Logarithmic perturbation expansion for the Dirac equation in one dimension: Application to the polarizability calculation

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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Supersymmetry aspects of the Dirac equation in one dimension with a Lorentz scalar potential

Physical Review A ◽

10.1103/physreva.47.1708 ◽

1993 ◽

Vol 47 (3) ◽

pp. 1708-1714 ◽

Cited By ~ 50

Author(s):

Y. Nogami ◽

F. M. Toyama

Keyword(s):

Dirac Equation ◽

Scalar Potential ◽

One Dimension ◽

Lorentz Scalar

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Reconstruction of Objects from their Projections Using Orthogonal Functions

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100071788 ◽

1973 ◽

Vol 31 ◽

pp. 268-269

Author(s):

Elrnar Zeitler

Keyword(s):

Unit Area ◽

Three Dimensional ◽

One Dimension ◽

Spatial Density ◽

Dimensional Representation ◽

Orthogonal Functions ◽

Object A ◽

Plane Normal ◽

Dimensional Object ◽

Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

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Quantitative Diameter Measurements of Chromatin and TMV Fibers in the Freeze-Dried State

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100102936 ◽

1988 ◽

Vol 46 ◽

pp. 170-171

Author(s):

B. D. Athey ◽

A. L. Stout ◽

M. F. Smith ◽

J. P. Langmore

Keyword(s):

Reliable Method ◽

Fiber Diameter ◽

Quantitative Agreement ◽

One Dimension ◽

Fiber Axis ◽

Two Dimensional ◽

Fiber Density ◽

Variable Diameter ◽

Freeze Dried ◽

Expected Values

Although there is general agreement that Inactive chromosome fibers consist of helically packed nucleosomes, the pattern of packing is still undetermined. Only one of the proposed models, the crossed-linker model, predicts a variable diameter dependent on the length of DNA between nucleosomes. Measurements of the fiber diameter of negatively-stained and frozen- hydrated- chromatin from Thyone sperm (87bp linker) and Necturus erythrocytes (48bp linker) have been previously reported from this laboratory. We now introduce a more reliable method of measuring the diameters of electron images of fibrous objects. The procedure uses a modified version of the computer program TOTAL, which takes a two-dimensional projection of the fiber density (represented by the micrograph itself) and projects it down the fiber axis onto one dimension. We illustrate this method using high contrast, in-focus STEM images of TMV and chromatin from Thyone and Necturus. The measured diameters are in quantitative agreement with the expected values for the crossed-linker model for chromatin structure

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