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is short range. The embedded atom method (EAM), for metals on metals, predicts
that the indirect interaction is repulsive and proportional to the number of shared
substrate atoms [
3
]. On very general grounds one expects
R
−
5
decay of the envelope
in (
2.2
). We caution that the distinction between electronic and elastic has long been
recognized as subtle [
40
] and that the asymptotic limit of the elastic interaction is
likely to be significantly modified at separations
O(
a
)
by more rapidly decaying
terms in the multipole series [
41
].
2.4 Refined Schemes for Extracting Interaction Energies
To extract estimates of interaction energies when there are many such parameters
are a delicate task typically one obtains a large number of simultaneous equations
by computing the total interaction of a large variety of different overlayer structures.
One should have more overlayer structures than interactions so as to be able to check
for robustness of the deduced values. While informal schemes had formerly seemed
sufficient [
42
], it is safer and sounder to use formal cross-validation schemes used
by several groups [
12
,
43
] to study overlayer systems.
In recent work [
44
] we used the leave-
n
v
-out cross-validation method [
45
]tofit
the computed energies for Cu(110) to the interaction parameters (cf. Sect.
2.6
). This
method is expected to perform better than the more commonly used leave-one-out
cross-validation scheme [
45
]. We calculated the interaction strengths in the follow-
ing way: for a particular supercell, we computed total energies for, say,
n
different
configurations of adatoms. In addition, we posit the number of significant interac-
tions
n
i
. We then use
n
i
(out of
n
) equations to solve for the interaction energies.
These interactions are then used to predict the energies of the remaining
n
n
i
equations. We then compute the root mean squared (rms) of the prediction error per
adatom for all configurations
j
(1
v
=
n
−
≤
j
≤
n
), each with
a
j
adatoms in it:
v
n
v
E
pred
E
VA S P
1
n
v
−
E
rms
=
1
(
E
j
)
2
E
j
=
(2.9)
a
j
j
=
We repeat this procedure for different partitions of (
n
,
n
i
), and sets of interactions
from only those partitions whose
E
rms
values are lower than a specified threshold
value (10meV/adatom in Ref. [
44
]) are considered for the final averaging of inter-
action values. Finally, we find the value of
n
i
that leads to the best convergence. As
a check, we perform this procedure on two different computational supercells.
2.5 Effect of Relaxations in Homoepitaxy with Direct
Lateral Interactions
When direct interactions play a significant role, such as for (1
1) homoepitax-
ial partial monolayers, one must be wary of relaxation-induced modification of
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