17 research outputs found
Discretization error cancellation in the plane-wave approximation of periodic Hamiltonians with Coulomb singularities
In solid-state physics, energies of molecular systems are usually computed
with a plane-wave discretization of Kohn-Sham equations. A priori estimates of
plane-wave convergence for periodic Kohn-Sham calculations with
pseudopotentials have been proved , however in most computations in practice,
plane-wave cut-offs are not tight enough to target the desired accuracy. It is
often advocated that the real quantity of interest is not the value of the
energy but of energy differences for different configurations. The computed
energy difference is believed to be much more accurate because of
`discretization error cancellation', since the sources of numerical errors are
essentially the same for different configurations. For periodic linear
Hamiltonians with Coulomb potentials, error cancellation can be explained by
the universality of the Kato cusp condition. Using weighted Sobolev spaces,
Taylor-type expansions of the eigenfunctions are available yielding a precise
characterization of this singularity. This then gives an explicit formula of
the first order term of the decay of the Fourier coefficients of the
eigenfunctions. It enables one to prove that errors on eigenvalue differences
are reduced but converge at the same rate as the error on the eigenvalue.Comment: 14 pages, 3 figures, improved result on main theorem, corrected typo
Variational projector-augmented wave method: a full-potential approach for electronic structure calculations in solid-state physics
In solid-state physics, energies of crystals are usually computed with a
plane-wave discretization of Kohn-Sham equations. However the presence of
Coulomb singularities requires the use of large plane-wave cut-offs to produce
accurate numerical results. In this paper, an analysis of the plane-wave
convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb
potentials using the variational projector-augmented wave (VPAW) method is
presented. In the VPAW method, an invertible transformation is applied to the
original eigenvalue problem, acting locally in balls centered at the
singularities. In this setting, a generalized eigenvalue problem needs to be
solved using plane-waves. We show that cusps of the eigenfunctions of the VPAW
eigenvalue problem at the positions of the nuclei are significantly reduced.
These eigenfunctions have however a higher-order derivative discontinuity at
the spheres centered at the nuclei. By balancing both sources of error, we show
that the VPAW method can drastically improve the plane-wave convergence of the
eigenvalues with a minor additional computational cost. Numerical tests are
provided confirming the efficiency of the method to treat Coulomb
singularities.Comment: 29 pages, 4 figure
Projector augmented-wave method: an analysis in a one-dimensional setting
In this article, a numerical analysis of the projector augmented-wave (PAW)
method is presented, restricted to the case of dimension one with Dirac
potentials modeling the nuclei in a periodic setting. The PAW method is widely
used in electronic ab initio calculations, in conjunction with
pseudopotentials. It consists in replacing the original electronic Hamiltonian
by a pseudo-Hamiltonian via the PAW transformation acting in
balls around each nuclei. Formally, the new eigenvalue problem has the same
eigenvalues as and smoother eigenfunctions. In practice, the
pseudo-Hamiltonian has to be truncated, introducing an error that is
rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue
are proved for the one-dimensional periodic Schr\"odinger operator with double
Dirac potentials.Comment: 31 pages, 4 figure
Dissociation limit of the H2 molecule in the particle-hole random phase approximation
In this work, we consider the particle-hole random phase approximation
(phRPA), an approximation to the correlation energy in electronic structure,
and show that the phRPA energy of the H2 molecule correctly dissociates. That
is, as the hydrogen atoms are pulled apart, the phRPA energy of the system
converges to twice the phRPA energy of a single hydrogen atom. Despite the
simplicity of the H2 system, the correct dissociation of H2 is known to be a
difficult problem for density functional approximations. As part of our result,
we prove that the phRPA correlation energy is well-defined
Radiofrequency Ablation of Benign Thyroid Nodules and Recurrent Thyroid Cancers: Consensus Statement and Recommendations
Thermal ablation using radiofrequency is a new, minimally invasive modality employed as an alternative to surgery in patients with benign thyroid nodules and recurrent thyroid cancers. The Task Force Committee of the Korean Society of Thyroid Radiology has developed recommendations for the optimal use of radiofrequency ablation for thyroid nodules. These recommendations are based on a comprehensive analysis of the current literature, the results of multicenter studies, and expert consensus
Projector augmented-wave method: an analysis in a one-dimensional setting
International audienceIn this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian by a pseudo-Hamiltonian PAW via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as and smoother eigenfunctions. In practice, the pseudo-Hamiltonian PAW has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schrödinger operator with double Dirac potentials
Analyse de la méthode projector augmented-wave pour les calculs de structure électronique en géométrie périodique
This thesis is devoted to the study of the PAW method (projector augmented-wave) and of a variant called the variational PAW method (VPAW). These methods aim to accelerate the convergence of plane-wave methods in electronic structure calculations. They rely on an invertible transformation applied to the eigenvalue problem, which acts in a neighborhood of each atomic site. The transformation captures some difficulties caused by the Coulomb singularities. The VPAW method is applied to a periodic one-dimensional Schr\"odinger operator with Dirac potentials and analyzed in this setting. Eigenfunctions of this model have derivative jumps similar to the electronic cusps. The derivative jumps of eigenfunctions of the VPAW eigenvalue problem are significantly reduced. Hence, a smaller plane-wave cut-off is required for a given accuracy level. The study of the VPAW method is also carried out for 3D periodic Hamiltonians with Coulomb singularities yielding similar results. In the PAW method, the invertible transformation has infinite sums that are truncated in practice. The induced error is analyzed in the case of the periodic one-dimensional Schrödinger operator with Dirac potentials. Error bounds on the lowest eigenvalue are proved depending on the PAW parameters.Cette thèse est consacrée à l'étude de la méthode PAW (projector augmented-wave) et d'une de ses modifications, baptisée méthode PAW variationnelle (VPAW), pour le calcul de l'état fondamental d'Hamiltoniens en géométrie périodique. Ces méthodes visent à améliorer la vitesse de convergence des méthodes d'ondes planes (ou méthodes de Fourier) en appliquant une transformation inversible au problème aux valeurs propres initial agissant au voisinage de chaque site atomique. Cette transformation permet de capter une partie des difficultés dues aux singularités coulombiennes. La méthode VPAW est analysée pour un opérateur de Schr\"odinger unidimensionnel avec des potentiels de Dirac. Les fonctions propres de ce modèle comprennent des sauts de dérivées similaires aux cusps électroniques. Le saut de dérivée des fonctions propres du problème aux valeurs propres issu de la méthode VPAW est réduit de façon importante. Cela entraîne une accélération de convergence en ondes planes du calcul des valeurs propres corroborée par une étude numérique. Une étude de la méthode VPAW est conduite pour des Hamiltoniens 3D périodiques avec des singularités coulombiennes, parvenant à des conclusions similaires. Pour la méthode PAW, la transformation inversible comporte des sommes infinies qui sont tronquées en pratique. Ceci introduit une erreur, qui est rarement quantifiée en pratique. Elle est analysée dans le cas de l'opérateur de Schrödinger unidimensionnel avec des potentiels de Dirac. Des bornes sur la plus basse valeur propre en fonction des paramètres PAW sont prouvées conformes aux tests numériques
Convergence analysis of adaptive DIIS algorithms with application to electronic ground state calculations
This paper deals with a general class of algorithms for the solution of
fixed-point problems that we refer to as \emph{Anderson--Pulay acceleration}.
This family includes the DIIS technique and its variant sometimes called
commutator-DIIS, both introduced by Pulay in the 1980s to accelerate the
convergence of self-consistent field procedures in quantum chemistry, as well
as the related Anderson acceleration which dates back to the 1960s, and the
wealth of techniques they have inspired. Such methods aim at accelerating the
convergence of any fixed-point iteration method by combining several iterates
in order to generate the next one at each step. This extrapolation process is
characterised by its \emph{depth}, i.e. the number of previous iterates stored,
which is a crucial parameter for the efficiency of the method. It is generally
fixed to an empirical value. In the present work, we consider two
parameter-driven mechanisms to let the depth vary along the iterations. In the
first one, the depth grows until a certain nondegeneracy condition is no longer
satisfied; then the stored iterates (save for the last one) are discarded and
the method "restarts". In the second one, we adapt the depth continuously by
eliminating at each step some of the oldest, less relevant, iterates. In an
abstract and general setting, we prove under natural assumptions the local
convergence and acceleration of these two adaptive Anderson--Pulay methods, and
we show that one can theoretically achieve a superlinear convergence rate with
each of them. We then investigate their behaviour in quantum chemistry
calculations. These numerical experiments show that both adaptive variants
exhibit a faster convergence than a standard fixed-depth scheme, and require on
average less computational effort per iteration. This study is complemented by
a review of known facts on the DIIS, in particular its link with the Anderson
acceleration and some multisecant-type quasi-Newton methods.Comment: Final version to appear in ESAIM:M2A