been described by Stratman et al. [ 109 ]. Wiberg et al. used this implementation to
study the effect of five functionals and five basis sets on the transition energies (the
UV absorption wavelengths) of formaldehyde, acetaldehyde, and acetone [ 110 ].
Satisfactory results were obtained, and the energies were not strongly dependent on
the functional, but B3P86 seemed to be the best and B3LYP the worst. The 6-311+
+G basis was recommended. Although these workers used MP2/6-311+G
geometries, the results in Table7.9indicate that AM1 geometries, which can be
calculated perhaps a thousand times faster, give transition energies that are nearly
as accurate (mean absolute errors of 0.12 and 0.18 eV, respectively). Table7.10
compares with experiment [ 111 ] the UV spectrum of methylenecyclopropene,
calculated by ab initio, semiempirical, and DFT methods. The best of the three is
the TDDFT calculation, which is the only one that reproduces the 308 nm band.
Jacquemin et al. obtained very accurate UV spectra of indigo dyes by taking solvent
into account with a polarizable continuum (in contrast to explicit solvent mole-
cules) model and employing TDDFT at the PBE0/6-311þG(2d,p) level [ 112 ]. Zhao
and Truhlar have presented their M06-HF functional as being particularly good for
electronic transitions of the Rydberg and charge-transfer type [ 113 ].
The HOMO-LUMO gap calculated with hybrid gradient-corrected functionals is
approximately equal to thep!p* UV transition of unsaturated molecules, and this
Table 7.9 UV spectra (as transition energies in eV) of acetone, acetaldehyde, and
formaldehyde, calculated by time-dependent DFT, using Gaussian 98 [ 78 ]. The results
of using MP2/6-311þG** [ 110 ] and (calculations by the author) AM1 geometries are
compared; both sets of calculations are single-point B3P86/6-311þþG**. For each
molecule only 6 transitions, all singlets, are shown. The number of positive and negative
deviations from experiment and the mean absolute errors are given
MP2 geometry AM1 geometry Experiment
Acetone 4.41 4.26 4.43
6.28 6.19 6.36
7.26 7.17 7.41
7.43 7.4 7.36
7.67 7.59 7.49
7.89 7.82 8.09
Acetaldehyde 4.29 4.14 4.28
6.76 6.69 6.82
7.29 7.26 7.46
7.7 7.68
7.89 7.98 7.75
8.35 8.16 8.43
Formaldehyde 3.95 3.83 4.1
6.98 6.97 7.13
7.93 7.95 8.14
8.09 8.07 7.98
8.81 8.84
9.23 8.87
5 þ, 10-mean of 15: 0.12 4þ, 11-mean of 15: 0.18492 7 Density Functional Calculations