2.3.5 Choice of an Atomic Basis Set
Similar to DFT, TD-DFT is relatively less sensitive to the size of the atomic basis
set than the corresponding highly-correlated wavefunction theories, though exceptions have been reported [57]. Irrespective of the agreement with experimental data,
reaching ES data which are converged with respect to the extension of the basis set
generally requires the selection of larger atomic basis sets than for GS properties.
For electronic transitions to low-lying excited-states in conjugated molecules, a
double-ζ (or, better, triple-ζ) polarized atomic basis set augmented with diffuse
orbitals appears to be a judicious choice. In other words, 6-31+G(d) or aug-ccpVDZ could be advised as reasonable compromises between computational cost
and accuracy for both E
vertÀa and E
vertÀ f . Of course, for Rydberg ES, a much larger
basis set may be necessary, e.g., aug-cc-pVTZ. When optimizing the geometry of a
given ES, one should also be cautious as the PES are often quite flat and
diffusionless basis sets could yield rather poor results but in strongly constrained
fluorophores. The interested reader can find elsewhere longer discussions regarding
basis set effects for both small [58] and large [15] molecules in the context of
TD-DFT spectroscopic investigations.
3 Benchmarks
In this section we present the results obtained in several benchmarks aiming to
pinpoint the most adequate XCF. Both 0–0 energies and band topologies, obtained
through the calculation of vibronic couplings, are discussed. A general statement, at
least applicable to low-lying ES of organic molecules, is that pure XCF which do
not include exact exchange (e.g., BLYP [59, 60] or PBE [61]) tend to provide much
poorer results than hybrid XCF. In global hybrid functionals (e.g., B3LYP [52],
PBE0 [53, 62], and M06-2X [63, 64]) the main parameter affecting the computed
E
AFCP is the mixing between the exact and DFT exchange, whereas in rangeseparated hybrids (e.g., CAM-B3LYP [54] and ωB97X-D [55]), the attenuation
parameter which defines the rate at which one goes from DFT to exact exchange is
the key parameter. We redirect the interested readers to [24] for a longer and more
general review of existing TD-DFT benchmarks.
3.1 AFCP Energies
In this section we focus on investigations treating the E
AFCP of large molecules [7,
13, 15, 65], though there are several works dealing with small gas-phase compounds for which the 0–0 band can be accurately measured [17, 19, 66–69]. First, as
ΔE
ZPVE is the most computationally expensive term, let us discuss its magnitude
Computational Molecular Electronic Spectroscopy with TD-DFT
355
Similar to DFT, TD-DFT is relatively less sensitive to the size of the atomic basis
set than the corresponding highly-correlated wavefunction theories, though exceptions have been reported [57]. Irrespective of the agreement with experimental data,
reaching ES data which are converged with respect to the extension of the basis set
generally requires the selection of larger atomic basis sets than for GS properties.
For electronic transitions to low-lying excited-states in conjugated molecules, a
double-ζ (or, better, triple-ζ) polarized atomic basis set augmented with diffuse
orbitals appears to be a judicious choice. In other words, 6-31+G(d) or aug-ccpVDZ could be advised as reasonable compromises between computational cost
and accuracy for both E
vertÀa and E
vertÀ f . Of course, for Rydberg ES, a much larger
basis set may be necessary, e.g., aug-cc-pVTZ. When optimizing the geometry of a
given ES, one should also be cautious as the PES are often quite flat and
diffusionless basis sets could yield rather poor results but in strongly constrained
fluorophores. The interested reader can find elsewhere longer discussions regarding
basis set effects for both small [58] and large [15] molecules in the context of
TD-DFT spectroscopic investigations.
3 Benchmarks
In this section we present the results obtained in several benchmarks aiming to
pinpoint the most adequate XCF. Both 0–0 energies and band topologies, obtained
through the calculation of vibronic couplings, are discussed. A general statement, at
least applicable to low-lying ES of organic molecules, is that pure XCF which do
not include exact exchange (e.g., BLYP [59, 60] or PBE [61]) tend to provide much
poorer results than hybrid XCF. In global hybrid functionals (e.g., B3LYP [52],
PBE0 [53, 62], and M06-2X [63, 64]) the main parameter affecting the computed
E
AFCP is the mixing between the exact and DFT exchange, whereas in rangeseparated hybrids (e.g., CAM-B3LYP [54] and ωB97X-D [55]), the attenuation
parameter which defines the rate at which one goes from DFT to exact exchange is
the key parameter. We redirect the interested readers to [24] for a longer and more
general review of existing TD-DFT benchmarks.
3.1 AFCP Energies
In this section we focus on investigations treating the E
AFCP of large molecules [7,
13, 15, 65], though there are several works dealing with small gas-phase compounds for which the 0–0 band can be accurately measured [17, 19, 66–69]. First, as
ΔE
ZPVE is the most computationally expensive term, let us discuss its magnitude
Computational Molecular Electronic Spectroscopy with TD-DFT
355