V
di
11 x; R
ð Þ= D 1 − e
− k x + x 0
ð
Þ
2
+ c,
ð6Þ
V
di
22 x; R
ð Þ= D 1 − e
k x − x 0
ð
Þ
2
+ c,
ð7Þ
V
di
12 x; R
ð Þ= A exp − bx
2
À
Á
.
ð8Þ
For hetero-molecular pairs, V
di
11 , V
di
22 and V
di
12 are defined as
V
di
11 r; R
ð Þ= D 1 1 − e
− k 1 r − r 0
ð
Þ
2
+ c,
ð9Þ
V
di
22 r; R
ð Þ= D 2 1 − e
− k 2 R − r − r
0
0
ð
Þ
2
+ c + D 3 ,
ð10Þ
V
di
12 r; R
ð Þ= A exp − b r − r c
ð
Þ
2
.
ð11Þ
We constructed the PES by optimization of the potential parameters in these
functions (V
di
11 , V
di
22 and V
di
12 ). For V
di
11 and V
di
22 , parameters D, D 1 , and D 2 are binding
energies; x 0 , r 0 , r
0
0 are the equilibrium bond lengths; k, k 1 , and k 2 are the decay
constants. For V
di
12 , parameter A is the amplitude of the Gaussian function;
parameter b corresponds to the spread of the function; r c is the point of the maximum value of the function.
2.3 Optimization of Potential Parameters
Optimization of the potential parameters was conducted according to the following
procedures.
First, we estimated the parameters of V
di
11 and V
di
22 . Parameters D, D 1 , and D 2
were estimated by dissociation energy of the proton corresponding to one side of
the molecular pairs, i.e., ammonium, imidazolium, and oxonium ions. x 0 , r 0 , r
0
0
were used the equilibrium bond lengths of these molecules. The estimated values of
these parameters are shown in Table 1. For hetero-molecular pairs, parameter D 3
was obtained from the difference between V
ad r 0 ; R
ð
Þ and V
ad r
0
0 ; R
À
Á
:
D 3 = V
ad r
′
0 ; R
À
Á − V
ad r 0 ; R
ð
Þ.
ð12Þ
The remaining parameters k, k 1 , and k 2 included in V
di
11 and V
di
22 are considered to
be dependent on the R coordinate. These parameters were estimated by comparison
with the adiabatic potential obtained from DFT calculations according to Eq. (2).
V
di
11 and V
di
22 describes molecular vibration for the reactant and product states,
respectively. Therefore, we assumed that V
di
11 and V
di
22 reproduced the adiabatic
Construction of a Potential Energy Surface …
183
di
11 x; R
ð Þ= D 1 − e
− k x + x 0
ð
Þ
2
+ c,
ð6Þ
V
di
22 x; R
ð Þ= D 1 − e
k x − x 0
ð
Þ
2
+ c,
ð7Þ
V
di
12 x; R
ð Þ= A exp − bx
2
À
Á
.
ð8Þ
For hetero-molecular pairs, V
di
11 , V
di
22 and V
di
12 are defined as
V
di
11 r; R
ð Þ= D 1 1 − e
− k 1 r − r 0
ð
Þ
2
+ c,
ð9Þ
V
di
22 r; R
ð Þ= D 2 1 − e
− k 2 R − r − r
0
0
ð
Þ
2
+ c + D 3 ,
ð10Þ
V
di
12 r; R
ð Þ= A exp − b r − r c
ð
Þ
2
.
ð11Þ
We constructed the PES by optimization of the potential parameters in these
functions (V
di
11 , V
di
22 and V
di
12 ). For V
di
11 and V
di
22 , parameters D, D 1 , and D 2 are binding
energies; x 0 , r 0 , r
0
0 are the equilibrium bond lengths; k, k 1 , and k 2 are the decay
constants. For V
di
12 , parameter A is the amplitude of the Gaussian function;
parameter b corresponds to the spread of the function; r c is the point of the maximum value of the function.
2.3 Optimization of Potential Parameters
Optimization of the potential parameters was conducted according to the following
procedures.
First, we estimated the parameters of V
di
11 and V
di
22 . Parameters D, D 1 , and D 2
were estimated by dissociation energy of the proton corresponding to one side of
the molecular pairs, i.e., ammonium, imidazolium, and oxonium ions. x 0 , r 0 , r
0
0
were used the equilibrium bond lengths of these molecules. The estimated values of
these parameters are shown in Table 1. For hetero-molecular pairs, parameter D 3
was obtained from the difference between V
ad r 0 ; R
ð
Þ and V
ad r
0
0 ; R
À
Á
:
D 3 = V
ad r
′
0 ; R
À
Á − V
ad r 0 ; R
ð
Þ.
ð12Þ
The remaining parameters k, k 1 , and k 2 included in V
di
11 and V
di
22 are considered to
be dependent on the R coordinate. These parameters were estimated by comparison
with the adiabatic potential obtained from DFT calculations according to Eq. (2).
V
di
11 and V
di
22 describes molecular vibration for the reactant and product states,
respectively. Therefore, we assumed that V
di
11 and V
di
22 reproduced the adiabatic
Construction of a Potential Energy Surface …
183