Adaptive models for electronic structure computations
Résumé
The first principle approach is mostly used nowadays to evaluate physical and mechanical properties of solids. In fact, the macroscopic behaviour depends on the type of binding through the electron density, derived from the quantum mechanics Schrödinger equation. It also allows designing new materials with cutting edge performances. When willing to solve the Schrödinger equation for a global system of Ne electrons, one faces a problem of partial differential equations in R^3Ne and needs to approximate the form of the wave function psi (or of the equation itself) one looks for. Among them is the density functional theory (DFT). But it includes a whole set of approximations, from the most basic one known as LDA (local density approximation) where the exchange correlation energy Exc is written as a functional of the electron density rho, to more elaborated ones such as general gradient approximation (GGA) and other meta-GGA where Exc is a functional of rho, and also of its gradient, its laplacian... This sequence of more elaborated approximations, referred as “Jacob's ladder”, shows that soon one wished to decrease the error due to the approximation as its complexity was increased. Nevertheless, the comparison is based on experimental values and not on error estimation. Apart from DFT, there exists another model known as the Hartree-Fock approximation (HF) where psi is written as a single Slater determinant. That model, mostly used for molecular systems, is based on a variational principle, and the global R^3Ne Schrödinger equation is rewritten as a system of Ne R^3 equations.