Di storted s-type orbital s: the $${\rm H}^{+}_{2}$$ problem revi sited

On the example of the molecular ion, we show that spherically distorted s-type orbitals possessing angular dependent orbital exponents, even in a minimal basis may lead to total energies the accuracy of which is comparable with the ones obtained by fully numerical (‘complete basis’) calculations.     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

Mathematical relations between codon bias indexes and their applications

Codon Adaptation Index (CAI), Effective Number of Codons as well as its modifications , can be used to measure gene codon bias. In this article, we prove is more efficient and unbiased than and by revisiting correlations of them with CAI in the level of individual amino acid’s codon bias. Correlations are studied by mathematical expressions rather than statistic methods, because the latter unavoidably depend on the data set used. Additionally, the immediate cause of correlations of with CAI (as well as those of RSCU with CAI) are also described in mathematical language. Perhaps, mathematics provides us a new way to study correlations between biological indexes.     

Unified treatment of complete orthonormal sets of nonrelativistic,quasirelativistic and relativistic sets of spinor wave functions,and Slater spinor orbitals in coordinate,momentum and four-dimensional spaces

Using the properties of tensor spherical harmonics introduced by the author in previous paper (Guseinov, Phys Lett A 372:44, 2007) and complete orthonormal scalar basis sets of nonrelativistic -exponential type orbitals ( -ETO), - momentum space orbitals ( -MSO) and z ^α-hyperspherical harmonics (z ^α-HSH) for particles with spin s = 0 the new analytical relations for the quasirelativistic and relativistic spinor wave functions and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces are derived, where α = 1, 0, −1, −2,.... The 2-component quasirelativistic and 4-component relativistic spinor wave functions obtained are complete without the inclusion of the continuum. The relativistic spinor wave function sets and Slater spinor orbitals are expressed through the corresponding quasirelativistic spinor wave functions and Slater spinor orbitals, respectively. The analytical formulas for overlap integrals over quasirelativistic and relativistic Slater spinor orbitals with the same screening constants in coordinate space are also derived.     

Unified treatment of complete orthonormal sets for exponential type vector orbitals of particles with spin 1 in coordinate, momentum and four-dimensional spaces

The new formulas are obtained for complete orthonormal sets of exponential type vector orbitals of a particle with spin 1 in coordinate, momentum and four-dimensional spaces using the properties of spherical vectors and complete orthonormal scalar basis sets of -exponential type orbitals ( -ETO), -momentum space orbitals ( -MSO) and -hyperspherical harmonics ( -HSH) introduced by the author for particles with spin s = 0, where These vector orbitals are complete without the inclusion of the continuum and, therefore, their group of transformation is the four-dimensional rotation group of O(4). For overlap integrals over vector Slater orbitals with the same screening constant the analytical relations in coordinate space are also derived. It should be noted that the new idea presented in this study is the combination of spherical vectors with complete orthonormal scalar sets for radial parts of -orbitals.     

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives and , all with m > n, are also evaluated.     

Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Interaction of a finite quantum system that contains ρ eigenvalues and eigenstates with an infinite quantum system that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system is developed. This system contains embedded eigenstates with continuous eigenvalues , and, in addition, it may contain isolated eigenstates with discrete eigenvalues . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system that interacts with the system can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems and . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system that interacts with the infinite system that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.     

Cluster Oxalate Complexes [W3(μ3-Q)(μ2-Q)3 (C2O4)3(H2O)3]2? (Q=S, Se): Mechanochemical Synthesis and Crystal Structure

Mechanochemical reaction of cluster coordination polymers (Q=S, Se) with solid leads to the cluster core excision with the formation of anionic complexes . Extraction of the reaction mixture with water followed by crystallization gives crystalline (main product) and (1) (minor product). In the case of the Se cluster, the complex could not be isolated, and the treatment of the aqueous extract with PPh₃ gave (2) in a low yield. Alternatively, it was obtained from and in high yield. Both 1 and 2 were characterized by X-ray structure analysis. Dedicated to Academician I. I. Moiseev on the occasion of his 75th birthday and in recognition of his outstanding contribution to cluster chemistry.     

Apparent Michaelis constant of the enzyme modified porous electrode

Plate-gap model of enzyme doped porous electrode was utilized in order to calculate apparent Michaelis constants () and apparent maximal currents () of modeled amperometric biosensor for the wide range of given reaction/diffusion parameters. It was found that of plate-gap biosensor linearly depends on when rates of enzymatic reaction are lower than critical. Theoretically predicted linear correlation between apparent parameters was observed experimentally for the case of carbon paste electrodes, which were modified by PQQ-dependent alcohol dehydrogenases. At overcritical rates (or apparent maximal currents), is practically independent on Michaelis constant of soluble enzyme. Therefore, apparent Michaelis constant can be regarded as biosensor’s topology representing parameter which, in fact, is not related to the specificity of enzyme kinetics. High and rate-independent values of indicate that reaction proceeds at substrate-exposed top layer of the gap. In this case, reaction–diffusion system formally is stratified into separate reaction (top) and diffusion (bottom) zones. Topology of such reaction–diffusion system reminds “inverted” planar electrode, which contains diffusion layer below reaction layer. The net effect of plate-gap topology of working electrode on apparent Michaelis constant is similar to the effect of diffusion layer covering enzymatic planar electrode.     

Unified treatment of nonrelativistic and quasirelativistic atomic integrals over complete orthonormal sets of $${\Psi^{\alpha}}$$ -exponential type orbitals

The new one- and two-electron nonrelativistic and quasirelativistic basic functions are introduced. The general analytical relations in terms of basic functions suggested are derived for the non- and quasi-relativistic atomic integrals over complete orthonormal sets of -exponential type orbitals introduced by the author, where α  = 1,0, − 1, − 2, . . . The relationships obtained are valid for the arbitrary values of quantum numbers and screening constants of orbitals.     

Identifiability of chemical reaction networks

We consider the dynamics of chemical reaction networks under the assumption of mass-action kinetics. We show that there exist reaction networks for which the reaction rate constants are not uniquely identifiable, even if we are given complete information on the dynamics of concentrations for all chemical species of . Also, we show that there exist reaction networks such that their dynamics are identical under appropriate choices of reaction rate constants, and present theorems that characterize the properties of , , that make this possible. We use these facts to show how we can determine dynamical properties of some chemical networks by analyzing other chemical networks.     

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

Time-dependent properties of a state that interacts with an infinite dimensional quantum system containing several one-parameter eigenvalue bands are considered. This is done by a new mathematical method that produces correct results, however strong the interaction between the state and the system . It is shown that in the case of the weak interaction one obtains standard results that are usually obtained within the formalism of the perturbation expansion method. In particular, if the eigenvalue E of the state is embedded inside the range of the unperturbed eigenvalues, time evolution of the state that is initially prepared in the state has typical exponential decay behavior. One also reproduces standard results concerning probabilities of the transition of the state at infinite time (t=∞) into various eigenvalue bands. However, if the interaction is strong, one finds much more complex and much more complicated behavior.     

On minimal energies of trees of a prescribed diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let denote the set of trees on n vertices and diameter d, . Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in with minimal energy. In this article, the trees in with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35     

Kinetics and Equilibria for Complex Formation between Palladium(II) and Chloroacetate

Kinetics and equilibria for the formation of a 1:1 complex between palladium(II) and chloroacetate were studied by spectrophotometric measurements in 1.00 mol HClO₄ at 298.2 K. The equilibrium constant, K, of the reaction

Use of Filter-Steinborn B and Guseinov $${Q_{ns}^q }$$ auxiliary functions in evaluation of two-center overlap integrals over Slater type orbitals

An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those obtained by other method for arbitrary principal quantum numbers and different screening constants. An erratum to this article can be found at     

Classical dynamics of 3D Hydrogen molecular ion in intense laser fields

The classical trajectory method is used to study the dynamics of 3D Hydrogen molecular ion interacting with intense laser fields. In the 3D classical model, a three-body Hamiltonian with one-dimensional nuclear motion restricted to the direction of the laser field is considered. The motion of electron and nucleus is described by the classical Hamiltonian canonical equations. The probabilities of ionization, dissociation and Coulomb explosion as functions of time are calculated and the average distances from electron to the mass-center for various laser parameters are implemented by symplectic method. The dynamics of in two-color laser fields are also investigated. We compare our results with the corresponding quantum-mechanical calculations and find they produce similar qualitative features in many cases.     

Direct and exchange contributions to inner and outer radii in many-electron atoms

Within the Hartree-Fock framework, the spinless two-electron density function Γ (r ₁, r ₂) consists of direct Γ_di (r ₁, r ₂) and exchange Γ_ex (r ₁, r ₂) parts. Accordingly, the inner and outer radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., and . It is generally shown that and , where is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms from He to Lr in their ground states find that the electron exchange works to decrease and increase . However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii.     

A second-order scheme for the “Brusselator” reaction–diffusion system

A second-order method is developed for the numerical solution of the initial-value problems , , and , , , in which the functions and , where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point ( , V* = A/B) provided . The approach adopted is extended to solve a class of non-linear reaction–diffusion equations in two-space dimensions known as the “Brusselator” system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs). This revised version was published online in July 2006 with corrections to the Cover Date.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35