A Benchmark Study of Electronic Couplings in DonorâBridge

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Quantum Electronic Structure

A Benchmark Study of Electronic Couplings in DonorBridge-Acceptor Systems with the FMR-B Method Alessandro Biancardi, and Marco Caricato J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00029 • Publication Date (Web): 09 Mar 2018 Downloaded from http://pubs.acs.org on March 12, 2018

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Journal of Chemical Theory and Computation

A Benchmark Study of Electronic Couplings in Donor–Bridge–Acceptor Systems with the FMR-B Method Alessandro Biancardi and Marco Caricato∗ Department of Chemistry, University of Kansas, 1251 Wescoe Hall Dr., Lawrence KS, 66045 E-mail: [email protected] Abstract We present a benchmarking study for the evaluation of electronic couplings in donorbridge-acceptor systems with the Fock matrix reconstruction-bridge (FMR-B) method. We compile a data set for the benchmark that contains 29 molecules for which reliable experimental coupling values are available: DBA29. This data set is general and includes different types of donor, acceptor, and bridge units as well as different bridge lengths, and it spans a range of couplings from 0.1 to 0.8 eV. We use DBA29 to test FMR-B with eleven density functionals belonging to different classes (pure, global hybrid, and range-separated) and the Hartree-Fock (HF) method. We also test a subset of these methods with nine basis sets from the Pople and Dunning families, which include a varying number of polarization and diffuse functions. We find that the best accuracy and lowest computational cost is obtained with range-separated functionals and compact basis sets. Global hybrids with a large amount of HF exchange also work well because of error cancellation between the approximate exchange-correlation kernel and the HF part. Pure functionals, although less accurate, still provide reasonable results

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with a consistent underestimation of the experimental values, and they can be used for larger and more computationally demanding systems.

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Introduction

Through-bond charge transfer plays a key role in many molecular and biomolecular systems. 1 For instance, the use of individual bridged molecules as components of electronic circuits may lead to better and smaller electronic devices. 2,3 The accurate treatment of charge transfer is thus important for the investigation of biological, chemical, and optoelectronic systems towards the development of more efficient catalytic processes and molecular devices. 4,5 Although the formulas to compute the transfer rates depend on the strength of the interaction between the donor (D) and the acceptor (A) moieties, 5–7 a key component that needs to be evaluated is the electronic coupling J. This is the off-diagonal element of the Hamiltonian matrix in some diabatic representation where the D/A groups can be described as independent units. J shows a near-exponential decay with the D–A distance: J (R) ∝ exp (−βR), 8 except at shorter distances where the change is non-exponential as a result of a complex interplay among through-bond and through-space pathways. A variety of methods have been proposed for the evaluation of the coupling, and they are collected in a recent review. 5 An approach that we find particularly appealing for its simplicity and efficiency is based on the work of Brédas 9,10 and Thoss 11 with mean-field methods, called Fock matrix reconstruction (FMR). In FMR, the coupling is obtained by projection of the Fock/Kohn-Sham matrix of the D/A supersystem onto fragment orbitals of the D/A groups. The difference between the Brédas and Thoss approaches is in how the fragment orbitals are obtained, and we refer the reader to Ref. 12 for more details. We have expanded the FMR approach to compute electronic couplings in extended systems with periodic boundary condition (PBC) density functional theory (DFT) methods (FMRPBC), 13,14 and for bridged systems (FMR-B). 12 The FMR-PBC method can be used to

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study interlayer transfer in films and solids, and the FMR-B can be used to study molecular nanowires. In our original work, 12 FMR-B compared well with available experimental data and alternative theoretical methods (we used the work of Gillet et al. 15 and Ramos and Pavanello, 16 as reference). Other works investigated the performance of approximate density functionals compared to higher level methods. 17–20 However, a more complete benchmark data set of experimental values is highly desirable for a more thorough testing of our and other methods. Experimental couplings can be obtained for symmetric molecules as half of the splitting between the two lowest ionization energies (IE):

J=

1 (IEn−1 − IEn ) 2

(1)

where n is the number of electrons. In the framework of Koopmans theorem, J is half of the splitting between the highest occupied molecular orbital (HOMO) and the HOMO-1. Thus, we compiled a data set of 29 experimentally determined couplings, called DBA29, which can now be used for testing any theoretical method for the calculation of electronic couplings. DBA29 contains different D/A groups, bridge types and lengths, and the couplings cover a large range of values (0.1–0.8 eV). We test FMR-B with the DBA29 data set using eleven density functionals as well as the Hartree-Fock (HF) method, and up to nine basis sets. Our results show that the FMR-B approach is robust with respect to the choice of method and basis set, and the best performance is obtained with range-separated functionals and compact basis sets. The paper is organized as follows. Section 2 contains a review of the theory and the computational details, Section 3 presents the results of the calculations and their analysis, and Section 4 summarizes our findings and provides concluding remarks.

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Theory and Computational Details

The FMR-B method requires a calculation on the entire system with a mean-field approach (HF or DFT) to obtain the Fock or Kohn-Sham (KS) matrix in atomic orbital (AO) basis. The AO basis is inherently localized, and it allows an easy partition of the basis functions into different molecular fragments. The Fock matrix is thus subdivided in blocks for the donor (D), bridge (B), and acceptor (A) part of the system: 

DD

DB

DA

F F  F  BD F= FBB FBA  F  FAD FAB FAA

     

(2)

The D −B and B −A blocks are then diagonalized to obtain the fragment orbitals, which are further orthogonalized between fragments. The couplings are computed by projecting the Fock matrix of the entire system on the fragment orbital space: B−A

Fb φ JijDBA ' FijDBA = φD−B i j =

(3)

γ D−B† εγ B−A i j

where ε is the diagonal matrix of the orbital energies of the full system, and γ D−B† /γ B−A i j are the projection vectors of the super-system orbitals onto the fragment orbitals. We refer the reader to Ref. 12 for more details. Although for symmetric systems one could calculate the coupling directly from half the energy difference between the HOMO and the HOMO-1, testing FMR-B and other methods on a benchmark set as DBA29 is important because one is often interested in non-symmetric systems where the half-splitting approach cannot be applied, and for which no experimental reference is available. The coupling evaluations were performed with a local code, while all super-system calculations were performed with a development version of the GAUSSIAN suite of programs. 21 The geometries of the test molecules were optimized with the B3LYP/6-311G(d,p) model

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chemistry, and they can be found in the supporting information (SI). We also optimized the structures with the B97D/6-311G(d,p) model chemistry, and noted only negligible difference in the coupling values, thus only the results with the B3LYP geometries are discussed. We tested HF and 11 generalized gradient approximation (GGA) functionals. The latter include a series of pure functionals: B97D, 22 M06L, 23 PBE, 24,25 and PW91; 26–28 a series of global hybrids with varying degree of HF exchange (in parentheses): B3LYP (20%), 29–31 M06 (27%), 32 PBE0 (25%), 33 BHandH (50%); 30,31,34,35 and a series of range-separated functionals with improved long-range behavior: 36 CAM-B3LYP (19% at short range and 62% at long range), 37 LC-ωPBE (0% at short range and 100% at long range), 38–40 and ωB97XD (40% at short range and 100% at long range); 41 the last three functionals are reported in the figures as CAM, LCPBE and wB97XD, respectively. The J calculations with all of these functionals were performed with the 6-311G(d,p) basis set, which provided the best performance in terms of computational cost/accuracy in our previous study. 12 We then chose one functional from each category (M06L, BHandH, and CAM-B3LYP) together with HF and the ubiquitous B3LYP to perform a more thorough basis set study. We examined nine basis sets from the Pople and Dunning families that include a growing number of polarization and diffuse functions: 6-311G, 6-311G(d,p), 6-311++G(d,p), 6-311G(df,pd), 6-311++G(df,pd), 6-311G(2d,2p), cc-pVDZ, aug-cc-pVDZ, and cc-pVTZ. 42–44 All experimental and theoretical values of the coupling are also available in the SI.

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Results and Discussion

The structures of the DBA29 data set are reported in Figure 1. DBA29 includes a variety of donor/bridge/acceptor moieties, classified according to the following notation. The number in bold font identifies the type of donor/acceptor groups: compounds 1-4 have C−C groups with different relative orientation, compounds 5 have C− −C groups, and compounds 6 have benzene groups. The number in parentheses identifies the distance between the donor and

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1(4a)

1(2a)

trans-1(3a)

a) a) 1(21(6

6(2a)

1(2ba) 1(6 ) 1(2c)

1(4a)

a) b) a) b) cis-1(3 1(4bb)) cis-1(3a) trans-1(3 cis-1(3 trans-1(3 trans-1(3

4(6a)

4(4)

6(4a)

4(4)

6(2a)

4(6b4(6 ) a)

1(2b) 1(2d) 1(4c) cis-1(3b)

b)b) 4(8 4(6

a) a) 6(66(4

c 1(2 1(2) e)

d)f 1(2 1(2 )

1(2f)

1(2e)

c) a b), R=CH a), R=H 2(4b), R=CH 2(4 2(4 3(3) b) ), R=H 2(4 1(4 3 3 1(4

5(3a)

5(4) 4(8b)

a) b 6(66(5 )

5(3ab)) 5(3

5(4)

6(5 1(5)b)

1(5)

Figure 1: Structures of the molecules in the DBA29 data set. The number in bold font identifies the type of donor/acceptor groups: compounds 1-4 have ethylene groups with different relative orientation, compounds 5 have acetylene groups, and compounds 6 have benzene groups. The number of bridging bonds is reported in parentheses (n), alongside a superscript for different bridge structures. acceptor groups in terms of single bonds. Finally, we use a superscript to distinguish between I) I) I) I) slightly of4(3) bridge structures. this set molecules 4(3types 5(3) 4(3 5(3 6(4),We R=H 5(3)chose 6(4I), R=CH 5(3 6(4)of , R=H 6(4I), R=CHbased on available 4(3)different 3

3

experimental data, and excluding systems with large uncertainty.

molec num 7 ha probabilmente 2 minimi… molec num 7 ha probabilmente 2 minimi… controllare controllare 0.8 Molec num 13 sembrerebbe simile 1, ma in realta’ e’ planare, e non … Molec num 13 sembrerebbe simile 1, ma in realta’ e’ planare, e non … 0.6 Molec 26 ha due minimi Molec 26 ha due minimi 0.4 0.2 0

1(2a ) 1(4a ) 1(6a ) 1(2b ) 1(2c ) 1(2d ) 1(2e ) 1(2f ) trans-1(3a ) cis-1(3a ) trans-1(3b ) cis-1(3b ) 1(4b ) 1(4c ) 2(4a ) 2(4b ) 3(3) 4(4) 4(6a ) 4(6b ) 4(8a ) 5(3a ) 5(4) 5(3b ) 6(2a ) 6(4a ) 6(6a ) 6(5b ) 1(5b )

Coupling (eV)

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

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Figure 2: Experimental electronic couplings (eV) for the compounds in the DBA29 data set. See text for the references from where the values were taken. The range of couplings is pretty large, from 0.1 to 0.8 eV, as shown in Figure 2, which illustrates the complex interplay of through-bond and trough-space interactions and the tuning effect of the bridge structure. Such interplay is exemplified by molecules 1(2a , 4a , 6a ), 6


3(3

5(3b)

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where the coupling slightly increases at short distances while decreasing at longer distances (J = 0.43, 0.44, and 0.16 eV, respectively). 45,46 Molecules 1(2b-f ) are (bi)cyclic dienes structurally related to molecule 1(2a ) where the coupling is significantly modulated by the nature of the bridge; J = 0.50, 47 0.32, 47 0.18, 47 0.45, 48 and 0.64 eV, 48 respectively. For instance, in molecule 1(2f ) the presence of the alkene group on the bridge results in a significant increase of the coupling, even without explicit π conjugation, due to the fact that the energy levels on the bridge are close to the site energies of the D/A groups. Trans- and cis-1(3a , 3b ) are isomers that highlight the strong effect of the bridge, as the trans isomers have larger couplings (0.50 eV for 1(3a ) and 0.80 eV for 1(3b )) than the cis isomers (0.15 eV for 1(3a ) and 0.13 eV for 1(3b )) even if in the latter the D/A pair is spatially closer to each other. The extra methylene bridging group in the 1(3b ) molecules also increases the value of the couplings because of its electron-donating properties. 49–51 The 1(4b ,4c ) compounds are structurally similar to 1(4a ), but the withdrawing groups in the bridge of 1(4b ), i.e., the O centers, reduce the coupling (0.29 eV) 52 compared to 1(4a ), while the complicated bridge structure of 1(4c ) produces a coupling (0.43 eV) 49,53 similar to that of 1(4a ). Such effect is likely due to a combination of reduced electron donation on the top part of the bridge (induced by the extra bond between the methylene groups), and an increase in electron donation of the carbonyl groups in the bottom part of the bridge. The complicated bridge structure makes 1(4c ) a hard case for many theoretical methods, as discussed later in this section. Molecules 2(4a,b ) are dienes connected by four propyl chains with and without additional substitutes. The couplings are similar and large (0.77 and 0.74 eV, respectively), 54 given that the electron has multiple through-bond pathways for the transfer. Molecule 3(3) has an interesting cage structure that allows the D/A groups to be in relatively close proximity. However, since the bridge is not spatially in-between the two groups, the coupling is small (0.20 eV) 55,56 because the interaction is mainly through-space. Molecules 4(4), 4(6a ), 4(6b ), 4(8b ) are faced bridged dienes. The coupling for the first one is large (0.63 eV) 57 probably due to a favorable combination of through-space and through-bond effects. The coupling for 4(6a ) is considerably

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smaller (0.09 eV) due to the longer distance between the D/A moieties; adding a methylene group on the bridge (4(6b )) increases the coupling (0.26 eV), similar to previous cases. Thus, the coupling of 4(8b ) (0.14 eV) is smaller than that of 4(6b ) but it is still larger than that of 4(6a ). 46,58,59 The molecules in the 5 subset have a different D/A group, i.e., acetylenes, 49 connected through C or Si centers. The coupling for 5(3a ) is smaller (0.20 eV) 60 than for 5(4) (0.80 eV) 61 even if in the latter case the A/D groups are further apart. The reason is that the D/A groups in 5(3a ) are not well aligned due to steric effects of the chains, while they face each other favorably in 5(4) thanks to more flexibility in the bridging chains. The replacement of the C-C bridge with a Si-Si bridge in 5(3b ) results in a significant increase of the coupling (0.46 eV) due to the larger polarizability of Si compared to C. 62 Molecules 6(2a ,4a ,6a ) are dibenzo analogues of molecules 1(na ), and they have smaller couplings: 0.30, 0.28, and 0.13 eV, respectively. 45,46,59 This may be related to the aromatic effect in benzene, which stabilizes the electron on the ring thus making the transfer less favorable. The same trend is shown by 6(5b ) and its diene analogue 1(5b ), with J = 0.16 and 0.21 eV. 59 We begin our analysis by comparing the results for all functionals with the 6-311G(d,p) basis set. The latter provided the best results at the lowest computational cost in our previous study, 12 and its use is justified also in this case as shown below. Correlation plots between experiment and theory are shown in Figure 3, where the dashed black line represents ideal correlation, and the full red line represents the best linear fit for the data. Thus, when the red line is above the black line, the theoretical method tends to underestimate experiment, and vice versa. The plots in Figure 3 show that, on average, DFT tends to underestimate experiment while HF tends to overestimate it. This underestimation is larger for pure functionals, and it becomes less pronounced with increasing amount of HF exchange, which indicates an error compensation between the approximate exchange-correlation functionals and the HF exchange. Indeed, the global hybrid functional with the largest amount of HF exchange, BHandH (50%), has among the best correlation with experiment. The range-separated functionals all perform very well, especially LC-ωPBE (which has correct

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Figure 3: Correlation plots of calculated and experimental electronic couplings (eV) for all methods and the 6-311G(d,p) basis set. long-range behavior), also indicating that the proper description of the electron density at long distance is very important. This is not surprising given that we are trying to reproduce the interaction between relatively distant donor and acceptor groups. At the same time, the poor performance of HF indicates that correlation effects are important and need to be taken into account. The performance of functionals that belong to the same class is very similar, and it seems to be regulated mostly by the amount of HF exchange: pure functionals provide similar results, global hybrids work increasingly better with more HF exchange, and range-separated functionals work best (for the latter, the exact amount of HF exchange at long range seems to be less relevant as long as its large). The one molecule where J is severely overestimated by the pure functionals is 1(4c ), while HF and BHandH underesti-

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mate it. This may be due to delocalization effects of the carbonyl groups on the bridge. If we removed this molecule from the test set, this would actually worsen the performance of all of these methods. For the other functionals, which have a more balanced description of correlation and exchange effects, this molecule is not problematic. 1.00

0.95

R2 0.90 0.85

0.80 B9 7 M D 06 L PB PW E 9 B3 1 L BH YP an dH M 06 PB E0 CA M LC P w BE B9 7X D HF

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

Figure 4: R2 correlation coefficients for the linear fits of the data in Figure 3 (all methods and the 6-311G(d,p) basis set). Figure 4 reports the correlation coefficients R2 for all methods, which gives a measure of the data spread around the linear fits in Figure 3. All methods have good correlation, as R2 > 0.80 across the board. In particular, correlation is high (0.95 and above) for the global hybrid and range-separated functionals. HF, B97D, and M06L have values of R2 between 0.85 and 0.90, and PBE and PW91 are slightly below 0.85. Figure 5 collects the results of a statistical analysis for the absolute error (in eV) and the relative absolute error (in %), i.e. mean and max errors, standard deviation (σ) and root mean square (RMS). The mean (relative) error is below 0.1 eV (< 25%) for all methods, and below 0.05 eV (∼ 10%) for the best methods (BHandH and the range-separated functionals). The largest error (Max in Figure 5) is driven by molecule 1(4c ) for the pure functionals and HF, but not for the other methods. For the range-separated functionals the max (relative) error is close to 0.1 eV (40%). The RMS and σ plots indicate that the spread of the data is small for global hybrids and range-separated functionals (10% for σ and < 20% for the RMS). The relative 10


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B97D B3LYP CAM

0.10

M06L BHandH LCPBE

25

PW91 PBE0 HF

20 15 10

0.05

5

0.00

0 Mean

σ

Mean

σ

0.3

100 80

0.2

60 40

0.1

20

0.0

Rel. Error (%)

Abs. Error (eV)

PBE M06 wB97XD

Rel. Error (%)

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


Abs. Error (eV)

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0 Max

RMS

Max

RMS

Figure 5: Statistical analysis of the absolute error (eV, left plots) and the relative absolute error (%, right plots) for all methods and the 6-311G(d,p) basis set. The plots include the average error (Mean), the maximum error (Max), the standard deviation (σ), and the error root mean square (RMS). errors for HF are larger than the other methods because HF has difficulties with molecules with small J values, e.g., for cis-1(3b ) J exp = 0.13 eV, and the calculated value with HF is 0.23 eV, which represents an 85% overestimation (obviously, the difference is small in terms of absolute error). The data in Figures 4–5 confirms that functionals belonging to the same class tend to perform similarly and that larger amounts of HF exchange are beneficial for improving accuracy. The evaluation of electronic couplings can be sensitive to the choice of basis sets. 12,15,63,64 Therefore, we also perform a basis set analysis with a representative method from each category (B3LYP and BHandH are both included because of the difference in HF exchange amount). The selected basis sets are presented in Section 2, and they were chosen to explore the effect of polarization and diffuse functions. The sets belong to two families, Pople’s and Dunning’s, where the former tend to have a more compact form. The linear correlation plots for the five methods and nine basis sets are reported in Figure 6. We have color-coded

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d) d)

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

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Figure 6: Correlation plots of calculated and experimental electronic couplings (eV) for a subset of methods and all basis sets. Green dots: molecules with two-bond D/A separation; pink dots: molecules with three-bond D/A separation; blues dots: molecules with four-bond D/A separation and higher; orange dot: problematic 1(4c ) case. 12


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the molecules so that we can better discuss the performance of each basis set: the green dots represent the molecules with two-bond separation between donor and acceptor, the pink dots represent three-bond separation, the blue dots represent four-bond separation and higher, and the only orange dot represents the problematic 1(4c ) case. The color-coding clearly shows that the compounds with two-bond separation are the most difficult cases both for too-small basis sets (6-311G with no polarization or diffuse functions) and for large basis sets (with too many polarization or diffuse functions). This result is expected since the D/A groups are very close in these molecules, and large basis sets produce too much overlap so that the Fock matrix partition discussed in Section 2 fails. Nevertheless, this is not a considerable issue for the FMR-B method even with relatively large basis sets, i.e., 6-311++G(d,p), 6-311G(df,pd), and 6-311++G(df,pd), when used with the best functionals (BHandH and the range-separated CAM-B3LYP). The three-bond separated systems are only a problem for the largest basis sets, i.e., 6-311G(2d,2p), aug-cc-pVDZ and cc-pVTZ, while the couplings for the molecules with larger D/A separation are well reproduced with all basis sets. The coupling for 1(4c ) is difficult with 6-311G and cc-pVTZ using B3LYP and CAM-B3LYP. Good performance is obtained in certain cases, for instance with the M06L/6-311G(2d,2p) combination (which otherwise provides a poor performance), and it can be attributed to fortuitous error cancellation. Each functional and HF tend to perform in the same way, on overage, with all basis sets: M06L and B3LYP tend to underestimate experiment, HF tends to overestimate it, and BHandH and CAM-B3LYP are very close to the ideal linear correlation. An interesting observation is that the poor performance of a basis set is only due to the small distance between the donor and acceptor groups, and not to the magnitude of the coupling. In other words, both small and large couplings are well reproduced with these basis sets as long as the D/A distance is larger than two bonds. A statistical analysis for the absolute error (in eV) and the relative absolute error (in %) is reported in Figures 7 and 8, respectively. All four parameters (mean and max error, σ and RMS) further support the analysis above: all methods behave similarly with all

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0.20

6-311G(d,p) 6-311++G(df,pd) aug-cc-pVDZ

6-311++G(d,p) 6-311G(2d,2p) cc-pVTZ

0.80 0.60

0.15

0.40

0.10 0.05

0.20

0.00

0.00

0.25

0.20

0.20

0.15

0.15

0.10

0.10 0.05

0.05

0.00

0.00 M06L B3LYP BH&H CAM

M06L B3LYP BH&H CAM

HF

σ (eV)

RMS (eV)

Mean (eV)

6-311G 6-311G(df,pd) cc-pVDZ

Max (eV)

HF

Figure 7: Statistical analysis of the absolute error (eV) for a subset of the methods and all basis sets. The plots include the average error (Mean), the maximum error (Max), the standard deviation (σ), and the error root mean square (RMS).

6-311G 6-311G(df,pd) cc-pVDZ

400 300

40

200

20

100

0

0

100

80

80

60

60

40

40

20

20

σ (%)

RMS (%)

Mean (%)

60

500

6-311G(d,p) 6-311++G(d,p) 6-311++G(df,pd) 6-311G(2d,2p) aug-cc-pVDZ cc-pVTZ

Max (%)

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

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0

0 M06L B3LYP BH&H CAM

M06L B3LYP BH&H CAM

HF

HF

Figure 8: Statistical analysis of the relative absolute error (%) for a subset of the methods and all basis sets. The plots include the average error (Mean), the maximum error (Max), the standard deviation (σ), and the error root mean square (RMS).

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basis sets, a poor performance is consistently obtained with 6-311G, 6-311G(2d,2p), aug-ccpVDZ, and cc-pVTZ, which are either too small or contain functions that are too diffuse. The FMR-B method is otherwise rather insensitive to the choice of basis set, with average errors < 20% and standard deviations around 10% (except for HF). Given these results, the best compromise between accuracy and computational cost is achieved with the 6-311G(d,p) and cc-pVDZ basis sets.

4

Summary and Conclusions

In this work, we present a benchmark of our recently proposed method, FMR-B, for the calculation of electronic couplings in bridged systems. We compile a test set of 29 molecules, DBA29, for which the coupling is evaluated from experiment (see Figure 1). This test set includes different donor/acceptor groups, bridge types, and bridge lengths, so that a large range of coupling values are includes (from < 0.1 eV to 0.8 eV, see Figure 2). DBA29 is general and can be used to test any theoretical method. With this set, we test 11 density functionals and the HF method. The functionals include all major classes: pure, global hybrid, and range-separated GGA functionals with varying amount of HF exchange. Finally, we also investigate the basis set effect by repeating the calculations for a member of each class of methods with nine basis sets that include an increasing number of polarization and diffuse functions. We perform linear fits of the results and present correlation plots (see Figures 3, 4, and 6) together with a complete statistical analysis of the data (see Figures 5, 7, and 8). Our results indicate that the best agreement with experiment is obtained with rangeseparated functionals, because they can better describe the long-range interaction between the donor and acceptor groups. Global hybrids tend to underestimate experiment, and a large amount of HF exchange tends to improve the performance. This effect is likely due to error compensation given that HF tends to overestimate experiment (lack of electron correlation) while pure functionals tend to underestimate experiment (poor long-range behavior).

15



Functionals within the same class behave similarly, and meta-GGAs do not outperform regular GGAs. Global hybrid and range-separated functionals also provide a smaller data spread (R2 > 0.95 and σ ∼ 10%). Note that experimental errors are not always exactly known, thus imperfect correlation between theoretical and measured data may be due to limitations in both approaches, which are not easily distinguishable. BHandH works well for most cases, with a performance similar to that of range-separated functionals, but it does not reproduce the experimental result for the difficult 1(4c ) case, where the large error inherited from HF is not well compensated. HF is the method with the largest standard deviation both for the absolute and for the relative errors. These results are consistent with previous findings for other methods to compute electronic couplings. 65 It is worth pointing out that although pure functionals do not perform as well as range-separated functionals, their overall performance is still reasonable, considering the uncertainty in the experimental data: R2 values are close to 0.90, the mean (relative) error is below 0.1 eV (∼ 20%), and σ is slightly above 0.05 eV (∼ 20%). This is an important result because these functionals are likely to be the preferred choice for larger systems due to computational limitations. FMR-B is virtually insensitive to the choice of basis set for systems where the D/A groups are four or more bonds apart, independently of the magnitude of the coupling. For molecules where these groups are closer, basis sets with diffuse d functions and higher (e.g., 6-311G(2d,2p), aug-cc-pVDZ, and cc-pVTZ) tend to perform poorly, but the other basis sets perform equally well as long as they have polarization functions. Basis sets with s and p diffuse functions do not present significant issues. In any case, all of the methods perform qualitatively in the same way with all basis sets, including the larger ones. The best compromise between accuracy and computational cost is reached with the 6-311G(d,p) and cc-pVDZ sets. In conclusion, this work shows that FMR-B is a reliable method to compute electronic coupling in bridged systems, and it can be effectively used to study non-symmetrical systems where experimental values are not available. Furthermore, the DBA29 data set can be

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utilized as a benchmark reference to test new method developments in the field.

Supporting Information Available See supplementary material for the B3LYP/6-311G(d,p) optimized structures of the molecules in the DBA29 data set, the experimental values of the electronic couplings, and the calculated values with all methods and basis sets.

This material is available free of charge via

the Internet at http://pubs.acs.org/.

Acknowledgement Support from startup funds provided by the University of Kansas is gratefully acknowledged.

References (1) Paddon-Row, M. N. Superexchange-mediated charge separation and charge recombination in covalently linked donor–bridge–acceptor systems. Austr. J. Chem. 2003, 56, 729–748. (2) Van Dyck, C.; Ratner, M. A. Molecular Rectifiers: A New Design Based on Asymmetric Anchoring Moieties. Nano Lett. 2015, 15, 1577–1584. (3) Van Dyck, C.; Ratner, M. A. Molecular Junctions: Control of the Energy Gap Achieved by a Pinning Effect. J. Phys. Chem. C 2017, 121, 3013–3024. (4) Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Brédas, J.-L. Charge Transport in Organic Semiconductors. Chem. Rev. 2007, 107, 926–952. (5) Oberhofer, H.; Reuter, K.; Blumberger, J. Charge Transport in Molecular Materials: An Assessment of Computational Methods. Chem. Rev. 2017, 117, 10319–10357.

17



(6) Marcus, R. A. On the theory of oxidation-reduction reactions involving electron transfer. I. J. Chem. Phys. 1956, 24, 966–978. (7) Marcus, R. A. Electrostatic free energy and other properties of states having nonequilibrium polarization. I. J. Chem. Phys. 1956, 24, 979–989. (8) Jordan, K. D.; Paddon-Row, M. N. Encyclopedia of Computational Chemistry; John Wiley and Sons, Ltd, 2002. (9) Valeev, E. F.; Coropceanu, V.; da Silva Filho, D. A.; Salman, S.; Brédas, J.-L. Effect of electronic polarization on charge-transport parameters in molecular organic semiconductors. J. Am. Chem. Soc. 2006, 128, 9882–9886. (10) Baumeier, B.; Kirkpatrick, J.; Andrienko, D. Density-functional based determination of intermolecular charge transfer properties for large-scale morphologies. Phys. Chem. Chem. Phys. 2010, 12, 11103–11113. (11) Kondov, I.; Cízek, M.; Benesch, C.; Wang, H.; Thoss, M. Quantum dynamics of photoinduced electron-transfer reactions in dye-semiconductor systems: first-principles description and application to coumarin 343-TiO2. J. Phys. Chem. C 2007, 111, 11970– 11981. (12) Biancardi, A.; Martin, S. C.; Liss, C.; Caricato, M. Electronic Coupling for DonorBridge-Acceptor Systems with a Bridge-Overlap Approach. J. Chem. Theory Comput. 2017, 13, 4154–4161. (13) Biancardi, A.; Caricato, M. Evaluation of Electronic Coupling in Solids from Ab Initio Periodic Boundary Condition Calculations: The Case of Pentacene Crystal and Bilayer Graphene. J. Phys. Chem. C 2016, 120, 17939–17948. (14) Biancardi, A.; Caraiani, C.; Chan, W.-L.; Caricato, M. How the Number of Layers and

18


Page 18 of 26

Page 19 of 26 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


Relative Position Modulate the Interlayer Electron Transfer in π-Stacked 2D Materials. J. Chem. Phys. Lett. 2017, 8, 1365–1370. (15) Gillet, N.; Berstis, L.; Wu, X.; Gajdos, F.; Heck, A.; de la Lande, A.; Blumberger, J.; Elstner, M. Electronic Coupling Calculations for Bridge-Mediated Charge Transfer Using Constrained Density Functional Theory (CDFT) and Effective Hamiltonian Approaches at the Density Functional Theory (DFT) and Fragment-Orbital Density Functional Tight Binding (FODFTB) Level. J. Chem. Theory Comput. 2016, 12, 4793–4805. (16) Ramos, P.; Pavanello, M. Quantifying environmental effects on the decay of hole transfer couplings in biosystems. J. Chem. Theory Comput. 2014, 10, 2546–2556. (17) Kubas, A.; Hoffmann, F.; Heck, A.; Oberhofer, H.; Elstner, M.; Blumberger, J. Electronic couplings for molecular charge transfer: Benchmarking CDFT, FODFT, and FODFTB against high-level ab initio calculations. J. Chem Phys. 2014, 140, 104105. (18) Mavros, M. G.; Voorhis, T. V. Communication: CDFT-CI couplings can be unreliable when there is fractional charge transfer. J. Chem Phys. 2015, 143, 231102. (19) Ramos, P.; Papadakis, M.; Pavanello, M. Performance of Frozen Density Embedding for Modeling Hole Transfer Reactions. J. Phys. Chem. B 2015, 119, 7541–7557. (20) Kim, H.; Goodson, T.; Zimmerman, P. M. Density Functional Physicality in Electronic Coupling Estimation: Benchmarks and Error Analysis. J. Phys. Chem. Lett. 2017, 8, 3242–3248. (21) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.;

19



Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A.; Jr.,; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian Development Version, Revision I.09. Gaussian Inc. Wallingford CT 2016. (22) Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comp. Chem. 2006, 27, 1787–1799. (23) Zhao, Y.; Truhlar, D. G. A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J. Chem. Phys. 2006, 125, 194101. (24) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (25) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett. 1997, 78, 1396–1396. (26) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1992, 46, 6671–6687. (27) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1993, 48, 4978–4978. 20


Page 20 of 26

Page 21 of 26 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


(28) Perdew, J. P.; Burke, K.; Wang, Y. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Rev. B 1996, 54, 16533– 16539. (29) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652. (30) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785–789. (31) Vosko, S. H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 1980, 58, 1200–1211. (32) Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc. 2008, 120, 215–241. (33) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158–6170. (34) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864– B871. (35) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133–A1138. (36) Cencek, W.; Szalewicz, K. On asymptotic behavior of density functional theory. J. Chem. Phys. 2013, 139, 024104. (37) Yanai, T.; Tew, D. P.; Handy, N. C. A new hybrid exchange-correlation functional using

21



the Coulomb-attenuating method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51 – 57. (38) Zhao, Y.; Truhlar, D. G. A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J. Chem. Phys. 2006, 125, 194101. (39) Vydrov, O. A.; Scuseria, G. E. Assessment of a long-range corrected hybrid functional. J. Chem. Phys. 2006, 125, 234109. (40) Vydrov, O. A.; Heyd, J.; Krukau, A. V.; Scuseria, G. E. Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals. J. Chem. Phys. 2006, 125, 074106. (41) Chai, J.-D.; Head-Gordon, M. Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615– 6620. (42) Jr., T. H. D. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (43) Kendall, R. A.; Jr., T. H. D.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796– 6806. (44) Woon, D. E.; Jr., T. H. D. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358–1371. (45) Balaji, V.; Ng, L.; Jordan, K.; Paddon-Row, M.; Patney, H. Study of long-range. pi.*,. pi.* interactions in rigid molecules by electron transmission spectroscopy. J. Am. Chem. Soc. 1987, 109, 6957–6969.

22


Page 22 of 26

Page 23 of 26 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


(46) Jordan, K. D.; Paddon-Row, M. N. Analysis of the interactions responsible for longrange through-bond-mediated electronic coupling between remote chromophores attached to rigid polynorbornyl bridges. Chem. Rev. 1992, 92, 395–410. (47) Heilbronner, E.; Martin, H.-D. The π-Orbital Sequence in Norbornadiene and Related Hydrocarbons. Helv. Chim. Acta 1972, 55, 1490–1502. (48) Haselbach, E.; Rossi, M. Physical and Chemical Interaction of Homoconjugated Dienes with Tetracyanoethylene. Helv. Chim. Acta 1976, 59, 2635–2652. (49) Gleiter, R.; Schaefer, W. Interactions between nonconjugated. pi.-systems. Acc. Chem. Res. 1990, 23, 369–375. (50) Gleiter, R.; Heilbronner, E.; Hekman, M.; Martin, H.-D. π-Orbital-Wechselwirkungen through space und through bond in Tricyclo [4.2. 0.0] octadienen. Chem. Ber. 1973, 106, 28–34. (51) Paquette, L. A.; Doecke, C. W.; Klein, G. Regiocontrolled hydrogenolysis of strained. sigma. bonds. Application to the synthesis of syn-and anti-tricyclo [4.2. 1.12, 5] decane and two elusive cyclopentadiene dimers. J. Am. Chem. Soc. 1979, 101, 7599–7604. (52) Eckert-Maksć, M.; Maksimović, L.; Hodošček, M. Electronic structure of fused 7oxanorbornenes. Photoelectronspectroscopic study. Tetrahedron Lett. 1993, 34, 4245– 4248. (53) Bartetzko, R.; Gleiter, R.; Muthard, J. L.; Paquette, L. A. Long-range electronic transmission in conformationally rigid. alpha.-diketones. J. Am. Chem. Soc. 1978, 100, 5589–5594. (54) Lange, H.; Schäfer, W.; Gleiter, R.; Camps, P.; Vázquez, S. π- π-Interactions in Pentacyclo-[8.2. 1.1. 2, 51.4, 718, 11] hexadeca-1, 7-diene. J. Org. Chem. 1998, 63, 3478–3480. 23



(55) Marchand, A. P.; Kaya, R.; Baker, A. D. A photoelectron spectroscopic study of proximity effects in 8, 11-disubstituted pentacyclo (5.4.0.02,6 .03,10 .05,9 ) Undecanes. Tetrahedron Lett. 1984, 25, 795–798. (56) Shen, K.-w.; Kuebler, N. A. Synthesis, reactions, and photoelectron spectrum of 8, 11-dimethylenepentacyclo [5.4.0.02,6 .03,10 .05,9 ] tridecane. Tetrahedron Lett. 1973, 14, 2145–2148. (57) Martin, H.-D.;

Schwesinger, R. Notiz über eine beobachtete through-bond-

Wechselwirkung über vier σ-Bindungen. Chem. Ber. 1974, 107, 3143–3145. (58) Prinzbach, H.; Sedelmeier, G.; Martin, H.-D. Geometrical Dependence of [2π+ 2σ] Photocycloaddition. Ang. Chem. Int. Ed. 1977, 16, 103–104. (59) Paddon-Row, M. N.; Patney, H. K.; Peel, J. B.; Willett, G. D. The observation of long-range benzene–benzene interactions. J. Chem. Soc., Chem. Commun. 1984, 0, 564–566. (60) Bieri, G.; Heilbronner, E.; Kloster-Jensen, E.; Schmelzer, A.; Wirz, J. Electronic States of 1, 5-Cyclooctadiyne Radical Cation and of Related Systems: The electronic structure of cis-bent carbon-carbon triple bonds. Helv. Chim. Acta 1974, 57, 1265–1283. (61) Gleiter, R.; Karcher, M.; Jahn, R.; Irngartinger, H. Darstellung, Struktur und Eigenschaften von 1, 6-Cyclodecadiin. Vergleich mit 1, 5-Cyclooctadiin und 1, 7Cyclododecadiin. Chem. Ber. 1988, 121, 735–740. (62) Gleiter, R.; Schaefer, W.; Sakurai, H. Evidence for a strong. sigma.,. pi. interaction in 3, 4, 7, 8-tetrasilacycloocta-1, 5-diyne and 3, 4, 7, 8, 11, 12-hexasilacyclododeca-1, 5, 9-triyne. J. Am. Chem. Soc. 1985, 107, 3046–3050. (63) Berstis, L.; Baldridge, K. K. DFT-based Green’s function pathways model for prediction

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of bridge-mediated electronic coupling. Phys. Chem. Chem. Phys. 2015, 17, 30842– 30853. (64) Herrmann, C.; Solomon, G. C.; Subotnik, J. E.; Mujica, V.; Ratner, M. A. Ghost transmission: How large basis sets can make electron transport calculations worse. J. Chem. Phys. 2010, 132, 024103. (65) Migliore, A. Full-electron calculation of effective electronic couplings and excitation energies of charge transfer states: application to hole transfer in DNA π-stacks. J. Chem. Phys. 2009, 131, 114113.

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Graphical TOC Entry 0.8 Jexp. (eV)

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 26 of 26

A J ijDBA = γ iD− B†εγ B− j

LC-!PBE

0.6

1(2a)

trans-1(3a)

0.4

1(4a)

a) a) 1(21(6

1(2ba) 1(6 ) 1(2c)

1(4a)

a) b) a) b) cis-1(3 1(4bb)) cis-1(3a) trans-1(3 cis-1(3 trans-1(3 trans-1(3

4(6a)

4(4)

4(4)

) a) 4(6b4(6

1(2b) 1(2d) 1(4c)

cis-1(3b)

c 1(2 1(2)e)

5(3a)

b)b) 4(8 4(6

1(2 1(2d)f)

c) a b), R=CH 2(4 b) ), R=H 2(4 1(4 3 1(4

5(4) 4(8b)

1(2f)

1(2e)

a), R=H 2(4b), R=CH 2(4 3(3) 3

5(3ab)) 5(3

5(4)

0.2 6(4a)

6(2a)

0

0

6(2a)

a) a) 6(66(4

a) b 6(66(5 )

6(5 1(5)b)

1(5)

0.2 0.4 0.6 0.8 Jcalc. (eV) 4(3)

4(3I)

4(3)5(3)

I) I) 4(3 5(3

6(4), R=H 5(3) 6(4I), R=CH 5(3I)3

6(4), R=H 6(4I), R=CH3

molec num 7 ha probabilmente 2 minimi… molec num 7 ha probabilmente 2 minimi… controllare controllare Molec num 13 sembrerebbe simile 1, ma in realta’ e’ planare, e non … Molec num 13 sembrerebbe simile 1, ma in realta’ e’ planare, e non … Molec 26 ha due minimi Molec 26 ha due minimi

26


3(3)

5(3b)

A Benchmark Study of Electronic Couplings in DonorâBridge

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