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Excitation energy transfer (EET) is a photophysical process where the electronic excitation energy is transferred from a donor fragment to an acceptor fragment. This process usually starts with a donor being optically excited. The excitation is subsequently transferred to a nearby acceptor, as depicted in Figure 1. This type of EET process is often referred to as singlet excitation energy transfer (SEET). There is another type of EET where the electronic excitation is in a triplet state. For example, the triplet excited state of a donor can be reached through intersystem crossing (ISC) from a singlet excited state, and the triplet excitation energy can be subsequently transferred to an acceptor fragment, forming a triplet excited acceptor (Fig. 2). Such EET processes are referred to as triplet excitation energy transfer (TEET).
EET is seen in many natural and artificial systems. For example, both organic light-emitting diodes and photovoltaics require exciton migration in the devices.[1-10] Plants and photosynthetic bacteria use the antenna system to capture solar energy and transfer the excitation to the reaction center.[11-13] In case of too much light absorption, one source of the oxidative damage is through the generation of triplet excited chlorophyll, as singlet oxygen may be sensitized. It is possible to take advantage of TEET to quench the triplet energy in the antenna system. The excess singlet excitation can also be regulated through SEET.[14-20] In addition, fluorescence resonance energy transfer (FRET) provides the basis for variety of applications for imaging[21, 22] and spectroscopy.[23, 24] Understanding EET process is a key to molecular design for improving the performance of molecular devices. It is also important for gaining insights in the generic mechanisms in the light-harvesting of photosynthesis. Theoretically predicting both the SEET and TEET has become routinely possible. In this work, we aim to introduce the physical origins and computational approaches for both SEET and TEET couplings, and theoretical models for obtaining EET rates.
SEET is the process behind the widely utilized Förster (fluorescence) resonance energy transfer and the light-harvesting in photosynthesis.[1-4, 11-13, 25-28] In photosynthesis, light-harvesting is the first step after a photon is absorbed, and the excitation energy is transferred to the reaction center of the photosynthetic apparatus. The efficiency and the regulation of this light-harvesting process are the focus of many studies. For a detailed understanding on the SEET rates, it is necessary to calculate (or estimate) the corresponding coupling values.[29-32] As the FRET efficiency decays as with R being the donor-acceptor distance, the structure and dynamics between the donor and acceptor can be experimentally probed with FRET.[21, 22, 33-35] This distance dependence arises mainly from the electronic coupling. The SEET coupling is traditionally modeled as using Förster's dipole approximation or the Coulomb coupling. To calculate the full coupling, the fragment excitation difference (FED) scheme can be used for a general class of systems. The computational aspects, physics and limitation of SEET coupling are discussed in the following sections.
TEET plays an important role in many artificial and natural systems.[5, 7, 8, 10, 14, 15, 37-42] In nature, the formation of triplet chlorophylls is dangerous, because they can sensitize oxygen and generate highly oxidative singlet oxygen. Carotenoids are believed to quench triplet chlorophylls, and it is one of its photoprotective functions.[15, 43] TEET has been utilized in phosphorescence-based light-emitting materials[5, 7, 8, 10] and in the low-energy up-conversion system.
The theory for TEET was first developed by Dexter. In the Dexter's theory, a TEET process is driven by an exchange coupling that accounts for the indistinguishability of electrons in many-electron wavefunctions. Other contributions, such as that from the overlap of donor and acceptor electronic densities, and the ionic configuration (IC) interactions were also discussed.[45-49] With quantum chemistry calculation, the exchange coupling values were reported for a number of TEET systems.[47, 50-54] However, the calculated exchange couplings are often much smaller than the plausible range that could yield the observed TEET rates or the exciton band gap.[47, 50, 53, 54] Scholes et al.[55, 56] have suggested that the overlap contribution can be more important than exchange coupling. In the studies of triplet exciton in aromatic crystals,[47, 50] it was demonstrated that the ICs have an important contribution to the TEET coupling and the results agreed with the observed Davydov splitting (the splitting of bands). In Scholes' work,[55, 56] the estimated IC interactions are comparable to the contribution of the other two components. Therefore, the exchange integral does not predict the TEET coupling well. It is important to obtain a full account of TEET coupling. In this work, the computational methods for TEET are introduced, and the roles of overlap and IC are discussed.
Theory of EET
For EET, one can model the process as a first-order reaction kinetics,
where D and A denote donor and acceptor fragments, respectively, and the superscript “ex” indicates where the electronic excitation is located. The EET problem is similar to the electron transfer problem,[57, 58] where the Born–Oppenheimer approximation breaks down near the transition. One can construct potential energy surfaces and describe EET reaction dynamics along with the reaction coordinates. Diabatic states are often used in construction of the potential energy surfaces because they are smooth functions of the nuclear coordinates. The diabatic states include the initial state representing and the final state representing . These states are with localized excitation, and they form the basis in describing the EET process.
The golden-rule rate theory
In the weak coupling limit, the rate for eq. (1) can be described by the Fermi's golden rule
where Ei (Ef) is the energy of the state ( ), and V is the electronic coupling factor. ensures that the process is energy conserved. The coupling is the off-diagonal Hamiltonian matrix element for the diabatic states,
where is the system Hamiltonian. The electronic coupling is an important factor in determining the EET rate. V can be calculated by quantum chemistry computation with a number of means, which will be discussed in this work.
In condensed phases, there is a continuous distribution of states due to the motion of nuclei, and so the energy conservation in eq. (2), becomes an overlap integral for the Franck–Condon weighted density of states (FCWD).[44, 61, 62] The golden-rule rate is
where µ and ν are indexes for donor vibrational states for the initial (i) and final (f) states, and are the same for the acceptor, is the µ-th vibration wavefunction for the donor fragment in the corresponding electronic state i, describes the initial distribution in the vibronic states, and is the difference of vibronic energy for the donor (or acceptor) fragment. The expressions in eqs. (5) and (6) are vibrational FCWD for electronic transitions for the donor and acceptor, respectively.
The FCWD can be obtained approximately from optical spectra. When the transition moment (transition dipole moment or the dipole operator involving spin-orbit coupling for TEET) is independent of the vibrational motion, the FCWD and the optical spectra are closely related[44, 61-63]:
where is the absorption cross section of acceptor
and is the spontaneous emission probability of donor
In both eqs. (9) and (10), n and ε are the refractive index and dielectric constant of the medium, respectively, and f represents the effective electric field correction for the electrons inside a media. is the transition dipole moment from state i to state f. In eqs. (7) and (8), the Einstein's energy dependences in the transition probabilities are removed for the underlying FCWD. For TEET, the phosphorescence spectra and its mirror image (as an approximated triplet “absorption” spectra) were used in a previous study.
The widely used Förster theory[44, 63] is a golden-rule based expression that uses experimental spectral information together with a dipole–dipole coupling model[63, 65]:
where n is the refractive index of the solvent, is the distance (in units of nm) between donor and acceptor, and κ is the orientation factor in dipole–dipole interaction,
where denotes the corresponding unit vector for v. and are the donor fluorescence quantum yield and fluorescence lifetime in the absence of the acceptor, respectively. J is the spectral overlap integral in the units of M−1cm3,
where is wavenumber in the units of cm−1, is the normalized fluorescence spectrum for the donor (in units of cm), and is the molar extinction coefficient for the acceptor (in units of M−1cm−1).
The expression in eq. (11) is connected to the golden-rule expression in eq. (4) by the following conversions: first, the coupling V in eq. (4) can be expressed as
where provides corrections for the medium, and the Gaussian units are assumed. The transition dipole strengths together with the density of states eqs. (7) and (8) are then obtained from experimental spectra eqs. (9) and (10). For the absorption spectrum, the molar extinction coefficient (in units of M−1cm−1) is directly related to the absorption cross section, (in units of cm2),
where Na is the Avogadro's constant. The total emission probability give rises to the radiative rate ,
which provides an experimental determination for the donor transition dipole strength. The normalized emission spectrum is kept in the spectral overlap. The radiative decay rate is obtained from experimentally measured fluorescence quantum yield ( ), divided by the fluorescence lifetime ( ),
as the excited-state lifetime reflects both radiative and nonradiative decays and
Therefore, the Förster SEET rate is based on the Fermi's golden rule and it is under the dipole approximation for the coupling.
We note that many aspects of the theoretical ground for EET are similar to those in the Marcus' electron transfer theory, as the latter is also based on the golden rule, but with the two (diabatic) electronic states strongly coupled to environmental fluctuation. In previous works, TEET rates were reported using the Marcus electron transfer rate with an reorganization energy, instead of a FCWD overlap.[66, 67] The Marcus' theory assumes full quadratic free energy surfaces for the two states involved, due to the strong coupling for the differently charged states with the polarization of the surrounding environment. In this case, the equivalent FCWD are essentially Gaussian functions. Compared to electron transfer, an excited molecule (fragment) imposes a much weaker perturbation to the environment, because there is little change on the distribution for the charges. The absorption and emission spectra in a condensed phase are often very different from a Gaussian function. Therefore, we believe that using the spectral profile as FCWD is a more general and reasonable approach for EETs.
The golden-rule rate theory may not be applicable to the exciton dynamics, particularly the SEET case. The excited molecule (fragment) perturbs the solvent or environment weakly. We cannot exclude the possibility of a large displacement in the vibrational modes of a large fragment and thus, a large reorganization energy. However, if there is only a small change on the bond lengths and angles, as with many rigid aromatic molecules with extended π-conjugated structures, the excitation would only lead to a small difference in the short-range van der Waals interactions between the system and its surrounding solvent molecules. Therefore, the EET dynamics falls into the regime of weak system-bath coupling. This is especially true for SEET, when the dipole coupling strength becomes comparable or even larger than the reorganization energy. In this case, the energy transfer may be coherent, and the golden-rule rate theory is not applicable.
Coherent exciton dynamics in light-harvesting complexes have been observed experimentally.[68-74] Through nonlinear spectroscopy techniques, it is observed that the exciton preserves its phases for at least 300 femtoseconds at room temperature. The long-lived quantum coherence plays an important role in achieving the remarkable quantum efficiency in photosynthetic light harvesting. Theoretical model for the exciton dynamics has been an important area of development recently.[75-77] We note that the electronic coupling is still an important parameter in these models. In testing for theoretical dynamics models, quantifying the coupling factor with first principle calculation reduces the uncertainty. Also, with the capability of dissecting the origin of electronic coupling, the role of the nondipole (or non-Coulomb) interaction in the theory would be clear.[30, 78]
In derivation of the coupling for SEET,[44-46, 48, 49, 52, 63] a first-order perturbation expansion is often used. In these approaches, it can be shown that the electronic coupling for SEET contains three terms,
where is the Coulomb coupling that arises from the Coulomb interaction between electronic transitions, is the Dexter's exchange coupling that accounts for the indistinguishability of the electrons in many-electron wavefunctions, and is a term arising from the overlap of donor-acceptor orbitals. Additional contribution through IC interactions to SEET coupling was also discussed.[47, 49]
To see different contributions in SEET coupling, we use a simplified four-orbital model as shown in Figure 3. In this model, we assume that all other electrons doubly occupy the lower-energy molecular orbitals (MOs) that are not shown, and they are not affected in SEET. Therefore, other MOs do not contribute to the coupling in this simplified model.
The initial and final states can be approximately written as
where denotes a Slater's determinant, and a short bar above an orbital ( ) denotes a β spin-orbital, whereas an orbital without the short bar ( ) is an α spin-orbital. We have omitted the details of the doubly occupied lower-energy orbitals in the determinant and assumed that they are the same in both states. It is assumed that the four orbitals are orthogonal, and so there is no overlap between donor and acceptor. As the initial and final states differ in the occupation of two spin orbitals, only two-electron terms survive in eq. (3), and the coupling becomes
where denotes the spin coordinates. The α and β spin-orbitals are denoted as and , respectively. The first term in the right-hand side of eq. (22) after integration over the spin coordinates is the Coulomb coupling. It can be rewritten as
which is very similar to a static Coulomb interaction energy between two charge densities and . The transition density is in general a one-particle density matrix formed from the ground state and an excited-state wavefunctions:
and the transition density we have in eq. (23) is the diagonal element of the density matrix:
The commonly seen dipole and multipole break-down for the Coulomb coupling is a spherical harmonic expansion of the Coulomb potential in eq. (23). The Coulomb potential corresponds to the transition density can be expanded as
where we have assumed that the transition density in the acceptor is the “source charge”. A multipole expansion for the acceptor is possible when , and are replaced with r1 and r2, respectively. It is performed to reproduce the electric potential outside a spherical boundary that encloses the source transition charge . Similar expansion can be performed for the donor. Therefore, to determine the interaction between two groups of charges by multipole expansion, the sphere enclosing each group of charges must not overlap.
As the ground and excited-state wavefunctions are always orthogonal, there is zero “total transition charge”. The leading order for the Coulomb coupling is the dipole–dipole interaction:
where is the transition dipole moment of the molecule, and is intermolecular distance between the center of the charge distributions. It is seen that this dipole–dipole term decays as . At a large donor-acceptor separation, the dipole–dipole interaction is the dominant contribution to . This dipole–dipole coupling gives the distance dependence in the well-known Förster theory.
The second term in the right-hand side of eq. (22) after integration over the spin coordinates is the exchange coupling:
which arises from the indistinguishability of the electrons in many-electron wavefunctions. We note that the expression in eq. (30) is for our simplified four-orbital model, whereas that in eq. (31) is for general transitions. It can be regarded as a simultaneous exchange of two electrons with different energy, as depicted in Figure 4. As first proposed by Dexter, the exchange coupling [eq. (31)] is like a Coulomb interaction between two orbital overlaps, and it decays exponentially as increasing donor-acceptor distances.
Compared to the Coulomb coupling, the exchange coupling decays faster and can be neglected at a large donor-acceptor separation. In most of SEET studies, the Coulomb coupling is often regarded as major contribution to the SEET coupling, and most works only focused on the Coulomb coupling. In applications, the Coulomb coupling is often approximated as the dipole–dipole interaction, as seen in eq. (29). Thus, the Förster theory is commonly used approach in studying SEET.
In the subsections for the short-range couplings below, we discuss the overlap and IC contributions that generally exist in both SEET and TEET. While most of the previous work reported these effects for TEET, the short-range couplings exist in SEET, but they are not expected to be very important.
For a TEET process (Fig. 5), the coupling can be written similarly as:
With a similar four-orbital model, we can express the initial and final states as:
The electronic coupling V can be obtained
In this case, the Coulomb coupling, the first term in the right-hand side of eq. (35) vanishes because the excitation from the singlet ground-state to a triplet excited state is spin forbidden. The transition density for a triplet excitation (β spin-orbital to α spin-orbital ) is zero:
The exchange term remains because the overlap density involves the orbitals with the same spin. Therefore, the TEET process is mainly the Dexter's exchange. It is expected that the TEET rates decay exponentially as increases, as observed in experiments.[80, 81]
Computation Methods for EET Coupling
In this section, we outline the computational methods to calculate either SEET or TEET couplings. The computational cost, and suitable systems to use with these methods, are also outlined below.
The energy-gap approach
In the two-state model, the Hamiltonian can be written in the basis state and the eigenstate representations:
where Ei (Ef) is the energy of the initial (final) state and V is the coupling. The eigenvalues for this two-state model are
In a resonance condition where , the coupling can be derived from a half of the energy gap,
The energy-gap scheme is a simple way to evaluate the coupling as long as the eigenstate energies at the resonance condition are obtained. It is applicable to both SEET and TEET, in which a symmetric and antisymmetric pair of singlet or triplet excited states are obtained through quantum chemical calculations.[48, 82-85] However, for general asymmetric systems, it may be difficult to search for the resonance condition. The difficulty of doing so increases rapidly with the size of the system. Unlike electron transfer problems, in which an external electric field can be used to create resonance, there is no general way to adjust for the resonance in the energy transfer system. The application of the energy-gap scheme for EET is, therefore, largely limited to small or symmetric systems.
To achieve resonance through symmetry, the donor and acceptor must be identical, and they must be mutually equivalent. As pointed out before, even dimers constructed from identical monomers may have different site energies, a situation that should be avoided in deriving coupling from energy gaps. This is not valid for general application, but such symmetric models nevertheless provide insights to the nature of the coupling.[48, 82-84]
The computational cost for the energy-gap based scheme is simply that for the excited-state calculation for the full system. With the low-cost configuration-interaction singles (CIS) and time-dependent density functional theory (TDDFT), the calculation is feasible for general molecules for practical applications.
The direct coupling scheme
Another approach directly calculates couplings from localized excited states, which we refer to the direct coupling (DC) scheme. The initial state (or the final state ) is with its electronic excitation localized to the donor (or the acceptor) fragment. DC assumes that the eigenstates are symmetric and antisymmetric linear combinations of these two excitation-localized states, , where is the overlap integral between the two states. We can write the electronic coupling in eq. (2) as half of the energy difference of these two eigenstates:
where is the matrix element of the Hamiltonian. Depending on their definition, the diabatic state wavefunctions and are generally not orthogonal to each other, and therefore, there is an overlap contribution for the coupling.
For TEET, the diabatic states can be modeled as the spin-localized states. As we are mostly interested in energy transfer of the lowest triplet excited states, an unrestricted Hatree–Fock (UHF) solution can often provide a good approximation for the lowest triplet state. We can model the spin-localized diabatic states by the broken-symmetry UHF wavefunctions.[85, 87] The spin-localized wavefunctions can be constructed by combining the donor and accepter's HF solution calculated separately, for example, with being a combination of donor triplet state and acceptor singlet ground state, and is constructed from donor singlet and acceptor triplet states. A further self-consistent field (SCF) calculation may be used to take into account for the relaxation of the orbital. Unrestricted Kohn–Sham solution is also seen. The constrained density functional theory[88-90] is one possible way to obtain the diabatic states.
Computational cost for DC approach in TEET coupling is generally quite mild. It typically involves four ground-state calculations, for the two isolated fragments in both initial and final states, and an SCF cycle to evaluate the necessary overlap and Hamiltonian matrix elements. As the diabatic states are based on ground state models, the application of this approach is limited to TEET involving the lowest-energy triplet states.
For SEET, similar DC scheme is possible but less seen. An excited state typically involves several configurations, and a DC would involves an account of many terms in configuration interactions, which increases the computational costs. Fink et al. have used Boys localization to construct localized HF orbitals. The evaluation of eq. (41) requires calculations of many configuration interactions and each configuration interaction is a time-consuming computation because the localized orbitals are not orthogonal. Consequently, this type of the DC model is not frequently seen.
Directly evaluating the Coulomb integrals for SEET
It is common to estimate the SEET coupling with the Coulomb coupling eq. (23).[29, 32, 52, 53, 92-101] Typically the transition densities of individual donor and acceptor fragments are used, and therefore, the computational cost is reduced to that for two excited state calculations, one for each fragment. This approach is equivalent to a direct computation of Hif approximately without considering the effect of overlap, and the exchange coupling is often ignored as well.
There are several computational approaches developed for evaluating the Coulomb integrals. The Coulomb coupling is in the form of electrostatic interaction between the transition densities of donor and acceptor. If the extent of the transition density distributions is much smaller than their separation, we can expand the Coulomb coupling term in eq. (23) with multipole expansion [eq. (27)]. The simplest model to estimate the Coulomb coupling is the dipole–dipole interaction, eq. (29). It is still the most commonly used way to estimate the SEET coupling.[32, 92-97]
Early work with semi-empirical Hamiltonians have already reported the Coulomb coupling for a number of systems.[29, 52-54, 98] With ab initio calculation, a full account for the Coulomb coupling was calculated by first recording the transition density in three-dimensional (3D) grids, followed by a 6D Riemann sum for the integration described in eq. (23).[29, 31] This approach requires a large amount of computer time, and the numerical quality is rather limited due to the steep cusp in the Coulomb kernel and the finite grid size. Nevertheless, it has been the simplest possible approach without having to change the quantum chemistry program, an option that is not always available to most research groups.
Madjet et al. have developed transition charges from electrostatic potential (TrEsp), a numerically efficient method for the calculation of eq. (23). In this TrEsp method, atom-centered transition charges are obtained from a fit of the 3D electrostatic potential of the ab initio transition density of a molecule. Equation (23) is calculated from the Coulomb interactions between the derived transition charges of the two molecules. The accuracy of this approach can be sensitive to the electrostatic potential fitting algorithm.
The Coulomb integration in eq. (23) can be calculated using the analytical integration routine of atomic orbitals in quantum chemistry computation programs.[36, 100, 102] This is the most efficient and accurate approach, because the Coulomb integration is composed of two-electron integrals, which is a time-consuming part in the quantum calculation that has been improved greatly for any reasonably efficient code. Therefore, when Coulomb coupling is implemented in the program package, the computational cost is greatly reduced and it becomes less than one round of excited state matrix element evaluation.
The Coulomb coupling may not follow the distance dependence when the interfragment separation is small. The first possibility is that the effect of a higher-order multipole may exist, and there may be a higher-order inverse polynomial distance dependence. For example, a quadrupole–dipole interaction decays as , leading to a in the SEET rate. The second possibility is, when the interfragment separation is smaller than the radius of the sphere enclosing the transition density , the spherical harmonic expansion is no longer hold. In this case, it is not possible to dissect the Coulomb coupling into multipole interactions.
In several studies, a deviation from a typical dipole–dipole form in short distances is reported.[103-108] At short distances, both the exchange and overlap contributions are expected. However, the dipole approximation may also break down in short donor-acceptor distances, and the full Coulomb coupling may not follow a typical dipole–dipole form. The spherical harmonic approximation may not be valid or the effects of higher-order multipole terms may be important.[29, 31] Therefore, observation of a distance dependence that is steeper than is not sufficient to conclude the role of the short-range couplings. We also note that the higher-order multipoles have a larger power distance dependence (i.e., when fitted to , n is larger than 3), but the overall Coulomb coupling may have a smaller apparent power dependence ( ) due to phases in these multipoles' contribution.
A direct evaluation of the full Hamiltonian interaction between the donor and acceptor excitations of the two fragments is less frequently seen, as the Coulomb coupling has been regarded as the major term in the coupling. There exist a limited number of methods to account for the full coupling, and one such useful approach is discussed below.
Directly evaluating the exchange integrals
Similar to the Coulomb coupling, the Dexter exchange coupling can be evaluated by the quantum chemical approach. The integration in eq. (31) can be calculated using the analytical integration routine of atomic orbitals in quantum chemistry computation programs. We note that the exchange coupling exists in both SEET and TEET, but the evaluation for exchange coupling is mostly performed for TEET problems.
Early work on TEET coupling reported the exchange integral values, and some of the resulting TEET rates are at least two orders of magnitude lower than those expected from experimental results. These are based on semi-empirical Hamiltonians that use predefined parameters and functions for the one- and two-electron integrals.[53, 54] Many two-electron integrals are also ignored due to their negligible contributions to the state energy. For a small quantity such as the exchange integral, it is important to use ab initio models.
Full ab initio account for the coupling improves the quality of the final exchange coupling values.[66, 85] In these works, it was found that the full TEET coupling magnitudes and attenuation rates with a series of fully stacked π-conjugated molecule pairs are very similar irrespective of their molecular size. However, such a property is not seen in the exchange coupling values. For a set of π-conjugated molecules, the value of the exchange coupling has only a minor correlation to the full TEET coupling. This result indicates that the exchange coupling may not be a good representation for the full TEET coupling, despite of its consistence with the physical intuition for TEET. The overlap and other contribution to the TEET coupling may also be important, as discussed below. A full account for the TEET coupling is a more reliable approach.
The fragment difference approaches
For charge-transfer coupling involving excited states, there exist two schemes that are useful: the generalized Mulliken–Hush (GMH) method and the fragment charge difference (FCD) scheme. In GMH and FCD, two eigenstates, and are used to form charge-localized states, and :
where θ is the transformation parameter. The linear transformation is determined by a diagonalization of either a dipole operator (GMH) or a charge difference operator (FCD). Similar schemes for EET coupling were developed before.[36, 66] Below we outline these schemes and discuss their features and applications.
The fragment spin difference scheme
Similar to the charge difference used in FCD, we define a fragment spin difference (FSD) matrix, with its matrix element being a measure of the spin localization for the triplet states:
where is the spin population that is the difference between one-electron α and β density matrices
Here ( ) is the alpha (beta) spin contribution of the transition density matrix . In a single excitation model such as the CIS, the transition density can be expressed in terms of MO basis. The α transition density between m-th and n-th CIS wavefunctions is given by
where is the ground-state density, is the coefficient matrix for the n-the CIS solution, and are occupied and virtual orbitals, respectively. The superscript α indicates that all these terms are associated with the α-spin orbitals. The definition for is similar.
The spin-localized states are thus the states that diagonalize the FSD matrix,
with the unitary transformation matrix
Then the transformation angle θ satisfies
We note that the transition density expressed in the atomic orbitals is not symmetric, which would lead to a non-Hermitian FSD matrix . Following previous work on FCD, we can take the Hermitian part of the original FSD matrix for FSD calculation:
Following the strategy used in FCD and FSD described above, the fragment difference schemes can be generalized for SEET if we can define a quantity as a measure of the excitation localization. In the analysis of electronic transition, the difference between one-electron densities of the excited and the ground states can be described as the difference between an attachment and a detachment densities.
where the detachment density is the negative definite part of , and it can be considered as a hole density associated with the electronic transitions, and the attachment density , the positive definite part of corresponds to the electron density associated with the additional electronic density in the transitions. Therefore, an excitation can be viewed as the creation of an electron-hole pair, and the corresponding “excitation density”, , can be defined as the sum of the attachment and detachment densities,
The diagonal terms are simply the detachment and attachment densities. The off-diagonal terms can be a direct generalization from the diagonal terms. For example, with the single-excitation theory, the state-to-state transition density is given by
where the first term in the right-hand side is composed of the occupied orbital contributions, corresponding to the negative detachment transition density, and the second term denotes the attachment transition density that is sum of the virtual orbital contributions.
Similar to the spin difference [eq. (44)], we now define the FED matrix as a measure of the excitation localization, and its matrix element is defined as:
The excitation-localized states are thus the states that diagonalize the FED matrix
with the same unitary transformation matrix U as defined in eq. (48). We note that the FED matrix is not symmetric if the off-diagonal excitation density is expressed in atomic orbital basis. In order for FED calculation, similar symmetrization of can be performed [eq. (50)].
Similar to eq. (49), the transformation angle θ satisfies
The coupling for SEET is then
We note several features of the FSD and FED couplings here. First, the coupling is extracted from the eigenvalues of the system Hamiltonian. Therefore, the final coupling contains all contributions from the Hamiltonian used. A special case is at the resonance condition, where the diagonal matrix elements are identical, the expression eqs. (51) and (58) are reduced to half energy gap, eq. (40). Second, the coupling is evaluated in terms of the eigenstate properties obtained from the standard quantum chemical calculations. No further (artificial) constrain on the excitation is required, which is advantageous over the DC approach. Finally, the FSD and FED coupling can be used for general systems, regardless of their symmetry. They have been utilized several works as they are developed.[62, 112-118]
Similar approaches are proposed recently. Subotnik et al.[67, 119, 120] extended Foster and Boys and Edmiston and Ruedenberg algorithms to construct localized excited states for EET. It is generally a multistate-based approach, and the electronic coupling for EET can be obtained similarly.
Effects of the Surrounding Medium
In the following, we briefly review the effects of surrounding medium studied in recent literature.
Implicit solvation models for SEET
The Coulomb coupling is in the form of Coulomb interaction energy between the donor and the acceptor's transition densities. Similar to the corresponding electrostatic problem, there is a “dielectric” effect from the surrounding medium, except that the optical dielectric constant should be used, and this effect is mainly due to the polarizability of the surrounding molecules.
The effect of solvent is accounted for in the original Förster's theory [eq. (11)], where the factor is like a direct dielectric screening. As the optical dielectric constant ( ) is n2, the Förster's theory has a basic screening effect in the coupling, and it is basically the polarizability of the solvent.
SEET has been treated with a classical formulation in situations in which planar dielectric interfaces are nearby.[123-125] Chance et al.[123, 124] treated the transition dipole as a harmonic oscillating dipole with damping to properly account for the lifetime. In this case, the dielectric medium interacts with the oscillating transition dipoles, and the effects of the medium are described using classical dielectric theory. Quantum electrodynamics (QED) theory offers another means by which to describe SEET with the inclusion of the medium influence. The Hamiltonian for the interaction between the transition dipoles and the quantized displacement field can be properly written within QED, and dipole–dipole resonant energy transfer can be derived from first principles.[126-131]
A general TDDFT was used to obtain the EET coupling, and the effect of surrounding medium was formulated. With illustrated models, it was shown that for two separated molecules, the medium adds a dielectric screening effect to the Coulomb coupling of their transitions ranging from for a dipole to in the limit of high-order multipoles. However, if the two molecules are so closely spaced that they effectively reside in a single cavity, the medium can either enhance or reduce the strength of the coupling depending on the orientation and the alignment of the two chromophores.
Following Ref. , a polarizable continuum model-based computation for the solvent effect is developed. It has been applied to the light-harvesting systems in photosynthesis.[133, 134] In these works, the EET couplings among pigments in photosynthetic proteins were obtained from the directly evaluating the Coulomb and exchange integrals with including the polarization effect. The results reveal the distance dependence of solvent screening on the EET coupling. At large donor-acceptor distances where the chromophores can be separated by solvent, the solvent effect is independent of the separation, and the screening factor is between that for point-charge model, and the Onsager dipole model, . At separations less than about 20 Å, it was found that the screening factor attenuates exponentially when the chromophores share a common cavity.
Bridge mediation effect in SEET
A similar polarization mediated effect for SEET was seen in bridge-linked systems. In Ref. , it was shown that the bridge may act as a dielectric block that modulates the SEET coupling. For the collinear case, where the donor and acceptor transition dipoles are aligned to the donor-bridge-acceptor (DBA) direction, the polarizable bridge was found to bring a large enhancement effect to the Coulomb coupling, whereas for the parallel case, where the transition dipoles are perpendicular to the DBA molecule, there is a small reduction in the bridge mediation effect.
Explicit solvent model for TEET
Dielectric continuum solvent models use classical electrostatics to approximate the solute–solvent interactions to the Coulomb couplings, which only exist in SEET. For TEET, there is no such dielectric solvent effect because there is no Coulomb coupling. The intervening medium allows a better electron tunneling, and the one-electron orbitals are with longer exponential penetration. Therefore, there is a superexchange effect[136, 137] in both the exchange and overlap terms. In this case, the solvent molecules should be explicitly considered in the quantum chemical calculations. In this way, the intervening solvent can be treated as a nonresonant bridge and the superexchange theory can be applied. A recent quantum mechanics/molecular dynamics study based on FED shows that an intervening solvent can modify the distance dependence of TEET in a stacked perylene dimer. Such results can be rationalized by using McConnell's superexchange model for electron transfer, where smaller triplet energy differences between the donor and the solvent lead to smaller coupling distance decay parameters.
The optical dielectric constants are typically small, and therefore, the solvent polarization effect in SEET coupling is modest in most cases. Systems with large polarizability are those with a low band gap, such as molecules with a large π-conjugation system. The polarization effect is likely to affect more when the environment is composed of molecules with large polarizability, and the aromatic residues in a protein matrix are likely important.
Another important character for the polarization effect is its large dependence on molecular orientation. The orientational dependence can be averaged in a disordered system, while in systems with fixed molecular orientation, such as the light-harvesting protein complexes, such an effect can be highly heterogeneous and can lead to special effect in the strength of exciton coupling and subsequent migration.
A Closer Look into the Short-Range Coupling
Nature of the short-range coupling in SEET
The formal difference between SEET and TEET [eqs. (19) and (32)] is the Coulomb coupling term. In an earlier work, we took the difference of the full SEET coupling (from the FED scheme) and the Coulomb coupling and found that it decays exponentially. The magnitude, exponential decay rate, and the basis set dependence are all very similar to the full TEET coupling. Therefore, this difference of SEET coupling and the Coulomb coupling can be regarded as the short-range coupling, which may contain the exchange and the overlap effects.
Here, we provide an example to see the short-range coupling in SEET. The FED and the Coulomb couplings for a pair of stacked naphthalenes were calculated in the CIS model. The low-lying excited states of a naphthalene include ( ), ( ) and ( ). All three states were considered in the calculations. In Figure 6, it is seen that at long distances, all FED couplings are identical with the Coulomb couplings. The deviation of the FED and Coulomb couplings at a small separation is the short-range contribution. The short-range coupling can be calculated by
and the results are plotted in Figure 7, as a function of intermolecular distances. In Figure 7, it is seen that the sizes and the decay rates of the short-range couplings are very similar among all three states, despite of their large difference in the Coulomb coupling. The Coulomb coupling arises from the long-range Coulomb interaction of the transition densities (and transition dipoles), whereas the exchange and overlap effects are from the interaction in the asymptotic region of the molecules. This result indicates a very small correlation for the Coulomb coupling and the short-range coupling.
The short-range coupling may be important, especially for systems where the molecules are in van der Walls contact. The short-range coupling in SEET may include the exchange and overlap effects as indicated in eq. (19), and it is very similar to the total TEET coupling.
The overlap contribution
In eq. (19), the term under a first-order approximation is the interaction arising from the overlap of donor and acceptor's transition densities that can be written as,
where is the transition energy of the monomers. It is usually assumed that the diabatic basis are excitation in individual isolated donor and acceptor fragments. An overlap generally arises as the two fragments approaches each other. We note that the result in eq. (60) is similar to the first-order term in Sij in eq. (41), and it is expected to decay exponentially as the interfragment distances increase.
If the MOs of the two diabatic states are not orthogonal, as the case for the DC with two UHF solutions, all the one- and two-electron terms in the Hamiltonian would have an overlap dependence.[139-141] If we follow the four-orbital example in the theory section but assume that there is orbital overlap between donor and acceptor, the two-electron integral eq. (22) becomes
where the two-electron integrals are
The first two terms in eq. (61) are equivalent to the expression in eq. (22), and the remaining terms are scaled by the square of the orbital overlap: . Harcourt et al. derived the overlap term explicitly with orbital overlap in the four-orbital model. The overlap contribution contains the orbital-overlap dependent one- and two-electron integrals [eqs. (21) and (22) in Ref. ]. In general, the overlap coupling is a short-range effect that takes place in a short donor-acceptor distance.
The overlap contribution was often neglected in the literature[44, 47, 50-54, 103, 142] because it was believed to be small. Its importance in EET coupling was demonstrated in two earlier quantum mechanical studies.[55, 56] In Scholes et al.'s work, the calculated and for a pair of stacking ethylene dimers at a distance 4 Å are 0.477 and 1.47 meV, respectively. We also note that and are not of the same sign (phase). It has been stated the exchange integral in Hif is cancelled by integrals that contribute to [eq. (41)]. The exchange and overlap effects have opposite contributions to the TEET coupling, and the overlap term has been reported to be larger than the exchange term.
Contribution from ICs
There have been discussions on the contribution of ICs in the TEET coupling calculation.[50, 55, 56, 143] If we assume that the set of diabatic wavefunctions can be projected to a set of localized wavefunctions , with small components in the ICs or :
where are the normalizing factors. λ and µ are mixing coefficients which can be obtained from first-order perturbation theory,
Here, we have assumed Ni and Nf are approximately one. is the energy difference between the neutral and ICs. The interactions of ICs as second-order contribution to the TEET coupling are,[49, 50]
Therefore, an apparent ionic contribution to the EET coupling may be obtained simply by projecting the diabatic states to a more localized basis.
The IC contribution was found to be larger than the exchange and overlap coupling values.[50, 55] In Ref. , the calculated mixing coefficients [λ and µ as in eqs. (63) and (64)] are less than 1% in population, but as the numerator in eq. (67) is composed of electron transfer coupling which is relatively large in value, the overall contribution from ICs become non-negligible. In another work, the authors used orthogonal localized MOs through block diagonalization in the Fock matrix. As a result, there is absolutely zero overlap between and . It was found that the contributes 90% of the full coupling values. With the FSD scheme, we have also checked for the IC component in the diabatic states. We tried to obtain the IC component from the natural transition orbitals[144, 145] and calculated the contribution to corresponding TEET coupling. For a pair of naphthalene molecules separated at 4 Å, the IC population was less than 0.1%, and there is virtually no detectable IC contribution for TEET in this case.
In general, the determination of the diabatic states and , as well as the localized wavefunctions and are not unique. Diabatic states arise from the need to treat the break down of Born–Oppenheimer approximation at the transition, and most of the strategies to construct diabatic states require some degree of localization in the excitation, as discussed in the subsections for computation methods above. Therefore, projecting the diabatic states, and , to a set of localized states, and , is like a comparison of two sets of diabatic states and the results from Refs. [   and  indicate that the partitions of the TEET coupling can be greatly different when one uses a more localized set of states. We note that the localized MOs for π-electrons used in Refs. [  and  were represented by the linear combinations of monomer atomic orbitals. The block-diagonalized MOs used in Ref.  are a set of strictly localized basis that has zero overlaps. In these works, are obtained with very restricted localized wavefunctions.
We also note that the localized MOs used in Refs. [  and  are a basis for the typical exchange integral evaluation, as described in the DC section. Early theoretical works reported exchange integrals that are often smaller TEET rates than those expected from experimental results.[53, 54] In addition that the semi-empirical Hamiltonians that tends to underestimate the exchange integral, we believe that the restricted localized wavefunctions used, contribution can be important in the full TEET coupling.
The diabatic state wavefunctions are generally not strictly localized. Different definition for the diabatic states may have different extent of localization, and an IC contribution may exist if the diabatic wavefunctions are projected to a differently localized basis. Conversely, the TEET coupling values are similar when they are obtained from half CIS energy gap, DC, or the FSD schemes.[66, 85] TEET rates derived from FSD couplings are shown to be consistent with experimental results. These results seem to indicate that different physical assumption may still lead to similar diabatic states and similar coupling values, as long as a reasonably large basis set is used.
As it has been shown that rigorous diabatic states do not exist for general polyatomic molecules, there is not yet a universal definition for the diabatic states, the zero-order state which the coupling is based on. From the break-down of the Born–Oppenheimer approximation, a set of ideal diabatic states should have zeros in the off-diagonal nuclear kinetic energy operator. It has been shown that an exact set of diabatic states is not possible in most cases. Therefore, it would be desirable to see how the diabatic states obtained from various techniques behave in the off-diagonal elements of the nuclear kinetic energy operator. Before that, we believe that diabatic states derived from the techniques discussed in this work are generally useful.
We have reviewed the theory and physics behind EET couplings for both singlet and triplet cases. The computational schemes are discussed, and the factors contributing to the SEET and TEET couplings are discussed, with a goal to aid future work involving calculation, prediction, and understanding of the EET coupling.
We gratefully acknowledge the support from the National Science Council, Republic of China and the Academia Sinica.
Zhi-Qiang You obtained his Ph.D. from National Tsing-Hua University in Taiwan in 2012, and he developed ab initio methods for excitation energy transfer coupling. He has also developed applications for triplet energy transfer in photosynthetic light-harvesting complexes. He is currently a postdoctoral fellow at Academia Sinica with Dr. Chao-Ping Hsu. He is currently working on the use of long-range corrected density functional theory in calculating electron transfer coupling. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Chao-Ping Hsu obtained her Ph.D. degree from California Institute of Technology in 1998 under the supervision of R. A. Marcus. She was a Miller Research Fellow in University of California at Berkeley, 1998–2002, where she used quantum chemistry methods to quantify the energy transfer rates in light-harvesting complexes, and it was done under the guidance of Graham R. Fleming, Martin Head-Gordon and Teresa Head-Gordon. She was an Assistant Research Fellow in Institute of Chemistry, Academia Sinica in Taiwan (2002–2007). She is currently an Associate Research Fellow in Institute of Chemistry, Academia Sinica (2007–current). She and her research group have developed or improved computational methods for electron transfer and excitation energy transfer couplings, which allows a first-principle prediction for the corresponding rates for these processes. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]