Dirac formalism

Eleig, T. and Visscher, L. (2005) Large-scale electron correlation calculations in the framework of the spin-free dirac formalism the Au2 molecule revisited. Chemical Physics, 311, 63. 

An isolated microscopic system is fully determined, in the quantum mechanical sense, when its state function y/ is known. In the Dirac formalism of quantum mechanics, the state function can be identified with a vector of state, 11//). (11) The system in the 1y/ state may equivalently be described by a Hermitian operator, the so-called density matrix p of a pure state,... 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

The following approach will show how the already proved quite reliable approach of path integrals is naturally needed within the Dirac formalism of quantum mechanics applied on many-particle systems, specific to chemical structures formed by many-electrons in valence state, by means of the celebrated density matrix formalism - from where there is just a step to the observable density functional theory of many-body systems. 

The lOTC method presented in this review is obviously related to earlier attempts to reduce the four-component Dirac formalism to computationally much simpler two-component schemes. Among the different to some extent competitive methods, the priority should be given to the Douglas-Kroll-Hess method [13,53,54]. It was Bernd Hess and his work which was our inspiration to search for the better solutions to the two-component methodology. 

Brooks, Johansson and Skriver (1984) investigated the band structure of UC and ThC by nonrelativistic and relativistic (based on the Dirac formalism) LMTO methods. They analysed the electron density changes in the compounds as compared with free atoms, as well as the influence of pressure on the band structure. Crystal pressures as a function of lattice constants (equations of state) were calculated as well as theoretical values of the lattice constants. The calculated trends in the variations of lattice constants and bulk moduli agree well with the available experimental data. Some of the most important results of these calculations are shown in Figs. 2.20 and 2.21. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

In concluding this section we briefly establish the connection between the Dirac theory for a single isolated free particle described in the previous section and the present formalism. If T> is the state vector describing a one-particle state, iV T> = 1 T> consider the amplitude... 

Treatment of the relativistic effects within the Douglass-Kroll-Hess (DIDirac equation for most atoms. 

Covariant elements, molecular systems modulus-phase formalism, Dirac theory electrons, 267-268... 

Dirac theory, molecular systems, modulus-phase formalism ... 

---