Computational Chemistry

risen to supremacy with MO theory) and summarized the essential concepts of AIM
theory [ 270 ]. Bader reviewed the subject a decade later [ 271 ], and summarized it a
few years after that in his comprehensive 1990 book [ 272 ] and in a review [ 273 ].
Popelier updated the subject in his 1999 book [ 274 ] and in the 2007 book edited by
Matta and Boyd, Bader’s “classic 1990 treatise” is again updated [ 275 ]. Bader has
condensed the theory into an optimistically titled “Everyman’s Derivation of the
Theory of Atoms in Molecules” of seven pages [ 276 ]. We now examine the theory
and some applications.
The AIM approach rests on analyzing the variation from place to place in a
molecule of the electron density function (electron probability function, charge
density function, charge density),r. This is a functionr(x,y,z) which gives
the variation of the total electron density from point to point in the molecule:
r(x,y,z)dxdydz¼r(x,y,z)dvis the probability of finding an electron in the
infinitesimal volumedvcentered on the point (x,y,z) (the probability of finding
more than oneelectron indvis insignificant). This probability is the same as the
charge indvif we take the charge on an electron as our unit of charge, hence the
name charge density for the electron density functionr. Sincerdvhas the “units” of
probability, a pure number, the functionr logically has the units volume#^1.
However, the probability we deal with here is the same as the number (or fraction)
of electrons indv, which is the charge indvin electronic units, so the units ofrcan
be taken more physically concretely as electrons volume#^1 or charge volume#^1. In
atomic units this is electrons bohr#^3. The electron density function can be calcu-
lated from the wavefunction. It is not, as one might have thought, simply |C|^2 ,
whereCis the multielectron wavefunction of space and spin coordinates (Sec-
tion 5.2.3.1).Thislatter is the probability function for finding in the region of (x,y,
z) electron 1 with a specified spin, electron 2 with a specified spin, etc. The function
ris the number of electrons in the molecule times the sum over all their spins of the
integral of the square of the molecular wavefunction integrated over the coordinates
of all but one of the electrons [ 277 ]. We can write it in the condensed notation

rðx;y;zÞ¼n

X

all spins

Zn

2

C^2 dr 2 ...drn ð 5 : 234 Þ

whereris vector notation for the coordinates of electrons. If we think of the
electrons as being smeared out in a fog around the molecule, then the variation of
rfrom point to point corresponds to the varying density of the fog, andr(x,y,z)
centered on a pointP(x,y,z) corresponds to the amount of fog in the volume
elementdxdydz¼dv. Alternatively, in a scatterplot of electron density (charge
density) in a molecule, the variation ofrwith position can be indicated by varying
the volume density of the points. The electron density functionris the “density” in
density functional theory, DFT (Chapter 7). Let us look at some properties ofrthat
are relevant to AIM, the theory of atoms in molecules.
Consider first the electron density functionraround an atom. As we approach the
nucleus this rises toward a maximum, or thenegativeof the electron density,#r,

354 5 Ab initio Calculations