Recursive Density-Matrix-Spectral-Moment Algorithm for Molecular Nonlinear Polarizabilities

Relations between chemical and electronic structure and linear and nonlinear optical polarizabilities of conjugated molecule have drawn much attention of chemists and physicists over the past 20 years [1–4]. Resonant nonlinear spectroscopy provides most valuable information on states that are inaccessible by linear techniques [5–9], whereas off-resonant measurements probe the collective response of many molecular eigenstates [1–3]. The size-scaling of various optical properties has been studied extensively, both theoretically [10–12] and experimentally [5,9,13,14]. Linear absorption of short oligomers with N < 12 − 20 carbon atoms shows an Ω ∼ N−μ scaling of the optical gap with μ ∼ 0.4 − 0.6 [5]. The Hückel model yields μ = 1 [12]. An important relation is the scaling of nonlinear susceptibilities with molecular size. The power scaling law γ ∼ N b for the third-order polarizability, where N is the number of carbon atoms, has been established experimentally in the early 70’th [15], and supported by theoretical calculations using the free electron model [16]. Numerous subsequent studies showed that for short chains, the exponent b can vary between 3 and 8, depending on the system and model, and eventually approaches 1 (saturates) for long molecules. The crossover between these two behaviors is related to the exciton coherence size [4,5,13]. Calculations performed using the Hückel model (which neglects Coulomb interactions) predict saturation at long chains (N ∼ 50) and b ∼ 5 − 9 [12,17], whereas calculations based on the Pariser-ParrPople (PPP) Hamiltonian which includes electronic correlations predict a shorter saturation size (N ∼ 20 − 30) and b ∼ 4−5 [4,18,1]. Sum-over-states calculations of γ in short oligomers yield a scaling exponents b ≈ 8 for symmetric linear cyanites and b ≈ 4 for linear polyenes [19]. Difficulties with the controlled synthesis and poor solubility of polyenic oligomers restricted early experimental studies to molecules with up to 30-40 carbon atoms [14,15,20] which showed no saturation. These problems have been overcome, and the saturation of γ has been observed experimentally at ∼ 200 double bonds [13], which is much larger than early estimates.