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The sampling of the configuration space in diffusion Monte Carlo (DMC) is done using walkers moving randomly. In a previous work on the Hubbard model [\href{https://doi.org/10.1103/PhysRevB.60.2299}{Assaraf et al.~Phys.~Rev.~B \textbf{60}, 2299 (1999)}], it was shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the on-state dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states. Here, we extend this idea to the general case of a walker trapped within domains of arbitrary shape and size. The equations of the resulting effective stochastic dynamics are derived. The larger the average (trapping) time spent by the walker within the domains, the greater the reduction in statistical fluctuations. A numerical application to the Hubbard model is presented. Although this work presents the method for finite linear spaces, it can be generalized without fundamental difficulties to continuous configuration spaces., 14 pages, 4 figures

Numerically cheap single-core subsamplings have been used to build improved estimators for molecular properties in the variational Monte Carlo framework. The resulting estimators depend only on the valence electron positions and can be thought of as an exact effective core potential for the total energy. We are proposing a spin-dependent core definition which enables exploiting these single-core subsamplings (or sidewalks) not only to decrease the variance of the estimators but also to restrict the main variational Monte Carlo dynamics to the valence region. This results mainly in a simplification of the algorithm and additionally in a gain in efficiency as illustrated on alkane chains and silicon clusters. An evaluation of the efficiency on transition metal systems is done using cobalt clusters, a gain of up to two orders of magnitude is achieved compared to a standard all-electron calculation.

We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations-the scaling of the statistical fluctuations as a function of the number of particles N-is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.

International audience; We propose a Monte Carlo method which exploits that core regions are physically independent in a molecule to almost remove their contribution to the numerical cost. The method is tantamount to computing an effective core potential on the fly, by efficiently subsampling the core regions with independent sidewalks. The removal of fluctuations in the core region enables also the dynamic in the valence region to be accelerated using a process with two time steps. As a function of the total number of electrons N the numerical overhead O(N) is negligible in comparison to the overall scaling O(N 3) (due to the evaluation of determinants). Tests are presented on atoms, alkane chains, and clusters of silicons. We report a transferability of the parameters of the method from atoms to molecules, enabling a calibration using only single atoms. These tests display a gain in numerical efficiency between one and two orders of magnitude for large N.

\textsc{Quantum Package} is an open-source programming environment for quantum chemistry specially designed for wave function methods. Its main goal is the development of determinant-driven selected configuration interaction (sCI) methods and multi-reference second-order perturbation theory (PT2). The determinant-driven framework allows the programmer to include any arbitrary set of determinants in the reference space, hence providing greater methodological freedoms. The sCI method implemented in \textsc{Quantum Package} is based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm which complements the variational sCI energy with a PT2 correction. Additional external plugins have been recently added to perform calculations with multireference coupled cluster theory and range-separated density-functional theory. All the programs are developed with the IRPF90 code generator, which simplifies collaborative work and the development of new features. \textsc{Quantum Package} strives to allow easy implementation and experimentation of new methods, while making parallel computation as simple and efficient as possible on modern supercomputer architectures. Currently, the code enables, routinely, to realize runs on roughly 2\,000 CPU cores, with tens of millions of determinants in the reference space. Moreover, we have been able to push up to 12\,288 cores in order to test its parallel efficiency. In the present manuscript, we also introduce some key new developments: i) a renormalized second-order perturbative correction for efficient extrapolation to the full CI limit, and ii) a stochastic version of the CIPSI selection performed simultaneously to the PT2 calculation at no extra cost., 21 pages, 8 figures

We present the extension of variational Monte Carlo (VMC) to the calculation of electronic excitation energies and oscillator strengths using time-dependent linear-response theory. By exploiting the analogy existing between the linear method for wave-function optimisation and the generalised eigenvalue equation of linear-response theory, we formulate the equations of linear-response VMC (LR-VMC). This LR-VMC approach involves the first-and second-order derivatives of the wave function with respect to the parameters. We perform first tests of the LR-VMC method within the Tamm-Dancoff approximation using single-determinant Jastrow-Slater wave functions with different Slater basis sets on some singlet and triplet excitations of the beryllium atom. Comparison with reference experimental data and with configuration-interaction-singles (CIS) results shows that LR-VMC generally outperforms CIS for excitation energies and is thus a promising approach for calculating electronic excited-state properties of atoms and molecules.

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter $\lambda$, and admitting a unique invariant measure for any value of $\lambda$ around $\lambda$ = 0. Our aim is to compute the derivative with respect to $\lambda$ of averages with respect to the invariant measure, at $\lambda$ = 0. We analyze a numerical method which consists in simulating the process at $\lambda$ = 0 together with its derivative with respect to $\lambda$ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $\lambda$ of the mean of an observable through Monte Carlo simulations.

International audience; We present an improved formalism for quantum Monte Carlo calculations of energy derivatives and properties (e.g., the interatomic forces), with a multideterminant Jastrow–Slater function. As a function of the number Ne of Slater determinants, the numerical scaling of O(Ne) per derivative we have recently reported is here lowered to O(Ne) for the entire set of derivatives. As a function of the number of electrons N, the scaling to optimize the wave function and the geometry of a molecular system is lowered to O(N3) + O(NNe), the same as computing the energy alone in the sampling process. The scaling is demonstrated on linear polyenes up to C60H62 and the efficiency of the method is illustrated with the structural optimization of butadiene and octatetraene with Jastrow–Slater wave functions comprising as many as 200 000 determinants and 60 000 parameters.

International audience; We propose a method for obtaining effective lifetimes of scattering electronic states for avoiding the artificially confinement of the wave function due to the use of incomplete basis sets in time-dependent electronic-structure calculations of atoms and molecules. In this method, using a fitting procedure, the lifetimes are extracted from the spatial asymptotic decay of the approximate scattering wave functions obtained with a given basis set. The method is based on a rigorous analysis of the complex-energy solutions of the Schrödinger equation. It gives lifetimes adapted to any given basis set without using any empirical parameters. The method can be considered as an ab initio version of the heuristic lifetime model of Klinkusch et al. [J. Chem. Phys. 131, 114304 (2009)]. The method is validated on the H and He atoms using Gaussian-type basis sets for calculation of high-harmonic-generation spectra.

No fancy statistical objects here, we go back to the computation of one of the most basic and fundamental quantities in the statistical mechanics of fluids, namely the pair distribution functions. Those functions are usually computed in molecular simulations by using histogram techniques. We show here that they can be estimated using a global information on the instantaneous forces acting on the particles, and that this leads to a reduced variance compared to the standard histogram estimators. The technique is extended successfully to the computation of three-dimensional solvent densities around tagged molecular solutes, quantities that are noisy and very long to converge, using histograms.

We present a simple and general formalism to compute efficiently the derivatives of a multi-determinant Jastrow-Slater wave function, the local energy, the interatomic forces, and similar quantities needed in quantum Monte Carlo. Through a straightforward manipulation of matrices evaluated on the occupied and virtual orbitals, we obtain an efficiency equivalent to algorithmic differentiation in the computation of the interatomic forces and the optimization of the orbital paramaters. Furthermore, for a large multi-determinant expansion, the significant computational gain recently reported for the calculation of the wave function is here improved and extended to all local properties in both all-electron and pseudopotential calculations., 15 pages, 3 figures

We explore the use in quantum Monte Carlo (QMC) of trial wave functions consisting of a Jastrow factor multiplied by a truncated configuration-interaction (CI) expansion in Slater determinants obtained from a CI perturbatively selected iteratively (CIPSI) calculation. In the CIPSI algorithm, the CI expansion is iteratively enlarged by selecting the best determinants using perturbation theory, which provides an optimal and automatic way of constructing truncated CI expansions approaching the full CI limit. We perform a systematic study of variational Monte Carlo (VMC) and fixed-node diffusion Monte Carlo (DMC) total energies of first-row atoms from B to Ne with different levels of optimization of the parameters (Jastrow parameters, coefficients of the determinants, and orbital parameters) in these trial wave functions. The results show that the reoptimization of the coefficients of the determinants in VMC (together with the Jastrow factor) leads to an important lowering of both VMC and DMC total energies, and to their monotonic convergence with the number of determinants. In addition, we show that the reoptimization of the orbitals is also important in both VMC and DMC for the Be atom when using a large basis set. These reoptimized Jastrow-CIPSI wave functions appear as promising, systematically improvable trial wave functions for QMC calculations., Comment: in Molecular Physics, Taylor \& Francis, 2016

We provide a pedagogical introduction to the two main variants of real-space quantum Monte Carlo methods for electronic structure calculations: variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). Assuming no prior knowledge on the subject, we review in depth the Metropolis–Hastings algorithm used in VMC for sampling the square of an approximate wave function, discussing details important for applications to electronic systems. We also review in detail the more sophisticated DMC algorithm within the fixed-node approximation, introduced to avoid the infamous Fermionic sign problem, which allows one to sample a more accurate approximation to the ground-state wave function. Throughout this review, we discuss the statistical methods used for evaluating expectation values and statistical uncertainties. In particular, we show how to estimate nonlinear functions of expectation values and their statistical uncertainties.

We show that the recently proposed correlated sampling without reweighting procedure extends the locality (asymptotic independence of the system size) of a physical property to the statistical fluctuations of its estimator. This makes the approach potentially vastly more efficient for computing space-localized properties in large systems compared with standard correlated methods. A proof is given for a large collection of noninteracting fragments. Calculations on hydrogen chains suggest that this behavior holds not only for systems displaying short-range correlations, but also for systems with long-range correlations.

We study the efficiency of quantum Monte Carlo (QMC) methods in computing space localized ground state properties (properties which do not depend on distant degrees of freedom) as a function of the system size N. We prove that for the commonly used correlated sampling with reweighting method, the statistical fluctuations σ2(N) do not obey the locality property. σ2(N) grow at least linearly with N and with a slope that is related to the fluctuations of the reweighting factors. We provide numerical illustrations of these tendencies in the form of QMC calculations on linear chains of hydrogen atoms.

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.

We present a general approach to greatly increase at little cost the efficiency of Monte Carlo algorithms. To each observable to be computed we associate a renormalized observable (improved estimator) having the same average but a different variance. By writing down the zero-variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived (zero-variance principle for each observable separately). We show, with several examples including classical and quantum Monte Carlo calculations, that the method can be very powerful., Comment: 9 pages, Latex, to appear in Phys. Rev. Lett

We investigate the metal-insulator transition of the one-dimensional $\mathrm{SU}(N)$ Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half filling ${(k}_{F}=\ensuremath{\pi}{/Na}_{0})$ is conjectured for $Ng2.$ 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 Green's 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 nonzero value of the Coulomb interaction. We find ${U}_{c}\ensuremath{\sim}2.2$ for $N=3$ and ${U}_{c}\ensuremath{\sim}2.8$ for $N=4.$

6 pages, 3 figures, 2 tables; When using Hartree-Fock (HF) trial wave functions in quantum Monte Carlo calculations, one faces, in case of HF instabilities, the HF symmetry dilemma in choosing between the symmetry-adapted solution of higher HF energy and symmetry-broken solutions of lower HF energies. In this work, we have examined the HF symmetry dilemma in hydrogen rings which present singlet instabilities for sufficiently large rings. We have found that the symmetry-adapted HF wave function gives a lower energy both in variational Monte Carlo and in fixed-node diffusion Monte Carlo. This indicates that the symmetry-adapted wave function has more accurate nodes than the symmetry-broken wave functions, and thus suggests that spatial symmetry is an important criterion for selecting good trial wave functions.

International audience; 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.

We present some exact results for the Hubbard model on d -dimensional hypercubes and fillings corresponding to ( N ↑(↓) = N , N ↓(↑) =1) with N arbitrary. Introducing a spin formalism associated with the symmetry operations of the hypercube it is shown that the Hubbard model can be rewritten as a spin Hamiltonian defined on a d ×( N +1) rectangular spin lattice. For the two-electron case ( N =1) a logarithmic reduction of the active part of the Hilbert space can be achieved. It is shown that a very important size reduction can also be achieved for N >1.

We construct improved quantum Monte Carlo estimators for the spherically- and system-averaged electron pair density (i.e. the probability density of finding two electrons separated by a relative distance u), also known as the spherically-averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth non-local operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations., 13 pages, 9 figures, published version

International audience; An alternative Monte Carlo estimator for the one-body density rho(r) is presented. This estimator has a simple form and can be readily used in any type of Monte Carlo simulation. Comparisons with the usual regularization of the delta-function on a grid show that the statistical errors are greatly reduced. Furthermore, our expression allows accurate calculations of the density at any point in space, even in the regions never visited during the Monte Carlo simulation. The method is illustrated with the computation of accurate variational Monte Carlo electronic densities for the Helium atom (one-dimensional curve) and for the water dimer (three-dimensional grid containing up to 51x51x51=132,651 points).

In this work we present a detailed study of the Fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies [M.H. Kalos and F. Pederiva, Phys. Rev. Lett. vol. 85, 3547 (2000)]. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact Diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose-Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full sign problem. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled., 49 pages with 4 postscript figures

We investigate dynamical symmetry enlargement in the half-filled SU(4) Hubbard chain using non-perturbative renormalization group and Quantum Monte Carlo techniques. A spectral gap is shown to open for arbitrary Coulombic repulsion $U$. At weak coupling, $U \lesssim 3t$, a SO(8) symmetry between charge and spin-orbital excitations is found to be dynamically enlarged at low energy. At strong coupling, $U \gtrsim 6t$, the charge degrees of freedom dynamically decouple and the resulting effective theory in the spin-orbital sector is that of the SO(6) antiferromagnetic Heisenberg model. Both regimes exhibit spin-Peierls order. However, although spin-orbital excitations are $incoherent$ in the SO(6) regime they are $coherent$ in the SO(8) one. The cross-over between these regimes is discussed., Comment: 4 pages, 2 figures

A simple and stable method for computing accurate expectation values of observable 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 observableby an improved or ``renormalized'' estimator. Using this estimator more accurate averages are obtained:Not only the statistical fluctuations are reduced but alsothe 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 (ZVZB) 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. Calculations of the entire force curve of the H$_2$,LiH, and Li$_2$ molecules are presented. Spectroscopic constants $R_e$ (equilibrium distance)and $\omega_e$ (harmonic frequency) are also computed.The equilibrium distances are obtained with a relative error smaller than 1$\%$, while the harmonic frequencies are computed with an error of about 10$\%$.

Quantum Monte Carlo (QMC) methods are powerful stochastic approaches to calculate ground-state properties of quantum systems. They have been applied with success to a great variety of problems described by a Schrodinger-like Hamiltonian (quantum liquids and solids, spin systems, nuclear matter, ab initio quantum chemistry, etc ... ). In this paper we give a pedagogical presentation of the main ideas of QMC. We develop and exemplify the various concepts on the simplest system treatable by QMC, namely a 2 x 2 matrix. First, we discuss the Pure Diffusion Monte Carlo (PDMC) method which we consider to be the most natural implementation of QMC concepts. Then, we discuss the Diffusion Monte Carlo (DMC) algorithms based on a branching (birth-death) process. Next, we present the Stochastic Reconfiguration Monte Carlo (SRMC) method which combines the advantages of both PDMC and DMC approaches. A very recently introduced optimal version of SRMC is also discussed. Finally, two methods for accelerating QMC calculations are sketched: (a) the use of integrated Poisson processes to speed up the dynamics (b) the introduction of “renormalized” or “improved” estimators to decrease the statistical fluctuations.

In this paper we discuss various aspects of diffusion Monte Carlo methods using a fixed number of walkers. First, a rigorous proof of the divergence of pure diffusion Monte Carlo (PDMC) methods (DMC without branching in which the weights are carried along trajectories) is given. Second, a bias-free Monte Carlo method combining DMC and PDMC approaches, and based on a minimal stochastic reconfiguration of the population, is discussed. Finally, some illustrative calculations for a system of coupled quantum rotators are presented.