# Journal of Mathematical Chemistry

## Current research reports and chronological list of recent articles..

The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals.

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## Journal of Mathematical Chemistry - Abstracts

Rubber blends: kinetic numerical model by rheometer experimental characterization

### Abstract

A robust kinetic numerical model for natural rubber NR–poly-butadiene PB blends vulcanized with sulfur is presented. For a preliminary experimental insight used then as benchmark, a 70% NR with 30% high-cis PB blend vulcanized in presence of sulfur and two accelerants (TBSS and DPG) in different concentrations is tested under standard vulcanization conditions (rheometer) at three different vulcanization temperatures, namely 150, 170 and 180 °C. The numerical model reproduces normalized experimental rheometer curves using a Han’s kinetic model for NR and a modification of the Han’s model proposed recently by the authors for PB. The model bases therefore on existing kinetic approaches that proved to be effective for NR and PB separately, but merging them linearly to cope with NR–PB peculiar reactions occurring in a blend. Han’s model depends on three rate constants, whereas the approach used for PB is characterized by four kinetic parameters. The linear interaction between PB and NR (with amount of rubber involved not a priori known) is again assumed ruled by a Han’s model. The mathematical approach proposed is therefore characterized by 10 rate constants plus an additional parameter represented by NR–PB concentration involved by the interaction. Such constants are estimated by standard best fitting against experimental data. Quite good match is found, showing that the procedure may be useful for practical purposes in all those cases where expensive experimental investigations are not possible.

Datum: 01.06.2018

On the linear algebra of biological homochirality

### Abstract

We look for structural properties of chemical networks giving place to homochiral phenomena. We found a necessary condition for homochirality that we call Frank inequality, and which is a linear inequality related to the entries of the jacobian matrices that occur at racemic steady states. We also investigate the existence of stronger conditions that can be formulated in a similar algebraic way. Those investigations lead us to introduce a homochirality degree for the racemic states of chiral neworks, which is intended to measure the probability of observing homochiral dynamics after perturbing those states. It is important to stress that all the introduced concepts and degrees are effective. The later fact allows us to develop an algorithm that can be used to, given a chiral network as input, compute large samples of steady states of different degrees.

Datum: 01.06.2018

Predicting specific surface areas of layered double hydroxides based on support vector regression integrated with a residual bootstrapping method

### Abstract

The layered double hydroxides (LDHs) with high specific surface areas (SSA) are of great benefit for their practical applications as catalysts, sensors or adsorbents and so on. So, it has instructional significance to effectively predict specific surface areas of LDHs for the syntheses of LDHs with desired properties. In the work, support vector regression (SVR) integrated with a residual bootstrapping method (BTSVR) is employed to construct the prediction models of SSA with the chemical compositions and technical parameters of LDHs. By comparison, the quantitative structure–property relationship (QSPR) model constructed by using BTSVR method has higher predictive accuracy, better reliability and extrapolation than the SVR model. The strategy expressed in the work can be broadly used to the controllable synthesis of the compounds, and further promotes study on data mining to assist the material science and engineering.

Datum: 01.06.2018

Quasi-steady state reduction for the Michaelis–Menten reaction–diffusion system

### Abstract

The Michaelis–Menten mechanism is probably the best known model for an enzyme-catalyzed reaction. For spatially homogeneous concentrations, QSS reductions are well known, but this is not the case when chemical species are allowed to diffuse. We will discuss QSS reductions for both the irreversible and reversible Michaelis–Menten reaction in the latter case, given small initial enzyme concentration and slow diffusion. Our work is based on a heuristic method to obtain an ordinary differential equation which admits reduction by Tikhonov–Fenichel theory. We will not give convergence proofs but we provide numerical results that support the accuracy of the reductions.

Datum: 01.06.2018

Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic

### Abstract

This article studies the pattern formation of reaction–diffusion Brusselator model along with Neumann boundary conditions arising in chemical processes. To accomplish this work, a new modified trigonometric cubic B-spline functions based differential quadrature algorithm is developed which is more general than (Mittal and Jiwari in Appl Math Comput 217(12):5404–5415, 2011; Jiwari and Yuan in J Math Chem 52:1535–1551, 2014). The reaction–diffusion model arises in enzymatic reactions, in the formation of ozone by atomic oxygen via a triple collision, and in laser and plasma physics in multiple couplings between modes. The algorithm converts the model into a system of ordinary differential equations and the obtained system is solved by Runge–Kutta method. To check the precision and performance of the proposed algorithm four numerical problems are contemplated and computed results are compared with the existing methods. The computed results pamper the theory of Brusselator model that for small values of diffusion coefficient, the steady state solution converges to equilibrium point $$( {\mu , \lambda /\mu })$$ if $$1-\lambda +\mu ^{2}>0$$ .

Datum: 01.06.2018

Comment on: “The kinetic energy partition method applied to quantum eigenvalue problems with many harmonic-oscillator potentials” by Y.-H. Chen and S. D. Chao

### Abstract

We show that the recently proposed kinetic energy partition method for the approximate solution of the Schrödinger equation is a particular case of the well known Rayleigh–Ritz variational method with an extremely restricted implementation. In addition to it, the quantum-mechanical example chosen by the authors to illustrate their approach is unsuitable for the proposed application.

Datum: 01.06.2018

Valence bond approach and Verma bases

### Abstract

The unitary group approach (UGA) to the many-fermion problem is based on the Gel’fand–Tsetlin (G–T) representation theory of the unitary or general linear groups. It exploits the group chain $$\mathrm {U}(n) \supset \mathrm {U}(n-1) \supset \cdots \supset \mathrm {U}(2) \supset \mathrm {U}(1)$$ and the associated G–T triangular tableau labeling basis vectors of the relevant irreducible representations (irreps). The general G–T formalism can be drastically simplified in the many-electron case enabling an efficient exploitation in either configuration interaction (CI) or coupled cluster approaches to the molecular electronic structure. However, while the reliance on the G–T chain provides an excellent general formalism from the mathematical point of view, it has no specific physical significance and dictates a fixed Yamanouchi–Kotani coupling scheme, which in turn leads to a rather arbitrary linear combination of distinct components of the same multiplet with a given orbital occupancy. While this is of a minor importance in molecular orbital based CI approaches, it is very inconvenient when relying on the valence bond (VB) scheme, since the G–T states do not correspond to canonical Rumer structures. While this shortcoming can be avoided by relying on the Clifford algebra UGA formalism, which enables an exploitation of a more or less arbitrary coupling scheme, it is worthwhile to point out the suitability of the so-called Verma basis sets for the VB-type approaches.

Datum: 01.06.2018

Port-Hamiltonian modeling of non-isothermal chemical reaction networks

### Abstract

Motivated by recent progress on the port-Hamiltonian formulation of isothermal chemical reaction networks and of the continuous stirred tank reactor, the present paper aims to develop a port-Hamiltonian formulation of chemical reaction networks in the non-isothermal case, and to exploit this for equilibrium and stability analysis.

Datum: 01.06.2018

Multivariate generalized Gram–Charlier series in vector notations

### Abstract

This article derives the generalized Gram–Charlier (GGC) series in multivariate that expands an unknown joint probability density function (pdf) of a random vector in terms of the differentiations of joint pdf of a known reference random vector. Conventionally, the higher order differentiations of a multivariate pdf and corresponding to it the multivariate GGC series use multi-element array or tensor representations. Instead, the current article derives them in vector notations. The required higher order differentiations of a multivariate pdf are achieved in vector notations through application of a specific Kronecker product based differentiation operator. The resultant multivariate GGC series expression is more compact and more elementary compare to the coordinatewise tensor notations as using vector notations. It is also more comprehensive as apparently more nearer to its counterpart for univariate. Same notations and advantages are shared by other expressions obtained in the article, such as the mutual relations between cumulants and moments of a random vector, integral form of a multivariate pdf, integral form of the multivariate Hermite polynomials, the multivariate Gram–Charlier A series and others. Overall, the article uses only elementary calculus of several variables instead of tensor calculus to achieve the extension of a specific derivation for the GGC series in univariate (Berberan-Santos in J Math Chem 42(3):585–594, 2007) to multivariate.

Datum: 01.06.2018

A topological characterization of the products arising from site-specific recombination on $$T(2,n) \# C(2,r)$$ T ( 2 , n ) # C ( 2 , r ) DNA substrates

### Abstract

A mathematical knot is a simple closed curve in 3-space. Following work of Buck, Flapan, and others, we model circular DNA as a knot to classify the possible knotted or linked products which can arise as a result of site-specific recombination on DNA substrates of the topological form $$T(2,n) \# C(2,r)$$ . We show that, given some reasonable biological assumptions, all of the possible products are contained in one of two families.

Datum: 01.06.2018

New five-stages two-step method with improved characteristics

### Abstract

In the present paper and for the first time in the literature, a new five-stages symmetric two-step method with improved characteristics is developed.

The main properties of the new method are:

• the new method is of symmetric type,

• the new method is of two-step algorithm,

• the new method is of five-stages,

• the new method is of twelfth-algebraic order,

• the new hybrid symmetric two-step method is based on the following approximations:

1. Approximation of the first layer on the point $$x_{n-1}$$ ,

2. Approximation of the second layer on the point $$x_{n-1}$$ ,

3. Approximation of the third layer on the point $$x_{n-1}$$ ,

4. Approximation of the fourth layer on the point $$x_{n}$$ and finally,

5. Approximation of the fifth (final) layer on the point $$x_{n+1}$$ ,

• the new method has vanished the phase-lag and its first, second and third derivatives,

• the new method has optimized stability properties for the general problems,

• the new method is of P-stable type since it has an interval of periodicity equal to $$\left( 0, \infty \right)$$ .

• For the new obtained method a full numerical analysis (error and stability analysis) is presented.

The evaluation of the efficiency of the new method is based on the numerical solution of systems of coupled differential equations of the Schrödinger type.

Datum: 01.06.2018

An analytical altitude- and wavelength-dependent expression of the j 3 rate coefficient governing the photolysis of the atmospheric Ozone

### Abstract

In this study we determine an analytical, effective expression of the rate coefficient j3 (governing the photolysis of the atmospheric Ozone) as a function of altitude h and wavelength λ of the incoming solar radiation, in the range 0 ≲ h ≲ 32 km.

Datum: 01.06.2018

The statistical properties of q-deformed Morse potential for some diatomic molecules via Euler–MacLaurin method in one dimension

### Abstract

In this paper, we present a closed-form expression of the vibrational partition function for the one-dimensional q-deformed Morse potential energy model. Through this function, the related thermodynamic functions are derived and studied in terms of the parameters of the model. Especially, we plotted the q-deformed vibrational partition function and vibrational specific heat for some diatomic molecule systems such as $$\text {H}_{2}$$ , HCl, LiH, and CO. The idea of a critical temperature $$T_{C}$$ is introduced in relation to the specific heat.

Datum: 01.06.2018

On the convergence of Mickens’ type nonstandard finite difference schemes on Lane-Emden type equations

### Abstract

In this paper, we analyse Mickens’ type non-standard finite difference schemes (NSFD) and establish their convergence. We then apply these schemes on Lane Emden equations. The numerical results thus obtained are compared with existing analytical solutions or with solutions computed with standard finite difference (FD) schemes. NSFD and FD solutions and their errors have also been compared graphically and observed that the errors in NSFD tends to zero as step size tends to zero. The result shows that the NSFD behave qualitatively in the same way as the original equations. NSFD approximate solution near singular point efficiently where FD fails to do so (Buckmire in Numer Methods Partial Differ Equ 19:380–398, 2003).

Datum: 01.06.2018

DNA, unnatural base pairs and hypercubes

Datum: 01.05.2018

Boolean Hypercubes as time representation holders

Datum: 01.05.2018

Approximation of the electron–proton mass ratio as a series in powers of $$\uppi$$ π

### Abstract

Eddington in 1923, first identified four dimensionless numbers, derived from combinations of the basic physical constants, which are known as the “Eddington constants”. In formulating these dimensionless numbers, Eddington, a leading physicist of his time, claimed that they are characteristic of the structure and dynamics of the Universe at large, on the microscopical scale and at the macroscopical scale. Recently, there has been suggested a possible way of accounting for the magnitude of one of these four dimensionless constants, indicated as the “fine structure constant”, $$\upalpha$$ , that first emerged from studies of the atomic line spectrum of H. A simple power series in the product $$\hbox {e}\cdot \uppi$$ has been proposed, that fits the measured value of the fine structure constant to better than 9999 parts in 10,000. Following along these lines, the authors here propose a simple power series expansion in $$\uppi$$ that agrees with the currently accepted measurement of the value of the electron–proton mass ratio (m/M), or $$\upbeta$$ , to better than 999 parts in 1000.

Datum: 01.05.2018

Atomic thermal voltage population distributions

Datum: 01.05.2018

Analytic second-order energy derivatives in natural orbital functional theory

### Abstract

The analytic energy gradients in the atomic orbital representation have recently been published (Mitxelena and Piris in J Chem Phys 146:014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.

Datum: 01.05.2018

Generalization of the concepts of seniority number and ionicity

### Abstract

We present generalized versions of the concepts of seniority number and ionicity. These generalized numbers count respectively the partially occupied and fully occupied shells for any partition of the orbital space into shells. The Hermitian operators whose eigenspaces correspond to wave functions of definite generalized seniority or ionicity values are introduced. The generalized seniority numbers afford to establish refined hierarchies of configuration interaction spaces within those of fixed ordinary seniority. Such a hierarchy is illustrated on the buckminsterfullerene C $$_{60}$$ molecule.

Datum: 01.05.2018

Category: Current Chemistry Research

Last update: 11.04.2018.