Chemistry Reference
In-Depth Information
methods if only equilibrium properties are investigated, because they (1) are
usually computationally faster than dynamical ones, (2) can probe a larger
part of the configurational space, and (3) are not affected by the spurious
wave reflections at the atomistic/continuum interface as happens with most
dynamical methods. The key difference between zero-temperature and finite-
temperature equilibrium methods is that in the former it is sufficient to mini-
mize the effective energy, while in the latter an effective free energy must be
constructed and then minimized. However, if dealing with the free energy
directly is computationally too expensive, alternative routes can also be used.
One of the most popular of these is the Monte Carlo (MC) method,
15-17
where
a random sampling of the phase space is used to reach equilibrium, instead of
directly minimizing the free energy. To conclude, it is worth noticing that most
of the methods discussed in this section can be considered as extentions of the
zero-temperature quasi-continuum (QC) method (see above).
Basic Approximations
The majority of the finite-temperature equilibrium methods that we are
about to discuss make use of a few key approximations in order to minimize
the computational load. To begin with, an harmonic approximation of the
atomistic potential is used,
110,146
both when computing the Helmholtz free
energy directly and when determining the effective energy to use in MC calcu-
lations. In the harmonic approximation of the potential, only terms up to the
second order are retained in the Taylor expansion of the total potential energy.
A second, very important, approximation is the local harmonic approx-
imation (LHA),
147-149
which states that all of the atoms in the system can be
treated as independent oscillators, i.e., the Einstein model can be used to
describe the material, all of the atoms have the same vibrational frequencies,
and the correlation between the vibrations of different atoms is neglected. The
introduction of this approximation greatly simplifies the calculations, but also
significantly diminishes the accuracy of the results. Therefore, the next three
computational methods are discussed in order of increasing theoretical accu-
racy, with the first method being the least accurate (and therefore the most
computationally tractable) because it uses the LHA for describing both the
representative and slave atoms. The second method is an intermediate treat-
ment (the LHA is used only for slave atoms), and the third method is the
most accurate since it does not use the LHA at all.
A third approximation often used is the high-temperature classical
limit,
147
where sinh
k
B
T
. Using such an approxima-
tion greatly simplifies computing the Helmholtz free energy or the effective
energy, but it is not valid at temperatures above one half of the melting
point.
146,149
Lastly, we would like to mention that the methods presented
here are just some of the proposed methodologies that expand the QC idea
to finite temperature (for others, see, e.g., Ref. 146).
ð
h
o
=
4
p
k
B
T
Þ
h
o
ik
=
4
p
ik
