Chapter 2, Harder Questions, Answers
Q6
What kind(s) of stationary points do you think a second-order saddle point
connects?
A second-order saddle point has two of its normal-mode vibrations
corresponding to imaginary frequencies, that is, two modes “vibrate” without a
restoring force, and each mode takes the structure on a one-way trip downhill on the
potential energy surface. Now compare this with a first-order saddle point (a
transition state); this has one imaginary normal-mode vibration: as we slide down-
hill along the direction corresponding to this vibration, the imaginary mode dis-
appears and the structure is transformed into a relative minimum, with no imaginary
vibrations. Correspondingly, as a second-order saddle structure moves downhill
along the path indicated by one of the imaginary vibrations, this vibration vanishes
and the structure is transformed into a first-order saddle point. Illustrations of this
are seen in Figs. 2.9 and 2.14 where the hilltops lead to saddle point by conforma-
tional changes.
Chapter 2, Harder Questions, Answers
Q7
If a species has one calculated frequency very close to 0 cm#^1 what does that tell
you about the (calculated) PES in that region?
First let us acknowledge a little inaccuracy here: frequencies are either positive,
imaginary (not negative), or, occasionally, essentially zero. Some programs desig-
nate an imaginary frequency by a minus sign, some byi(the symbol for
pffi
#1).
Frequencies are calculated from the force constants of the normal vibrational
modes, and the force constant of a vibrational mode is equal to the curvature of
the PES along the direction of the mode (¼the second derivative of the energy with
respect to the geometric change involved). Whether a frequency is positive or
imaginary depends qualitatively on the curvature. A minimum has positive curva-
ture along the direction of all normal-mode vibrations, a first-order saddle point has
negative curvature along the direction of one normal-mode vibration and positive
curvature along all other normal-mode directions, and analogously for a second-,
third-order etc. saddle point. Positive curvature corresponds to positive force con-
stants and positive frequencies, and negative curvature to negative force constants
and, taking square roots, imaginary frequencies. A zero frequency, then, corre-
sponds to a zero force constant (
pffi
0 ¼0) and zero curvature of the potential energy
surface along that direction. Moving the atoms of the structure slightly along that
direction leads to essentially no change in the energy, since the curvature of the
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