Simpler is often better: Computational efficiency of explicitly correlated two-electron basis sets generated by the...

A measure Δ of computational efficiency of the explicitly correlated basis sets (XCBSs) generated by the regularized Krylov sequences of Nakatsuji (also known as "the free complement" or "the free iterative CI" method) is derived from the convergence characteristics of the underlying iterative process. A complete mathematical definition of this process, which includes a crucial projection operator undefined in previous publications, is provided. Comparison of the values of Δ pertaining to several XCBSs designed for the helium isoelectronic series (for which Δ = -K ^-1/3 ln ϵ, where ϵ is the error in the computed energy and K is the number of the basis functions comprising the basis set) leads to a surprising conclusion that, among sufficiently large XCBSs, those stemming from the seed basis sets (SBSs) independent of the interelectron distance are the most efficient ones. Consequently, taking into account the simplicity of the resulting matrix elements of diverse quantum-mechanical operators, the XCBSs generated from the {exp(-ζs), s ^1/2 exp(-ζs)} SBS (where s is the first of the Hylleraas coordinates) appear to be best suited for extremely accurate electronic structure calculations on helium-like species.