Computational Chemistry

falls toward a minimum (Fig.5.40). Viewing the electron distribution in terms of#r
rather thanris useful because it more easily enables us to discern analogies
between the variation ofrin a molecule (arvs location-in-molecule graph), and
a potential energy surface (PES, an energy vs geometry graph), with which we are
familiar fromChapter 2. Examine the distribution of#rin a homonuclear diatomic
molecule X 2 (Fig.5.41). This shows a plot of#rversus two of the three Cartesian
coordinates needed to assign positions to all the points in the molecule. The graph
retains the internuclear axis (by convention thez-axis) and one other axis, sayy; the
molecule is symmetrical with respect to reflection in theyzplane. The negative of
the electron density,#r, dips toward a minimum at the atomic nuclei (rgoes
toward a maximum), analogously to the occurrence of a minimum on a PES. The
analogy is not perfect, because the nuclei do not correspond to true stationary
points: the point is acusp, where∂r/∂qis discontinuous rather than zero (unlike
∂E/∂qfor a stationary point on a PES;qis a geometric parameter) [ 278 ]. This is not
the death knell for our analogy, because there is always a function “similar” –
technically, homeomorphic – to r(x, y, z) for which the nuclear positions
are stationary points [ 278 ]. With the caveat that strictly speaking the derivatives
apply to the homeomorphic function, we can write for points at the nuclei:

@ð#rÞ

@z

¼ 0 ;

@ð#rÞ

@y

¼ 0 ;

@ð#rÞ

@x

¼ 0 ð 5 : 235 Þ

and

@^2 ð#rÞ

@z^2

> 0 ;

@^2 ð#rÞ

@y^2

> 0 ;

@^2 ð#rÞ

@x^2

> 0 ð 5 : 236 Þ

x

y

z

a b c

.

nucleus

x

y

. x

y

r r

Fig. 5.40 The distribution of electron density (charge density)rfor an atom; the nucleus is at the
origin of the coordinate system. (a) Variation ofrwith distance from the nucleus. Moving away
from the nucleusrdecreases from its maximum value and fades asymptotically toward zero. (b)
Variation of#rwith distance from the nucleus;#rbecomes less negative and approaches zero as
we move away from the nucleus. The#rpicture is useful for molecules (Fig.5.41) because it
makes clearer analogies with a potential energy surface. (c) A “4-D” picture (rvsx,y,z) of the
variation ofrin an atom: the density of the dots (number of dots per unit volume) indicates
qualitatively electron densityrin various regions

5.5 Applications of the Ab initio Method 355