[ 102 ]. As withthermodynamicenergy differences, i.e. energy differences not involv-
ing a transition state, consistently obtaining with some confidence activation energies
accurate to 10–20 kJ mol"^1 may require a high-accuracy multistep method like CBS-
QB3. For some barriers the problem seems to be with the functionals: Merrill et al.
found that for the fluoride ion-induced elimination of HF from CH 3 CH 2 F none of the
11 functionals tested (including B3LYP) was satisfactory, by comparison with high-
level ab initio calculations. Transition states were often looser and stabler than
predicted by ab initio, and in several cases a transition state could not even be
found. They concluded that hybrid functionals offer the most promise, and that
“the ability of density functional methods to predict the nature of TS’s demands a
great deal more attention than it has received to date” [ 38 ].
More recent references to the accuracy of DFT in calculating barriers are the
extensive 2007 compilations noted above for thermochemistry, namely [ 44 – 46 ]. Of
the functionals considered in reference [ 44 ] only B3LYP is among the few on which
we have focussed (B3LYP, M06, and TPSS), and it, scrutinized throughout the
review because of its popularity, was well down on the barrier accuracy list, with
typical errors of ca. 16 kJ mol"^1 ; the star functionals in this regard were MPW1K
and BB1K with errors typically of ca. 5 kJ mol"^1. Reference [ 45 ] documents a
litany of shortcomings of B3LYP and extols the virtues of the M06-class of
functionals. For barriers (kinetics) it recommends “M06-2X, BMK, and M05-2X
for main-group thermochemistry and kinetics”, and “M06-2X, M05-2X, and M06
for systems where main-group thermochemistry, kinetics, and noncovalent inter-
actions are all important”. M06, the general-purpose M06-class functional, appar-
ently has an error of about 0.63–2.2 kcal mol"^1 (2.6–9.2 kJ mol"^1 , depending on the
database used to test it. The rather extensive tests by Riley et al. ([ 46 ], summarized
in their Figs. 16–19) of functionals and their partner basis sets indicated, as far as
this wealth of data can be encapsulated into a few words, that the best functionals
for barriers were BBB1K, B1B95, and B1LYP (with B3LYP being only very
slightly less accurate than this latter), and with no clear advantage to either the
Pople or the Dunning basis sets. Typical barrier errors for these functionals were ca.
3–5 kcal mol"^1 (13–21 kJ mol"^1 ).
7.3.3 Frequencies and Vibrational Spectra.............................
The general remarks and theory about frequencies that were given in Section 5.5.3
apply to DFT frequencies also. As with ab initio frequency calculations, but unlike
semiempirical, one reason for calculating DFT frequencies is to get zero-point
energies to correct the frozen-nuclei energies. The frequencies are also used to
characterize the stationary point as a minimum, transition state, etc., and to predict
the IR spectrum. As usual the wavenumbers (“frequencies”) are the mass-weighted
eigenvalues of the Hessian, and the intensities are calculated from changes in dipole
moment incurred by the vibrations.
484 7 Density Functional Calculations