Review of quantum Monte Carlo methods and their applications

Abstract

Correlation energy makes a small but very important contribution to the total energy of an electronic system. Among the traditional methods used to study electronic correlation are coupled clusters (CC), configuration interaction (CI) and manybody perturbation theory (MBPT) in quantum chemistry, and density functional theory (DFT) in solid state physics. An alternative method, which has been applied successfully to systems ranging from the homogeneous electron gas, to atoms, molecules, solids and clusters is quantum Monte Carlo (QMC). In this method the Schrödinger equation is transformed to a diffusion equation which is solved using stochastic methods. In this work we review some of the basic aspects of QMC in two of its variants, variational (VMC) and diffusion Monte Carlo (DMC). We also review some of its applications, such as the homogeneous electron gas, atoms and the inhomogeneous electron gas (jellium surface). The correlation energy obtained by Ceperley and Alder (D.M. Ceperley and B.J. Alder, Physical Review, 45 (1980) 566), as parameterized by Perdew and Zunger (J.P. Perdew and A. Zunger, Phys. Rev. B23 (1980) 5469), is one of the most used in DFT calculations in the local density approximation (LDA). Unfortunately, the use of the LDA in inhomogeneous systems is questionable, and better approximations are desired or even necessary. We present results of the calculations performed on metallic surfaces in the jellium model which can be useful to obtain better approximations for the exchange and correlation functionals. We have computed the electronic density, work function, surface energy and pair correlation functions for a jellium slab at the average density of magnesium ( r s = 2.66). Since there is an exact expression for the exchange and correlation functional in terms of the pair correlation functions, the knowledge of such functions near the edge of the surface may be useful to obtain exchange and correlation functionals valid for inhomogeneous systems. From the exchange and correlation functional we can conclude that the exchange-correlation hole is nearly spherical in the bulk region but elongated in the direction perpendicular to the surface as the electron approaches the edge of the surface, showing the anisotropic character of the electronic correlation near the surface.