Nonadiabatic instanton calculation of multistate electron transfer reaction rate: Interference effects in three and four states systems

A personal perspective of the present status and future challenges facing thermal reaction rate theory

Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

Nonadiabatic instanton rate theory beyond the golden-rule limit

Fermi's golden rule describes the leading-order behaviour of the reaction rate as a function of the diabatic coupling. Its asymptotic (ℏ →0) limit is the semiclassical golden-rule instanton rate theory, which rigorously approximates nuclear quantum effects, lends itself to efficient numerical computation and gives physical insight into reaction mechanisms. However the golden rule by itself becomes insufficient as the strength of the diabatic coupling increases, so higher-order terms must be additionally considered. In this work we give a first-principles derivation of the next-order term beyond the golden rule, represented as a sum of three components. Two of them lead to new instanton pathways that extend the golden-rule case and, among other factors, account for the effects of recrossing on the full rate. The remaining component derives from the equilibrium partition function and accounts for changes in potential energy around the reactant and product wells due to diabatic coupling. The new semiclassical theory demands little computational effort beyond a golden-rule instanton calculation. It makes it possible to rigorously assess the accuracy of the golden-rule approximation and sets the stage for future work on general semiclassical nonadiabatic rate theories.

Frequency-dependent current noise in quantum heat transfer: A unified polaron calculation

To investigate frequency-dependent current noise (FDCN) in open quantum systems at steady states, we present a theory which combines Markovian quantum master equations with a finite time full counting statistics. Our formulation of the FDCN generalizes previous zero-frequency expressions and can be viewed as an application of MacDonald's formula for electron transport to heat transfer. As a demonstration, we consider the paradigmatic example of quantum heat transfer in the context of a non-equilibrium spin-boson model. We adopt a recently developed polaron-transformed Redfield equation which allows us to accurately investigate heat transfer with arbitrary system-reservoir coupling strength, arbitrary values of spin bias, and temperature differences. We observe a turn-over of FDCN in the intermediate coupling regimes, similar to the zero-frequency case. We find that the FDCN with varying coupling strengths or bias displays a universal Lorentzian-shape scaling form in the weak coupling regime, and a white noise spectrum emerges with zero bias in the strong coupling regime due to distinctive spin dynamics. We also find that the bias can suppress the FDCN in the strong coupling regime, in contrast to its zero-frequency counterpart which is insensitive to bias changes. Furthermore, we utilize the Saito-Utsumi relation as a benchmark to validate our theory and study the impact of temperature differences at finite frequencies. Together, our results provide detailed dissections of the finite time fluctuation of heat current in open quantum systems.

Path integral molecular dynamics for exact quantum statistics of multi-electronic-state systems

An exact approach to compute physical properties for general multi-electronic-state (MES) systems in thermal equilibrium is presented. The approach is extended from our recent progress on path integral molecular dynamics (PIMD), Liu et al. [J. Chem. Phys. 145, 024103 (2016)] and Zhang et al. [J. Chem. Phys. 147, 034109 (2017)], for quantum statistical mechanics when a single potential energy surface is involved. We first define an effective potential function that is numerically favorable for MES-PIMD and then derive corresponding estimators in MES-PIMD for evaluating various physical properties. Its application to several representative one-dimensional and multi-dimensional models demonstrates that MES-PIMD in principle offers a practical tool in either of the diabatic and adiabatic representations for studying exact quantum statistics of complex/large MES systems when the Born-Oppenheimer approximation, Condon approximation, and harmonic bath approximation are broken.

Optimization of exciton trapping in energy transfer processes

In this paper, we establish optimal conditions for maximal energy transfer efficiency using solutions for multilevel systems and interpret these analytical solutions with more intuitive kinetic networks resulting from a systematic mapping procedure. The mapping procedure defines an effective hopping rate as the leading order picture and nonlocal kinetic couplings as the quantum correction, hence leading to a rigorous separation of thermal hopping and coherent transfer useful for visualizing pathway connectivity and interference in quantum networks. As a result of these calculations, the dissipative effects of the surrounding environments can be optimized to yield the maximal efficiency, and modulation of the efficiency can be achieved using the cumulative quantum phase along any closed loops. The optimal coupling of the system and its environments is interpreted with the generic mechanisms: (i) balancing localized trapping and delocalized coherence, (ii) reducing the effective detuning via homogeneous line-broadening, (iii) suppressing the destructive interference in nonlinear network configurations, and (iv) controlling phase modulation in closed loop configurations. Though these results are obtained for simple model systems, the physics thus derived provides insights into the working of light harvesting systems, and the approaches thus developed apply to large-scale computation.

Path-integral simulations beyond the adiabatic approximation

Within the adiabatic approximation, it is trivial to generalize existing imaginary time path-integral techniques to the case of multiple electronic surfaces. However, there are many times where nonadiabatic effects can play an important role. To this end, we reformulate the well-known path-integral expressions to incorporate multiple potential surfaces, without necessitating the adiabatic approximation. We show that the resulting expression, like its adiabatic counterpart, can be interpreted in terms of a simple classical isomorphic system and thus is amenable to simulation through Monte Carlo techniques. We derive simple expressions to compute expectation values of a general operator in both the nuclear coordinate and electronic state, and demonstrate the existence of a simple internal diagnostic that can be used to evaluate the magnitude of equilibrium nonadiabatic effects.

Including quantum effects in the dynamics of complex (i.e., large) molecular systems

The development in the 1950s and 1960s of crossed molecular beam methods for studying chemical reactions at the single-collision molecular level stimulated the need and desire for theoretical methods to describe these and other dynamical processes in molecular systems. Chemical dynamics theory has made great strides in the ensuing decades, so that methods are now available for treating the quantum dynamics of small molecular systems essentially completely. For the large molecular systems that are of so much interest nowadays (e.g., chemical reactions in solution, in clusters, in nanostructures, in biological systems, etc.), however, the only generally available theoretical approach is classical molecular dynamics (MD) simulations. Much effort is currently being devoted to the development of approaches for describing the quantum dynamics of these complex systems. This paper reviews some of these approaches, especially the use of semiclassical approximations for adding quantum effects to classical MD simulations, also showing some new versions that should make these semiclassical approaches even more practical and accurate.

Matching-pursuit split-operator Fourier-transform simulations of excited-state intramolecular proton transfer in 2-(2′-hydroxyphenyl)-oxazole

The excited-state intramolecular proton-transfer dynamics associated with the keto-enolic tautomerization reaction in 2-(2(')-hydroxyphenyl)-oxazole is simulated according to a numerically exact quantum-dynamics propagation method and a full-dimensional excited-state potential energy surface, based on an ab initio reaction surface Hamiltonian. The reported simulations involve the propagation of 35-dimensional wave packets according to the recently developed matching-pursuit/split-operator-Fourier-transform (MP/SOFT) method by Wu and Batista. The underlying propagation scheme recursively applies the time-evolution operator as defined by the Trotter expansion to second order accuracy in dynamically adaptive coherent-state expansions. Computations of time-dependent survival amplitudes, photoabsorption cross sections, and time-dependent reactant(product) populations are compared to the corresponding calculations based on semiclassical approaches, including the Herman-Kluk semiclassical initial value representation method. The reported results demonstrate the capabilities of the MP/SOFT method as a valuble computational tool to study ultrafast reaction dynamics in polyatomic systems as well as to validate semiclassical simulations of complex (nonintegrable) quantum dynamics in multidimensional model systems.

Quantum dynamics of complex molecular systems

This Perspective presents a broad overview of the present status of theoretical capabilities for describing quantum dynamics in molecular systems with many degrees of freedom, e.g., chemical reactions in solution, clusters, solids, or biomolecular environments.

Stationary phase evaluations of quantum rate constants

We compute the quantum rate constant based on two extended stationary phase approximations to the imaginary-time formulation of the quantum rate theory. The optimized stationary phase approximation to the imaginary-time flux-flux correlation function employs the optimized quadratic reference system to overcome the inaccuracy of the quadratic expansion in the standard stationary phase approximation, and yields favorable agreements with instanton results for both adiabatic and nonadiabatic processes in dissipative and nondissipative systems. The integrated stationary phase approximation to the two-dimensional barrier free energy is particularly useful for adiabatic processes and demonstrates consistent results with the imaginary-time flux-flux correlation function approach. Our stationary phase methods do not require calculation of tunneling paths or stability matrices, and work equally well in the high-temperature and the low-temperature regimes. The numerical results suggest their general applicability for calibration of imaginary-time methods and for the calculation of quantum rate constants in systems with a large number of degrees of freedom.

Theoretical investigations on Azotobacter vinelandii ferredoxin I: effects of electron transfer on protein dynamics

Structural, energetic, and dynamical studies of Azotobacter vinelandii ferredoxin I are presented for native and mutant forms. The protein contains two iron-sulfur clusters, one of which ([3Fe-4S]) is believed to play a central role in the electron-coupled proton transfer. Different charge sets for the [3Fe-4S] cluster in its reduced and oxidized state are calculated with broken symmetry ab initio density functional theory methods and used in molecular dynamics (MD) simulations. The validity of the ab initio calculations is assessed by comparing partially optimized structures of the [3Fe-4S] clusters with x-ray structures. Possible proton transfer pathways between the protein and the iron-sulfur cluster are examined by both MD simulations and ab initio calculations. The MD simulations identify three main-chain hydrogen atoms--HN(13), HN(14), and HN(16)--that are within H-bonding distance of the [3Fe-4S] cluster throughout the MD simulations. They could thus play a role in the proton transfer from the protein to the iron-sulfur cluster. By contrast, the HD2(15) atom of the Asp-15 is seldom close enough to the [3Fe-4S] cluster to transfer a proton. Poisson-Boltzmann calculations indicate that there is a low, but nonzero probability, that Asp-15 is protonated at pH 7; this is a requirement for it to serve as a proton donor. Ab initio calculations with a fragment model for the protein find similar behavior for the transfer of a proton from the OH of the protonated side chain and the main-chain NH of Asp-15. The existence of a stable salt bridge between Asp-15 and Lys-84 in the D15E mutant, versus its absence in the wild-type, has been suggested as the cause of the difference in the rate of proton transfer. Extensive MD simulations were done to test this idea; the results do not support the proposal. The present findings, together with the available data, serve as the basis for an alternative proposal for the mechanism of the coupled electron-proton transfer reaction in ferredoxin I.