This is the second in a series of four closely related papers by the same author. These works introduce a new concept of second-order directional derivative for nonsmooth functions, and demonstrate its applicability to the study of necessary and sufficient conditions for optimality in finite-dimensional mathematical programming. The first paper [J. Math. Anal. Appl. (to appear)] defines this derivative and states its elementary properties. The current paper (second in the series) describes the derivative's role in necessary conditions for the unconstrained local minimization of a ``semismooth'' function over \({\mathbb{R}}^ n\), and offers a detailed comparison of the results with those given earlier by Ioffe, Ben-Tal and Zowe, and Womersley. The third paper gives second-order necessary conditions for constrained semismooth minimization in \({\mathbb{R}}^ n\) [SIAM J. Control Optimization 25, 1072-1081 (1987; Zbl 0635.49013)]. The fourth gives sufficient conditions for both constrained and unconstrained problems [Math. Oper. Res. (to appear)]. For maximum benefit, all four papers should be studied together. Also, the reader should note that the author has used different second-order directional derivatives in the past [Nonlinear Anal., Theory Methods Appl. 9, 1189-1209 (1985; Zbl 0575.49005)], and considers those defined in this series of papers to be better.