Quantitative Structure-Activity Relationships Study on the Rate Constants of Polychlorinated Dibenzo-p-Dioxins with OH Radical

Abstract

The OH-initiated reaction rate constants (k_OH) are of great importance to measure atmospheric behaviors of polychlorinated dibenzo-p-dioxins (PCDDs) in the environment. The rate constants of 75 PCDDs with the OH radical at 298.15 K have been calculated using high level molecular orbital theory, and the rate constants (k_α, k_β, k_γ and k_OH) were further analyzed by the quantitative structure-activity relationships (QSAR) study. According to the QSAR models, the relations between rate constants and the numbers and positions of Cl atoms, the energy of the highest occupied molecular orbital (E_HOMO), the energy of the lowest unoccupied molecular orbital (E_LUMO), the difference ΔE_HOMO-LUMO between E_HOMO and E_LUMO, and the dipole of oxidizing agents (D) were discussed. It was found that E_HOMO is the main factor in the k_OH. The number of Cl atoms is more effective than the number of relative position of these Cl atoms in the k_OH. The k_OH decreases with the increase of the substitute number of Cl atoms.

1. Introduction

Polychlorinated dibenzo-p-dioxins and polychlorinated dibenzofurans (PCDD/Fs) in the atmosphere mainly derive from the processes of waste combustion, including carbon source and chlorine [1,2,3,4]. Due to the fact that PCDD/Fs tend to bioaccumulate and biomagnify in the food chain, PCDD/Fs have been a frequent topic in the public discourse [5,6]. The OH radical reactions with PCDD/Fs are considered a dominant removal process in the atmosphere [7,8].

The OH radical reaction rate constants (k_OH) play a significant role in measuring the atmospheric behaviors of organic pollutants in the environment, and are indispensable for environmental risk assessment of organic chemicals. The k_OH can be used to calculate the half lives of most organic compounds in the air. Thus, it is critical to know the k_OH. Unfortunately, due to the high toxicities and rather low vapor pressure of PCDD/Fs, direct kinetic measurements of the reactions of PCDD/Fs and OH radical are scarce. Only a few experimental reports are available for the gas-phase reactions between PCDD/Fs, for example, Brubaker and Hites [9] measured the k_OH of dibenzo-p-dioxin (DD), 2,7-dichlorodibenzo-p-dioxin (2,7-DCDD), dibenzofuran (DF) and 2,8-dichlorodibenzofuran (2,8-DCDF) in a small, heated quartz reaction chamber sampled by on-line mass spectrometry. Taylor et al. [10] obtained the k_OH of DD, 2-chlorodibenzo-p-dioxin (2-CDD), 2,3-dichlorodibenzo-p-dioxin (2,3-DCDD), 2,7-DCDD, 2,8-dichlorodibenzo-p-dioxin (2,8-DCDD), 1,2,3,4-tetrachlorodibenzo-p-dioxin (1,2,3,4-TCDD) and octachlorodibenzo-p-dioxin (OCDD) using the pulsed laser photolysis/pulsed laser-induced fluorescence (PLP/PLIF) technique. In recent years, computational chemistry has emerged as a potent tool to identify rate constants. Altarawneh et al. [11] obtained a rate constant of 2.70 × 10⁻¹¹ cm³·molecule⁻¹·s⁻¹ for DF with OH radicals using conventional transition state theory (TST). Wang and Tang [12] have estimated the k_OH of DD, 1,4-dichlorodibenzo-p-dioxin (1,4-DCDD), 1,6-dichlorodibenzo-p-dioxin (1,6-DCDD), 1,9-DCDD, 2,3-DCDD, 2,7-DCDD, 2,8-DCDD, 1,4,6,9-tetrachlorodibenzo-p-dioxin (1,4,6,9-TCDD), 2,3,7,8-tetrachlorinated dibenzo-p-dioxin (2,3,7,8-TCDD) and OCDD with the same method, which showed that the k_OH of these PCDD/Fs were found closely related to the number and position of chlorine (Cl).

Therefore, this study was aimed to obtain the k_OH database of PCDDs at 298 K. Based on our findings, we employed the quantitative structure-activity relationships (QSAR) model to establish the relation between the k_OH and the molecular structure of the PCDDs. The technology of QSAR have been developed to establish relationships between the properties and the molecular structure of the compounds [13,14,15] and predict the rate constants for the reactions of organic compounds with OH radical [16,17]. Additionally, the calculated k_OH were applied to the theoretical consideration of persistence of PCDDs.

2. Results and Discussion

2.1. Reactions with OH Radicals

There are two channels in the reactions of OH radical with PCDDs, i.e., OH radical addition and H atom abstraction. Generally, the H atom abstraction is not dominant for PCDDs and can be negligible [18]. So here we focus on the addition reactions between PCDDs and OH radical. The scheme for OH radical addition pathways are shown in Figure 1. Three different carbon sites, C_α (1, 4, 6, 9), C_β (2, 3, 7, 8), and C_γ (A, B, C, D), can be added by OH radical.

Figure 1. Schematic pathways for OH radical addition of PCDDs.

Figure 1. Schematic pathways for OH radical addition of PCDDs.

2.2. Relation between Rate Constants and the Configuration Parameters

The configuration parameters of PCDDs include the numbers and positions of Cl atoms, the energy of the highest occupied molecular orbital (E_HOMO), the energy of the lowest unoccupied molecular orbital (E_LUMO), and the dipole of oxidizing agents (D).

The number and the positions of Cl substitution are defined as follows: the number of Cl atoms at positions α (2, 3, 7, 8) and positions β (1, 4, 6, 9) are denoted as N_α and N_β, the number of Cl atoms at positions 1 and 9 (or 4 and 6) and positions 2 and 8 (or 3 and 7) are symbolized as N_1,9 and N_2,8, and the pair numbers of ortho-, meta- and para- positions are defined as N_o, N_m and N_p where both Cl atoms are located on one benzene ring. The total number of Cl atoms is defined as N. After the frequency at the MPWB1K/6-31+G(d,p) level are calculated, the E_HOMO, E_LUMO and D can be obtained.

The GFA (genetic function approximation) calculations are performed, the correlation of rate constants and the configuration parameters can be obtained, and the results are given in the following.

2.2.1. k_α, k_β and k_γ with the Configuration Parameters

k_α, k_β and k_γ for all 75 PCDDs with OH radical at 298.15 K are listed in Table 1, Table 2 and Table 3. After performing GFA calculations, we arrive at expressions connecting logk_α, logk_β and logk_γ to the configuration parameters which are referred to as model (1), model (2) and model (3), respectively. These equations show how the number and positions of Cl atoms, E_HOMO, E_LUMO and D influence rate constants. The observed and predicted values are compared in Figure 2, Figure 3 and Figure 4, which can suggest the relativity between the actual values and predicted ones.

Table 1. The rate constants (k_α, k_β, k_γ) of MCDDs, DCDDs and Tri-CDDs with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
MCDDs
1	1.35 × 10−13	9.15 × 10−12	7.40 × 10−12	1	0	1	0	0	0	0	1	−0.247	−0.003	0.245	1.63
2	3.18 × 10−13	1.69 × 10−11	1.12 × 10−11	0	1	0	1	0	0	0	1	−0.246	−0.003	0.243	2.07
DCDDs
1,2	8.66 × 10−14	3.98 × 10−12	8.21 × 10−12	1	1	1	1	1	1	0	2	−0.251	−0.010	0.242	2.86
1,3	4.44 × 10−13	5.66 × 10−12	7.97 × 10−12	1	1	1	0	0	1	0	2	−0.253	−0.011	0.242	2.29
1,4	8.23 × 10−14	1.13 × 10−11	5.50 × 10−12	2	0	1	0	0	0	1	2	−0.255	−0.011	0.244	0.88
1,6	5.25 × 10−13	1.54 × 10−11	5.62 × 10−12	2	0	1	0	0	0	0	2	−0.255	−0.010	0.245	0.00
1,7	2.34 × 10−13	8.03 × 10−12	6.81 × 10−12	1	1	1	0	0	0	0	2	−0.253	−0.011	0.243	1.52
1,8	2.72 × 10−13	7.81 × 10−12	8.25 × 10−12	1	1	1	1	0	0	0	2	−0.253	−0.011	0.242	2.79
1,9	2.85 × 10−13	8.39 × 10−12	5.57 × 10−12	2	0	2	0	0	0	0	2	−0.255	−0.010	0.245	2.99
2,3	3.07 × 10−13	4.58 × 10−12	7.92 × 10−12	0	2	0	1	1	0	0	2	−0.251	−0.009	0.242	3.16
2,7	2.45 × 10−13	3.90 × 10−12	1.10 × 10−11	0	2	0	1	0	0	0	2	−0.252	−0.012	0.241	0.00
2,8	1.99 × 10−13	4.63 × 10−12	1.05 × 10−11	0	2	0	2	0	0	0	2	−0.252	−0.012	0.240	1.79
Tri-CDDs
1,2,3	2.11 × 10−13	2.46 × 10−12	7.40 × 10−12	1	2	1	1	2	1	0	3	−0.256	−0.015	0.241	3.42
1,2,4	1.04 × 10−13	6.06 × 10−12	7.66 × 10−12	2	1	1	1	1	1	1	3	−0.258	−0.018	0.240	2.40
1,2,6	8.76 × 10−14	1.85 × 10−12	5.61 × 10−12	2	1	1	1	1	0	0	3	−0.259	−0.017	0.242	1.59
1,2,7	9.32 × 10−14	2.22 × 10−12	7.55 × 10−12	1	2	1	1	1	0	0	3	−0.257	−0.017	0.240	1.14
1,2,8	9.77 × 10−14	1.81 × 10−12	8.14 × 10−12	1	2	1	2	1	0	0	3	−0.257	−0.017	0.240	2.89
1,2,9	7.05 × 10−14	2.66 × 10−12	5.68 × 10−12	2	1	2	1	1	0	0	3	−0.258	−0.016	0.242	3.76
1,3,6	4.03 × 10−13	4.99 × 10−12	7.66 × 10−12	2	1	1	0	0	1	0	3	−0.260	−0.018	0.242	1.96
1,3,7	2.16 × 10−13	1.57 × 10−12	8.17 × 10−12	1	2	1	0	0	1	0	3	−0.259	−0.019	0.240	0.48
1,3,8	3.04 × 10−13	4.63 × 10−12	1.04 × 10−11	1	2	1	1	0	1	0	3	−0.259	−0.019	0.240	1.51
1,3,9	1.90 × 10−13	2.38 × 10−12	5.80 × 10−12	2	1	2	0	0	1	0	3	−0.260	−0.018	0.242	2.66
1,4,6	7.72 × 10−14	2.60 × 10−12	3.95 × 10−12	3	0	1	0	0	0	1	3	−0.262	−0.017	0.245	1.48
1,4,7	1.17 × 10−13	4.45 × 10−12	5.93 × 10−12	2	1	1	0	0	0	1	3	−0.260	−0.018	0.242	1.27
2,3,6	2.98 × 10−13	2.92 × 10−12	5.63 × 10−12	1	2	0	1	1	0	0	3	−0.258	−0.016	0.242	3.06
2,3,7	1.82 × 10−13	1.04 × 10−12	7.47 × 10−12	0	3	0	1	1	0	0	3	−0.257	−0.017	0.240	1.58

Table 1. The rate constants (k_α, k_β, k_γ) of MCDDs, DCDDs and Tri-CDDs with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
MCDDs
1	1.35 × 10−13	9.15 × 10−12	7.40 × 10−12	1	0	1	0	0	0	0	1	−0.247	−0.003	0.245	1.63
2	3.18 × 10−13	1.69 × 10−11	1.12 × 10−11	0	1	0	1	0	0	0	1	−0.246	−0.003	0.243	2.07
DCDDs
1,2	8.66 × 10−14	3.98 × 10−12	8.21 × 10−12	1	1	1	1	1	1	0	2	−0.251	−0.010	0.242	2.86
1,3	4.44 × 10−13	5.66 × 10−12	7.97 × 10−12	1	1	1	0	0	1	0	2	−0.253	−0.011	0.242	2.29
1,4	8.23 × 10−14	1.13 × 10−11	5.50 × 10−12	2	0	1	0	0	0	1	2	−0.255	−0.011	0.244	0.88
1,6	5.25 × 10−13	1.54 × 10−11	5.62 × 10−12	2	0	1	0	0	0	0	2	−0.255	−0.010	0.245	0.00
1,7	2.34 × 10−13	8.03 × 10−12	6.81 × 10−12	1	1	1	0	0	0	0	2	−0.253	−0.011	0.243	1.52
1,8	2.72 × 10−13	7.81 × 10−12	8.25 × 10−12	1	1	1	1	0	0	0	2	−0.253	−0.011	0.242	2.79
1,9	2.85 × 10−13	8.39 × 10−12	5.57 × 10−12	2	0	2	0	0	0	0	2	−0.255	−0.010	0.245	2.99
2,3	3.07 × 10−13	4.58 × 10−12	7.92 × 10−12	0	2	0	1	1	0	0	2	−0.251	−0.009	0.242	3.16
2,7	2.45 × 10−13	3.90 × 10−12	1.10 × 10−11	0	2	0	1	0	0	0	2	−0.252	−0.012	0.241	0.00
2,8	1.99 × 10−13	4.63 × 10−12	1.05 × 10−11	0	2	0	2	0	0	0	2	−0.252	−0.012	0.240	1.79
Tri-CDDs
1,2,3	2.11 × 10−13	2.46 × 10−12	7.40 × 10−12	1	2	1	1	2	1	0	3	−0.256	−0.015	0.241	3.42
1,2,4	1.04 × 10−13	6.06 × 10−12	7.66 × 10−12	2	1	1	1	1	1	1	3	−0.258	−0.018	0.240	2.40
1,2,6	8.76 × 10−14	1.85 × 10−12	5.61 × 10−12	2	1	1	1	1	0	0	3	−0.259	−0.017	0.242	1.59
1,2,7	9.32 × 10−14	2.22 × 10−12	7.55 × 10−12	1	2	1	1	1	0	0	3	−0.257	−0.017	0.240	1.14
1,2,8	9.77 × 10−14	1.81 × 10−12	8.14 × 10−12	1	2	1	2	1	0	0	3	−0.257	−0.017	0.240	2.89
1,2,9	7.05 × 10−14	2.66 × 10−12	5.68 × 10−12	2	1	2	1	1	0	0	3	−0.258	−0.016	0.242	3.76
1,3,6	4.03 × 10−13	4.99 × 10−12	7.66 × 10−12	2	1	1	0	0	1	0	3	−0.260	−0.018	0.242	1.96
1,3,7	2.16 × 10−13	1.57 × 10−12	8.17 × 10−12	1	2	1	0	0	1	0	3	−0.259	−0.019	0.240	0.48
1,3,8	3.04 × 10−13	4.63 × 10−12	1.04 × 10−11	1	2	1	1	0	1	0	3	−0.259	−0.019	0.240	1.51
1,3,9	1.90 × 10−13	2.38 × 10−12	5.80 × 10−12	2	1	2	0	0	1	0	3	−0.260	−0.018	0.242	2.66
1,4,6	7.72 × 10−14	2.60 × 10−12	3.95 × 10−12	3	0	1	0	0	0	1	3	−0.262	−0.017	0.245	1.48
1,4,7	1.17 × 10−13	4.45 × 10−12	5.93 × 10−12	2	1	1	0	0	0	1	3	−0.260	−0.018	0.242	1.27
2,3,6	2.98 × 10−13	2.92 × 10−12	5.63 × 10−12	1	2	0	1	1	0	0	3	−0.258	−0.016	0.242	3.06
2,3,7	1.82 × 10−13	1.04 × 10−12	7.47 × 10−12	0	3	0	1	1	0	0	3	−0.257	−0.017	0.240	1.58

Table 2. The rate constants (k_α, k_β, k_γ) of TCDDs with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
TCDDs
1,2,3,4	6.62 × 10−14	2.07 × 10−12	7.06 × 10−12	2	2	1	1	3	2	1	4	−0.260	−0.022	0.239	3.30
1,2,3,6	1.91 × 10−13	1.35 × 10−12	5.38 × 10−12	2	2	1	1	2	1	0	4	−0.263	−0.022	0.241	2.77
1,2,3,7	2.00 × 10−13	1.34 × 10−12	7.56 × 10−12	1	3	1	1	2	1	0	4	−0.261	−0.023	0.239	1.42
1,2,3,8	2.39 × 10−13	7.45 × 10−13	7.97 × 10−12	1	3	1	2	2	1	0	4	−0.261	−0.023	0.239	2.40
1,2,3,9	1.34 × 10−13	1.70 × 10−12	5.23 × 10−12	2	2	2	1	2	1	0	4	−0.262	−0.022	0.241	3.71
1,2,4,6	4.89 × 10−14	1.79 × 10−12	4.88 × 10−12	3	1	1	1	1	1	1	4	−0.265	−0.024	0.241	2.05
1,2,4,7	6.42 × 10−14	1.09 × 10−12	7.42 × 10−12	2	2	1	1	1	1	1	4	−0.263	−0.025	0.239	0.61
1,2,4,8	7.46 × 10−14	2.02 × 10−12	7.10 × 10−12	2	2	1	2	1	1	1	4	−0.263	−0.025	0.238	1.48
1,2,4,9	3.81 × 10−14	2.16 × 10−12	4.97 × 10−12	3	1	2	1	1	1	1	4	−0.265	−0.024	0.241	2.68
1,2,6,7	6.42 × 10−14	1.18 × 10−12	6.06 × 10−12	2	2	1	1	2	0	0	4	−0.262	−0.023	0.239	0.00
1,2,6,8	1.47 × 10−13	2.36 × 10−12	8.51 × 10−12	2	2	1	2	1	1	0	4	−0.263	−0.024	0.239	1.41
1,2,6,9	5.65 × 10−14	1.92 × 10−12	4.16 × 10−12	3	1	2	1	1	0	1	4	−0.265	−0.023	0.241	2.17
1,2,7,8	1.45 × 10−13	8.84 × 10−13	6.35 × 10−12	1	3	1	2	2	0	0	4	−0.261	−0.023	0.239	2.28
1,2,7,9	1.93 × 10−13	1.51 × 10−12	8.04 × 10−12	2	2	2	1	1	1	0	4	−0.263	−0.024	0.240	2.49
1,2,8,9	6.79 × 10−14	1.53 × 10−12	7.22 × 10−12	2	2	2	2	2	0	0	4	−0.261	−0.023	0.239	3.88
1,3,6,8	4.41 × 10−13	2.69 × 10−12	9.96 × 10−12	2	2	1	1	0	2	0	4	−0.265	−0.025	0.240	0.00
1,3,6,9	2.71 × 10−13	2.35 × 10−12	5.68 × 10−12	3	1	2	0	0	1	1	4	−0.267	−0.025	0.242	1.39
1,3,7,8	2.25 × 10−13	1.38 × 10−12	6.97 × 10−12	1	3	1	1	1	1	0	4	−0.263	−0.024	0.239	1.08
1,3,7,9	2.57 × 10−13	2.41 × 10−12	7.57 × 10−12	2	2	2	0	0	2	0	4	−0.265	−0.025	0.240	1.08
1,4,6,9	3.93 × 10−16	3.70 × 10−12	3.20 × 10−12	4	0	2	0	0	0	2	4	−0.269	−0.024	0.244	0.00
1,4,7,8	1.24 × 10−13	1.35 × 10−12	4.06 × 10−12	2	2	1	1	1	0	1	4	−0.264	−0.024	0.241	2.21
2,3,7,8	1.72 × 10−13	1.35 × 10−15	5.20 × 10−12	0	4	0	2	2	0	0	4	−0.261	−0.022	0.239	0.00

Table 2. The rate constants (k_α, k_β, k_γ) of TCDDs with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
TCDDs
1,2,3,4	6.62 × 10−14	2.07 × 10−12	7.06 × 10−12	2	2	1	1	3	2	1	4	−0.260	−0.022	0.239	3.30
1,2,3,6	1.91 × 10−13	1.35 × 10−12	5.38 × 10−12	2	2	1	1	2	1	0	4	−0.263	−0.022	0.241	2.77
1,2,3,7	2.00 × 10−13	1.34 × 10−12	7.56 × 10−12	1	3	1	1	2	1	0	4	−0.261	−0.023	0.239	1.42
1,2,3,8	2.39 × 10−13	7.45 × 10−13	7.97 × 10−12	1	3	1	2	2	1	0	4	−0.261	−0.023	0.239	2.40
1,2,3,9	1.34 × 10−13	1.70 × 10−12	5.23 × 10−12	2	2	2	1	2	1	0	4	−0.262	−0.022	0.241	3.71
1,2,4,6	4.89 × 10−14	1.79 × 10−12	4.88 × 10−12	3	1	1	1	1	1	1	4	−0.265	−0.024	0.241	2.05
1,2,4,7	6.42 × 10−14	1.09 × 10−12	7.42 × 10−12	2	2	1	1	1	1	1	4	−0.263	−0.025	0.239	0.61
1,2,4,8	7.46 × 10−14	2.02 × 10−12	7.10 × 10−12	2	2	1	2	1	1	1	4	−0.263	−0.025	0.238	1.48
1,2,4,9	3.81 × 10−14	2.16 × 10−12	4.97 × 10−12	3	1	2	1	1	1	1	4	−0.265	−0.024	0.241	2.68
1,2,6,7	6.42 × 10−14	1.18 × 10−12	6.06 × 10−12	2	2	1	1	2	0	0	4	−0.262	−0.023	0.239	0.00
1,2,6,8	1.47 × 10−13	2.36 × 10−12	8.51 × 10−12	2	2	1	2	1	1	0	4	−0.263	−0.024	0.239	1.41
1,2,6,9	5.65 × 10−14	1.92 × 10−12	4.16 × 10−12	3	1	2	1	1	0	1	4	−0.265	−0.023	0.241	2.17
1,2,7,8	1.45 × 10−13	8.84 × 10−13	6.35 × 10−12	1	3	1	2	2	0	0	4	−0.261	−0.023	0.239	2.28
1,2,7,9	1.93 × 10−13	1.51 × 10−12	8.04 × 10−12	2	2	2	1	1	1	0	4	−0.263	−0.024	0.240	2.49
1,2,8,9	6.79 × 10−14	1.53 × 10−12	7.22 × 10−12	2	2	2	2	2	0	0	4	−0.261	−0.023	0.239	3.88
1,3,6,8	4.41 × 10−13	2.69 × 10−12	9.96 × 10−12	2	2	1	1	0	2	0	4	−0.265	−0.025	0.240	0.00
1,3,6,9	2.71 × 10−13	2.35 × 10−12	5.68 × 10−12	3	1	2	0	0	1	1	4	−0.267	−0.025	0.242	1.39
1,3,7,8	2.25 × 10−13	1.38 × 10−12	6.97 × 10−12	1	3	1	1	1	1	0	4	−0.263	−0.024	0.239	1.08
1,3,7,9	2.57 × 10−13	2.41 × 10−12	7.57 × 10−12	2	2	2	0	0	2	0	4	−0.265	−0.025	0.240	1.08
1,4,6,9	3.93 × 10−16	3.70 × 10−12	3.20 × 10−12	4	0	2	0	0	0	2	4	−0.269	−0.024	0.244	0.00
1,4,7,8	1.24 × 10−13	1.35 × 10−12	4.06 × 10−12	2	2	1	1	1	0	1	4	−0.264	−0.024	0.241	2.21
2,3,7,8	1.72 × 10−13	1.35 × 10−15	5.20 × 10−12	0	4	0	2	2	0	0	4	−0.261	−0.022	0.239	0.00

Table 3. The rate constants (k_α, k_β, k_γ) of Penta-CDDs, Hexa-CDDs, Hepta-CDDs and OCDD with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
Penta-CDDs
1,2,3,4,6	4.48 × 10−14	9.83 × 10−13	5.07 × 10−12	3	2	1	1	3	2	1	5	−0.267	−0.028	0.240	3.14
1,2,3,4,7	4.09 × 10−14	6.84 × 10−13	7.14 × 10−12	2	3	1	1	3	2	1	5	−0.265	−0.028	0.237	1.70
1,2,3,6,7	1.34 × 10−13	2.34 × 10−13	6.10 × 10−12	2	3	1	1	3	1	0	5	−0.266	−0.028	0.238	1.51
1,2,3,6,8	2.05 × 10−13	1.09 × 10−12	7.71 × 10−12	2	3	1	2	2	2	0	5	−0.267	−0.029	0.238	1.12
1,2,3,6,9	1.10 × 10−13	6.93 × 10−13	4.24 × 10−12	3	2	2	1	2	1	1	5	−0.269	−0.028	0.240	2.49
1,2,3,7,8	1.49 × 10−13	1.24 × 10−15	5.21 × 10−12	1	4	1	2	3	1	0	5	−0.265	−0.028	0.237	1.06
1,2,3,7,9	1.66 × 10−13	7.56 × 10−13	6.46 × 10−12	2	3	2	1	2	2	0	5	−0.267	−0.029	0.238	1.84
1,2,3,8,9	1.01 × 10−13	5.26 × 10−13	6.51 × 10−12	2	3	2	2	3	1	0	5	−0.266	−0.028	0.238	3.18
1,2,4,6,7	3.66 × 10−14	1.98 × 10−12	5.72 × 10−12	3	2	1	1	2	1	1	5	−0.268	−0.030	0.239	1.39
1,2,4,6,8	2.38 × 10−13	2.50 × 10−12	1.17 × 10−11	3	2	1	2	1	2	1	5	−0.267	−0.029	0.238	1.12
1,2,4,6,9	7.33 × 10−16	1.56 × 10−12	4.79 × 10−12	4	1	2	1	1	1	2	5	−0.271	−0.030	0.241	1.51
1,2,4,7,8	7.27 × 10−14	5.78 × 10−13	4.89 × 10−12	2	3	1	2	2	1	1	5	−0.267	−0.030	0.238	0.93
1,2,4,7,9	8.16 × 10−14	2.05 × 10−12	7.50 × 10−12	3	2	2	1	1	2	1	5	−0.270	−0.031	0.239	1.02
1,2,4,8,9	3.71 × 10−14	1.12 × 10−12	6.03 × 10−12	3	2	2	2	2	1	1	5	−0.268	−0.030	0.238	2.41
Hexa-CDDs
1,2,3,4,6,7	1.60 × 10−14	2.54 × 10−13	6.16 × 10−12	3	3	1	1	4	2	1	6	−0.270	−0.033	0.237	2.26
1,2,3,4,6,8	1.10 × 10−13	8.46 × 10−13	7.87 × 10−12	3	3	1	2	3	3	1	6	−0.271	−0.034	0.237	1.19
1,2,3,4,6,9	7.74 × 10−16	7.82 × 10−13	4.45 × 10−12	4	2	2	1	3	2	2	6	−0.273	−0.034	0.239	2.36
1,2,3,4,7,8	6.63 × 10−14	9.51 × 10−16	5.04 × 10−12	2	4	1	2	4	2	1	6	−0.269	−0.033	0.236	0.19
1,2,3,6,7,8	2.56 × 10−13	1.15 × 10−15	6.27 × 10−12	2	4	1	2	4	2	0	6	−0.270	−0.033	0.237	0.00
1,2,3,6,7,9	1.07 × 10−13	6.23 × 10−13	6.05 × 10−12	3	3	2	1	3	2	1	6	−0.272	−0.034	0.237	0.97
1,2,3,6,8,9	1.20 × 10−13	5.56 × 10−13	6.32 × 10−12	3	3	2	2	3	2	1	6	−0.272	−0.034	0.237	1.71
1,2,3,7,8,9	1.46 × 10−13	1.13 × 10−15	6.19 × 10−12	2	4	2	2	4	2	0	6	−0.269	−0.033	0.237	2.01
1,2,4,6,7,9	6.30 × 10−16	6.19 × 10−13	6.07 × 10−12	4	2	2	1	2	2	2	6	−0.274	−0.036	0.238	0.00
1,2,4,6,8,9	6.55 × 10−16	1.22 × 10−12	5.86 × 10−12	4	2	2	2	2	2	2	6	−0.274	−0.036	0.238	1.02
Hepta-CDDs
1,2,3,4,6,7,8	7.21 × 10−14	9.28 × 10−16	5.71 × 10−12	3	4	1	2	5	3	1	7	−0.273	−0.038	0.236	0.95
1,2,3,4,6,7,9	6.81 × 10−16	4.15 × 10−13	5.41 × 10−12	4	3	2	1	4	3	2	7	−0.276	−0.039	0.237	1.06
OCDD
1,2,3,4,6,7,8,9	7.95 × 10−16	7.51 × 10−16	5.98 × 10−12	4	4	2	2	6	4	2	8	−0.277	−0.043	0.235	0.00

Table 3. The rate constants (k_α, k_β, k_γ) of Penta-CDDs, Hexa-CDDs, Hepta-CDDs and OCDD with parameters.

Compound	kα	kβ	kγ	Nα	Nβ	N1,9	N2,8	No	Nm	Np	N	EHOMO	ELUMO	ΔEHOMO-LUMO	D
Penta-CDDs
1,2,3,4,6	4.48 × 10−14	9.83 × 10−13	5.07 × 10−12	3	2	1	1	3	2	1	5	−0.267	−0.028	0.240	3.14
1,2,3,4,7	4.09 × 10−14	6.84 × 10−13	7.14 × 10−12	2	3	1	1	3	2	1	5	−0.265	−0.028	0.237	1.70
1,2,3,6,7	1.34 × 10−13	2.34 × 10−13	6.10 × 10−12	2	3	1	1	3	1	0	5	−0.266	−0.028	0.238	1.51
1,2,3,6,8	2.05 × 10−13	1.09 × 10−12	7.71 × 10−12	2	3	1	2	2	2	0	5	−0.267	−0.029	0.238	1.12
1,2,3,6,9	1.10 × 10−13	6.93 × 10−13	4.24 × 10−12	3	2	2	1	2	1	1	5	−0.269	−0.028	0.240	2.49
1,2,3,7,8	1.49 × 10−13	1.24 × 10−15	5.21 × 10−12	1	4	1	2	3	1	0	5	−0.265	−0.028	0.237	1.06
1,2,3,7,9	1.66 × 10−13	7.56 × 10−13	6.46 × 10−12	2	3	2	1	2	2	0	5	−0.267	−0.029	0.238	1.84
1,2,3,8,9	1.01 × 10−13	5.26 × 10−13	6.51 × 10−12	2	3	2	2	3	1	0	5	−0.266	−0.028	0.238	3.18
1,2,4,6,7	3.66 × 10−14	1.98 × 10−12	5.72 × 10−12	3	2	1	1	2	1	1	5	−0.268	−0.030	0.239	1.39
1,2,4,6,8	2.38 × 10−13	2.50 × 10−12	1.17 × 10−11	3	2	1	2	1	2	1	5	−0.267	−0.029	0.238	1.12
1,2,4,6,9	7.33 × 10−16	1.56 × 10−12	4.79 × 10−12	4	1	2	1	1	1	2	5	−0.271	−0.030	0.241	1.51
1,2,4,7,8	7.27 × 10−14	5.78 × 10−13	4.89 × 10−12	2	3	1	2	2	1	1	5	−0.267	−0.030	0.238	0.93
1,2,4,7,9	8.16 × 10−14	2.05 × 10−12	7.50 × 10−12	3	2	2	1	1	2	1	5	−0.270	−0.031	0.239	1.02
1,2,4,8,9	3.71 × 10−14	1.12 × 10−12	6.03 × 10−12	3	2	2	2	2	1	1	5	−0.268	−0.030	0.238	2.41
Hexa-CDDs
1,2,3,4,6,7	1.60 × 10−14	2.54 × 10−13	6.16 × 10−12	3	3	1	1	4	2	1	6	−0.270	−0.033	0.237	2.26
1,2,3,4,6,8	1.10 × 10−13	8.46 × 10−13	7.87 × 10−12	3	3	1	2	3	3	1	6	−0.271	−0.034	0.237	1.19
1,2,3,4,6,9	7.74 × 10−16	7.82 × 10−13	4.45 × 10−12	4	2	2	1	3	2	2	6	−0.273	−0.034	0.239	2.36
1,2,3,4,7,8	6.63 × 10−14	9.51 × 10−16	5.04 × 10−12	2	4	1	2	4	2	1	6	−0.269	−0.033	0.236	0.19
1,2,3,6,7,8	2.56 × 10−13	1.15 × 10−15	6.27 × 10−12	2	4	1	2	4	2	0	6	−0.270	−0.033	0.237	0.00
1,2,3,6,7,9	1.07 × 10−13	6.23 × 10−13	6.05 × 10−12	3	3	2	1	3	2	1	6	−0.272	−0.034	0.237	0.97
1,2,3,6,8,9	1.20 × 10−13	5.56 × 10−13	6.32 × 10−12	3	3	2	2	3	2	1	6	−0.272	−0.034	0.237	1.71
1,2,3,7,8,9	1.46 × 10−13	1.13 × 10−15	6.19 × 10−12	2	4	2	2	4	2	0	6	−0.269	−0.033	0.237	2.01
1,2,4,6,7,9	6.30 × 10−16	6.19 × 10−13	6.07 × 10−12	4	2	2	1	2	2	2	6	−0.274	−0.036	0.238	0.00
1,2,4,6,8,9	6.55 × 10−16	1.22 × 10−12	5.86 × 10−12	4	2	2	2	2	2	2	6	−0.274	−0.036	0.238	1.02
Hepta-CDDs
1,2,3,4,6,7,8	7.21 × 10−14	9.28 × 10−16	5.71 × 10−12	3	4	1	2	5	3	1	7	−0.273	−0.038	0.236	0.95
1,2,3,4,6,7,9	6.81 × 10−16	4.15 × 10−13	5.41 × 10−12	4	3	2	1	4	3	2	7	−0.276	−0.039	0.237	1.06
OCDD
1,2,3,4,6,7,8,9	7.95 × 10−16	7.51 × 10−16	5.98 × 10−12	4	4	2	2	6	4	2	8	−0.277	−0.043	0.235	0.00

Figure 2. The observed values and predicted values of −logk_α for the training and validation set.

Figure 2. The observed values and predicted values of −logk_α for the training and validation set.

Figure 3. The observed values and predicted values of −logk_β for the training and validation set.

Figure 3. The observed values and predicted values of −logk_β for the training and validation set.

Figure 4. The observed values and predicted values of −logk_γ for the training and validation set.

Figure 4. The observed values and predicted values of −logk_γ for the training and validation set.

logk_α = −12.50 − 0.064 × N_α − 0.880 × N_p, R² = 0.761, R²(CV) = 0.715, Q²ext = 0.716.

(1)

logk_β = 42.950 − 0.660 × N_β − 0.467 × N_o − 222.406 × ΔE_HOMO-LUMO + 0.258 × D, R² = 0.835, R²(CV) = 0.821, Q²ext = 0.733.

(2)

logk_γ = 5.179 − 0.062 × N_o + 0.080 × N_m + 18.188 × E_HOMO − 48.247 × ΔE_HOMO-LUMO, R² = 0.878, R²(CV) = 0.866, Q²ext = 0.743.

(3)

From model (1), it can be shown that the correlation coefficient R² and the cross validation R²(CV) are not ideal, but still show some useful information. The coefficient of N_α is 0.064, while the coefficient of N_p is 0.880. The effect of relative positions of these Cl atoms is more important than the number of these Cl atoms. As described in the algorithm and the methodology section, we have separately performed steps 1 to 4 and calculated S_new(k) value for training and test set compounds. All the PCDDs in the training set are not X-outlier, and all the PCDDs in the test set are within applicability domain.

As shown by the R² and R²(CV) in model (2) and model (3), the k_β and k_γ are well correlated to the variables. The relativity between the actual values and predicted values is excellent as shown in the Figure 3 and Figure 4. The k_β is positively correlated with N_β, N_o and ΔE_HOMO-LUMO. The effect of relative positions and the number of these Cl atoms is in the order of N_β > N_o. The k_γ is positively correlated with E_HOMO and negatively correlated with ΔE_HOMO-LUMO. But the k_γ is not related to the number of these Cl atoms (N_α, N_β and N) and it has little to do with the number of relative position for these Cl atoms (N_o, N_m, N_p, N_1,9 and N_2,8) because its coefficient is less than those of other variables. The applicability domain of the model (2) and model (3) were checked; the training set compound OCDD was considered as an outlier, and the S_new value of this compound was 3.02.

2.2.2. The Total Rate Constant k_OH with the Configuration Parameters

The total rate constant of the addition reaction between PCDDs and the OH radical is k_OH = k_α + k_β + k_γ. Using GFA calculations, the correlation of k_OH with the configuration parameters is given in model (4), where the R² is 0.865 and the R²(CV) is 0.852. As revealed in Figure 5, the actual values and predicted values seem pretty relevant.

Figure 5. The observed values and predicted values of –logk_OH for the training and validation set.

Figure 5. The observed values and predicted values of –logk_OH for the training and validation set.

logk_OH = −10.594 + 0.118 × N_o − 0.144 × N, R² = 0.865, R²(CV) = 0.852, Q²ext = 0.806.

(4)

Several important conclusions can be drawn from the model (4). First, the main influence factor of k_OH is the N, and k_OH is positively correlated with the N_o. Second, the number of Cl atoms is more effective than the number of relative position for these Cl atoms because the coefficients of N is greater than that of N_o. Finally, the Cl atom located at the ortho-position exerts more effect on the k_OH. The applicability domain of the model (4) have been checked, the training set compound OCDD is considered as an outlier, and the S_new value of this compound is 3.02.

3. Computational Methods

3.1. Geometry Optimization

Using the Gaussian 03 programs [19], high-level ab initio molecular orbital calculations were carried out for the 75 PCDDs and OH radical. The geometrical parameters of the 75 PCDDs, transition states (TS), and OH-adducts are optimized at the MPWB1K/6-31+G(d,p) level. The MPWB1K method is a hybrid density functional theory model with excellent performance for thermochemistry and thermochemical kinetics [20,21]. This computational approach has been successfully used in TCDD and TCDF [22,23]. All open-shell species were treated with an unrestricted approach, while closed-shell species were described with a restricted method. Structure optimizations were performed in Cartesian coordinates with an energy convergence criterion of 10⁻⁶ Hartree. Vibrational frequencies, calculated at the same level, were scaled by a standard scaling factor of 0.934 to remove systematic errors [24]. Each transition state was verified to connect the designated reactants and products by performing an intrinsic reaction coordinate (IRC) analysis [25].

3.2. Kinetic Calculation

Based on the quantum chemical information, rate constants are computed using the Rice–Ramsperger–Kassel–Marcus (RRKM) theory [26]. The microcannonical rate constants were calculated using RRKM:

$$k_i\left(E\right)={\text{α}}_i {\mathrm{\kappa }}_i\sqrt{\frac{I_i^{\ne }}{I_j^{IM}} }\frac{N\left(E-E_i^{\ne }\right)}{h{\text{ρ}}_j\left(E\right)}, i = 1,2,3\dots ; j = 1,2,3\dots$$

(5)

where α_i is the statistical factor for the path degeneracy, κ_i is the tunneling factor, $I_i^{\ne }$ and $I_J^{IM}$ are the moments of inertia (IaIbIc) of transition sate i and intermediate j, and N(E − $E_i^{\ne }$) is the number of states with energy above the barrier height $E_i^{\ne }$ for i. The Beyer–Swinehart algorithm was employed to calculate the density and number of states.

3.3. Quantitative Structure–Activity Relationship

The genetic function approximation (GFA) [27] in the Materials studio package is used to analyze the relationship between the OH radical addition rate constants and configuration parameters [28]. In order to improve the correlation of mathematic model, the exceptional values in test data are eliminated after outlier analysis.

In absence of any new data, for all the datasets, we picked one compound out of every five to constitute validation sets and the rest constituted the training set. The list of chemicals used for developing models along with the division of training and validation sets is given in Tables S1 and S2.

The external predictive ability of the models may be shown by using different metric values [29]. In this study, it was tested through test set and evaluated by cross-validation coefficient Q²ext between predicted and observed values. Q²ext = 1 − PRESS/SD, where PRESS is the sum of squared differences between the experimental values and the predicted value for each molecule in the test set, and SD is the sum of squared deviations between the experimental values for each molecule in the test set and the mean experimental value of the training set [30].

The reliability of any QSAR model depends on the confident predictions of these new compounds based on the applicability domain of the model. The algorithm and methodology for the proposed approach are discussed below [31]:

(1) First, all the descriptors appearing in the developed model (for training and test set compounds) are standardized using the following formula:

$$S_{ki}=\frac{\left|X_{ki}-\overline{X_i}\right|}{{\text{σ}}_{X_i}}$$

(6)

where k is the number of compounds, i is the number of descriptors, S_ki is standardized descriptor i for compound k (from the training or test set), X_ki is the original descriptor i for compound k (from the training or test set), $\overline{X_i}$ is the mean value of the descriptor X_i for the training set compounds only, and ${\text{σ}}_{X_i}$ is standard deviation of the descriptor X_i for the training set compounds only.

(2) Thereafter, one needs to compute the maximum S_i(k) value ([S_i]_max(k)) for the compound k. If [S_i]_max(k) is lower than or equal to 3, then that compound is not an X-outlier (if in the training set) or is within the applicability domain (if in the test set).

(3) If [S_i]_max(k) is above 3, then one should compute [S_i]_min(k). If [S_i]_min(k) > 3, then the compound is an X-outlier (if in the training set) or is not within applicability domain (if in the test set).

(4) If [S_i]_max(k) > 3and [S_i]_min(k) < 3, then one should compute S_new(k) from the following equation:

$$S_{new(k)}=\overline{S_k} + 1.28 \times {\mathrm{\sigma }}_{S_k}$$

(7)

where S_new(k) is the S_new value for the compound k, $\overline{S_k}$ is the mean of S_i(k) value for the compound k, and ${\text{σ}}_{S_k}$ is the standard deviation of S_i(k) values of the compound k.

If the calculated S_new(k) value is lower than or equal to 3, then that compound is not an X-outlier (if in the training set) or is within applicability domain (if in the test set).

4. Conclusions

The atmospheric chemical mechanism of all 75 PCDDs initiated by the OH radical was investigated. Rate constants of the elementary reactions at 298.15 K were calculated. In addition, the rate constants (k_α, k_β, k_γ and k_OH) were further quantified by the study on QSAR. According to the QSAR models, some valuable conclusions can be drawn:

(1) The k_α is negatively correlated with the N_α and the k_β is negatively correlated with the N_β, while the k_γ is not related to the number of these Cl atoms (N_α, N_β and N).

(2) The number of Cl atoms is the main influence factor and is positively correlated with the k_OH. For the k_OH, the number of Cl atoms is more effective than the relative position for these Cl atoms. The k_OH decreases with the increase of the substitute number of Cl atoms.