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U
M
would not change so much. If the two phases compete with each other, then the
effect of
dominates for intermediate
d
MXM
and that of
U
M
for large
d
MXM
.This
partially explains why the CDW phase appears for intermediate
d
MXM
and the CP
phase for large
d
MXM
. This variation of the chargeordered phases can be explained
also by the competition between the short and longrange electronelectron
interactions, as discussed below.
b
12.5 ShortRange Versus LongRange ElectronElectron
Interactions
Here, the longrange interactions
V
MM
,
V
MXM,
and
V
2
are included. The competi
tion is easily understood in the strongcoupling limit,
t
MM
¼ t
MXM
¼ a ¼
0. The
contribution from each interaction term to the total energy per binuclear unit is
listed on the righthand side of Fig.
12.5
. The bondchargedensitywave (BCDW)
phase is introduced at the top here just to explain the competition, although it is
never realized. Charge ordering is formally represented by M
2+
M
4+
XM
2+
M
2+
X
there. Both the bond and sitecharge densities are modulated in this phase. If the
sitediagonal electronlattice coupling
were so strong that it dominated over the
onsite repulsion
U
M
, the BCDW phase would be stable, forming a bipolaron
lattice. The energy gain from the electronlattice coupling
b
b
is the largest, approxi
mately given by 3

y
 per unit, though the magnitudes of the lattice displacements
are not uniform in the lowestenergy configuration. In any case, the BCDW phase is
not experimentally observed because it is destabilized by the strong onsite repul
sion
U
M
. The energy loss is also the largest, (3/2)
U
M
. For a fixed magnitude of the
lattice displacements, all of the CDW, ACP, and CP phases gain energy by
b

y
 and
lose it by
U
M
. Then, the longrange interactions differentiate their energies. The
CDW phase loses energy by (5/2)
V
MM
, the ACP phase by (5/2)
V
MXM
, and the CP
phase by 5
V
2
. Otherwise, the energy loss is given by 2
V
MM
þ
b
4
V
2
. When
the nearestneighbor repulsion within the unit
V
MM
is dominant, the CDW phase is
unstable. Since we reasonably expect
U
M
>V
MM
>V
MXM
>V
2
, the CP phase is the
most stable in the strongcoupling limit if
2
V
MXM
þ
is weak enough.
In R
4
[Pt
2
(pop)
4
I]
n
H
2
O, the CDW phase appears for intermediate
d
MXM
, and the
CP phase appears for large
d
MXM
[
7
]. First of all, as long as
K
MXM
is infinitely large,
the ACP phase was not realized in our calculations. As
d
MXM
increases, the next
nearestneighbor repulsion
V
2
would become weak. Meanwhile, the distance
between the nearestneighbor M ions within the unit is almost unchanged, so that
the corresponding nearestneighbor repulsion
V
MM
would not change so much in
comparison with
V
2
. Then, as
d
MXM
increases, the CP phase becomes more stable
relative to the CDW phase. This would be the other reason why the CDW phase is
changed into the CP phase with increasing
d
MXM
. So far, we have limited
the discussions to the case of infinitely large
K
MXM
, prohibiting displacements of
M ions and keeping
d
MXM
constant.
b
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