[63, 64] and CAM-B3LYP [54] provide more consistent values, i.e., larger correlation coefficients with respect to experimental results (than B3LYP [52] or PBE0
[53, 62]) and can therefore be valued if design is sought: they overshoot the
transition energies in a rather systematic way. The best results for the set of
molecules of Fig. 2 are obtained with the optimally-tuned LC-PBE*, but at the
cost of a systematic (non-empirical) optimization of the attenuation parameter.
Grimme and coworkers also performed a series of benchmarks [7, 13, 65] with a
similar focus on “real-life” structures, and the results are collected in Table 1. In
their first contribution, they evaluated 3 XCF (BP86 [59, 73], B3LYP [52], and
BHLYP [74]) on 30 singlet–singlet and 13 doublet–doublet transitions in aromatic
and radical dyes, respectively. Solvent effects were empirically accounted for by
applying a standard correction to the experimental 0–0 energies. These authors
concluded that global hybrids with 30–40% exact exchange emerged the best
compromises [7]. More recently, the same group treated 12 molecules,
transforming the measured energies in reference vertical values thanks to
TD-DFT calculations. With this model, they could obtain deviations smaller than
0.2 eV with a recent global hybrid (BMK [75]), a range-separated hybrid
(CAM-B3LYP [54]), and their double hybrid (B2GPPLYP [65]).
In short, the typical TD-DFT errors for E
AFCP are of the order of 0.2–0.3 eV,
when hybrid XCF are used. It should also be noted that XCF including a large share
of exact exchange (ca. 50%) deliver too large transition energies but tend to yield a
good consistency (large correlation coefficient) with experiment. The most accurate
results are obtained with double-hybrids or optimally-tuned range-separated XCF
but for an increased computational effort.
3.2 Band Shapes
The accuracy of the band topologies obtained with several XCF has been evaluated
by several groups [7, 16, 27, 71]. For the sake of consistency with the E
AFCP works
presented above, we discuss here the two latter investigations which relied on a set
of 20 conjugated molecules belonging to the same families as the one shown in
Fig. 1. The selected protocol also relied on the 6-31+G(d) atomic basis set and
included environmental effects thanks to the PCM approach. Selected key statistical data are given in Table 2. As all vibronic calculations have been performed on
the basis of GS and ES vibrations obtained in the harmonic approximation, the clear
trend is to overestimate the separation between the different vibronic peaks,
irrespective of the selected XCF, an error which could be reduced by including
anharmonic effects [16, 76, 77]. It is also obvious that the average absolute errors
are smaller for absorption (ca. 100 cm
À1 ) than for emission (ca. 250 cm
À1 ). All
XCF, apart from LC-PBE, provide rather similar deviations, and it is therefore
difficult to select an unambiguously more accurate hybrid functional. Nevertheless,
it should be noted that the obtained accuracy is significantly system dependent, e.g.,
most XCF are able to reproduce accurately the characteristic multi-peak structure of
358
D. Jacquemin and C. Adamo
[53, 62]) and can therefore be valued if design is sought: they overshoot the
transition energies in a rather systematic way. The best results for the set of
molecules of Fig. 2 are obtained with the optimally-tuned LC-PBE*, but at the
cost of a systematic (non-empirical) optimization of the attenuation parameter.
Grimme and coworkers also performed a series of benchmarks [7, 13, 65] with a
similar focus on “real-life” structures, and the results are collected in Table 1. In
their first contribution, they evaluated 3 XCF (BP86 [59, 73], B3LYP [52], and
BHLYP [74]) on 30 singlet–singlet and 13 doublet–doublet transitions in aromatic
and radical dyes, respectively. Solvent effects were empirically accounted for by
applying a standard correction to the experimental 0–0 energies. These authors
concluded that global hybrids with 30–40% exact exchange emerged the best
compromises [7]. More recently, the same group treated 12 molecules,
transforming the measured energies in reference vertical values thanks to
TD-DFT calculations. With this model, they could obtain deviations smaller than
0.2 eV with a recent global hybrid (BMK [75]), a range-separated hybrid
(CAM-B3LYP [54]), and their double hybrid (B2GPPLYP [65]).
In short, the typical TD-DFT errors for E
AFCP are of the order of 0.2–0.3 eV,
when hybrid XCF are used. It should also be noted that XCF including a large share
of exact exchange (ca. 50%) deliver too large transition energies but tend to yield a
good consistency (large correlation coefficient) with experiment. The most accurate
results are obtained with double-hybrids or optimally-tuned range-separated XCF
but for an increased computational effort.
3.2 Band Shapes
The accuracy of the band topologies obtained with several XCF has been evaluated
by several groups [7, 16, 27, 71]. For the sake of consistency with the E
AFCP works
presented above, we discuss here the two latter investigations which relied on a set
of 20 conjugated molecules belonging to the same families as the one shown in
Fig. 1. The selected protocol also relied on the 6-31+G(d) atomic basis set and
included environmental effects thanks to the PCM approach. Selected key statistical data are given in Table 2. As all vibronic calculations have been performed on
the basis of GS and ES vibrations obtained in the harmonic approximation, the clear
trend is to overestimate the separation between the different vibronic peaks,
irrespective of the selected XCF, an error which could be reduced by including
anharmonic effects [16, 76, 77]. It is also obvious that the average absolute errors
are smaller for absorption (ca. 100 cm
À1 ) than for emission (ca. 250 cm
À1 ). All
XCF, apart from LC-PBE, provide rather similar deviations, and it is therefore
difficult to select an unambiguously more accurate hybrid functional. Nevertheless,
it should be noted that the obtained accuracy is significantly system dependent, e.g.,
most XCF are able to reproduce accurately the characteristic multi-peak structure of
358
D. Jacquemin and C. Adamo