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Electronic Spectral Prope'rties of Chlorophyll Aggregates
Quantum Mechanical Formalism for Computation of the Electronic Spectral Properties of Chlorophyll Aggregates' Lester L. Shlpman, James R. Norrls, and Joseph J. Katz* Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received October 23, 1975) Publication costs assisted by Argonne National Laboratory
An improved exciton formalism is presented that permits more accurate evaluation of environmental (solvent and/or neighboring chlorophyll molecules) effects on the electronic transition energies of chlorophyll aggregates. This formalism, based on well-established exciton theory for molecular crystals, clearly shows that the environmental shifts and transition density-transition density shifts of the electronic transition energies of molecular aggregates are not additive (except in special cases) as has been generally assumed previously. Explicit equations have been derived for (a) the lower excited singlet state wave functions of chlorophyll dimer, and (b) transition energies and transition dipoles for the ground to lower excited singlet state transitions in chlorophyll dimer. The formalism is applied to a discussion of the nature of antenna chlorophyll and the electronic relationship between antenna chlorophyll and photoreaction trap chlorophyll. Much of the formalism in this paper is not restricted to chlorophyll aggregates and is applicable in general to aggregates containing a mixture of light-absorbing (chromophore) molecules and solvent molecules.
I. Introduction Frenke12-4 was the first to introduce the concept of excitation waves (excitons) to explain the electromagnetic spectra of crystals. Later, the concept of excitons was developed in detail for molecular crystals by D a ~ y d o v Much .~ of the f o r m a l i ~ m ~for - ~excitons in crystals can also be applied directly to noncrystalline (Le., nonperiodic) molecular aggregates. Our interest in exciton theory arose from our concern with the primary events of light collection and conversion in photosynthesis. These primary events are generally believed to occur within a photosynthetic unitg-ll in which several hundred or more associated chlorophyll (Chl) molecules act cooperatively to absorb, transport, and convert light energy. In the photosynthetic unit light energy is absorbed by antenna Chl (which consists of the great majority of the Chl molecules present) where it is converted into electronic excitation energy. This excitation energy is then transferred very efficiently to a few special12 chlorophyll molecules in a photoreaction center13 where oxidizing and reducing capacity is generated. Exciton theory has been used14-20 to calculate the electronic spectral properties of Chl aggregates. Unfortunately, the formulations1a,21r22applied in these previous studies14-20can be shown to be inapplicable either because the effects of the environment have been totally neglected, or because identical environmental transition energy shifts have been assumed for all chromophore molecules. In the present paper we present an improved exciton formalism for Chl aggregates that explicitly includes the effects of the environment. The formalism is not restricted to Chl aggregates, however, and is applicable in general to aggregates containing a mixture of light-absorbing (chromophore) molecules and solvent molecules. We are well aware that exciton states have been treated correctly for the case of molecular crystal^,^-^ and thus the basic physics of our formalism is not novel. However, previous applications of exciton theory to molecular aggregates have been subjected to simplifying assumptions regarding environmental effects, and these may seriously affect the validity of conclusions
drawn about the electronic structure of molecular aggregates. We believe that the formalism presented here is more appropriate than earlier ones for many problems of particular interest to chemists.
11. Chlorophyll Aggregates Nuclear magnetic r e ~ o n a n c e ~and ~ - ~infrared ~ spectroscopic23~24~27~2a studies have shown that the central magnesium atom of Chl is coordinatively unsaturated when assigned a coordination number of 4, as is the case in the structural formula as usually written (Figure 1). Thus, there is a strong tendency for one or both of the Mg axial positions to contain an electron donor The donor ligand can be a typical Lewis base (e.g., a polar solvent such as acetone, pyridine, or diethyl ether that acts as a monofunctional donor), in which case monomeric Chl species, ChlSL1 and C h l ~ L 2are , ~ ~formed. If the donor is a bifunctional ligand (e.g., dioxane, pyrazine), then cross-linked Chl species, which may be of colloidal dimensions, can be f ~ r m e d .Water ~ ~ ? is ~ ~a particularly important bifunctional ligand, for the oxygen atom can be coordinated to the Mg atom of one Chl molecule, and the two hydrogen atoms are then available for hydrogen bonding to one (or possibly two) other Chl m o l e ~ u l e s . ~ In~ the t ~ ~absence of other donors a Chl molecule can act as donor (via its keto C=O function at position 9 in ring V, Figure 1)to the Mg atom of another Chl.35 The keto C=O--Mg self-aggregation can lead to the formation of dimers or depending upon conditions. Spectroscopic comparisons of in vivo Chl with various Chl species that can be prepared in vitro have led to the tentative identification of hydrated Chl dimer with photoreaction Ch1,37-43and the Chl oligomer (Chl), with antenna Chl.44 111. Electronic Spectral Properties of Molecular Aggregates In this section we present a formalism for calculating the electronic spectral properties of chromophore-containing molecular aggregates of arbitrary structure and composiThe Journal of Physical Chemistry, Voi. 80, No. 8, 1976
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L. L. Shipman, J. R. Norris, and J. J. Katz H
individual molecules with one or more of the molecules in excited states. For example
I
\ =O
O\CH3
0 R' phytyl =
is the (zeroth-order) wave function that represents the nonstationary excited state with the ith molecule excited to its first excited state with all other molecules remaining in their ground states. Additional &'s can be constructed that correspond (a) to just one molecule excited to a singlet state other than the first excited singlet state, (b) to two or more molecules excited, or (c) to charge transfer between molecules. Quasi-stationary electronic states are formed by taking linear combinations of the 4i's;the linear coefficients are determined by minimizing the total energy with respect to the linear coefficients (variational method). The problem of determining the linear coefficients reduces to the problem of finding the unitary matrix, U, that diagonalizes the matrix, H, where
ethyl = C H z C H 3
Figure 1. Molecular structures for chlorophyll a (Chl a) and ethyl chlorophyllide a: R = phytyl for chlorophyll a; R = ethyl for ethyl chlorophyllide a.
tion. In the present paper we will consider only the case where the Born-Oppenheimer a p p r o ~ i m a t i o nholds ~ ~ for the aggregate. Consider an aggregate of N molecules composed of one or more types of chromophore molecules along with some solvent molecules.46 Let pik denote the electronic wave function (under the Born-Oppenheimer approximation) for the k t h electronic state of the ith molecule in the aggregate. If the overlap between the molecules is the ground electronic state wave function, fig, for the aggregate may be approximated by the (zeroth-order) wave function
where plo is the ground state electronic wave function for the ith molecule when isolated. The Hamiltonian operator, k,for the system contains electronic kinetic energy, electron-nuclear Coulombic interaction, electron-electron Coulombic interaction, and nuclear-nuclear Coulombic interaction terms and may be partitioned as follows:
k=
N
i= 1
k ;+
and
The quasi-stationary states, order48) are given by ($1,
+i,
of energy
fiz,. . .) = (41, 4 2 , . . .)U
q
(to first (9)
It should be pointed out that the states, 4i, are nonstationary; if a single molecule is excited, the excitation energy will not remain localized on that molecule, but spreads in time to other molecules. The states, fii, are called quasistationary here because they are purely electronic wave functions, and coupling to the nuclear motions would bring about a transfer of energy from electronic excitation energy to nuclear motions (i.e., heat). Consider the lowest energy excited singlet state of the aggregate. We construct this state by taking a linear combination of the nonstationary states, &, as defined in eq 5. The elements of H simplify to
N
x
i
