Fermi's Golden Rule Rate Research Articles

Cavity-modified Fermi's golden rule rate constants: Beyond the single mode approximation.

We extend our recently proposed theoretical framework for estimating cavity-modified equilibrium Fermi's golden rule (FGR) rate constants beyond the single cavity mode case to cases where the molecular system is coupled to multiple cavity modes. We show that the cumulative effect of simultaneous coupling to multiple modes can enhance FGR rate constants by orders of magnitude relative to the single mode case. We also present an analysis of the conditions necessary for maximizing this effect in the Marcus limit of FGR-based rate theory.

A Computational Study of the Electronic Energy and Charge Transfer Rates and Pathways in the Tetraphenyldibenzoperiflanthene/Fullerene Interfacial Dyad.

The electronic transition rates and pathways underlying interfacial charge separation in tetraphenyldibenzoperiflanthene:fullerene (DBP:C70) blends are investigated computationally. The analysis is based on a polarization-consistent framework employing screened range-separated hybrid functional in a polarizable continuum model to parametrize Fermi's golden rule rate theory. The model considers the possible transitions within the 25 lowest excited states of a DBP:C70 dyad that are accessible by photoexcitation. The different identified pathways contributing to charge carrier generation include electron and hole transfer and backtransfer, exciton transfer, and internal relaxation steps. The larger density of states of C70 appears to explain the previously observed larger efficiency for charge separation through hole transfer mechanism. We also analyze the validity of the high-temperature and short-time semiclassical approximations of the FGR theory, where both overestimated and underestimated Marcus theory based constants can be affected.

Modified Fermi's golden rule rate expressions.

Fermi's golden rule (FGR) serves as the basis for many expressions of spectroscopic observables and quantum transition rates. The utility of FGR has been demonstrated through decades of experimental confirmation. However, there still remain important cases where the evaluation of a FGR rate is ambiguous or ill-defined. Examples are cases where the rate has divergent terms due to the sparsity in the density of final states or time dependent fluctuations of system Hamiltonians. Strictly speaking, assumptions of FGR are no longer valid for such cases. However, it is still possible to define modified FGR rate expressions that are useful as effective rates. The resulting modified FGR rate expressions resolve a long standing ambiguity often encountered in using FGR and offer more reliable ways to model general rate processes. Simple model calculations illustrate the utility and implications of new rate expressions.

Two bridge-particle-mediated RET between chiral molecules.

The problem of resonance energy transfer between a pair of chiral molecules mediated by two electrically polarizable bridging particles is solved using molecular quantum electrodynamics theory. In this framework, a single virtual photon propagates between any two-coupled entities and is responsible for the conveyance of excitation energy from the emitter to the absorber. Electric dipole and quadrupole, and magnetic dipole couplings linear in the Maxwell fields are employed for donor and acceptor, while each mediator scatters two virtual photons and responds quadratically to the electric displacement field via its electric dipole polarizability. This enables fourth-order diagrammatic perturbation theory to be used to compute the probability amplitude for the process. Individual multipole moment contributions to the Fermi golden rule rate are then extracted for oriented and isotropic systems. Discriminatory transfer rates arise when either the donor or the acceptor are electric-magnetic dipole and the other has a pure multipole moment, or when both are chiral, with mixed electric dipole-quadrupole contributions vanishing in the fluid phase. The bridge-mediated transfer rate is found to be a maximum for a collinear geometry. Moreover, a multi-level model of the mediator is necessary for energy migration. Asymptotically limiting rates for arbitrary and collinear geometries are also obtained for one center purely electric dipolar and the other purely quadrupolar, or both donor and absorber purely quadrupolar. Understanding is gained of radiationless and radiative transfer mechanisms between chiral moieties in a dielectric medium.

An Accurate Linearized Semiclassical Approach for Calculating Cavity-Modified Charge Transfer Rate Constants.

We show that combining the linearized semiclasscial approximation with Fermi's golden rule (FGR) rate theory gives rise to a general-purpose cost-effective and scalable computational framework that can accurately capture the cavity-induced rate enhancement of charge transfer reactions that occurs when the molecular system is placed inside a microcavity. Both partial linearization with respect to the nuclear and photonic degrees of freedom and full linerization with respect to nuclear, photonic, and electronic degrees of freedom (the latter within the mapping Hamiltonian approach) are shown to be highly accurate, provided that the Wigner transforms of the product (WoP) of operators at the initial time is not replaced by the product of their Wigner transforms. We also show that the partial linearization method yields the quantum-mechanically exact cavity-modified FGR rate constant for a model system in which the donor and acceptor potential energy surfaces are harmonic and identical except for a shift in the equilibrium energy and geometry, if WoP is applied.

Resonance energy transfer mediated by a chiral molecule.

The problem of resonant energy transfer (RET) between an electric dipole donor, D, and an electric dipole acceptor, A, mediated by a passive, chiral third-body, T, is considered within the framework of molecular quantum electrodynamics theory. To account for the optical activity of the mediator, magnetic dipole and electric quadrupole coupling terms are included in addition to the leading electric dipole interaction term. Fourth-order diagrammatic time-dependent perturbation theory is used to obtain the matrix element. It is found that the Fermi golden rule rate depends on pure multipole moment polarizabilities and susceptibilities of T, as well as on various mixed electric and magnetic multipole moment response functions. The handedness of T manifests through mixed electric-magnetic dipole and mixed electric dipole-quadrupole polarizabilities, which affect the rate and, respectively, require the use of fourth-rank and sixth-rank Cartesian tensor averages over T, yielding non-vanishing isotropic rate formulae applicable to a chiral fluid medium. Terms of a similar order of magnitude proportional to the product of electric dipole polarizability and either magnetic dipole susceptibility or electric quadrupole polarizability of T are also computed for oriented and freely tumbling molecules. Migration rates dependent upon the product of the pure electric dipole or magnetic dipole polarizability with the mixed electric-magnetic or electric dipole-quadrupole analogs, which require fourth- and fifth-rank Cartesian tensor averaging, vanish for randomly oriented systems. Asymptotically limiting rate expressions are also evaluated. Insight is gained into RET occurring in complex media.

A computational study of the vibronic effects on the electronic spectra and the photophysics of aza[7]helicene.

We report a computational study on vibronic effects in the spectroscopy, photoinduced processes and decay back to the ground state of aza[7]helicene, a helicene with an unusually high fluorescence quantum yield (QY = 0.39). In a first step, we compute and assign the absorption and electronic circular dichroism (ECD) spectra in its full frequency range from 2.7 to 5.0 eV, accounting for nonadiabatic effects. Then we compute the quantum dynamics of the cascade of ultrafast internal conversions of the highly-excited singlet states to the lowest-energy one S1. Finally we adopt Fermi golden rule rates to compute the QY of the dye, taking into account the competition between the radiative decay and the nonradiative decays to the ground state and to the energy-accessible triplet states. We use time-dependent density functional theory (TD-DFT), including solvent (dichloromethane) effects within the polarizable continuum model, to parameterize a linear vibronic coupling (LVC) model involving the first lowest 12 singlet states and all the normal coordinates. Nonadiabatic spectra and internal conversions dynamics are then computed through wavepacket propagations with the Multilayer (ML) extension of the Multiconfigurational Time Dependent Hartree method (ML-MCTDH). We highlight the molecular vibrations playing a major role in determining the shape of the spectra and analyse the effect of inter-state couplings. At the same time we report a breakdown of perturbative Herzberg-Teller approach. The computed QY is in perfect agreement with experiment and allows us to ascertain that intersystem crossings are the processes limiting the fluorescence from S1. They involve the three lowest triplet states and are made effective by spin-orbit coupling and vibronic effects.

Computational modeling of charge hopping dynamics along a disordered one-dimensional wire with energy gradients in quantum environments.

This computational study investigates the effects of energy gradients on charge hopping dynamics along a one-dimensional chain of discrete sites coupled to quantum bath, which is modeled at the level of Pauli master equation (PME). This study also assesses the performance of different approximations for the hopping rates. Three different methods for solving the PME, a fourth order Runge-Kutta method, numerical diagonalization of the rate matrix followed by analytic propagation, and kinetic Monte Carlo simulation method, are tested and confirmed to produce virtually identical values of time dependent mean square displacement, diffusion constant, and mobility. Five different rate expressions, exact numerical evaluation of Fermi's Golden Rule (FGR) rate, stationary phase interpolation (SPI) approximation, semiclassical approximation, classical Marcus rate, and Miller-Abrahams rate, are tested to help understand the effects of approximations in representing quantum environments in the presence of energy gradients. The results based on direct numerical evaluation of FGR rate exhibit transition from diffusive to non-diffusive behavior with the increase in the gradient and show that the charge transport in the quantum bath is more sensitive to the magnitude of the gradient and the disorder than in the classical bath. Among all the four approximations for the hopping rates, the SPI approximation is confirmed to work best overall. A comparison of two different methods to calculate the mobility identifies drift motion of the population distribution as the major source of non-diffusive behavior and provides more reliable information on the contribution of quantum bath.

Charge transfer rate constants for the carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran: The spin-boson model vs the linearized semiclassical approximation.

Charge transfer rate constants were calculated for the carotenoid-porphyrin-C60 (CPC60) molecular triad dissolved in explicit tetrahydrofuran. The calculation was based on mapping the all-atom anharmonic Hamiltonian of this system onto the spin-boson Hamiltonian. The mapping was based on discretizing the spectral density from the time correlation function of the donor-acceptor potential energy gap, as obtained from all-atom molecular dynamics simulations. Different spin-boson Hamiltonians were constructed for each of the possible transitions between the three excited electronic states in two different triad conformations. The rate constants of three possible transitions were calculated via the quantum-mechanically exact Fermi's golden rule (FGR), as well as a progression of more approximate expressions that lead to the classical Marcus expression. The advantage of the spin-boson approach is that once the mapping is established, the quantum-mechanically exact FGR and the hierarchy of approximations are known in closed form. The classical Marcus charge transfer rate constants obtained with the spin-boson Hamiltonians were found to reproduce those obtained from all-atom simulations with the linearized semiclassical approximation, thereby confirming the equivalence of the two approaches for this system. Within the spin-boson Hamiltonian, we also found that the quantum-mechanically exact FGR rate constants were significantly enhanced compared to the classical Marcus theory rate constants for two out of three transitions in one of the two conformations under consideration. The results confirm that mapping to the spin-boson model can yield accurate predictions for charge transfer rate constants in a system as complex as CPC60 dissolved in tetrahydrofuran.

Polariton mediated resonance energy transfer in a fluid.

The focus of this work is on a microscopic quantum electrodynamical understanding of cumulative quantum effects in resonance energy transfer occurring in an isotropic and disordered medium. In particular, we consider quantum coherence, defined in terms of interferences between Feynman pathways, and analyze pure-amplitude and phase cross terms that appear in the Fermi golden rule rate equation that results from squaring the matrix element for mediated energy transfer. It is shown that pure-amplitude terms dominate in the near-zone when chromophores are close in proximity to one another (within a few nanometers), and phase cross terms dominate toward the far-zone when phase differences between different Feynman pathways begin to emerge. This can be understood in terms of physical attributes of the mediating photon, whose character becomes more real at long distances, coinciding with vanishing longitudinal components of the field, as transverse components begin to dominate.

RET in a dielectric medium: insights from molecular QED theory

Molecular quantum electrodynamics (QED) theory is used to study the challenging problem of resonance energy transfer (RET) in a dielectric medium characterised by a complex refractive index. First, pair transfer is briefly reviewed, in which the propagation of a single virtual polariton is responsible for conveying energy from the donor to the acceptor species. Well-known free-space Fermi golden rule rates are shown to follow in the dilute medium approximation. Next, an inert, polarisable third-body is introduced, and its role in mediating transfer is examined for the three particles in fixed relative orientation. Three coupling mechanisms involving the mediator now arise, and the total rate comprises direct and third-body mediated terms, as well as interferences between direct and indirect routes. Coherences arising from amplitude and phase terms, environment-induced decoherence due to dissipation in the medium, decay arising from damping terms featuring in the polarisability of the scatterer, in addition to distance and orientational factors, and their separate effect in modifying the rate, are all explored. Physical insight is gained into the subtle interplay of the various factors that permit enhancement or suppression of the total rate.

Efficient Charge Generation via Hole Transfer in Dilute Organic Donor-Fullerene Blends.

Efficient organic photovoltaics (OPVs) require broadband charge photogeneration with near-unity quantum yield. This can only be achieved by exploiting all pathways that generate charge. Electron transfer from organic donors to acceptors has been well-studied and is considered the primary path to charge photogeneration in OPVs. In contrast, much less is known about the hole transfer pathway. Here we study charge photogeneration in an archetypal system comprising tetraphenyldibenzoperiflanthene:C70 blends using our recently developed multispectral two-dimensional electronic spectroscopy (M-2DES), supported by time-dependent density functional theory and fully quantum-mechanical Fermi's golden rule rate calculations. Our approach identifies in real time two rapid charge transfer pathways that are confirmed through computational analysis. Surprisingly, we find that both electron and hole transfer occur with comparable rates and efficiencies, facilitated by donor-acceptor electronic interactions. Our results highlight the importance of the hole transfer pathway for optimizing the efficiency of OPV devices employing small-molecule heterojunctions.

Mediation of resonance energy transfer by two polarisable particles.

The molecular quantum electrodynamics theory is employed to calculate the matrix element and Fermi golden rule rate for resonant transfer of electronic excitation energy between a donor and an acceptor in the vicinity of two neutral electric dipole polarizable particles, which play the role of bridging species. The emitter and absorber couple linearly to the electric displacement field via their electric dipole moments, while each mediator interacts quadratically with this field through its dynamic polarizability. This form of interaction Hamiltonian enables fourth-order perturbation theory to be used to compute the probability amplitude together with summation over 24 time-ordered diagrams representing a single virtual photon exchange between each pair of coupled particles. Expressions for the migration rate mediated by two inert molecules are obtained for an arbitrary arrangement of the four species that are in fixed mutual orientation or are freely tumbling. These formulae are valid for all interparticle separation distances outside the orbital overlap region. From the general result, rate equations applicable to an equidistant collinear configuration of the four bodies are evaluated. Near- and far-zone limiting forms of the transfer rate for the relay pathway are also calculated and exhibit inverse sixth and inverse square dependences on relative separation distances between pairs of particles, confirming the short-range (radiationless) and long-range (radiative) energy transfer mechanisms associated with two-body theory. The distance behavior of interference terms between two-, three-, and four-body terms is also examined, and the relative importance of each contribution to the total transfer rate is discussed.

Nonadiabatic quantum transition-state theory in the golden-rule limit. I. Theory and application to model systems.

We propose a new quantum transition-state theory for calculating Fermi's golden-rule rates in complex multidimensional systems. This method is able to account for the nuclear quantum effects of delocalization, zero-point energy, and tunneling in an electron-transfer reaction. It is related to instanton theory but can be computed by path-integral sampling and is thus applicable to treat molecular reactions in solution. A constraint functional based on energy conservation is introduced which ensures that the dominant paths contributing to the reaction rate are sampled. We prove that the theory gives exact results for a system of crossed linear potentials and show numerically that it is also accurate for anharmonic systems. There is still a certain amount of freedom available in generalizing the method to multidimensional systems, and the suggestion we make here is exact in the classical limit but not rigorously size consistent in general. It is nonetheless seen to perform well for multidimensional spin-boson models, where it even gives good predictions for rates in the Marcus inverted regime.

A comparative study of different methods for calculating electronic transition rates.

We present a comprehensive comparison of the following mixed quantum-classical methods for calculating electronic transition rates: (1) nonequilibrium Fermi's golden rule, (2) mixed quantum-classical Liouville method, (3) mean-field (Ehrenfest) mixed quantum-classical method, and (4) fewest switches surface-hopping method (in diabatic and adiabatic representations). The comparison is performed on the Garg-Onuchic-Ambegaokar benchmark charge-transfer model, over a broad range of temperatures and electronic coupling strengths, with different nonequilibrium initial states, in the normal and inverted regimes. Under weak to moderate electronic coupling, the nonequilibrium Fermi's golden rule rates are found to be in good agreement with the rates obtained via the mixed quantum-classical Liouville method that coincides with the fully quantum-mechanically exact results for the model system under study. Our results suggest that the nonequilibrium Fermi's golden rule can serve as an inexpensive yet accurate alternative to Ehrenfest and the fewest switches surface-hopping methods.

Computational study on hole conduction in normal alkanes: Anisotropy and effect of dynamic disorder

Despite its importance, carrier conduction in electrical insulators is poorly understood. This work presents a computational study of hole conduction in single crystalline alkanes (n-C18H38 and n-C36H74). Hole mobilities are computed with the combination of molecular dynamics simulation, quantum chemical calculation, and the kinetic Monte Carlo method. The hole hopping rate is computed by the Fermi golden rule rate kernel without high temperature approximation. A strong correlation between the anisotropy of hole mobility and crystalline morphology is found. Hole mobilities in the direction of the c axis are more than an order of magnitude larger than those in the a − b plane. At room temperature, hole mobility is increased by roughly a factor of 10 due to the thermal motion of molecules. Computed anisotropic hole mobilities are in reasonable agreement with experimental values when the effect of dynamic disorder is taken into account. The results strongly indicate that hole transfer in crystalline alkane occurs in the phonon-assisted transport regime.

Can classical marcus theory describe hole transfer in polyethylene?

Recently, hole mobilities in polyethylene (PE) oligomers has been investigated by means of Marcus theory. Although it is well known that Marcus theory is successfully applied to various organic semiconductors, strictly speaking, charge transfer in organic molecules requires quantum mechanical treatment to account for quantum nuclear tunneling between vibrational levels. We evaluate hole transfer rates between PE oligomers by Fermi's golden rule (FGR) rate expression and by classical Marcus rate expression at various temperature and energetic disorder. The results indicate that Marcus theory can be a good approximation for describing hole transfer between long polyethylene chain with small reorganization energy or in highly energetically disordered system. However, hole hopping rate between C10H22 molecules computed by Marcus theory was roughly 100 times smaller than that computed through FGR rate expression. High-temperature approximation must be applied with caution especially for quantitative evaluation of hole transfer rates in PE.

Non-Condon equilibrium Fermi's golden rule electronic transition rate constants via the linearized semiclassical method.

In this paper, we test the accuracy of the linearized semiclassical (LSC) expression for the equilibrium Fermi's golden rule rate constant for electronic transitions in the presence of non-Condon effects. We do so by performing a comparison with the exact quantum-mechanical result for a model where the donor and acceptor potential energy surfaces are parabolic and identical except for shifts in the equilibrium energy and geometry, and the coupling between them is linear in the nuclear coordinates. Since non-Condon effects may or may not give rise to conical intersections, both possibilities are examined by considering: (1) A modified Garg-Onuchic-Ambegaokar model for charge transfer in the condensed phase, where the donor-acceptor coupling is linear in the primary mode coordinate, and for which non-Condon effects do not give rise to a conical intersection; (2) the linear vibronic coupling model for electronic transitions in gas phase molecules, where non-Condon effects give rise to conical intersections. We also present a comprehensive comparison between the linearized semiclassical expression and a progression of more approximate expressions. The comparison is performed over a wide range of frictions and temperatures for model (1) and over a wide range of temperatures for model (2). The linearized semiclassical method is found to reproduce the exact quantum-mechanical result remarkably well for both models over the entire range of parameters under consideration. In contrast, more approximate expressions are observed to deviate considerably from the exact result in some regions of parameter space.

Mean field ring polymer molecular dynamics for electronically nonadiabatic reaction rates.

We present a mean field ring polymer molecular dynamics method to calculate the rate of electron transfer (ET) in multi-state, multi-electron condensed-phase processes. Our approach involves calculating a transition state theory (TST) estimate to the rate using an exact path integral in discrete electronic states and continuous Cartesian nuclear coordinates. A dynamic recrossing correction to the TST rate is then obtained from real-time dynamics simulations using mean field ring polymer molecular dynamics. We employ two different reaction coordinates in our simulations and show that, despite the use of mean field dynamics, the use of an accurate dividing surface to compute TST rates allows us to achieve remarkable agreement with Fermi's golden rule rates for nonadiabatic ET in the normal regime of Marcus theory. Further, we show that using a reaction coordinate based on electronic state populations allows us to capture the turnover in rates for ET in the Marcus inverted regime.

Equilibrium Fermi's Golden Rule Charge Transfer Rate Constants in the Condensed Phase: The Linearized Semiclassical Method vs Classical Marcus Theory.

In this article, we present a comprehensive comparison between the linearized semiclassical expression for the equilibrium Fermi's golden rule rate constant and the progression of more approximate expressions that lead to the classical Marcus expression. We do so within the context of the canonical Marcus model, where the donor and acceptor potential energy surface are parabolic and identical except for a shift in both the free energies and equilibrium geometries, and within the Condon region. The comparison is performed for two different spectral densities and over a wide range of frictions and temperatures, thereby providing a clear test for the validity, or lack thereof, of the more approximate expressions. We also comment on the computational cost and scaling associated with numerically calculating the linearized semiclassical expression for the rate constant and its dependence on the spectral density, temperature, and friction.