Chaotic versus Nonchaotic Stochastic Dynamics in Monte Carlo Simulations: A Route for Accurate Energy Differences in N-Body Systems Roland Assaraf, 1 Michel Caffarel, 2 and A. C. Kollias 3 1 Laboratoire de Chimie The ´orique, CNRS-UMR 7616 et Universite ´ Pierre et Marie Curie, 75252 Paris Cedex, France 2 Laboratoire de Chimie et Physique Quantiques, CNRS-UMR 5626, IRSAMC et Universite ´ de Toulouse, 31062 Toulouse Cedex, France 3 United States Patent and Trademark Office, 600 Dulany Street, Alexandria, Virginia 22314-5796, USA (Received 27 July 2010; published 11 April 2011) We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved. DOI: 10.1103/PhysRevLett.106.150601 PACS numbers: 05.10.Ln, 02.70.Ss, 05.10.Gg, 31.15.A At the heart of the N-body quantum problem in physics and chemistry is the evaluation of excitation energies corresponding to small energy differences associated with two close Hamiltonians. Evaluating such excitations allows detailed knowledge of both the nature of the ground state and of the low-energy properties of the system. Denoting H as the reference N-body Hamiltonian defined over N degrees of freedom and H  some small perturbation of it (including eventually a slight change in N), where  is a small parameter connecting both Hamiltonians, we are interested in the excitation energies defined as the differ- ence   E  0  E 0 in ground-state energies. Depending on the domain of application (nuclear physics, quantum liquids, solid-state physics, quantum chemistry, etc.), such excitations may appear under various names such as, e.g., phonon, plasmon, spinon, or charge excitations, bound states, reaction barriers, electronic affinities, bind- ing energies, etc. We may also be interested in evaluating infinitesimal energy differences such as in the important case of computing properties other than the energy via the Hellman-Feynman theorem (energy derivative) or in the case of the thermodynamic limit where energy differences (for example, a one-particle gap) usually scale at most as a tiny fraction of order 1=N of the total energy. Although total energies are usually computed with high accuracy in quantum Monte Carlo (QMC) calculations, it is in most cases not sufficient to compensate for the tremendous loss of relative accuracy when small differences are computed by using independent calculations of each energy compo- nent. Today, a number of approaches have been devised to tackle the problem of computing small differences and derivatives of energies, e.g., [1–5]. The most popular strat- egy employed up to now is to introduce a scheme based on the idea of correlated sampling [1,6–8]. This idea has been implemented by introducing common stochastic dynamics for the two close Hamiltonians H  and H and by taking into account their difference via modified estimators in- cluding weighting factors related to the difference, i.e., reweighting techniques. Here, it is emphasized that intro- ducing such weights is problematic since they are respon- sible for the occurrence of large statistical or in some case infinite fluctuations; see, for example, [3]. These fluctua- tions not only completely or partially cancel the benefit of the correlation but also introduce systematic errors, in particular, for the important case of fermionic systems where the change of nodal topology is difficult to take into account [1,2,9]. The purpose of this Letter is to present a method which circumvents such difficulties. In short, we rely on a stabil- ity property versus chaos which states that stochastic tra- jectories having different initial conditions but a common noise meet exponentially fast in time. We show that be- cause of this property two slightly different processes sharing the same noise will produce two trajectories that will remain close forever. This situation is clearly particu- larly favorable for computing small energy differences efficiently, since we keep the advantage of the correlation without the drawback of reweighted estimators. Note that the observation that configurations can coalesce after some time in a Monte Carlo scheme when a common set of random numbers is used is not new [10–13]. The ideas will be presented here in the case of two popular QMC approaches, namely, the variational Monte Carlo and the fixed-node diffusion Monte Carlo methods. However, we emphasize that such ideas can be extended without diffi- culties to other types of Monte Carlo methods including classical Monte Carlo simulations (e.g., by sampling the Boltzmann density with a continuous Langevin-type pro- cess). Finally, we present a few applications of the method to long chains of hydrogen atoms and molecules. In any Monte Carlo algorithm, one evaluates a property, for example, the energy, as the average of some function PRL 106, 150601 (2011) PHYSICAL REVIEW LETTERS week ending 15 APRIL 2011 0031-9007= 11=106(15)=150601(4) 150601-1 Ó 2011 American Physical Society