FIG. 1 presents the PILD infrared spectra for the HCHO molecule at $ T $ = 300 K and $ T $ = 100 K. Table Ⅰ then depicts comparison of the exact vibrational frequencies to the spectra obtained by PILD at $ T $ = 100 K and 300 K and by the normal mode analysis (NMA). In comparison to the exact values of this molecular system, the NMA results are reasonable for low-frequency vibrational modes, but significantly blue-shifted for high-frequency modes. It indicates that nuclear quantum effects are remarkably strong for C-H bond stretching motions in the high-frequency region. The peak position of the symmetric stretching mode for the two C-H bonds is only about 50 cm-1 from that of the asymmetric stretching mode. The two peaks are readily distinguished in the PILD spectrum because the full width at half maximum (FWHM) of the PILD peak is about 10-15 cm-1. The peak positions of PILD spectrum are relatively insensitive to the temperature---no more than 5 cm-1 difference between the results for 300 K and for 100 K. This is in good agreement with the fact that exact vibrational frequencies of an isolated small molecule are independent of the temperature change.
Figure 1. The IR spectra for HCHO at both 300 K and 100 K. Purple dotted lines represent exact frequencies.
Table Ⅰ. Peak positions of the HCHO molecule at different temperatures. PES and exact results from Ref. (unit: cm−1).
FIG. 2 and Table Ⅱ then demonstrate the spectra and peak positions of the isotope molecules (HCDO and DCDO) for 300 K. All peaks are effectively sharp and their positions agree well with the exact values (the difference is no more than 11 cm-1). When the hydrogen atoms are substituted with the deuterium atoms in the formaldehyde molecule, the peak positions in HCDO and in DCDO are red-shifted from the corresponding ones in HCHO. The red-shift data are shown in Table Ⅲ. For instance, the red shifts for O-H stretching motions for the exact vibrational frequencies are 689 cm-1 in HCDO, and 725.9 cm-1, 684.2 cm-1 in DCDO, respectively. By comparison, the red shifts for the PILD results are 713 cm-1 in HCDO, and 737 cm-1, 693 cm-1 in DCDO. The differences between PILD and exact red shifts are small, no more than 24 cm-1. As a contrast, the NMA red shifts are 780.5 cm-1 in HCDO, and 795.8 cm-1, 758.2 cm-1 in DCDO, respectively. That is, NMA considerably overestimates the red shift, of which the deviation from the exact value is always greater than 70 cm-1. Comparison of the PILD peak positions of HCHO, HCDO, and DCDO to the exact vibrational frequencies in Table Ⅱ indicates that PILD robustly captures isotope effects for this system.
Figure 2. The IR spectra for isotope molecules of HCHO for 300 K. Purple dotted lines represent exact frequencies.
Table Ⅱ. Peak positions of isotope molecules of the HCHO molecule at 300 K. PES and exact results from Ref. (unit: cm−1).
FIG. 3 shows the IR spectra obtained by PILD at $T = 300 $K and $T = 100$ K and Table Ⅳ lists the peak positions in comparison to exact and NMA results. In addition, experimental values are listed. Both PILD and NMA yield reasonable results that are close to exact vibrational frequencies in the low-frequency region. In the high-frequency region, PILD is much superior to NMA. While the error of the PILD peak position for the highest frequency is no more than ~25 cm-1 at 300 K and ~ 11 cm-1 at 100 K, that of the NMA result is as great as ~ 187 cm-1. Any PILD peak position for 100 K is no more than 16 cm-1 different from that for 300 K, which demonstrates that PILD is reasonably stable as the temperature changes.
Figure 3. The IR spectra for H2O2 at both 300 K and 100 K. Purple dotted lines represent exact frequencies.
FIG. 4 and Table Ⅴ demonstrate the results of the isotope molecules of H2O2 (HDO2, D2O2, and H218O2). All PILD peak positions show reasonable agreement with the exact results for these isotope molecules. The peak of the O-O stretching mode (the second fundamental, 864.5/884.9 cm-1 for H2O2, 825.7 cm-1 for H218O2, or 877.4 cm-1 for D2O2) is extremely weak in the infrared spectrum of H2O2, H218O2, or D2O2 (Refs.[21, 25, 26]. Since HDO2 breaks the symmetry of H2O2, one more distinct peak appears around 880 cm-1, as well as the two O-H/O-D bond stretching peaks become noticeably distinguishable. The isotopic substitution yields a significant red shift from the O-H stretching peak to the O-D one. Table Ⅵ lists the red-shift values of the peak positions of D2O2 in comparison to H2O2. PILD faithfully describes the red shift with an error no larger than 27 cm-1, while the difference between the NMA red shift and the exact value can be as large as 89 cm-1. The characteristic features in the spectrum of D2O2 are similar to those in the spectrum of H2O2. When the hydrogen atoms are replaced by the heavier deuterium atoms, the symmetry and asymmetry stretching modes (with highest frequencies) are considerably red-shifted from those of H2O2. The deviations of the PILD results from the exact red shift values are smaller than 3 cm-1, and in contrast, the error of the red shift caused by NMA is larger than 84 cm-1. Isotope effects of the substitution for the oxygen element are less significant for the high-frequency stretching vibrational modes.
Figure 4. The IR spectra for isotope molecules of H2O2 for 300 K. Purple dotted lines represent exact frequencies.
Table Ⅵ. Corresponding peak position red-shifts of the D2O2 molecule from the H2O2 molecule at 300 K. Calculated based on Table Ⅳ and Table Ⅴ (unit: cm−1). Except for the first fundamental corresponding to the torsion mode, the exact red-shift value is obtained by using the average of the two tunneling splitting peak positions of the fundamental of the H2O2 molecule as the reference.
It is worth pointing out that PILD fails to capture true long-time quantum coherence effects  in the correlation function, so it is not able to faithfully describe the tunneling splitting when its value is smaller than 25 cm-1. PILD semi-quantitatively captures the tunneling splitting of the torsion motion (during 200-400 cm-1) in the spectrum, which is relatively large and ranges from 39.3 to 120.3 cm-1 for the PES as shown in Tables IV and V. But PILD is not able to semi-quantitatively depict the tunneling splitting of any other fundamental as shown in Table Ⅳ. Neither is PILD capable of describing the difference between the exact vibrational frequencies around 3614-3632 cm-1 for H2O2 or those around 2676 cm-1 for D2O2 corresponding to the symmetric and asymmetric O-H (or O-D) stretching modes that are near-degenerate, e.g., the difference is less than 3 cm-1 for D2O2. This drawback of PILD can hardly be improved in the present framework, because no unique definition of trajectory in the phase space exists in quantum mechanics.
Finally, the overtone bands of the torsion mode appear in the region of about 500-800 cm-1 in the PILD spectra. For example, the exact first overtone 776.2 cm-1 is reasonably accurately captured in the IR spectra of H2O2 in FIG. 3. As the intensity is relatively low, we do not explicitly discuss it in the paper.