Electrons relativistic kinetic energy

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. ... 

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),... 

A time-of-flight (TOF) analyser measures the time t required for a particle to travel a fixed distance d. If applied to electron spectrometry, non-relativistic electrons with kinetic energy kin have a velocity v... 

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] ... 

Determine the relativistic momentum of an electron and its relativistic kinetic energy if the electron is moving at a speed of 0.9c. 

Compare the electron s kinetic energy for two cases and compare them with the electron s rest mass the kinetic energy is 51 eV and 0.51 MeV the electron s rest mass is just 0.51 MeV. Therefore, the second case corresponds to a relativistic one. The wavelengths for these cases are A = 2nhl t U = 172 and Aj = 2nh/i3mQC = 1.4 pm. 

Free-Electron Lasers. The free-electron laser (EEL) directly converts the kinetic energy of a relativistic electron beam into light (45,46). Relativistic electron beams have velocities that approach the speed of light. The active medium is a beam of free electrons. The EEL, a specialized device having probably limited appHcations, is a novel type of laser with high tunabiHty and potentially high power and efficiency. 

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)). 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

With the purpose of evaluate not only the energy but also the electron density itself, Ashby and Holzman [15] performed calculations in which the relativistic TF density was replaced at short distancies from the nucleus from the one obtained for the 1 s Dirac orbital for an hydrogenic atom, matched continuously to the semiclassical density at a switching radius rg where the kinetic energy density of both descriptions also match. 

Before the progress with the relativistic gradient expansion of the kinetic energy took place, and due to a growing interest of applying the Kohn-Sham scheme of density functional theory [19] in the relativistic framework, an explicit functional for the exchange energy of a relativistic electron gas was found [20,21] ... 

The scalar ZORA method has been implemented in the standard non relativistic Ab Initio electronic structure program GAMESS-UK [8]. The technical details of this implementation will be given in the following section. Comparing the Schrodinger equation with the ZORA equation (7) one sees that application of the ZORA method has resulted in a potential dependent correction on the kinetic energy term. 

The second moment (p ) is twice the electronic kinetic energy, and the fourth moment p ) is proportional to the correction to the kinetic energy due to the relativistic variation of mass with velocity [174—178]. 

The full N-electron non-relativistic Hamiltonian H discussed earlier in this text involves the kinetic energies of the electrons and of the nuclei and the mutual coulombic interactions among these particles... 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

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