Stagnation graphs and separatrices of local bifurcations in velocity and current density planar vector fields

Topological data analysis of vortices in the magnetically-induced current density in LiH molecule

A novel strategy for extracting axial (AV) and toroidal (TV) vortices in the magnetically-induced current density (MICD) in molecular systems is introduced, and its pilot application to LiH molecule is demonstrated. It exploits differences in the topologies of AV and TV cores and involves two key steps: selecting a scalar function that can describe vortex cores in MICD and its subsequent topological analysis. The scalar function of choice is Ω^B_α based on the velocity-gradient Ω method known in research on classical flows. The Topological Data Analysis (TDA) is then used to analyze the Ω^B_α scalar field. In particular, TDA robustly assigns distinct topological features of this field to different vortex types in LiH: AV to saddle-maximum separatrices which connect maxima to 2-saddles located on the domain's boundary, TV to a 1-cycle of the super-level sets of the input data. Both are extracted as the most persistent features of the topologically-simplified Ω^B_α scalar field.

A natural scheme for the quantitative analysis of the magnetically induced molecular current density using an oriented flux-weighted stagnation graph. I. A minimal example for LiH

A new natural scheme is introduced to analyze quantitatively the magnetically induced molecular current density vector field, J. The set of zero points of J, which is called its stagnation graph (SG), has been previously used to study the topological features of the current density of various molecules. Here, the line integrals of the induced magnetic field along edges of the connected subset of the SG are calculated. The edges are oriented such that all weights, i.e., flux values become non-negative, thereby, an oriented flux-weighted (current density) stagnation graph (OFW-SG) is obtained. Since in the exact theoretical limit, J is divergence-free and due to the topological characteristics of such vector fields, the flux of all separate vortices (current density domains) and neighbouring connected vortices can be determined exactly by adding the weights of cyclic subsets of edges (i.e., closed loops) of the OFW-SG. The procedure is exemplified by the minimal example of LiH for a weak homogeneous external magnetic field, B, perpendicular to the chemical bond. The OFW-SG exhibits one closed loop (formally decomposed into two edges), and an open line extending to infinity on both of its ends. The method provides the means of accurately determining the strength of the current density even in molecules with a complicated set of distinct vortices.

Origin Independent Current Density Vector Fields Induced by Time-Dependent Magnetic Field. I. The LiH molecule

The electronic current density, induced in a molecule by the optical magnetic field associated to a frequency dependent monochromatic plane wave, assumed to be spatially uniform within the electric quadrupole approximation, has been studied by a theoretical method based on continuous translation of its origin. The induced electronic current density vector field designated by this procedure, invariant of the origin for any point of the molecular domain, is obtained via a computational scheme formally annihilating the diamagnetic contribution of the conventional common-origin approach based on perturbation theory. In a preliminary application of the theoretical methods outlined in the present work, the simple molecule of lithium hydride has been investigated. Particular attention has been paid to the structure of induced electronic current density for several values of the magnetic field frequency, by investigating equilibrium points of four different types, organized in stagnation lines which constitute its stagnation graph, i.e., a topological fingerprint of the vector field conveying complete information in condensed form, to verify the fulfillment of fundamental requirements, e.g., the continuity equation and the Poincaré-Hopf theorem on spherical and toroidal surfaces.

Current density and molecular magnetic properties

We give an overview of the molecular response to an external magnetic field perturbing quantum mechanical systems. We present state-of-the-art methods for calculating magnetically-induced current-density susceptibilities. We discuss the essence and properties of current-density susceptibilities and how molecular magnetic properties can be calculated from them. We also review the theory of spin-current densities, how relativity affects current densities and magnetic properties. An overview of the magnetic ring-current criterion for aromaticity is given, which has implications on theoretical and experimental research. The recently reported theory of antiaromaticity and how molecular symmetry affects the magnetic response are discussed and applied to closed-shell paramagnetic molecules. The topology of magnetically induced current densities and its consequences for molecular magnetic properties are also presented with twisted and toroidal molecules as examples.

Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs

Stagnation graphs provide a useful tool to analyze the main topological features of the often complicated vector field associated with magnetically induced currents. Previously, these graphs have been constructed using response quantities appropriate for modest applied magnetic fields. We present an implementation capable of producing these graphs in arbitrarily strong magnetic fields, using current-density-functional theory. This enables us to study how the topology of the current vector field changes with the strength and orientation of the applied magnetic field. Applications to CH4, C2H2 and C2H4 are presented. In each case, we consider molecular geometries optimized in the presence of the magnetic field. The stagnation graphs reveal subtle changes to this vector field where the symmetry of the molecule remains constant. However, when the electronic state and symmetry of the corresponding equilibrium geometry changes with increasing field strength, the changes to the stagnation graph are extensive. We expect that the approach presented here will be helpful in interpreting changes in molecular structure and bonding in the strong-field regime.