Towards a molecular scale interpretation of excitation energy transfer in solvated bichromophoric systems. II. The through-bond contribution

In this paper we present a quantum-mechanical investigation on the mechanisms which promote intramolecular EET coupling. This investigation is done by using a new computational strategy in which we combine a configuration-interaction and a linear response approach. The combined use of these two methods allows a direct identification and a quantification of both "direct" (coulomb and exchange) and through-bond (superexchange and charge-transfer) contributions. In addition, solvent effects are introduced using the polarizable continuum model. The method is applied to a family of naphthalene-bridge-naphthalene and naphthalene-bridge-anthracene systems, and the results obtained are compared with experiments. The results found suggest that the through-bond charge-transfer effects are not significant when the EET goes through permitted excitations on distant chromophores (see DN4 and DN6) while they become as important as (or even more important than) the covalent terms for EETs involving weakly allowed excitations (see A6N). By contrast, the presence of a very short bridge (in DN2) allows a very efficient delocalization of the excitation energy which is also largely modified by the presence of a solvent.     

Photophysical properties of natural light-harvesting complexes studied by subsystem density functional theory

In this study, we investigate the excited states and absorption spectra of a natural light-harvesting system by means of subsystem density functional theory. In systems of this type, both specific interactions of the pigments with surrounding protein side chains as well as excitation energy transfer (EET) couplings resulting from the aggregation behavior of the chromophores modify the photophysical properties of the individual pigment molecules. It is shown that the recently proposed approximate scheme (J. Chem. Phys. 2007, 126, 134116) for coupled excitations within a subsystem approach to time-dependent DFT is capable of describing both effects in a consistent manner, and is efficient enough to study even the large assemblies of chromophores occurring in the light-harvesting complex 2 (LH2) of the purple bacterium Rhodopseudomonas acidophila. A way to extract phenomenological coupling constants as used in model calculations on EET rates is outlined. The resulting EET coupling constants and spectral properties are in reasonable agreement with the available reference data. Possible problems related to the effective exchange-correlation kernel are discussed.     

Ligand-length dependence of the intramolecular electron transfer through-bond coupling parameter

For a donor-bridging-ligand-acceptor molecule excited in a non-ionized state, a Bloch transformation is used to decompose the time-dependent Schrödinger equation and to construct an effective evolution with the Bloch effective hamiltonian. The effective through-bond coupling is calculated in a tight-binding approximation as a function of the number of ligand active units. The precision of the effective coupling is tested by a comparison between exact and effective transfer rate. A class of hamiltonians is given where this coupling does not decrease with an increase of the ligand length.     

Theory and calculation for the electronic coupling in excitation energy transfer

Excitation energy transfer (EET) is a process where the electronically excitation is transferred from a donor to an acceptor. EET is widely seen in both natural and in artificial systems, such as light‐harvesting in photosynthesis, the fluorescence resonance energy transfer technique, and the design of light‐emitting molecular devices. In this work, we outline the theories describing both singlet and triplet EET (SEET and TEET) rates, with a focus on the physical nature and computational methods for the electronic coupling factor, an important parameter in predicting EET rates. The SEET coupling is dominated by the Coulomb coupling, and the remaining short‐range coupling is very similar to the TEET coupling. The magnitude of the Coulomb coupling in SEET can vary much, but the contribution of short‐range coupling has been found to be similar across different excited states in naphthalene. The exchange coupling has been believed to be the major physical contribution to the short‐range coupling, but it has been pointed out that other contribution, such as the orbital overlap effect is similar or even larger in strength. The computational aspects and the subsequent physical implication for both SEET and TEET coupling values are summarized in this work. © 2013 The Authors. International Journal of Quantum Chemistry Published by Wiley Periodicals, Inc.     

A measure of the error in wave functions

The quantity ? = (Φ||(H ? E)Φ|) gives a measure of the error in the approximate solution, Φ (with corresponding energy expectation value E), to an eigenfunction of the Hamiltonian operator H of the system under consideration; this quantity vanishes for the exact function ψ. In a percentage scale (with 0% error for the exact function and 100% for a reference, approximate function), the error of Φ may be expressed as 100(?/?_r), where ?_r corresponds to the reference function (e.g., obtained with a minimal basis set). This approach eliminates the need of knowing beforehand the exact solution in order to have an estimate of the error of an approximate solution.     

Weak distance dependence of through-bond interactions in tetrahydro-4H-thiopyran-4-ylidene end-capped oligo(cyclohexylidenes); a computational survey

Calculations on members of the oligo(cyclohexylidene) series [(n), n = 1-5)] and related tetrahydro-4H-thiopyran end-capped analogues [(n), n = 1-4)] show a strong through-bond coupling between their pi bonds and sulfur lone pairs (Lp(pi)S). This coupling is mediated by an interaction between the H(ax)-C-C-H(ax) structural sub-units and the pi bonds connecting the cyclohexyl moieties. A comparison of the length dependency of the through-bond coupling via an oligo(cyclohexylidene) and an alkane bridge [divinyl alkanes (n)] shows that oligo(cyclohexylidenes) are more efficient in mediating through-bond couplings over large distances. Oligo(cyclohexylidene) bridges exhibit molecular wire characteristics.     

How solvent controls electronic energy transfer and light harvesting: toward a quantum-mechanical description of reaction field and screening effects

This paper presents a quantum-mechanical study of electronic energy transfer (EET) coupling on over 100 pairs of chromophores taken from photosynthetic light-harvesting antenna proteins. Solvation effects due to the protein, intrinsic waters, and surrounding medium are analyzed in terms of screening and reaction field contributions using a model developed recently that combines a linear response approach with the polarizable continuum model (PCM). We find that the screening of EET interactions is quite insensitive to the quantum-mechanical treatment adopted. In contrast, it is greatly dependent on the geometrical details (distance, shape, and orientation) of the chromophore pair considered. We demonstrate that implicit (reaction field) as well as screening effects are dictated mainly by the optical dielectric properties of the host medium, while the effect of the static properties is substantially less important. The empirical distance-dependent screening function we proposed in a recent letter (Scholes, G. D.; Curutchet, C.; Mennucci, B.; Cammi, R.; Tomasi, J. J. Phys. Chem. B 2007, 111, 6978-6982) is analyzed and compared to other commonly used screening factors. In addition, we show that implicit medium effects on the coupling, resulting from changes in the transition densities upon solvation, are strongly dependent on the particular system considered, thus preventing the possibility of defining a general empirical expression for such an effect.     

Exchange coupling mediated through-bonds and through-space in conformationally constrained polyradical scaffolds: calix[4]arene nitroxide tetraradicals and diradical

Calix[4]arenes constrained to the 1,3-alternate conformation and functionalized at the upper rim with four and two tert-butylnitroxides have been synthesized and characterized by X-ray crystallography, magnetic resonance (EPR and (1)H NMR) spectroscopy, and magnetic studies. The 1,3-alternate nitroxide tetraradical and diradical provide unique polyradical scaffolds for dissection of the through-bond and through-space intramolecular exchange couplings. In addition, detailed magnetic studies of the previously reported calix[4]arene nitroxide tetraradical, which possesses cone conformation in solution, reveal conformational dependence of exchange coupling. Through-bond coupling between the adjacent nitroxide radicals is mediated by the nitroxide-m-phenylene-CH(2)-m-phenylene-nitroxide coupling pathway, and through-space coupling is found between the diagonal nitroxide radicals at the conformationally constrained N...N distance of 5-6 A. Magnetic studies of the calix[4]arene polyradical scaffolds in frozen solutions show that the through-bond exchange coupling in the 1,3-alternate calix[4]arene tetraradical is antiferromagnetic, while that in cone calix[4]arene tetraradical is ferromagnetic. The through-space exchange couplings are antiferromagnetic in both cone and 1,3-alternate calix[4]arene tetraradical, as well as in the 1,3-alternate calix[4]arene diradical. The exchange coupling constants (|J/k|) are of the order of 1 K.     

Interpolant polynomial technique applied to the PPP model. II. Testing the interpolant technique on the Hubbard model

One is often led, in quantum mechanics, to a perturbative solution of an eigenvalue problem, which is defined by a given Hamiltonian. The perturbative series for the energy which results will be a function of a coupling constant which appears in the Hamiltonian. In this article, the perturbative series for the energy of a state of a cyclic polyene ring which are valid for the small and large coupling limit of the model are used to construct algebraic functions. These algebraic functions are defined in terms of polynomials which are given as a function of the energy variable and coupling parameter and can be solved to give the energy as a function of coupling. It is found that relatively small polynomials give very good agreement with the exact values and that the accuracy of the results increases rapidly as the degree of the polynomial increases. The final goal of this and subsequent articles is to study energy levels in PPP models of planar conjugated hydrocarbons. In this article, we test an interpolant technique on the case of the one-dimensional Hubbard model, where an exact solution can be obtained by solving a system of nonlinear equations. In the case of the Hubbard model, the correlation effects are overestimated. Therefore, if the technique works for the Hubbard model, it is reasonable to assume that the technique would work even better for the PPP model. © 1996 John Wiley & Sons, Inc.     

Ab initio characterization of electron transfer coupling in photoinduced systems: generalized Mulliken-Hush with configuration-interaction singles

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.     

Advanced theory of excitation energy transfer in dimers

Previously, we developed a unified theory of the excitation energy transfer (EET) in dimers, which is applicable to all of the cases of excitonic coupling strength (Kimura, A.; Kakitani, T.; Yamato, T. J. Phys. Chem. B 2000, 104, 9276). This theory was formulated only for the forward reaction of the EET. In the present paper, we advanced this theory so that it might include the backward reaction of the EET as well as the forward reaction. This new theory is formulated on the basis of the generalized master equation (GME), without using physically unclear assumptions. Comparing the present result with the previous one, we find that the excitonic coupling strengths of criteria between exciton and partial exciton and between hot transfer and hopping (F?rster) mechanisms are reduced by a factor of 2. The critical coherency eta c is also reduced significantly.     

The centrifugally decoupled exponential distorted wave (CDEDW) approximation for the calculation of rotationally inelastic molecular collision cross se

A new approximate method is presented for the rapid calculation of rotationally inelastic molecular collision cross sections. The method is called the centrifugally decoupled exponential distorted wave (CDEDW) approximation and involves the combination of two well known approximations. The first approximation is the neglect of the off-diagonal coupling terms which arise from the orbital angular momentum operator in the coupled differential equations in the body-fixed axis system. The second approximation is to treat the remaining coupling terms, which arise from the interaction potential, using a unitary perturbation approximation. The CDEDW method is applied to the calculation of total and partial rotationally inelastic cross sections in the ArN₂ system, and detailed comparisons are made with exact and several other types of approximate calculations. Agreement with exact calculations is good and often comparable with the coupled states and p-helicity decoupled approximations. The CDEDW method requires a similar amount of computational effort to the infinite order sudden (IOS) approximation, and we show that for the present system the CDEDW method gives more reliable results.     

Excitation energy transfer in ion pairs of polymethine cyanine dyes: efficiency and dynamics

The present work deals with singlet excitation energy transfer (EET) occurring in contact ion pairs (CIPs) of several anionic oxonol analogues (acting as EE donors) and cationic cyanines (acting as acceptors) characterized by off resonance individual transitions. Combining conductometric and spectroscopic measurements with decreasing solvent polarity, we were able to observe a progressive ion pairing leading first to solvent-separated ion pairs (SSIPs) and then to CIPs. Analysis of the absorption spectra of three selected salts (A2,C1, A2,C2, and A1,C4) in chloroform-toluene mixtures showed that the transformation of SSIP into CIP involves the appearance of a certain exciton coupling, the extent of which decreases regularly with increasing gap between the local excitation energies. Fluorescence excitation spectra showed that EET occurs in CIP, and EET efficiencies were evaluated with a procedure expressly devised for weakly emitting donors. These were between 0.2 and 0.65 for the examined ion pairs involving anions A1 and A2. The spectroscopic study was complemented by a theoretical investigation aimed at establishing the dynamic regime of the observed EET. From classical MD simulations and local full geometry optimizations, A2,C1 and A2,C2 were found to form rather stable sandwich-type CIP structures with interchromophore distances (R) of about 0.45-0.50 nm. The donor-acceptor electronic coupling was calculated in terms of Coulombic interactions between atomic transition charges. For CIP, the electronic coupling was decidedly beyond the limit of the weak coupling required for an incoherent F?rster-type mechanism. Thus, we tried to arrange the EET dynamics within the theory developed by Kimura, Kakitani, and Yamato (J. Phys. Chem. B 2000, 104, 9276) for the intermediate coupling case, which provides analytical expressions of time-dependent occupation probability, EET rate, and coherency in terms of two basic quantities: the electronic coupling and a correlation time related to the Franck-Condon factor. The latter was shown to be primarily modulated by F?rster's spectral overlap integral (related in turn to the excitation energy gap). Calculations were carried out for the three sample systems using three values of the electronic coupling roughly corresponding to CIP, 1.0, and 2.0 nm interchromophore distances. At the CIP distance, EET in both A2,C1 and A2,C2 was predicted to occur with a partial exciton mechanism, very short transfer times (about 10 fs), and high degree of coherence. In A1,C4 (having the largest energy gap), EET was found to occur with a hot-transfer mechanism. More or less hot-transfer dynamics appeared to be retained by all three systems at R = 1.0 nm. Fully incoherent EET appeared to become operative only at distances larger than 2.0 nm.     

Orbital Theory of Heterolytic Fragmentation and Remote Effects on Nitrogen Inversion Equilibria

From a molecular orbital study of model systems we derive the electronic requirements for the Grob fragmentation. The necessary condition for an allowed fragmentation in an X-C₁-C₂-C₃-N system, or the amino cation ⁺C₁-C₂-C₃-N is the level ordering A below S . This in turn is set by maximal through-bond coupling of the empty cation orbital and the nitrogen lone pair. The conformational dependence of through-bond coupling is exactly that derived by Grob, namely parallel orientation of the cation orbital (or the C-X bond), the C₂-C₃-σ-bond, and the N-lone-pair. When the C₁-C₂-C₃ and C₂-C₃-N angles are small, the through-space interaction dominates, reversing the level ordering to S below A , and consequently makes the fragmentation forbidden even though the conformational requirements for it are met. Ring closure becomes allowed. Some examples exploiting this result are presented, as well as procedures for enhancing through-bond coupling and thus fragmentation. The through-bond-effect has also kinetic consequences, allowing the definition of a new type of remote neighbouring group participation operative through bonds and not by direct overlap. The position of equilibria in nitrogen inversion processes should also be influenced by remote substituents which are π-acceptors or donors.     

Quantum investigation of linear triatomic exchange reactions. A computational method

The model hamiltonian for a linear triatomic exchange reaction is derived in natural reaction coordinates, and a method is developed to solved the system of wave equations in close coupling approximation. The interaction zone is divided into subintervals, in each of which the coefficients of the system are assumed to be constant; this assumption provides an exact analystical solution over the interval. Special boundary conditions which account for the derivatives of the coefficients are used to match the solutions for adjacent intervals. The method converges to the exact solution with an increasing number of intervals of decreasing length. Results obtained for certain model system illustrate the merits of the method and its convergence.     

Linking the historical and chemical definitions of diabatic states for charge and excitation energy transfer reactions in condensed phase

Marcus theory of electron transfer (ET) and Fo?rster theory of excitation energy transfer (EET) rely on the Condon approximation and the theoretical availability of initial and final states of ET and EET reactions, often called diabatic states. Recently [Subotnik et al., J. Chem. Phys. 130, 234102 (2009)], diabatic states for practical calculations of ET and EET reactions were defined in terms of their interactions with the surrounding environment. However, from a purely theoretical standpoint, the definition of diabatic states must arise from the minimization of the dynamic couplings between the trial diabatic states. In this work, we show that if the Condon approximation is valid, then a minimization of the derived dynamic couplings leads to corresponding diabatic states for ET reactions taking place in solution by diagonalization of the dipole moment matrix, which is equivalent to a Boys localization algorithm; while for EET reactions in solution, diabatic states are found through the Edmiston-Ruedenberg localization algorithm. In the derivation, we find interesting expressions for the environmental contribution to the dynamic coupling of the adiabatic states in condensed-phase processes. In one of the cases considered, we find that such a contribution is trivially evaluable as a scalar product of the transition dipole moment with a quantity directly derivable from the geometry arrangement of the nuclei in the molecular environment. Possibly, this has applications in the evaluation of dynamic couplings for large scale simulations.     

Consequences of a through-bond interaction in tetracyclo-[5.3.0.0.0]-deca-4,8-diene (“Hypostrophene”)

Photoelectron spectroscopy is used to demonstrate the mechanistic consequences of the level ordering in a given molecule on its reactivity, using the recently synthesized hypostrophene, which contains two CC double bonds in a rigid, cisoid conformation, as an example. The inability of this molecule to close photochemically to the saturated analog is traced to the presence of an exceptionally high-lying σ level which is ideally oriented for an effective through-bond coupling of the two π orbitals. Contrary to the norbornadiene case, this through-bond coupling overrides the direct through-space interaction, placing the in-phase combination of the two π orbitals above the out-of-phase combination, and thus converts the _π2_s+_π2_s photocycloaddition from a symmetry-allowed to a symmetry-forbidden reaction.     

Non-Born-Oppenheimer treatment of the H2 Hookean molecule

We show that the exact non-Born-Oppenheimer Schrodinger equation for the Hookean diatomic molecule H2 (a two-proton, two-electron system where the electron-proton interaction is harmonic while the proton-proton and electron-electron interactions are Coulombic) can be decoupled into equations describing the relative motion of the electrons, the relative motion of nuclei, the motion of a collective mode representing a three-dimensional harmonic oscillator, and the motion of a free particle expressed as a linear combination of the individual center-of-mass coordinates of the nuclei and electrons. Analytic solutions to the relative motion of electrons can be readily obtained for the given values of the harmonic coupling constant. However, exact analytic solutions to the equation for the relative motion of the nuclei cannot be obtained simultaneously due to the fact that the harmonic constants in these two equations are coupled. For this reason, we present for the relative nuclear motion approximate analytic wave functions, one of them obtained variationally and the other by a series solution where the coefficients are determined recursively. We also explore a variational solution to the Taylor-series expansion of the nuclear interaction potential. Properties of the electronic and nuclear intracule densities are examined at different values of the coupling constant. An interesting result of the present non-Born-Oppenheimer treatment of this harmonic model is the fact that the relative nuclear motion occurs in a highly correlated regime. This leads in a natural way to a spatial localization of the nuclei akin to Wigner electronic crystallization.