July 26, 2002


Publications in Chemistry by Richard F.W. Bader et al. :

BibTeX references.

Richard F.W. Bader

Professor Emeritus, McMaster, Canada


Topology of Electron Density and Open Quantum Systems

Richard F.W. Bader

in "Density functional theory", edited by Eberhard K.U. Gross and Reiner M. Dreizler,
NATO ASI series. Series B, Physics ; v. 337,
New York : Plenum Press, pp.237-272, 1995.

Proceedings of a NATO Advanced Study Institute on Density Functional Theory,
held August 16-27, 1993, in Il Ciocco, Italy.

Theory of Atoms in Molecules

R. Bader, 1995

Web links:


The presence of a local maxima of the electron density, rho(r), at the position of the nuclei, is the general and dominant topological property. The nuclei are the attractors of the gradient vector field, Grad(rho(r)).

Space around and including a molecule is dijointly and exhuastively partitioned into "basins" between and joining nuclei. Basins are separated via interatomic surfaces. The latter can be defined as critical paths: sets of points being critical points of the density function, i.e., Grad(rho(r)) = 0.

Lines of maximum density are used to construct a molecular graph, which recovers the network of chemical bonds.

The Topological Atom is the Quantum Atom

Topological and quantum definitions of an atom coincide.

The interatomic surface is defined by the set of trajectories that terminate at a point where Grad(rho(r)) = 0. Thus an interatomic surface satisfies the "zero-flux" boundary condition :

< Grad(rho(r_s)) , n(r_s) > = 0 ,  for every point r_s on the surface    Surf(r_s)      (1)

where n(r_s) is the unit vector normal to the surface at r_s. In words, the surface is not crossed by any trajectories of Grad(rho(r_s)). An atom, as a constituent of some larger system, is itself an open system subject to fluxes in charge and momentum through its bounding surface.

The Principle of Stationary Action (Shwinger, 1950+) equates the change in action to the infinitesimal transformations caused by the generators acting in the space-like and time-like surfaces that bound the space-time volume swept out by a system, as well as to displacements in these surfaces. A time-like surface describes the temporal evolution of the spatial boundary enclosing a portion of some total system.

As a consequence, only an open system bounded by surfaces satisfying the `zero-flux' boundary condition stated in equation (1) yields an expression for the change in action that is the same in form and content to that for an isolated system and in addition, yields equations of motion for the observables that are identical to those predicted by the field equation. Thus the definition of an open system at the atomic level is not open to choice but is determined by physics.

Atoms in Molecules - A Quantum Theory

R. F. W. Bader,
Oxford University Press, Oxford, 1990.
The International Series of Monographs on Chemistry, No 22


The molecular structure hypothesis--that a molecule is a collection of atoms linked by a network of bonds-- provides the principal means of ordering and classifying observations in chemistry. However this hypothesis is not related directly to the physics which governs the motions of atomic nuclei and electrons. It is the purpose of this important new book to show that a theory can be developed to establish the molecular structure hypothesis, demonstrating that the atoms in a molecule are real, with properties predicted and defined by the laws of quantum mechanics, and that the structure their presence imparts to a molecule is indeed a consequence of the underlying physics. As a result, the classification based upon the concept of atoms in molecules is freed from its empirical constraints and the full predictive power of quantum mechanics can be incorporated into the resulting theory--a theory of atoms in molecules.

Table of Contents

  1. Atoms in Chemistry
  • Atoms and the Topology of the Charge Density
  • Molecular Structure and its Change
  • Mathematical Models of Structural Change
  • The Quantum Atom
  • The Mechanics of an Atom in a Molecule
  • Chemical Models and the Laplacian of the Charge Density
  • The Action Principle for a Quantum Subsystem

  • BACK

    Page created & maintained by Frederic Leymarie, 2001-2.
    Comments, suggestions, etc., mail to: leymarie@lems.brown.edu