*3.3. Comparison of HF and DFT Methods*

Different methods have different calculation accuracy for different substances, and the accuracy of the Raman shifts calculated by HF and DFT needs to be verified by comparison with experimental data. From the literature [20–22], it is clear that theoretical studies have been carried out on PAEs, but only one or two methods are selected for study, and no comparative studies have been conducted. The Raman spectra calculated by some theoretical methods will have many spurious peaks that are not present in the experimental spectra, which will cause interference and errors in the analysis of experimental data; so, it needs to find the suitable theoretical method for PAEs.

Figure 4 shows the theoretical Raman spectra of the five PAEs (DMP, DEP, DBP, DEHP, and DINP) calculated by HF and DFT with the 6-31G(d) basis set. Because many theoretical spectra have some offset errors from experimental spectra, it is necessary to use the scale factors from the database of frequency scale factors for electronic model chemistries [38] to correct the theoretical spectra in order to eliminate the offset error to the greatest extent [39,40]. From the scale factors database, it can be seen that the scale factors of HF 6-31G(d), DFT B3LYP 6-31G(d), and DFT B3PW91 6-31G(d) are 0.885, 0.952, and 0.947, respectively. In this study, the theoretical spectra are the spectra after correction.

**Figure 4.** Theoretical Raman spectra of five phthalic acid esters based on HF and DFT: (**a**) DMP; (**b**) DEP; (**c**) DBP; (**d**) DEHP; (**e**) DINP. HF: Hartree–Fock method; DFT: density functional theory; DMP: dimethyl phthalate; DEP: diethyl phthalate; DBP: dibutyl phthalate; DEHP: di(2-ethyl)hexyl phthalate; DINP: diisononyl phthalate.

As shown in Figure 4, the theoretical Raman spectra calculated by HF and DFT show good agreement as a whole with the experimental Raman spectra, but there are large differences in individual Raman peaks. Comparing the theoretical Raman spectra calculated by HF with the experimental Raman spectra, it is found that the peaks of the five PAEs have common differences. The wide peaks of 1284 and 1450 cm−<sup>1</sup> all become sharp, and the strong peaks of 1726 cm−<sup>1</sup> become much weaker. The theoretical Raman peaks all have a red shift in the band of 300~800 cm−1, while all have a blue shift in the band of 1500~2000 cm−1. In addition to the above common differences, the theoretical Raman spectra of the five PAEs also individually have a lot of spurious Raman peaks. Among them, the theoretical Raman spectra of DMP, DEP, and DBP have more spurious peaks. In terms of DMP, the peak of 818 cm−<sup>1</sup> is divided into peaks of 790 and 806 cm−1, the peak of 1120 cm−<sup>1</sup> is divided into peaks of 1130 and 1146 cm−1, and there is a spurious peak of 1080 cm−1. In terms of DEP, the peak of 1120 cm−<sup>1</sup> is divided into peaks of 1086, 1102, and 1126 cm<sup>−</sup>1, and there is a spurious peak of 992 cm−1. In terms of DBP, the peaks of 940 and 962 cm−<sup>1</sup> are shifted to 970 and 990 cm−1, and the peak of 1120 cm−<sup>1</sup> is divided into peaks of 1086 and 1110 cm−1. Therefore, it can be seen that the theoretical Raman spectra calculated by HF 6-31G(d) have so many errors. This may be because HF ignores most of the electronic correlations [41], which makes the theoretical spectra of PAEs inaccurate.

There are also some errors in the theoretical Raman spectra calculated by DFT. However, the theoretical Raman spectra calculated by B3LYP and B3PW91 of DFT both have fewer spurious peaks than the theoretical Raman spectra calculated by HF. Therefore, DFT is more applicable to theoretical studies of PAEs. Then, the theoretical Raman spectra

calculated by B3LYP and B3PW91 are further compared. It is found that the theoretical Raman spectra calculated by the B3PW91 have slightly more spurious peaks. In terms of DMP, there are two spurious peaks of 320 and 330 cm−1. In terms of DEHP, there is a spurious peak of 800 cm<sup>−</sup>1. Peak of 1726 cm−<sup>1</sup> in five PAEs are all divided into two Raman peaks. In addition, the theoretical calculation time of the two methods is not significantly different. So, it can be concluded that the DFT B3LYP method is more applicable to the theoretical study of PAEs.

## *3.4. Different Basis Sets with DFT B3LYP*

The above results show that the DFT B3LYP method is more suitable for the study of PAEs. Different basis sets of DFT B3LYP have different calculation accuracy, and further theoretical studies are needed to select the suitable basis set.

Figure 5 shows the theoretical Raman spectra of five PAEs (DMP, DEP, DBP, DEHP, and DINP) calculated by DFT B3LYP, with 3-21G, 6-31G(d), 6-311G(d, p), and 6-311G+(d, p) basis sets. From the scale factors database, it can be seen that the scale factors of DFT B3LYP 6-311G(d, p) is 0.9708. The scale factors of DFT B3LYP 6-31G(d) is as above, and the scale factors of 3-21G and 6-311G+(d, p) are not found in the scale factors database.

**Figure 5.** Theoretical Raman spectra of five phthalic acid esters calculated by DFT B3LYP with different basis sets: (**a**) DMP; (**b**) DEP; (**c**) DBP; (**d**) DEHP; (**e**) DINP. DFT: density functional theory; DMP: dimethyl phthalate; DEP: diethyl phthalate; DBP: dibutyl phthalate; DEHP: di(2-ethyl)hexyl phthalate; DINP: diisononyl phthalate.

As shown in Figure 5, compared with the other three basis sets, the theoretical Raman spectra of the 3-21G basis set have significantly more spurious peaks. Comparing the theoretical Raman spectra calculated by the 3-21G basis set with the experimental Raman spectra, it is found that the peaks of the five PAEs have common differences. The peaks of 650 cm−<sup>1</sup> are all shifted to 680 cm−1; the peaks of 1120 cm−<sup>1</sup> are all divided into peaks of 1138 and 1156 cm−1; the peaks of 1160 cm−<sup>1</sup> are all shifted to 1190 cm−1; and the wide peaks of 1450 cm−<sup>1</sup> are all divided into peaks of 1548 and 1570 cm−1. In addition to the above common differences, the theoretical Raman spectra of the five PAEs also individually have many spurious Raman peaks. This may be because the 3-21G basis set has only three original functions fitting per nuclear orbital basis function [42], which makes the theoretical spectra of PAEs inaccurate. Therefore, the 3-21G basis set is not applicable to the theoretical study of PAEs.

The theoretical Raman spectra calculated by the 6-31G(d) basis set have slightly more spurious peaks than the theoretical Raman spectra calculated by the 6-311G(d, p) and 6-311G+(d, p) basis sets. Comparing the theoretical Raman spectra calculated by 6-31G(d) with the experimental Raman spectra, it is found that the peaks of the five PAEs have common differences. The wide peaks of 1450 cm−<sup>1</sup> all turn into a sharp peak and the intensity increases too much. The peaks of 1726 cm−<sup>1</sup> are all divided into peaks of 1706 and 1720 cm<sup>−</sup>1. In addition to the above common differences, the theoretical Raman spectra of the five PAEs also have some spurious peaks. It may be because the 6-31G(d) basis set is represented by two basis functions per valence orbit, which is one function less than the other two basis groups; so, the accuracy of the 6-31G(d) basis set is a bit worse for PAEs.

The difference between the theoretical spectra calculated by 6-311G(d, p) and 6- 311G+(d, p) basis sets is extremely small, and theoretical spectra are both in good agreement with the experimental spectra. However, the scale factors of the 6-311G+(d, p) basis set is not found in the scale factors database. Therefore, compared with the Raman peaks of the experimental spectra, the Raman peaks of the theoretical spectra calculated by the 6-311G+(d, p) basis set are all blue shifted as a whole. In addition, because the 6-311G+(d, p) basis set has more plus dispersion functions on heavy atoms than 6-311G(d, p), the calculation of 6-311G+(d, p) takes nearly three times longer time than 6-311G(d, p). Therefore 6-311G(d, p) is more appropriate for the theoretical study of PAEs.

In summary, the DFT B3LYP 6-311G(d, p) is most suitable for the theoretical study of PAEs. However, the theoretical spectra calculated by DFT B3LYP 6-311G(d, p) still have some differences with experimental Raman spectra in some details. Therefore, the theoretical Raman spectra obtained by DFT B3LYP 6-311G(d, p) were further analyzed. Table 2 shows the common Raman peaks in the theoretical and experimental Raman spectra of the five PAEs. There are some differentiated peaks in theoretical Raman spectra. Compared with the results calculated using the DFT B3LYP 6-311G (d, p) method in the literature, the results of DEHP in this study are basically consistent with the results of DEHP in the literature [37]. In the literature, the experimental and theoretical Raman peaks of DEHP are 399, 653, 1043, 1127, 1167, 1585, 1605, 1731, and 385, 645, 1043, 1134, 1163, 1583, 1608, 1742, 1751 cm<sup>−</sup>1, respectively.


**Table 2.** Common Raman peaks in theoretical and experimental Raman spectra of the five PAEs.

PAEs: phthalic acid esters; DMP: dimethyl phthalate; DEP: diethyl phthalate; DBP: dibutyl phthalate; DEHP: di(2 ethyl)hexyl phthalate; DINP: diisononyl phthalate.

In addition to the peaks in the Table 2, there are still a few other peaks in the Raman spectra of PAEs. Compared with the experimental spectrum, the theoretical spectrum of DMP has two more peaks of 782 and 948 cm−1, which are extremely weak and negligible. The theoretical spectrum of DEP has four more Raman peaks of 342, 834, 870, and 990 cm−<sup>1</sup> than the experimental spectrum. The peak of 342 cm−<sup>1</sup> can be regarded as the differentiated peak from the peak of 360 cm−1, peaks of 834 and 870 cm−<sup>1</sup> can be regarded as the differentiated peaks from the peak of 850 cm−1, and peak of 990 cm−<sup>1</sup> can be ignored because its peak strength is small. The theoretical spectrum of DEHP has two more Raman peaks of 924 and 982 cm−<sup>1</sup> than the experimental spectrum, and peaks of 924 and 982 cm−<sup>1</sup> can be regarded as the differentiated peaks from the peak of 958 cm−1. The theoretical spectrum of DINP has one more Raman peak of 858 cm−<sup>1</sup> than the experimental spectrum, which can be regarded as the differentiated peak from the peak of 822 cm<sup>−</sup>1. The peaks of the theoretical spectrum of DINP are blue-shifted by nearly 40 cm<sup>−</sup>1, while the theoretical Raman peaks of the other four PAEs are all shifted approximately 0~20 cm−<sup>1</sup> relative to the experimental Raman peaks.

From the above results, it can be seen that the theoretical Raman spectra of PAEs still have some differences from the experimental Raman spectra. These differences may be caused by the following reasons. First, the Raman instrument has accuracy problems. Second, the DFT may take the electronic correlation too much into account, leading to calculation errors [20]. Third, theoretical studies generally calculate the structure of individual molecules, while the substances detected experimentally are multimolecular [23]. There are interactions between molecules, and this leads to errors between theory and experiment.
