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A DPF analysis yields accurate analytic potentials for Li$_2(a\,^3Σ^+_u)$ and Li$_2(1\,^3 Σ^+_g)$ that incorporate 3-state mixing near the $1\,^3 Σ^+_g$-state asymptote

A combined-isotopologue direct-potential-fit (DPF) analysis of optical and photoassociation spectroscopy data for the $a\,^{3}Σ_u^+$ and $1\,^3Σ_g^+$ states of Li$_2$ has yielded accurate analytic potential energy functions for both states. The recommended M3LR$_{5,3}^{8.0}(3)$ potential for the $a\,^3Σ_u^+$ of $^{7,7}$Li$_2$ has a well depth of $\,{\mathfrak D}_e= 333.758(7)\,{\rm cm^{-1}}$ and equilibrium distance of $\,r_e= 4.17005(3)~$Å, and the associated scattering lengths are $\,a_{\rm SL}= -14.759(9)\,$Å for $^{7,7}$Li$_2$ and $-1906(50)\,$Å for $^{6,6}$Li$_2$. For the $1\,^3Σ_g^+$ state, in spite of a gap of $\,\sim 5200\,{\rm cm}^{-1}$ (from $\,v(1\,^3Σ_g^+)= 8-61$) for which there are no data, the DPF procedure has no difficulty determining an accurate overall potential. The $1\,^3Σ_g^+$ state of the $^{7,7}$Li$_2$ isotopologue has a well depth of $\,{\mathfrak D}_e= 7092.417(33)\,{\rm cm^{-1}}$ and equilibrium distance of $\,r_e= 3.06524(9)~$Å. The long-range tail of the recommended M3LR$_{6,3}^{3.6}(9)$ potential energy function for the $1\,^3Σ_g^+$ state is defined by the lowest eigenvalue of a $3\times 3$ long-range interstate coupling matrix to take into account the 3-state mixing near its asymptote.