Introduction to the Variational Monte Carlo Method in Quantum Chemistry and Physics

Abstract

Variational Monte Carlo (VMC) methods are a powerful set of quantum Monte Carlo (QMC) methods that may not only be used to determine the variational energy of a fully parameterized wave function, but to optimize wave functions as well. Because they can provide highly accurate trial wave functions for more advanced quantum Monte Carlo methods at a comparatively low cost, they may be viewed as the foundation upon which modern quantum Monte Carlo methods are built. In this chapter, I provide a basic introduction to VMC methods intended for beginning graduate students and workers in the fields of quantum physics and chemistry unfamiliar with the topic. I begin with a general introduction to quantum Monte Carlo methods and then describe how VMC methods fit into this larger context. I then describe how VMC may be used to determine the variational energy of a given wave function and subsequently detail how this algorithm can be modified to optimize wave functions. After illustrating how a number of basic VMC algorithms work, I elucidate how two of the most important modern VMC methods—the Linear and Stochastic Reconfiguration methods—work. To provide context, I present a few recent, novel applications of VMC methods to important problems in chemistry and physics, including the Hubbard model, excited state chemistry, and the calculation of accurate atomization energies. I end with a discussion of possible future directions for VMC algorithms.