In this work, we systematically investigate single-electron capture process in the collision between N⁶⁺(1s) ions and H(1s) atoms in a wide energy range from 0.25 to 225 keV/u by using a two-electron semiclassical asymptotic-state close-coupling method. Spin-averaged and spin-resolved total cross sections, as well as n-resolved and $n\ell $-resolved partial cross sections, are calculated and comprehensively compared with existing experimental measurements and theoretical predictions. The results show at low energies (<10 keV/u), energy dependence of the total cross section is weak, and it follows a monotonically decreasing trend at higher energies. The analysis of $n\ell $-resolved cross sections reveals the strong coupling effects between various channels at low energies, while at high energies the relative $\ell $ distributions in each $n\ell $-resolved cross section approximately follow the statistical $\ell $ distribution, for which the electrons are therefore mainly captured into subshells of the maximum $\ell $. The present study demonstrates the importance of a two-electron treatment taking into account electronic correlation and the use of extended basis sets in the close-coupling scheme. However, substantial discrepancies exist among theoretical approaches at low energies. It is clear that further experimental and theoretical efforts are required to draw definite conclusions. Our work provides a complete and consistent set of cross sections in a broad range of collision energies, which can be used for various plasma diagnosis and modeling. The datasets presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j00213.00143.