Fe phthalocyanine (FePc) has a typical two-dimensional, plane-symmetric structure that results in a symmetric electron distribution in the described \(F_{e}N_{4}\) compound. The molecule contains two types of nitrogen atoms denoted \(N_{1}\) and \(N_{2}\), whereby the latter one has no direct bond to the Fe center2.The order and size of the structure of Fe phthalocyanine (FePc) are \(m’=55n+2\) and \(n’=68n\), respectively. It has four types of vertices, of degrees 1, 2, 3, 4 respectively. Table 1 shows the edge partition. The unit structure of FePc(n) is shown in Fig. 2. For more details about this structure of FePc(n) see Figs. 3, 4, and 5. The order and size of Phthalocyanine are \(55n+2\) and 68n, respectively.
Table 1 Edge partition of Fe phthalocyanine \(FePc[{\mathfrak {c}},{\mathfrak {d}}]\).Figure 2The structure of Phthalocyanine FePc(n) for \(n=1\).Figure 3The structure of Phthalocyanine FePc(n) for \(n=2\).Figure 4The structure of Phthalocyanine FePc(n) for \(n=3\).Figure 5The structure of Phthalocyanine FePc(n) for \(n=4\).Main results for Fe phthalocyanineIn this section, the degree-based topological indices have been computed. Using the above-defined formulas of the topological indices and Table 1, we compute the following indices:$$\begin{aligned} ABC(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\sqrt{\frac{ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})-2}{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}}\\= & {} \left( \sqrt{\frac{2}{3}}\right) (12n+4)+\left( \sqrt{\frac{3}{6}}\right) (12n-4) +\left( \sqrt{\frac{4}{9}}\right) (40n)+\left( \sqrt{\frac{5}{12}}\right) (4n)\\= & {} 47.531896n+0.437559. \end{aligned}$$$$\begin{aligned} GA(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\frac{2\sqrt{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}}{ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})}\\= & {} \left( \frac{2\sqrt{3}}{4}\right) (12n+4)+\left( \frac{2\sqrt{6}}{5}\right) (12n-4) +\left( \frac{2\sqrt{9}}{6}\right) (40n)+\left( \frac{2\sqrt{12}}{7}\right) (4n)\\= & {} 66.108829n-0.455082. \end{aligned}$$$$\begin{aligned} F(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}( \omega ({\mathfrak {c}})^2+ \omega ({\mathfrak {d}})^2)\\= & {} (10)(12n+4)+(13)(12n-4)+(18)(40n)+(25)(4n)\\= & {} 1096n – 12. \end{aligned}$$$$\begin{aligned} AZI(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\bigg (\frac{ \omega {({\mathfrak {c}})}\times \omega ({\mathfrak {d}})}{ \omega {({\mathfrak {c}})}+ \omega ({\mathfrak {d}})-2}\bigg )^3\\= & {} \left( \frac{4}{3}\right) ^3(12n+4)+\left( \frac{8}{4}\right) ^3(12n-4) +\left( \frac{16}{6}\right) ^3(40n)+\left( \frac{32}{10}\right) ^3(4n)\\= & {} 647.421n-18.5. \end{aligned}$$$$\begin{aligned} M_1(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}{ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})}\\= & {} (4)(12n+4)+(5)(12n-4) +(6)(40n)+(7)(4n)\\= & {} 496n-4. \end{aligned}$$$$\begin{aligned} M_2(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}\\= & {} (3)(12n+4)+(6)(12n-4) +(9)(40n)+(12)(4n)\\= & {} 516n-12. \end{aligned}$$$$\begin{aligned} HM(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}({ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})})^{2}\\= & {} (16)(12n+4)+(25)(12n-4) +(36)(40n)+(49)(4n)\\= & {} 2128n-36. \end{aligned}$$$$\begin{aligned} J(FePc)= & {} \frac{m^{‘}}{m^{‘}-n^{‘}+2}\sum _{i=1}^{4}{\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}}{\frac{1}{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}}\\= & {} \left( \frac{68n}{13n-2}\right) \left( \frac{1}{3}(12n+4)+\frac{1}{6}(12n-4) +\left( \frac{1}{9}\right) (40n)+\left( \frac{1}{12}\right) (4n)\right) \\= & {} \left( \frac{68n}{13n-2}\right) (10.78n+0.67). \end{aligned}$$$$\begin{aligned} ReZG_1(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\frac{ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})}{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}\\= & {} \left( \frac{4}{3}\right) (12n+4)+\left( \frac{5}{6}\right) (12n-4) +\left( \frac{6}{9}\right) (40n)+\left( \frac{7}{12}\right) (4n)\\= & {} 55n+2. \end{aligned}$$$$\begin{aligned} ReZG_2(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\frac{ \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}})}{ \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}})}\\= & {} \left( \frac{4}{5}\right) (12n+4)+\left( \frac{8}{6}\right) (12n-4) +\left( \frac{16}{8}\right) (40n)+\left( \frac{32}{12}\right) (4n)\\= & {} 90.26n-1.8. \end{aligned}$$$$\begin{aligned} ReZG_3(FePc)= & {} \sum _{i=1}^{4}\sum _{\mathfrak {cd}\mathfrak {E_{i}}(FePc)}\bigg (( \omega ({\mathfrak {c}})\times \omega ({\mathfrak {d}}))\times ( \omega ({\mathfrak {c}})+ \omega ({\mathfrak {d}}))\bigg )\\= & {} (12)(12n+4)+(30)(12n-4) +(54)(40n)+(84)(4n)\\= & {} 83000n-72. \end{aligned}$$The numerical comparison of ABC(FePc), GA(FePc), F(FePc), and AZI(FePc) is shown in Table 2, and the graphical representation for each of these indices is shown in Fig. 6. The table and figure that correspond to these indices provide a thorough examination of both the numerical and visual components.
Table 2 Numerical analysis of ABC(FePc), GA(FePc), F(FePc) and AZI(FePc).Figure 6Geometrical analysis of ABC(FePc), GA(FePc), F(FePc) and AZI(FePc).The numerical analysis for \(M_1(FePc)\), \(M_2(FePc)\), HM(FePc), and J(FePc) is shown in Table 3, and Fig. 7a shows a graphical depiction of their behaviours. The numerical and visual properties of these indices are thoroughly explored in both the table and the accompanying graphic.
Table 3 Numerical analysis of \(M_1(FePc)\), \(M_2(FePc)\), HM(FePc), J(FePc).Figure 7Graphical comparison between (a) \(M_1(FePc)\), \(M_2(FePc)\), HM(FePc) and J(FePc); (b) \(ReZG_1(FePc)\), \(ReZG_2(FePc)\), \(ReZG_3(FePc)\).The numerical analysis of each redefined Zagreb index is shown in Table 4, and Fig. 7b provides graphical depictions of these index behaviours. This analysis includes a detailed look at the redesigned Zagreb indices from a numerical and visual standpoint.
Table 4 Numerical analysis of \(ReZG_1(FePc)\), \(ReZG_2(FePc)\) and \(ReZG_3(FePc)\).HOF phthalocyanineFor Fe phthalocyanine, the degree-based topological indices were calculated for the following unit cell configurations: F(FePc), J(FePc), \(M_{1}(FePc)\), and ABC(FePc) etc. These indices show relationships with important Fe phthalocyanine thermodynamic parameters such as heat of formation (HOF). Fe phthalocyanine’s standard molar enthalpy is found to be \(-87.9{\text { kJmol}}^{-1}\). The enthalpy of the cell can be calculated by multiplying this value by the number of formula units in the cell. Notably, the HOF of Fe phthalocyanine inversely decreases with the number of crystal structures and the size of its crystals, as shown in Table 5.
Table 5 Values of HOF for FePc.Standard framework for HOF vs indicesIn this section, we develop mathematical models to establish relationships between the Fe phthalocyanine Heat of Formation (HoF), as found in Sect. 2.2, and all the topological indices calculated in Part 2.1. Figures 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 show graphical representations of the fitted curves for these connections. These curves mean and standard deviations, denoted by \(\Xi\) and \(\Psi\) correspondingly, provide information about the variability and general trends of the established relationships.Generic framework between ABC(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 6. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term, and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 8 plots HoF(x)vsABC(FePc). An additional variable under investigation is ABC(FePc). The plot indicates that HoF(x) and ABC(FePc) have a positive association. This implies that ABC(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=ABC(G)\) is classified by \(\Xi =166.8\) and \(\psi =88.92.\)
Table 6 HoF vs ABC(FePc).Figure 8Generic framework between GA(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 7. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 9 plots HoF(x)vsGA(FePc). An additional variable under investigation is GA(FePc). The plot indicates that HoF(x) and GA(FePc) have a positive association. This implies that GA(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=GA(FePc)\) is classified by \(\Xi =230.9\) and \(\psi =123.7.\)
Figure 9Generic framework between F(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 8. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term, and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 10 plots HoF(x)vsF(FePc). An additional variable under investigation is F(FePc). The plot indicates that HoF(x) and F(FePc) have a positive association. This implies that F(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{(p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4})}{x + q1}, \end{aligned}$$where \(x=F(FePc)\) is classified by \(\Xi = 3824\) and \(\psi =2050.\)
Figure 10Generic framework between AZI(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 9. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 11 plots HoF(x)vsAZI(FePc) . An additional variable under investigation is AZI(FePc). The plot indicates that HoF(x) and AZI(FePc) have a positive association. This implies that AZI(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=AZI(FePc)\) is classified by \(\Xi =2247\) and std \(\psi =1211\)
Table 9 HoF vs AZI(FePc).Figure 11Generic framework between \(M_1(FePc)\) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 10. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 12 plots \(HoF(x) vs M_1(FePc)\). An additional variable under investigation is \(M_1(FePc)\). The plot indicates that HoF(x) and \(M_1(FePc)\) have a positive association. This implies that \(M_1(FePc)\) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=M_{1}(FePc)\) is classified by \(\Xi =1732\) and std \(\psi =927.9.\)
Table 10 HoF vs \(M_1(FePc)\).Figure 12Generic framework between \(M_2(FePc)\) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 11. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 13 plots \(HoF(x) vs M_2(FePc)\). An additional variable under investigation is \(M_2(FePc)\). The plot indicates that HoF(x) and \(M_2(FePc)\) have a positive association. This implies that \(M_2(FePc)\) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=M_{2}(FePc)\) is classified by \(\Xi =1794\) and \(\psi =965.3.\)
Table 11 HoF vs \(M_2(FePc)\).Figure 13Generic framework between HM(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 12. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 14 plots HoF(x)vsHM(FePc). An additional variable under investigation is HM(FePc). The plot indicates that HoF(x) and HM(FePc) have a positive association. This implies that HM(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=HM(FePc)\) is classified by \(\Xi =7412\) and \(\psi =3981.\)
Table 12 HoF vs HM(FePc).Figure 14Generic framework between J(FePc) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 13. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 15 plots HoF(x)vsJ(FePc). An additional variable under investigation is J(FePc). The plot indicates that HoF(x) and J(FePc) have a positive association. This implies that J(FePc) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4}}{x + q1}, \end{aligned}$$where \(x=J(FePc)\) is classified by \(\Xi =210.4\) and std \(\psi =104.9.\)
Figure 15Generic framework between \(ReZG_1(FePc)\) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 14. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 16 plots \(HoF(x) vs ReZG_1(FePc)\). An additional variable under investigation is \(ReZG_1(FePc)\). The plot indicates that HoF(x) and \(ReZG_1(FePc)\) have a positive association. This implies that \(ReZG_1(FePc)\) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{(p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4})}{x + q1}, \end{aligned}$$where \(x=ReZG_1(FePc)\) is classified by \(\Xi =194.5\) and \(\psi =102.9.\)
Table 14 HoF vs \(ReZG_1(FePc)\).Figure 16Generic framework between \(ReZG_2(FePc)\) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 15. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 17 plots \(HoF(x) vs ReZG_2(FePc)\). An additional variable under investigation is \(ReZG_2(FePc)\). The plot indicates that HoF(x) and \(ReZG_2(FePc)\) have a positive association. This implies that \(ReZG_2(FePc)\) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{(p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4})}{x + q1}, \end{aligned}$$where \(x=ReZG_2(FePc)\) is classified by \(\Xi =314.1\) and \(\psi =168.8\)
Table 15 HoF vs \(ReZG_2(FePc)\).Figure 17Generic framework between \(ReZG_3(FePc)\) and HoF
The values of \(p_{i}\) and \(q_{i}\) for various values of i are displayed in Table 16. In the expansion of HoF(x), \(p_{i}\) is the coefficient of the i-th term and qi is its mean. The confidence intervals (CIs) for \(p_{i}\) and \(q_{i}\) are also displayed in the table. The range of numbers that \(p_{i}\) is most likely to fall inside, given the evidence, is known as the confidence interval (CI). The range of numbers that \(q_{i}\) is most likely to fall inside, given the data, is its confidence interval (CI). Figure 18 plots \(HoF(x) vs ReZG_3(FePc)\). An additional variable under investigation is \(ReZG_3(FePc)\). The plot indicates that HoF(x) and \(ReZG_3(FePc)\) have a positive association. This implies that \(ReZG_3(FePc)\) increases along with HoF(x).$$\begin{aligned} HoF(x)=\frac{(p_{1}x^3 + p_{2}x^2 + p_{3}x + p_{4})}{x + q1}, \end{aligned}$$where \(x=ReZG_3(G)\) is classified by \(\Xi =1.043e+04\) and \(\psi =5612.\)
Table 16 HoF vs \(ReZG_3(FePc)\).Figure 18Table 17 provides a quality of fit for the framework fitted between HoF and each index computed in section 2.1. The selection of fits between\(R_{-\frac{1}{2}}(FePc)\), HM(FePc) and J(FePc) has been made considering \(R^2\) etc. and adjusted \(R^{2}\) as well.
Table 17 Quality of Fit for Indices for FePc vs HoF.
On physical analysis of phthalocyanine iron (II) using topological descriptor and curve fitting models
