Analytic potentials and vibrational energies for Li${}_2$ states dissociating to $Li\left(2S\right)+Li\left(3P\right)$. Part 1: The ${}^{2S+1}{\mathrm{\Pi }}_{u\mathrm{/}g}$ states

Analytic potentials are built for all four ${}^{2S+1}{\mathrm{\Pi }}_{u\mathrm{/}g}$ states of Li${}_2$ dissociating to Li$(2S)$ + Li$(3P)$: $3b(3^3{\mathrm{\Pi }}_u)$, $3B(3^1{\mathrm{\Pi }}_u)$, $3C(3^1{\mathrm{\Pi }}_g),$ and $3d(3^3{\mathrm{\Pi }}_g)$. These potentials include the effect of spin-orbit coupling for large internuclear distances, and include state of the art long-range constants. This is the first successful demonstration of fully analytic diatomic potentials that capture features that are usually considered too difficult to capture without a point-wise potential, such as multiple minima, and shelves. Vibrational energies for each potential are presented for the isotopologues ${}^{6,6}$Li${}_2$, ${}^{6,7}$Li${}_2$, ${}^{7,7}$Li${}_2$, and the elusive `halo nucleonic molecule’ ${}^{11,11}$Li${}_2$. These energies are claimed to be accurate enough for new high-precision experimental setups such as the one presented in {[}Sebastian \emph{et al.} Phys. Rev. A, $\mathbf{\text{90}}$, 033417 (2014){]} to measure and assign energy levels of these electronic states, all of which have not yet been explored in the long-range region. Measuring energies in the long-range region of these electronic states may be significant for studying the \emph{ab initio} vs experiment discrepancy discussed in {[}Tang \emph{et al.} Phys. Rev. A, $\mathbf{\text{84}}$, 052502 (2014){]} for the $C_3$ long-range constant of Lithium, which has significance for improving the SI definition of the second.