Computation of Entropy Measures for Metal-Organic Frameworks

Entropy is a thermodynamic function used in chemistry to determine the disorder and irregularities of molecules in a specific system or process. It does this by calculating the possible configurations for each molecule. It is applicable to numerous issues in biology, inorganic and organic chemistry, and other relevant fields. Metal–organic frameworks (MOFs) are a family of molecules that have piqued the curiosity of scientists in recent years. They are extensively researched due to their prospective applications and the increasing amount of information about them. Scientists are constantly discovering novel MOFs, which results in an increasing number of representations every year. Furthermore, new applications for MOFs continue to arise, illustrating the materials’ adaptability. This article investigates the characterisation of the metal–organic framework of iron(III) tetra-p-tolyl porphyrin (FeTPyP) and CoBHT (CO) lattice. By constructing these structures with degree-based indices such as the K-Banhatti, redefined Zagreb, and the atom-bond sum connectivity indices, we also employ the information function to compute entropies.


Introduction
Molecular organic frameworks are compounds composed of a central metal ion or atom surrounded by one or more organic ligands [1]. These ligands are typically organic molecules with a functional group that can bind to the metal center through covalent or coordinate bonds. The resulting structure is a complex in which the metal ion or atom is coordinated to the ligands and surrounded by a coordination sphere [2]. Molecular organic frameworks have many applications [3], including catalysis [4], sensing [5], and molecular recognition [6]. For example, some metalloenzyme active sites are molecular organic frameworks, and the coordination of the metal ion or atom to the ligands plays a critical role in the enzyme's function. In addition to their practical applications, molecular organic frameworks are also studied for their fundamental chemical properties and as models for more complex systems. The structures of molecular organic frameworks can be determined using techniques such as X-ray crystallography, and their reactivity and stability can be studied through various chemical and spectroscopic methods [7]. Molecular organic frameworks have a wide range of applications due to their unique properties, such as catalytic HB 2 (T, s) = ∑ g 1 ∼ġ 2 s (wġ 1 ×wġ 2 ) 2 HB 2 (T) = ∑ g 1 ∼ġ 2 (wġ 1 × wġ 2 ) 2 (4) The concept of Redefined Zagreb indices was initiated by Ranjini in [38], and Shanmukha in [39] and defined as ReZG 1 (T, s) = ∑ g 1 ∼ġ 2 s wġ 1 +wġ 2 wġ 1 ×wġ 2 ReZG 2 (T, s) = ∑ g 1 ∼ġ 2 s wġ 1 ×wġ 2 wġ 1 +wġ 2 The third redefined Zagreb index was defined as s (wġ 1 ×wġ 2 )(wġ 1 +wġ 2 ) The notion of atom-bond connectivity index and sum connectivity index gathered by Ali et al., and initiated the new molecular descriptor named as the atom-bond sumconnectivity index in [40]: The idea of entropy was initiated by Shannon in 1948 [41]. The quantity of thermal energy per unit temperature in a system that is not accessible for meaningful work is measured by entropy [42,43]. The system's molecular disorder is also measured by Entropy [44,45]. In this article, we have computed entropies of metal organic frameworks of T(g, h) [46][47][48].

Entropy Measure of FeTPyP-Co T(g, h)
The FeTPyP-Co MOFs, also known as iron(III) tetra-p-tolyl porphyrin (FeTPyP) frameworks coordinated with cobalt (Co) ligands, are a type of molecular organic framework. The structure of FeTPyP-Co MOFs consist of a central iron(III)ion coordinated with four ptolylporphyrin (TPyP) ligands and one Co ligand. The TPyP ligands provide a tetradentate coordination, while the Co ligand provides a monodentate coordination. The properties of FeTPyP-Co MOFs exhibit catalytic activity for a variety of reactions, including oxidation reactions and cyclohexane oxidation. The Co ligand can modulate the redox properties of the iron center, enhancing its ability to oxidize substrates [50]. FeTPyP-Co MOFs have been studied for their magnetic properties, which are influenced by the coordination environment of the iron center. The TPyP ligands can induce antiferromagnetic coupling between the iron centers, while the Co ligand can modulate the magnitude of the coupling. FeTPyP-Co MOFs have also been investigated for their optical properties, which arise from the TPyP ligands. The TPyP ligands can absorb visible light and undergo photoinduced electron transfer, leading to the generation of reactive intermediates with potential applications in photocatalysis. Overall, FeTPyP-Co MOFs are a promising class of molecular organic frameworks with diverse applications in catalysis, electrocatalysis, magnetism, and optics. T(g, h) is a graph of FeTPyP-Co (TPyP ¼ Tetrakis pyridyl porphyrin) metal-organic frameworks, which embodies cells in rows and embodies cells in columns. The molecular graph of FeTPyP-Co is given in Figure 1. There are total 74gh vertices and 88gh − 2g − 2h + 1 edges. In this article, we tried to explain T(g, h), with a total atom count of 74gh; as described in Figure 1.  Table 1 represents the atom-bond partitions of T(g, h) derived from these results. Table 1. Atom-bond partition of FeTPyP-Co.

Types of Atom Bonds
The first K-Banhatti entropy measure of T(g, h) Table 1 and Equation (1) imply: After differentiating Equation (18), we obtain the first K-Banhatti index at s = 1.
The first K-Banhatti entropy measure of T(g, h) is obtained using Equation (19) and Table 1 in Equation (10): •

The second K-Banhatti entropy measure of T(g, h)
In view of Table 1 and Equation (2), we obtain After differentiating Equation (20) at s = 1, we obtain the second K-Banhatti index The second K-Banhatti entropy measure of T(g, h) is obtained in view of Equation (21), Table 1 and Equation (11): 12 . •

The first K-hyper Banhatti entropy measure of T(g, h)
The Equation (3) and Table 1 gives: After differentiating Equation (22) at s = 1, we obtain the first K-hyper Banhatti index: The first K-hyper Banhatti entropy measure of T(g, h) is obtained in view of Equation (23), Table 1, and Equation (13): •

The second K-hyper Banhatti entropy measure of T(g, h)
In view of Table 1 and Equation (4), we obtain: After differentiating Equation (24) at s = 1, we obtain the second K-hyper Banhatti index: The second K-hyper Banhatti entropy measure of T(g, h) is obtained in view of Equation (25), Table 1, and Equation (13): •

The first redefined Zagreb entropy measure of T(g, h)
Using Equation (5) and Table 1, we get: After differentiating Equation (26) at s = 1, we obtain The first redefined Zagreb entropy measure is obtained in view of Equation (27), Table 1, and Equation (14): •

The second redefined Zagreb entropy measure of T(g, h)
In view of Table 1 and Equation (6), we have: After differentiating Equation (28) at s = 1, we obtain The second redefined Zagreb entropy measure is obtained in view of Equation (29), Table 1, and Equation (15): . •

Comparison
In this section, comparison (numerical in Table 2 and graphical in Figure 2) of various computed K-Banhatti and the redefined Zagreb indices is presented.

Entropy Measure of CoBHT (CO) Lattice
The CoBHT (CO) lattice refers to a type of molecular organic framework in which cobalt (Co) is coordinated with 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (TCNQ) ligands and carbon monoxide (CO) ligands. The structure of the CoBHT (CO) lattice consists of a one-dimensional array of Co atoms coordinated with TCNQ and CO ligands. Each Co atom is coordinated with four TCNQ ligands and two CO ligands, forming an octahedral coordination geometry. The TCNQ ligands stack along the one-dimensional axis, forming a charge transfer complex with the Co atoms, and the properties of the CoBHT (CO) lattice exhibits interesting magnetic properties, including spin-crossover behavior and long-range magnetic ordering. The TCNQ ligands provide a highly anisotropic electronic structure, which can result in highly directional exchange interactions between the Co atoms. The CO ligands can modulate the magnetic properties of the Co atoms by influencing their coordination environment and electronic structure. The CoBHT (CO) lattice has potential applications in magnetic data storage, spintronics, and molecular electronics. Overall, the CoBHT (CO) lattice is a promising molecular organic framework with unique magnetic properties and potential applications in various fields.
The C(g, h), a graph of CoBHT (CO) lattice, denotes the unit cell in the column and g denotes the unit cell in a row. The structure of the molecular graph of CoBHT (CO) lattice is shown in Figure 3, where the portion in a square shows the unit structure of CoBHT (CO) lattice. The T(g, h) has 27gh vertices and 36gh − 2(g + h) edges. In Figure 3 two-dimensional 3 × 3 CoBHT(CO) lattice structure is shown. • The 1st K-Banhatti entropy measure of CoBHT (g,h) Let CoBHT (g,h) be a metal-organic framework. In view of Table 3 and Equation (1), we obtain After differentiating Equation (34) at s = 1, we obtain Table 3. Atom-bonds partition of CoBHT (g,h) .

Types of Atom Bonds
Cardinality of Atom bonds 2(g + h) The first K-Banhatti entropy measure of (C(g, h)) in view of Equations (10) and (35), Table 3: • The second K-Banhatti entropy measure of C(g, h) The Equation (1) and Table 3, gives After differentiating Equation (36) at s = 1, we have The second K-Banhatti entropy measure of C(g, h) is obtained in view of Equations (11) and (37), Table 3: •
After differentiating Equation (38) at s = 1, we get The first K-hyper Banhatti entropy measure of C(g, h) in view of Equations (12) and (39), Table 3: •
After differentiating Equation (40) at s = 1, we have The second K-hyper Banhatti entropy measure of C(g, h) is obtained in view of Equation (41) Table 3 and Equation (13): This gives •

The first redefined Zagreb entropy measure of C(g, h)
In view of Table 3 and Equation (5), we have After differentiating Equation (43) at s = 1, we obtain the first redefined Zagreb index The first redefined Zagreb entropy measure is obtained in view of Equation (44) Table 3 and Equation (14): •

Atom-bond sum connectivity entropy measure of C(g, h)
In view of Table 1 and Equation (8), the atom-bond sum connectivity polynomial is After differentiating Equation (49) at s = 1, we have ABS(C(g, h)) = √ 2(g + h) + 2 3 5 (g + h) The third redefined Zagreb entropy measure is obtained in view of Equation (49), Table 3 and Equation (17):

Comparison
In this section, we present a comparison (numerical in Table 4 and graphical in Figure 4) of various K-Banhatti and redefined Zagreb indices for C(g, h).

Conclusions
MOFs' allure stems from their distinct qualities, which can be predicted and modified. MOF synthesis and analysis employ a diverse set of current scientific methodologies and procedures. Because of the amazing structural diversity observed in MOFs, these methods allow scientists to predict and regulate the properties of synthesised materials. The ability to tailor the structure of MOFs enables the development of materials with specialised properties for certain applications. The amazing optical attributes of metallic nanoparticles have piqued the curiosity of researchers and scientists of this era. In this study, the CoBHT (CO) lattice and the iron(III) tetra-p-tolyl porphyrin (FeTPyP), two significant metal-organic frameworks, have been investigated and using the atom-bond partitioning strategy, the precise formulas of numerous significant valency-based topological indices have been determined. The CoBHT (CO) lattice has potential applications in magnetic data storage, spintronics, and molecular electronics. Overall, the CoBHT (CO) lattice is a promising molecular organic framework with unique magnetic properties and potential applications in various fields. In this study, we also looked at the distance-based entropies related to a novel information function and evaluated the association between degreebased topological indices and degree-based entropies in light of Shannon's entropy and Chen et al.'s entropy. This has been utilized to determine the complexity of molecules and molecular ensembles as well as their electrical structure, signal processing, physicochemical reactions, and complexity. The K-Banhatti entropy may be utilized in combination with thermodynamic entropy, chemical structure, energy, and mathematics to fill in gaps across various fields of study and build the foundation for new interdisciplinary research. This will open up new avenues for research in this field, as we plan to apply this concept to diverse metal organic frameworks in the future. Data Availability Statement: All data generated or analyzed during this study are included in this article.