Transport coefficients of He$^{\mathrm{+}}$ ions in helium

New experimental mobilities of $^{\mathrm{4}}$He$^{\mathrm{+}}$ in $^{\mathrm{4}}$He at 298.7 K, as a function of E/N, have been determined. Uncertainties in the mobilities were reduced to about 1{\%} by using a shuttered drift tube. Comparison with previously measured values show that only one set of previous data is reliable. We demonstrate that the mobilities and diffusion coeffcients of $^{\mathrm{4}}$He$^{\mathrm{+}}$ in $^{\mathrm{4}}$He can be calculated over wide ranges of E/N with high precision if accurate potential energy curves are available for the X$^{\mathrm{2}}\Sigma_{\mathrm{u}}^{\mathrm{+}}$ and A$^{\mathrm{2}}\Sigma_{\mathrm{g}}^{\mathrm{+}}$ states, and if one takes into account resonant charge transfer and corrects for quantum-mechanical effects. Potentials, obtained by extrapolation of results from d-aug-cc-pVXZ (X$=$6,7) basis sets using the CASSCF$+$MRCISD approach were found to be in exceptionally close agreement with the best potentials available (separately) and with experiment, and those were subsequently used in a new computer program to determine semi-classical phase shifts and transport cross sections, from which the gaseous ion transport coefficients are determined. A new set of data for the mobilities of alpha particles (He$^{\mathrm{2+)}}$ ions was obtained as a byproduct of the experiment, but the transport theory has not yet been completed.