Michel Caffarel

Zero-variance zero-bias principle for observables in quantum Monte Carlo: Application to forces

A simple and stable method for computing accurate expectation values of observables with variational Monte Carlo (VMC) or diffusion Monte Carlo (DMC) algorithms is presented. The basic idea consists in replacing the usual “bare” estimator associated with the observable by an improved or “renormalized” estimator. Using this estimator more accurate averages are obtained: Not only the statistical fluctuations are reduced but also the systematic error (bias) associated with the approximate VMC or (fixed-node) DMC probability densities. It is shown that improved estimators obey a zero-variance zero-bias property similar to the usual zero-variance zero-bias property of the energy with the local energy as improved estimator. Using this property improved estimators can be optimized and the resulting accuracy on expectation values may reach the remarkable accuracy obtained for total energies. As an important example, we present the application of our formalism to the computation of forces in molecular systems. Cal...

Computing forces with quantum Monte Carlo

We present a simple and stable quantum Monte Carlo approach for computing forces between atoms in a molecule. In this approach we propose to use as Monte Carlo estimator of the force the standard Hellmann–Feynman expression (local force expressed as the derivative of the total potential energy with respect to the internuclear coordinates). Invoking a recently introduced zero-variance principle it is shown how the infinite variance associated with the Hellmann–Feynman estimator can be made finite by introducing some suitably renormalized expression for the force. Practical calculations for the molecules H2, Li2, LiH, and C2 illustrate the efficiency of the method.

Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism

This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time‐averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many‐fermion systems, where the Pauli principle must be taken into account, we can use the fixed‐node approximation, but we also discuss the potentially exact release‐node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), ass...

Electron pair localization function: A practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called electron pair localization function, is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.

Second‐order exchange effects in intermolecular interactions. The water dimer

A new method of deriving explicit formulas for the calculation of second‐order exchange contributions (induction as well as dispersion) within the framework of symmetry‐adapted perturbation theories is presented. It is shown how exchange contributions can be expressed as a combination of electrostatic interaction energies between suitably generalized charge distributions (overlap intermolecular charge distributions). Each of these contributions are derived within the Hartree–Fock approximation (neglect of all electron correlation effects within the noninteracting molecules) and by considering only single‐electron exchange between interacting molecules. Numerical calculations for the interaction of two water molecules are presented. In the region of the equilibrium geometry, it is found that the complete second‐order exchange contribution accounts for about 20% of the total intermolecular interaction energy. This contribution is essentially dominated by the exchange induction component which is found to re...

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

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 belo...

Quantum Monte Carlo method for some model and realistic coupled anharmonic oscillators

A new quantum Monte Carlo (QMC) method of evaluating low lying vibrational levels for coupled modes is presented. We use a modified fixed‐node (FN) approach in which an extremum principle for energy levels is invoked. In this way, the nodal hypersurfaces of the nuclear wave function are parametrized and then optimized for each excited state. The method is tested on the fundamental excitations of some two‐dimensional model potentials and is applied to the case of realistic coupled modes of the CO molecule adsorbed on a palladium cluster. The effect of an external electric field is also examined. The quantum Monte Carlo results are compared with those obtained in the conventional variational treatment of the nuclear Schrodinger equation for coupled vibrations. The QMC results give the exact values with an error which is in general less than 1 cm−1 . In all cases (even in the case of strong coupling) the use of our procedure leads to ‘‘optimal’’ nodal lines (in the sense of the extremum principle used in thi...

Metal-insulator transition in the one-dimensional SU ( N ) Hubbard model

We investigate the metal-insulator transition of the one-dimensional SU(N) Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half-filling (k_F=pi/Na_0) is conjectured for N > 2. Expressions for the charge and spin velocities as well as for the Luttinger liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Greens function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger liquid parameters as a function of the Coulomb interaction U are given for the cases N=3 and N=4. Our results strongly support the existence of a Mott-Hubbard transition at a {it non-zero} value of the Coulomb interaction. We find

A pedagogical introduction to Quantum Monte-Carlo

U_c sim 2.2

Spin-stiffness and topological defects in two-dimensional frustrated spin systems

for N=3 and