arXiv subjects

Michel Caffarel

Publications and source records attributed to Michel Caffarel.

At least 37 records · Page 2Linked to original sources

arXiv2019

Self-Consistent Electron-Nucleus Cusp Correction for Molecular Orbitals

We describe a method for imposing the correct electron-nucleus (e-n) cusp in molecular orbitals expanded as a linear combination of (cuspless) Gaussian basis functions. Enforcing the e-n cusp in trial wave functions is an important asset in quantum Monte Carlo calculations as it significantly reduces the variance of the local energy during the Monte Carlo sampling. In the method presented here, the Gaussian basis set is augmented with a small number of Slater basis functions. Note that, unlike other e-n cusp correction schemes, the presence of the Slater function is not limited to the vicinity of the nuclei. Both the coefficients of these cuspless Gaussian and cusp-correcting Slater basis functions may be self-consistently optimized by diagonalization of an orbital-dependent effective Fock operator. Illustrative examples are reported for atoms (\ce{H}, \ce{He} and \ce{Ne}) as well as for a small molecular system (\ce{BeH2}). For the simple case of the \ce{He} atom, we observe that, with respect to the cuspless version, the variance is reduced by one order of magnitude by applying our cusp-corrected scheme.

physics.chem-ph↗

arXiv2018

Reference Energies for Double Excitations

Excited states exhibiting double excitation character are notoriously difficult to model using conventional single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC). In the present work, we provide accurate reference excitation energies for transitions involving a substantial amount of double excitation using a series of increasingly large diffuse-containing atomic basis sets. Our set gathers 20 vertical transitions from 14 small- and medium-size molecule. Depending on the size of the molecule, selected configuration interaction (sCI) and/or multiconfigurational calculations are performed in order to obtain reliable estimates of the vertical transition energies. In addition, coupled cluster approaches including at least contributions from iterative triples are assessed. Our results clearly evidence that the error in CC methods is intimately related to the amount of double excitation character of the transition. For "pure" double excitations (i.e. for transitions which do not mix with single excitations), the error in CC3 can easily reach 1 eV, while it goes down to few tenths of an eV for more common transitions (like in \emph{trans}-butadiene) involving a significant amount of singles. As expected, CC approaches including quadruples yield highly accurate results for any type of transitions. The quality of the excitation energies obtained with multiconfigurational methods is harder to predict. We have found that the overall accuracy of these methods is highly dependent of both the system and the selected active space. The inclusion of the $σ$ and $σ^*$ orbitals in the active space, even for transitions involving mostly $π$ and $π^*$ orbitals, is mandatory in order to reach high accuracy. A theoretical best estimate (TBE) is reported for each transition.

physics.chem-ph↗

arXiv2018

Selected configuration interaction dressed by perturbation

Selected configuration interaction (sCI) methods including second-order perturbative corrections provide near full CI (FCI) quality energies with only a small fraction of the determinants of the FCI space. Here, we introduce both a state-specific and a multi-state sCI method based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm. The present method revises the reference (internal) space under the effect of its interaction with the outer space via the construction of an effective Hamiltonian, following the shifted-Bk philosophy of Davidson and coworkers. In particular, the multi-state algorithm removes the storage bottleneck of the effective Hamiltonian via a low-rank factorization of the dressing matrix.

physics.chem-ph↗

arXiv2018

A Mountaineering Strategy to Excited States: Highly-Accurate Reference Energies and Benchmarks

Striving to define very accurate vertical transition energies, we perform both high-level coupled cluster (CC) calculations (up to CCSDTQP) and selected configuration interaction (sCI) calculations (up to several millions of determinants) for 18 small compounds. By systematically increasing the order of the CC expansion, the number of determinants in the CI expansion as well as the size of the one-electron basis set, we have been able to reach near full CI (FCI) quality transition energies. These calculations are carried out on CC3/aug-cc-pVTZ geometries, using a series of increasingly large atomic basis sets systematically including diffuse functions. In this way, we define a list of 110 transition energies for states of various characters to be used as references for further calculations. Benchmark transition energies are provided at the aug-pVTZ level as well as with additional basis set corrections, in order to obtain results close to the complete basis set limit. These reference data are used to benchmark a series of twelve excited-state wave function methods accounting for double and triple contributions, namely ADC(2), ADC(3), CIS(D), CIS(D$_\infty$), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, CC3, CCSDT and CCSDTQ. It turns out that CCSDTQ yields a negligible difference with the extrapolated CI values with a mean absolute error as small as $0.01$ eV, whereas the coupled cluster approaches including iterative triples are also very accurate (mean absolute error of $0.03$ eV). Consequently, CCSDT-3 and CC3 can be used to define reliable benchmarks. This observation does not hold for ADC(3) that delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version, ADC(2). Finally, we discuss the possibility to use basis set extrapolation approaches so as to tackle more easily larger compounds.

physics.chem-ph↗

arXiv2018

Excitation energies from diffusion Monte Carlo using selected Configuration Interaction nodes

Quantum Monte Carlo (QMC) is a stochastic method which has been particularly successful for ground-state electronic structure calculations but mostly unexplored for the computation of excited-state energies. Here, we show that, within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using selected configuration interaction (sCI) expansions, one is able to obtain accurate excitation energies at the fixed-node diffusion Monte Carlo (FN-DMC) level. This evidences that the fixed-node errors in the ground and excited states obtained with sCI wave functions cancel out to a large extent. Our procedure is tested on two small organic molecules (water and formaldehyde) for which we report all-electron FN-DMC calculations. For both the singlet and triplet manifolds, accurate vertical excitation energies are obtained with relatively compact multideterminant expansions built with small (typically double-$ζ$) basis sets.

physics.chem-ph↗

arXiv2017

Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS

In diffusion Monte Carlo (DMC) methods, the nodes (or zeroes) of the trial wave function dictate the magnitude of the fixed-node (FN) error. Within standard DMC implementations, they emanate from short multideterminant expansions, \textit{stochastically} optimized in the presence of a Jastrow factor. Here, following a recent proposal, we follow an alternative route by considering the nodes of selected configuration interaction (sCI) expansions built with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm. In contrast to standard implementations, these nodes can be \textit{systematically} and \textit{deterministically} improved by increasing the size of the sCI expansion. The present methodology is used to investigate the properties of the transition metal sulfide molecule FeS. This apparently simple molecule has been shown to be particularly challenging for electronic structure theory methods due to the proximity of two low-energy quintet electronic states of different spatial symmetry. In particular, we show that, at the triple-zeta basis set level, all sCI results --- including those extrapolated at the full CI (FCI) limit --- disagree with experiment, yielding an electronic ground state of $^{5}Σ^+$ symmetry. Performing FN-DMC simulation with sCI nodes, we show that the correct $^{5}Δ$ ground state is obtained if sufficiently large expansions are used. Moreover, we show that one can systematically get accurate potential energy surfaces and reproduce the experimental dissociation energy as well as other spectroscopic constants.

physics.chem-ph↗

arXiv2017

Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory

A hybrid stochastic-deterministic approach for computing the second-order perturbative contribution $E^{(2)}$ within multireference perturbation theory (MRPT) is presented. The idea at the heart of our hybrid scheme --- based on a reformulation of $E^{(2)}$ as a sum of elementary contributions associated with each determinant of the MR wave function --- is to split $E^{(2)}$ into a stochastic and a deterministic part. During the simulation, the stochastic part is gradually reduced by dynamically increasing the deterministic part until one reaches the desired accuracy. In sharp contrast with a purely stochastic MC scheme where the error decreases indefinitely as $t^{-1/2}$ (where $t$ is the computational time), the statistical error in our hybrid algorithm displays a polynomial decay $\sim t^{-n}$ with $n=3-4$ in the examples considered here. If desired, the calculation can be carried on until the stochastic part entirely vanishes. In that case, the exact result is obtained with no error bar and no noticeable computational overhead compared to the fully-deterministic calculation. The method is illustrated on the F$_2$ and Cr$_2$ molecules. Even for the largest case corresponding to the Cr$_2$ molecule treated with the cc-pVQZ basis set, very accurate results are obtained for $E^{(2)}$ for an active space of (28e,176o) and a MR wave function including up to $2 \times 10^7$ determinants.

physics.chem-ph↗

arXiv2016

Using CIPSI nodes in diffusion Monte Carlo

Several aspects of the recently proposed DMC-CIPSI approach consisting in using selected Configuration Interaction (SCI) approaches such as CIPSI (Configuration Interaction using a Perturbative Selection done Iteratively) to build accurate nodes for diffusion Monte Carlo (DMC) calculations are presented and discussed. The main ideas are illustrated with a number of calculations for diatomics molecules and for the benchmark G1 set.

physics.chem-ph↗

arXiv2016

Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule

All-electron Fixed-node Diffusion Monte Carlo (FN-DMC) calculations for the nonrelativistic ground-state energy of the water molecule at equilibrium geometry are presented. The determinantal part of the trial wavefunction is obtained from a perturbatively selected Configuration Interaction calculation (CIPSI method) including up to about 1.4 million of determinants. Calculations are made using the cc-pCV$n$Z family of basis sets, with $n$=2 to 5. In contrast with most QMC works no re-optimization of the determinantal part in presence of a Jastrow is performed. For the largest cc-pCV5Z basis set the lowest upper bound for the ground-state energy reported so far of -76.43744(18) is obtained. The fixed-node energy is found to decrease regularly as a function of the cardinal number $n$ and the Complete Basis Set limit (CBS) associated with {\it exact nodes} is easily extracted. The resulting energy of -76.43894(12) -in perfect agreement with the best experimentally derived value- is the most accurate theoretical estimate reported so far. We emphasize that employing selected CI nodes of increasing quality in a given family of basis sets may represent a simple, deterministic, reproducible, and systematic way of controlling the fixed-node error in DMC.

physics.chem-ph↗

arXiv2015

Quantum Monte Carlo with very large multideterminant wavefunctions

An algorithm to compute efficiently the first two derivatives of (very) large multideterminant wavefunctions for quantum Monte Carlo calculations is presented. The calculation of determinants and their derivatives is performed using the Sherman-Morrison formula for updating the inverse Slater matrix. An improved implementation based on the reduction of the number of column substitutions and on a very efficient implementation of the calculation of the scalar products involved is presented. It is emphasized that multideterminant expansions contain in general a large number of identical spin-specific determinants: for typical configuration interaction-type wavefunctions the number of unique spin-specific determinants $N_{\rm det}^σ$ ($σ=\uparrow,\downarrow$) with a non-negligible weight in the expansion is of order ${\cal O}(\sqrt{N_{\rm det}})$. We show that a careful implementation of the calculation of the $N_{\rm det}$-dependent contributions can make this step negligible enough so that in practice the algorithm scales as the total number of unique spin-specific determinants, $\; N_{\rm det}^\uparrow + N_{\rm det}^\downarrow$, over a wide range of total number of determinants (here, $N_{\rm det}$ up to about one million), thus greatly reducing the total computational cost. Finally, a new truncation scheme for the multideterminant expansion is proposed so that larger expansions can be considered without increasing the computational time. The algorithm is illustrated with all-electron Fixed-Node Diffusion Monte Carlo calculations of the total energy of the chlorine atom. Calculations using a trial wavefunction including about 750 000 determinants with a computational increase of $\sim$ 400 compared to a single-determinant calculation are shown to be feasible.

physics.chem-ph↗

arXiv2014

Fixed-Node Diffusion Monte Carlo potential energy curve of the fluorine molecule F2 using selected configuration interaction trial wavefunctions

The potential energy curve of the F$_2$ molecule is calculated with Fixed-Node Diffusion Monte Carlo (FN-DMC) using Configuration Interaction (CI)-type trial wavefunctions. To keep the number of determinants reasonable (the first and second derivatives of the trial wavefunction need to be calculated at each step of FN-DMC), the CI expansion is restricted to those determinants that contribute the most to the total energy. The selection of the determinants is made using the so-called CIPSI approach (Configuration Interaction using a Perturbative Selection made Iteratively). Quite remarkably, the nodes of CIPSI wavefunctions are found to be systematically improved when increasing the number of selected determinants. To reduce the non-parallelism error of the potential energy curve a scheme based on the use of a $R$-dependent number of determinants is introduced. Numerical results show that improved FN-DMC energy curves for the F$_2$ molecule are obtained when employing CIPSI trial wavefunctions. Using the Dunning's cc-pVDZ basis set the FN-DMC energy curve is of a quality similar to that obtained with FCI/cc-pVQZ. A key advantage of using selected CI in FN-DMC is the possibility of improving nodes in a systematic and automatic way without resorting to a preliminary multi-parameter stochastic optimization of the trial wavefunction performed at the Variational Monte Carlo level as usually done in FN-DMC.

physics.chem-ph↗

arXiv2014

Spin density distribution in open-shell transition metal systems: A comparative post-Hartree-Fock, Density Functional Theory and quantum Monte Carlo study of the CuCl2 molecule

We present a comparative study of the spatial distribution of the spin density (SD) of the ground state of CuCl2 using Density Functional Theory (DFT), quantum Monte Carlo (QMC), and post-Hartree-Fock wavefunction theory (WFT). A number of studies have shown that an accurate description of the electronic structure of the lowest-lying states of this molecule is particularly challenging due to the interplay between the strong dynamical correlation effects in the 3d shell of the copper atom and the delocalization of the 3d hole over the chlorine atoms. It is shown here that qualitatively different results for SD are obtained from these various quantum-chemical approaches. At the DFT level, the spin density distribution is directly related to the amount of Hartree-Fock exchange introduced in hybrid functionals. At the QMC level, Fixed-node Diffusion Monte Carlo (FN-DMC) results for SD are strongly dependent on the nodal structure of the trial wavefunction employed (here, Hartree-Fock or Kohn-Sham with a particular amount of HF exchange) : in the case of this open-shell system, the 3N -dimensional nodes are mainly determined by the 3-dimensional nodes of the singly occupied molecular orbital. Regarding wavefunction approaches, HF and CASSCF lead to strongly localized spin density on the copper atom, in sharp contrast with DFT. To get a more reliable description and shed some light on the connections between the various theoretical descriptions, Full CI-type (FCI) calculations are performed. To make them feasible for this case a perturbatively selected CI approach generating multi-determinantal expansions of reasonable size and a small tractable basis set are employed. Although semi-quantitative, these near-FCI calculations allow to clarify how the spin density distribution evolves upon inclusion of dynamic correlation effects. A plausible scenario about the nature of the SD is proposed.

physics.chem-ph↗

arXiv2012

Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond

Various strategies to implement efficiently QMC simulations for large chemical systems are presented. These include: i.) the introduction of an efficient algorithm to calculate the computationally expensive Slater matrices. This novel scheme is based on the use of the highly localized character of atomic Gaussian basis functions (not the molecular orbitals as usually done), ii.) the possibility of keeping the memory footprint minimal, iii.) the important enhancement of single-core performance when efficient optimization tools are employed, and iv.) the definition of a universal, dynamic, fault-tolerant, and load-balanced computational framework adapted to all kinds of computational platforms (massively parallel machines, clusters, or distributed grids). These strategies have been implemented in the QMC=Chem code developed at Toulouse and illustrated with numerical applications on small peptides of increasing sizes (158, 434, 1056 and 1731 electrons). Using 10k-80k computing cores of the Curie machine (GENCI-TGCC-CEA, France) QMC=Chem has been shown to be capable of running at the petascale level, thus demonstrating that for this machine a large part of the peak performance can be achieved. Implementation of large-scale QMC simulations for future exascale platforms with a comparable level of efficiency is expected to be feasible.

cs.PF↗

arXiv2011

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field (VBSCF) and breathing-orbital valence bond (BOVB) wave functions. Complemented by a Jastrow factor, these Jastrow-Valence-Bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C$_2$, N$_2$, O$_2$, and F$_2$ in both variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlation. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.

physics.chem-ph↗

arXiv2010

Quantum Monte Carlo calculations of electronic excitation energies: the case of the singlet $n \to π^*$ (CO) transition in acrolein

We report state-of-the-art quantum Monte Carlo calculations of the singlet $n \to π^*$ (CO) vertical excitation energy in the acrolein molecule, extending the recent study of Bouabça {\it et al.} [J. Chem. Phys. {\bf 130}, 114107 (2009)]. We investigate the effect of using a Slater basis set instead of a Gaussian basis set, and of using state-average versus state-specific complete-active-space (CAS) wave functions, with or without reoptimization of the coefficients of the configuration state functions (CSFs) and of the orbitals in variational Monte Carlo (VMC). It is found that, with the Slater basis set used here, both state-average and state-specific CAS(6,5) wave functions give an accurate excitation energy in diffusion Monte Carlo (DMC), with or without reoptimization of the CSF and orbital coefficients in the presence of the Jastrow factor. In contrast, the CAS(2,2) wave functions require reoptimization of the CSF and orbital coefficients to give a good DMC excitation energy. Our best estimates of the vertical excitation energy are between 3.86 and 3.89 eV.

physics.chem-ph↗

arXiv2006

The Fermion Monte Carlo revisited

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.

cond-mat.str-el↗

arXiv2006

Efficient Monte Carlo Calculations of the One-Body Density

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 (1D curve) and for the water dimer (3D grid containing up to 51x51x51=132651 points).

physics.comp-ph↗

arXiv2006

An efficient sampling algorithm for Variational Monte Carlo

We propose a new algorithm for sampling the $N$-body density $|Ψ({\bf R})|^2/\int_{\mathbb{R}^{3N}} |Ψ|^2$ in the Variational Monte Carlo (VMC) framework. This algorithm is based upon a modified Ricci-Ciccotti discretization of the Langevin dynamics in the phase space $({\bf R},{\bf P})$ improved by a Metropolis acceptation/rejection step. We show through some representative numerical examples (Lithium, Fluorine and Copper atoms, and phenol molecule), that this algorithm is superior to the standard sampling algorithm based on the biased random walk (importance sampling).

cond-mat.other↗