On physical analysis of topological indices for iron disulfide network via curve fitting model

A statistical method called curve fitting is used to design a curve that most accurately depicts the relationship between a collection of data points. A method known as rational curve fitting can be used to mathematically estimate a set of data points using a curve that is represented by a rational function. In this paper, we represent the rational curve fitting by \(\textit{r}_{c}{f}\). The concept of the illumination of the heat (Enthalpy) HOF of iron disulfide FeS\(_{2}\) is presented in this section of the article. The enthalpy of formation, sometimes referred to as the heat of formation, is a thermodynamic quantity that indicates the change in enthalpy when one mole of a compound is produced from its constituent elements in their standard states under standard conditions typically 298.15 K and 1 atm pressure48. To calculate the formation of heat (HOF) multiply the molar standard Enthalpy of FeS\(_{2}\)’s, which is approximately 523 (kJ/mol) at temperature 298.15K, by the number of formula units in the cell and then divide the obtained values by the Avogadro’s number.$$\begin{aligned} HOF = \frac{Standard \,\ Molar\,\ HOF\times \,\ Formula \,\ Units}{Avogadro’s \,\ Number} \end{aligned}$$
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Where, \(Avogadro’s \,\ Number = 6.02214076 \times 10^{23} mol^{-1}\). A mole of a substance such as an atom, molecule, or ion is equivalent to \(6.022 \times 10^{23}\) of that substance.Finding the rational function that best fits the data points where the indices are the independent variable (x) and the heat of formation values are the dependent variable (y) is the first step in fitting a rational curve between indices and the heat of formation. Next, we collect the data into two lists, one for the indices and another for the heat of formation to fit a rational function. After that determine the polynomials P(x) and Q(x)’s degrees, m and n, respectively. Starting with low degrees (e.g., m=1, n=1) and increasing as needed is a popular option. Fit the rational function to the data using numerical techniques. Evaluate the fit quality by utilizing both visual examination and statistical measurements. The accuracy measures used are \(R^2\), the sum of squared errors (SSE), and the mean squared error (RMSE) which is shown in Tables 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and graphical behavior presented in Figs. 6, 7, 8, 9, 10, 11, 12 and 13.$$\begin{aligned} f(R_1) = \frac{(p_1\times R_1^2 + p_2\times R_1 +p_3)}{ (R_1^5 + q_1+R_1^4 + q_2+R_1^3 + q_3+R_1^2 + q_4+R_1 + q_5)} \end{aligned}$$where the mean \(1.904e+04\) and standard deviation \(1.75e+04\) normalise \(R_1\). These are the Coefficients:\(p_1=5.747(3.222, 8.271)\), \(p_2=2.1 (1.392, 2.807)\), \(p_3=-3.354 (-5.219, -1.49)\), \(q_1=-1.529 (-2.107, -0.9511)\), \(q_2=-1.952 (-2.258, -1.645)\), \(q_3=-0.4058 (-0.7617, -0.04992)\), \(q_4=-0.4694 (-0.8079, -0.1309)\), \(q_5=0.4747 (0.2299, 0.7195)\).The confidence bound in each curve fitting is \(95 \%\).Table 7 HOF and \(R_{1}({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(R_{-1}) = \frac{(p_1\times {R_1}^5 + p_2\times {R_1}^4 +p_3\times {R_1}^3 + p_4\times {R_1}^2 + p_5\times {R_1} + p_6) }{({R_1}^3 + q_1\times {R_1}^2 + q_2\times {R_1} + q_3)} \end{aligned}$$This normalizes \(R_{-1}\) using the mean (102.7) and standard deviation (90.9. The following are the coefficients:$$\begin{aligned}{} & {} p_1=0.975 (-1.285, 3.236), p_2=-2.089 (-5.943, 1.765), p_3=-3.645 (-8.174, 0.8836), p_4=4.164 (-5.79, 14.12),\\{} & {} p_5=2.195 (-0.8135, 5.204), p_6=-2.348 (-6.923, 2.228), q_1= 0.03154 (-1.412, 1.475), q_2=-0.6427 (-1.384, 0.09889),\\{} & {} q_3=0.3164 (-0.3611, 0.9939). \end{aligned}$$Table 8 HOF and \(R_{-1}({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f({R_\frac{1}{2})} = \frac{(p_1\times R_\frac{1}{2}^3 + p_2\times R_\frac{1}{2}^2 +p_3\times R_\frac{1}{2}+p_4)}{(R_\frac{1}{2}^5 + q_1\times {R_\frac{1}{2}}^4 + q_2\times {R_\frac{1}{2}}^3+ q_3\times {R_\frac{1}{2}}^2 +q_4\times {R_\frac{1}{2}} +q_5)} \end{aligned}$$This normalizes \(R_\frac{1}{2}\) using the mean 4947 and standard deviation 4516. The following are the coefficients:$$\begin{aligned}{} & {} p_1=-0.3383 (-9.546, 8.869), p_2=5.419 (-8.148, 18.99), p_3=2.233 (-2.196, 6.663), p_4=-3.239 (-11.3, 4.82),\\{} & {} q_1=-1.487 (-3.929, 0.9549), q_2=-1.818 (-5.605, 1.97), q_3=-0.307 (-3.197, 2.583), q_4=-0.49 (-1.852, 0.8717),\\{} & {} q_5=0.4573 (-0.6537, 1.568). \end{aligned}$$Table 9 HOF and \(R_{\frac{1}{2}}({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f({R_\frac{-1}{2})} = \frac{(p_1\times R_\frac{-1}{2}^3 + p_2\times R_\frac{-1}{2}^2 +p_3\times R_\frac{-1}{2}+p_4)}{(R_\frac{-1}{2}^5 + q_1\times {R_\frac{-1}{2}}^4 + q_2\times {R_\frac{-1}{2}}^3+ q_3\times {R_\frac{-1}{2}}^2 +q_4\times {R_\frac{-1}{2}} +q_5)} \end{aligned}$$This normalizes \(R_\frac{-1}{2}\) using mean 362.6 and standard deviation 324.9. The following are the coefficients:$$\begin{aligned}{} & {} p_1=-0.5478 (-8.913, 7.818), p_2=5.363 (-6.666, 17.39), p_3=2.345 (-1.875, 6.564), p_4=-3.288 (-10.57, 3.991),\\{} & {} q_1=-1.485 (-3.661, 0.6917), q_2=-1.728 (-5.22, 1.764), q_3=-0.2357 (-2.887, 2.416), q_4=-0.522 (-1.851, 0.8067),\\{} & {} q_5=0.4648 (-0.5425, 1.472). \end{aligned}$$Table 10 HOF and \(R_{-\frac{1}{2}}({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f(ABC) = \frac{(p_1\times {ABC}^2 + p_2\times {ABC} + p_3)}{({ABC}^5 + q_1 \times {ABC}^4 + q_2 \times {ABC}^3 + q_3 \times {ABC}^2 + q_4 +ABC + q_5)} \end{aligned}$$In which ABC is normalized by mean 845.8 and std 764.1. The Coefficients:$$\begin{aligned}{} & {} p_1=5.942 (3.195, 8.69), p_2=2.134 (1.367, 2.901), p_3=-3.517 (-5.543, -1.491), q_1=-1.562 (-2.187, -0.9362),\\{} & {} q_2=-1.943 (-2.265, -1.62), q_3=-0.3908 (-0.7671, -0.01441), q_4=-0.4933 (-0.8653, -0.1212),\\{} & {} q_5=0.4993 (0.2307, 0.7679). \end{aligned}$$Table 11 HOF and \(ABC ({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(GA) = \frac{(p_1 \times GA^3 + p_2 \times GA^2 + p_3 \times GA + p_4)}{(GA ^5+ q_1 \times GA^4 + q_2 \times GA^3 + q_3 \times GA ^2 + q_4 \times GA + q_5 )} \end{aligned}$$In which GA is normalized by mean 1273 and std 1151. The Coefficients are:$$\begin{aligned}{} & {} p_1=-0.459 (-9.212, 8.294), p_2=5.396 (-7.319, 18.11), p_3=2.297 (-2.033, 6.628), p_4=-3.278 (-10.92, 4.366),\\{} & {} q_1=-1.489 (-3.781, 0.803), q_2=-1.766 (-5.394, 1.863), q_3=-0.2641 (-3.023, 2.495), q_4=-0.509 (-1.858, 0.8398),\\{} & {} q_5=0.4631 (-0.5935, 1.52). \end{aligned}$$Table 12 HOF and \(GA ({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f(M_1) = \frac{(p_1\times {M_1}^3 + p_2\times {M_1}^2 + p_3\times {M_1} + p_4)}{({M_1}^5 + q_1\times {M_1}^4 + q_2\times {M_1}^3 + q_3 \times {M_1}^2 + q_4\times {M_1} + q_5)} \end{aligned}$$where mean \(1.031e+04 \) and standard \(1.031e+04 \) are used to normalise \(M_1\). The following are the coefficients:$$\begin{aligned}{} & {} p_1=-0.3245 (-9.623, 8.974), p_2=5.423 (-8.298, 19.15), p_3=2.226 (-2.232, 6.685), p_4=-3.236 (-11.38, 4.904),\\{} & {} q_1=-1.487 (-3.956, 0.9814), q_2=-1.824 (-5.645, 1.998), q_3=-0.3119 (-3.228, 2.604), q_4=-0.488 (-1.857, 0.8806),\\{} & {} q_5=0.4569 (-0.6651, 1.579). \end{aligned}$$Table 13 HOF and \(M_1({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(M_2) = \frac{(p_1\times M_2^2 + p_2\times M_2 + p_3)}{({M_2}^5 +q_1 \times {M_2}^4 + q_2 \times {M_2}^3 + q_3 \times {M_2}^2 + q_4 \times {M_2} + q_5)} \end{aligned}$$where mean \(1.904e+04\) and standard \(1.75e+04\) are used to normalise \(M_2\). The following are the coefficients:$$\begin{aligned}{} & {} p_1=5.747 (3.222, 8.271), p_2=2.1 (1.392, 2.807), p_3=-3.354 (-5.219, -1.49), q_1=-1.529 (-2.107, -0.9511),\\{} & {} q_2=-1.952 (-2.258, -1.645), q_3=-0.4058 (-0.7617, -0.04992), q_4=-0.4694 (-0.8079, -0.1309),\\{} & {} q_5=0.4747 (0.2299, 0.7195). \end{aligned}$$Table 14 HOF and \(M_2({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f{(HM)} = \frac{(p_1\times HM^3 + p_2\times HM^2 + p_3\times HM + p_4)}{(HM^5 + q_1\times HM^4 + q_2\times HM^3 + q_3\times HM^2 + q_4 \times HM + q_5 )} \end{aligned}$$In which HM is normalized by mean \(8.331e+04 \) and std \(7.671e+04\). The Coefficients are:$$\begin{aligned}{} & {} p_1=-0.227 (-9.973, 9.519), p_2=5.451 (-9.062, 19.97), p_3=2.176 (-2.394, 6.746), p_4=-3.214 (-11.75, 5.323),\\{} & {} q_1=-1.489 (-4.094, 1.116), q_2=-1.865 (-5.843, 2.112), q_3=-0.3456 (-3.385, 2.694), q_4=-0.4738 (-1.859, 0.9111),\\{} & {} q_5=0.4536 (-0.7211, 1.628). \end{aligned}$$Table 15 HOF and \(HM({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(PM_1) = \frac{(p_1 \times PM_1^5 + p_2\times PM_1^4 + p_3\times PM_1^3 + p_4\times PM_1^2 + p_5\times PM_1 + p_6 )}{(PM_1^3 + q_1 \times PM_1^2 + q_2 \times PM_1 + q_3 )} \end{aligned}$$which uses the mean \(1.031e+04\) and standard deviation 9418 to normalise \(PM_1\). The following are the coefficients:$$\begin{aligned}{} & {} p_1=1.047 (-1.171, 3.265), p_2=-2.242 (-6.111, 1.627), p_3=-3.727 (-8.011, 0.5571), p_4=4.431 (-5.339, 14.2),\\{} & {} p_5=2.226 (-0.678, 5.131), p_6=-2.441 (-6.864, 1.981), q_1=0.0071 (-1.409, 1.423), q_2=-0.6433 (-1.375, 0.08855),\\{} & {} q_3=0.3259 (-0.3247, 0.9764). \end{aligned}$$Table 16 HOF and \(PM_1 ({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f(PM_2) = \frac{(p_1 \times PM_2^4 + p_2 \times PM_2^3 + p_3 \times PM_2^2 + p_4\times PM_2 + p_5)}{(PM_2^4 + q_1\times PM_2^3 + q_2\times PM_2^2 + q_3\times PM_2 + q_4)} \end{aligned}$$where mean \(1.661e+11\) and standard deviation \(2.243e+11\) are used to normalise \(PM_2\). The following are the coefficients:$$\begin{aligned}{} & {} p_1=-1.272 (-18.21, 15.66), p_2=-3.797 (-24.81, 17.22), p_3=-2.972 (-44.24, 38.3), p_4=-0.05547 (-4.655, 4.544), p_5=0.4284 (-8.439, 9.295), q_1=1.881 (-2.644, 6.406), q_2=1.22 (-5.886, 8.325), q_3=0.2259 (-1.513, 1.965),\\{} & {} q_4=-0.03794 (-0.9377, 0.8618). \end{aligned}$$Table 17 HOF and \(PM_2({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f{(F)} = \frac{(p_1\times {F}^3 + p_2\times {F}^2 + p_3\times {F} + p_4)}{({F}^5 + q_1\times {F}^4 + q_2 \times {F}^3 + q_3 \times {F}^2 + q_4 \times {F} + q_5)} \end{aligned}$$where mean \(4.527e+04\) and standard deviation \(4.174e+04\) are used to normalise F. The following are the coefficients:$$\begin{aligned}{} & {} p_1=-0.2051 (-10.05, 9.644), p_2=5.458 (-9.238, 20.15), p_3=2.165 (-2.431, 6.76), p_4=-3.209 (-11.84, 5.418),\\{} & {} q_1=-1.489 (-4.125, 1.147), q_2=-1.875 (-5.889, 2.139), q_3=-0.3531 (-3.421, 2.715), q_4=-0.4707 (-1.859, 0.9177),\\{} & {} q_5=0.4529 (-0.7339, 1.64). \end{aligned}$$Table 18 HOF and \(F({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f{(J)} = \frac{(p_1\times J^2 + p_2\times J^1 + p_3)}{(J^5 + q_1 + J^4 + q_2 + J^3 + q_3 + J^2 + q_4 + J + q_5 )} \end{aligned}$$This normalizes J using the mean 362.6 and standard deviation 324.9. The following are the coefficients:$$\begin{aligned}{} & {} p_1=6.029 (3.131, 8.927), p_2=2.149 (1.342, 2.956), p_3=-3.589 (-5.724, -1.453), q_1=-1.576 (-2.234, -0.9181),\\{} & {} q_2=-1.939 (-2.275, -1.604), q_3=-0.3844 (-0.7764, 0.007551),\\{} & {} q_4=-0.504 (-0.898, -0.1099), q_5=0.5102 (0.2259, 0.7945). \end{aligned}$$Table 19 HOF and \(J({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(ReZG_1) =\frac{(p_1\times ReZG_1^3 + p_2\times ReZG_1^2 + p_3\times ReZG_1 + p_4)}{(ReZG_1^5 + q_1 + ReZG_1^4 + q_2 + ReZG_1^3 + q_3 + ReZG_1^2 + q_4 + ReZG_1 + q_5 )} \end{aligned}$$where the mean 749 and standard deviation 671.7 are used to normalise \(ReZG_1\).These are the coefficients:$$\begin{aligned}{} & {} p_1=-0.5288 (-8.969, 7.912), p_2=5.368 (-6.799, 17.53), p_3=2.334 (-1.905, 6.574), p_4=-3.284 (-10.63, 4.067),\\{} & {} q_1=-1.485 (-3.686, 0.7154), q_2=-1.736 (-5.255, 1.783), q_3=-0.2421 (-2.915, 2.431), q_4=-0.519 (-1.851, 0.8133),\\{} & {} q_5=0.4641 (-0.5527, 1.481). \end{aligned}$$Table 20 HOF and \(ReZG_1({\text{FeS}}_{2})\) by curve fitting.
$$\begin{aligned} f(ReZG_2) =\frac{(p_1\times ReZG_2 + p_2)}{(ReZG_2^5 + q_1\times ReZG_2^4 + q_2\times ReZG_2^3 + q_3\times ReZG_2^2 + q_4\times ReZG_2 + q_5)} \end{aligned}$$In which \(ReZG_2\) is normalized by mean 2376 and standard deviation 2167. The coefficients are:$$\begin{aligned}{} & {} p_1=-1.218 (-5.154, 2.718), p_2=-1.162 (-5.013, 2.69), q_1=1.863 (-0.3929, 4.119), q_2=1.082 (-2.779, 4.943),\\{} & {} q_3=-0.0212 (-1.318, 1.275), q_4=-0.08763 (-0.5496, 0.3743), q_5=0.1467 (-0.3322, 0.6257). \end{aligned}$$Table 21 HOF and \(ReZG_2({\text{FeS}}_{2})\) by curve fitting.$$\begin{aligned} f(ReZG_3) =\frac{(p_1 \times ReZG_3^3 + p_2 \times ReZG_3^2 + p_3 \times ReZG_3 + p_4)}{(ReZG_3^5 + q_1 \times ReZG_3^4 + q_2 \times ReZG_3^3 + q_3 \times ReZG_3^2 + q_4 \times ReZG_3 + q_5 )} \end{aligned}$$In which \(ReZG_3\) is normalized by mean \(1.573e+05 \) and standard deviation \(1.455e+05\). The coefficients are:$$\begin{aligned}{} & {} p_1=-0.1701 (-10.19, 9.848), p_2=5.468 (-9.523, 20.46), p_3=2.147 (-2.49, 6.783), p_4=-3.202 (-11.98, 5.572),\\{} & {} q_1=-1.489 (-4.176, 1.197), q_2=-1.89 (-5.962, 2.183), q_3=-0.3653 (-3.479, 2.749), q_4=-0.4657 (-1.86, 0.9283),\\{} & {} q_5=0.4517 (-0.7545, 1.658). \end{aligned}$$Table 22 HOF and \(ReZG_3({\text{FeS}}_{2})\) by curve fitting.Figure 6(a) HOF of \(R_{1}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\); (b) HOF of \(R_{-1}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 7(a) HOF of \({R_{\frac{1}{2}}({\text{FeS}}_{2})}\) via \(\textit{r}_{c}{f}\); (b) HOF of \(R_{-\frac{1}{2}}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 8(a) HOF of \({ABC}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\); (b) HOF of \({GA}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 9(a) HOF of \({M_1}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\) (b) HOF of \({M_2}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 10(a) HOF of \({HM}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\) (b) HOF of \({PM_1}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 11(a) HOF of \({PM_2}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\) (b) HOF of \({F}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 12(a) HOF of \({J}({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\); (b) HOF of \(ReZG_1({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).Figure 13(a) HOF of \(ReZG_2({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\) ; (b) HOF of \(ReZG_3({\text{FeS}}_{2})\) via \(\textit{r}_{c}{f}\).