Behavior of the Position–Spread Tensor in Diatomic Systems

Near equivalence of polarizability and bond order flux metrics for describing covalent bond rearrangements

Identification of the breaking point for the chemical bond is essential for our understanding of chemical reactivity. The current consensus is that a point of maximal electron delocalization along the bonding axis separates the different bonding regimes of reactants and products. This maximum transition point has been investigated previously through the total position spread and the bond-parallel components of the static polarizability tensor for describing covalent bond breaking. In this paper, we report that the first-order change of the Wiberg and Mayer bond index with respect to the reaction coordinate, the bond flux, is similarly maximized and is nearly equivalent with the bond breaking points determined by the bond-parallel polarizability. We investigate the similarites and differences between the two bonding metrics for breaking the nitrogen triple bond, twisting around the ethene double bond, and a set of prototypical reactions in the hydrogen combustion reaction network. The Wiberg-Mayer bond flux provides a simpler approach to calculating the point of bond dissociation and formation and can yield greater chemical insight through bond specific information for certain reactions where multiple bond changes are operative.

When Is a Bond Broken? The Polarizability Perspective

The question of when a chemical bond can be said to be broken is of fundamental chemical interest but has not been widely studied. Herein we propose that the maxima of static polarizability along bond dissociation coordinates naturally define cutoff points for bond rupture, as they represent the onset of localization of shared electron density into constituent fragments. Examples of computed polarizability maxima over the course of bond cleavage in main-group and transition metal compounds are provided, across covalent, dative and charge-shift bonds. The behavior along reaction paths is also considered. Overall, the static polarizability is found to be a sensitive reporter of electronic structure reorganization associated with bond stretching, and thus can serve as a metric for describing bond cleavage (or diagnose the absence of a chemical bond).

Cumulant mapping as the basis of multi-dimensional spectrometry

Cumulant mapping employs a statistical reconstruction of the whole by sampling its parts. The theory developed in this work formalises and extends ad hoc methods of 'multi-fold' or 'multi-dimensional' covariance mapping. Explicit formulae have been derived for the expected values of up to the 6th cumulant and the variance has been calculated for up to the 4th cumulant. A method of extending these formulae to higher cumulants has been described. The formulae take into account reduced fragment detection efficiency and a background from uncorrelated events. Number of samples needed for suppressing the statistical noise to a required level can be estimated using Matlab code included in Supplemental Material. The theory can be used to assess the experimental feasibility of studying molecular fragmentations induced by femtosecond or X-ray free-electron lasers. It is also relevant for extending the conventional mass spectrometry of biomolecules to multiple dimensions.

The localization spread and polarizability of rings and periodic chains

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.

An unexpected bridge between chemical bonding indicators and electrical conductivity through the localization tensor

While the modern theory of the insulating state shows that the conducting or insulating properties of a system can be extracted solely from the ground state properties via the so-called localization tensor (LT), no chemical reading of this important quantity has ever been offered. Here, a remarkable link between the LT and the bond orders as described by the delocalization indices (DIs) of chemical bonding theory is reported. This is achieved through a real space partition of the LT into intra- and interatomic contributions. We show that the convergence or divergence of the LT in the thermodynamic limit, which signals the insulating or conducting nature of an extended system, respectively, can be nailed down to DIs. This allows for the exploitation of traditional chemical intuition to identify essential and spectator atomic groups in determining electrical conductivity. The thermodynamic limit of the LT is controlled by the spatial decay rate of the interatomic DIs, exponential in insulators and power-law in conductors. Computational data of a few selected toy systems corroborate our results.

Decay rate of real space delocalization measures: a comparison between analytical and test systems

We examine in this contribution the possible relation between the spatial decay rate of real space delocalization measures and the insulating- or metallic-like character of molecular and extended systems. We first show that in simple one-electron models, like the Hückel or tight binding approximations, delocalization indices (DIs) are intimately linked to the first-order reduced density matrix (1RDM), whose decay rate is known to be exponential in gapped systems and algebraic in gapless ones. DIs are shown to behave equivalently, with wild oscillations in gapless 1D, 2D and 3D models that do only persist in one-dimensional real cases, as computed at the Hartree-Fock or Kohn-Sham levels. Oscillations are shown to be directly related to Pauling resonant structures and chemical mesomerism. DIs in insulating-like moieties decay extremely fast. We propose that examining the decay of DIs along different directions in real materials may be used to detect facile and non-facile conductivity channels.

The spin-partitioned total position-spread tensor: An application to Heisenberg spin chains

The spin partition of the Total Position-Spread (TPS) tensor has been performed for one-dimensional Heisenberg chains with open boundary conditions. Both the cases of a ferromagnetic (high-spin) and an anti-ferromagnetic (low-spin) ground-state have been considered. In the case of a low-spin ground-state, the use of alternating magnetic couplings allowed to investigate the effect of spin-pairing. The behavior of the spin-partitioned TPS (SP-TPS) tensor as a function of the number of sites turned to be closely related to the presence of an energy gap between the ground-state and the first excited-state at the thermodynamic limit. Indeed, a gapped energy spectrum is associated to a linear growth of the SP-TPS tensor with the number of sites. On the other hand, in gapless situations, the spread presents a faster-than-linear growth, resulting in the divergence of its per-site value. Finally, for the case of a high-spin wave function, an analytical expression of the dependence of the SP-TPS on the number of sites n and the total spin-projection Sz has been derived.