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Academic Journal of Chemistry

Interacting Subsystems and Their Molecular Ensembles

Roman F. Nalewajski

Published Apr 25, 2020• 7 citations
Publisher

Abstract

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.

DOI
10.32861/ajc.54.25.30

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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.
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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.
Overall descriptors of the information (determinicity) and entropy (uncertainty) content of complex molecular states are reexamined. These resultant concepts combine the classical (probability) contributions of Fisher and Shannon, and the relevant nonclassical supplements due to the state phase/current. The information-theoretic principles determining equilibria in molecules and their fragments are explored and the nonadditive part of the global entropy is advocated as a descriptor of the classical index of the quantum entanglement of molecular subsystems. Affinities associated with the probability and phase fluxes are identified and the criterion of vanishing overall information-source is shown to identify the system stationary electronic states. The production of resultant density of the gradient-information is expressed in terms of the conjugate affinities (forces, perturbations) and fluxes (currents, responses). The Schrödinger dynamics of probability and phase components of molecular electronic states is used to determine the temporal evolution of the overall gradient information and complex entropy. The global sources of the resultant information/entropy descriptors are shown to be of purely nonclassical origin, thus identically vanishing in real electronic states, e.g., the nondegenerate ground state of a molecule.
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.
We present a reduced density operator for electronically open molecules by explicitly averaging over the environmental degrees of freedom of the composite Hamiltonian. Specifically, we include the particle-number nonconserving (particle-breaking) interactions responsible for the sharing of electrons between the molecule and the environment, which are neglected in standard formulations of quantum statistical mechanics. We propose an unambiguous definition of the partial trace operation in the composite fermionic Fock space based on composite states in a second quantization framework built from a common orthonormal set of orbitals. Thereby, we resolve the fermionic partial trace ambiguity. The common orbital basis is constructed by spatial localization of the full orbital space, in which the full composite Hamiltonian naturally partitions into a molecule Hamiltonian, an environment Hamiltonian, and an interaction Hamiltonian. The new reduced density operator is based on the approximation of commutativity between the subsystem Hamiltonians (i.e., molecule and environment Hamiltonians) and the interaction Hamiltonian, but our methodology provides a hierarchical approach for improving this approximation. The reduced density operator can be viewed as a generalization of the grand canonical density operator. We are prompted to define the generalized chemical potential, which aligns with the standard interpretation of the chemical potential, apart from the possibility of fractional rather than strictly integer electron transfer in our framework. In contrast to standard approaches, our framework enables an explicit consideration of the electron occupancy in the environment at any level of theory, irrespective of the model used to describe the molecule. Specifically, our reduced density operator is fully compatible with all possible level-of-theory treatments of the environment. The approximations that render our reduced density operator identical to the grand canonical density operator are (i) restriction of excitations to occur within the same orbitals and (ii) assumption of equal interaction with the environment for all molecule spin orbitals (i.e., the wide-band approximation).
The theory of open quantum systems (OQSs) is applied to partition the squared spin operator into fragment (local spin) and interfragment (spin-coupling) contributions in a molecular system. An atomic or fragment subsystem is described by a quantum mechanical mixed density operator composed of sectors, characterized by different integer number of electrons that appear with specific probabilities. The OQS fragment spin operators coincide with those defined by Clark and Davidson in their paper on local spins (J. Chem. Phys., 2001, 115, 7382) and are fully consistent with the theory of local operators by Stollhoff and Fulde (J. Chem. Phys., 1980, 73, 4548). OQSs provide a unique way to rationalize the non-zero values of local spins found in closed-shell molecules, a fact that has led to a large number of modified definitions being proposed, which we show suffer from inconsistencies. The OQS viewpoint makes it easy to build models for localized and itinerant spins. These models are used to classify possible local spin arrangements. The role of electron correlation is also studied through the analysis of the Hubbard Hamiltonian in small chains. Local spins result from a game played differently by localized and delocalized electrons. A number of examples exemplifying the ability of the OQS local spin perspective to uncover simple chemical patterns are examined.