  In high magnetic field calculations, anisotropic Gaussian type orbital (AGTO)
basis functions are capable of reconciling the competing demands of the
spherically symmetric Coulombic interaction and cylindrical magnetic ($B$
field) confinement. However, the best available {\it a priori} procedure for
composing highly accurate AGTO sets for atoms in a strong $B$ field [Phys.\
Rev. A {\bf 90}, 022504 (2014)] yields very large basis sets. Their size is
problematical for use in any calculation with unfavorable computational cost
scaling. Here we provide an alternative constructive procedure. It is based
upon analysis of the underlying physics of atoms in $B$ fields that allows
identification of several principles for the construction of AGTO basis sets.
Aided by numerical optimization and parameter fitting, followed by fine tuning
of fitting parameters, we devise formulae for generating accurate AGTO basis
sets in an arbitrary $B$ field. For the hydrogen iso-electronic sequence, a set
depends on $B$ field strength, nuclear charge, and upon orbital quantum
numbers. For multi-electron systems, the basis set formulae also include
adjustment to account for orbital occupations. Tests of the new basis sets for
atoms H through C ($1 \le Z \le 6$), and ions Li$^+$, Be$^+$, and B$^+$, in a
wide $B$ field range ($0 \le B \le 2000$ a.u.), show an accuracy better than a
few $\mu$H for single-electron systems, and a few hundredths to a few mHs for
multi-electron atoms. The relative errors are similar for different atoms and
ions in a large $B$ field range, from a few to a couple of tens of millionths,
thereby confirming rather uniform accuracy across the nuclear charge $Z$ and
$B$ field strength values. Residual basis set errors are two to three orders of
magnitude smaller than the electronic correlation energies in muti-electron
atoms ...
