Two new classes of (HCN)(n)...Li and Li...(HCN)(n) (n = 1, 2, 3) clusters with the electride characteristic are formed in theory by the metal Li atom attaching to the (HCN)(n) (n = 1, 2, 3) clusters. Because of the interaction between the Li atom and the (HCN)(n) part, the 2s valence electron of the Li atom becomes a loosely bound excess electron. Our high-level ab initio calculations show that these new clusters with the excess electron have large first hyperpolarizabilities, for example, beta(0) = -15 258 au for (HCN)...Li and beta(0) = -3401 an for Li...(HCN) at the QCISD/6-311++G(3df,3pd) level (only beta(0) = -2.8 au for HCN monoiner(26)). Obviously, the excess electron from the Li atom plays a crucial role in the large first hyperpolarizabilities of these clusters. The beta(0) value of (HCN)(n)...Li (beta(0) > 10(4) au, from sigma --> pi* transition) is larger than that of Li..(.)(HCN)(n), (beta(0) > 10(3) au, from sigma --> sigma* transition) for n = 1, 2, or 3. In addition, two interesting rules have been observed. They are that \beta(0)\ decreases with lengthening of the HCN chain for (HCN)(n)...Li clusters and that \beta(0)\ increases as n increases for Li..(.)(HCN)(n), clusters. In this paper, we discuss two classes of clusters that are highly similar to the electride structure model, of which the structural characteristics are that alkali metal atoms ionize to form cations and trapped electrons under the action of other polar molecules. Thus, the investigation on the large first hyperpolarizabilities of (HCN)(n)...Li and Li...(HCN)(n) (n = 1, 2, 3) may prompt one to study the unusual nonlinear optical responses of some electrides.