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CP
CDW
D
E(
β
)
0
β
/(2K
MX
)
*
E
CDW
(
β
)
u
c
β
/(2K
MX
)
*
E
CP
(
β
)
D
E(
β
)
2

β
/(4K
MX
)
X
X
Fig. 12.12 Diabatic potentials and schematic electronic structures in the CP (
left
) and CDW
(
right
) phases [
24
]
We consider the two electronic configurations shown at the bottom of Fig.
12.12
,
where
n
1
2 and
n
4
1 . The total energies are given in the secondorder
perturbation theory with respect to
t
MM
and
t
MXM
[
24
]:
2
2
4
K
MX
þ
D
EðbÞþ
b
2
K
MX
b
M
2
u
1
;
E
CDW
ðu
1
; u
1
Þ¼K
MX
u
1
(12.4)
2
2
4
K
MX
þ
b
2
K
MX
b
M
2
u
1
;
E
CP
ðu
1
; u
1
Þ¼K
MX
u
1
þ
(12.5)
where the energies are equally shifted to have
E
CP
(0,0)
¼
0 and the energy differ
ence between the CDW and CP states
D
E
(
b
) is given by
8
p
t
2
MM
2
t
2
MM
K
MX
b
4
t
2
MM
D
EðbÞ¼
U
M
p
K
MX
U
3
M
ðb
c
bÞ;
(12.6)
2
1
=
2
, as a function of
with
. The diabatic potentials,
E
CDW
(
u
1
,0)
and
E
CP
(
u
1
,0), are plotted in Fig.
12.12
. The CDW and CP states become degenerate
at
b
c
¼ðK
MX
U
M
=
2
Þ
b
b ¼ b
c
and
DE
(
b
c
)
¼
0. The crossing point
u
c
is defined by the relation
E
CDW
ð
u
c
;
0
Þ¼E
CP
u
c
;
ð
0
Þ
and given by
u
c
¼ DEðbÞ=ð
2
bÞ
. The barrier heights are given
by
2
2
b
2
K
MX
;
K
MX
4
D
EðbÞ
b
E
CDW
ðbÞ¼E
CDW
ðu
c
;
0
ÞE
CDW
0
¼
;
(12.7)
2
K
MX
b
2
2
b
2
K
MX
;
0
K
MX
4
D
EðbÞþ
b
E
CP
ðbÞ¼E
CP
ðu
c
;
0
ÞE
CP
¼
:
(12.8)
2
K
MX
b
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