Journal of Mathematical Chemistry

The information-theoretic (IT) approach to chemical bonds, based upon the molecular probability-scattering channels in orbital resolution, is developed to cover both the overall promotion of atomic orbitals (AO) and the partial communication systems due to typical intermediate stages of reconstructing the system electronic structure, due to the orbital hybridization, orthogonalization, mixing into the molecular orbitals (MO), etc. The geometric and physical orbital promotion channels are distinguished, with the former taking into account solely the elementary channels due to the orbital-mixing, and the latter additionally including the MO-occupation term, which effectively projects-out the orbital promotion via the occupied MO only. This IT development generates the complementary descriptors of the molecular communication channel, viz., its conditional entropy (IT-covalency) and mutual-information (IT-ionicity), which add up to the total IT bond-order. These components provide a transparent description of the covalent/ionic competition for the valence electrons of constituent atoms. Their orbital-promotion contributions, from the elementary sub-channels reflecting the familiar intermediate stages of the bond-formation process, offer a novel information-scattering perspective on the relative roles played by these partial transformations of orbitals in shaping the resultant entropy/information descriptors of the system chemical bonds. The combination (grouping) rules for the consecutive and parallel arrangements of the elementary sub-channels are summarized and the stage-additivity of the IT bond- descriptors in the molecular sequential cascade of the elementary sub-channels for the intermediate sets of orbitals is examined in a more detail. A distinction between the molecular information channels describing the separated atoms and the free-atoms in the system atomic promolecule, respectively, is stressed and their entropy/information descriptors are briefly summarized. The associated difference descriptors of the overall IT bond-orders with respect to the promolecular reference are introduced and similar displacement measures are designed for the molecular promotion of intermediate orbitals. The illustrative results for the simplest model of a single chemical bond originating from an interaction between two overlapping atomic orbitals are presented. In particular, the bond-increments due to the orthogonalization and de-orthogonalization sub-channels of the overall AO-promotion cascade will be investigated.

A conjuctured bound for the number of internal vertices in a simply connected mono-q-polyhex is disproved

It is shown that the diatomic potential energy functions of Dunham, SPF and Ogilvie can be easily converted from one another when their coefficients are related. Through Maclaurin expansion and comparison of terms, the coefficients can be related by using the Pascal Triangle. In this paper, the coefficients were related up to the tenth order of δr/R for HX (X = H, Ga, Cl, I). Comparison of all three potential energy curves shows very good agreement for r ≤ 1.5R, thereby verifying the formulated relations. Observation of the plotted potential energy curves for r > 1.5R shows that the difference of the three potential function definitions is not reflected as any consistent trend arising from the related potential functions.

We calculate zeta and normal zeta functions of space groups with the point group isomorphic to the cyclic group of order 2. The obtained results are applied to determine the number of subgroups, resp. normal subgroups, of a given index for each of these groups.

The system of Emden-Fowler equations occurs frequently in physical and natural sciences. These system of equations are very difficult to handle due to their singularity and the nonlinearity. We provide a numerically robust technique for approximating the solution of the system of Emden-Fowler equations. This algorithm is based on the Bernstein polynomial accompanied by the collocation method. First, we transform the system of Emden-Fowler equations into their integral form. Then we implement the Bernstein polynomial approximation and the collocation technique to acquire a nonlinear system of equations. We apply the Newton-Raphson method to analyze the resulting nonlinear system of equations. We also discuss the error analysis of this method. We provide the numerical results of the $$L_{\infty }$$ error, the $$L_{2}$$ -norm error, and the residual error of some numerical supporting problems to examine the accuracy of the current technique. The advantage of the present method is exhibited by comparing the numerical results obtained by the proposed technique and other known techniques.

In this paper, we present generating functions for counting constitutional isomers and stereo isomers of acyclic saturated compounds consisting of O, C and H for the first time, and obtain the numbers of constitutional isomers and stereo isomers of acyclic saturated hydroxyl ethers. The numerical results are tabulated.

The global dynamics of a deterministic model in wastewater treatment has been investigated in Zhang (J Math Chem 50:2239–2247, 2012). The stochastic version, which can be used for continuous flow bioreactor and membrane reactor is presented in this study. Precisely, we assume there is some uncertainty in the part describing the recycle, which results in a set of stochastic differential equations with white noise. We first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout equilibrium and non-washout equilibrium are stochastically stable. At the end, we carry out some numerical simulations, which support our theoretical conclusions well.

The kinetics are investigated for diffusion-controlled reactions in a fractal medium described by a Smoluchowski equation. A class of model potentials are taken into account so as to solve the time-dependent Smoluchowski equation. Corrections to the standard solution are proposed for the fractal geometry of the medium.

The phase lag and its first, second, third, fourth, and fifth derivatives are all accounted for in a phase-fitting method. The new system uses the fewest possible function evaluations per integration step (FEvs) to achieve the highest possible algebraic order (AOR) and is hence P-stable. PF5DPFN2SPS is the symbol for the new method. The proposed method can be used for many different problems with periodic and/or oscillating solutions. We used the unique approach to the well-known problem in quantum chemistry posed by Schrödinger-type coupled differential equations. The new method is part of the economic algorithms since it uses 5 FEvs at each stage to achieve a 12 AOR.