Highly accurate O(N) method for delocalized systems

The elongation method, developed in our groups, is an ab initio method approaching order O(N) type scalability with high efficiency and high accuracy (error <10^-8 au/atom in total energy compared to the conventional calculation) that can be applied to any one-dimensional (polymer), two-dimensional (surface) or three-dimensional (solid material) systems. For strongly delocalized systems, however, the accuracy of the original elongation method for the targeted entire systems declines by approximately two orders of magnitude in the total energy as compared to the value obtained by the earlier implemented version of the elongation method for nondelocalized systems. The relatively small differences (10^-6-10^-8 au) between the elongation method and conventional method total energies have caused more serious errors in the second hyperpolarizability, γ, especially in nano-scale systems which have accompanying strong delocalization. In order to solve this problem, we have incorporated a simple correction technique based on an additional "orbital basis" to the "region basis" in our original elongation method procedures. Some not so-well-localized orbitals are incorporated into the interaction with the attacking molecule. This treatment has been applied to some model nano- and bio-systems that previously have shown strong delocalization, and the high accuracy in the energy obtained for nonstrongly delocalized systems was retained even for the strongly delocalized systems, both for the energies and for the second hyperpolarizabilities. This is a major breakthrough and now expands the systems for which the elongation method can be used to calculate and predict second-order nonlinear optical properties for delocalized systems.