Multiplet Theory for Symmetric Wave Functions

Multiplet theory for a single configuration for symmetric wave functions is developed and applied to the $s^2$, $\mathrm{sp}$, $p^2$, $s^3$, $s^{2}p$, $s{p}^2$, $p^3$, $s^4$, $s^{3}p$, $s^2{p}^2$, $s{p}^3$, and $p^4$ configurations. Orthonormal eigenfunctions of $L^2$ and $S^2$ are given for these configurations in terms of symmetric products of one-particle functions with $\Sigma m_l=\Sigma m_s=0\mathrm{}$. With these eigenfunctions, the deuteron structure of molecules with up to four deuterons can be studied. The multiplets of the deuterides are compared with the multiplets of the hydrides. The difference between the multiplet structures imply that ${\mathrm{D}}_2$O can make three deuteronic electric dipole transitions where ${\mathrm{H}}_2$O can make only one protonic electric dipole transition; N${\mathrm{D}}_3$ can make three deuteronic electric dipole transitions and two deuteronic electric quadrupole transitions where N${\mathrm{H}}_3$ can make only one protonic electric dipole transition; and C${\mathrm{D}}_4$ can make three deuteronic electric dipole transitions and two deuteronic electric quadrupole transitions where C${\mathrm{H}}_4$ has no allowed protonic transitions.