Computational Chemistry

For the#r/qsurface of Fig.5.41the number of positive and negative eigenva-
lues for a nuclear critical point are 3 and 0, and for a bond critical point, 2 and 1.
Thus for ther/qsurface to which the Hessian of Eq. (5.238) refers (the mirror
image of the#r/qsurface), the number of positive and negative eigenvalues is,
respectively, 0 and 3 (for a nucleus), and 1 and 2 (for a bond critical point).
The behavior of the second derivative ofr, the Laplacian ofr,(∂^2 /∂x^2 þ∂^2 /∂y^2 þ
∂^2 /∂z^2 )r¼r^2 r, is a key concept in AIM theory.
The minimum (#r)path (maximumrpath) from one X nucleus to the other is
thebond path; with certain qualifications this can be regarded as a bond. It is
analogous to the minimum-energy path connecting a reactant and its products, i.e.
to the intrinsic reaction coordinate. Such a bond is not necessarily a straight line: in
strained molecules it may be curved (bent bonds). The bond passes through the
bond critical point, which for a homonuclear diatomic molecule X 2 is the midpoint
of the internuclear line. Now consider Fig.5.42, which shows in the X 2 molecule
another characteristic of the electron density function. The contour lines represent
electron density, which rises as we approach a nucleus and falls off as we go to and
beyond the van der Waals surface. If it is true that the molecule can be divided into
atoms, then for X 2 the dividing surface S (represented as a vertical line in Fig.5.42)
must lie midway between the nuclei, with the internuclear line being normal to S

C

X X

S

Fig. 5.42 Contour lines forr, the electron density distribution, in a homonuclear diatomic
molecule X 2. The lines originating at infinity and terminating at the nuclei and at the bond critical
pointCare trajectories of the gradient vector field (the lines of steepest increase inr; two
trajectories alsooriginateatC). The lineSrepresents the dividing surface between the two
atoms (the line is where the plane of the paper cuts this surface).Spasses through the bond critical
point and is not crossed by any trajectories

5.5 Applications of the Ab initio Method 357