In this work, we analyzed two simple chemical reactions using Fermi–Dirac’s entropy defined in terms of the Löwdin’s occupation numbers, this definition of entropy has not been applied or explored enough in the field of quantum chemistry, in this vein, Fermi–Dirac’s entropy maybe a good option to perform works in which the main purpose will be the study or analysis of the effect of the electron interactions. The results presented in this work were complemented by the analysis of the kinetic energy, potential energy, and the variation of the electric potential, along the reaction path, the results obtained, suggest that Fermi-Driac’s entropy presented in this work can be a useful tool to perform and complement studies about electron–electron interactions.

A minimal mathematical approach is used to state Landauer’s principle in a precise, general way. The results are obtained by means of a rigorous development which is based on the use of quantum statistical mechanics. A mathematical form of the principle results as an equality rather than an inequality. The equality version does imply the original statement of the principle as introduced by Landauer.

The paradigm of Markov jump process is widely employed in disparate contexts, like for instance in ecology, epidemiology and chemistry, any time the state-space is discrete and a tracked entity (the system) stochastically jumps from site to site. The classical master equation describes the system’s evolution in terms of site occupation probabilities starting from a given initial condition associated with the initial knowledge about the system. For the master equation of Markov processes admitting stationary occupation probabilities, it is here derived an equivalent quadratic format in an extended space of mutually interrelated variables having physical dimension of inverse of time. The evolution in the probability space is thus mirrored by the evolution in the extended space. This universal format potentially allows one to unveil general traits underlying the master equation dynamics. Here we specifically consider the emergence of an intrinsic rate which, behaving as a state function in the probability space, introduces a timing during the relaxation process. This specific feature has to be taken an empirical discovery which derives from the analysis of numerical calculations; a possible direction towards a formal proof is however proposed. The conjecture made here is that such intrinsic timing is a typical trait (i.e., normally present) of the Markov jump processes.

Adsorption integral equations are used to compute adsorption energy distributions by measured total isotherms. Since these types of equations are Fredholm integral equations of the first kind on bounded domains, they are unstable or ill-posed, respectively. Hence, there is a need for regularization. In this work, we present general regularizations based on the fourier transform for a special kernel, the Langmuir kernel. The regularization parameters are chosen as zeros or minimizers of simple functions depending on the mean absolute error of a transformed total isotherm. In difference to many other solutions proposed, an explicit error and convergence analysis is made. Additionally, we consider adsorption energy distributions with sharp peaks or for ideal adsorbents. Here, we construct a regularization for computing averages of the adsorption energy distribution and the maximal approximation error is estimated uniformly.

The aim of this paper is to introduce a two-dimensional discrete-time chemical model, identify its fixed points, as well as investigate one- and two-parameter bifurcations. Numerical normal forms are used in bifurcation analysis. For this model, the Neimark–Sacker and strong resonance bifurcations are observed. Based on the critical normal form coefficients, the bifurcation scenarios can be identified. Based on numerical continuation methods, we use the MATLAB package MatContM to verify the analytical results.

In this work, we analyze the reaction \(H_{2} + H\) using Shannon’s entropy defined in terms of spin density in position and momentum spaces. We analyzed the changes in the trends obtained in terms of the first derivative of Shannon’s entropy with respect to the electron number and with respect to spin density, in the first case, we show that this result is related to Fukui’s function, while in the second case, we found that the functional derivative is related to the so-called generalized moments.

A mathematical–chemical approach using alkyl biradicals has been used to enumerate isomers of alkenes, alkylcyclopropanes, and alkyl triradicals for alkylcyclobutadienes. In this non-recursive enumeration method for the number of constitutional isomers of methyl alkanes (branched-chain alkanes with all branches to be CH3), a methyl alkane of n carbon atoms is formed from a number of methyldiyl radicals:CH2, b number of 1,1-ethanediyl radicals:CH–CH3, c number of 2,2-propanediyl radical:C–(CH3)2, and 2 methyl radicals ·CH3, where a, b, and c are solutions of a Diophantine equation a + 2b + 3c + 2 = n. This algorithm does not have to use any previous data on alkyl biradicals and alkanes. Intuitively, in a hydrocarbon isomer series, the number of constitutional isomers of a hydrocarbon of n + 1 carbon atoms should be larger than that of having n carbon atoms, except at the beginning of the series. A graphical proof showed that the conjecture is erroneous for symmetrical methyl alkanes. In addition, the graphical proof showed that even and odd isomer series of symmetrical methyl alkanes are equivalent, i.e., having the equal number of isomers and form equivalent pairs with each pair containing the same number of symmetrical isomers. To my knowledge, this characteristic of a hydrocarbon isomer series has not been reported in the literature since Cayley’s publication on the mathematical theory of isomers.

The preceding Part IV of this series has described the half-size-subgroup method for type-itemized enumeration of stereoisograms by adopting epoxide (oxirane) derivatives as probes. As a continuation of this series, the present Part V has proposed the factor-group method for type-itemized enumeration of stereoisograms. The two methods are compared by applying to epoxide (oxirane) derivatives as common probes. The merit of the factor-group method is emphasized by adopting the same mark table during the treatments of Category 1 to Category 3 for categorizing type-I to type-V stereoisograms. Thus, the coset decomposition of the RS-stereoisomeric group \(\textbf{C}_{2v\widetilde{\sigma }\widehat{I}}\) by the subgroup \(\textbf{C}_{2}\) is calculated so as to give the corresponding factor group. After the mark table of the factor group is obtained, type-itemized enumerations of epoxide derivatives are conducted, where five types of stereoisograms are classified into the three categories. Isomer-classification schemes and isomer-classification diagrams are introduced in order to illustrate epoxide derivatives of respective compositions.

The exact solutions of 1D Schrödinger equation subject to a complex periodic potential \(V(x)=-[i\,a\,\sin (b\,x)+c]^2\) (\(a, b, c\in R\)) are found as a confluent Heun function (CHF) \(H_{C}(\alpha ,\beta ,\gamma ,\delta ,\eta ;z)\). The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes \(a\rightarrow -a\) or \(b\rightarrow -b\) or \(c\rightarrow -c\) due to the \(\mathcal{P}\mathcal{T}\) symmetry with the property \(V(x)=V(-x)^{*}\).

In this manuscript, we postulate a possible relationship of philicity within a molecular ensemble. The philicity has been studied by the Chattaraj et al. (J Phys Chem A 107(25):4973–4975, 2003). This important concept was obtained using the electrophilicity index and Fukui functions. In this work, was related the philicity with Molecular Quantum Similarity framework, to obtain a possible way to relate the philicity between molecules A and B. This methodology, can open a possible way to find systematic relationships in a given molecular ensemble using the selectivity defined in the Density Functional Theory context.

Cycloarenes such as kekulenes, septulenes and extended kekulenes are of considerable interest as they are precursors to holey nanographene with pores of variable sizes with interesting electronic and magnetic properties. These systems have also been of interest due to aromaticity, ring currents arising from \(\pi \)-electrons and intriguing topological properties. This study’s primary objective is to determine the topological properties including entropies through analytical expressions of degree-based entropy metrics for two well-known classes of tessellations of kekulenes. The various topological indices and entropies obtained here to shed light on the underlying topological connectivities and molecular structural features. It is shown that the rhomboidal tessellations exhibit greater entropies compared to triangular tessellations. Applications of the developed techniques to machine learning of NMR and ESR spectra of these tessellations are pointed out.

The analytical methods for solving Schrödinger equation are essential and effective tools with which we can investigate the spectroscopic properties, the electronic structure, and the energetic properties of the diatomic molecules (DMs). Accordingly, in this work, we used the Nikiforov-Uvarov (NU) method to solve the three-dimensional nonrelativistic Schrödinger equation with the molecular Kratzer-Feus (KF) potential and obtain the exact analytical bound state energy eigenvalues as well as their corresponding normalized eigenfunctions. The effective KF diatomic molecular potential well is investigated and represented graphically for several well-known DMs. The bound state energy levels are tabulated numerically for arbitrary values of the vibrational and rotational quantum numbers. The results obtained in this work are found to be in excellent agreement with the already-existing results in the literature.

The pre-industrial state of the global carbon cycle is a significant aspect of studies related to climate change. In this paper, we recall the power law kinetic representations of the pre-industrial models of Schmitz (Chem Eng Educ 36(4):296–309, 2002) and Anderies et al. (Environ Res Lett 8(4):044–048, 2013) from our earlier work. The power law kinetic representations, as uniform formalism, allow for a more extensive analysis and comparison of the different models for the same system. Using the mathematical theories of chemical reaction networks (with power-law kinetics), this work extends the analysis of the kinetic representations of the two models and assesses the similarities and differences in their structural and dynamic properties in relation to model construction assumptions. The analysis includes but is not limited to the coincidence of kinetic and stoichiometric spaces of the networks, capacity for equilibria multiplicity and co-multiplicity, and absolute concentration robustness in some species. We bring together previously published results about the power law kinetic representations of the two models and consolidate them with new observations here. We also illustrate how the pre-industrial model of Anderies et al. may serve as a building block in the analysis of a kinetic representation of a global carbon cycle with carbon dioxide removal intervention.

This paper deals with the design and analysis of a robust numerical scheme based on an improvised quartic B-spline collocation (IQBSC) method for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The convergence analysis of the method is studied by means of Green’s function approach. It should be pointed out that the numerical order of convergence of standard quartic B-spline collocation (SQBC) scheme for second-order boundary value problems (BVPs) is four, however, our proposed IQBSC method is shown to be sixth order convergence. To illustrate the applicability and accuracy of the method, we consider eight test problems. The obtained results are compared to those from some existing numerical schemes in order to show the advantage of present method. It is shown that the rate of convergence of present numerical scheme is higher than that of some of existing numerical methods. The CPU time of the present numerical method is provided.

Similarity analysis of chemical elements has long been studied from various perspectives. With the help of complex network theory, we study similarities from the network perspective using an undirected chemical network with 97 elements and 2198 edges. We proposed a new similarity index using the number of common neighbor elements as well as common non-neighbor elements. It is shown that this similarity index can be used to measure similarities between elements as well as group similarities among elements, and most similar elements are located near each other in the Periodic Table. Moreover, we find a similarity relation for 19 elements that have eight surrounding elements. This relation indicates that the tendency of the central element to form binary compounds is closely related to the surrounding eight elements, which can be used to predict potential binary compounds, thus provides a new way to study chemistry.

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs which are extremal with respect to them. Also, a QSPR study of this index and its exponential extension was performed.

This paper considers a chemical reaction-diffusion model for studying pattern formation with the Sel’kov–Schnakenberg model. Firstly, the stability conditions of the positive equilibrium and the existing conditions of the Hopf bifurcation are established for the local system. Then, Turing instability (diffusion-driven), which causes the spatial pattern is investigated and the existing condition of the Turing bifurcation is obtained. In addition, the dynamic behaviors near the Turing bifurcation are also studied by employing the method of weakly nonlinear analysis. The theoretical analysis shows that spatio-temporal patterns change from the spot, mixed (spot-stripe) to stripe with the variation of parameters, which can be verified by a series of numerical simulations. These numerical simulations give a visual representation of the evolution of spatial patterns. Our results not only explain the evolution process of reactant concentration, but also reveal the mechanism of spatio-temporal patterns formation.

Although the electron density has the largest value at the nuclear site, the associated Coulomb singularity does not allow its determination in the numerical solution of the Euler-Lagrange equation in density functional theory (DFT) through the corresponding single density equation, or equivalently, in the orbital-free DFT. The problem may be bypassed by using the cusp condition, which, in its conventional form, may turn out to be inadequate for the related mixed boundary value problem. For this purpose, a new generalized cusp condition has been derived.

This article presents a uniformly convergent numerical technique for a time-dependent reaction-dominated singularly perturbed system, including the same diffusion parameters multiplied with second-order spatial derivatives in all equations. Boundary layers are observed in the solution components for the small parameter. The proposed numerical technique consists of the Crank–Nicolson scheme in the temporal direction over a uniform mesh and quadratic \(\mathbb {B}\)-splines collocation technique over an exponentially graded mesh in the spatial direction. We derived the robust error estimates to establish the optimal order of convergence. Numerical investigations confirm the theoretical determinations and the proposed method’s efficiency and accuracy.