Journal of Mathematical Chemistry

The main purpose of this work is to develop a new multi-way decomposition technique by considering statistical structure of target multi-way array. To this end enhanced multivariance products representation (EMPR), which is an expansion extended from high dimensional model representation (HDMR), is used. EMPR provides quite successful results on representation and approximation of multivariate functions when they have high level multiplicativity. Hence this urges us to reconstruct EMPR as a multi-way array decomposer. This paper presents this decomposition technique with all reconstruction formulations and numerical experiments on synthetic and real-life data sets to denote EMPR’s efficiency as a decomposer and also presents a combined method Reductive-EMPR (R-EMPR) as a multi-way array decomposition technique.

According to the main result of W. Feit and G. M. Seitz (Illinois J Math 33(1):103–131, 1988), the Thompson group Th is non-rational or unmatured group (S. Fujita in Bull Chem Soc Jpn 71:2071–2080, 1998). Using the concept of markaracter tables proposed by S. Fujita (Bull Chem Soc Jpn 71:1587–1596, 1998), we are able to obtain tables of integer-valued characters for finite unmatured groups. In this paper, the integer-valued character for Thompson group is successfully derived for the first time.

We examine the singlet stability of symmetry adapted, restricted Hartree-Fock (RHF) solutions and the implied symmetry breaking for various planar, π-electron systems with conjugated double bonds as described by the semiempirical Pariser-Parr-Pople Hamiltonian. In particular, we explore the energy and charge- density waves (CDWs) in various real and hypothetical structures that result by a systematic deformation of the nuclear framework: we start with a highly symmetric cyclic polyene C N H N having a nondegenerate ground state (N = 2n = 4ν + 2, ν = 1, 2,...), whose sites form a regular N-gon (D Nh point group), and proceed to structures with lower symmetry (D 6h , D 3h , D 2h point groups), or with only the planar symmetry of the conjugated π-electron system (C 1h ). The objective of this study is to explore the phenomenon that could be referred to as a breaking of an approximate symmetry or an implied symmetry breaking.

The construction of vertex-decorated graphs can be used to produce derived graphs with specific eigenvalues from undecorated graphs, which themselves do not have such eigenvalues. An instance of a decorated graph is the rooted product G(H) of graphs G and H. Let $$F = (V, E)$$ be a molecular graph with the vertex set V and the edge set E $$(|V|=n; |E|=m)$$ , and let $$n_{+}=n_{-}$$ $$(n_{+}+n_{-}=n)$$ , where $$n_{+}$$ and $$n_{-}$$ are the numbers of positive and negative eigenvalues, respectively. Then, in the spectrum of the eigenvalues of F, two minimum-modulus eigenvalues, positive $$\lambda _{+}$$ and negative $$\lambda _{-}$$ , are of special interest because the value $$\delta =\lambda _{+}-\lambda _{-}$$ determines the energy gap. In quantum chemistry, the energy gap $$\delta $$ is associated with the energy of an electron transfer from the highest occupied molecular orbital to the lowest unoccupied molecular orbital of a molecule. As an example, we consider obtaining a (molecular) graph $$F=G(H)$$ whose median eigenvalues $$\lambda _{+}$$ and $$\lambda _{-}$$ are predictably close to 0.

The Fermi holes are presented as a new means of analysis and visualisation of molecular structure. Based on these quantities it is possible to get clear and highly visual insight into the structure of molecular fragments (functional groups) even in molecules with complex bonding patterns like multicenter bonding, hypervalence, etc. In addition to allowing the detection and localization of multicenter bonding, the new approach also brings some new interesting possibilities for the quantitative evaluation of molecular similarity.

This paper is devoted to the study of local controllability of chemical reactions. After a brief introduction we give a sufficient condition of local controllability of chemical reactions. We show the use of our condition on a series of examples. In the last section we mention some unsolved problems.

We discuss the application of perturbation theory to a system of particles confined in a spherical box. A simple argument shows that the particles behave almost independently in sufficiently strong confinement. We choose the helium atom with a moving nucleus as a particular example and compare results of first order with those for the nucleus clamped at the center of the box. We provide a suitable explanation for some numerical results obtained recently by other authors.

The existing factor rotational methods (an assortment of orthogonal, oblique and simplex techniques) are assessed for their potential to be generalized for use at higher dimensions (i.e. above threefold factor space). An answer is sought to the problem of why the application of an orthogonal rotation results in mathematical solutions (those containing negative entries) rather than all positive solutions having physical and chemical meaning. Such positive solutions will be within the limits of experimental error if non-perfect data is used. An evaluation of a methodological improvement to an algorithm on factor analysis based on the optimization of them(m - 1) independent variables of a transformation matrix T′ ofm factors is made, and a case is presented for its introduction as a realistic alternative to the established methods.