J. Phys. Chem. 1996, 100, 1445-1448
1445
Analysis of the Intervalence Band in the Oxidized Photosynthetic Bacterial Reaction Center Zbigniew Gasyna and Paul N. Schatz* Chemistry Department, UniVersity of Virginia, CharlottesVille, Virginia 22901 ReceiVed: NoVember 10, 1995X
Electronic and vibronic coupling parameters of the oxidized photosynthetic bacterial reaction center (P+) of Rhodobacter sphaeroides are estimated by analyzing its ≈2600 cm-1 intervalence band which was recently measured by Breton et al. A two-mode model is used which pictures P+ as an unsymmetrical mixed-valence dimer. One mode (q1) has even and the other mode (q2) has odd interchange symmetry with respective frequencies ν1 ≈ 154 cm-1 and ν2 ≈ 1630 cm-1. The vibronic coupling is strong in the former and weak in the latter and in sum is larger than the electronic coupling. The asymmetry parameter which appears in the model is estimated from the ESR data of Rautter et al. The energy difference between the minima of the two coupled potential surfaces is ≈850 cm-1, whereas the Franck-Condon transition energy is ≈2100 cm-1. An analysis of the potential surfaces and probability distribution in configuration space show that P+ is a strongly delocalized mixed-valence system.
The special pair (P) is the primary electron donor in the photosynthetic reaction center. It consists of a pair of bacteriochlorophyll molecules (PL and PM) which are nearly parallel, are separated by ≈3.3 Å and are offset so that there is an overlap of only one pyrrole ring of each molecule.1 When P is excited, it rapidly transfers an electron to a bacteriophenophytin (BPh) and is converted into P+, a dimeric mixed-valence species. Any information that can be adduced about P+ is clearly of great interest because it may give further insight into the electronic structure of P and the electron-transfer mechanism. We here estimate (Table 1) electronic and vibronic coupling parameters of P+ from a detailed fit of its intervalence band2 (IVB) around 2600 cm-1 using a formalism which was recently developed and applied to the LuPc2 molecule.3 We find that the system is delocalized with a subunit (PL, PM) electronic coupling ∆ (i.e., splitting into bonding and antibonding MO) of ≈1010 cm-1. A reasonable fit (Figure 1) of both the broad and resolved features of the IVB is obtained using just two vibrational modes, namely, a totally symmetric mode (q1) with frequency ν1 ≈ 154 cm-1, and an antisymmetric combination (q2) of a totally symmetric breathing coordinate on each of the two subunits, with frequency ν2 ≈ 1630 cm-1. There is strong vibronic coupling in q1, but the vibronic coupling in q2 is very weak precluding any possibility of valence trapping. We show (Figure 1a) the experimental2 midinfrared band of P+ obtained at 260 K by taking the light-minus-dark difference spectrum of the Rhodobacter sphaeroides reaction center. Our “best-fit” stick spectrum (stick height directly proportional to absorbance) and its Gaussian convolution are shown in Figure 1b (dashed curve) where we have used ad hoc Gaussian line widths of 150 and 45 cm-1 above and below ν2 respectively. We have for the moment ignored the features labeled 1 and 2 in Figure 1assee later. Figure 1c shows a comparison of the experimental (solid curve) and calculated (dashed curve) spectra at a temperature of 260 K. The calculated spectrum is identical to Figure 1b except that the sticks have been omitted for clarity. The transition dipole has been chosen to scale the calculated absorbance to experiment, and a sufficiently large vibronic basis has been used (n ) 30)3 to assure complete convergence of the calculation. P+ is an example of an unsymmetrical mixed-valence dimer since it is well-known2,4 that the PL and PM “monomers” have X
Abstract published in AdVance ACS Abstracts, January 15, 1996.
0022-3654/96/20100-1445$12.00/0
TABLE 1: Simulation Parametersa parameter
valuec
valued
∆/cm-1 λ(2) 1 ≡ λ1 ν1/cm-1 ≡ λ2 λ(1,2) 2 ν2/cm-1 R δsharpb/cm-1 δbroadb/cm-1
1010 ( 10 3.05 ( 0.01 154 ( 1 0.0919 ( 0.01 1630 ( 10 0.157 ( 0.01 45 ( 5 150 ( 5
1070 ( 5 3.05 ( 0.01 156 ( 1 0.0884 ( 0.01 1780 ( 10 0.157 ( 0.01 45 ( 5 145 ( 5
a
Notation as in previous work.3 b Gaussian line width in convolution 2 2 using f(e) ) e-(e-ei) /δ /δxπ. c Excluding lines 1 and 2ssee text. d Fitting line 1 as a Fano resonancessee text.
different energies and different charge distributions.5 In our formalism,3 the asymmetry is measured by the parameter R. Our estimated value (R ) (0.157) is obtained from the esr data of Rautter et al.5 in the following way. These authors report that the electron spin density of P+ is 65:35 in favor of the PL half of the dimer. Our ground- and excited-state MO wave functions are respectively3
φ2 ) |γγ*2〉
φ1 ) |γ2γ*〉,
(1)
where the bonding (γ) and antibonding (γ*) MOs are given by
|γ〉 ) c1|b1〉 + c2|a2〉;
|γ*〉 ) -c2|b1〉 + c1|a2〉
c12 + c22 ) 1;
|c1| . |c2|
|b1〉 ) (1/x2)(a1(M) + a1(L)) |a2〉 ) (1/x2)(a1(M) - a1(L))
(2)
where a1(M) and a1(L) are the respective PM and PL monomer HOMOs. Clearly the spin density in P+ arises from the unpaired electron which is in the antibonding (γ*) MO if electronic coupling is larger than vibronic coupling and is in the bonding MO (γ) if vice versa. Using eq 2 and grouping terms © 1996 American Chemical Society
1446 J. Phys. Chem., Vol. 100, No. 5, 1996
Letters
Figure 1. Intervalence band (IVB) of Rhodobacter sphaeroides: (a) Experimental spectrum2 at 260 K. (b) Calculated spectrum using the parameters in the second column of Table 1. The dashed curve is at 260 K, and the solid curve is at 100 K. Sticks are shown only for 260 K. Each stick is convoluted with a Gaussian whose area is directly proportional to stick height; the latter in turn is directly proportional to absorbance. Gaussian line widths above and below 1630 cm-1 are 150 and 45 cm-1, respectively. (c) Comparison of measured (a) and calculated (b) spectra at 260 K. (d) Comparison of measured and calculated spectra at 260 K using the parameters in the third column of Table 1. Line 1 has been convoluted with a Fano line shape. Gaussian line widths above and below 1780 cm-1 are 145 and 45 cm-1, respectively.
||x〉|2 ) a12(M)(1/2 - c1c2) + a12(L) (1/2 ( c1c2) ( a1(L) a1(M)(c22 - c12) (3) noting that all upper or all lower signs apply depending on whether x is γ* or γ, respectively. We drop the last term in eq 3 since it is relatively small and makes a symmetrical contribution. We then equate the ratio of the coefficients of a12(L) and a12(M) to the ESR spin density ratio:
1 ( 2c1c2 ) 65/35 1 - 2c1c2
(4)
from which we obtain c1 ) (0.988, c2 ) (0.152 or c1 ) (0.988, and c2 ) -0.152 (recalling that3 |c1| . |c2|). From eqs 23 and 24 of our earlier paper3 it follows that R ) c1c2/(c22 - c12), and substituting the values above, we obtain R ) (0.157. The theoretical spectrum is invariant to the sign of R. Since there are many theoretical parameters, we briefly discuss our fitting method and the sensitivity of the parameters to the final “best fit”. Inspecting the experimental intervalence band (IVB) spectrum (Figure 1a), we note an evident division into a broad slightly structured higher energy (>1700 cm-1) band followed by a considerably more sharply structured region to the red. Let us suppose we start the simulation at low temperature (4.2 K) and set both λ2 (the q2 mode vibronic coupling parameter) and R to zero. Then to get some resemblance to experiment, we must choose ∆ (the electronic coupling parameter) around 1000 cm-1 and ν1 around 150 cm-1 with a large vibronic coupling parameter (λ1 ≈ 3). In that case, we get a single totally symmetric progression in ν1 peaking near the Franck-Condon (FC) transition energy (≈2100 cm-1). This progression roughly aligns with the observed structure on the red side of the actual IVB. Next we turn λ2 up from zero, and as we do so, a gap begins to appear in the vicinity of ν2 (which
we choose to be about 1600 cm-1). This gap becomes steadily more pronounced with increasing λ2, and a subsidiary maximum grows in on the high-energy side which we subsequently identify with the experimentally observed shoulder at ≈2100 cm-1. In addition, as λ2 is increased, additional transitions grow in, mainly in the broad higher energy region. Thus the simulation begins to show considerable resemblance to the observed spectrum. If we now turn R up from zero, a very distinct gap develops between about 1600 and 1800 cm-1, and many additional transitions appear in the broad region; in general the resemblance to experiment improves significantly. We next turn the temperature up to 260 K (the experimental value), and many smaller hot bands grow in at the expense of the original “cold” bands. Finally, we make relatively small further adjustments in various of the parameters to obtain the “best fit” shown in Figure 1b,c. The parameters are summarized in the middle column of Table 1 along with estimated uncertainties which are inevitably somewhat subjective. They reflect limits beyond which the fit clearly deteriorates based on a very large number of trials. The experimental features labeled 1 and 2 (Figure 1a) have a derivative-like dispersion form which requires further consideration. We note that this type of feature also occurs2,6 at about the same frequency in the light-minus-dark difference spectra of closely related dimer species where no broad intervalence absorption at all is seen in the 2600 cm-1 region and which also lack the three distinct absorptions further to the red (≈1285, 1470, 1550 cm-1) which invariably accompany the broad absorption. This suggests that features 1 and 2 are not associated with the P+ intervalence band, and we have made this assumption in our fits in Figure 1b,c and in the middle column of Table 1. On the other hand, it is possible to obtain a derivative-like dispersion form when a sharp line is overlapped by a broad continuum-like absorption from another transition. Such features are termed Fano (or antiresonance) lines.7 Indeed, such resonances have previously been observed in strongly delocalized mixed-valence linear chain systems, first
Letters by Rice and co-workers8,9 in (TCNQ)2- and later by Collman et al.10 in mixed-valence OsOEP polymers. We therefore present an alternate fit (Figure 1d) in which it is assumed that line 1 is a Fano resonance. The resulting parameters are summarized in the right column of Table 1. In this fit, the calculated line corresponding to feature 1 is convoluted with a Fano rather than a Gaussian line shape; Gaussians are still used for all other lines. We have not attempted to calculate the Fano line-shape parameters7 but have simply chosen them to simulate the observed feature. We have however ascertained from an analysis of our eigenfunctions that line 1 satisfies the necessary condition for a Fano resonance with lines from the broader spectral region starting at ≈2000 cm-1. (We have not attempted to simulate line 2 which is considerably weaker.) Comparing the second and third columns in Table 1, we note that most of the parameters change relatively little, the largest change being for ν2 which increases by about 10%. We will use the parameters in the second column in our subsequent discussion, but our qualitative conclusions apply equally for either choice, and even the numerical values quoted in the discussion below are little changed by the alternate choice. Our rationale in Figure 1b,c (and 1d) for using a larger ad hoc Gaussian width above ν2 ≈ 1630 cm-1 (1780 cm-1) is the supposition that the resolved lines to the red are dominated by a single effective progression in ν1 whereas a variety of modes couple to ν2 (which is also an effective mode) and share the intensity in the broad spectral region. The overall fit of the experimental spectrum is quite reasonable including the fact that the shoulder at ≈2100 cm-1 is clearly suggested in our simulations which also predict that members of the ν1 progression to the red of ≈1200 cm-1 should be observed if the experimental measurements can be extended to lower energy. We note that the electronic coupling parameter (∆) is at least a factor of 2 smaller than the value suggested by Breton et al.2 The reason is that their treatment is purely electronic whereas we take explicit account of the Vibronic coupling. Indeed, when we examine our eigenfunctions, we find that the electronic contribution to the ground vibronic state of the system is essentially pure φ2 (see eq 1). This means that the vibronic coupling is significantly larger than the electronic coupling. The energy difference between the minima in the two potential surfaces is 849 cm-1, and the FranckCondon transition energy from the minimum in the lower surface to its vertical intersection with the upper surface is 2118 cm-1ssee Figures 2 and 3. Our conclusion that P+ is a delocalized system follows from Figure 2 where we plot the potential energy surfaces corresponding to the parameters in the second column of Table 1. The crucial question is whether there is valence trapping in the system, that is, is there more than one potential minimum such that at one minimum the system is predominantly PL+PM and at the other minimum (at a different energy since the system is unsymmetrical) the system is predominantly PLPM+? The coordinate that measures this is q2, because if one imagines transferring the hole adiabatically (i.e., slowly while remaining on the ground surface) say from PL+ to PM, q2 will vary continuously to accommodate this change, that is, PL breathes out and PM breathes in (or vice versa) as we go from PL+PM to PLPM+. We emphasize that q1 is insensitive to such a change since it is totally symmetric. Inspection of Figures 2 and 3 suggests immediately that there is no valence trapping since a cut (Figure 3a) in the potential energy surface at a fixed value of q1 in the region of the potential minimum (or anywhere else for that matter) yields a parabolic-like variation of potential energy with q2, i.e., there is clearly only a single minimum. A
J. Phys. Chem., Vol. 100, No. 5, 1996 1447
Figure 2. Calculated potential energy surfaces using the parameters in the second column of Table 1. Contour plots of the upper and lower surfaces are shown respectively at the top and bottom. The two minima in the lower surface have energies, -527 and -1.45 cm-1 at q1 ) -4.39, q2 ) -0.020, and q1 ) -0.015, q2 ) -0.044, respectively. The single minimum on the upper surface has energy 323 cm-1 at q1 ) -1.49, q2 ) 0.13.
Figure 3. Cuts in the potential surfaces of Figure 2 passing through the lower surface minimum. The Franck-Condon energy (vertical arrow) is 2118 cm-1.
quantitative expression of the delocalized nature of P+ is provided by the plot of P(q2) vs q2 shown in Figure 3a. P(q2) is obtained11 by squaring the vibronic eigenfunction of each thermally occupied state, integrating over q1 and all electronic coordinates, multiplying by the appropriate Boltzmann population factor, and summing. P(q2) is thus the probability distribution in q2 space, and it shows clearly that there is only one distinct maximum, near q2 ) 0. We thus conclude that in P+, q2 assumes an equilibrium value intermediate between the extremes which would characterize PL+PM and PLPM+, and the electronic charge distribution does likewise. This is the description of a delocalized system. The fact that the hole distribution is weighted roughly 2:1 in favor of the PM center
1448 J. Phys. Chem., Vol. 100, No. 5, 1996 in no way contradicts this conclusion. It is simply a consequence of the unsymmetrical (R * 0) nature of this delocalized system. We note that the single-potential minimum on the upper surface (Figure 2) is also close to q2 ) 0, so one cannot argue, for example, that P+ is primarily PL+PM at the lower surface minimum and primarily PLPM+ at the upper surface minimum. On the other hand, it is immediately apparent from Figures 2 and 3 that there are indeed two minima in the lower potential surface. These two minima differ in energy by ≈525 cm-1, are separated by a minimum barrier of ≈42 cm-1 and occur at widely differing values of q1 (but almost the same value of q2). Thus our model suggests that there is a metastable configuration of P+ which lies beyond the equilibrium value along the q1 coordinate. Perhaps such a shallow trap could play a role in the back electron transfer to re-form P. We note that in subsequent work, Nabedryk et al.6 report a P+ spectrum of Rhodobacter sphaeroides at 100 K. However, this spectrum has a maximum absorbance more than an order of magnitude smaller than the 260 K spectrum which we have fit. The main difference between the experimental spectra at the two temperatures is that the 2100 cm-1 shoulder region has considerably lower absorption relative to the 2600 cm-1 peak region in the 100 K spectrum. We cannot reproduce this behavior, as is seen in our calculated 100 K spectrum (solid curve, Figure 1b), where the main effect is a sharpening in the region of broad absorption and some overall increase in integrated intensity when the temperature is lowered from 260 to 100 K. Finally, we stress that our treatment is based on a much simplified model employing only two vibrational coordinates. These should be regarded as effective modes since it seems highly likely that many modes contribute to the actual spectrum. Resonance Raman (RR) studies12-15 on reaction centers have revealed a rich structure in both the low (
