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which is, by construction, only dependent on the representative atoms (all the
slave degrees of freedom have been integrated out). In this model, the displa-
cements of the slave atoms are assumed to be
X
u
j
N
j
ð
X
i
Þþ
g
i
u
i
¼
½
22
j
which means that they are not completely determined in terms of the displace-
ments of the representative atoms by finite-element interpolation [
u
i
¼
P
j
u
j
N
j
ð
X
i
Þ
], as they were in the zero-temperature QC Eq. [5], but they also
depend on a fluctuational variable,
g
i
, which accounts for temperature-driven
random fluctuations in the finite-element shape functions. Using Eq. [22],
expanding
H
in terms of the random fluctuations, terminating the expansion
to the second order, and going through a rather involved derivation (for details
we refer to Ref. 153), an approximate form of the effective energy function is
obtained:
(
)
3
k
B
T
X
1
=
6
h
½
det
D
ð
F
e
Þ
H
QC
ðf
u
i
g;
H
QC
ðf
u
i
g;
s
T
Þ¼
0
Þþ
e
n
e
ln
½
23
k
B
T
where the first term accounts for the internal energy of the deformed crystal,
while the second term contains the entropic contribution due to the slave
degrees of freedom that were integrated out. In Eq. [23]
D
, is the local
dynamical matrix of an atom in a crystal undergoing an homogeneous defor-
mation gradient
F
, and
ð
F
e
Þ
s
e
is the number of slave atoms in element
e
. In obtain-
ing Eq. [23], several approximations were made, the most important one being
the use of the local harmonic approximation (LHA; see discussion above), i.e.,
slave atoms are assumed to have harmonic displacements around their equili-
brium positions. This means that all of the slave atoms in one element have the
same vibrational frequencies, i.e., the correlation between the vibrations of
different atoms is neglected. However, while the introduction of the local har-
monic approximation in deriving Eq. [23] has notably expedited the calcula-
tion of the effective energy function, the energy determination still requires a
larger computational effort than performing QC calculations at zero tempera-
ture. The most important limitation of this approach comes from the use of the
local harmonic approximation as well: Because of it, the method is reliable
only at low temperatures and has a tendency to underestimate the temperature
dependence of defect free energies.
n
QC k-space Quasi-harmonic Model
An alternative approach to the calculation of the Helmholtz free energy
is the k-space quasi-harmonic model (QC-QHMK)
110,154,155
introduced in
2001 by Aluru et al. This method, still a generalization of the quasi-continuum
