Entropy analysis of nickel(II) porphyrins network via curve fitting techniques

Nickel(II) porphyrins typically adopt a square planar coordination geometry, with the nickel atom located at the center of the porphyrin ring and the coordinating atoms arranged in a square plane. The additional atoms or groups coordinated to the nickel atom in nickel(II) porphyrins are called ligands. Porphyrins have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells and be used as sensors for a variety of analytes. Nickel(II) porphyrins are relatively stable compounds, with high thermal and chemical stability. They can be stored in a solid state or in solution without significant degradation. In this study, we compute several connectivity indices, such as general Randi’c, hyper Zagreb, and redefined Zagreb indices, based on the degrees of vertices of the chemical graph of nickel porphyrins. Then, we compute the entropy and heat of formation NiP production, among other physical parameters. Using MATLAB, we fit curves between various indices and the thermodynamic properties parameters, notably the heat of formation and entropy, using various linearity- and non-linearity-based approaches. The method’s effectiveness is evaluated using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2$$\end{document}R2, the sum of squared errors, and root mean square error. We also provide visual representations of these indexes. These mathematical frameworks might offer a mechanism to investigate the thermodynamical characteristics of NiP’s chemical structure under various circumstances, which will help us understand the connection between system dimensions and these metrics.


Entropy analysis of nickel(II) porphyrins network via curve fitting techniques
Muhammad Talha Farooq 1,2 , Thiradet Jiarasuksakun 2,3 & Pawaton Kaemawichanurat 1,2* Nickel(II) porphyrins typically adopt a square planar coordination geometry, with the nickel atom located at the center of the porphyrin ring and the coordinating atoms arranged in a square plane.The additional atoms or groups coordinated to the nickel atom in nickel(II) porphyrins are called ligands.Porphyrins have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells and be used as sensors for a variety of analytes.Nickel(II) porphyrins are relatively stable compounds, with high thermal and chemical stability.They can be stored in a solid state or in solution without significant degradation.In this study, we compute several connectivity indices, such as general Randi'c, hyper Zagreb, and redefined Zagreb indices, based on the degrees of vertices of the chemical graph of nickel porphyrins.Then, we compute the entropy and heat of formation NiP production, among other physical parameters.Using MATLAB, we fit curves between various indices and the thermodynamic properties parameters, notably the heat of formation and entropy, using various linearity-and non-linearity-based approaches.The method's effectiveness is evaluated using R 2 , the sum of squared errors, and root mean square error.We also provide visual representations of these indexes.These mathematical frameworks might offer a mechanism to investigate the thermodynamical characteristics of NiP's chemical structure under various circumstances, which will help us understand the connection between system dimensions and these metrics.
Transition metal (TM) porphyrins are widely used in a variety of technological applications, including sensors, pigment applications, cancer therapy, synthetic photosynthesis, nonlinear optics, and nanomaterials, as a result of their special features 1 .Their value for catalysis and biological significance is directly tied to this interest.The coordination characteristics and conformational flexibility of porphyrins have been extensively used over the past ten years in the quest for potential porphyrin isomers that can provide enhanced functionality in particular technological applications 2 .The nitrogen-confused porphyrins (NCPs), a unique and promising class of porphyrins with enhanced capabilities for application as acid catalysts and anion/cation sensors, are one such significant class of porphyrins 3 .The chemical structure, physical characteristics, and coordination properties of these porphyrin isomers are significantly different from those of the parent porphyrins.Such structures are great candidates for use in photodynamic therapy because of their effective singlet-oxygen sensitization 4 .
The information content of complex networks 5 and graphs based on Shannon's entropy 6 work was first studied by researchers in the late 1990s.In discrete mathematics, computer science, information theory, statistics, chemistry, biology, and other domains, a large range of quantitative methods for studying complex networks have been developed 7,8 .For instance, graph entropy measurements have been extensively employed in the fields of mathematical chemistry, biology, and computer science to characterize the structure of graph-based systems 9,10 .To measure the structural complexity of graphs, the idea of graph entropy, created by Rashevsky 11 and Trucco, has been employed 12 .Chemical indices are valuable resources for researching various physico-chemical characteristics of molecules without having to perform several tests.Quantitative structure-activity relationships (QSAR) are used in the study of drugs to understand the chemical properties using mathematical calculations 13,14 .The entropy of a graph was first described as an information-theoretic property by Mowshowitz 15 .Here, the complexity is clear.As stated by Shannon, uncertainty and information are two sides of the coin: a reduction in uncertainty is the same as the reception of a certain amount of information.Distinguished researchers have developed numerous techniques for efficiently computing structural descriptors, aimed at optimizing computational efficiency.Among these techniques, the polynomial representation of structural descriptors has garnered significant attention and widespread acceptance in the scientific literature 16,17 .Entropy has emerged as a comprehensive and overarching concept across a wide spectrum of disciplines, spanning from logic and biology to physics and engineering.It serves as the link between the ideas of randomness and uncertainty, connecting them with physical processes that are viewed as channels for the transformation of information 18 .
The visible-light-induced photo redox catalyst nickel(II) tetraphenyl porphyrin (NiTPP) is given as a reliable, affordable, and effective catalyst.Recently, it was demonstrated that a library of Ni(II) ligand-to-ligand charge transfer complexes has useful features as photosensitizers, but their use in conventional photoredox catalysis is still unexplored 19,20 .Porphyrins have a wide range of applications in various fields, including: Porphyrins have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells.They are also used in photodynamic therapy (PDT), a treatment method that uses light-sensitive compounds to destroy cancer cells 21 .Porphyrins and their derivatives have been widely studied as catalysts in organic chemistry, with applications in hydrogenation, oxidation, and cycloaddition reactions.Porphyrins have been used as sensors for a variety of analytes, including metal ions, pH, and gases 22 .Porphyrins have been used for the detection and removal of heavy metal ions from contaminated water 23 .Porphyrins have been used as natural pigments in plant breeding and as a growth regulator in crops 24 .Porphyrins have applications in biotechnology such as biosensors, bioimaging, and biocatalysis 25 .
Several well-known topological indices, or the values that help characterize a structure's topological properties after it has been replicated, are used to calculate a structure's degree-based entropy.Zhdanov examined the chemical processes involving organic compounds using entropy values 26 .Chen 27 first defined the entropy of an edge-weighted graph in 2014.The information entropy is defined as: In Eq. ( 1) η e is the edge set, η v is the vertex set, and φ(ab) is the edge weight of the edge (ab) in η and η = NiP be a molecular graph of Nickel Porphyrin and logarithm to be presumed to be based 10.

Structure of nickel(II) porphyrins
Nickel(II) porphyrins are a class of coordination complexes composed of a central nickel atom coordinated to four nitrogen atoms of a porphyrin ring and two additional atoms or groups 35 .These compounds are known for their stability, strong absorption in the visible region of the electromagnetic spectrum, and potential applications in catalytic and biomedical fields.They have been studied as catalysts in a variety of organic reactions, such as hydrogenation, oxidation, and cycloaddition reactions.In addition, they have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells.The additional atoms or groups coordinated to the nickel atom in nickel(II) porphyrins are called ligands.The ligands can vary depending on the synthesis method and the specific compound.Common ligands include water, chloride, and various organic groups 36 .Nickel(II) porphyrins have strong absorption in the visible region of the electromagnetic spectrum, with the absorption maximum typically around 400 nm.This property is due to the porphyrin ring and is used in applications such as imaging and photodynamic therapy 37 .Nickel(II) porphyrins have a low-spin electron configuration and have no unpaired electrons, so they have no net magnetic moment.Porphyrins and similar tetrapyrrolic macrocycles are found abundantly in nature and serve vital roles across a diverse range of disciplines, spanning from medicine to materials science.These compounds, particularly their metal complexes known as metalloporphyrins, serve as essential active centers in numerous enzymes 38 .
Since Küsterover 39 initially postulated the porphyrin macrocyclic structure a century ago, study in the area has increased significantly, leading to a massive body of literature that is still growing quickly.To give you an idea, the "Handbook of Porphyrin Science" series, 40 which was started in 2010, currently consists of 44 volumes and 214 chapters.The use of X-ray crystallography (including synchrotron) and neutron crystallography to determine the crystal structures of porphyrins has greatly aided the development to date.Currently, the Cambridge Structural Database has far more than 4000 porphyrin crystal structures (CSD) 41 .The structure of the Ni-metallated version, which has an interlayer spacing of 3.347 and Ni that is coplanar with the macrocycle, is otherwise comparable to that of Porphyrins.Using nanoelectrodes, Yoon 42 assessed the electrical conductivity of two varieties of porphyrin wires.One type included 48 Ni(II) porphyrin moieties in directly meso-meso-connected Ni(II) porphyrin arrays.Because of the orthogonal arrangement of these arrays, consecutive porphyrins are aligned along the chain at right angles to one another.By using X-ray diffraction and a combination of single-crystal and solution resonance Raman studies, the structure of nickel(II) [Ni(P)] has been identified.Both resonance Raman spectroscopy and X-ray diffraction are approaches that are effective for examining porphyrin structure.

Methodology
Firstly, we find the degree of all types of vertices for the structure of nickel(II) porphyrins like we have three types of vertices: degree 2,3 and 4 and then we formulate general formulas for [m, n] dimensions and by utilizing these provided formulas in Table 1 we can compute vertices for any cell and by using same method we calculate the edge partition of nickel(II) porphyrins is shown in Table 3 and we have 4 types of edge partitions.The order and size of nickel(II) porphyrins for [m, n] is 37mn + 6m + 6n and 48mn + 6m + 6n respectively.Furthermore, degree-based topological indices that are mentioned in Table 2 are computed for nth cell of nickel(II) porphyrins and entropy for these calculated indices by using Table 3 and explain it with numerical and graphical representation.And after that, we built-in function in MATLAB is used to create models between the Heat of Formation and each information entropy because it provides the lowest RMSE value, which indicates the best match.The Numerical Integrity of fit for entropy versus indices of Ni(II) porphyrins is depicted in Table 8.The unit www.nature.com/scientificreports/structure of nickel(II) porphyrins (NiP) is shown in Fig. 1 and for more details about the structure of NiP(II) see the Figs. 2 and 3.

Computation of degree based indices and entropy of nickel(II) porphyrins [m,n] The Randić index and Randić entropy for Ni(II) porphyrins
Using Tables 2, 3 and Eq.(1) the Randic index 28 and corresponding entropy is:

The atom bond connectivity index and atom bond connectivity entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the Atom bond connectivity index 29,30 and corresponding entropy is: −1 9 −1 12

The geometric arithmetic index and geometric arithmetic entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the Geometric Arithmetic index 29,30 and corresponding entropy is:

The first Zagreb index and first Zagreb entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the first Zagreb index 29,30 and corresponding entropy is: The second Zagreb index and second Zagreb entropy for Ni(II) porphyrins By using Tables 2, 3 and Eq.(1) the second Zagreb index 29 and corresponding entropy is:

The forgotten index and forgotten entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the Forgotten index 32 and corresponding entropy is:

The augmented Zagreb index and augmented Zagreb entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the Augmented Zagreb index 33 and corresponding entropy is:

The first redefined Zagreb index and first redefined Zagreb entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the first redefined Zagreb index 34 and corresponding entropy is:

The second redefined Zagreb index and second redefined Zagreb entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the second redefined Zagreb index 34 and corresponding entropy is:

The third redefined Zagreb index and third redefined Zagreb entropy for Ni(II) porphyrins
By using Tables 2, 3 and Eq.(1) the third redefined Zagreb index 34 and corresponding entropy is:

Numerical and graphical representation of computed results
In this section, we represent the numerical and graphical representation of the computed results.In Table 4 we represent the numerical results and in Figs. 4, 5, 6, 7 we represent the graphical comparison of the Randić entropies for different values of (α = 1, −1, 1 2 and −1 2 ).Table 5 shows the numerical results and in Figs. 8, 9, 10 and 11 we represent the graphical comparison of E ABC , E GA , E M 1 and E M 2 entropies.The numerical results for E HM , E F and E AZI entropies are depicted in Table 6, and graphical comparisons are shown in Figs. 12, 13 and 14 and Table 7 shows the numerical results of redefined Zagreb entropies and graphical comparisons are shown in Figs.15,16 and 17.We have examined degree-based molecular descriptors for the nickel(II) porphyrins Network.The Randic index has a strong correlation with a variety of physicochemical characteristics of alkanes, including chromatographic retention times, surface area, vapor pressure, and boiling temperature variables in the Antonie equation.The atom-bond connectivity (ABC) index proves highly effective in calculating the strain energy of molecules through correlation.A good quantitative structural property relationship (QSPR) model is created when the temperature of alkane production is described by using the ABC index with a high correlation coefficient (r = 0.9970).Moreover, the geometric arithmetic index is a stronger correlation coefficient across a range of physicochemical parameters for octanes.Zagreb indices have been utilized to investigate complexity and hetero systems.These indices are also applied for constructing multilinear regression models and are instrumental in studies related to Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Activity Relationship (QSAR) 43,44 .The Forgotten index has demonstrated associations with numerous chemical attributes of molecules.The Augmented Zagreb index proves to be more effective in correlating with the measurement of strain energy [m,n] [1,1] [ the y-axes.These graphs reveal the differences between each entropy for these topological indices for a specific structure.The computational outcomes underscore that the estimates of degree-based indices are significantly influenced by the values of m and n, or in other words, by the molecular structure.[ [m,n]

Rational curve fitting between heat of formation and entropy of their corresponding indices
In this part, we explain the ideas of Information Entropy and Heat of Formation (Enthalpy) of nickel(II) porphyrins (NiP).The standard molar enthalpy (HoF) of Porphyrins is +629.8kjmole −1 .The following is a mathematical     [  formula to determine Heat of Formation (HOF) for various formula units:    Based on the characteristics of its chemical graph structure, this could offer us an effective technique to comprehend the molecular structure of Ni(II) porphyrins.The rational built-in function in MATLAB is used to create models between the Heat of Formation and each information entropy because it provides the lowest RMSE value, which indicates the best match.The Numerical Integrity of fit for entropy versus indices of Ni(II) porphyrins is depicted in Table 8.Enthalpy is a property or state function that resembles energy as a result, it has the same dimensions as energy and is measured in joules or ergs. (2)

Figure 8 .
Figure 8. Graphically representation of E ABC .

Figure 9 .
Figure 9. Graphically representation of E GA .

Table 1 .
Vertex partition of the Ni(II) porphyrins.

Table 2 .
Topological characteristics and the edge's (mn) weight are shown together.

Table 3 .
Edge partition of the Ni(II) porphyrins.

Table 5 .
Numerical comparison of E ABC , E GA , E M 1 and E M 2 .

Table 6 .
Numerical comparison of E HM , E F and E AZI .

Table 8 .
Goodness of fit for HoF versus entropy of indices for nickel(II) porphyrins.

Table 10 .
Rational curve fitting of HoF versus E R −1 .