3.2 Potential Energy Surfaces and Spectroscopical Parameters

In Figures 2, 3, 4, and 5, the potential energy surface of LiH is shown as computed with different basis sets. For each basis set, MP4, F4, [2/2], $\Pi$2 and CCSD(T) results are presented. In this particular example, the $\Pi$2 methods is closest to the CCSD(T) result up to R $\approx$ 5 bohr where criterion (8) is satisfied. For larger distances R, this is no longer the case, and in the case of Figures 2 and 4, some unphysical behavior of the $\Pi$2 surface is seen in this region where however criterion (8) indicates that the perturbative results are not acceptable. Comparing the figures, it is also observed that in this case, adding polarization functions slightly enlarges the region where criterion is satisfied, while adding diffuse functions does not have a large effect.

In order to further judge the quality of the method, a Morse potential and polynomials of degree 3 and 5 were fitted to the data as described in Section 2. The results are displayed in Tables 18 to 22 and Figures 6 (LiH), 7 (HF), 8 (HCl) and 9 ( N₂). The color code is blue for MP4, red for $\Pi$2, and green for CCSD(T) results. Broken lines represent Morse fits, and full lines the polynomial fit of third degree to the data points that are displayed also. It is seen that in the equilibrium region, the Morse fits often are lying below the data although they reproduce the overall shape of the curves nicely. This leads to the assumption that the vibrational parameters will be acceptable for the Morse fit. The third degree polynomial fits are in most cases more accurate in the region of the minimum but differ significantly from the potential curve for bigger distances. Hence, it is expected that they will produce more accurate estimates for the harmonic frequencies but worse values for the other spectroscopic constants which should be calculated using a 5th degree fitting polynomial.

This is indeed the case. In Table 18 we report the results for the harmonic frequencies, the anharmonic correction term obtained from the Morse fit, the values of the resulting fit parameters $\ensuremath{\mathbf{D}_e}$ and $\beta$ and experimental data from Refs. [29] and (in parentheses) [30]. The results for the polynomial fits, 8 Parameter for each molecule, are shown in Tables 19 (LiH),20 (HF), 21 (HCl) and 22 ( N₂).

In this context, it should be noted that all results are relatively sensitive to the selection of the data points used in the fitting procedure. In the fits, we usually included all points that are compatible with criterion (8) in case of the Morse potential - beware that not all these data points are displayed in the figures in order to be able to simultaneously display the Morse and polynomial fits with sufficient resolution -, and 8 points around the minimum for the polynomial fit. The selection of the latter points is plain from the corresponding figures.

In the case of N₂, the MP4 values for R > 2.6 bohr were unphysical and excluded from the fit. Also, criterion (8) was not satisfied in this region. However, since it was observed that the $\Pi$2 results agreed rather closely the CCSD(T) results even in this region, we included some points from this region also in order to produce the corresponding Morse fits. The results for the harmonic frequency as obtained from a second Morse fit using only points for which criterion (8) was satisfied was close to the result displayed in Table 18 while the value for the anharmonicity constant was worse.

The main results using $\Pi$2/6 - 311G and CCSD(T)/6 - 311G levels of theory are that

• the $\Pi$2 results are sufficiently close to the (more expensive) CCSD(T) method using the same fit methodology,
• harmonic frequencies are in most cases better (accuracy 1-3 percent) calculated via polynomial fits of third degree than using Morse fits that can be competitive if these fits of the potential surface are in the equilibrium region of high accuracy, as in the case of the HF molecule, or at least have the same shape as the polynomial fit, as in the case of LiH,
• Morse fits are able to reproduce experimental anharmonic parameter x_e with an accuracy of about 10-30 percent. For this parameter the 5th degree polynomial fit results are of similar accuracy whereas for the remaining three spectroscopic constants the accuracy lies around 5-10 percent.