and meeting S at the bond critical point. The electron density defines agradient
vector field, the totality of thetrajectorieseach of which results from starting at
infinity and moving along the path of steepest increase inr. Figure5.42shows that
only two of the trajectories (of those in the plane of the paper) that originate at
infinity do not end at the nuclei; these end at the bond critical point. These two
trajectories define the intersection of S with the plane of the paper. None of the
trajectories cross S, which is thus called azero-fluxsurface (the gradient vector field
is analogous to an electric field whose “flux lines” point along the direction of
attraction of a positive charge toward a central negative charge). Because X 2 is
homonuclear, the zero-flux surface is a plane. For a molecule with different nuclei,
the zero-flux surfaces are curved, convex in one direction, concave in the other. The
space within a molecule bounded by one (for a diatomic molecule) or more zero-
flux surfaces is anatomic basin. Away from the nuclei toward the outside of the
molecule the basin extends outward to infinity, becoming shallower as the electron
density fades toward zero. The nucleus and the electron density in an atomic basin
constitute an atom in a molecule. Even for molecules other than homonuclear
diatomics, atoms are still defined by atomic basins partitioned off by unique zero-
flux surfaces, as illustrated in Fig.5.43.
In the AIM method, the charge on an atom is calculated by integrating the
electron density functionr(x,y,z) over the volume of its atomic basin. The charge is
the algebraic sum of the electronic charge and the nuclear charge (the atomic
number of the nucleus minus the number of electrons in the basin). An AIM bond
SCA BFig. 5.43 Heteronuclear (as well as homonuclear; cf. Fig.5.42) molecules can be partitioned into
atoms.Srepresents a slice through the zero-flux surface that defines the atoms A and B in a
molecule AB. The lines with arrows are the trajectories of the gradient vector field.Spasses
through the bond critical pointCand is not crossed by any trajectory lines
358 5 Ab initio Calculations