A solution to the problem of variational collapse for the one-particle Dirac equation

Hill, Robert Nyden; Krauthauser, Carl

doi:10.1103/physrevlett.72.2151

Cited by 53 publications

(60 citation statements)

References 20 publications

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“…with γ = κ 2 − (Zα) 2 , and the equivalent expression for g. To avoid nonlinear terms in the eigensystem (31), only the contribution independent of has been kept. This is a good approximation for bound states for which mc 2 .…”

Section: Discretisation Methodsmentioning

confidence: 99%

“…Other variational techniques are based on a correspondence between the eigenvalues of A and those of T (A), for some operator function T , like the inverse function T x = x −1 (see [31]) or the function T x = x 2 (see [59,2]). Finally, some authors solve the variational problem in a subspace of the domain in which the operator is bounded from below and 'avoids' the negative continuum.…”

Section: Linear Dirac Equationsmentioning

confidence: 99%

“…The practical difficulty arises from the need to compute complicated matrix representations. Hill and Krautkauser [31] use the Rayleigh-Ritz variational principle applied to the inverse of the Dirac Hamiltonian, 1/H. A difficulty arises here in the computation of the matrix elements for the inverse operator.…”

Section: Direct Variational Approachesmentioning

confidence: 99%

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Computational approaches of relativistic models in quantum chemistry

Desclaux¹,

Dolbeault

Esteban

et al. 2003

Handbook of Numerical Analysis

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This chapter is a review of some methods used for the computation of relativistic atomic and molecular models based on the Dirac equation. In the linear case, we briefly describe finite basis set approaches, including ones that are generated numerically, perturbation theory and effective Hamiltonians procedures, direct variational methods based on nonlinear transformations, min-max formulations and constrained minimizations. In the atomic case, we describe the MCDF method and some ways to solve numerically the homogeneous and inhomogeneous Dirac-Fock equations. Finally, we describe also some numerical methods relevant to the case of molecules.

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Section: Discretisation Methodsmentioning

confidence: 99%

Section: Linear Dirac Equationsmentioning

confidence: 99%

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Computational approaches of relativistic models in quantum chemistry

Desclaux¹,

Dolbeault

Esteban

et al. 2003

Handbook of Numerical Analysis

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“…formulas (89) and (92). Therefore, when one represents the Dirac sea by the projector P 0 −,c , one describes formally the vacuum as an infinite Slater determinant…”

Section: The Mean-field Approximation In Quantum Electrodynamicsmentioning

confidence: 99%

“…For instance, it was proposed to minimize the Rayleigh quotient for the squared Hamiltonian (D c + V ) 2 (see, e.g. [167,16]) or later on, to maximize the Rayleigh quotients for the "inverse Hamiltonian" [92]). Before we go further, let us recall some useful inequalities which are usually used to control the external field V and show that D c + V is essentially self-adjoint.…”

Section: Linear Dirac Equations For An Electron In An External Fieldmentioning

confidence: 99%

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

Dolbeault

Esteban

Séré

2000

Lecture Notes in Chemistry

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Abstract. This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow us to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or KleinGordon-Dirac equations.The second part is devoted to the presentation of min-max principles allowing us to characterize and compute the eigenvalues of linear Dirac operators with an external potential in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them are new kinds of Hardy-like inequalities and a stable algorithm to compute the eigenvalues.In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit.In the last part, we present a more involved relativistic model from Quantum Electrodynamics in which the behavior of the vacuum is taken into account, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

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Relativistic Effects of the Superheavy Elements

Schwerdtfeger

Seth

1998

Encyclopedia of Computational Chemistry

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A solution to the problem of variational collapse for the one-particle Dirac equation

Cited by 53 publications

References 20 publications

Computational approaches of relativistic models in quantum chemistry

Computational approaches of relativistic models in quantum chemistry

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

Relativistic Effects of the Superheavy Elements

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