Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

Dolbeault, Jean; Esteban, Maria J.; Séré, Éric

doi:10.1007/978-3-642-57237-1_10

Cited by 6 publications

(3 citation statements)

References 107 publications

(188 reference statements)

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“…excited states of the Dirac Hamiltonian that are separated from lowenergy states in the Dirac sea and high-energy states in the Fermi sea), since the spectrum of the Dirac Hamiltonian is unbounded both from above and below. This is sometimes known as the variational collapse problem [70]. A solution to this problem is precisely with classical shift-invert diagonalization techniques [71][72][73] -shiftinversion maps the bound states of the Dirac Hamiltonian to the ground state of a shift-inverted Hamiltonian.…”

Section: Appendix E: Existing Algorithms For the Computation Of Excit...mentioning

confidence: 99%

Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

Shao-Hen¹,

Kwek²

2023

Preprint

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We propose a natural application of Quantum Linear Systems Problem (QLSP) solvers such as the HHL algorithm to efficiently prepare highly excited interior eigenstates of physical Hamiltonians in a variational manner. This is enabled by the efficient computation of inverse expectation values, taking advantage of the QLSP solvers' exponentially better scaling in problem size without concealing exponentially costly pre/post-processing steps that usually accompanies it. We detail implementations of this scheme for both fault-tolerant and near-term quantum computers, analyse their efficiency and implementability, and discuss applications and simulation results in many-body physics and quantum chemistry that demonstrate its superior effectiveness and scalability over existing approaches.

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Section: Appendix E: Existing Algorithms For the Computation Of Excit...mentioning

confidence: 99%

Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

Shao-Hen¹,

Kwek²

2023

Preprint

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“…Basically, the minimization has to be replaced by adequate saddle-point methods. Some new minimax characterizations have been established by Esteban and Séré (2002), Dolbeault, Esteban and Séré (2000a), Desclaux et al (2003), and have given rise to new algorithmic techniques to compute the eigenfunctions and eigenvalues of the Dirac operator in molecules: see Dolbeault, Esteban, Séré and Vanbreugel (2000b), Dolbeault, . Likewise, in the many-electron case where models such as the Dirac- Fock model, introduced in Swirles (1935, 1936, play the role of the Hartree-Fock model, adequate definitions of the ground state need to be derived.…”

Section: Relativistic Modelsmentioning

confidence: 99%

Computational chemistry from the perspective of numerical analysis

Bris

2005

Acta Numerica

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Link to this article: http://journals.cambridge.org/abstract_S096249290400025XHow to cite this article: Claude Le Bris (2005). Computational chemistry from the perspective of numerical analysis.We present the field of computational chemistry from the standpoint of numerical analysis. We introduce the most commonly used models and comment on their applicability. We briefly outline the results of mathematical analysis and then mostly concentrate on the main issues raised by numerical simulations. A special emphasis is laid on recent results in numerical analysis, recent developments of new methods and challenging open issues. CONTENTS1 Outline 364 2 A short introduction to modelling for molecular simulation 365 3 Discretization of molecular models 376 4 The condensed phase 408 5 Time-dependent problems 411 6 Current and future trends 430 References 431 364 C. Le Bris Computational chemistry 365decade, but a lot remains to be done. We conjecture that a better theoretical knowledge of existing techniques will lead to their enhancement, and will therefore improve their applicability, as demonstrated by the history of scientific computing in the engineering sciences. The purpose of the present article is to overview the numerical difficulties encountered in computational chemistry, to show how they are solved within the current state of the art, and to indicate the needs for further improvements. Of course, whenever they exist, we will indicate the results of numerical analysis that help to give a sound grounding for these techniques. But most often the article will deal with descriptions of techniques rather than with statements of theorems. We hope that this bias will stimulate further works.The article is organized in a rather narrative way, without any ordering of scientific priority and/or importance. We begin with a description of the models and the discretization techniques for small systems in a static picture. Then, we progressively and (hopefully) pedagogically proceed to the modelling and simulation of more difficult situations: larger systems, systems in situ, time-dependent settings. Of course, as the size of the system increases, the models need to be coarse-grained, otherwise they cannot be tackled in practice. Therefore, the article also proceeds from the finest models to the coarsest ones.The present state-of-the-art survey focuses on issues in numerical analysis for a readership familiar with such questions in other settings. A more detailed description may be found in the book of Le Bris, ed. (2003), and also, to a smaller extent, in the proceedings volume by Defranceschi and Le Bris, eds (2000). For readers with a background in chemistry, we refer to two survey articles, Defranceschi and Le Bris (1997) and Defranceschi and Le Bris (1999), for an introduction to the mathematical and numerical analysis. On the other hand, questions related to the mathematical analysis of models are overviewed in Le Bris and Lions (2005).Computational chemistry 367 state of minimal energy) of the system under consider...

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“…These kinds of minimax characterizations have given rise to new algorithmic techniques to compute the eigenfunctions and eigenvalues of the Dirac operator in molecules [87,84,88,83]. But, as is the case in the nonrelativistic setting, it is only possible to attack a system of N electrons modelled by the Dirac Hamiltonian when N is dramatically small.…”

Section: Theorem 34 [85 86] Minimax Characterization Of the Eigenvmentioning

confidence: 99%

From atoms to crystals: a mathematical journey

Bris

Lions²

2005

Bull. Amer. Math. Soc.

112

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Abstract. We present an overview of some works on the models of computational quantum chemistry. We examine issues such as the existence of ground states (both for the electronic structure and the configuration of nuclei), the foundations of the models of the crystalline phase, and the macroscopic limits. We emphasize the connections between the physical modelling, the numerical concerns and the mathematical analysis of the problems.

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Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

Cited by 6 publications

References 107 publications

Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

Computational chemistry from the perspective of numerical analysis

From atoms to crystals: a mathematical journey

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