Introduction to the Diffusion Monte Carlo Method

Making and using polaritonic states (i.e., hybrid electron-photon states) for chemical applications have recently become one of the most prominent and active fields that connects the communities of chemistry and quantum optics. Modeling of such polaritonic phenomena using ab initio approaches calls for new methodologies, leading to the reinvention of many commonly used electronic structure methods, such as Hartree-Fock, density functional, and coupled cluster theories. In thi...

Computational codes based on the Diffusion Monte Carlo method can be used to determine the quantum state of two-electron systems confined by external potentials of various nature and geometry. In this work, we show how the application of this technique in its simplest form, that does not employ complex analytic guess functions, allows to obtain satisfactory results and, at the same time, to write programs that are readily adaptable from one type of confinement to another. Thi...

This is a book chapter soon to appear (2002) in the "Handbook for Numerical Analysis" volume dedicated to "Computational Chemistry" edited by Claude Le Bris. The series editors are P.G. Ciarlet and J. L. Lions. [North Holland/Elservier]. This review deals with some of the methods known under the umbrella term quantum Monte Carlo (QMC), specifically those that have been most commonly used for electronic structure.

The Diffusion Monte Carlo method with constant number of walkers, also called Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used Monte Carlo methodology for computing the ground-state energy and wave function of quantum systems. In this study, we present the first mathematically rigorous analysis of this class of stochastic methods on non necessarily compact state spaces, including linear diffusions evolving in quadratic absorbing potentials, yield...

The diffusion Monte Carlo method with symmetry-based state selection is used to calculate the quantum energy states of H$_2^+$ confined into potential barriers of atomic dimensions (a model for these ions in solids). Special solutions are employed permitting one to obtain satisfactory results with rather simple native code. As a test case, $^2\Pi_u$ and $^2\Pi_g$ states of H$_2^+$ ions under spherical confinement are considered. The results are interpreted using the correlati...

We present an elementary and self-contained account of the analogies existing between classical diffusion and the imaginary-time evolution of quantum systems. These analogies are used to develop a new quantum simulation method which allows to calculate the ground-state expectation values of local observables without any mixed estimates nor population-control bias, as well as static and dynamic (in imaginary time) response functions. This method, which we name Reptation Quantu...

The quantum Monte Carlo methods represent a powerful and broadly applicable computational tool for finding very accurate solutions of the stationary Schroedinger equation for atoms, molecules, solids and a variety of model systems. The algorithms are intrinsically parallel and are able to take full advantage of the present-day high-performance computing systems. This review article concentrates on the fixed-node/fixed-phase diffusion Monte Carlo method with emphasis on its ap...

The self-healing diffusion Monte Carlo algorithm (SHDMC) [Phys. Rev. B {\bf 79}, 195117 (2009), {\it ibid.} {\bf 80}, 125110 (2009)] is shown to be an accurate and robust method for calculating the ground state of atoms and molecules. By direct comparison with accurate configuration interaction results for the oxygen atom we show that SHDMC converges systematically towards the ground-state wave function. We present results for the challenging N$_2$ molecule, where the binding...

The auxiliary-field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time-independent Schroedinger equation in atoms, molecules, solids, and a variety of model systems by stochastic sampling. We introduce the theory and formalism behind this framework, briefly discuss the key technical steps that turn it into an effective and practical computational method, present several illustrative results, and conclude with comments on the prospects o...

By decomposing the important sampled imaginary time Schr\"odinger evolution operator to fourth order with positive coefficients, we derived a number of distinct fourth order Diffusion Monte Carlo algorithms. These sophisticated algorithms require higher derivatives of the drift velocity and local energy and are more complicated to program. However, they allowed very large time steps to be used, converged faster with lesser correlations, and virtually eliminated the step size ...