Supporting Information for
Impact of Dielectric Constant on the Singlet-Triplet Gap in
Thermally Activated Delayed Fluorescence Materials
Haitao Sun†,‡,*, Zhubin Hu†, Cheng Zhong‡, Xiankai Chen‡,¶,
Zhenrong Sun†,§, and Jean-Luc Brédas‡,¶,*
†
State Key Laboratory of Precision Spectroscopy
School of Physics and Materials Science
East China Normal University
Shanghai 200062, P. R. China
‡
Laboratory for Computational and Theoretical Chemistry of Advanced Materials
Physical Science and Engineering Division
King Abdullah University of Science and Technology
Thuwal 23955-6900, Kingdom of Saudi Arabia
§ Collaborative
Innovation Center of Extreme Optics
Shanxi University
Taiyuan, Shanxi 030006, P. R. China
*Corresponding authors: [email protected]; [email protected]
¶
New permanent address: School of Chemistry and Biochemistry, Georgia Institute of
Technology, Atlanta, Georgia 30332-0400
Computational Details
In range-separated (RS) functionals such as the ωB97X functional employed in the
present work, a general expression for the separation of the exchange term into a shortrange domain and a long-range domain is defined by the interelectronic distance r12 and
the error function erf(x)1 as 𝑟12 −1 = 𝑟12 −1 erfc(ω𝑟12 ) + 𝑟12 −1 erf(ω𝑟12 ) . Thus, the
exchange term is split into a long-range exact exchange (eX) component derived from
Hartree-Fock and a short-range DFT component. The range-separation parameter ω
represents the inverse of the distance r12 at which the exchange term switches from DFTlike to HF-like. In exact Kohn-Sham (KS) theory,2 the negative HOMO energy −εH(N)
for an N-electron system should be equal to the vertical ionization potential (IP). Baer,
Kronik, and collaborators have utilized this non-empirical criterion to determine the
optimal ω in RS functionals.3
The concept of “optimal tuning” corresponds to adjusting ω to fulfill a fundamental
property that the exact functional must obey. For a given neutral system, the optimal ω
value can be determined by minimizing the following equations4:
𝐽 = |𝜀𝐻 (𝑁) + 𝐼𝑃(𝑁)|
(1)
A more refined target functional as shown below was proposed particularly for “better”
description of HOMO−LUMO gap or transport gap:
1
𝐽2 = ∑𝑖=0[𝜀𝐻 (𝑁 + 𝑖) + 𝐼𝑃(𝑁 + 𝑖)] 2
(2)
The latter expression, which simultaneously applies the above criterion for both neutral
(N) and anion (N + 1) systems, is referred to as the “gap-tuning” procedure and has been
2
demonstrated to yield a good agreement between the negative HOMO/LUMO energies
and the corresponding IPs/EAs.5-9 The “gap-tuning” scheme that is particularly relevant
to donor-acceptor systems, is employed throughout this work. The Tamm-Dancoff
Approximation (TDA) scheme is employed for all the calculations of excitation energies
because of its lower computational cost and its avoidance of the triplet instability issue
compared to full TDDFT.10,
The adiabatic singlet-triplet gap ΔEST* were also
11
calculated since the experimental singlet-triplet gaps are estimated from delayed
fluorescence and phosphorescence measurements in solution or film, which involve
excited-state geometry relaxations. The geometries of the S1 and T1 states were optimized
at the TDA-ωB97X*/cc-pVDZ level.
The dielectric constant ε of an amorphous mCP thin film was evaluated via the Clausius–
Mossotti relation12,13, which reads:
ε − 1 4π 𝛼
=
ε+2
3 𝑉
(3)
where V is the volume occupied by a single molecule, which is obtained by applying the
Multiwfn software.14,15 The α term denotes the isotropic component of the molecular
polarizability. The volume (499 Å3) and polarizability (50.1 Å3) were calculated for the
mCP molecule at the B3LYP/6-31G(d) level using the Gaussian 09 software.16
The distance (Δr) between the centroids of hole and electron is defined as14,15:
∆𝑟 =
2
∑𝑖,𝑎 𝐾𝑖𝑎
|⟨𝜑𝑎 |𝒓|𝜑𝑎 ⟩−⟨𝜑𝑖 |𝒓|𝜑𝑖 ⟩|
2
∑𝑖,𝑎 𝐾𝑖𝑎
3
(4)
where ⟨𝜑𝑎 |𝒓|𝜑𝑎 ⟩ and ⟨𝜑𝑖 |𝒓|𝜑𝑖 ⟩ are the centroids of holes and electrons, Ki,a is the
configuration coefficient corresponding to the excitation i → a within TDA and φ is
orbital wavefunction.
Figure S1. Schematic representation of vertical ΔEST and adiabatic ΔEST*using potential
energy surfaces. S0: singlet ground state; S1: lowest singlet excited state; T1: lowest triplet
excited state.
4
Figure S2. Calculated adiabatic ΔEST* of TXO-TPA, TXO-PhCz, CBP, and PXZ-TRZ
as a function of dielectric constant.
5
Figure S3. Distributions of hole and electron wavefunctions in the lowest singlet (S1) and
triplet state (T1) of CBP and PXZ-TRZ in both gas phase and solid film (ε = 3.0), as
calculated at the PCM-tuned ωB97X*/cc-pVDZ level. Blue and green isosurfaces (iso.
val. for density = 0.0004) refer to hole and electron, respectively. CT and LE denote
charge-transfer and localized excitations, respectively. Superscripts “1” or “3” indicate a
singlet or triplet state. The dominant character of the excited state is indicated in bold.
6
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