Quantum states of physical domains in molecular systems: A three-state model approach.

The physical regions (domains or basins) within the molecular structure are open systems that exchange charge between them and, consequently, house a fractional number of electrons (net charge). The natural framework describing the quantum states for these domains is the density matrix (DM) in its grand-canonical version, which corresponds to a convex expansion into a set of basis states of an integer number of electrons. In this report, it is shown that the solution for these quantities is supported by the DM expansion into three states of different numbers of particles: the neutral and two (edge) ionic states. The states and the average number of particles in the domains (fractional occupation population) are determined by the coefficients of the expansion in terms of the fundamental transference magnitudes, revealing the donor/acceptor character of the domains by which the quantum accessible states are discussed.

Topological description of complex patterns of bonding, charge transference and structural changes in chemical reactions: SN2 type reactions, a case study.

In this report we deeply explore the electron rearrangements for a chemical reaction case study of the SN2 type OH- + CH3-CN ↔ HO-CH3 + CN-, within the topological formalism of QTAIM model in its local and the non-local or integrated form implementations. This is part of our main interest of a subtle description of the electron distribution at both, equilibrium and non-equilibrium conformations for the state function at correlated level of approximation. The emphasis is mainly placed on the determination of complex patterns of interaction of two or four electrons in three centers (2e-3c, 4e-3c). These results confirm them as a measure of electron excess or electron deficiency effects and revels their crucial importance in the charge transference process and its relationship with these complex patterns. All electronic magnitudes were followed along the configurations from reactants to products passing through the corresponding transition state TS. The results obtained throughout this case study reveals the high performance of the topological formalism of Bader type in the subtle description of electron rearrangements in a chemical transformation.

Topological population analysis and pairing/unpairing electron distribution evolution: Atomic B3+ cluster bending mode, a case study.

Local and non-local topological treatment of electronic distributions are applied to a simple out of equilibrium case of an electron-deficient three-atom cluster, B3+. The bending movement is described in detail through the onset and disappearance of critical points defining two kinds of molecular structures, characterizing a transition state (TS) and predicting two stable equilibrium geometries. All points in this rich evolution and the structural change in the out of equilibrium conformations has been featured and distinguished by the behavior of the population magnitudes and of the paired and unpaired electron densities within the non-local and local points of view of the topological formalism. The unpaired or electron hole density appears as relevant in both versions, the non-local or integrated one, in which it is sometimes called free-valence and also for its complementary counterpart, the local one, to describe and to quantify the interatomic interactions. The stability of the cluster B3+ is characterized in terms of a topologically defined ring structure and the highest total two- and three-center populations, thus showing the role of the geometry, the covalence, and the complex patterns. Consideration of the electron correlation effects constitutes the basement of the results gathered, thus displaying their influence in the formation and breaking of boron bonding interactions.

Do Organometallic CH4-Me(+p) Adducts and X4H(+) (X = P, As) Clusters Undergo Two-Electron Three-Center Interactions? Some Aspects of Discussion.

Most of the systems possessing true two-electron three-center interactions are electron deficient compounds like boron hydrids, closo-boranes, and some organic ions such as butonium cations. In this work, we perform a detailed study of the electron distribution for two different types of systems to which likewise interactions has been adjudicated: organometallic CH4-Me(+p) (p = 1, 2) adducts with Me, alkaline and earth alkaline metallic ions of Li, Na, K, Be, Mg, Ca in their stable gaseous phase and X4H(+) (X = P, As) simple clusters. For this purpose, topological analysis of the electron density decomposed into its effectively paired and unpaired contributions has been carried out looking for complex interactions.

Pairing and unpairing electron densities in organic systems: two-electron three center through space and through bonds interactions.

Two-electron three-center bonding interactions in organic ions like methonium (CH5(+)), ethonium (C2H7(+)), and protonated alkanes n - C4H11(+) isomers (butonium cations) are described and characterized within the theoretical framework of the topological analysis of the electron density decomposition into its effectively paired and unpaired contributions. These interactions manifest in some of this type of systems as a concentration of unpaired electron cloud around the bond paths, in contrast to the well known paradigmatic boron hydrids in which it is not only concentrated close to the atomic nucleus and the bond paths but out of them and over the region defined by the involved atoms as a whole. This result permits to propose an attempt of classification for these interactions based in such manifestations. In the first type, it is called as interactions through bonds and in the second type as interactions through space type.

Communication: reduced density matrices in molecular systems: grand-canonical electron states.

Grand-canonical like descriptions of many electron atomic and molecular open systems which are characterized by a non-integer number of electrons are presented. Their associated reduced density matrices (RDMs) are obtained by introducing the contracting mapping for this type of distributions. It is shown that there is loss of information when connecting RDMs of different order by partial contractions. The energy convexity property of these systems simplifies the description. Consequently, this formulation opens the possibility to a new look for chemical descriptors such as chemical potential and reactivity among others. Examples are presented to discuss the theoretical aspects of this work.

Note: Energy convexity and density matrices in molecular systems.

A novel appropriate definition for the density matrix for an interacting Coulombic driven atomic or molecular system with non-integer number of particles is given. Our approach leads to a direct derivation of the proposal reported by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)] and points out its suitability and perspective advances.

Determination of Local Spins by Means of a Spin-Free Treatment.

This work describes a Mulliken-type partitioning of the expectation value of the spin-squared operator corresponding to an N-electron system. Our algorithms, which are based on a spin-free formulation, predict appropriate spins for the molecular fragments (at equilibrium geometries and at dissociation limits) and can be applied to any spin symmetry. Numerical determinations performed in selected closed- and open-shell systems at correlated level are reported. A comparison between these results and their counterpart ones arising from other alternative approaches is analyzed in detail.

Electronic Structure and Effectively Unpaired Electron Density Topology in closo-Boranes: Nonclassical Three-Center Two-Electron Bonding.

This article provides a detailed study of the structure and bonding in closo-borane cluster compounds X2B3H3 (X = BH(-), P, SiH, CH, N), with particular emphasis on the description of the electron distribution using the topology of the quantum many-body effectively unpaired density. The close relationship observed between the critical points of this quantity and the localization of the electron cloud allows us to characterize the nonclassical bonding patterns of these systems. The obtained results confirm the suitability of the local rule to detect three-center two-electron bonds, which was conjectured in our previous study on boron hydrides.

Relationships between cumulant and spin-density matrices: application to the decomposition of spin.

This paper reports the derivation of a relationship between some elements of the cumulant matrix of the second-order reduced density matrix and the elements of the spin-density matrix. This relationship turns out to be very useful to determine local spins through the partitioning of the spin expectation value of an N-electron system. The procedure enables expression of both one- and two-center contributions only in terms of one-electron matrix elements, the elements of the spin-density matrix. We report numerical determinations of local spins in the Hilbert space of atomic orbitals in selected molecules and radicals in triplet and doublet states.

Topology of the electron density in open-shell systems.

This work describes the decomposition of the electron density field of open-shell molecular systems into physically meaningful contributions. The new features that the open-shell nature of the wave function introduces into these fields are topologically studied and discussed, showing the charge concentration and depletion regions within the system. The localized character (field concentration only close to the nuclear positions of the system) or the nonlocalized character (concentration in other regions of the system) is used to study the reliability of the Lewis model of bonding to describe chemical bonding phenomenon in open-shell systems. Numerical examples are reported for molecular systems at a correlated level of approximation, in the single-double configuration interaction wave function approach.

Topology of the Effectively Paired and Unpaired Electron Densities for Complex Bonding Patterns: The Three-Center Two-Electron Bonding Case.

Our previously reported local formalism of the electron density decomposition into effectively paired and unpaired densities is applied to electron deficient molecular systems possessing complex bonding patterns. It is shown that the unpaired density is not only near the nuclear positions, like in classical bonds, but also spills out over the bonding regions, to compensate the electron deficiency. Topological information obtained from the effectively unpaired density, which may not be directly observed from the total density, allows us to establish a procedure to detect complex interactions. This study is complemented with results arising from nonlocal formalism of topological population analyses. The conclusions from both formalisms are in complete agreement and permit to interpret the well-known structural information from Lipscomb styx numbers going beyond it in cases where the electronic description becomes ambiguous, pointing out the subtle information contained in the unpaired density. Numerical results for three-center two-electron bondings in the boranes B2H6, B4H10, B5H9, and B5H11 are reported.

Determination of energies and electronic densities of functional groups according to partitionings in the physical space.

This work formulates our previously reported partitionings of the first-order reduced density matrix and the molecular electronic energy using for both quantities an identical mathematical framework. The procedure provides a consistent and rigorous scheme for extending our algorithms to unions of atomic domains, in order to describe molecular fragments which can be identified as functional groups. Numerical determinations, performed in several series of organic compounds and clusters, support the reliability of our methodology to describe properties of atomic groups.

Covalent bond orders revisited: the open-shell case.

This article states the concept of covalent bond order for open-shell systems from the invariance properties of the first- and second-order reduced density matrices for all the components of a multiplet state. A general bond order definition is formulated in the framework of the electronic population analyses in the Hilbert space of atomic orbitals.

Energy decompositions according to physical space partitioning schemes: treatments of the density cumulant.

This article is a continuation of our previous paper on schemes of energy decompositions of molecular systems in the real space [D. R. Alcoba et al., J. Chem. Phys. 122, 074102 (2005)] now using correlated state functions. We study, according to physical arguments, the appropriate management of the density cumulant arising from the second-order reduced density matrix at correlated level, whose contributions can be assigned to one-center or to two-center terms in the energy partitioning. Our treatments are applied within two physical space partitioning schemes: the Bader partitioning into atomic basins and the fuzzy atom procedure. The results obtained in selected molecules are analyzed and discussed in detail.

Laplacian field of the effectively unpaired electron density: determination of many-body effects on electron distributions.

This work carries out the study of the Laplacian field function of the electron density L(r) = -nabla2rho(r) splitted in two contributions rho(r) = rho(p)(r) + rho(u)(r), which correspond to the effectively paired and effectively unpaired electron densities, respectively. The visualization of the concentration and depletion of these fields and their spatial localization show no contribution of the effectively unpaired electrons to the conventional bonding among two centers, but the field -nabla2rho(u)(r) provides an interesting structure. We also study the reliability of the information contained in the partitioning of this electron density field function for describing nonclassical bondings as the three-center two-electron ones.

Decomposition of the first-order reduced density matrix: an isopycnic localization treatment.

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.

An orbital localization criterion based on the theory of "fuzzy" atoms.

This work proposes a new procedure for localizing molecular and natural orbitals. The localization criterion presented here is based on the partitioning of the overlap matrix into atomic contributions within the theory of "fuzzy" atoms. Our approach has several advantages over other schemes: it is computationally inexpensive, preserves the sigma/pi-separability in planar systems and provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and atomic occupancies. The corresponding algorithm has been implemented and its efficiency tested on selected molecular systems.

A study of the partitioning of the first-order reduced density matrix according to the theory of atoms in molecules.

This work describes a simple spatial decomposition of the first-order reduced density matrix corresponding to an N-electron system into first-order density matrices, each of them associated to an atomic domain defined in the theory of atoms in molecules. A study of the representability of the density matrices arisen from this decomposition is reported and analyzed. An appropriate treatment of the eigenvectors of the matrices defined over atomic domains or over unions of these domains allows one to describe satisfactorily molecular properties and chemical bondings within a determined molecule and among its fragments. Numerical determinations, performed in selected molecules, confirm the reliability of our proposal.

Electron-density topology in molecular systems: paired and unpaired densities.

This work studies the partitioning of the electron density into two contributions which are interpreted as the paired and the effectively unpaired electron densities. The topological features of each density field as well as of the total density are described localizing the corresponding critical points in simple selected molecules (local formalism). The results show that unpaired electron-density concentrations occur out of the topological bonding regions whereas the paired electron densities present accumulations inside those regions. A comparison of these results with those arising from population analysis techniques (nonlocal or integrated formalisms) is reported.