4928
J . Phys. Chem. 1986, 90, 4928-4931
well ordered and coherent copper layer. It is clear that the X-ray standing wave technique is ideally suited for following the course of electrochemical processes involving the deposition of metallic layers and we are currently involved in the in-situ study of the deposition of layers of copper, silver, and lead. By measuring the phase and amplitude of the modulation of the iodine fluorescence intensity we will be able to characterize the thickness and uniformity of the underlying deposit.
In addition, by combining surface EXAFS with the X-ray standing wave information, we hope to further elucidate (in-situ) the three-dimensional structure of electrodeposited layers.
Acknowledgment. This work was generously supported by the National Science Foundation (DMR-84-12465), the National Institutes of Health (RRO1646-02), the Materials Science Center at Cornell University, and by the Army Research Office. CHESS is a facility supported by the National Science Foundation.
Photochemical Hole Burning and Strong Electron-Phonon Coupling: Primary Donor States of Reaction Centers of Photosynthetic Bacteria J. M. Hayes and G . J . Small* Ames Laboratory-USDOE and Department of Chemistry, Iowa State University, Ames, Iowa 5001 1 (Received: August I , 1986)
A theory for photochemical hole burning valid for arbitrarily strong linear electron-phonon coupling is developed and applied to the P-870 and P-960 states of Rh. sphaeroides and uiridis. The mechanism responsible for the unusually large hole widths of these states is established.
Very recently, transient photochemical hole burning (PHB) has been reported for the P-870132and P-9603 absorption bands of the primary electron donor states of isolated reaction centers of Rhodopseudomonas sphaeroides and viridis, respectively. These states correspond to the lowest excited singlet state (Q,) of the special bacteriochlorophyll (Bchl) pair. Time domain studies had previously shown that electron transfer from the special pair to a bacteriopheophytin occurs in about 2 and 5 ps for Rh. sphaeroides and Rh. uiridis, respectively, at cryogenic temperat u r e ~ . ~These . ~ decays correspond to homogeneous line broadenings of 2.5 and 1 cm-'. By contrast, the PHB experiments yielded hole widths at T 1.5 K equal to -400 cm-l, a significant fraction of the absorption line widths for P-870 and P-960. Given the current interest in the rate of the initial charge separation process of photosyntheand the electronic structures of P-870 and P-960," the
-
( I ) Boxer, S. G.; Lockhart, D. J.; Middendorf, T. R. Chem. Phys. Lett. 1986, 123, 476.
(2) Meech, S. R.; Hoff, A. J.; Wiersma, D. A. Chem. Phys. Lett. 1985, 121, 287. (3) Boxer, S. G.; Middendorf, T. R.; Lockhart, D. J. FEBS Lett. 1986. 200, 237. (4) Parson, W. W.; Woodbury, N. W. T.; Becker, M.; Kirmaier, C.; Holten, D. In Antennas and Reaction Centers of Photosyntheric Bacteria, Michel-Beyerle, M. E., Ed.; Springer-Verlag: Berlin, 1985; Spinger Series in Chemical Physics, Vol. 42, pp 278-285. (5) Holten, D.; Windsor, M. W.; Parson, W. W.; Thornber, J. P. Biochim. Biophys. Acta 1978, 501, 112. (6) Rockley, M. G.; Windsor, M. W.; Cogdell, R. J.; Parson, W. W. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 2251. (7) Kaufmann, K. J.; Dutton, P. L.; Netzel, T. L.; Leigh, J. S.; Rentzepis, P. M. Science 1975, 188, 1301. (8) Holten, D.; Hoganson, C.; Windsor, M. W.; Schenck, C. C.; Parson, W W.; Migus, A.; Fork,R. L.; Shank, C. V. Biochim. Biophys. Acta 1980. 592, 46 1. (9) Woodbury, N. W.; Becker, M.; Middendorf, D.; Parson, W. W. Biochemistry 1985, 24, 7516. (10) Martin, J. L.; Breton, J.; Hoff, A . J.; Migus, A,; Antonetti, A. Proc. Nail. Acad. Sci. U.S.A. 1986, 83, 957. (1 1) Parson, W. W.; Scherz, A,; Warshel, A. In Antennas and Reaction Centers of Photosynthetic Bacteria, Michel-Beyerle, M. E., Ed.; SpringerVerlag: Berlin. 1985; Springer Series in Chemical Physics, Vol. 42, pp 122-130.
question of whether or not the hole widths represent homogeneous broadening is very important. Meech et aL2 interpreted the hole width of P-870 in terms of an ultrafast (25 fs) dynamical charge separation process within the special pair prior to electron transfer to the bacteriopheophytin. The same interpretation was offered by Boxer et al.'33 who also mentioned the possibility that the hole is inhomogeneously broadened due to strong electron-phonon coupling. However, the question of which interpretation is correct was left open and no quantitative theoretical discussion was presented. The purpose of this Letter is twofold: to present the key results of a new theory for PHB valid for arbitrarily strong linear electron-phonon coupling; and to demonstrate with this theory that the aforementioned holes are essentially what one would expect on the basis of the absorption and fluorescence spectra of the primary donor states. That is, the holes are inhomogeneously broadened. The relationship between these spectra and the hole profiles was not considered in ref 1-3. An important point is that for both P-870 and P-960 the Stokes shift (between absorption ' ~ Rh. and fluorescence maxima) is large, -500 ~ m - ' . ' ~ , For viridis the T dependence of the absorption and fluorescence line widths and Stokes shift have been studied between 2 and 300 K.12 The Stokes shift is weakly dependent on temperature while the absorption and fluorescence maxima undergo comparable red shifts with decreasing T. Although less complete, the data for P-870 show similar behavior.13 A method of moments analysis (based on strong linear electron-phonon coupling) of the P-960 line widths, which exhibit a dependence, revealed that the Huang-Rhys factor (S) is very large, -8.12 Furthermore, the average phonon frequency associated with this factor is -30 cm-'. In this Letter we use these facts as a guide to interpret the hole burning data for Rh. sphaeroides since the S / N for these data is higher than for Rh. viridis. The reaction center pigment ~
~~
(12) Scherer, P. 0. J.; Fischer, S. F.; Horber, J. K. H.; Michel-Beyerle, M. E. In Antennas and Reaction Centers of Photosynthetic Bacteria, Michel-Beyerle, M. E., Ed.; Springer-Verlag: Berlin, 1985; Springer Series in Chemical Physics, Vol. 42, pp 131-137. (13) Woodbury, N. W. T.; Parson, W. W. Biochim. Biophys. Acta 1984, 767, 345.
0022-3654 I86 12090-4928SO1.50 1'0 0 1986 American Chemical Societv
The Journal of Physical Chemistry, Vol, 90, No. 21, 1986 4929
Letters
r
-1000
I
-750
-500
-250
0
250
500
750
1000
1250
WAVEN U MBERS Figure 1. Computed absorption spectra for a single site and for a Lorentzian distribution of such sites. For the single-site absorption: y = 2.5; = 30;S = 8; r = 30. For the overall absorption urn = 0; rinh = 400.
structures for the two organisms are very ~ i m i l a r . ’ ~ ,A ’ ~detailed account of the theory, more extensive calculations for R h . sphaeroides, and application of the theory to Rh. viridis will be forthcoming.I6 For the development of a theory of PHB valid for arbitrarily strong linear electron-phonon coupling it is first necessary to derive the single site absorption profile. Our approach is based on a time correlation function method” and employs the Condon approximation. Defining u as the single-site zero-phonon transition frequency and w, as the maximum frequency of the one-phonon absorption profile (relative to v), the single-site absorption profile takes the form
w,
for the hole spectrum in the short burn time limitIs for which the data of ref 1 and 2 are applicable:
I
r, + r,’ (0- wB
+ w,(r’-
r ) ) 2+
((r,+ r f ) / 2 ) 2
I+
-
in the T 0 K limit of interest here. The values of r = 0, 1, 2, ... correspond to zero-, one-, etc. phonon processes and the Huang-Rhys factor for arbitrary temperature is S=
dw A ( w ) coth
(g)
Here, A ( w ) is the product of the phonon density of states and the frequency-dependent electron-phonon coupling matrix element. The r-phonon profile is the result of convolving the one-phonon profile r times itself. If the one-phonon profile is taken to be a Gaussian or a Lorentzian with width r, the r-phonon profile has a width of r1I2rand r r , respectively, and is centered at rw,. In order to present an expression for the hole profile in a closed and physically transparent form, we will utilize Lorentzians for l,(r 2 1) with a width given by r*I*I’.The important features of our calculations will not be affected by utilization of more realistic line shapes, e.g., asymmetric Gaussians. For the zero-phonon line (ZPL), lo can be taken to be a Lorentzian with a line width y. We note that eq 1 is valid for coupling to either a “sea” of delocalized phonons or a single pseudolocalized phonon. The approach taken to derive the hole line shape function with eq 1 is, with suitable modification for strong electron-phonon coupling, that of Friedrich et a1.18 We present here only the result (14) Deisenhofer, J.; Epp, 0.; Miki, K.; Huber, R.;Michl, H. Nature (London) 1985, 318, 618. (15) Norris, J., private communication. (16) Hayes, J. M.; Small, G. J., manuscript in preparation. (17) Pyrce, M.H. L. In Phonom in Perfect Luttices and in Lattices with Point Defects, Stevenson, R. W. H., Ed.; Oliver and Boyd: London, 1965; p 403. (18) Friedrich, J.; Swalen, J. D.; Haarer, D. J . Chem. Phys. 1980, 73, 705.
the desired result. For compactness we have defined r, y for r = 0. For r 2 1, r, = r 1 / 2 r .The width r l n h is that of the site excitation energy distribution function (SDF) determined by the disorder. The S D F has a maximum at the frequency u,. The terms A . and A , represent the absorption spectrum prior to and after the burn, respectively. Considering the complexity of the problem, eq 3 is gratifyingly simple in form as well as physically transparent. For example, by inspection one can see that for S > 1 and the laser burn frequency, wB,located on the low and high energy tails of the absorption profile the hole maximum can be shifted to the blue and red of wB,respectively. We return to this point below. For the calculations to be presented, the Huang-Rhys factor S and the frequency corresponding to the maximum of the onephonon process (a,) were chosen as 8.0 and 30 cm-’. These values are consistent with the results of ref 12 and, in addition, phonon frequencies in the vicinity of 30 cm-l are typically seen in linenarrowed fluorescence spectra of glassy systems exhibiting weak electron-phonon coupling (S < 1) where the one-phonon process dominates higher order processes. For the single-site, zero-phonon line width, y, a value of 2.5 cm-I (consistent with the reported employed was 2-ps lifetime of P-870) was used. The value of rlnh
4930 The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
Letters
_ _ 200 _ _300
I
u
1
~
1
L
-200
-400
-
.
.
i
-
200
0
1
I
I
400
600
800
1 000
WAVENUMBERS Figure 2. Computed holes burnt at various wavelengths across the inhomogeneously broadened spectrum of Figure 1. The holes have been normalized to unit maximum hole depth.
_ _ 400
_ _ 600 -800
-600
-400
-200
0
200
400
600
800
WAVENUMBERS Figure 3. Computed holes burnt for wavelengths remote from the maximum of the site distribution function.
400 cm-], consistent with the inhomogeneously broadened absorption line width of Bchl monomer Q, transitions in glasses and polymers. The other parameter required, viz., r, the one-phonon absorption profile width, was allowed to vary within reasonable limits in order to produce results in essential agreement with the Rh. Sphaeroides hole burning data. A more systematic treatment of the effect of varying this and other parameters will be included in ref 16. However, we stress that the data of ref 12 and 13 mandate large S values, Le. strong coupling. Figure 1 shows the computed absorption band for r = 30 cm-I. The wavelength axis is measured relative to the maximum of the S D F at 0.0 cm-]. Note that the absorption maximum is shifted 200 cm-' to the blue of the peak of the distribution. Also note that the band is asymmetric, tailing more to the blue than to the red. This tailing is evident in the Rh. sphaeroides absorption spectrum but is usually ascribed to overlap with the next band in the spectrum. Although there is indeed some overlap, the overlap region does not appear to be simply the convolution of two symmetric bands. Figure 1 also shows the shape of the single-site absorption for the parameters given above. Note that the zero-phonon line is barely detectable. The ZPL carries
-
1/3000 of the total absorption strength for S = 8.0. Also note that there is considerable absorption to lower energy of the ZPL. This is a consequence of the width of the phonon absorption. Friedrich et a].'* in their computations of hole shapes, eliminated this effect by using asymmetric Gaussians in their calculations. In addition to being somewhat arbitrary and artificial, this procedure would also complicate eq 3 to the point that the physical interpretation would no longer be apparent. Figure 2 shows holes burnt at various wavelengths across the inhomogeneously broadened spectrum of Figure 1. The holes have beer-,normalized to the same value so that comparison with Figure 3 of ref 1 is facilitated. The following properties of the holes were noted: the hole maxima do not correspond with the burn wavelength; the shift of the hole maximum from the burn wavelength varies from -0 in the vicinity of the maximum of the SDF, to several hundred cm-'; the hole widths vary across the absorption band; and the holes are asymmetric. All of these observations are consistent with those for hole burning of P-870 in Rh. sp h aeroides. The burn frequencies, wB,chosen for Figure 2 were chosen to be similar to those used by Boxer et al.' in their study of the wavelength dependence of hole burning in Rh. sphaeroides. As
J . Phys. Chem. 1986, 90, 4931-4938 indicated above, the parameters chosen do reproduce the hole properties reported in ref 1. Additional information about the nature of the holes is obtained by extending the burn frequency to wavelengths remote from the peak of the SDF. Results of such burns are shown in Figure 3. From this figure the cause of the observed peak broadening and the shift of hole maximum become apparent. Each hole has two major components: one centered near the maximum of the S D F and the other peaked near the burn frequency. The component near the S D F maximum arises from the large number of absorbers around this wavelength, each absorbing through its broad phonon side band. Because of the large density of absorbers in the vicinity of the S D F maximum, the contributions of this term dominate the holes until wB is shifted several hundred wavenumbers from the S D F maximum. At large shifts from this frequency the other term begins to become significant and the hole begins to split. In these simulations the splitting is easily seen due to normalizing of the holes to the same value rather than scaling the hole maxima to the strength of the absorption. In an actual sample the splitting may not be detectable due to the weak absorption at the wB at which splitting will be seen. In summary, our theory for PHB together with linear electron-phonon coupling dataI2J3establish that the unusually large widths observed'q3 for P-870 of Rh. sphaeroides are the result of a multiphonon broadening process stemming from strong electron-phonon coupling. That is, the widths are not a manifestation of an ultrafast charge separation within the special Bchl pair
4931
following preparation (by excitation) of a neutral excitonic state.l,2 The same conclusion holds for P-960 of Rh. ~ i r i d i s Nevertheless, .~ the question of why P-870and P-960 exhibit Huang-Rhys factors comparable to those of charge-transfer states associated with a-molecular donoracceptor complexedg remains and is important. We note that theoretical calculations by Parson et al." indicate that both of the above dimer or excitonic states carry significant charge-transfer character (intradimer). A related question is why the primary electron donor state, P-700, of reaction centers of green algaeZoand spinachZ1yield photochemical holes which are more than 3 orders of magnitude sharper than those considered here. Sharp nonphotochemical holes have also been observed for the lowest excited state of self-aggregated dimers of Chl a.22
Acknowledgment. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This research was supported by the Director for Energy Research, Office of Basic Energy Science. (19) Haarer, D.; Philpott, M. R. In Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, Agranovich, V. M., Hochstrasser, R. M., Eds, North-Holland: Amsterdam, 1983; Modern Problems in Condensed Matter Sciences Vol. 4, pp 27-82. (20) Maslov, V. G.; Chunaev, A. S.; Tugarinov, V. V. Mol. Bioi. 1981, 15, 788. (21) Golbeck, J.; Gillie, J. K.; Fearey, B. L.; Small, G. J., unpublished results. (22) Carter, T. P.; Small, G. J. J . Phys. Chem. 1986, 90, 1997.
FEATURE ARTICLE Pure Dephasing of a Two-Level System J. L. Skinner* and D. Hsu \
Department of Chemistry, Columbia University, New York, New York 10027 (Received: May 16, 1986)
The relevance of dephasing to many different branches of spectroscopy is discussed. Very general results for both quantum mechanical and stochastic formulations of the dephasing problem are presented and are shown to be intimately related. Our results are nonperturbative in nature, in contrast to the commonly accepted theories. Our results for specific quantum mechanical models are evaluated, and usefui expressions for the analysis of optical and vibrational dephasing experiments are presented.
I. Introduction In many branches of spectroscopy, the description of a system by only two quantum states plays a ubiquitous and fundamental role in our understanding of the absorption of radiation. For example, in magnetic resonance spectroscopy of electrons or spin nucleii, where the application of a static external magnetic field lifts the degeneracy of the two spin levels, many phenomena for isolated spins such as free induction decay or spin echoes can be understood simply by considering the dynamics of the two spin levels in applied time-dependent magnetic In vibrational relaxation or infrared spectroscopy experiments one can often (1) Abragam, A. Principles of Nuclear Magnetism; Oxford: London, 1961. ( 2 ) Slichter, C. P. Principles of Mugnetic Resonance; Springer-Verlag: Berlin, 1980; 2nd ed. ( 3 ) Farrar, T. C.; Becker, E. D.Pulse and Fourier Transform N M R , Academic: New York. 1971.
0022-3654/86/2090-4931$01.50/0
describe a molecule by only two vibrational levels4 This description is useful if the vibrational frequencies of the different modes are sufficiently different, and if either the temperature is low enough so that only the ground state is appreciably populated, or if the anharmonicity is great enough so that individual transitions for a single mode are well separated. In optical experiments one has traditionally analyzed both line shapes and quantum optics phenomena by considering only two electronic levels of atoms or molecule^.^ As long as the particular optical transition of interest is well separated from other transitions, this approximation is adequate. Other examples of two-level systems (TLS) that can be probed spectroscopically include tunnel-split levels in molecules (ammonia, for example), certain proton-transfer reactions, and configurational tunneling in glasses. (4) Oxtoby, D. Adu. Chem. Phys. 1979, 10, 1. 1981, 47 (part 2), 487. (5) Allen, L.; Eberly, J. H. Optical Resonance and Two-Level Atoms; Wiley: New York, 1975.
0 1986 American Chemical Society
