The process of creating a curve, or mathematical function, that has the best fit to a set of data points, possibly subject to constraints, is known as a curve fitting. When fitting a curve, one of two methods can be used: either smoothing, which creates an approximation “smooth” function to suit the data, or interpolation, which fits the data exactly. A closely related subject is regression analysis, which concentrates primarily on statistical inference-related issues like the degree of uncertainty in a curve fitted to data seen with random errors. Fitted curves can be used to summarise the relationships between two or more variables, infer values for a function in the absence of data, and assist with data visualization. The use of a fitted curve beyond the range of the observed data is known as extrapolation, and it carries some degree of risk because the curve may represent the construction process just as much as the observed data.Gibbs energy is a thermodynamic potential that expresses the greatest work a thermodynamic system may accomplish at constant temperature and pressure. It is sometimes referred to as the Gibbs function, Gibbs free energy, or free enthalpy. The standard molar-free Gibb’s energy of carbon nitride is \(-55.5\) KJmol\(^{-1}\). The free Gibbs energy for various formula units can be found using mathematical formulas that combine thermodynamic principles, theoretical computations, and experimental evidence. The generic versions of these formulas are as follows:$$\begin{aligned} GE=\Delta _{f}G^\circ =\frac{Standard\, Molar \Delta _{f}G^\circ }{Avogadro’s\, Number}\times Formula\, Units \end{aligned}$$Where, \(Avogadro’s\, Number=6.022 \times 10^{23}mol^{-1}\). The relationship between structural features and thermodynamic properties of materials can be better understood through curve fitting between indices and Gibbs energy using statistical measurements like R-squared \((R^{2})\), adjusted R-squared \((R^{2}_adj)\), and root mean square error (RMSE). Within this framework, scientists employ mathematical models, most commonly polynomial regression, to determine relationships between structural indices, like K-Banhatti indices, and the system’s free Gibbs energy. The percentage of the variance in the Gibbs energy that the regression model explains is expressed as the coefficient of determination, or R-squared. There is a substantial correlation between the response variable (Gibbs energy) and the predictors (indices) when the R-squared value is high. Adjusted R-squared, which penalizes model complexity to account for the number of predictors, is crucial because R-squared alone may not always provide a clear picture, particularly when dealing with many predictors. Additionally, by quantifying the average variation between the predicted and observed Gibbs energy values, RMSE and modified RMSE function as measurements of the accuracy of the model and offer insights into its goodness of fit. Through the examination of these statistical measures, scientists can evaluate the validity of the curve-fitting model and acquire a more profound understanding of the structural elements affecting the thermodynamic stability of materials. This information can then be utilized to direct the creation of new materials with customized characteristics for particular uses. Goodness values of fit between indices and heat of formation are shown in Table 4 and parametric values are shown in Tables 5, 6, 7 and 8. The curve fitting between indices and Gibb’s energy is shown in Figs. 12, 13, 14, 15, 16, 17, 18, 19 and 20.Table 4 Goodness values of fit between indices and Gibb’s energy.The goodness-of-fit values for semiconducting carbon nitride networks between different indices and Gibbs energy are shown in Table 4. For \(B_2(C_3{N_4})\) and \(HB_2(C_3{N_4})\), the SSE values vary from 0.00221 to 3.004. A model that fits the data better is indicated by lower SSE values. Particularly low SSE values are shown by indicators like \(B_2(C_3{N_4})\) and \(\eta R^{2}(C_3{N_4})\), indicating that these models closely match the observed data. A perfect match is shown by the values of \(R^{2}\) and adjusted \(R^{2}\) being consistently equal to 1 across all indices. This implies that the models account for all of the variation in Gibbs energy, but it may also be a sign of overfitting, especially for more sophisticated models.$$\begin{aligned} GE(B_1) = \frac{p_1({B_1})^2 + p_2({B_1}) + p_3}{{B_1} + q_1} \end{aligned}$$The coefficients are \(p_1 = -0.01739\) with \(C_q=(-0.01747, -0.01731)\) are \(p_2 = 17.48\) with \(C_q=(-4.931, 39.89)\) are \(p_3 = 1787\) with \(C_q=(435.8, 3137)\) are \(q_1 = -1111\) with \(C_q=(-2361, 138.9)\) and Mean: 8058.0 Standard Deviation: 7239.935911318553$$\begin{aligned} GE(B_2) = \frac{p_1{(B_2)}^2 + p_2{(B_2)} + p3}{{B_2} + q_1} \end{aligned}$$The coefficients are \(p_1 =-0.01159\) with \(C_q=(-0.01162, -0.01156)\) are \(p_2 =-60.81\) with \(C_q=(-81.1, -40.53)\) are \(p3 =-3877\) with \(C_q=(-6249, -1504)\) are \(q_1 =4612\) with \(C_q=(2945, 6279)\) and Mean: 11854.0 Standard Deviation: 10730.255293638947.$$\begin{aligned} GE(HB_1) = \frac{p_1{(HB_1)}^2 + p_2{(HB_1)} + p3 }{ {(HB_1)} + q_1} \end{aligned}$$The coefficients are \(p_1 =-0.2959\) with \(C_q=(-0.2966, -0.2952)\) are \(p_2 =-58.59\) with \(C_q=(-78.49, -38.68)\) are \(p_3 =196.4\) with \(C_q=(85.83, 307.1)\) are \(q_1 =227\) with \(C_q=(156.2, 297.8)\) and Mean: 48300.6 Standard Deviation: 43718.68089958799.$$\begin{aligned} GE(HB_2) = \frac{p_1{(HB_2)} + p_2 }{ {(HB_2)} + q_1} \end{aligned}$$The coefficients are \(p_1 = -1.541e+04\) with \(C_q=(-2.222e+04, -8594)\) are \(p_2 = -9.255e+06\) with \(C_q=(-1.632e+07, -2.193e+06)\) are \(q_1 = 5.225e+06\) with \(C_q=(2.861e+06, 7.589e+06)\) and Mean: 112250.0 Standard Deviation: 102398.72424118487.$$\begin{aligned} GE(H_b) = \frac{p_1{(H_b)} + p_2 }{ {(H_b)} + q_1} \end{aligned}$$The coefficients are \(p_1 = -1.092e+04\) with \(C_q=(-1.582e+04, -6026)\) are \(p_2 = -2.124e+07\) with \(C_q=(-3.782e+07, -4.665e+06)\) are \(q_1 = 8.594e+06\) with \(C_q=(4.615e+06, 1.257e+07)\) and Mean: 497.0619 Standard Deviation: 432.00395580093425.$$\begin{aligned} GE(\eta R_1) = \frac{p_1{(\eta R_1)}^2 + p_2{(\eta R_1)} + p_3 }{ {(\eta R_1)} + q_1} \end{aligned}$$The coefficients are \(p_1 = -0.02689\) with \(C_q=(-0.02701, -0.02677)\) are \(p_2 = -56.53\) with \(C_q=(-75.38, -37.68)\) are \(p_3 = 3707\) with \(C_q=(1496, 5919)\) are \(q_1 = 2768\) with \(C_q=(1990, 3545)\) and Mean: 5666.0 Standard Deviation: 4830.135401828814.$$\begin{aligned} GE(\eta R_2) = \frac{p_1{(\eta R_2)}^2 + p_2{(\eta R_2)} + p_3 }{ {(\eta R_2)} + q_1} \end{aligned}$$The coefficients are \(p_1 = -0.06076\) with \(C_q=(-0.06145, -0.06006)\) are \(p_2 = -49.94\) with \(C_q=(-66.95, -32.93)\) are \(p_3 = 5222\) with \(C_q=(2110, 8334)\) are \(q_1 = 1976\) with \(C_q=(1583, 2369)\) and Mean: 2941.0 Standard Deviation: 2314.0567552820885.$$\begin{aligned} GE(\eta R_3) = \frac{p_1{(\eta R_3)}^2 + p_2{(\eta R_3)} + p_3 }{ {(\eta R_3)} + q_1} \end{aligned}$$The coefficients are \(p_1 = -0.3023\) with \(C_q=(-0.3029, -0.3017)\) are \(p_2 = -57\) with \(C_q=(-76.44, -37.55)\) are \(p_3 = 279.3\) with \(C_q=(151.2, 407.4)\) are \(q_1 = 214.6\) with \(C_q=(147.4, 281.8)\) and Mean: 486.0 Standard Deviation: 421.89414154105845.$$\begin{aligned} GE(R_1) = \frac{p_1{(R_1)}^4 + p_2{(R_1)}^3 + p_3{(R_1)}^2 + p4{(R_1)} + p5}{{(R_1)}^2 + q_1{(R_1)} + q_2} \end{aligned}$$The coefficients are \(p1 = -3613\) with \(C_q=(-1.456e+08, 1.456e+08)\) are \(p_2 = 3.961e+04\) with \(C_q=(-1.598e+09, 1.599e+09)\) are \(p_3 = -1.78e+05\) with \(C_q=(-7.2e+09, 7.199e+09)\) are \(p_4 = 3.363e+05\) with \(C_q=(-1.365e+10, 1.365e+10)\) are \(p_5 = -2.72e+05\) with \(C_q=(-1.109e+10, 1.109e+10)\) are \(q_1 = -4606\) with \(C_q=(-1.845e+08, 1.845e+08)\) are \(q_2 = 5.046e+04\) with \(C_q=(-2.026e+09, 2.026e+09)\) and Mean: 6.77414 Standard Deviation: 1.4760628902590838.Figure 12Curve fitting of GE and \(B_1\) of \(({C_3}{N_4})\).Figure 13Curve fitting of GE and \(B_2\) of \(({C_3}{N_4})\).Figure 14Curve fitting of GE and \(HB_1\) of \(({C_3}{N_4})\).Figure 15Curve fitting of GE and \(HB_2\) of \(({C_3}{N_4})\).Figure 16Curve fitting of GE and Hb of \(({C_3}{N_4})\).Figure 17Curve fitting of GE and \(\eta R_1\) of \(({C_3}{N_4})\).Figure 18Curve fitting of GE and \(\eta R_2\) of \(({C_3}{N_4})\).Figure 19Curve fitting of GE and \(\eta R_3\) of \(({C_3}{N_4})\).Table 5 Parametric fit values between Gibb’s energy and index values.Table 6 Parametric fit values between Gibb’s energy and index values.Table 7 Parametric fit values between Gibb’s energy and index values.Table 8 Parametric fit values between Gibb’s energy and index values.Figure 20Curve fitting of GE and \(R_1\) of \(({C_3}{N_4})\).
