Chemistry Reference
In-Depth Information
energy. An obvious, but computationally expensive solution to this problem is
to only consider very small (ideally atomic size) elements very close to highly
inhomogeneous features.
QC-Mixed: Coupled Local/Nonlocal QC
A way to exploit the accuracy of
the nonlocal QC method in regions where atomic resolution is needed or
inhomogeneities are present, without paying the high computational price
that this formulation requires, is to couple it to the local formulation. Thus,
the local approach is used where the elastic displacement field is changing
slowly, and the nonlocal method is used only in proximity to defects,
surfaces, interfaces, and so on. This general prescription leads to a very
effective methodology, the coupled local/nonlocal QC method. Here, each
repatom is treated either as being local or nonlocal, and the total energy of
the system is given by the sum of the repatom energies:
X
X
N
nonloc
N
loc
E
tot;
h
¼
n
a
E
a
ð
u
h
Þþ
1
e
e
e
ð
F
e
Þ
½
10
a
¼
1
¼
e
The disadvantage of this formulation is the creation of a local/nonlocal
interface, i.e., a region across which the computational methodology is
abruptly changed. This does not happen in the two previous formulations.
Even though a detailed prescription is given (Ref. 95, e.g.) for avoiding double
counting the energy contributions from interface repatoms, it is not possible
to avoid the formation of unphysical forces (ghost forces) if a unique energy
functional is considered for the whole system (e.g., as in eq. [10]).
Ghost Forces
If a coupled local/nonlocal QC method is used in its standard
formulation (with a unique energy functional for the whole system), then the
mismatch between the local (continuum) and nonlocal (atomistic)
formulations produces nonphysical forces in the hand-shake region. As a
result, atoms close to the local/nonlocal interface have nonzero forces acting
on them even for a perfect crystal in its ground state, i.e., local displacements
at the interface will unphysically lower the total energy of the system. These
forces, named ghost forces, come into play because of the asymmetry in the
contributions to the repatom energies between the local and nonlocal cases.
The origin of such an asymmetry is the following. Since a
nonlocal
repatom
treats all of the repatoms in its crystallite as neighbors (even those that do not
belong to adjacent elements), their displacements affect the nonlocal repatom's
energy. However, the energy of a
local
repatom depends only on the
displacements of the adjacent repatoms. This asymmetry leaves unbalanced
forces. A more explicit treatment of these forces can be found in Ref. 71.
Ghost Force Reduction Methods (QC-GFC, SCF-HACG,
...
)
Several
methods have been developed to alleviate or eliminate the effect of ghost
