## 3 Results and Discussion
### Strategies for Upgrading the Dielectric Strength Prediction Protocol
#### 3.2.3 Strategy III: Importance of Original Factors in Dielectric Strength Predictions  
It is interesting to consider whether an equation-based protocol can be developed by combining factors that are highly correlated with dielectric strength (i.e., Equation (4)). Accordingly, the impact of this distinct protocol design on the prediction accuracy was explored (Figure 6; Table 1). To achieve this goal, chemical (i.e., formation energy, polarizability, ionization energy, the number of electrons, HOMO, HOMO-LUMO gap, and electronic spatial extent gap) and physical (i.e., molecular weight and molar volume) descriptors were further investigated for the 168 PFC and non-PFC compounds (Figure S3). The chemical descriptors were computed by the DFT method. For instance, the formation energy, which is defined as the energy required for the formation of a compound from its elemental constituents, was predicted by the DFT-calculated energies of the compound and its elemental constituents. The DFT-calculated frontier orbital analyses were employed for HOMO and LUMO values, while the DFT computation of the dipole-based electric field was used for polarizability. The DFT-calculated energy change associated with oxidation was utilized for ionization energy. The electronic spatial extent gap is a measure of how widely the electronic density is distributed within a molecule. It reveals the av  
Figure 4: (**a**) Comparison of re-PIM-predicted and experimental dielectric strengths for 168 PFC and non-PFC compounds. (**b**) Distribution of relative errors for the re-PIM-predicted dielectric strengths. (**c**) re-PIM-predicted dielectric strengths as functions of the core factors (polarizability, ionization energy, and molecular weight). erage distance between the electrons and the nucleus in a molecule, providing insight into its electronic structure and properties. The electronic spatial extent gap of a molecule was computed by \(\int~{}\varphi(r)^{2}r^{2}dv\), where \(\varphi(r)\) represents the electron wave function of the molecule, \(r\) is the distance between an electron and its molecular center, and \(dv\) is a 3D element of space, in other words, the integration of \(\varphi(r)\) and \(r^{2}\) over three-dimensional space. Direct correlations of these descriptors with experimental dielectric strength were further analyzed for the 168 PFC and non-PFC compounds (Figure S3). As seen in the figures, the correlation degree depends on the introduced descriptor. Consequently, five distinct factors (polarizability, the number of electrons, MW, formation energy, and molar volume) were chosen as the most highly ranked factors in terms of correlation with dielectric strength (Figure S3). Notably, the IE, which is a main contributor in the original PIM protocol, as well as crucial core factors, such as the HOMO-LUMO gap and HOMO, were excluded from this equation-based protocol in PIMN. Unexpectedly, the parameterization of the developed equation-based protocol resulted in a deterioration in prediction accuracy (relative error = 17.70%; RMSD = 0.340) compared with PIMN (_via HOMO-LUMO gap_ and _HOMO_) (Figure 7a,b). This result implies that the synergistic effect of appropriately selected core factors, such as polarizability, IE, MW, and HOMO-LUMO gap, is more impactful for predicting dielectric strength than the correlating ability of an individual factor. Figure 5: _Cont._  
Figure 5: (**a**) Comparison of PIMN-predicted (based on five distinct novel factors) and experimental dielectric strengths for 168 PFC and non-PFC compounds. (**b**) Distribution of relative errors for the PIMN-predicted dielectric strengths. In (**b**), the distribution for each core factor has blocks of 0–10, 10–20, 20–30, 30–40, 40–50, 50–60, and 60–70. (**c–f**) PIMN-predicted dielectric strength as functions of the core factors (polarizability, ionization energy, molecular weight, and novel factors). Figure 6: (**a**) Comparison of OPT-predicted and experimental dielectric strengths for 168 PFC and non-PFC compounds.