7.1 Introduction
The magnetic response of diamagnetic atoms, molecules and clusters, i.e., typical
quantum mechanical systems, can be effectively interpreted and visualized via the
laws of classical electrodynamics, allowing for functions of position r which
describe the electronic charge density qðr Þ, a scalar property, and the electronic
current density J(r), a vector field, evaluated by quantum mechanical methods.
An introduction to magnetic properties merging classical relationships with
quantum mechanical computational recipes constitutes a trait d’union between
classical and quantum mechanics rather interesting from the epistemological point
of view. In fact, qðr Þ and J(r) are subobservables [1], that is, expectation values of
corresponding quantum mechanical operators: if these expectation values are
known as functions in R
3 , a number of molecular electromagnetic properties can be
evaluated without the explicit use of electronic wave functions.
The central aim of this chapter is to give a simple, self-contained approach to a
set of molecular magnetic properties in terms of induced current densities and
related property density maps, via classical relationships combined with quantum
mechanical definitions, and computational procedures. Some efforts are made to
document the effectiveness of such a theoretical treatment, in the attempt to
rationalize the phenomenology and to form a mental image of the mechanisms
underlying the electronic interaction with static magnetic perturbations.
Particular prominence is given to J
B
ðrÞ and J
m I ðrÞ, describing respectively the
current density induced in the electron cloud by a spatially uniform static magnetic
field B and by a permanent magnetic dipole m I at nucleus I. Emphasis is placed on
the practical advantages arising from the use of functions defined within the
3-dimensional space for interpreting experimental data, e.g., the parameters of
nuclear magnetic resonance (NMR) spectroscopy, by means of two-dimensional
maps and perspective representations of three-dimensional fields.
Besides, the mathematical properties of J
B and J
m I as dynamical systems
constitute an object of investigation per se quite appealing from the purely mathematical point of view. The structure of these vector fields can be studied by the
tools of differential topology. The phase portraits giving a geometric representation
of the trajectories in the vicinity of points at which the modulus of the current
density vanishes are particularly interesting.
Terminology and notation adopted in previous papers and reviews [2–5] are
used, allowing for standard tensor formalism, e.g., summation over repeated Greek
indices is implied according to the Einstein convention.
152
P. Lazzeretti
The magnetic response of diamagnetic atoms, molecules and clusters, i.e., typical
quantum mechanical systems, can be effectively interpreted and visualized via the
laws of classical electrodynamics, allowing for functions of position r which
describe the electronic charge density qðr Þ, a scalar property, and the electronic
current density J(r), a vector field, evaluated by quantum mechanical methods.
An introduction to magnetic properties merging classical relationships with
quantum mechanical computational recipes constitutes a trait d’union between
classical and quantum mechanics rather interesting from the epistemological point
of view. In fact, qðr Þ and J(r) are subobservables [1], that is, expectation values of
corresponding quantum mechanical operators: if these expectation values are
known as functions in R
3 , a number of molecular electromagnetic properties can be
evaluated without the explicit use of electronic wave functions.
The central aim of this chapter is to give a simple, self-contained approach to a
set of molecular magnetic properties in terms of induced current densities and
related property density maps, via classical relationships combined with quantum
mechanical definitions, and computational procedures. Some efforts are made to
document the effectiveness of such a theoretical treatment, in the attempt to
rationalize the phenomenology and to form a mental image of the mechanisms
underlying the electronic interaction with static magnetic perturbations.
Particular prominence is given to J
B
ðrÞ and J
m I ðrÞ, describing respectively the
current density induced in the electron cloud by a spatially uniform static magnetic
field B and by a permanent magnetic dipole m I at nucleus I. Emphasis is placed on
the practical advantages arising from the use of functions defined within the
3-dimensional space for interpreting experimental data, e.g., the parameters of
nuclear magnetic resonance (NMR) spectroscopy, by means of two-dimensional
maps and perspective representations of three-dimensional fields.
Besides, the mathematical properties of J
B and J
m I as dynamical systems
constitute an object of investigation per se quite appealing from the purely mathematical point of view. The structure of these vector fields can be studied by the
tools of differential topology. The phase portraits giving a geometric representation
of the trajectories in the vicinity of points at which the modulus of the current
density vanishes are particularly interesting.
Terminology and notation adopted in previous papers and reviews [2–5] are
used, allowing for standard tensor formalism, e.g., summation over repeated Greek
indices is implied according to the Einstein convention.
152
P. Lazzeretti