Chapter 7
Topology of Quantum Mechanical
Current Density Vector Fields Induced
in a Molecule by Static Magnetic
Perturbations
P. Lazzeretti
Abstract It is shown that the quantum mechanical theory of static magnetic
properties can be reformulated in terms of electronic current densities induced by an
external magnetic field and permanent magnetic dipole moments at the nuclei.
Theoretical relationships are reported to evaluate magnetizability, nuclear magnetic
shielding and nuclear spin-spin coupling via the equations of classical electromagnetism, assuming that the current density is evaluated by quantum mechanical
methods. Emphasis is placed on the advantage of the proposed formulation, as an
alternative to procedures based on perturbation theory, as regards interpretation of
response allowing for the ideas of current density tensor and current susceptibility
vector. Visualisation of the electronic interaction with a magnetic field and
intramolecular perturbations, e.g., nuclear magnetic dipoles, is made possible via
current density maps, nuclear shielding density maps and plots of nuclear spin-spin
coupling density. Topological analysis of the quantum mechanical current density
in terms of Gomes stagnation graphs is shown to yield fundamental information for
understanding magnetic response. Examples are given for a few archetypal molecules. A topological definition of delocalized electron currents is proposed.
Keywords Electronic current densities induced by magnetic fields and nuclear
magnetic dipoles Á Molecular magnetic response Á Property densities Á Gauge
invariance Á Charge conservation Á Current density vectors and current density
tensors Á Current susceptibility vectors Á Topology of current density vector fields Á
Bifurcations Á Topological definition of ring current
P. Lazzeretti (&)
Dipartimento di Scienze chimiche e geologiche,
Università degli Studi di Modena e Reggio Emilia,
Via Campi 213/b, 41125 Modena, Italy
e-mail: lazzeret@unimore.it
© Springer International Publishing Switzerland 2016
R. Chauvin et al. (eds.), Applications of Topological Methods
in Molecular Chemistry, Challenges and Advances in Computational
Chemistry and Physics 22, DOI 10.1007/978-3-319-29022-5_7
151
Topology of Quantum Mechanical
Current Density Vector Fields Induced
in a Molecule by Static Magnetic
Perturbations
P. Lazzeretti
Abstract It is shown that the quantum mechanical theory of static magnetic
properties can be reformulated in terms of electronic current densities induced by an
external magnetic field and permanent magnetic dipole moments at the nuclei.
Theoretical relationships are reported to evaluate magnetizability, nuclear magnetic
shielding and nuclear spin-spin coupling via the equations of classical electromagnetism, assuming that the current density is evaluated by quantum mechanical
methods. Emphasis is placed on the advantage of the proposed formulation, as an
alternative to procedures based on perturbation theory, as regards interpretation of
response allowing for the ideas of current density tensor and current susceptibility
vector. Visualisation of the electronic interaction with a magnetic field and
intramolecular perturbations, e.g., nuclear magnetic dipoles, is made possible via
current density maps, nuclear shielding density maps and plots of nuclear spin-spin
coupling density. Topological analysis of the quantum mechanical current density
in terms of Gomes stagnation graphs is shown to yield fundamental information for
understanding magnetic response. Examples are given for a few archetypal molecules. A topological definition of delocalized electron currents is proposed.
Keywords Electronic current densities induced by magnetic fields and nuclear
magnetic dipoles Á Molecular magnetic response Á Property densities Á Gauge
invariance Á Charge conservation Á Current density vectors and current density
tensors Á Current susceptibility vectors Á Topology of current density vector fields Á
Bifurcations Á Topological definition of ring current
P. Lazzeretti (&)
Dipartimento di Scienze chimiche e geologiche,
Università degli Studi di Modena e Reggio Emilia,
Via Campi 213/b, 41125 Modena, Italy
e-mail: lazzeret@unimore.it
© Springer International Publishing Switzerland 2016
R. Chauvin et al. (eds.), Applications of Topological Methods
in Molecular Chemistry, Challenges and Advances in Computational
Chemistry and Physics 22, DOI 10.1007/978-3-319-29022-5_7
151