Chemistry Reference
In-Depth Information
Fago et al.'s
249-251
implementation of OFDFT inside a QC method is
straightforward. They utilize a local approach in the QC part (LQC, see above
for details) and use OFDFT to compute the energy at each quadrature point.
Therefore, the pros and cons of this method are the same as those for the stan-
dard LQC, with the difference that the energy evaluation is now more accurate
than if a classical potential was used. However, even if local electronic effects
are accounted for, no coupling between neighbor unit cells is included in the
calculations because, in LQC, the energy of each quadrature point is calcu-
lated using the Cauchy-Born rule. Such a rule states that the energy of a quad-
rature point can be found by computing the energy of a perfect, infinite crystal
undergoing the same uniform deformation that is applied to the quadrature
point in question. Also, as pointed out earlier, the Cauchy-Born rule holds
well for slowly varying elastic deformation, but breaks down in the vicinity
of defects, impurities, and the like. This methodology has been applied to
the study of dislocation nucleation during indentation of Al.
To bypass the limitations of the Cauchy-Born rule, in 2006, Lu et al.
253
proposed a more involved scheme to couple standard DFT to quasi-continuum
calculations. In their method, the part of the system far away from the zone of
interest is described using a classical (nonquantum) quasi-continuum approach
(see discussion above on QC for details), i.e., considering both local (conti-
nuum) and nonlocal (atomistic) terms. Classical potentials (EAM
9,10
in the
applications presented) are used to evaluate the energy within the QC calcula-
tions. A third region is considered as well, covering the part of the system that
needs a more detailed description. It is in this region that density functional
theory is used.
In this methodology, the coupling between the continuum and the
atomistic regions is handled in the same way as it is in the standard formula-
tion of the mixed quasi-continuum. However, the coupling scheme used to
connect the quantum region to the atomistic one is that proposed by Choly
et al.,
241
which we described earlier. In particular, Choly's approach offers
two ways of evaluating the hand-shake energy term, and Lu et al.
253
adopted
the one where the interaction energy is computed using only classical evalua-
tions (Eq. [60]). For details on the evaluation of the forces and the relaxation
scheme, we refer to the original paper.
252
The method has been applied to the
investigation of the core structure of an edge dislocation in aluminum, with
and without H impurities.
An even more involved scheme, which does not assume the Cauchy-Born
rule and which allows seamless incorporation of defects, was proposed in
2007 by Gavini et al.
252
Here, OFDFT is used for the quantum evaluation
of the energies, and three levels of coarsening are considered in the QC mesh-
ing of the system. Following the naming convention suggested by the authors
(see Figure 14), they are:
T
h
1
, the coarse grid,
T
h
3
, the intermediate grid, and
T
h
2
, the finer one. The
T
h
1
mesh is the atomic mesh, i.e., the standard meshing
used in QC methods: Selected atoms are chosen as representative, and, in this
