In an attempt to discover and analyze trends in distance relationships and properties at the bond critical point (BCP) in linear or near-linear N-H . . .N hydrogen bonds, the geometry of such bonds in a large number of suitable simple chemical species was optimized at the RHF/6-31G** level. The results for 67 of these are reported here; the geometry of 19 of them was optimized also at the MP2/6-31G** level. Correlations between the internuclear N-H, H . . .N, and N . . .N separations as well as between the N BCP and H BCP distances for these data sets and for different model functions are described in detail. The special case of symmetric N-H-N bonds is discussed; comparison with available experimental evidence shows that the correlation functions derived from the-ab initio data have useful predictive value for crystallographic determinations involving short N-H-N bonds. Analysis of the correlation between the d(N-H) distance and the electron density rho (c) at the BCP has shown that although acceptable d,rho (c) representations are obtained when rho (c) is fitted over the entire d(N-H) range by a single model function, significantly better fits result for both the 6-31G** and the MP2/6-31G** set when two separate regression functions of the same type are used, one for the covalent and another for the H . . .N bonds. The implications of these findings are discussed. The results of the correlation analysis of the curvatures lambda (i), the Laplacians del (2)rho (c), and the kinetic energy densities at the BCP, based on the data presented in this paper, will be reported in a subsequent paper, together with some aspects of the energy of formation of the N-H . . .N bonds.