``excPZ'' and ``excpPZ''

double excPZ(double rs)

double excpPZ(double rs)

Return values:

• exc: value of Perdew-Zunger form for $\epsilon_{xc}(r_s)$
• excp: value of Perdew-Zunger form for $\epsilon_{xc}'(r_s)\equiv d\epsilon_{xc}/d r_s$

Input:

• rs: value of $r_s \equiv \left(\frac{3}{4 \pi}\right)^{1/3} n^{-1/3}$

As described in lecture, the literature specifies the exchange-correlation energy per electron $\epsilon_{xc}\left(r_s(n)\right)$ as a function of the average distance between electrons $r_s(n) \equiv \left(\frac{3}{4 \pi}\right)^{1/3} n^{-1/3}$. Perdew and Zunger (Phys. Rev. B 23, 5048 (1981).) gave one of the earlier, simplest forms for $\epsilon_{xc}$,

$$\epsilon_{xc} = -\frac{\alpha}{r_s} + \left\{\begin{array}{l... ...ac{A}{1+B \sqrt{r_s} + C r_s} & r_s \ge 1 \end{array}\right.,$$ (3)

where Table 1 gives the values of the constant coefficients.

Table 1: Table of coefficients for Perdew-Zunger form (3) of the local exchange correlation functional.

$\alpha$	$\frac{3}{4} \left(\frac{3}{2\pi}\right)^{2/3}$
a	0.0311
b	-0.0480
c	0.0020
d	-0.0116
A	-0.1423
B	1.0529
C	0.3334

Write routines of the two above two prototypes which compute $\epsilon_{xc}(r_s)$ and $\epsilon'_{xc}(r_s) \equiv d \epsilon_xc/d r_s$ for the Perdew-Zunger form.