Table 1 Diabatic potential energy parameters for (a) AmH
+
-Am, (b) ImH
+ -Im, (c) ImH
+ -Am,
and (d) AmH
+ -Wat
(a) AmH
+
-Am
(b) ImH
+ -Im
D(kJ mol
−1
)
r 0 (Å)
D(kJ mol
−1
)
r 0 (Å)
664.758
1.027
668.81
1.0148
(c) ImH
+ -Am
D 1 (kJ mol
−1
)
D 2 (kJ mol
−1 )
r 0 (Å)
r
0
0 (Å)
668.81
664.758
1.015
1.027
(d) AmH
+ -Wat
D 1 (kJ mol
−1
)
D 2 (kJ mol
−1 )
r 0 (Å)
r
0
0 (Å)
725.69
593.731
1.027
0.977
potential, when the proton position was closer to a binding atom than a stable point
(r < r 0 ). Here, we estimated the parameter k by comparing them with the V
di
22 and
adiabatic potential at x = 2x 0 for homo-molecular pairs for fixed R. In the same way,
for hetero-molecular pairs, k 1 and k 2 were determined by comparing with the
adiabatic potential at r = 2r 0 − R ̸ 2 ≡ r k 1 and r = 3R ̸ 2 − 2r
0
0 ≡ r k 2 , respectively, for
fixed R. These relations are explicitly defined as
V
di
22 2x 0
ð Þ= V
ad 2x 0
ð Þ,
ð13Þ
V
di
11 r k 1
ð Þ= V
ad r k 1
ð Þ,
ð14Þ
V
di
22 r k 2
ð Þ= V
ad r k 2
ð Þ.
ð15Þ
for homo- and hetero-molecular pairs, respectively. Parameters k, k 1 , and k 2 are
then denoted as
k =
1
x 0
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ad 2x 0
ð Þ− c
D
r
!
,
ð16Þ
k 1 =
1
r 0 − r k 1
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V ad r k 1
ð Þ− c
D 1
s
0
@
1
A ,
ð17Þ
k 2 =
1
r k 2 + r
0
0 − R
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ad r k 2
ð Þ− c − D 3
D 2
s
0
@
1
A .
ð18Þ
Subsequently, parameters A and b of V
di
12 depending on the coordinate R were
estimated. These parameters were also estimated by comparison with the adiabatic
potential obtained from DFT calculations and using obtained diabatic potentials
(V
di
11 and V
di
22 ). Determination of these parameters was conducted according to the
following steps for each pairs.
184
Y. Hori et al.
+
-Am, (b) ImH
+ -Im, (c) ImH
+ -Am,
and (d) AmH
+ -Wat
(a) AmH
+
-Am
(b) ImH
+ -Im
D(kJ mol
−1
)
r 0 (Å)
D(kJ mol
−1
)
r 0 (Å)
664.758
1.027
668.81
1.0148
(c) ImH
+ -Am
D 1 (kJ mol
−1
)
D 2 (kJ mol
−1 )
r 0 (Å)
r
0
0 (Å)
668.81
664.758
1.015
1.027
(d) AmH
+ -Wat
D 1 (kJ mol
−1
)
D 2 (kJ mol
−1 )
r 0 (Å)
r
0
0 (Å)
725.69
593.731
1.027
0.977
potential, when the proton position was closer to a binding atom than a stable point
(r < r 0 ). Here, we estimated the parameter k by comparing them with the V
di
22 and
adiabatic potential at x = 2x 0 for homo-molecular pairs for fixed R. In the same way,
for hetero-molecular pairs, k 1 and k 2 were determined by comparing with the
adiabatic potential at r = 2r 0 − R ̸ 2 ≡ r k 1 and r = 3R ̸ 2 − 2r
0
0 ≡ r k 2 , respectively, for
fixed R. These relations are explicitly defined as
V
di
22 2x 0
ð Þ= V
ad 2x 0
ð Þ,
ð13Þ
V
di
11 r k 1
ð Þ= V
ad r k 1
ð Þ,
ð14Þ
V
di
22 r k 2
ð Þ= V
ad r k 2
ð Þ.
ð15Þ
for homo- and hetero-molecular pairs, respectively. Parameters k, k 1 , and k 2 are
then denoted as
k =
1
x 0
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ad 2x 0
ð Þ− c
D
r
!
,
ð16Þ
k 1 =
1
r 0 − r k 1
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
V ad r k 1
ð Þ− c
D 1
s
0
@
1
A ,
ð17Þ
k 2 =
1
r k 2 + r
0
0 − R
ln 1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ad r k 2
ð Þ− c − D 3
D 2
s
0
@
1
A .
ð18Þ
Subsequently, parameters A and b of V
di
12 depending on the coordinate R were
estimated. These parameters were also estimated by comparison with the adiabatic
potential obtained from DFT calculations and using obtained diabatic potentials
(V
di
11 and V
di
22 ). Determination of these parameters was conducted according to the
following steps for each pairs.
184
Y. Hori et al.