## Solution of the equations

We are now prepared to outline a general strategy for finding the total energy . To evaluate the various terms in (12), we need the correct orbitals , which we can find from using a Schrödinger-solver routine to solve (16). To determine according to (17), we need the nuclear potential, which we know from (2); the potential from the electrons, which we get from a Poisson-solver routine to solve (4) for a given ; and a simple subroutine to evaluate the function for the values of .

The minimum of obtains only when all of the above equations hold simultaneously. In particular, the density must be self-consistent: the density which we input must lead to a potential which gives rise to a set of orbitals that sum to a final density equal to the input density. (See Figure 2.) Viewing the contents of the dashed box in Figure 2 as a function which takes the density as input and produces a new density as output, self-consistency is the condition that , a set of non-linear equations for the value of the charge density at each point in space. There are quite power numerical methods for solving such equations given the capability of computing . Eventually, we shall discuss these techniques, but first we shall develop the ability to compute .

Tomas Arias 2004-01-26