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Journal of Mathematical Chemistry

On phases and interference of local communications in molecules

Roman F. Nalewajski

Published Sep 12, 2013• 2 citations
Publisher

Abstract

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 $$\mathrm{H}_{2}^{+}$$ H 2 + and $$\mathrm{H}_{2}$$ H 2 .

DOI
10.1007/s10910-013-0246-4

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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 H2+\mathrm{H}_{2}^{+}H2+​ H 2 + and H2\mathrm{H}_{2}H2​ H 2 .
The classical Information Theory (IT) deals with entropic descriptors of the probability distributions and probability-propagation (communication) systems, e.g., the electronic channels in molecules reflecting the information scattering via the system chemical bonds. The quantum IT additionally accounts for the non-classical (current/phase)-related contributions in the resultant information content of electronic states. The classical and non-classical terms in the quantum Shannon entropy and Fisher information are reexamined. The associated probability-propagation and current-scattering networks are introduced and their Fisher- and Shannon-type descriptors are identified. The non-additive and additive information descriptors of the probability channels in both the Atomic Orbital and local resolution levels are related to the network conditional-entropy and mutual-information, which represent the IT covalency and ionicity components in the classical communication theory of the chemical bond. A similar partition identifies the associated bond indices in the molecular current/phase channels. The resultant bond descriptors combining the classical and non-classical terms, due to the probability and current distributions, respectively, are proposed as generalized communication-noise (covalency) and information-flow (iconicity) concepts in the quantum IT.
The atomic orbitals (AO) contributed by bonded atoms of molecular systems emit or receive the “signals” of electronic allocations to these basis functions, thus acting as the signal source (input) or receiver (output), respectively, in the associated communication network. Each orbital simultaneously participates in both the through-space and through-bridge probability propagations: the former involve direct communications between two AO while the latter are realized indirectly via orbital intermediates. This work examines the interference effects of the amplitudes of molecular probability scatterings, and introduces the operator representation of AO communications. The eigenvalue problem of the associated Hermitian operator combining the forward and reverse information propagations defines the stationary modes (“standing” waves) of the molecular propagation of electronic conditional probabilities. The combined effect of interference between the multiple (direct and indirect) information scatterings, which establishes the stationary distribution of electronic probabilities, is probed. The wave-superposition principle for the conditional-probability amplitudes of the generalized through-bridge information propagation is linked to the idempotency relations of the system density matrix. It explicitly demonstrates that the resultant effect of the probability propagations involving bridges containing all basis functions, at arbitrary bridge orders indeed generates the (stationary) molecular distribution of conditional probabilities.
Abstract: The quantum entanglement of molecular fragments in reactive systems is approached. The "external" (inter-fragment) and “internal” (intra-fragment) correlation energies are expressed in terms of the DFT average correlation holes resulting from the coupling constant integration of the scaled electron repulsion terms in the electronic Hamiltonian. Information networks in the local and configuration resolutions are examined, and their conditional entropy (covalency) and mutual information (iconicity) descriptors are summarized. The local channels in the single Slater determinant approximation of HF theory are explored in some detail. The multisite events in the bond system for the specified molecular state are tackled, cascade (bridge) propagations are examined, and the Fermi (exchange) correlation of HF theory is discussed. The partial density matrices of interacting fragments are introduced, and their role in shaping the ensemble averages of physical observables and effective communications within reactants is examined.
The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrödinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
The classical (modulus/probability) and nonclassical (phase/current) components of molecular states are reexamined and their information contributions are summarized. The state and information continuity relations are discussed and a nonclassical character of the resultant gradient information source is emphasized. The states of noninteracting and interacting subsystems in the model donor-acceptor reactive system are compared and configurations of the mutually-closed and -open equidensity orbitals are tackled. The density matrices for subsystems in reactive complexes are used to describe the entangled molecular fragments and electron communications in donor-acceptor systems which determine the entropic multiplicity and composition of chemical bonds between reactants.
The use of the mutual information (MI) as a measure of the entanglement in quantum systems has gained a consensus in recent years, even if there is an ongoing effort to distinguish the classical and quantum contributions contained therein. This quantity has been first introduced in condensed matter physics, in particular, in studies based on the density matrix renormalization group method. This method has been successfully adapted to quantum chemistry problems, opening the way to compute MI also in molecular systems. A key aspect of this quantity is its dependence on the one-electron (orbital) basis set, even for wave functions that are invariant under unitary transformation of the orbitals. In this work, we investigate the role of the orbital basis set (delocalized or localized, following different strategies) for wave functions expressed as linear combinations of Slater determinants and we give the analytic expression for the MI for a few special cases. This study aims to improve the knowledge of the relationship between the characteristics of the chemical bond (considering a few paradigmatic molecules, H 2, F 2, N 2, and short linear polyenes) and the properties of interest in the field of quantum information theory.