An Optimal Way for Computation of Non-Linear Optical Properties of Crystals and Large Conjugated Molecules within Cluster Models

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Cluster models are of importance in quantum-mechanical computations of Nonlinear Optical (NLO) properties of macroscopic objects such as crystals and large conjugated molecules, since they preserve important electronic structure features, topology and symmetry of the nuclear frame. In a cluster approach the NLO properties such as polarizability (α), first (β) and second (γ) hyperpolarizability tensors are computed for a series of clusters large enough for further extrapolation of the calculated values to infinite cluster dimensions. For realistic models the most appropriate and acceptable is the finite field method, when α,β,γ are derived by differentiation of either induced dipole moment d(ɛ) or total energy E(ɛ) with respect to the intensity of a trial uniform electric field (ɛ). In this study, we apply non-finite-difference formulae of numerical differentiation with interpolating Lagrangian polynomials on regular evenand odd-point grids. Estimates of absolute errors (8) are obtained for derivatives of any order, including the number of points in the grid, the e value, the values of the higher derivatives of d(ɛ) and E(ɛ) with respect to e. Errors in α,β,γ are minimized with an optimal e that can be found a priori from model systems. Our approach is more than just a technical improvement: it yields the maximum size of a cluster that can be studied with a finite-field method. In particular, the dependence of the δ interval on the spatial length and extent of delocalization of valence electrons is established. Of special notice are the cases where the finite-field method fails due to divergence. It is found for the variational methods of quantum chemistry that other things being equal, the errors are decreased by two orders of magnitude by differentiation of E(ɛ) rather than d(ɛ), even though the order of derivative is lower by one in the latter case. New analytical expressions are presented for the higher order NLO properties of conjugated molecular systems.