Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. II. Applications to simple systems

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Abstract
We have described in part I of this work the theoretical basis of a quantum Monte Carlo method based on the use of a pure diffusion process and of the so-called full generalized Feynman–Kac (FGFK) formula. In this second part, we present a set of applications (one-dimensional oscillator, helium-like systems, hydrogen molecule) with the purpose of illustrating in a systematic way the various aspects pertaining to the practical implementation of this method. We thus show how energy and other observables can be obtained, and we discuss the various sources of biases occurring in the different procedures (notably the so-called short-time approximation pertaining to the generation of the sample trajectories of the diffusion process, and the numerical integration pertaining to the evaluation of the ‘‘Feynman–Kac factor’’). After having thus considered the case of the genuine ‘‘bosonic’’ ground state, we illustrate the various proposals for dealing with some ‘‘relative’’ ground state (namely the lowest state belonging to some prescribed symmetry), one of the most important cases being obviously the physical ground state of many-fermion systems (owing to the Pauli principle requirements). More specifically, we consider the so-called fixed-node approximation (FNA), on one hand, and two variants of a potentially exact procedure, the so-called simple projection (SP) and release-node projection (RNP) methods, on the other hand. Finally, some perspectives concerning future developments are outlined.