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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 charge-ordered phases can be explained
also by the competition between the short- and long-range electron-electron
interactions, as discussed below.
b
12.5 Short-Range Versus Long-Range Electron-Electron
Interactions
Here, the long-range interactions
V
MM
,
V
MXM,
and
V
2
are included. The competi-
tion is easily understood in the strong-coupling 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 right-hand side of Fig.
12.5
. The bond-charge-density-wave (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+
-X-M
2+
M
2+
-X-
there. Both the bond- and site-charge densities are modulated in this phase. If the
site-diagonal electron-lattice coupling
were so strong that it dominated over the
on-site repulsion
U
M
, the BCDW phase would be stable, forming a bipolaron
lattice. The energy gain from the electron-lattice 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 lowest-energy configuration. In any case, the BCDW phase is
not experimentally observed because it is destabilized by the strong on-site 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 long-range 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 nearest-neighbor 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 strong-coupling 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-
nearest-neighbor repulsion
V
2
would become weak. Meanwhile, the distance
between the nearest-neighbor M ions within the unit is almost unchanged, so that
the corresponding nearest-neighbor 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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