Kinetic energy relativistic correction

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)... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

This problem can be illustrated by estimating, for a weak relativistic atom, the first correction to the kinetic energy. Using the usual expansion... 

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.

Two types of corrections to the Thomas-Fermi-Dirac non-relativistic energy density appear. The first is the correction to the kinetic energy given by the mass-variation term ... 

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. 

We evaluate now the contribution of //a/f, which provides the first order relativistic correction to the kinetic energy ... 

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 relativistic correction to the fermion kinetic energy is represented as a potential. The Breit-Fermi interaction includes the effects of transverse photon exchange as well as relativistic corrections to Coulomb photon exchange. The potentials are given with the assumption that the states acted on are S states with total spin 1. 

It is clear that with a high density of states the theoretical investigations of electronic spectra must sometimes go beyond a traditional BO and non-relativistic analysis that only refers to energy criteria, and that in the description of spectroscopic properties smaller terms of the Hamiltonian must be accounted for. The major corrections to the BO electrostatic Hamiltonian is the non-adiabatic coupling induced by the nuclear kinetic energy operator, and the electronic SOC treated in the present review. 

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

We can now turn to a discussion of how relativistic effects will modify the molecular energies, Eab (or AEab )r as well as their functional dependence on the interatomic distance, Rab-There are two non-zero Relativistic corrections from the first order Hamiltonian, Hi, of Eq. (5). One is the first order correction, T to the kinetic energy from the mass-velocity term, Hmv /and... 

Fig. E.I. Relativistic corrections to kinetic energy densities. Eq. (B.44)—solid line (GEO), Eq. (E.16)—dashed line (GE2)...

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... 

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