2.4 Adiabatic Potential Energy Calculation
To obtain the adiabatic PES V
ad x, R
ð Þ or V
ad r, R
ð Þ for the proton transfer reaction,
quantum chemical calculations were performed with stepwise movement of the
proton and intermolecular distances by 0.02 and 0.05 Å, respectively. The
geometries of the molecules were then kept in the minimum energy structures,
except for the proton position (x or r) and the intermolecular distance (R). Minimum
energy structures, as shown in Fig. 1, were determined by geometrical optimization. Finally, the adiabatic potential energies were fitted using a polynomial series
function of the fourth order with respect to x or r and of the third order with respect
to R [31, 32]. In previous work [33, 34], the proton transfer of ImH
+
-Im systems
was discussed with the B3LYP approach. Therefore, all DFT calculations were
performed at the B3LYP/aug-cc-pVDZ level using the Gaussian-09 package [35].
3 Results and Discussion
We first estimated the potential parameters k, k 1 , k 2 , and D 3 of V
di
11 and V
di
22 using
Eqs. (12) and (16)–(18) and obtained parameters (Table 1) for (a) AmH
+
-Am,
(b) ImH
+
-Im, (c) ImH
+
-Am, and (d) AmH
+
-Wat. The parameters A and b were then
estimated using Eqs. (20), (21), (23), and (24). Figure 3 shows the computed V
di
11 ,
V
di
22 , and V
di
12 values using obtained the potential parameters at some intermolecular
distance for (a) AmH
+
-Am, (b) ImH
+
-Im, (c) ImH
+
-Am and (d) AmH
+
-Wat,
respectively. To compare with the adiabatic potential by DFT calculation, Figs. 4
and 5 show the transformed adiabatic potential derived from the diabatic potential
matrix elements (V
di
11 , V
di
22 , and V
di
12 ) using Eq. (2) and one obtained from DFT.
Although the potentials using diabatic model not reproduce the ones using DFT
around the local minimum of the potentials, however, the figures show that the
transformed adiabatic potentials are qualitatively in good agreement with those
calculated by DFT calculations for all proton transfer systems at various intermolecular distance R. Furthermore, to compare with the present work and DFT data
Table 2 shows the coefficient of determination (R-squared). Because the values of
R-squared for all models were close to unity, we confirmed the validity of our
approach. Thus, it is indicates that PES for various proton transfer systems can
qualitatively reproduce by using V
di
11 and V
di
22 with Morse potential described the
vibrational motion and V
di
12 with the Gaussian function by assumption of two-state
VB wave functions as a diabatic basis.
In our approach, parameters k, A, and b can be uniquely determined not necessary to fit the potential energy. Especially, the construction procedures of the
186
Y. Hori et al.
To obtain the adiabatic PES V
ad x, R
ð Þ or V
ad r, R
ð Þ for the proton transfer reaction,
quantum chemical calculations were performed with stepwise movement of the
proton and intermolecular distances by 0.02 and 0.05 Å, respectively. The
geometries of the molecules were then kept in the minimum energy structures,
except for the proton position (x or r) and the intermolecular distance (R). Minimum
energy structures, as shown in Fig. 1, were determined by geometrical optimization. Finally, the adiabatic potential energies were fitted using a polynomial series
function of the fourth order with respect to x or r and of the third order with respect
to R [31, 32]. In previous work [33, 34], the proton transfer of ImH
+
-Im systems
was discussed with the B3LYP approach. Therefore, all DFT calculations were
performed at the B3LYP/aug-cc-pVDZ level using the Gaussian-09 package [35].
3 Results and Discussion
We first estimated the potential parameters k, k 1 , k 2 , and D 3 of V
di
11 and V
di
22 using
Eqs. (12) and (16)–(18) and obtained parameters (Table 1) for (a) AmH
+
-Am,
(b) ImH
+
-Im, (c) ImH
+
-Am, and (d) AmH
+
-Wat. The parameters A and b were then
estimated using Eqs. (20), (21), (23), and (24). Figure 3 shows the computed V
di
11 ,
V
di
22 , and V
di
12 values using obtained the potential parameters at some intermolecular
distance for (a) AmH
+
-Am, (b) ImH
+
-Im, (c) ImH
+
-Am and (d) AmH
+
-Wat,
respectively. To compare with the adiabatic potential by DFT calculation, Figs. 4
and 5 show the transformed adiabatic potential derived from the diabatic potential
matrix elements (V
di
11 , V
di
22 , and V
di
12 ) using Eq. (2) and one obtained from DFT.
Although the potentials using diabatic model not reproduce the ones using DFT
around the local minimum of the potentials, however, the figures show that the
transformed adiabatic potentials are qualitatively in good agreement with those
calculated by DFT calculations for all proton transfer systems at various intermolecular distance R. Furthermore, to compare with the present work and DFT data
Table 2 shows the coefficient of determination (R-squared). Because the values of
R-squared for all models were close to unity, we confirmed the validity of our
approach. Thus, it is indicates that PES for various proton transfer systems can
qualitatively reproduce by using V
di
11 and V
di
22 with Morse potential described the
vibrational motion and V
di
12 with the Gaussian function by assumption of two-state
VB wave functions as a diabatic basis.
In our approach, parameters k, A, and b can be uniquely determined not necessary to fit the potential energy. Especially, the construction procedures of the
186
Y. Hori et al.