arXiv subjects

Michel Caffarel

Publications and source records attributed to Michel Caffarel.

At least 19 recordsLinked to original sources

arXiv2026

Self-Healing Diffusion Monte Carlo applied to a simple fermionic model: A critical assessment of the method

We investigate the Self-Healing Diffusion Monte Carlo (SHDMC) method using a one-dimensional model with periodic boundary conditions. An inversion symmetry is introduced to mimic the antisymmetry property of fermionic wavefunctions, with the bosonic and fermionic sectors being modeled by the even and odd eigenstates, respectively. As in realistic fermionic systems, the nodal structure is only partially constrained by symmetry, making this model a non-trivial testbed for nodal optimization algorithms such as SHDMC. We show that the nodal evolution under SHDMC iterations can be cast into a dynamical system exhibiting both attractive and repulsive fixed points. In the standard formulation of SHDMC applied to this model, the fixed-node energy is found to increase upon iteration, and the node converges to a wrong value, indicating that SHDMC does not always converge to the correct solution. We further show that this problem is partially cured by modifying the nodal update criterion to give more importance to the nodal region. Achieving convergence in the general case very likely requires the use of a localized basis set, as is the case for this model.

cond-mat.str-el↗

arXiv2026

Thermodynamics of quantum oscillators

In this work, we present a compact analytical approximation for the quantum partition function of systems composed of quantum oscillators. The proposed formula is general and applicable to an arbitrary number of oscillators described by a rather general class of potential energy functions (not necessarily polynomials). Starting from the exact path integral expression of the partition function, we introduce a temperature-dependent Gaussian approximation for the high-temperature propagator and, then, invoke a principle of minimal sensitivity to minimize the error. This leads to a system of coupled nonlinear equations whose solution yields the optimal parameters of the Gaussian approximation. The resulting approximate partition function accurately reproduces thermodynamic quantities such as the free energy, average energy, and specific heat -- even at zero temperature -- with typical relative errors in the range of about 1\%--5\%. The accuracy deteriorates only moderately when the anharmonicity and coupling strengths are increased. We illustrate the performance of our analytical formula with numerical results for systems of up to ten coupled anharmonic oscillators. These results are compared to "exact" numerical results obtained via Hamiltonian diagonalization for small systems and Path Integral Monte Carlo simulations for larger ones.

cond-mat.stat-mech↗

arXiv2024

On the Mathematical foundations of Diffusion Monte Carlo

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, yielding what seems to be the first result of this type for this class of models. We present a novel and general mathematical framework with easily checked Lyapunov stability conditions that ensure the uniform-in-time convergence of Diffusion Monte Carlo estimates towards the top of the spectrum of Schrödinger operators. For transient free evolutions, we also present a divergence blow up of the estimates w.r.t. the time horizon even when the asymptotic fluctuation variances are uniformly bounded. We also illustrate the impact of these results in the context of generalized coupled quantum harmonic oscillators with non necessarily reversible nor stable diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force.

math.ST↗

arXiv2024

Path integral for the quartic oscillator: An accurate analytic formula for the partition function

In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential $V(x) = \frac{1}{2} ω^2 x^2 + g x^4$ is presented. Using a path integral formalism, the exact partition function is approximated by the partition function of a harmonic oscillator with an effective frequency depending both on the temperature and coupling constant $g$. By invoking a Principle of Minimal Sensitivity (PMS) of the path integral to the effective frequency, we derive a mathematically well-defined analytic formula for the partition function. Quite remarkably, the formula reproduces qualitatively and quantitatively the key features of the exact partition function. The free energy is accurate to a few percent over the entire range of temperatures and coupling strengths $g$. Both the harmonic ($g\rightarrow 0$) and classical (high-temperature) limits are exactly recovered. The divergence of the power series of the ground-state energy at weak coupling, characterized by a factorial growth of the perturbational energies, is reproduced as well as the functional form of the strong-coupling expansion along with accurate coefficients. Explicit accurate expressions for the ground- and first-excited state energies, $E_0(g)$ and $E_1(g)$ are also presented.

quant-ph↗

arXiv2024

Analytic Model for the Energy Spectrum of the Anharmonic Oscillator

In a recent work we have proposed an original analytic expression for the partition function of the quartic oscillator. This partition function, which has a simple and compact form with {\it no adjustable parameters}, reproduces some key mathematical properties of the exact partition function and provides free energies accurate to a few percent over a wide range of temperatures and coupling constants. In this work, we present the derivation of the energy spectrum of this model. We also generalize our previous study limited to the quartic oscillator to the case of a general anharmonic oscillator. Numerical application for a potential of the form $V(x)=\frac{ω^2}{2} x^2 + g x^{2m}$ show that the energy levels are obtained with a relative error of about a few percent, a precision which we consider to be quite satisfactory given the simplicity of the model, the absence of adjustable parameters, and the negligible computational cost.

quant-ph↗

arXiv2024

On the Particle Approximation of Lagged Feynman-Kac Formulae

In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman-Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operators as found, for example, in physics or particle simulation of rare-events. We focus on a novel \emph{lagged} approximation of this invariant measure, based upon the introduction of a ratio of time-averaged Feynman-Kac marginals associated with a positive operator iterated $l \in\mathbb{N}$ times; a lagged Feynman-Kac formula. This estimator and its approximation using Diffusion Monte Carlo (DMC) have been extensively employed in the physics literature. In short, DMC is an iterative algorithm involving $N\in\mathbb{N}$ particles or walkers simulated in parallel, that undergo sampling and resampling operations. In this work, it is shown that for the DMC approximation of the lagged Feynman-Kac formula, one has an almost sure characterization of the $\mathbb{L}_1$-error as the time parameter (iteration) goes to infinity and this is at most of $\mathcal{O}(\exp\{-κl\}/N)$, for $κ>0$. In addition a non-asymptotic in time, and time uniform $\mathbb{L}_1-$bound is proved which is $\mathcal{O}(l/\sqrt{N})$. We also prove a novel central limit theorem to give a characterization of the exact asymptotic in time variance. This analysis demonstrates that the strategy used in physics, namely, to run DMC with $N$ and $l$ small and, for long time enough, is mathematically justified. Our results also suggest how one should choose $N$ and $l$ in practice. We emphasize that these results are not restricted to physical applications; they have broad relevance to the general problem of particle simulation of the Feynman-Kac formula.

math.PR↗

arXiv2024

Go Green: Selected Configuration Interaction as a More Sustainable Alternative for High Accuracy

Recently, a new distributed implementation of the full configuration interaction (FCI) method has been reported [Gao et al. J. Chem Theory Comput. 2024, 20, 1185]. Thanks to a hybrid parallelization scheme, the authors were able to compute the exact energy of propane (\ce{C3H8}) in the minimal basis STO-3G. This formidable task involves handling an active space of 26 electrons in 23 orbitals or a Hilbert space of \SI{1.3d12} determinants. This is, by far, the largest FCI calculation reported to date. Here, we illustrate how, from a general point of view, selected configuration interaction (SCI) can achieve microhartree accuracy at a fraction of the computational and memory cost, via a sparse exploration of the FCI space. The present SCI calculations are performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm, as implemented in a determinant-driven way in the \textsc{quantum package} software. The present study reinforces the common wisdom that among the exponentially large number of determinants in the FCI space, only a tiny fraction of them significantly contribute to the energy. More importantly, it demonstrates the feasibility of achieving comparable accuracy using more reasonable and sustainable computational resources, hence reducing the ever-growing carbon footprint of computational chemistry.

physics.chem-ph↗

arXiv2022

Diffusion Monte Carlo using domains in configuration space

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.

cond-mat.str-el↗

arXiv2022

Ground- and Excited-State Dipole Moments and Oscillator Strengths of Full Configuration Interaction Quality

We report ground- and excited-state dipole moments and oscillator strengths (computed in different ``gauges'' or representations) of full configuration interaction (FCI) quality using the selected configuration interaction method known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI). Thanks to a set encompassing 35 ground- and excited-state properties computed in 11 small molecules, the present near-FCI estimates allow us to assess the accuracy of high-order coupled-cluster (CC) calculations including up to quadruple excitations. In particular, we show that incrementing the excitation degree of the CC expansion (from CCSD to CCSDT or from CCSDT to CCSDTQ) reduces the average error with respect to the near-FCI reference values by approximately one order of magnitude.

physics.chem-ph↗

arXiv2021

A quantum Monte Carlo study of systems with effective core potentials and node nonlinearities

We study beryllium dihydride (BeH$_2$) and acetylene (C$_2$H$_2$) molecules using real-space diffusion Monte Carlo (DMC) method. The molecules serve as perhaps the simplest prototypes that illustrate the difficulties with biases in the fixed-node DMC calculations that might appear with the use of effective core potentials (ECPs) or other nonlocal operators. This is especially relevant for the recently introduced correlation consistent ECPs (ccECPs) for $2s2p$ elements. Corresponding ccECPs exhibit deeper potential functions due to higher fidelity to all-electron counterparts, which could lead to larger local energy fluctuations. We point out that the difficulties stem from issues that are straightforward to address by upgrades of basis sets, use of T-moves for nonlocal terms, inclusion of a few configurations into the trial function and similar. The resulting accuracy corresponds to the ccECP target (chemical accuracy) and it is in consistent agreement with independent correlated calculations. Further possibilities for upgrading the reliability of the DMC algorithm and considerations for better adapted and more robust Jastrow factors are discussed as well.

physics.comp-ph↗

arXiv2021

Accurate full configuration interaction correlation energy estimates for five- and six-membered rings

Following our recent work on the benzene molecule [\href{https://doi.org/10.1063/5.0027617}{J.~Chem.~Phys.~\textbf{153}, 176101 (2020)}], itself motivated by the blind challenge of Eriksen \textit{et al.} [\href{https://doi.org/10.1021/acs.jpclett.0c02621}{J.~Phys.~Chem.~Lett.~\textbf{11}, 8922 (2020)}] on the same system, we report accurate full configuration interaction (FCI) frozen-core correlation energy estimates for twelve five- and six-membered ring molecules in the standard correlation-consistent double-$ζ$ Dunning basis set (cc-pVDZ). Our FCI correlation energy estimates, with estimated error smaller than 1 millihartree, are based on energetically optimized-orbital selected configuration interaction (SCI) calculations performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm. Having at our disposal these accurate reference energies, the respective performance and convergence properties of several popular and widely-used families of single-reference quantum chemistry methods are investigated. In particular, we study the convergence properties of i) the Møller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the iterative approximate coupled-cluster series CC2, CC3, and CC4, and iii) the coupled-cluster series CCSD, CCSDT, and CCSDTQ. The performance of the ground-state gold standard CCSD(T) as well as the completely renormalized CC model, CR-CC(2,3), are also investigated.

physics.chem-ph↗

arXiv2021

Excited States From State Specific Orbital Optimized Pair Coupled Cluster

The pair coupled cluster doubles (pCCD) method (where the excitation manifold is restricted to electron pairs) has a series of interesting features. Among others, it provides ground-state energies very close to what is obtained with doubly-occupied configuration interaction (DOCI), but with polynomial cost (compared with the exponential cost of the latter). Here, we address whether this similarity holds for excited states, by exploring the symmetric dissociation of the linear \ce{H4} molecule. When ground-state Hartree-Fock (HF) orbitals are employed, pCCD and DOCI excited-state energies do not match, a feature that is assigned to the poor HF reference. In contrast, by optimizing the orbitals at the pCCD level (oo-pCCD) specifically for each excited state, the discrepancies between pCCD and DOCI decrease by one or two orders of magnitude. Therefore, the pCCD and DOCI methodologies still provide comparable energies for excited states, but only if suitable, state-specific orbitals are adopted. We also assessed whether a pCCD approach could be used to directly target doubly-excited states, without having to resort to the equation-of-motion (EOM) formalism. In our $Δ$oo-pCCD model, excitation energies were extracted from the energy difference between separate oo-pCCD calculations for the ground state and the targeted excited state. For a set comprising the doubly-excited states of \ce{CH+}, \ce{BH}, nitroxyl, nitrosomethane, and formaldehyde, we found that $Δ$oo-pCCD provides quite accurate excitation energies, with root mean square deviations (with respect to full configuration interaction results) lower than CC3 and comparable to EOM-CCSDT, two methods with much higher computational cost.

physics.chem-ph↗

arXiv2021

QUESTDB: a database of highly-accurate excitation energies for the electronic structure community

We describe our efforts of the past few years to create a large set of more than 500 highly-accurate vertical excitation energies of various natures ($π\to π^*$, $n \to π^*$, double excitation, Rydberg, singlet, doublet, triplet, etc) in small- and medium-sized molecules. These values have been obtained using an incremental strategy which consists in combining high-order coupled cluster and selected configuration interaction calculations using increasingly large diffuse basis sets in order to reach high accuracy. One of the key aspect of the so-called QUEST database of vertical excitations is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating theoretical cross comparisons. Following this composite protocol, we have been able to produce theoretical best estimate (TBEs) with the aug-cc-pVTZ basis set for each of these transitions, as well as basis set corrected TBEs (i.e., near the complete basis set limit) for some of them. The TBEs/aug-cc-pVTZ have been employed to benchmark a large number of (lower-order) wave function methods such as CIS(D), ADC(2), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, ADC(3), CC3, NEVPT2, and others (including spin-scaled variants). In order to gather the huge amount of data produced during the QUEST project, we have created a website [https://lcpq.github.io/QUESTDB_website] where one can easily test and compare the accuracy of a given method with respect to various variables such as the molecule size or its family, the nature of the excited states, the type of basis set, etc. We hope that the present review will provide a useful summary of our effort so far and foster new developments around excited-state methods.

physics.chem-ph↗

arXiv2020

Towards a Systematic Improvement of the Fixed-Node Approximation in Diffusion Monte Carlo for Solids -- A Case Study In Diamond

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for \textit{ab initio} electronic structure calculations, in practice the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros) must be used. This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced.

physics.chem-ph↗

arXiv2020

Systematic Comparison and Cross-validation of Fixed-Node Diffusion Monte Carlo and Phaseless Auxiliary-Field Quantum Monte Carlo in Solids

Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this comparative study between DMC and AFQMC. By focusing on a set of canonical finite size solid state systems, we show that both QMC methods can be made to systematically converge towards the same energy once basis set effects and systematic biases have been removed. AFQMC shows a much smaller dependence on the trial wavefunction than DMC while simultaneously exhibiting a much larger basis set dependence. We outline some of the remaining challenges and opportunities for improving these approaches.

physics.comp-ph↗

arXiv2019

Evaluating two-electron-repulsion integrals over arbitrary orbitals using Zero Variance Monte Carlo: Application to Full Configuration Interaction calculations with Slater-type orbitals

A Monte Carlo method for evaluating multi-center two-electron-repulsion integrals over any type of orbitals (Slater, Sturmian, finite-range, numerical, etc.) is presented. The approach is based on a simple and universal (orbital-independent) gaussian sampling of the two-electron configuration space and on the use of efficient zero-variance Monte Carlo estimators. Quite remarkably, it is shown that the high level of accuracy required on two-electron integrals to make Hartree-Fock (HF) and configuration interaction (CI) calculations feasible can be achieved. A first zero-variance estimator is built by introducing a gaussian approximation of the orbitals and by evaluating the two-electron integrals using a correlated sampling scheme for the difference between exact and approximate orbitals. A second one is based on the introduction of a general coordinate transformation. The price to pay for this simple and general Monte Carlo scheme is the high computational cost required. However, we argue that the great simplicity of the algorithm, its embarrassingly parallel nature, its ideal adaptation to modern computational platforms and, most importantly, the possibility of using more compact and physically meaningful basis sets make nevertheless the method attractive. HF and near full CI (FCI) calculations using Slater-type orbitals (STO) are reported for Be, CH4 and [H2N(CH)NH2]^+ (a simple model of cyanine). To the best of our knowledge, our largest FCI calculation involving 18 active electrons distributed among 90 orbitals for the cyanine molecule, is the most extensive molecular calculation performed so far using pure STO orbitals (no gaussian approximation, even for the challenging four-center two-electron integrals).

physics.chem-ph↗

arXiv2019

Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs

\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.

physics.comp-ph↗

arXiv2019

Influence of Pseudopotentials on Excitation Energies From Selected Configuration Interaction and Diffusion Monte Carlo

Due to their diverse nature, the faithful description of excited states within electronic structure theory methods remains one of the grand challenges of modern theoretical chemistry. Quantum Monte Carlo (QMC) methods have been applied very successfully to ground state properties but still remain generally less effective than other non-stochastic methods for electronically excited states. Nonetheless, we have recently reported accurate excitation energies for small organic molecules at the fixed-node diffusion Monte Carlo (FN-DMC) within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using the selected configuration interaction (sCI) algorithm known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). Albeit highly accurate, these all-electron calculations are computationally expensive due to the presence of core electrons. One very popular approach to remove these chemically-inert electrons from the QMC simulation is to introduce pseudopotentials (also known as effective core potentials). Taking the water molecule as an example, we investigate the influence of Burkatzki-Filippi-Dolg (BFD) pseudopotentials and their associated basis sets on vertical excitation energies obtained with sCI and FN-DMC methods. Although these pseudopotentials are known to be relatively safe for ground state properties, we evidence that special care may be required if one strives for highly accurate vertical transition energies. Indeed, comparing all-electron and valence-only calculations, we show that using pseudopotentials with the associated basis sets can induce differences of the order of 0.05 eV on the excitation energies. Fortunately, a reasonable estimate of this shift can be estimated at the sCI level.

physics.chem-ph↗