Corrections relativistic energy

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are additional contributions from the molecular bonds and associated zero-point energies. The experimental value for the total energy of H2O is —76.480 a.u., and the estimated contribution from relativistic effects is —0.045 a.u. Including a mass correction of 0.0028 a.u. (a non-Bom-Oppenheimer effect which accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated at —76.438 0.003 a.u. ... 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

Comparison of atomic and molecular estimates of relativistic energy corrections... 

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 ... 

Note that replacing c 2 by zero in this procedure, we find all the TFD expresions, and if we make C 2 = 0 we supress the relativistic exchange corrections. When we include these in our calculations we have always chosen the value of 3jr/2, the one which matches the weak relativistic limit of the fully relativistic energy functional. 

Our method of calculation is based on an idea by Ivanov-Ivanova [11]. In an atomic system, the radiative shift and the relativistic part of the energy are, in principle, determined by one and the same physical field. It may be assumed that there exists some universal function that connects the self-energy correction and the relativistic energy. The self-energy correction for the states of a hydrogen-like ion was presented by Mohr [1] as ... 

Apart from these relativistic energy corrections, the treatment is non-relativistic regarding the theoretical formulation and one-electron wave function basis, d) The above treatment is also applied to the case of two core holes in the same main shell. The self-energies and hole-hole repulsion in Eq. (31) are then approximated by static monopole relaxation and screening, evaluated through the zlSCF method (cf. Sect. 3.6)... 

The result quoted here uses the relativistic energy from [75] with QED corrections from [68]... 

Recently we developed a new approach which improves the sensitivity to a variation of a by more than an order of magnitude [1,2]. The relative value of any relativistic corrections to atomic transition frequencies is proportional to a2. These corrections can exceed the fine structure interval between the excited levels by an order of magnitude (for example, an s-wave electron does not have the spin-orbit splitting but it has the maximal relativistic correction to energy). The relativistic corrections vary very strongly from atom to atom and can have opposite signs in different transitions (for example, in s-p and d-p transitions). Thus, any variation of a could be revealed by comparing different transitions in different atoms in cosmic and laboratory spectra. 

Goudsmit and Uhlenbeck electron spin. Thomas spin-orbit energy. Heisenberg and Jordan relativistic correction to energy. Net result recovery of Sommerfeld energy levels, different quantum numbers. 

Burned that the total atomic relativistic energy is almost independent of the atomic electronic state and the chemical environment in molecules, which implies cancellation of relativistic effects in chemical processes. According to Kolas the correction of the total energy of Hit for relativistic effects amounts to -1.60 cm whereas with the dissociation energy it is only 0,14 cm. Data for H2 are presented in Table 1,3, This table also gives us evidence on the validity of... 

We comment first on the last point. As we have learned in Chapter 1 the relativistic energies are large for inner shells and their values increase with the atomic number. This trend is reflected in the relativistic corrections. For example, for the inner-shell ionization potentials the relativistic corrections are 0,1 for CH, 0,8... 

This gives sufficient resolution to study the geometric structure of molecules. [Since 40-keV electrons travel at a significant fraction of the speed of light, the relativistic energy-momentum relation must be used. The corrected de Broglie wavelength is actually 6.016 x 10- m.]... 

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