Zero‐phonon line profiles in Markovian and non‐Markovian schemes in high‐temperature systems: Applications to glassy water and ethanol

Abstract

ABSTRACT:This article mainly presents applications of previously derived formulas by the author to experimental systems whereby the Markovian and nonMarkovian multimode Brownian oscillator (MBO) model and their consequent dynamics are explored. These applications include computing the zero-phonon line (ZPL) widths of aluminum phthalocyanine tetrasulphonate (APT) in glassy films of water and ethanol, which are compared to those of the Ohmic MBO model-calculated ZPL widths at different temperatures. The analytical forms of the ZPL width and Franck-Condon factors (FCF) derived from the high-temperature limit underdamped MBO model absorption line shape (Toutounji, Chem Phys, 2003, 293, 311) are recovered from the finite-temperature MBO model, which includes Matsubara terms (Toutounji and Small, J Chem Phys, 2002, 117, 3848). As the applicability of the Ohmic MBO model at low temperatures is questionable, the corresponding low-temperature (T) Markovian dynamics is discussed. A formula for the Ohmic MBO model ZPL FCF at T ϭ 0 is presented. It is established that this formula reduces to e ϪS , S is Huang-Rhys factor, at T ϭ 0, which further ratifies our previous conclusion (Toutounji and Small, J Chem Phys, 2002, 117, 3848) that the bath modes are completely thwarted from contributing to the ZPL profile at this T. Hayes-Small theory of linear absorption and hole-burning line shapes is discussed and compared to that of the MBO model and other line shapes. Overdamped Ohmic MBO model is briefly discussed. Illustrative calculations are presented.

Anharmonic Electron−Phonon Coupling in Condensed Media: 1. Formalism

“…Another drawback that may be inferred by inspecting both Figures 1 and 2 is the ZPL and PSB bands have the same width (γ = 5 cm -1 ), which is unphysical. 5,7,13,65,66 This is tantamount to saying that vibrational and electronic relaxations occur on the same time scales. Part 2 of this work looks at this problem and suggests a solution using eqs 33 and 50.…”

“…2,3 Furthermore, eq 33 will prove very useful should the need arise for adopting a particular model to account for electronic dephasing, as was rigorously done in refs 5, 6, 13, 65, and 66. As such, the ZPL FCF in eq 33 may be eliminated to be substituted by that of the MBO model 13, [65][66][67] or the models that appeared in refs 5 and 6. Despite the apparent complexity of eq 34 it is feasibly calculable and can be used in further computing linear and nonlinear spectral signals.…”

“…(The profile of the ZPL obtained using the MBO model of Mukamel 1 has a similar mathematical structure, infinite sum.) 13,65,66 The FCF for anharmonic progression members, which are responsible for vibrational relaxation and eventually form the phonon side band of the absorption profile, are…”

Does ℏ Play a Role in Multidimensional Spectroscopy? Reduced Hierarchy Equations of Motion Approach to Molecular Vibrations

“…The SL coupling strength is then replaced by V SL /( m ω c / ℏ ) 1/2 → V SL . Although we can deal with any form of potential, here we consider the Morse potential system, as has been studied from a variety of approaches. − For the dissociation energy D e and the curvature α, the Morse potential is expressed as U false( q̂ false) = D normale false( 1 − e − normalα q̂ false) 2 The v th eigenenergy of the Morse oscillator systems is given by ℏ normalω v = ℏ normalω normalc [ ( v + 1 2 ) − ℏ 2 m normalω normalc normalα 2 true( v + 1 2 true) 2 ] where ω c = (2 D e α 2 / m ) 1/2 . Then the anharmonicity Δ anh ≡ ω 10 − ω 21 and the fundamental frequency ω 10 of Morse potential are given by Δ anh = ℏ α 2 / m and ω 10 = ω c − Δ anh , respectively.…”

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

“…4-Wave mixing experiments such as photon echo, hole-burning, and pump−probe signals are effective tools to eliminate this inhomogeneous broadening, thereby unmasking useful information about molecular motion, homogeneous dephasing, and relaxation dynamics of the system of interest. Since the primary purpose of doing nonlinear spectroscopy is eliminating inhomogeneous broadening in order probe electronic dephasing and vibrational structure, this work would be the perfect juncture to present as to how one can employ nonlinear spectroscopy to study anharmonic dynamics molecules in condensed media, as the anharmonicity becomes more important as T rises which in turn will increase the dephasing. ,− While part 1 (ref ) of this study focuses on deriving only linear DMTCFs and the corresponding lineshapes, part 2 makes use of them in accounting for electronic dephasing and calaculating nonlinear signals, such as hole-burning and photon echo.…”