"Inverse sodium hydride" (AdzH(+)Na(-)) is an alkalide compound synthesized in recent experiments containing the unusual charge distribution H+ and Na- (inverse charge state). The new class of compounds interests scientists to investigate their especial structures and properties. In this paper, the structures of three alkalides compounds, (Me)(3)NH+Na-, AdzH(+)Na(-), and AdzLi(+)Na(-), have been obtained in theory. Especially, the structure of AdzLi(+)Na(-) is still researched by experimental scientists. We investigated the NLO properties of the alkalides complexes for the first time and found that inverse sodium hydride AdzH(+)Na(-) has a considerably large NLO response beta(0) = 5.7675 x 10(4) an by density functional theory (DFT). To understand the essential features of the large NLO properties, four related systems have been also calculated. Their first hyperpolarizabilities are beta(0) = 7.357 x 10(3) au for (Me)(3)NH+Na-, beta(0) = 3.9 au for (Me)(3)NH+, beta(0) = 1.10 x 10(2) au for (Me)(3)NH+Cl-, and beta(0) = 6.20681 x 10(5) an for AdzLi(+)Na(-), respectively. By comparing, we found that, first, the Na- anion plays a crucial role in the considerably large first hyperpolarizability of inverse sodium hydride and, second, the first hyperpolarizability of inverse sodium hydride increases with the charge value of the sodium anion. The above results are useful for designing potential NLO materials.