Perturbation expansion of the Dirac equation

In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent nonrelativistic methods, from which it is apparent that four-component calculations will be considerably more expensive than the corresponding nonrelativistic calculations—perhaps two orders of magnitude more expensive. For this reason, there have been many methods developed that make approximations to the Dirac equation, and it is to these that we turn in this part of the book. There are two elements of the Dirac equation that contribute to the large amount of work: the presence of the small component of the wave function and the spin dependence of the Hamiltonian. The small component is primarily responsible for the large number of two-electron integrals which, as will be seen later, contain all the lowest-order relativistic corrections to the electron–electron interaction. The spin dependence is incorporated through the kinetic energy operator and the correction to the electronic Coulomb interaction, and also through the coupling of the spin and orbital angular momenta in the atomic 2-spinors, which form a natural basis set for the solution of the Dirac equation. Spin separation has obvious advantages from a computational perspective. As we will show for several spin-free approaches below, a spin-free Hamiltonian is generally real, and therefore real spin–orbitals may be employed for the large and small components. The spin can then be factorized out and spin-restricted Hartree–Fock methods used to generate the one-electron functions. In the post-SCF stage, where the no-pair approximation is invoked, the transformation of the integrals from the atomic to the molecular basis produces a set of real molecular integrals that are indistinguishable from a set of nonrelativistic MO integrals, and therefore all the nonrelativistic correlation methods may be employed without modification to obtain relativistic spin-free correlated wave functions. In most cases, spin–free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order.

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Finite Size of the Electron and Runaway Solutions

Zeitschrift für Naturforschung A ◽

10.1515/zna-1977-0502 ◽

1977 ◽

Vol 32 (5) ◽

pp. 383-389 ◽

Cited By ~ 1

Author(s):

J. Petzold ◽

W. Heudorfer ◽

M. Sorg

Keyword(s):

Dirac Equation ◽

Free Electron ◽

Constant Velocity ◽

Initial Conditions ◽

Perturbation Expansion ◽

Finite Size ◽

Equation Of Motion ◽

Local Equation ◽

Causality Violation ◽

Non Local

Abstract The problem of runaway solutions is studied within the framework of a non-local equation of motion for the classically radiating electron. It is found that the force-free electron oscillates down to a constant velocity under emission of radiation, if certain restrictions on the initial conditions are imposed. Causality violation is not present in this model, but penetrates into the theory as consequence of a false perturbation expansion leading to the notorious Lorentz-Dirac equation of motion.

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Dirac equation

Modern Particle Physics ◽

10.1017/cbo9781139525367.022 ◽

2018 ◽

pp. 515-522

Keyword(s):

Dirac Equation

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Bosonic Symmetries, Solutions, and Conservation Laws for the Dirac Equation with Nonzero Mass

Ukrainian Journal of Physics ◽

10.15407/ujpe58.06.0523 ◽

2013 ◽

Vol 58 (6) ◽

pp. 523-533 ◽

Cited By ~ 12

Author(s):

V.M. Simulik ◽

◽

I.Yu. Krivsky ◽

I.L. Lamer ◽

◽

...

Keyword(s):

Dirac Equation ◽

Conservation Laws

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SPIN-SPIN INTERACTION IN ONE PARTICLE DIRAC EQUATION

Scientific Herald of Uzhhorod University Series Physics ◽

10.24144/2415-8038.1999.4.66-68 ◽

1999 ◽

Vol 4 (0) ◽

pp. 66-68

Author(s):

І. І. Гайсак ◽

В. С. Морохович

Keyword(s):

Dirac Equation ◽

Spin Interaction

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First order perturbation expansion for systems of convex molecules

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19742333 ◽

1974 ◽

Vol 39 (9) ◽

pp. 2333-2343 ◽

Cited By ~ 19

Author(s):

T. Boublík

Keyword(s):

Perturbation Expansion ◽

Order Perturbation ◽

First Order ◽

First Order Perturbation

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Introduction to Relativistic Quantum Chemistry

10.1093/oso/9780195140866.001.0001 ◽

2007 ◽

Cited By ~ 30

Author(s):

Kenneth G. Dyall ◽

Knut Faegri

Keyword(s):

Dirac Equation ◽

Quantum Chemistry ◽

Relativistic Theory ◽

Density Functional ◽

Relativistic Quantum ◽

Relativistic Effects ◽

Basis Sets ◽

Spin Orbit ◽

Approximate Methods ◽

Nonrelativistic Theory

This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.

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