input an equilateral triangle structure, probably with bond lengths about those of an
O–O single bond.
In the propenol example, the PES scan suggested that to obtain the global
minimum we should start with an input structure resembling 1 , but the exact values
of the various bond lengths and angles were unknown (the exact values of even the
dihedrals was not known with certainty, although general chemical knowledge
made H–O–C–C¼H–C-C¼C¼ 0 seem plausible). The actual creation of input
structures is usually done nowadays with an interactive mouse-driven program,
in much the same spirit that one constructs plastic models or draws structures
on paper. An older alternative is to specify the geometry by defining the various
bond lengths, angles and dihedrals, i.e. by using a so-called Z-matrix (internal
coordinates).
To move along the PES from the input structure to the nearest minimum is
obviously trivial on the one-dimensional PES of a diatomic molecule: one simply
changes the bond length till that corresponding to the lowest energy is found.
On any other surface, efficient geometry optimization requires a sophisticated
algorithm. One would like to know in which direction to move, and how far in
that direction (Fig.2.16). It is not possible, in general, to go from the input structure
to the proximate minimum in just one step, but modern geometry optimization
algorithms commonly reach the minimum within about ten steps, given a reason-
able input geometry. The most widely-used algorithms for geometry optimization
[ 14 ] use the first and second derivatives of the energy with respect to the geometric
parameters. To get a feel for how this works, consider the simple case of a
one-dimensional PES, as for a diatomic molecule (Fig.2.17). The input structure
is at the point Pi(Ei, qi) and the proximate minimum, corresponding to the
optimized structure being sought, is at the pointPo(Eo,qo). Before the optimization
energygeometryTSB
AB′A′several steps
several stepsFig. 2.15 Geometry optimization to a minimum gives the minimum closest to the input structure.
The input structure A^0 is moved toward the minimum A, and B^0 toward B. To locate a transition
state a special algorithm is usually used: this moves the initial structure A^0 toward the transition
state TS. Optimization to each of the stationary points would probably actually require several
steps (see Fig.2.16)
26 2 The Concept of the Potential Energy Surface