The electronic structure of an inorganic compound with a spin gap ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{-}{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ has been studied by first-principles calculations with the generalized gradient approximation. We have calculated the total energy of the crystal structure with ${\mathrm{P}}_{{\mathrm{mn}2}_{1}}$ and ${\mathrm{P}}_{\mathrm{mmn}}$ symmetries, respectively, which is still controversial in experiments. Insulating states are reproduced by both antiferromagnetic and ferromagnetic calculations, although metallic states are obtained within the nonmagnetic ones. After structure optimization, the crystal structure with the ${\mathrm{P}}_{{\mathrm{mn}2}_{1}}$ symmetry is unstable and the optimized atomic positions form the structure with the ${\mathrm{P}}_{\mathrm{mmn}}$ symmetry. The antiferromagnetic state with the ${\mathrm{P}}_{\mathrm{mmn}}$ symmetry has the lowest energy. Our numerical results for the antiferromagnetic case can explain many experimental results of the insulating state of ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{-}{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ above the charge ordering temperature.