3050
J . Phys. Chem. 1994,98, 3050-3055
Vibronic Mixing in the Strong Electronic Coupling Limit. Spectroscopic Effects of Forbidden Transitions Elizabeth J. P. Lathrop and Richard A. Friesner' Department of Chemistry, Columbia University, New York, New York 10027 Received: August 17, 1993; In Final Form: December 22, 1993"
We consider an excited-state manifold in which a n allowed transition is strongly coupled electronically to a forbidden (dark) state and use nonperturbative methods to determine the effects of this coupling on the optical properties. W e find that the strong coupling limit is qualitatively different from the usual weak coupling case; in particular, the bright state spectrum is substantially altered, displaying renormalized Franck-Condon factors which can be observed in absorption, hole burning, and resonance Raman experiments. We map out the magnitude of this renormalization as a function of various parameters in the theory and note that there is a phase-transition-like behavior as one passes from the weak to the strong coupling regimes. Finally, we suggest that this mechanism provides a straightforward explanation for the hole-burning spectra observed for the primary donor in the bacterial photosynthetic reaction center and determine a region of parameter space compatible with the experimental results.
Introduction Vibronic borrowing is an ubiquitous phenomenon in molecular spectroscopy. In the usual case, a dark state (Le., one with no independent oscillator strength) acquires intensity from a bright state by mixing with excited vibrational levels of the latter. The manifestation of this phenomenon in the spectrum is then a peak at the energetic location of a forbidden electronic transition. The standard theory of vibronic coupling assumes that the matrix element between the bright and dark states is small compared to the energy gap between them and to the vibrational frequency of the promoting mode. In this case, perturbation theory in electronic coupling is valid, and simple analytical formulas involving Franck-Condon overlaps of the diabatic harmonic oscillator states of the bright and dark surfaces can be constructed. Recently, we have carried out detailed vibronic modeling of the optical spectra from the photosyntheticreactioncenter (RC).' In the course of this work, we encountered a case of vibronic coupling that was qualitatively different from the situation described above. In particular, the electronic coupling between the bright and dark states is quite strong, and the effects of the optical spectrum are rather different. The situation we have investigated is in fact more analogous to mixed valence compounds in which electronic matrix elements between excited-state configurations can be substantially greater than thevibrational modes involved in thevibronic mixing. There has been a considerable body of theoretical work on such system, most prominently by Hush and co-workers.2 These workers have noted a similar qualitative transition in the nature of the vibronic wave function as one moves from the weak to the strong electronic coupling regime. This subject was also considered in detail by Friesner and Silbey,3 who provided analytical models of the transition and compared these models with exact numerical results for the simple case of two states coupled to a single vibrational mode. The present paper explores the same phenomenon, but with a different focus. We are concerned here with the results of several important experimental techniques when applied to strongly coupled vibronic systems. We examine not the borrowing of intensity by the forbidden state but rather the effect of the dark state on the optical properties of the bright state. These properties Abstract published in Aduance ACS Abstracts. February 1, 1994.
are manifested in absorption, hole-burning, and resonance Raman (RR) experiments. For weak electronic coupling the dark state has little or no effect on the bright state other than a decrease in the overall intensity; the line shape remains more or less invariant. In the strong coupling regime, however, a number of unusual effects can be detected. The width of the absorption band can increase, its temperature dependence can change markedly, and R R and hole-burning amplitudes can be dramatically altered. The explanation lies in that only in strong coupling is the nature of the vibrational wave function coupled to the optical transition altered substantially. In limiting cases, this can easily be shown via analytical arguments. For complicated intermediate parameter values, numerical computations are required to determine the magnitude of the effect. A simplified description of what occurs is as follows. The Franck-Condon factors of the bright state are renormalized in the direction of those of the dark state. If the dark state has much larger dimensionless displacements A(n) for a given vibrational mode n (or equivalently, Huang-Rhys factors S,) than the bright state, the vibronic interactions will enhance the "effective" displacements of modes in the bright state, leading to the spectroscopic effects just described. A simple way to characterize the effect is to define an effective Huang-Rhys factor S$$ for the bright state. In the weak coupling limit, S g is equal to the original S for the bright state before vibronic mixing. In strong coupling however, S B can be dramatically altered. We determine S$) phenomenologically by carrying out absorption simulations and evaluating the ratio of the intensity of the zerophonon line (ZPL) to the remainder of the spectrum; we then ascertain the value of S that produces such a ratio in a standard Franck-Condon calculation with a single excited electronic state. The actual homogeneous absorption line shape is in fact considerably more complicated than that generated by a single harmonic progression, due to chaotic mixing of higher vibronic levels. However, the effects of inhomogeneous broadening and acoustic phonons are to wash out the fine details of this structure. The effective S is the observable most readily seen in the experiments (other signatures of the vibronic coupling could be visible in the R R excitation profiles). Model Hamiltonian and Parameters The objective of this paper is to point out basic phenomena observed and to illustrate them by numerical computations on a
0022-3654/94/2098-3050$04.50/0 0 1994 American Chemical Society
Spectroscopic Effects of Forbidden Transitions few simple models that are relevant to the reaction center calculations in the paper that follows.1 It is not our intention to present an exhaustive characterization of all relevant parameter regimes; we reserve such work for another publication. Furthermore, we study here the effect on the absorption and holeburning spectra; investigations of RR spectra (which obviously will be significantly affected) are deferred to another paper. We present below a brief description of the effective vibronic Hamiltonian employed for this work. Previous efforts on the study of different electronic coupling regimes for the two excited electronic level system (TLS) coupled to a single vibrational mode by Friesner and Silbey3(which are essential to the understanding of the present work) are also briefly summarized here. For the first part of the paper, we shall restrict the discussion to a twolevel excited-state manifold, in which one of the electronic transitions is a forbidden dark state with zero oscillator strength. A similar dark state also exists in the three-level reaction center model system, as will be explored in detail in the latter part of this paper and in the paper following this one.’ In the diabatic representation, the effective Hamiltonian for general multilevel multimode system in the linear vibronic coupling model of the excited-state surfaces can be written as
where N is the number of excited electronic levels and M , the number of harmonic vibrational modes. Here, li) is the diabatic representation of an excited-state surface with energy eii. The parameter Ji, is the coupling matrix element for electronic interactions between states l i ) and b), and wn, the harmonic oscillator vibrational frequency of mode n with the corresponding vibronic coupling parameter 8;). The vibronic coupling term 8;) is related to the dimensionless normal coordinate shift A;) by the relation A!”)= fic9t)/wnfor a given vibrational mode n. The operators 8 a n d b are boson creation and annihilation operators, respectively. In this model, we assume a single common electronic ground state for all excited state surfaces. We take the zero energy to be ( l / N ) X E l e i i -EB+ ‘ / 2 x E l w n ,where h is 1 and E, is the ground-state electronic energy. The most significant parameter for our purposes is the dimensionless ratio y (=J/w),which distinguishes the weak from the strong electronic coupling. When y S-.f g , O 0.8 -
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Figure 2. (a) Homogeneous line shape where the value of S$$ approaches that of S$. Parameter A is fixed at 3.00, J at 1500 cm-1, and AE at 2000 cm-I. (B) Homogeneous line shape where the value of S$ is represented by a mixed contributions from both bright and dark transitions. Parameter A is fixed at 3.00, J at 1500 cm-', and AE at 200
cm-1. bands are negligible, S$ approaches an S value of 0.594 calculated from 1.090for All. This is true for all values of A, with the exception of A = 1.OO,in which case S,, is calculated exactly. We show in Figure 2 examples of the homogeneous absorption spectra in the two different regions. In Figure 2A where the behavior in S g represents that of ,we see that a large e bright transition as fraction of the total intensity comes expected. In the case where S$ contains mixed contributions from the two transitions, we see comparable contributions from both intensities as shown in Figure 2B. We present the results of effective S versus hE for a fixed A (A = 3.00)but with variable Jvalues in Figure 3. Here, we have shown only g8)and S$ for easy visualization. The turning point is fixed &r different values of J. However, the general observation that the behavior in represents that in S$ for a good part of the parameter space remains unchanged. We further notice that for weaker electronic coupling J values, the behavior of approaches more quickly to that of with increasing AE values. In the case where J = 400 cm-1 and AE = 0.0 cm-l, the observed lower value of S!$ is in fact caused by stronger contribution from the dark transition. For a large part of the parameter space investigated, the ZPL in the bright transition can be thought to represent the total ZPL for the system. For the remaining part of this discussion, we intent to concentrate on this particular region only and refer to as the effective S value unless otherwise specified. Cons g r i n g both Figures 1 and 3, one sees that as the dark state is
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Figure 4. Effective S values versus the vibronic coupling scaling factor A for various values of J, of the one-mode TLS model. Here, w is 94 cm-1, A11 is 1.090, and AE is 2835 cm-'. 20-
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Figure 5. Effective S values in the zero AE, large J limit for a single-
mode TLS system, for various values of A. Here, A11 is again 1.090. The solid lines represent S$ and lines with symbols represent ,!$$for the corresponding A values. moved farther away from the bright transition position, smaller effective S values are observed, reflecting in sharper ZPL amplitudes. Figure 4 shows the effective values of S versus parameter A for various values of J. Parameter hE is fixed at 2835 cm-1. For J
