Hartree equation

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential v ( r ) , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the self-consistent field method.

In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radial function P ( r ) / r and a spherical harmonic with an angular quantum number ℓ , namely ψ = ( 1 / r ) P ( r ) S ℓ ( θ , ϕ ) . The equation for the radial function was

In mathematics, the Hartree equation, named after Douglas Hartree, is

in R d + 1 where

and

The non-linear Schrödinger equation is in some sense a limiting case.