Review of background

Within the local density approximation to density functional theory, the total energy of an electronic system is

$$E_{tot} = T + E_{el-nuc}+E_{Hartree}+E_{xc},$$

with

$$\begin{eqnarray*} T & \equiv & \sum_i f_i \int \psi_i^*(\vec x) \left( -\frac{1}... ... \, \, dV \\ E_{xc} & \equiv & \int f_{xc}(n(\vec x)) \, \, dV, \end{eqnarray*}$$

where

$$n(\vec x) \equiv \sum_i f_i \vert\psi_i(\vec x)\vert^2$$

and

$$\nabla^2 \phi(\vec x) = - 4 \pi n(\vec x).$$

Minimizing the above total energy is equivalent to solving the Schrödinger equation

$$-\frac{1}{2} \nabla^2 \psi_i(\vec x) + V(\vec x) \psi_i(\vec x) = E_i \psi_i(\vec x),$$

with

$$V(\vec x) \equiv V_{nuc}(\vec x) + \phi(\vec x) + V_{xc}(\vec x),$$

where

$$V_{xc}(\vec x) \equiv f'_{xc}(n(\vec x)).$$

In terms of the electron energies $E_i$ above, an alternative form for the total energy is

$$E_{tot} = \sum_i f_i E_i - E_{Hatree} + \int \left[ f_{xc}(n(\vec x))-f'_{xc}(n(\vec x)) n(\vec x) \right] \, dV.$$

This first problem of this homework develops the software to compute the exchange-correlation terms in the above equations. The literature always gives $f_{xc}(n)$ in the form $f_{xc}(n) = \epsilon_{xc}(n) n$. In terms of $\epsilon_{xc}$, the quantities we shall need are

$$V_{xc}(\vec x) \equiv f'_{xc}(n(\vec x))$$

$$= \epsilon_{xc}'(n(\vec x)) n(\vec x) + \epsilon_{xc}(n(\vec x))$$ (1)

and the exchange-correlation correction energy,

$$\begin{eqnarray*} \Delta E_{xc} & \equiv & \int \left[ f_{xc}(n(\vec x))-f'_{xc}... ...\equiv & \int \Delta\epsilon_{xc}(n(\vec x)) n(\vec x) \, \, dV, \end{eqnarray*}$$

where

$$\Delta\epsilon_{xc}(n(\vec x)) \equiv \frac{1}{n(\vec x)} \left[ f_{xc}(n(\vec x))-f'_{xc}(n(\vec x)) n(\vec x) \right]$$

$$= \epsilon_{xc}(n(\vec x))-f'_{xc}(n(\vec x))$$

$$= \epsilon_{xc}(n(\vec x))-(\epsilon_{xc}'(n(\vec x)) n(\vec x) + \epsilon_{xc}(n(\vec x)))$$

$$= -\epsilon_{xc}'(n(\vec x)) n(\vec x)$$ (2)