Dipole Moment Time Correlation Function Research Articles

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 in mixed quantum–classical molecular systems using the spin-boson model

Pure electronic dephasing is investigated using the spin-boson Hamiltonian in mixed quantum–classical environment. The spin-boson model used here is a composite system made up of a quantum subsystem, an electronic 2-level subsystem linearly coupled to harmonic vibrations, interacting with a classical bath. Experimental results for a multitude of molecular systems indicate that the zero-phonon line (ZPL) profile is determined by electronic dephasing, which is not accounted for in the multimode Brownian oscillator (MBO) model due to the unphysical contribution from the MBO bath modes to the ZPL profile. Mixed quantum–classical dynamics formalism of non-equilibrium systems is employed to assess the contribution of the bath modes to pure electronic dephasing by probing the ZPL profile when coupled to a classical bath in the mixed quantum–classical condensed systems. Pure electronic dephasing is discussed in the context of mixed quantum–classical dynamics formalism which starts with mixed quantum–classical Liouville equation in a mixed quantum–classical environment. It is noteworthy, however, that the fundamental difference between the fully quantum MBO model and the mixed quantum–classical Brownian oscillator, is that the zero-phonon line calculated by the former shows unphysical asymmetry on the low-energy side as it has not been observed in real systems, whereas the ZPL reported herein eliminates this asymmetry. A systematic approach using matrix mechanics is developed to treat this phenomenon. To this end, a closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in a dissipative media are derived. Linear absorption spectra and 4-wave mixing signals at various temperatures showing a sound thermal broadening, temporal decay, and accurate pure dephasing further ratify the applicability and correctness of the mixed quantum–classical dynamics approach to spectroscopy and dynamics are computed.

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.

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.

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.

Linear and nonlinear Herzberg-Teller vibronic coupling effects. I: Electronic photon echo spectroscopy

The dependence of the transition dipole moment on nuclear coordinates gives rise to non-Condon interaction, signaling that the adiabatic Herzberg-Teller vibronic coupling should be considered. This coupling manifests itself in linear spectroscopy through the breakdown of the mirror image symmetry between the absorption and emission spectra. The stronger the non-Condon interaction, the more pronounced is the absorption-emission asymmetry. A nuclear exponential function is used as a simple model to represent the electronic transition dipole moment, the purpose of which is to avoid the use of eigenstate-approach, convergence issues, and finally be able to extend its applicability to nonlinear spectroscopy inexpensively while accounting for linear and nonlinear non-Condon effects (Herzberg-Teller vibronic coupling). The non-Condon effects on linear and nonlinear spectroscopic signals (e.g. 2-pulse photon echo) are examined. Closed-form expressions of the non-Condon electronic transition dipole moment time correlation functions, accounting for Herzberg-Teller vibronic coupling effects, are derived. It is shown that nonlinear optical signals in time domain such as 2-pulse photon echo signal, conspicuously reflects the degree of the non-Condon interaction (mostly constructive interference in this work) despite the inhomogeneous broadening disguising the homogeneous structural aspects of the molecular system of interest. As such, this study should be useful in relation to extracting structural and dynamical information of condensed molecular systems. Model calculations of absorption and emission spectra and 2-pulse photon echo signals of Herzberg-Teller vibronically coupled systems are presented. The method presented here seems to more efficient computationally, fast and stable compared to the methods which involve perturbation theory and eigenstate expansion.

Collective excitations and ultrafast dipolar solvation dynamics in water-ethanol binary mixture.

In order to understand the intermolecular vibrational spectrum and the collective excitations of water-ethanol binary mixture, we investigate the density of states and the power spectrum using computer simulations aided by theory. We investigate in particular the spectra at intermediate to low frequencies (a few hundreds to few tens of cm-1) by calculating (i) the density of states from quenched normal modes, (ii) the power spectrum from velocity time correlation function, and (iii) the far infrared and dielectric spectra (that is, the Cole-Cole plot) from the total dipole moment time correlation function. The different spectra are in broad agreement with each other and at the same time reveal unique characteristics of the water-ethanol mixture. Inverse participation ratio reveals several interesting features. Libration of pure ethanol is more localized than that of pure water. With increasing ethanol content, we observe localization of the collective libration mode as well as of the hindered translational and rotational mode. An interesting mixing between the libration of water and ethanol is observed. Solvation dynamics of tryptophan measured by equilibrium energy fluctuation time correlation function show surprisingly strong non-linear dependence on composition that can be tested against experiments.

Influence of hydrogen bonds and temperature on dielectric properties.

Dielectric properties are evaluated by means of molecular dynamics simulations on two model systems made up of dipolar molecules. One of them mimics methanol, whereas the other differs from the former only in the ability to form hydrogen bonds. Static dielectric properties such as the permittivity and the Kirkwood factor are evaluated, and results are analyzed by considering the distribution of relative orientations between molecular dipoles. Dipole moment-time correlation functions are also evaluated. The relevance of contributions associated with autocorrelations of molecular dipoles and with cross-correlations between dipoles belonging to different molecules has been investigated. For methanol, the Debye approximation for the overall dipole moment correlation function is not valid at room temperature. The model applies when hydrogen bonds are suppressed, but it fails upon cooling the nonassociated liquid. Important differences between relaxation times associated with dipole auto- versus cross-correlations as well as their relative relevance are at the root of the Debye model breakdown.

Electronic energy gap correlation function and spectral density of anharmonic molecules at low temperatures. II: Spectroscopy

A simple line broadening function of anharmonic molecules that may be fit into the Brownian oscillator model and directly be used in calculating nonlinear spectroscopic signals without actually evaluating nonlinear response/correlation functions over different intervals is presented. Dipole moment time correlation functions and their corresponding linear absorption spectra of model systems are calculated at zero temperature. Franck–Condon factors of the zero-phonon line using both an approximate and exact expressions for comparison purposes. Pump-probe spectra of the same systems are calculated to test the reliability, utility and applicability of the derived line broadening function in extracting the homogeneous structure of the systems due to local environmental heterogeneities. The agreement is very good.

Two-phonon absorption operator in spectroscopy

While the raising operator aˆ+ does not have eigenvectors, coherent states are eigenvectors of the lowering operator aˆ. Therefore, operating with the operator exp[kaˆ+2] on coherent states poses a challenge, where k is an arbitrary constant. The result of this operation on coherent states has not yet been reported in a closed form. A closed-form expression for this problem is derived. The applicability and correctness of the result are tested by calculating the electronic dipole moment time correlation function and the corresponding Franck–Condon factors for systems with distorted Hamiltonian which arises in case of quadratic electron–phonon coupling which is significant in quantifying electronic dephasing.

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.

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen-Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed-form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two- and four-point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Modeling the Infrared and Circular Dichroism Spectroscopy of a Bridged Cyclic Diamide

Density functional theory based molecular dynamics simulations are used to study the structure, infrared (IR) spectroscopy, circular dichroism (CD) spectroscopy, and coupling between the amide I vibrations of a bridged cyclic diamide in the gas phase and in aqueous solution. IR spectra computed via the dipole moment time correlation function show a large red-shift of 30 cm(-1) in the amide I vibration in solution compared to the gas phase, and are in good agreement with experiment. Conformationally averaged CD spectra computed using the CIS(D) method are highly sensitive to the structures used, and structures sampled in the aqueous phase simulation are required to obtain qualitatively correct CD spectra. Analysis of the coupling between the amide I modes shows that in the aqueous phase there is an increased localization of the vibrations on the individual peptide groups and a reduction in the mode coupling parameter compared to the gas phase. Overall, the results illustrate the significance of incorporating molecular dynamics in the simulation of IR and CD spectra.

Anharmonic Electron−Phonon Coupling in Condensed Media: 2. Application to Electronic Dephasing, Hole-Burning, and Photon Echo

While different expressions of homogeneous linear electronic dipole moment time correlation functions and their respective absorption lineshapes of anharmonic systems were derived and discussed at length in part I [Toutounji, M. J. Phys. Chem. B 2010, in press] of this study, the electronic dephasing (zero-phonon line width), caused by pseudolocal phonons, and vibrational relaxation time scales were treated equally, hence, unphysical. In this study electronic dephasing and vibrational structure are correctly accounted for in harmonic and anharmonic multimode systems. Model calculations are presented. Anharmonic hole-burned spectra are calculated using linear anharmonic absorption lineshapes. An analytical expression of hole-burned absorption line shape in terms of the linear dipole moment time correlation function, leading to a hole-burned absorption line shape expressed in time-domain, is reported. An elementary method is utilized to treat nonlinear spectra in anharmonic molecules. The link between a hole-burned spectrum and a 2-pulse photon echo profile is established by combining the theoretical framework of Mukamel and that of Hays-Small, thereby calculating photon echo signals of anharmonic systems. Model photon echo calculations of a solvable harmonic model and an anharmonic system are reported. Applications to Al-phthaolcyanine tetrasulphonate (APT) in hyperquenched glassy ethanol and special pair bacterial reaction center are presented and discussed in detail.

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.

Experimental and quantum chemical study of pyrrole self-association through N–H⋯π hydrogen bonding

An FT-IR study of pyrrole self-association in CCl 4 solutions was carried out. According to the IR measurements, pyrrole forms self-associated dimeric species via N–H⋯π hydrogen bonding. This was also confirmed by quantum chemical calculations for pyrrole monomer and dimer at B3LYP/6-31++G(d,p) level of theory. A T-shaped minimum was located on B3LYP/6-31++G(d,p) PES of pyrrole dimer characterized with a hydrogen bond of an N–H⋯π type, with centers-of-mass separation of monomeric units of 4.520 Å, H⋯π distance of 2.475 Å, the interplanar angle between the two monomeric units being 72.9°. The anharmonic vibrational frequency shift upon dimer formation calculated on the basis of 1D DFT vibrational potentials is in excellent agreement with the experimental data (84 vs. 87 cm −1). Harmonic vibrational analysis predicts somewhat smaller shift (68 cm −1). On the basis of NIR spectroscopic data, anharmonicity constants for the 2 ν(N–H) and 2 ν(N–H⋯π) vibrational transitions were calculated. The orientational dynamics of monomeric and self-associated pyrrole species was studied within the framework of the transition dipole moment time correlation function formalism. The period of essentially free rotation in the condensed phase reduces from 0.05 ps for the monomeric pyrrole to 0.02 ps for the proton-donor molecule within the dimer.

Mixed quantum-classical dynamics response function approach to spectroscopy

Mixed quantum-classical dynamics formulation of Kapral and co-workers has been successfully employed to systems composed of a quantum subsystem coupled to an environment with classical degrees of freedom to study the dynamics of condensed many-body systems. In this formalism the quantum subsystem and the bath dynamics obey the full quantum mechanics, classical mechanics, respectively, whereas the coupling term dynamics is governed by mixed quantum-classical equations. To this end, the linear response function approach in mixed quantum-classical systems is used to derive the optical linear electronic dipole moment time-correlation function of a two-level system coupled to harmonic vibrations in condensed media. The fact that this is an exactly solvable model using full quantum mechanics allows us to test the applicability of the presented approach. An alternative approach to the aforementioned method is also developed as a second method to further test the applicability of the linear response function approach in mixed quantum-classical systems, and to confirm the correctness of the end result when using mixed quantum-classical dynamics formulation of Kapral and co-workers. Both approaches are found to yield identical results. These results are compared to those of the full quantum results in the high temperature limit. Model application of electronic absorption spectra is presented. Optical nonlinear response functions are also obtained in mixed quantum-classical systems with only linear electron–phonon coupling.