J. Phys. Chem. 1994,98, 7337-7341
7337
A Theory for the Temperature Dependence of Hole-Burned Spectra J. M. Hayes,'
P. A. Lyle, and C . J. Small'
Department of Chemistry and Ames Laboratory-USDOE, Iowa State University, Ames, Iowa 50011 Received: February 4, 1994; In Final Form: April 21, 1994"
A theory is presented for the temperature dependence of the absorption and hole-burned spectra of chromophores imbedded in solids characterized by structural heterogeneity. The theory is applicable for arbitrarily strong linear electron-phonon coupling and describes the overall hole profile which consists of the zero-phonon hole and its associated phonon sideband hole structure. A novel and convenient form for the thermally averaged Franck-Condon factors is employed. Illustrative calculations are presented which pertain to the temperature dependence of the absorption band of the primary electron donor of the photosynthetic bacterial reaction center and the design of high-temperature hole-burning materials for high-density frequency domain optical storage. I. Introduction Understanding the temperature dependence of the optical absorption profile of a chromophore imbedded in a solid host is a problemof longstanding14 and, still today, much interest. There are several facets to the problem: the linear electron-phonon interaction which governs the intensity of the zero-phonon line (ZPL) relative to the phonon sideband; the importance of pseudolocalized phonons relative to bulk phonons in the coupling; the mechanisms that determine the homogeneous width of the ZPL which, for amorphous hosts at low temperatures, can be distinct- from those operativein crystalline hosts; the conditions under which the heterogeneity-induced profile of the ZPL is nonGaussian;lO and the extent to which spectral diffusion competes with pure dephasing in contributing to the width of the ZPL." From an experimental point of view, fluorescence line narrowinglz13and hole burning14.15 have proven most useful for revealing the profile of the ZPL and phonon sideband in disordered hosts where the site inhomogeneous broadening is severe. It is the temperature dependence of the hole profile that is the focus of this paper. The development of a theory16.17 for the burn-wavelength dependence of the overall hole profile in the low-temperature limit was stimulated by the first reported photochemical holeburned spectralGz0of the special bacteriochlorophyll pair of the photosynthetic bacterial reaction center.21J2 It is the lowest excited 1 m * state (P*) of the special pair that serves as the primary electron donor for the initial phase of charge separation which culminates in the fixation of carbon.23 From the above spectra and Stokes shift data it was immediately apparent that P* was characterized by strong linear electron-phonon coupling. Thus, the theory of refs 16 and 17 is valid for arbitrarily strong coupling. By necessity, it utilizes a shape for the one-phonon profile that is guided by experimental data; Le., it is still not possible to predict the shape (intensity) of the one-phonon profile in any system. The theory is also generally applicable in the sense that it does not depend on the physics that leads to inhomogeneous broadening. Most recently, it was successfully tested24.25 against hole-burned spectra of the bacterial reaction centers (RC), which are far superior to earlier reported spectra. Reviews which discuss the theory are a ~ a i l a b l e . ~In~ these .~~ reviews the importance of describing both the burn-wavelength dependence of the hole-burned spectra and the absorption spectrum is emphasized. A shortcomingof the theory is that it is valid only in the lowtemperaturelimit. Aside from the desire todescribethe absorption and hole-burned spectra of photosynthetic protein complexes at *Abstract published in Aduunce ACS Absrracrs, July 1, 1994.
0022-3654/94/2098-7337%04.50/0
elevated temperatures, there were two other reasons that stimulated us to develop an appropriate theory. The first has to do with the problem of the temperature dependence of F6rster energy transfer in 'glasslike" photosynthetic antenna protein complexes.28*29 The second has to do with the design of materials which are useful for frequency domain optical storage and imaging'0.31 by spectral hole burning at high temperatures. The form of the thermally averaged Franck-Condon factors used in the theory given below was recently derived32 in the context of energy and electron transfer and provides for greater case of computation than earlier f ~ r m s . ~ ~ J ~
II. Theory Previously, we have ~ h o w n that ~ ~ Jin~ the low-temperature limit the absorption spectrum following hole burning at a frequency WB for a time 7 may be represented as
A,(Q) = Jdv No(v - vm) L(Q - v ) exp(-[uP#.rL(o,-
v)])
(1)
Here, L(O - v) is the absorption profile of a single absorber (site) of the inhomogeneous distribution with a ZPL frequency equal to v. No(v - v,) is the Gaussian distribution of ZPL frequencies with a width of rid and centered at the frequency vm. The single site absorption profile in the low-temperature limit is
for the case where the coupling is restricted to a distribution of host phonons with a mean frequency of wm The values R = 0, 1,2, ... correspond to the zero-phonon, one-phonon, two-phonon, etc., transitions with 10 being a Lorentzian of homogeneous width y for the ZPL. The coefficients of the 1~ are the Franck-Condon factors with S being the Huang-Rhys factor. Our modeling of the one-phonon profile ( 1 1 ) is discussed later. We emphasize, however, that the normalized 1~ (R1 2) are obtained from 11 by proper folding of 1, with itself R tima.35 Returning to eq 1, u is the absorption cross section, P the spectral photon flux, and 4 the hole-burning quantum yield. The form of L(0 - v) appropriate for the inclusion of higher frequency localized or pseudolocalized modes is given in ref 17. In this paper, we are interested in the temperature dependence of the hole and absorption spectra. In this case one must take into account thermally populated phonon levels and account for both phonon creation and annihilation processes. This is done through inclusion of the phonon occupation number, A = [exp(ho/ 0 1994 American Chemical Society
Hayes et al.
7338 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
kT) - 11-1, where w is the phonon frequency. This problem has been considered by numerous workers, not only for spectral line shapes14 but also for electron transfer,33J4 which also depends upon the thermally averaged Franck-Condon factors, FC. The result is usually presented in terms of a Bessel function, I,,,,,as
FC =
exp[-iEt/h
+ iuk(R - R')t - iwfl'r]
(12)
Finally, performing the integration, we obtain
The details of the derivation of eq 3 can be found in refs 33 and 34. Though mathematically convenient, eq 3 does not give a physical picture of how the phonons contribute to the line shape. Therefore, a more physical derivation will be given. We begin with a formula for the FC generalized for multiple frequencies,
k,
m-dtf(t)e-iE'/*
FC(E) = 1
(4)
where
+; (a
Tuk[R-2R1)
(13)
+
The (Rk 1) and fik thermal factors are associated with phonon creation and annihilation, respectively. In the limit as T -.,0 K, eq 13 reduces to eq 2 when the mean phonon frequency approximation is introduced and the delta functions are replaced by realistic line shape functions, 1 ~ That . eq 13 is equivalent to eq 3 is shown in ref 32. With eq 13, eq 1 for A, can be cast into a convenient form valid for arbitrary temperature:
A,(fl,T) =
At) = exp[-G + G+(t) + G-(t)]
(5)
(7) and
G = G+(O) + GJO) = T S k ( 2 h k+ 1)
(8)
Equation 5 is the form of the generating function suitable for harmonic oscillators with no frequency change which was first developed by K ~ b and o ~by~ Lax.2 The assumption of a small frequency change for phonons is generally an excellent one. G+ and G-represent respectively phonon creation and annihilation during the absorption process; e.g., G-tends to zero as T ---c 0 K. The exponential of the G's forms a sum of all possiblecombinations of loss and gain of phonon energy. E is the difference between the energy of the ZPL and that of the optical transition. Note that t need not represent time as it is simply a dummy variable in the integration. The generating function can be expanded in a Taylor series in the G+ and G-terms, giving
Here, I R , represents ~ the normalized line shape function of the (R - 2R')-fold term in the phonon progression. As noted in the Introduction, it is necessary to model the shape of these terms. Guided by experimental low-temperature one-phonon profiles of molecular chromophores imbedded in glasses, polymers, and proteins, we have recently employed an asymmetric one-phonon profile having a Gaussian shape on the low-energy side and a Lorentzian on the high-energy side.24~25 This corresponds to 1, in eq 2. Typically the mean phonon frequency, Om, in the above systems is -20-30 cm-l with a one-phonon profile width (I') of 30-40 cm-l. This profile is governed by g(w)D(o),where g(o) is the host phonon density of states and D(o) is the electronphonon coupling term which it is still not possible to calculate. Interestingly,pseudolocalized phonons have proven to be generally unimportant in the above systems. One notable exception is the special pair of the bacterial RC (uide infra). Pseudolocalized or localized phonons must be treated separately from the host ph0nons.3~Guided by our earlier work, we employ for computations involving delocalized phonons
-
for v
> (R - 2R')wm, and
The power of R can then be expanded with the binomial theorem
which, when substituted into eq 9, gives
Substituting eq 11 into eq 4 and expanding the G terms gives
for vl (R-2R?om. FornonzeroR,rRisdefinedasIR-2R'Il/2I', where I' is the width of the one-phonon profile. When R = 0, 10.0 is the zero phonon line: a Lorentzian of width, 7 . Equations 15aand 15bareusedineq 14which,inthemeanphononfrequency approximation, reduces to
Jdv N,(v- v,,,) e-up~TL(wsu) lRx(fl - v - (R - 2R')o,,,) (16)
Temperature Dependence of Hole-Burned Spectra
The Journal of Physical Chemistry, Vol. 98, No. 30, I994 7339
TABLE 1: Parameters Used for tbe Calculation of Rps.
with
viridis Absorption Spectra.
l R p ( w B - Y - (R - 2R')")
(17)
Here fi is the phonon occupation number for mean frequency om and S is defined so that Sw, (optical reorganization energy) = Cfiwk. The preburn absorption spectrum is obtained from eq 16 by setting 7 = 0.
Calculations and Discussion A computer program to calculate the absorption and holeburned spectra from eq 16 has been written. The integral is calculated numerically using Bode's approximation. The infinite sum is replaced by a sum over sufficient terms to account for 99% of the absorption intensity. Because of the first illustration given below, the program allows for inclusion of the contribution from one higher-frequency pseudolocalized mode to the spectra. We first apply the theory to the calculation of the temperature dependence of an absorption profile. To this end we will use the P960 band of the Rps. viridis RC. P960* is the primary electron donor state of the RC; cf. Introduction. As shown previo~sly,~~*~ the structure underlying the broad and essentially featureless P960 band (fwhm 430 cm-1 at 4.2 K) can be revealed by photochemical hole burning (PHB). The burn frequencydependent PHB spectra directly establish" that a broad distribution of phonons with om = 30 cm-1 and a special pair pseudolocalizedmode with alp= 145cm-1 dominate the electronphonon coupling of the P* P optical transition. The values of the theoretical variables determined by Reddy et al.25are given in Table 1. Note that the coupling is strong with S + Ssp= 3.1. The inhomogeneousbroadening (I'inh) of 140cm-1 is quite typical of those determined for the lowest ITT* (Q,,) state of chlorophylls in other photosynthetic protein complexes.27 As discussed in ref 25, the PHB spectra indicate that the special pair pseudolocalized mode at 145cm-1 undergoes dephasing in 200 fs, corresponding toa homogeneouswidth of rlP = 50 cm-1. Utilizing the parameter values of Table 1, we have calculated the P960 absorption profile over the temperature range between 0 and 250 K. Figure 1shows several representative spectra. Also shown is the calculated lowtemperature single site absorption profile with a ZPL frequency equal to vm (set equal to0 cm-1 in the figure). The 10 K absorption profile is very similar to the experimental profile.25 As expected, sinceonly linear electron-phonon couplingis included in the model, the maximum of the P960 profile exhibits only a weak dependence on temperature. Experimental data show a much larger shift, -200 cm-1 over the same range?** which would be due to quadratic terms in the electron-phonon coupling and/or intermolecular anharmonicity. Of particular interest to us was whether the analysis of Reddy et al.,25 which led to Table 1, could account for the thermal broadening of P960 from linear electron-phonon coupling. Figure 2 shows the temperature dependence of the width of P960 along with the experimental data of Tang et al.I7 The agreement is gratifyingly good for T 5 175 K. Significant deviations occur for higher temperatures. The deviations could be due to a number of processes which onset in this temperature region and which are not included in our model, e.g., population of thermally accessible and distinct conformational substates of the protein41.42 which are believed to be ubiquitous in proteins and dynamically important at -200 K42;contributions of modes with frequencies >150 cm-1; softening of the glycerol: H2O glass used in the experiments; and anharmonicity. The point is that for the temperature range where the model would be expected to be valid, the values of the linear electron-phonon coupling and other parameters (determined by hole burning) when used with eq 16 (embellished for the 145-cm-1 special pair marker mode)
-
-
r
Y wm
3 cm-1 3Ocm-I
wIp
S
2.1
S,
55 cm-l 145 cm-I 1.0
rlP
so ~ m - 1
140cm-l
rbh
y is the zero-phonon line width; wm,the mean phonon frequency; S, the electron-phonon coupling; r, the one-phonon bandwidth; wgp the special pair marker mode; S, the Huang-Rhys factor for this mode; rip, the wdith of the special pair mode; and ru the inhonogeneous bandwidth.
0
-io0
200
600
400
Wavenumber (cm")
Figure 1. Simulated absorptionspectraof Rps.uiridis at the temperatures shown. The spectra were calculated with eq 13 with T = 0 and the parameters used for simulation of hole spectra of Rps. uiridis in Table 1. The slight change in slopeon the high-wavenumber side of the highertemperature spectra is due to truncating the integration range in the computation. The abscissa is in wavenumber units relative to the center of the site distribution function. Also shown is the single-siteabsorption profile at 5 K for the same parameters.
-
I
50
I 1w
I
I
150
200
J m
(K)
Figure 2. Absorption width as a function of temperature of the data shown in Figure 1 (+). The circles are experimental data from ref 17.
lead to an accurate accounting of P960's thermal broadening. To the best of our knowledge, this is the first time that a theory has been used with low-temperature hole-burning data to account for the thermal broadening of an absorption profile characterized by strong linear electron-phonon coupling. This was our only objective since it was already apparentI7 that our linear coupling theory could not account for the temperature dependence of the band shift of P960. We turn next to the use of eq 16 to simulate temperaturedependent hole spectra. Although there is considerable interest in the possibility of high-temperature hole burning for practical applications, there have been relatively few published reports of the hole profiles as a function of temperature. Rather than try to fit the scant experimental data available, then, we simply show somecalculatedspectra and from these draw conclusions regarding the requirements for a practical high-temperature hole-burning material. Figures 3 and 4 show hole spectra calculated at TB=
Hayes et al.
7340 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
TABLE 2 ZPL Widths and Fraction of ZPH Relative to Total Hole for the Simulated Holes of Figure 3 (S = 0.5, w = 30 cm-1) 5 25 50 80
0.15 1.2 2.5 5.0
0.368 0.239 0.086 0.022
TABLE 3: ZPL Widths and Fraction of ZPH Relative to the Total Hole for the Simulated Holes of Figure 4 (S = 0.3, w = 15 cm-1)
Q u e 3. Simulated hole spectra at 5 (a), 25 (b), 50 (c), and 80 K (d). Thespectrawerecalculatedwithq 13withy=O.l5at5Kandincreasing with temperature according to a dependence. Other parameters are ri = 400 cm-1, S = 0.5, ho = 30 cm-l, and r = 30 cm-l. The spectral intensitiesare scaled as shown in the figure. The bars on the right of each spectrum indicate a 1% or l/2% hole depth. For the 50 and 80 K spectra the simulated spectra are overlaid with random noise.
5.0
5.0
Figure 4. Simulated hole spectra at 5 (a), 25 (b), 50 (c), and 80 K (d). Thespectrawerecalculatedwitheq 13withy sO.lSat5Kandincreasing with temperature according to a F . 3 dependence. Other parameters are ri 400 cm-1, S = 0.3, hw = 15 cm-1, and r = 30 cm-1. The spectral intensitiesare scaled as shown in the figure. The bars on the right of each spectrum indicate a 1% or 1/2% hole depth. For the 50 and 80 K spectra the simulated spectra are overlaid with random noise.
-
5,25, 50, and 80 K for S = 0.5, w,,, = 30 cm-1 and for S = 0.3, 15 cm-l. For these holes a ZPL width of 0.15 cm-1 at 5 K was used and F . 3 dependence for its broadening11143was
wm =
T
7
5 25 50 80
0.15 1.2 2.5 5 .O
&%2M.')
0.540 0.229 0.059 0.01 1
assumed. An inhomogeneous width, rid, of 400 cm-1 was used. The values of the linear electron-phonon coupling parameters and I'inb are appropriate for molecular chromophores imbedded in glasses, polymers, and proteins and which are characterized by weak coupling. The low-temperature holes, (a), are similar to those shown in ref 17, being dominated by the zero phonon hole (ZPH). The burn frequency, WE, was chosen to be 300 cm-I to the red of the center of the inhomogeneous distribution of ZPL frequencies(um). At this position, thereis a noticeableasymmetry between the pseudophonon sideband hole and the real phonon sideband hole. This is most evident in the 25 K spectra. Although the holes of Figures 3a and 4a are relatively deep (-8%), these holes are still within the short burn-time, low-temperature limit. Thus, the intensity of the ZPH relative to the total hole intensity is approximately e-=. At the higher temperatures the fraction of hole in the ZPH is given by e-2s(2n+1).The fractional ZPH is tabulated in Tables 2 and 3 for the holes of Figures 3 and 4, respectively. Also, shown are the ZPL widths. The burn intensities used in the simulations were chosen to give a 1% hole at 50 K. This intensity was used for the 25, 50, and 80 K simulations; one-tenth of this intensity was used for the 5 K simulations. Note that the 5 K holes are shown on a reduced intensity scale. For the 50 and 80 K simulations, the holes are shown both with and without added noise. Although the ZPH are quite clearly evident in the noise free 50 K simulations, they are barely detectable in the presence of noise such that a signal-to-noise ratio of three is obtained. This seems a suitable amount of noise to add to a hole that represents a 1% change in absorption. Note that the advantage in ZPH detectability, which is expected from a decrease in S, is more than offset by a decrease in w, (Figures 3c and 4c). Although the values of w,,, and S used give roughly equal ZPH intensities (see Tables 2 and 3), the lower phonon frequency of Figure 4 results in more overlap between the phonon sideband holes and the ZPH and, therefore, a less detectable ZPH. Thus, for high-temperature hole burning, selectionof materials with low S and high wm are necessary for maximum hole detectability. In addition, materials with narrow ZPL widths also enhance detectability since the ZPH intensity is distributed over a narrower spectral region. The F . 3 ZPL broadening used here is probably an underestimate as the contribution of the twolevel system relaxation characteristic of glassy materials at low temperatures is likely dominated by other processes at temperatures above 10 K. For example, Littau et al.42 found that for glycerol glasses the line width was fit by the combination of a F,3 process and an Arrhenius term. Such a fit was first proposed by Jackson and Silbey.43 Inclusion of this additional term in the ZPL width of the simulations would result in a further decrease in detectability of the ZPH at elevated temperatures.
-
Temperature Dependence of Hole-Burned Spectra
Acknowledgment. This article was made possible by support from the Division of Materials Research of the National Science Foundation under Grant DMR-9307034. References and Notes (1) Huang, K.; Rhys, A. Proc. R . Soc. London 1950,204A, 406. (2) Lax, M. J. Chem. Phys. 1952, 20, 1752. (3) Kubo, R.; Toyozawa, Y. Prog. Theor. Phys. 1955.13, 160. (4) Markham, J. J. Reu. Mod. Phys. 1959, 31,956. (5) Lyo, S. K.; Orbach, R. Phys. Rev. B 1980, 22,4223. (6) Hegarty, J.; Yen, W. M. Phys. Rev. Lett. 1974,33, 1126. (7) Reinecke, T. L. Solid Stare Commun. 1979, 32, 1103. (8) Hayes, J. M.; Stout, R. P.; Small, G. J. J. Chem. Phys. 1980,83, 4129. (9) Hayes, J.M.;Janlrowiak,R.;Small,G. J. In TopicsinCurrentPhysics, Persistent Spectral Hole Burning Science and Applications, Moerner, W. E., Ed.; Springer-Verlag: New York, 1987; Vol. 44, p 153. (10) Saven, J. G.; Skinner, J. L. J. Chem. Phys. 1993, 99,4591. (1 1) Narasimhan, L. R.; Littau, K. A.; Pack, D. W.; Bai, Y. S.;Elschner, A,; Fayer, M. D. Chem. Rev. 1990,90,439. (12) Szabo, A. Phys. Reu. Lett. 1970, 25,924. (13) Personov, R. I.; AI'Shits, E. I.; Bykovskaya, L. A. Opt. Commun. 1972, 6, 169. (14) Kharlamov, E. M.;Personov, R. I.; Bykovskaya, L. A. Opt. Commun. 1974, 12, 191. (15) Gorokhovskii, A. A,; Kaarli, R. K.; Rebane, L. A. JETP Lett. 1974, 20, 216. (16) Haves. J. M.: Small. G. J. J. Phvs. Chem. 1986.90.4928. (17) Haies; J. M.;Gillie, J. K.; Tang, D.; Small, G. J. Eikhim. Biophys. Acra 1988, 932, 287. (18) Boxer, S.G.; Lockhart, D. J.; Middendorf, T. R. Chem. Phys. Lett. 1986,123,476. (19) Boxer. S.G.; Middendorf. T. R.: Lockhart, D. J. FEBS Lett. 1986. 2Od,237. (20) Meech, S. R.; Hoff, A. J.; Wiersma, D. A. Proc. Natl. Acad. Sci. U.S.A. .~ 1986.83. --.- - ,9464. (2l)-Deisenhofer, J.; Epp, 0.; Miki, K.; Huber, R.; Michl, H. Nature 1985, 318, 618. (22) Allen, J. P.; Feher, G.; Yeates, T. 0.;Deisenhofer, J.; Michel, H.; Huber, R. Proc. Natl. Acad. Sci. U.S.A. 1986,83, 8589.
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