Electronic Coupling for Donor-Bridge-Acceptor Systems with a Bridge

Article pubs.acs.org/JCTC

Electronic Coupling for Donor-Bridge-Acceptor Systems with a Bridge-Overlap Approach Alessandro Biancardi,† Seth C. Martin,† Cameron Liss,†,‡ and Marco Caricato*,† †

Department of Chemistry, University of Kansas, 1251 Wescoe Hall Drive, Lawrence, Kansas 66045, United States Department of Biological, Chemical and Physical Sciences, Roosevelt University, 430 South Michigan Avenue, Chicago, Illinois 60605, United States

‡

S Supporting Information *

ABSTRACT: Understanding the modulation of the electronic coupling in donor− acceptor systems connected through an aliphatic bridge is crucial from a fundamental point of view as well as for the development of organic electronics. In this work, we present a first-principles approach for the calculation of the electronic coupling (or transfer integrals) in such systems via a block-diagonalization of the Fock/Kohn−Sham matrix of the supersystem, followed by a projection on the basis of the fragment orbitals of the donor and acceptor groups. The strength of the approach is that the bridge is shared by the donor and acceptor blocks in the diagonalization step, so that throughspace and through-bond couplings are obtained simultaneously. The method is applied to two test sets: a series of fused-ring bridged systems and G(T)nG DNA oligomers. The results for the first set are compared to experiment and show an average error lower than 10%. For the DNA set, we show that the coupling may be significantly larger (and the decay with length slower) when the entire backbone is included.

1. INTRODUCTION Charge transfer plays a key role in many biological, chemical, and optoelectronic processes.1−4 Long-range charge transfer in the case of molecular systems has received increasing attention in recent decades due to their potential use as nanowires.5,6 Much work has also been done on DNA because of its ability to form self-assembled nanostructures;7 for instance, Barton and co-workers showed a significant charge transport even through a distance of 34 nm.5 As proposed in the pioneering work of Aviram and Ratner in the case of molecular rectifiers,8 the use of individual molecules as components of electronic circuits may be the ultimate step toward faster and smaller electronic devices, since the transfer happens between molecular fragments that are just a few nanometers apart. Nevertheless, this goal is still out of reach because of the incomplete understanding of the transfer process in single molecule devices.9 Different mechanisms have been proposed to account for the conductivity in nanowires built with a donor-bridgeacceptor structure (D-B-A), namely direct and multistep transfer.6,10,11 In the former case, the energy levels of the bridge are distant enough from those of the donor and the acceptor groups that the occupation of the bridge orbitals during the transfer is only virtual,12 whereas in the latter case the bridge orbitals are close enough in energy to be somewhat occupied during the transfer.10 In addition, several mechanisms have been proposed to account for the subtle impact of the structural dynamics on the transfer properties (see for instance refs 13−16). A key ingredient for understanding these systems and designing new materials is the development of theoretical methods to simulate the charge transfer. Although an explicit © XXXX American Chemical Society

quantum dynamical description is in principle desirable, this is not often feasible or even necessary because of the large computational cost and the availability of more approximate approaches. These approaches are based on the definition of an electronic coupling (J) between D/A fragment orbitals, which is a fundamental term in Marcus’ formula for the transfer rate17−19 k=

|J |2 ℏ

⎛ (ΔE − λ)2 ⎞ π exp⎜ − ⎟ 4λkBT ⎠ λkBT ⎝

(1)

where ΔE is the site energy difference, and λ is the reorganization energy. Although the charge transfer rate is given by the sum of through-space and through-bond contributions (TS and TB, respectively),20−23 the latter is mainly responsible for the enhanced transfer through nanowires.6,24 However, the lack of a well-defined boundary between D and A groups in D-B-A systems poses theoretical and practical challenges, and thus most efforts have been devoted to unbound D/A systems.25,26 Indeed, the bridge is not only a mere spacer constraining the D/A distance, but it actively modulates the short- and longrange transfer. For large distances between the D/A pair, where the TS contribution is weak, the transfer is dominated by the TB bridge-mediated contribution, and the total coupling displays a near-exponential decay27 JDA (R ) ∝ exp( −βR )

(2)

Received: April 27, 2017 Published: July 24, 2017 A

DOI: 10.1021/acs.jctc.7b00431 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

focus in particular on the methods of Brédas25 and Thoss,31 as they represent the basis for our extension to bridged systems. We also review the approach of Larsson,36,41,42 as it is used for the numerical comparisons in Section 3. 2.1. From Isolated to Bridged Fragments. The approach of Brédas and co-workers,25 later revisited in ref 43, is based on the Fock/Kohn−Sham (KS) matrix reconstruction (FMR) in the basis of isolated fragment orbitals. This method has been extensively applied to noncovalently bonded D/A molecules28,29 and periodic systems.26,30 The coupling is related to the off-diagonal elements of the projected Fock/KS matrix

where R is the D/A effective distance. Conversely, at short distances where the TS contribution is large, the change of the J magnitude is nonexponential, and the transfer maximum may occur at some intermediate R. Several approaches have been proposed in recent years for the fast evaluation of the electronic coupling: for instance, the work of Brédas and co-workers,28,29 which we extended to periodic boundary conditions (PBC),26,30 is particularly suited for nonbonded fragments; Thoss and co-workers31 used a similar approach, based on the block-diagonalization of the partitioned Fock/KS matrix, to study systems where the donor and acceptor groups are directly covalently bonded; Van Voorhis and co-workers32 used an approach based on constrained density functional theory (CDFT); Mo, Gao, and co-workers33 used a valence bond approach based on blocklocalized DFT orbitals;34 Pavanello and co-workers35 proposed an approach based on frozen density embedding (FDE). Recently, Gillet et al.36 reported a comprehensive comparison among different methods for the calculation of the electronic coupling, namely constrained density functional theory (CDFT),37 fragment-orbital density functional tight-binding (DFTB),38 and an effective Hamiltonian technique39 (Heff) relying on the Pipek-Mezey orbital localization.40 This comparison showed qualitatively consistent trends with respect to experiment, with the best performance offered by the Heff method. In this work, we focus on the development of a novel approach for the calculation of the electronic coupling in bridged systems, based on the definition of fragment orbitals obtained via block diagonalization of the Fock matrix where the orbital contribution of the bridge is shared between the D and A fragments. This approach can be seen as a combination of the Brédas’25 and Thoss’ approaches31 that allows for a unified treatment of the TS and TB contributions to the coupling. The paper is organized as follows. A review of relevant previous approaches, followed by the presentation of our method, is presented in Section 2. The performance of the methods is tested against experimental and alternative theoretical values in Section 3. Final considerations are reported in Section 4.

FijDA = ⟨ϕi D|F |̂ ϕjA ⟩

(4)

where F̂ is the Fock operator, and is the orbital k of fragment X. By expanding the orbitals in a common atomic basis set (see refs 26 and 43 for additional details), eq 4 can be rewritten in matrix form ϕXk

FijDA = γiD †εγjA

(5)

where ε is the diagonal matrix of the supersystem orbital energies, and γXk is the projector onto the fragment orbital k of fragment X γkX = C†CkX

(6)

where C and CXk are the matrices of the supersystem and fragment molecular orbital (MO) coefficients, respectively. The supersystem MOs are orthonormalized with a Lö wdin orthogonalization.44 In general, the fragment orbitals are also not orthogonal, where the overlap is given by SijDA = γiD †γjA

(7)

Thus, an additional Löwdin orthogonalization is performed to ensure the orthogonality between the fragment orbitals:25 F̃ DA = (SDA )−1/2 FDA (SDA )−1/2

(8)

Finally, the off-diagonal elements of the transformed Fock matrix represent the electronic coupling:

2. THEORY The calculation of the couplings in bridged systems is still challenging, and several approaches have been proposed having as key differences the construction of the D/A fragment orbitals representing the initial and final states of the transfer and the specific theoretical framework used to obtain the working equations.36 Within the single-particle approximation and Koopmans’ theorem, i.e., assuming only one relevant orbital per fragment and neglecting orbital-relaxation effects, the hole/ particle transfer takes place between the highest-occupied/ lowest-unoccupied orbitals, respectively. As a result, the coupling can be simply obtained as half of the orbital splitting,41 e.g. in the case of the hole coupling 1 J DA = (εHOMO − εHOMO − 1) (3) 2 where ε HOMO and ε HOMO−1 are the energies of the corresponding frontier orbitals. However, this simple relation only holds for symmetric systems,25 thus more advanced techniques are needed for nonsymmetric systems, still relying on the single-particle and Koopmans’ theorem framework. In this section, we review some of the most popular approaches to address this issue, and we then introduce our contribution. We

DA JijDA = Fij̃ ,

i ∈ D,

j∈A

(9)

This approach requires three separate calculations to obtain the MO coefficients and energies: one for the supersystem and one for each fragment. It has the advantage of being rather simple as it does not require any additional implementation in order to obtain the fragment orbitals, and the calculation of the couplings only involves matrix multiplications of quantities that are easily obtained from any standard quantum chemistry software. An important limitation is the inability to account for the presence of covalent bonds, which restricts the evaluation of the coupling only to the TS contribution. Thoss and co-workers proposed a more approximate approach for the definition of the fragment orbitals that does not require a QM calculation on the fragments.31 Approximate fragment orbitals are obtained from the block-diagonalization of the partitioned Fock/KS matrix in atomic orbital (AO) basis, rearranged in the following D/A block structure:

⎛ FDD FDA ⎞ ⎟⎟ F = ⎜⎜ ⎝ F AD F AA ⎠ B

(10) DOI: 10.1021/acs.jctc.7b00431 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Figure 1. Test molecules used in this work. The donor and acceptor groups are the terminal ethylene units, and the number of bridging bonds is reported in parentheses (n).

This approach takes advantage of the fact that the AOs based on Gaussian functions are inherently localized, so that the definition of a fragment is easy even in the presence of covalent bonds.31 The D/A fragment orbitals are obtained by a separate diagonalization of the corresponding sub-blocks (FDD and FAA, respectively) X † XX

X

C F C =ε

X

Once the orbitals for the D-B and B-A fragments are computed, the coupling is calculated with the same procedure as for the Brédas method

FijDBA = γiD ‐ B †εγjB ‐ A

where ε is the diagonal matrix of the supersystem orbital energies, γXk is the projector calculated as in eq 6, and the coupling is defined as in eq 9. The orthogonalization of the fragment orbitals (eqs 7 and 8) is extremely important for this method because the bridge is shared between the fragments. There are many advantages to FMR-B: only one QM calculation on the supersystem is necessary, and the coupling evaluation only requires some matrix manipulations as in Thoss’ approach; the ambiguity on how to split the bridge is removed; the method treats independent and bridged systems uniformly within the same theoretical framework; throughspace and through-bond couplings are obtained in one step. 2.2. The Heff Approach. In the Larsson approach,41,42 the through-bridge coupling is obtained from the definition of an effective Hamiltonian (Heff) that includes a block for the D/A pair and a block for the bridge

(11)

where X identifies the D or A sub-block, and CX is the matrix of the corresponding orthogonalized MO coefficients. Once the fragment MO coefficients are obtained as in eq 11, J can be evaluated with the same procedure described for the Brédas approach (eqs 5−9). The two methods usually provide close results for nonbonded fragments, but the Thoss approach only requires one QM calculation on the supersystem (at the price of two diagonalizations of the Fock matrix sub-blocks), and it can be used for bonded systems. Although the Thoss approach has been applied to fragments that are directly bonded,31 the simple two-block partition becomes ambiguous in the presence of a bridge. In fact, initial tests on the systems studied in Section 3 where the cut was made at different points on the bridge revealed that the couplings significantly depend on the chosen partition. The reason is that the tail of the electron density for the donor and acceptor orbitals extends on the bridge, and an arbitrary cut affects the results uncontrollably. The main goal of this work is to overcome this difficulty with a well-defined partitioning scheme of the supersystem Fock/KS matrix, in order to maintain the good features of the Thoss approach while avoiding the ambiguities. We will refer to this extension as “FMR-B”, where the B stands for bridge. The first step is to partition the Fock matrix in AO basis into sub-blocks for the donor (D), bridge (B), and acceptor (A): ⎛ FDD FDB FDA ⎞ ⎜ ⎟ F = ⎜ FBD FBB FBA ⎟ ⎜ AD AB AA ⎟ ⎝F F F ⎠

⎛ FDA − ε IDA ⎞⎛ CDA ⎞ V ⎜⎜ ⎟⎟⎜⎜ ⎟⎟ = 0 ⎝ V† FB − ε IB ⎠⎝ CB ⎠ AD

⎛ FBB FBA ⎞ ⎟⎟ FB ‐ A = ⎜⎜ ⎝ F AB F AA ⎠

(16)

B

where F and F are the sub-blocks belonging to D/A and the bridge, respectively, and V is the off-diagonal block between D/ A and the bridge. The linear system of equations is solved to obtain CB, which is then substituted in the equation for CDA to obtain an effective equation for D/A: [FDA + V(ε IB − FB)−1V†]CDA = CDA ε

(17)

Finally, the coupling is † JijDA,eff (ε) = JijDA + ViDBGB(ε)V BA j

(18)

(13)

where GB(ε) = (εIB−FB)−1 is the bridge Green’s function evaluated at the tunneling energy ε, with i and j labeling the bridge-localized orbitals. VAB and VBD are off-diagonal subblocks between acceptor and bridge and between bridge and donor, respectively. As the tunneling energy (ε) is not welldefined for D/A orbitals with different energies, different strategies have been proposed, i.e. taking the average of the D/ A orbital energies,35 or solving eq 17 self-consistently.36

(14)

3. RESULTS AND DISCUSSION In this section, we consider the calculation of the electronic coupling in bridged systems. We evaluate the couplings for

(12)

The fragment orbitals are then obtained by a separate diagonalization of the D-B (FD‑B) and B-A (FB‑A) sub-blocks: ⎛ FDD FDB ⎞ ⎟⎟ FD ‐ B = ⎜⎜ ⎝ FBD FBB ⎠

(15)

C


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basis sets because they are fairly compact, and they represent a good compromise between accuracy and computational cost for JDA calculations. The most challenging case is at the shortest distance (n = 2), due to the close proximity of the D/A groups. The hole coupling shows significant dependence on the choice of polarization functions: good agreement with experiment is obtained with (d,p) and (df,pd) functions, while a significant overestimation appears with (2d) functions or more. Additionally, the absence of polarization functions (6-311G) results in a strong underestimation of J. This behavior seems to be related to limitations in the block diagonalization when a large direct D/A interaction is allowed by the basis set, which is consistent with the analysis in ref 48. On the other hand, the electron coupling is less sensitive to the type of polarization functions included in the basis set, except for 6-311G, for which the coupling is again underestimated. Diffuse functions do not present a problem for the FMR-B method, contrary to what was reported for Heff.39 For longer distances (n > 2), which are relevant in molecular electronics, FMR-B seems rather insensitive to the choice of basis set for both hole and electron coupling, providing a systematic agreement with respect to the experimental estimates (Figure 2). For the remainder of the section, we present results with the 6-311G(d,p) basis set as it is the smallest one with a reliable performance. Figure 3 reports the hole and electron couplings for the 1(n) set, including a comparison with other theoretical methods36 and with experiment.46 The figure also reports the TS coupling computed with FMR-B, where the bridge is removed and the D/A groups are kept in the same geometry as in the real molecule (with the open valencies capped with H atoms). An example of the fragment MOs used for the coupling calculation is reported in Figure 4, which shows the tail of the donor orbital on the bridge. The figure also indicates where the cut at the border between the bridge and the acceptor group is performed for the definition of the sub-blocks of the Fock/KS matrix (eqs 12−14). The hole coupling allows a comparison of all methods and experiment. J slightly increases from n = 2−4, and it decreases exponentially at longer distances. This behavior, due to the change in interaction between the D and A groups with distance as discussed in the Introduction, is well reproduced by FMR-B and Heff, which slightly over- and underestimate J, respectively. DFTB qualitatively reproduces the trend, but the J magnitude is severely underestimated at small n values, while CDFT seems to predict an incorrect coupling at short distance. This may be the result of fractional charge transfer, which can make CDFT unreliable.49 The TS coupling is very large at short distances and decreases rapidly at n ≥ 4, indicating that the mediating role of the bridge is significant. For the electron coupling, FMR-B follows the experimental data closely and provides a smooth decay at larger n. Since these are symmetric molecules, the couplings can be computed with the orbital splitting (eq 3); these values are reported in Tables S2 and S3 of the SI and show that FMR-B is able to reproduce these trends accurately. Results for molecules 1(4), 2(4), and 3(4) are reported in Figure 5. These systems are similar in structure, but the D/A groups are in different orientation and at slightly different distance. The hole couplings are well reproduced by the FMRB and Heff methods, although only the former provides a consistent underestimation of the experimental estimates. DFTB and CDFT provide large errors, and DFTB calculations did not converge for 2(4).39 FMR-B shows the same behavior for the electron coupling, i.e. the qualitative trend of the

several test systems using FMR-B and compare them with other calculated couplings and experimental estimates (when available). All DFT calculations were performed with a development version of the GAUSSIAN suite of programs,45 and the FMR-B couplings were calculated with a postprocessing code that we developed. Although our implementation is general and can be applied to DFT as well as to Hartree−Fock (HF), the choice of density functionals follows refs 35 and 36 for a direct comparison with the methods employed in these works. 3.1. Fused-Ring Bridged Systems. In this section, we focus on the distance decay of the coupling in unsaturated D/A fragments covalently linked by aliphatic fused-ring bridges of increasing length. The test set is shown in Figure 1, following the comprehensive experimental work of Paddon-Row and coworkers.46 The experimental estimates were obtained by using the splitting approach (eq 3), and the orbital energies were measured via photoelectron and electron transmission spectroscopy.46 These are symmetric species, except for 2(4), and the D/A groups are the ethylene units. We also compare FMRB results with previous calculations of Gillet et al.36 based on different approaches, namely the Heff technique (Section 2.2), CDFT, and DFTB. Therefore, we use the molecular structures reported in ref 36 and the B97D functional.47 Before discussing the comparison with the other methods, we present a basis set study for the first three members of the 1(n) set, where the FMR-B results are compared to experiment. In fact, the calculation of the couplings can be sensitive to the choice of basis set, e.g. in refs 36 and 39 it is reported that diffuse functions are responsible for poor performance with the Heff method. Therefore, the right balance in the choice of basis functions is necessary to obtain reliable results. Figure 2 shows the comparison among calculated and experimental hole and electron couplings for molecules 1(n) as a function of the number of σ bonds (n) between D and A (the numerical values of the couplings for double- and triple-ζ basis sets are reported in Table S1 of the Supporting Information, SI). We use Pople

Figure 2. Comparison of calculated hole (a) and electron (b) couplings for molecules 1(n) using different basis sets and the B97D functional. D


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Figure 3. Comparison of experimental estimates (Exp.) and calculated hole and electron couplings for molecules 1(n). The Heff, CDFT, and DFTB results are from ref 36, and the experimental estimates are from ref 46. The through-space coupling calculated with FMR-B is also shown (TS).

Figure 4. Fragment orbitals used for the calculation of the hole and electron couplings for molecule 1(8). Top: side view; bottom: top view. The red line indicates the block partition of the Fock/KS matrix.

Figure 6. Plots of RMSE and MAE (in %) compared to experiment for all four computational approaches, considering molecules 1(2,4,6), 2(4), and 3(4).

experimental values is reproduced, and the couplings are slightly but consistently underestimated. To summarize the overall performance of the various schemes, we calculated the average of the relative errors in the form of root-mean-square and mean absolute errors (RMSE and MAE, respectively) compared to experiment for molecules 1(2,4,6), 2(4), and 3(4), shown in Figure 6. FMR-B and Heff have the smallest errors, showing a comparable accuracy in accounting for the effects of the bridge. However, although Heff shows smaller

errors, FMR-B seems to provide a more consistent trend for the error (Figure 5), little basis set dependence for n > 2 (Figure 2), and it does not require any preliminary orbital localization procedure. This is clearly an advantage since it has been reported that the Pipek-Mezey localization is not always applicable.36

Figure 5. Comparison of experimental estimates (Exp.) and calculated hole/electron couplings for molecules 1(4), 2(4), and 3(4). The Heff, CDFT, and DFTB values are from ref 36, and the experimental estimates are from ref 46. E


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Figure 7. Comparison between FMR-B and CDFT calculated hole couplings for molecules 4(n) and 5(n). The CDFT values are from ref 36.

The last set of values we report are the hole couplings for molecules 4(n) and 5(n) shown in Figure 7. For these molecules, the only comparison is with CDFT36 and orbital splitting approaches. The agreement between FMR-B and CDFT is quite good for 4(n), while CDFT seems to underestimate the coupling for 5(n). The key difference between the two molecular sets is the presence of the bicyclo[2.2.0]hexane group(s). This group significantly enhances the electronic coupling,46 which is well reproduced by FMR-B. Since these are symmetric molecules, the coupling can be also computed as half of the orbital splitting (eq 3). These values are reported in Tables S5 and S6 of the SI, which show that FMR-B is able to effectively reproduce the smooth decay of the coupling with increasing n. 3.2. DNA Oligomers. The next test set involves the evaluation of the decay with distance of the hole coupling in DNA oligomers. We consider guanine (G) to guanine hole transfer in the presence of thymines (T) for several single strand G(T)nG model systems, with n = 0−3, where the coupling is mediated by the presence of the thymines and of the backbone. These systems are based on previous work of Ramos and Pavanello,35 who used FDE-DFT for the definition of localized D/A states. In ref 35, the phosphate linker groups were removed, and the dangling bonds were capped with hydrogen atoms. The resulting G(T)nG model systems were considered both with and without the deoxyribose group bound to each nucleobases. When thymine(s) are present in between the D/A fragments, the Heff approach was used in combination with FDE-DFT to obtain the effective couplings. Here, we consider the model systems with and without the deoxyribose group as in ref 35, and we also include the phosphate linker groups and corresponding counterions, so that the entire backbone is included. We calculate the couplings at the same level of theory (PW91/TZP) and geometries as in ref 35. For the calculations with the full backbone, the phosphate + counterions were relaxed, while the other atoms were kept fixed, still using the PW91/TZP model chemistry. An example of the partition of the donor-bridge and bridgeacceptor orbitals for the GTG system is shown in Figure 8, with and without the backbone. This figure shows the active role of the thymine, which carries considerable orbital density. However, the backbone is also important, even if the explicit orbital contribution is small, because it modulates the amount of electron density on the thymine. The natural logarithm of the coupling for the G(T)nG systems computed with the FMRB and FDE methods is reported in Figure 9 (the numerical

Figure 8. Fragment orbitals used for the calculation of the hole coupling for model system GTG, with (below) and without (above) the backbone. The green line is added to define the block partition of the Fock/KS matrix.

Figure 9. Comparison between the natural logarithm of the calculated hole couplings (in meV) for the G(T)nG model systems with and without the backbone, i.e. the ribose groups (+R), and the phosphate linker groups with the corresponding counterions (+RP). The FDE results are from ref 35.

values are reported in Table S7 of the SI). The two methods agree very well for the smallest model (GG), where the effect of the backbone is negligible. The agreement worsens by F


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increasing the number of T bases. The FMR-B values follow a nearly linear decay, consistently with the exponential decay of the coupling (eq 2), while FDE+Heff only shows this behavior for n > 0, which may be due to the approximate tunneling energy for Heff, chosen as the average of the D/A orbital energies in ref 35. FDE predicts a faster decay with distance than FMR-B and a similar rate with and without the deoxyribose groups. On the other hand, FMR-B predicts a slower decay with the deoxyribose groups than for the isolated bases and an even slower decay when the full backbone is included. This behavior is consistent with what is expected comparing TS and TB coupling at longer D/A distances. This trend is also reproduced with the BHandH functional (50% of HF exchange), suggested by Migliore for coupling calculations,50 and it compares well with the couplings computed with the Mulliken-Hush model and the complete active space self-consistent field (CASSCF) method with and without second order perturbation corrections (CASPT2),51 as shown in Table S7 in the SI.

Alessandro Biancardi: 0000-0003-0697-8758 Marco Caricato: 0000-0001-7830-0562 Funding

Support from startup funds provided by the University of Kansas is gratefully acknowledged. C.L. also thanks the NSF Research Experiences for Undergraduates (REU) Grant CHE1560279. Notes

The authors declare no competing financial interest.

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(1) Carroll, R. L.; Gorman, C. B. The genesis of molecular electronics. Angew. Chem., Int. Ed. 2002, 41, 4378−4400. (2) Weibel, N.; Grunder, S.; Mayor, M. Functional molecules in electronic circuits. Org. Biomol. Chem. 2007, 5, 2343−2353. (3) Ratner, M. A brief history of molecular electronics. Nat. Nanotechnol. 2013, 8, 378−381. (4) Kim, W. Y.; Kim, K. S. Tuning molecular orbitals in molecular electronics and spintronics. Acc. Chem. Res. 2010, 43, 111−120. (5) Slinker, J. D.; Muren, N. B.; Renfrew, S. E.; Barton, J. K. DNA charge transport over 34 nm. Nat. Chem. 2011, 3, 230−235. (6) Weiss, E. A.; Wasielewski, M. R.; Ratner, M. A. Molecules as Wires: Molecule-Assisted Movement of Charge and Energy. Molecular Wires and Electronics; Springer Berlin Heidelberg: Berlin, Heidelberg, 2005; pp 103−133; DOI: 10.1007/b136068. (7) Seeman, N. C. Nucleic acid junctions and lattices. J. Theor. Biol. 1982, 99, 237−247. (8) Aviram, A.; Ratner, M. A. Molecular rectifiers. Chem. Phys. Lett. 1974, 29, 277−283. (9) Peplow, M. Rebooting the molecular computer. ACS Cent. Sci. 2016, 2, 874−877. (10) Muren, N. B.; Olmon, E. D.; Barton, J. K. Solution, surface, and single molecule platforms for the study of DNA-mediated charge transport. Phys. Chem. Chem. Phys. 2012, 14, 13754−13771. (11) Genereux, J. C.; Barton, J. K. Mechanisms for DNA charge transport. Chem. Rev. 2010, 110, 1642−1662. (12) Marcus, R. A.; Sutin, N. Electron transfers in chemistry and biology. Biochim. Biophys. Acta, Rev. Bioenerg. 1985, 811, 265−322. (13) O’Neil, M. A.; Barton, J. K. DNA charge transport: conformationally gated hopping through stacked domains. J. Am. Chem. Soc. 2004, 126, 11471−11483. (14) O’Neill, M. A.; Becker, H.-C.; Wan, C.; Barton, J. K.; Zewail, A. H. Ultrafast dynamics in DNA-mediated electron transfer: base gating and the role of temperature. Angew. Chem., Int. Ed. 2003, 42, 5896− 5900. (15) Conwell, E. M. Charge transport in DNA in solution: the role of polarons. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 8795−8799. (16) Henderson, P. T.; Jones, D.; Hampikian, G.; Kan, Y.; Schuster, G. B. Long-distance charge transport in duplex DNA: the phononassisted polaron-like hopping mechanism. Proc. Natl. Acad. Sci. U. S. A. 1999, 96, 8353−8358. (17) Marcus, R. Chemical and electrochemical electron-transfer theory. Annu. Rev. Phys. Chem. 1964, 15, 155−196. (18) Marcus, R. A. Electron transfer reactions in chemistry. Theory and experiment. Rev. Mod. Phys. 1993, 65, 599−610. (19) Marcus, R. A. On the theory of oxidation-reduction reactions involving electron transfer. I. J. Chem. Phys. 1956, 24, 966−978. (20) Hoffmann, R.; Imamura, A.; Hehre, W. J. Benzynes, dehydroconjugated molecules, and the interaction of orbitals separated by a number of intervening sigma bonds. J. Am. Chem. Soc. 1968, 90, 1499−1509. (21) Hoffmann, R. Interaction of orbitals through space and through bonds. Acc. Chem. Res. 1971, 4, 1−9. (22) Paddon-Row, M. N. Some aspects of orbital interactions through bonds: physical and chemical consequences. Acc. Chem. Res. 1982, 15, 245−251.

4. CONCLUSIONS In this work, we propose an efficient and reliable method to evaluate the electronic coupling between a donor and an acceptor moiety in covalently bonded bridged systems, based on first-principles electronic structure calculations. FMR-B combines and extends two previous approaches by Brédas25 and Thoss,31 where the coupling is obtained as a projection of the Fock/KS matrix of the supersystem on the basis of the fragments orbitals. Our development maintains the advantages of these approaches, e.g. only matrix manipulations are required, while removing some ambiguities in the partitioning of the Fock matrix in the presence of a bridge. The FMR-B method also provides some advantages compared to other similar methods, such as Heff, as the TS and TB contributions to the coupling are obtained simultaneously, and no preliminary orbital localization procedure is necessary. FMR-B only requires quantities that are routinely computed by any standard quantum chemistry packages and can be easily implemented. We applied this method to two prototypical sets of bridged systems for charge transfer processes: fused-ring bridged systems and G(T)nG DNA oligomers. We find that FMR-B performs as well as or better than other available methods. Our results for stacked DNA bases also show that merely looking at the effect of π-stacking and ignoring the backbone (including phosphate linker groups and counterions) may provide underestimated couplings. We are currently working on combining the FMR-B approach with PBC-DFT methods,26 which will permit, for instance, the study of electron transfer processes of chromophores attached to semiconducting materials.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00431. Values of the electronic couplings, and the molecular coordinates the G(T)nG test systems with backbone (PDF)

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