Relativistic methods

For the present discussion, it is of importance that a relativistic NMR computation is carried out if E in Eq. (4) is defined and calculated within a relativistic framework, otherwise it is nonrelativistic. Another approach is to treat relativity as an additional perturbation and to evaluate relativistic corrections to aA and Kab individually in the form of derivatives of Eqs. (6a) and (6b) with respect to a suitably chosen relativistic perturbation parameter (usually c-2), i.e. first a nonrelativistic NMR observable is calculated and then correction terms are obtained in the form of 

The computation of NMR observables within a relativistic formalism has some important consequences. The most obvious is that the Hamiltonian of the 

The central theme in relativity is that the speed of light, c, is constant in all inertia frames (coordinate systems that move with respect to each other). Augmented with the requirement that physical laws should be identical in such frames, this has as a consequence that time and space coordinates become equivalent . A relativistic description of a particle thus requires four coordinates, three space and one time coordinate. The latter is usually multiplied by c to have units identical to the space variables. 

A change between different coordinate systems can be described by a Lorentz transformation, which may mix space and time coordinates. The postulate that physical laws should be identical in all coordinate systems is equivalent to the requirement that equations describing the physics must be invariant (unchanged) to a Lorentz transformation. Considering the time-dependent Schrbdinger equation (8.1), it is clear that it is not Lorentz invariant since the derivative with respect to space coordinates is of second order, but the time derivative is only first order. The fundamental structure of the Schrbdinger equation is therefore not relativisticaUy correct. 

For use below, we have elected here to explicitly write the electron mass as m, although it is equal to one in atomic units. 

One of the consequences of the constant speed of light is that the mass of a particle, which moves at a substantial fraction of c, increases over the rest mass mo. 

Introduction to Computational Chemistry, Second Edition, trank Jensen. 2007 John Wiley Sons, Ltd 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E = —T = 1/2 F). In atomic units the classical velocity of a Is-electron 

Although an in-depth treatment is outside the scope of this book, it may be iiisLmetive to point out some of the features and problems in a relativistic quantum description of atoms and molecules. 

For a free electron Dirac proposed that the (time-dependent) Schrodinger equation should be replaced by  

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

The a function is an eigenfunction of the operator with an eigenvalue of 1 jl, and the 0 function similarly has an eigenvalue of —1 /2. 

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Relativistic methods can be extended to include electron correlation by methods analogous to those for the non-relativistic cases, e.g. Cl, MCSCF, MP and CC. Such methods are currently at the development stage. ... 

Relativistic Methods for Molecular Calculations and Diatomic Gold Compounds... 

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

Bioinorganic systems often contain heavy elements that need to be treated with an explicit relativistic method. It is now possible to carry out an explicit relativistic electronic structure calculation on the fly (152). The scalar-relativistic Douglas - Kroll - Hess method was implemented by us recently in the BOMD simulation framework (152). To use the relativistic densities in a non-relativistic gradient calculations turned out to be a valid approximation of relativistic gradients. An excellent agreement between optimized structures and geometries obtained from numerical gradients was observed with an error smaller than 0.02 pm. 

Some Aspects of the Theory Approximate Relativistic Methods... 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

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