Solution of the equations

We are now prepared to outline a general strategy for finding the total energy $E$. To evaluate the various terms in (12), we need the correct orbitals $\psi_i(\vec x)$, which we can find from $V(\vec x)$ using a Schrödinger-solver routine to solve (16). To determine $V(\vec x)$ 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 $n(\vec
x)$; and a simple subroutine to evaluate the function $f'_{xc}(\ldots)$ for the values of $n(\vec
x)$.

Figure 2: Stages of solving the Kohn-Sham equations
\scalebox{0.75}{\includegraphics{depend.eps}}

The minimum of $E$ obtains only when all of the above equations hold simultaneously. In particular, the density must be self-consistent: the density $n_{in}(\vec x)$ which we input must lead to a potential $V(\vec x)$ which gives rise to a set of orbitals $\psi(\vec x)$ that sum to a final density $n_{out}(\vec x)$ equal to the input density. (See Figure 2.) Viewing the contents of the dashed box in Figure 2 as a function $F[n(\vec x)]$ which takes the density as input and produces a new density as output, self-consistency is the condition that $F[n(\vec
x)]-n(\vec x)=0$, 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 $F[n(\vec x)]-n(\vec x)$. Eventually, we shall discuss these techniques, but first we shall develop the ability to compute $F[\ldots]$.

Tomas Arias 2004-01-26