New Avenues for Quantum Monte Carlo Techniques

The Holy Grail of condensed matter physics is the ability to solve the 78 year old Schrödinger equation for “real-life” systems. The great difficulty is that the motions of the electrons in molecules and crystals are correlated because of the strong repulsion between their negative charges. The solution of the manyelectron Schrödinger equation (the wavefunction) is therefore a complicated function of 3N variables, where N is the number of electrons in the system. A boost to the field was given by Hohenberg and Kohn (HK) 40 years ago [1], with the introduction of density functional theory (DFT). They showed that it was possible to reformulate quantum mechanics in such a way that the important physical quantity is the electron density, rather than the electronic wavefunction, reducing the complexity of the problem from 3N to 3. Of course, free lunches are seldom available, and the price to pay for the great simplification of HK was the introduction of a new quantity, called exchange-correlation (XC) energy, which is an (as yet) unknown functional of the electron density. To make the theory work, HK suggested a simple form for the XC functional, known as the local-density approximation (LDA). The LDA is exact in a system with a homogeneous electron density, and is only an approximation in (real-life) inhomogeneous systems. Forty years later the LDA is still widely used, and has been the main factor for the great success of DFT, however, there are a number of cases where more accuracy is needed than the LDA can provide. Physicists have struggled for decades to find better approximations for the XC energy, and come up with a number of improvements to the LDA, but there are still several ’difficult’ cases for which no available approximation for the XC is really satisfactory.