We have theoretically examined a series of organic molecules that exhibit large two-photon absorption cross sections in the visible region and that have been synthesized in different laboratories. One- and two-photon absorption cross sections of the four lowest excited states of each molecule have been calculated at the same theoretical level using ab initio response theory. It is found that the molecular length and the one-photon absorption intensity are quite strongly correlated factors, but that a corresponding correlation for the two-photon absorption is much weaker or is missing. In contrast, a most crucial role for large two-photon absorption is played by the pi center. For molecules with a given pi center a symmetrical structure with strong donor groups can result in a maximum two-photon absorption cross section. Our theoretical findings are consistent with some recent experimental observations. The chromophore based on dithienothiophene as pi center attached with symmetrical N,N-diphenylamine donors is found to have the largest two-photon cross section in the visible region among all known one-dimensional two-photon organic materials that have been reported in the literature.