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Electron transfer coupling is a critical factor in determining electron transfer rates. This coupling strength can be sensitive to details in molecular geometries, especially intermolecular configurations. Thus, studying charge transporting behavior with a full first-principle approach demands a large amount of computation resources in quantum chemistry (QC) calculation. To address this issue, we developed a machine learning (ML) approach to evaluate electronic coupling. A prototypical ML model for an ethylene system was built by kernel ridge regression with Coulomb matrix representation. Since the performance of the ML models highly dependent on their building strategies, we systematically investigated the generality of the ML models, the choice of features and target labels. The best ML model trained with 40 000 samples achieved a mean absolute error of 3.5 meV and greater than 98% accuracy in predicting phases. The distance and orientation dependence of electronic coupling was successfully captured. Bypassing QC calculation, the ML model saved 10-10

In describing the dynamics of electron transfer or charge transport, the reorganization energy and the spectral density function describe the influence of nuclei motion to the transporting electron. The spectral density can be obtained using various theoretical approaches: from a model of dielectric response, or calculated with various computational means. With the vast advancement of modern computational techniques, many details in vibronic coupling can be obtained, including those described in the early literature. In this work, we provide a comprehensive understanding of the nature of vibronic coupling in light of some early literature. The theoretical connection among different quantities for vibronic coupling is discussed, followed by a brief review of the spectral density function. Various approaches and some of the results for the spectral density function are also reviewed. The importance of low-frequency bands in nonpolar systems that can be overlooked is also discussed, for both Holstein and Peierls types of electron–phonon couplings.

The density functional theory (DFT) with commonly used functionals is known to be incorrect for charge-transfer problems. With long-range-corrected (LC) density functionals, the asymptotic exchange potential is gradually switched to the Hartree-Fock exchange at a long range, and the prediction for charge-transfer states is greatly improved. In this work, we test LC-DFT's performance on charge-transfer couplings. The range-separation parameter can be tuned nonempirically for properties of a generalized DFT. We propose to minimize the difference of highest-occupied Kohn-Sham orbital energy and the ionization potential (for hole transfer) or the lowest-unoccupied orbital energy and the electron affinity (for electron transfer). For photoinduced charge transfer, the minimum in the sum of such differences for the donor and the acceptor is proposed. With the range-separation parameters optimized, we found that ET couplings derived from the LC-DFT are close to those derived from coupled cluster with singles and doubles. When compared with experimentally derived Mulliken-Hush couplings, LC-DFT couplings are greatly improved as well. We also found that the couplings from BNL and LC-BLYP functionals are generally better than those from LC-ωPBE and LC-ωPBE0. LC-DFT is suitable for calculating ET coupling, especially with this nonempirical approach for the range-separation parameter.

A series of new benzo[ghi]perylenetriimide (BPTI) derivatives has been synthesized and characterized. These remarkably soluble BPTI derivatives show strong optical absorption in the range of =300-500nm and have a high triplet-state energy of 1.67eV. A cyanophenyl substituent renders BPTI such a strong electron acceptor (E-red=-0.11V vs. the normal hydrogen electrode) that electron-trapping reactions with O-2 and H2O do not occur. The BPTI radical anion on a fluorine-doped tin oxide|TiO2 electrode is persistent up to tens of seconds (t(1/2)=39s) in air-saturated buffer solution. As a result of favorable packing, theoretical electron mobilities (10(-2)approximate to 10(-1)cm(2)V(-1)s(-1)) are high and similar to the experimental values observed for perylene diimide and C-60 derivatives. Our studies show the potential of the cyanophenyl-modified BPTI compounds as electron acceptors in devices for artificial photosynthesis in water splitting that are also very promising nonfullerene electron-transport materials for organic solar cells.

Here, we introduced benzo[ghi]perylenetriimide (BPTI) derivatives including monomer and twisted dimer (t-BPTI) as an alternative electron-transport layer (ETL) material to replace the commonly used PC61 BM in inverted planar heterojunction perovskite solar cells (PSCs). Moreover, the double ETL was applied in our PSCs with structure of glass/ITO/PEDOT:PSS/perovskite/BPTI/C60 or PDI-C4/BCP/Al. The use of a double ETL structure can effectively eliminate the leakage current. The devices with the t-BPTI/C60 double ETL yield an average power conversion efficiency of 10.73 % and a maximum efficiency of 11.63 %. The device based on the complete non-fullerene electron acceptors of t-BPTI/PDI-C4 as double ETL achieved maximum efficiency of 10.0 %. Moreover, it was found that the utilization of alloy t-BPTI+BPTI as ETL can effectively reduce the hysteresis effect of PSCs. The results suggest that BPTI-based electron-transport materials are potential alternatives for widely used fullerene acceptors in PSCs.

Photovoltaic devices have great potential in harvesting solar energy. In organic solar cells, the role of the donor–acceptor heterojunction is critically important; here optical excitation leads to charge separation (CS) and subsequently photocurrent generation. Charge recombination (CR) may also happen, which reduces the overall efficiency of the device. After light absorption, the singlet excitation of a pentacene can achieve CS at the interface, or it may also undergo singlet fission and produce two triplet excitons, which may further undergo CS at the interface. In this work, we study charge-transfer processes at the pentacene–C60 interface, including CS rates in both singlet and triplet surfaces, and the CR rate back to the ground state. We first constructed two different pentacene–C60 interfacial structures by force field optimization. With pentacene–C60 molecular pairs derived from the structures, we calculate the electronic couplings of CS and CR reactions. We found that the electron-transfer coup...

We calculate the electron transfer (ET) rates for a series of heptacyclo[6.6.0.02,6.03,13.014,11.05,9.010,14]-tetradecane (HCTD) linked donor–acceptor molecules. The electronic coupling factor was calculated by the fragment charge difference (FCD) [19] and the generalized Mulliken–Hush (GMH) schemes [20]. We found that the FCD is less prone to problems commonly seen in the GMH scheme, especially when the coupling values are small. For a 3-state case where the charge transfer (CT) state is coupled with two different locally excited (LE) states, we tested with the 3-state approach for the GMH scheme [30], and found that it works well with the FCD scheme. A simplified direct diagonalization based on Rust's 3-state scheme was also proposed and tested. This simplified scheme does not require a manual assignment of the states, and it yields coupling values that are largely similar to those from the full Rust's approach. The overall electron transfer (ET) coupling rates were also calculated.

The transport of charge via electrons and the transport of excitation energy via excitons are two processes of fundamental importance in diverse areas of research. Characterization of electron transfer (ET) and excitation energy transfer (EET) rates are essential for a full understanding of, for instance, biological systems (such as respiration and photosynthesis) and opto-electronic devices (which interconvert electric and light energy). In this Account, we examine one of the parameters, the electronic coupling factor, for which reliable values are critical in determining transfer rates. Although ET and EET are different processes, many strategies for calculating the couplings share common themes. We emphasize the similarities in basic assumptions between the computational methods for the ET and EET couplings, examine the differences, and summarize the properties, advantages, and limits of the different computational methods. The electronic coupling factor is an off-diagonal Hamiltonian matrix element between the initial and final diabatic states in the transport processes. ET coupling is essentially the interaction of the two molecular orbitals (MOs) where the electron occupancy is changed. Singlet excitation energy transfer (SEET), however, contains a Frster dipole-dipole coupling as its most important constituent. Triplet excitation energy transfer (TEET) involves an exchange of two electrons of different spin and energy; thus, it is like an overlap interaction of two pairs of MOs. Strategies for calculating ET and EET couplings can be classified as (1) energy-gap-based approaches, (2) direct calculation of the off-diagonal matrix elements, or (3) use of an additional operator to describe the extent of charge or excitation localization and to calculate the coupling value. Some of the difficulties in calculating the couplings were recently resolved. Methods were developed to remove the nondynamical correlation problem from the highly precise coupled cluster models for ET coupling. It is now possible to obtain reliable ET couplings from entry-level excited-state Hamiltonians. A scheme to calculate the EET coupling in a general class of systems, regardless of the contributing terms, was also developed. In the past, empirically derived parameters were heavily invoked in model description of charge and excitation energy drifts in a solid-state device. Recent advances, including the methods described in this Account, permit the first-principle quantum mechanical characterization of one class of the parameters in such descriptions, enhancing the predictive power and allowing a deeper understanding of the systems involved.

We propose to use Spin-Flip (SF) methods for calculating electron transfer coupling strengths in terms of energy gaps between adiabatic states. The SF variant of Configuration-Interaction-Singles (SF-CIS) significantly improves descriptions of adiabatic states in charge-transfer systems and yields reliable energy splittings at the electron-transfer (ET) transition state. The SF-CIS scheme is more robust than the Hartree–Fock Koopmans Theorem, and its coupling dependence on intermolecular separation is correct even in diffuse basis sets. The SF approach allows for the systematic inclusion of dynamical correlation effects in electron transfer couplings. Our results indicate a small dynamical correlation effect for intermolecular electron transfer couplings.

The electron transfer rate constant at electrode/liquid interfaces is theoretically described on the basis of the Anderson–Newns–Schmickler model. A compact formula for the rate constant is derived in the nonadiabatic limit, which is expressed in terms of the spectral density of surrounding media, the density of states of electrons in the electrode, and the weighted electronic coupling constant between the electrode and the redox couple in the liquid. The outer-sphere spectral density is then related to the experimentally accessible data on the frequency-dependent dielectric response functions of the solvent and the electrode with the aid of the dielectric continuum approximation. The derived formula provides a quantum-mechanical extension of the conventional nonadiabatic expression for the heterogeneous electron transfer reactions at electrode/liquid interfaces, taking into account the quantum effects associated with the high-frequency modes of both outer and inner spheres. On this basis, the quantum cor...

A sequential formula developed by Hsu and Marcus [J. Chem. Phys., 106 (1997) 584] is applied to calculate the electronic coupling between a gold electrode and each of the molecules ( η 5 -C 5 H 5 )Fe( η 5 -C 5 H 4 )CO 2 (CH 2 ) n SH and ( η 5 -C 5 H 5 )Fe( η 5 -C 5 H 4 )(CH 2 ) n SH ( n = 3 to 50). Most of the calculated coupling strengths, if converted to rate constants according to a high-temperature non-adiabatic expression, agree well with the experimental rate constants. The exponential coefficients are obtained by a linear fit of the logarithm of the coupling strengths to the chain length. The calculated exponential coefficient is 1.05 per methylene unit for both molecules and for large n . For shorter chains the exponential coefficient is slightly larger with an irregularity caused by an oscillation of coupling strengths between even and odd numbers of units. The physical origin of such oscillation among short chains is discussed.

A recursion relation is formulated for the Green's function for calculating the effective electron coupling in bridge-assisted electronic transfer systems, within the framework of the tight-binding Hamiltonian. The recursion expression relates the Green's function of a chain bridge to that of the bridge that is one unit less. It is applicable regardless of the number of orbitals per unit. This method is applied to the system of a ferrocenylcarboxy-terminated alkanethiol on the Au(111) surface. At larger numbers of bridge units, the effective coupling strength shows an exponential decay as the number of methylene(–CH2–) units increases. This sequential formalism shows numerical stability even for a very long chain bridge and, since it uses only small matrices, requires much less computer time for the calculation. Identical bridge units are not a requirement, and so the method can be applied to more complicated systems.

The charge mobility is a key property in many electro-optical materials, with charge transfer (CT) taking place in a solid matrix of molecules. Large intermolecular electronic interaction is one of the key factors for a good CT rate, which is dependent on both intra- and intermolecular structures. The connection of the molecular structure with the intermolecular CT property would facilitate the search for a new material with desirable CT property, but currently it is still quite limited by the lack of knowledge for intermolecular configurations. In the present work, we study factors influencing the intermolecular configurations, and subsequently the CT property, in tris(8-hydroxyquinolinato) aluminium(III) (AlQ(3)) from all currently available crystal structures. We found that there exists a pair of CH-π interactions in a good majority of the π-π stacked bimolecular configurations. Such CH-π and π-π interacting structures are also seen in the crystal structures of many other similar molecules. With both experimental and simulated structures, we show that the CH-π interaction stabilizes the bimolecular configurations, and drives the structure towards a region with a higher electron transfer coupling and lower hole transfer coupling. This effect likely affects the electron transport property of AlQ(3), since it is consistent with recent experimental results, where AlQ(3) analogs with their CH-π interaction blocked either require a higher operating voltage in light-emitting devices [Sapochak et al., J. Am. Chem. Soc., 2001, 123, 6300], or become bipolar in their charge mobilities [Liao et al., J. Am. Chem. Soc., 2009, 131, 763]. CH-π interaction is commonly seen in aromatic molecules, which are frequently used as building blocks in molecules for electro-optical applications. Our work points out a possible way to enhance the desired CT property in the design of new materials.

To calculate the electronic couplings in both inter- and intramolecular triplet energy transfer (TET), we have developed the "fragment spin difference" (FSD) scheme. The FSD was a generalization from the "fragment charge difference" (FCD) method of Voityuk et al. [J. Chem. Phys. 117, 5607 (2002)] for electron transfer (ET) coupling. In FSD, the spin population difference was used in place of the charge difference in FCD. FSD is derived from the eigenstate energies and populations, and therefore the FSD couplings contain all contributions in the Hamiltonian as well as the potential overlap effect. In the present work, two series of molecules, all-trans-polyene oligomers and polycyclic aromatic hydrocarbons, were tested for intermolecular TET study. The TET coupling results are largely similar to those from the previously developed direct coupling scheme, with FSD being easier and more flexible in use. On the other hand, the Dexter's exchange integral value, a quantity that is often used as an approximate for the TET coupling, varies in a large range as compared to the corresponding TET coupling. To test the FSD for intramolecular TET, we have calculated the TET couplings between zinc(II)-porphyrin and free-base porphyrin separated by different numbers of p-phenyleneethynylene bridge units. Our estimated rate constants are consistent with experimentally measured TET rates. The FSD method can be used for both intermolecular and intramolecular TET, regardless of their symmetry. This general applicability is an improvement over most existing methodologies.

The transport of charge via electrons and the transport of excitation energy via excitons are two processes of fundamental importance in diverse areas of research. Characterization of electron transfer (ET) and excitation energy transfer (EET) rates are essential for a full understanding of, for instance, biological systems (such as respiration and photosynthesis) and opto-electronic devices (which interconvert electric and light energy). In this Account, we examine one of the parameters, the electronic coupling factor, for which reliable values are critical in determining transfer rates. Although ET and EET are different processes, many strategies for calculating the couplings share common themes. We emphasize the similarities in basic assumptions between the computational methods for the ET and EET couplings, examine the differences, and summarize the properties, advantages, and limits of the different computational methods. The electronic coupling factor is an off-diagonal Hamiltonian matrix element between the initial and final diabatic states in the transport processes. ET coupling is essentially the interaction of the two molecular orbitals (MOs) where the electron occupancy is changed. Singlet excitation energy transfer (SEET), however, contains a Frster dipole-dipole coupling as its most important constituent. Triplet excitation energy transfer (TEET) involves an exchange of two electrons of different spin and energy; thus, it is like an overlap interaction of two pairs of MOs. Strategies for calculating ET and EET couplings can be classified as (1) energy-gap-based approaches, (2) direct calculation of the off-diagonal matrix elements, or (3) use of an additional operator to describe the extent of charge or excitation localization and to calculate the coupling value. Some of the difficulties in calculating the couplings were recently resolved. Methods were developed to remove the nondynamical correlation problem from the highly precise coupled cluster models for ET coupling. It is now possible to obtain reliable ET couplings from entry-level excited-state Hamiltonians. A scheme to calculate the EET coupling in a general class of systems, regardless of the contributing terms, was also developed. In the past, empirically derived parameters were heavily invoked in model description of charge and excitation energy drifts in a solid-state device. Recent advances, including the methods described in this Account, permit the first-principle quantum mechanical characterization of one class of the parameters in such descriptions, enhancing the predictive power and allowing a deeper understanding of the systems involved.

To calculate electronic couplings for photoinduced electron transfer (ET) reactions, we propose and test the use of ab initio quantum chemistry calculation for excited states with the generalized Mulliken-Hush (GMH) method. Configuration-interaction singles (CIS) is proposed to model the locally excited (LE) and charge-transfer (CT) states. When the CT state couples with other high lying LE states, affecting coupling values, the image charge approximation (ICA), as a simple solvent model, can lower the energy of the CT state and decouple the undesired high-lying local excitations. We found that coupling strength is weakly dependent on many details of the solvent model, indicating the validity of the Condon approximation. Therefore, a trustworthy value can be obtained via this CIS-GMH scheme, with ICA used as a tool to improve and monitor the quality of the results. Systems we tested included a series of rigid, sigma-linked donor-bridge-acceptor compounds where "through-bond" coupling has been previously investigated, and a pair of molecules where "through-space" coupling was experimentally demonstrated. The calculated results agree well with experimentally inferred values in the coupling magnitudes (for both systems studied) and in the exponential distance dependence (for the through-bond series). Our results indicate that this new scheme can properly account for ET coupling arising from both through-bond and through-space mechanisms.

The charge transport properties of mer-tris(8-hydroxyquinolinato)aluminum(III) (mer-Alq), which is the most widely used electron transport material in OLED, were investigated by quantum chemistry calculations within the framework of the charge hopping model and Marcus electron transfer theory. Internal reorganization energies of 0.276 and 0.242 eV were calculated by the DFT-B3LYP method employing a 6-31 G* basis set for the electrons lambdai(e) and holes lambdai(h), respectively. The relative distances and orientations of Alq molecules in amorphous film were simulated by those in the beta-phase. The intermolecular charge-transfer integrals, Hda(h) and Hda(e), along all 14 hopping pathways were then calculated by the Koopmans Theorem in conjunction with the Hartree-Fock method employing a 6-31 G* basis set as well as by the direct coupling method. The results showed that there were some Hda(e) that were 1 order of magnitude larger than any Hda(h), because hopping pathways with effective overlaps of LUMOs can occur and, thus, large Hda(e). On the other hand, effective overlap of HOMO was absent in all pathways, resulting in a relatively small Hda(h). This difference in the magnitudes of Hda(e) and Hda(h) would predict a 2 orders of magnitude difference in the electron-transfer rate constants and account for the observed 2 orders of magnitude difference in the mobilities of electrons and holes.

The effect of solvent fluctuations on the rate of electron transfer reactions is considered using linear response theory and a second-order cumulant expansion. An expression is obtained for the rate constant in terms of the dielectric response function of the solvent. It is shown thereby that this expression, which is usually derived using a molecular harmonic oscillator ("spin-boson") model, is valid not only for approximately harmonic systems such as solids but also for strongly molecularly anharmonic systems such as polar solvents. The derivation is a relatively simple alternative to one based on quantum field theoretic techniques. The effect of system inhomogeneity due to the presence of the solute molecule is also now included. An expression is given generalizing to frequency space and quantum mechanically the analogue of an electrostatic result relating the reorganization free energy to the free energy difference of two hypothetical systems [J. Chem. Phys. 39, 1734 (1963)]. The latter expression has been useful in adapting specific electrostatic models in the literature to electron transfer problems, and the present extension can be expected to have a similar utility.

The dynamical correlation effect in electron transfer (ET) coupling was studied in this work, for cases where electrons tunnel through a many-electron environment. The ET couplings for three different bridge-mediated model systems were calculated: (I) trans-alkyl chains [H2C-(CH2)n-CH2, n = 2-10], (II) two isomers of trans-1,4-dimethylenecyclohexane, and (III) two ethylenes spaced by a saturated ethane molecule. The couplings were calculated as half energy gaps of the two lowest adiabatic states. The dynamical correlation was included with spin-flip (SF) and ionization potential or electron affinity coupled-cluster singles and doubles (SF-CCSD and IP/EA-CCSD) and a DeltaCCSD scheme. The direct coupling (DC) scheme is also used as a way to obtain a solution with nondynamical correlation, since DC uses approximated eigenstates that are symmetry-restoring linear combinations of two symmetry-broken unrestricted Hartree-Fock configurations. For all cases tested except for one, results from the DC scheme closely follow the CCSD data, indicating that the dual-configuration solutions can be a good approximation of wave functions with nondynamical correlation included, but there exist exceptions. Comparing the DC results with SF-CCSD and IP or EA-CCSD data, we concluded that the dynamical correlation effect is small for most of the cases we tested.

Triplet-triplet TT energy transfer requires two molecular fragments to exchange electrons that carry different spin and energy. In this paper, we analyze and report values of the electronic coupling strengths for TT energy transfer. Two different methods were proposed and tested: 1 Directly calculating the off-diagonal Hamiltonian matrix element. This direct coupling scheme was generalized from the one used for electron transfer coupling, where two spin-localized unrestricted Hartree-Fock wave functions are used as the zero-order reactant and product states, and the off-diagonal Hamiltonian matrix elements are calculated directly. 2 From energy gaps derived from configuration-interaction-singles CIS scheme. Both methods yielded very similar results for the systems tested. For TT coupling between a pair of face-to-face ethylene molecules, the exponential attenuation factor is 2.59 A 1 CIS/6-311+G ** , which is about twice as large as typical values for electron transfer. With a series of fully stacked polyene pairs, we found that the TT coupling magnitudes and attenuation rates are very similar irrespective of their molecular size. If the polyenes were partially stacked, TT couplings were much reduced, and they decay more rapidly with distance than those of full-stacked systems. Our results showed that the TT coupling arises mainly from the region of close contact between the donor and acceptor frontier orbitals, and the exponential decay of the coupling with separation depends on the details of the molecular contacts. With our calculated results, nanosecond or picosecond time scales for TT energy-transfer rates are possible. © 2006 American Institute of Physics. DOI: 10.1063/1.2155433