On physical analysis of phthalocyanine iron (II) using topological descriptor and curve fitting models

Abstract

A new area of applied chemistry called chemical graph theory uses combinatorial techniques to explain the complex interactions between atoms and bonds in chemical systems. This work investigates the use of edge partitions to decipher molecular connection patterns. The main goal is to use topological indices that capture important topological features to create a connection between the thermodynamic properties and structural characteristics of chemical molecules. We specifically examine the complex web of atoms and links that make up the Fe phthalocyanine chemical graph. Moreover, our study demonstrates a relationship between the calculated topological indices and the thermodynamic properties of Fe phthalocyanine (Phthalocyanine Iron (II)). This work offers insight into the thermodynamic consequences of molecule structures. It advances the subject of chemical graph theory, providing a useful perspective for future applications in catalysis and materials science.

Introduction

Graph theory has broad applications in many fields of study, including computer science, chemistry, and biology. It is vital to study the chemical characteristics built into certain molecular structures. In this context, the fundamental representation is a graph, which is defined as an ordered pair consisting of a vertex set and an edge set. The degree of a vertex $\mathfrak{c}$ in the chemical graph is denoted by $\omega (c)$ , which is the number of edges incident to that vertex. Let G be a graph consisting of a vertex set and an edge set. A graph invariant is a statistical value that can only be found by the graph itself in the context of chemical graphs, in which atoms and their bonds are represented by vertices and edges, respectively. This characteristic, which is different from a particular graph version, is discussed in the context of chemical graph theory¹. Any graphical configuration can be analyzed thanks to graph invariants, which reflect fundamental graph structures. Understanding the network’s topology makes it easier to investigate various chemical properties inherent in the graph structure.

In this investigation, topological connectivity indices, which are graph invariants, are essential for deciphering the intricate details of molecular graphs. The methodology of the whole manuscript is shown in Fig. 1.

Figure 1 [Images not available. See PDF.]

Methodology

A potential catalyst for the oxygen reduction reaction is fe phthalocyanine, or (FePc). The energy effectiveness of metal-air batteries and fuel cells is directly influenced by oxygen reduction. Phthalocyanine molecules with a (FePc) core. These compounds’ interactions with various substrates and their impact on magnetic characteristics have been researched². The exchange-correlation function is highly dependent on the electronic structure and separation of the molecule from the substrate. Due to weak $O_2$ adsorption and activation, FePc with a plane-symmetric $F_e{N}_4$ site typically exhibits mediocre ORR activity. This hypothesis is realized by coordinating (FePc) with an oxidized carbon. Mössbauer spectra and synchrotron X-ray absorption confirm the Fe–O coordination between carbon and (FePc). Different phthalic acid derivatives, such as phthalonitrile, phthalic anhydride, and phthalimides, are cyclotetramerized to produce phthalocyanine. When FePc was first discovered, dyes and pigments were essentially its only known applications.

Estrada et al.^3,4 established the atom bond connectivity index as follows:

1

$$\begin{array}{c}\hfill ABC(G)=\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}\sqrt{\frac{\omega (\mathfrak{c})+\omega (\mathfrak{d})-2}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}}\end{array}$$

Vukic Evic et al.⁵ presented the geometric arithmetic index as follows:

2

$$\begin{array}{c}\hfill GA(G)=\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}\frac{2\sqrt{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}}{\omega (\mathfrak{c})+\omega (\mathfrak{d})}\end{array}$$

Gutman et al⁶ and Furtula et al.⁷ presented the forgotten connectivity index as follows:

3

$$\begin{array}{cc}\hfill F(G)=& \sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\omega {(\mathfrak{c})}^2+\omega {(\mathfrak{c})}^2)\hfill \end{array}$$

Furtula et al.,^8,7,9 presented augmented zagreb is as follows:

4

$$\begin{array}{cc}\hfill AZI(G)=& \sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\frac{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}{\omega (\mathfrak{c})+\omega (\mathfrak{d})-2}{)}^3\hfill \end{array}$$

Gutman et al.^10,11 presented Zagreb index as follows:

5

$$\begin{array}{c}\hfill M_1(G)=\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\omega (\mathfrak{c})+\omega (\mathfrak{d}))\end{array}$$

6

$$\begin{array}{c}\hfill M_2(G)=\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\omega (\mathfrak{c})\times \omega (\mathfrak{d}))\end{array}$$

In 2013, Shirdel et al.⁹ introduced the Hyper-Zagreb index:

7

$$\begin{array}{c}\hfill HM(G)=\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\omega (\mathfrak{c})+\omega (\mathfrak{d}){)}^2\end{array}$$

The Balaban index^12,13 is presented as follows:

8

$$\begin{array}{cc}\hfill J(G)=& \frac{m_{}^{_{}^{\prime }}}{m_{}^{_{}^{\prime }}-n_{}^{_{}^{\prime }}+2}\sum _{\mathfrak{cd}\in \mathfrak{E}(G)}\frac{1}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}\hfill \end{array}$$

Ranjini et al. in¹⁴ reformulated versions are as follows:

9

$$\begin{array}{cc}\hfill ReZ{G}_1(G)=& \sum _{\mathfrak{cd}\in \mathfrak{E}(G)}\frac{\omega (\mathfrak{c})+\omega (\mathfrak{d})}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}\hfill \end{array}$$

10

$$\begin{array}{cc}\hfill ReZ{G}_2(G)=& \sum _{\mathfrak{cd}\in \mathfrak{E}(G)}\frac{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}{\omega (\mathfrak{c})+\omega (\mathfrak{d})}\hfill \end{array}$$

11

$$\begin{array}{cc}\hfill ReZ{G}_3(G)=& \sum _{\mathfrak{cd}\in \mathfrak{E}(G)}(\omega (\mathfrak{c})\times \omega (\mathfrak{d}))(\omega (\mathfrak{c})+\omega (\mathfrak{d}))\hfill \end{array}$$

Structure of phthalocyanine FePc

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 $m^{\prime }=55n+2$ and $n^{\prime }=68n$ , 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 $FePc[\mathfrak{c},\mathfrak{d}]$ .

( $\omega (\mathfrak{c})$ , $\omega (\mathfrak{d})$ )	Frequency	Set of edges
(1, 3)	$12\text{n}+4$	${\mathfrak{E}}_{\mathfrak{1}}$
(2, 3)	$12\text{n}-4$	${\mathfrak{E}}_{\mathfrak{2}}$
(3, 3)	40n	${\mathfrak{E}}_{\mathfrak{3}}$
(3, 4)	4n	${\mathfrak{E}}_{\mathfrak{4}}$

Figure 2 [Images not available. See PDF.]

The structure of Phthalocyanine FePc(n) for $n=1$ .

Figure 3 [Images not available. See PDF.]

The structure of Phthalocyanine FePc(n) for $n=2$ .

Figure 4 [Images not available. See PDF.]

The structure of Phthalocyanine FePc(n) for $n=3$ .

Figure 5 [Images not available. See PDF.]

The structure of Phthalocyanine FePc(n) for $n=4$ .

Main results for Fe phthalocyanine

In 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:

• Atom bond connectivity of FePc

$$\begin{array}{cc}\hfill ABC(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\sqrt{\frac{\omega (\mathfrak{c})+\omega (\mathfrak{d})-2}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}}\hfill \\ \hfill =& \left(\sqrt{\frac{2}{3}}\right)(12n+4)+\left(\sqrt{\frac{3}{6}}\right)(12n-4)+\left(\sqrt{\frac{4}{9}}\right)(40n)+\left(\sqrt{\frac{5}{12}}\right)(4n)\hfill \\ \hfill =& 47.531896n+0.437559.\hfill \end{array}$$

• Geometric arithmetic of FePc

$$\begin{array}{cc}\hfill GA(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\frac{2\sqrt{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}}{\omega (\mathfrak{c})+\omega (\mathfrak{d})}\hfill \\ \hfill =& \left(\frac{2\sqrt{3}}{4}\right)(12n+4)+\left(\frac{2\sqrt{6}}{5}\right)(12n-4)+\left(\frac{2\sqrt{9}}{6}\right)(40n)+\left(\frac{2\sqrt{12}}{7}\right)(4n)\hfill \\ \hfill =& 66.108829n-0.455082.\hfill \end{array}$$

• Forgotten index of FePc

$$\begin{array}{cc}\hfill F(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}(\omega {(\mathfrak{c})}^2+\omega {(\mathfrak{d})}^2)\hfill \\ \hfill =& (10)(12n+4)+(13)(12n-4)+(18)(40n)+(25)(4n)\hfill \\ \hfill =& 1096n-12.\hfill \end{array}$$

• Augmented Zagreb Index of FePc

$$\begin{array}{cc}\hfill AZI(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}(\frac{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}{\omega (\mathfrak{c})+\omega (\mathfrak{d})-2}{)}^3\hfill \\ \hfill =& {\left(\frac{4}{3}\right)}^3(12n+4)+{\left(\frac{8}{4}\right)}^3(12n-4)+{\left(\frac{16}{6}\right)}^3(40n)+{\left(\frac{32}{10}\right)}^3(4n)\hfill \\ \hfill =& 647.421n-18.5.\hfill \end{array}$$

• First Zagreb Index of FePc

$$\begin{array}{cc}\hfill M_1(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\omega (\mathfrak{c})+\omega (\mathfrak{d})\hfill \\ \hfill =& (4)(12n+4)+(5)(12n-4)+(6)(40n)+(7)(4n)\hfill \\ \hfill =& 496n-4.\hfill \end{array}$$

• Second Zagreb Index of FePc

$$\begin{array}{cc}\hfill M_2(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\omega (\mathfrak{c})\times \omega (\mathfrak{d})\hfill \\ \hfill =& (3)(12n+4)+(6)(12n-4)+(9)(40n)+(12)(4n)\hfill \\ \hfill =& 516n-12.\hfill \end{array}$$

• Hyper Zagreb Index of FePc

$$\begin{array}{cc}\hfill HM(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}{(\omega (\mathfrak{c})+\omega (\mathfrak{d}))}^2\hfill \\ \hfill =& (16)(12n+4)+(25)(12n-4)+(36)(40n)+(49)(4n)\hfill \\ \hfill =& 2128n-36.\hfill \end{array}$$

• Balban Index of FePc

$$\begin{array}{cc}\hfill J(FePc)=& \frac{m_{}^{_{}^{\prime }}}{m_{}^{_{}^{\prime }}-n_{}^{_{}^{\prime }}+2}\sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\frac{1}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}\hfill \\ \hfill =& \left(\frac{68n}{13n-2}\right)\left(\frac{1}{3},(12n+4),+,\frac{1}{6},(12n-4),+,\left(\frac{1}{9}\right),(40n),+,\left(\frac{1}{12}\right),(4n)\right)\hfill \\ \hfill =& \left(\frac{68n}{13n-2}\right)(10.78n+0.67).\hfill \end{array}$$

• First Redefined Zagreb Index

$$\begin{array}{cc}\hfill ReZ{G}_1(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\frac{\omega (\mathfrak{c})+\omega (\mathfrak{d})}{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}\hfill \\ \hfill =& \left(\frac{4}{3}\right)(12n+4)+\left(\frac{5}{6}\right)(12n-4)+\left(\frac{6}{9}\right)(40n)+\left(\frac{7}{12}\right)(4n)\hfill \\ \hfill =& 55n+2.\hfill \end{array}$$

• Second Redefined Zagreb Index

$$\begin{array}{cc}\hfill ReZ{G}_2(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}\frac{\omega (\mathfrak{c})\times \omega (\mathfrak{d})}{\omega (\mathfrak{c})+\omega (\mathfrak{d})}\hfill \\ \hfill =& \left(\frac{4}{5}\right)(12n+4)+\left(\frac{8}{6}\right)(12n-4)+\left(\frac{16}{8}\right)(40n)+\left(\frac{32}{12}\right)(4n)\hfill \\ \hfill =& 90.26n-1.8.\hfill \end{array}$$

• Third Redefined Zagreb Index

$$\begin{array}{cc}\hfill ReZ{G}_3(FePc)=& \sum _{i=1}^4\sum _{\mathfrak{cd}{\mathfrak{E}}_{\mathfrak{i}}(FePc)}((\omega (\mathfrak{c})\times \omega (\mathfrak{d}))\times (\omega (\mathfrak{c})+\omega (\mathfrak{d})))\hfill \\ \hfill =& (12)(12n+4)+(30)(12n-4)+(54)(40n)+(84)(4n)\hfill \\ \hfill =& 83000n-72.\hfill \end{array}$$

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).

[n]	ABC(FePc)	GA(FePc)	F(FePc)	AZI(FePc)
[1]	47.969455	65.653847	1084	100.0954
[2]	95.501351	131.762776	2180	936.8361
[3]	143.033247	197.871705	3276	2508.3917
[4]	190.565143	263.980634	4372	4814.7620
[5]	238.097039	330.089563	5468	7855.9472
[6]	285.628935	396.198492	6564	11,631.9472
[7]	333.160831	462.307421	7660	16,142.7620
[8]	380.692727	528.41635	8756	3,574,675.4383
[9]	428.224623	594.525279	9852	3,574,675.4383
[10]	475.756519	660.634208	10,948	3,647,524.0295

Figure 6 [Images not available. See PDF.]

Geometrical 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).

[n]	$M_1(FePc)$	$M_2(FePc)$	HM(FePc)	J(FePc)
[1]	492	504	2092	70.78181818
[2]	988	1020	4220	125.97
[3]	1484	1536	6348	182.0010811
[4]	1980	2052	8476	238.2176
[5]	2476	2568	10,604	294.5047619
[6]	2972	3084	12,732	350.8263158
[7]	3468	3600	14,860	407.167191
[8]	3964	4116	16,988	463.52
[9]	4460	4632	19,116	519.8806957
[10]	4956	5148	21,244	576.246875

Figure 7 [Images not available. See PDF.]

Graphical comparison between (a) $M_1(FePc)$ , $M_2(FePc)$ , HM(FePc) and J(FePc); (b) $ReZ{G}_1(FePc)$ , $ReZ{G}_2(FePc)$ , $ReZ{G}_3(FePc)$ .

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 $ReZ{G}_1(FePc)$ , $ReZ{G}_2(FePc)$ and $ReZ{G}_3(FePc)$ .

[n]	$ReZ{G}_1(FePc)$	$ReZ{G}_2(FePc)$	$ReZ{G}_3(FePc)$
[1]	57	88.46	2928
[2]	112	178.72	5928
[3]	167	268.98	8928
[4]	222	359.24	11,928
[5]	277	449.5	14,928
[6]	332	539.76	17,928
[7]	387	630.02	20,928
[8]	442	720.28	23,928
[9]	497	810.54	26,928
[10]	552	900.8	29,928

HOF phthalocyanine

For 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.

[m, n]	Formula units	HOF $\times {10}^{-22}kJ$
[1, 1]	4	$-27.5124$
[2, 2]	16	$-110.0498$
[3, 3]	36	$-247.6120$
[4, 4]	64	$-440.1992$
[5, 5]	100	$-687.8113$
[6, 6]	144	$-990.4483$
[7, 7]	196	$-1348.1102$

Standard framework for HOF vs indices

In 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 $\mathrm{\Xi }$ and $\mathrm{\Psi }$ 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=ABC(G)$ is classified by $\mathrm{\Xi }=166.8$ and $\psi =88.92.$ Table 6

HoF vs ABC(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.1745$	$(-7.057,6.708)$
$i=2$	$-72.89$	$(-213.5,67.75)$	–	–
$i=3$	$-58.18$	$(-584.4,468.1)$	–	–
$i=4$	12.48	$(-479.8,504.7)$	–	–

Figure 8 [Images not available. See PDF.]

HoF vs ABC(FePc).

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=GA(FePc)$ is classified by $\mathrm{\Xi }=230.9$ and $\psi =123.7.$ Table 7

HoF vs GA(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.1986$	$(-9.597,9.2)$
$i=2$	$-72.4$	$(-264.5,119.6)$	–	–
$i=3$	$-56.34$	$(-774.9,662.3)$	–	–
$i=4$	14.2	$(-658,686.4)$	–	–

Figure 9 [Images not available. See PDF.]

HoF vs GA(FePc).

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{(p_1{x}^3+p_2{x}^2+p_{3}x+p_4)}{x+q1},\end{array}$$ where $x=F(FePc)$ is classified by $\mathrm{\Xi }=3824$ and $\psi =2050.$ Table 8

HoF vs F(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.1418$	$(-6.641,6.358)$
$i=2$	$-73.56$	$(-206.4,59.25)$	–	–
$i=3$	$-60.68$	$(-557.6,436.3)$	–	–
$i=4$	10.14	$(-454.7,475)$	–	–

Figure 10 [Images not available. See PDF.]

HoF vs F(FePc).

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=AZI(FePc)$ is classified by $\mathrm{\Xi }=2247$ and std $\psi =1211$ Table 9

HoF vs AZI(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.7058$	$(-16.7,15.29)$
$i=2$	$-62.04$	$(-388.9,264.8)$	–	–
$I=3$	$-17.56$	$(-1240,1205)$	–	–
$I=4$	50.48	$(-1093,1194)$	–	–

Figure 11 [Images not available. See PDF.]

HoF vs AZI(FePc).

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=M_1(FePc)$ is classified by $\mathrm{\Xi }=1732$ and std $\psi =927.9.$ Table 10

HoF vs $M_1(FePc)$ .

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.08164$	$(-6.73,6.567)$
$i=2$	$-74.79$	$(-210.7,61.07)$	–	–
$i=3$	$-65.28$	$(-573.6,443.1)$	–	–
$i=4$	5.839	$(-469.7,481.4)$	–	–

Figure 12 [Images not available. See PDF.]

HoF vs $M_1(FePc)$ .

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=M_2(FePc)$ is classified by $\mathrm{\Xi }=1794$ and $\psi =965.3.$ Table 11

HoF vs $M_2(FePc)$ .

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.2163$	$(-7.768,7.335)$
$i=2$	$-72.04$	$(-226.4,82.28)$	–	–
$i=3$	$-54.98$	$(-632.4,522.4)$	–	–
$i=4$	15.47	$(-524.6,555.6)$	–	–

Figure 13 [Images not available. See PDF.]

HoF vs $M_2(FePc)$ .

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=HM(FePc)$ is classified by $\mathrm{\Xi }=7412$ and $\psi =3981.$ Table 12

HoF vs HM(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	$-0.6593$	$(-28.49,27.17)$
$i=2$	$-62.99$	$(-631.7,505.8)$	6.945	$(-5.705,19.6)$
$i=3$	$-21.11$	$(-2149,2107)$	–	–
$i=4$	47.16	$(-1943,2038)$	–	–

Figure 14 [Images not available. See PDF.]

HoF vs HM(FePc).

Generic 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). $$\begin{array}{c}\hfill HoF(x)=\frac{p_1{x}^3+p_2{x}^2+p_{3}x+p_4}{x+q1},\end{array}$$ where $x=J(FePc)$ is classified by $\mathrm{\Xi }=210.4$ and std $\psi =104.9.$ Table 13

HoF vs J(FePc).

	$p_i$	CI	$q_i$	CI
$i=1$	$-17.35$	$(-5415,5380)$	$-193.3$	$(-3.681\text{e}+05,3.677\text{e}+05)$
$i=2$	3825	$(-7.422\text{e}+06,7.429\text{e}+06)$	–	–
$i=3$	1.468e + 04	$(-2.807\text{e}+07,2.81\text{e}+07)$	–	–
$i=4$	1.386e + 04	$(-2.638\text{e}+07,2.64\text{e}+07)$	–	–

Figure 15 [Images not available. See PDF.]

HoF vs J(FePc).

Generic framework between $ReZ{G}_1(FePc)$ 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 $HoF(x)vsReZ{G}_1(FePc)$ . An additional variable under investigation is $ReZ{G}_1(FePc)$ . The plot indicates that HoF(x) and $ReZ{G}_1(FePc)$ have a positive association. This implies that $ReZ{G}_1(FePc)$ increases along with HoF(x). $$\begin{array}{c}\hfill HoF(x)=\frac{(p_1{x}^3+p_2{x}^2+p_{3}x+p_4)}{x+q1},\end{array}$$ where $x=ReZ{G}_1(FePc)$ is classified by $\mathrm{\Xi }=194.5$ and $\psi =102.9.$ Table 14

HoF vs $ReZ{G}_1(FePc)$ .

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	0.7951	$(-56.46,58.05)$
$i=2$	$-92.71$	$(-1263,1077)$	–	–
$i=3$	$-132.3$	$(-4510,4246)$	–	–
$i=4$	$-56.86$	$(-4152,4038)$	–	–

Figure 16 [Images not available. See PDF.]

HoF vs $ReZ{G}_1(FePc)$ .

Generic framework between $ReZ{G}_2(FePc)$ 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 $HoF(x)vsReZ{G}_2(FePc)$ . An additional variable under investigation is $ReZ{G}_2(FePc)$ . The plot indicates that HoF(x) and $ReZ{G}_2(FePc)$ have a positive association. This implies that $ReZ{G}_2(FePc)$ increases along with HoF(x). $$\begin{array}{c}\hfill HoF(x)=\frac{(p_1{x}^3+p_2{x}^2+p_{3}x+p_4)}{x+q1},\end{array}$$ where $x=ReZ{G}_2(FePc)$ is classified by $\mathrm{\Xi }=314.1$ and $\psi =168.8$ Table 15

HoF vs $ReZ{G}_2(FePc)$ .

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.43$	$(-20.45,-20.42)$	$-1.299$	$(-1.961,-0.6377)$
$i=2$	$-49.91$	$(-63.43,-36.38)$	–	–
$i=3$	27.81	$(-22.75,78.38)$	–	–
$i=4$	92.92	(45.61, 140.2)	–	–

Figure 17 [Images not available. See PDF.]

HoF vs $ReZ{G}_2(FePc)$ .

Generic framework between $ReZ{G}_3(FePc)$ 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 $HoF(x)vsReZ{G}_3(FePc)$ . An additional variable under investigation is $ReZ{G}_3(FePc)$ . The plot indicates that HoF(x) and $ReZ{G}_3(FePc)$ have a positive association. This implies that $ReZ{G}_3(FePc)$ increases along with HoF(x). $$\begin{array}{c}\hfill HoF(x)=\frac{(p_1{x}^3+p_2{x}^2+p_{3}x+p_4)}{x+q1},\end{array}$$ where $x=ReZ{G}_3(G)$ is classified by $\mathrm{\Xi }=1.043e+04$ and $\psi =5612.$ Table 16

HoF vs $ReZ{G}_3(FePc)$ .

	$p_i$	CI	$q_i$	CI
$i=1$	$-20.44$	$(-20.44,-20.44)$	0.8707	$(-16.47,18.21)$
$i=2$	$-94.25$	$(-448.6,260.1)$	–	–
$i=3$	$-138.1$	$(-1464,1188)$	–	–
$i=4$	$-62.27$	$(-1302,1178)$	–	–

Figure 18 [Images not available. See PDF.]

HoF vs $ReZ{G}_3(FePc)$ .

Table 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.

Index	Fit type	SSE	$R^2$	Adjusted $R^2$	RMSE
$R_1(FePc)$	rat13	1.78e−17	1	1	4.219e−09
$R_{-1}(FePc)$	rat22	1.163e−15	1	1	3.41e−08
$R_{\frac{1}{2}}(FePc)$	rat22	7.229e−16	1	1	2.689e−08
$R_{-\frac{1}{2}}(FePc)$	rat21	1.78e−17	1	1	4.219e−09
ABC(FePc)	rat22	1.618e−17	1	1	4.022e−09
GA(FePc)	rat22	7.223e−17	1	1	8.499e−09
$M_1(FePc)$	rat22	2.992e−17	1	1	5.47e−09
$M_2(FePc)$	rat12	1.823e−17	1	1	4.27e−09
HM(FePc)	rat13	7.798e−17	1	1	8.83e−09
J(FePc)	rat23	7.078e−05	1	1	0.008413
AZI(FePc)	rat21	2.451e−17	1	1	4.951e−09
F(FePc)	rat23	1.478e−17	1	1	3.844e−09
$ReZ{G}_1(FePc)$	rat23	2.742e−15	1	1	5.236e−08
$ReZ{G}_2(FePc)$	rat21	2.325e−07	1	1	0.0004821
$ReZ{G}_3(FePc)$	rat12	2.466e−15	1	1	4.966e−08

Conclusion

This work began with a thorough examination of Fe phthalocyanine, first investigating indices based on topological degrees and then calculating thermodynamic characteristics. Strong mathematical models were produced by using fitting curves to find relationships between each topological index and thermodynamic attribute. This analysis covered an important category of thermochemical properties heat of formation (HOF). The application of a curve-fitting strategy carried out via MATLAB software enabled a sophisticated comprehension of the complex connection between the molecular topology and thermodynamic characteristics of Fe phthalocyanine. This coordinated strategy not only improves our understanding of chemical processes but also highlights the value of mathematical modeling in clarifying intricate relationships between molecules.

Author contributions

Muhammad Farhan Hanif contributed to the Investigation, analyzing the data curation, designing the experiments, data analysis, computation, funding resources, calculation verifications, and wrote the initial draft of the paper. Hasan Mahmood contributed to the computation and investigated and approved the final draft of the paper. Shahbaz Ahmad contributed to supervision, conceptualization, Methodology, Matlab calculations, Maple graphs improvement project administration. Mohamed Abubakar Fiidow contributes to formal analyzing experiments, software, validation, and funding. All authors read and approved the final version.

Funding

There is no funding to support this article.

Data availibility

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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