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Classical and nonclassical contributions to Author's resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$ , due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, are shown to determine the real and imaginary parts of the state complex Shannon entropy, a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, and the gradient entropy: [ABSTRACT FROM AUTHOR]

Nanostructures may be fabricated from metal nanoclusters such as gold or platinum, which are of interest for catalytic and structural characteristics, or from nano forms of carbon allotropes. Here, informational thermodynamic properties such as free energy, enthalpy, and entropy are calculated using a graph network model at T = 298.15 K. We calculate the partition function using Euclidean adjacency matrices from the Hamiltonian and estimated bond energies. The summed atomic displacement from the Kirchhoff index has power law behavior, while the thermodynamic properties exhibit large $$N$$ logarithmic behavior: however, the data shows structurally related anomalies. [ABSTRACT FROM AUTHOR]

The basic variables of the information theory of quantum systems (e.g., frequency or entropic moments, Rényi and Tsallis entropies) can be expressed in terms of $$L_{q}$$ norms of general hypergeometrical polynomials. These polynomials are known to control the radial and angular parts of the wavefunctions of the quantum-mechanically allowed states of numerous physical and chemical systems. The computation of the $$L_{q}$$ norms of these polynomials is presently an interesting issue per se in the theory of special functions; moreover, these quantities are closely related to the frequency moments and other information-theoretic properties of the associated Rakhmanov probability density. In this paper we calculate the unweighted and weighted $$L_{q}$$ -norms $$(q=2k, k\in {\mathbb {N}})$$ of general hypergeometric real orthogonal polynomials (Hermite, Laguerre and Jacobi) and some entropy-like integrals of Bessel polynomials, in terms of $$q$$ and the parameters of the corresponding weight function by using their explicit expression and second order differential equation. In addition, the asymptotics $$(q\rightarrow \infty )$$ of the unweighted $$L_{q}$$ norms of the Jacobi polynomials is determined by the Laplace method. [ABSTRACT FROM AUTHOR]

The role of phases in local Communication Theory of the Chemical Bond is investigated. Probability amplitudes of such molecular ( fine-grained) information systems originate from the superposition principle of quantum mechanics involving the projection onto the bond system defined by the subspace of the state occupied Molecular Orbitals. They are explicitly phase-dependent, thus being capable of interference effects. The phase factors of the local direct and indirect (bridge, cascade) channels are examined and the associated amplitude/probability sum rules are established. The entropic descriptors of the local channels, providing the system 'covalent' (communication- noise) and 'ionic' (information- flow) components, are investigated using prototype one-electron systems. The competition between these information-theoretic measures of the chemical bond covalency (electron delocalization) and ionicity (electron localization) is illustrated in $$\mathrm{H}_{2}^{+}$$ and $$\mathrm{H}_{2}$$ . [ABSTRACT FROM AUTHOR]

The Orbital Communication Theory of the chemical bond, in which molecules are treated as information systems transmitting 'signals' of electron allocations to Atomic Orbitals, is extended to cover the local resolution level of electron distributions and the Configuration-Interaction (CI, multi-determinantal) description of molecular states. These communication systems generate the information-theoretic measures of both the absolute and relative multiplicities of chemical bonds, as well as the bond covalent (communication-noise) and ionic (information-flow) components. The orbital/local communications via the CI ensembles of the occupied molecular orbitals in such generalized molecular states are investigated. Illustrative two-orbital model and its prototype Valence-Bond structures are examined in a more detail. [ABSTRACT FROM AUTHOR]

The standard deviations and Shannon information entropies of the probability densities for a particle in a quantum corral are compared and contrasted to determine their effectiveness in measuring particle (de)localization. We illustrate how the two measures emphasize different aspects of the underlying distributions which can lead to inconsistent interpretations. Among these, we show that the Shannon entropy is able to distinguish between the presence of an attractive or repulsive effective potential in the radial Schrödinger equation while the standard deviation does not. The analysis of this radial model is then extended to momentum space where the dependence of the measures, entropic sum and uncertainty product on the effective potential, is examined. [ABSTRACT FROM AUTHOR]

We investigate the complexity of the hydrogenic identity S2 exchange reaction by means of information-theoretic functionals such as disequilibrium ( D), exponential entropy ( L), Fisher information ( I), power entropy ( J) and joint information-theoretic measures, i.e., the I-D, D-L and I-J planes and the Fisher-Shannon (FS) and López-Mancini-Calbet (LMC) shape complexities. The several information-theoretic measures of the one-particle density were computed in position ( r) and momentum ( p) spaces. The analysis revealed that the chemically significant regions of this reaction can be identified through most of the information-theoretic functionals or planes, not only the ones which are commonly revealed by the energy, such as the reactant/product (R/P) and the transition state (TS), but also those that are not present in the energy profile such as the bond cleavage energy region (BCER), the bond breaking/forming regions (B-B/F) and the charge transfer process (CT). The analysis of the complexities shows that the energy profile of the identity S2 exchange reaction bears no simple behavior with respect to the LMC and FS measures. Most of the chemical features of interest (BCER, B-B/F and CT) are only revealed when particular information-theoretic aspects of localizability ( L or J), uniformity ( D) and disorder ( I) are considered. [ABSTRACT FROM AUTHOR]

The main information-theoretic measures of a one-dimensional particle-in-a-box (also known as the infinite potential well or the infinite square well) in both position and momentum spaces, as well as their associated uncertainty relations, are calculated and discussed. The power and entropic moments, the Shannon, Renyi and Tsallis entropies and the Fisher information together with two composite measures (Fisher-Shannon and LMC shape complexities) are considered. Moreover, the associated information-theoretic spreading lengths, which characterize the spread/delocalization of the particle beyond (but complementarily) the standard deviation, and their corresponding uncertainty relations are given and mutually compared. It is found, in particular, that the Fisher length is the proper measure of uncertainty for the infinite well, mainly because it grasps the oscillatory nature of the wavefunctions. Finally, this study is extended to a D-dimensional box. [ABSTRACT FROM AUTHOR]

The use of multi-dimensional 'chemistry spaces' to represent large compound collections has become widespread in pharmaceutical research. In such spaces compounds are treated as points. Points in close proximity represent similar compounds, while distant points represent dissimilar compounds. Assessing the diversity of a compound collection, thus, is tantamount to characterizing the distribution of points in chemistry space. To facilitate many procedures such as selecting subsets of compounds for screening, for compound acquisition and designing combinatorial libraries, chemistry spaces have been partitioned into sets of non-overlapping, multi-dimensional cells, which are generated by dividing each axis into a number of equal-sized bins. This leads to a lattice of $${(N_{bins})^{N_{\rm dim}}}$$ cells, where Nis the number of bins on each axis and N is the dimensionality of the space. One diversity measure that is typically used in cell-based chemistry spaces is identical in form to Shannon entropy, $${D_{N_{cpd}}^{cpd}}$$ A normalized measure of this Shannon entropy given by, $${D_{rel}^{cpd}}$$ enables comparison between compound collections that occupy different number of occupied cells. Although $${D_{rel}^{cpd}}$$ characterizes the uniformity and 'spreadout' of the corresponding compound collection, it treats cells as positionally independent. Some of the positional information lost can be recaptured by another diversity measure, which is also related in form to Shannon entropy. This new measure $${D_{N_{bin}}^{cell} (\lambda)}$$ characterizes the distribution of occupied cells along each axis of chemistry space. The normalized measure $${\left\langle {D_{rel}^{cell}}\right\rangle}$$ over all axes is given then by the average. Examples illustrating the applicability of these two Shannon-like measures to compound collections are presented. [ABSTRACT FROM AUTHOR]

The information systems of the local spatial “events” of two-electrons in the stockholder atoms-in-molecules (AIM) are established for the three-representative states of the hydrogen molecule in the minimum basis set description: the ground ( bonding, singlet) state, the singly-excited ( non-bonding, triplet) state describing the free atoms of the system “promolecule”, and the doubly-excited ( anti-bonding, singlet) state. The Hirshfeld atoms reflect the molecular electron densities thus reflecting the bonding and anti-bonding polarizations of AIM in the corresponding molecular states. Their integral entropy-covalency and information-ionicity descriptors are expressed in terms of the relevant Shannon entropies of the molecular/promolecular and atomic probability distributions of electrons, the numerical values of which have been previously reported. The resulting estimates of the information-theoretic bond indices provide the entropic “fingerprints” of the bonding conditions in these three prototype electron configurations, which reflect upon the associated displacements in the information contained in the molecular electronic distribution relative to the initial, promolecular reference. A brief discussion of such changes due to the AIM contraction and/or promotion in the molecule and their mutual polarizations implied by the bonding/antibonding character of the molecular state is presented. This analysis uncovers the information origins of the covalent bond in this prototype molecular system. [ABSTRACT FROM AUTHOR]

The molecular similarity problem is addressed within the recently proposed information-theoretic (IT) approach to molecular electronic structure. In this Communication Theory of the chemical bond the direct criteria of the overall similarity are formulated in terms of the conditional-entropy (average noise, IT-covalency) and mutual-information (information flow, IT-ionicity) descriptors of the compared molecular communication systems, and the associated variational principles for their maximum entropy/information resemblance are formulated and discussed. Implications for molecular similarity from the parallel and sequential arrangements of the compared information channels of molecules or their fragments are investigated and tested on illustrative π-electron systems. [ABSTRACT FROM AUTHOR]