H$_{2}$N: Part 1. Hyperfine energies

H$_{2}$N, (from frozen, irradiated ammonia), is the smallest of the large group of $\pi $ (or p)-electron free radicals. With $^{1}$H, $^{2}$H, $^{14}$N and $^{15}$N there are 4 H$_{2}$N isotopes, with corresponding sets of hyperfine interactions, available for measurements. In a simple model of H$_{2}$N, 1.0 electron spin is in a Slater N2p-wave perpendicular to the molecular plane and -0.033 electron spin density in 1s waves on each H; the small effects of 0.066 electron spin (in other waves) required for net unit electron spin can be added. In a more complex model, the electronic structure is expressed with the 19 function 6-31G$^{\ast }$ basis. Nuclear spin-state mixing arises from linear combinations of dipolar off-diagonal matrix elements, e.g. M$_{xx} \quad \equiv \quad \sigma \kappa <\Psi \vert \Sigma $(S$_{kz}$x$^{2}_{kn}$/r$^{5}_{kn }$+ S$_{k\mbox{'}z}$x$^{2}_{k\mbox{'}n}$/r$^{5}_{k\mbox{'}n})\vert \Psi >$ (Airne and Brill, Phys. Rev.A \textbf{63} 052511). The M's are calculated in a molecular coordinate system with formulas applicable to any basis. Euler angles transform from molecular to lab spherical polar angles giving \textbf{B} with respect to the principle hyperfine axes at each nucleus. It is now shown that the principle hyperfine A-values can be expressed in terms of the M's, e.g. A$_{zz}$ = A$_{Fermi}$- (4/3$\sigma )$( M$_{xx}$ + M$_{yy}$ - 2 M$_{zz})$, thereby simplifying the energy matrices.