Hylleraas wave functions revisited

Optimality of the combination of N terms used in the Hylleraas N‐term wave function is examined for several two‐electron atoms. It is pointed out that the N‐term wave function given in the literature is not always the best N‐term function for a fixed N. For example, the known Hylleraas six‐term function Ψ_6=exp(−ζs) [1+{c_1}u+{c_2}t^2+{c_3}s+{c_4}s^2+{c_5}u^2] yields the energy of −2.903 329 354 a.u. for the helium atom, whereas a new six‐term function Ψ_{6,opt}=exp(−ζs) [1+{c_1}u+{c_2}t^2+{c_3}u^2+{c_4}s^2u+{c_5}s^3u] gives a much lower energy of −2.903 452 763 a.u. It is also shown that the best combination of N terms depends largely on the nuclear charge