Exchange & Correlation Beyond the LDA Sample Clauses

Exchange & Correlation Beyond the LDA. ‌ Whilst the LDA is a reasonable approximation, it does systematically underestimate the equilibrium lattice parameters in metals [170]. This is one of the primary reasons for the roughly 10% discrepancy between LDA elastic constants and experiment. One method to go beyond the LDA is to add derivatives of the density into the exchange- correlation function [64, 65] xc EGGA [n (r)] = ∫ dr n (r) fxc (r) (2.34) where fxc is some unknown function of n (r) and ∇n (r), analogous to εxc. There have been a number of different proposals for fxc [64, 170]. A ubiquitous parameterisations of fxc was given by Xxxxxx, Xxxxx and Xxxxxxxxx (PBE) [65, 170] whereby xc f PBE [n, ∇n] = εxFxc (rs, s) (2.35) with rs = (3/4πn)1/3 as the electron radius and s = |∇n|/2kF n. The use of the GGA within the context of the Generalised Pseudopotential Theory (GPT) [99, 115, 116] is restricted to transition and actinide metals, thanks to the presence of localised d- and f -states. For the valence s- and p-states, the GPT uses many-body perturbation theory of the interacting electron gas to go beyond the LDA. Consider a perturbation in the electron density δn(q), the resulting change in the potential [160] is given by δV (q) = 4πe2 [1 ( )] ( ) (2.36) − G q δn q ,2 q F