A dynamic theory that connects electronic motions and the nonlinear optical response of conjugated polyenes is developed by introducing the concept of electronic normal modes. A useful picture for the mechanism of optical nonlinearities is obtained by identifying the few dominant modes. This quasi-particle electron-hole representation establishes a close analogy with small semiconductor particles (quantum dots) and is very different from the traditional approach based on electronic eigenstates. The effective conjugation length (coherence size), which controls the scaling and saturation of the static third-order susceptibility X((3)) with the number of double bonds, is related to the coherence of the relative motion of electron-hole pairs created upon optical excitation.

achieved after 1000 cycles at 3.4 A g −1 (or 20 A g −1 ), with a capacity retention rate of ≈84% (≈88%), without the use of any binder or conductive agent. Remarkably, they can survive an extremely fast charging rate at 70 A g −1 for 35 runs (corresponding to one full cycle in 30 s) and recover 88% capacity. This novel alloy-nanotube structure could represent an ideal candidate to fulfi ll the true potential of Si-loaded LIB applications.

The nonlinear optical response of conjugated polymers is calculated using an electron-hole (exciton) representation. Equations of motion are derived which map the calculation of # (3) onto the dynamics of TV(2TV -1) nonlocal coupled anharmonic oscillators representing electrons and holes, TV being the number of double bonds. The scaling of the static # (3) with size and Coulomb interactions is shown to be directly correlated with the exciton coherence size associated with the relative electron-hole motion. PACS numbers: 36.20.Kd, 42.65.-k, 71.35.+Z, 78.65.Hc Conjugated polymers with extended (delocalized) electronic states constitute an important class of materials with interesting nonlinear optical properties [1-7]. The magnitude of the off-resonant nonlinear response and its scaling with size have received considerable attention [2-5]. A power scaling law on molecular size % {3)~~Nb has been established [3], TV being the number of double bonds. Estimates of the scaling exponent b vary from 6 to 3.5 for the size range TV = 2-12. Values of 5.257 [4(a)] and 4.32 [4(b)] were reported for the Hiickel (single electron) model. Similar values were found when Coulomb interactions are incorporated, b =4.6 [5(a)] and b =5.27 [5(b)]. The scaling is expected to saturate for large TV where the thermodynamic limit implies that b~\.Understanding the origin of the nonlinear response and the factors determining the magnitude and the response time scale of large polyenes, and their scaling and saturation with size, constitutes an important experimental and theoretical challenge. A major obstacle in the theoretical modeling of these phenomena is the lack of an efficient method for computing the nonlinear response, particularly for large polyenes, where the conventional sum over molecular eigenstates expressions becomes prohibitively tedious.In this Letter we develop an anharmonic-oscillator picture for the nonlinear optical response of conjugated polymers using equations of motion for two-particle (electron-hole) variables. The method allows a very efficient calculation of the nonlinear response over a broad range of sizes and Coulomb interactions and resolves the ambiguity regarding the scaling exponent. The most notable result of the present study is the clear identification of the elementary excitations as charge-transfer excitons which i are intermediate between the molecular (Frenkel) and the semiconductor (Wannier) excitons. The coherence size determining # (3) is shown to be related to the exciton size associated with the relative electron-hole motion. We start with the Pariser-Parr-Pople (PPP) Hamiltonian which consists of the tight-binding single-electron (Su-Schrieffer-Heeger or Hiickel) Hamiltonian with the addition of Coulomb interactions [1]. The Hiickel model represents a linear chain with a single 2p z orbital per site and with alternating exchange couplings Pi=p(\-~S), Pi^fiil+S) and TV repeat units. The Coulomb interaction between two electrons located at positions x and x is modeled using the Ohno formula, Z(x -x...

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