Rest energy subtraction

Quasi-relativistic approximations usually start from re-writing the Dirac equation in two-component form. Starting from the Dirac equation (with rest energy subtracted)... 

We find the rate of change of the internal energy for an observer at rest by subtracting Eq. (3.109) from the total energy conservation relation Eq. (3.99) and using Eq. (3.102)... 

The initial conditions for the electron motion mentioned above are consistent with the choice s — yo me fio and = — yomc, respectively (Salamin and Faisal 1996). Given this result for S the energy of the electron can be derived via E(rj) = —3S/dt. Subtracting the rest energy the kinetic energy of the electron can be expressed as... 

Later, when making comparisons with nonrelativistic calculations, we subtract the electron rest energy mc firom e. ) The choice of the potential U r) is more or less arbitrary one important choice being the (Dirac) Hartree-Fock potential. Eigenstates of Eq. (1) fall into three classes bound states with —mc < k < rn< , continuum states with > m( , and negative energy (positron) states ej < —me . Since contributions from virtual electron-positron pairs are projected out of the no-pair Hamiltonian, we will be concerned primarily with bound and continuum electron states. 

A quite rigorous and practicable approach for electronic structure calculations of atoms and molecules is based on the Dirac-Coulomb Hamiltonian, i.e., the exact relativistic Dirac Hamiltonian (Dirac 1928a,b, 1929) is used for the one-electron terms (the rest energy has been subtracted). 

The consideration of the nonrelativistic limit of the Dirac energy eigenvalue for the hydrogen-like atom with a Coulombic potential for the electron-nucleus attraction, Eq. (6.3), demonstrates the effect of subtracting the rest energy mgC and leads us to a discussion of the reference energy in the following section 6.7. 

From this equation we understand that the nonielativistic Schrodinger energy eigenvalue for hydrogen-like atoms with point-like Coulomb nucleus, Eq. (6.19), is obtained only after subtraction of the rest energy,... 

Second, we will shift the energy scale by subtracting the rest energy, and define the kinetic energy operator by... 

Finally, the xc-energy Exc, in which all many-body aspects beyond the Pauli principle are absorbed, is defined by (4.17) (the rest mass of the electrons has been subtracted from Etot). As the existence theorem (4.4) is equally valid for noninteracting particles,... 

The nonrelativistic limit of (116) is easily computed. We subtract the rest-mass rrK from the energy, that is, we write... 

The energy turnover associated with contraction can be divided into several components, more or less interrelated. The most obvious division is in "resting" and "activated" energy turnover, where typically the activated is two to three times the resting value in living smooth muscle. Subtracting the resting from the... 

This different behaviour allows an easy separation of the asymmetric part of the spectrum from the rest, just depending upon which parent molecule (unoriented HI or oriented HXel) is photodisso-ciated. Normally, the distribution due to the asymmetric part (the shaded area in the spectrum) is subtracted so that the two symmetric halves of the remaining spectrum give identical results, when transformed to the translational energy distribution. 

Table 17.15 shows results obtained from the application of various bulk and surface analysis methods to lithium metal at rest or after cyclization experiments, as well as at noble metal and carbon electrodes after cathodic polarization. Several surface and elemental analysis methods are applied, including X-ray photoelectron spectroscopy (XPS, ESCA/XPS), energy dispersive analysis of X-rays (X-ray microanalysis, EDAX), Eourier transform infrared spectroscopy (ETIR), Auger electron spectroscopy (AES), ellipsometry (E), electro-modulated infrared reflectance spectroscopy (EMIRS), double modulation Fourier transform infrared spectroscopy (DMFTIR), subtractively normalized interfacial Fourier transform infrared spectroscopy (SNIFTIRS), gas chromatography (GC), IR spectroscopy. X-ray diffraction (XRD), and atomic force microscopy (AFM). 

To align the zero of the relativistic energy scale with that of the non-relativistic one, the rest mass of the electron is subtracted through the substitution... 

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