All-electron local-density-functional (LDF) total-energy calculations are used to study the interaction between icosahedral C60 and endohedral lithium, sodium, and potassium ions and atoms. LDF potential energies as a function of radial displacement from the center along the 5-fold and 3-fold axes of C60 suggest that the carbon shell-alkali interaction is spherically symmetric to a good approximation. At equilibrium Li+, Na+, and K+ are displaced radially outward 1.4, 0.7, and 0.0 angstrom, respectively, from the center of the ball in the ground states of both the neutral and positively-charged complexes. Excited intramolecular charge-transfer states of the neutral molecules exist in which the endohedral alkali ion is neutralized. For these electronically excited neutral molecules the equilibrium position of the alkali atom is at or very near the center of the C60 shell. A minimum in the spacing between totally-symmetric endohedral vibrational energy levels indicates a potential energy maximum at the center of the shell. The height of the potential energy maximum lies between the two corresponding vibrational energies.