Supporting Information for
Full-dimensional multilayer multiconfigurational time-dependent Hartree
study of electron transfer at the anthracene/C60 complex
Yu Xie,1, 2 Jie Zheng, 1, 2 Zhenggang Lan, 1, 2 *
Key Laboratory of Biobased Materials, Qingdao Institute of Bioenergy and Bioprocess
Technology, Chinese Academy of Sciences, Qingdao, 266101 Shandong, P. R. China
University of Chinese Academy of Sciences, Beijing 100049, P. R. China
1
Key Laboratory of Biobased Materials, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese
Academy of Sciences, Qingdao, 266101 Shandong, P. R. China
2
University of Chinese Academy of Sciences, Beijing 100049, P. R. China
1
Contents
 Figure S1: Energy levels of the investigated complex (anthracene/C60), the isolated
anthracene and the isolated C60.
 Figure S2: The time-dependent populations of reduced model investigated by the standard
quantum dynamics, MCTDH and ML-MCTDH.

Figure S3: The time-dependent populations of reduced model with different number of
modes obtained by MCTDH and ML-MCTDH.

Figure S4: The time-dependent populations within the 4S model including all vibrational
modes. Electronic structures calculations were performed using different DFT functionals.

Figure S5: The time-dependent population dynamics within the 2S model when propagation
time is increased to 1 ps.

Figure S6: The distribution of the frequency interval Δω between two adjacent modes.

Figure S7: Time-dependent donor state populations in the 4S model including all modes at
different temperatures.

Figure S8: Vibrational modes.

Table S1: The relative energies of the diabatic states and the donor-acceptor coupling
elements calculated with different functionals.
2
Figure S1. Energy levels of the investigated complex (anthracene/C60), the isolated anthracene
and the isolated C60. From left to right: the MOs of the isolated anthracene, the LMOs of the
donor (  dj ), the MOs of the anthracene/C60 complex, the LMOs of the acceptor (  aj ), and the
MOs of the isolated C60. The correlations between selected orbital levels are indicated.
3
Figure S2. As our first efforts to utilize MCTDH and ML-MCTDH to study the quantum
evolution of complex systems, the standard wave-packet propagation method was used to
simulate the 2S-2D (2 electronic states, 2 vibrational degrees of freedom) model for comparison
to make sure the correct employment of MCTDH and ML-MCTDH. The time-dependent
populations of the donor state (D) and acceptor state (A1) only considering two important modes
(ν15 and ν26) were calculated using standard quantum dynamics method (a), MCTDH (b) and
ML-MCTDH (c), respectively.
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Figure S3. The time-dependent populations of the donor state (D) and acceptor state 1 (A1)
including 7 modes with xi > 1 obtained with MCTDH (a) and ML-MCTDH (b). The
dynamics with the model including 11 modes with xi > 0.5 obtained with MCTDH (c) and
ML-MCTDH (d). The slight different results are obtained for the reduced model with 44 modes
( xi > 0.2) by using MCTDH (e) and ML-MCTDH (f).
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Figure S4. The ML-MCTDH calculation results based on diabatic models at different
electronic-structure levels.
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Figure S5. When time is increased to 1 ps, the diabatic populations still keep the asymptotic
equilibrium values.
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Figure S6. The distribution of the frequency interval Δω between two adjacent modes.
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Figure S7. Time-dependent donor state populations in the 4S model including all modes at
different temperatures.
The precise estimation of the temperature effect should be done in the below procedure. (1)
Several initial conditions at different vibrational levels were created by the Monte-Carlo
sampling of the vibrational level for each normal mode according to the Boltzmann distribution
at the certain temperature. (2) The quantum dynamics were performed for each initial condition.
(3) The ensemble average of the quantum evolution of all initial conditions finally gives the
quantum ET dynamics at the certain temperature. Since this approach may involve the large
number of initial conditions, we use a rather approximated way to estimate the temperature effect.
At a certain temperature, the probability of the energy level for the particular vibrational mode
satisfies the Boltzmann distribution
pi 
exp   Ei / kT 
 exp   E
j
/ kT 
.
j
The average energy level of the particular vibrational mode is defined as
n   ni pi . Then,
i
we chose the rounded value of
n
(an integer) as the initial energy level of the vibrational
modes to generate the initial wavepacket in the ML-MCTDH calculation. The results obtained at
two temperatures (300 K and 600 K) were comparable to our previous results at (0 K). The
minor difference indicates that the temperature effect on the current ET process is very weak.
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Figure S8. Vibrational modes referred in our paper are given below. Low frequency (<100 cm-1)
modes ν1, ν2,…, ν6 are attributed to intermolecular vibration. Modes ν15 (261.60 cm-1) and ν26
(391.92 cm-1) display the largest vibronic couplings and do not belong to low-frequency
intermolecular vibrations. Modes ν218, ν219, ν220 and ν221 are with frequencies close to Rabi
frequency for the D-A1 case. High-frequency (>3000 cm-1) modes ν237, ν238,…, ν246 are
corresponding to C-H stretching vibrations.
ν1: 26.51 cm-1
ν2: 35.41 cm-1
ν3: 51.88 cm-1
ν4: 57.39 cm-1
ν5: 59.86 cm-1
ν6: 76.73 cm-1
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ν15: 261.60 cm-1
ν26: 391.92 cm-1
ν218: 1515.25 cm-1
ν219: 1537.19 cm-1
ν220: 1537.68 cm-1
ν221: 1538.21 cm-1
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ν237: 3099.53 cm-1
ν238: 3102.18 cm-1
ν239: 3104.18 cm-1
ν240: 3107.09 cm-1
ν241: 3109.36 cm-1
ν242: 3112.18 cm-1
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ν243: 3119.93 cm-1
ν244: 3123.14 cm-1
ν245: 3130.12 cm-1
ν246: 3133.82 cm-1
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Table S1. To estimate the influence of the DFT functionals for the anthrancene/C60 complex, we
also performed further electronic-structure calculations with the B3LYP/SVP functional using
Turbomole program, and the long range-corrected LC-ωPBE/6-31G* functional using Gaussian
09 program. Next the new diabatic Hamiltonian was re-constructed by replacing the zero order
terms (Vdd, Vaa and Vda in Eq. 2.2-2.3 in the man manuscript) by the corresponding values
obtained with these hybrid functinals.
Table S1a. The relative energies of the diabatic states and the energy of D state is set to zero.
states
D
A1
A2
A3
Vii, PBE/SVP(ev)
0
-0.160
-0.127
-0.102
Vii, B3LYP/SVP(ev)
0
-0.201
-0.162
-0.135
Vii, LC-ωPBE/6-31G*(ev)
0
-0.264
-0.222
-0.185
Table S1b. The donor-acceptor coupling elements.
D-A1
D-A2
D-A3
Vda, PBE/SVP(ev)
0.050
0.038
-0.018
Vda, B3LYP/SVP(ev)
-0.080
0.058
0.015
Vda, LC-ωPBE/6-31G*(ev)
-0.060
0.045
-0.018
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