Second-order contributions to the fine structure of helium from all intermediate states

For the theoretical assessment of the $2{}_{}{}^3{}_{}{}^{}P$ helium fine structure to become comparable to the precision measurements that have been made, it is necessary that the theory be calculated through order ${\alpha }^{6}m{c}^2$. In particular, the second-order contribution from the Breit and mass-polarization operators must be evaluated to an accuracy of 1% or so. In this work, for each of five possible intermediate state symmetries the Dalgarno-Lewis method is used to obtain the first-order perturbed wave function, from which the second-order energy follows by integration. Both the perturbed and unperturbed wave functions are expanded in Hylleraas-type series with a progressively larger number of terms, the second-order energies being computed at each stage; up to 455 terms are used for ${}_{}{}^3{}_{}{}^{}P$ intermediate states and up to 286 for ${}_{}{}^1{}_{}{}^{}P$, ${}_{}{}^3{}_{}{}^{}D$, ${}_{}{}^1{}_{}{}^{}D$, and ${}_{}{}^3{}_{}{}^{}F$. The sequence of second-order energy results for each symmetry is extrapolated to the limit of an infinite number of basis functions to arrive at a final result. The ${}_{}{}^3{}_{}{}^{}P$, ${}_{}{}^1{}_{}{}^{}P$, and ${}_{}{}^3{}_{}{}^{}D$ states will contribute to both the larger and the smaller fine-structure intervals ${\nu }_{01}$ and ${\nu }_{12}$, respectively, while ${}_{}{}^3{}_{}{}^{}F$ and ${}_{}{}^1{}_{}{}^{}D$ states affect only ${\nu }_{12}$. The total theoretical result, up to order ${\alpha }^{6}m{c}^2$, for ${\nu }_{01}$ is much more accurate than that for ${\nu }_{12}$, allowing the fine-structure constant $\alpha$ to be determined very precisely by comparison of theory to experiment, with the result ${\alpha }^{-1\mathrm{}}=137.036\mathrm{}08\left(13\right)\mathrm{}$, good to 0.94 ppm.