Theory of Anisotropic Circular Dichroism of Excitonically Coupled

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Theory of Anisotropic Circular Dichroism of Excitonically Coupled Systems: Application to the Baseplate of Green Sulfur Bacteria Dominik Lindorfer, and Thomas Renger J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b12832 • Publication Date (Web): 08 Feb 2018 Downloaded from http://pubs.acs.org on February 15, 2018

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The Journal of Physical Chemistry

Theory of Anisotropic Circular Dichroism of Excitonically Coupled Systems: Application to the Baseplate of Green Sulfur Bacteria Dominik Lindorfer∗ and Thomas Renger∗ Institut f¨ ur Theoretische Physik, Johannes Kepler Universität Linz, Altenberger Str. 69, 4040 Linz, Austria E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract A simple exciton theory for the description of anisotropic circular dichroism (ACD) spectra of multichromophoric systems is presented that is expected to be of general use for the analysis of structure-function relationships of molecular aggregates such as photosynthetic light-harvesting antennae. The theory is applied to the baseplate of green sulfur bacteria. It is demonstrated that only the combined analysis of ACD and CD spectra for the present baseplate bacteriochlorophyll (BChl) a dimer allows for an unambiguous determination of the parameters of the exciton Hamiltonian from experimental data. The analysis of experimental absorption and linear dichroism spectra suggests that either the NMR structure has to be refined, or in addition to the dimers seen in the NMR structure and in the CD and ACD spectra, BChl a monomers are present in the baseplate carotenosome sample. A refined dimer structure is presented, explaining all four optical spectra.

Introduction The excitonic coupling between chromophores in molecular aggregates leads to a partial delocalization of excited states, that shifts the optical bands and redistributes their intensity. 1,2 This effect has been used to predict the molecular structures of J-aggregates 3–5 and of molecular dimers in pigment-protein complexes. 6–8 Standard optical spectra included in such analyses are linear absorption (OD), circular dichroism (CD) and linear dichroism (LD). Krausz and coworkers measured, in addition, the circular polarization of fluorescence and were able to verify and refine exciton Hamiltonians of two light-harvesting pigment-protein complexes. 9,10 Whereas OD and CD are measured on isotropic samples, containing randomly oriented complexes, in LD spectroscopy the complexes are partially oriented, before the absorption of linear polarized light is measured. This partial orientation is achieved, e.g., by a flow in liquid environment in case of tubular J-aggregates 11 or by introducing the complexes in a gel and squeezing the latter in a certain direction. 12,13 The polarization of the light 2


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is then chosen parallel or perpendicular to this direction, termed in the present context as face and edge directions, respectively. CD spectra of molecular aggregates are particularly sensitive to the intermolecular orientations because often the intrinsic CD of the isolated chromophores is negligibly small compared to the CD arising from the interaction between chromophores. Recently, Lambrev and coworkers 8 reported a new signal of molecular aggregates - the circular dichroism of an oriented pigment-protein complex, the baseplate complex of green sulfur bacteria. These baseplate complexes consist of arrays of bacteriochlorophyll (BChl) a dimers with strong intra dimer and weak inter dimer couplings (Figure 1). The baseplate connects the outer light-harvesting antenna, the chlorosomes, to the FMO protein, which itself is connected to the reaction center complexes, where the light-energy is finally converted into chemical energy (Figure 1). In the study by Nielsen et. al. 8 the baseplate structure was determined by NMR and CD spectroscopy and the pigment orientations were verified afterwards by calculations of ACD spectra and comparison with experimental data. In the ACD calculations, an earlier theory by Hansen and collaborators 14,15 was used that includes the full spatial variation of the electromagnetic field and is referred to as fully retarded description. In the present work we want to simplify the earlier analysis in order to obtain a method that can be routinely applied to describe anisotropic circular dichroism of molecular aggregates. In particular, the new theory is simple enough to be directly included in the optimization of structural models, as will be demonstrated. Whereas isotropic CD just contains contributions from electric transition dipoles and magnetic transition dipoles, in the case of ACD spectra, in addition, electric transition quadrupoles have to be included for a meaningful result that is independent on the origin of the coordinate system used. 16,17 In the isotropic average the electric quadrupole contributions become zero, and the standard Rosenfeld equation 18–20 for the rotational strength is obtained. To the best of our knowledge all text book derivations of the Rosenfeld equation for the rotational strength implicitly make use of this result by neglecting the electric quadrupole terms already from the beginning. 2,20,21 As proven by Jacob and coworkers 22 in their theoretical work on

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X-ray spectroscopy, the oscillator-strength of an electronic transition becomes independent on the origin of the coordinate system, if in the multipole expansion of the system-field coupling all terms up to a given order in the wave vector k of the external field are included in the theory. We make use of this result by taking into account in our derivation all terms in the light-matter interaction up to first order in k . The remaining of this work is organized in the following way. We start with a general molecular Hamiltonian that includes the coupling to an external monochromatic electromagnetic field in a semi-classical way. This Hamiltonian is afterwards expanded in terms of singly excited states of the molecular aggregate and the rate constant for optical excitation of these states is expanded with respect to the wavevector of the external field. The ACD spectrum is obtained from the first-order terms in this expansion. Finally the theory is applied to the baseplate complex of green sulfur bacteria and used to refine a recent structural model.

Figure 1: In the left part a schematic representation of the photosynthetic apparatus of green sulfur bacteria is shown, consisting of an outer BChl c antenna (chlorosomes) which is connected via the baseplate and the FMO proteins to the inner membrane reaction center complexes. The right upper part shows the arrangement of BChl a pigments as arrays of dimers in the baseplate. 8 An enlarged view on one of these dimers is given in the lower right part. The unit vectors n ⊥ and n k denote the propagation directions of circularly polarized light in face and edge configuration considered in ACD spectroscopy.

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Theory The Hamiltonian of the molecular aggregate consisting of Ne electrons and Nn nuclei including a semi-classical interaction with an external electromagnetic field reads

H=

2 X 1 e pî − A (rr i , t) + V (rr 1 , · · · , r Ne , R) + Tn 2me c i

(1)

∇i its momentum, e the electron charge, where r i is the coordinate of the ith electron, pî = i~∇ R 1 , · · · , R Nn } comprises all nuclear degrees of freedom, V contains the Coulomb R = {R coupling involving electrons and nuclei and Tn denotes the kinetic energy of nuclei. The vector potential

A (rr i , t) = −A0 cos(kk · r − ωt)

(2)

∇ · A (rr i , t) = 0) of the electromagnetic monochromatic field is expressed in Coulomb gauge (∇ and the electric and magnetic fields follow as 1∂ A (rr i , t) = −A0 k sin(kk · r − ωt) c ∂t

(3)

B (rr , t) = ∇ × A (rr i , t) = A0 (kk × ) sin(kk · r − ωt)

(4)

E (rr , t) = −

and

respectively, with amplitudes E0 = B0 = A0 k and polarization unit vector of the electric field. The light propagates along the unit vector n = k /k, where k = 2π/λ = ω/c is related to the wavelength of light λ, and to the angular frequency ω and velocity c. In the following we assume that the chromophores are non-covalently bound in the complex and that electron exchange between the chromophores and their environments can be neglected. In this case the Hamiltonian H in eq 1 can be expanded with respect to product states of the isolated

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chromophores. 23 For the linear optical spectra, considered in this work, it is sufficient to include besides the ground state

ψ0 (r) = hr|0 >=

Y

(0)

hrk |ϕk >=

Y

k

(0)

ϕk (rk )

(5)

k

the singly excited states (k6=m)

k6=m

ψm (r) = hr|m >=

hrm |ϕ(e) m

>

Y

(0) hrk |ϕk

>=

ϕ(e) m (rm )

k

Y

(0)

ϕk (rk )

(6)

k

where chromophore m is excited and all other chromophores k 6= m are in their electronic ground state. Please note that in the above wavefunctions r = {rr 1 , · · · , r Ne } comprises all (l)

(l)

electrons of the aggregate, whereas rl = {rr 1 , · · · , r Ne } just stands for the subset of electrons that belong to chromophore l. Expanding H in this basis gives

H = H00 (R) |0ih0| +

X

(Hm0 (R) |mih0| + h.c.) +

X

Hmn (R) |mihn|

(7)

m,n

m

where h.c. denotes the hermitian conjugate and the electronic matrix elements of H depend on the nuclear degrees of freedom R of the aggregate. In Heitler-London approximation, 24 due to the large energy difference between excited and ground state of the monomers (as compared to the inter-chromophore Coulomb coupling) the contribution of V in eq 1 to Hm0 (R) can safely be neglected and only the coupling to the external field, arising from the mixed term in the first expression on the r.h.s. of eq 1, needs to be taken into account. Applying, in addition, a rotating wave approximation, where the transition |0i → |mi between the ground and excited state of the aggregate leads to the annihilation of a photon and the reverse transition creates a photon gives Hm0 = hm0 e−iωt that defines the exciton-radiation interaction Hamiltonian

Hex−rad =

X

hm0 e−iωt + h.c.

m

6


(8)

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with

hm0 =

X (m) eA0 (m) hm| eikk ·rr j · pˆj |0i 2me c j

(9)

The electronic wavefunctions ψ0 (r) = ψ0 (r, R) and ψm (r) = ψm (r, R) in general depend parametrically on the nuclear coordinates R of the aggregate. We choose as a reference point (0)

(0)

(0)

R 1 , · · · , R Nn } the equilibrium coordinates of nuclei in the electronic ground state of R0 = {R (0)

the complex. Taylor expansion of H00 (R) and Hmn (R) around R0 up to second-order in the displacement of nuclear coordinates from their equilibrium values in the electronic ground state and a subsequent normal mode analysis 25 results in the standard exciton Hamiltonian (choosing as a zero point of energy H00 (R0 ) = 0)

H = Hex + Hex−vib + Hvib + Hex−rad

(10)

with the exciton Hamiltonian m6=n

Hex =

X

Em |mihm| +

m

X

Vm,n |mihn|

(11)

m,n

where the site energy Em denotes the optical transition energy of chromophore m in the absence of the excitonic couplings Vmn to other chromophores n. The exciton-vibrational coupling Hamiltonian Hex−vib describes the modulation of site energies and excitonic couplings by the dynamics of nuclei and Hvib is simply a sum over independent harmonic oscillator Hamiltonians of the different normal modes. Explicit expressions for Hex−vib and Hvib are given in the SI. If the excitonic coupling is large compared to the exciton-vibrational coupling, delocalized electronic states

|M i =

X

) c(M m |mi

m

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(12)


(M )

are excited in the aggregate, where the coefficients cm

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and corresponding energies EM =

~ωM 0 are obtained by diagonalizing Hex . The rate constant k0→M for an optical excitation of the M th exciton state is given by Fermi’s golden rule

k0→M

Here

P

(M ) m cm hm0

2π = 2 ~

2 X ) c(M h δ(ω − ωM 0 ) m0 m

(13)

m

= hM 0 is the exciton representation of the system-field coupling Hamilto-

nian hm0 in eq 9. Taking into account the exciton-vibrational coupling leads to a replacement of the δ-function by a realistic line-shape function DM (ω) that has been derived before 26 and is given explicitly in the SI. The matrix elements of eq 13 are calculated by using a multipole expansion 22 in the wave vector k . Truncating this expansion in first-order leads to (0)

(1)

hm0 ≈ hm0 + hm0

(14)

with (0)

hm0 =

X eA0 (m) hm| · pî |0i 2me c i

(15) (16)

and (1)

hm0 = i

X eA0 (m) (m) · pî k · rî hm| |0i 2me c i

(17)

By using the standard relation 20,22,27 , also proven in the SI, (m) hm|ˆ p i |0i = i

me (m) (m) Em hm|ˆ r i |0i ≈ iωme hm|ˆ r i |0i ~

8


(18)

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which contains the local transition energy Em of pigment m, that has been approximated by the photon energy ~ω, the 0th-order contribution becomes (0)

hm0 = i

where µ m0 = hm|

P

i

E0 ( · µ m0 ) 2

(19)

(m)

e rî |0i is the transition dipole moment of molecule m. The first-order

(1)

contribution hm0 is split into two parts 17,22,27 that are symmetric and antisymmetric with respect to the interchange of k and (1)

(1,m)

hm0 = hm0 (1,m)

The first part hm0

(1,Q)

+ hm0

(20)

is related to the magnetic transition dipole moment

m m0 =

X (m) e (m) hm| rî × pî |0i 2me c i

(21)

and reads (1,m)

hm0

=i

X eA0 1 (m) (m) (m) (m) |0i hm| k · rî k · pî · pî − · rî 2me c 2 i

=i

E0 n × ) · m m0 (n 2

(22)

where the vector identity

(a · c)(b · d) − (b · c)(a · d) = (a × b) · (c × d)

(23)

(m) (m) valid for arbitrary vectors a, b, c and d, was used. Please note that · pî and k · rî ˆ (1,Q) commute since and k are orthogonal. The second part h results from the electric m

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transition quadrupole and is given as (1,Q)

X eA0 1 (m) (m) (m) (m) |0i k · pî + · rî · pî hm| k · rî 2me c 2 i eA0 X X (m) (m) (m) (m) kα εβ hm|ˆ ri,α pî,β + rî,β pî,α |0i =i 4me c i αβ

hm0 = i

(24)

As shown in the SI, the matrix element on the r.h.s can be expressed in terms of the matrix ˆ (m) |0i of the quadrupole operator Q ˆ (m) = e P rˆ(m) rˆ(m) and the local transition element hm|Q αβ αβ i i,α i,β energy Em . Approximating the latter again by the photon energy ~ω results in (1,Q)

hm0 = −

X E0 X (αβ) ˆ (m) |0i = − E0 kα εβ hm|Q kα εβ Qm0 αβ 4 α,β 4 α,β

(25)

The rate constant k0→M can now be expressed in terms of contributions that are in 0th and 1st order in the wavevector k (0)

(1)

k0→M ≈ k0→M + k0→M

(26)

(0)

where k0→M is obtained, using eq 19, as (0)

k0→M =

E02 π | · µ M 0 |2 DM (ω − ωM 0 ) 2 2~

with the exciton transition dipole moment µ M 0 = (1) k0→M

(M ) m cm µ m0

P

(27)

and

2π X (M ) (M ) (0) ∗(1) ∗(0) (1) c c hn0 hm0 + hn0 hm0 DM (ω − ωM 0 ) = 2 ~ m,n m n

(28)

follows from eqs 19, 20, 22 and 25 as, (1)

k0→M

( !) X E02 π X (M ) (M ) 1 (αβ) n × ) · m m0 − = c c < −i (∗ · µ n0 ) i (n kα εβ Qm0 DM (ω − ωM 0 ) 2~2 m,n m n 2 α,β (29) 10


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where < denotes the real part. The magnetic transition dipole moment m m0 can be related (m) to the electric transition dipole moment µ m0 by expressing the coordinate r i in terms of 0

(m) the center position R m of pigment m and the relative coordinate r i with respect to this

center as 0

(m) (m) r i = Rm + r i

(30)

The magnetic transition dipole m m0 then becomes, using eq 18,

m m0 =

ik R m × µ m0 ) + m (intr) (R m0 2

(31)

where the intrinsic magnetic transition dipole reads (intr) m m0 =

X 0(m) e (m) hm| rî × pî |0i 2me c i

(32)

In the following we will concentrate on such molecular aggregates, which exhibit a conserR∞ vative isotropic circular dichroism signal CD(ω), that is, 0 CD(ω) dω = 0. In this case the rotational strength of the aggregate is dominated by exciton effects and the intrinsic (intr) rotational strength of the isolated chromophores can be neglected by setting m m0 ≈ 0. (αβ)

The transition quadrupole moment Qm0 is decomposed, using eq 30, according to (αβ)

(β)

(α)

0 (αβ)

(α) (β) Qm0 = Rm µm0 + Rm µm0 + Qm0

0 (αβ)

where Qm0

= hm|e

0(m) 0(m) î,α rî,β |0i ir

P

(33)

is the intrinsic quadrupole transition moment of pigment

m, defined with respect to a coordinate system with origin at the center R m of this pigment, introduced in eq 30.

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(1)

By using eq 33 and the vector identity in eq 23, the rate k0→M is simplified to (1) k0→M

( !) X 0 (αβ) 1 E02 π X (M ) (M ) n · R m ) ( · µ m0 ) + c c < i (∗ · µ n0 ) k (n = kα εβ Qm0 DM (ω − ωM 0 ) 2~2 m,n m n 2 α,β (34) (0)

(1)

In the following the rate constants k0→M and k0→M derived above will be used to obtain expressions for the CD and ACD spectra.

Isotropic and Anisotropic Circular Dichroism The frequency-dependent absorption coefficient α(ω) is related to the rate constant k0→M by 1

α(ω) =

X 8π nmol ~ω k0→M ≈ α(0) (ω) + α(1) (ω) 2 cE0 M

(35)

where nmol is the density of molecular aggregates in a homogeneous sample and the 0th-order (0)

contribution α(0) (ω) results from k0→M in eq 27 and the first-order contribution α(1) (ω) is (1)

obtained from the k0→M in eq 29. CD is defined as the difference in absorption between left- and right circularly polarized light. Assuming light-propagation in Z-direction, that is, n = (0, 0, 1), the left- and right circular polarizations are described by the polarization vectors r =

√1 (1, −i, 0) 2

and l =

√1 (1, i, 0). 2

(0) Since r = ∗l , the 0th-order terms αl/r (ω) in

eq 35 are identical and, hence, do not contribute to CD and the latter is obtained from the first-order contributions. Using eqs 23, 29, 35 and k =

ω c

we obtain

(1)

CD0 (ω) = α(ω)l − α(ω)(1) r 2 2 X X −4π ω nmol 1 ∗ (M ) (M ) 0 µn0 × µ m0 ) (n n · R m ) + µ n0 × (Qm0 · n ) = cm cn = (l × l ) · (µ ~c2 2 M m,n (36) P n and writing the n · R m ) as Noting that (∗l × l ) = in m,n · · · containing the term (n P P P 1 and interchanging the names of summation variables m,n · · · = 2 m,n · · · + m,n · · · 12


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m and n in the second sum we arrive at

CD0 (ω) =

(Z) 0 2π 2 ω 2 nmol X X (M ) (M ) (Z) (Z) R µ · n × µ c c (R (µ × µ ) + Q )DM (ω − ωM 0 ) n0 m0 n0 m0 m n mn ~c2 M m,n (37)

(Z) (Z) (Z) where R mn = R m − R n is the Z-component of the vector that connects the centers of

pigments m and n. Since only relative coordinates enter, the above expression is independent on the origin of the coordinate system, as expected. Note that the prime in CD0 (ω) indicates that no orientational average was included yet. To describe measurements on isotropic or anisotropic samples, orientational averages of eq 37 are carried out by using Euler angles. 28 For detailed calculations please see the SI. The isotropic average of eq 37 yields the isotropic CD signal

CD(ω) =

2π 2 ω 2 nmol X X (M ) (M ) R mn · (µ µm0 × µ n0 )) DM (ω − ωM 0 ) cm cn (R 3~c2 M m,n

(38)

which is well known from the literature. 2,19–21 Please note that the quadrupole contribution 0

Qm0 averages to zero in the isotropic case as shown in the SI. In the case of anisotropic circular dichroism (ACD) the complexes in the sample are oriented, and, therefore, the orientational average only has to be performed with respect to rotation around the symmetry axis of the complex. We choose the laboratory-fixed coordinate system such that its z-axis is oriented n = n ⊥ = (0, 0, 1)) this alongside this symmetry axis. If light propagates along this axis (n configuration is called face-configuration (see Figure 1). Since eq 37 is already invariant with respect to rotation of the complex around the Z-axis, the rotational average has no effect and the ACD signal for the face configuration is given by this equation, which is rewritten

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slightly using the propagation vector n ⊥ of the external field

ACD(ω)face =

2π 2 ω 2 nmol X X (M ) (M ) R mn · n ⊥ ) (µ µm0 × µ n0 ) · n ⊥ cm cn ((R ~c2 M m,n + ((Q0m0 · n ⊥ ) × µ n0 ) · n ⊥ )DM (ω − ωM 0 )

(39)

For light propagation along a direction perpendicular to the symmetry axis of the complex the configuration is called edge-configuration. In the following we assume light propagation nk = (1, 0, 0)) where the unit vectors of the right- and left-hand circularly along the X-axis (n polarized light are given as r =

√1 (0, 1, −i) 2

and l =

√1 (0, 1, i), 2

respectively. In this case,

the superscript Z in eq 37 has to be replaced by X. The orientational average of this signal with respect to rotation of the complex in the X-Y plane gives, as shown in the SI,

ACD(ω)edge =

π 2 ω 2 nmol X X (M ) (M ) R mn · (µ µm0 × µ n0 ) − (R R mn · n ⊥ ) · (µ µm0 × µ n0 ) · n ⊥ cm cn (R ~c2 M m,n − ((Q0m0 · n ⊥ ) × µ n0 ) · n ⊥ )DM (ω − ωM 0 )

(40)

The anisotropic CD signals ACD(ω)face and ACD(ω)edge are related to the isotropic signal CD(ω), using eqs 38, 39 and 40 by

CD(ω) =

ACD(ω)face + 2 · ACD(ω)edge 3

(41)

as expected. Eqs 39 and 40 represents the main theoretical result of the present work.

Results Application to the Baseplate BChl a Dimer In the following, we apply the present theory to the baseplate complex in the photosynthetic green sulfur bacterium Cba. tepidum. A recently proposed NMR structure 8 (PDB code 5LCB) shows that the repeating functional

14


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unit of the baseplate is a BChl a dimer, whose bacteriochlorin planes are oriented almost orthogonal to the baseplate plane (Figure 1). The conservative nature of the CD spectrum of R∞ the baseplate, 8 0 CD(ω) dω = 0, shows that the intrinsic CD of BChl a is negligibly small compared to the excitonic contribution. Hence, neglecting the intrinsic magnetic transition (intr) dipole moment m m0 in eq 31 is justified and the present theory can be applied. The

excitonic coupling, that is, the Coulomb coupling between the transition densities, of the two BChl a chromophores is calculated by the Poisson-TrEsp method. 29,30 The latter takes into account the solvent environment as a homogeneous dielectric with optical dielectric constant = 2.0. The pigments are represented by molecule-shaped cavities containing atomic transition charges. The latter were obtained in ref. 36 by a fit of the electrostatic potential (ESP) of the transition density calculated with time-dependent density functional theory (TD-DFT) with the B3LYP exchange-correlation (XC) functional on the isolated BChl a pigment. In order to correct for uncertainties in the absolute magnitude of the transition density originating from the quantum chemical calculations, the transition charges are rescaled with a constant factor to yield the vacuum transition dipole moment for BChl a of 6.1 D, that was extrapolated from experiments in different solvents by Knox and Spring. 31 Numerical values of the (unscaled) transition charges are given in the SI of ref. 36 and in Figure 4. A value of V = 52 cm−1 is obtained for the excitonic coupling in the baseplate BChl a dimer in this way. In order to describe the optical spectra, besides the excitonic coupling, we need to know the local transition energies Em (site energies) of the two monomers, the spectral density J(ω) of the modulation of the site energies by the protein dynamics, static disorder in site energies and the transition dipole and quadrupole moments µ m0 and Q0m0 , respectively of the pigments. The transition moments are obtained from the atomic transition charges qI (m) that are placed at the position R I of the Ith atom of pigment m. The transition dipole P (m) 0(αβ) moment µ m0 of pigment m is given as µ m0 = I qI R I and the component Qm0 of the

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quadrupole transition moment Q0m0 reads 0(αβ)

Qm0 =

X

(m) (m) (m) qI Rα,I − Rα(m) Rβ,I − Rβ

(42)

I

where R (m) denotes the molecular center of pigment m. Static disorder in site energies is taken into account by a Monte Carlo variation of individual site energies according to a Gaussian distribution of width ∆inh . For the present homo dimer we have equal (mean) site energies, E1 = E2 = E0 , where E0 is determined from the overall position of the optical spectra along the energy (wavelength) axis. The experiments on the baseplate can be best described by assuming an E0 that corresponds to a wavelength of 800 nm. For the functional form of the spectral density J(ω) we take an expression that has been extracted from fluorescence line narrowing spectra of the B777 complex 26 and we determined the R integral coupling strength S = J(ω) dω, the Huang-Rhys factor, from the temperature dependence of the circular dichroism spectrum. As shown in the SI the best fit is obtained for S = 1.2. With the excitonic coupling V = 52 cm−1 and the other parameters, described above, the experimental CD spectrum of the baseplate BChl a dimer can be quantitatively described if static disorder in site energies is taken into account by a ∆inh = 450 cm−1 (Figure 2, left, blue dashed-dotted line). In contrast, using these parameters gives only semi-quantitative agreement between calculated and measured ACD spectra in face configuration (Figure 2, right, blue dashed-dotted line). Whereas the experimental ACD spectrum shows a bleaching maximum at 808 nm the calculated bleaching peaks at 805 nm. In addition, the calculated ACD line shape is somewhat too broad. The different peak positions indicate that the excitonic coupling used in the simulation has to be adjusted. Indeed, increasing the excitonic coupling from V = 52 cm−1 (obtained from Poisson-TrEsp) to V = 85 cm−1 and adjusting ∆inh from 450 cm−1 to 400 cm−1 gives quantitative agreement with both experimental spectra, ACD and CD (Figure 2, red-dashed lines). In case of CD alone, it is obviously not possible to find a unique

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Figure 2: Circular dichroism (left) and anisotropic circular dichroism in face configuration (right) of the baseplate. The experimental data (black solid lines) are compared to exciton simulations on the BChl a dimer of the baseplate using different values for the excitonic coupling and the inhomogenous width ∆inh of the site energy distributuion (dashed and dot-dashed colored lines), as indicated in the figure legends. The calculated spectra are based on the recent NMR structural model 8 and were scaled independently to obtain equal amplitudes, for better comparison. combination of ∆inh and V from a fit of the spectrum. Due to the conservative nature of CD there are always overlapping bands with different sign that partially cancel each other in the center of the spectrum. Because of this cancellation, the positive and negative peaks in the CD spectrum are splitted more than the underlying exciton states (SI, Figure S3, top left). Therefore, a too small excitonic coupling can be compensated by a too large ∆inh , since the latter will increase the cancellation effects and thereby the splitting between the positive and negative bands. A perfect fit of the CD spectrum can even be obtained by using a coupling as small as 10 cm−1 , if the inhomogeneous broadening is increased accordingly (Figure 2 left, green dotted line). In the ACD spectrum mainly one exciton state contributes (SI, Figure S4, top right) and, therefore, the above error compensation is avoided (Figure 2 right, green dotted line).

Using the exciton Hamiltonian determined above, we calculated the linear dichroism (LD) and absorption (OD) spectrum of the baseplate dimer by using the well-known expressions P P µM 0 |2 DM (ω − ωM 0 ) and LD(ω) ∝ ω M |µ µM 0 |2 (1 − 3 cos2 (θM )) DM (ω − OD(ω) ∝ ω M |µ

17



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Figure 3: Linear dichroism (left) and linear absorption (right) of the baseplate. The experimental spectra (black solid lines) are compared to exciton simulations on the BChl a dimers of the baseplate (red dashed lines) using the NMR structural model. 8

Figure 4: Atomic transition charges of BChl a in units of e/10 obtained in ref. 36 from a fit of the electrostatic potential of the ab-initio transition density calculated with TD-DFT using the B3LYP XC functional. These charges are used to calculate the quadrupole transition moments in eq 42, which are needed for the simulation of the ACD spectrum. ωM 0 ), where θM is the angle between the transition dipole moment µ M 0 and the normal on the plane of the baseplate (described by the unit vector n ⊥ ). LD is defined as the difference in absorption of light polarized linearly along two orthogonal axes in edge configuration. Please note that the LD in face configuration is zero because of the random in plane orientation of the complexes. The calculated OD and LD spectra are compared in Figure 3 to the experimental data. 8 The calculated LD spectrum exhibits a large positive positive at 808 nm and a smaller negative peak at 785 nm corresponding to the two exciton transition dipole moments parallel and orthogonal to the baseplate plane, respectively, in striking contrast to the experimental LD spectrum, which shows a single negative band around 795 nm (Figure

18


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3, left part). The experimental absorption spectrum has a maximum at 800 nm, whereas the calculated spectrum of the baseplate dimer peaks at 808 nm (Figure 3, right part). Obviously, the NMR structural model needs some extension / refinement as will be discussed in detail below.

Discussion A simple exciton theory for the description of ACD spectra of multichromophoric systems with conservative isotropic CD spectrum has been developed. The new expressions in eqs 39 and 40 describe the CD of the complexes in face and edge orientation with respect to the propagation direction of the light. These expressions contain, besides quantities like the transition dipole moments of the chromophores, their mutual distances and the exciton coefficients, one new parameter not needed for other spectra: the quadrupole transition moment of the chromophores defined with respect to their center (eq 42). This quantity can be obtained directly from quantum chemical calculations of the chromophore’s transition density. In order to avoid artefacts from a distorted structure, the quantum chemical calculations in ref. 36 were performed on the optimized geometry of the isolated chromophore. The ESP of the ab-initio transition density was fitted by that of atomic transition charges (Figure 4), which are placed here onto the chromophore structure in the aggregate, in order to evaluate the quadrupole transition moment. Once the transition charges of a chromophore have been determined, the ACD spectra of all molecular aggregates formed by this type of chromophore can be calculated without having to repeat the quantum chemical calculations. Therefore, we expect that the present theory is simple enough to find widespread applications. For molecular aggregates with non-conservative isotropic CD spectrum two possible contributions, neglected in the present approach, might have to be included: (i) the intrinsic magnetic transition dipole moment of the chromophores and (ii) the excitonic coupling with higher excited states of the chromophores. Extension (i) is readily incorporated in the present ap-

19



proach, as discussed in the SI, where expressions for the additional contributions to CD and ACD resulting from the intrinsic magnetic transition dipoles of the pigments are presented. The incorporation of higher excited states into the exciton Hamiltonian has been used to explain the non-conservative nature of the isotropic CD spectrum of the LH2 ligh-harvesting complex of purple bacteria 32 and the CP29 light-harvesting complex of plant photosystem II. 33 The present approach could be expanded along these lines. The present theory has been applied to the baseplate of green sulfur bacteria, using a NMR structural model proposed recently. 8 According to this structure the BChl a chromophores are arranged in dimers with strong intra- and weak inter-dimer excitonic couplings. Since the coupling between pigments in neighboring dimers (V = 20 cm−1 ) is much smaller than the reorganisation energy of the local exciton-vibrational coupling (λ = 100 cm−1 ) the exciton states of the baseplate are expected to be localized in the dimers and the optical spectrum of the baseplate is a sum over independent dimer spectra. In other words by restricting our theory to the BChl a dimer we implicitly take into account the dynamic localization of exciton states. Note, however, that in the present case we find that the artificial delocalization of exciton states between different dimers practically has no effect on the spectra. An important result that goes beyond the present application is that the parameters of the dimer Hamiltonian, in particular the excitonic coupling, can be unambiguously determined from a combined analysis of the CD and the ACD spectra, whereas the CD spectrum alone suffers from error compensation effects due to overlapping bands with equal magnitudes and opposite sign. The excitonic coupling V = 85 cm−1 is about 40% larger than the coupling obtained from the NMR structure using the Poisson-TrEsp method. Since the latter contains already a correction for uncertainties in the absolute magnitude of the quantum chemical transition charges, the deviations could be due to the description of the environment as a homogeneous dielectric with optical dielectric constant n2 . Taking into account inhomogeneous atomic polarizabilities within a polarizable atomic force field model Curutchet et. al. 34 reported de-

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viations with respect to a homogeneous model of a factor 4 in the rate constant of a weakly coupled system. This factor corresponds to a factor two in the coupling, in qualitative agreement with the present deviations. Whereas the orientation of the transition dipoles of the pigments suggested by the NMR structure lead to a perfect agreement between calculated and experimental CD as well as ACD spectra (Figure 2), a striking deviation is obtained for the LD spectrum (Figure 3, left part), that needs to be discussed in detail. By just diagonalizing the exciton Hamiltonian of the dimer, assuming equal site energies and leaving out static disorder, we obtain a low energy exciton state at 805.5 nm with a transition dipole moment of µ− = 1.25 µ0 and a high energy exciton state at 794.5 nm nm with a transition dipole moment of µ+ = 0.66 µ0 , where µ0 is the magnitude of the transition dipole moment of the isolated BChl a pigment. The angle of the first transition dipole moment with respect to to n ⊥ is θ− = 90.25◦ and that of the second exciton state is θ+ = 1.73◦ giving rise to µ+ |2 = −0.87 |µ0 |2 and 0.99 |µ µ− |2 = 1.55 |µ0 |2 . These values are in LD strengths of −1.99 |µ strong contrast to the experimental LD spectrum, which shows just one negative band. An obvious question is: can we refine the structural model for the BChl a dimer in the baseplate in order to explain all four experimental spectra? Assuming that the orientation and relative positions of pigment planes obtained from the NMR study are correct we have considered only in-plane rotations of the pigments by the same angle α towards the normal on the baseplate plane for both pigments, as illustrated in Figure 5. All four optical spectra (CD, ACD, OD and LD) were calculated as a function of this angle α. For α = 90◦ we obtained a perfect fit of the experimental CD, OD and LD spectra but a sign-inverted ACD Spectrum (SI, Figure S4) demonstrating how useful the present ACD theory is in order to discriminate between different structural models. Indeed, we find that only for α = 25◦ − 30◦ it is possible to reach agreement between all four calculated and measured spectra, as demonstrated in Figure 6 where the spectra calculated for α = 30◦ are compared to the experimental data. The overall agreement is excellent, except for a slight blue shift of the calculated OD and LD peaks. For the new pigment orientations,

21



the upper exciton state carries a larger oscillator strength (µ+ = 1.18 µ0 , θ+ = 1.44◦ ) than the low energy exciton state (µ− = 0.78 µ0 , θ− = 90.3◦ ) and dominates the LD spectrum µ+ |2 = −2.78 |µ0 |2 , (1 − 3 cos2 θ− )|µ µ− |2 = 0.61 |µ0 |2 ), in agreement with the ((1 − 3 cos2 θ+ )|µ experimental data. Since the experimental absorption spectrum peaks approximately in the middle between the two exciton energies, calculated, an alternative explanation could be that, in addition to the BChl a dimers seen in the NMR structure and in the ACD and CD spectra, BChl a monomers are present in the carotenosome sample. Assuming that those monomeric BChl a pigments have the same site energy as those in the dimer in the baseplate and leaving also the remaining parameters (except for the excitonic coupling) unchanged gives a good quantitative description of the experimental absorption spectrum (SI, Figure S5, left). In the case of the LD spectrum, we have to assume that the monomeric BChl a have a different orientation of their transition dipole moment than the BChl a in the dimers of the NMR structure, in order to explain the negative sign of the spectrum (SI, Figure S5, right). The fact that the experimental LD spectrum peaks 4 nm to the blue of the experimental absorption spectrum could indicate that there is a subpopulation ob BChl a monomers at the high energy (short wavelength) side of the ensemble that exhibits a smaller angle θ of the transition dipole moment with respect to the normal on the plane and thereby a larger contribution to the negative LD signal. Since only the excitonically coupled pigments contribute to the CD and ACD spectra, these spectra are not affected by the additional BChl a monomers. The latter could, however, dominate the experimental absorption and LD spectra. It has indeed been proposed before 35 that the excited states of the BChl a pigments in the baseplate have monomeric character. However, the CD spectrum measured in the latter study does not show any excitonic signature, in striking contrast to the experiments analyzed in the present study. Obviously, different structural organisations exist for the baseplate.

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Figure 5: Refinement of the structure. The pigments of the original structural model, 8 shown in atomic detail, are rotated in plane of their macrocycles around the central Mg atom (shown in yellow) by an angle α in the direction towards the normal on the baseplate plane n ⊥ . The black dashed arrows indicate the orientations of the original monomer transition dipole moments and the red-dashed arrows those of the refined model. The black and red solid arrows illustrate the transition dipole moments of the exciton states in the original (α = 0◦ ) and in the refined model (α = 30◦ ), respectively.

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Figure 6: Optical spectra calculated for a refined structural model of the BChl a dimer with α = 30◦ (red-dashed lines), illustrated in Figure 5, are compared to experimental data (black solid lines).

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Conclusions A simple exciton theory of anisotropic circular dichroism spectra was developed that is expected to be very helpful in the verification / extraction of the exciton Hamiltonians of molecular aggregates. Three essential simplifications with respect to an earlier theory by Hansen 14,15 are: (i) We consider the small molecule limit in which it is assumed that the molecular aggregates (or the delocalization length of excited states) are small compared to the wavelength of visible light (ii) The intrinsic rotational strength of the monomer of the aggregate is neglected, an approximation that is justified for aggregates that exhibit a R∞ conservative isotropic circular dichroism spectrum CD(ω), that is 0 CD(ω) dω = 0. In this case the calculation of the intrinsic magnetic transition dipole moments of the pigments can be avoided. (iii) Atomic transition charges obtained in ref. 36 from a fit of the 3D ESP of the ab-initio transition density of the monomers (known in the literature as TrESP charges 36 ) are used in order to evaluate the electric dipole and electric quadrupole transition moments of the monomers, needed to calculate the ACD spectrum. Here, this theory was used to refine a recent NMR structural model of the BChl a dimers in the baseplate complex of green sulfur bacteria. In order to decide whether this refinement or the alternative model, assuming that in addition to the BChl a dimers also monomers (not seen in the NMR structure) are present, is valid, it is important to check whether the present refined dimer model is not only in agreement with all optical spectra but also with the constraints set by the NMR data.

Acknowledgement We would like to thank Petar Lambrev for pointing out the need for a simple exciton theory of ACD and for stimulating discussions.

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Supporting Information available Expressions for exciton-vibrational coupling Hamiltonian of Hex−vib , vibrational Hamiltonian Hvib and lineshape function DM (ω), derivation of eq 18 and eq 24, orientational averages of the isotropic and anisotropic CD signals, determination of the Huang-Rhys factor S from the broadening of CD spectra at different temperatures, high- and low-energy contributions to CD, ACD and LD spectra, expressions for the contributions of the intrinsic magnetic transition dipole moments of the pigments to the CD and ACD spectra, optical spectra for alternative structural model with α = 90◦ , OD and LD spectra assuming monomeric BChl a.

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References (1) May, V.; K¨ uhn, O. Charge and Energy Transfer Dynamics in Molecular Systems; WileyVCH: Berlin, 2011. (2) Van Amerongen, H.; Van Grondelle, R.; Valkunas, L. Photosynthetic Excitons; World Scientific, 2000. (3) Kriete, B.; Bondarenko, A. S.; Jumde, V. R.; Franken, L. E.; Minnaard, A. J.; Jansen, T. L. C.; Knoester, J.; Pshenichnikov, M. S. Steering Self-Assembly of Amphiphilic Molecular Nanostructures via Halogen Exchange. J. Phys. Chem. Lett. 2017, 8, 2895–2901. (4) Friedl, C.; Renger, T.; Berlepsch, H. v.; Ludwig, K.; Schmidt am Busch, M.; Megow, J. Structure Prediction of Self Assembled Dye Aggregates from Cryogenic Transmission Electron Microscopy, Molecular Mechanics, and Theory of Optical Spectra. J. Phys. Chem. C 2016, 120, 19416–19433. (5) Didraga, C.; Pugzlys, A.; Hania, P. R.; von Berlepsch, H.; Duppen, K.; Knoester, J. Structure, Spectroscopy, and Microscopic Model of Tubular Carbocyanine Dye Aggregates. J. Phys. Chem. B 2004, 108, 14976–14985. (6) Hughes, J. L.; Razeghifard, R.; Logue, M.; Oakley, A.; Wydrzynski, T.; Krausz, E. Magneto Optic Spectroscopy of a Protein Tetramer Binding Two Exciton-Coupled Chlorophylls. J. Am. Chem. Soc. 2006, 128, 3649–3658. (7) Renger, T.; Trostmann, I.; Theiss, C.; Madjet, M. E.; Richter, M.; Paulsen, H.; Eichler, H. J.; Knorr, A.; ; Renger, G. Refinement of a Structural Model of a Pigment Protein Complex by Accurate Optical Line Shape Theory and Experiments. J. Phys. Chem. B 2007, 111, 10487–10501. (8) Nielsen, J. T. e. a. In Situ High-Resolution Structure of the Baseplate Antenna Complex in Chlorobaculum Tepidum. Nat. Commun. 2016, 7, 12454. 26


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Theory of Anisotropic Circular Dichroism of Excitonically Coupled

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