Computational Chemistry

Moving along the internuclear line we find a point in a saddle-shaped region,
analogous to a transition state, where the surface again (caveat) has zero slope (all
first derivatives zero), and is negatively curved along thez-axis but positively
curved in all other directions (Fig.5.41), i.e.

@^2 ð#rÞ

@z^2

< 0 ;

@^2 ð#rÞ

@y^2

> 0 ;

@^2 ð#rÞ

@x^2

> 0 ð 5 : 237 Þ

This transition-state-like point is called abond critical point. All points at which
the first derivatives are zero (caveat above) are critical points, so the nuclei are also
critical points. Analogously to the energy/geometry Hessian of a potential energy
surface, an electron density function critical point (a relative maximum or mini-
mum or saddle point) can be characterized in terms of its second derivatives by
diagonalizing ther/qHessian(q¼x,y, orz) to get the number of positive and
negative eigenvalues:

r=qHessian¼

@^2 r=@x^2 @^2 r=@xy @^2 r=@xz

@^2 r=@yx @^2 r=@y^2 @^2 r=@yz

@^2 r=@zx @^2 r=@zy @^2 r=@z^2

0

@

1

A ð 5 : 238 Þ

• r
  
  
  • r
  
  
  
  

y

.. z

nucleus

.

saddle point

.

bond critical point nucleus

surface

Fig. 5.41 The distribution of the electron density (charge density)rfor a homonuclear diatomic
molecule X 2. One nucleus lies at the origin, the other along the positivez-axis (thez-axis is
commonly used as the molecular axis). Thexzplane represents a slice through the molecule along
thez-axis. The#r¼f(x,z) surface is analogous to a potential energy surfaceE¼f(nuclear
coordinates), and has minima at the nuclei (maximum value ofr) and a saddle point,
corresponding to a bond critical point, along thezaxis (midway between the two nuclei since
the molecule is homonuclear)

356 5 Ab initio Calculations