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Finding the optimal exchange-correlation functional to describe the excited state
properties of push-pull organic dyes designed for thermally activated delayed
fluorescence
Cardeynaels, Tom; Paredis, Simon; Deckers, Jasper; Brebels, Sonny; Vanderzande, Dirk;
Maes, Wouter; Champagne, Benoît
Published in:
Physical Chemistry Chemical Physics

DOI:
10.1039/d0cp02409k

Publication date:
2020

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Publisher's PDF, also known as Version of record

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Citation for pulished version (HARVARD):
Cardeynaels, T, Paredis, S, Deckers, J, Brebels, S, Vanderzande, D, Maes, W & Champagne, B 2020, 'Finding
the optimal exchange-correlation functional to describe the excited state properties of push-pull organic dyes
designed for thermally activated delayed fluorescence', Physical Chemistry Chemical Physics, vol. 22, no. 28,
pp. 16387-16399. https://doi.org/10.1039/d0cp02409k

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Cite this:Phys.Chem.Chem.Phys.,

2020, 22, 16387

Finding the optimal exchange–correlation
functional to describe the excited state properties
of push–pull organic dyes designed for thermally
activated delayed fluorescence†

Tom Cardeynaels, abc Simon Paredis,bc Jasper Deckers, bc Sonny Brebels,bc

Dirk Vanderzande, bc Wouter Maes bc and Benoı̂t Champagne *a

To gauge the suitability of an organic dye for thermally activated delayed fluorescence (TADF), its

excited state properties are often calculated using density functional theory. For this purpose, the choice

of the exchange–correlation (XC) functional is crucial as it heavily influences the quality of the obtained

results. In this work, 19 different XC functionals with various amounts of Hartree–Fock (HF) exchange

and/or long-range correction parameters are benchmarked versus resolution-of-the-identity second-

order coupled cluster (riCC2) calculations for a set of 10 prototype intramolecular donor–acceptor

compounds. For the time-dependent density functional theory (TD-DFT) calculations, LC-BLYP(o =

0.20) and M06-2X are the better performing XC functionals when looking at singlet and triplet excitation

energies, respectively. For the singlet–triplet energy gap, LC-BLYP(o = 0.17), LC-oPBE(o = 0.17) and a

hybrid LC-BLYP(o = 0.20)/M06-2X method give the smallest mean average errors (MAEs). Using the

Tamm–Dancoff approximation (TD-DFT/TDA), the MAEs are further reduced for the triplet vertical

excitation energies and the singlet–triplet energy gaps.

1 Introduction

Thermally activated delayed fluorescence (TADF) has gained
significant interest since its first application to the field of
organic light-emitting diodes (OLEDs) in 2009.1 The prospect of
a more efficient and green light-producing technology has
prompted researchers worldwide to investigate the intricate
properties of TADF and hundreds of new materials have been
designed for this purpose.2–8 Unlike the phosphorescent emit-
ters applied in the 2nd generation OLEDs, TADF-based OLEDs
are emissive from the singlet excited state via fluorescence.
However, they operate under similar circumstances. Due to
spin statistics, the excited states within the active layer of the

OLED device are formed in a 75/25 ratio for triplet (T1) with
respect to singlet (S1) states. These triplet states have to be
upconverted to the singlet state to achieve external quantum
efficiencies (EQEs) that can compete with those obtained for
the phosphorescent counterparts. Reversed intersystem cross-
ing (rISC) allows the triplet excitons to be converted to the
singlet state, a process that is facilitated when the two states are
close in energy and thermal energy enables to overcome this
small energy difference. Since the S1–T1 energy gap (DEST) is
governed by the overlap between (generally) the highest occu-
pied molecular orbital (HOMO) and the lowest unoccupied
molecular orbital (LUMO), efficient rISC is achieved by spatially
separating the HOMO and LUMO on the organic emitter
molecule, giving rise to charge transfer absorption and emis-
sion. This can be achieved by coupling ‘donor’ (electron-rich)
and ‘acceptor’ (electron-poor) subunits, either directly (giving
rise to large steric hindrance) or via a spiro or other types of
sp3-hybridized junction.9–12 Moreover, since we are dealing with
spin changes, changing the spin angular momentum has to be
paired with a change in orbital angular momentum (spin–orbit
coupling, SOC) and this is forbidden between 1CT and 3CT states by
the El-Sayed rule13,14 when they have the same orbital character.
To overcome this, a second triplet state (locally excited, LE, or of
different CT nature) is required to allow mixing through a process
known as vibronic coupling.15–19

a University of Namur, Laboratory of Theoretical Chemistry, Theoretical and

Structural Physical Chemistry Unit, Namur Institute of Structured Matter, Rue de

Bruxelles 61, 5000 Namur, Belgium. E-mail: benoit.champagne@unamur.be
b Hasselt University, Institute for Materials Research (IMO-IMOMEC), Design &

Synthesis of Organic Semiconductors (DSOS), Agoralaan 1, 3590 Diepenbeek,

Belgium
c IMOMEC Division, IMEC, Wetenschapspark 1, 3590 Diepenbeek, Belgium

† Electronic supplementary information (ESI) available: Signed error plots,
detailed information about all TD-DFT and TD-DFT/TDA calculations, optimized
geometries, HOMO and LUMO topologies, synthesis procedures, 1H and 13C NMR
spectra, UV-VIS and fluorescence and coordinates of the optimized geometries
can be found. See DOI: 10.1039/d0cp02409k

Received 4th May 2020,
Accepted 6th July 2020

DOI: 10.1039/d0cp02409k

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http://orcid.org/0000-0001-7883-3393
http://orcid.org/0000-0003-3678-8875
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16388 | Phys. Chem. Chem. Phys., 2020, 22, 16387--16399 This journal is©the Owner Societies 2020

Computational chemistry has been instrumental to help
resolving the mechanism behind these processes,18–24 showing
that even by considering simple properties like structural
parameters, it can help to understand the excited state proper-
ties of any given emitter. With density functional theory (DFT),
the ground-state geometry optimization of small (o25 atoms)
to large (4100 atoms) molecules can be performed at low to
moderate computational cost. From this, one can already
determine the dihedral angle between the donor and acceptor
units, as well as the spatial separation of the frontier orbitals.
Adiabatic linear-response time-dependent DFT (TD-DFT) can
then be used to investigate the excitation energies of the singlet
and triplet states, from which the singlet–triplet energy gap
(DEST) can be deduced. Although higher level theoretical calcu-
lations do exist, such as the second-order approximate coupled
cluster singles and doubles model CC2, the computational cost
increases rapidly, as Nx with N a measure of the size of the
compound (x = 5 for CC2), making these methods inviable to
calculate the excited state properties of larger systems for
screening purposes.25 However, the choice of the exchange–
correlation (XC) functional in (TD-)DFT calculations is of
crucial importance to obtain accurate and reliable results.
Several papers have already described the use of TD-DFT or
the Tamm–Dancoff approximation to TD-DFT (TD-DFT/TDA) to
obtain accurate results for the first singlet and triplet vertical
excitation energies of small organic conjugated compounds26,27

and larger organic dyes for solar cell applications.28 In the work
of Jacquemin et al.,26 34 different XC functionals (XCFs) were
investigated and benchmarked versus excitation energies
obtained using high level methods such as MS-CASPT2
[i.e., multi-state complete active space self-consistent field
(CASSCF) corrected with second-order perturbation theory],
CC2, and CC3 (i.e., approximate coupled cluster singles,
doubles, and triples model). Among the best performing func-
tionals in their study are BMK (42% HF exchange) and M06-2X
(54% HF exchange). Brückner et al.29 investigated the singlet
and triplet excitation energies for the application of singlet
fission in organic solar cells using 14 different XC functionals
benchmarked against MS-CASPT2 calculations. On top of reg-
ular TD-DFT calculations, the Tamm–Dancoff approximation
was also assessed. The M06-2X XCF was found to give the best
results when TD-DFT singlet excitations are combined with
TDA triplet values. Wong et al.28 investigated the behavior of
the long-range corrected functional LC-BLYP for different
values of the range-separating parameter o on a series of
coumarin dyes for dye-sensitized solar cells. Their results using
the B3LYP and modified LC-BLYP functionals were bench-
marked versus CC2 calculations and the authors concluded
that the modified LC-BLYP functional significantly improves
the accuracy of the singlet excitation energies with respect to
B3LYP. Of particular interest are the works of Huang et al.,30

Penfold,31 Sun et al.,32 and Moral et al.,33 who investigated the
behavior of various functionals for TADF active materials and
host materials. The earliest investigation by Huang et al.30

benchmarks a series of XCFs to experimental values. The
investigation includes long-range separated functionals, but

these were found to severely overestimate the S1 vertical excita-
tion energy due to the lack of tuning of the range-separated
parameters. Moral et al.33 used PBE0 within the TDA to evaluate
the S1 vertical excitation energies with respect to experimental
values, not aiming to provide a benchmark but rather to deduce
structure–property relationships. Additionally, they employed
the so-called double hybrids B2-PLYP and B2GP-PLYP to inves-
tigate the effects of non-local exchange and correlation on the
excitation energies. Although the double hybrid functionals are
noted to have a higher computational cost with similar overall
results to their calculations with PBE0, they present a slight
improvement for the compounds with a larger charge transfer
character. Sun et al.32 and Penfold31 opted to modify the range-
separating parameter in LC-oPBE and LC-BLYP, respectively.
In both works, the calculated excitation energies were bench-
marked against experimental values for the singlet excitation
energies and singlet–triplet energy gaps. With o values from
0.14 to 0.20 (given in Bohr�1 units throughout the paper) for
LC-oPBE and 0.15 to 0.19 for LC-BLYP, the authors found very
small errors (MAE o 0.15 eV) with respect to the experimental
values for both properties under investigation.

In this work, the adiabatic TD-DFT method is employed to
evaluate the first singlet and triplet excitation energies and
the corresponding singlet–triplet energy gaps for a set of 10
compounds (Fig. 1) and a number of functionals from different
rungs of a ladder leading to quantum chemical accuracy34 are
tested. Our goal is to find a method with a good trade-off
between computational cost and accuracy while being applicable
to an as large as possible set of compounds. The 10 compounds
were chosen from TADF literature or devised from ongoing work
within our groups. They were selected because they span a broad
range of experimental (and theoretical) singlet–triplet energy
splitting values, together with fluorescence characteristics going
all the way from blue to red. Furthermore, the compounds
were designed to consist of varying donor and acceptor groups
to allow generalization. For compounds 1, 4, 5, 6, 9, and 10, TADF
properties have been observed experimentally. For compound 3,
it was shown that no TADF is present. To our knowledge,
compounds 2, 7, and 8 have not been investigated yet in literature
and no experimental data on their TADF behavior are available.
The synthesis of compounds 2, 7, and 8 is therefore reported,
together with their UV-VIS absorption and emission spectra.
For the sake of consistency, we carried out similar optical
characterizations for the other compounds to provide a data
set for comparison with the quantum chemical predictions.
Our set of XCFs includes those that were already previously
tested with good results, such as M06-2X, LC-BLYP(o = 0.17),
and LC-oPBE(o = 0.17), the two latter with modified range-
separating parameters. We opted for averages of the optimal o
values obtained by Sun et al.32 and Penfold,31 because this
is more convenient for screening purposes than having to
optimize this parameter for every compound, either to get
the smallest errors on the excitation energies or to satisfy
Koopmans’ theorem.35 These XC functionals are benchmarked
against resolution-of-the-identity second-order coupled cluster
(riCC2)36 calculations. Furthermore, this work expands the

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investigation on the accurate determination of the singlet and
triplet excitation energies beyond the first excited states. This
could lead to useful insights in other fields outside that of
TADF such as upper state photophysics,37 singlet fission,38 and
photodynamic therapy (PDT).39–41 Accurate determination of
the second triplet excitation energy could be particularly useful
for (image-guided) PDT applications41–43 or when screening
materials for their potential as efficient TADF emitters according
to the vibronic mechanism, without going as far as calculating
the spin–orbit coupling or vibronic interactions.

2 Materials and methods
2.1 Synthesis and optical characterizations

Compounds 1,9 3–6,44–46 and 947 were synthesized according
to their respective literature procedures. Compound 10 was
obtained from Lumtech Inc (LT-N545). Syntheses of 2, 7,
and 8 were performed following the procedures presented
in Scheme 1. TPA-DK9 and 1,4-dibromo-2,3-difluoro-5,6-
dinitrobenzene48 were synthesized according to their respective
literature procedures. Details with respect to the synthesis and
characterization of these compounds are provided in the ESI.†

All electronic absorption spectra were measured on a
Varian Cary 5000 UV-Vis-NIR spectrophotometer from Agilent
Technologies. Corrected steady-state emission spectra were
recorded on a Horiba-Jobin Yvon Fluorolog-3 spectrofluoro-
meter equipped with a 450 W xenon lamp as light source, with
excitation at the ascending slope of the charge transfer absorp-
tion band. Freshly prepared samples in 1 cm quartz cells were
used to perform all UV-Vis-NIR absorption and fluorescence
measurements. Fluorescence measurements were performed
using 10 mm optical path length cuvettes under a right-angle
arrangement. All spectroscopic measurements were done in
non-degassed samples at 20 1C in spectroscopic grade toluene.

2.2 Theoretical and computational details

The ground-state geometries of all compounds were fully
optimized at the M06/6-311G(d) level.49 All vibrational frequencies
are real, demonstrating the optimized geometries correspond to
minima on the potential energy surface. For compound 8, the
triisopropysilyl (TIPS) groups were substituted by H atoms to
reduce the computational cost. Adiabatic TD-DFT calculations
were performed using a variety of XC functionals (Table 1),
ranging from GGA, meta-GGA and global hybrid to range-
separated hybrid XC functionals using the 6-311G(d) basis set.
For the LC-BLYP and LC-oPBE range-separated hybrids, the
range-separating parameter o was modified. The modified
versions of LC-BLYP and LC-oPBE are henceforth denoted with
the respective value of o in the name: LC-BLYP33 [which is a
condensed notation for LC-BLYP(o = 0.33)], LC-BLYP20,
LC-BLYP17, and LC-oPBE17. All excited state calculations were
performed using the PCM model (cyclohexane) to account for
an apolar environment. In addition to TD-DFT calculations, its
Tamm–Dancoff approximation was also applied for all func-
tionals and all compounds, allowing the assessment of its
performance with respect to TD-DFT. From the excited state
calculations, the vertical excitation energies to the S1 (DES0–S1

=
ES1
� ES0

), S2 (DES2–S0
= ES2

� ES0
), T1 (DET1–S0

= ET1
� ES0

) and T2

(DET2–S0
= ET2

� ES0
) states, the oscillator strengths of the

corresponding S0 - S1 and S0 - S2 transitions ( fS0–S1
and

fS0–S2
), and the dominant one-electron transitions (molecular

orbital pairs) for S0 - S1 and S0 - T1 transitions were
obtained. From these values, DES1–T1

= DES1–S0
� DET1�S0

(also
referred to as DEST) and DET2–T1

= DET2–S0
� DET1–S0

were
calculated. All DFT and TD-DFT/TDA calculations were per-
formed using the Gaussian09 program.50

The vertical excitation properties obtained with TD-DFT and
TD-DFT/TDA were benchmarked using the resolution-of-
the-identity approximation of the second-order approximated
coupled-cluster model36 using the Turbomole program (Version
7.3.1).51 The resolution of the identity approximation for two-
electron integrals reduces the CPU time needed to calculate these
integrals. In addition, the method uses a partitioned form of the
CC2 equations, eliminating the need to store double excitation
cluster amplitudes. Using the riCC2 method allows computation
of much larger systems such as the ones used in this work,
whereas they would be difficult to perform with the unaltered
CC2 method. Schreiber et al.52 showed that the vertical excitation

Fig. 1 Overview of the donor–acceptor compounds studied in this work.

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Scheme 1 Synthesis procedures for compounds 2, 7, and 8: (i) ethanol/acetic acid (19/1) at reflux for 1 h; (ii) Pd(OAc)2, XPhos and Na(OtBu) in toluene at
110 1C for 24 h; (iii) bis(pinacolato)diboron, [Ir(OMe)(COD)]2, 4,40-di-tert-butyl-2,20-bipyridine in dry cyclohexane at 80 1C for 24 h; (iv) Oxonet in THF/
H2O (10/1) at room temperature for 3 h; (v) n-BuLi, dry THF at 0 1C for 1 h, (vi) TIPSCl, dry THF at 65 1C for 16 h; (vii) n-BuLi, dry THF at �78 1C for 1 h;
(viii) CuBr, LiBr in dry THF at 0 1C for 1 h; (ix) oxalyl chloride at 0 1C for 1 h; (x) FeCl3, CH3NO2 in CH2Cl2 at room temperature for 3 h; (xi) iron powder in
acetic acid at 45 1C for 6 h; (xii) acetic acid at 75 1C for 16 h; (xiii) Pd(PPh3)4, K2CO3 in DMF/H2O (4/1) at 130 1C for 24 h.

Table 1 Overview of the selected XC functionals grouped by their position on Jacob’s ladder

XC functional % HF exchangea Range-separating parameter ob (Bohr�1)

GGA BLYP
BPW91
PBE

Meta-GGA M06L
TPSS

Global hybrid GGA B3LYP 20
B971 21
PBE0 25

Global hybrid meta-GGA TPSSh 10
M06 27
M06-2X 54

Long-range separated hybrid GGA/meta-GGA oB97 0–100 0.40 (1.323)
oB97X 16–100 0.30 (1.764)
CAM-B3LYP 19–65 0.33 (1.604)
LC-oPBE17 0–100 0.17 (3.113)
LC-BLYP17 0–100 0.17 (3.113)
LC-BLYP20 0–100 0.20 (2.646)
LC-BLYP33 0–100 0.33 (1.604)
LC-BLYP 0–100 0.47 (1.126)

a For long-range corrected functionals, the % HF exchange is given at interelectronic distance r12 = 0 and N. b Corresponding length L = 1/o (Å) in
parentheses.

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energies obtained with CC2 are in good agreement with those
obtained with even higher levels of approximation such as the
third-order approximation of the coupled cluster (CC3) and
coupled cluster singles, doubles and triples (CCSDT) when the
excitations exhibit a single excitation nature. Indeed, in these
benchmarks on small- and medium-size closed-shell organic
molecules, for singlet excitation energies, the mean absolute errors
(MAEs) with respect to the CASPT2 reference data amount to
0.32 eV, 0.50 eV, and 0.22 eV for the CC2, CCSD, and CC3 methods,
whereas for the triplet excitation energies, the MAEs given in the
same order are 0.19 eV, 0.16 eV, and 0.08 eV, respectively.
An alternative choice for the benchmark calculations might have
been the spin-scaled-component version of CC2, SCS-CC2,53 but
recent investigations on the prediction of the valence excitation
energies of closed-shell organic chromophores did not demonstrate
an advantage for employing or not the spin-scaled-component
scheme.29,53–55 The same ground-state geometries were used for
the riCC2 calculations as for the TD-DFT and TD-DFT/TDA calcula-
tions to exclude any geometry dependent differences. For these
Turbomole calculations, def2-TZVP was chosen both as the main
and auxiliary basis set as it also comprises split-valence triple zeta
basis functions with polarization functions for the second- and
third-row atoms. Additional TD-DFT/def2-TZVP calculations were
performed to demonstrate that the use of different basis sets for
the riCC2 and TD-DFT calculations has a negligible impact. Indeed,
at the TD-DFT/LC-BLYP17 level, differences of excitation energies
between the def2-TZVP and 6-311G* basis sets are always smaller
than 0.08 eV and the mean absolute difference amounts to 0.03 eV.
Moreover, the impact of the ground state geometries on the first
singlet and triplet excitation energies has been assessed by
performing additional TD-DFT/LC-BLYP17 excitation energy calcu-
lations on geometries optimized at the oB97X-D/6-311G(d) level of
approximation. The oB97X-D56 XC functional accounts explicitly
for London dispersion forces by using empirical expressions,
whereas M06 was parameterized to account, implicitly, for
London dispersion interactions. The differences of TD-DFT/
LC-BLYP17 excitation energies between the M06 and oB97X-D
optimized geometries is negligible, with a mean absolute
difference of 0.04 eV.

The mean absolute errors (MAEs), mean signed errors
(MSEs), and standard deviations are calculated for all XCFs
and for both TD-DFT and TD-DFT/TDA schemes in comparison
to the reference riCC2 results (difference = TD-DFT � riCC2).
The absolute average errors help to objectively determine the
most accurate XC functional, whereas the sign-dependent
average errors yield some insight to whether the chosen XC
functional tends to over- or underestimate the given property.

3 Results and discussion
3.1 Experimental spectroscopic results and earlier
theoretical data

Results from the literature are summarized in Table 2 for the
electronic and optical properties of compounds 1, 3–6, and
9–10. These include experimental data as well as TD-DFT
results related to the first singlet and triplet excited states.
Our experimental UV-Vis absorption and emission data are also
given for the whole list of compounds. Compound 7 was found
to be non-emissive. Although the differences between literature
results and ours are generally small (a few tenths of eV), the
DEmax of absorption of 9 differs by more than one eV.
As discussed below, the new value is much more consistent
with the quantum-chemical calculations. This new experi-
mental value of 9 substantiates the fact that we enacted our
own set of optical characterizations.

3.2 Reference riCC2 calculations

The benchmark riCC2 values used to reference the TD-DFT and
TD-DFT/TDA calculations are given in Table 3. The agreement
between the first vertical singlet excitation energies (DES0–S1

)
calculated using riCC2 and the experimental absorption
maxima was verified by constructing a correlation plot between
the two properties (Fig. 2). The slope of the correlation plot is
slightly larger than 1, indicating that the riCC2 vertical excita-
tion energy increases more rapidly with increasing absorption
maximum. The R2 value of 0.95 points to a very good correlation
between the two properties, proving the high level of predict-
ability of riCC2 calculations in comparison to experimental

Table 2 Experimental and theoretical properties of the selected compounds

DEmax, eV
(lmax, nm) abs.a

DEmax, eV
(lmax, nm) em.a

DEmax, eV
(lmax, nm) abs.b

DEmax, eV
(lmax, nm) em.b

DET1–S0

d,
eV

DEST
d,

eV
Calc. DET1–S0

e,
eV

XC functional and
basis sete

19 2.95 (420) 2.25 (552) 2.95 (421) 2.15 (578) — — 0.11 B3LYP/6-31G(d)
2 — — 3.37 (368) 2.55 (486) — — — —
345 2.43 (510) 1.91 (648) 2.43 (510) 1.86 (668) — — 0.69 LC-oPBE/6-31G(d)f

445 2.57 (482) 1.94 (638) 2.58 (481) 1.92 (645) 2.12 0.11 0.02 LC-oPBE/6-31G(d)f

546 3.32 (373) 2.65 (467) 3.31 (375) 2.62 (473) 2.84 0.13 0.32 B3LYP/6-31G
644 3.38 (367) 2.99 (414) 3.43 (362) 2.99 (414) 2.97 0.26 0.13 B3LYP/6-31G(d,p)
7 — — 2.02 (613) —c — — — —
8 — — 2.88 (430) 2.24 (553) — — — —
947 3.67 (338) 2.10 (590) 2.58 (480) 2.06 (602) 1.95 0.10 0.68 B3LYP/6-31G(d)
1057 B3.18 (B390) 2.45 (507) 3.14 (395) 2.39 (519) 2.65 0.08 0.03 B3LYP/6-31G(d)

a Experimental DEmax (lmax) for the absorption (S0 - S1) and emission (S1 - S0) in toluene as taken from the respective references. b Experimental
DEmax (lmax) for the absorption (S0 - S1) and emission (S1 - S0) in toluene as measured in this work. c No fluorescence observed. d Experimental
values obtained from the corresponding literature. e TD-DFT values taken from the respective references. f The authors make no reference to
modification of the range-separating parameter, which is therefore expected to be default (o = 0.33).

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16392 | Phys. Chem. Chem. Phys., 2020, 22, 16387--16399 This journal is©the Owner Societies 2020

values. Although the number of points is reduced, a similarly
good correlation between riCC2 and experiment is observed for

the S0 - T1 transition energies. For DEST no valid correlation
could be drawn, probably because the experimental values are
not obtained from the difference between DES0–S1

and DES0–T1
,

but rather from the onsets of the fluorescence (at room tem-
perature) and phosphorescence (at 77 K) CT emission peaks.

3.3 TD-DFT and TD-DFT/TDA versus riCC2 calculations

Singlet excitation energies and oscillator strengths. For the
vertical singlet excitation energies and the corresponding oscil-
lator strengths, the choice was made to only consider states
that have the same nature in both the riCC2 and the TD-DFT or
TD-DFT/TDA calculations. With the exception of compound 7
(HOMO�1 - LUMO), the dominant nature of the first excited
state of all compounds is of HOMO - LUMO character for
the riCC2 calculations. Given the near degeneracy (Table 3) of
the first and second vertical excitation energies, and to have the
comparison between the compounds more in line with each
other, we have chosen to take the HOMO - LUMO transition
as the first state and the HOMO�1 - LUMO transition as the
second state for compound 7. In a few compound-functional
combinations, the nature of the first singlet excited state
calculated with TD-DFT or TD-DFT/TDA does not correspond
to a HOMO - LUMO transition. We then opted to use the
second (HOMO - LUMO) excited state to compare states of
the same nature. Indeed, comparison between two states of
different nature may lead to faulty conclusions about the XCFs
ability to correctly predict the targeted properties. These statis-
tical analyses are provided in Table 4 for TD-DFT values and in
Table 5 for the TD-DFT/TDA ones. The full list of all compounds
and their excited state data for the first two states can be found
in the ESI.†

In Fig. 3, each colored line represents a single XCF and the
sign-dependent error with respect to the riCC2 calculation is
plotted for every compound. For the first singlet excitation

Table 3 riCC2 excitation energies, oscillator strengths, and energy gaps
for compounds 1–10

DES0–S1
(eV)

DES0–S2
(eV) fS0–S1

fS0–S2

DES0–T1
(eV)

DES0–T2
(eV)

DET1–T2
(eV)

DES1–T1
(eV)

1 2.969 2.996 0.170 0.098 2.903 2.948 0.045 0.067
2 3.468 3.496 0.157 0.131 3.337 3.354 0.016 0.130
3 2.474 3.355 0.231 0.008 2.081 3.256 1.175 0.393
4 2.456 2.525 0.000 0.000 2.457 2.527 0.069 �0.001
5 3.524 3.718 0.396 0.004 3.227 3.411 0.185 0.297
6 3.565 3.646 0.207 0.001 3.494 3.502 0.009 0.071
7 1.923 1.882 0.094 0.002 1.852 1.853 0.001 0.029
8 3.076 3.150 0.083 0.005 2.575 2.961 0.386 0.501
9 2.692 3.488 1.011 0.585 2.159 2.809 0.651 0.534
10 3.022 3.031 0.032 0.010 3.010 3.019 0.009 0.012

Fig. 2 Correlation plots between the riCC2 vertical excitation energies
and the experimental values for maximum absorption DES0–S1

and DES0–T1
.

Table 4 Statistical analysis including mean absolute errors (MAEs, eV) and standard deviations (eV) obtained from the comparison between TD-DFT
calculations with different XCFs and riCC2 reference values

DES0–S1
(eV) fS0–S1

DES0–S2
(eV) fS0–S2

DES0–T1
(eV) DES0–T2

(eV) DET1–T2 (eV) DES1–T1
(eV)

MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev

BLYP 1.242 0.213 0.116 0.144 1.329 0.231 0.072 0.173 1.153 0.265 1.224 0.190 0.139 0.161 0.113 0.144
BPW91 1.236 0.216 0.113 0.143 1.322 0.230 0.072 0.173 1.153 0.266 1.220 0.193 0.138 0.160 0.111 0.142
PBE 1.238 0.216 0.113 0.143 1.322 0.229 0.072 0.173 1.153 0.268 1.225 0.195 0.144 0.168 0.112 0.142
M06L 1.021 0.202 0.109 0.138 1.102 0.237 0.072 0.173 0.952 0.224 1.005 0.178 0.130 0.145 0.100 0.130
TPSS 1.166 0.204 0.110 0.139 1.244 0.223 0.069 0.173 1.090 0.251 1.151 0.188 0.137 0.155 0.105 0.138
TPSSh 0.850 0.149 0.095 0.119 0.904 0.191 0.072 0.173 0.820 0.158 0.845 0.157 0.115 0.137 0.078 0.083
B3LYP 0.596 0.110 0.074 0.089 0.628 0.172 0.070 0.173 0.610 0.086 0.605 0.112 0.076 0.090 0.054 0.048
B971 0.555 0.102 0.068 0.083 0.583 0.169 0.070 0.173 0.566 0.083 0.563 0.105 0.072 0.084 0.053 0.046
PBE0 0.430 0.089 0.053 0.064 0.445 0.164 0.119 0.229 0.526 0.068 0.482 0.080 0.049 0.059 0.096 0.070
M06 0.379 0.106 0.050 0.057 0.399 0.179 0.068 0.173 0.460 0.089 0.430 0.091 0.035 0.037 0.081 0.068
M06-2X 0.283 0.102 0.084 0.107 0.257 0.109 0.024 0.050 0.162 0.081 0.133 0.095 0.081 0.081 0.239 0.112
oB97 0.882 0.239 0.254 0.233 0.850 0.238 0.088 0.150 0.388 0.196 0.351 0.243 0.362 0.311 1.263 0.161
oB97X 0.740 0.215 0.225 0.224 0.713 0.203 0.060 0.135 0.311 0.184 0.300 0.221 0.332 0.287 1.044 0.142
CAM-B3LYP 0.311 0.124 0.101 0.145 0.316 0.118 0.047 0.084 0.332 0.177 0.234 0.133 0.213 0.162 0.643 0.119
LC-oPBE17 0.141 0.102 0.051 0.101 0.182 0.086 0.048 0.117 0.344 0.099 0.302 0.086 0.050 0.057 0.209 0.127
LC-BLYP17 0.157 0.106 0.050 0.098 0.192 0.085 0.042 0.115 0.314 0.089 0.287 0.079 0.039 0.042 0.156 0.102
LC-BLYP20 0.107 0.074 0.086 0.138 0.075 0.074 0.052 0.128 0.215 0.129 0.157 0.076 0.072 0.072 0.270 0.136
LC-BLYP33 0.650 0.192 0.210 0.211 0.600 0.195 0.058 0.137 0.231 0.161 0.248 0.179 0.245 0.220 0.868 0.125
LC-BLYP 0.978 0.264 0.270 0.236 0.956 0.262 0.117 0.204 0.623 0.239 0.492 0.248 0.400 0.354 1.601 0.187
LC-BLYP20-M06-2X — — — — — — — — — — — — — — 0.155 0.090

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energy (Fig. 3a), we can clearly see some functionals heavily
overestimating (LC-BLYP, oB97, oB97X, LC-BLYP33) and some
functionals underestimating (PBE, BPW91, BLYP, TPSS, M06L,
TPSSh, B3LYP, B971, PBE0, M06) the singlet excitation energy.
Overestimation of the singlet excitation energy typically occurs
when there is too much HF exchange, i.e., for range-separated
hybrids when the range-separating parameter is not properly
tuned (o is too large). Underestimation happens when there is
not enough HF exchange present, as is the case for the GGA and
meta-GGA functionals, but even for some of the hybrid
functionals. This originates from the ultra-locality of the XCF
and the related self-interaction error, leading to a poor description
of the CT excitations.58,59 We also observe some variations of the
errors as a function of the nature of the compounds, but most XC
functionals seem to follow the same trend. The absolute values of
the errors depicted in Fig. 3 are averaged per functional in Table 4
and are provided together with their standard deviation. From
Table 4, the functionals that show a relatively small error are
CAM-B3LYP (0.311 eV) and M06-2X (0.283 eV), but they are
outperformed by LC-oPBE17 (0.141 eV), LC-BLYP17 (0.157 eV)
and LC-BLYP20 (0.141 eV). From the difference between the
signed and absolute errors (Table S1, ESI†), it is also clear that
LC-BLYP20 tends to either slightly over- or underestimate the
singlet excitation energy for a given compound. This is likely
due to the small size of the error, as the other functionals all
show the same amplitudes for the signed and absolute errors.
Taking the LC-BLYP XCFs, the signed errors range from�0.157 eV
(o = 0.17, the amount of HF exchanges grows the least rapidly
with r12) to 0.056 eV (o = 0.20), 0.650 eV (o = 0.33), and 0.978 eV
(o = 0.47, the amount of HF exchanges grows the fastest with r12),
demonstrating the key role of long-range exchange. Using the
Tamm–Dancoff approximation has a minor impact in the sense
that large underestimations or overestimations of DES0–S1

remain
when employing that approximation.

The second vertical singlet energy, DES0–S2
, follows the same

trend as the first vertical singlet energy. A decrease in the MAE
is observed with increasing HF exchange until a minimum is
reached and the error increases again as the amount of HF
exchange becomes too high. The most notable XC functionals
are the same as for the first vertical singlet energy, LC-oPBE17
(0.182 eV), LC-BLYP17 (0.192 eV) and LC-BLYP20 (0.182 eV).
Again, there is little difference between the TD-DFT and
TD-DFT/TDA results. For both singlet excitation energies,
LC-BLYP20 gives the smallest MAEs. The standard deviations
on the MAEs for both properties are related to the MAEs
themselves. If the MAE decreases, so does the standard
deviation and vice versa. A slight exception to this is M06-2X
(0.102 eV), which has relatively small standard deviations,
still comparable to LC-BLYP17 (0.106 eV) and LC-oPBE17
(0.102 eV), despite having a MAE that is nearly twice as large.

For the oscillator strengths, the analysis is less straight-
forward because their amplitudes cover more than one order of
magnitude [even after neglecting those states with very small
( f o 0.05) values]. Moreover, as shown in Table 3 (riCC2
values), with exception of compound 9, all compounds show
relatively small to very small oscillator strengths. This is most
definitely true for the oscillator strength of the S0 - S2

transition. As a matter of fact, for these very small oscillator
strengths, most XCFs perform well because they reproduce the
dominant character of the transition, which determines the
negligible f values. This is illustrated by compound 4, where the
S0 - S1 and S0 - S2 transitions have a CT character, between
the donor and acceptor groups that are perpendicular to each
other (and therefore of different symmetry). To a given extent,
the same trend as for the first singlet excitation energies
roughly holds. This is expected as a good representation of
the excitation energy should give a good description of its
oscillator strength. Then, as already observed for benzene

Table 5 Statistical analysis including mean absolute errors (MAEs, eV) and standard deviations (eV) obtained from the comparison between TD-DFT/TDA
calculations with different XCFs and riCC2 reference values

DES0–S1
(eV) fS0–S1

DES0–S2
(eV) fS0–S2

DES0–T1
(eV) DES0–T2

(eV) DET1–T2
(eV) DES1–T1

(eV)

MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev MAE StdDev

BLYP 1.220 0.226 0.098 0.118 1.318 0.242 0.071 0.173 1.148 0.269 1.222 0.191 0.142 0.165 0.112 0.137
BPW91 1.214 0.229 0.096 0.117 1.310 0.242 0.071 0.173 1.146 0.272 1.217 0.194 0.141 0.164 0.111 0.136
PBE 1.215 0.229 0.096 0.116 1.311 0.241 0.071 0.173 1.147 0.273 1.218 0.196 0.142 0.165 0.112 0.136
M06L 1.001 0.213 0.093 0.117 1.092 0.245 0.072 0.173 0.932 0.240 0.992 0.184 0.135 0.153 0.105 0.136
TPSS 1.144 0.216 0.094 0.114 1.236 0.236 0.071 0.173 1.079 0.259 1.143 0.193 0.138 0.160 0.108 0.135
TPSSh 0.828 0.160 0.076 0.097 0.894 0.198 0.070 0.173 0.784 0.190 0.818 0.171 0.129 0.151 0.094 0.116
B3LYP 0.574 0.120 0.055 0.071 0.618 0.177 0.069 0.173 0.560 0.122 0.580 0.122 0.102 0.108 0.074 0.081
B971 0.533 0.113 0.050 0.065 0.573 0.173 0.068 0.173 0.520 0.116 0.540 0.115 0.097 0.101 0.072 0.077
PBE0 0.407 0.099 0.036 0.047 0.435 0.168 0.067 0.173 0.436 0.079 0.444 0.082 0.068 0.064 0.064 0.056
M06 0.356 0.113 0.035 0.045 0.389 0.182 0.067 0.173 0.372 0.079 0.371 0.113 0.080 0.083 0.061 0.062
M06-2X 0.322 0.088 0.100 0.137 0.289 0.116 0.017 0.026 0.144 0.127 0.177 0.101 0.052 0.045 0.187 0.104
oB97 0.966 0.208 0.287 0.274 0.903 0.234 0.051 0.118 0.236 0.183 0.385 0.269 0.155 0.172 0.731 0.103
oB97X 0.812 0.187 0.255 0.265 0.758 0.199 0.047 0.114 0.210 0.166 0.322 0.213 0.129 0.135 0.612 0.100
CAM-B3LYP 0.356 0.105 0.119 0.171 0.351 0.109 0.037 0.047 0.151 0.090 0.127 0.103 0.067 0.074 0.303 0.127
LC-oPBE17 0.128 0.087 0.065 0.125 0.153 0.089 0.064 0.120 0.273 0.078 0.264 0.089 0.043 0.038 0.171 0.109
LC-BLYP17 0.141 0.092 0.066 0.123 0.164 0.090 0.026 0.069 0.262 0.075 0.261 0.079 0.036 0.021 0.137 0.090
LC-BLYP20 0.118 0.097 0.107 0.171 0.088 0.072 0.036 0.088 0.149 0.066 0.123 0.052 0.044 0.040 0.226 0.130
LC-BLYP33 0.721 0.168 0.241 0.254 0.645 0.194 0.047 0.113 0.175 0.149 0.255 0.184 0.100 0.098 0.563 0.127
LC-BLYP 1.073 0.230 0.305 0.280 1.030 0.254 0.061 0.119 0.199 0.181 0.351 0.307 0.171 0.186 0.883 0.125
LC-BLYP20-M06-2X — — — — — — — — — — — — — — 0.146 0.097

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16394 | Phys. Chem. Chem. Phys., 2020, 22, 16387--16399 This journal is©the Owner Societies 2020

Fig. 3 Signed errors on each property for each individual XC functional for all 10 compounds obtained using TD-DFT. (a) First vertical singlet excitation
energy, DES0–S1

. (b) Oscillator strength for the first singlet excitation, fS0–S1
. (c) Second vertical singlet excitation energy, DES2–S0

. (d) Oscillator strength for
the second vertical singlet excitation, fS0–S2

. (e) First vertical triplet excitation energy, DET1–S0
. (f) Second vertical triplet excitation energy,

DET2–S0
. (g) Triplet–triplet energy gap for the first two triplet excited states, DET2–T1

. (h) Singlet–triplet gap for the first singlet and triplet excited states,
DES1–T1

.

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This journal is©the Owner Societies 2020 Phys. Chem. Chem. Phys., 2020, 22, 16387--16399 | 16395

derivatives, XCFs with a small amount of HF exchange under-
estimate both the excitation energies and oscillator strengths.60

For the S0 - S1 transition, a large number of functionals
are within 0.2 of each other and give relatively consistent
results. Functionals such as LC-BLYP17 (0.050 eV), LC-oPBE17
(0.051 eV), LC-BLYP20 (0.086 eV), M06-2X (0.084 eV), CAM-
B3LYP (0.101 eV), M06 (0.050 eV), PBE0 (0.053 eV), B971
(0.068 eV), B3LYP (0.074 eV) and TPSSh (0.095 eV) all have
errors below 0.1, albeit their standard deviations are larger than
the errors themselves (since the errors originate from a few
excited states only). Using the Tamm–Dancoff approximation
again barely changes these general conclusions, although, in
more detail, its impact depends on the XCF and on the excited
state (S1 or S2). Indeed, the absolute average error decreases
slightly for most functionals, except for M06-2X and the range-
separated functionals when looking at the S0 - S1 transition.
For the S0 - S2 transition, the MAEs decrease slightly for all
functionals except for LC-oPBE17 and LC-BLYP. The best
performing functionals for both transitions are M06-2X,
LC-BLYP17, LC-oPBE17, and LC-BLYP20.

First and second triplet excitation energies. To achieve
efficient intersystem crossing, the 3CT and 3LE state (or 3CT
state of a different nature) need to be close in energy for
efficient vibronic coupling to take place. Therefore, this work
not only focuses on calculating the first, but also the second
vertical triplet excitation energies (Fig. 3 and Tables 4, 5).
Looking at the data obtained using the TD-DFT approach,
the error on the first triplet excitation energy, DES0–T1

, is the
smallest for M06-2X (0.162 eV). This is consistent with the
results of Brückner et al.9a in their report on the singlet–triplet
gap for triplet–triplet annihilation. When looking at the
other functionals that perform well, LC-BLYP20 (0.215 eV),
LC-BLYP33 (0.231 eV), oB97X (0.311 eV), LC-BLYP17 (0.314 eV),
CAM-B3LYP (0.332 eV), and LC-oPBE17 (0.344 eV) all have
absolute average errors within 0.35 eV. It is apparent that a
substantially high amount of HF exchange is necessary to accu-
rately describe the triplet energy as is observed for the LC-BLYP
functionals with o = 0.20 and 0.33 which outperform those with
o = 0.17 and LC-oPBE17. On the other hand, the unmodified
LC-BLYP has a much higher MAE, indicating a too large value for
o. The MAEs for the second vertical triplet excitation energy,
DES0–T2

, follow the same trend as for the first excitation energy and
are even a bit smaller for all functionals.

When comparing TD-DFT with TD-DFT/TDA, all functionals
show a more consistent behavior throughout the series (Fig. S1,
ESI†). The accuracy of all functionals increases as can be seen
by a decrease of the absolute average error for all functionals.
Most notable are CAM-B3LYP (0.332 eV - 0.151 eV) and
LC-BLYP (0.623 eV - 0.199 eV), which gain significant accuracy
and have smaller standard deviations when using the TD-DFT/
TDA approach. Upon using the TDA, a number of functionals,
comprising M06-2X (0.144 eV), CAM-B3LYP (0.151 eV),
LC-BLYP20 (0.149 eV), LC-BLYP33 (0.175 eV), and LC-BLYP
(0.199 eV), have absolute average errors within 200 meV. The
sudden decrease in MAE for the vertical triplet excitation
energies using functionals such as CAM-B3LYP and LC-BLYP

is attributed to the ground state triplet instability problem for
these systems, which is known to occur when using XCFs with
a large amount of HF exchange.58,59,61,62 In the TDA, only
excitation between occupied-virtual orbital pairs is allowed as
opposed to conventional TD-DFT, where virtual-occupied
de-excitation contributions are also allowed. The form of the
TDA eigenvalue equation precludes the occurrence of imagi-
nary excitation energies and hence circumvents the triplet
instability issues that can arise. Still, the triplet instability
problem has smaller impact for our set of selected compounds
than in the case of cyanines.59,62

The error on the second triplet excitation energy is the
smallest for M06-2X (0.133 eV) and furthermore it has also
the smallest standard deviation. Second best is LC-BLYP20
(0.157 eV), which, despite its relatively poor accuracy with
respect to M06-2X, has a similar standard deviation and is
closely followed by CAM-B3LYP (0.234 eV). Other functionals
that are noteworthy are LC-oPBE (0.302 eV), LC-BLYP17
(0.287 eV), and LC-BLYP33 (0.248 eV), all having absolute
average errors within 0.3 eV. The overall performance is
enhanced when using the TDA, except for M06-2X, in which
the absolute average error goes up by 25% (0.133 - 0.177 eV).
Using TDA, LC-BLYP20 (0.123 eV) and CAM-B3LYP (0.127 eV)
perform similarly.

Due to the vibronic enhancement, the T1–T2 energy gap can
help to explain a compound’s potential for TADF (Fig. 3 and
Tables 4 and 5). The largest absolute average error is 0.400 eV
(LC-BLYP), meaning that the errors are generally smaller than
when looking at the energies of the individual states. This leads
us to conclude that a large number of functionals benefit from
a cancellation of two relatively large errors. Therefore, the
discussion will mainly focus on the functionals that did well
in estimating the transition energies to T1 and T2. The best
performing functional is LC-BLYP17 (0.039 eV), followed by
LC-oPBE17 (0.050 eV), LC-BLYP20 (0.072 eV), and M06-2X
(0.081 eV). Applying the Tamm–Dancoff approximation, the
absolute average errors have significantly decreased further
for the aforementioned functionals. Furthermore, CAM-B3LYP
(0.213 - 0.067 eV) performs a whole lot better under the TDA,
as was also shown for the T1 and T2 energies.

Singlet–triplet energy gap. The first property that is typically
considered when analyzing a compound for TADF, is the
singlet–triplet energy gap. Its amplitude is correlated, with
good approximation, to the spatial overlap between the HOMO
and LUMO because the first singlet and triplet transitions are
dominated by the HOMO-to-LUMO configuration. This is
evidenced in Fig. S2 (ESI†) at the representative M06/
6-311G(d) level of approximation. One would expect that the
XC functionals that are accurately describing both the singlet
and triplet excitation energies also perform well at describing
the energy gap between them. Therefore, from the previous
sections, it follows that LC-oPBE17 (0.209 eV), LC-BLYP17
(0.156 eV), LC-BLYP20 (0.270 eV) and M06-2X (0.239 eV)
are reliable functionals to use under the regular TD-DFT
formalism, despite not having the smallest absolute errors or
standard deviations. Ideally, TD-DFT/TDA is used and the error

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on the energy gaps for LC-oPBE17 (0.171 eV), LC-BLYP17
(0.137 eV), LC-BLYP20 (0.226 eV), and M06-2X (0.187 eV) is
reduced. CAM-B3LYP (0.643 - 0.303 eV) still shows a relatively
large error on DEST due to its larger average error on the singlet
excitation energy than the other two functionals.

Surprisingly, most functionals, including the GGA and
meta-GGA functionals without HF exchange (BLYP, BPW91,
PBE, TPSS, M06L), perform well for the singlet–triplet energy
gap. These data also explain why one of the most used func-
tionals in literature (B3LYP), has not been found to give
erroneous results, despite its poor ability to describe both the
singlet and triplet excitation energies. The same goes for the
other functionals. When looking at the singlet and triplet
energies individually, they fail dramatically in predicting accu-
rate values. The corresponding underestimations of both types
of excitation energies have the same origin, which is related to
the CT character of the excitations.58 Therefore, their ability to
accurately predict DEST is due to the compensation of two large
errors, as was also seen for the T1–T2 energy gap. Ultimately,
since they are obtained with the wrong underlying quantities,
these functionals are not recommended, even though they
might give a quantitatively correct answer for a certain set of
compounds. Because of the large errors on the singlet and
triplet excitation energies when using these functionals, one
can never be sure that the calculated singlet–triplet energy
gap is trustworthy, nor whether related quantities like the
spin–orbit couplings could be accurate since they depend on
the nature of the excited state.

Lastly, the possibility to use a ‘hybrid’ approach in which the
vertical singlet and triplet excitation energies are calculated
using two different functionals can be taken into consideration,
i.e., (i) LC-BLYP20 that gives the best results for the singlet
excitation energies and (ii) M06-2X that provides the best
results for the triplet excitation energies. Combining the results
from both functionals shows mean absolute errors on
the singlet–triplet energy gap of 0.155 eV using TD-DFT and
0.146 eV using TD-DFT/TDA (Tables 4 and 5). These values are
comparable to those obtained with single functional calcula-
tions such as LC-BLYP17, but give better performance than
functionals such as M06-2X and LC-BLYP20 separately.

Further discussion on the performance of the TD-DFT
scheme with respect to riCC2. To further evaluate the XC
functionals used in this work, the consistency of the TD-DFT
and TD-DFT/TDA results versus the results obtained using
riCC2 for the first excited singlet and triplet states was checked
by constructing correlation plots. In Fig. 4, the correlation is
given for LC-BLYP20, which is the best performing functional
for the vertical singlet excitation energies and also performs
well for the triplet vertical excitation energies. In Fig. S3 (ESI†),
the correlation plots for all functionals used in this work can be
found. While each of the functionals shows a linear correlation,
the slopes vary widely from 0.66 to 1.07, with R2 values varying
from 0.73 to 0.98. XC functionals without HF exchange (BLYP,
BPW91, PBE, M06L, and TPSS) reproduce the riCC2 values
poorly, with slopes much smaller than 1 and R2 values below
0.85. The group of functionals with a small percentage of

HF exchange (TPSSh, B3LYP, B971, PBE0, M06, and M062X)
performs significantly better, especially with increasing
amount of HF exchange. Apart from TPSSh (with only 10% of
HF exchange), these functionals are showing correlations with
R2 values above 0.95, but with slopes that are still only around
0.9. As the amount of HF exchange increases to 54% for
M06-2X, slopes of around 1.0 with R2 values of around 0.95
are found. Finally, the group of range-separated functionals
(oB97, oB97X, CAM-B3LYP, LC-oPBE17, LC-BLYP17, LC-BLYP20,
LC-BLYP33, and LC-BLYP) shows varying behavior, depending on
their range-separated parameter. Due to the, generally, higher
amount of HF exchange, the slopes of the correlation plots are all
in the range of 0.95–1.07, but the R2 values change heavily. In the
cases where the range-separating parameter is too large, the
amount of HF exchange increases rapidly and the R2 values get
smaller, as is the case for oB97, oB97X, CAM-B3LYP, LC-BLYP33,
and LC-BLYP. For the tuned range-separating functionals such as
LC-oPBE17, LC-BLYP17, and LC-BLYP20, both the slopes and R2

values obtained are close to 1.0. Utilizing the Tamm–Dancoff
approximation shows little influence on the correlation of the
vertical singlet excitation energies, as was also shown above. For
the correlation on the triplet excitation energies, the TD-DFT/TDA
approach leads to a slight decrease in the slope and R2 values for
the hybrid and non-hybrid functionals and leads to a slight
increase or decrease in the slope depending on whether the slope
was smaller or larger than 1.0 and a slight increase in the R2

values for the range-separated functionals. Overall, it means
the TDA approach gives a better correlation (closer to 1) for the
range-separated functionals with an increased R2. It is further
interesting to analyze the intercepts at the origin for the different
first singlet and triplet excitation energies, which are directly
related to a comparison of the regression lines. For GGA, mGGA,
and hybrids with small amount of HF exchange, these intercepts
vary little from the singlet to triplet excitation energies or from the
TD-DFT to the TD-DFT/TDA calculations (typically by less than
0.2 eV). This gives a kind of impression (at the scale of the plots)
that a unique regression line might describe the 4 sets of data.
On the other hand, with large amount of HF exchange (global
or range-separated hybrids), the intercepts at the origin are

Fig. 4 Correlation between the LC-BLYP17 and riCC2 calculations for the
first vertical singlet and triplet excitations.

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This journal is©the Owner Societies 2020 Phys. Chem. Chem. Phys., 2020, 22, 16387--16399 | 16397

systematically larger for the DES0–S1
values than the DES0–T1

ones.
In addition, for the triplets, there is a difference between the
TD-DFT and TD-DFT/TDA results, with smaller intercepts at the
origin using the TD-DFT scheme.

These results substantiate our findings in the previous
sections. The functionals that tend to predict the vertical
singlet and triplet excitation energies better, give better correla-
tions to the riCC2 results. The Tamm–Dancoff approximation
again gives improved results for the vertical triplet excitation
energies, proving that, owing to correcting for the errors related
to triplet instabilities,58 it is a very useful approximation to get
the most accurate results from the TD-DFT calculations.

4 Conclusions

In this work, the adiabatic TD-DFT method has been employed
to evaluate the first singlet and triplet excitation energies and
the corresponding singlet–triplet energy gaps on a set of
10 compounds that were designed to exhibit thermally acti-
vated delayed fluorescence. These compounds have been
selected because they cover a broad range of singlet–triplet
energy splitting values, together with fluorescence characteris-
tics going from blue to red, while presenting a clear chemical
diversity in terms of donor and acceptor units. By using a broad
variety of XC functionals, our goal was to highlight the best
ones in comparison to benchmark results evaluated at the
resolution-of-the-identity second-order coupled cluster (riCC2)
level, while the performance of the latter method against
experimental data was preliminarily demonstrated.

We have shown that two functionals stand out when aiming
at predicting the vertical singlet and triplet excitation energies,
being LC-BLYP20 and M06-2X. These functionals show steady
behavior and minimal errors on the excited state energies in
comparison to riCC2 calculations. LC-BLYP20 tends to predict
the singlet energies more accurately than the triplet energies,
whereas the opposite is true for M06-2X. Therefore, the singlet–
triplet energy gaps that follow from these calculations have
similar errors for a given compound. Furthermore, when looking
at the singlet–triplet energy gaps, LC-oPBE17 and LC-BLYP17
outperform LC-BLYP20 and M06-2X. The question of which
functional is best to use then falls back to the primary focus of
the investigation. If the focus is on the singlet state energies,
LC-BLYP20 followed by LC-oPBE17 and LC-BLYP17 would be
the optimal functionals. If the focus is rather on the position of
the triplet states and their mutual energy difference, M06-2X
would be the optimal functional. Using the Tamm–Dancoff
approximation has no significant influence on the singlet
excitation energies. It does, however, decrease the errors
obtained using LC-BLYP20, LC-BLYP17, LC-oPBE17, and M06-
2X for the triplet excitation energies and consequently for the
singlet–triplet energy gap. Furthermore, using TDA increases
the precision of the obtained triplet energy errors within the set
of 10 compounds. Although we have shown in this work that
small errors for the singlet–triplet energy gap can be obtained
with a large number of functionals, their accuracy lies in a

cancellation of two large errors. These functionals are therefore
not trustworthy.

We therefore propose the use of the Tamm–Dancoff approxi-
mation in combination with either LC-BLYP17, LC-oPBE17,
LC-BLYP20, or M06-2X when looking at the DEST quantities for
a given compound. Ultimately, a hybrid approach using the
TDA approach and taking the DEST from the singlet and triplet
excitation energies obtained with LC-BLYP20 and M06-2X,
respectively, gives errors on the singlet–triplet gap that are
roughly the same as those obtained with the best functional
LC-BLYP17, but with better estimates for the singlet and triplet
excitation energies respectively.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

This work is supported by the University of Namur and Hasselt
University (PhD BILA scholarship T. Cardeynaels). The authors
also thank the Research Foundation – Flanders (FWO
Vlaanderen) for financial support (project G.0877.18N,
Hercules project GOH3816NAUHL, and SB PhD scholarship
S. Paredis). The calculations were performed on the computers
of the ‘‘Consortium des équipements de Calcul Intensif
(CÉCI)’’ (http://www.ceci-hpc.be), including those of the
‘‘UNamur Technological Platform of High-Performance
Computing (PTCI)’’ (http://www.ptci.unamur.be), for which we
gratefully acknowledge the financial support from the FNRS-
FRFC, the Walloon Region, and the University of Namur
(Conventions No. 2.5020.11, GEQ U.G006.15, U.G018.19,
1610468, and RW/GEQ2016).

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