  We demonstrate that basis sets suitable for electronic structure calculations
can be obtained from simple accuracy considerations for the hydrogenic
one-electron ions $Y^{(Y-1)+}$ for $Y\in[1,Z]$, necessitating no
self-consistent field calculations at all. It is shown that even-tempered basis
sets with parameters from the commonly-used universal Gaussian basis set (UGBS)
[E. V. R. de Castro and F. E. Jorge, J. Chem. Phys. 108, 5225 (1998)] reproduce
non-relativistic spin-restricted spherical Hartree-Fock total energies from
fully numerical calculations to better accuracy than UGBS, which is shown to
exhibit huge errors for some elements, e.g. 0.19 $E_{h}$ for Th$^+$ and 0.13
$E_{h}$ for Lu as it has been parametrized for a single atomic configuration.
Having shown the feasibility of the one-electron approach, partially
energy-optimized basis sets are formed for all atoms in the periodic table,
$1\leq Z\leq118$, by optimizing the even-tempered parameters for $Z^{(Z-1)+}$.
As the hydrogenic Gaussian basis sets suggested in the present work are built
strictly from first principles, also polarization shells can be obtained in the
same fashion in contrast to previous approaches. The accuracy of the polarized
basis sets is demonstrated by calculations on a small set of molecules by
comparison to fully numerical reference values, which show that chemical
accuracy can be reached even for challenging cases like SF$_6$. The present
approach is straightforward to extend to relativistic calculations, and could
facilitate studies beyond the established periodic table.
