Calculation of Charge-Transfer Electronic Coupling with

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C: Energy Conversion and Storage; Energy and Charge Transport

Calculation of Charge-Transfer Electronic Coupling with Nonempirically Tuned Range-Separated Density Functional Hirotaka Kitoh-Nishioka, and Koji Ando J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11343 • Publication Date (Web): 16 Apr 2019 Downloaded from http://pubs.acs.org on April 16, 2019

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The Journal of Physical Chemistry

Calculation of Charge-Transfer Electronic Coupling with Nonempirically Tuned Range-Separated Density Functional Hirotaka Kitoh-Nishioka∗,†,‡ and Koji Ando∗,¶ †JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan ‡Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan ¶Department of Information and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zenpukuji, Suginami-ku, Tokyo 167-8585, Japan E-mail: [email protected]; ando [email protected]

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Abstract Computation of charge-transfer coupling energy with a nonempirically tuned rangeseparated density functional is examined. The results are assessed by comparing with the high-level ab initio benchmark data sets, HAB11 [Kubas et al. J. Chem. Phys. 140, 104105 (2014)] of eleven cation radical homo-dimers and HAB7− [Kubas et al. Phys. Chem. Chem. Phys. 17, 14342 (2015)] of seven anion radical homo-dimers. The mean relative unsigned error (MRUE) of the charge-transfer coupling energy was 3.2 % for the HAB11 set and 7.3 % for the HAB7− set. The MRUE of the exponential decay constant along the face-to-face intermolecular distance was 2.2 % for the HAB11 set and 4.9 % for the HAB7− set. The errors were always smaller than those from the popular B3LYP functional, and in most cases smaller than those reported in previous studies. We also found nearly linear correlations between the tuned range-separated parameter µ and the energies of highest-occupied and lowest-unoccupied orbitals of the monomers, and between 1/µ and the number of double-bonds in the monomers.

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1. INTRODUCTION Intermolecular charge-transfer (CT) is of fundamental importance in broad arena of chemistry, biochemistry, and materials science. 1–4 Such a CT reaction often occurs in the nonadiabatic regime because the electronic coupling energy between localized donor and acceptor sites separated at several ˚ A distances is very weak. From Fermi’s golden rule based on the time-dependent first-order perturbation theory, the non-adiabatic CT rate is determined by the electronic coupling energy and the thermally averaged Franck-Condon factor. 4,5 However, separation of these electronic and nuclear factors in an experimental analysis is not a trivial task. In this regard, theoretical modeling will offer essential support. 6–8 This work is on the computational modeling of the electronic coupling energy which depends critically on the molecular specifics. Most CT reactions are properly described as one-electron processes; the CT coupling can be approximately calculated in terms of the Hartree-Fock (HF) or Kohn-Sham (KS) orbitals with Koopmans’ theorem. (See Ref. 9 and references therein.) Since HF and density functional theory (DFT) are based on a single Slater determinant, the one-electron scheme to the CT ignores the static electron correlation for the two-state description of nearly degenerate initial and final states. On the other hand, since Koopmans’ theorem 10 is based on the frozen orbital approximation, the one-electron scheme also ignores the orbital relaxation effects associated with the considered CT. However, the errors arising from static electron correlation and orbital relaxation in the CT coupling tend to cancel each other for many ET systems, which supports the applicability of the one-electron scheme. (See Ref. 11 and references therein.) In contrast, the dynamic correlation should be non-negligible for quantitative description, as indicated by the previous study 12 based on post HF methods. In this respect, the KS orbitals and energies of the DFT are expected to provide a useful model, even though the orbitals are in principle the only subsidiary in the DFT, which is related to the validity of Koopmans’ theorem in DFT described later. During an intermolecular CT, one electron is removed from the donor molecule; on the 3



other hand, one electron is added to the acceptor molecule. Energetics of molecules relevant to CT processes are, therefore, the ionization potential (IP) of the electron donor and the electron affinity (EA) of the acceptor. Under the condition that the charge localized donor and acceptor diabatic states come into resonance, the CT coupling energy for cation hole transfer (D-A+ → D+ -A) can be often approximated as the half-energy splitting between the first and second vertical IPs obtained for the neutral donor-acceptor pair (D-A). 9 Similarly, the CT coupling energy for anion electron transfer (D− -A → D-A− ) can be often approximated as the half-energy splitting between the first and second vertical EAs obtained for the neutral D-A pair. 9 From Koopmans’ theorem, highest-occupied (HO) and lowest-unoccupied (LU) molecular orbitals (MOs) of each neutral D/A molecule dominantly contribute to the half-energy splittings for cation hole and anion electron transfers, respectively. Therefore, models that give proper IP and EA of individual molecules are expected to provide proper CT coupling energies. For the construction of a proper model of the CT on the basis of the KS orbitals and their energies, we have used long-range corrected (LC) density functionals 13–31 that correct the exchange functionals of conventional DFTs for long-range electron-electron exchange interactions by activating exact HF exchange asymptotically with the preservation of LDA (local-density-approximation) or GGA (generalized-gradient-approximation) exchange at short range. It has been reported that LC-DFT solves or remarkably improves a wide variety of problems of conventional DFT calculations, including charge-transfer and Rydberg excitation energies, polarizabilities in conjugated systems, and van der Waals bonds. ( See Refs. 32,33 and references therein.) One of the most significant features of LC-DFT is to give valence orbital energies quantitatively, which rationalizes Koopmans’ theorem in DFT. 34,35 In this regard, we have been studying a model of ‘nonempirically tuned (NET) range-separated (RS) DFT’, 36,37 in which the range-separation parameter µ that separates the two-electron interaction as 1/r = (1 − erf(µr))/r + erf(µr)/r is tuned such that the energies of highestoccupied (HO) and lowest-unoccupied (LU) KS orbitals reproduce the theoretical IP and EA,

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the differences of independently optimized total energies of neutral and charged molecules. We have reported that the method gives accurate CT coupling energies for cation hole transfer in furane and imidazole dimers 38 that were chosen from the benchmark data set HAB11 39 for being mostly related to the nucleobase molecules in DNA studied in that work. Around the same time, You et al. 40 have reported that the CT couplings derived from the NETRS DFT for cation hole and anion electron transfers in ethylene, pyrrole, and naphthalene homo-dimers are in good agreement with those derived from the equation-of-motion couplecluster model with single and double substitutions (EOM-CCSD). The present work follows to cover the entire data sets of HAB11 39 and HAB7− 41 that contain benchmark results from high-level ab initio wave function calculations of eleven cation and seven anion radical homo-dimers. The HAB11 benchmark provides the results from the multireference configuration interaction with quadratic correction (MRCI+Q) 42–44 and n-electron valence state perturbation theory (NEVPT2), 45 whereas the HAB7− employed the spin-component scaled approximate coupled-cluster (SCS-CC2) method. 46 The data sets have been used for testing fragment orbital DFT (FO-DFT), 47,48 constrained DFT-configuration interaction (CDFTCI), 49–51 frozen density-embedding (FDE), 52 and many others. 53–56 In this work, we also examine the optimized values of the range-separation parameter in relation to the orbital energies and the chemical structures. The computational details are described in section 2. The optimized values of the rangeseparation parameter and the results of electronic coupling energy are discussed in section 3. The performance of several existing LC-DFTs and the effect of diffuse basis functions are also examined. Section 4 concludes.

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2. COMPUTATIONAL DETAILS The eleven and seven molecules in the HAB11 and HAB7− data sets are listed in Tables 1 and 2. The nuclear coordinates in the ideal face-to-face stacking configurations were taken from the original papers. 39,41 We optimized the range-separation parameter µ in the long-range corrected Becke-LeeYang-Parr (LC-BLYP) functional 16,57,58 to minimize either of the following two functions, 36,37 (N )

JH2 (µ) = [εHO (µ) + IP(N ) (µ)]2 (N +1)

+ [εHO

(µ) + EA(N ) (µ)]2 ,

(1)

(N )

JL2 (µ) = [εHO (µ) + IP(N ) (µ)]2 (N )

+ [εLU (µ) + EA(N ) (µ)]2 ,

(2) (N )

in which N is the number of electrons in the neutral molecule, εHO/LU (µ), IP(N ) (µ), and EA(N ) (µ) are the HO or LU orbital energy, ionization potential, and electron affinity of N -electron molecule calculated with the parameter µ, respectively. IP(N ) (µ) and EA(N ) (µ) are derived from the ∆SCF procedure, which satisfies the relation EA(N ) (µ) = IP(N +1) (µ). Both JH2 (µ) and JL2 (µ) optimize the HO orbital of neutral molecule. In addition, JH2 (µ) optimizes the singly-occupied orbital of anion, whereas JL2 (µ) optimizes the LU orbital of neutral molecule. Note that, in contrast to an relation εHO = −IP, 59,60 there is not an exact relation εLU = −EA, which might raise doubts about the optimization with eq 2. As concerns the Kohn-Sham LUMO, Perdew and Levy 61 have shown an equation ∆xc = {IP − EA} − {εLU − εHO }, where ∆xc is the discontinuity of exchange-correlation potentials that lead to the underestimation of the HOMO-LUMO energy gap from conventional DFT (N +1)

calculations. Moreover, Sham and Schl¨ uter 62 deduced an equation ∆xc = εHO

(N )

− εLU .

Therefore, minimizing the function JL2 (µ) makes ∆xc be as close to zero as possible, which can

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be expected to accurately describe the MOs using LC-DFT with the optimized µ parameter. Since the target dimers in HAB11 and HAB7− are symmetric donor and acceptor CT systems, we calculated the electronic coupling energy from the splitting of KS orbital energies for the dimer, whose charge state is set to neutral in accord with the idea of the NET-RS scheme (eqs 1 and 2), as follows: 9,11,12,40 ( ) (dim) (dim) ′ TDA = ε− − ε+ /2. (dim)

We, here, define ϕ+

(dim)

and ϕ−

(dim)

tively. In this case, the ϕ+

(dim)

as the KS orbitals corresponding to ε+ (dim)

and ϕ−

(dim)

follows: (dim)

ϕ±

(dim)

(dim)

and ε−

, respec-

are written by the symmetric and antisymmetric

superpositions of donor and acceptor diabatic orbitals, ϕd

For each dimer in HAB11, ϕd

(3)

(dim)

and ϕa

, in the dimer, as

( ) √ (dim) = ϕd ± ϕ(dim) / 2. a (dim)

and ϕa

(4)

are almost identical to the HOMOs of the (dim)

corresponding donor and acceptor molecules, respectively. For each dimer in HAB7−, ϕd (dim)

and ϕa

are almost identical to the LUMOs of the corresponding donor and acceptor (dim)

molecules, respectively. Therefore, it is easy to select the proper ϕ+

(dim)

and ϕ−

from the

KS orbitals of the target dimer judging by HOMO or LUMO of the corresponding monomer with eq 4. Note that eq 3 is inapplicable to asymmetric CT systems where the donor and acceptor molecules are dissimilar. In that case, the FO-DFT, 39,41 the fragment molecular orbital (FMO), 38 or the GMH method 63,64 is useful. (See Ref. 38 and references therein). ′ For example, in the one-electron approximation, the GMH method offers the TDA calculation

from the orbital energy splitting scaled by the dipole moment matrix element with respect to the two adiabatic KS orbitals. 38,65 We used the program GAMESS 66 for the electronic coupling calculations with NET-LCBLYP and Gaussian 09 67 for the optimization of the NET-RS parameter. The results of ′ TDA were first compared to those with the widely used B3LYP functional 68 in Sec. 3.2. For

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further assessment, we then tested another three standard range-separated functionals, LRCωPBEh, 31 CAM-B3LYP, 20 and ωB97X-D, 30 and two standard hybrid GGA (generalizedgradient-approximation) functionals, PBE0 69 and BH&HLYP, 57,58 which is presented in Sec. ′ 3.3. GAMESS was also used for the calculations of TDA with these functionals except for

the LRC-ωPBEh; NWChem 6.5 70 was used only for the LRC-ωPBEh. The cc-pVTZ basis set 71 was used for the heavy atoms and the cc-pVDZ set 71 for hydrogens. We also employed the aug-cc-pVTZ set 72 for the heavy atoms in the HAB7− set to check the effect of diffuse functions in Sec. 3.4.

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3. RESULTS 3.1. Optimization of Range-separation Parameter As an illustration of the parameter tuning procedure, we display in Figure 1 the computed JH2 (µ) and JL2 (µ) for anthracene. The corresponding plots for the other molecules are in Figures S1 and S2 of the Supporting Information. Table 1 lists the numerical values of µ, (N )

(N +1)

εHO , εHO

(N )

, εLU , IP(N ) , and EA(N ) obtained for the HAB11 set of molecules. The vertical

IPs from the photoelectron spectroscopy 50,73,74 are also included. Table 2 is the same as Table 1 but for the HAB7− set of molecules, where the experimental IPs were taken from Refs. 75, 76, 77, and 78. Table 2 includes for comparison the vertical EAs calculated with the coupled cluster (CC) methods 50 (to our knowledge, there is no experimental report on (N )

their vertical EAs.) As with the previous studies, 37,79 the resultant values of −εHO from the NET-RS method reasonably reproduces the experimental IPs with the maximum errors of 0.25 eV for phenol in HAB11 and 0.38 eV for perfluoroanthracene in HAB7−. The values of (N )

−εLU agree with the EAs from the CC method within 0.14 eV for the molecules in HAB7−. Figure 1 indicates that JH2 (µ) of eq 1 and JL2 (µ) of eq 2 both have a sharp single minimum at which the optimal values of µ are identical within the scanning increment of 0.005 bohr−1 . Similar behavior was observed for the other molecules, with an exception of Pyrrole for which the values from eqs 1 and 2 deviated by 0.015 bohr−1 . For the molecules in HAB7−, we also carried out calculations with only the second terms of eqs 1 and 2, since they are more relevant in the anion charge transfer and the absolute values of EA are generally smaller than those of IP so that the contribution of the second term could have been masked by the first term. Interestingly, however, Figure 1 indicates that these latter calculations gave the optimal values of µ identical within the range of ±0.005 bohr−1 . The same was found for the other molecules in the HAB7− set, as shown in Table S1 of the Supporting Information. Note that the resultant EA(N) ’s have negative values for all the molecules of HAB11,

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as listed in Table 1. Previous studies 80,81 have reported that the EA(N) based on DFT is applicable to the calculation of negative vertical EA of neutral N -electron molecule only when the basis set is compact, such as the cc-pVTZ basis sets used in this study. The extra electron of an anion molecule is artificially bound by the basis set employed. As the basis set is further increased, the extra electron tends to leave the molecule (namely, EA(N) will approach zero). Therefore, when using the large basis sets, especially including diffuse functions, we probably had better optimize the RS parameter through only the first term of eq 1. (N )

(N )

Figure 2 (a) plots εHO and εLU at the optimized values of µ for all the molecules of HAB11 and HAB7− sets. There seem to exist approximately linear correlations for both (N )

(N )

εHO and εLU with µ. Figure 2 (b) plots the number of double bonds in the monomer along 1/µ, in which a linear correlation appears to exist. The linear correlation between µ and the orbital energies seems reasonable in a sense that the dimension of µ is the inverse of length to which the Coulomb energy is proportional. Similarly, as the number of double bonds may relate to the delocalization (conjugation) length, its correlation with 1/µ sounds sensible. (N )

Figure 3 looks more closely the molecules in the HAB7− set. Figure 3 (a) plots −εHO

that corresponds to the vertical IP at the optimized µ. There seems to exist a positive linear (N )

correlation between −εHO and µ for the oligoacenes; anthracene, tetracene, and pentacene. Perylene marks close to this line, while porphin and perfluoroanthracene are slightly off the line. Perylene diimide is apparently off. More systematic linear positive correlation is seen in Figure 3 (b) which plots the number of double-bonds against 1/µ. The correlation between 1/µ and the number of conjugation length in simple polyenes and oligoacenes has been examined by Körzdörfer et al. 79 The present results generalize the picture for different types of conjugated molecules. The corresponding analysis was performed for the HAB11 set and plotted in Figure 4. Almost linear positive correlations are observed, but the deviation and dispersion are more apparent. In particular, the five molecules (cyclopentadiene, furane, pyrrole, thiophene, and

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imidazole) that share the five-membered ring structure but contain different heteroatoms have a range of values of optimal µ. Some other quantities or adjustments that account for these differences should be examined to find better correlations.

3.2. Electronic Coupling Energy ′ The computed values of electronic coupling energy TDA along the dimer face-to-face distance

R are listed in Tables 3, 4, and 5. The mean relative unsigned error (MRUE) from the ∑ ′ reference, MRUE = ( ni=1 | Xcalc,i − XRef,i |/XRef,i ) /n where X = TDA , was computed for the set of R for each molecule, as well as for the entire sets of HAB11 and HAB7−. The overall MRUE with the NET-RS LC-BLYP functional were 3.2 % for the HAB11 and 7.3 % for the HAB7−, compared to those with the B3LYP functional, 18.5 % for HAB11 and 15.4 % for HAB7−. ′ along R, of the four cases in which the NET-RS LCFigure 5 plots the log of TDA

BLYP functional gave the best and the worst MRUE in each of the HAB11 and HAB7− sets: ethylene (0.9 %), cyclobutadiene (7.1 %), tetracene (2.5 %), and perylene (12.0 %). The B3LYP functional gave the MRUE of 20.6 %, 23.3 %, 12.5 %, and 21.2 % for these molecules. The plots for the remaining fourteen molecules are displayed in Figures S3-S4 of the Supporting Information. The linearity of the log plots in Figures 5 and S3-S4 indicates that the coupling energy ′ TDA (R) is well described by an exponential function with the decay constant β,

′ ′ TDA (R) = TDA (0) exp(−βR/2).

(5)

′ The physical reason for the validity of eq 5 is that TDA (R) is proportional to the overlap (dim)

between donor and acceptor diabatic orbitals, ϕd

(dim)

and ϕa

, around their exponentially

attenuated tail region. The values of β determined by the least-squares fitting are listed in Table 6. The MRUE of β with the NET-RS LC-BLYP functional was 2.2 % for HAB11 and

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4.9 % for HAB7−. As seen in Figure 5, the B3LYP functional gives proper decay constant even when the absolute values were notably inaccurate: the MRUE of β was 3.2 % for HAB11 and 5.4 % for HAB7−. The present calculation is from the splitting of orbital energies (eq 3) of the neutral dimers rather than that of state energies of cation or anion dimers. Although the latter is in principle more appropriate for the electronic coupling, it was found to give larger errors in practice. For instance, Manna et al. 56 computed the electronic coupling energies of the HAB11 set from the first adiabatic excitation energies with the NET-RS time-dependent LCDFT (TD-LC-DFT) and found the MRUE of 11.3 %. Félix and Voityuk 65 studied π stacks of nucleobases with the generalized Mulliken-Hush (GMH) method 63,64 and found that the TD-DFT gives less accurate electronic coupling energies than those using the KS orbitals because the spin-unrestricted DFT overestimates the hole delocalization in radical cations. Our results confirm that the well-balanced description of the orbital energies is essential for the accurate electronic coupling energies. We now note the previous benchmark studies. Schober et al. performed a critical analysis of the FO-DFT method 48 by constructing the dimer Hamiltonians from B3LYP functional on BLYP densities and found the best MRUEs of 4.4 % for HAB11 and 11.8 % for HAB7−. Hwang et al. have developed the fragment-orbital tunneling current (FOTC) method, 55 and found the best MRUEs of 17.1 % for HAB11 and 5.4 % for HAB7−. Ramos et al. have found that the FDE method with PW91 functional with PW91k non-additive kinetic energy functional yields the MRUE of 6.9 % for HAB11. 52 Kim et al. proposed an error metric η to select the adequate density functional for calculating CDFT-CI electronic couplings, 50 and found the best MRUEs of 14.7 % for HAB11 and 12.4 % for HAB7− with the BH&HLYP functional. The previous best results were 5.3 % for HAB11 39 and 8.2 % for HAB7− 41 by the CDFT/50 functional.

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3.3. Comparison with Another Existing LC-DFTs We tested another three standard RS functionals, LRC-ωPBEh, CAM-B3LYP, and ωB97X′ D, and two hybrid functionals, PBE0 and BH&HLYP, for the TDA calculations. In LC-BLYP

functional, the RS Coulomb operator 1/r = (1 − erf(µr))/r+erf(µr)/r is applied to the B88 57 GGA exchange functional. On the other hand, for the other three LC-DFTs, RS operator is applied to the hybrid exchange functionals, which incorporates a fraction of short-range HF exchange. In CAM-B3LYP functional, the following generalized RS-Coulomb operator is used, 20 1 1 − [αCAM + βCAM erf(µr)] αCAM + βCAM erf(µr) = + . r r r

(6)

Yanai et al. gave a parameter set αCAM = 0.19, βCAM = 0.46, and µ = 0.33 bohr−1 as its standard one, which was optimized using the atomization energy of the G2 set. 20 Note that the CAM-B3LYP with the standard parameter set does not provide 100% HF exchange in the long-range because of αCAM + βCAM ̸= 1. For LRC-ωPBEh and ωB97X-D, the RS exchange-correlation energy is expressed by, 29–31

SR SR LR ExLC−DFT = Ec + (1 − CHF )Ex,local + CHF Ex,HF + CHF Ex,HF ,

(7)

LR SR represent the HF exchange energies calculated with short- and longand Ex,HF where Ex,HF

range components (namely, (1 − erf(µr)) /r and erf(µr)/r) of the RS operator, respectively. SR Ex,local represents the local DFT exchange energy calculated with the short-range component

of the RS operator. Equation 7 guarantees the use of 100% HF exchange in the long-range. Note that the RS parameter is expressed by ω instead of µ in Refs. 29–31. For ωB97XD, Chai and Head-Gordon gave a parameter set CHF = 0.222036 and µ = 0.2 bohr−1 as its standard one, which was optimized from a systematic fitting procedure to 412 accurate experimental and theoretical results. 30 For LRC-ωPBEh, Rohrdanz et al. gave a parameter set CHF = 0.2 and µ = 0.2 bohr−1 as its standard one, which was optimized using both ground-state and excited-state properties. 31 The degrees of HF exchange in the global hybrid 13



density functionals, B3LYP, PBE0, and BH&HLYP, are 20%, 25%, and 50%, respectively. Table 7 compares the statistical quantities, MUE, MRSE, MRUE, and MAX, obtained ′ from the TDA calculations of the HAB11 set: MUE is the mean unsigned error from the ∑ ′ reference, MUE = ( ni=1 | Xcalc,i − XRef,i |) /n where X = TDA , was computed for the set

of R for each molecule; similarly, MRSE is the mean unsigned error from the reference, ∑ MSRE = ( ni=1 ( Xcalc,i − XRef,i )) /n, and MAX is the maximum unsigned error, MAX = ′ | Xcalc,i − XRef,i |. The computed TDA -values are listed in Tables S2 and S3 of the Supporting

Information. As mentioned in Sec. 3.2, Manna et al. 56 optimized the RS parameter of LRCωPBEh functional for each molecule in HAB11 by minimizing the function JH2 (µ) of eq 1 with the constant fraction of short-range HF exchange CHF = 0.2. To compare the performance between NET-LC-BLYP and NET-LRC-ωPBEh, we used the NET-RS parameters given in Ref. 56 instead of its standard one (µ = 0.2 bohr−1 ). On the other hand, we used the corresponding standard parameter sets for CAM-B3LYP and ωB97X-D. From Table 7, we can see that the NET-LC-BLYP offered the best performance for the HAB11 set among the functionals used. The NET-LRC-ωPBEh showed the comparable performance to those of CAM-B3LYP and ωB97X-D involving the corresponding constant default values as their RS parameters. The performance of BH&HLYP was comparable with those of NET-LRCωPBEh, CAM-B3LYP, and ωB97X-D, while the other B3LYP and PBE0 functional resulted in the inferior performances. ′ Table 8 lists MUE, MRSE, MRUE, and MAX obtained from the TDA calculations of the ′ HAB7− set. The computed TDA -values are listed in Table S4 of the Supporting Information.

We used the standard RS parameters for LRC-ωPBEh (µ = 0.2 bohr−1 ), CAM-B3LYP (µ = 0.33 bohr−1 ), and ωB97X-D (µ = 0.2 bohr−1 ). The CAM-B3LYP showed slightly better performance than that of NET-LC-BLYP for the HAB7− set. On the other hand, the performance of LRC-ωPBEh was slightly worse than those of the other three LC-DFTs. Interestingly, BH&HLYP showed the best performance among the density functionals we tested for the HAB77- set.

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So far, various types of global hybrid and LC DFTs have been reported and their standard parameter sets, including the degree of HF exchange and/or RS parameter, were optimized using each training data set. The difficulty in choosing the proper density functional among various existing ones for investigating the target physical quantity of the target molecule is a common problem in the DFT-based study. As listed in Tables 7 and 8, the LC-BLYP with the NET-RS parameter shows the best performance for the HAB11 set and shows the reasonably good performance for the HAB7− set among the seven density functionals we used. It is extremely easier to use the NET-RS scheme developed by Stein et al. 36 than it is to tune empirically the RS parameter for your target CT system with the preparation of high-level ab initio wave function results and experimental IPs or EAs. Although our investigation only tested a limited number of density functionals, the results indicate that ′ with eq 3 for both cation hole and the NET-RS LC-BLYP is the first choice to calculate TDA

anion electron transfers in the intermolecular CT reactions. Our results also indicate that ′ BH&HLYP offers reasonably good estimations of TDA for both the HAB11 and HAB7− sets.

Therefore, the global hybrid functionals including similar degree (ca. 50%) of HF exchange, such as M06-2X, 82 might work well. Such a study is in progress and would be reported in due course. The comparable performances obtained using NET-LC-ωPBEh, CAM-B3LYP, and ωB97XD for HAB11 seem to indicate that the NET-RS scheme to hybrid exchange functionals does ′ not work for the TDA calculations with eq 3. The ineffectiveness of the NET-RS scheme to

LC-ωPBEh may be due to the use of the improper fraction of short-range HF exchange. In the similar context, when applying the NET-RS scheme to CAM-B3LYP, Okuno et al. optimized the parameters, αCAM , βCAM , and µ in eq 6 on the same footing; 83 namely, in the case, JH2 of eq 1 is regarded as a function of αCAM , βCAM , and µ. Okuno et al. have reported that the NET-RS CAM-B3LYP can reproduce the excitation energies of photochromic diarylethene derivatives when the condition αCAM + βCAM = 1 that guarantees the correct asymptotic behavior of the exchange functional (100% HF exchange at long range) is satisfied. In the

15



future work, we should examine whether the NET-RS scheme using JH2 (µ, αCAM , βCAM ) for CAM-B3LYP or JH2 (µ, CHF ) for LRC-ωPBEh is preferable to the NET-RS LC-BLYP using JH2 (µ).

3.4. Effect of Diffuse Basis Functions Because the anion electron distributions are generally extended in space, it is sometimes needed to add diffuse basis functions for quantitative accuracy. However, the diffuse functions often cause instability in the calculations. Because of this and of our aim toward applications to larger systems, we mainly employed the cc-pVTZ and cc-pVDZ sets. Nevertheless, it would be worthwhile to check the effect of diffuse basis functions, particularly for anions. The optimized µ and orbital energies and the theoretical and experimental IP’s for the HAB7− set of molecules are listed in Table S5 of the Supporting Information. In general, the optimal µ with the diffuse basis functions are smaller than those without them by 0.005∼0.01 bohr−1 , except for the case of porphin where the former is larger by 0.02 bohr−1 . The computed CT coupling energies are listed in Table S6 of the Supporting Information. For anthracene, tetracene, and pentacene, the MRUE was larger than those with the smaller basis set. Nevertheless, the notable reduction of MRUE was found for perylene (from 12.0 % to 2.6 %) and porphin (from 9.8 % to 2.0 %), and consequently the overall MRUE was reduced from 7.3 % to 5.6 %.

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4. CONCLUSION The MRUEs of the CT coupling energy with the NET-RS LC-BLYP functional were 3.2 % for the HAB11 set and 7.3 % for the HAB7− set, whereas the B3LYP functional gave 18.5 % (HAB11) and 15.4 % (HAB7−). We compared our results with the previous benchmark studies by the other groups. The NET-LC-BLYP method with eq 3 is computationally less expensive, but the accuracy is generally comparable to or better than those from the more intensive calculations. The simple orbital-energy splitting method, eq 3, is rather approximate, replacing the true adiabatic excitation energy by KS orbital energies, not carrying out the calculations for the correct system charge, etc. However, the accuracy comes from the well-balanced description of the orbital energies with the NET-RS scheme. The errors arising from static electron correlation and orbital relaxation tend to, fortunately, cancel each other, as reported in the previous studies. 9,11,12 The KS orbitals obtained from the NET-RS scheme, satisfying Koopmans’ theorem in DFT, can take properly into account the electron dynamic correlation, leading to quantitatively accurate evaluation of the CT coupling energy. We also tested another three RS functionals, LRC-ωPBEh, CAM-B3LYP, and ωB97XD, and two hybrid functionals, PBE0 and BH&HLYP. From the statistical quantities of the ′ -values listed in Tables 7 and 8, we can see that the NET-LC-BLYP gives the obtained TDA

best performance for the HAB11 set and the reasonably good performance for the HAB7− set among the seven density functionals. It is generally difficult to know whether the LC functional selected from among various existing ones will show the favorable performance for your target CT system by judging from its previous performances on the different systems. When the selected LC functional with its standard RS parameter set shows the strong-system ′ dependence of the TDA calculation, the NET-RS scheme has a possibility to circumvent the

problem with a rather inexpensive computational cost. Although our investigation only tested a limited number of density functionals, the results indicate that the NET-LC-BLYP is the first choice. From the moderate performance of the NET-LRC-ωPBEh, the application of 17



the NET-RS scheme to the hybrid exchange functionals, such as CAM-B3LYP and ωB97XD, likely requires to optimize the RS parameter and the fraction of the short-range HF exchange on the same footing by using the function JH2 (eq 1) or JL2 (eq 2). BH&HLYP ′ also gave reasonably good estimations of TDA for both the HAB11 and HAB7− sets, which

corresponds to the good performances obtained by the CDFT/50 calculations. This paper also examined the optimized NET-RS parameter µ in relation to the chemical structures of the molecules and found a linear correlation between 1/µ and the number of double bonds in the monomer. In experimental studies, it is a useful way to extend the conjugation length of the constituent molecules with chemical substituent to regulate the CT coupling. The accurate estimation of the CT coupling from the NET-RS LC-DFT method and the observed dependence of the optimized µ on the chemical structures will offer vital information for the efficient design of conductive materials.

Acknowledgement The authors acknowledge support from KAKENHI No. 20108017 (“π-space”). H. K.-N. also acknowledges support from Collaborative Research Program for Young Scientists of ACCMS and IIMC, Kyoto University and from JST, PRESTO Grant Number JPMJPR17G4.

Supporting Information Available ′ Data for the optimization of the parameter µ and the calculation of TDA for all the molecules

in HAB11 and HAB7−, results of an alternative optimization method for µ, results of another three LC-DFTs and two hybrid DFTs, and the effect of diffuse basis functions on HAB7−.

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Table 1: Optimized parameter µ and orbital energies, with the theoretical and experimental IP’s (IP(N) and IPRef. ) and theoretical EA (EA(N) ), for the HAB11 set of molecules with the cc-pVTZ basis set for heavy atoms and cc-pVDZ for hydrogen. Molecule Ethylene Acetylene Cyclopropene Cyclobutadiene Cyclopentadiene Furane Pyrrole Thiophene Imidazole Benzene Phenol

(N )

(N +1)

µH µL IP(N ) EA(N ) εHO εHO 0.385 0.390 10.61 -2.43 -10.57 2.36 0.430 0.435 11.34 -3.33 -11.31 3.27 0.365 0.365 9.81 -2.34 -9.74 2.24 0.335 0.340 8.01 -0.62 -7.96 0.57 0.310 0.310 8.51 -1.48 -8.50 1.45 0.330 0.330 8.98 -2.29 -8.95 2.25 0.325 0.340 8.31 -3.25 -8.31 3.23 0.305 0.305 8.98 -1.53 -8.97 1.53 0.330 0.335 8.99 -2.72 -8.96 2.68 0.300 0.300 9.35 -1.69 -9.32 1.67 0.300 0.300 8.52 -1.58 -8.50 1.53 2 −1 µH/L were optimized with JH/L in bohr . IP, EA, and ε’s are in eV. a See the references in Ref. 50. b See the references in Ref. 73. c See the references in Ref. 74.

29


(N )

εLU IPRef. 2.47 10.50a 3.35 11.43a 2.40 9.82b 0.64 8.24a 1.50 8.53a 2.31 8.90a 3.09 8.23a 1.52 8.85a 2.74 8.96c 1.69 9.23a 1.61 8.75a


Page 30 of 40

Table 2: Same as Table I but for the HAB7− set of molecules. For comparison, the vertical EAs calculated with the high-level coupled cluster (CC) methods, 50 EARef. , are also listed. Molecule Anthracene Tetracene Pentacene Perfluoroanthracene Perylene Perylene Diimide Porphin

µH 0.235 0.215 0.195 0.240 0.215 0.195 0.205 µH/L

(N )

(N +1)

µL IP(N ) EA(N ) εHO εHO 0.235 7.23 0.38 -7.21 -0.39 0.215 6.67 0.94 -6.66 -0.96 0.195 6.27 1.34 -6.26 -1.35 0.240 7.92 1.38 -7.90 -1.39 0.215 6.74 0.79 -6.73 -0.81 0.195 7.66 2.55 -7.66 -2.55 0.205 6.69 1.33 -6.71 -1.30 2 were optimized with JH/L in bohr−1 . IP, EA, and ε’s are in eV.

30


(N )

εLU -0.37 -0.93 -1.34 -1.38 -0.79 -2.55 -1.35

IPRef. 7.42 75 6.94 75 6.59 75 8.28 76 6.97 77 NA 6.9 78

EARef. 0.24 0.80 1.20 1.24 0.70 NA 1.24

Page 31 of 40

Anthracene

10 0 10−1

J 2 (eV2 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−2 10−3 Eq.(1) Eq.(2) Eq.(1)−1 Eq.(2)−1

10−4 10−5 0.1

0.2

µ

0.3 –1 (bohr )

0.4

Figure 1: Plots of JH2 (eq 1, purple) and JL2 (eq 2, green) as a function of range-separation (N +1) parameter µ for Anthracene. Second terms of eqs 1 and 2, [εHO (µ) + EA(N ) (µ)]2 and (N ) [εLU (µ) + EA(N ) (µ)]2 are also plotted by light-blue and orange solid lines, respectively.

31



Table 3: Electronic coupling energy for hole transfers in cation dimers of the HAB11 set. The reference is from the MRCI+Q calculation. 39 MRUE is the mean relative unsigned error. R (˚ A) Ethylene 3.5 4.0 4.5 5.0 MRUE(%) Acetylene 3.5 4.0 4.5 5.0 MRUE(%) Cyclopropene 3.5 4.0 4.5 5.0 MRUE(%) Cyclobutadiene 3.5 4.0 4.5 5.0 MRUE(%) Cyclopentadiene 3.5 4.0 4.5 5.0 MRUE(%) Furane 3.5 4.0 4.5 5.0 MRUE(%) Pyrrole 3.5 4.0 4.5 5.0 MRUE(%)

32

′ (meV) TDA LC-BLYP B3LYP Ref. 39 516.4 425.0 519.2 270.6 215.4 270.8 137.3 107.9 137.6 66.5 53.2 68.5 0.9 20.6 — 460.8 376.7 460.7 233.0 184.9 231.8 114.1 89.7 114.8 53.5 42.8 56.6 1.6 21.2 — 544.2 464.0 536.6 260.7 215.1 254.0 122.4 98.9 118.4 55.8 45.0 54.0 2.7 15.5 — 441.6 376.5 462.7 227.1 187.1 239.1 113.8 92.1 121.7 54.7 44.6 62.2 7.1 23.3 — 456.7 390.3 465.8 234.1 192.3 234.4 116.9 93.6 114.3 56.1 44.8 53.4 2.3 17.1 — 428.2 363.2 440.3 214.0 174.7 214.9 104.5 83.1 101.8 49.3 39.2 46.0 3.3 17.3 — 445.8 377.8 456.3 226.9 185.0 228.6 113.0 89.8 111.3 54.4 43.2 52.2 2.2 18.2 —


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Table 4: Same as Table III but with the reference from NEVPT2 calculation. 39 The last line is the total MRUE for all the eleven cation dimer of the HAB11 set. R (˚ A) Thiophene 3.5 4.0 4.5 5.0 MRUE(%) Imidazole 3.5 4.0 4.5 5.0 MRUE(%) Benzene 3.5 4.0 4.5 5.0 MRUE(%) Phenol 3.5 4.0 4.5 5.0 MRUE(%) Total MRUE(%)

′ TDA (meV) LC-BLYP B3LYP Ref. 39 452.5 386.8 449.0 228.3 186.9 218.9 112.2 89.1 106.5 53.1 41.9 54.4 3.2 16.9 — 422.3 357.8 411.6 211.2 172.2 202.8 103.3 82.2 99.1 49.0 38.9 49.7 3.1 16.7 — 452.7 386.5 435.2 227.4 185.6 214.3 111.3 87.8 104.0 52.5 41.1 51.7 4.7 15.2 — 358.9 311.4 375.0 173.6 144.1 179.6 82.3 65.8 85.2 38.5 30.3 41.3 4.5 21.5 — 3.2 18.5 —

33



Table 5: Electronic coupling energy for electron transfers in anion dimers of the HAB7− set. The reference is from the SCS-CC2 calculation. 41 R (˚ A) Anthracene 3.5 4.0 4.5 5.0 MRUE(%) Tetracene 3.5 4.0 4.5 5.0 MRUE(%) Pentacene 3.5 4.0 4.5 5.0 MRUE(%) Peryfluoroanthracene 3.5 4.0 4.5 5.0 MRUE(%) Perylene 3.5 4.0 4.5 5.0 MRUE(%) Perylene Diimide 3.5 4.0 4.5 5.0 MRUE(%) Porphin 3.5 4.0 4.5 5.0 MRUE(%) Total MRUE(%)

34

′ TDA (meV) LC-BLYP B3LYP Ref. 41 408.3 374.5 421.1 206.0 182.0 212.3 102.7 87.5 106.1 50.1 41.6 52.3 3.4 15.9 — 400.9 372.4 417.2 199.2 178.5 204.3 97.6 84.4 97.9 46.8 39.5 45.4 2.5 12.5 — 392.0 368.9 411.0 192.9 175.6 198.0 93.5 82.3 92.4 44.4 38.1 41.0 4.2 9.9 — 288.5 265.9 310.9 133.6 117.6 139.1 62.3 52.0 59.9 29.6 23.7 24.0 9.6 11.1 — 399.5 373.0 423.7 202.2 182.8 220.7 101.4 89.0 116.6 49.7 42.8 62.8 12.0 21.2 — 337.6 321.4 373.8 160.9 148.4 179.2 75.3 67.3 84.1 34.9 30.3 38.0 9.6 17.9 — 348.8 329.0 374.5 168.6 153.8 182.9 80.4 70.8 89.4 37.7 32.4 44.1 9.8 18.8 — 7.3 15.3 —


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Table 6: The computed values of decay constant β of Eq. (5). β (˚ A−1 ) LC-BLYP B3LYP Ref. 39 or 41 HAB11 Ethylene Acetylene Cyclopropene Cyclobutadiene Cyclopentadiene Furane Pyrrole Thiophene Imidazole Benzene Phenol MRUE(%) HAB7− Anthracene Tetracene Pentacene Peryfluoroanthracene Perylene Perylene Diimide Porphin MRUE(%)

2.73 2.87 3.03 2.78 2.79 2.88 2.80 2.86 2.87 2.87 2.98 2.2

2.77 2.90 3.11 2.84 2.88 2.97 2.89 2.96 2.96 2.99 3.11 3.2

2.70 2.80 3.06 2.68 2.89 3.01 2.89 2.82 2.82 2.85 2.95 —

2.79 2.86 2.90 3.04 2.78 3.03 2.97 4.9

2.93 2.99 3.03 3.23 2.89 3.15 3.09 5.4

2.78 2.96 3.07 3.41 2.55 3.05 2.85 —

Table 7: Total mean unsigned error MUE, mean relative signed error MRSE, mean relative unsigned error MRUE, and maximum unsigend error MAX obtained using several RS and hybrid functionals for all the eleven cation dimer of the HAB11 set.

a

b

MUE (meV) MRSE(%) MRUE(%) MAX(meV) Functional NET-LC-BLYP 5.44 0.146 3.23 21.1 ab NET-LRC-ωPBEh 14.3 −8.46 8.46 37.6 CAM-B3LYP 14.8 −7.61 7.61 45.6 ωB97X-D 15.8 −8.66 8.66 45.4 B3LYP 36.4 −18.5 18.5 94.2 PBE0 35.1 −18.6 18.6 88.0 BH&HLYP 13.9 −7.54 7.57 40.2 The cc-pVTZ basis sets were used for all atoms; NET-RS parameters were taken from Ref. 56. Furane and Benzene cation dimers were excluded because the corresponding NET-RS parameters are not given in Ref. 56. 35



Table 8: Same as Table 7 but for for all the seven anion dimer of the HAB7− set. Functional MUE (meV) MRSE(%) MRUE(%) MAX(meV) NET-LC-BLYP 10.9 −4.44 7.29 36.2 LRC-ωPBEh 13.7 −8.28 9.27 29.5 CAM-B3LYP 6.25 −2.19 5.44 14.7 ωB97X-D 8.36 −5.42 6.72 18.3 B3LYP 24.9 −15.3 15.3 52.4 PBE0 22.7 −14.6 14.6 46.0 BH&HLYP 5.10 2.18 5.09 12.1

Graphical TOC Entry

36


Page 36 of 40

Page 37 of 40

ε HO / LU (eV)

4

(a)

0 HAB11 HO LU HAB7− HO LU

−4

−8

−12

15

Number of Double Bond

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

0.2

0.3

μ (bohr−1)

0.4

(b) HAB11 HAB7−

5

0 2

3

4

5

1/μ (bohr) (N )

(N )

Figure 2: (a) Plots of εHO and εLU at the nonempirically tuned (NET) values of range seperated (RS) parameter µ for all the molecules in HAB11 and HAB7−. (b) Plots of the number of double bonds at the inverse of NET-RS optimized µ for all the molecules in HAB11 and HAB7−.

37



8

(a)

−ε HO (eV)

PFA PDI 7

Por

Ant

Per Tet

Pen 6

0.2

0.22

0.24

−1

μ (bohr ) 15


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 40

(b) PDI

Per

10

Por

Pen

Tet

Ant PFA 5

4

4.5

5

5.5

1/μ (bohr) (N )

Figure 3: (a) Plots of −εHO at the nonempirically tuned (NET) values of rangeseparated (RS) parameter µ for the molecules in HAB7−. (b) Plots of the number of double bonds at the inverse of NET-RS optimized µ for the molecules in HAB7−. Ant=Anthracene, Tet=Tetracene, Pen=Pentacene, PFA=Perfluoroanthracene, Per=Perylene, PDI=Perylenediimide, Por=Porphin.

38


Page 39 of 40

12

(a) 2

−ε HO (eV)

11

1

10

10 8

9

11

5

8 7

0.3

3

6,9 7

4 0.35

0.4

−1

0.45

μ (bohr )


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(b) 3

10 11 2

4 67 5 8 9 2

1 3

1 2

3

4

1/μ (bohr) (N )

Figure 4: (a) Plots of −εHO at the nonempirically tuned (NET) values of range-separated (RS) parameter µ for the molecules in HAB11. (b) Plots of the number of double bonds at the inverse of NET-RS optimized µ for the molecules in HAB11. 1. Ethylene, 2. Acetylene, 3. Cyclopropene, 4. Cyclobutadiene, 5. Cyclopentadiene, 6. Furane, 7. Pyrrole, 8. Thiophene, 9. Imidazole, 10. Benzene, 11. Phenol.

39



103

(a)

Ethylene

Transfer Integral (meV)


103

102

LC−BLYP B3LYP MRCI+Q 101

3.5

4

4.5


5


102


4.5

4

4.5

5

103

(c)

Tetracene

3.5

3.5

Intermolecular Distance (Å)

103

101

(b)

Cyclobutadiene

102



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 40

102


5

(d)

Perylene


3.5

4

4.5

5


Figure 5: Electronic coupling energies from NET-RS LC-BLYP and B3LYP calculations as a function of intermolecular distance for (a) ethylene, (b) cyclobutadiene, (c) tetracene, and (d) perylene homo-dimer. The reference data (MRCI+Q or SCS-CC2) taken from Refs. 39 and 41 are also plotted.

40


Calculation of Charge-Transfer Electronic Coupling with

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