Fine Structure Constants of MetastableH2in thecΠu3State

The expectation values of spin-orbit ($A_1$) spin-other-orbit ($A_2$) and spin-spin ($B_0$ and $B_2$) interactions have been evaluated over the electronic wave function for the $c{}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ state of ${\mathrm{H}}_2$. $A_1=5.737\mathrm{}$ kMc/sec, $A_2=-9.8\mathrm{}$ kMc/sec, $B_0=-1.33\mathrm{}$ kMc/sec, and $B_2=-3.84\mathrm{}$ kMc/sec. These values agree well with those obtained from Lichten's experimental fine-structure splittings. The electronic wave function used is a linear combination of Heitler-London-type and ionic-type wave functions. By expressing the gradient operators as components of a spherical tensor, the spin-other-orbit interaction can be expressed as a linear combination of irregular solid spherical harmonics of the first order, while the space part of the spin-spin interaction consists of irregular solid spherical harmonics of the second order. The method of expansion and evaluation of these irregular solid spherical harmonics for a two-electron system is discussed in detail. The small second-order perturbations due to the nearby rotational states are also calculated.