# Exchange and correlation in many-electron systems

This dissertation first reviews some basic and popular methods of solving a many-electron problem, and the physical principles underpinning these methods. Following these reviews are two thematic chapters which cover different but related topics and ideas: (1) For a uniform electron gas of density n = n$\sb\uparrow$ + n$\sb\downarrow$ = 3/4$\pi$r$\sb{\rm s}\sp3 = \pi$k$\sb{\rm s}\sp6$/192 and spin polarization $\zeta$ = (n$\sb\uparrow - n\sb\downarrow)$/n, we study the Fourier transform $\bar\rho\sb{\rm c}$(k,r$\sb{\rm s},\zeta$) of the correlation hole, as well as the correlation energy $\epsilon\sb{\rm c}$(r$\sb{\rm s}$, $\zeta) = \int\sbsp{\rm o}{\infty}$ dk $\bar\rho\sb{\rm c}/\pi$. In the high-density (r$\sb{\rm s}$ $\to$ 0) limit, we find a simple scaling relation k$\sb{\rm s}\bar\rho\sb{\rm c}/\pi$g$\sp2$ $\to$ f(z,$\zeta$), where z = k/gk$\sb{\rm s}$, g = $\lbrack (1 + \zeta)\sp{2/3}$ + $(1 - \zeta)\sp{2/3}$) /2, and f(z,1) = f(z,0). The function f(z,$\zeta$) is only weakly $\zeta$-dependent, and its small-z expansion $-$3z/$\pi\sp2$ + 4$\sqrt 3$ z$\sp2/\pi\sp2$ + ... is also the exact small-wavevector (k $\to$ 0) expansion for any r$\sb{\rm s}$ or $\zeta$. Motivated by these considerations, and by a discussion of the large-wavevector and low-density limits, we present two Pade representations for $\bar\rho\sb{\rm c}$ at any k, r$\sb{\rm s}$, or $\zeta$, one within and one beyond the random phase approximation (RPA). We also show that $\bar\rho\sbsp{\rm c}{\rm RPA}$ obeys a generalization of Misawa's spin scaling relation for $\epsilon\sbsp{\rm c}{\rm RPA},$ and that the low-density (r$\sb{\rm s}\to\infty)$ limit of $\epsilon\sbsp{\rm c}{\rm RPA}$ is $\sim$r$\sb{\rm s}\sp{-3/4}$; (2) The Harbola-Sahni exchange potential is the work needed to move an electron against the electric field of its hole charge distribution. We prove that it is not the exact exchange potential of density functional theory, by showing that it yields the wrong second-order gradient expansion in the slowly-varying limit. But we also discover that it yields the correct local density approximation. Thus the Harbola-Sahni potential is a more physically-correct version of the Slater potential, one that is better suited for molecular and solid-state applications. As a step in our derivation, we present the third-order gradient expansion of the exchange hole density, and discuss its structure. We also describe a new version of the Harbola-Sahni potential which corrects its path-dependence. The exact exchange potential for an atom is given by the Optimized Potential Model (OPM) of Talman and Shadwick. By using enhanced numerics, we confirm that the OPM potential satisfies the Levy-Perdew virial relation and exhibits correct $-{1\over\rm r}$ behavior at large r. Numerical calculations also show that the intershell maxima in the exact exchange potential are needed to lower the total energy. These 'bumps' are missing from the Harbola-Sahni and Slater potentials. In the last chapter a few possible future works are proposed