Mean-Field Ring Polymer Rates Using a Population Dividing Surface.

Mean-field ring polymer molecular dynamics offers a computationally efficient method for the simulation of reaction rates in multilevel systems. Previous work has established that, to model a nonadiabatic state-to-state reaction accurately, it is necessary to ensure reactive trajectories form kinked ring polymer configurations at the dividing surface. Building on this idea, we introduce a population difference coordinate and a reactive flux expression modified to only include contributions from kinked configurations. We test the accuracy of the resulting mean-field rate theory on a series of linear vibronic coupling model systems. We demonstrate that this new kMF-RP rate approach is efficient to implement and quantitatively accurate for models over a wide range of driving forces, coupling strengths, and temperatures.

GPU-accelerated on-the-fly nonadiabatic semiclassical dynamics.

GPU-accelerated on-the-fly nonadiabatic dynamics is enabled by interfacing the linearized semiclassical dynamics approach with the TeraChem electronic structure program. We describe the computational workflow of the "PySCES" code interface, a Python code for semiclassical dynamics with on-the-fly electronic structure, including parallelization over multiple GPU nodes. We showcase the abilities of this code and present timings for two benchmark systems: fulvene solvated in acetonitrile and a charge transfer system in which a photoexcited zinc-phthalocyanine donor transfers charge to a fullerene acceptor through multiple electronic states on an ultrafast timescale. Our implementation paves the way for an efficient semiclassical approach to model the nonadiabatic excited state dynamics of complex molecules, materials, and condensed phase systems.

Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics.

This review focuses on a recent class of path-integral-based methods for the simulation of nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.

Non-linear correlation functions and zero-point energy flow in mixed quantum-classical semiclassical dynamics.

Mixed quantum classical (MQC)-initial value representation (IVR) is a recently introduced semiclassical framework that allows for selective quantization of the modes of a complex system. In the quantum limit, MQC reproduces the semiclassical Double Herman-Kluk IVR results, accurately capturing nuclear quantum coherences and conserving zero-point energy. However, in the classical limit, although MQC mimics the Husimi-IVR for real-time correlation functions with linear operators, it is significantly less accurate for non-linear correlation functions with errors even at time zero. Here, we identify the origin of this discrepancy in the MQC formulation and propose a modification. We analytically show that the modified MQC approach is exact for all correlation functions at time zero, and in a study of zero-point energy (ZPE) flow, we numerically demonstrate that it correctly obtains the quantum and classical limits as a function of time. Interestingly, although classical-limit MQC simulations show the expected, unphysical ZPE leakage, we find that it is possible to predict and even modify the direction of ZPE flow through selective quantization of the system, with the quantum-limit modes accepting energy but preserving the minimum quantum mechanically required energy.

A Semiclassical Framework for Mixed Quantum Classical Dynamics.

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.

Singularity-free internal conversion golden-rule rate with application to correlated triplet pair recombination in bipentacenes.

We introduce a singularity-free golden-rule rate expression for internal conversion (IC), a spin-conserved radiationless relaxation process, expressed as the product of a nonadiabatic coupling term and the time integral of a vibration correlation function. For a set of small polyatomic molecules (acenes and azulene), we show that our calculated rates are in near quantitative agreement with the rates derived from experiments. Interestingly, we find that our rates do not agree with previous golden-rule-based theoretical efforts; detailed analysis shows that while the level of electronic structure theory can play a role, the more significant error is from not fully converging the numerical time integral over the oscillatory vibration correlation function. We then use our singularity-free IC rate expression to compute the rate of recombination of the correlated triplet pair state generated by intramolecular singlet fission in a trio of bipentacenes. We show that the recombination rates are in good agreement with experimentally observed trends-specifically, the rates are robust to temperature changes and decrease rapidly with increasing inter-monomer dihedral angle. Finally, we use a pair-wise breakdown of normal mode contributions to the rate to identify the key vibrational modes that drive recombination in bipentacenes.

Multistate ring polymer instantons and nonadiabatic reaction rates.

We present two multistate ring polymer instanton (RPI) formulations, both obtained from an exact path integral representation of the quantum canonical partition function for multistate systems. The two RPIs differ in their treatment of the electronic degrees of freedom; while the Mean-Field (MF)-RPI averages over the electronic state contributions, the Mapping Variable (MV)-RPI employs explicit continuous Cartesian variables to represent the electronic states. We compute both RPIs for a series of model two-state systems coupled to a single nuclear mode with electronic coupling values chosen to describe dynamics in both adiabatic and nonadiabatic regimes. We show that the MF-RPIs for symmetric systems are in good agreement with the previous literature, and we show that our numerical techniques are robust for systems with non-zero driving force. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPI uniquely reports on the changes in the electronic state populations along the instanton path. In both cases, we analytically demonstrate the existence of a zero-mode, and we numerically find that these solutions are true instantons with a single unstable mode as expected for a first order saddle point. Finally, we use the MF-RPI to accurately calculate rate constants for adiabatic and nonadiabatic model systems with the coupling strength varying over three orders of magnitude.

Anticipating Acene-Based Chromophore Spectra with Molecular Orbital Arguments.

Recent synthetic studies on the organic molecules tetracene and pentacene have found certain dimers and oligomers to exhibit an intense absorption in the visible region of the spectrum that is not present in the monomer or many previously studied dimers. In this article we combine experimental synthesis with electronic structure theory and spectral computation to show that this absorption arises from an otherwise dark charge-transfer excitation "borrowing intensity" from an intense UV excitation. Further, by characterizing the role of relevant monomer molecular orbitals, we arrive at a design principle that allows us to predict the presence or absence of an additional absorption based on the bonding geometry of the dimer. We find this rule correctly explains the spectra of a wide range of acene derivatives and solves an unexplained structure-spectrum phenomenon first observed over 70 years ago. These results pave the way for the design of highly absorbent chromophores with applications ranging from photovoltaics to liquid crystals.

Semiclassical dynamics in the mixed quantum-classical limit.

The semiclassical double Herman-Kluk initial value representation is an accurate approach to computing quantum real time correlation functions, but its applications are limited by the need to evaluate an oscillatory integral. In previous work, we have shown that this "sign problem" can be mitigated using the modified Filinov filtration technique to control the extent to which individual modes of the system contribute to the overall phase of the integrand. Here, we follow this idea to a logical conclusion: we analytically derive a general expression for the mixed quantum-classical limit of the semiclassical correlation function-analytical mixed quantum-classical-initial value representation (AMQC-IVR), where the phase contributions from the "classical" modes of the system are filtered while the "quantum" modes are treated in the full semiclassical limit. We numerically demonstrate the accuracy and efficiency of the AMQC-IVR formulation in calculations of quantum correlation functions and reaction rates using three model systems with varied coupling strengths between the classical and quantum subsystems. We also introduce a separable prefactor approximation that further reduces computational cost but is only accurate in the limit of weak coupling between the quantum and classical subsystems.

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation.

We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), a semiclassical method for computing real-time correlation functions, to electronically nonadiabatic systems using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian in order to treat electronic and nuclear degrees of freedom (dofs) within a consistent dynamic framework. We introduce an efficient symplectic integration scheme, the MInt algorithm, for numerical time evolution of the phase space variables and monodromy matrix under the non-separable MMST Hamiltonian. We then calculate the probability of transmission through a curve crossing in model two-level systems and show that MQC-IVR reproduces quantum-limit semiclassical results in good agreement with exact quantum methods in one limit, and in the other limit yields results that are in keeping with classical limit semiclassical methods like linearized IVR. Finally, exploiting the ability of the MQC-IVR to quantize different dofs to different extents, we present a detailed study of the extents to which quantizing the nuclear and electronic dofs improves numerical convergence properties without significant loss of accuracy.

Tuning Singlet Fission in π-Bridge-π Chromophores.

We have designed a series of pentacene dimers separated by homoconjugated or nonconjugated bridges that exhibit fast and efficient intramolecular singlet exciton fission (iSF). These materials are distinctive among reported iSF compounds because they exist in the unexplored regime of close spatial proximity but weak electronic coupling between the singlet exciton and triplet pair states. Using transient absorption spectroscopy to investigate photophysics in these molecules, we find that homoconjugated dimers display desirable excited-state dynamics, with significantly reduced recombination rates as compared to conjugated dimers with similar singlet fission rates. In addition, unlike conjugated dimers, the time constants for singlet fission are relatively insensitive to the interplanar angle between chromophores, since rotation about σ bonds negligibly affects the orbital overlap within the π-bonding network. In the nonconjugated dimer, where the iSF occurs with a time constant >10 ns, comparable to the fluorescence lifetime, we used electron spin resonance spectroscopy to unequivocally establish the formation of triplet-triplet multiexcitons and uncoupled triplet excitons through singlet fission. Together, these studies enable us to articulate the role of the conjugation motif in iSF.

Validating and implementing modified Filinov phase filtration in semiclassical dynamics.

The Mixed Quantum-Classical Initial Value Representation (MQC-IVR) is a recently introduced approximate semiclassical (SC) method for the calculation of real-time quantum correlation functions. MQC-IVR employs a modified Filinov filtration (MFF) scheme to control the overall phase of the SC integrand, extending the applicability of SC methods to complex systems while retaining their ability to accurately describe quantum coherence effects. Here, we address questions regarding the effectiveness of the MFF scheme in combination with SC dynamics. Previous work showed that this filtering scheme is of limited utility in the context of semiclassical wavepacket propagation, but we find that the MFF is extraordinarily powerful in the context of correlation functions. By examining trajectory phase and amplitude contributions to the real-time SC correlation function in a model system, we clearly demonstrate that the MFF serves to reduce noise by damping amplitude only in regions of highly oscillatory phase leading to a reduction in computational effort while retaining accuracy. Further, we introduce a novel and efficient MQC-IVR formulation that allows for linear scaling in computational cost with the total simulation length, a significant improvement over the more-than quadratic scaling exhibited by the original method.

A mapping variable ring polymer molecular dynamics study of condensed phase proton-coupled electron transfer.

We investigate the mechanisms of condensed phase proton-coupled electron transfer (PCET) using Mapping-Variable Ring Polymer Molecular Dynamics (MV-RPMD), a recently developed method that employs an ensemble of classical trajectories to simulate nonadiabatic excited state dynamics. Here, we construct a series of system-bath model Hamiltonians for the PCET, where four localized electron-proton states are coupled to a thermal bath via a single solvent mode, and we employ MV-RPMD to simulate state population dynamics. Specifically, for each model, we identify the dominant PCET mechanism, and by comparing against rate theory calculations, we verify that our simulations correctly distinguish between concerted PCET, where the electron and proton transfer together, and sequential PCET, where either the electron or the proton transfers first. This work represents a first application of MV-RPMD to multi-level condensed phase systems; we introduce a modified MV-RPMD expression that is derived using a symmetric rather than asymmetric Trotter discretization scheme and an initialization protocol that uses a recently derived population estimator to constrain trajectories to a dividing surface. We also demonstrate that, as expected, the PCET mechanisms predicted by our simulations are robust to an arbitrary choice of the initial dividing surface.

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.

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation.

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Direct simulation of electron transfer in the cobalt hexammine(ii/iii) self-exchange reaction.

We present an atomistic simulation of the cobalt hexammine(ii/iii) self-exchange reaction using path integral (PI) methods. We construct a simple force field for the system in its reactant state that includes parameters for both atom-atom interactions, and interactions with an explicit transferring electron represented in the PI framework. We then calculate the outer sphere free energy barrier due to solvent reorganization from a PI molecular dynamics simulation and we obtain the dynamic transmission coefficient using ring polymer molecular dynamics. Combining these calculated values with literature values for the inner sphere reorganization energy, we obtain a reaction rate in good agreement with experimental measurements. The protocol introduced here circumvents the need for complex, system-specific force field parameterization along an assumed reaction coordinate making it sufficiently accurate, efficient, and broadly applicable to the study of both adiabatic and nonadiabatic charge transfer reactions in transition metal complexes.

Dynamically consistent method for mixed quantum-classical simulations: A semiclassical approach.

We introduce a new semiclassical (SC) framework, the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), that can be tuned to reproduce existing quantum-limit and classical-limit SC approximations to quantum real-time correlation functions. Applying a modified Filinov transformation to a quantum-limit SC formulation leads to the association of a Filinov parameter with each degree of freedom in the system; varying this parameter from zero to infinity controls the extent of quantization of the corresponding mode. The resulting MQC-IVR expression provides a consistent dynamic framework for mixed quantum-classical simulations and we demonstrate its numerical accuracy in the calculation of real-time correlation functions for a model 1D system and a model 2D system over the full range of quantum- to classical-limit behaviors.

The low-lying electronic states of pentacene and their roles in singlet fission.

We present a detailed study of pentacene monomer and dimer that serves to reconcile extant views of its singlet fission. We obtain the correct ordering of singlet excited-state energy levels in a pentacene molecule (E (S1) < E (D)) from multireference calculations with an appropriate active orbital space and dynamical correlation being incorporated. In order to understand the mechanism of singlet fission in pentacene, we use a well-developed diabatization scheme to characterize the six low-lying singlet states of a pentacene dimer that approximates the unit cell structure of crystalline pentacene. The local, single-excitonic diabats are not directly coupled with the important multiexcitonic state but rather mix through their mutual couplings with one of the charge-transfer configurations. We analyze the mixing of diabats as a function of monomer separation and pentacene rotation. By defining an oscillator strength measure of the coherent population of the multiexcitonic diabat, essential to singlet fission, we find this population can, in principle, be increased by small compression along a specific crystal direction.

Seeking small molecules for singlet fission: a heteroatom substitution strategy.

We design theoretically small molecule candidates for singlet fission chromophores, aiming to achieve a balance between sufficient diradical character and kinetic persistence. We develop a perturbation strategy based on the captodative effect to introduce diradical character into small π-systems. Specifically, this can be accomplished by replacing pairs of not necessarily adjacent C atoms with isoelectronic and isosteric pairs of B and N atoms. Three rules of thumb emerge from our studies to aid further design: (i) Lewis structures provide insight into likely diradical character; (ii) formal radical centers of the diradical must be well-separated; (iii) stabilization of radical centers by a donor (N) and an acceptor (B) is essential. Following the rules, we propose candidate molecules. Employing reliable multireference calculations for excited states, we identify three likely candidate molecules for SF chromophores. These include a benzene, a napthalene, and an azulene, where four C atoms are replaced by a pair of B and a pair of N atoms.

Tuning spin-states of carbynes and silylynes: a long jump with one leg.

The challenge motivating this paper is to induce, by chemical substitution, a silylyne, SiR, or a congeneric carbyne, CR, to adopt the high-spin quartet rather than the low-spin doublet as its ground state. The difficulty is seen in the preference for the doublet of the parent SiH (doublet-quartet energy difference ∼39 kcal/mol, favoring the doublet) or CH (∼17 kcal/mol). Strategies for having high-spin ground state parallel those for silylenes and carbenes: greater electropositivity (σ-donation) and π-acceptance of the single substituent favor the high-spin state. The electronegativity trend can be understood from an ions in molecules way of thinking already present in the literature in the works of Boldyrev and Simons, and of Mavridis and Harrison; i.e., the quartet ground state spin of some CR/SiR species is largely determined by the ground state spin of C(-)/Si(-). In this study, we provide a diabatization analysis that solidly confirms the ions in molecules picture and explains the difference in the equilibrium internuclear distances for the two spin states. In general, electronegativity dominates the ordering of spin states. π-Acceptors also help to lower the quartet state energy of the many carbynes (silylynes) examined, whose range of doublet-quartet differences calculated is impressive, 120 (100) kcal/mol. The qualitative understanding gained leads to the prediction of some quartet-ground state carbynes (CMgH, CAlH2, CZnH, CSiH3, CSiF3, etc.) and a smaller number of silylynes (SiMgH, SiMgF, SiBeH, etc.). A beginning is made on the energetics of approach geometries of the fragments in the highly exoergic dimerization of CH to acetylene; it should proceed for the ground state doublet CH through C2h-like trajectories, with no activation energy.