arXiv subjects

Geneviève Dusson

Publications and source records attributed to Geneviève Dusson.

27 records · Page 2Linked to original sources

arXiv2021

Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{ö}dinger operators

In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schr{ö}dinger operators, by combining the Feshbach-Schur perturbation theory with the spectral Fourier discretization. In order to analyze the method, we establish an abstract framework of Feshbach-Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new spectral Fourier discretization method. Finally, we present some numerical results that underline the theoretical findings.

math.NA↗

arXiv2021

Grassmann extrapolation of density matrices for Born-Oppenheimer molecular dynamics

Born-Oppenheimer Molecular Dynamics (BOMD) is a powerful but expensive technique. The main bottleneck in a density functional theory bomd calculation is the solution to the Kohn-Sham (KS) equations, that requires an iterative procedure that starts from a guess for the density matrix. Converged densities from previous points in the trajectory can be used to extrapolate a new guess, however, the non-linear constraint that an idempotent density needs to satisfy make the direct use of standard linear extrapolation techniques not possible. In this contribution, we introduce a locally bijective map between the manifold where the density is defined and its tangent space, so that linear extrapolation can be performed in a vector space while, at the same time, retaining the correct physical properties of the extrapolated density using molecular descriptors. We apply the method to real-life, multiscale polarizable QM/MM.

physics.chem-ph↗

arXiv2021

An adaptive planewave method for electronic structure calculations

We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn--Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.

physics.comp-ph↗

arXiv2021

The Feshbach-Schur map and perturbation theory

This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and it has advantages with respect to the standard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.

math-ph↗

arXiv2020

Atomic Permutationally Invariant Polynomials for Fitting Molecular Force Fields

We introduce and explore an approach for constructing force fields for small molecules, which combines intuitive low body order empirical force field terms with the concepts of data driven statistical fits of recent machine learned potentials. We bring these two key ideas together to bridge the gap between established empirical force fields that have a high degree of transferability on the one hand, and the machine learned potentials that are systematically improvable and can converge to very high accuracy, on the other. Our framework extends the atomic Permutationally Invariant Polynomials (aPIP) developed for elemental materials in [Mach. Learn.: Sci. Technol. 2019 1 015004] to molecular systems. The body order decomposition allows us to keep the dimensionality of each term low, while the use of an iterative fitting scheme as well as regularisation procedures improve the extrapolation outside the training set. We investigate aPIP force fields with up to generalised 4-body terms, and examine the performance on a set of small organic molecules. We achieve a high level of accuracy when fitting individual molecules, comparable to those of the many-body machine learned force fields. Fitted to a combined training set of short linear alkanes, the accuracy of the aPIP force field still significantly exceeds what can be expected from classical empirical force fields, while retaining reasonable transferability to both configurations far from the training set and to new molecules.

physics.chem-ph↗

arXiv2020

Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schr{ö}dinger operator with periodic boundary conditions of the form $--$Δ$ + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

math.NA↗

arXiv2020

An approximation strategy to compute accurate initial density matrices for repeated self-consistent field calculations at different geometries

Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix, expanded in a set of localized basis functions, within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are changed along a few user-specified collective variables, such as the molecule's normal modes. Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value.The method allows non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold.The so-called Grassmann exponential and logarithm map the manifold onto the tangent space and thus provides the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself.Numerical tests on different amino acids show promising initial results.

physics.chem-ph↗

arXiv2019

Regularised Atomic Body-Ordered Permutation-Invariant Polynomials for the Construction of Interatomic Potentials

We investigate the use of invariant polynomials in the construction of data-driven interatomic potentials for material systems. The "atomic body-ordered permutation-invariant polynomials" (aPIPs) comprise a systematic basis and are constructed to preserve the symmetry of the potential energy function with respect to rotations and permutations. In contrast to kernel based and artificial neural network models, the explicit decomposition of the total energy as a sum of atomic body-ordered terms allows to keep the dimensionality of the fit reasonably low, up to just 10 for the 5-body terms. The explainability of the potential is aided by this decomposition, as the low body-order components can be studied and interpreted independently. Moreover, although polynomial basis functions are thought to extrapolate poorly, we show that the low dimensionality combined with careful regularisation actually leads to better transferability than the high dimensional, kernel based Gaussian Approximation Potential.

physics.comp-ph↗

arXiv2017

Discretization error cancellation in electronic structure calculation: a quantitative study

It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. The purpose of this paper is to provide a quantitative study of discretization error cancellation. The latter is the error component due to the fact that the model used in the calculation (e.g. Kohn-Sham LDA) must be discretized in a finite basis set to be solved by a computer. We first report comprehensive numerical simulations performed with Abinit on two simple chemical systems, the hydrogen molecule on the one hand, and a system consisting of two oxygen atoms and four hydrogen atoms on the other hand. We observe that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cut-off goes to infinity. We then analyze a simple one-dimensional periodic Schrödinger equation with Dirac potentials, for which analytic solutions are available. This allows us to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.

cond-mat.mtrl-sci↗