Numerical spin-unrestricted Hartree-Fock (SUHF) wave functions have been computed for the neutral open-shell atoms Li to Br and the transition-metal ions Sc2+ to Cu2+. These wave functions have been used to obtain accurate values of the SUHF spin densities. These calculations are used to make a critical review of the numerical accuracy of previously reported SUHF spin densities and to compare our accurate values of the SUHF spin densities with those extracted from experimental data. A large majority of previous SUHF calculations have been performed using the analytic-expansion method. The spin density at the origin computed from analytic-expansion calculations is sensitive to the choice of basis functions, and it is very difficult to obtain a precise measure of its accuracy. Our numerical SUHF calculations yield the SUHF value of the spin density χ to an accuracy conservatively estimated to be ±0.01 a.u. Comparison with analytic-expansion calculations indicates that the use of large, carefully selected basis sets usually yields spin densities within 20% of the numerical SUHF values reported here. The calculated SUHF spin densities are compared with available experimental values and it is found that (a) for the alkali atoms Li, Na, and K, the SUHF spin densities are 97, 86, and 77% of experiment; (b) for the first-row atoms N,O, and F, the SUHF spin densities are 193, 172, and 186% of experiment; (c) the SUHF spin density for P is opposite in sign and 153% of the absolute value of the experimental spin density; (d) the SUHF spin densities of the 4s23dn transition-metal atoms Ti, V, Mn, and Co are 82, 75, 72, and 67% of experiment; and (e) the SUHF spin density of As is 231% of experiment. It is concluded that, although some correlations are possible between SUHF and experimental values, the SUHF method should be used with extreme caution for the prediction of spin densities at the origin. This work provides a guide to the reliability of the SUHF predictions. © 1970 The American Physical Society.