Relativistic formalism

Electrons (and many other particles) have associated with them an intrinsic angular momentum that has come to be called spin . One of the greatest successes of relativistic quantum mechanics is that spin is seen to arise naturally within the relativistic formalism, and does not need to be added post facto as it is in non-relativistic treatments. As with orbital angular momentum, spin angular momentum has x, y, and z components, and the operators 5, Sy, and S, together with orthonormal eigenfunctions a and fi of electron spin, are defined from ... 

The relativistic formalism arises as a natural consequence of relaxing the constraint of a stationary universe. The formalism is not considered in any detail here but, briefly, its development can be described as follows. If the universe is not stationary, then it is evolving—and this implies the need for a... 

We will show in the following how the non-relativistic formalism of equation 1 can be augmented to approximately account for relativistic effects. 

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

There are also a number of volumes dealing with the problem of describing atoms and molecules within a relativistic formalism. Again in reverse chronological order these are ... 

First, in the relativistic formulation the number of particles is not conserved. Electron-positron creation processes, which conserve the total charge of the system but not the number of particles, are permitted in the relativistic formalism. The use of second quantized methodology is therefore mandatory in a fully relativistic formulation of the molecular structure problem. 

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. 

This equation is solved in complete analogy to the non-relativistic formalism. 

It will be shown that the Dirac equation for the tree electron m an external electromagnetic field is leading to the spin concept. Thus, in relativistic theory, the spin angular momentum spears in a natural way, whereas in the non-relativistic formalism, it was the subject of a postulate of quantum mechanics (p. 26). 

Spin-orbit interaction is only included in the density functional framework if the proper fully relativistic formalism is invoked (Strange, 1998). Often an approximate treatment of relativistic effects is implemented, by solving for the kinetic energy in the scalar-relativistic approximation (Skriver, 1983a), and adding to the total energy functional the spin-orbit term as a perturbation of the form... 

The most advanced relativistic approach in relativistic calculations of X-ray spectra, is most likely that based on the Dirac-Coulomb-Breit Hamiltonian and quantum electrodynamic contributions accounted for. In addition, one should also carry out the corresponding correlated-level calculation within these relativistic formalism. To illustrate the role and size of relativistic and QED corrections the core and valence ionisation potentials and excitation energies of noble gases are shown. The relativistic fOTC CASSCE/CASPT2 method together with the restricted active space... 

Solid-state calculations were performed on the Rf metal [191]. The structural and electronic properties were evaluated by the first principles DFT in scalar relativistic formalism with and without SO coupling and compared with its 5d homolog Hf. It is found that Rf should crystalhze in the hexagonal close packed stmcture as Hf. However, under pressure, it should have a different sequence of phase transitions than Hf hep bcc instead of hep co bee. An explanation is offered for this difference in terms of the competition between the band structure and the Ewald energy contributions. 

Even for molecules containing light atoms, properties depending on the wave function close to the nuclear region can require the use of a relativistic formalism. 

---