The Hartree method allows us to to change the
dimensional Schrödinger equation
electrons in 3 dimensions)
into a 3 dimensional equation for each electron.
This equation depends on the wavefunctions of the other electrons but can be solved
in a self consistent way using the variational principle and iterating.
In the Hartree equation above,
represents the energy contribution of
represents the potential due to the other electrons
in which electron
In this equation we can formally see the effect of screening by the other electrons.
The equation is derived (see Gasiorowicz pp 309-311) from the Schrödinger equation
Since we will not apply these equations to solve problems, we will not go into the
derivation, however, it is useful to know how one might proceed to solve more difficult problems.
An improved formalism known as the Hartree-Fock equations,
accounts for the required antisymmetry and gives slightly different results.