Vibration assisted electron tunnelling in COVID-19 infection using quantum state diffusion.

The spread of the COVID-19 virus has become a global health crisis, and finding effective treatments and preventions is a top priority. The field of quantum biology primarily focuses on energy or charge transfer, with a particular emphasis on photosynthesis. However, there is evidence to suggest that cellular receptors such as olfactory or neural receptors may also use vibration-assisted electron tunnelling to enhance their functions. Quantum tunnelling has also been observed in enzyme activity, which is relevant to the invasion of host cells by the SARS-CoV-2 virus. Additionally, COVID-19 appears to disrupt receptors such as olfactory receptors. These findings suggest that quantum effects could provide new insights into the mechanisms of biological systems and disease, including potential treatments for COVID-19. We have applied the open quantum system approach using Quantum State Diffusion to solve the non-linear stochastic Schrödinger equation (SSE) for COVID-19 virus infection. Our model includes the mechanism when the spike protein of the virus binds with an ACE2 receptor is considered as dimer. These two entities form a system and then coupled with the cell membrane, which is modelled as a set of harmonic oscillators (bath). By simulating the SSE, we find that there is vibration-assisted electron tunnelling happening in certain biological parameters and coupling regimes. Furthermore, our model contributes to the ongoing research to understand the fundamental nature of virus dynamics. It proposes that vibration-assisted electron tunneling could be a molecular phenomenon that augments the lock-and-key process for olfaction. This insight may enhance our understanding of the underlying mechanisms governing virus-receptor interactions and could potentially lead to the development of novel therapeutic strategies.

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.

Homogeneous Dephasing in Photosynthetic Bacterial Reaction Centers: Time Correlation Function Approach.

A new tractable linear electronic transition dipole moment time correlation function (ETDMTCF) that accurately accounts for electronic dephasing, asymmetry, and width of 1-phonon profile, which the zero-phonon line (ZPL) contributes to it, in Rhodopseudomonas viridis bacterial reaction center is derived. This time correlation function proves to be superior to other frequency-domain expressions in case of strong electron-phonon coupling (which is often the case in bacterial RCs and pigment-protein complexes), many vibrational modes involved, and high temperature, whereby more vibronic and electronic (sequence) transitions would arise. The Fourier transform of this ETDMTCF leads to asymmetric multiphonon profiles composed of Lorentzian distribution and Gaussian distribution on the high- and low-energy sides, respectively, whereby the overtone widths fold themselves with that of the one-phonon profile. This ETDMTCF also features expedient computation in large systems using asymmetric phonon profiles to account correctly for dephasing and pigment-protein interaction (electron-phonon coupling). The derived ETDMTCF allows computing all nonlinear optical signals in both time and frequency domains, through the nonlinear dipole moment time correlation functions (as guided by nonlinear optical response theory) in line with the eight Liouville space pathways. The linear transition dipole moment time correlation function is of a central value as the nonlinear transition dipole moment time correlation function is expressed in terms of the linear transition dipole moment time correlation function, derived herein. One of the great advantages of presenting this ETDMTCF is its applicability to nonlinear transition dipole moment time correlation functions in line with the eight Liouville space pathways needed in computing nonlinear signals. As such, there is more to the utility and applicability of the presented ETDMTCF besides computational expediency and efficiency. Results show good agreement with the reported literature. The intimate connection between a one-phonon profile and the corresponding bath spectral density in photosynthetic complexes is discussed.

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.

Excited state distribution function for probing Herzberg-Teller vibronic coupling using linear optical response theory: Application to glassy pheophytin a.

The goal of the present work is to develop an excited-state distribution function that can be used to calculate electronic transition dipole moment time correlation functions at a considerably low computational cost. An additional merit of the distribution function is its capability to probe the Hertzberg-Teller vibronic coupling effect in terms of the previously reported Condon correlation functions in the literature without having to start from the equilibrium density operator to probe spectral non-Condon effects, thereby exploring Hertzberg-Teller vibronic coupling by building on the Condon regime. It is easily extendable to anharmonic systems. Model calculations are reported to show the high degree of accuracy and computational efficiency of the presented approach. Application to a photosynthetic system such as pheophytin a in triethylamine is provided.

Rosa canina L. Can Restore Endoplasmic Reticulum Alterations, Protein Trafficking and Membrane Integrity in a Dextran Sulfate Sodium-Induced Inflammatory Bowel Disease Phenotype.

Rosa canina L. is a natural polyphenol-rich medicinal plant that exhibits antioxidant and anti-inflammatory activities. Recent in vivo studies have demonstrated that a methanol extract of Rosa canina L. (RCME) has reversed an inflammatory bowel disease (IBD)-like phenotype that has been triggered by dextran sulfate sodium (DSS) in mice. In the current study, we investigated the effects of RCME on perturbations of cellular mechanisms induced by DSS-treatment of intestinal Caco-2 cells, including stress response in the endoplasmic reticulum (ER), protein trafficking and sorting as well as lipid rafts integrity and functional capacities of an intestinal enzyme. 6 days post-confluent cells were treated for 24 h with DSS (3%) or simultaneously with DSS (3%) and RCME (100 µg/mL) or exclusively with RCME (100 µg/mL) or not treated. The results obtained demonstrate the ability of RCME to counteract the substantial increase in the expression levels of several ER stress markers in DSS-treated cells. Concomitantly, the delayed trafficking of intestinal membrane glycoproteins sucrase-isomaltase (SI) and dipeptidyl peptidase 4 (DPP4) induced by DSS between the ER and the Golgi has been compromised by RCME. Furthermore, RCME restored the partially impaired polarized sorting of SI and DPP4 to the brush border membrane. An efficient sorting mechanism of SI and DPP4 is tightly associated with intact lipid rafts structures in the trans-Golgi network (TGN), which have been distorted by DSS and normalized by RCME. Finally, the enzymatic activities of SI are enhanced in the presence of RCME. Altogether, DSS treatment has triggered ER stress, impaired trafficking and function of membrane glycoproteins and distorted lipid rafts, all of which can be compromised by RCME. These findings indicate that the antioxidants in RCME act at two major sites in Caco-2 cells, the ER and the TGN and are thus capable of maintaining the membrane integrity by correcting the sorting of membrane-associated proteins.

Exact solutions of the harmonic oscillator plus non-polynomial interaction.

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x 2 + b x 2/(1 + c x 2) (a > 0, c > 0) are given by the confluent Heun functions H c (α, ß, γ, δ, η;z). The minimum value of the potential well is calculated as V min ( x ) = - ( a + | b | - 2 a | b | ) / c at x = ± [ ( | b | / a - 1 ) / c ] 1 / 2 (|b| > a) for the double-well case (b < 0). We illustrate the wave functions through varying the potential parameters a, b, c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c. However, we find that the wave peaks are concave to the origin as the parameter |b| is increased.

Dextran Sodium Sulfate-Induced Impairment of Protein Trafficking and Alterations in Membrane Composition in Intestinal Caco-2 Cell Line.

A key morphological feature of inflammatory bowel disease (IBD) is the loss of the barrier function of intestinal epithelial cells. The present study investigates endoplasmic reticulum (ER) stress in addition to alterations in protein and membrane trafficking in a dextran sulfate sodium (DSS)-induced IBD-like phenotype of intestinal Caco-2 cells in culture. DSS treatment significantly reduced the transepithelial electric resistance (TEER) and increased the epithelial permeability of Caco-2 cells, without affecting their viability. This was associated with an alteration in the expression levels of inflammatory factors in addition to an increase in the expression of the ER stress protein markers, namely immunoglobulin-binding protein (BiP), C/EBP homologous protein (CHOP), activation transcription factor 4 (ATF4), and X-box binding protein (XBP1). The DSS-induced ER-stress resulted in impaired intracellular trafficking and polarized sorting of sucrase-isomaltase (SI) and dipeptidyl peptidase-4 (DPPIV), which are normally sorted to the apical membrane via association with lipid rafts. The observed impaired sorting was caused by reduced cholesterol levels and subsequent distortion of the lipid rafts. The data presented confirm perturbation of ER homeostasis in DSS-treated Caco-2 cells, accompanied by impairment of membrane and protein trafficking resulting in altered membrane integrity, cellular polarity, and hence disrupted barrier function.

Spectroscopy of Vibronically Coupled and Duschinskcally Rotated Polyatomic Molecules.

Electron-vibration coupling (or vibronic coupling) of polyatomic molecules in the condensed phase has received considerable attention experimentally and theoretically using linear spectroscopy and four-wave mixing techniques in an effort to probe the structure and dynamics of the system at hand. For this reason, a detailed study of vibronic coupling in harmonic polyatomic molecules featuring a greater degree of computational efficiency is presented. A full treatment of non-Condon systems whereupon linear and nonlinear Herzberg-Teller vibronic coupling and Duschinsky mixing effects in harmonic systems taking place upon electronic excitation is provided. The utilization of an exponential function to express the nuclear dependence of the electronic transition dipole moment, thereby avoiding the finite sum and eigenstate representation that is normally used in computing a non-Condon interaction, leads to a simpler electronic transition dipole moment time correlation function with rapid convergence and better numerical stability than previously reported works. A closed-form expression is obtained for the electronic transition dipole moment time correlation function of polyatomic molecules in which linear and nonlinear Herzberg-Teller vibronic coupling and Duschinsky mixing effects are accounted for. An important numerical observation regarding dealing with branch cuts, which manifest themselves as discontinuities in the function itself or its first derivative, and are often exhibited by complex-valued correlation functions, is noted and treated using the Riemann surface approach. The resultant dipole moment correlation function is in turn employed to calculate linear absorption and hole-burning signals, accounting for the aforementioned spectroscopic effects. A link between wavelets and the electronic transition dipole moment time correlation function in multidimensional harmonic systems is made in the concluding remarks.

Anharmonic nuclear dynamics in the mixed quantum-classical limit.

This study employs mixed quantum-classical dynamics (MQCD) formalism to evaluate the linear electronic dipole moment time correlation function (DMTCF) in which a Morse oscillator serves to model the associated vibrations in a mixed quantum-classical (MQC) environment. While the main purpose of this work is to study the applicability of MQCD formalism to anharmonic systems in condensed phase, approximate schemes to physically evaluate the mathematically divergent integrals have been developed in order to deal with the essential singularities that arise while evaluating the Morse oscillator canonical partition function and the DMTCF in MQC systems in the classical limit. The motivation for numerically and analytically evaluating these divergent integrals is that a partition function of any system should lead to a finite value at any temperature and therefore this divergence is unphysical. Additionally, since a partition function is to signify the number of accessible states to the system at hand, divergent results are not physically acceptable. As such, straightforward approximate analytic expressions, at different levels of rigor, for both the classical Morse oscillator partition function and the DMTCF in MQC systems are derived, for the first time. Calculations of Morse oscillator partition function values using different approaches at various temperatures for CO, HCl, and I(2) molecules, showing good results, are presented to test the expressions derived herein. It is found that this divergence, due to singularity, diminishes upon lowering the temperature and only arises at high temperatures. The gradual diminishing of the singularity upon lowering the temperature is sensible since the Morse potential fits the parabolic potential at low temperatures. Model calculations and discussion of the DMTCF and linear absorption spectra in MQC systems using the molecular constants of CO molecule are provided. The linear absorption lineshape is derived by two methods, one of which is asymptotic expansion.

Exploring Anharmonic Nuclear Dynamics and Spectroscopy Using the Kratzer Oscillator.

The Kratzer oscillator is useful in modeling anharmonic molecular vibrations; therefore, its underlying theory is briefly explored in this study. The linear dipole moment time correlation function, within the Condon approximation, is analytically evaluated, and linear absorption lineshapes are calculated at different temperatures. An important integral formula of Landau and Liftshitz is, for the first time, utilized to evaluate the anharmonic Franck-Condon factor (FCF) resulting from modeling the initial and final states by Kratzer potentials. In addition, an exact closed-form expression of the FCF for the linearly displaced and shape-distorted final state energy curve, with respect to the ground state, is reported. Within the context of Mukamel formalism, nonlinear spectral/temporal lineshapes, such as hole-burning, photon echo, and pump-probe signals, may not be calculated without nonlinear response theory using the so-called "four-point dipole moment time correlation function". The above FCFs will be employed to calculate optical linear and nonlinear spectra at different temperatures utilizing a previously developed formula [Toutounji, M. J. Phys. Chem. C2010, in press], whereby a hole-burned absorption lineshape may be found using a linear dipole moment time correlation function.

Anharmonic electron-phonon coupling in condensed media: 1. Formalism.

Three different schemes for calculating anharmonic line shape functions are reported and discussed for the first time in this article using eigenstate representation. First, the linear dipole-moment time correlation function (DMTCF), homogeneous (single-site) absorption line shape function, and the respective Franck-Condon factors (FCF) are derived and explored as a molecule makes a transition from a harmonic to an anharmonic (Morse potential) electronic state. Second, the linear DMTCF, homogeneous absorption line shape function, and FCFs are also derived as a molecule makes a transition from one anharmonic to another linearly displaced anharmonic state; FCFs in this case are reported in an exact closed-form expressed in terms of Appell's hypergeometric function. Third, same as the latter set of results are reported but with both linearly displaced and distorted shape of the upper Morse potential. FCFs of the zero-phonon line in all three cases are reported. The first case is rather mathematically complex as a result of taking the overlap integral of the Morse oscillator eigenfunctions, whose spatial decay is a simple exponential, with those of harmonic oscillator, whose decay is a Gaussian. This form of a functional disparity gives rise to some challenges. Model calculations are presented and discussed.

Empirical relaxation function and spectral density for underdamped vibrations at low temperatures.

A new relaxation function which accounts for electronic dephasing (electronic phase loss and excited state lifetime) is presented, whose applicability for underdamped motion at low temperatures is examined in detail. This new empirical relaxation function phi(t) yields linear and nonlinear spectral/temporal profiles that render accurate dephasing time in the underdamped regime. The relaxation function phi(t) is normally expressed in terms of the coupling functions M(j) (') and M(j) (") on which the time evolution of the vibrational modes in question depends. The corresponding spectral density, which is a central quantity in probing dynamics, is derived and compared to that of the multimode Brownian oscillator model. Derivation and discussion of the new position and momentum autocorrelation functions in terms of our new spectral density are presented. While the position autocorrelation function plays a key role in representing solvation structure in polar or nonpolar medium, the momentum correlation function projects out the molecular vibrational motion. The Liouville space generating function (LGF) for harmonic and anharmonic systems is expressed in terms of our new empirical phi(t) and spectral density, leading to more physical observation. Several statistical quantities are derived from the position and momentum correlation function, which in turn contribute to LGF. Model calculations reflecting the infinite population decay in the low temperature limit in linear and nonlinear spectroscopic signals are presented. The herein quantum dipole moment correlation function is compared to that derived in [M. Toutounji, J. Chem. Phys. 118, 5319 (2003)] using mixed quantum-classical dynamics framework, yielding reasonable results, in fact identical at higher temperatures. The results herein are found to be informative, useful, and consistent with experiments.

Algebraic approach to electronic spectroscopy and dynamics.

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.

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.

Optical nonlinear response function with linear and diagonal quadratic electron-vibration coupling in mixed quantum-classical systems.

While an optical linear response function of linearly and quadratically coupled mixed quantum-classical condensed-phase systems was derived by Toutounji [J. Chem. Phys. 121, 2228 (2004)], the corresponding analytical optical line shape is derived. The respective nonlinear correlation functions are also derived. Model calculations involving photon-echo, pump-probe, and hole-burning signals of model systems with both linear and quadratic coupling are provided. Hole-burning formula of Hayes-Small is compared to that of Mukamel in mixed quantum-classical systems.

Mixed quantum-classical Redfield master equation.

Redfield master equation is derived from mixed quantum-classical Liouville equation using product initial conditions. Simple two-level system example is given and comparison with Fermi golden rule is made.

Optical linear response function with linear and diagonal quadratic electron-vibration coupling in mixed quantum-classical systems.

Optical linear response function of linearly and quadratically coupled mixed quantum-classical condensed phase systems is derived. The linear response function is derived using Kapral's formalism of statistical mechanics in mixed quantum-classical systems. Our mixed quantum-classical linear dipole moment correlation function J(t) is compared with the full quantum J(t) [Y. J. Yan and S. Mukamel, J. Chem. Phys. 85, 5908 (1986)] in the high temperature limit. Model calculations and discussion of our results are presented. Various formulas of Franck-Condon factors for both linear and quadratic coupling are discussed.