Nature of Highly Efficient Thermally Activated Delayed Fluorescence in OLED Emitters: Non-adiabatic Effect between Excited States Xian-Kai Chen1, Shou-Feng Zhang1, Jian-Xun Fan1 and Ai-Min Ren*,1 1
State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun, 130023, P. R. China *E-mail:
[email protected]
1
A. The selection of DFT functionals and transition characters of excited states For the TADF molecules shown in Figure 1, density functional theory (DFT) was adopted in the optimization of their ground-state (S0) geometric structures and calculation of the vibrational frequencies. Time-dependent DFT (TDDFT) was employed to calculate the vertical excitation energies from S0 minima to S1 or T1, optimize their excited-state geometries and calculate the vibrational frequencies. The molecular structures were optimized without any symmetry constraints. All these computational works were performed in Gussian 09 (D.01) program package.1 Effective spin-orbit coupling (SOC) matrix elements between singlet and triplet excited states (see the following Equation S8) were calculated by using two-component relativistic TDDFT method implemented in the Beijing Density Functional (BDF) program.2-4 TDDFT methods have been employed to calculate SOC matrix elements in many studies,5-7 and the results are in line with those given by DFT-MRCI method.5 For TADF organic emitters with different degree of charge transfer, different functionals should be used for exact description of the optical absorption and emission spectra observed in experiments.8 Here, some popular hybrid functionals with different Hartree-Fock (HF) exchange percentage, e.g. B3LYP (20%HF), PBE0 (25%HF), M06 (27%HF), MPW1B95 (31%HF), BMK (42%HF) and M062X (54%HF), were tested for all the studied molecules. Standard long-range corrected functionals induce triplet instability in the ground state,9 so they aren’t employed in the present work. Many previous investigations about TADF emitters have shown that
2
the choice of the basis sets has no large effect on excitation energies.8,10 Thus, to save the computational cost, the basis set 6-31G* was used in the whole study. For molecule 1, taken as an example, the equilibrium geometries of its S0, S1 and T1 states were optimized by TDDFT method using the foregoing functionals, and the S0→S1 or T1 vertical and adiabatic excitation energies were calculated. Our calculation results are presented in Figure S1 (a) and compared with the experimental spectra results. From Figure S1 (a), it can be found that both the vertical and adiabatic excitation energies given by BMK functional are the best agreement with the experimental values; with the increase of HF exchange percentage, both the calculated vertical and adiabatic excitation energies are enhanced. Because its HOMO and LUMO are almost completely localized at the donor and acceptor molecular fragments (see Figure S1 (b)), respectively, inducing zero electronic exchange integral, most of the functionals give very small gaps between S0→S1 and S0→T1 adiabatic excitation energies, which is very consistent with the experimental results. Here, it should be emphasized that the energy gaps between S0→S1 and S0→T1 vertical excitation energies are not responsible for S1 ↔ T1 ISC or RISC rate expressed by Equation (2), but rather the gaps between the adiabatic ones. Through testing DFT functionals, proper functionals are found for other three molecules shown in Figure 1 and additional D-A-D TADF emitters shown in Figure S3. Their calculated S0→S1 or T1 vertical and adiabatic excitation energies and S1-T1 energy gaps are listed in Table S1. It can be found that our computational results are very consistent with the experimental ones, which indicates the validity of the selected functionals.
3
Furthermore, the natural transition orbital (NTO) analysis of the optimized S1 and T1 states for D-A molecule 1 and D-A-D molecule 4 were carried out, and the results are shown in Figure S2. In these intra-molecule CT systems, the most major electron configurations of their S1 states arise from HOMO→LUMO transitions, so they are assigned as the states with CT transition character. (see Figure S2) In contrast to the S1 states, the electron configurations of the T1 states are sometimes more complex. Our NTO analysis results demonstrate that the transition character of T1 for molecule 1 is assigned as LE+CT transitions, which is good agreement with the experimental spectra results11. As reflected by the experimental phosphoresce spectra of molecule 4 with clear vibronic peaks,12 the NTO analysis of the T1 emissive state not only verifies its LE character, but also clearly shows that the electron configurations of the T1 state stem from both the n→π* and π→π* transitions. (HOMO of molecule 4 is composed of both non-bonding orbital induced by the two oxygen atoms with lone electron pairs and π bonding orbital, see Figure S1(b)) In addition, Figure S4 presents the optimized geometric structures of S0, S1 and T1 states for all the studied molecules. Our results show that the major differences between the equilibrium geometries of S1 and T1 states are distributed in the rotation angles between donor and acceptor molecular fragments, up to 45°. (see Figure S4)
B. Potential energy surfaces of excited states and conical intersection points independent of the choice of DFT functionals In D-A(-D) molecules, internal rotation around the single bond between donor
4
and acceptor molecular fragments is the first choice vibrational mode for a change in the electronic structure of excited states.13 Figure 2 and S4 show the displacement vectors for the representative low-frequency vibrational normal modes of molecule 1-4 and 5-6, respectively. Potential energy surfaces (PESs) of their excited states (e.g. T1, T2, S1 and S2) are scanned through changing the rotation angles between donor and acceptor fragments, via vertical excitation from ground to excited states by using TDDFT methods. Excited-state PESs of molecule 1/3 and 2, taken as the examples, are shown in Figure 3 and S6, respectively. The absolute energies of their S0 equilibrium geometries are used as the benchmarks in the excited-state PESs. Although the relaxation of PESs gives rise to the reduction of potential energies, it has no large effect on the shapes of the whole PESs. The D-A rotation angles in the equilibrium geometries of molecule 1/3 and 2 optimized by TDDFT (see Figure S4) are exactly corresponding to those in their no-relaxed PESs (see Figure 3 and S6). To further confirm that the relaxation of PESs has no effect on the emergence of conical intersection (CI) points between low-lying excited states in D-A-D molecules, the relaxed excited-state energies of molecule 3 taken as an example, are calculated through constrained optimization by freezing the D-A rotation angles of 90°, employing TDDFT/BMK/6-31G* method. Our results demonstrate that the emergence of its S1-S2 and T1-T2 CI points can not be influenced (the relaxed excited-state energies of S1, S2, T1 and T2 states are 2.71, 2.71, 2.7 and 2.7 eV, respectively). Furthermore, we also examine the effects of polar solvent environment (e.g. ethanol) on these CI points with Polarizable Continuum Model (PCM) and
5
demonstrate that their appearance can not be changed. In addition, to know whether the appearance of CI points depends on the choice of DFT functionals in the studied D-A-D molecules, three functionals with different HF exchange percentage are employed to examine the excited-state energies at their geometries with the D-A rotation angles of 90°. Full TDDFT can not give correct ordering of triplet excited states due to the ground state triplet instability problem in some molecular systems, especially in the case when the energies of triplet excited states are close.14 TDDFT using the Tamm-Dancoff approximation (TDA) does not suffer from these issues.9,14,15 Therefore, TDA-TDDFT method is employed here and the results are listed in Table S2. Remarkably, it can be found that the energies of the singlet and triplet excited states are enhanced with the increase of HF exchange percentage. However, the emergence of S1-S2 and T1-T2 CI points for all the studied D-A-D molecules is not changed. From Table S2, we also find that for D-A-D molecule 3, full TDDFT and TDA-TDDFT methods give the consistent result that the S1-S2 and T1-T2 CI points at its geometry with the D-A rotation angles of 90° exist.
C. The derivative details of T1→S1 RISC rate formula Within the framework of time-dependent second-order perturbation theory and under the Born-Oppenheimer (B-O) adiabatic approximation, the thermal average T1→S1 RISC rate constant from the initial electronic state i with the vibrational quantum numbers v to the final electronic state f with the vibrational quantum numbers v′ 16
6
ki → f
2π = h
2
∑∑ P
iv
H′ ′ H′ + ∑ fv ,nµ nµ , iv δ (∆Eif + Evi - Evf′ ) Eiv - Enµ nµ
H ′fv ′,iv
v′
v
S1
Here, the interaction Hamiltonians can be considered as Hˆ ′ ψiv (q,Q ) = ( Hˆ SO + Hˆ BO ) ψiv (q,Q ) = Hˆ SO Φ i ( q,Q ) Θ iv (Q ) + Hˆ BO Φ i ( q,Q) Θ iv (Q )
S2
The B-O operator Hˆ BO arising from electron-vibration coupling is precisely written by ∂ Θiv
∂ Φi Hˆ BO ψiv (q,Q) = −h 2 ∑ ∂Q j j
∂Q j
−
∂ 2 Φi h2 Θiv ∑ 2 j ∂Q 2j
S3
The second term in most cases is much smaller than the first one. Therefore, the coupling matrix can be expressed as ∂Φ i ∂Θiv ψ fv ′ (q,Q ) Hˆ BO ψiv (q,Q) = −h 2 ∑ Φ f Θvf′ = ∑ Φ f Θvf′ (Pˆj Φ i )(Pˆj Θiv ) ∂Q j ∂Q j j j
S4
The expression formula of the rate constant only including the first item of Equation S1 is given by ki0→ f =
2π h
∑∑ P
2
H ′fv′,iv δ ( ∆Eif + Evi - Evf′ )
iv
S5
v′
v
If T1→S1 RISC process is considered here, Equation S5 becomes kT01 → S1 =
2π h
∑∑ P v
ΘTv1 T1 Hˆ SO S1 ΘvS′1
T1v
v′
2
δ (∆ET S + EvT - EvS′ ) 1
1 1
1
S6
∆ET1 S1 (= ET1 - ES1 ) is the energy difference between the minima of T1 and S1 states Due to the spin symmetry requirement, the Hˆ BO term in Equation S2 does not make any contribution to RISC rate given by Equation S6. Therefore, under the Condon approximation, we have kT01 → S1 =
2π T1S1 H SO h
2
∑∑ P v
v′
T1v
ΘTv1 ΘvS′1
7
2
δ (∆ET S + EvT - EvS′ ) 1
1 1
1
S7
At high temperatures, e.g. 300 K, effective SOC matrix element between S1 and T1 states is adopted and calculated by averaging over the interactions between S1 and the three triplet substates T1m ( m = −1, 0, 1 )17,18 T1 S1 H SO =
1 ∑ T1m Hˆ SO S1 3 m = −1, 0, 1
S8
Under the harmonic oscillator model, N-vibrational mode states of T1 and S1 states consist
of
a
collection
ΘTv1 (Q) = ∏ χ vTl1 (QvTl1 )
of
harmonic
oscillator
and ΘvS′1 (Q) = ∏ χvSk′1 (QvS′k1 )
l
,
and
eigen-wavefunctions, the
one-dimensional
k
harmonic oscillator Hamiltonians are
( ) (
)
S9
( ) (
)
S10
2 2 1 Hˆ lT1 = PˆlT1 + ωlT1 QlT1 2
2 2 1 Hˆ kS1 = PˆkS1 + ωkS1 QkS1 2
where, for example, PˆlT1 and QlT1 are the lth mass-weighted nuclear normal momentum operator and normal coordinate, respectively. The eigenvalue of harmonic 1 oscillator Hamiltonian Hˆ lT1 is ElT1 = hωlT1 (v + ) . The delta function can be Fourier 2 transformed as
1 2π
δ (∆ET S + EvT - EvS′ ) = 1
1
1 1
where τ ≡
∫
+∞
−∞
dτ e
i∆ET
1 S1
τ
e
i(E vT1 - E vS′1 )τ
S11
t . Substituting Equation S9−S11 into Equation S7, the following rate h
formula expressed by vibration correlation function is given by
kT01 → S1 =
1 T1S1 H SO h
2
∫
+∞
−∞
dτ e
i∆ET1S1τ
ρT0 → S (τ , β ) 1
1
Z T1
The thermal vibration correlation function are defined as
8
S12
ρ T0 → S (τ , β ) ≡ Tr(e 1
-iτ S1 Hˆ S1
1
e
-iτ T1 Hˆ T1
) = ∑∑ ΘTv1 e
-iτ S1 E vS′1
Θ vS′1 Θ vS′1 e
-iτ T1 E vT1
Θ Tv1
S13
v′
v
where τ = τ S1 , τ T1 = −τ S1 − iβ and β =
1 . The vibrational coordinates of T1 state kbT
can be given as linear combinations of the vibrational coordinates of S1 state: N
QlT1 = ∑ SlTk1 ← S1 QkS1 + Dl
S14
k
where S lTk1 ← S1 is the element of Duschinsky rotation matrix ST1 ← S1 which defines the degree of the intermixing between the normal modes of T1 and S1 states; Dl is the component of shift vector D between the equilibrium geometries of T1 and S1 states. Therefore, Equation S13 is rewritten by19
ρ
0 T1 → S1
det(aT1 a S1 ) i T (τ , β ) = exp{ [ D ES(B − A)−1 GS T D]} det(K) h
S15
B A S1 S1 S1 S1 T T1 T T1 where K = and , A =a +S a S , B = b +S b S , G = b −a A B E = bT1 − aT1 where
a
and
b
are
diagonal
matrixes
with
elements
akS1 /T1 = ωkS1 /T1 sin(hωkS1 /T1τ S1 /T1 ) ； bkS1 /T1 = ωkS1 /T1 tan(hωkS1 /T1τ S1 /T1 ) . Last, the second item of Equation S1 is derived as followed,
∑µ n
H ′fv′,nµ H n′µ ,iv Eiv - Enµ
=∑ nµ
fv′ H ′ nµ nµ H ′ iv Eiv - Enµ
S16
Substituting Equation S2 into Equation S16, it is rewritten by
∑µ n
fv′ H SO nµ nµ H BO iv fv′ H BO nµ nµ H SO iv +∑ Eiv - Enµ Eiv - Enµ nµ
S17
Under the Condon approximation and Plazcek approximation ( Eiv - Enµ ≈ Ei - En ), Equation S16 becomes
9
∑µ n
H ′fv′,nµ H n′µ ,iv Eiv - Enµ
=∑ nµ
fv′ H ′ nµ nµ H ′ iv Eiv - Enµ
n′i H fn Φ Pˆ Φ Φ f Pˆj Φ n′ H SO = ∑ v′ Pˆj v ∑ SO n j i + ∑ n Ei - En Ei - En′ j n′
S18
fn n′i c j H SO c j H SO ∂ ∂ = ∑∑ Φn Φ i + ∑∑ Φ f Φ n′ E -E ′ ∂Q j ∂Q j n j Ei - En n′ j i n
where c j = −h 2 v′
∂ v ∂Q j
and Φ n Pˆ j Φ i = −ih Φ n
∂ Φi ∂Q j
In the present work, the T1→S1 RISC rates were calculated by using our own code.
(a)
10
(b) Figure S1. (a) Vertical excitation energies from S0 minima to S1 or T1 and adiabatic
excitation energies between the minima of S0 and S1 or T1 calculated by TDDFT methods, using B3LYP (20%HF), PBE0 (25%HF), M06 (27%HF), MPW1B95 (31%HF), BMK (42%HF) and M062X (54%HF) with the basis set 6-31G* for molecule 1; “Exp” denotes the experimental results. (b) electron density contours of HOMOs and LUMOs for molecule 1 and 3.
11
Figure S2. Natural transition orbital analysis of excited states for molecule 1 and 4.
Figure S3. Molecular structures of D-A-D molecule 5 and 6.
12
Figure S4. Optimized geometric structures of S0, S1 and T1 states for all the studied
molecules. The red numbers are the rotation angles between donor and acceptor molecular fragments.
13
Figure S5. Diagrammatic illustrations of the displacement vectors for the
representative low-frequency vibrational normal modes, 11 cm-1and 11 cm-1 for molecule 5 and 6, respectively. The left and right pictures are from front and side views, respectively.
Figure S6. Potential energy surfaces of the excited states T1, T2 and S1 calculated by
TDDFT/BMK/6-31G* method for molecule 2 14
Figure S7. Contour map of Duschinsky rotation matrix elements (absolute value)
between the S1 and T1 normal modes for molecule 1 and 3. The scale is shown at the right-hand side with interval of 0.125.
Table S1. Calculated vertical and adiabatic excitation energies (in eV) from S0 to S1
or T1 and S1-T1 gaps (in eV) of all studied TADF molecules by employing TDDFT methods with suitable functionals. “v” and “a” denote vertical and adiabatic excitations, respectively. The values in parentheses are the experimental results. molecule
1
2
3
4
5
6
functional
BMK
BMK
BMK
MPW1B95
MPW1B95
M062X
S0→S1 (v)
2.71(2.76)11
3.09(3.18)20
3.09(3.12)20
3.45(3.5)12
3.7(3.5)21
3.63(3.5)22
S0→T1 (v)
2.7
3.02
3.02
2.93
3.24
3.34
S0→S1 (a)
2.37(2.4)11
2.79(2.76)20
2.7(2.76)20
3.13(3.1)12
3.42(3.35)21
3.4
S0→T1 (a)
2.33(2.38)11
2.5
2.46
2.61(2.61)12
3.06(2.99)21
3.23
15
S1-T1 gap (a)
0.04
0.29
0.24
0.52
0.36
0.17
Table S2. Calculated excited-state energies (in eV) at the geometries with the D-A
rotation angles of 90° for D-A-D molecule 3-6 by TDA-TDDFT methods, using different functionals. molecule functional T1 T2 S1 S2
3 B3LYP
BMK
M062X
2.23
2.97
3.28
3.15
3.48
3.9
2.23
2.97
3.28
3.15
3.48
3.9
2.24
2.98
3.29
3.16
3.49
3.91
2.24
2.98
3.29
3.16
3.49
3.92
molecule functional T1 T2 S1 S2
4 B3LYP MPW1B95
5
M062X
6
B3LYP
MPW1B95
M062X
B3LYP
BMK
M062X
3.3
3.72
4.16
2.43
3.34
3.61
3.3
3.72
4.16
2.43
3.35
3.62
3.31
3.73
4.36
2.44
3.38
3.64
3.31
3.73
4.36
2.44
3.38
3.64
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