Abstract
Concise information on the general features of the quantum-mechanical current density induced in the electrons of a molecule by a spatially uniform, time-independent magnetic field is obtained via a stagnation graph that shows the isolated singularities and the lines at which the current-density vector field vanishes. Stagnation graphs provide compact description of current-density vector fields and help the interpretation of molecular magnetic response, e.g., magnetic susceptibility and nuclear magnetic shielding. The stagnation graph of six cyclic, planar aromatic molecules has been obtained at the Hartree-Fock level via a procedure based on continuous transformation of the origin of the current density formally annihilating the diamagnetic contribution. Some common distinctive elements observed for cyclic aromatic rings ${\mathrm{C}}_{n}{\mathrm{H}}_{n}$, with $n=3,4,\dots{},8$, in the presence of a magnetic field normal to the molecular plane, are discussed. The results can be used for a general discussion of diatropism in aromatic systems.
Highlights
In his fourth paper on quantization as an eigenvalue problem1͔, Schrödinger gave a definition of quantummechanical current density satisfying a continuity equation formally identical to that of classical electrodynamics
The magnetic response of a molecule to an external magnetic field and to intramolecular magnetic dipole moments at the nuclei can be investigated by analyzing the currentdensity induced in the electron cloud
A precise, rational, and compact description of the current density is obtained by the stagnation graph defined by the singularities of the vector field, i.e., isolated points and onedimensional manifolds that are referred to as vortex and saddle lines
Summary
In his fourth paper on quantization as an eigenvalue problem1͔, Schrödinger gave a definition of quantummechanical current density satisfying a continuity equation formally identical to that of classical electrodynamics. A few months later in 1926, Madelung proposed an alternative foundation of quantum theory allowing for a hydrodynamical analogy2͔. Within the hydrodynamical approach to quantum mechanics, the continuity condition and a vector equation, with the same form as the Hamilton-Jacobi equation of motion of classical mechanics, replace the wave equation5–11͔. Bohm showed that the hydrodynamical representation of quantum mechanics offers a broader conceptual framework. It is deterministic and provides an interpretation of physical reality alternative to that of the Copenhagen School5,6͔. Besides providing powerful interpretative tools, maps of current-density field are interesting by themselves for more general reasons. The singularities determine the topological structure of the vector field, which is described in compact form by a “stagnation graph” ͑SGthat conveys essential information31–33͔ for understanding magnetic response.
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