J. Phys. Chem. 1994,98, 7270-7273
7270
Spectral Effects of Exciton Splitting in “Statistical Pairs” Robert S. b o x ‘ AFRC Photosynthesis Group, Department of Biochemistry, Imperial College of Science, Technology, and Medicine, London SW7 2AY, U.K. Received: December 27, 1993; In Final Form. April 5. 1994.
For fixed intermolecular distances, exciton states of molecular pairs have a range of splittings and oscillator strengths. Correlation of these two quantities reveals that for random angular distributions the lower lying of the pairs of exciton transitions have, on the average, lower oscillator strength. For distributions with steric constraints typical of planar molecules, this effect is enhanced. The implication of these findings for spectra of planar molecules in polar solvents is that close pairs, normally expected to be recognizable by an exciton splitting, actually exhibit a net blue shift, which tends to mask their presence. A numerical example relating to chlorophyll a spectra is presented.
1. Introduction
.-I-.
At the concentration of chlorophyll in vivo, about 0.1 M,a polar solution of chlorophyll exhibits considerable fluorescence quenching with no apparent concomitant effect on its absorption spectrum.l.2 This quenching has been attributed2 to transfer of excitation to “statistical pairs”, which are molecular pairs whose separations are 1.0 nm or less in the case of a 0.1 M solution. Such quenching pairs would normally be expected to broaden the chlorophyll absorption spectrum as a result of their exciton interaction. How do the statistical pairs quench the bulk fluorescence? In the case of nonpolar solvents the formation of dimers is quitevivid,3-5 and they may be expected to form efficient traps for diffusing excitation. There is also a question of how these dimers consistently avoid fluorescence once excited. However, in this paper, we limit ourselves to one aspect of the somewhat more subtle statistical pair problem. Beddard and Porter2 dealt with the quenching of fluorescence in dense solutions with a model based on the F6rstefi excitation transfer process. They assumed that migratingexcitation arriving at a statistical pair having a separation equal to or less than a certain interchromophore separation would be quenched. The characteristic separation best fitting the data for chlorophyll in ether1 was 1.0 nm.2 Taken literally, the Beddard-Porter model actually contains no mechanism for quenching. What happens after the excitation arrives at the first member of a statistical pair is merely an assumed demise. On F6rster theory alone, the excitation would stay trapped at the statistical pair, hopping back and forth within it, leaving on the same path and at the same rate at which it came, to the site whence it came. One must therefore search not only for a quenching mechanism within this pair but also for a reason that the excitation does not leave the statistical pair more rapidly than it is quenched. In this paper we do not attempt to answer the ultimate quenching question, but we examine the circumstance that very little disturbanceof the absorptionspectrum is seen to accompany the quenching phenomenon.2 The distribution of oscillator strengths associated with the close molecular pairs is examined, and we show, on the basis of molecular exciton theory: that on average the lower-lying state of the two molecular pair states has the smaller oscillator strength. In thecaseof planar chromophores that cannot sustain a completely random orientation at small separation, this asymmetry of the oscillator strength distribution is accentuated by the steric constraint. Thus we argue here that Present address: Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171. Abstract published in Aduunce ACS Abstracts, June 15, 1994.
0022-3654/94/209~-727oS04.50/0
Figure 1. Transition dipoles of two chromophores whosc centers lie at (O,O,O) and (O,O,R). Not shown are the azimuthal angles 41 and 42, measured in the standard way from the x axis. In a random solution of thtse chromophores, a sampling of all relative orientations of interest is obtained by allowing 42 - 41, sin 81, and sin 82 to have random values in the ranges 0-2r, 0-1, and 0-1, respectively. For each relative orientation among thee there exist two states corresponding to 2-14pl - p z ) and 2-1/2(p1 + p2) with oscillator strengths and energies are given by eqs 1 and 2. The cone of angle 80 is a region of exclusion used to simulate a steric constraint (see text, section 2b).
on average, the quenching pairs have trapping states that are potentially effective but ‘dark”. 2. PairSpectn a. Freely Random Orientptiolrs. For a pair of identical chromophores whose transition dipoles p1 and 8 2 are oriented in accordance with Figure 1, imbedded in a medium of dielectric constant e, two states exist with the following relative oscillator
where EOi_sthe transition energy in the absence of inte!action ( R infinite), w is a unit vector corresponding to pi and r is a unit vector in thedirectionofR (thevector separatingthechromophore centers). The distribution of oscillator strengths and energies in several random samplings of 50 OOO relative dipole orientations has been determined. All are nearly identical, and a typical one is shown in Figure 2. It is clear from this plot that states of lower energy are likely to be associated with lower oscillator strength. Q 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7271
Exciton Splitting in "Statistical Pairs"
TABLE 1: Properties of the Assumed Radial Distribution of Chlorophylls in a 0.1 M Solutlon radial segment (nm)
(deg)
(nm)
RP
Jo(&) (meV)
75 65
0.6 0.86 1.17 1.44
64.0 21.7 8.61 4.62 2.96 2.17 1.69
Bob
0.0
04.6 0.6-0.7 0.7-1.0 1.CL1.3 1.3-1.55 1.55-1.75 1.75-1.93 1.93-2.02
Figure 2. Effect of random relative angular distribution of transition dipoles on oscillator strengths (relative to the monomer) and energies (relativeto Eo; see text). The distance between the chromophoresis kept fixed at R, and JO is defined as fi1p2/CR3. The vertical coordinate is proportional to the number of cam (out of 50000 with orientations selected at random) occurring in bins at the resolution shown.
fraction of population (ab) 3.6 14.7 21.3 19.3 13.9 10.1 4.0 13.1
2.02-up
55 45 35 25 10
1.67 1.85 2.01
0
0.00
R, is a 'representative" shell radius used to calculate the fiducial energy parameter for that shell,Jo(R,), defined in the text. 00, maximum "sandwich angle". 0
*
A
Figure 4. Same as Figure 2, but referring to a distribution for which the region 8 < 80 = 75O is excluded. With this constraint, the rblative orientationof the transition moments is determined essentiallyby a single parameter, 42 - 41.
3. Effect on Absorption Spectrum -
0 -2.0
-1.0
0
1.0
2.0
Energy shift (units of 4)
Figure 3. Partial spectra contributed by pairs of chromophoresat a fixed distance but randomly oriented. In a, the strengths are represented by sticks whose heights are proportional to the sums Over the distributions of Figure 2 at fixed energies and weighted by oscillator strength. In b, the sticks are given Gaussian shapes with fwhm = 0.5 in units of the horizontal scale (on which EO= 0 and JO= 1). The asymmetry clearly reflects the oscillator strength asymmetry inherent in Figure 2.
(The case of the typical reaction center, in which dipoles are rather parallel and axial, is an exceptional geometry in the random solution; it corresponds to the "front corner" of the plot.) Thespectrum produced by this distribution when each transition contributes a 'stick spectrum" weighted by these oscillator strengthsis shown in Figure 3a. In Figure 3b a smoothed spectrum is shown. b. StericallyHindered Orientations. Because of the extended planar nature of chlorophyll, the chromophores we used to exemplify our theory, those pairs closer than some characteristic distance (taken to be about 2.0 nm in chlorophyll; see Table 1) are unable to have a completely random mutual orientation. The transition moments of chlorophylls are known to lie in the plane of the chromophore. We impose the simplestpossible constraint: the vectors pl and p2 may not lie outside a plane perpendicular to R by more than some angle 90° - 00 (see Figure l), where, with decreasing separation, t90 increases from Oo (random case) and becomes 90° at some limiting separation. In section 3 the connection between 00 and separation will be discussed and made explicit. As 00 increases, the oscillator strengtheigenvalue distribution becomes narrower and is confined to a "straight line" in the plane, limited by fJo, where JO= plp2/eR3. This behavior, illustrated for the case 00 = 75O in Figure 4, reflects the fact that the relative orientations of the momentsis in this casedetermined by essentially one parameter, their relative 4 coordinate.
The transition moment for the Qv bands of chlorophyll u has been estimated from the molar extinctioncoefficient by Sauer et a1.3 Their value8 of p2 = 31 D2 corresponds to p = 0.1 16 e-nm, implying a dipole-dipoleinteraction JO= plpz/tR3 of 19.4/t meV at R = 1.0 nm (e is the optical dielectric constant). This value of JOis characteristic of pairs having parallel moments that are perpendicularto the chromophoreseparation; W Ois the maximal splitting for "sandwich" pairs as discussed above, and it is onehalf of the maximal exciton splitting for parallel, axial transition moments. In a medium of effective dielectric constant 1.4, JOat 1.0 nm is reduced to 13.8 meV. Pairs with the most-probable spacings (1.38 nm at 0.1 M) have the characteristic interaction Jo = 5.3 meV. The spectrumof a 0.1 M chlorophyllsolution is modeled crudely as follows: we divide the pair distribution into spherical shells: 0-0.6 nm, no pairs; 0.6-2.0 nm, pairs whose spectra are distanceand orientation-dependent;above 2.0 nm, normal spectra. The radial distribution from 0.6 nm outward is furthermore assumed to be random, with characteristics as shown in Table 1. The angular distribution is assumed to be random with a simple steric constraint, also shown in Table 1, determined according to the following simple argument: consider two porphyrin rings whose centers lie on a separation vector at distance R. If their "radius" is u, and they tilt toward one another, they will bump into each other when R = 2( 1.2a)sin 00. Here a factor of 20% is allowed for externalligands. Wechoosea = 0.85 nm. This is an extremely crude approximation,but we believe it will convey the important effects of the steric constraint. With each segment of the radial distribution we now associatea spectralcontributionwhoseenergy scale is determined by the separation shown in Table 1 and whose shape depends on the steric constraint, computed again by randomly sampling 50 000 cases with the angle 00 appropriately constrained. The fraction of the population within each segment is computed by using the probability that the nearest neighbor of a chromophore is within a volume u is given by 1 exp(-w), where no is the concentration.
7272 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
I
.
,'
w ~ t hexciton effect
$'
;
x',
,/
Knox
Isotropic average
1.0 2.0 3.0 Relatie Forster transfer rate
4.0
Figure 6. Distribution of FOrster transfer ratesdue to orientationaleffects (with nosteric hindrance) at fied intermolecularseparation. The random saTplin8 is the one used for Figures 2-5. The abscissa is K~ = [PI& 3(~1-P)(&)]~, in the same notation as that of q s 1 and 2. The "isotropic average" is z/j, often used as a "man value for all rates at fixed separation. The ordinate is the number of samples (of 50 OOO) falling within bins of size 0.1 on the horizontal scale. ,
unperturbed spectrum
+
Energy --+ Figure 5. Effects of (a) exciton interaction and (b) steric constraints on a model Gaussian absorptionspectrumfor the main red band ofchlorophyll in a 0.l.Msolution. In c, the cumulativeeffect is shown. Energy increases to the right (for scale see insert). In each case the dotted line is the reference spectrum as marked, the dashed line is the corrected spectrum as marked, and the solid line is the difference between them. In Figure 5 the effectsof exciton interactionand stericconstraint are shown for an assumed typical chlorophyll Q, band. The monomer is given a Gaussian shape with fwhm = 10 nm, and the chlorophylls interact as shown in Table 1. The width and shape of the monomer spectrum are not seriously affected. Long tails develop on each side (see Figure sa), and no clear shoulders do. The asymmetry in the exciton effect is seen in the difference curve. The steric effect mainly clears out oscillator strength on the low side of the band and moves it to the high side, which means that the statistical-pair spectral effect is easily lost in the higher vibronic bands of the chlorophylls. We have deliberatelychosen parameters conservatively (slightly narrow band, low dielectric constant) to avoid deemphasizing the spectral effect. In all runs of 50 000 randomly chosen pairs of angles, the same general shape of the distributions resulted. Rather than combine them into one large sample, which would not have changed our predictions in the least, we show single-sample results so that the statistical error may be assessed simply by observing the lack of smoothness in the distributions of Figures 2 and 3a.
shouldprovidea pathway for both todecaynonradiatively. While these authors' model did not provide transitions between the pair states themselves, recent work on weakly coupled pairs has indicated rapid population transfer between coupled states that lies in the femtosecond region (e.g., Beck and Sauer,ll Zhu et a1.'2). If similar rates are found for chlorophyll a pairs then the basic question about statistical-pair quenching in polar solvents is reduced completely to verifying that (and explaining why) the lower pair state has a radiationless decay time of the order of tens of picoseconds or less. This assumes that "radiationless" means, in practice, the inability to see infrared photons at a yield of 1% or more. This work cannot and does not rule out the possibility of excimer effects as the quenching mechanism (as, for example, in the case of perylenel3), but there is at present no evidence for this in the case of ~hlorophyll.~4
5. Concluding Remarks It remains to determine corrections to our small-distanceradial distribution, as might be occasioned by physical interactions between close solvent pairs and by more highly specific steric constraints than the elementary ones we have assumed here. We have examined the effect of exciton interactions on the spectrum of hypothetical systems in which no true dimer formation occurs. If there are some dimers formed in the polar solutions and the number of such dimers is not so great as to distort the joint radial distribution, spectral blue shifts due to the statistical pairs should still dominate in the large. Any red-shifted states of true dimers should then be even easier to observe than one might have expected. An interesting corollary of this work is the determination of the distribution of Farster rates at fixed distance. The Farster rate is proportional to the square of the dipoledipole interaction appearing in eqs 1 and 2 and to many other factors that need not be mentioned heres6 Figure 6 shows how badly skewed the distribution of rates about the angular average of 2/3 really is. The standard deviation
4. Quenching of the Excitation Statistical pairs whose separation is 1.0 nm or less are likely to genuinely trap the excitation because the lower state is, on average, at least 2 J o = 27.6 meV > kT below the upper (and more likely emitting) state. The downward rate between the states is thusat least 3 times theupward rate, accordingtodetailed balance. A completely quantitative mechanism for radiationless decay in statistical pairs is apparently still elusive. One of the earliest postulates (Brody and Brodyg) involved the observation that when the pair states are split, the lower of the two has a smaller energy gap to the triplet state and hence a higher intersystem crossing rate. Gillbro and Sundstromlo showed that one could readily find vibrational states that allowed radiationless transitions to the ground state from the unallowed optical state; in a pair in which both transitions share oscillator strength this mechanism
wheref = K~ = [;1.;2-3(;1.P)(~z.P)]2 and the brackets indicate the angular average, has the not surprisingly large value of 0.7 18. Acknowledgment. Fruitful conversations with George Porter at the Centre for Photomolecular Sciences at Imperial College on several occasions were the stimulus for this work. The author is indebted to Porter, Robert Pearlstein, and Kenneth Sauer for their helpful comments on early drafts of the manuscript. The author extends thanks to J. Barber for his hospitality during an extended visit to the Imperial College Department of Biochemistry, to the Agricultural and Food Research Council (Swindon, Wilts., U.K.) for partial support of the research, to the Royal Society for a guest research fellowship, and to the Fulbright
Exciton Splitting in "Statistical Pairs" Commission for a travel grant. The work was also supported in part by the U. S. Department of Agriculture under Grant 9237306-7696.
References and Notes R. J. Chem. Phys. 1950,18, 802. (2) Beddard, G. S.; Porter, G. Nature 1976, 260, 366. (3) Sauer, K.; Lindsay-Smith, J. R.; Schultz, A. J. J. Am. Chem. Soc. 1966.88. 2681. (1) Watson, W. F.; Livington,
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 1273 (6) Fanter, T. Ann. Phys. ( k i p z i g ) 1948, 2, 55. (7) McRae, E.G.; Kasha, M.J. Chem, Phys. 19S8,28,721. ( 8 ) Sauer, K. Private communication, 1993. The entire Qv oscillator strength as measured in ref 3 has the value 31 Dz,the dipole strength reported in ref 3 is 24.65 D2,corresponding to the 0-0 vibronic band. (9) Brody, S. S.; Brody, M . Blochim. Biophys. Acta 1961, 54, 495. (IO) Sundstrom, V.; Gillbro, T. J. Chem. Phys. 1985,83, 2733. (11) Beck, W. F.; Sauer, K. J. Phys. Chem. 1992, 96, 4658. (12) Zhu, F.; Galli, C.; Hochstrasser, R. M.J. Chem. Phys. 1993, 98, 1042. (13) Ferreira, J. A,; Porter, G. J. Chem. Soc. Faraday Trans. 2 1977,73, 340. (14) Porter, G. Private communication, 1993.
