methods give good bond angles (to within ca. 1; ca. 2for some CH 3 NH 2 angles).
For bond lengths positive deviations are very roughly twice as numerous as
negative, except for TPSS where they are about eight times as frequent, and for
bond angles (where errors are usually trivial) there are slightly more positive than
negative deviations.
Table7.2presents for examination ten dihedral angles from eight molecules. For
each of these the dihedral was calculated by B3LYP, M06, TPSS, and MP2 (with
the 6-31G* basis in each case). The mean absolute deviations (arithmetic mean of
the unsigned errors), MAD, are:
B3LYP M06 TPSS MP2
1.9 2.0 2.3 2.3Because of the periodic (sinusoidal) nature of the energy-dihedral angle func-
tion, the direction of deviation from experiment, the number of positive versus
negative deviations, is not meaningful, provided these are small (under 10), as they
are here. The calculated dihedral angles are all within ca. 3, except FCH 2 CH 2 OH
(HOCC by B3LYP. M06, TPSS) and ClCH 2 CH 2 OH (HOCC by MP2), where they
are ca. 8–6. In view of the soft nature of the energy-dihedral function (energies do
not rise or fall steeply with small changes in dihedrals, unlike changes in bond
lengths or angles), and of possible errors in the experimental values, this is not
serious. All four methods seem to be satisfactory for dihedrals.
B3LYP, M06, TPSS and MP2(fc) geometries (and relative energies) are com-
pared, using the 6-31G basis, for the species shown in four reaction profiles in
Fig.7.2. These correspond to the ab initio comparisons of Fig. 5.21 and the
semiempirical comparisons of Fig. 6.3. Since experimental geometries are not
available for any of the transition states and also not for cyclopropylidene, we
content ourselves with some simple comparisons among the calculated geometries.
For the reactants and products the DFT bond length deviations from the MP2
geometries, which latter we tacitly take to be reasonably good (Section 5.51), are
not more than 0.016 A ̊(for the CO bond of ethenol), and for the transition states not
more than 0.073 (for the partial NC bond of the CH 3 NC transition state). The DFT
angles do not deviate by more than 3.5(for the CH 3 NC transition state) from the
MP2 values. The consistency of the three DFT methods and their good agreement
with MP2 suggest that these DFT methods are quite comparable to MP2/6-31G in
calculating transition state geometries.
Geometry errors for 108 molecules were reported by Scheiner et al. [ 74 ], compar-
ing several ab initio and DFT methods. They found that Becke’s original three-
parameter function (which they denote ACM, for adiabatic connection method ;
B3LYP was developed as a modification of this [ 58 ]), with a 6-31G*-type and with
the 6-31G** basis sets, gave average bond length errors of about 0.01 A ̊and bond
angle errors of about 1.0. They concluded that of the methods they examined ACM is
the best choice for both geometries and reaction energies. St-Amant et al. [ 52 ] also
compared ab initio and DFT methods and found average dihedral angle errors of ca. 3
for 11 molecules using a perturbative gradient-corrected DFT method with an
474 7 Density Functional Calculations