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Journal of Mathematical Chemistry

On interference of orbital communications in molecular systems

Roman F. Nalewajski

Published Dec 2, 2010• 28 citations
Publisher

Abstract

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.

DOI
10.1007/s10910-010-9777-0

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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.
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 molecular density-partition problem is reexamined and the information-theoretic (IT) justification of the stockholder division rule is summarized. The ensemble representations of the promolecular and molecular mixed states of constituent atoms are identified and the electron probabilities in the isoelectronic stockholder atoms-in-molecules (AIM) are used to define the molecular-orbital ensembles for the bonded Hirshfeld atoms. In the pure quantum state of the whole molecular system its interacting (entangled) fragments are described by the subsystem density operators, with the subsystem physical properties being generated by the partial traces involving the fragment density matrices.
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.
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.
In photodissociation of a homonuclear diatomic molecule, conservation of linear and angular momenta requires that each atomic fragment follow identical but oppositely directed trajectories with respect to the molecular center of mass. For certain values of the incident photon energy, one of the atoms will be electronically excited while the other will be in its ground state. When a detector records the undispersed fluorescence from the decay of the excited atom it cannot distinguish which trajectory the excited atom followed and hence whether the photon possesses a blue or red Doppler shift corresponding to recoil towards or away from the detector. According to the principle of superposition in quantum mechanics, the wave function of the photon must be a sum over two paths corresponding to emission from an excited atom in either of the two recoil directions. Detection of such photons gives rise to an interference between the probability amplitudes for the two paths that is manifested in the output from the photodetector as a beat frequency. Since photodissociation of an ensemble of molecules gives rise to a distribution of recoil directions relative to the electric vector of the incident radiation, a range of beat frequencies must be present in the detected signal. This paper gives a calculation of expected output voltage waveforms from a photodetector viewing atomic fluorescence excited by molecular photodissociation. Expressions are given for partial Fourier transforms of the photodetector signals. The results show that the shape of the voltage waveforms depends strongly on the asymmetry parameter for recoil of the photofragments and on whether or not the magnetic sublevels of the excited atomic state are created in a coherent superposition.