Low-temperature zero phonon lineshapes with various Brownian oscillator spectral densities

Empirical asymmetric phonon sideband due to phonons in the protein matrix present in photosynthetic complexes: Time-domain response theory

The phonon spectral density plays a key role in probing the dynamical and spectral behavior of molecular aggregates. One may utilize the intimate connection between the one-phonon profile and the phonon spectral density to extract a plausible form of the spectral density of media with rich structure using advanced optical spectroscopy. The excitonic transition is normally accompanied by a broad, asymmetric phonon-side band due to the coupling to the phonons in the surrounding protein matrix present in photosynthetic complexes. The asymmetry in the one-phonon profile of a homogeneous absorption spectrum and other experiments performed on photosynthetic bacterial reaction centers (BRCs) led the Small group to employ a half-Gaussian distribution function on the red side and half-Lorentzian distribution function on the blue side of the absorption lineshape to account for the one-phonon profile asymmetrical shape and relaxation effects contributing to spectroscopy and dynamics of BRCs at hand. Different research groups successfully employed the theory of Small to simulate their photosynthetic spectral data so they could calculate the homogeneous absorption and hole-burned spectra of photosynthetic complexes. Although this report does not directly use the formulae of homogeneous absorption, hole-burning, and fluorescence line-narrowed spectra of BRCs, and photosynthetic complexes, developed by Hayes-Small, it builds on their idea of the phonon sideband asymmetric shape in deriving an accurate and computationally efficient linear electronic transition dipole moment time correlation function. Besides the compelling tractability and efficiency of this correlation function, it accounts for excitonic coupling and eliminates all the inconsistencies arising in the Hayes-Small theory.

Mixed Quantum-Classical Liouville Equation Treatment of Electronic Spectroscopy of Condensed Systems: Harmonic and Anharmonic Electron-Phonon Coupling

This Review integrates the use of electronic optical response function theory and the mixed quantum-classical (MQC) Liouville equation (MQCLE), thereby leading to electronic spectroscopy in MQC media. It further sheds light on the applicability, utility, and efficiency of the mixed quantum-classical dynamics (MQCD) formalism, which starts off with the MQCLE, in probing spectroscopy and dynamics of condensed systems, whereby quantum mechanics and classical mechanics are combined systematically. The author has been exploring and implementing MQCD to investigate electron-phonon coupling effects on electronic dephasing in harmonic and anharmonic systems by calculating linear and nonlinear optical transition analytically and numerically dipole moment time correlation functions in an MQC environment, thereby presenting an in depth spectral profile analysis and their shape and symmetry. The distinctive capability of the MQC time correlation functions is that ergodicity and stationarity properties are inherently satisfied as part of the mixed quantum-classical dynamics (MQCD) framework, unlike classical correlation functions. While some research groups have applied MQCLE to calculate vibrational spectra to study hydrogen-bonded complexes in a MQC environment and other groups calculated Optical response function to probe electron transfer dynamics using the basis mapping technique, the approach, purpose, rigor, applications, and path to the end results reported herein are different. Finally, the same framework is employed to study dissipative systems in the MQC limit, whereby the zero-phonon line adopts the correct width and eliminates its asymmetry. While the full quantum mechanical model, like the multimode Brownian oscillator (MBO) model, yields the correct width and inaccurate shape in the low-temperature limit, the MQCD formalism seems to produce an accurate zero-phonon profile. Nonlinear optical signals are also reviewed in MQC media to show the applicability and utility of this approach. The vibronic optical response functions developed here will account for geometry change, frequency change, and anharmonicity upon electronic excitation to accurately probe electronic dephasing, electron-phonon coupling, shape, and symmetry of profiles and present differences and similarities to the MBO model on pure electronic dephasing. Frequency change and anharmonicity are vitally crucial for accurately assessing electron-phonon coupling upon electronic excitation. This is an additional unique result obtained by the author to further demonstrate the applicability and utility of this approach over other approximation schemes in probing electronic dephasing including that of the MBO model.

Low-temperature vibronic spectroscopy of condensed chromophore exhibiting inhomogeneous distribution of vibrational frequencies in a mixed quantum-classical environment

This work has been motivated by the recent paper by the author [M. Toutounji, Phys. Chem. Chem. Phys., 2021, 23, 21981] whereby a mixed quantum-classical Liouville equation was used to probe the spectroscopy and dynamics of a spin-boson system. A mixed quantum-classical Liouville equation treats the system of interest quantum mechanically, the bath classically, and the coupling term mixed quantum-classical mechanically. This paper offers a two-fold advantage: correcting the treatment of the electronic transition decay (width in frequency domain) and assessing the local heterogeneous vibrational structure. The homogeneous linear absorption spectrum of a chromophore embedded in a mixed quantum-classical environment at low temperature is composed of a sharp peak called a zero-phonon line (ZPL) and a broad phonon sideband (PSB), whereby the ZPL and the PSB are assimilated by a Lorentzian function and Voigt profiles, respectively. The PSB, in this case, is characterized by a local heterogeneous structure due to a dispersive medium of vibrations, modeled by vibrational Gaussian distributions to represent the arising inhomogeneous broadening and Lorentzians to model the homogeneous vibrations. This description seems to model proteins and amorphous solids exhibiting a local heterogeneous structure as both electronic and vibrational inhomogeneous broadening seems to be large in these media. This work provides a derivation of linear absorption lineshape and vibronic transition dipole moment time correlation functions, both of which account for pure electronic dephasing (ZPL width) the Voigt profile description of the phonon profiles (PSB) in dispersive media.

Electronic dephasing of polyatomic molecules interacting with mixed quantum-classical media

This paper offers an expedient, efficient, and unique treatment of multimode quantum subsystems (polyatomic molecules) interacting with a classical environment in which the time evolution of the coupling term is governed by the algebraic rules of statistical mechanics in mixed quantum-classical systems developed by Kapral and Nielsen [S. Nielsen, R. Kapral, and G. Ciccotti, J. Chem. Phys., 2001, 115, 5805]. This unique time evolution of the coupling term is neither quantal nor classical but rather something different that relies heavily on Wigner transform, thereby leading to non-Newtonian mechanics. As such, an argument is presented that the approach provided herein for treating polyatomic molecular systems in a mixed quantum-classical environment is new and different as opposed to the many other schemes of semiclassical dynamics that are normally employed to study such systems. The merits of expediency and efficiency of the herein mixed quantum-classical dynamics calculations emanate from avoiding using integrals for time evolutions, and, instead, employing matrix mechanics whereby LU decomposition and singular value decomposition (SVD) numerical techniques are utilized for diagonalization. An electronic 2-level subsystem interacting with a classical bath through the spin-boson model to render accurate pure electronic dephasing in multimode molecular systems by eliminating the unphysical asymmetry in the line shape of the zero-phonon line (ZPL) exhibited by other models is exploited. This work has a superior advantage over the single-mode spin-boson model, published previously, whereby a multitude of types of vibrational modes (slow, fast, or both) of the quantum subsystem may readily be handled using different spectral densities. The spin-boson model used here is a composite system made up of a quantum subsystem, i.e., a subsystem bilinearly coupled to a multidimensional harmonic oscillator (representing the intermediate quantum vibrational modes between the electronic subsystem and the bath), interacting with a classical bath, where the coupling term is governed by the mixed quantum-classical Liouville equation. A multidimensional coherent-state approach is employed to deal with the time evolution of the quantum subsystem. A closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in mixed quantum-classical dissipative media is derived. Pure electronic dephasing is probed using the aforementioned approach. Linear absorption spectra and 4-wave mixing signals (e.g., photon echo and pump-probe) are calculated showing a reasonable thermal broadening, temporal decay, and accurate pure dephasing.

Analytic and numerical vibronic spectra from quasi-classical trajectory ensembles

The truncated Wigner approximation to quantum dynamics in phase space is explored in the context of computing vibronic line shapes for monomer linear optical spectra. We consider multiple model potential forms including a shifted harmonic oscillator with both equal and unequal frequencies on the ground and excited state potentials as well as a shifted Morse potential model. For the equal-frequency shifted harmonic oscillator model, we derive an analytic expression for the exact vibronic line shape that emphasizes the importance of using a quantum mechanical distribution of phase space initial conditions. For the unequal-frequency shifted harmonic oscillator model, we are no longer able to obtain an exact expression for the vibronic line shape in terms of independent deterministic classical trajectories. We show how one can rigorously account for corrections to the truncated Wigner approximation through nonlinear responses of the line shape function to momentum fluctuations along a classical trajectory and demonstrate the qualitative improvement in the resulting spectrum when the leading-order quantum correction is included. Finally, we numerically simulate absorption spectra of a highly anharmonic shifted Morse potential model. We find that, while finite quantization and the dissociation limit are captured with reasonable accuracy, there is a qualitative breakdown of the quasi-classical trajectory ensemble's ability to describe the vibronic line shape when the relative shift in Morse potentials becomes large. The work presented here provides clarity on the origin of unphysical negative features known to contaminate absorption spectra computed with quasi-classical trajectory ensembles.

Singlet fission dynamics and optical spectra of pentacene and its derivatives

A multimode Brownian oscillator model is employed to investigate the absorption spectra of pentacene and its derivatives in solution and thin films. Excellent agreement has been obtained between simulated and measured absorption spectra. Furthermore, using parameters obtained from fitting the absorption spectra of these pentacene derivatives, the singlet fission dynamics and two-dimensional electronic spectra of an ab initio Hamiltonian are investigated by Dirac-Frenkel time-dependent variation with multiple Davydov trial states. It is found that the periodic wave packet motion induced in the displaced excited state, and the accompanying vibrational relaxation, can be visualized by two-dimensional electronic spectra at short times.

Dynamics of coherence, localization and excitation transfer in disordered nanorings

Self-assembled supramolecular aggregates are excellent candidates for the design of efficient excitation transport devices. Both artificially prepared and natural photosynthetic aggregates in plants and bacteria present an important degree of disorder that is supposed to hinder excitation transport. Besides, molecular excitations couple to nuclear motion affecting excitation transport in a variety of ways. We present an exhaustive study of exciton dynamics in disordered nanorings with long-range interactions under the influence of a phonon bath taking the LH2 system of purple bacteria as a model. Nuclear motion is explicitly taken into account by employing the Davydov ansatz description of the polaron and quantum dynamics are obtained using a time-dependent variational method. We reveal an optimal exciton-phonon coupling that suppresses disorder-induced localization and facilitate excitation de-trapping. This excitation transfer enhancement, mediated by environmental phonons, is attributed to energy relaxation toward extended, low-energy excitons provided by the precise LH2 geometry with anti-parallel dipoles and long-range interactions. An analysis of localization and spectral statistics is followed by dynamic measures of coherence and localization, transfer efficiency and superradiance. Linear absorption, 2D photon-echo spectra and diffusion measures of the exciton are examined to monitor the diffusive behavior as a function of the strengths of disorder and exciton-phonon coupling.

Luminescence Anisotropy and Thermal Effect of Magnetic and Electric Dipole Transitions of Cr3+ Ions in Yb:YAG Transparent Ceramic

In this article, we present an in-depth optical study on luminescence spectral features and the thermal effect of the magnetic dipole (MD) transitions (e.g., the R lines of ²E → ⁴A₂) and the associated electric dipole transitions (e.g., phonon-induced sidebands of the R lines) of Cr³⁺ ions in ytterbium-yttrium aluminum garnet polycrystalline transparent ceramic. The doubly split R lines predominately due to the doublet splitting of the ²E level of the Cr³⁺ ion in an octahedral crystal field are found to show a very large anisotropy in both emission intensity and thermal broadening. The large departure from the intensity equality between them could be interpreted in terms of large difference in coupling strength with phonons for the doubly split states of the ²E level. For the large anisotropy in thermal broadening, very different effective Debye temperatures for the two split states may be responsible for it. Besides the ²E excited state, the higher excited states, for example, ⁴T₁ and ⁴T₂ of the Cr³⁺ ion, also exhibit a very large inequality in coupling strength with phonons at room temperature. By examining the Stokes phonon sidebands of the MD R lines at low temperatures with the existing ion-phonon coupling theory, we reveal that they indeed carry fundamental information of phonons. For example, their broad background primarily reflects Debye density of states of acoustic phonons. These new results significantly enrich our existing understanding on interesting but challenging luminescence mechanisms of ion-phonon coupling systems.

Elucidating the enhancement in optical properties of low band gap polymers by tuning the structure of alkyl side chains

We carry out a computational study of optical properties of two novel 5,6-difluorobenzo[c][1,2,5]-thiadiazole-based polymers, PFBT-T20TT and PFBT-T12TT, to elucidate the surprisingly superior performance of polymer solar cells based on the former, when it differs from the latter only in the alkyl side chains. Density Functional Theory (DFT) based geometry optimization at the B3LYP/6-31G(d) level reveals differences in internal coordinates, which are important in tuning the electronic and optical properties. We further calculate the electronic structure at room temperature by employing molecular dynamics (MD) simulations in combination with DFT techniques. The energies of the highest occupied molecular orbital (HOMO) are found to be in reasonable agreement with the available experimental data and the HOMO-lowest unoccupied MO energy gap is found to be similar for both the molecules. The electronic density of the HOMO in PFBT-T20TT is, however, found to be significantly more delocalized along the backbone, which is proposed to be conducive for the formation of charge separated states leading to an improved device performance. Furthermore, via fitting the absorption spectra calculated with the multi-mode Brownian Oscillator model, we have also extracted a weaker exciton-phonon coupling parameter in PFBT-T20TT, consistent with the trends revealed via the DFT-MD results.

The temperature dependence of vibronic lineshapes: linear electron-phonon coupling

We calculate the effect of a linear electron-phonon coupling on vibronic transitions of dye molecules of arbitrary complexity. With the assumption of known vibronic frequencies (for instance from quantum-chemical calculations), we give expressions for the absorption or emission lineshapes in a second-order cumulant expansion. We show that the results coincide with those obtained from generalized Redfield theory if one uses the time-local version of the theory and applies the secular approximation. Furthermore, the theory allows to go beyond the Huang-Rhys approximation and can be used to incorporate Dushinsky effects in the treatment of the temperature dependence of optical spectra. We consider both, a pure electron-phonon coupling independent of the molecular vibrations and a coupling bilinear in the molecular vibrational modes and the phonon coordinates. We discuss the behavior of the vibronic density of states for various models for the spectral density representing the coupling of the vibronic system to the harmonic bath. We recover some of the results that have been derived earlier for the spin-boson model and we show that the behavior of the spectral density at low frequencies determines the dominant features of the spectra. In case of the bilinear coupling between the molecular vibrations and the phonons we give analytical expressions for different spectral densities. The spectra are reminiscent of those obtained from the well known Brownian oscillator model and one finds a zero-phonon line and phonon-side bands located at vibrational frequencies of the dye. The intensity of the phonon-side bands diminishes with increasing vibrational frequencies and with decreasing coupling strength (Huang-Rhys factor). It vanishes completely in the Markovian limit where only a Lorentzian zero-phonon line is observed.

Mixed quantum-classical description of spectroscopy of dissipative systems

Mixed quantum-classical statistical mechanics is employed to calculate dipole moment correlation function and linear absorption spectra. A quantum two-level subsystem interacting with quantum vibrations (primary oscillators) which in turn are coupled to a classical bath composed of infinite set of harmonic oscillators is used as a dissipative system. Starting with mixed quantum-classical Liouville equation for the evaluation of the mixed quantum-classical dipole moment correlation function and using coherent states and the inverse of Baker-Campbell-Hausdorf formula to evaluate the trace over the primary oscillators, whereby, a closed analytical expression for the electronic dipole moment correlation function is obtained. Illustrations of several absorption spectra at different temperatures are provided. An approximate optical four-point correlation is obtained in the high temperature limit. A strategy for deriving an exact optical four-point correlation is suggested.

Green Luminescence Band in ZnO: Fine Structures, Electron−Phonon Coupling, and Temperature Effect

The green emission band of ZnO has been investigated by both experimental and theoretical means. Two sets of equally separated fine structures with the same periodicity (close to the longitudinal optical (LO) phonon energy of ZnO) are well resolved in the low-temperature broad green emission spectra. As the temperature increases, the fine structures gradually fade out and the whole green emission band becomes smooth at room temperature. An attempt to quantitatively reproduce the variable-temperature green emission spectra using the underdamped multimode Brownian oscillator model taking into account the quantum dissipation effect of the phonon bath is done. Results show that the two electronic transitions strongly coupled to lattice vibrations of ZnO lead to the observed broad emission band with fine structures. Excellent agreement between theory and experiment for the entire temperature range enables us to determine the dimensionless Huang-Rhys factor characterizing the strength of electron-LO phonon coupling and the coupling coefficient of the LO and bath modes.

Evidence of the key role of metal-molecule bonding in metal-molecule-metal transport experiments

The transport properties of two oligothiophene derivatives, that differ only by the chemical group coupling to gold, are compared. It is shown that the role of the coupling group in the transport properties can be decoupled from that of the conjugated body of the molecules and that Se is a better electronic coupling group than S. These results are accounted for semiquantitatively within the frame of the scattering theory of transport, using results from ultraviolet photoemission spectroscopy experiments as inputs for the position in energy of the molecular orbitals with respect to the Fermi level of the electrodes.