;
b
Þ
is the force constant matrix,
a
is the
;
k
representative atom (center atom), and
loops over all the atoms in the crystal.
For more detail on this derivation, we refer to the original study.
110,154,155
The free energy of a representative atom
b
a
can therefore be written as
!
ð
k
B
T
ð
X
X
6
s
¼1
6
1
2
V
B
1
2
U
0
e
h
o
s
ð
k
Þ=
k
B
T
F
a
¼
a
þ
h
o
s
ð
k
Þ
dk
þ
ln
ð
1
Þ
dk
½
25
k
k
s
¼1
where
U
0
a
evaluated using the equilibrium
position of the system,
V
B
is the volume of the first Brillouin zone of the recipro-
cal lattice, and the factor 2 is due to the fact that the silicon lattice is made by
two interpenetrating Bravais lattices. More importantly,
is the total potential energy of atom
a
p
l
s
ð
k
Þ
o
s
ð
k
Þ¼
,
where
6 dynamical matrix
D
and
s
is the
index of the polarization for the crystal. In deriving Eq. [25], the authors also
used the fact that
k
can be assumed continuous for a bulk material. Lastly, in
the QC-QHMK model, the vibrational component of the Helmholtz free energy
(second term on the right-hand side in Eq. [25]) is evaluated for all the atoms
corresponding to the continuum nodes by considering a bulk, nonlocal silicon
crystal subjected to a homogeneous deformation given by the local deformation
gradient (Cauchy-Born rule
158,159
).
l
s
ð
k
Þ
are the eigenvalues of the 6
Dynamical Methods
In this section, we discuss hybrid methodologies that explore the dyna-
mical evolution of systems composed of a continuum region (usually described
using finite-element methods) coupled to a discrete one [modeled using mole-
cular dynamics (MD) algorithms and semiempirical classical potentials].
