We present a comparative study of numerical methods for computingelectronic densities of states (DOS) in periodic systems. We provide a detailed analysis of the domain of validity of the Brillouincomplex deformation (BCD), a recently-proposed method promising exponential convergence without need for smearing. We compare on a range of systems the BCD with several methods, including the standard smearing and linear tetrahedron methods, as well as an adaptive integration method. Our results establish clear performance regimes for each method, offering practical guidance for DOS computations across a range of systems and accuracy requirements.

The temperature dependence of most solid-state properties is dominated by lattice vibrations, but metals display notable purely electronic effects at low temperature, such as the linear specific heat and the linear entropy, that were derived by Sommerfeld for the noninteracting electron gas via the low-temperature expansion of Fermi-Dirac integrals. Here we treat temperature as a perturbation within density-functional perturbation theory (DFPT). For finite temperature, we show how self-consistency screens the bare, temperature-induced density change obtained in the noninteracting picture: the inverse transpose of the electronic dielectric operator, that includes Adler-Wiser and a term related to the shift in Fermi level, links the self-consistent density response to the bare thermal density change. This approach is implemented in DFTK, and demonstrated by the computation of the second-order derivative of the free energy, and the first-order derivative of entropy for aluminum. Then, we examine the T -> 0 limit. The finite temperature formalism contains divergences, that we cure using the Sommerfeld expansion to analyze metallic systems at 0 K. The electronic free energy is quadratic in T provided the Fermi level is not at a Van Hove singularity of the density of states. If the latter happens, another temperature behavior might appear, depending on the type of Van Hove singularity, that we analyze. Our formulation applies to systems periodic in one, two, or three dimensions, and provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within density-functional theory and DFPT.

The convergence of the self-consistent field iterations in Kohn-Sham density functional theory can be significantly hindered by the presence of small eigenvalues in the dielectric matrix, which are often associated with electronic phase transitions in magnetic systems. In this work, we study this type of convergence issues and propose a new preconditioning scheme to mitigate them. Our preconditioning scheme is inspired by the Stoner model and based on a non-interacting susceptibility that neglects orbital variations. We demonstrate the effectiveness of our approach on a range of ferromagnetic systems, showing that it can significantly reduce the number of iterations required to achieve convergence in the vicinity of magnetic phase transitions.

We consider the electrons of a molecule in the adiabatic time-dependent density functional theory approximation. We establish the well-posedness of the time evolution and its linear response close to a non-degenerate ground state, and prove the appearance of resonances at relevant frequencies. The main mathematical difficulty is due to the structure of the linearized equations, which are not complex-linear. We bypass this difficulty by reformulating the linearized problem as a real Hamiltonian system, whose stability is ensured by the second-order optimality conditions on the energy.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific N$N$-representability problem inherent to DMET.

The acceleration of material property calculations while maintaining ab initio accuracy (1 meV/atom) is one of the major challenges in computational physics. In this paper, we introduce a Python package enhancing the computation of (finite temperature) material properties at the ab initio level using machine learning interatomic potentials (MLIP). The Machine-Learning Assisted Canonical Sampling (MLACS) method, grounded in a self-consistent variational approach, iteratively trains a MLIP using an active learning strategy in order to significantly reduce the computational cost of ab initio simulations. MLACS offers a modular and user-friendly interface that seamlessly integrates Density Functional Theory (DFT) codes, MLIP potentials, and molecular dynamics packages, enabling a wide range of applications, while maintaining a near-DFT accuracy. These include sampling the canonical ensemble of a system, performing free energy calculations, transition path sampling, and geometry optimization, all by utilizing surrogate MLIP potentials, in place of ab initio calculations. This paper provides a comprehensive overview of the theoretical foundations and implementation of the MLACS method. We also demonstrate its accuracy and efficiency through various examples, showcasing the capabilities of the MLACS package.

We study the decay of the interatomic force constants (equivalently, the smoothness properties of the dynamical matrix) in perfect crystals both at finite electronic temperature, and for insulators at zero temperature, within the reduced Hartree-Fock approximation (also called Random Phase Approximation). At finite temperature the electrons are mobile, leading to exponential decay of the force constants. In insulators, there is incomplete screening, leading to an algebraic decay of dipole-dipole interaction type.

With the aim of progressing toward a practical implementation of an effective quantum-electrodynamics (QED) theory of atoms and molecules, which includes the effects of vacuum polarization through the creation of virtual electron-positron pairs but without the explicit photon degrees of freedom, we study a one-dimensional effective QED model of the hydrogen-like atom with delta-potential interactions. This model resembles the three-dimensional effective QED theory with Coulomb interactions while being substantially simpler. We provide some mathematical details about the definition of this model, calculate the vacuum-polarization density, and the Lamb-type shift of the bound-state energy, correcting and extending results of previous works. We also study the approximation of the model in a finite plane-wave basis, and in particular we discuss the basis convergence of the bound-state energy and eigenfunction, of the vacuum-polarization density, and of the Lamb-type shift of the bound-state energy. We highlight the difficulty of converging the vacuum-polarization density in a finite basis and we propose a way to improve it. The present work could give hints on how to perform similar calculations for the three-dimensional effective QED theory of atoms and molecules.

We introduce a new numerical method to compute resonances induced by localized defects in crystals. This method solves an integral equation in the defect region to compute analytic continuations of resolvents. Such an approach enables one to express the resonance in terms of a “resonance source”, a function that is strictly localized within the defect region. The kernel of the integral equation, to be applied on such a source term, is the Green function of the perfect crystal, which we show can be computed efficiently by a complex deformation of the Brillouin zone, named Brillouin Complex Deformation (BCD), thereby extending to reciprocal space the concept of complex coordinate transformations.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact at first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions of the DMET equations. We moreover introduce and discuss a specific N-representability problem inherent to DMET.

This paper is concerned with the numerical analysis of linear and nonlinear Schrödinger equations with periodic analytic potentials. We prove that, for linear equations, when the potential is analytic in a strip of width A of the complex plane, the solution is analytic in the same strip, ensuring an exponential convergence of the planewave discretization of the equation with rate A. On the other hand, for nonlinear equations, we find that the solution may be analytic only in a strip of width smaller than A. This behavior is illustrated by two examples using a combination of numerical and analytical arguments.

Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature, these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix, this involves a choice of gauge. Taking a numerical analysis point of view, we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods, we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved.

We present a method to compute the photoionization spectra of atoms and molecules in linear-response, time-dependent density functional theory. The electronic orbital variations corresponding to ionized electrons are expanded on a basis set of delocalized functions, obtained as the solution of the inhomogeneous Helmholtz equation, with gaussian basis set functions as the right-hand side. The resulting scheme is able to reproduce the photoionization spectra without any need for artificial regularization or localization. We demonstrate that this Green's function-based approach is able to produce accurate spectra for semilocal exchange-correlation functionals, even using relatively small standard gaussian basis sets.

We present two open-source implementations of the locally optimal block preconditioned conjugate gradient ( lobpcg ) algorithm to find a few eigenvalues and eigenvectors of large, possibly sparse matrices. We then test lobpcg for various quantum chemistry problems, encompassing medium to large, dense to sparse, well-behaved to ill-conditioned ones, where the standard method typically used is Davidson’s diagonalization. Numerical tests show that while Davidson’s method remains the best choice for most applications in quantum chemistry, LOBPCG represents a competitive alternative, especially when memory is an issue, and can even outperform Davidson for ill-conditioned, non-diagonally dominant problems.

Finite temperature calculations, based on ab initio molecular dynamics (AIMD) simulations, are a powerful tool able to predict material properties that cannot be deduced from ground state calculations. However, the high computational cost of AIMD limits its applicability for large or complex systems. To circumvent this limitation we introduce a method named machine learning assisted canonical sampling, which accelerates the sampling of the Born-Oppenheimer potential surface in the canonical ensemble. Based on a self-consistent variational procedure, the method iteratively trains a machine learning interatomic potential to generate configurations that approximate the canonical distribution of positions associated with the ab initio potential energy. By proving the reliability of the method on anharmonic systems, we show that the method is able to reproduce the results of AIMD with an ab initio accuracy at a fraction of its computational cost.

We propose accurate computable error bounds for quantities of interest in plane-wave electronic structure calculations, in particular ground-state density matrices and energies, and interatomic forces. These bounds are based on an estimation of the error in terms of the residual of the solved equations, which is then efficiently approximated with computable terms. After providing coarse bounds based on an analysis of the inverse Jacobian, we improve on these bounds by solving a linear problem in a small dimension that involves a Schur complement. We numerically show how accurate these bounds are on a few representative materials, namely silicon, gallium arsenide and titanium dioxide.

We propose a novel adaptive damping algorithm for the self-consistent field (SCF) iterations of Kohn-Sham density-functional theory, using a backtracking line search to automatically adjust the damping in each SCF step. This line search is based on a theoretically sound, accurate and inexpensive model for the energy as a function of the damping parameter. In contrast to usual SCF schemes, the resulting algorithm is fully automatic and does not require the user to select a damping. We successfully apply it to a wide range of challenging systems, including elongated supercells, surfaces and transition-metal alloys.

We explore the merits of linear-response range-separated time-dependent density-functional theory (TDDFT) for the calculation of photoionization spectra. We consider two variants of range-separated TDDFT, namely, the time-dependent range-separated hybrid (TDRSH) scheme, which uses a global range-separation parameter, and the time-dependent locally range-separated hybrid (TDLRSH), which uses a local range-separation parameter, and compare with standard time-dependent local-density approximation (TDLDA) and time-dependent Hartree-Fock (TDHF). We show how to calculate photoionization spectra with these methods using the Sternheimer approach formulated in a non-orthogonal B-spline basis set with appropriate frequency-dependent boundary conditions. We illustrate these methods on the photoionization spectrum of the Be atom, focusing, in particular, on the core resonances. Both the TDRSH and TDLRSH photoionization spectra are found to constitute a large improvement over the TDLDA photoionization spectrum and a more modest improvement over the TDHF photoionization spectrum.

We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present applications to Langevin-like dynamics and Fokker–Planck equations, as well as the linear Boltzmann equation (which is also the generator of randomized Hybrid Monte Carlo in molecular dynamics). In particular, we make precise the dependence of the resolvent bounds on the parameters of the dynamics and on the dimension. We also highlight the relationship of our method with other hypocoercive approaches.

We consider the calculations of photoionization spectra and core resonances of open-shell systems using range-separated time-dependent density-functional theory. Specifically, we use the time-dependent range-separated hybrid (TDRSH) scheme, combining a long-range Hartree-Fock exchange potential and kernel with a short-range potential and kernel from a local density-functional approximation, and the time-dependent locally range-separated hybrid (TDLRSH) scheme, which uses a local range-separation parameter. To efficiently perform the calculations, we formulate a spin-unrestricted linear-response Sternheimer approach in a non-orthogonal B-spline basis set using appropriate frequency-dependent boundary conditions. We illustrate this approach on the Li atom, which suggests that TDRSH and TDLRSH are adequate simple methods for estimating the single-electron photoionization spectra of open-shell systems.