Fine Structure of Helium

Under the expectation that experiments will soon give values for the fine structure intervals of the $2{}_{}{}^3{}_{}{}^{}P$ state of helium to an accuracy of $\frac{1}{{10}^6}$, we have undertaken a program of calculations which, it is hoped, will lead to a new determination of the fine-structure constant $\alpha$. This paper gives a brief survey of the over-all program, and a detailed report of the successful completion of the first task: The construction of approximate solutions to the Schrödinger equation which lead to average values of the leading fine structure operators accurate to about one part per million.