Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics

Short-time accuracy and intra-electron correlation for nonadiabatic quantum-classical mapping approaches

Nonadiabatic quantum-classical mapping approaches have significantly gained in popularity over the past several decades because they have acceptable accuracy while remaining numerically tractable even for large system sizes. In the recent few years, several novel mapping approaches have been developed that display higher accuracy than the traditional Ehrenfest method, linearized semiclassical initial value representation (LSC-IVR), and Poisson bracket mapping equation (PBME) approaches. While various benchmarks have already demonstrated the advantages and limitations of those methods, unified theoretical justifications of their short-time accuracy are still demanded. In this article, we systematically examine the intra-electron correlation, as a statistical measure of electronic phase space, which has been first formally proposed for mapping approaches in the context of the generalized discrete truncated Wigner approximation and which is a key ingredient for the improvement in short-time accuracy of such mapping approaches. We rigorously establish the connection between short-time accuracy and intra-electron correlation for various widely used models. We find that LSC-IVR, PBME, and Ehrenfest methods fail to correctly reproduce the intra-electron correlation. While some of the traceless Meyer-Miller-Stock-Thoss (MMST) approaches, partially linearized density matrix (PLDM) approach, and spin partially linearized density matrix (spin-PLDM) approach are able to sample the intra-electron correlation correctly, the spin linearized semiclassical (spin-LSC) approach, which is a specific example of the classical mapping model, and the other traceless MMST approaches sample the intra-correlation faithfully only for two-level systems. Our theoretical analysis provides insights into the short-time accuracy of semiclassical methods and presents mathematical justifications for previous numerical benchmarks.

How Sophisticated Are Neural Networks Needed to Predict Long-Term Nonadiabatic Dynamics?

Nonadiabatic dynamics is key for understanding solar energy conversion and photochemical processes in condensed phases. This often involves the non-Markovian dynamics of the reduced density matrix in open quantum systems, where knowledge of the system's prior states is necessary to predict its future behavior. In this study, we explore time-series machine learning methods for predicting long-time nonadiabatic dynamics based on short-time input data, comparing these methods with the physics-based transfer tensor method (TTM). To understand the impact of memory time on these approaches, we demonstrate that non-Markovian dynamics can be represented as a linear map within the Nakajima-Zwanzig generalized quantum master equation framework. We further propose a practical method to estimate the effective memory time, within a given tolerance, for reduced density matrix propagation. Our predictive models are applied to various physical systems, including spin-boson models, multistate harmonic (MSH) models with Ohmic spectral densities and for a realistic organic photovoltaic system composed of a carotenoid-porphyrin-fullerene triad dissolved in tetrahydrofuran. Results indicate that the simple linear-mapping fully connected neural network (FCN) outperforms the more complicated nonlinear-mapping networks including the gated recurrent unit (GRU) and the convolutional neural network/long short-term memory (CNN-LSTM) in systems with short memory times, such as spin-boson and MSH models. Conversely, the nonlinear CNN-LSTM and GRU models yield higher accuracy in the triad MSH systems characterized by long memory times. These findings offer valuable insights into the role of effective memory time in non-Markovian quantum dynamics, providing practical guidance for the application of time-series machine learning models to complex chemical systems.

Reduced density matrix dynamics in multistate harmonic models via time-convolution and time-convolutionless quantum master equations with quantum-mechanical and semiclassical kernels

In this work, we explore the electronic reduced density matrix (RDM) dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) that are based on perturbative electronic couplings within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonian consistently incorporates the electronic-vibrational correlations between all pairs of states by satisfying the pairwise reorganization energies directly obtained from all-atom simulations, representing the globally heterogeneous environments that couple to the multiple states differently. We derive the exact quantum-mechanical and a hierarchy of semiclassical approximate expressions for the kernels in TC and TCL QMEs that project the full RDM for general shifted harmonic systems, including the MSH model. These QMEs are applied to simulate RDM dynamics of photoinduced charge transfer (PICT) in organic photovoltaic carotenoid-porphyrin-fullerene triad solvated in tetrahydrofuran solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes from C. tepidum and P. aestuarii. Our results show that while both TC and TCL QMEs capture similar phenomena in PICT and EET processes, TC QME generally provides more accurate results than TCL QME, particularly in the initial oscillation of EET population dynamics. This study highlights the effectiveness of the TC and TCL QMEs in modeling RDM dynamics of nonadiabatic processes, offering insights for realistic condensed phase systems.

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

Combining the generalized quantum master equation approach with quasiclassical mapping Hamiltonian methods to simulate the dynamics of electronic coherences

The generalized quantum master equation (GQME) approach provides a powerful general-purpose framework for simulating the inherently quantum mechanical dynamics of a subset of electronic reduced density matrix elements of interest in complex molecular systems. Previous studies have found that combining the GQME approach with quasiclassical mapping Hamiltonian (QC/MH) methods can dramatically improve the accuracy of electronic populations obtained via those methods. In this paper, we perform a complimentary study of the advantages offered by the GQME approach for simulating the dynamics of electronic coherences, which play a central role in optical spectroscopy, quantum information science, and quantum technology. To this end, we focus on cases where the electronic coherences predicted for the spin-boson benchmark model by direct application of various QC/MH methods are inaccurate. We find that similar to the case of electronic populations, combining the QC/MH methods with the GQME approach can dramatically improve the accuracy of the electronic coherences obtained via those methods. We also provide a comprehensive analysis of how the performance of GQMEs depends on the choice of projection operator and electronic basis and show that the accuracy and feasibility of the GQME approach can benefit from casting the GQME in terms of the eigen-basis of the observable of interest.

Semiclassical dynamics in Wigner phase space I: Adiabatic hybrid Wigner dynamics

The Wigner phase space formulation of quantum mechanics is a complete framework for quantum dynamic calculations that elegantly highlights connections with classical dynamics. In this series of two articles, building upon previous efforts, we derive the full hierarchy of approximate semiclassical (SC) dynamic methods for adiabatic and non-adiabatic problems in Wigner phase space. In Paper I, focusing on adiabatic single surface processes, we derive the well-known double Herman-Kluk (DHK) approximation for real-time correlation functions in Wigner phase space and connect it to the linearized SC (LSC) approximation through a stationary phase approximation. We exploit this relationship to introduce a new hybrid SC method, termed Adiabatic Hybrid Wigner Dynamics (AHWD) that allows for a few important "system" degrees of freedom (dofs) to be treated at the DHK level, while treating the rest of the dofs (the "bath") at the LSC level. AHWD is shown to accurately capture quantum interference effects in models of coupled oscillators and the decoherence of vibrational probability density of a model I2 Morse oscillator coupled to an Ohmic thermal bath. We show that AHWD significantly mitigates the sign problem and employs reduced dimensional prefactors bringing calculations of complex system-bath problems within the reach of SC methods. Paper II focuses on extending this hybrid SC dynamics to nonadiabatic processes.

Semiclassical dynamics in Wigner phase space II: Nonadiabatic hybrid Wigner dynamics

We present an approximate semiclassical (SC) framework for mixed quantized dynamics in Wigner phase space in a two-part series. In the first article, we introduced the Adiabatic Hybrid Wigner Dynamics (AHWD) method that allows for a few important "system" degrees of freedom to be quantized using high-level double Herman-Kluk SC theory while describing the rest (the "bath") using classical-limit linearized SC theory. In this second article, we extend our hybrid Wigner dynamics to nonadiabatic processes. The resulting Nonadiabatic Hybrid Wigner Dynamics (NHWD) has two variants that differ in the choice of degrees of freedom to be quantized. Specifically, we introduce NHWD(E) where only the electronic state variables are quantized and the NHWD(V) where both electronic state variables and a handful of strongly coupled nuclear modes are quantized. We show that while NHWD(E) proves accurate for a wide range of scattering models and spin-boson models, systems where a few nuclear modes are strongly coupled to electronic states require NHWD(V) to accurately capture the long-time dynamics. Taken together, we show that AHWD and NHWD represent a new framework for SC simulations of high-dimensional systems with significant quantum effects.

Benchmarking various nonadiabatic semiclassical mapping dynamics methods with tensor-train thermo-field dynamics

Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.

Quantum Quality with Classical Cost: Ab Initio Nonadiabatic Dynamics Simulations Using the Mapping Approach to Surface Hopping

Nonadiabatic dynamics methods are an essential tool for investigating photochemical processes. In the context of employing first-principles electronic structure techniques, such simulations can be carried out in a practical manner using semiclassical trajectory-based methods or wave packet approaches. While all approaches applicable to first-principles simulations are necessarily approximate, it is commonly thought that wave packet approaches offer inherent advantages over their semiclassical counterparts in terms of accuracy and that this trait simply comes at a higher computational cost. Here we demonstrate that the mapping approach to surface hopping (MASH), a recently introduced trajectory-based nonadiabatic dynamics method, can be efficiently applied in tandem with ab initio electronic structure. Our results even suggest that MASH may provide more accurate results than on-the-fly wave packet techniques, all at a much lower computational cost.

Nonadiabatic Field with Triangle Window Functions on Quantum Phase Space

Recent progress on the constraint coordinate-momentum phase space (CPS) formulation of finite-state quantum systems has revealed that the triangle window function approach is an isomorphic representation of the exact population-population correlation function of the two-state system. We use the triangle window (TW) function and the CPS mapping kernel element to formulate a novel useful representation of discrete electronic degrees of freedom (DOFs). When it is employed with nonadiabatic field (NaF) dynamics, a new variant of the NaF approach (i.e., NaF-TW) is proposed. The NaF-TW expression of the population of any adiabatic state is always positive semidefinite. Extensive benchmark tests of model systems in both the condensed phase and gas phase demonstrate that the NaF-TW approach is able to faithfully capture the dynamical interplay between electronic and nuclear DOFs in a broad region, including where the states remain coupled all the time, as well as where the bifurcation characteristic of nuclear motion is important.

All-Atom Photoinduced Charge Transfer Dynamics in Condensed Phase via Multistate Nonlinear-Response Instantaneous Marcus Theory

Photoinduced charge transfer (CT) in the condensed phase is an essential component in solar energy conversion, but it is challenging to simulate such a process on the all-atom level. The traditional Marcus theory has been utilized for obtaining CT rate constants between pairs of electronic states but cannot account for the nonequilibrium effects due to the initial nuclear preparation. The recently proposed instantaneous Marcus theory (IMT) and its nonlinear-response formulation allow for incorporating the nonequilibrium nuclear relaxation to electronic transition between two states after the photoexcitation from the equilibrium ground state and provide the time-dependent rate coefficient. In this work, we extend the nonlinear-response IMT method for treating photoinduced CT among general multiple electronic states and demonstrate it in the organic photovoltaic carotenoid-porphyrin-fullerene triad dissolved in explicit tetrahydrofuran solvent. All-atom molecular dynamics simulations were employed to obtain the time correlation functions of energy gaps, which were used to generate the IMT-required time-dependent averages and variances of the relevant energy gaps. Our calculations show that the multistate IMT could capture the significant nonequilibrium effects due to the initial nuclear state preparation, and this is corroborated by the substantial differences between the population dynamics predicted by the multistate IMT and the Marcus theory, where the Marcus theory underestimates the population transfer. The population dynamics by multistate IMT is also shown to have a better agreement with the all-atom nonadiabatic mapping dynamics than the Marcus theory does. Because the multistate nonlinear-response IMT is straightforward and cost-effective in implementation and accounts for the nonequilibrium nuclear effects, we believe this method offers a practical strategy for studying charge transfer dynamics in complex condensed-phase systems.

Consistent Construction of the Density Matrix from Surface Hopping Trajectories

Proper construction of the density matrix based on surface hopping trajectories remains a difficult problem. Due to the well-known overcoherence in traditional surface hopping simulations, the electronic wave function cannot be used directly. In this work, we propose a consistent density matrix construction method, which takes the advantage of occupation of active states to rescale the coherence calculated by wave functions and ensures the intrinsic consistency of the density matrix. This new trajectory analysis method can be used for both Tully's fewest switches surface hopping (FSSH) and our recently proposed branching corrected surface hopping (BCSH). As benchmarked in both one- and two-dimensional standard scattering models, the new approach combined with BCSH trajectories achieves highly accurate time-dependent spatial distributions of adiabatic populations and coherence compared to exact quantum results.

Exciton dynamics from the mapping approach to surface hopping: comparison with Förster and Redfield theories

We compare the recently introduced multi-state mapping approach to surface hopping (MASH) with the Förster and Redfield theories of excitation energy transfer. Whereas Förster theory relies on weak coupling between chromophores, and Redfield theory assumes the electronic excitations to be weakly coupled to fast chromophore vibrations, MASH is free from any perturbative or Markovian approximations. We illustrate this with an example application to the rate of energy transfer in a Frenkel-exciton dimer, showing that MASH interpolates correctly between the opposing regimes in which the Förster and Redfield results are reliable. We then compare the three methods for a realistic model of the Fenna-Matthews-Olson complex with a structured vibrational spectral density and static disorder in the excitation energies. In this case there are no exact results for comparison so we use MASH to assess the validity of Förster and Redfield theories. We find that Förster theory is the more accurate of the two on the picosecond timescale, as has been shown previously for a simpler model of this particular light-harvesting complex. We also explore various ways to sample the initial electronic state in MASH and find that they all give very similar results for exciton dynamics.

Detailed balance in mixed quantum-classical mapping approaches

The violation of detailed balance poses a serious problem for the majority of current quasiclassical methods for simulating nonadiabatic dynamics. In order to analyze the severity of the problem, we predict the long-time limits of the electronic populations according to various quasiclassical mapping approaches by applying arguments from classical ergodic theory. Our analysis confirms that regions of the mapping space that correspond to negative populations, which most mapping approaches introduce in order to go beyond the Ehrenfest approximation, pose the most serious issue for reproducing the correct thermalization behavior. This is because inverted potentials, which arise from negative electronic populations entering the nuclear force, can result in trajectories unphysically accelerating off to infinity. The recently developed mapping approach to surface hopping (MASH) provides a simple way of avoiding inverted potentials while retaining an accurate description of the dynamics. We prove that MASH, unlike any other quasiclassical approach, is guaranteed to describe the exact thermalization behavior of all quantum-classical systems, confirming it as one of the most promising methods for simulating nonadiabatic dynamics in real condensed-phase systems.

Multistate Reaction Coordinate Model for Charge and Energy Transfer Dynamics in the Condensed Phase

Constructing multistate model Hamiltonians from all-atom electronic structure calculations and molecular dynamics simulations is crucial for understanding charge and energy transfer dynamics in complex condensed phases. The most popular two-level system model is the spin-boson Hamiltonian, where the nuclear degrees of freedom are represented as shifted normal modes. Recently, we proposed the general multistate nontrivial extension of the spin-boson model, i.e., the multistate harmonic (MSH) model, which is constructed by extending the spatial dimensions of each nuclear mode so as to satisfy the all-atom reorganization energy restrictions for all pairs of electronic states. In this work, we propose the multistate reaction coordinate (MRC) model with a primary reaction coordinate and secondary bath modes as in the Caldeira-Leggett form but in extended spatial dimensions. The MRC model is proven to be equivalent to the MSH model and offers an intuitive physical picture of the nuclear-electronic feedback in nonadiabatic processes such as the inherent trajectory of the reaction coordinate. The reaction coordinate is represented in extended dimensions, carrying the entire reorganization energies and bilinearly coupled to the secondary bath modes. We demonstrate the MRC model construction for photoinduced charge transfer in an organic photovoltaic caroteniod-porphyrin-C60 molecular triad dissolved in tetrahydrofuran as well as excitation energy transfer in a photosynthetic light-harvesting Fenna-Matthews-Olson complex. The MRC model provides an effective and robust platform for investigating quantum dissipative dynamics in complex condensed-phase systems since it allows a consistent description of realistic spectral density, state-dependent system-bath couplings, and heterogeneous environments due to static disorder in reorganization energies.

A mapping approach to surface hopping

We present a nonadiabatic classical-trajectory approach that offers the best of both worlds between fewest-switches surface hopping (FSSH) and quasiclassical mapping dynamics. This mapping approach to surface hopping (MASH) propagates the nuclei on the active adiabatic potential-energy surface, such as in FSSH. However, unlike in FSSH, transitions between active surfaces are deterministic and occur when the electronic mapping variables evolve between specified regions of the electronic phase space. This guarantees internal consistency between the active surface and the electronic degrees of freedom throughout the dynamics. MASH is rigorously derivable from exact quantum mechanics as a limit of the quantum-classical Liouville equation (QCLE), leading to a unique prescription for momentum rescaling and frustrated hops. Hence, a quantum-jump procedure can, in principle, be used to systematically converge the accuracy of the results to that of the QCLE. This jump procedure also provides a rigorous framework for deriving approximate decoherence corrections similar to those proposed for FSSH. We apply MASH to simulate the nonadiabatic dynamics in various model systems and show that it consistently produces more accurate results than FSSH at a comparable computational cost.

Quasiclassical approaches to the generalized quantum master equation

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin-boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.

Effects of Heterogeneous Protein Environment on Excitation Energy Transfer Dynamics in the Fenna-Matthews-Olson Complex

The Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria has been serving as a prototypical light-harvesting protein for studying excitation energy transfer (EET) dynamics in photosynthesis. The most widely used Frenkel exciton model for FMO complex assumes that each excited bacteriochlorophyll site couples to an identical and isolated harmonic bath, which does not account for the heterogeneous local protein environment. To better describe the realistic environment, we propose to use the recently developed multistate harmonic (MSH) model, which contains a globally shared bath that couples to the different pigment sites according to the atomistic quantum mechanics/molecular mechanics simulations with explicit protein scaffold and solvent. In this work, the effects of heterogeneous protein environment on EET in FMO complexes from Prosthecochloris aestuarii and Chlorobium tepidum, specifically including realistic spectral density, site-dependent reorganization energies, and system-bath couplings are investigated. Semiclassical and mixed quantum-classical mapping dynamics were applied to obtain the nonadiabatic EET dynamics in several models ranging from the Frenkel exciton model to the MSH model and their variants. The MSH model with realistic spectral density and site-dependent system-bath couplings displays slower EET dynamics than the Frenkel exciton model. Our comparative study shows that larger average reorganization energy, heterogeneity in spectral densities, and low-frequency modes could facilitate energy dissipation, which is insensitive to the static disorder in reorganization energies. The effects of the spectral densities and system-bath couplings along with the MSH model can be used to optimize EET dynamics for artificial light-harvesting systems.

All-Atom Nonadiabatic Semiclassical Mapping Dynamics for Photoinduced Charge Transfer of Organic Photovoltaic Molecules in Explicit Solvents

Direct all-atom simulation of nonadiabatic dynamics in disordered condensed phases like liquid solutions and amorphous solids has been challenging. The first all-atom simulation of the photoinduced charge-transfer dynamics of a prototypical organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad in explicit tetrahydrofuran is presented. Based on the Meyer-Miller mapping Hamiltonian, various semiclassical and mixed quantum-classical dynamics are employed, including the linearized semiclassical, symmetrical quasiclassical, mean-field Ehrenfest, classical mapping model, and spin-mapping model approaches. The all-atom nonadiabatic dynamics were compared to multi-state harmonic models with a globally shared bath, and the models built using the ensemble averages on the initial electronic state could reproduce the all-atom results. The solvent effect was found to be critical for the photoinduced charge transfer, and the time-dependent solute-solvent radial distribution functions revealed that only the nonadiabatic dynamics started with the effective forces on the initial electronic state could capture the correct nuclear dynamics. The proposed strategy for modeling condensed-phase nonadiabatic dynamics with atomistic details is readily applied to complex condensed-phase systems.

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.

Electronic Absorption Spectra from Off-Diagonal Quantum Master Equations

Quantum master equations (QMEs) provide a general framework for describing electronic dynamics within a complex molecular system. Off-diagonal QMEs (OD-QMEs) correspond to a family of QMEs that describe the electronic dynamics in the interaction picture based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. The fact that OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrodinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrodinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrodinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained in this way is studied in the context of a biexciton benchmark model, by comparing spectra calculated based on time-local and time-nonlocal OD-QMEs to spectra calculated based on a Redfield-type QME and the non-perturbative and quantum-mechanically exact hierarchical equations of motion (HEOM) method.

2D electronic-vibrational spectroscopy with classical trajectories

Two-dimensional electronic-vibrational (2DEV) spectra have the capacity to probe electron–nuclear interactions in molecules by measuring correlations between initial electronic excitations and vibrational transitions at a later time. The trajectory-based semiclassical optimized mean trajectory approach is applied to compute 2DEV spectra for a system with excitonically coupled electronic excited states vibronically coupled to a chromophore vibration. The chromophore mode is in turn coupled to a bath, inducing redistribution of vibrational populations. The lineshapes and delay-time dynamics of the resulting spectra compare well with benchmark calculations, both at the level of the observable and with respect to contributions from distinct spectroscopic processes.

NQCDynamics.jl: A Julia package for nonadiabatic quantum classical molecular dynamics in the condensed phase

Accurate and efficient methods to simulate nonadiabatic and quantum nuclear effects in high-dimensional and dissipative systems are crucial for the prediction of chemical dynamics in the condensed phase. To facilitate effective development, code sharing, and uptake of newly developed dynamics methods, it is important that software implementations can be easily accessed and built upon. Using the Julia programming language, we have developed the NQCDynamics.jl package, which provides a framework for established and emerging methods for performing semiclassical and mixed quantum-classical dynamics in the condensed phase. The code provides several interfaces to existing atomistic simulation frameworks, electronic structure codes, and machine learning representations. In addition to the existing methods, the package provides infrastructure for developing and deploying new dynamics methods, which we hope will benefit reproducibility and code sharing in the field of condensed phase quantum dynamics. Herein, we present our code design choices and the specific Julia programming features from which they benefit. We further demonstrate the capabilities of the package on two examples of chemical dynamics in the condensed phase: the population dynamics of the spin-boson model as described by a wide variety of semiclassical and mixed quantum-classical nonadiabatic methods and the reactive scattering of H₂ on Ag(111) using the molecular dynamics with electronic friction method. Together, they exemplify the broad scope of the package to study effective model Hamiltonians and realistic atomistic systems.

Two-dimensional vibronic spectroscopy with semiclassical thermofield dynamics

Thermofield dynamics is an exactly correct formulation of quantum mechanics at finite temperature in which a wavefunction is governed by an effective temperature-dependent quantum Hamiltonian. The optimized mean trajectory (OMT) approximation allows the calculation of spectroscopic response functions from trajectories produced by the classical limit of a mapping Hamiltonian that includes physical nuclear degrees of freedom and other effective degrees of freedom representing discrete vibronic states. Here, we develop a thermofield OMT (TF-OMT) approach in which the OMT procedure is applied to a temperature-dependent classical Hamiltonian determined from the thermofield-transformed quantum mapping Hamiltonian. Initial conditions for bath nuclear degrees of freedom are sampled from a zero-temperature distribution. Calculations of two-dimensional electronic spectra and two-dimensional vibrational–electronic spectra are performed for models that include excitonically coupled electronic states. The TF-OMT calculations agree very closely with the corresponding OMT results, which, in turn, represent well benchmark calculations with the hierarchical equations of motion method.

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.

Calculating Multidimensional Optical Spectra from Classical Trajectories

Multidimensional optical spectra are measured from the response of a material system to a sequence of laser pulses and have the capacity to elucidate specific molecular interactions and dynamics whose influences are absent or obscured in a conventional linear absorption spectrum. Interpretation of complex spectra is supported by theoretical modeling of the spectroscopic observable, requiring implementation of quantum dynamics for coupled electrons and nuclei. Performing numerically correct quantum dynamics in this context may pose computational challenges, particularly in the condensed phase. Semiclassical methods based on calculating classical trajectories offer a practical alternative. Here I review the recent application of some semiclassical, trajectory-based methods to nonlinear molecular vibrational and electronic spectra.

Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

A partially linearized spin-mapping approach for simulating nonlinear optical spectra

We present a partially linearized method based on spin-mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin partially linearized density matrix method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is, indeed, the case by calculating the nonlinear optical response functions relevant for the pump-probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna-Matthews-Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiments.

Quantum Entanglement from Classical Trajectories

A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.

Forecasting nonadiabatic dynamics using hybrid convolutional neural network/long short-term memory network

Modeling nonadiabatic dynamics in complex molecular or condensed-phase systems has been challenging, especially for the long-time dynamics. In this work, we propose a time series machine learning scheme based on the hybrid convolutional neural network/long short-term memory (CNN-LSTM) framework for predicting the long-time quantum behavior, given only the short-time dynamics. This scheme takes advantage of both the powerful local feature extraction ability of CNN and the long-term global sequential pattern recognition ability of LSTM. With feature fusion of individually trained CNN-LSTM models for the quantum population and coherence dynamics, the proposed scheme is shown to have high accuracy and robustness in predicting the linearized semiclassical and symmetrical quasiclassical mapping dynamics as well as the mixed quantum-classical Liouville dynamics of various spin-boson models with learning time up to 0.3 ps. Furthermore, if the hybrid network has learned the dynamics of a system, this knowledge is transferable that could significantly enhance the accuracy in predicting the dynamics of a similar system. The hybrid CNN-LSTM network is thus believed to have high predictive power in forecasting the nonadiabatic dynamics in realistic charge and energy transfer processes in photoinduced energy conversion.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

A bosonic perspective on the classical mapping of fermionic quantum dynamics

We consider the application of the original Meyer-Miller (MM) Hamiltonian to mapping fermionic quantum dynamics to classical equations of motion. Non-interacting fermionic and bosonic systems share the same one-body density dynamics when evolving from the same initial many-body state. The MM classical mapping is exact for non-interacting bosons, and therefore, it yields the exact time-dependent one-body density for non-interacting fermions as well. Starting from this observation, the MM mapping is compared to different mappings specific for fermionic systems, namely, the spin mapping with and without including a Jordan-Wigner transformation and the Li-Miller mapping (LMM). For non-interacting systems, the inclusion of fermionic anti-symmetry through the Jordan-Wigner transform does not lead to any improvement in the performance of the mappings, and instead, it worsens the classical description. For an interacting impurity model and for models of excitonic energy transfer, the MM and LMM mappings perform similarly, and in some cases, the former outperforms the latter when compared to a full quantum description. The classical mappings are able to capture interference effects, both constructive and destructive, that originate from equivalent energy transfer pathways in the models.

Multi-state harmonic models with globally shared bath for nonadiabatic dynamics in the condensed phase

Model Hamiltonians constructed from quantum chemistry calculations and molecular dynamics simulations are widely used for simulating nonadiabatic dynamics in the condensed phase. The most popular two-state spin-boson model could be built by mapping the all-atom anharmonic Hamiltonian onto a two-level system bilinearly coupled to a harmonic bath using the energy gap time correlation function. However, for more than two states, there lacks a general strategy to construct multi-state harmonic (MSH) models since the energy gaps between different pairs of electronic states are not entirely independent and need to be considered consistently. In this paper, we extend the previously proposed approach for building three-state harmonic models for photoinduced charge transfer to the arbitrary number of electronic states with a globally shared bath and the system-bath couplings are scaled differently according to the reorganization energies between each pair of states. We demonstrate the MSH model construction for an organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad dissolved in explicit tetrahydrofuran solvent. Nonadiabatic dynamics was simulated using mixed quantum-classical techniques, including the linearized semiclassical and symmetrical quasiclassical dynamics with the mapping Hamiltonians, mean-field Ehrenfest, and mixed quantum-classical Liouville dynamics in two-state, three-state, and four-state harmonic models of the triad system. The MSH models are shown to provide a general and flexible framework for simulating nonadiabatic dynamics in complex systems.

Ab initio symmetric quasi-classical approach to investigate molecular Tully models

We perform on-the-fly non-adiabatic molecular dynamics simulations using the symmetrical quasi-classical (SQC) approach with the recently suggested molecular Tully models: ethylene and fulvene. We attempt to provide benchmarks of the SQC methods using both the square and triangle windowing schemes as well as the recently proposed electronic zero-point-energy correction scheme (the so-called γ correction). We use the quasi-diabatic propagation scheme to directly interface the diabatic SQC methods with adiabatic electronic structure calculations. Our results showcase the drastic improvement of the accuracy by using the trajectory-adjusted γ-corrections, which outperform the widely used trajectory surface hopping method with decoherence corrections. These calculations provide useful and non-trivial tests to systematically investigate the numerical performance of various diabatic quantum dynamics approaches, going beyond simple diabatic model systems that have been used as the major workhorse in the quantum dynamics field. At the same time, these available benchmark studies will also likely foster the development of new quantum dynamics approaches based on these techniques.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Commutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics

We show that a novel, general phase space mapping Hamiltonian for nonadiabatic systems, which is reminiscent of the renowned Meyer-Miller mapping Hamiltonian, involves a commutator variable matrix rather than the conventional zero-point-energy parameter. In the exact mapping formulation on constraint space for phase space approaches for nonadiabatic dynamics, the general mapping Hamiltonian with commutator variables can be employed to generate approximate trajectory-based dynamics. Various benchmark model tests, which range from gas phase to condensed phase systems, suggest that the overall performance of the general mapping Hamiltonian is better than that of the conventional Meyer-Miller Hamiltonian.

Generalized discrete truncated Wigner approximation for nonadiabatic quantum-classical dynamics

Nonadiabatic molecular dynamics occur in a wide range of chemical reactions and femtochemistry experiments involving electronically excited states. These dynamics are hard to treat numerically as the system's complexity increases, and it is thus desirable to have accurate yet affordable methods for their simulation. Here, we introduce a linearized semiclassical method, the generalized discrete truncated Wigner approximation (GDTWA), which is well-established in the context of quantum spin lattice systems, into the arena of chemical nonadiabatic systems. In contrast to traditional continuous mapping approaches, e.g., the Meyer-Miller-Stock-Thoss and the spin mappings, GDTWA samples the electron degrees of freedom in a discrete phase space and thus forbids an unphysical unbounded growth of electronic state populations. The discrete sampling also accounts for an effective reduced but non-vanishing zero-point energy without an explicit parameter, which makes it possible to treat the identity operator and other operators on an equal footing. As numerical benchmarks on two linear vibronic coupling models and Tully's models show, GDTWA has a satisfactory accuracy in a wide parameter regime, independent of whether the dynamics is dominated by relaxation or by coherent interactions. Our results suggest that the method can be very adequate to treat challenging nonadiabatic dynamics problems in chemistry and related fields.

Simulating energy transfer dynamics in the Fenna-Matthews-Olson complex via the modified generalized quantum master equation

The generalized quantum master equation (GQME) provides a general and formally exact framework for simulating the reduced dynamics of open quantum systems. The recently introduced modified approach to the GQME (M-GQME) corresponds to a specific implementation of the GQME that is geared toward simulating the dynamics of the electronic reduced density matrix in systems governed by an excitonic Hamiltonian. Such a Hamiltonian, which is often used for describing energy and charge transfer dynamics in complex molecular systems, is given in terms of diabatic electronic states that are coupled to each other and correspond to different nuclear Hamiltonians. Within the M-GQME approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic density matrix is fully captured by a memory kernel superoperator, which can be obtained from short-lived (compared to the time scale of energy/charge transfer) projection-free inputs. In this paper, we test the ability of the M-GQME to predict the energy transfer dynamics within a seven-state benchmark model of the Fenna-Matthews-Olson (FMO) complex, with the short-lived projection-free inputs obtained via the Ehrenfest method. The M-GQME with Ehrenfest-based inputs is shown to yield accurate results across a wide parameter range. It is also found to dramatically outperform the direct application of the Ehrenfest method and to provide better-behaved convergence with respect to memory time in comparison to an alternative implementation of the GQME approach previously applied to the same FMO model.

On-the-Fly Symmetrical Quasi-Classical Dynamics with Meyer-Miller Mapping Hamiltonian for the Treatment of Nonadiabatic Dynamics at Conical Intersections

The on-the-fly version of the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian (SQC/MM) is implemented to study the nonadiabatic dynamics at conical intersections of polyatomic systems. The current on-the-fly implementation of the SQC/MM method is based on the adiabatic representation and the dressed momentum. To include the zero-point energy (ZPE) correction of the electronic mapping variables, we employ both the γ-adjusted and γ-fixed approaches. Nonadiabatic dynamics of the methaniminium cation (CH₂NH₂⁺) and azomethane are simulated using the on-the-fly SQC/MM method. For CH₂NH₂⁺, both ZPE correction approaches give reasonable and consistent results. However, for azomethane, the γ-adjusted version of the SQC/MM dynamics behaves much better than the γ-fixed version. Further analysis indicates that it is always recommended to use the γ-adjusted SQC/MM dynamics in the on-the-fly simulation of photoinduced dynamics of polyatomic systems, particularly when the excited state is well separated from the ground state in the Franck-Condon region. This work indicates that the on-the-fly SQC/MM method is a powerful simulation protocol to deal with the nonadiabatic dynamics of realistic polyatomic systems.

Two-dimensional vibrational-electronic spectra with semiclassical mechanics

Two-dimensional vibrational-electronic (2DVE) spectra probe the effects on vibronic spectra of initial vibrational excitation in an electronic ground state. The optimized mean trajectory (OMT) approximation is a semiclassical method for computing nonlinear spectra from response functions. Ensembles of classical trajectories are subject to semiclassical quantization conditions, with the radiation-matter interaction inducing discontinuous transitions. This approach has been previously applied to two-dimensional infrared and electronic spectra and is extended here to 2DVE spectra. For a system including excitonic coupling, vibronic coupling, and interaction of a chromophore vibration with a resonant environment, the OMT method is shown to well approximate exact quantum dynamics.

Three-state harmonic models for photoinduced charge transfer

A widely used strategy for simulating the charge transfer between donor and acceptor electronic states in an all-atom anharmonic condensed-phase system is based on invoking linear response theory to describe the system in terms of an effective spin-boson model Hamiltonian. Extending this strategy to photoinduced charge transfer processes requires also taking into consideration the ground electronic state in addition to the excited donor and acceptor electronic states. In this paper, we revisit the problem of describing such nonequilibrium processes in terms of an effective three-state harmonic model. We do so within the framework of nonequilibrium Fermi's golden rule (NE-FGR) in the context of photoinduced charge transfer in the carotenoid-porphyrin-C₆₀ (CPC₆₀) molecular triad dissolved in explicit tetrahydrofuran (THF). To this end, we consider different ways for obtaining a three-state harmonic model from the equilibrium autocorrelation functions of the donor-acceptor, donor-ground, and acceptor-ground energy gaps, as obtained from all-atom molecular dynamics simulations of the CPC₆₀/THF system. The quantum-mechanically exact time-dependent NE-FGR rate coefficients for two different charge transfer processes in two different triad conformations are then calculated using the effective three-state model Hamiltonians as well as a hierarchy of more approximate expressions that lead to the instantaneous Marcus theory limit. Our results show that the photoinduced charge transfer in CPC₆₀/THF can be described accurately by the effective harmonic three-state models and that nuclear quantum effects are small in this system.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Cavity-Modified Molecular Dynamics

Recent experimental realizations of strong coupling between optical cavity modes and molecular matter placed inside the cavity have opened exciting new routes for controlling chemical processes. Simulating the cavity-modified dynamics of complex chemical systems calls for the development of accurate, flexible, and cost-effective approximate numerical methods that scale favorably with system size and complexity. In this Letter, we test the ability of quasiclassical mapping Hamiltonian methods to serve this purpose. We simulated the spontaneous emission dynamics of an atom confined to a microcavity via five different variations of the linearized semiclassical (LSC) method. Our main finding is that recently proposed LSC-based methods which use a modified form of the identity operator are reasonably accurate and perform significantly better than the Ehrenfest and standard LSC methods, without significantly increasing computational costs. These methods are therefore highly promising as a general purpose tool for simulating cavity-modified dynamics of complex chemical systems.

Non-adiabatic Matsubara dynamics and non-adiabatic ring-polymer molecular dynamics

We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future.

Analysis of bath motion in MM-SQC dynamics via dimensionality reduction approach: Principal component analysis

The system-plus-bath model is an important tool to understand the nonadiabatic dynamics of large molecular systems. Understanding the collective motion of a large number of bath modes is essential for revealing their key roles in the overall dynamics. Here, we applied principal component analysis (PCA) to investigate the bath motion in the basis of a large dataset generated from the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian nonadiabatic dynamics for the excited-state energy transfer in the Frenkel-exciton model. The PCA method clearly elucidated that two types of bath modes, which either display strong vibronic coupling or have frequencies close to that of the electronic transition, are important to the nonadiabatic dynamics. These observations were fully consistent with the physical insights. The conclusions were based on the PCA of the trajectory data and did not involve significant pre-defined physical knowledge. The results show that the PCA approach, which is one of the simplest unsupervised machine learning dimensionality reduction methods, is a powerful one for analyzing complicated nonadiabatic dynamics in the condensed phase with many degrees of freedom.

Two-oscillator mapping modification of the Poisson bracket mapping equation formulation of the quantum-classical Liouville equation

Mapping basis solutions provide efficient ways for simulating mixed quantum-classical (MQC) dynamics in complex systems by matching multiple quantum states of interest to some fictitious physical states. Recently, various MQC methods were devised such that two harmonic oscillators are employed to represent each electronic state, showing improvements over one-oscillator-based methods. Here, we introduce and analyze newly modified mapping approximations of the quantum-classical Liouville equation (QCLE) using two oscillators for each electronic state. We design two separate mapping relations that we can adopt toward simulating dynamics and computing expectation values. Through the process, two MQC methods can be constructed, one of which actually reproduces the population dynamics of the forward and backward trajectory solution of QCLE. By applying the methods to spin-boson systems with a range of parameters, we find out that the choice of mapping relations greatly affects the simulation results. We also show that further improvement is possible through using modified identity operator formulations. Our findings may be helpful in constructing improved MQC methods in the future.

Spectroscopic response theory with classical mapping Hamiltonians

Exact quantum dynamics with a time-independent Hamiltonian in a discrete state space can be computed using classical mechanics through the classical Meyer-Miller-Stock-Thoss mapping Hamiltonian. In order to compute quantum response functions from classical dynamics, we extend this mapping to a quantum Hamiltonian with time-dependence arising from a classical field. This generalization requires attention to time-ordering in quantum and classical propagators. Quantum response theory with the original quantum Hamiltonian is equivalent to classical response theory with the classical mapping Hamiltonian. We elucidate the structure of classical response theory with the mapping Hamiltonian, thereby generating classical versions of the two-sided quantum density operator diagrams conventionally used to describe spectroscopic processes. This formal development can provide a foundation for new semiclassical approximations to spectroscopic observables for models in which classical nuclear degrees of freedom are introduced into a mapping Hamiltonian describing electronic states. Calculations of the temperature-dependence of two-dimensional electronic spectra for an exciton dimer using two semiclassical approaches are compared with benchmark calculations using the hierarchical equations of motion method.