Crystallography in Canada 

QTAIM: quantum theory of atoms in molecules

Richard F.W. Bader

Department of Chemistry, McMaster University, Hamilton, Ont., Canada
Reprinted from CNCC Newsletter No 1 (Canadian National Committee for Crystallography) September 2009

Fig 1: Three representations of the structure
of the guanine-cytosine base pair:
(a) classical structure (b) the network of
bond paths linking neighbouring atoms
that defines the molecular graph
(c) contour map of the electron density
showing the bond paths and the
intersections of the zero-flux interatomic
surfaces with the plane of the diagram.
The atomic symbols in (b) denote the
positions of the nuclei: H(w),C (b), O(r), N(blue).
The small yellow dots denote ring critical
points, the red dots the positions of the
bond critical points: the origin of the unique
pair of trajectories of grad ρ, each of which
terminates at a neighbouring nucleus, and
the terminus for the set of trajectories that
define the interatomic surface. There is a
wealth of information in the molecular graph.
Note that it provides a faithful mapping of the
classical structure onto the topology of the
density showing in addition to the three
hydrogen bonds, a weak interaction between
a methyl H and the keto O in cytosine that
is not indicated in the classical structure.
The presence of the bond path is but a useful
way of depicting and summarizing which pairs
of atoms share an interatomic surface, as
demonstrated in (c). That this shorthand notation
mimics the way in which the same information is
conveyed by the structures that evolved from
experimental chemistry is surely one of the most
powerful of all the physical vindications of the
zero-flux boundary condition for the definition
of a quantum open system.

Science is based on observation. Among the most important of the quantities accessible to measurement is the distribution of charge - nuclear and electronic - that constitutes matter and determines its properties. We are indeed fortunate to live in an age wherein the accurate measurement of the charge distribution became a reality, with each year witnessing an increase in our technical ability to determine its form and interpret its physical consequences.


The topology of the charge distribution is dominated by the electron-nuclear force, causing the electron density ρ  to exhibit maxima at the nuclear positions, thereby imposing the atomic form on the structure of matter. The resulting topology of the electron density provides the physical basis for the partitioning of space into atomic regions. The connectivity of the atomic regions defined by the lines of maximum density linking neighbouring atoms - the bond paths - yield all of the structural concepts of chemistry: open, cyclic and cage structures. Changes in the topology of ρ  occasioned by motions of the nuclei cause changes in structure yielding a theory of both structure and structural stability.


The primary purpose in postulating the existence of atoms in molecules or crystals is a consequence of the observation that atoms or functional groupings of atoms exhibit characteristic sets of static, reactive and spectroscopic properties which in general vary between relatively narrow limits. Thus the knowledge of chemistry is ordered, classified and understood by assigning properties to functional groupings of atoms, properties which are then used to identify the presence of a given group or to understand the behaviour of some total system. It follows that the topological definition of an atom in a molecule is of no physical substance unless it enables the extension of quantum mechanics to an atom in a molecule enabling the definition of its properties. The necessary boundary condition for the extension of quantum mechanics to an open system - to an atom in a molecule - is a natural consequence of the topological definition of an atom as determined by the dominance of the nuclear-electron force. An atom is a region of space bounded by a surface not crossed by any gradient vectors of the density - a 'surface of zero-flux in grad ρ '. The resulting theory is called the quantum theory of atoms in molecules, QTAIM. As is well documented, the atomic and group properties predicted by QTAIM agree with the additive group contributions measured experimentally, agreement with experiment being the only test of theory.


QTAIM, by providing the quantum basis for an atom in a molecule, necessarily recovers all of the related concepts of experimental chemistry. Thus in addition to the definition of atoms and molecular structure, QTAIM provides the physical basis for the Lewis model and its associated chemical concepts. The pairing of electrons and the associated concepts of electron localization / delocalization, are determined by the atomic expectation value of the exchange density and given physical expression in the topology of the Laplacian of the electron density, the quantity 2ρ .


Thus the measurable electron density provides the link between the primary concepts of experimental chemistry and quantum mechanics, providing the prediction and understanding of the properties of matter at the atomic level. It is to be understood that every question capable of expression in terms of quantum observables - energy, force, pressure, current, etc - can be both asked and answered. QTAIM provides an entirely new way of asking and answering questions concerning structure and its relation to measured properties that previously were matters of intense debate, the very definition of structure - of bonds being present or absent - being a source of controversy. Now it only remains for the investigator of the density to ask proper and meaningful questions.


 The acceptance by the experimental community of the topological theory of structure was not immediate. One could not find in the displays of the total density the accumulations of density corresponding to the bonded and lone pairs anticipated on the basis of the Lewis or orbital models. The field of accurate determination of ρ  was instead dominated by the construction of density difference maps, Δρ , obtained by subtracting the promolecule density, ρ from the 'true' density ρ  determined by a modelling of the measured structure factors. An interatomic surface defining the common boundary of two bonded atoms cuts a 'bond' (as depicted in the Δρ  maps) in two making it difficult for some to accept that the primary structural unit is the atom, not the bond and thus to turn the equation around to study ρ  = Δρ  + ρo , rather than Δρ .


The turning point in the acceptance of the topological theory of structure by the x-ray crystallographers came with the demonstration that the topology of the Laplacian of the density provides a remarkable pictorial mapping of the Lewis electron pair concept onto real space, the bonded and lone pairs being defined in terms of concentrations of charge appearing as local maxima in -2ρ . This work was presented at a GRC on Electron density and bonding in 1983. The topological properties of the Laplacian have been shown to be a direct consequence of the properties imposed by the Pauli principle on the pair density and onto real space through the spatial localization/delocalization of the density of the Fermi hole. With the Laplacian providing a bridge between the old and the new, the topological theory of molecular structure rapidly gained in acceptance to the dominant position it enjoys to-day amongst an ever growing audience of chemists and in particular, x-ray crystallographers. An interested reader my view a video of a talk reviewing the development of QTAIM that I presented last January in the 'Frontiers in Chemistry Series' at Case Western Reserve University to be found at: