Franck–Condon and Herzberg–Teller Signatures in Molecular Absorption and Emission Spectra

Abstract

Some molecules of chemical and biological significance possess vibrations with significant Herzberg−Teller (HT) couplings, which render the Franck−Condon (FC) approximation inadequate and cause the breakdown of the well-known mirror-image symmetry between linear absorption and emission spectra. Using a model two-state system with displaced harmonic potential surfaces, we show analytically that the FC-HT interference gives rise to asymmetric intensity modification, which has the same sign for all transitions on one side of the 0−0 absorption line and the opposite sign in the equivalent fluorescence transitions, while the trend is exactly reversed for all transitions on the other side the 0−0 line. We examine the dependence of the absorption−emission asymmetry on the mode frequency, Huang−Rhys factor, and dipole moment parameters to show the recovery of symmetry with particular combinations of parameters and a crossover from fluorescence to absorption dominance. We illustrate the analytical predictions through numerically exact calculations in models of one and two discrete vibrational modes and in the presence of a harmonic dissipative bath. In addition to homogeneous broadening effects, we identify large asymmetric shifts of absorption and emission band maxima, which can produce the illusion of unequal frequencies in the ground and excited potential surfaces as well as a nontrivial modulation of spectral asymmetry by temperature, which results from the enhancement of transitions on one side of the 0−0 line. These findings will aid the interpretation of experimental spectra in HT-active molecular systems.

The dominant nature of Herzberg-Teller terms in the photophysical description of naphthalene compared to anthracene and tetracene

“…This methodology was used by Dong and co-workers 72 , Guo and co-workers 73 , and Wykes and co-workers 8 ; in all cases providing a significant degree of clarity to the results. For reported treatments straying from the FCCLASSES methodology; Kunda and co-workers 74 developed an exact treatment using a path integral formalism. Despite achieving positive results, they were limited to a small number of discrete normal modes.…”

Interference between Franck–Condon and Herzberg–Teller Terms in the Condensed-Phase Molecular Spectra of Metal-Based Tetrapyrrole Derivatives

“… 23 Very few instances of the systematic inclusion of finite-temperature HT effects exist in the literature. 20 , 30 , 31 Our recent work 32 identified analytical symmetry arguments dictating the trends of FC–HT interference within the normal mode approximation and reported numerical calculations incorporating temperature dependence and homogeneous broadening. 32 Non-Condon signatures were also recently found in excitation energy transfer 33 , 34 as well as electron transfer 35 , 36 dynamics.…”

“… 20 , 30 , 31 Our recent work 32 identified analytical symmetry arguments dictating the trends of FC–HT interference within the normal mode approximation and reported numerical calculations incorporating temperature dependence and homogeneous broadening. 32 Non-Condon signatures were also recently found in excitation energy transfer 33 , 34 as well as electron transfer 35 , 36 dynamics. A theoretical study by Zhang et al 33 , 34 suggested that non-Condon vibronic coupling plays a central role in facilitating electronic–vibrational energy transfer and enhancing the quantum yield in contrast to FC vibronic coupling.…”

“…To this end, we illustrate our simple model framework ( Figure 3 a). 32 Expanding the transition dipole moment (in the direction of the applied electric field) along the normal mode coordinates q ≡ { q k } about a particular geometry { q k eq } of the molecule, we write where μ k (1) is the first derivative of the dipole moment along the particular mode. The FC approximation involves a truncation at the zeroth-order, electronic dipole term μ (0) ; thus, FC transitions follow Kasha’s rule, 41 producing mirror-image symmetry between absorption and emission spectra.…”

“…Trends similar to those observed in ZnTPP were recovered in our recent FC–HT model study in various parameter regimes. 32 …”