The exciton scattering (ES) approach attributes excited electronic states in quasi-1D branched polymer molecules to standing waves of quantum quasiparticles (excitons) scattered at the molecular vertices. We extract their dispersion and frequency-dependent scattering matrices at termini, ortho, and meta joints for -conjugated phenylacetylene-based molecules from atomistic time-dependent density-functional theory (TD DFT) calculations. This allows electronic spectra for any structure of

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... size within the considered molecular family to be obtained with negligible numerical effort. The agreement is within 10 -20 meV for all test cases, when comparing the ES results with the reference TD DFT calculations. Theoretical understanding and simulation of electronic spectra in nanosized molecular systems are important areas of research in physics and chemistry. Commonly used quantum-chemical methodologies, such as time-dependent density-functional theory (TD DFT), are generally able to provide quantitatively correct results [1, 2] . However, the substantial numerical cost of such approaches [ON 2 ÿ ON 4 , N being the system size] makes computations for large molecules with thousands of atoms prohibitively expensive. This calls for development of multiscale ladder-type approaches. Recently we proposed an exciton scattering (ES) approach [3] designed for molecular systems where excited states can be described as strongly bound electron-hole pairs (excitons) [4 -6]. This picture holds for lowdimensional (quasi-1D) geometries, where strong Coulomb correlations result in exciton binding energies comparable to the optical gap [4, 5] . Conjugated organic macromolecules and molecular wires featuring delocalized -electron states, which are promising for numerous technological applications, are examples of such systems [7] [8] [9] [10] [11] . However, explicit computation of excited states in large conjugated molecules is not feasible with quantumchemical approaches (e.g., TD DFT) that properly account for electron exchange and correlations. In this Letter we develop an efficient multiscale modeling procedure based on the ES approach and demonstrate its remarkable accuracy in reproducing the electronic spectra of test polymers. The excitation quasimomentum k in an infinite polymer chain is well defined due to its discrete translational symmetry. The excitations are characterized by the spectrum !k that relates the excitation frequency ! to the quasimomentum, as well as by their size l e , i.e., a typical distance between the electron and the hole. At length scales longer than l e , the conjugated polymer molecules are represented by graphs whose edges correspond to linear chains [8] [9] [10] [11] [12] [13] , which allows for the development of a phenomenological theory. The main idea behind the ES approach is the representation of excited states in a branched conjugated molecule by states of quasiparticles (excitons) on the corresponding graph (see Fig. 1 ) [3, 14] . The picture of plane waves scattered at the molecular vertices is asymptotically exact in long linear segments, l l e . The computation of the excited state electronic structure in large molecules is divided into two steps: (i) retrieving the exciton properties in the molecular building blocks and (ii) solving a generalized "particle in a box" problem. Step (i) involves quantum-chemical calculations in simple molecular fragments and requires limited numerical effort. The numerical effort of step (ii) depends on the number of linear segments rather than on the number of single-electron orbitals, which substantially reduces computations in macromolecules. In a segment between vertices a and b an exciton wave function is given by a superposition of plane waves: x a ;ab expik ;ab x a ;ba expik ;ba x ; (1) where an integer x labels repeat units in the segment. The quasimomenta k ;ab and k ;ba are related to the excitation frequency ! through the exciton spectrum: !k ;ab !k ;ba !. Time reversal invariance in the absence of magnetic field results in the symmetric spectrum !ÿk !k; hence, k ;ab ÿk ;ba . Denoting the values of the outgoing and incoming plane waves at a joint a by a and ÿ a , respectively, we have the following relation in the segment of l repeat units [15]