Kinetic energy relativistic

The relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

Table 1 Energies (in KeV) of single positive ions evaluated with (AH) a full relativistic kinetic energy functional without exchange [15] the c -order semi-relativistic functional (Eq. 46) without (1) and with (2) the relativistic exchange correction ((f-term), all using near-nuclear corrections, compared to Dirac-Fock (DF) values.

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

Nevertheless, when we include the near nuclear corrections (where the fully relativistic kinetic energy is used), the truncation of the energy functional only in the outer region up to order both in the kinetic and exchange energies turns out to be an adequate approximation. 

For a quasi-relativistic framework as relevant to chemistry (21), we may neglect the magnetic retardation between the electrons and the nuclei and therefore employ standard Coulombic interaction operators for the electrostatic interaction. The interaction between the electrons and the nuclei is not specified explicitly but we only describe the interactions by some external 4-potential. For the sake of brevity this 4-potential shall comprise all external contributions. Explicit expressions for the interaction between electrons and nuclei will be introduced at a later stage. Furthermore, we can neglect the relativistic nature of the kinetic energy of the nuclei and employ the non-relativistic kinetic energy operator denoted as hnuc(I),... 

Herein, Ei denotes the relativistic kinetic energy operator... 

Fig. 1. The NRQED instantaneous potentials (a) four-fermion contact, (b) four-fermion derivative, (c) Coulomb, (d) relativistic kinetic energy, and (e) Breit-Fermi...

The so-called mass-velocity term Hmv /which represents the first order (in a ) relativistic correction to the non-relativistic kinetic energy operator... 

Figure 1.Non-relativistic kinetic energy, T, the first order relativistic kinetic energy, T, and the total kinetic energy, as a function of the absolute momentum p ...

The lowest order term 7 f °[n], the relativistic kinetic energy in the Thomas-Fermi limit, has first been calculated by Vallarta and Rosen [12], In the second order contribution (which is given in a form simplified by partial integration) explicit vacuum corrections do not occur after renormalisation. Finite radiative corrections, originating from the vacuum part of the propagator (E.5), first show up in fourth order, where the term in proportion ll to... 

If the relativistic kinetic energy functional T[p] is completely local, then one can rewrite the general chemical potential equation of DFT as... 

The various terms (99) can be interpreted physically. Expanding the classical relativistic kinetic energy into powers of 1 /c gives the power series... 

Combining this with the (non-relativistic) kinetic energy T = one gets the Lagrangean... 

If Eq. (93a) could be solved with Eq. (93b), the solution to the Dirac equation can be obtained exactly. However, Eq. (93a) has the total and potential energies in the denominator, and an appropriate approximation is needed. In our strategy, E — V in the denominator is replaced by the classical relativistic kinetic energy (relativistic substitutive correction)... 

COMMENT. This calculation uses the non-relativistic kinetic energy, which is only about 3 percent less than the accurate (relativistic) value of 1,52 x10-15 J. In this exercise, however, finding is a small difference of two larger numbers, so a small error in the kinetic energy results in a larger error in binding the accurate value is binding = 1.26 x10-16J. 

Here the first term is the non-relativistic kinetic energy. The second term, hereafter H°h( ), is transferred to the higher order of (1/c) and will be discussed later. 

The superscript (4) indicates the Dirac four-component picture of operators and wave functions. is the relativistic kinetic energy functional of the Dirac-Kohn-Sham (DKS) reference system of non-interacting electrons with ground state density yO [45] ... 

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