Dynamics of an excitation-transfer trimer: Interference, coherence, Berry's phase development, and vibrational control of non-adiabaticity.

We detail several interesting features in the dynamics of an equilaterally shaped electronic excitation-transfer (EET) trimer with distance-dependent intermonomer excitation-transfer couplings. In the absence of electronic-vibrational coupling, symmetric and antisymmetric superpositions of two single-monomer excitations are shown to exhibit purely constructive, oscillatory, and purely destructive interference in the EET to the third monomer, respectively. In the former case, the transfer is modulated by motion in the symmetrical framework-expansion vibration induced by the Franck-Condon excitation. Distortions in the shape of the triangular framework degrade that coherent EET while activating excitation transfer in the latter case of an antisymmetric initial state. In its symmetrical configuration, two of the three single-exciton states of the trimer are degenerate. This degeneracy is broken by the Jahn-Teller-active framework distortions. The calculations illustrate closed, approximately circular pseudo-rotational wave-packet dynamics on both the lower and the upper adiabatic potential energy surfaces of the degenerate manifold, which lead to the acquisition after one cycle of physically meaningful geometric (Berry) phases of π. Another manifestation of Berry-phase development is seen in the evolution of the vibrational probability density of a wave packet on the lower Jahn-Teller adiabatic potential comprising a superposition of clockwise and counterclockwise circular motions. The circular pseudo-rotation on the upper cone is shown to stabilize the adiabatic electronic state against non-adiabatic internal conversion via the conical intersection, a dynamical process analogous to Slonczewski resonance. Strategies for initiating and monitoring these various dynamical processes experimentally using pre-resonant impulsive Raman excitation, short-pulse absorption, and multi-dimensional wave-packet interferometry are outlined in brief.

Monitoring the evolution of intersite and interexciton coherence in electronic excitation transfer via wave-packet interferometry.

We detail an experimental strategy for tracking the generation and time-development of electronic coherence within the singly excited manifold of an energy-transfer dimer. The technique requires that the two monomers have nonparallel electronic transition-dipole moments and that these possess fixed orientations in space. It makes use of two-dimensional wave-packet interferometry (WPI or whoopee) measurements in which the A, B, C, and D pulses have respective polarizations e, e, e, and e'. In the case of energy-transfer coupling that is weak or strong compared to electronic-nuclear interactions, it is convenient to follow the evolution of intersite or interexciton coherence, respectively. Under weak coupling, e could be perpendicular to the acceptor chromophore's transition dipole moment and the unit vector e' would be perpendicular to the donor's transition dipole. Under strong coupling, e could be perpendicular to the ground-to-excited transition dipole to the lower exciton level and e' would be perpendicular to the ground-to-excited transition dipole to the upper exciton level. If the required spatial orientation can be realized for an entire ensemble, experiments of the kind proposed could be performed by either conventional four-wave-mixing or fluorescence-detected WPI methods. Alternatively, fluorescence-detected whoopee experiments of this kind could be carried out on a single energy-transfer dimer of fixed orientation. We exhibit detailed theoretical expressions for the desired WPI signal, explain the physical origin of electronic coherence detection, and show calculated observed-coherence signals for model dimers with one, two, or three internal vibrational modes per monomer and both weak and strong energy-transfer coupling.

Quantum dynamics and spectroscopy of dihalogens in solid matrices. II. Theoretical aspects and G-MCTDH simulations of time-resolved coherent Raman spectra of Schrödinger cat states of the embedded I2Kr18 cluster.

This study presents quantum dynamical simulations, using the Gaussian-based multiconfigurational time-dependent Hartree (G-MCTDH) method, of time-resolved coherent Raman four-wave-mixing spectroscopic experiments for the iodine molecule embedded in a cryogenic crystal krypton matrix [D. Picconi et al., J. Chem. Phys. 150, 064111 (2019)]. These experiments monitor the time-evolving vibrational coherence between two wave packets created in a quantum superposition (i.e., a "Schrödinger cat state") by a pair of pump pulses which induce electronic B Πu30+⟵XΣg+1 transitions. A theoretical description of the spectroscopic measurement is developed, which elucidates the connection between the nonlinear signals and the wave packet coherence. The analysis provides an effective means to simulate the spectra for several different optical conditions with a minimum number of quantum dynamical propagations. The G-MCTDH method is used to calculate and interpret the time-resolved coherent Raman spectra of two selected initial superpositions for a I2Kr18 cluster embedded in a frozen Kr cage. The time- and frequency-dependent signals carry information about the molecular mechanisms of dissipation and decoherence, which involve vibrational energy transfer to the stretching mode of the four "belt" Kr atoms. The details of these processes and the number of active solvent modes depend in a non-trivial way on the specific initial superposition.

Quantum dynamics and spectroscopy of dihalogens in solid matrices. I. Efficient simulation of the photodynamics of the embedded I2Kr18 cluster using the G-MCTDH method.

The molecular dynamics following the electronic BΠu30+⟵XΣg+1 photoexcitation of the iodine molecule embedded in solid krypton are studied quantum mechanically using the Gaussian variant of the multiconfigurational time-dependent Hartree method (G-MCTDH). The accuracy of the Gaussian wave packet approximation is validated against numerically exact MCTDH simulations for a fully anharmonic seven-dimensional model of the I2Kr18 cluster in a crystal Kr cage. The linear absorption spectrum, time-evolving vibrational probability densities, and I2 energy expectation value are accurately reproduced by the numerically efficient G-MCTDH approach. The reduced density matrix of the chromophore is analyzed in the coordinate, Wigner and energy representations, so as to obtain a multifaceted dynamical view of the guest-host interactions. Vibrational coherences extending over the bond distance range 2.7 Å < RI-I < 4.0 Å are found to survive for several vibrational periods, despite extensive dissipation. The present results prepare the ground for the simulation of time-resolved coherent Raman spectroscopy of the I2-krypton system addressed in Paper II.

Mixed quantum/semiclassical wave-packet dynamical method for condensed-phase molecular spectroscopy signals.

We report the successful application of a recently developed mixed quantum/semiclassical wave-packet dynamical theory to the calculation of a spectroscopic signal, the linear absorption spectrum of a realistic small-molecule chromophore in a cryogenic environment. This variational fixed vibrational basis/Gaussian bath (FVB/GB) theory avails itself of an assumed time scale separation between a few, mostly intramolecular, high-frequency nuclear motions and a larger number of slower degrees of freedom primarily associated with an extended host medium. The more rapid, large-amplitude system dynamics is treated with conventional basis-set methods, while the slower time-evolution of the weakly coupled bath is subject to a semiclassical, thawed Gaussian trial form that honors the overall vibrational ground state, and hence the initial state prepared by its Franck-Condon transfer to an excited electronic state. We test this general approach by applying it to a small, symmetric iodine-krypton cluster suggestive of molecular iodine embedded in a low-temperature matrix. Because of the relative simplicity of this model complex, we are able to compare the absorption spectrum calculated via FVB/GB dynamics using Heller's time-dependent formula with one obtained from rigorously calculated eigenenergies and Franck-Condon factors. The FVB/GB treatment proves to be accurate at approximately 15-cm-1 resolution, despite the presence of several thousand spectral lines and a sequence of various-order system-bath resonances culminating at the highest absorption frequencies in an inversion of the relative system and bath time scales.

Broad-Band Pump-Probe Spectroscopy Quantifies Ultrafast Solvation Dynamics of Proteins and Molecules.

In this work, we demonstrate the use of broad-band pump-probe spectroscopy to measure femtosecond solvation dynamics. We report studies of a rhodamine dye in methanol and cryptophyte algae light-harvesting proteins in aqueous suspension. Broad-band impulsive excitation generates a vibrational wavepacket that oscillates on the excited-state potential energy surface, destructively interfering with itself at the minimum of the surface. This destructive interference gives rise to a node at a certain probe wavelength that varies with time. This reveals the Gibbs free-energy changes of the excited-state potential energy surface, which equates to the solvation time correlation function. This method captures the inertial solvent response of water (∼40 fs) and the bimodal inertial response of methanol (∼40 and ∼150 fs) and reveals how protein-buried chromophores are sensitive to the solvent dynamics inside and outside of the protein environment.

Ultrafast transient absorption revisited: Phase-flips, spectral fingers, and other dynamical features.

We rebuild the theory of ultrafast transient-absorption/transmission spectroscopy starting from the optical response of an individual molecule to incident femtosecond pump and probe pulses. The resulting description makes use of pulse propagators and free molecular evolution operators to arrive at compact expressions for the several contributions to a transient-absorption signal. In this alternative description, which is physically equivalent to the conventional response-function formalism, these signal contributions are conveniently expressed as quantum mechanical overlaps between nuclear wave packets that have undergone different sequences of pulse-driven optical transitions and time-evolution on different electronic potential-energy surfaces. Using this setup in application to a simple, multimode model of the light-harvesting chromophores of PC577, we develop wave-packet pictures of certain generic features of ultrafast transient-absorption signals related to the probed-frequency dependence of vibrational quantum beats. These include a Stokes-shifting node at the time-evolving peak emission frequency, antiphasing between vibrational oscillations on opposite sides (i.e., to the red or blue) of this node, and spectral fingering due to vibrational overtones and combinations. Our calculations make a vibrationally abrupt approximation for the incident pump and probe pulses, but properly account for temporal pulse overlap and signal turn-on, rather than neglecting pulse overlap or assuming delta-function excitations, as are sometimes done.

Variational mixed quantum/semiclassical simulation of dihalogen guest and rare-gas solid host dynamics.

A variational mixed quantum-semiclassical theory for the internal nuclear dynamics of a small molecule and the induced small-amplitude coherent motion of a low-temperature host medium is developed, tested, and used to simulate the temporal evolution of nonstationary states of the internal molecular and surrounding medium degrees of freedom. In this theory, termed the Fixed Vibrational Basis/Gaussian Bath (FVB/GB) method, the system is treated fully quantum mechanically while Gaussian wave packets are used for the bath degrees of freedom. An approximate time-dependent wave function of the entire model is obtained instead of just a reduced system density matrix, so the theory enables the analysis of the entangled system and bath dynamics that ensues following initial displacement of the internal-molecular (system) coordinate from its equilibrium position. The norm- and energy-conserving properties of the propagation of our trial wave function are natural consequences of the Dirac-Frenkel-McLachlan variational principle. The variational approach also stabilizes the time evolution in comparison to the same ansatz propagated under a previously employed locally quadratic approximation to the bath potential and system-bath interaction terms in the bath-parameter equations of motion. Dynamics calculations are carried out for molecular iodine in a 2D krypton lattice that reveal both the time-course of vibrational decoherence and the details of host-atom motion accompanying energy dissipation and dephasing. This work sets the stage for the comprehensive simulation of ultrafast time-resolved optical experiments on small molecules in low-temperature solids.

How fissors works: observing vibrationally adiabatic conformational change through femtosecond stimulated Raman spectroscopy.

With the help of a two-dimensional model system comprising a slow conformational degree of freedom and a higher-frequency vibration, we investigate the molecular-level origin and dynamical information content of femtosecond stimulated Raman spectroscopy (fissors) signals. Our treatment avails itself of the time scale separation between conformational and vibrational modes by incorporating a vibrationally adiabatic approximation to the conformational dynamics. We derive an expression for the fissors signal without resort to the macroscopic concepts of light- and phonon-wave propagation employed in prior coupled-wave analyses. Numerical calculations of fissors spectra illustrate the case of relatively small conformational mass (still large enough that conformational motion does not induce any change in the vibrational quantum number) in which conformational sidebands accompany a central peak in the Raman gain at a conformationally averaged vibrational transition frequency, and the case of a larger conformational mass in which the sidebands merge with the central peak and the frequency of the latter tracks the time-evolving conformational coordinate.

Studies of impulsive vibrational influence on ultrafast electronic excitation transfer.

We investigated electronic energy-transfer dynamics in three model dimers within which coherent intramonomer nuclear motion had been induced by impulsive Raman excitation using an optimized, electronically preresonant control pulse. Calculations of the donor-survival probability, the ultrafast pump-probe signal, and the pump-probe difference signal are presented for dithia-anthracenophane and homodimers of 2-difluoromethylanthracene and 2-trifluoromethylanthracene. Survival probabilities and signals, along with phase-space analyses, elucidated the mechanisms, extent, and spectroscopic manifestations of external vibrational or torsional control over electronic excitation transfer.

Numerical tests of a fixed vibrational basis/gaussian bath theory for small molecule dynamics in low-temperature media.

A recently framed quantum/semiclassical treatment for the internal nuclear dynamics of a small molecule and the induced small-amplitude coherent motion of a low-temperature host medium (Chapman, C. T.; Cina, J. A. J. Chem. Phys.2007,127, 114502) is further analyzed and subjected to initial tests of its numerical implementation. In the illustrative context of a 1D system interacting with a 1D medium, we rederive the fixed vibrational basis/gaussian bath (FVB/GB) equations of motion for the parameters defining the gaussian bath wave packet accompanying each of the energy eigenkets of the quantum mechanical system. The conditions of validity for the gaussian-bath approximation are shown to coincide with those supporting approximate population conservation. We perform initial numerical tests of the FVB/GB scheme and illustrate the semiclassical description it provides of coherent motion in the medium by comparing its predictions with the exact results for a high-frequency system harmonic oscillator bilinearly coupled to a lower-frequency bath oscillator. Linear vibronic absorption spectra or, equivalently, ultrafast wave packet interferometry signals are shown to be readily and accurately calculable within the FVB/GB framework.

Using wave-packet interferometry to monitor the external vibrational control of electronic excitation transfer.

We investigate the control of electronic energy transfer in molecular dimers through the preparation of specific vibrational coherences prior to electronic excitation, and its observation by nonlinear wave-packet interferometry (nl-WPI). Laser-driven coherent nuclear motion can affect the instantaneous resonance between site-excited electronic states and thereby influence short-time electronic excitation transfer (EET). We first illustrate this control mechanism with calculations on a dimer whose constituent monomers undergo harmonic vibrations. We then consider the use of nl-WPI experiments to monitor the nuclear dynamics accompanying EET in general dimer complexes following impulsive vibrational excitation by a subresonant control pulse (or control pulse sequence). In measurements of this kind, two pairs of polarized phase-related femtosecond pulses following the control pulse generate superpositions of coherent nuclear wave packets in optically accessible electronic states. Interference contributions to the time- and frequency-integrated fluorescence signals due to overlaps among the superposed wave packets provide amplitude-level information on the nuclear and electronic dynamics. We derive the basic expression for a control-pulse-dependent nl-WPI signal. The electronic transition moments of the constituent monomers are assumed to have a fixed relative orientation, while the overall orientation of the complex is distributed isotropically. We include the limiting case of coincident arrival by pulses within each phase-related pair in which control-influenced nl-WPI reduces to a fluorescence-detected pump-probe difference experiment. Numerical calculations of pump-probe signals based on these theoretical expressions are presented in the following paper [J. D. Biggs and J. A. Cina, J. Chem. Phys. 131, 224302 (2009)].

Calculations of nonlinear wave-packet interferometry signals in the pump-probe limit as tests for vibrational control over electronic excitation transfer.

The preceding paper [J. D. Biggs and J. A. Cina, J. Chem. Phys. 131, 224101 (2009)] (referred to here as Paper 1), describes a strategy for externally influencing the course of short-time electronic excitation transfer (EET) in molecular dimers and observing the process by nonlinear wave-packet interferometry (nl-WPI). External influence can, for example, be exerted by inducing coherent intramolecular vibration in one of the chromophores prior to short-pulse electronic excitation of the other. Within a sample of isotropically oriented dimers having a specified internal geometry, a vibrational mode internal to the acceptor chromophore can be preferentially driven by electronically nonresonant impulsive stimulated Raman (or resonant infrared) excitation with a short polarized "control" pulse. A subsequent electronically resonant polarized pump then preferentially excites the donor, and EET ensues. Paper 1 investigates control-pulse-influenced nl-WPI as a tool for the spectroscopic evaluation of the effect of coherent molecular vibration on excitation transfer, presenting general expressions for the nl-WPI difference signal from a dimer following the action of a control pulse of arbitrary polarization and shape. Electronic excitation is to be effected and its interchromophore transfer monitored by resonant pump and probe "pulses," respectively, each consisting of an optical-phase-controlled ultrashort pulse-pair having arbitrary polarization, duration, center frequency, and other characteristics. Here we test both the control strategy and its spectroscopic investigation-with some sacrifice of amplitude-level detail-by calculating the pump-probe difference signal. That signal is the limiting case of the control-influenced nl-WPI signal in which the two pulses in the pump pulse-pair coincide, as do the two pulses in the probe pulse-pair. We present calculated pump-probe difference signals for (1) a model excitation-transfer complex in which two equal-energy monomers each support one moderately Franck-Condon active intramolecular vibration; (2) a simplified model of the covalent dimer dithia-anthracenophane, representing its EET dynamics following selective impulsive excitation of the weakly Franck-Condon active nu(12) anthracene vibration at 385 cm(-1); and (3) a model complex featuring moderate electronic-vibrational coupling in which the site energy of the acceptor chromophore is lower than that of the donor.

Wave-packet interferometry and molecular state reconstruction: spectroscopic adventures on the left-hand side of the Schrödinger equation.

This article summarizes theoretical studies of molecular state determination by wave-packet interferometry (WPI) and recounts some recent experimental applications of molecular WPI. Calculations predict that two-color nonlinear WPI data can be used to reconstruct a rovibronic target wave packet evolving under an incompletely characterized nuclear Hamiltonian. This can be accomplished by the isolation via phase cycling or wave-vector matching of an exhaustive collection of overlaps between the unknown target and the members of a family of reference wave packets whose form is known by construction. This review highlights recent experiments employing WPI to gain amplitude-level information about the photoexcited-state dynamics of small molecules in the gas phase and in rare-gas crystals. I briefly describe a new semiclassical theory for condensed-phase WPI and other coherence-spectroscopy measurements, such as time-resolved coherent anti-Stokes Raman scattering, and mention our initial studies of nonlinear WPI from electronic energy-transfer complexes.

Semiclassical treatments for small-molecule dynamics in low-temperature crystals using fixed and adiabatic vibrational bases.

Time-resolved coherent nonlinear optical experiments on small molecules in low-temperature host crystals are exposing valuable information on quantum mechanical dynamics in condensed media. We make use of generic features of these systems to frame two simple, comprehensive theories that will enable the efficient calculations of their ultrafast spectroscopic signals and support their interpretation in terms of the underlying chemical dynamics. Without resorting to a simple harmonic analysis, both treatments rely on the identification of normal coordinates to unambiguously partition the well-structured guest-host complex into a system and a bath. Both approaches expand the overall wave function as a sum of product states between fully anharmonic vibrational basis states for the system and approximate Gaussian wave packets for the bath degrees of freedom. The theories exploit the fact that ultrafast experiments typically drive large-amplitude motion in a few intermolecular degrees of freedom of higher frequency than the crystal phonons, while these intramolecular vibrations indirectly induce smaller-amplitude--but still perhaps coherent--motion among the lattice modes. The equations of motion for the time-dependent parameters of the bath wave packets are fairly compact in a fixed vibrational basis/Gaussian bath (FVB/GB) approach. An alternative adiabatic vibrational basis/Gaussian bath (AVB/GB) treatment leads to more complicated equations of motion involving adiabatic and nonadiabatic vector potentials. Computational demands for propagation of the parameter equations of motion appear quite manageable for tens or hundreds of atoms and scale similarly with system size in the two cases. Because of the time-scale separation between intermolecular and lattice vibrations, the AVB/GB theory may in some instances require fewer vibrational basis states than the FVB/GB approach. Either framework should enable practical first-principles calculations of nonlinear optical signals from molecules in cryogenic matrices and their semiclassical interpretation in terms of electronic and vibrational decoherence and vibrational population relaxation, all within a pure-state description of the macroscopic many-body complex.

Nonlinear wave-packet interferometry and molecular state reconstruction in a vibrating and rotating diatomic molecule.

We formulate two-color nonlinear wave-packet interferometry (WPI) for application to a diatomic molecule in the gas phase and show that this form of heterodyne-detected multidimensional electronic spectroscopy will permit the reconstruction of photoinduced rovibrational wave packets from experimental data. Using two phase-locked pulse pairs, each resonant with a different electronic transition, nonlinear WPI detects the quadrilinear interference contributions to the population of an excited electronic state. Combining measurements taken with different phase-locking angles isolates various quadrilinear interference terms. One such term gives the complex overlap between a propagated one-pulse target wave packet and a variable three-pulse reference wave packet. The two-dimensional interferogram in the time domain specifies the complex-valued overlap of the given target state with a collection of variable reference states. An inversion procedure based on singular-value decomposition enables reconstruction of the target wave packet from the interferogram without prior detailed characterization of the nuclear Hamiltonian under which the target propagates. With numerically calculated nonlinear WPI signals subject to Gaussian noise, we demonstrate the reconstruction of a rovibrational wave packet launched from the A state and propagated in the E state of Li2.

Molecular state reconstruction by nonlinear wave packet interferometry.

We show that time- and phase-resolved two-color nonlinear wave packet interferometry can be used to reconstruct the probability amplitude of an optically prepared molecular wave packet without prior knowledge of the underlying potential surface. We analyze state reconstruction in pure- and mixed-state model systems excited by shaped laser pulses and propose nonlinear wave packet interferometry as a tool for identifying optimized wave packets in coherent control experiments.