Computational Chemistry

Figure2.1represents a one-dimensional PES in the two-dimensional graph of
Evs.q. A diatomic molecule AB has only one geometric parameter for us to vary,
the bond lengthqAB. Suppose we have a molecule with more than one geometric
parameter, for example water: the geometry is defined by two bond lengths and a
bond angle. If we reasonably content ourselves with allowing the two bond lengths
to be the same, i.e. if we limit ourselves to C2vsymmetry (two planes of symmetry
and a two-fold symmetry axis; seeSection 2.6) then the PES for this triatomic
molecule is a graph ofEversus two geometric parameters,q 1 ¼the O–H bond
length, andq 2 ¼the H–O–H bond angle (Fig.2.3). Figure2.3represents a two-
dimensional PES (a normal surface is a 2-D object) in the three-dimensional graph;
we could make an actual 3-D model of this drawing of a 3-D graph ofEversus
q 1 andq 2.
We can go beyond water and consider a triatomic molecule of lower symmetry,
such as HOF, hypofluorous acid. This has three geometric parameters, the H–O and
O–F lengths and the H–O–F angle. To construct a Cartesian PES graph for HOF
analogous to that for H 2 O would require us to plotEvs.q 1 ¼H–O,q 2 ¼O–F, and
q 3 ¼angle H–O–F. We would need four mutually perpendicular axes (forE,q 1 ,q 2 ,
q 3 , Fig.2.4), and since such a four-dimensional graph cannot be constructed in our
three-dimensional space we cannot accurately draw it. The HOF PES is a 3-D
“surface” of more than two dimensions in 4-D space: it is a hypersurface, and
potential energy surfaces are sometimes called potential energy hypersurfaces.
Despite the problem of drawing a hypersurface, we can define theequation E¼f
(q 1 ,q 2 ,q 3 ) as the potential energy surface for HOF, wherefis the function that
describes howEvaries with theq’s, and treat the hypersurface mathematically. For
example, in the AB diatomic molecule PES (a line) of Fig.2.1the minimum

energy

0

quadratic curve

.

.

.

vibrational levels

true molecular

potential energy

curve

bond length, q

qe

Fig. 2.2 Actual molecules do not sit still at the bottom of the potential energy curve, but instead
occupy vibrational levels. Also, only nearqe, the equilibrium bond length, does the quadratic curve
approximate the true potential energy curve

2.1 Perspective 11