Dirac equation Relativity

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

Einstein granted that the (Dirac) equation was "the most logically perfect presentation" of quantum mechanics yet found, but not that it got us any closer to the "secret of the Old One". It neither described the real world phenomena that he wanted to understand nor proposed new concepts that would make the real world accessible to understanding. Furthermore, Dirac s unification of quantum mechanics with special theory of relativity left out Einstein s later success with general relativity and the gravitational field. 

In other words, it is the distance, ct, travelled by light in a given time interval which fulfills the role of the fourth coordinate, rather than the time interval itself. The special theory of relativity requires that after a Lorentz transformation the new form of all laws of physics is the same as the old form. The Dirac equation, for example, is invariant under a Lorentz transformation. 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

The Dirac equation corresponds to satisfying the requirements of special relativity in connection with the quantum behaviour of the electron. Special relativity considers only systems which move with a constant velocity with respect to each other, which hardly can be considered a good approximation for the movement of an electron around a... 

A significant point here is that it is not the squared invariant ds2 that is to underlie the covariance of the laws of nature. It is rather the linear invariant ds that plays this role. How, then, do we proceed from the squared metric to the linear metric That is to say, how does one take the square root of ds2l The answer can be seen in Dirac s procedure, when he factorized the Klein-Gordon equation to yield the spinor form of the electron equation in wave mechanics -the Dirac equation. Indeed, Dirac s result indicated that by properly taking the square root of ds2 in relativity theory, extra spin degrees of freedom are revealed that were previously masked. 

Another big discovery of the early 20th century was the theory of relativity. One of the most novel discoveries was that particles moving with a speed near the speed of light behaved in different ways than more mundane objects like cars or apples. Notions such as time dilation , the twin paradox , and space-time continuum became well known. Many times, you do not have to bother with using relativistic equations for the description of particle movements, but in some cases you do, e.g. when trying to describe particles in big accelerators, and then one has to use the relativistic version of the Schrodinger equation, known as the Dirac equation. In fact, this is what is implemented in the computer codes I will describe later, but notations become very complicated when dealing with the... 

Relativity is relevant to computational chemistry because it must often be explicitly taken into account in accurate calculations involving atoms heavier than about chlorine or bromine (see below) and because, strictly speaking, the Schrbdinger equation, the fundamental equation of quantum chemistry, is an approximation to a relativistic equation, the Dirac equation. 

In this section we briefly review the main properties of the Dirac equation that is the basic equation to start with to build a relativistic effective Hamiltonian for atomic and molecular calculations. This single particle equation, as already stated in the introduction, was established in 1928 by P.A.M Dirac [1] as the Lorentz invariant counterpart of the Schrodinger equation. On a note let us recall that the first attempts to replace the Schrodinger equation by an equation fulfilling the requirements of special relativity started just after quantum... 

As required by special relativity, space and time variables should appear in a symmetric way and this requirement is most obvious in the covariant form of the Dirac equation ... 

The Dirac equation describes the quantum-mechanical motion of particles with spin-1/2 according to the requirements of the special theory of relativity. Correspondingly, it contains the following scalar parameters Planck s constant h, which sets the scale of quantum phenomena, the velocity of light c, which sets the scale for relativistic effects, and m, the rest-mass of the particle. 

The basic theory of second quantization is found in most advanced textbooks on quantum mechanics but inclusion of relativity is not often considered. A good introduction to this topic is given by Strange [10] in his recent textbook on relativistic quantum mechanics. We will basically follow his arguments but make the additional assumption that a finite basis of Im Kramers paired 4-spinors is used to expand the Dirac equation. This brings the formalism closer to quantum chemistry where use of an (infinite) basis of plane waves, as is done in traditional introductions to the subject, is impractical. 

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