Graph derivative indices interpretation from the quantum mechanics perspective

This contribution examines the interpretation of the Graph Derivative Indices (GDIs) from the Quantum Mechanics (QM) perspective and its relation with the Hückel molecular orbital (HMO) method. The different elements used for calculating Graph Discrete Derivatives over atom-pairs are related to the QM integrals appearing in the Hückel Determinant. The relation between the Coulomb and Resonance Integrals was estimated by the topological way and quantitative values for resonance energies were calculated. Some GDI calculations were performed to the topological interpretation of the aromaticity, expressing the frequencies as probabilities. Starting from topological interpretation for aromaticity, GDI calculations were performed. There is the possibility of expressing the frequencies as probabilities. It allows explaining the atypical formation of cyclobutadiene from the entropic point of view. Considering this hypothesis, experimental resonance energies for 14 molecules were correlated with GDI-topological resonance energies by the same structures. Taking into consideration the regularity and coherence stablished in experiments performed with the GDIs, it is possible to assure that GDIs have interpretations in QM terms.


Introduction
Graph theory (GT) can be traced back from Euler time who was asked to find a nice path across the seven Konigsberg bridges [1]. Since that time, GT has greatly evolved. GT has proven to be a useful tool with a wide variety of applications in the real world. Among them, it has been applied to Chemistry. GT is a subject that deals with molecular structures and is thus a valuable mathematical tool for efficient design and operation of large-scale systems, including chemical processes. The Extended author information available on the last page of the article concepts of the graph and molecular graph have been widely used in mathematical chemistry [2][3][4].
Graph theoretical approach appears to be an important alternative to computeraided molecular design methods and other specific questions from cheminformatics and bioinformatics. Molecular process descriptions require mathematical codifications for chemical structures. Some more attractive hypotheses to quantitatively codify the chemical structures are defined as Molecular Descriptor (MD) [5].
MDs nature depends on the procedure followed for its corresponding definitions and based on the kind of representation used in its formulation. They can be grouped from 0 to 4D [6]. These viable numbers of theoretical sources related to chemical information descriptors will be useful to a better understanding of the relations between the molecular structure and the experimental evidence, allowing MDs to play a significant role in future scientific development [5]. Grouping the different MDs over specific definitions makes organized studies functional. However, many MDs can be, equally, located in several classification groups.
The versatility of several ways to mathematically describe the chemical structures makes possible that MDs can be written in terms of well-known scientific concepts. In case of MDs, those which quantify structural information contained in the bidimensional (2D) structural representation are Topological Indices (TIs). At the same time, they can be classified into topo-structural (collecting only adjacency and/or length information) or topo-chemical (also include physical-chemical features of atoms and/or fragmentations) [5,7,8]. Using different algorithms, TIs quantify the size, ramifications, multiplex bonds, symmetry, heteroatoms presence, and others. These characteristics can be directly related to the molecular electronic features. Therefore, they have been related to other MDs group named Quantum-Chemical MDs, which describe the electronic features of the molecules based on the use of the molecular wave function. There are several varieties of problems on Quantum Chemistry, in which the molecular Hamiltonian can be described by topological matrix way [9].
Despite of the success of ab initio calculations some elemental methods such as Hückel molecular orbitals (HMO) still are used. Actually, the use of the Hückel approximation is justified because through it is possible to illustrate some quantum fundamental concepts at the price of little calculation [10,11].
In general way, Hückel Determinant (H) can be expressed as [9,12]: where, n and m are the empirical constants of Hückel molecular orbital theory, A k (G) is the adjacency matrix of the Hückel graph constructed for the π-electron network of conjugated hydrocarbons, and I is the identical matrix. In this manner, characteristic polynomials entered in the field of GT. It can be proved there is a coincidence between the own topological vector matrix and the Hückel molecular orbitals (HMO), which is derived from the fact that H and A commute ([H, A] = 0), reason why HMOs are also named topological orbitals [9,10]. Due to its very drastic approximations, Hückel theory results offer a qualitative overview. However, simplicity of Hückel model is an excellent tool to illustrate some Quantum definitions. (1) The popularity of Hückel method came from the fact that the Hamiltonian matrix of the HMO theory is a simple linear function of the adjacency matrix of the corresponding molecular graph. Thus, each p-electron energy level is a linear function of the corresponding zero of the characteristic polynomial of molecular graph [9][10][11].
Recently, new families of TIs have been defined and applied on modeling a great variety of chemical-physics and biological properties, with diverse purposes [13][14][15][16][17]. Specifically, the Graph Derivative Indices (GDIs) has been applied, with great accomplishment, to theoretical description of different chemical-physics and pharmaceutical process [18][19][20][21][22]. The present authors have reported a paper with a novel approach that shown some GDI calculations as a quantitative expression of reactivity of atoms and molecules in organic compounds [17]. Following this original idea, it was demonstrated a lineal relation between the GDI-Local vertices invariant (GDI-LOVI) and the chemical shifts (chemical shifts in nuclear magnetic resonance) of respective atoms [17]. The chemical shifts are an expression of electronic environment of each atom into the molecular structure [10,23,24], therefore, the GDI-Total Indices were correlated with global molecular electronic properties, like resonance energy [17].
The main aim of this contribution is to achieve an understanding of parameters and algorithm utilized by GDIs for codification the chemical structures. In the simplest case, the GDI calculation coincides with the semi-quantitative results obtained from HMO, when both are applied to the same molecular systems. When some traditional quantum chemistry questions are explained with GDI method, it gets to: (i) understand the topological parameters from GDI in Quantum Mechanical terms, (ii) explain in quantitative way, the thermodynamic stability of conjugated molecules (having into account energetic and entropic factors), and (iii) propose a new quantitative and more general way of approaching the concept of aromaticity, a key concept in chemistry.

First close approximation between GDI and quantum mechanics
The Hückel matrix can be considered similar to adjacency matrix (A 1 ) of first order with some modifications on its main diagonal. As any other adjacency matrix, it can be written in terms of an incidence matrix Q: Being, Q T transposed matrix of Q and the order of Q determine the k order of Adjacency matrix (A k ). The product Q T Q results in a squared and symmetric matrix, which is named Relations Frequency Matrix (F) [18,25]. Previously, obtained results had used the n-dimensional F matrix as base of organizing the information of each individual atom, as well as the n-uples formed by them (sub-structures) [19,26]. Only first order matrices will be used below. Earlier analysis show that Frequency matrix (F) is similar to the Hückel secular determinant (H), in which each column and row can be identified with each p orbitals. Defining the Eq. 2 in terms of the matrix H and F, the next mathematical relation can be denoted as: being, fi each atom's own participating frequencies, and ε allowed values for the energies corresponding to the different molecular orbitals. Analyzing the expression 3, it can verify the combination of Topological and Quantum Mechanic information to understand the conjugated molecule's structure into the same procedure. This matrix relation also suggests a relation among the elements from matrices, which were created from different theories and procedures, destined to quantify the chemical structure, but all with high similarity degrees.

Getting topological flexibility of strong approximations in HMO method, using the GDIs methodologies
The 1,3-butadiene molecule is a good example for the introduction of this methodology because it shows how the information about electronic distribution can be evaluated, recognizing and using topological information codified by GDIs.
The corresponding H and F matrixes (quantifying only atomic frequencies of first order molecular fragments) for the molecule are: There is a similarity between both matrices, in general because it is used a frequency matrix of first order. If a term-by-term comparison is established, some coincidences in the algebraic structure can be deduced, such as: (i) Only possess values the contiguous atoms, in distance of one step. In Hückel model, only the overlap between contiguous p-orbitals, are taking into account. The interactions among p-electrons far by two steps, or more, are suppressed. On the other side, the F matrix of order one is only the simplest option that can be used to organize the intensity of the participation of atoms on the molecular [20,27]. In general way, the GDIs consider each atom's simultaneous connection with the rest, allowing to have a complete information about the molecular structure and each one of its particularities. (ii) The resonance and Coulomb integrals ( i and ij ) are not all necessarily the same, even though all the atoms of the same chemical element (carbon atoms in this case). Frequency matrix recognizes equal intensities acting simultaneously for all connected atom pairs, which is a coincidence with the Hückel approximations; although, same direct electronic environment can be defined for the four carbon atoms in equivalent way to matrix of order one. On GDIs F matrix, 2 and 3 carbon atoms possess a higher participation intensity value than 1 and 4 atoms, highlighting these atoms (2 and 3) are surrounded by a superior electronic density than the external atoms. A mathematical way to relate the parameters from both theories (Quantum Mechanic Integrals and topological inclusion frequencies or intensities) has been described until now.
Standardization of the F matrix can be achieved by converting each term to a corresponding probability value, which does not affect neither its structure nor its information. By definition, the elements f i on main diagonal of F matrix, are defined as own frequencies.
Each f i quantify the number of times that element i is included into a given true event [20,25,27]. If the event is the molecular formation based on structural fragments organized by specific criteria, each f i element quantifies the number of times the i atom participates or appears into the molecular structure fragmentation set. Analogously, in bi-dimensional F matrixes, f ij elements represent the inclusion frequency of atoms-pair ij ( f ijk , f ijkl … f ijkl…n frequencies in 3D, 4D…nD matrices, correspond with the quantification of the topological inclusion of 3, 4…n atoms, simultaneously) [19,25,26]. Bearing in mind the total frequency as equivalent of the trace of F matrix then, the multiplication of each element into the F per 1/f T , it becomes F in a probability matrix Fp. The resulting matrix for 1,3-butadiene, would be the next one: Each main diagonal element can be related with the topological probability of i electron belong at C i atom. The external elements out of the main diagonal express the probability that possess any electron to describe an orbital around the atoms ij. At this point, there are theoretical (qualitative and quantitative) tools to find a direct relation between the QM terms in HMO and topological terms in the calculation of GDIs. The relations founded and previously mentioned are the basis for a novel interpretation by GDIs.
Based on Eq. 3 and substituting the frequencies at the corresponding probabilities, the result is: Developing the Eq. 4 for the molecule of 1,3-butadiene: The probabilities value p ij for atom-pairs connected by two steps, or more, are zero.
The comparative analysis about the equality among the previous matrices suggests a similitude in their terms: The c a and c b constants can be denoted as proportionality constants. It can be assumed that these constants take the same value for all atoms and adjacent atompairs from molecular system. Besides, values equal to one are assigned for previous constants. In fact, the relations 5 and 6 will allow making a full description of the conjugated molecules with topological information. Codifying different conjugated systems, it is possible to establish a quantitative explanation about stability, reactivity, aromaticity, etc., by topological approaches. Noticeable, Coulomb and Resonance integrals should not necessarily assume the same value by mathematical simplicity, as it is presumed on HMO method. When molecular description is made together with GDI method, a more reasonable way to evaluate these integrals is developed by quantifying the topological connection/perturbation of each atom (or of each electron if there is only considering the π system).

Electronic distribution in molecular graph discrete derivative terms
The Discrete Derivative is the main equation in the GDI algorithm of the mathematical codification of chemical structures. The graphs derivation theory was introduced by V. A. Gorbátov in his book Discrete Mathematics Fundamental [25], and then it was extended to mathematical codification of chemical structures in previous manuscripts [18][19][20]. The general expression to find the discrete derivative over atom-pairs in a molecular system is shown at the Eq. 7: where, G∕ E derivative [for duplex or a pair of vertices (v i , v j ) in this case] of a molecular graph (G) regarding to an event (E). If the standardization (multiplication by inverse simultaneous frequency 1∕f ij ) in Eq. 7 is changed by multiplication for inverse total frequency 1∕f T , then, Eq. 7 is become in Eq. 8.
Set theory, Information theory and discrete derivative itself show how the coefficient located in front of simultaneous probabilities (or frequencies) appears as result of avoiding the unnecessary repetition of information [13]. In other terms, it is a measure of the number of times that it is necessary to delete the overlapping degree among orbitals to obtain the separated orbitals into adjacent atoms.
The overlapping degree of atomic orbitals determinates the shape of molecular orbital and its final energy. This coefficient emerged from the group theory and can be directly related with Hückel solution ( H ) for any conjugated π-system, which is a topological solution [9].
Starting from all the connected derivative-pairs, which at the same time compose the set of different molecular orbitals; Eq. 8 can be generalized as the Minkowsky Norm 1(N 1 ): Taking into account the Eqs. 5 and 6, then the Eq. 9 can be written as: A careful analysis of the Eq. 10 shows its analogy with the expression proposed by HMO method to qualitatively express the energy for each molecular orbital, like function of Coulomb and Resonance integrals (denoted α and β respectively). The main difference is that HMO method does not develop an addition of all the corresponding QM integrals due to this method considers all them with the same value.
This non-real approximation is solved to include the influence of all the factors involved in π-system formation, whit a minimum of approximations. These can be expressed by means of Molecular Graph Discrete Derivative, which is the essence of the mathematical codification by the GDIs of chemical structures and can be expressed in terms of energy. In the specific case of unsaturated molecules with and without electronic delocalization of π electron systems, discrete derivative represents the energy of the formed molecular orbitals. This energy can be directly related to the electronic distribution proposed when Hückel matrix is resolved.
Applying the Eq. 10 and relating the derivatives G E with each orbital energies MOi , in a way that G E = C ⋅ MOi , being C a proportionality constant, which is function of the c a and c b constants.
For the 1,3-butadiene molecule: For each H value, there is an energy value for each orbital. The electronic distribution found with the values calculated of each molecular orbital energy from Eq. 10b, is showed in Fig. 1.

Linearity between resonance energy and GDIs
According to the proposed method, the resonance energy of a set of 14 conjugated molecules, to which the experimental resonance energy has been reported, was estimated. A direct correlation between their experimental resonance energy [28,29] (see Table 1) and the values calculated of their topological resonance energies, referred as discrete derivatives of the molecular graph, was observed. This implicates they are able of recognizing electronic densities and their delocalization capacities. The Statistical 8.0 software [30] was used in this study, and there was developed a linear regression analysis. The mathematical model that relates both resonance energies is: This model shows a good correlation with the experimental results, which confirms that there is a lineal relation between MDs and experimental resonance energies of this molecule set. The Eq. 11 explains more than 98% of the experimental property variation. In Table 1  In Table 1, system energy E (GDI) was calculated as a lineal combination of orbital energy values. The orbital energy values were weighted with corresponding number of electrons (n) in each orbital and were obtained by Eq. 10. This theoretical experiment suggests a possible interpretation of GDI as a measure of electronic environment energy, when it is applied for codification molecular graphs.

A topological understanding of aromaticity
Some conjugated cyclic systems could have properties that are quite different from those open-chain polyenes. Aromaticity is a special electronic property of some cyclic organic molecules with alternation among single and double bonds. In aromatic molecules, -electrons are delocalized around the ring, increasing the molecule stability. These kind of molecules cannot be represented by one structure, but rather a resonance hybrid of different structures, such as with the two resonance structures of benzene [28,31].
Benzene is the smallest of all the aromatic hydrocarbons participating in this study (Table 1). It supposes that resonance energy value exhibited by benzene could be considered as critical value for showing aromatic properties in a molecule. However, as HMO consequence application, 2 delocalized electrons also could confer aromaticity to ring molecules. In the way to find an explanation to this apparent contradiction, it must be resolved the calculations proposed by the theory of GDI energetic interpretation applied for an aromatic system with a minor number of delocalized π-electrons.
An analysis of Hückel numbers allows inferring always, with less than 6 π-electrons, that a cyclic system can only be aromatic if it has 2 π-electrons. For instance, the topological resonance energy for 2 π-electrons system can be located into a cycle of 3 carbon atoms. The calculation was made over the cation, the radical and the anion and the outcomes are shown in Fig. 3 and Table 2.
A detailed analysis of Table 2 shows only the cyclopropenyl cation possesses a topological resonance energy value that contributes with the formation of π-delocalized system. This one has a desirable coincidence with the GDI resonance energy found for benzene, which was previously assigned as the critical energy value necessary to attribute aromaticity into a structure. Therefore, the results suggest cyclopropenyl cation is aromatic, which can be experimentally corroborated due to the stable salts it is able to form [32]. Radical and anionic systems increase their energy during their formation. For this reason, they are catalogued as not stable. Until this moment, synthetizing stable salts from this anion has not been possible so far [33] due to the structure of cyclopropenyl anion is considered as an extremely reactive bi-radical [34].
As consequence of this study, it can be summarized that one aromatic system, besides of having a (4n + 2π) number of delocalized electrons; since GDI criteria it should: (i) the Resonance energy associated with any -system formation should be superior to one unit of GDI topological resonance energy (ii) the aromatic features of each molecule are more evident if their resonance energy obtained by GDI are integer value numbers. It does not mean the molecules with not-entire topological resonance energy values would not present aromatic features, but just that these features will be less evident and they always should fulfill the i and ii aspects. It is also well known that many systems possess a Hückel number of delocalized electrons and they do not show aromatic features because of the loss of planarity, which does not allow the accomplishment of the i item. The above-mentioned statements are in correspondence with the experimental results but from the GDI topological approach. Table 2 Resonance energy, Kekulé energy and GDI-energy associated with delocalized cyclopropenyl systems Name Structure

The interesting features of cyclobutadiene since GDIs point of view
Cyclobutadiene is a particular molecule with a cyclic structure composed by four carbon atoms with a sp 2 hybridation state, which possesses a π electron system not energetically favored by delocalization. If the electron delocalization into π molecular orbitals does not energetically support the structure's formation, in such case: is this molecule stable? How it could be explained? The most applied theory to make clear the energetic structure of π-system in cyclobutadiene is the HMO method. Resolving the corresponding determinant, following values: 1 H = 2, 2 H = 0, 3 H = 0 and 4 H = −2 were obtained. The consequent electronic distribution (Fig. 4) using these values reveals the formation of four molecular orbitals. This electronic distribution suggests the formed molecule should have high reactivity due to the presence of a not paired electron-pair, becoming the molecule in a double radical. This outcome is perfectly corroborated because the usual polymerization of cyclobutadiene molecules is known. Stabilization by resonance has some contradictions; due to that the HMO method as well as GDIs predict zero value for resonance energy. This fact would implicate that the molecule does not have electronic delocalization, coinciding with its Kekulé structure. However, this affirmation is not corroborated by the own electronic distribution (Fig. 4) resulting from HMO method.
The fact of evaluating GDIs in terms of probabilities not only allows finding the system energies, but also its corresponding Shannon´s entropy values using Eq. 12. Many previous researches have proven the equivalence between Shannon and Boltzmann entropies [35][36][37], which is very useful to understand how the entropic variation makes influence during the molecule formation.  Table 3 GDI-resonance energy, Kekulé energy and GDI-energy associated with cyclobutadiene molecule Name Structure Another studies have proven that Shannon entropy and others entropies derived on the information theory can be used as molecular indices [35,38]. Therefore, it is possible to evaluate the entropy variation (variation the information content) during the formation of π-system into cyclobutadiene molecule (the same evaluation can be made for any other molecule applying this approach) ( Table 3).
Considering that cyclobutadiene structure possesses two alternant double bonds, but in rigid positions without electronic delocalization (Kekulé structure), each electron will be localized around the domain of two carbon atoms. Therefore, the resonance integrals play an important role, which have been related to the simultaneous probability (Eq. 6). Following the proposed method and applying the GDI algorithm, the π system with rigid hypothetical bonds can be described using matrices shown at Fig. 5.
There is only π connection on this first case between the atoms 1-4 and 2-3. With the previous information, entropy can be calculated taking as calculation base simultaneous probabilities, which indicate π connection in Kekulé-structure for cyclobutadiene. The calculation of Shannon entropy (the information contained in the rigid double bonds) for cyclobutadiene Kekulé-structure is developed by Eq. 13: On the expression 13, the p k ij represent that the simultaneous probabilities of i and j electrons would be delocalized in a π connection among two adjacent atoms. This expression is known as union entropy [38,39]. For π-system of structure in Fig. 5, the probabilities different of zero are: p k 14 = p k 23 = 1∕4. Assigning the value 1 to k proportionality constant, substituting each probability by its value and resolving Eq. 13 for both connections, the result is S k = 0.6931 . This value represents the global information content (π entropy) that four p electrons offer when they form two independent π-bonds.
Supposing a possible π electron delocalization in cyclobutadiene structure, new calculation of Shannon entropy is carried out. The corresponding matrices are shown as follow: Fig. 5 A π connections in cyclobutadiene Kekulé-structure. B Frequency matrix. C Probabilities matrix Then, entropy corresponding to the four p electrons delocalization into the π system formed by four atoms can be evaluated using simultaneous probabilities in the same way to the previous procedure The value of π-topological entropy is different for each previous description used to represent the cyclobutadiene. Knowing this difference value allows having an idea of the measure that the information production (entropy) support (or not) the molecule formation.
Equation 15 is the expression of a simple thermodynamic analysis taking as initial state the structure with rigid double bonds, and as final state the structure with possible π electronic delocalization in the whole molecule. The increase of information production (entropy) indicates a rising stabilization of the system caused by this phenomenon.
Equation 16 can be used to evaluate the free energy variation associated to the electronic delocalization process since a topological optic.
This equation can be used to evaluate the spontaneity degree in the formation of any π-conjugated system, taking as base only molecule topological information in the way it has been described until this moment. For the specific case of cyclobutadiene, with value zero for the resonance energy, the expression 1 is transformed: suggest that entropic increase gives stability to possible delocalization by -electrons, in available range of temperatures. In fact, despite of the high torsional tension into the ring, the molecule can be conceived. Equally, the molecule is highly reactive due to its electronic configuration as it was previously described. These calculations have illustrated the use and interpretation of discrete derivatives in terms of energy and the use of inclusion frequencies (GDIs calculation element) to evaluate entropies.

Graph theory and molecular orbitals: on topological resonance energy
In the past, highly successful computational and theoretical tools have been developed for the analysis of molecules, among which the valuable contributions made by I. Gutman stand out [40][41][42][43][44][45]. Both Incremental Derivative Graphs (GDIs) and indices conceived by I. Gutman offer valuable instruments for the mathematical description and interpretation of chemical structures. While GDIs focus on theoretical concepts of graphs and discrete derivatives, I. Gutman's indices encompass a broader spectrum of molecular descriptors, enabling a comprehensive analysis of molecular properties [40][41][42][43].
The approach of GDIs, based on graph theory and quantum mechanics, aims to accurately describe and interpret the electronic distribution and properties of molecules. These calculations are based on discrete derivatives computed over pairs of atoms in a molecular graph, providing crucial information about connectivity, resonance, and aromaticity [40][41][42][43]. On the other hand, the indices developed by I. Gutman encompass a wide range of molecular descriptors that go beyond graph derivatives. These indices encompass various topological characteristics, such as size, symmetry, and the presence of heteroatoms, and can be classified as topo-structural or topo-chemical, depending on whether they consider only adjacency and length information or also incorporate physicochemical features.
Although this particular study did not employ I. Gutman's indices, it is imperative to recognize that GDIs and indices conceived by I. Gutman are complementary approaches that provide complementary perspectives in the analysis of chemical structures [40][41][42][43][44][45]. The combination of both approaches enables a comprehensive and detailed understanding of molecular properties, exploring topological, physicochemical, and quantum facets. This integration of approaches enriches the analysis and interpretation of electronic and structural characteristics of molecules, providing a global and encompassing view of chemical systems.

Conclusion
The outcomes described in this contribution show a mathematical analogy between the formalisms of GDIs and HMO. The adequate explanation of that mathematical analogy allows establishing connections among the calculation parameters of both methods, rendering a quantitative evaluation with topological information of some QM calculation integrals. The possibility of extending the use of quantitative evaluation of Coulomb and Resonance integrals into the semi-empirical molecular calculation approaches stays opened. The mathematical expression of Graphs Discrete Derivatives (base of GDI calculation) was written in terms of the previously mentioned integrals, making it possible to relate the energies from π molecular orbitals with the solutions of different discrete derivatives in molecular graph. Therefore, the knowledge of the orbital energies (in topological unities standardized by GDI metric) allows quantifying π-energies and resonance energies from each molecular system. In this scientific report was proposed a chemical-physical interpretation of GDI calculation, when it is related to the particular order 1-GDI calculation with HMO method. However, GDI can be used, not only to describe π electrons of conjugated molecules, but also to quantify information related to all bonds (π and σ) from any covalent molecular system. This research not only presents an interpretation of the discrete derivative in terms of energy, but also explains the particularities of some peculiar molecular systems with structures and stability, which had only been referred since the point of view of HMO method. Quantitative vision offered by GDI in terms of probabilities together with tools of Information Theory explain the existence of some chemical species with poor stability in normal atmospheric conditions and offers a general and quantitative vision of the aromaticity characteristics.