3.2 THE [HeHNe] + SYSTEM
3.2.3 Global analytical PES
−70
−65
−60
−55
−50
−45
−40
−35
−30
0 2 4 6 8 10 12 14
Energy (kcal/mol)
Reaction Coordinate
180° 165° 150° 135° 120°105° 90°
Figure 3.3: Minimum energy pathways for different He-H-Ne angles. Reaction coor- dinate has an arbitrary unit of length. Zero of energy is set to the all atom dissociation
limit energy for the system.
In Figure 3.3, the minimum energy pathways for different He-H-Ne angles are plot- ted. The MEP for the collinear arrangement goes through the deepest potential energy well, which is the global minimum of the [HeHNe]+ system. The depth of the well is about 12.01 kcal/mol with respect to the He+NeH+ asymptote and it is located at RHeH = 1.804 a.u. andRNeH = 2.102 a.u. The well depth decreases to 9.867 kcal/mol when the He-H-Ne angle becomes 150◦. For 105◦, a barrier is seen in the MEP and this barrier height increases for 90◦. Harmonic vibrational frequencies for the most stable structure of [HeHNe]+ are calculated numerically by means of ab initiocomputation as well as from the analytical PES. The structural and energetic details of the equilibrium geometries for the diatoms and the triatomic system obtained via ab initio computa- tions and calculated from the analytical PES are tabulated in Table 3.5 along with the harmonic vibrational frequencies of the triatom at the equilibrium geometry. The fundamental modes with degenerate energies correspond to the bent vibrational modes and the highest stretching frequency corresponds to the asymmetric stretching mode, where two bonds vibrate simultaneously with opposite atomic motions (while one bond is elongating the other is contracting). A very good agreement between the parameters
obtained from ab initio computation and from the analytical PES is quite obvious in Table 3.5.
Table 3.5: Equilibrium bond lengths (in a.u.) and energies (in kcal/mol) at equi- librium geometries of diatoms and triatom and harmonic vibrational frequencies (in cm−1) of the triatom at the most stable geometry. Zero of energy is the same as Figure
3.3.
ab initio analytical PES
HeH+ Req 1.464 1.464
Energy -47.066 -47.066
NeH+ Req 1.872 1.872
Energy -53.433 -53.433
HeNe Req 5.696 5.684
Energy -0.0484 -0.0485
[HeHNe]+ Req(HeH) 1.804 1.804
Req(NeH) 2.102 2.102
∠HeHNe 180.0 180.0
Energy -65.447 -65.447
Harmonic vibrational frequencies
853.8, 853.8, 848.8, 848.8, 855.8, 1602.5 856.0, 1605.3
r (a.u.)
1 3 5 7
9 0° 15°
1 3 5 7 9
2 4 6 8 10
30°
R (a.u.)
2 4 6 8 10
45°
Figure 3.4: Contour diagrams of the analytical PES in reactant Jacobi coordinates for the He+NeH+ reactive system for different θ. The spacing between the contour
lines is 7.5 kcal/mol.
r (a.u.) 1 3 5 7 9
11 0° 15°
1 3 5 7 9 11
2 4 6 8 10 12
30°
R (a.u.) 2 4 6 8 10 12 45°
Figure 3.5: Same as Figure 3.4 but in terms of Ne+HeH+reactant Jacobi coordinates.
R (a.u.)
2 4 6 8
0 30 60 90 120 150 (a)
θ°
0 30 60 90 120 150 180 (b)
Figure 3.6: Contour diagrams of the analytical PES in reactant Jacobi coordinates. r is kept constant atreq. (a) He+NeH+system,req = 1.872 a.u. (b) Ne+HeH+ system,
req = 1.464 a.u. The spacing between the contour lines is 2.5 kcal/mol.
In Figures 3.4, 3.5 and 3.6, the contour diagrams of the analytical PES are presented in reactant Jacobi coordinates for the two reactive collision systems. It is clear from these figures that both the He+NeH+and Ne+HeH+systems prefer collinear and near- collinear paths for a reactive collision. While the He + NeH+ → Ne + HeH+ reaction has a late barrier (barrier height corresponds to the endothermicity of the reaction), Ne + HeH+ → He + NeH+ reaction is exothermic and barrierless in nature. For larger attacking angles, both the reactions fail to occur.
The energies obtained from the analytical PES for different geometries which include short range and long range and potential well regions are plotted along with their ab initio counterparts in Figure 3.7. As can be seen in Figure 3.7, there is an excellent agreement between theab initio and analytical energies. Hence, it is worth mentioning that the analytical surface successfully describes the asymptotic regions as well as the interaction regions of the [HeHNe]+ system with high accuracy. To further check the accuracy of the analytical PES, ab initio energies of 422 randomly distributed points are calculated and compared with the analytical values. Figure 3.8 shows the difference between theab initio and analytical energies with respect to the total energies for those random points. The largest deviation has been found to be 0.083 kcal/mol and the rms error is less than 0.023 kcal/mol.
As mentioned earlier, two local potential minima corresponding to collinear He- Ne-H and Ne-He-H configurations were characterized by ab initio calculations. Those two minima are also found in the analytical PES. The local minimum [HeNeH]+ is energetically 4050.9 cm−1 higher than the most stable structure. This is in excellent agreement with theab initioenergy difference (4047.5 cm−1) between the local minimum [HeNeH]+ and the global minimum. The HeNe and NeH+ bond lengths in [HeNeH]+ are 4.822 and 1.871a0, respectively. The HeNe and HeH+bond distances for the second local minimum [NeHeH]+ are 4.128 and 1.458 a0, respectively. This local minimum is located at 5959.1 cm−1 high in energy with respect to the global minimum and it is close to theab initiovalue 5954.1 cm−1.
−65
−60
−55
−50
−45
0 2 4 6 8 10 12 14
(a) MEP at ∠HeHNe = 180°
Energy (kcal/mol)
Reaction Coordinate
−10 0 10 20 30 40 50
(b) r = 2.0 a0, θ = 20°
Energy (kcal/mol)
0 5 10 15
2 4 6 8 10
(c) r = 2.0 a0, θ = 90°
Energy (kcal/mol)
−4
−3
−2
−1 0 1
4 8 12 16 20
(d)
R (a.u.) r = req, θ = 0°
−20 0 20 40 60
(e) r = 1.6 a0, θ = 20°
0 10 20 30 40
2 4 6 8 10
(f) r = 1.6 a0, θ = 90°
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0
0.2 (g) r = req, θ = 90°
−5
−4
−3
−2
−1 0
4 8 12 16 20
(h)
R (a.u.)
r = req, θ = 0°
Figure 3.7: Comparison between ab initio energies (square points) and analytical energies (solid lines): (a) along the collinear MEP as presented in Figure 3.3, (b, c, d) in He+NeH+ reactant Jacobi coordinates and (e, f, g, h) in Ne+HeH+ reactant Jacobi coordinates. Zero of energy for (a) is the same as defined in Figure 3.3 and it
corresponds to the reactant asymptote for the others.
−0.003
−0.002
−0.001 0 0.001 0.002 0.003
−3 −2 −1 0 1 2 3
∆E (eV)
Total Energy (eV)
Figure 3.8: ∆E, the differences in energies between analytical andab initio values, are plotted against totalab initioenergy at some random configurations. Zero is set at
all atom dissociation limit.