Assessment of density functional methods for exciton binding energies and related
optoelectronic properties†
Jui-Che Lee,aJeng-Da Chai*band Shiang-Tai Lin*a
The exciton binding energy, the energy required to dissociate an excited electron–hole pair into free charge carriers, is one of the key factors to the optoelectronic performance of organic materials. However, it remains unclear whether modern quantum-mechanical calculations, mostly based on Kohn–Sham density functional theory (KS-DFT) and time-dependent density functional theory (TDDFT), are reliably accurate for exciton binding energies. In this study, the exciton binding energies and related optoelectronic properties (e.g., the ionization potentials, electron affinities, fundamental gaps, and optical gaps) of 121 small- to medium-sized molecules are calculated using KS-DFT and TDDFT with various density functionals. Our KS-DFT and TDDFT results are compared with those calculated using highly accurate CCSD and EOM-CCSD methods, respectively. The uB97, uB97X, and uB97X-D functionals are shown to generally outperform (with a mean absolute error of 0.36 eV) other functionals for the properties investigated.
I. Introduction
An exciton, a bound electron–hole pair, can be generated when light is absorbed in a photoactive material. The elec- tron–hole pair may be located at the same molecular unit (e.g., a Frenkel exciton) or different molecular units (e.g., an intermolecular charge-transfer exciton).1 The Coulomb interaction that stabilizes the exciton with respect to free electrons and holes is known as the exciton binding energy.2 Materials with small exciton binding energies usually exhibit high charge separation efficiency,3which are oen desirable for photovoltaic applications, whereas the opposite may be favorable for light-emitting devices.1 Covalently bounded inorganic semiconductors have delocalized charge carriers and broad valence and conduction bands with exciton binding energies being as small as a few millielectron volts (meV). In contrast, the electronic properties of organic semiconductors are dominated by the localized charge on individual molecules and the large polarizabilities.4,5 As a result, the exciton binding energies of organic semi- conductors can be as large as 0.1–1 eV. Therefore, it is important to develop a comprehensive understanding of the
relevant parameters controlling the exciton binding energies of organic materials for the design of ideal photoactive material.
While a direct measurement of the exciton binding ener- gy may be challenging, the exciton binding energy can be obtained from the difference between the fundamental and optical gaps, each of which can be measured directly.4,6–9 On the other hand, the fundamental and optical gaps can also be calculated using quantum-mechanical methods.
Nayak et al. calculated the exciton binding energies of small organic conjugated molecules in vacuum and in thin
lms10,11 using Kohn–Sham density functional theory (KS-
DFT)12,13 and time-dependent density functional theory
(TDDFT)14,15 with the B3LYP functional.16,17 However, the overall accuracy of KS-DFT and TDDFT with conventional density functionals is not comprehensively examined on exciton binding energies.
While intermolecular charge-transfer excitons are also important, in this work, we only focus on the Frenkel excitons.
Specically, we examine the accuracy of exciton binding energies and related optoelectronic properties on a diverse range of molecules using KS-DFT and TDDFT with several widely used density functionals. The rest of this paper is organized as follows. In Section II, we describe our test sets and computational details. The exciton binding energies and related optoelectronic properties obtained from density func- tional methods are compared with those obtained from high- level ab initio methods in Section III. Our conclusions are given in Section IV.
aDepartment of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan. E-mail: [email protected]
bDepartment of Physics, Center for Theoretical Sciences, Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan. E-mail:
†Electronic supplementary information (ESI) available. See DOI:
10.1039/c5ra20085g
Cite this: RSC Adv., 2015, 5, 101370
Received 29th September 2015 Accepted 18th November 2015 DOI: 10.1039/c5ra20085g www.rsc.org/advances
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II. Test sets and computational details
To evaluate the performance of the functionals on exciton binding energies and related optoelectronic properties (e.g., vertical ionization potentials, vertical electron affinities, fundamental gaps, optical gaps, and exciton binding ener- gies), we collect a test set, which consists of experimental vertical ionization potentials of 121 small- to medium-sized molecules in the experimental geometries. The geometries are taken from the IP131 database.18 The nine molecules (C4H5N, C6H6, CF3CN, CH2ClCH2CH3, CH3CH(CH3)CH3, CH3CONH2, CH3SOCH3, N(CH3)3, SF6) were excluded because of the excessive computational resources needed. The coupled-cluster theory with iterative singles and doubles (CCSD)19–21 and equation-of-motion CCSD (EOM-CCSD),22–30 are employed as the benchmarks for the electronic and optical properties, respectively.
We examine the vertical ionization potentials (IP), vertical electron affinities (EA), fundamental gaps (Eg), optical gaps (Eopt), and exciton binding energies (Eb) of 121 mole- cules using various density functional methods. The IP, EA, and Eg calculations follows the procedure detailed in our previous work.31 For the KS-DFT and TDDFT calculations, we adopt popular density functionals, involving the local- density-approximation (LDA32,33) functional, a generalized- gradient-approximation (GGA) functional (PBE34,35), a meta- GGA functional (M06-L36), a global hybrid GGA functional (B3LYP16,17), three long-range corrected (LC) hybrid GGA functionals (uB97,37 uB97X,37 and uB97X-D38), and two global hybrid meta-GGA functionals (M06-HF36,39 and M06-2X40).
All calculations are performed using the Gaussian09 program.41Results are computed using the aug-cc-pVQZ basis set with the ultrane grid, EML (99 590), consisting of 99 Euler–
Maclaurin radial grid points42 and 590 Lebedev angular grid points.43–45 All the calculated results are provided in the ESI (Tables S1–S12†). The error for each entry is dened as error ¼ theoretical value " reference value. The notations used for characterizing statistical errors are as follows: mean signed errors (MSEs), mean absolute errors (MAEs), and root-mean- square (RMS) errors.
A. Vertical ionization potentials
The vertical ionization potential (IP) of a neutral molecule is dened as the energy difference between the cationic and neutral charge states,
IP(1)¼ Etot(cation)" Etot(neutral). (1) For the exact KS-DFT, the vertical IP of a neutral molecule is the same as the minus HOMO (highest occupied molecular orbital) energy of the neutral molecule,32,46
IP(2)¼ "EHOMO(neutral) (2) Therefore, IP(2) is the same as IP(1) for the exact KS-DFT. For approximate density functional methods, IP(1) and IP(2) may be
adopted to examine the accuracy of the predicted total energies and HOMO energies, respectively.
B. Vertical electron affinities
The vertical electron affinity (EA) of a neutral molecule is dened as the energy difference between the neutral and anionic charge states,
EA(1)¼ Etot(neutral)" Etot(anion). (3) By comparing eqn (1) with (3), the vertical EA of a neutral molecule is identical to the vertical IP of the anion, which is, for the exact KS-DFT, the minus HOMO energy of the anion,
EA(2)¼ "EHOMO(anion). (4) For the exact KS-DFT, EA(2) is identical to EA(1). Therefore, EA(1) and EA(2) may be adopted to examine the accuracy of approximate density functional methods on total energies and HOMO energies, respectively.
In addition, the vertical EA of a neutral molecule is conventionally approximated by the minus LUMO (lowest unoccupied molecular orbital) energy of the neutral molecule,18 EA(3)¼ "ELUMO(neutral). (5) However, even for the exact KS-DFT, a difference exists between EA(3) and vertical EA due to the derivative disconti- nuity (DD) of the exchange–correlation functional.47–50 Recent study shows that DD is close to zero for LC hybrid functionals,51 so the EA(3) calculated by a LC hybrid functional should be close to the true vertical EA.
C. Fundamental gaps
The fundamental gap (Eg) of a molecule is dened as the difference between the vertical IP and vertical EA of the molecule,
Eg¼ IP " EA. (6)
Since there are many different ways of calculating vertical IP and vertical EA from density functional methods, we have mainly three ways of calculating Eg:
Eg(1)¼ IP(1) " EA(1) ¼ Etot(cation) + Etot(anion)
" 2Etot(neutral) (7) Eg(2)¼ IP(2) " EA(2) ¼ EHOMO(anion)
" EHOMO(neutral) (8) Eg(3)¼ IP(2) " EA(3) ¼ ELUMO(neutral)
" EHOMO(neutral) (9) Note that Eg(3) is the so-called Kohn–Sham (KS) gap or HOMO–LUMO gap in KS-DFT.18,52
As mentioned previously, for LC hybrid functionals, EA(3) should be close to vertical EA, and hence Eg(3) should be close to the true Eg.
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D. Optical gaps
The optical gap (Eopt) of a molecule, dened by a neutral exci- tation, is the energy difference between the lowest dipole- allowed excited state and the ground state,
Eopt¼ Etot(neutral, excited)" Etot(neutral). (10) As Eopt is an excited-state property, it cannot be directly calculated using CCSD or KS-DFT. To be consistent with the ground-state calculations, we adopt EOM-CCSD22–30 and TDDFT53–60 (with the same density functionals for the ground- state calculations) to calculate the Eopt.
E. Exciton binding energies
The exciton binding energy (Eb) of a molecule is dened as the difference between the fundamental and optical gaps,
Eb¼ Eg" Eopt. (11) As there are three ways of calculating Eg(see eqn (7)–(9)), Eb
can also be calculated in three ways, i.e.,
Eb(1)¼ Eg(1)" Eopt (12) Eb(2)¼ Eg(2)" Eopt (13) Eb(3)¼ Eg(3)" Eopt (14) It is important to note that Eb(1), Eb(2), and Eb(3) obtained with density functional methods are closely related to the calculated Eg(1), Eg(2), and Eg(3), respectively. For the exact KS- DFT, as IP(2) is identical to IP(1), and EA(2) is identical to EA(1), Eg(2) is the same as Eg(1), the exact Eg. Therefore, for the exact KS-DFT and TDDFT, Eb(2) should be identical to Eb(1), the exact Eb. Consequently, the errors in the calculated Eg(1), Eg(2), Eb(1), and Eb(2) are directly related to the accuracy of the exchange–
correlation functional adopted in KS-DFT and TDDFT. However, due to the lack of the DD in the calculated KS gap, Eg(3) tends to be smaller than Eg(1) (the true fundamental gap), even for the exact KS-DFT. Accordingly, when the calculated KS gap (Eg(3)) is less than Eopt, one obtains an unphysical negative value of Eb. We show that this is indeed frequently observed for local and semilocal density functional methods. However, as the DD is close to zero for LC hybrid functionals (which belong to the generalized Kohn–Sham (GKS) method, not the “pure” KS-DFT), Eg(3) and Eb(3) calculated using LC hybrid functionals are ex- pected to be close to the true Egand Eb, respectively.
III. Results and discussion
A. Benchmark methods
The IP(1) calculated from CCSD and CCSD(T) are compared to experimental values for the 121 molecules (see Fig. S1 and Table S13 in ESI†). Both methods are very accurate with MAE being 0.17 eV from CCSD and 0.14 eV from CCSD(T). The difference between these two methods are around 0.1 eV for IP(1) and Eg(1), and around 0.05 eV for EA(1) (see Fig. S1 and Table S14
ESI†). Such differences are much smaller than those associated with density functional methods (usually greater than 0.5 eV).
Therefore, we believe that the CCSD results can be taken as benchmarks. In this study, CCSD and EOM-CCSD (i.e., for consistency with CCSD) are adopted as the benchmarking methods for the ground-state and excited-state properties, respectively.
B. Accuracy of density functional methods
Five optoelectronic properties, including the vertical ionization potentials, vertical electron affinities, fundamental gaps, optical gaps, and exciton binding energies of 121 molecules are adop- ted to examine the accuracy of various density functionals in KS- DFT and TDDFT.
The accuracy of various density functionals with respect to experimental values on IP(1) and IP(2) are summarized in Table S15 in ESI.† Despite the fact that different measures of accuracy yield slightly different ranks in performance of IP(1), M06-2X and uB97X-D are among the most accurate density func- tionals, with an MAE of around 0.18 eV. LDA is the least accu- rate with an MAE of 0.56 eV, which is more than three times the error of the best functional. It is also interesting to note that while M06-HF and M06-2X are both hybrid meta-GGA func- tionals, the MAE of M06-HF (0.46 eV) is more than twice that of M06-2X (0.18 eV).
There is a signicant variation in the performance when the ionization potentials are estimated from the HOMO energy (IP(2)). The uB97 functional is the best functional with an MAE of 0.42 eV. PBE is the least accurate with an MAE of 4.44 eV, which is more ten times higher than that of the best functional here. The MAE difference in IP(1) between M06-HF and M06-2X is more than twice, but the performance of M06-HF (0.99 eV) and M06-2X (1.51 eV) is similar for IP(2). In general, the MAE in IP(2) is 2 to 13 times larger than those for IP(1).
Since there are no comprehensive experimental data avail- able for the other properties, the results from CCSD are taken as the reference values for evaluating the performance of density functional methods. Table 1 shows such comparison for the vertical ionization potentials, vertical electron affinities, fundamental gaps, optical gaps, and exciton binding energies.
Fig. 1–5 illustrates the MAE in IP, EA, Eg, Eopt, and Eb with different denitions. For IP(1), M06-2X agrees best with CCSD with an MAE difference of 0.13 eV. The uB97 series also provide quite consistent results with CCSD with MAE difference being about 0.16 eV. For EA(1), the performance of M06-2X and the uB97 series are among the best, with the uB97X-D being the best functional (MAE ¼ 0.16 eV). The uB97X and M06-2X show the best performance for Eg(1) are shown that uB97X (MAE ¼ 0.26 eV) and M06-2X (MAE ¼ 0.27 eV) are the best functionals.
The LDA yields the worst results for IP(1) and EA(1); however, the PBE is the least accurate for Eg(1) because it generally underestimates IP(1) (MAE ¼ 0.42 eV, MSE ¼ "0.31 eV) but overestimates EA(1) (MAE ¼ 0.34 eV, MSE ¼ 0.32 eV). In summary, M06-2X and the uB97 series (uB97, uB97X, and uB97X-D) are the best density functionals for the ground-state properties (with an accuracy of 0.3 eV or less). Furthermore,
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as in the case of IP, the MAE from the least accurate methods (oen LDA and GGAs) can be more than twice that of the best functional.
The optical gap is a measure of performance in describing the excited state. The uB97 and uB97X functionals perform the best for optical gaps. PBE shows surprisingly poor results for Table 1 Statistical errors (in eV) of 9 density functional methods for various properties (with respect to the CCSD or EOM-CCSD data)
Error LDA PBE M06L B3LYP uB97 uB97X uB97X-D M06-2X M06-HF
IP(1) MSE 0.39 "0.31 "0.36 "0.09 "0.07 "0.07 "0.09 0.03 0.35
MAE 0.58 0.42 0.39 0.26 0.16 0.17 0.20 0.13 0.37
RMS 0.66 0.61 0.56 0.37 0.23 0.24 0.28 0.18 0.46
IP(2) MSE "3.92 "4.50 "4.35 "3.25 "0.34 "0.59 "1.12 "1.57 0.84
MAE 3.92 4.50 4.35 3.25 0.41 0.59 1.13 1.57 0.91
RMS 4.05 4.61 4.45 3.32 0.67 0.82 1.28 1.63 1.04
EA(1) MSE 0.78 0.32 "0.07 0.30 0.02 0.07 0.13 0.05 0.17
MAE 0.79 0.34 0.24 0.31 0.20 0.18 0.16 0.21 0.33
RMS 0.86 0.46 0.38 0.43 0.34 0.32 0.31 0.37 0.42
EA(2) MSE "1.27 "1.62 "1.92 "1.22 0.13 0.12 0.00 "0.75 0.34
MAE 1.31 1.67 1.95 1.28 0.24 0.20 0.24 0.80 0.55
RMS 1.60 1.97 2.20 1.47 0.39 0.35 0.39 0.87 0.76
EA(3) MSE 3.15 2.53 2.87 1.95 "0.07 "0.03 0.24 1.25 1.75
MAE 3.15 2.53 2.87 1.96 0.29 0.34 0.42 1.26 1.80
RMS 3.44 2.81 8.90 2.15 0.37 0.41 0.56 1.80 5.57
Eg(1) MSE "0.39 "0.63 "0.29 "0.39 "0.09 "0.14 "0.22 "0.02 0.18
MAE 0.51 0.65 0.44 0.45 0.28 0.26 0.30 0.27 0.33
RMS 0.71 0.88 0.62 0.62 0.42 0.41 0.44 0.40 0.46
Eg(2) MSE "5.46 "2.88 "2.43 "2.03 "0.47 "0.71 "1.13 "0.82 0.49
MAE 6.14 2.88 2.43 2.03 0.58 0.73 1.13 0.86 0.75
RMS 8.08 3.14 2.70 2.22 0.82 0.94 1.30 1.01 0.95
Eg(3) MSE "7.07 "7.03 "7.22 "5.19 "0.27 "0.56 "1.36 "2.82 "0.92
MAE 7.07 7.03 7.22 5.19 0.51 0.64 1.39 2.82 1.57
RMS 7.32 7.25 11.25 5.35 0.77 0.93 1.61 3.21 5.66
Eopt MSE "0.91 "1.05 "0.61 "0.63 "0.13 "0.15 "0.33 "0.38 "0.44
MAE 0.97 1.12 0.70 0.76 0.31 0.32 0.46 0.46 0.68
RMS 1.15 1.29 0.90 0.92 0.55 0.63 0.71 0.54 1.05
Eb(1) MSE 0.52 0.42 0.32 0.24 0.04 0.01 0.12 0.36 0.62
MAE 0.66 0.66 0.57 0.53 0.39 0.38 0.40 0.48 0.80
RMS 0.88 0.88 0.74 0.78 0.71 0.71 0.71 0.61 1.16
Eb(2) MSE "1.74 "1.83 "1.82 "1.39 "0.34 "0.56 "0.79 "0.43 0.93
MAE 1.77 1.85 1.86 1.42 0.53 0.66 0.87 0.57 1.17
RMS 2.05 2.13 2.12 1.66 0.81 0.92 1.11 0.71 1.49
Eb(3) MSE "6.16 "5.98 "6.62 "4.56 "0.13 "0.41 "1.03 "2.44 "0.48
MAE 6.16 5.98 6.62 4.56 0.48 0.59 1.11 2.44 1.69
RMS 6.48 6.28 10.91 4.80 0.74 0.96 1.46 2.86 5.69
Total MSE "1.84 "1.88 "1.87 "1.36 "0.14 "0.25 "0.47 "0.63 0.32
MAE 2.75 2.47 2.47 1.83 0.36 0.42 0.65 0.99 0.91
RMS 4.08 3.45 5.50 2.57 0.60 0.69 0.96 1.52 2.93
Fig. 1 Comparison of MAE in ionization potential (IP) from 9 DFT methods with respective to CCSD. IP(1) (blue solid line) is calculated from eqn (1) and IP(2) (red dashed line) from eqn (2).
Fig. 2 Comparison of MAE in electron affinity (EA) from 9 DFT methods with respective to CCSD. EA(1) (blue solid line) is calculated from eqn (3), EA(2) (red dashed line) from eqn (4), and EA(3) from eqn (5).
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Eopt, with an MAE (1.12 eV) that is three times higher than the best method. It is useful to note that the electron–hole attrac- tion which stabilizes the exciton is not explicitly present in the TDDFT kernel associated with the standard local or semilocal density functional (e.g., LDA or PBE). There are efforts made to include electron–hole attraction and binding energies based on the Bethe–Salpeter formalism61,62or on developing new TDDFT kernels.63–65
Since uB97 and uB97X perform the best for both Eg(1) and Eopt, unsurprisingly, they are the most accurate methods for Eb(1). uB97X-D has 0.1 eV more than uB97 and uB97X for Eopt, but uB97X-D also has good performance as uB97 and uB97X.
The MAE in Eb(1) from M06-HF (0.80 eV) is more than twice larger than that from uB97X. It is interesting to note that while B3LYP is inaccurate for Eg(1) (MAE ¼ 0.45 eV) and Eopt(MAE ¼ 0.76 eV), it is reasonably accurate for Eb(1) (MAE ¼ 0.53 eV), which is a result of systematic underestimation IP(1) (MAE ¼ 0.26 eV, MSE ¼ 0.09 eV), EA(1) (MAE ¼ 0.31 eV, MSE ¼ 0.30 eV), Eg(1) (MAE ¼ 0.45 eV, MSE ¼ "0.39 eV), and Eopt(MAE ¼ 0.76 eV, MSE ¼ "0.63 eV). In summary, the uB97 functional provides the most reliable predictions for total energy of a molecule both in the ground state and in the excited state.
The performance in IP(2), EA(2), Eg(2), and Eb(2) is an indi- cation of the quality of HOMO energy. In general, the MAE for IP(2) is 3 to 9 times larger than those for IP(1). uB97 is the best functional for IP(2) with an MAE of 0.41 eV (compared to 0.16 for IP(1)). The uB97X-D underestimated the HOMO energy, resulting in a large error in IP(2) (MAE ¼ 1.13 eV and MSE ¼
"1.12 eV), Eg(2) and Eb(2). The MAE errors from uB97 methods for EA(2) are quite similar to those for EA(1). However, LDA, PBE, M06L, and B3LYP show signicantly increased inaccuracy in EA(2). As a result, these methods are also poor for Eg(2) and Eb(2). The uB97 is the best functional for Eg(2) (MAE ¼ 0.58 eV) and Eb(2) (MAE ¼ 0.53 eV). The global hybrid MGGA func- tionals, M06-2X and M06-HF, also provide reasonable accuracy in the HOMO energies. However, since M06-2X underestimates for Eoptand Eg, it is accurate for Eb(2) which is also a result of systematic underestimation.
The performance in EA(3), Eg(3), and Eb(3) implies the description accuracy in the LUMO energy. The performance of uB97 series in these properties are quite similar to the corre- sponding EA(2), Eg(2), and Eb(2). However, all other methods, including hybrid MGGA methods, show signicantly increased MAEs here. For example, the MAE of EA(3) from M06-HF (1.80 eV) is almost 3 times higher than that of EA(2) (0.55 eV).
Therefore, the uB97 series provide reliable description for the LUMO energy.
It is noteworthy that in some cases the calculated LUMO energies are so small that Eg(3) becomes smaller than optical gap, yielding qualitatively incorrect exciton binding energies (i.e., Ebcan be negative!). 115 out of 121 from LDA, 116 out of 121 from PBE, 112 out of 121 from M06L, 38 out of 121 from B3LYP, 10 out of 121 from M06-2X and 7 out of 121 from M06- HF, 2 out of 121 from uB97 and uB97X, 5 out of 121 from uB97X-D show negative Eb(3) (see Table S12† for a complete list of all the Eb(3)). In some cases, the calculated LUMO energies are even lower than the calculated HOMO energies from M06 methods (M06L, M06-HF, M06-2X), resulting in incorrect negative Eg(3) ("87.12 eV (hydrogen atom) from M06L, "5.00 eV (hydrogen atom) from M06-2X and "40.36 eV (hydrogen atom),
"18.76 eV (lithium atom), "13.04 eV (sodium atom) from M06- HF).
It is worthwhile to note that LC hybrid functionals with non- empirical optimization of the range-separation parameter u have been recently developed,66,67 and have yielded accurate Fig. 3 Comparison of MAE in fundamental gap (Eg) from 9 DFT
methods with respective to CCSD. Eg(1) (blue solid line) is calculated from eqn (7), Eg(2) (red dashed line) from eqn (8), and Eg(3) from eqn (9).
Fig. 4 Comparison of MAE in optical gap (Eoptfrom eqn (10)) from 9 DFT methods with respective to EOM-CCSD.
Fig. 5 Comparison of MAE in binding energy (Eb) from 9 DFT methods with respective to CCSD and EOM-CCSD. Eb(1) (blue solid line) is calculated from eqn (12), Eb(2) (red dashed line) from eqn (13), and Eb(3) from eqn (14).
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exciton binding energies and quasiparticle energies with respect to experiment.68,69While the performance of these LC hybrid functionals have not been examined in this work, we expect that these LC hybrid functionals should also yield accurate results for the properties studied.
IV. Conclusions
In conclusion, we have examined the performance of several density functional methods on the exciton binding energies and related optoelectronic properties of 121 small- to medium-sized molecules. Relative to the highly accurate CCSD and EOM-CCSD methods, uB97, uB97X, and uB97X-D exhibit the best accuracy for these properties. However, when intermolecular charge- transfer excitons are involved, uB97X-D, which includes dispersion corrections, is expected to be essential.
Acknowledgements
We thank the support from the Ministry of Science and Tech- nology of Taiwan (Grant No. MOST104-2221-E-002-186-MY3 and MOST104-2628-M-002-011-MY3), National Taiwan University (Grant No. NTU-CDP-104R7876 and NTU-CDP-104R7818), and the National Center for Theoretical Sciences of Taiwan. The computational resources from the National Center for High- Performance Computing of Taiwan and the Computing and Information Networking Center of the National Taiwan University are acknowledged.
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