Uniform description of the helium isoelectronic series down to the critical nuclear charge with explicitly correlated basis sets derived from regularized Krylov sequences.

An efficient computational scheme for the calculation of highly accurate ground-state electronic properties of the helium isoelectronic series, permitting uniform description of its members down to the critical nuclear charge Zc, is described. It is based upon explicitly correlated basis functions derived from the regularized Krylov sequences (which constitute the core of the free iterative CI/free complement method of Nakatsuji) involving a term that introduces split length scales. For the nuclear charge Z approaching Zc, the inclusion of this term greatly reduces the error in the variational estimate for the ground-state energy, restores the correct large-r asymptotics of the one-electron density ρ(Z; r), and dramatically alters the manifold of the pertinent natural amplitudes and natural orbitals. The advantages of this scheme are illustrated with test calculations for Z = 1 and Z = Zc carried out with a moderate-size 12th-generation basis set of 2354 functions. For Z = Zc, the augmentation is found to produce a ca. 5000-fold improvement in the accuracy of the approximate ground-state energy, yielding values of various electronic properties with between seven and eleven significant digits. Some of these values, such as those of the norms of the partial-wave contributions to the wavefunction and the Hill constant, have not been reported in the literature thus far. The same is true for the natural amplitudes at Z = Zc, whereas the published data for those at Z = 1 are revealed by the present calculations to be grossly inaccurate. Approximants that yield correctly normalized ρ(1; r) and ρ(Zc; r) conforming to their asymptotics at both r → 0 and r → ∞ are constructed.