Metal metal multiply bonded complexes in their singlet state have been predicted to form a novel class of "sigma-dominant" third-order nonlinear optical compounds based on the results of dichromium(II) and dimolybdenum(II) systems (H. Fukui et al. J. Phys. Chem. Lett. 2011, 2, 2063) whose second hyperpolarizabilities (gamma) are enhanced by the contribution of the d sigma electrons with an intermediate diradical character. In this study, using the spin-unrestricted coupled-cluster method with singles and doubles as well as with perturbative triples, we investigate the dependences of gamma on the group and on the period of the transition metals as well as on their atomic charges in several open-shell singlet dimetallic systems. A significant enhancement of gamma is observed in those dimetallic systems composed of (i) transition metals with a small group number, (ii) transition metals with a large periodic number, and (iii) transition metals with a small positive charge. From the decomposition of the gamma values into the contributions of d sigma, d pi., and d delta electrons, the gamma enhancements are shown to originate from the d sigma contribution, because it corresponds to the intermediate diradical character region. Furthermore, the amplitude of d sigma contribution turns out to be related to the size of the d(z)(2) atomic orbital of the transition metal, which accounts for the dependence of gamma on the group, on the period, and on the charge of the metal atoms. These dependences provide a guideline for an effective molecular design of highly efficient third-order nonlinear optical (NLO) systems based on the metal-metal bonded systems.