2208
J. Phys. Chem. 1988, 92, 2208-2214
Simuiatton of Optical Spectra from the Reaction Center of Rhodopseudomonas virkls Youngdo Won and Richard A. Friesner* Department of Chemistry, The University of Texas at Austin, Austin, Texas 7871 2 (Received: July 28, 1987; In Final Form: October 27, 1987)
The optical spectra of the reaction center (RC) from Rhodopseudomonasoiridis including absorption, polarized light absorption, and circular dichroism are simulated by employing an effective Hamiltonian of a vibronically coupled exciton manifold, which explicitly includes vibronic coupling to intra- and intermolecular vibrational modes. Thermally averaged Green's function matrix elements of the RC model Hamiltonian are computed by utilizing an approximate matrix continued fraction expansion, from which all single-photon optical line shapes are well-reproduced at various temperatures. The results support an analysis of photochemical hole-burning experiments (see the following paper) using an identical effective Hamiltonian.
I. Introduction The crystallization of reaction centers of Rps. viridis' and subsequent elucidation of the structure via X-ray diffractionz4 have provided a major impetus in the understanding of the primary charge separation events in photosynthesis. However, an accurate, detailed underlying model for the electron-transfer reactions and molecular interactions has yet to be produced. This state of affairs should not be surprising; there are many systems, both biological and nonbiological, for which the structure has been known for a long time but substantial mechanistic questions remain. In the absence of a reliable method for solving the electronic structure problem for large molecules in condensed phases, a direct theoretical attack of these questions is quite difficult, and one must usually resort to model construction based on experimentally derived information. Spectroscopic measurements can provide a probe of the relevant interaction parameters, although unfortunately not in the detail one would like. With the positions and orientations of the porphyrin chromophores obtained from the X-ray structure, one can use various levels of theory to calculate optical spectra, e.g., absorption, CD, and polarized absorption. The simplest approach is to use electronic exciton theory and model all other interactions via adjustable line width parameters. For these calculations, one must evaluate only the electronic site energies and exchange matrix elements explicitly; the latter can be obtained from a point dipole or point monopole approximation. Such an approach has been adopted by Fischer and co-worke r ~who , ~have ~ succeeded in reproducing a variety of spectroscopic results with reasonable agreement. However, there are a large number of phenomenological adjustable parameters in their calculations which are not easily interpretable physically. In particular, one would like to model the vibrational potential surfaces of the various excited states, which are important in controlling the electron-transfer dynamics. In this paper, we study a model which is one step beyond the simple exciton picture mentioned above. A set of strongly coupled molecular vibrational modes are explicitly included in Hamiltonian and utilized in calculating optical spectra. Intermolecular lowfrequency modes of the special pair also incorporated. Acoustic (1) Michel, H. J . Mol. Biol. 1982, 158, 567. (2) Deisenhofer, J.; Epp, 0.;Miki, K.; Huber, R.; Michel, H. J . Mol. B i d . -1984. - - -,180. 385. - - - 7
(3) Deisenhofer, J.; Epp, 0.;Miki, K.; Huber, R.; Michel, H. Nature (London) 1985, 318, 618. (4) Deisenhofer, J.; Michel, H.; Huber, R. Trends Biochem. Sci. (Pers. Ed.) 1985, IO, 243. (5) Knapp, E. W.; Scherer, P. 0.J.; Fischer, S . F. Biochim. Biophys. . . Acta 1986, 852,-295. (6) Knapp, E. W.; Fischer, S. F.; Zinth, W.; Sander, M.; Kaiser, W.; Deisenhofer, J.; Michel, H. Proc. Natl. Acad. Sci. U.S.A. 1985. 82. 8463. ( 7 ) Knapp, E. W.; Fischer, S . F. In Antennas and Reaction Centers of Photosynthetic Bacteria - Structures, Interactions and Dynamics; MichelBeyerle, M. E., Ed.; Springer Series in Chemical Physics; Springer-Verlag: Berlin, 1985; Vol. 42, pp 103-108.
0022-365418812092-2208$01.50/0
phonons and inhomogeneous broadening are treated phenomenologically, as in the simple exciton calculations. An approximate Green's function theory which has been extensively tested against converged basis set result^^-^ is used to efficiently generate the optical line shapes. Our approach has several objectives. First, we demonstrate that a vibronic coupling model is capable of reproducing several different experiments (including temperature dependence) with a relatively small number of adjustable parameters. We confirm the assignment by Breton and co-workersI0 of the shoulder on the 850-nm band as an exciton component of the special pair of BChl molecules. Second, we attempt to determine the coupling of the excited states to various vibrational modes as manifested in the optical line shapes. The values of vibronic coupling parameters obtained from monomeric spectra provide quite good agreement with the aggregate line shapes without explicit adjustment. The primary purpose of this paper, however, is to provide support for our analysis of the hole-burning experiments of Meech et al." and Boxer and c o - ~ o r k e r s . ' ~ JThe ~ present calculations show that the model used in the hole-burning simulations can also reproduce the remaining available optical data. A detailed treatment of the hole-burning data (which are more sensitive to vibronic coupling parameters than the present spectra) can lead to reasonable semiquantitative estimates of the intermolecular vibronic parameters not directly accessible from the monomer spectra. This will be carried out in the article following this one. The construction of a vibronic coupling model of various excited-state potential surfaces is an important task in constructing a realistic model for charge separation dynamics. This work represents an initial attempt in that direction. 11. Reaction Center Model The R C from Rps. viridis contains a large number of components. The major chromophores are four bacteriochlorophyll b (BChl) and two bacteriopheophytin b (BPh) molecules. According to the X-ray structural d e t e r m i n a t i ~ nthese , ~ ~ pigments are arranged in two branches, making an approximate C2 rotation symmetry. On the Cz axis two BChls are in close proximity, with an overlapping of their pyrrole ring I moieties. These BChls are called the special pair (P) and denoted as BCMp and BCLp according to whether the branch is associated with the protein subunit M or L. Next to P are the accessory BChls (B), BCMA and BCL,. Two BPhs (H) are positioned at the end of the branches, BPM and BPL. ~
~~
~~
~~
~
~~~
~~
~~
~
~~
~
~
(8) Lagos, R.; Friesner, R. A. J . Chem. Phys. 1984, 81, 5899. (9) Won, Y.;Lagos, R.; Friesner, R. J . Chem. Phys. 1986, 84, 6567. (10) Breton, J. Biochim. Biophys. Acta 1985, 810, 235. (1 1) Meech, S. R.; Hoff, A. J.; Wiersma, D. A. Chem. Phys. Lett. 1985, 121, 281. (12) Boxer, S. G.; Lockhart, D. J.; Middendorf, T. R. Chem. Phys. Lett. 1986, 123, 476. (13) Boxer, S . G.;Middendorf, T. R.; Lockhart, D. J. FEBS Lett. 1986, 200. 237.
0 1988 American Chemical Society
Reaction Center of Rhodopseudomonas uiridis
The Journal of Physical Chemistry, Vol. 92, No. 8. 1988 2209
While a variety of electronic configurations and interactions can be included in a model system of the RC, only the Qy transitions of the above six pigment molecules are considered in this work. The dominant optical properties in the 750-1050-nm region of the optical spectra are associated with these six transitions, which are coupled via excitonic and vibronic interactions. Most of the remaining electronic states and the protein environment are assumed to exhibit perturbative interactions with the Q, manifold and are treated by utilizing effective parameter values in our model Hamiltonian. Since the other neutral excited states of the chromophores (Q,, E,, and E,) are well-separated from the Qy transitions in energy, a perturbative treatment of these states is justified in an effective Hamiltonian approach. The only exception to this analysis are nearly resonant charge transfer (CT) states. We assume that these states possess negligible oscillator strength and affect the spectral line shapes only via vibronic borrowing. Such interactions have a significant effect on photochemical hole spectra but not on the inhomogeneously broadened line shapes discussed here, at least for the medium range values of the exchange interaction utilized below. (This has been confirmed by explicit computations.) Consequently, they are not included explicitly in this paper but are treated in detail elsewhere. 4 ~ 1 Each chromophore molecule has a set of normal modes which are displaced in its Qy excited state. Only linear vibronic coupling is considered here; this gives the dominant contributions to vibronic mixing and determines the energy level distributions and oscillator strengths of the aggregate excited-state manifold. From the site-selection absorption spectra of chlorophyll a and pheophytin a at 4.2 K, we extract six observable normal modes for BChl and five for BPh.16 There are many minor vibronic peaks found in the site-selection absorption data. These molecular modes of small vibronic intensities and the high-frequency modes reported in Warshel's QCFF-PI c a l c ~ l a t i o n sare ~ ~assumed to make a small contribution to the displacement of the potential surface of the excited-state manifold and are not explicitly included in the model system. In the formalism described in section 111, all the omitted low-frequency modes, the acoustic medium phonons, and quadratic and higher order couplings are incorporated in the calculation by assigning a small intrinsic line width to each vibronic peak (via a phenomenological line-broadening factor). The BChls of P are disposed with their macrocyclic ring planes approximately antiparallel with a center-to-center distance of 3 A.3 The intermolecular relative motion of the P molecules can strongly modulate the exchange interaction between excited dimer configurations. We include two low-frequency intermolecular modes of frequency 50 and 100 cm-I, which we suggest are important in explaining the anomalously large P absorption bandwidth. Photochemical hole-burning simulations show that strong coupling to these (or other) low-frequency modes is necessary to reproduce the extraordinarily broad hole spectra of the longwavelength P band.15 The use of two modes, rather than one, is not crucial in obtaining the results which follow. Two additional medium effects are considered in our RC modeling: coupling to acoustic phonons and inhomogeneous broadening. The effect of acoustic medium phonons is incorporated in the homogeneous line width by adjusting the phenomenological constant y (the imaginary part of the energy) in the Green's function calculation. Inhomogeneous broadening can be incorporated in several different ways. One can explicitly calculate inhomogeneous line shapes by convoluting homogeneous line shapes with some site energy distribution function such as a Guassian. Alternatively, inhomogeneous effects can be incorporated into the above-mentioned phenomenological line-broadening factor. Since a configurational average over a inhomogeneous distribution of parameters is computationally time-con-
-
(14) Won, Y.; Friesner, R. A. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 5511. (15) Won, Y.; Friesner, R.A. J . Phys. Chem., following paper in this issue. (16) Platenkamp, R. J.; Den Blanken, H. J.; Hoff, A. J. Chem. Phys. Lett. 1980, 76, 35. (17) Warshel, A. Proc. Natl. Acad. Sci. U.S.A. 1980, 77, 3105.
suming, we employ the phenomenological constant y as an adjustable parameter which includes all line-broadening effects. Representative test calculations (not shown below) indicate that the alternative approach provides results that are not very different from those presented here. While the explicit inclusion of vibronic coupling in the excited-state manifold requires a large number of parameters in the effective Hamiltonian, the approach described here actually has fewer arbitrary adjustable parameters than previous The simultaneous fit of various optical properties provides a test of the innate consitency of the vibronic coupling model. 111. Theory
The excited-state Hamiltonian for the model system consisting of N electronic excited states and M vibrational modes described in section I1 can be written as H =
As in elementary exciton theory, a local excitation at pigment i is characterized by a single state li) of energy e, and any two such states li) and b) are coupled via the interaction matrix element 0, is the vibrational frequency of mode n, and 0;) is the corresponding vibronic coupling parameter. The operators bt and b are respectively the usual boson creation and annihilation operators. The electronic ground state is assumed to be a single harmonic surface with unaltered frequencies. Optical spectra due to the Hamiltonian in eq 1 cannot be obtained exactly; no analytical solution exists for the multilevel multimode interaction Hamiltonian. We utilize an approximate Green's function formalism for calculating the optical line shapes8 The continued fraction method for evaluating one-exciton Green's function matrix elements has been extensively tested against converged basis set results for smaller analogous model Hamilt o n i a n ~ .Following ~ the prescription of previous work, the thermally averaged Green's function matrix G ( E ) for the effective Hamiltonian in eq 1 is given as
v,.
where Q is the appropriate partition function and m is the number of vibrational quanta in the ground electronic state for mode n. (mlD(E)lm)can be expressed as
Go(E)can be easily calculated as G0-'(E) = [ E + i y l l - t
(4)
where t is the exciton interaction matrix with elements eii = ei and eij = l$)(E) and I ' t ) ( E ) represent the finite and infinite continued fraction branches
vj.
r g ) ( E )= Go-*(E+ w,) - ( m -
1)Ee, n
Go-'(E
I
+ 2 ~ , )- ...en
where 8, denotes the vibronic coupling matrix for mode n. Once the exciton Green's function matrix elements Go(@ have been computed, various optical properties are evaluated from the knowledge of the transition dipole moments w, and the positions
2210
Won and Friesner
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988
TABLE 111: Chromophore Electronic Energies and SP Exchange Interaction Energy4
TABLE I: Qy Transition Dipole Unit Vectors of the RC Chromothores'
chromophore BPM BCMA BCMP BCLP BCLA BPL
clx
PY
clz
0.616617 -0.913473 0.643 991 -0.963 994 0.460 165 0.157 352
0.498643 0.316914 -0.716 394 0.239 023 -0.887 812 0.533 105
0.066872 0.255210 0.268 432 0.1 I6 548 0.006 159 -0.569 538
4.2 12930 12930 297 12930
100
TABLE 11: Exciton Coupling Matrix of the RC Model4 BPM
BCMA
e,
95.88
95.88 -4.85 17.95 -6.98 3.96
€2
BCMp -4.85 -16.05
-16.05 -94.00 19.60 -7.29
425.35 -104.05 19.96
tj
B C L ~ BCLA 17.95 -6.98 -94.00 19.60 425.35 -104.05 €4 2.79 €5 2.79 -6.57 123.11
BPL 3.96 -7.29 19.96 -6.57 123.11 t6
Electronic exchange interaction energies are evaluated by using the dipole approximation. Energies are given in wavenumbers. The relevant formulas are
of the chromophore given below: (i) Absorption intensity 1
[ ( E ) = - -&iij
Im Gij(E)
7r ij
(7)
(ii) Polarized light absorption. For the polarization of the electric field specified relative to the crystal z axis
1
1
Z,(E) = --c-[jiix.jijx + jii,..jijy] Im G i j ( E ) 7r jj 2
11000
11055 11240
11000 11055 11240
12240 12240 12240
12570 872 12570 807 12570 677
Energies are given in wavenumbers.
Amplitudes are scaled to reproduce the experimental intensities as described in the text.
BPM BCMA BCMp BCLp BCL, BPL
12240 12240 12240
(9)
(iii) Circular dichroismZo
where % ( E ) is the rotational strength.
IV. Model Parameters The pigment coordinates are obtained from the X-ray crystallographic structure of Deisenhofer and co-workers.2 The monomer Qy transition dipoles are taken to be along the axis running from the N atom of ring I to the N atom of ring I11 of the monomer.21 We employ dipole strengths of 30 and 19 D2 for accessory BChl and BPh, respectively.22 The P BChl Qydipole strength is set to 25.5 D2, which gives an excellent reproduction of the P and B band intensity ratio in the 4.2 K absorption spectrum.23 The norms of the transition moments are scaled to that of the BCA Qy transition moment, which is set to be a unit vector. Table I lists the six chromophore Qy transition dipole moments included in our model. Much larger dipole strengths have been reported in recent experiment^.^^ Knapp and Fischer have investigated the sensitivity (18) Tinoco, I., Jr. Radiat. Res. 1963, 20, 133. (19) Pearlstein, R. M. In Photosynthesis; Govindjee, Ed.; Academic: New York, 1982; Vol. 1, pp 293-330. (20) Heminger, R. P. J . Chem. Phys. 1978, 68, 1722. (21) Gouterman, M. J . Mol. Spectrosc. 1961, 6, 138. (22) Shuvalov, V. A,; Asadov, A. A. Biochim. Biophys. Acta 1979, 545, 296. (23) Vermeglio, A.; Paillotin, G. Biochim. Biophys. Acta 1982, 681, 32. (24) Parson, W. W.; Scherz, A,; Warhsel, A. In Antennas and Reactions Centers of Photosynthetic Bacteria-Structures, Interactions and Dynamics; Michel-Beyerle, M. E., Ed.; Springer Series in Chemical Physics; SpringerVerlag: Berlin, 1985; Vol. 42, pp 122-130.
of optical properties to an increase of parameters to 45 and 30 D2 for BChl and BPh, respectively.' The main difference between the calculated spectra with the larger and the smaller dipole strengths is the absolute value of the absorption intensity and rotational strength. Since we calculate normalized optical line shapes instead of absolute intensities, the actual magnitudes of the parameters do not make much difference in the resulting spectra. Note that the two sets of dipole strengths have the same ratio for the BChl and BPh values. The exciton coupling matrix elements V,, in Table I1 are calculated with the dipole a p p r o x i m a t i ~ n , ~using ~ ~ ' ~the dipole moments obtained from the crystal coordinates. Although the dipole approximation is not a highly accurate method, it gives reasonable estimates of all of the exciton interaction energies except VMp,Lp, the P excited-state exchange interaction energy (J.). The extended dipole method gives a similar exciton matrix.6 The diagonal energies in the exciton matrix are empirically obtained from the optical spectra by reproducing peak positions. As explained in section 11, we assume a perturbative treatment of all the states outside of the model Hamiltonian and do not explicitly investigate the chromophore excited-state shifts. The P site energy is adjusted together with the interaction energy, J*, to obtain the correct peak positions in the absorption and CD spectra for the P+ and P- spectral components (see the Results section for details). For example, the shoulder at 850 nm in the 4.2 K absorption spectrum is quite sensitive to these parameters. Table I11 lists all the diagonal energy parameters for different temperatures together with J. values. Note that different J I and SP site energy values are used for each temperature; this will be discussed subsequently. No explicit information on the vibrational modes of the BChl and BPh molecules is currently available. We adopt the data from the site-selection absorption spectra of chlorophyll a and pheophytin a in an n-octane matrix a t 4.2 K.16 The distinctive fundamentals in the Q band of the site-selection spectra are at 97.0, 227.0, 237.0, 340.0, and 662.0 cm-I for pheophytin a and 140.0, 260.0, 348.0, 388.0, 479.0, and 569.0 cm-' for chlorophyll a. Although there are structural differences between RC chromophores and the chlorophyll a derivatives, each set of obtained vibrational frequencies should be a reasonable qualitative representation of the actual chromophore excited-state manifold. The vibronic coupling strength 0 for each mode is estimated from the intensity ratio of the corresponding vibronic peak and the zero phonon line via the relation
where w is the frequency of the mode and Z/Zo is the intensity ratio. Since the intensity of the zero-phonon line is not accurately measured in the site-selection spectra,I6 the intensity ratios cannot be used directly to evaluate vibronic coupling strengths. Therefore, we first estimate the coupling parameters for the 140-cm-' mode in BChl and the 97-cm-I mode in BPh from the intensity of the 0-1 vibronic peak of the 148-cm-' mode in the supersonic jet spectrum of free-base porphin reported by Even and JortnerZ5and then scale the rest of the coupling constants to these values. The corresponding vibronic coupling parameters obtained at 44.8, 77.4, 72.2, 136.3, and 238.7 cm-I for BPh and 64.6, 94.3, 149.4, 132.6, 116.8, and 116.2 cm-' for BChl. The intramolecular vibronic coupling matrices are diagonal in the electronic-state representation of the Hamiltonian in eq 1, Le., e$')= 6(")6,, if the mode n belongs ( 2 5 ) Even, U.; Magen, J.; Jortner,
J. Chem. Phys Lett.
1982, 88, 131
Reaction Center of Rhodopseudomonas viridis
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 1.0
to the chromophore associated with the state i and zero otherwise. One vibronic contribution likely to be important is the intermolecular relative motion of the P molecules, which strongly modulates the exchange interaction. Two intermolecular vibrational modes of frequency 50 and 100 cm-’ are utilized, and the corresponding coupling constants (estimated from our analysis of photochemical hole-burning experiment^'^) are set to 50 and 100 cm-I, respectively. The intermolecular vibronic coupling is manifested as by off-diagonal coupling matrix elements and
Jt~1207
2211
1
043).
The phenomenological parameter y is empirically obtained by fitting the overall spectral line shapes. y generally increases with increasing temperature due to the enhanced importance of interaction with medium phonons. At low temperature, a single value of y for the entire spectrum yields excellent agreement with experiment. At higher temperatures, two values of y (one in the BPh/BCA region and one in the P- region) are necessary, with the intermediate region connected by linear interpolation. Note that this is a smaller number of adjustable line width parameters than that employed by Fischer and c o - w o r k e r ~ ~in- ~their simple exciton calculations, where a large number of adjustable asymmetric Gaussian functions were demanded by the model.
V. Results and Discussion The model described in sections I1 and IV contains 6 excited electronic levels and 36 vibrational modes. This represents only a moderately demanding computational problem for the formalism presented in section 111, whereas direct matrix diagonalization would clearly be impossible. Consequently, we have been able to carry out an extensive study of the parameter space in reasonable computation times. For the broad featureless line shapes obtained here, the Green’s function method has invariably been accurate to within a few percent in all of our test calculations. The Green’s function matrices are computed with the University of Texas Center for High Performance Computing Cray X-MP/24 supercomputer. The IMSL package is used for the complex matrix inversion and eigenvalue routines involved. The continued fraction program was originally developed on CDC 170/750, and the program size is small enough to be handeled in a smaller computer. Thirty continued fractions are employed in the infinite continued fraction branch in eq 7); usually 15 continued fractions are enough to get converged solutions. The experimental data are collected from various literature articles and digitized on a HP85A microcomputer equipped with a HP7225A plotter. Each spectrum is normalized to facilitate comparison with computed optical properties; we emphasize agreement of the relative spectral intensities, rather than absolute intensities. A . The Exciton Components of the Special Pair. In our vibronically coupled exciton model, the strongly interacting special pair has two exciton components. The long-wavelength band around 990 nm in the 4.2 K absorption spectrum is assigned to the low-energy P.. component. The position of the higher energy component, P+, has been the subject of extensive discuss i o n ~ Shuvalov . ~ ~ and ~ ~Asadov’s ~ ~ photochroism ~ ~ ~ ~ and CD study assigns the 830-nm band to BCLA and the 850-nm spectral feature to BCMA.22A recently reported linear dichroism spectrumlo shows a distinctive 850-nm negative band, which is bleached upon the oxidation of the special pair; this suggests an assignment of the 850-nm band to the P+ state, in agreement with Vermeglio et a1.k assignment based on low-temperature a b s o r p t i ~ n .The ~ ~ present work also focuses on the assignment of the 850-nm shoulder. As in ref 7, the exciton coupling between the P states is varied with their site energy, keeping the center of the P- state at the correct position. Aside from the 850-nm shoulder at 4.2 K, the absorption spectra are not much affected by the P+ position, Le., by J. variation. On the other hand, the C D spectra turn out to be very sensitive to the exciton coupling strength. Figure 1 shows the effect of variation of J. and 9, the P site energy, on the 100 K CD spectra. The P+ band has to be at around 850 nm, slightly (26) Parson, W. W. Annu. Reu. Biophys. Bioeng. 1982, 11, 5 7 .
-0 5 1
,v, 800
900 Wavelength ( n m i
1000
Figure 1. CD spectra as a function of the SP interaction energy J.. The P-exciton component is kept at the center of the long-wavelength band, and single line-broadening factor y 100 cm-I is used. The experimental data of Shuvalov and Asadov22are also shown for comparison.
Wavelength [ nm)
Figure 2. Comparison of the 4.2 K absorption spectra calculated from eq 7 (solid line) with the experimental result of Vermeglio et (dotted
line). lower than the BCA energies, in order to reproduce the experimental C D measurement.22 N o other J , and degenerate P site energy combination can reproduce the CD results. The roomtemperature CD spectra calculations lead to a similar conclusion (data not shown). For the case of 100 K CD, the 849-nm negative band is assigned to the P+ state with J. = 807 cm-l and c3 = 11 055 cm-I. The P site energies and J , values at other temperatures are obtained by reproducing the 850-nm shoulder in the low-temperature absorption spectra or the 846-nm negative band in the room-temperature CD spectrum. Table I11 displays the chromophore site energies and J. for the three temperatures of interest. Diagonalization of the resultant 6 X 6 exciton matrix shows that the P+ component has 44.5% BCMp, 45% BCLp, 5.4% BCMA, 4.4% BCLA, and negigible BPh character. Due to the exciton interaction with the accessory BChls, the intensity ratio of the two exciton components (P+/P-) is increased to 0.24 from 0.14, the ratio based on an exciton dimer model of the P BChls in which the transition dipole moments are disposed with an angle of 139’. B. Absorption Spectra. ( 1 ) 4.2 K Absorption Spectrum. The absorption spectrum calculated from the Green’s function matrix for the Hamiltonian in eq 1 is compared to the experimental data from ref 23, which is measured for an R C suspension at 4.2 K (Figure 2). The calculation correctly reproduces most of the spectral features in the experimentally measured absorption spectrum. The 790-nm band is attributed to BPM, and the 810-nm band is assigned to BPL. We assign the 830-nm band to both accessory BChl molecules and the 850-nm shoulder to the P+ component. The P- component contributes to the long-wavelength band around 990 nm. Excellent agreement for B and H band line widths are obtained by setting y to 55 cm-I. The asymmetric bandshapes are due to underlying vibronic progressions. The overemphasized vibronic contributions in the high-energy side of the P- band possibly indicate overestimation of intramolecular vibronic coupling strengths. The broad 990-nm bandwidth of -400 cm-’ is due
2212
Won and Friesner
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988
,
.
._.’ ’...._...
0
Wavelenqth
(
800
nm i
900
..._._ 1000
Wavelength ( n m )
Figure 3. Comparison of the 100 K absorption spectra calculated from eq 7 (solid line) with the experimental result of Shuvalov and Asadov2* (dotted line).
Figure 4. Comparison of the 297 K absorption spectra calculated from eq 7 (solid line) with the experimental result of Phillipson and Sauer2* (dotted line).
to strong vibronic interactions with the low-frequency intermolecular mode supplementing the phenomenological line broadening of 55 cm-’. One feature of the experimental results in Figure 2 which are nor reproduced by our simulation is the weak shoulder on the red side of the 990-nm band. We have been unable to obtain this shoulder by including interacting C T states (with zero oscillator strength) in the model. Assuming that the shoulder is not an experimental artifact, there are two possible explanations. One is that the shoulder represents a distinct chromophore/protein configuration which is sometimes frozen in at low temperature, depending upon the experimental method of preparation. In other words, our calculation is actually for a “site-selected” spectrum and the shoulder arises from a second sometimes accessible site is ’also of the SP. This explanation, suggested recently by H ~ f f , ~ consistent with the fluctuating observation of the feature experimentally. We could easily simulate the overall spectrum by adding together two spectra with appropriate amplitudes and different SP site energies; this would, however, be an uninformative exercise in parameter adjustment. An alternative possibly is that the shoulder could be reproduced by specific values of the vibronic coupling constants. There is some indication that this is feasible, and it is currently under investigation. ( 2 ) 100 K Absorption Spectrum. We calculate the thermally averaged Green’s function matrix elements at 100 K from eq 2, employing y of 100 cm-’ for B and H bands (880 nm). For the region 836-880 nm y is lineary interpolated from 100 to 200 cm-l. In contrast to the 4.2 K absorption spectrum, a larger y for the P- band region is necessary to reproduce the experimental results. Several explanations for this are possible, including enhanced interactions with acoustic phonons, thermally induced inhomogeneity in the exchange interaction J., and a need for additional explicit lowfrequency modes in the model. The overall line shapes from the Green’s function matrix elements agree well with the experimental spectrum obtained from ref 22 (Figure 3). The humps in the high-energy side of the 830-nm band are correctly obtained, and the 850-nm shoulder is merged into the main 830-nm peak. The spectral features on the high-energy side of 830 nm could be more accurately generated by fine tuning of B and H site energies from the values used in the 4.2 K calculation. For the present calculations, we retained the site energies determined from the resolved low-temperature simulations. ( 3 ) 297 K Absorption Spectrum. The absorption spectrum at 297 K was calculated by using the same scheme for line broadening as at 100 K except that y = 250 cm-’ for y > 880 nm and y = 110 cm-’ for X
