Origin of the enhancement of the second hyperpolarizabilities in open-shell singlet transition-metal systems with metal-metal multiple bonds

Using the spin-unrestricted coupled-cluster method, we explore the origin of the second hyperpolarizabilities (γ) of singlet dichromium(II) and dimolybdenum(II) model systems with various bond lengths as a function of the diradical characters of the dσ, dπ, and dδ orbitals. Both systems exhibit enhanced γ values in the intermediate diradical character region, but by using a partitioning scheme, the dσ electrons are shown to play the essential role in contrast with the π-electrons of conventional organic π-conjugated systems. Then, in the equilibrium bond length region, the γ values are still governed by dσ electrons in the dichromium(II) system, although by dδ/dπ electrons in the dimolybdenum(II) system.