The numerical solution of a partial differential equation on the IBM type 701 electronic data processing machines

In 1927 Thomas¹, and in 1928 Fermi², obtained a partial differential equation now known as the Thomas - Fermi equation. From the solution of this equation with suitable boundary conditions may be obtained the charge and energy distribution of electrons in their ground state in a potential field. In 1930 Dirac³ obtained an equation somewhat similar to the Thomas-Fermi equation but with a term added to account in part for “exchange forces”, a quantum-mechanical effect. This equation is called the Thomas- Fermi-Dirac equation. Except for one-dimensional cases, this equation has never been solved (except, perhaps, very approximately), although since 1930 it has stood as a potentially powerful mathematical model for complicated systems such as molecules and atomic lattices, where the number of degrees of freedom prohibits the solution of the Schroedinger equation. During August, 1952, solutions of the Thomas-Fermi- Dirac equation for the diatomic Nitrogen Molecule were computed on the IBM Type 701 Electronic Data Processing Machines located at Poughkeepsie, New York. Different solutions were for different internuclear distances. The solutions obtained were very satisfactory in that the initially desired accuracy was actually obtained on the calculator. Here we describe the computing form used, the organization of the problem for the IBM Type 701 Electronic Data Processing Machines, and the speeds obtained. The physical interpretation of results and numerical methods used are to be discussed elsewhere