the self - organization of supramolecular structures driven by weak 
 intermolecular interactions is one of the most fascinating thermodynamic 
 phenomena in biology , chemistry , and physics .
because the interactions 
 are weak , many different structures can be formed with the same type 
 of building blocks . in biology ,
20 different amino acids serve as 
 building blocks for many unique three - dimensional structures .
each 
 of these structures has been optimized by evolution in order to perform 
 a specific function , e.g. , to catalyze a chemical reaction or to create 
 an efficient solar energy converting photosystem .
an artificial chemical system with the same goal is an 
 organic solar cell . in some bulk - heterojunction solar cells
an interpenetrating 
 network of a polymer phase and a fullerene phase is formed spontaneously . 

again , weak interactions between the side groups of the polymer and 
 the fullerene determine the morphology of the solar cell that is key 
 to its function .
another example is self - assembled 
 dye aggregates , called chlorosomes , that function as light - harvesting 
 complexes in green sulfur bacteria .
the present work deals with j - aggregates of self - assembled cyanine 
 dyes .
while the optical properties of 
 cyanine dyes are mainly determined by the polymethine chromophore , 
 their solution behavior and aggregate morphology can decisively be 
 influenced by the substituents .
cyanines 
 with long alkyl substituents ( amphiphilic cyanines ) form bilayers 
 in aqueous solution with the side chains facing inward and the polar / charged 
 part of the molecule facing outward , thus stabilizing the aggregate 
 by increasing the entropy of the surrounding water molecules . besides 
 this so - called hydrophobic interaction , dispersive and electrostatic 
 couplings are important for the formation of the aggregate .
they are significant for cyanine dyes due to the large 
 polarizability of their delocalized -electron systems .
since 
 the dye molecules are charged , electrostatic interactions between 
 dye molecules among each other and also between dye molecules and 
 solvated counterions are important as well .
the interplay of 
 all of the above interactions determines the three - dimensional 
 structure of the aggregate .
their relative weight can be changed by 
 varying the character of the substituents , the solvent properties , 
 or even the preparation conditions .
it 
 is a challenging thermodynamical problem to predict the structure 
 of a j - aggregate .
most likely , the free energy surface is highly frustrated , 
 that is , it consists of many local minima , and slight changes in the 
 above - discussed interactions can shift the global minimum and thereby 
 the equilibrium structure of the aggregate . in recent years
v. berlepsch 
 and co - workers were able to elucidate 
 the morphologies of aggregates formed by many different cyanine dyes 
 by using cryogenic transmission electron microscopy ( cryo - tem ) .
cryo - tem 
 provides highly resolved direct images of supramolecular structures 
 in their native environment that are free of the drying artifacts 
 due to the specific sample preparation technique . detected architectures of dye aggregates range from thread - like 
 fibers across ribbons to fascinating single- and double - walled tubes . 

however , due to the smallness of dye molecules the resolution of cryo - tem 
 is generally not sufficient to directly resolve their mutual orientation 
 within the aggregates .
probably the simplest way to distinguish 
 different structures is 
 to measure their optical spectra .
the coulomb interaction between 
 the delocalized -electrons in one cyanine monomer and those 
 in another monomer gives rise to excitonic interaction between the 
 local optical excitations . for a pair of monomers subject to excitonic 
 interaction , if one monomer is excited and another one in the ground 
 state , the first one gets de - excited and simultaneously the second 
 one gets excited .
if the excitonic coupling is strong compared to 
 static and dynamic disorder , the excited states of the aggregate are 
 delocalized , the excitation energies of the delocalized states of 
 the aggregate are shifted , and the oscillator strength is redistributed 
 with respect to those of the isolated monomers .
the redistribution of oscillator strength depends on the mutual 
 geometry of optical transition dipole moments of the monomers . in 
 the classical aggregates of jelley and
scheibe the transition dipoles are oriented in line , and the oscillator 
 strength in this case is shifted to the lowest exciton state .
the 
 corresponding spectral band is very narrow and red shifted with respect 
 to that of the isolated monomers .
because all of the oscillator strength 
 is redistributed to this narrow band it is also said to possess 
scheibe the red - shifted 
 band is called a j - band and self - aggregated cyanine dyes possessing 
 a dominant j - band are called j- or scheibe aggregates . if the 
 mutual geometry of neighboring dye molecules is such that 
 the line connecting molecular centers is perpendicular to the transition 
 dipole moments of both dyes , the oscillator strength is shifted to 
 the exciton state with the highest energy , which is blue shifted with 
 respect to the excitation energy of the isolated monomer . due to this 
 hypsochromic shift this band
is called a h - band , in contrast to the 
 red - shifted j - band discussed before . in any case , there is a critical 
 dependence of the optical spectra of the aggregate on the mutual orientation 
 of the transition dipole moments of the monomers that allows one to 
 use the optical spectra for structure prediction . a subtlety 
 in the interpretation of spectra of dye aggregates concerns 
 the coupling of excitons to intramolecular vibrations of the chromophores . 
 the vibrational progression that is observed in optical spectra of 
 monomers in solution is distinctly changed by the excitonic coupling 
 in the aggregate
whereas j - bands do show weaker vibrational side 
 lines , in the case of h - bands these side lines get stronger than those 
 of the isolated chromophores .
this remarkable redistribution of oscillator 
 strength was explained by spano with 
 a model that included one effective vibrational mode per chromophore 
 in the exciton hamiltonian , which was numerically diagonalized .
an 
 alternative description of this effect was given by briggs and co - workers 
 using a greens - function approach . here
, 
 we will use the same approach as spano and in addition investigate 
 a simplified model ( assuming the same local electronic transition 
 energies of the chromophores ) that can explain the observed redistribution 
 of oscillator strength semianalytically .
of course , the optical 
 excitation energy of the monomers is not 
 only shifted by excitonic coupling but also by dispersive and electrostatic 
 couplings between monomers in the aggregate .
however , the effect of 
 those couplings can be taken into account by shifting the energies 
 of the localized ( one - exciton ) states of the aggregate . in the simplest 
 approximation
these localized energies are assumed to be equal to 
 some reference value   that describes at which energy a monomer 
 in the aggregate would absorb if it was not coupled excitonically 
 to the other monomers . in this case   can be obtained by comparison of calculated 
 and measured spectra , since variation of   just shifts the calculated spectrum along 
 the energy ( wavelength ) axis .
pioneering work in the relation 
 of optical spectra to the structure 
 of chromophore aggregates was provided by kuhn and later continued by knoester and co - workers . in the models 
 for simple sheet - like aggregates the dye molecule
is represented by 
 a brick containing the transition dipole that is described by two 
 atomic partial charges at a certain distance .
the discovery of tubular 
 dye aggregates has led to increasingly complex theoretical models . 

created tubular aggregates by wrapping a brick - layer lattice in certain 
 directions , where the optimal direction was determined by comparing 
 the resulting absorbance and linear dichroism spectra with experimental 
 data , and the radius was estimated by cryo - tem .
of course , the final 
 geometry depends also on the organization of the layer used to create 
 the tube , and it has to be checked whether the proposed organization 
 is also consistent with the minimum free energy of the molecular aggregate .
again , pioneering work was provided by knoester and co - workers , 
 who studied the spontaneous self - aggregation of an alkyl - substituted 
 pseudoisocyanine ( amphi - pic ) in aqueous solution by molecular dynamics 
 simulations .
cylindrical and ribbon - like 
 conformations were suggested as possible candidates for the equilibrium 
 structure , but due to the finite simulation times ( 200700 
 ns ) it could not be decided which of the two structures is more likely . 
 in contrast , cryo - tem images suggest that 
 the investigated amphi - pic dye forms double - walled tubes and multilamellar 
 spherical aggregates , at least at high dye concentrations .
it seems 
 that there is a need for either longer simulation times or some type 
 of preselection of possible structures that are closer to the equilibrium 
 structure .
knoester and co - workers suggested to take for this purpose 
 the phenomenological dipole models that describe the optical spectra . in the case of the amphiphilic cyanine dye c8s3 , 
 which forms double - walled tubular aggregates , megow et al .
successfully modified 
 the phenomenological dipole model of knoester et al . by incorporating information from high - resolution cryo - tem 
 data and running md simulations , arriving at 
 a model of a stable structure that still describes the optical spectra . 

a crucial step in the description of optical spectra was to take into 
 account the dispersive shift in site energies that was found to be 
 different for molecules in the outer and the inner cylinder of the 
 aggregate . in the present work
dye 
 molecular crystals as a basis for generating starting structures for 
 md simulations .
the rationale behind this approach is the expectation 
 that the three - dimensional arrangement of dye molecules in the respective 
 single crystals reflects their intermolecular interactions .
the question 
 to be addressed is whether and if so to what degree these interactions 
 are varied if soft colloidal structures , tubes , or sheets are formed 
 in solution .
nevertheless , we believe that the packing in a dye aggregate 
 is closer to the packing in a molecular crystal than to a random geometry . as an example , we study the cyanine derivative 5,5,6,6-tetrachloro-1,1,3,3- 
 tetraethyl - benzimidacarbocyanine ( ttbc ) ( figure 1 ) .
single - walled tubes have been observed 
 in ref ( 30 ) in aqueous 
 solutions for this dye after aggregation with cl or i counterions under different solvent conditions . 

a cryo - tem image of fibrous aggregates of ttbc - cl is reproduced here 
 ( figure 2 ) together 
 with an averaged density profile across a single aggregate , which 
 proves its tubular architecture .
ttbc - cl forms aggregates with an 
 absorption spectrum ( type i ) that contains a strong broad h - band and 
 a weaker and more narrow j - band , independent of the preparation conditions . 
 in the recent literature this type of aggregate
is referred to as 
 hybrid hj - aggregate . in the case of ttbc - i , tubular 
 aggregates with the same diameter as for ttbc - cl
are seen in cryo - tem , 
 but the measured absorption spectrum depends strongly on the solvent 
 conditions . besides the type i spectrum observed at high ph values , 
 an absorption spectrum with a single j - band ( in the following referred 
 to as type ii ) was reported at low ph values .
structure 
 of the ttbc molecule with color - coded atoms ( light blue , 
 c ; dark blue , n ; green , cl ; white , h ) .
( a ) network of single 
 fibrous aggregates and bundles of fibers at low magnification .
interestingly , for ttbc 
 there exist also two distinct single - crystal 
 structures that were obtained for two different 
 solvents . in the case of a 2:1 methanol : ttbc - i solvate , ttbc organizes 
 in separated layers where the molecules in each layer form a herringbone - like 
 arrangement .
the structure can be described as a crystal lattice with 
 two molecules per unit cell . referring to its solvent ,
it was termed 
 dyem . in the alternative dyea structure ( an 1:1 acetonitrile : ttbc - i 
 solvate )
just a single molecule per unit cell is found and the molecules 
 inside crystal sheets are arranged in a brick - layered fashion . compared 
 to dyem the distances between layers of ttbc molecules in dyea crystals 
 are much smaller .
details about these crystal structures are given 
 in the supporting information ( si ) .
a simple explanation of the two types of optical spectra found 
 for the j - aggregates of ttbc discussed above has been suggested in 
 the experimental work .
the explanation 
 has been that the two peaks found in the type i spectrum just represent 
 a davydow split spectrum , as expected for crystals with two molecules 
 per unit cell as found in dyem .
in fact , the present type i spectrum 
 of ttbc aggregates has been explained by assuming a 2d flat herringbone 
 structure of transition dipole moments in a phenomenological model 
 by glen and co - workers . at that time , no information 
 from tem was available , which later revealed a tubular structure of 
 the aggregate , as discussed above . here , 
 based on our assumption of a crystal - like arrangement
, we want to 
 find out whether a tubular herringbone structure can also explain 
 the optical spectra and if such a structure is stable .
an alternative 
 brick - layered structure with one molecule per unit cell as found in 
 dyea crystals could then be responsible for the spectrum with the 
 single j - band ( type ii ) . from analytical work of knoester and 
 co - workers it is known that tubes with 
 rotational symmetry and one 
 chromophore per unit cell in general exhibit two bands , where one 
 is polarized in parallel to the symmetry axis of the tube and the 
 other is polarized perpendicular to this axis .
thus far only tubes 
 have been found , where both transitions are red shifted with respect 
 to the transition energy of the isolated monomer . in the present work 

we show that the analytical model also contains a solution with one 
 red - shifted and one blue - shifted exciton band that is a hybrid hj - aggregate , 
 which seems to be realized in monowalled ttbc tubes with a type i 
 spectrum .
during the rest of our considerations we will focus 
 on the latter 
 type i spectrum , since it contains a rich substructure that we want 
 to understand on a molecular basis .
for this purpose we will use the 
 crystal structures of dyem and dyea to generate tubes , the resulting 
 optical spectra of which will be compared to experimental data .
interestingly , 
 in both cases tubes can be created which explain the absorption and 
 linear dichroism spectra .
we will use those structures as starting 
 geometries for molecular dynamics simulations in order to determine 
 their relative stability and , in an iterative procedure , provide a 
 proposal for the molecular structure of the ttbc - cl single - walled 
 hybrid hj - aggregate that can explain all available experimental spectroscopic 
 data and is stable within the time range of our md simulations ( 100 
 ns ) .
we treat the 
 intramolecular vibrational degrees of freedom of the monomers quantum 
 mechanically by introducing one effective vibrational mode per monomer 
 with frequency vib and huang  rhys factor s. the coupling of electronic excitations to these modes 

is treated within the one - particle approximation , that is , vibrational excitations are taken into account 
 only in the electronic excited state of the chromophores .
the rationale 
 behind this approximation is that the energy of the quanta of this 
 intramolecular mode is large compared to the thermal energy and , therefore , 
 an electronic transition induced by an external field will always 
 start from the vibrational ground state of the electronic ground state . 

however , the coulomb coupling between the vibronic transitions of 
 different monomers mixes the excited intramolecular states of the 
 electronic ground state of the monomers into the excited eigenstates 
 of the aggregate .
all intermolecular degrees of freedom are treated classically 
 by molecular dynamics simulations . neglecting electron exchange between 
 the monomers results in the following frenkel 
exciton hamiltonian1the   are the local excitation energies 
 of the 
 monomers between the electronic and intramolecular vibrational ground 
 state and the electronic excited state with  intramolecular 
 vibrational quanta .
the off - diagonal element   describes 
 the coupling of the transition 
 between the electronic and the intramolecular ground state and the 
 electronic and the th intramolecular excited vibrational state 
 of monomer m with the transition between the electronic 
 excited state with  vibrational excitations of monomer n and its electronic and vibrational ground state . in the 
 condon approximation this coupling , termed 
 excitonic coupling ,
condon factors 
 of the monomers ff , that is2where the franck  condon 
 factor f is given as3for the present system we estimated s = 0.83 and 
 vib = 800 cm for the 
 effective intermolecular vibrational mode 
 from optical spectra of isolated monomers , as described in detail 
 later .
please note that the explicit 
 time dependence of the above hamiltonian results from the classical 
 treatment of the intermolecular vibrational degrees of freedom .
the 
 interaction of the aggregate with a monochromatic external field e(t ) = e0e cos t is described in rotating wave 
 approximation by the hamiltonian4where m0 is the local electronic 
 transition dipole moment of monomer m , f is the franck 
condon 
 factor introduced in eq 3 , and h.c . denotes the hermitian conjugate of the first expression 
 on the righthand side .
the optical 
 spectra are obtained after diagonalization of hexc(t ) ( eq 1 ) , resulting in exciton states5with excitation energies m ( t ) .
the influence of intermolecular 
 degrees of freedom determines the time - dependence of the above eigenstates 
 and energies . in the calculation of the absorbance ( ) 
 this time dependence is taken into account in the static limit , where 
 it leads to inhomogeneous broadening6with the transition dipole moment7the average over inhomogeneous conformations 
 is denoted as   and is performed 
 taking into account the 
 time - dependent trajectories resulting from the classical propagation 
 of intermolecular degrees of freedom . in the case of linear dichroism , 
 the difference in absorbance of light polarized parallel and perpendicular 
 to the symmetry axis of the tube is measured reading8where 
 m(t ) is the angle 
 between the exciton transition 
 dipole moment dm(t ) and the symmetry axis of the tube . in the following , 

we consider a simplified hamiltonian for which it is possible to disentangle 
 the diagonalization of hexc(t ) ( eq 1 ) into a diagonalization 
 of the bare exciton hamiltonian and a subsequent diagonalization of 
 the exciton 
vibrational hamiltonian , separately for every eigenstate 
 of the bare exciton hamiltonian . in this way a simple explanation 
 of the different redistribution of oscillator strength by the exciton  vibrational 
 coupling for j- and h - bands will be obtained .
we consider a simplified 
 hamiltonian without static disorder , that is , the hamiltonian in eq 1 is assumed to be time 
 independent . in addition , we assume that all local electronic transition 
 energies of the chromophores are equal , i.e. , we have , where   is the transition energy between the vibrational 
 ground states of the electronic excited and ground state . the coefficients cm(m ) of the eigenstates |m ( eq 5 ) then fulfill the following equation9where em is the eigenenergy of |m. in addition , we investigate the 
 bare exciton hamiltonian hexc that 
 does not include the intramolecular 
 vibrational mode of the chromophores10where   is the local vertical transition 
 energy 
 of the chromophore in the tube taken at the equilibrium position of 
 nuclei in the electronic ground state11with the huang  rhys 
 factor s introduced in eq 3 .
the eigenstates |k 
 of hexc read12the coefficients cm(k ) are obtained 
 from the solution of the eigenvalue 
 problem13with the eigenenergy k of hexc .
we rewrite the second expression in eq 9 as14where 
 we used the orthogonality 
 of wave functions , that is , kcm(k)cm(k ) = m ,  m and ncn(k)cn(l ) = k , 
we proceed by introducing coefficients15hence , it holds that16using the above expressions , eq 9 can be rewritten as17multiplying this equation on both sides by cm(l ) and summation over l then results in the final eigenvalue problem18the optical transition dipole moment dm then follows from the eigenvalue 
 coefficients cm(m ) ( eq 9 ) using eq 16 as19where dk = mcm(k)dm is the transition 
 dipole 
 moment of the bare exciton state |k ( eq 12 ) , f is the franck 
condon factor defined in eq 3 , and ck(m ) is obtained from the solution 
 of the eigenvalue problem in eq 18 .
since eq 18 is diagonal with respect to k , no mixing of different exciton states k and l occurs due to the intramolecular exciton 
vibrational hamiltonian hex obtained for n chromophores taking into 
 account nvib vibrational levels per chromophore 
 can be arranged into n subgroups |mk for k = 1 ... n .
the eigenstates in each subgroup 
 are obtained by solving eq 18 , that is , diagonalizing a nvib times nvib matrix that describes the 
 vibrational progression of a given exciton state k. therefore , the quantum number m in eqs 18 and 19 can 
 be replaced by mk , and 
 the respective transition dipole moments of eigenstate subgroup mk are then obtained as20the resulting optical 
 spectrum reads21where mk = emk/ 
 is the optical transition 
 energy that is obtained from the eigenenergy em = emk in eq 18 and the respective eigenvectors contain 
 the coefficients ck(mk ) which determine whether there is 
 constructive or destructive overlay of different vibronic transition 
 dipole moments fdk of a given bare exciton state k. the above equations also show that only those bare exciton states k contribute to the absorbance spectrum , which have a nonvanishing 
 bare exciton transition dipole moment dk , that is , there is no mixing with dark exciton states induced 
 by the coupling to intramolecular vibrational modes .
therefore , in 
 the present homogeneous case we can analyze the optical spectrum in 
 two steps , first considering the bare exciton spectrum only and in 
 the second step including the influence of the intramolecular vibrational 
 mode .
the latter effect can be qualitatively understood by a 
 perturbative 
 treatment of the off - diagonal couplings in eq 18 that couple different vibrational states 
  and . for this purpose we can identify the eigenstate 
 |mk in the vibronic 
 manifold of the kth exciton state with a state |k that for absent off - diagonal coupling equals 
 |k|. taking into account the 
 off - diagonal coupling   in eq 18 up to first order , the 
 coefficient ck(k ) is obtained as22the resulting transition dipole moment dmk = dk then reads 
 using eq 2023where the 
 coefficient c( ) mixing 
 different transitions    is given as24the franck 
rhys 
 factor s < 1 , decreases with increasing . 
 therefore , 
 in the case of the low - energy exciton states ( ) we have c( ) > 
 0 for  > . in particular , it holds that c(0 ) > 
 0 for all  = 1 ...
nvib in eq 23 , where nvib denotes the number of vibrational levels included 
 per chromophore .
hence , there is constructive superposition of vibronic 
 transition dipole moments dkf and dkc()f in the optical transition to the vibrational ground state (  
 = 0 ) of the low - energy exciton states ( with nonvanishing bare exciton 
 transition dipole moment dk ) . in the case of high - energy exciton states we have   and obtain 
 constructive superposition , 
 that is , c( ) > 0 , if it holds that .
the magnitude of c( ) goes 
 rapidly to zero for large  or  , because the respective 
 franck  condon factors are small .
hence , in general , the mixing 
 of vibronic transitions goes to zero in this case .
therefore , the 
 redistribution toward higher vibronic transitions , as may occur in 
 the case of high - energy exciton states , will never be as strong as 
 the redistribution toward low - energy vibronic transitons in the case 
 of low - energy exciton states .
this effect in essence explains the 
 difference in the vibrational progression of j- and h - bands . concerning the shift in energies due to the off - diagonal coupling 
 in eq 18 we note that 
 the first nonvanishing contribution is obtained in second order in 
 this coupling and the resulting energy e reads25with26for large  , the 
 respective franck 
condon 
 factors f are practically zero 
 and , that is , there is no influence of the 
 excitonic interactions and the aggregate absorbs light as the monomers . 
 for small 
, however , there is a -dependent change of 
 the vibronic transition energies , which become nonequidistant and 
 experience a further shift e( ) by coupling with 
 other vibronic transitions . along the same line of discussion 
 used above , we see that for low - energy 
 exciton states ( ) and huang 
rhys factors s < 1 it holds that e( ) < 
 0 for  <  , that is , there is always a red shift due 
 to the coupling with higher energy vibronic transitions . in the case 
 of high - energy exciton states
( ) it depends on the relative magnitude 
 of 
 vib and   whether the shift goes to the 
 red or to 
 the blue . in the following
it will be discussed how the aggregate 
 hamiltonian 
 in eq 1 is parametrized 
 by combining quantum chemical , electrostatic , and molecular mechanics 
 calculations .
the diagonal elements of the exciton hamiltonian 
 ( eq 1 ) describe the energies 
 at which 
 the chromophores would absorb light if they were not coupled excitonically . 
 these diagonal elements   contain the vertical excitation 
 energy   of a ttbc molecule taken at the equilibrium 
 position of nuclei in the electronic ground state and a shift   that 
 is described by taking into account 
 intermolecular electrostatic coupling and intermolecular london dispersive 
 interactions , respectively .
the electrostatic transition energy 
 shift of chromophore m is obtained as27where jmk(eg , ge ) denotes the coulomb 
 coupling between the charge densities of the electronic excited state 
 of chromophore m and that of the electronic ground 
 state of the chromophore k , which may represent another 
 chromophore or a solvent molecule including the counterions , and jmk(gg , gg ) is the same type of coulomb coupling but between the 
 ground state charge densities of chromophore m and 
 its environment .
the charge density couplings are evaluated by performing 
 quantum chemical calculations on the isolated chromophore and fitting 
 the electrostatic potentials by that of atomic partial charges , which 
 are then used to calculate the couplings28here qi(m)(a , a ) is the partial charge placed 
 at atom i of chromophore m for the 
 ground ( a = g ) and excited state 
 ( a = e ) and qj(k)(g , g ) is the 
 partial charge at atom j of the environmental molecule k in its electronic ground state .
we used ( time - dependent ) 
 density functional theory and the b3lyp exchange correlation functional 
 for the calculation of the electrostatic potential of the ground ( excited ) 
 state of the isolated chromophores .
the numerical values of the atomic 
 partial charges that fit the respective potentials are given in the si . in
inductive shifts arise due to the polarization 
 of the wave function of a molecule by the charge density of its environment 
 and vice versa .
we follow our previous work on j - aggregates and neglect the inductive shifts , noting , however , 
 that for some systems these shifts were reported to be significant . in order to allow for a microscopic description 
 of static disorder 
 we will include besides the   in eq 27
a description of the dispersive transition 
 energy 
 shifts , introduced in our previous work as29where vmk(t ) is the 
 coupling of two effective extended 
 dipoles carrying unit charges and being located at molecules m and k. the distance between the two charges 
 on a given chromophore should reflect the extent of the -electron 
 system of the chromophore . for the present ttbc molecule
the factor q in the above equation is a constant that was derived using 
 second - order perturbation theory in the intermolecular interaction 
 and contains information about energies and oscillator strengths of 
 higher excited states .
this factor can be obtained from experiments 
 on transition energy shifts of the isolated chromophores in different 
 solvents . describing the latter by a homogeneous dielectric of refractive 
 index nr2 and the solute as a sphere of radius r results in a dispersive shift of30with the prefactor   that contains the difference in the variance 
 of the permanent dipole moment between the excited and the ground 
 state of the solute . from the absorption peaks of ttbc in methanol 
 and water we obtain   and extraplolate a vacuum transition energy 
 of the chromophore . by taking many different tube geometries 
 and averaging the dispersive shift ( eq 29 ) of a molecule with its neighbors
one should obtain 
 a similar result as from the continuum approximation in eq 30 .
hence , q follows 
 as31where < >orient denotes an average over many tube geometries and over 
 time and   is taken from 
 the experimental shift measured 
 in different solvents . with this
q the dispersive 
 shifts in eq 29 can 
 be calculated for every monomer m if the refractive 
 index nr is known . for 
 a monolayer of a related compound nr
= 1.64 was measured in ref ( 45 ) . as discussed before , the main effect of the 
 diagonal elements of the exciton hamiltonian
is to shift the center 
 of the spectrum along the energy ( wavelength ) axis . in the present 
 work
, we find that for nr  1.5 we can explain the shift of the absorbance spectrum 
 of our final suggestion for the ttbc tube with respect to that of 
 the isolated monomer in methanol ( nr = 1.33 ) .
the deviations between the above nr = 1.5 and nr = 1.64 measured in ref ( 45 ) could be due to the difference in morphology 
 ( tube versus plane monolayer ) , but it could also reflect the neglect 
 of inductive transition energy shifts , discussed above . using nr  1.5
we obtain q = 2.28 ev  from eq 31 , which is used to obtain dispersive shifts 
 from eq 29 along the 
 md trajectories , that will be used for a microscopic description of 
 static disorder . the excitonic coupling between the transition 
 densities of the 
 monomers is obtained from the poisson - tresp method , which allows us to take into account polarization effects of the 
 environment .
the environment in the present case comprises the transitions 
 between the ground state and higher ( than first ) excited states of 
 the other monomers and the excitations of solvent molecules .
the excitonic 
 coupling jmn between 
 monomers m and n is obtained in 
 the following way .
first , the transition density of the isolated ttbc 
 monomer is calculated with time - dependent density functional theory 
 using the b3lyp exchange - correlation functional .
the electrostatic 
 potential of this transition density in vacuum is fitted by atomic 
 transition charges qi(g , e ) .
a poisson equation 
 is solved in order to obtain the electrostatic potential m(r ) of the transition charges 
 of monomer m32where (r ) equals 1 if r points into the cavities of monomers m or n and (r ) = nr2 otherwise .
please note that in the original 
 poisson - tresp method 
 all chromophore cavities ( and not just those of chromophores m and n ) have  = 1 and the protein 
 solvent environment has  = 2 . in the present case there is 
 no protein environment forming binding pockets for the chromophore , 
 but the chromophores self - aggregate .
since the excitonic coupling 
 between the low - energy excitations is taken into account explicitly 
 in the hamiltonian a certain amount of polarization is included in 
 this way .
it is this residual polarization 
 that contributes to the screening in the present possion - tresp calculations . 

we take nr = 1.5 as has 
 been estimated from the dispersive shift in monomer transition energies 
 discussed above .
the excitonic coupling jmn then follows as33 in order to simplify the above 
 procedure , the above coupling is 
 compared with the coupling in vacuum34and a screening / local field correction factor fmn = jmn / jmnvac is determined . 

as will be shown below fmn varies between 0.50 and 0.60 and depends on the mutual geometries 
 of monomers in the tube aggregate .
the variation in fmn by the nuclear dynamics is found to 
 be small in the md simulations ( see below ) .
therefore , time - independent 
 factors fmn can be assumed . 
 in rough estimations of optical spectra
, we take an average screening / local 
 field correction factor f  0.55 , determined 
 as described in detail later . since different quantum chemical 
 methods give a similar shape of 
 the transition density but magnitudes that may differ by a factor 
 of up to 1.5 , it is necessary to rescale the quantum chemical transition 
 density by a constant factor that is chosen such as to reveal the 
 experimental transition dipole moment . as shown by knox and spring the dipole strength d ( in units of d ) of a chromophore in a certain spectral 
 region 
is given as 9.186  10nra( ) , where nr is the refractive index of 
 the medium and a( ) 
 = d()/ , with the absorption coefficient ( ) . 

we measured the absorption coefficient of ttbc in dmso , which has 
 a refractive index nr = 1.48 that is practically 
 identical to the one estimated for the tube and obtained a dipole 
 strength of 14.4 d. taking into account an average screening / local 
 field correction factor of f  0.55 results 
 in an effective transiton dipole moment of deff =   = 10.7 d , which is close to the deff = 11.4 d that has been used before for this 
 type of chromophore .
please note that 
 the present estimate of deff is an upper 
 bound , since the d in poisson - tresp refers to the 
 vacuum rather than the medium transition dipole moment of the chromophore . 

smaller deff however did not lead to any 
 reasonable agreement between calculated and measured spectra .
the 
 franck  condon factor s of the effective vibrational 
 mode has been estimated from optical spectra of ttbc monomers in methanol . 

the monomer absorption is given as35with the 
 franck  condon factor f that 
 is related to the huang 
for comparison with experimental data we have dressed the 
 -functions in eq 35 with gaussians of width  = (  + 1 ) 
 0 in order to take into account inhomogeneous broadening . 

the larger broadening of the higher vibrational states corrects for 
 the limitations of the effective mode model .
the absorption spectrum 
 calculated for s = 0.83 , vib = 
 800 cm and a 00 transition energy   corresponding to a wavelength of 515 nm , 
 is compared in figure 3 to the experimental data , revealing good agreement .
in addition , 
 we also show a homogeneous spectrum , calculated by dressing the sticks 
 in eq 35 by lorentzians 
 of width 30 cm , in order to visualize the vibrational 
 progression that will be important later when we will discuss how 
 it is changed by the excitonic coupling in the aggregate .
experimental 
 absorption spectrum of ttbc monomers in methanol ( black 
 dashed line ) compared to an effective mode model that is inhomogeneously 
 ( red line ) or homogeneously ( blue line ) broadened . for a direct consistency check of our final structural 
 model , we 
 also simulated the tem images . to this end , we estimated the actual 
 degree of noise in the raw data and determined the contrast transfer 
 function ( ctf ) of the used microscope .
the latter data allow one to 
 calculate projection images of the theoretical molecular model structures 
 with a resolution and degree of noise as found for the raw data .
as 
 raw cryo - tem image data are of poor contrast we used the single - particle 
 analysis approach to construct a noise - reduced sum image of some tube 
 segments . to get an impression of the reached consistency this sum 
 image and the simulated images , the class sum image of tubes or only 
 their density profiles
technical details of the image 
 processing and simulation procedures are given in the si .
cryo - tem images show thin elongated 
 thread - like structures which are interpreted as a projection of single - walled 
 dye tubes .
this interpretation becomes clear at high magnification , 
 where a pattern of fine lines with a spacing of about 19  is 
 visible for isolated individual fibers .
a line scan perpendicular 
 to the long axis of a single fiber reveals a cylindrical structure , 
 where the parallel lines with high contrast correspond to the walls 
 of a tubular aggregate .
the tem data provide one constraint for 
 our structure model ; the compatibility with the crystal lattice of 
 dyea or dyem provides another one .
these constraints will be fulfilled 
 simultaneously by using the following lattice - wrapping procedure .
first , a crystal layer is defined . because of its dense packing the 
 crystal plane ( 100 ) is taken for both dyea and dyem .
then a wrapping 
 vector c is chosen , the length of which becomes the circumference 
 of the tube after wrapping . to be compatible with the crystal lattice , c has to be a lattice vector itself c = n1c + n2b . from the range of likely tube diameters
each of them can be described by lattice indices 
 ( n1 , n2 ) . 

the two half circles in the ( 100 ) plane define the 
 range of possible tube circumferences , and the different vectors c define the wrapping directions considered .
the inner circle 
 corresponds to dmin = 16  and the 
 outer one to dmax = 26  .
choice of wrapping vectors c = n1c + n2b in the ( 100 ) plane of dyea ( left ) and dyem ( right ) .
each wrapping 
 vector is depicted as a solid line ; its label corresponds to its indices 
 ( n1 , n2 ) . 

minimum and maximum circumference as estimated from cryo - tem data 
 are drawn as half circles .
examples of tubes generated from the ( 100 ) plane 
 of dyea crystals 
 ( left ) and dyem crystals ( right ) applying the wrapping vector c = 6c + 0b .
this procedure may fail to provide a 
 reasonable 
 structure , because it creates steric overlaps between dye molecules 
 in some cases .
some of the overlapping structures can be  repaired  
 by increasing the tube diameter or increasing the spacing between 
 dye molecules in the direction along the tube axis .
this change corresponds 
 to a distortion of the primitive unit cell of the wrapped lattice 
 which increases the unit cell area and decreases the packing density . 

for the wrapping directions of figure 4 the resulting lattice parameters ( after removing steric 
 overlaps ) are given in the si .
after wrapping a representative subset of tubes 
 for dyea and dyem , linear optical spectra were calculated according 
 to eq 6 .
the following 
 simplifications , which turned out to be uncritical for a qualitative 
 interpretation of the spectra , were applied in order to save computational 
 time .
( i ) static disorder was described by applying a gaussian distribution 
 function for the local transition energies of the monomers .
a width 
 of 118 mev ( fwhm ) was estimated for this distribution function based 
 on a comparison of the calculated and measured low - energy absorbance 
 peak .
( ii ) the 
 mean local electronic transition energy   of all monomers was set equal to , corresponding 
 to a wavelength of 557 nm , 
 based on a comparison of calculated and experimental spectra . (
iii ) 
 an average screening local field correction factor f = 0.55 was used for the calculation of excitonic couplings with 
 the tresp method resulting in an effective 
 transition dipole moment of 10.7 d , as discussed before .
these 
 approximations allow us to obtain the optical spectra from a static 
 structure instead of a whole set of structures generated by md simulations . 
 as will be shown later the inhomogeneous spectra obtained by evaluating 
 the variations of site energies from the md trajectories are very 
 similar to the spectra obtained with the present simple method . here
, 
 the average over static disorder in local optical transition energies 
 is performed by generating random combinations of transition energies 
 and averaging the resulting optical spectra .
we find that sufficient 
 convergence of the spectra is obtained if 10  000 configurations 
 of disorder are taken into account in the monte carlo average .
we 
 varied the length of the tube and found that taking into account 
 around 100 chromophores leads to convergence of the absorption spectra .
the calculated linear optical spectra resulting from the different 
 wrapping directions are compared to experimental data in figures 6 and 7 for dyea and dyem tubes , respectively .
we see that for structures 
 generated from dyea only the wrappings in ( 6,0 ) and ( 5,5 ) 
 directions qualitatively match the experimentally determined linear dichroism and absorbance spectra .
the 
 intensity ratio between the main peaks is somewhat closer to the experimental 
 value for the ( 6,0 ) tube
. for dyem the wrapping direction ( 6,0 ) fits 
 best , although the red - shifted peak carries somewhat too little intensity . 

we take into account the ( 6,0)-dyea and ( 6,0)-dyem structures for 
 further optimization .
absorbance ( left ) and linear dichroism ( right ) spectra 
 of dyea 
 tubes that were generated by wrapping the ( 100 ) plane of the dyea 
 crystal in directions as described by the wrapping vector c = n1c + n2b denoted by ( n1 , n2 ) in comparison to experimental data 
 ( bottom ) .
vertical dashed lines are placed at the experimental maxima 
 for easy comparison with the calculations .
absorbance ( left ) and linear dichroism ( right ) spectra of dyem 
 tubes that were generated by wrapping the ( 100 ) plane of the dyem 
 crystal in directions as described by the wrapping vector c = n1c + n2b denoted by ( n1 , n2 ) in comparison to experimental data 
 ( bottom ) .
vertical dashed lines are placed at the experimental maxima 
 for easy comparison with the calculations . to test the stability of the 
 ( 6,0)-dyea and ( 6,0)-dyem candidate structures ,
md simulations were 
 carried out using the namd program package , the amber force field , and the gaff 
 parameter set .
the particle mesh ewald method was utilized in order to establish periodic 
 boundary conditions .
two snapshots for ( 6,0)-dyea taken at 60 
 ps and 2.7 ns are shown in figure 8a and 8b , respectively .
the 
 instability can be attributed to intermolecular distances being too 
 large in the initial structure .
the large intermolecular distances 
 stem from removing steric clashes as described before . at 2.7
ns a 
 well - ordered tube is recovered in the central region of the simulation 
 box .
snapshots of structures during md simulation with initial structures 
 obtained from wrapped crystal planes : ( a ) ( 6,0)-dyea after 60 ps , 
 ( b ) ( 6,0)-dyea after 2.7 ns , and ( c ) ( 6,0)-dyem after 0.5 ns . in the case of the alternative 
 (
6,0)-dyem starting geometry we 
 find that the structure breaks apart in the course of the md simulations . 
 at 0.5
ns the molecules partially realign and form a structure which 
 is similar to the ( 6,0)-dyea tube discussed above ( figure 8c ) .
hence , we conclude that 
 the ( 6,0)-dyem structure is further away from the stable tube structure 
 than the one of ( 6,0)-dyea . in a next step ,
the partially ordered 
 central segment of the md - simulated dyea tube at 2.7 ns ( figure 8b ) was used to determine 
 new structure parameters .
a least - squares fit between the disordered 
 tube and an ideal helical tube revealed the primitive lattice vectors 
 of the latter , given in the si , which deviate 
 from the lattice vectors of the original crystal plane of dyea .
next , 
 these new lattice vectors were used to create a new ( 6,0 ) tube .
this 
 geometry was found to be stable within a new md run which took 100 
 ns .
( a ) md snapshot of the 
 final tube structure viewed along the symmetry 
 axis of the tube ( top ) and perpendicular to this direction ( bottom ) . 

( b ) space - filling model of the final ideal tube structure with lattice 
 parameters c = 13.582  , b = 
 8.832  ,  = 140.344 , and  = 74.000 
 extracted from the final md structure by a least - squares fit .
the structure obtained after 100 
 ns md simulation was used again 
 to determine primitive lattice vectors used to create an ideal tube , 
 which represents our suggestion for the structure of the single - walled 
 tube of ttbc molecules with type i spectrum .
the lattice parameters 
 of this final ( 6,0 ) tube are c = 13.582  , b = 8.832  ,  = 140.344 , and  = 
 74.000 , where c and b are 
 the magnitudes of unit cell vectors c and b ( figure 4 ) ,  
 is the angle between these vectors , and the angle  denotes 
 the angle between the transition dipole moment of the dye molecule 
 and the symmetry axis of the tube .
the optical spectra resulting for this structure are compared 
 in figure 10 with 
 the experimental data revealing good agreement for both linear absorbance 
 and linear dichroism .
close inspection reveals a somewhat too large 
 intensity of the calculated spectra in the intermediate wavelength 
 region ( 515560 nm ) .
linear absorbance ( solid lines ) and linear 
 dichroism ( dashed lines ) 
 spectra calculated for the final structural model for the ttbc - cl 
 tube ( green lines ) compared to experimental data ( black lines ) . for a qualitative understanding 
 of the structure of the spectrum
, 
 we neglected static disorder and dressed the exciton stick spectra 
 with lorentzians of width 10 mev ( fwhm ) .
as before , a mean local transition 
 energy , corresponding 
 to a wavelength of 557 nm , 
 has been assumed for the monomers in the tube .
the resulting absorbance 
 spectra are shown in figure 11 ( green and red solid lines ) together with the experimental 
 data ( black solid line ) . for comparison
, we also included the homogeneous 
 spectra resulting from the bare exciton hamiltonian , that is , without 
 including the effective vibrational mode ( green and red dashed lines ) .
homogeneous 
 absorption spectra obtained by diagonalizing the bare 
 exciton hamiltonian ( dashed lines ) and the full hamiltonian that includes 
 the coupling to one effective intramolecular vibrational mode per 
 chromophore ( solid green and red lines ) . for illustrative purposes , 
 stick spectra were broadened by loretzians of width 10 mev ( fwhm ) . 

red lines indicate a polarization parallel to the symmetry axis of 
 the tube , and green lines mark those transitions which are polarized 
 perpendicular to this axis . for comparison
, we also show the experimental 
 absorption spectrum ( black solid line ) . in the homogeneous spectrum
we find a single low - energy peak 
 at 
 around 590 nm ( the j - band ) , the transition dipole moment of which 
 is polarized along the symmetry axis of the tube , and a number of 
 peaks between 450 and 550 nm ( the h - band ) with transition dipole moments 
 that are polarized perpendicular to this axis . in the bare exciton 

spectra it is seen that part of the broadening of the h - band is due 
 to the presence of multiple exciton transitions with nonzero transition 
 dipole moments . in the case of the j - band
switching on the coupling 
 of excitons to the effective intramolecular vibrational mode changes 
 the h - band qualitatively but leaves the j - band unchanged , except for 
 a 10 nm red shift and a slight reduction in intensity .
interestingly , 
 practically no vibrational progression is visible for the j - band . 

this remarkable behavior is explained by our analytical treatment 
 and the perturbation theory . as shown by the analytical treatment 
 of the homogeneous spectrum
, the coupling to the effective vibrational 
 mode does not lead to a mixing between different eigenstates of the 
 bare exciton hamiltonian .
the perturbation theory explains how for 
 every eigenstate of the bare exciton hamiltonian the vibrational progression , 
 which one would naively expect , is changed by the off - diagonal elements 
 of the exciton  vibrational coupling .
whereas in the case of 
 the j - band the majority of the oscillator strength is redistributed 
 to the lowest energy transition , that is , to the  = 0 state 
 of the lowest exciton state , in the case of the transitions in the 
 h - band , there is redistribution of oscillator strengths toward transitions 
 with larger . for the present system we have s < 1 , and therefore , all of the mixing coefficients c(0 ) ( eq 24 ) for  = 0 in 
 the case of the j - band are positive . in addition
, we have   for all 
 h - band exciton states  with 
 strong transition dipole moments , and therefore , it holds that c( ) > 0 for  < . hence , there is 
 redistribution 
 of oscillator strength toward the states with larger . this 
 effect is illustrated in figure 12 , where for the bare exciton transition with the largest 
 oscillator strength of the h - band and that of the j - band the redistribution 
 of oscillator strength by the exciton 
as seen in figure 12 the exciton  vibrational 
 coupling ( the first part in eq 18 ) leads to a vibrational progression in the exciton spectrum 
 ( figure 12b ) , which 
 is , however , qualitatively changed by the off - diagonal elements ( the 
 second part in eq 18 ) of the exciton 
whereas 
 for the exciton transitions in the j - band practically all oscillator 
 strength is redistributed to the  = 0 transition , for the transitions 
 in the h - band it is redistributed toward several vibronic transitions 
 with larger  , as expected .
the perturbative analysis already 
 explains the effect qualitatively as figure 12c ( perturbation theory ) in comparison to figure 12b ( no redistribution 
 of oscillator strength between different vibronic transitions of the 
 same excitonic transition ) and figure 12d ( exact numerical diagonalization ) demonstrates .
the most intense transitions of the h - band and the j - band 
 are marked by a green and red triangle , respectively .
those two transitions 
 are investigated further in c and d. ( b ) including the coupling to 
 the effective vibrational mode but neglecting the off - diagonal parts 
 of this coupling ( the sum over  in eq 18 ) .
vibrational coupling in first - order perturbation 
 theory ( eqs 23 and 25 ) .
we note that the present finding of a different redistribution 
 of oscillator strength for h- and j - bands is in line with earlier 
 numerical results by spano .
we are , however , 
 not aware of such a simple analytical explanation of the effect , as 
 presented in this work .
experimentally this effect is known since 
 the work of jelley and scheibe on the classical j - aggregates .
we also have 
 to note that spano went beyond the single - particle approximation , 
 used here for the inclusion of vibrational states , and found that 
 the redistribution can be somewhat less if a two - particle approximation 
 is used .
an alternative way to relate the vibronic line shape of the 
 isolated monomer to that of the aggregate was offered by briggs and 
 co - workers using a greens - function approach termed coherent exciton scattering 
 ( ces ) .
this approach was recently shown to be equivalent to a direct 
 diagonalization of the exciton 
we want to exploit the rotational symmetry of 
 our structural model to explain the general structure of its linear 
 absorption and dichroism spectra by applying analytical solutions 
 of the bare exciton problem worked out by knoester and co - workers 
 before in different approximations .
we 
 keep in mind that the inclusion of exciton  vibrational coupling 
 will lead to additional redistribution of oscillator strength and 
 splitting of exciton vibrational eigenstates but independently for 
 every eigenstate of the bare exciton hamiltonian , as has been shown 
 above .
our best - fitting structure model possesses 
 6-fold rotational symmetry with respect to its symmetry axis .
the 
 exciton states |k in such a structure can 
 be written as36where |n1 , n2 denotes an excited state 
 that is localized 
 at the n2th chromophore in the n1th ring of the tube .
the rotational symmetry 
 demands that the wave function k(n1 , n2 ) 
 possesses bloch form in ring direction .
using 2 = 
 2/n2 , where n2 denotes the number of chromophores within a ring , and counting 
 the eigenstate of the ring by k2 , we get 
 for k2 > 0 ( see eq 12 in ref ( 23))37hence , the eigenstate |k 
 = |k1k2 
 of the tube in eq 36 can be written as38where39denotes the contribution of the excited state 
 that is localized at the k2th eigenstate 
 of the n1th ring to the overall exciton 
 state |k1k2. in particular , it holds that the transition dipole moment40is only nonzero for those k2 values for which the transition - dipole moment 0 ,  n1  k2 of the isolated rings is nonzero .
transforming 
 the bare - exciton hamiltonian from the localized basis |n1n2 to that of delocalized 
 states of the rings |n1k2 gives rise to off - diagonal elements , that is , 
 coupling j(n1 , k2 ) between the eigenstates of different rings 
 in the same k2 state , that are n1 rings apart .
this coupling reads41here j(n1 , n2 ) is the excitonic coupling 
 between a pair of chromophores in the two rings , where the position 
 inside the rings differs by n2 .
the prime 
 in the sum indicates that for n1 = 0 we 
 do not sum over n2 = 0 . for n1 = 0
both chromophores are in the same ring and the excitation 
 energy of the k2 state of the isolated 
 ring follows as42where   denotes the local vertical exciton 
 energy 
 of the chromophore in the tube ( eq 11 ) .
the transition dipole moments 0 ,  n1  k2 of the delocalized states in the rings are obtained 
 as43thereby it is seen that only k2 = 0 and k2 = 1 lead 
 to a nonzero transition dipole moment 0,n1k2 of the 
 ring .
if the transition dipole moment of the chromophores in the ring 0 ,  n1n2 have an angle  with respect to 
 the symmetry axis of the tube , the parallel component 0 ,  n1n2 = 0 ,  n1n2 cos  adds up constructively only for k2 = 0 in eq 43 , whereas the perpendicular components 0 ,  n1n2 = 0 ,  n1n2 sin  
 of the single chromophores give a nonzero transition dipole moment 
 of the ring only for k2 = 1 because 
 of the rotational symmetry of arrangement of transition dipole moments . 

the linear absorption spectrum is obtained from the transition dipole 
 moments of exciton states 0 ,  k1k2 in eq 40 .
taking into account 
 the rotational symmetry in the location of chromophore transition 
 dipole moments 0,n1n2 and that of the eigenstates 
 in the rings , where eigenstates with k2 = 1 and those with k2
= 1 are 
 degenerate , and performing an average over random orientations of 
 tubes with respect to the polarization of the external light field 
 gives for the absorption spectrum44where the oscillator strength 
 reads45here , the angle 
  describes the relative 
 rotation of neighboring rings around the symmetry axis of the tube . 
 using the parameters n1 , n2 , c , b , and  , 
 which define the unit cell and the circumference vector , this
can 
 be written as46for the 
 present system we find  = 30. 
 contrary to the rest of this section , in the above equation n1 and n2 stand for 
 the components of the circumference vector as described in the si . in the case of linear dichroism , the 
 tubes are oriented in a flow 
 cell and the absorption is measured with light polarized parallel 
 and perpendicular to the symmetry axis of the tube .
linear dichroism is defined as the difference between these 
 two absorbances ( parallel minus perpendicular ) .
taking into account 
 an average over random orientations of the complexes with respect 
 to rotation around the symmetry axis yields the linear dichroism spectrum47 as in the case of isotropic absorption , bands with nonzero 
 oscillator 
 strength are created for the three values of k2 = 0 , 1 .
the internal states of the bands are counted 
 by k1 . as discussed above , due to the 
 rotational symmetry of the tube ,
the k2 = 0 band is polarized parallel to the symmetry axis of the tube 
 and the k2 = 1 bands are polarized 
 perpendicular to this axis .
the oscillator strengths m ( k1 , k2 ) 
 ( eq 45 ) of the different 
 transitions will be calculated below in different approximations . before a general conclusion
can be drawn about the angle  
 between the chromophore s transition dipole moments and the 
 symmetry axis of the tube . as seen from the expression for ld( ) in eq 47 ,  
equals the ratio between the negative and 
 the positive part of the ld spectrum .
hence , within the present approximations 
 of neglecting static disorder and homogeneous broadening we can get 
 an estimate for the angle  directly from the positive and negative 
 parts of the experimental ld spectrum . analyzing this spectrum in 
 this way results in   70 , which is close to the 
  = 74 obtained indirectly by optimizing the structure 
 and calculating optical spectra with a theory that includes static 
 disorder . due to the presence of static disorder the rotational symmetry 
 in the rings is broken and a mixing between transitions that are polarized 
 in parallel and perpendicular directions occurs , which changes the 
 relatives oscillator strengths in the ld spectrum . integrating the 
 oscillator strength calculated in the presence of disorder and using 
 the same analysis as applied to the experimental ld spectrum results 
 in an angle 
 71 , which is practically identical 
 to the   70 inferred above .
the oscillator 
 strength m ( k1 , k2 ) in eq 45 can be obtained analytically using either 
 a nearest neighbor approximation for the inter - ring couplings or periodic 
 boundary conditions for the ends of the tube .
inspection of the inter - ring couplings for the present system ( table 1 ) suggests that in 
 particular for the k2 = 1 bands 
 the coupling between rings falls off too weakly with increasing n1 to justify a nearest neighbor approximation . 

indeed , we find large deviations between the spectrum obtained in 
 this approximation and the one resulting from an exact numerical diagonalization , 
 as discussed in the si .
for periodic boundary conditions the eigenenergies 
 of the tube 
 are obtained as48 the oscillator strengths 
 of the transitions in the k2 = 0 band 
 read49and that of the k2 = 1 bands are50hence , in the case of k2 = 0 all oscillator 
 strength is concentrated in the k1 = 0 
 transition , which occurs at a transition 
 energy51by applying the values for j(n1 , k2 = 
 0 ) in table 1 it is 
 seen that e0 = 2.145 ev ( 578 nm ) is strongly 
 red shifted with respect to the monomer excitation energy   since 
 all j(n1 , k2 = 0 ) are negative . in the case of the k2 = 1 bands , 
 the oscillator strength is distributed over several transitions and 
 the one with the largest oscillator strength occurs at a transition 
 energy that is close to52 as shown in ref ( 23 ) , for very long tubes and 
 the periodic boundary conditions considered , 
 this transition is the only one carrying oscillator strength for k2 = 1 . for the present system ( table 1 )
we obtain eh = 2.460 ev ( 504 nm ) , which 
 is blue shifted with respect to the monomer transition energy .
as seen in tables 1 and 2 , the inter - ring 
 couplings are responsible for the blue shift . in figure 13
the absorption spectrum resulting 
 from numerical diagonalization of the bare exciton hamiltonian is 
 compared to the one obtained analytically by using periodic boundary 
 conditions , described above .
the qualitative behavior of the numerical 
 exact spectrum is well reproduced , in particular , the position and 
 intensity of the k2 = 0 transition that 
 is polarized parallel to the symmetry axis of the tube and the splitting 
 between this transition and the strongest transition of the k2 = 1 bands .
there are however quantitative 
 deviations between the intensities of the different transitions in 
 the k2 = 1 bands obtained in the 
 two different calculations .
the quantity re[j*(n1 ; k2)e ] in the 
 last column is used in eq 52 to estimate the energy of the most intense transition of 
 the h - band .
homogeneous absorption 
 spectrum resulting from the bare exciton 
 hamiltonian by numerical diagonalization ( black ) or analytical theory 
 ( eqs 48 , 49 , and 50 ) using periodic boundary conditions 
 ( green ) .
stick spectra ( vertical lines ) were dressed with lorentians 
 of fwhm 10 mev .
dashed lines represent transitions that are polarized 
 perpendicular to the symmetry axis of the tube and solid lines those 
 which are polarized parallel to this axis . vertical dashed 
finally , we use the md trajectories for 
 a microscopic description 
 of static disorder in transition energies and excitonic couplings . 
 for this purpose
500 snapshots between 22 and 72 ns of the md trajectories 
 were taken , and for each of the nuclear configurations the fluctuation 
 in transition energies   ( eq 27 ) was calculated . 

in addition , we used this parametrization 
 to explain the redshift between the vacuum transition energy   estimated for 
 the monomer in methanol and 
 the   ( 557 nm ) estimated 
 above from the calculations 
 of the optical spectra of the tube . for this purpose , we take the   and explicitly calculate the 
 electrostatic 
 ( eq 27 ) and dispersive 
 ( eq 29 ) transition energy 
 shift induced by the interchromophore coulomb coupling in the tube . concerning the excitonic couplings , an analysis of the screening 
 factors fmn of selected 
 snapshots revealed that the fluctuations of fmn along the md trajectories are small ( figure 14 ) .
the screening 
 factors fmn , however , 
 depend on the mutual orientation of chromophores , and 6 major groups 
 of screening factors are identified . for these groups we determined 

averaged screening factors that range between 0.58 for group 1 down 
 to 0.47 for group 2 , the overall average being 0.55 used before . in 
 the calculation of static disorder along the md trajectories
we distinguish 
 between the different groups and apply these average screening values 
 to the calculation of excitonic couplings with the tresp method53where the fluctuation of vmn(t ) is given solely 
 by the variation of ri(m)(t ) and rj(n)(t ) and not by that of fmn , which is taken from the respective group of screening factors the 
 chromophore pair ( m , n ) belongs 
 to .
the resulting optical spectra are compared in figure 15 to the experimental data 
 and show only slightly less agreement than the spectra obtained by 
 the phenomenological model of static disorder , discussed above ( figure 11 ) .
the calculated 
 spectrum was shifted by 80 mev to the red in order to implicitly take 
 into account the dispersive shift by the solvent molecules .
this shift 
 is within the range of 50100 mev estimated before for this 
 type of interaction .
screening factors of 
 excitonic couplings , obtained by the poisson - tresp 
 method , as a function of the vacuum coupling , taking into account 
 nuclear coordinates from 5 different snapshots along the md trajectories 
 ( at 37 , 42 , 47 , 52 , and 57 ns ) .
screening factors are grouped by the 
 vertical lines , and dashed horizontal lines denote the average screening 
 factor for a given group .
measured absorption ( black solid line ) and linear dichroism ( black 
 dashed line ) spectra are compared to calculated spectra ( red and green 
 lines , respectively ) . in the calculations
static disorder in site 
 energies and excitonic couplings was described microscopically taking 
 into account md trajectories . for comparison of theoretical model structures 
 with experimental cryo - tem data , projection images of the model structures 

were simulated , filtered , and normalized in the same way like the 
 experimental data . at the available resolution
the visual comparison 
 demonstrates a high level of consistency ( figure s11 in the si ) .
however , due to the small size of dye monomers 
 and the high degree of statistical noise details of the molecular 
 packing within the tubes can not be resolved by cryo - tem and are also 
 invisible in the mimicked model tem images . to provide a quantitative 
 assessment we generated cross - sectional density profiles of both the 
 simulated images and the sum image of the tem data .
as shown in figure 16 the curve progression 
 of the density profile derived from the experimental data ( red line ) 
 corresponds very well to the progression of the density plots of the 
 simulated micrographs derived from tomographic projection views of 
 the ( 6,0 ) dyea tube model ( black lines ) . in other words , within the 
 achievable resolution the deviations within the model projections 
 due to different orientations are greater than the differences between 
 the experimental structure and the theoretical model .
quite similar 
 results were obtained for other tube models with suitable diameter 
 and suggest that a clear proof of one particular structure model against 
 others is not possible in this way .
calculated cryo - tem 2d - density profiles 
 of the ( 6,0 ) dyea tube 
 model ( black lines ) for 10  tomographic projections  . 
 experimental density profile from the sum image of cryo - tem of the 
 tube ( red line ) .
horizontal straight line corresponds to the mean 
 gray value of the background ( 121.6  11.0 ( std .
therefore , in the present study , 
 the optical spectra and the molecular 
 stability were analyzed in addition to the cryo - tem data .
the structure 
 proposed on these grounds is fully consistent with the experimental 
 cryo - tem data , as figure 16 demonstrates .
a systematic 
 procedure for the structure prediction of self - organized 
 single - walled tubes of cyanine dyes has been developed and successfully 
 applied to ttbc .
the method combines information from cryo - tem with 
 that from molecular crystals of ttbc from molecular mechanics and 
 optical spectra of the monomer and the aggregate .
the frenkel exciton 
 hamiltonian of the aggregate , used to calculate optical spectra , is 
 parametrized by using quantum chemical / electrostatic methods , developed 
 before .
the arrangement of molecules in dyea crystals , containing 
 one molecule per unit cell , is found to be closer to the final structure 
 than that of the alternative dyem containing two molecules per unit 
 cell .
thus far we have not found any dyem tube that can explain the 
 optical spectra and is also stable during the md simulations .
in contrast , 
 our iterative refinement of the ( 6,0 ) dyea tube , which effectively 
 adjusted the unit cell , finally resulted in a stable tube .
although 
 this result at first glance is somewhat counterintuitive , it sheds 
 some light on our strategy of choosing the starting structure for 
 the md simulations . obviously the interaction in a plane in a 3d crystal 
 does not always contain the major interaction of a solvent - exposed 
 single - walled tube .
the proposed structure of the ( 6,0 ) dyea 
 tube consists of rings 
 of 6 chromophores with rotational symmetry .
the transition dipole 
 of each chromophore forms an angle of  = 74 with respect 
 to the symmetry axis of the tube .
neighboring rings are rotated by 
 an angle  = 30 with respect to each other .
this structure 
 explains all experimental data available so far and allows for a detailed 
 investigation of the effect of interchromophore couplings on the optical 
 properties of the aggregate that after all allowed for the prediction 
 of the structure . from the monomer spectrum in methanol
a mean 
 00 transition 
 energy corresponding to a wavelength of 515 nm , an effective vibrational 
 frequency of 800 cm , and a huang 
the interchromophore coupling 
 in the tube leads to a striking change in the optical properties . 

two main absorption bands , the transition - dipole of which are orthogonally 
 polarized , appear : a sharp low - energy j - band that is polarized along 
 the symmetry axis of the tube and a much broader high - energy h - band . 
 such hybrid h- and j - aggregate - like properties have been reported 
 before for semiconducting polymers and 
 terrylene dye crystals and have led to 
 the classification as hj - aggregates .
the difference in widths of the 
 two bands is shown to result from a redistribution of oscillator strengths 
 among vibronic transitions that is induced by exciton delocalization 
 and by coupling of excitons to the effective vibrational mode . an 
 analytical model in combination with perturbation theory
how 
 unique the present proposal is , additional 
 optical experiments like pump  probe and 2d electronic spectra 
 would be helpful
. it will be interesting to study exciton transfer / relaxation 
 between the h- and the j - bands in this system .
we expect that the 
 inclusion of exciton  relaxation - induced lifetime broadening 
 will further improve the agreement between calculated and measured 
 spectra .
 cryogenic transmission electron microscopy 
 ( cryo - tem ) studies suggest 
 that ttbc molecules self - assemble in aqueous solution to form single - walled 
 tubes with a diameter of about 35  . in order to reveal the arrangement 
 and mutual orientations of the individual molecules in the tube , we 
 combine information from crystal structure data of this dye with a 
 calculation of linear absorbance and linear dichroism spectra and 
 molecular dynamics simulations . 
 we start with wrapping crystal planes 
 in different directions to obtain tubes of suitable diameter . 
 this 
 set of tube models is evaluated by comparing the resulting optical 
 spectra with experimental data . 
 the tubes that can explain the spectra 
 are investigated further by molecular dynamics simulations , including 
 explicit solvent molecules . from the trajectories of the most stable 
 tube models , the short - range ordering of the dye molecules 
 is extracted 
 and the optimization of the structure is iteratively completed . the 
 final structural model is a tube of rings with 6-fold rotational symmetry , 
 where neighboring rings are rotated by 30 and the transition 
 dipole moments of the chromophores form an angle of 74 with 
 respect to the symmetry axis of the tube . 
 this model is in agreement 
 with cryo - tem images and can explain the optical spectra , consisting 
 of a sharp red - shifted j - band that is polarized parallel to to the 
 symmetry axis of the tube and a broad blue - shifted h - band polarized 
 perpendicular to this axis . 
 the general structure of the homogeneous 
 spectrum of this hybrid hj - aggregate is described by an analytical 
 model that explains the difference in redistribution of oscillator 
 strength inside the vibrational manifolds of the j- and h - bands and 
 the relative intensities and excitation energies of those bands . in 
 addition to the particular system investigated here 
 , the present methodology 
 can be expected to aid the structure prediction for a wide range of 
 self - assembled dye aggregates . 
Introduction
Theory
Results and Discussion
Conclusions