Improved Energy Bounds and Basis Set Construction Strategies for Lithium

Improved nonrelativistic energy bounds for the low-lying states of lithium are presented using the variational method in Hylleraas coordinates. For example, the nonrelativistic energies for the infinite nuclear mass case are $-7.478\,060\,323\,910\,147(1)$~a.u. for $1s^22s\, ^2\!S$, $-7.354\,098\,421\,444\,37(1)$~a.u. for $1s^23s\, ^2\!S$, $-7.318\,530\,845\,998\,91(1)$~a.u. for $1s^24s\, ^2\!S$, $-7.410\,156\,532\,652\,4(1)$~a.u.~for $1s^22p\, ^2\!P$, and $-7.335\,523\,543\,524\,688(3)$ ~a.u.~for $1s^23d\, ^2\!D$. The completeness of the angular momentum and spin configurations is investigated and examples presented for the $2P$ and $3D$ states to demonstrate the effect of different coupling schemes. In particular, the so-called second spin function (i.e.\ coupled to form an intermediate triplet state) is shown to have no effect on the final converged results, even for the expectation values of spin-dependent operators.