First, parameter A was estimated. Because this parameter represents the amplitude of the Gaussian type function, parameter b vanishes at x = 0 for
homo-molecular pairs. Thus, the relationship between the adiabatic and diabatic
potentials at x = 0 and fixed R is given by
V
di
11 0
ð Þ + V
di
22 0
ð Þ
2
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V di
11 0
ð Þ − V di
22 0
ð Þ
2
2
+ V di 2
12 0
ð Þ
s
= V
ad 0
ð Þ.
ð19Þ
Parameter A was then estimated by using obtained V
di
11 and V
di
22 , which was
expressed as
A = V
di
11 0
ð Þ − V
ad 0
ð Þ,
ð20Þ
for homo-molecular pairs. For hetero-molecular pair, parameter A was estimated at
the cross point (r = r c ) of V
di
11 and V
di
22 substituted D 3 = 0. Parameter A was then
expressed as
A = V
di
11 r c
ð Þ− V
ad r c
ð Þ,
ð21Þ
for hetero-molecular pairs.
Next, the remaining parameter b of V
di
12 was estimated. This parameter corresponds to the spread of the Gaussian type function. For homo-molecular pair, the
midpoint between the local minimum (x 0 ) and local maximum x = 0
ð
Þis assumed to
define the spread of V
di
12 . Therefore, the relationship between the adiabatic and
diabatic potentials at x = x 0 ̸ 2 and fixed R is also given by
V
di
11
x 0
2
À Á
+ V
di
22
x 0
2
À Á
2
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V di
11
x 0
2
À Á − V di
22
x 0
2
À Á
2
2
+ V di 2
12
x 0
2
s
= V
ad x 0
2
.
ð22Þ
According to this relation, parameter b is expressly defined by
b =
2
x 2
0
lnA
2
− ln
V
di
11
x 0
2
À Á
+ V
di
22
x 0
2
À Á
2
− V
ad x 0
2
2
−
V
di
11
x 0
2
À Á − V
di
22
x 0
2
À Á
2
2
(
)
"
#
.
ð23Þ
In the same way, for hetero-molecular pairs, b was determined by comparing
with the adiabatic potential at r = r c + r 0
ð
Þ ̸ 2 ≡ r b for fixed R, which was defined by
b =
1
r b − r c
ð
Þ
2
ln A
2
−
1
2
ln
V
di
11 r b
ð Þ+ V
di
22 r b
ð Þ
2
− V
ad r b
ð Þ
2
−
V
di
11 r b
ð Þ− V
di
22 r b
ð Þ
2
2
(
)
"
#
.
ð24Þ
Construction of a Potential Energy Surface …
185
homo-molecular pairs. Thus, the relationship between the adiabatic and diabatic
potentials at x = 0 and fixed R is given by
V
di
11 0
ð Þ + V
di
22 0
ð Þ
2
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V di
11 0
ð Þ − V di
22 0
ð Þ
2
2
+ V di 2
12 0
ð Þ
s
= V
ad 0
ð Þ.
ð19Þ
Parameter A was then estimated by using obtained V
di
11 and V
di
22 , which was
expressed as
A = V
di
11 0
ð Þ − V
ad 0
ð Þ,
ð20Þ
for homo-molecular pairs. For hetero-molecular pair, parameter A was estimated at
the cross point (r = r c ) of V
di
11 and V
di
22 substituted D 3 = 0. Parameter A was then
expressed as
A = V
di
11 r c
ð Þ− V
ad r c
ð Þ,
ð21Þ
for hetero-molecular pairs.
Next, the remaining parameter b of V
di
12 was estimated. This parameter corresponds to the spread of the Gaussian type function. For homo-molecular pair, the
midpoint between the local minimum (x 0 ) and local maximum x = 0
ð
Þis assumed to
define the spread of V
di
12 . Therefore, the relationship between the adiabatic and
diabatic potentials at x = x 0 ̸ 2 and fixed R is also given by
V
di
11
x 0
2
À Á
+ V
di
22
x 0
2
À Á
2
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V di
11
x 0
2
À Á − V di
22
x 0
2
À Á
2
2
+ V di 2
12
x 0
2
s
= V
ad x 0
2
.
ð22Þ
According to this relation, parameter b is expressly defined by
b =
2
x 2
0
lnA
2
− ln
V
di
11
x 0
2
À Á
+ V
di
22
x 0
2
À Á
2
− V
ad x 0
2
2
−
V
di
11
x 0
2
À Á − V
di
22
x 0
2
À Á
2
2
(
)
"
#
.
ð23Þ
In the same way, for hetero-molecular pairs, b was determined by comparing
with the adiabatic potential at r = r c + r 0
ð
Þ ̸ 2 ≡ r b for fixed R, which was defined by
b =
1
r b − r c
ð
Þ
2
ln A
2
−
1
2
ln
V
di
11 r b
ð Þ+ V
di
22 r b
ð Þ
2
− V
ad r b
ð Þ
2
−
V
di
11 r b
ð Þ− V
di
22 r b
ð Þ
2
2
(
)
"
#
.
ð24Þ
Construction of a Potential Energy Surface …
185